C H A P T E R 1 1

Instruments

“Provando e riprovando”[Trying and trying again]

Accademia del Cimento (1657–1667)

11.1 PHYSICAL SEPARATIONS: FRACTIONALCRYSTALLIZATION AND DISTILLATION

The partition of any molecule of interest M between two mutually distinct(usually thismeansmutually immiscible) phases (call them phases 1 and 2) isgoverned, as all equilibriumprocesses, by an equilibrium constant (the law ofGuldberg1 and Waage2):

K ¼ ½Mðphase 1Þ�=½Mðphase 2Þ� ð11:1:1Þ

Here, phase 1 would be the “mobile phase,” while phase 2 represents a“stationary phase.” The fundamental idea is that, after M had equilibratedin one location or region of the interface, the molecules M would travel to aphysically different location of the fairly long “phase 1 | phase 2” interfaceand equilibrate again, so that the net ratio would become first {[M(phase 1)]/[M(phase 2)]}2, then {[M(phase 1)]/[M(phase 2)]}3, and so on, until n suchsequential equilibria would bring the overall equilibrium constant to

f½Mðphase 1Þ�=½Mðphase 2Þ�gn ¼ Kn ð11:1:2Þ

The Physical Chemist’s Toolbox, Robert M. Metzger.� 2012 John Wiley & Sons, Inc. Published 2012 by John Wiley & Sons, Inc.

1 Cato Guldberg (1836–1902).2 Peter Waage (1833–1900).

647

The separation between two successive equilibria is called a theoretical plate.If this is a height H along some physical column, it is called the height-equivalent theoretical plate (HETP).

Fractional crystallization (or differential crystallization) is a processwhereby two chemically compounds that form crystalswith slightly differentsolubilities in some solvent (e.g., water) can be separated by a “tree-like”process. One should remember the herculean work by Marie Curie3, who byfractional crystallization isolated 0.1 g of intensely radioactive RaCl2 from1 ton of pitchblende (a black mixture of many other salts, mainly oxides ofuranium, lead, thorium, and rare earth elements).

Fractional distillation refers to separating two miscible liquids A and B,whose boiling points Tb are relatively close (� 25�C apart) at 1 atm (Fig. 11.1).One example would be A¼water (Tb¼ 100�C) and B¼ ethanol (Tb¼ 78.5�C).The idea is to boil the original AþB solution in the round-bottom flask; thevapor above it will be richer in the more volatile liquid (here, B). This vaporcondenses as a liquid slightly richer in B on one of the glass beads or cups inthe fractionating column above the flask; the liquid, heated from below, willevaporate one more time, but become even richer in B; this second vaporwill condense on a second surface, and evaporate again, richer yet in B. Theevaporation follows the “steps,” two of which are shown for the nonidealethanol–water mixture in Fig. 11.2. The distance between the glass beads,where a true re-equilibration takes place, is HETP. If all goes well, thethermometer will first indicate the boiling temperature of pure B; and when

Thermometer

FractionatingColumn

Round-bottomflask

Bunsen burner

Water out

Water in

Condenser

FIGURE 11.1

Fractional distillation setup.

3Marie Sklodowska Curie (1867–1934).

648 11 INSTRUMENTS

B is gone, the temperature will rise further to the boiling temperature of pureA. If the mixture is nonideal (as is the case for the ethanol–water mixture),one may reach a constant-boiling mixture, beyond which enrichment of thevapor in the more volatile compound B is no longer possible.

11.2 CHROMATOGRAPHY

Gas chromatography (GC), or, more precisely, gas–liquid chromatography(GLC), or, more loosely, vapor-phase chromatography, was discovered byTsvet 6 in 1901. The term “chromatography,” or “color writing,” was coinedby Tsvet. Sideline. Tsvet’s own last name, in Russian, means “color”! Atpresent, chromatography in general denotes any separation method formolecules of different chemical composition by their color or by some otherphysical property, by repeated exposures to quasi-equilibrium conditions.The various types of chromatography are listed in Table 11.1.

Prior 7developedsolid-stategas chromatography in1947.Martin8developedliquid–liquid chromatography in 1941, then paper chromatography in 1944, andfinally laid the foundations for, andperfectedgas-liquidchromatography in1950.

In chromatography, an equation by Van Deemter,9 Zuiderweg,10 andKlinkenberg11 [1] describes the height-equivalent theoretical plate H as a

0 atom% ethanol100% water

T

100% ethanol0 atom% water

A B

100°C

78.5°C

95.6% ethanol4.4% waterconstant-boiling mixture

78.2°C

Liquid composition:Raoult's law:PA

liq = XAliqPA°

Vapor composition: Dalton's law of partial pressures: PA

v = XAvPTOT

XA

P = 1 atm

VAPOR

LIQUID

BOILING POINT CURVE

or LIQUIDUS

CONDENSATION CURVE

L

M N

Q

TIE-LINE

TIE-LINE R

FIGURE 11.2

Temperature (T) versus atom per-cent (X) liquid–vapor diagram forethanol–water mixtures, whichfollow Dalton’s4 law for partialpressures in the vapor phase, andRaoult’s5 law for ideal solutions,but only approximately. One theo-retical plate is the segment LMþMN; another is NQþQR. At point Rthere is a constant-boilingmixture.This is a variation of Fig. 4.7

4 John Dalton (1766–1844).5 Francois-Marie Raoult (1830–1901).6Mikhail Semenovich Tsvet (1872–1919).7 Fritz Prior (ca. 1920– ).8Archer John Porter Martin (1910–2002).9 Jan Josef van Deemter (1918– )10 F. J. Zuiderweg (���).11A. Klinkenberg (���).

11.2 CHROMATOGRAPHY 649

function of the average linear mobile-phase velocity u:

H ¼ EþD=uþ Ru ð11:2:1Þ

where E is the contribution of eddy diffusion to peak broadening, D repre-sents the molecular diffusion in along the column or plate, and R representsthe resistance to mass transfer. In a general chromatogram (Fig 11.3) we notethree peaks, A, B, and C.

In gas chromatography (Fig. 11.4) the carrier gas is usually He, while thestationary phase is typically a coiled Cu tubing (typically 2- to 4-mm innerdiameter, and 1 to 10m long) coated on the inside by some polymer orparticulate solid (with specific surfaces around 1m2 g�1) impregnated with achemically active viscous liquid (usually commercial and proprietary).To improve separations, the Cu coil is usually placed in an oven, whosetemperature is controlled. The mixture of molecules to be analyzed(“analyte”), dissolved in a volatile solvent, is introduced into the carrier gas

Table 11.1 Nomenclature of Chromatography

ChromatographyMobile phase: GasStationary phase: LiquidConfiguration: Column

Name: Gas–liquid (GLC; GC); capillary gas (CGC)Stationary phase: Solid (absorbent or molecular sieve)Configuration: Column

Name: Gas–solid (GSC)Mobile phase: liquidStationary phase: LiquidConfiguration: Column

Name: Liquid–liquid (LLC); column (CC)High-pressure liquid (HPLC)

Configuration: PlaneName: Thin –layer (TLC)

Stationary phase: Solid absorbentConfiguration: Plane

Name: Thin-layer (TLC); paper (PC)Configuration: column

Name: Liquid–solid (LSC)Stationary phase: Bonded solid

Configuration: Plane or columnName: Bonded-phase (BPC)

Stationary phase: Solid resinConfiguration: Column

Name: Ion-exchange (IEC) or ion (IC)Stationary phase: “Solid” gel

Configuration: ColumnName: Size-exclusion (SEC) or gel-permeation (GPC)

Mobile phase: Supercritical fluidStationary phase: Liquid or absorbent solid or bonded solidConfiguration: Column

Name: Supercriticalfluid (SFC)

650 11 INSTRUMENTS

by using a syringe through a rubber septum. The crucial component is thestationary liquid or polymeric support: van der Waals,12 or polar, or elec-trostatic bonds will occur between the analyte and the support. There is nouniversal column for all analytes; rather, a specialized, empirical, and com-mercial effort has developed materials that are best for separating certainnarrow classes of compounds. A small amount of ambient air is also intro-duced from the tip of the syringe needle. At the far end of the Cu tubing is adetector, of which there are several types: (i) a thermal conductivity detectorthatmeasures the thermal conductivity of theHe-analytemixture; (ii) a flameionization detector, where the analyte is mixed with hydrogen, burned, andthe ions generated are collected as a current; (iii) a flame ionization detector;(iv) a flame photometric detector; and (v) an electron capture detector (ECD),where a radioactive source (usually 28Ni63) emits hot electrons, which ionizethe carrier gas, and produce a shower of 100 to 1000 thermalized electronseach, all of whom are detected; when some of these thermalized electrons arecaptured by the analyte, massive ions are created, thus reducing the ther-malized electron signal. The ECD is the most sensitive. In some cases, twocolumns are run in parallel. In someGC experiments a single column is used,but two detectors, one at input and one at output, are used; so-calledfrontalgrams are obtained, and valuable thermodynamic parameters can beobtained. The chromatogram consists of the detector signal versus time. Theair peak usually comes out first (peak B in Fig. 11.3). The retention time ofeach type of analyte (tR in Fig. 11.3) is characteristic of the analyte/stationary

Intensity

Time

B

W

W1/2

Retention time tR

tD t'

A:begin

B: Unretainedcomponent(air peak) C: Retained

peak

FIGURE 11.3

Schematic chromatogram: Point A is the time at which the syringe pierced theseptum;peakB is the “air”peak thatdenotes a component thatdidnot adsorbandequilibrateon the stationaryphase; peakC is thepeak thatdid equilibratewith thestationary phase;W1/2 is its full peakwidth at half-height,whereasW is a graphicalestimate of the total peak width. tR is the retention time for peak C from thebeginning of the experiment.

CarrierGas

FlowController

Injection Port

Column Detector Computer

Oven

FIGURE 11.4

Typical gas chromatograph.

12 Johannes Diderick van der Waals (1837–1923).

11.2 CHROMATOGRAPHY 651

phase combination, and the peak intensity is proportional to the amount ofanalyte introduced. If the separation did not work well, the peak will bedistorted by tailing. Wall-coated open tubular (WCOT) or capillary gaschromatography, invented by Golay13 in 1956, replaces the metal columnsof conventional gas–liquid chromatographs by very thin glass capillaries(0.2-mm inside diameter, at least 25 to 50m long), precoated with activematerials (solid or liquid) and then pulled through an oven hot enough tosoften glass to reduce their diameter.

Liquid–liquid chromatography (LLC) comes in two varieties: ambient-pressure column chromatography and high-pressure or high-performanceliquid chromatography (HPLC).

Columnchromatographyuses anopenvertical glass column,with a solidsupport (e.g., silica gel or alumina) which can interact weakly with a mixtureof analytes dissolved in a liquid eluant, which is fed through the column bygravity. It is used as an inexpensive separation technique, even on a prepar-ative scale.

High-performance liquid chromatography does similar things withmore sophisticated instrumentation. It can separate closely related chemicalcompounds on a research scale or on a preparative scale: liquid solvents, ormixtures of several solvents under positive pressure, replace the “carrier gas”of Fig. 11.3. The solid support must have small particle sizes (3- to 10-mmdiameter), so that relatively high pressures can be sustained throughout thecolumn, and it is at the interface between the liquid eluant and the solidparticles that the chromatographic separation is accomplished.

If the chemistry does not allow any inorganic impurities on the solidsupport particles, then countercurrent chromatography is a bulky anddifficult alternative: Two immiscible solvents with no solid supports areplaced in 10 to 20 connected glass containers, allowing for n¼ 20 partitionsin the sense of Eq. (11.1.2). After the analyte has reached equilibrium incontainer 1, the two liquid phases (one of which is now “partially enriched”in, say, analyte A, the other in analyte B) meet the same two pure liquidphases in container 2, thus allowing for more partial enrichment. Thecumbersome hardware led to replacing glassware by very thin (1-mmdiameter) polytetrafluoroethylene (Teflon�) tubing in the Chromatotron�,which spins many meters of tubing containing segments of liquid phases 1and 2, separated by air gaps, and allows for the same chemical separation asin countercurrent chromatography, but with much economy in time, space,and solvent.

Ion-exchange (or ion) chromatography uses vertical columns loadedwith ionic resins with either mobile anions or mobile cations (typically acidiccations and aminium anions) to separate ionic salts dissolved in water. Theseresins can separate even rare earth salts from each other andwould have beena godsend toMarie Curie! The charge, polarizability, and size of the solvatedion and the properties of the anionic or cationic resins are factors thatinfluence the separability.

Paper chromatography uses paper (cellulose) as the stationary phase, toseparate slightly polar organic compounds from each other; it is usedparticularly in undergraduate laboratory experiments.

13Marcel J. E. Golay (1902–1989).

652 11 INSTRUMENTS

Size-exclusion chromatography (SEC) or gel-permeation chromato-graphy (GPC), introduced by Moore14 in 1964, uses size as a way toseparate polymeric analytes: If the analyte particle is too large, it will notbe absorbed, while if it is small enough, or even smaller than the averagepore diameter of the gel, then absorption occurs. Polymers can thusbe separated. Polymers can be described by several measures of polymersize: number-average molar mass (Mn), the mass-average molar mass(Mw), size-average molar mass (Mz), or the viscosity molar mass (Mv).GPC can measure both Mv and (Mn/Mw), which also known as thepolydispersivity index.

Supercritical fluid chromatography (SFC) is very similar in principle to,and is as convenient as, high-performance liquid chromatography, but it usesas the high-pressure eluant fluid CO2 (or other fluid) above its critical point(for CO2: TC¼ 31.3�C, PC¼ 7.38MPa, rC¼ 0.448 g cm�3). SPC can separaterelatively small and/or thermally labile molecules. The analyte is introducedas a solution in methanol. Small amounts of organic solvents can be added as“modifiers.” Any CO2 brought out with the analyte to ambient atmospherewill evaporate harmlessly.

11.3 BIOCHEMICAL SYNTHESIZERS

While chemical synthesis is mostly an art, with specialized reactions for bothinorganic and inorganic synthesis, the complexities of biochemistry havenurtured specialized instruments that can split or assemble biomolecules.Anautomatedsolid-phasepeptide synthesizerwas introducedbyMerrifield15

in 1963: this allows for the facile synthesis of oligopeptides (up to 100 aminoacid units) [2]. The enzymeDNApolymerase I was discovered by Kornberg16

in 1957: this allowed the assembly of DNA from fragments [3].

11.4 ELEMENTAL ANALYSIS

A traditional and essential tool in synthetic chemistry is the determination ofthe mass fraction of chemical elements in supposedly pure samples. Organicchemists, in particular, require precision in an empirical formula CxHyOzNw;the percent composition (e.g., 100x, 100y, etc.) must correspond to theexpected structural formula to within better than 0.3%. The traditionalcombustion analysis uses maybe 20-mg samples (weighed to� 0.1 mg), burnsthem in excess oxygen, and adsorbs the released H2O vapor as a mass gainin anhydrous Mg(ClO4)2, thus determining y, and the released CO2 gas ismeasured as amass gain inLiOH,which is partially converted to Li2CO3, thusdetermining x. Nitrogen is analyzed by the Kjeldahl method (digestionof organic sample in hot concentrated H2SO4, which converts all N into(NH4)2SO4; cooling, andneutralizing inNaOH,which yieldsNH3gas; the gas

14 John C. Moore (���).15 Robert Bruce Merrifield (1921–2006).16Arthur Kornberg (1918–2007).

11.4 ELEMENTAL ANALYSIS 653

is collected in a measured excess of a standard acid solution; the excess acidis back-titrated with NaOH). For inorganic samples, there is a traditionalsystematic and qualitative separation scheme of elements, which usuallyrequires samples of about 50mg and can be made quantitative if the manydetailed reactions are carried outwith consummate care (Scheme 11.1). Thesedays, such old schemes are replaced (i) by spectrophotometricmeasurements(when interfering species are chemically removed), (ii) by atomic absorptionspectroscopy (which is extremely sensitive, but requires careful calibration!),or (iii) by X-ray fluorescence spectroscopy.

One starts by fusing the starting material to constant weight, then fusingit with excess NaOH. NaNO3, and Na2CO3, and adding water; the residuewill contain (1.0) the basic elements {Fe2O3, MnO2, TiO2, BaCO3, CaCO3,Mg(OH)2, Ag2O, Ag, CuO, NiO, PbO2, and SnO2}, the solution willcontain as soluble sodium salts: (2.0) the amphoteric elements

{PbðOHÞ2�4 , CuðOHÞ3�3 , AsO3�4 , SnðOHÞ2�6 , AlðOHÞ2�4 , ZnðOHÞ�3 , CrO2�

4 , and

VO�3 } and also (3.0) the acidic elements {I�, Br�, Cl�, PO3�

4 , AsO3�4 , SO2�

4 , F�,

NO�3 , CO

2�3 , and SiO4�

4 }. Na and K must be analyzed separately, since so

much Na is added in this scheme.Dissolving the basic element precipitates (1.0) with HNO3 and

NaClO4, followed by neutralization and addition of sodium acetateprecipitates the titanium subgroup (1.1) {Fe2O3, MnO2, TiO2, PbO2, &SnO2}; the remaining solution (1.2) is treated with NH4OH and(NH4)2CO3, which precipitates the alkaline earth subgroup (1.3){BaCO3,CaCO3, and MgCO3}; the remaining solution is treated with NaOH andboiled to remove NH3: the precipitate contains themetal amine subgroup(1.4) {Ag2O, CuO, and Ni(OH)2}.

The amphoteric element solution (2.0) is partly neutralized with HCl, made0.2M in HNO3, and heated to remove NO and NO2. Adding CH3CSNH2,heating, adding (NH4)2SO4, and heating again precipitates (2.1) the sulfidesubgroup {PbS, CuS, As2S3, SnS2, and S}; the solution (2.2) will consist ofthe aluminum–chromium group (Al3þ, Zn2þ, Cr3þ, and VðOHÞ2þ2 }. Thealuminum–chromium group is separated by treating with NH4(C6H5COO) andNaHS; the precipitate (2.3) is dissolved in HCl; HNO3 is added, then the solutionis diluted, H2O2 and NaOH are added, and the solution is saturated with CO2:the precipitate will be (2.4) the aluminum subgroup {Al(OH)3, ZnCO3}; thesolution (2.5) will be the chromium subgroup {CrO2�

4 , VO�3 }. The sulfide group

(2.1) is separated by treatment with NaHS and NaOH: the precipitate (2.6) is thelead subgroup {CuS, PbS} while the solution, treated with HCl, yields as aprecipitate (2.7) the arsenic subgroup {As2S5, SnS2, S}.

The acidic element solution (3.0) is saturated with CO2: (3.1) H4SiO4 pre-cipitates; the solution (3.2) is treated with HNO3 and AgNO3: the precipitate (3.3)is the halogen subgroup {AgI, AgBr, AgCl}. The solution (3.4) is neutralizedwith Na2CO3 and treated with CH3COOH, NH4CH3COO, and AgNO3: theprecipitate (3.5) is the phosphorus subgroup {Ag3PO4, Ag3AsO4}; the solution(3.6) is the sulfur and fluorine subgroup {HSO�

4 , HF�2 }.The cited reference discusses the further detailed and necessary separations

within each of the specified subgroups: titanium (1.1), alkaline earth (1.3), metalamine (1.4), sulfide (2.1), aluminum (2.4), chromium (2.5), lead (2.6), arsenic (2.7),halogen (3.3), phosphorus (3.5), sulfur and fluorine (3.6).

SCHEME 11.1

Inorganic qualitative analysisscheme, from Swift [4].

654 11 INSTRUMENTS

11.5 MASS SPECTROMETRY

The first deflection by a magnet of an ion “canal ray” beam inside a Crookstubewas obtained byWien17 in 1899, and the firstmodernmass spectrumwasmeasured by Dempster18 in 1918 and by Aston19 in 1919. Mass spectrometrytypically consists of a sample inlet, an ion source, a mass analyzer (either amagnet or an electrical quadrupole, or a time-of-flight path), and a detector.As shown in Fig. 11.5, the sample inlet, the ion source, themass analyzer, andthe detector are all enclosed in a high-vacuum system (10�4 torr to 10�10 torr).The equation formotion of an ion of charge Z|e| and velocity v in amagneticfield B is given by

F ¼ mðdv=dtÞ ¼ Zjejv� B ð11:5:1Þ

where F is the Lorentz20 force and |e| is the electronic charge. If themagneticfield is constant, the centripetal force (m v2/r) will equal the Lorentz force:

mv2=r ¼ ZjejvB ð11:5:2Þ

The kinetic energy of an ion of charge Z|e| and mass m accelerated to apotential V is given by

ZjejV ¼ ð1=2Þmv2 ð11:5:3Þ

Therefore, finally the mass-to-charge ratio (m/Z) is given by

m=Z ¼ B2r2jej=2V ð11:5:4Þ

Furthermore, v/r is the angular frequency o (radians s�1); thus the ion cantravel a circular path, at the cyclotron frequency n (Hz):

n ¼ o=2p ¼ v=2pr ¼ ZjejB=m ð11:5:5Þ

For a bare electron of charge -|e| in afield of 0.0875 T, the cyclotron frequencyis 2.45GHz.

SampleInlet

System

Ion Source

MassAnalyzer Detector

Vacuum System (10–4 to 10–10 torr)

Computer

FIGURE 11.5

Mass spectrometer.

17Wilhelm Carl Werner Otto Fritz Franz Wien (1864–1928).18Arthur Jeffrey Dempster (1886–1950).19 Francis William Aston (1877–1945).20Hendrik Antoon Lorentz (1853–1928).

11.5 MASS SPECTROMETRY 655

The mass m is usually measured in Daltons (Da) or atomic mass units(amu; mass of one 6C

12 nucleus 12.0Da 12.0 amus, by definition;1Da¼ 1.660538782� 10�27 kg). Mass spectrometers can resolve isotopesrather well, and they can measure them to high precision (sometimes to 1part in 107, but not in most routine or commercial instruments). Their massrange can be huge, from 1 amu to 100 kDa (the wider ranges have lowerresolution, but always below� 1Da). The sample sizes are of the order ofmicrograms to nanograms.

Mass spectra can be measured for cations (Z > 0) or, less frequently, foranions (Z < 0), provided that they can survive in a high vacuum. The fouressential components are as follows:

(a) The sample inlet system may be: (a1) a septum for a liquid sample,introduced by syringe; (a2) a valve system to introduce a meteredamount of a gaseous sample; (a3) a substrate from which a polymericsample, or a biological macromolecule, can be liberated by meansdescribed below.

(b) The ion source may be as Follows:

For atoms:

(b1) inductively coupled plasma (ICP) of hot Arþ ions;

(b2) direct-current plasma (DCP) of hot Arþ ions;

(b3) microwave-induced plasma (MIP) of hot Arþ ions;

(b4) spark source (SS);

(b5) thermal ionization (TI) by an electrically heated plasma;

(b6) glow discharge (GD) plasma;

(b7) laser microprobe (LM) using a focused laser beam;

(b8) secondary ion (SI) using bombardment by accelerated ions.

For easily vaporized molecules:

(b9) electron impact ionization (EI), using a 70-V electron source(heated W or Rh filament);

(b10) chemical ionization (CI) using ions derived from a source gas(CH4) to ionize the analyte;

(b11) field ionization (FI) using 10-mm-diameter W wire ending in avery sharp tip.

For macromolecules:

(b12) Field desorption (FD), where several W wires ending in sharptips are coated by the solution of the analyte, then introducedinto the vacuum chamber, where high voltage will allow themacromolecule ion to desorb from the W;

(b13) electrospray ionization (ESI) (invented by Fenn21 in 1984) of asolid, dissolved in a volatile solvent, and passed through a thincapillary that is held at several kiloelectronvolts; the resultingionize spray loses its volatile solvent molecules, whereafter thecharged macromolecule can be mass-analyzed;

21 John Bennett Fenn (1917– ).

656 11 INSTRUMENTS

(b14) matrix-assisted desorptionþionization (MALDI) invented in1988, where the matrix consists of organic acids (e.g., succinicacid) or bases (e.g., 2-amino-4-methyl-5-nitropyridine) which“dissolve” sensitive analytes, such as proteins; under theinfluence of a strong laser beam (CO2 at 10.6 mm, Nd-YaG at266 or 355 nm, N2 (337 nm), in the vapor, some analyte mole-cules get protonated at the expense of the matrix molecules;these protonated analytes are then injected in the massspectrometer;

(b15) plasma desorption (PD);

(b16) fast-atom bombardment (FAB) where a sample trapped in a gelor other matrix is sputtered by energetic (keV) but neutral Xe orAr atoms;

(b17) secondary-ion mass spectrometry (SIMS);

(b18) thermospray ionization.

(c) The mass analyzer can be:

(c1) a magnet which turns the beam in a circular path by a quarterturn (90�), or by a half turn (180�);

(c2) a time-of-flight (TOF) tube,which first accelerates a pulse of ionsto the same voltage and thus the same kinetic energy (103 to104V); these ionswill drift through the tubebut at different rates,because the ions of smallermassmwill havehigher speeds v; (c3)an electric quadrupole mass analyzer (particularly popular forstudying atoms);

(c4) a double-focussing analyzer, which uses a quadrupole massanalyzer followed by a magnetic mass analyzer.

(d) The detector may be:

(d1) a photomultiplier;

(d2) a Faraday22 cup;

(d3) a microchannel plate;

(d4) a micro-Faraday array transducer;

(d5) a Daly23 detector.

(d6) an Orbitron�

Mass spectrometry has three major uses: (1) determining the massspectrum of new compounds (the crucial datum for synthetic chemist is themolar mass M/z for the analyte, plus maybe an extra proton furnished insample injection port, (2) determining how a molecule breaks up into frag-ments after its first anion or cation is produced: the fragmentation pattern canreveal some aspects of bonding within the molecule; (3) following certainreactions and establishing the order of reactivity (in protonation, electrondetachment, electron attachment, etc.).

A mass spectrometer with large mass range but moderate resolution(�0.1Da) will see overlap between, say, C12H and C13 (which should be 0.01

22Michael Faraday (1791–1867).23Norman Richard Daly (ca. 1930– ).

11.5 MASS SPECTROMETRY 657

times as intense as the corresponding peak with C12, because the isotopicabundance of C13 in carbon is 1%); a high-resolution MS can detect the massdifference between these peaks.Ameasure of this is the resolutionR¼m/Dm,where m is the mass (in Da) of interest, and Dm is the mass difference (in Da)that wants to be resolved (i.e., two “peaks” with a difference Dmmust have a“valley” between them that is no more than one-tenth of the peak height).Instruments with R from 500 to 500,000 are available.

A technique related to mass spectrometry and instrumentally verysimilar to electron paramagnetic resonance is ion cyclotron resonance (ICR)spectrometry, invented by Baldeschwieler24 and others at Varian Associatesin 1964; the Fourier transform version (FT-ICR) was invented by Marshall25

and Comisarow26 in 1974. ICR uses essentially the same instrumentation aselectron paramagnetic resonance (EPR), but concentrates on the detection ofions produced externally that under moderate vacuum in a microwavecavity undergo cyclotron resonance at a frequency n¼Z|e|B/2pm (typicallydetected by varying B at fixed n 9GHz).

Application (Developed by C. Cassady): Analysis of Peptides by MALDI/TOF-MS. Matrix-assisted laser desorption ionization (MALDI)produces protonated molecular ions, [MþH]þ, for most small peptidesand is usually combined with a time-of-flight (TOF) mass selection.MALDI/TOF can be used to identify an oligopeptide in the positive-ionmode: All ions of interest will have a charge of þ1. Because MALDI is arelatively gentle ionization technique, little fragmentation occurs.However, by increasing the power of the bombarding laser, it ispossible to generate fragments that can be used to sequence peptides.This technique is known as post-source decay (PSD). At higher laserpowers, MALDI can impart excess energy to a parent ion and inducefragmentation. Ions are usually stable for the 200 to 500 ns that they spendin the source area, but do fragment during their microsecond time-scaletravel within the flight tube. The fragment “daughter” ions, with their ownunique mass-to-charge ratio m/Z, are formed after the protonatedmolecular ions leave the source and its accelerating voltage V. Givenhow they are formed, these daughter ions have the velocity v thatwould be expected for their parent ions. Since TOF measures flight timet, which relates to velocity and the length of the flight tube L, in a linear TOFsystem these fragment ions would appear at the same flight time as theirparent ions and would be seen as a single undifferentiated peak along withthe parent ion. However, an electric field E, added to the center of the flightpath, serves as a second mass analyzer (Fig. 11.6).

This electric field, known as a reflector or reflectron, separates thefragment ions by m/Z, so that they can be detected (the reflectron voltagesrange between 1 and 25 kV). Because only a narrow range of m/Z values canmake it through the reflectron at each setting of E, the complete range offragment ions can only be obtained by acquiring a series of 14 individualmass spectra over narrow m/Z ranges. These spectra are then “stitched”together by the software to form a post-source decay spectrum. The PSD

24 John Dickson Baldeschwieler (1933– ).25Alan George Marshall (1944– ).26Melvin Barnet Comisarow (1941– ).

658 11 INSTRUMENTS

fragment masses are measurable to� 0.5Da. PSD usually yields usefulinformation only when the parent m/Z � 5 kDa; it can be performed forpeptides, oligonucleotides, carbohydrates, polymers, and so on. Theanalysis of a mixture involves parent ion pre-selection, which is based onflight-time differences, as different parent ions leave the source area. Adeflection voltage only allows ions through a specific time window of�10 ns. At m/Z¼ 1 kDa, this corresponds to a window of� 10Da. The21 most common a-amino acids are shown in Fig. 11.7, and their residuemass data are listed in Table 11.2.

Of these 21 a-amino acids, nine have nonpolar “R” groups (Ala, Val, Leu,Ile, Pro, Hyp, Phe, Trp,Met), seven have polar “R” groups (Gly, Ser, Thr, Cys,Tyr, Asn, Gln), two are acidic (Asp, Glu: negatively charged at pH 6.0), andthree are basic (Lys, Arg, His: positively charged at pH 6.0). Two of theseamino acids contain S: Met and Cys. There exist also b-amino acids. Of these21 amino acids, adult humans can produce only 11 (Ala, Asn, Asp, Cys, Glu,Gln, Gly, Hyp, Pro, Ser, Tyr). The other 10 (Arg, His, Ile, Leu, Lys, Lys, Met,Phe, Thr, Trp, Val) must be supplied by continuously eating and digestinganimal or vegetable protein.

The acid equilibrium constant K1 involves the COOH termination; thebase equilibrium constant K2 involves the amine termination. At the iso-electric point (pH¼ (1/2)(pK1þpK2) the amino acid is a zwitterion or betaine:The proton is detached from the carboxylate end and is attached at the amineend. Dipeptides are formed by the addition of any two amino acids and a lossof a water molecule, for example:

TyrþArg! “Tyr-Arg”þH2O

and so on for all polypeptides. The polypeptides are named startingfrom the amine end (the N-terminus; on the left) and ending at thecarboxylic acid end (the C-terminus). Oligopeptides also have an iso-electric point.

Protonated peptides fragment in ways that are now well established(Fig. 11.8). Note that the “a,” “b,” and “c” ions incorporate the peptide’sN-terminus, while “x,” “y,” and “z” ions incorporate the peptide’s C-terminus. The numerical subscript refers to the number of residues thatthe ion contains. “c” and “y” ions involve rearrangements that add to theions two more hydrogens than they would have from simple cleavagealone. Figure 11.9 shows the CID spectrum of the protonated molecular ion

FIGURE 11.6

The electric field E of the reflectron(energy 1 to 25 kV) acts as a secondmass analyzer.

11.5 MASS SPECTROMETRY 659

CH3C

H

NH2

COOH

Alanine (C3H7O2N, Ala, A)

CCH

H

NH2

COOH

H3C

H3C

Valine (C5H11O2N, Val, V)

CH2C

H

NH2

COOHCH

H3C

H3C

Leucine (C6H13O2N, Leu, L)

CHC

H

NH2

COOHH2C

Isoleucine (C6H13O2N, Ile, I)

CH3

H3CH2C

H2CNH

CH

H2C

COOH

Proline (C5H9O2N, Pro, P)

NH

CHC

CH2C

H

NH2

COOH

Tryptophan (C11H12O2N2, Trp, W)

CH2C

H

NH2

COOHH2C

Methionine (C

(a)

5H11O2NS, Met, M)

SH3C C

H

NH2

COOH

Glycine (C2H5O2N, Gly, G)

H

CH2C

H

NH2

COOH

Serine (C3H7O3N, Ser, S)

HO CHC

H

NH2

COOH

Threonine (C4H9O3N, Thr, T)

H3C

OH

CH2C

H

NH2

COOH

Cysteine (C3H7O2NS, Cys, C)

HS

C

H

NH2

COOH

Tyrosine (C9H11O3N, Tyr, Y)

HOH2C

Hydroxyproline (C5H9O3N, Hyp, O)

CH2C

H

NH2

COOHC

H2N

O

Glutamine (C5H10O3N2, Gln, Q)

H2C

HC

H2CNH

CH

H2C

COOH

HO

C

H

NH2

COOH

Phenylalanine (C9H11O2N, Phe, F)

H2C

CH2C

H

NH2

COOHC

Histidine (C6H9O2N3, His, H)

CH2C

H(b)

NH2

COOHHOOC

Aspartic acid (C4H7O4N, Asp, D)

CH2C

H

NH2

COOHHOOC

Glutamic acid (C5H9O4N, Glu, E)

H2C C

H2C

H

NH2

COOHH2C

H2C

H2CH2N

Lysine (C6H14O2N2, Lys, K)

CH2C

H

NH2

COOHH2C

H2C

HNC

Arginine (C6H14O2N4, Arg, R)

H2N

NH

HC

N

HCNH

CH2C

H

NH2

COOHC

Asparagine (C4H8O3N2, Asn, N)

O

H2N

FIGURE 11.7

The 21 common a-amino acids.

660 11 INSTRUMENTS

(MþH)þ of a derivative of the linear pentadecapeptide and antibioticVal-gramicidicin A, a transmembrane protein; this derivative has structureformyl-NH-V-G-A-L-A-V-V-V-W-L-W-L-W-L-W-ethanolamine. The CID(collision-induced dissociation) spectrum is very similar to a MALDI/TOFspectrum. Figure 11.10 shows the PSD of ACTH, the adenocorticotropichormone.

Table 11.2 The Twenty-One Common a-Amino Acids, Their Chemical Formulae, Isotopic Molar Masses[Using H1 (99.99%)¼ 1.007825, C12 (98.91%)¼ 12.0000, N14 (99.6%)¼ 14.0031, O16 (99.76%)¼ 15.9949,S32 (95.02%): 31.9721], Chemical Molar Masses [Using C¼ 12.011, H¼ 1.0079, O¼ 15.9994, N¼ 14.0067,S¼ 32.066], Three-Letter Codes, Single-Letter Codes, and Residue Mass Values (Daltons) for FragmentNH-CH(R)-C(O)- [¼Neutral Isotopic Molar Mass�One Molecule of Water (18.0106), Using H1

(99.99%)¼ 1.007825, C12 (98.91%)¼ 12.0000, N14 (99.6%)¼ 14.0031, O16 (99.76%)¼ 15.9949, S32

(95.02%): 31.9721]

Amino AcidChemicalFormula

IsotopicMolar

Mass(Da)

ChemicalMolar Mass

(Da)IsoelectricPoint (pH)

Three-LetterCode

Single-LetterCode

ResidueMass

(Daa/amu(or Daltons)

Glycine C2H5O2N 75.032 (75.067) 5.97 Gly G 57.021Alanine C3H7O2N 89.048 (89.094) 6.02 Ala A 71.037Serine C3H7O3N 105.043 (105.093) 5.68 Ser S 87.032Proline C5H9O2N 115.063 (115.132) 6.30 Pro P 97.052Valine C5H11O2N 117.079 (117.147) 5.96 Val V 99.068Threonine C4H9O3N 119.058 (119.120) 6.53 Thr T 101.043Cysteine C3H7O2NS 121.020 (121.149) 5.07 Cys C 103.009Hydroxyproline C5H9O3N 131.058 (131.131) 5.83 Hyp O 113.047Isoleucine C6H13O2N 131.095 (131.116) 6.02 Ile I 113.084Leucine C6H13O2N 131.095 (131.116) 5.98 Leu L 113.084Asparagine C4H8O3N2 132.054 (132.119) 5.41 Asn N 114.043Aspartic acid C4H7O4N 133.037 (133.104) 2.97 Asp D 115.027Lysine C6H14O2N2 146.106 (146.189) 9.74 Lys K 128.095Glutamine C5H10O3N2 146.069 (146.156) 5.65 Gln Q 128.059Glutamic acid C5H9O4N 147.053 (147.130) 3.22 Glu E 129.043Methionine C5H11O2NS 149.051 (149.214) 5.74 Met M 131.040Histidine C6H9O2N3 155.069 (155.156) 7.58 His H 137.059Phenylalanine C9H11O2N 165.079 (165.191) 5.48 Phe F 147.068Arginine C6H14O2N4 174.112 (174.202) 10.76 Arg R 156.101Tyrosine C9H11O3N 181.074 (181.191) 5.65 Tyr Y 163.063Tryptophan C11H12O2N2 204.090 (204.228) 5.89 Trp W 186.079

Fragmentation pattern of generic tetrapeptide, and its four amino acid residues

HCNH

C

R1 OHN C

HC

R2 OHN C

HC

R3 OHN C

HC

R4

x3

a1

y3

b1

x3

c1

x2

a2

y2

b2

x3

c2

x1

a3

y1

b3

z1

c3

+2H+2H +2H

+2H+2H +2H

O

OHH

Residue 1 Residue 4Residue 2 Residue 3FIGURE 11.8

Fragmentation pattern of ageneric tetrapeptide.

11.5 MASS SPECTROMETRY 661

Application: Ionization and Electron Affinity. To measure the adiabatic first,second, and other ionization potentials I1, I2, I3 for a molecule D:

D!Dþ þ e� DE ¼ I1 ð11:5:4Þ

Dþ !D2þ þ e� DE ¼ I2 ð11:5:5Þ

Dþþ !D3þ þ e� DE ¼ I3 ð11:5:6Þ

many techniques can be used,most oftenmass spectrometry or ion cyclotronresonance spectroscopy, where the vertical excitation energy necessary togenerate themolecularmonoanionAþ is readilymeasured, but not so readilyconverted to the adiabatic energy I1; these measurements are “bracketed,” bycomparing appearance potentials for the product(s) with appearance poten-tials in the same instrument for a reaction whose energies are well known.

FIGURE 11.9

CID mass spectrum of the protonated molecular ion (m/Z¼ 1882.1) of a chemicalderivative of the pentadecapeptide Va1-gramicidin A. This linear pentadecapep-tide has structure formyl-NH-V-G-A-L-A-V-V-V-W-L-W-L-W-L-W-C(O)-ethanolamine. However, the derivative studied here is HC(O)-V-G-A-L-A-[V-V-V]-[W-L-W-L-W-L]-W-C(O)-NH(CH2)2OH. This derivative is viewed as{V-G-A-L-A-[V-V-V]-[W-L-W-L-W-L]-W}, but at the left N-terminus the group HCO-is added, while at the right “C-terminus” the group -NH(CH2)2OH is added, plusone more proton. The mass is thus given by 12þ 16þ 1þ {99.07þ 57.02þ 71.04þ 113.08þ 71.04þ 3 � 99.07þ 3 � (186.08þ 113.08)þ 186.08}þ 14þ 1þ 2 � (12þ 2)þ 16þ 1þ [1]¼ 12þ 16þ 1þ {1792.02}þ 14þ 1þ 2�(12þ 2)þ 16þ 1þ [1]¼ 1882;indeed the observed peak is at 1882.1. The fragment b2 corresponds to HC(O)-HN-R1CH-CO-NH-R2CH-CO, with R1¼CH(CH3)2 and R2¼H, that the fragment corre-sponds to HC(O)-Val-Gly or HC(O)-V-G, with mass (12þ 16þ 1)þ {57.02þ 99.07}¼ (29)þ {156.09}¼ 185 (the observed peak is at �185); fraction b3 corresponds tothe fraction HCO-VþGþAwith mass (12þ 16þ 1)þ {57.02þ 99.07þ 71.04}¼ (29)þ {227.13}¼ 256.13 (peak is at �255); the fraction y14 corresponds to {G-A-L-A-V-V-V-W-L-W-L-W-L-W}-C(O)-NH(CH2)2OHþ 2H¼ {57.02þ 71.04þ 113.08þ 71.04þ3�99.07 þ 3 � (186.08 þ 113.08) þ 186.08} þ 14 þ 1 þ 2 � (12 þ 2) þ 16þ 1þ [2]¼{1692.95}þ 14þ 1þ 2�(12þ 2)þ 16þ 1þ [2]¼ 1752.25; the observed peak is at�1755.

662 11 INSTRUMENTS

FIGURE 11.10

PSD of the tail end (residues 18 through 39) of a-ACTH (adrenocorticotropic hormone, or corticotropin). Theobserved protonated molecular ion is at 2447.2 Da. The full structure of a-ACTH is the much larger oligopeptide,which consists of 39 amino acids {[Ser-Tyr-Ser-Met-Glu-His-Phe-Arg-Trp-Gly-Lys-Pro-Val-Gly-Lys-Lys-Arg-Arg-Pro-Val-Lys-Val-Tyr-Pro]-[Asp-Ala-Gly-Glu-Asp-Gln-Ser-Ala-Glu-Ala-Phe-Pro-Leu-Glu-Phe]}, or {[S-Y-S-M-E-H-F-R-W-G-K-P-V-G-K-K-R-R-P-V-K-V-Y-P]-[D-A-G-E-D-Q-S-A-E-A-F-P-L-E-F]}, whose molar mass is 1þ {[87.03þ 163.06þ 87.03þ131.04þ 128.06 þ 137.06 þ 147.07 þ 156.10þ 186.07þ 57.02þ 128.10þ 97.05þ 99.07þ 57.02þ 128.10þ 128.10þ156.10 þ 156.10 þ 97.05 þ 99.07 þ 128.10þ 99.07þ 163.06þ 97.05]þ [115.03þ 71.04þ 57.02þ 129.04þ 115.03þ128.06 þ 87.03 þ 71.04 þ 129.04 þ 71.04 þ 147.07 þ 97.05 þ 113.08 þ 129.04 þ 147.07]}þ 17¼ 1þ {[2912.58]þ[1606.68]}þ 17¼ 4537.3, which is too big. Instead, the much shorter 22-amino acid residue 18–39 is just {Arg-Pro-Val-Lys-Val-Tyr-Pro-Asp-Ala-Gly-Glu-Asp-Gln-Ser-Ala-Glu-Ala-Phe-Pro-Leu-Glu-Phe]}, or {R-P-V-K-V-Y-P-D-A-G-E-D-Q-S-A-E-A-F-P-L-E-F}, whose molar mass is {156.10þ 97.05þ 99.07þ 128.10þ 99.07þ 163.06þ 97.05þ 115.03þ 71.04þ 57.02 þ 129.04 þ 115.03þ 128.06þ 87.03þ 71.04þ 129.04þ 71.04þ 147.07þ 97.05þ 113.08þ 129.04þ147.07}¼ 2446.18, which is close enough to what is shown. Thepeakat70.1¼ 1þ {97.1}�12�16correspondstoa1forproline. The peak at 353.2 may be due to the LEF right end, but this is too big, or to the RPV left-end (b3) as 1þ {156.10þ 97.05þ 99.07}¼ 1þ 352.22: this works. The peak at 464.3 may correspond to the left-end RPVK with b4¼ 1þ {156.10þ 97.05þ 99.07þ 128.10}¼ 1þ {480.32} which is too large, or else it may correspond with the right-end PLEF withy4¼ 2þ {97.05þ 113.08þ 129.04þ 147.07}þ 17¼ 503.24, which is too large, or with x3¼{97.05þ 113.08þ 129.04þ147.07� 28}þ 17¼ 465.24, which is close enough.

It is not easy to measure directly the adiabatic first and second electronaffinities A1 and A2, and so on, of a molecule:

Aþ e� !A� DE ¼ A1 ð11:5:7Þ

A� þ e� !A2� DE ¼ A2 ð11:5:8Þ

11.5 MASS SPECTROMETRY 663

Once again, it is difficult to be sure that both the reagent and the productare in their electronic, vibrational, and rotational ground states. Indirectly,one can measure electron affinities by using crossed atomic and molecularbeams—for example, Cs beam collisional ionization:

Aþ Cs!A� þ Csþ DE ¼ I1 ð11:5:9Þ

where the energy of the scattered Csþ ion is measured (but it is not clear thatthe daughter anion A� was created in its ground state). Electron affinitieswere first measured by themagnetronmethod, but it was never clear whichstates were involved, so this method is obsolete. One can measure electronaffinities by mass spectrometry or ion cyclotron resonance spectroscopy,or by X-ray photoelectron spectroscopy. As for the ionization potentialestimates discussed above, these electron affinity measurements are“bracketed,” by comparing appearance potentials for the product(s) withappearance potentials in the same instrument for a reaction whose energiesare well known. Figure 11.11 shows some ionization potentials and electronaffinities.

Combination Instruments. In order to enable good analyses for complicatedsamples, mass spectrometers or ion cyclotron resonance spectrometersare often front-ended with separation instruments: with input gaschromatographs (GC-MS) or liquid chromatographs (LC-MS), or GC-ICR,or LC-ICR, and so on.

graphite

DDQ, 8TCNQ, 7

TMPD, 1

f

Au(111)

Al(111)

TMPD, 1

TTF, 2

BEDT-TTF, 3

benzene, 4

10 eV

5 eV

vacuum level

TTF, 2

BQ, 5 TCNQ, 7 DDQ, 8

0 eV

BEDT-TTF, 3

C60, 6

N(CH3)2

N(CH3)2

410 eV

5 eV

S

S

S

S

S

S

S

S

S

S S

S CN

O

O

CNCl

Cl

CNNC

CNNC

O

O

BQ, 5, C60, 6

AA

Pt

Mg

ID

0 eVFIGURE 11.11

Someorganicone-electrondonorsD(1–4) with their first gas-phase ioni-zationpotentials ID (¼ 6.25eV forN,N,N0,N0-tetramethyl-para-phenyle-nediamine (TMPD, 1),¼ 6.83eV fortetrathiafulvalene (TTF, 2),¼ 7.6eVfor bisethylendithio-tetrathiafulva-lene (BEDT-TTF, 3),¼ 9.38eV forbenzene (4)) some one-electron ac-ceptors A (5–8) with their gas-phaseelectron affinities AA (¼ 1.9eV forpara-benzoquinone (BQ,5),¼ 2.6 to2.8 eVforbuckminsterfullerene(C60,6),¼ 2.8 eV for 7,7,8,8-tetracyano-quinodimethan, (TCNQ,7), thencor-rected to 3.3eV,¼ 3.13eV for 2,3-chloro-5,6,-dicyano-para-benzoqui-none(DDQ,8),andsomemetalswiththeir bulk work functions f (¼ 3.66eV for Mg,¼ 4.24eV for Al(111),¼ 5.31eV for Au(111),¼ 5.7eV for Pt(111), and¼ 4.3 eV for thesemimetal graphite).

664 11 INSTRUMENTS

11.6 SPECTROSCOPY

Bunsen27 and Kirchhoff28 launched spectroscopy in 1859, by findingthat each element or compound emits light of several characteristic wave-length(s): its spectrum (Latin for ghost). For the next 60 years, spectroscopywallowed in empiricism; atoms were found to emit or absorb “linespectra,” while molecules had much broader and more diffuse “bandspectra.” All these spectra were measured and catalogued, but wereunderstood only when the quantum theory of atoms and molecules ex-plained everything in the 1920s. The early spectrographs in the nineteenthcentury used a stone or concrete slab, black felt paper to block ambientlight, a high-intensity lamp or spark source, a set of slits to narrow thebeam, a dispersion unit (prism or grating), and photographic film asthe detector.

11.7 VISIBLE–ULTRAVIOLET (V–UV) SPECTROSCOPY

Before the 1980s, when digital microcomputers (PCs) were wedded to mostinstruments, most typical research-level visible–ultraviolet (V–UV) spec-trometers had been dual-beam instruments (e.g., the historic Cary� 14)(Fig. 11.12), which covered the wavelength range 200–900 nm (or200–1100 nm) with a single source (W lamp) and, in some cases, with asecond source, (D2 lamp) which extended the range to 2600 nm. These dual-beam instruments usually have one source, one set of entry slits, a focusinglens, and a chopper; they have one sample chamber and one referencechamber, a second lens, mirrors, and one detector (Si diode), and they usethe chopper to divide the source beam between the two sample chambers.Using the same optics and a reference container (filled with an “empty”sample cuvette with solvent but no solute), the difference spectrum for thesolute (sample – reference) takes care of most nonlinearities in the lampoutput and detector efficiencies, and of solvent absorption, to yield the“true” absorption spectrum of the solute in the sample chamber. With PCscapable of monitoring the data collection, a reverse trend has developedtoward single-beam instruments: There is only one sample chamber; thespectra for the reference are collected in a separate run and subtracted bythe PC software.

The absorbance intensity IL is expected to follow Beer’s29 law:

IL ¼ I0expð�e1! uðnÞcBÞ ðð3:31:7ÞÞ

where I0 is the input light intensity at frequency n,B is the path length (cm), c isthe concentration of the absorbing species (mol per liter), and el! u(n) is the

27 Robert Wilhelm Eberhard Bunsen (1811–1899).28Gustav Robert Kirchhoff (1824–1887).29August Beer (1825–1863).

11.7 VISIBLE–ULTRAVIOLET (V–UV) SPECTROSCOPY 665

decadic molar extinction coefficient. If the sample is not homogeneous, or ifit contains microparticles or micelles that scatter light, then deviations fromBeer’s law will be seen.

Enclosing the sample chamber in a thermostat, one can follow theabsorbance as a function of absolute temperature and thus perform kineticstudies of reactions in solution.

Solvents used for visible–ultraviolet spectroscopy may be used only forwavelengths greater than someultraviolet cutoffwavelength lc, belowwhichthe solvent absorbs strongly. These cutoffwavelengths lc are listedwith someother useful data in Tables 11.3 and 11.4.

The sensitivity of an absorbance measurement is limited by how wella detector can discriminate between I0 and IL: ratios (IL/I0)¼ 10�5 can bemeasured with ease. This means that concentrations of 1 mmol/L can bestudied, if el! u(n) is large enough. It should be noted that the overallabsorbance of an electronic transition at room temperature is typicallybroadened by vibronic and rotational modes: The full spectrum could

H2 Lamp

Tungsten Lamp

PbS Cell

A Ba

b

c

C

f

e

d

D

F

H

J

E

G

I

K LM N O P

S'R'

T'

U' V'

W'

W

X

Y

V

Z

U

T

Sam.SR

Ref.

FIGURE 11.12

Cary� 14 diagram (ca. 1953): The arrows on the optical diagram trace the path oftheUVand vis radiation through the instrument. Radiation from theD2 orW lampis directed to the monochromator entrance slit D by appropriate lenses andmirrors. From mirror E it travels to prism F where it is refracted, then to mirrorG which reflects it to variable-width intermediate slit H. Mirror I reflects theradiation to grating J and from there the monochromatic beam is directed tomirrorK andexits themonochromator through slit L. SemicircularmirrorO, drivenby motor Q, chops the beam at 30Hz and alternately sends half the beam to thereference and half to the sample. Elements V, V1, W, and W1 pass the separatedbeams to the phototube. The light pulses of the two beams are out of phase witheach other so that the phototube receives light from only one beam at a time.The photomultiplier for UV–vis work is shown at X and the NIR detector for700–2600nm is shown at Y.

666 11 INSTRUMENTS

Table 11.3 Ultraviolet Cutoff Wavelength lc (at which the Solvent in a Cell of Path Length L¼ 1 cm hasan Absorbance of 1 unit), Dielectric Constant «, Scalar Refractive Index nD (Measured at 589nm, the NaD-line), Dipole Moment m (Debyes30), and Reichardt’s31 Solvent Polarity Index ET

Solvent lc (m) e nD m(D) ET

Acetic acid 260 6.17 1.3719 1.68 0.648Acetone 330 20.56 1.3587 2.88 0.355Acetonitrile 190 37.5 1.3441 3.92 0.460Benzene 280 2.27 1.5011 0.00 0.1111-Butanol 210 17.51 1.3003 1.74 0.6022-Butanol 260 16.56 1.3971 1.65 0.506Butyl acetate 254 5.1 1.3930 1.84 —Carbon disulfide 380 2.64 1.6275 0.0 0.065Carbon tetrachloride 265 2.23 1.4602 0.0 0.0521-Chlorobutane 220 7.28 1.4014 1.90 —Chloroform (stabilized with ethanol) 245 4.81 1.4459 1.14 0.259Cyclohexane 210 2.02 1.4262 0.0 0.0061,2-Dichloroethane 226 10.37 1.4448 1.83 0.3271,2-Dimethoxyethane 240 7.20 1.3796 1.71 0.231N,N-Dimethylacetamide 268 37.78 1.4384 3.72 0.401N,N-Dimethylformamide 270 36.74 1.4305 3.24 0.404Dimethylsulfoxide 265 46.45 1.4793 3.96 0.4441,4-Dioxane 215 2.21 1.4224 0.45 0.164Diethyl ether 218 4.20 1.3524 1.14 0.117Ethanol 210 24.55 1.3614 1.74 0.8542-Ethoxyethanol (“ethyl cellosolve”) 210 — 1.407 — —Ethyl acetate 255 6.02 1.3724 1.83 0.228Ethylene chloride 228 4.6 1.4242 1.80 —Glycerol 207 42.5@77�F 1.4722 — —n-Heptane 197 1.92 1.3876 0.0 0.012Hexadecane 200 2.04 1.435 — —n-Hexane 210 1.88 1.3749 0.0 0.009Methanol 210 32.66 1.3284 1.71 0.7622-Methoxyethanol 210 — 1.402 — —Methylcyclohexane 210 — 1.423 — —Methylene chloride¼dichloromethane 235 8.93 1.4242 1.60 0.309Methyl ethyl ketone 330 18.4 1.3788 2.76 —Methyl isobutyl ketone 335 13.11 1.393 — —2-Methyl-1-propanol 230 — 1.3945 — —1-Methylpyrrolidin-2-one 285 32.2 1.4700 4.08 0.355Nitromethane 380 35.8 1.3819 3.57 0.481n-Pentane 210 1.84 1.3575 0.0 0.009Pentyl acetate 212 4.68 1.4000 — —1-Propanol 210 20.45 1.3856 1.65 0.6172-Propanol 210 19.92 1.3772 1.65 0.546Pyridine 330 12.91 1.5102 2.37 0.302Tetrachloroethylene (stabilized with thymol) 290 2.5 1.505 0.00 —Tetrahydrofuran 220 7.58 1.4072 1.74 0.207Toluene 286 2.38 1.4969 0.36 0.0991,1,2-Trichloro-1,2,2-trifluoroethane 215 2.41 — — —2,2,4-Trimethylpentane (iso-octane) 290 1.9 1.391 0 —o-Xylene 290 2.4 1.50545 — —Water 190 78.30 1.3330 1.77 1.000

30 Peter Joseph William Debye (1884–1966).31 Christian Reichardt (1934– ).

11.7 VISIBLE–ULTRAVIOLET (V–UV) SPECTROSCOPY 667

extend from, say, 300 nm to 500 nm. It is very narrow only if the vibronicmodes are suppressed—for example, in very rigid molecules like porphyr-ins and phthalocyanines in solution, whose sharp visible absorptionspectra are known as Soret32 bands. If the temperature is lowered to77K or to 4.2 K, the vibronic substructure can often be resolved, but thisis rarely done. The vibronic structure is better resolved if the analyte ismeasured in the vapor phase (but the signals are quite small). Somesolvents afford better resolution of the vibronic sublevels than others.Often two overlapping electronic spectra, each with its own vibronicstructure, make resolution difficult. Therefore, since the 1960s, full UV–Visspectra are rarely published; all one finds is values for lmax and el! u(n)at lmax.

Table 11.4 Magnetic Permeability m/m0, Dielectric Constant (at 0Hz) «/«0 and Scalar Index of RefractionnD (Measured at the Yellow Na D-Line: 590nm) Or Tensor Components of the Index of Refraction n0,ne¼nb, ng for Some Gases, Liquids, and Solids

Material m/m0 e/e0 nD or n0 ne¼ nb ng

Air 1.00000036 1.000295 1.000294

CO2 (gas) — 1.000473 1.000449

C6H6 (liquid) — 1.489 1.482

He (gas) — 1.000034 1.000036

H2 (gas) — 1.000132 1.000131

Water H2O (diamagnetic liquid) 0.99999 78.30 1.3330

Silver (diamagnetic solid) 0.99998

Copper (diamagnetic solid) 0.99999

Aluminum (paramagnetic solid) 1.000021

Iron (ferromagnetic solid) 5000.

Nickel (ferromagnetic solid) 600.

Ice H2O (diamagnetic uniaxial xtal) 1.309 1.313

Quartz (SiO2, diamagnetic uniaxial xtal) 1.54424 1.55335

Wurtzite (ZnS, (diamagnetic uniaxial xtal) 2.356 2.378

Cinnabar (HgS, (diamagnetic uniaxial xtal) 2.854 3.201

Calcite (CaCO3, (diamagnetic uniaxial xtal) 1.658 1.486

Tourmaline (diamagnetic uniaxial crystal) 1.669 1.638

Sapphire (diamagnetic uniaxial crystal) 1.7681 1.7599

Sellaite (MgF2) (diamagnetic uniaxial crystal) 1.378 1.390

Tridymite (SiO2, diamagnetic biaxial crystal) 1.469 1.47 1.473

Mica (muscovite, diamagnetic biaxial crystal) 1.5601 1.5936 1.5977

Turquoise (diamagnetic biaxial crystal) 1.612 1.62 1.627

Topaz (diamagnetic biaxial crystal) 1.619 1.62 1.627

Sulfur (S8, diamagnetic biaxial crystal) 1.95 2.043 2.240

Borax (diamagnetic biaxial crystal) 1.447 1.47 1.472

Lanthanite (diamagnetic biaxial crystal) 1.52 1.587 1.613

Stibnite (Sb2S3, (diamagnetic biaxial crystal) 3.194 4.303 4.46

32 Jacques–Louis Soret (1827–1890).

668 11 INSTRUMENTS

V–UV Application: First Excited State of Linear Polyenes. The first electronicabsorption band of perfect linear aromatic polyenes (CH)x, or perfectpolyacetylene shifts to the red (to lower energies) as the molecule becomeslonger, and the bond length alternation (BLA) would be zero. This wasdiscussed as the free-electron molecular orbital theory (FEMO) in Section3.3. If this particle-in-a-box analysis were correct, then as x!1, the energy-level difference between ground and first excited state would go to zero.This does not happen, however; first, because BLA $ 0, next, because theselinear polyenes do not remain linear, but are distorted from planarity andlinearity for x � 6.

V-UV Application: Solvatochromism. Molecules undergo shifts of theirabsorption bands in solution, as a function of the “polarity” of the solvent,if the absorption connects molecular electronic states with different moleculardipole moments. Most molecular excited states have higher dipole momentsthan thegroundstate, and theyhave“regular” or “bathochromic”or “positive”solvatochromism; that is, their bands shift to lowerwavelengths as the solventpolarity increases. Conversely, a hypsochromic, or negative solvatochromism,indicates that the ground-state dipole moment is larger than that of its firstexcited state (Fig. 11.13). The excited state always has a different BLA than theground state. In polymethine dyes, the polyenes with a measurable BLA mayhave anundissociated stateD-¼-Awith a lower dipolemoment, a zwitterionicstate Dþ¼-A� with a higher dipole moment, and also an intermediateresonance form, with an intermediate dipole moment and zero BLA.

V–UV Application: Polarization of V–UV Absorption. The transitionmoment in solution is difficult to measure, given the isotropy of thesolution, but in oriented Langmuir33–Blodgett34 multilayers one can

D A D A D A

0 -+ Bond length alternation

Zwitterionic stateUndissociated state Resonant state

Dip

ole

mom

ent /

Deb

yes

FIGURE 11.13

Theoretical trend of dipole mo-ment indonor-andacceptor-bisub-stituted oligoacetylene, with vary-ing bond length alternation, (BLA)from the undissociated state (D¼-¼-A) where, by convention, BLA >0, through a resonant state withequal bond lengths (cyanine dyeD. . .. . .A: BLA¼ 0), to the zwitter-ionic state (Dþ-¼-¼A�: BLA < 0).Here ¼- indicates a one or severalcarbon-carbon double bonds andsingle bonds.

33 Irving Langmuir (1881–1957).34 Katharine Burr Blodgett (1898–1979).

11.7 VISIBLE–ULTRAVIOLET (V–UV) SPECTROSCOPY 669

determine the polarization of a V–UV absorption, by using polarizing filtersin the sample cavity, thus differentiating between intramolecular andintermolecular charge-transfer processes [5].

V–UV Application: Specular Reflection by Crystal Faces. If one focuses theincoming beam I0 onto a single crystal face, the specular UV–vis reflectivity(and its polarization) can be measured. The crystal is mounted on agoniometer head; the orientation of the crystal axes relative to theinstrumental axes must be predetermined separately on an X-raydiffractometer.

Diffuse reflectance spectroscopy is used when the sample consists ofsmall particles that scatter the incoming beam (Fig. 11.14). Routine diffuse IRreflectance spectroscopy is becoming popular, because instead of formingpellets (0.1mg of analyte ground together with KBr or Nujol� and com-pressed to 10 to 100 atm), one can deposit a powder of the pure analyte on asurface and collect its diffuse reflectance spectrum.

The diffuse reflectance intensities are somewhat skewed, and theymust be corrected by the 1931 formula of Kubelka35 and Munk:36 F(R)¼(1�R2)/2R¼ k/c, where R is the ratio of the measured reflectance of thesample to the reflectance of a chemically unrelated nonabsorbing standard(e.g., ground KBr), c is the molar concentration of the sample, and k is itsmolar absorption.

Optical Conductivity. By measuring the specular reflectance of highlyabsorbing crystals, their optical conductivity can be measured, using amirror to correct for fluctuations in incoming beam intensity, thendepositing an Au film atop the crystal to compensate for crystal surfaceimperfections.

VacuumUltraviolet Spectroscopy. Forwavelengths between 10 and 200 nm(i.e., for electronic energy levels above 4 eV), there is too much absorptionby atmospheric O2 (in the range 150–200 nm) and also by most organicsolvents, so the sample chamber must be evacuated, and organic solventscan no longer be used. Few commercial instruments exist in thiswavelength domain, so most frequently components are assembledas needed: sources, monochromators (typically a grating), and detectors(usually an Si diode).

SPECULARREFLECTION

DIFFUSEREFLECTION

INPUT BEAMFIGURE 11.14

The incident light beam is partiallytransmitted through the sample(not shown here); the rest is in partspecularly reflected, and also (par-ticularly if the sample is microcrys-talline) scattered at other angles(diffuse reflection).

35 Paul Kubelka (1900– ).36 Franz Munk (1900–1964).

670 11 INSTRUMENTS

11.8 ATOMIC ABSORPTION, ATOMIC EMISSION, ANDATOMIC FLUORESCENCE SPECTROSCOPIES

Atomic absorption spectroscopy (AAS) was practiced in themid-nineteenthcentury by passing a small sample into a flame and noting the color of theflame. Compared to molecular absorption, atomic absorption lines arevery narrow. The linewidth is defined as the width of the signal at half-height Dl1/2, which for atoms is of the order of 0.002–0.005 nm. Dl1/2 consistsof the natural linewidth plus the Doppler37 linewidth.

The natural linewidth Dl1/2,nat is defined by the lifetime Dt of (usually)the upper energy level; the uncertainty principle assigns a Dn Dt (1/2) h,whence

Dl1=2;nat ¼ l2h=2cDv ð11:8:1Þ

PROBLEM 11.8.1. Prove Eq. (11.8.1).

If the sample moves toward or away from the detector at speed V, then thelinewidth of an absorption or emission at wavelength l0 is broadened byDoppler effect:

Dl1=2;Doppler ¼ V=cð Þl0 ð11:8:2Þ

In a gas of atoms at finite temperature, the atoms move according to theMaxwell38–Boltzmann39 distribution of speeds, which collectively cause aDoppler broadening Dl1/2,Doppler that is typically two orders of magnitudegreater than the natural linewidth Dl1/2,nat.

To observe the spectrum of isolated atoms, the sample must be“atomized,” avoiding the formation of clusters and chemical reactionproducts (which would prevent “quantitation”): this can be done by dis-solving it in a solvent, using a nebulizer to produce a dilute vapor, andaiming it at the hot center of a flame source (the outer parts of the flamewould be cooler and induce vapor-phase formation of carbides or clusters).Other ways of producing isolated atoms for study are: (i) electrochemicalvaporization, (ii) ICP, (iii) microwave-induced Ar plasma, (iv) glow-dis-charge plasma, (v) electric arc, and (vi) electric spark. If a solid source isused, then (i) laser ablation, (ii) electric arc ablation, (iii) electric sparkablation, or (iv) glow-discharge sputtering (using energetic Ar ions) arepossibilities.

The very narrow linewidth of atomic spectra presents a funny problem:Monochromators (prisms or gratings) have a much larger bandwidth, thusmaking quantitation difficult and nonlinear. The analytical expedient is touse a separate source for every element studied (!)—for example, using a

37Christian Andreas Doppler (1803–1853).38 James Clerk Maxwell (1831–1879).39 Ludwig Boltzmann (1844–1906).

11.8 ATOMIC ABSORPTION, ATOMIC EMISSION, AND ATOMIC FLUORESCENCE SPECTROSCOPIES 671

hollow-cathode lamp or a discharge lamp, which emits a source wavelengththat will correspond to the absorption line for the analyte element. Thedetection limits for specific elements for AAS, atomic emission spectroscopy(AES), and atomic fluorescence spectroscopy (AFS) are given in Table 11.5.Casual inspection shows the obvious: AES and AFS are generally moresensitive than AAS.

11.9 INFRARED AND NEAR-INFRARED SPECTROSCOPY

The dispersive infrared spectrometer emerged in the 1940s, and it helped tospread the use of infrared spectroscopy as a common analytical techniquefor organic compound characterization in laboratories (e.g., Beckmann� DU)(Fig. 11.15). Dispersive infrared instruments are sometimes called grating orscanning spectrometers. A dispersive infrared instrument also has a sourceand mirrors, but the similarities to an FT-IR end there. The source energy issent through both a sample and a reference path, through a chopper tomoderate the energy reaching the detector, and directed to a diffractiongrating. Each wavelength is measured one at a time, with the slit monitoringthe spectral bandwidth, and the grating moving to select the wavelengthbeing measured. The x axis of a dispersive infrared spectrum is typicallynanometers, which can be converted to wavenumbers. An external wave-length calibration is required.

Table 11.5 Detection Limits (ng/mL¼ppb) for Selected Elements byAtomic Absorption Spectroscopy (AAS), Atomic Emission Spectroscopy(AES), and Atomic Fluorescence Spectroscopy (AFS) [6]

AAS AAS AES AES AFSElement Flame Electrothermal Flame ICP Flame

Al 30 0.1 5 0.2 5As 200 0.5 NAa 2 15Ca 1 0.25 0.1 0.0001 0.4Cd 1 0.01 2000 0.07 0.1Cr 4 0.03 5 0.08 0.6Cu 2 0.05 10 0.04 0.2Fe 6 0.25 50 0.09 0.3Hg 500 5 NAa NA 5Mg 0.2 0.002 5 0.003 0.3Mn 2 0.01 NA 0.01 1Mo 5 0.5 100 0.2 8Na 0.2 0.02 0.1 0.1 0.3Ni 3 0.5 600 0.2 0.4Pb 8 0.1 200 1 5Sn 15 5 200 NA 200V 25 1 200 0.06 20Zn 1 0.005 50,000 0.1 0.1

aNA, not available.

672 11 INSTRUMENTS

Fourier transform infrared (FT–IR) spectrometers were developed inthe 1960s in the wake of the Cooley40–Tukey41 fast Fourier transform (seeSection 2.16). An FT-IR spectrometer typically uses a Michelson42 inter-ferometer to collect a spectrum (Fig. 11.16): this consists of a source, slits,lenses, one beam splitter, two mirrors, a laser, and a detector. The energygoes from the source to the beamsplitter, which splits the beam into twoalmost equal parts. One half is transmitted to amovingmirror; the other halfis reflected by a fixed mirror. The moving mirror moves back and forth at aconstant velocity (it is typically suspended in a compressed air bearing).This velocity is timed by a very precise visible laser wavelength in thesystem (typically He–Ne), which also acts as a high-precision internalwavelength calibration.

Source

Reference

Grating

Slit

Detector

Chopper

Sample

FIGURE 11.15

Dispersive spectrometer diagram.

FIGURE 11.16

FTIRwithMichelsoninterferometer.

40 James William Cooley (1926– ).41 John Wilder Tukey (1915–2000).42Albert Abraham Michelson (1851–1931).

11.9 INFRARED AND NEAR-INFRARED SPECTROSCOPY 673

The desired resolution n1�n2 of the instrument is the reciprocal of theoptical retardation d, which is defined as twice the range of motion R of themoving mirror:

n1�n2 ¼ 1=d ¼ 2=R ð11:9:1Þ

Thus for instance for a 0.01-cm�1 resolution, R must be 50 cm. RoutineFT-IRs have a resolution n1� n2¼ 2 cm�1, while research-level FT-IRs haven1� n2¼ 0.001 to 0.25 cm�1.

After reflection from the two mirrors, the two beams are recombined atthe beam splitter. The beam from the moving mirror has traveled a differentdistance than the beam from the fixedmirror.When the beams are combined,an interference pattern (Fig. 11.17) is created, since some of the wavelengthsrecombine constructively and some destructively. This interference pattern iscalled an interferogram. This interferogram then goes from the beam splitterto the sample, where some energy is absorbed and some is transmitted.The transmittedportion reaches thedetector. Toobtain the infrared spectrum,the detector signal is sent to a computer, and a fast Fourier transformalgorithmconverts the interferogram intoa single-beamspectrum(Fig. 11.18).A reference or “background” single-beam interferogram is also collectedwithout a sample; and the sample single beam is ratioed to (i.e., divided by)the background single beam, to produce a transmittance or “%T” spectrum.This transmittance spectrum can be converted to absorbance by taking thenegative of the Naperian43 logarithm (log10) of the data points. The x axis of

140

25

4

3

2

1

0

–1

–2

–3

0

–5

1

0

–1

–2

Volts

Volts

Volts

120 100 80Data Points Data Points

Data Points

Two wavelengths Multiple wavelengths

Infrared interferogram

60 40 20 0 300 250 200 150 100 50 0

5000 4000 3000 2000 1000 0

FIGURE 11.17

Interference patterns [7].

43 John Napier of Murchiston (1550–1617).

674 11 INSTRUMENTS

the FT-IR spectrum is typically displayed in “wavenumbers,” or cm�1, orkaysers (three names for the same thing).

FT-IRAdvantages. There are threemajor advantages for FT-IR spectrometerover a dispersive IR spectrometer: (1) The multiplex or Fellgett44 advantage.Since each point in the interferogram contains information from eachwavelength of light being measured, and every cycle of the moving mirrorequals one scan of the entire infrared spectrum, combining each scan improvesthe spectrum of the sample. In contrast, a dispersive spectrometer uses slitsand measures each narrow range of wavelengths separately, sequentiallyand rather slowly. (2) The throughput advantage. An FT-IR instrumentdoes not use slits and has fewer mirrors than a dispersive instrument; thusmore energy reaches the detector, improving the signal-to-noise ratio andalso the resolution. (3) The precision advantage. By using a control laser,an FT-IR spectrometer has better internal precision and accuracy of thewavelength.

IR absorptions involve elastic or Rayleigh45 or constant-energy scatteringof light: inmore detail, the electric field vector E of the input lightmust couplewith the transition electric dipole moment mif as E mif. If E?mif, then no IRtransition is seen. Allowed IR transitions require that the transition momentvector mif be nonzero—i.e., that is, that the static electric dipole moment m ofthe molecule change during the IR absorption.

In contrast, Raman46 absorption involves inelastic scattering,with gain orloss of energy; in detail, the Raman transition involves the polarizability

Sample lgm Sample Single Beam

% Transmittance Spectrum

Absorbance Spectrum

Background Single BeamBackground lgm

smp:FFT

bkg:FFT

RATIO

–log10

FIGURE 11.18

The process of obtaining an FT-IR spectrum [7].

44 P. B. Fellgett (ca. 1922–2008).45 John William Strutt, first baron Rayleigh (1842–1919).46 Sir Chandrasekhara Venkata Raman (1888–1970).

11.9 INFRARED AND NEAR-INFRARED SPECTROSCOPY 675

Table 11.6 Typical Infrared Maxima (cm�1)a

Diatomic Stretches (for commonest isotopes) Mid-IR

HH 4401, DD 3115, TT 2547,HB 2367, HC 2858, HN 3283, HO 3738, HF 4138,HNa 1172,HP 2365, HS 2712, HCl 2891,HK 984, HBr 2649,BN 1515, OO 1580

Group id-IR

CH Stretches: AlkaneGroups

CH3-C methyl 3000–2910 s, 2910–2850 s, 1430–1880 s, 1400–1350mCH3-(C¼O) methyl 3010–2920 s, 2950–2870 s, 1440–1380m, 1390–1330 s-CH2- methylene 2970–2930 s, 2930–2900 s, 1480–1430 s-CH2-CH2-CH2-CH2- 740–710mCH2-(C¼O), CH2-(CN) 3010–2920 s, 2950–2870 s, 1440–1380s->CH- 2970–2800m, 1360–1300wethyl 3000–2900 s, 2970–2870 s, 2880–2810m,

1470–1420 s, 1400–1300m, 1100–1000m, 900–840wn-propyl 3000–2900 s, 2950–2870 s, 2880–2810m,

1480–1430 s, 1400–1300m, 1100–1000m, 990–920w,880–820w

isopropyl 3000–2910 s, 2950–2880m, 2880–2810m, 2850–2780w,1470–1430 s, 1400–1370m, 1370–1350m, 1350–1310w,1200–1140m, 1150–1100m, 970–900w

tert-butyl 3000–2920 s, 2950–2880m, 2890–2810m,1470–1430 s, 1400–1370m, 1380–1350 s,1270–1230m, 1250–1170m, 950–860w

Alkene Groups-CH¼CH2 Vinyl 3090–3010m, 3010–2900 s, 1850–1800m, 1670–1630m,

1450–1390 s, 1330–1270w, 1120–1260w, 1000–950 s,950–900 s, 700–530 s

H-C¼C-H (trans) 3030–2940 s, 1680–1630m, 1320–1270w, 1000–950 sH-C¼C-H (cis) 3030–2940 s, 1680–1630m, 1410–1350m, 830–700 s>C¼CH2 3110–3030m, 3030–2930 s, 1800–1750m, 1680–1630m,

1450–1390 s, 1130–1060w, 930–870 s, 700–530 s

Alkyne Groups-CCH 3300–3200m, 2070–1990m, 700–600 s-CC- 2180–2070w

AromaticMono-substitutedbenzene

3130–3080w, 3060–3040w, 3050–2990m, 1660–1570m,1590–1560m, 1530–1470m, 1190–1120m (sharp),1105–1060m (sharp), 1060–1020w (sharp), 780–730 s,710–680 s

Ortho disubstitutedbenzene

3130–3080w, 3060–3040w, 3050–2990m, 1660–1570m,1590–1560m, 1530–1470m, 1190–1120m (sharp),1105–1060m (sharp), 1060–1020w (sharp), 780–730 s,710–680 s

a In dilute solution in nonpolar solvents: s, strong; m, medium; w, weak; v, variable.

676 11 INSTRUMENTS

tensoraif asE aif E; Ramanprocesses are typically only 10�7 times as intenseas IR transitions, and they require that the static electric polarizability tensoraof the molecule change during the Raman absorption.

For obscure historical reasons, IR spectra are usually plotted not as% absorbances, as in most other spectroscopies (stalagmites, risingfrom 0% absorbance), but as % transmittances (stalactites, falling from100% transmittance).

The prevalent organic IR absorption spectral maxima are listed inTable 11.6. Normal modes of vibrations are identified for some simplemolecules in Table 11.7. Overtone bands (higher excited states) for thevibrations listed in Table 11.6 occur in the near-infrared region.

Very thin films (monolayer or multilayer) can be studied by IR spectros-copy, provided that the IR beam crosses a reasonably large fraction of themonolayer; this can be done by (i) grazing-angle techniques (whereby the IRbeam angles of incidence and reflection are of the order of 1� to 3�) or(ii)multiple internal reflections of the IR beam inside themonolayer (infraredreflectance and absorbance spectroscopy (IRRAS). Fig. 11.19 shows thechemical structure of an analyte (Fullerene-bis-[ethylthio-tetrakis(3,4-dibutyl-2-thiophene-5-ethenyl)-5-bromo-3,4-dibutyl-2-thiophene] malonate),

Table 11.7 Normal Modes of Several Moleculesa

Methane: n1(a1: symmetric stretch) at 2917 cm�1: Raman-active, IR-inactive;CH4 (T d) n2 (e: symmetric bend; 2� degree) at 1534 cm�1: Raman-active, IR-inactive;(9 modes) n3 (f2: asymmetric stretch, 3�degree) at 3019 cm�1: Raman-inactive, IR-active;

n4 (f2; deformation 3� degree) at 1306 cm�1: Raman-inactive, IR-active.

Ethylene: n1(b1u) (CH symmetric stretch) at 3217 cm�1: Raman-inactive, IR-active;C2H4 (D2h) n2(ag) (CH symmetric stretch) at 3026 cm�1: Raman-active, IR-inactive;(12 modes) n3(b2u) (CH asymmetric stretch) at 3185 cm�1: Raman-inactive, IR-active;

n4(b3g) (CH asymmetric stretch) at 3153 cm�1: Raman-active, IR-inactive;n5(ag) (CC stretch) at 1623 cm�1: Raman-active, IR-inactive;n6(b1u) (H-C-H in-plane scissor) at 1413 cm�1: Raman-inactive, IR-active;n7(ag) (H-C-H in plane scissor) at 1342 cm�1: Raman-active, IR-inactive;n8(b3g) (C-C-H in-plane rocking) at 1167 cm�1: Raman-active, IR-inactive;n9(b3u) (H-C-H out-of-plane wag) at 1068 cm�1: Raman-inactive, IR-activen10(b2g) (H-C-H out-of-plane wag) at 1057 cm�1: Raman-active, IR-inactive;n11(ag) (H-C-H out-of-plane twist) at 875 cm�1: Raman-inactive, IR-inactive;n12(b1g) (CH in-plane rock) at 1236 cm�1: Raman-inactive, IR-active;n��(au) (CH twist) at 1023 cm�1;n��(b1g) (CH stretch) at 3013 cm�1

aMolecules with N atoms have 3N� 6 normal modes if nonlinear and 3N� 5 modes if linear.

OO

OS

Bu Bu

SBr

Bu Bu

OS

Bu BuS

Bu Bu

SS

Bu BuS

BuBu

S Br

BuBuS

BuBuS

BuBu

SS

BuBu

FIGURE 11.19

Chemical structure of Fullerene-bis-[ethylthio-tetrakis(3,4-dibutyl-2-thiophene-5-ethenyl)-5-bromo-3,4-dibutyl-2-thiophene] malonate [8].

11.9 INFRARED AND NEAR-INFRARED SPECTROSCOPY 677

its IR spectrum (KBrmull) in Fig. 11.20 and its RAIRS spectrum in Fig. 11.21.Fig. 11.22 shows the gas-phase IR spectrumof carbonmonoxide, determinedin a high-resolution (0.0025 cm�1) FTIR instrument: The individual rota-tional states are resolved, as is the contribution due to the (1%) minorityisotope 6C

138O

16.

1000

700797

923 1229

11811101

1019

1457

13731746

2854

2954

3015

2921

16461593

1540

0

0.2

0.4

0.6

0.8

1

0.08 0.12 0.16 0.2 0.24

Energy (eV)

Abs

orba

nce

0.28 0.32 0.36

1500 2000

Wavenumber (cm–1)

2500 3000

FIGURE 11.20

“Bulk” IR spectrum of Fullerene-bis-[ethylthio-tetrakis(3,4-dibutyl-2-thiophene-5-ethenyl)-5-bromo-3,4-dibutyl-2-thiophene] malonate (structure is shown inFig. 11.19) [8].

FIGURE 11.21

RAIRS spectrum of Fullerene-bis-[ethylthio-tetrakis(3,4-dibutyl-2-thiophene-5-ethenyl)-5-bromo-3,4-dibutyl-2-thiophene] malonate (structure is shown inFig. 11.19) [8].

678 11 INSTRUMENTS

11.10 RAMAN SPECTROSCOPY

The inelastic scattering of light by molecules was predicted by Smekal47 in1923 and observed by Raman, Krishnan,48 Landsberg49 and Mandelstam50

in 1928. Early instruments used sunlight or Hg lamps, but the advent ofpowerful lasers made the rather weak “spontaneous” Raman effect moreeasily measured. The [Smekal]–Raman effect occurs when a photon in thevisible range interacts with the wavefunction of a molecule, excitingthe molecule to a short-lived virtual excited state, from which the moleculerelaxes to a different vibrational or rotational state than the initial state, byemitting a new photon in the visible range, of either slightly decreasedenergy (Stokes51 line) or of slightly increased energy (anti-Stokes line),depending on whether the transition is to a more excited rovibrationalstate or to a less excited rovibrational state, respectively. Again, the Ramanabsorption and emission requires a change in the molecular polarizabilitytensor a.

Present-day Raman spectrometers use a laser source, a sample holder,a lens, a monochromator (a holographic grating in early instruments, aCzerny52–Turner53 grating in recent instruments), and a detector (formerly,a photomultiplier tube, now usually a CCD detector). Wavelengths close tothe laser line, due to elastic and very intense Rayleigh scattering, must befilteredoutwith anotchfilter or an edgefilter.Normal, “spontaneous”Raman

FIGURE 11.22

Part of the high-resolution rotatio-nal–vibrational FTIR spectrum ofCO(g), showing the P(<2140 cm�1)and R (>2140 cm�1) bands, andthe contributions from the C13 iso-tope [9].

47Adolf Gustaf Stephan Smekal (1895–1959)48 Padma Bushan Sir Kariamanickam Srinavasa Krishnan (1989–1961).49Grigory Samuilovich Landsberg (1890–1957).50 Leonid Isaakovich Mandelstam (1879–1944).51 Sir George Gabriel, first baronet Stokes (1819–1903).52M. Czerny (���).53A. F. Turner (���).

11.10 RAMAN SPECTROSCOPY 679

spectrometry is sensitive down to only 0.1M solutions (10�1mol/L).FT Raman spectroscopy has also been introduced.

By using an exciting light close to a real electronic transition, resonanceRaman is achieved, with intensity increases by factors of 102 to 106.

Certain roughened surfaces (Ag or Au colloids) exhibit another niceintensification of the Raman effect of 103 to 106 by exciting surface plasmonsin the colloid particles: this is surface-enhanced Raman, first seen byFleischmann54, and explained by van Duyne.55 Combining resonance andsurface-enhanced effects in surface-enhanced resonance Raman spectros-copy (SERRS), the Raman intensity can increase by factors as large as 1012, sothat solutions of concentration down to 10�12M can be detected.

Other methods include tip-enhanced Raman using 20- to 30-nm diam-eter Au or Ag tips, polarized Raman, stimulated Raman, micro-Ramanspectroscopy, and coherent anti-Stokes Raman spectroscopy (CARS),where two laser beams are combined to generate an anti-Stokes beam, andso on.

11.11 INELASTIC ELECTRON TUNNELING SPECTROSCOPY

In investigating howCooper pairs56 in a superconductor (Pb)may retain theircoherence if they penetrated, across a thin tunneling gap (a film of poly-methylmethacrylate, PMMA) into a conventional nonsuperconductingmetal(Al), Jaklevic57 and Lambe58 discovered accidentally in 1966 that steps in thecurrent, when displayed as the second derivative d2I/dV2 versus V, wereactually due to vibrations in polar molecules (IR) and also vibrations in

V / Volts

I vs V

dI/dV vs V

d2I/dV2 vs V

Elastic current

Inelastic current jumps

I, dI/dV, or d2I/dV2

FIGURE 11.23

Schematic representation of howinelastic (absorption) effects mod-ify the IV curve.

54Martin Fleischmann (1927– ).55 Richard P. van Dyne (194�– ).56 Leon Cooper (1930– ).57 Robert C. Jaklevic (���).58 John Lambe (���).

680 11 INSTRUMENTS

nonpolarmolecules (Raman) peaks of PMMA; thus inelastic electron tunnel-ing spectroscopy (IETS) was born [10]. This is sketched in Fig 11.23.

The IETS spectrum is caused by electrons tunneling through (or veryclose to) molecules that activate intramolecular vibrations, without theelectric dipole selection rules that dominate the absorption of photons (thusIR and Raman lines occur together in the IETS spectrum). Specializedequipment was traditionally built to measure IETS (Fig. 11.23), but IETSsoftware has been added to recent commercial magnetometers.

The relative smallness of IIETS suggests that measuring I at a DC potentialV is facilitated by superposing onto it a small AC modulation signal Mcos (otþj):

I V þMð Þ ¼ IðV þMcos otþ jð ÞÞ ð11:11:1Þ

which in a Taylor series expanded around V¼V0 yields

I V þMð Þ ¼ IðVÞ þ ½dI=dV�V¼VMcos otþ jð Þþ 1=2ð Þ d2I=dV2

� �V¼V

M2cos2 otþ jð Þ þ ð11:11:2Þ

Using the trigonometric identity cos2a¼ (1/2)(1þcos(2a)), one sees the argu-ment for second harmonic detection:

I V þMð Þ ¼ IðVÞ þ ½dl=dV�V¼VMcos otþ jð Þþ 1=4ð Þ d2I=dV2

� �V¼V

M2cos 2otþ 2jð Þ þ . . .ð11:11:3Þ

The signal-to-noise ratio is further improved if phase-sensitive second-harmonic detection is performed at a fixed reference frequencyoref is used—hence the need for a lock-in amplifier (Fig. 11.24).

The total current I through a molecule has an inelastic contribution IIETSfromexcitation ofmolecular vibrations (themaingoal of IETS) (Fig. 11.25) andcan have also an elastic contribution, which consists of two parts: the off-resonance elastic ohmic IR (shown in the center part of Fig. 11.26) and aresonant elastic contribution IOMT between the Fermi59 level of the relevantmetal electrode andanunoccupiedmolecular orbital of themolecule; this parthas been dubbed orbital-mediated tunneling (OMT) (Fig. 11.26) [11]:

I ¼ IIETS þ IR þ IOMT ð11:11:4Þ

The IETS spectrum (Fig. 11.25) is very similar to the IRRAS spectrum ofFig. 11.21; however, some peaks are seen that are usually only seen inconventional Raman spectra. The IETS spectrum is severely affected bytemperature, by ambient electrical disturbances, and bymechanical vibrations.The full-width-at-half-maximum signal linewidthWFWHM (in volts) is consid-erably larger than the natural linewidth WNL:

WFWHM ¼ WNL2 þ 5:4kBTð Þ2 þ 1:7Mð Þ2

h i1=2ð11:11:5Þ

59 Enrico Fermi (1901–1954).

11.11 INELASTIC ELECTRON TUNNELING SPECTROSCOPY 681

where kB isBoltzmann’s constant. Thismeans thatmost vibration signals canbeseen in IETS conveniently only for T < 20 K.

When IOMTwas notmeasured, it was guessed that IIETS/IR 0.01, but laterwork [11] suggested that (IIETSþ IOMT)/IR 0.3.

FIGURE 11.24

Components of a home-built IETSspectrometer. DCV¼DC voltage;ACV¼AC voltage; DMM¼digitalmultimeter; GPIB-IEEE¼GeneralPurpose Interface Bus, according toInstituteofElectrical andElectronicEngineers Standard 488 [8].

FIGURE 11.25

IETS spectrum of fullerene-bis-[ethylthio-tetrakis(3,4-dibutyl-2-thiophene-5-ethenyl)-5-bromo-3,4-dibutyl-2-thiophene] malo-nate [8]. The chemical structure isshown in Fig. 11.19. The peaks at572 cm�1 and 1784 cm�1 corre-spond to Raman transitions.

682 11 INSTRUMENTS

11.12 FLUORESCENCE SPECTROSCOPY

(Molecular) fluorescence spectroscopy (also known as spectrofluorometry,fluorometry, or fluorimetry) analyzes fluorescence (or rapid light emission)froma sample. It uses an input light beamof light, typically fromaXearc lamp(fairly constant output energy from 300 nm to 800 nm, and acceptable outputbetween 200 nmand 300 nm) or aHgvapor lamp (line spectrum); rarely, froma laser (intense, but fixed wavelength) to promote electronic excitation inmolecules, and measuring the rapid fluorescent emission at lower energiesthan the light absorbed; it is complementary to absorption spectroscopy andusually much more sensitive. The light emission intensity If is given by

If ¼ I0 GQEexp �ecLð Þ ð11:12:1Þ

where I0 is the input light intensity,QE is the quantum efficiency (0�QE< 1)of the emission process (the other ways the electronically excited state candecay are phonons (heat) or phosphorescence), the factor exp (�e c L) is part ofBeer’s law, e is thewavelength-dependentmolar extinction coefficient, c is theconcentration of the analyte (mol L�1), L is the cell length (cm), and G is afactor that includes several effects: (i) what fraction of the total possible solidangle of 4p radians is being measured; (ii) other instrumental limitations(monochromator nonlinearity, scattering; detector nonlinearity, etc.);(iii) Rayleigh and Raman scattering by the sample (Rayleigh scattering atthe same wavelength, Raman scattering when a Raman mode is excited, andthe fluorescence output is red-shifted by a Raman transition). G can bemeasured with considerable difficulty, but most often a relative standard isused instead—that is, a molecule for which the quantum efficiency is reliablyknown.

Usually, the fluorescent emission is red-shifted (“Stokes shift”) from theparent absorption process. The vibrational substructure in the absorptionband is similar to that in the fluorescence band, but the two energy bands arelike mirror images of each other. This is because the absorption from the

FIGURE 11.26

IETSþOMT spectrum of Fullerene-bis-[ethylthio-tetrakis(3,4-dibutyl-2-thiophene-5-ethenyl)-5-bromo-3,4-dibutyl-2-thiophene] malonate [8](structure is shown in Fig. 11.19).

11.12 FLUORESCENCE SPECTROSCOPY 683

electronic ground state (v¼ 0) goes to several vibronically excited substates ofthe electronically excited state (v’ � 0), while the fluorescent emission isfrom the lowest (v’¼ 0) excited state to several vibrationally excited sublevels(v � 0) of the ground state.

A fluorescence spectrometer (Fig. 11.27) typically uses an input mono-chromator (with prism or grating) to select one particular frequency for theinput light beam, and then it uses an outputmonochromator (alsowith prismor grating) to monitor the emission wavelength. If one uses a multichannelanalyzer instead of a simple diode detector, then the output monochromatoris not needed.

By observing at a fixed and large angle (typically around 90�) from theinput wavevector, the emission detector can bemuchmore sensitive than thedetector used in absorption; a single-photon detector can also be used,making fluorescence spectroscopy an instrument of unsurpassed sensitivity.Since fluorescence in solution will randomly radiate in all directions,measuring in the neighborhood of 90� from the incident beam means mea-suring only a small component of the output energy, so quantitation offluorescence efficiency can be difficult and requires the knowledge of G,which shouldmostly be the solid angle of detection divided by the total solidangle (4p radians).

One can (and does) measure the emission as a function of varying theinput beam energy (moving only the input monochromator); this gives theexcitation spectrum (which will qualitatively resemble the absorption spec-trum). By choosing a fixed input frequency (i.e., fixing the input monochro-mator) and moving the output monochromator, one obtains the fluorescentemission spectrum.

Sometimes a very fluorescent molecule (fluoroprobe) is used to measureits microenvironment [fluorescence quenching, fluorescence polarization(developed by Weber60), and fluorescence probe for hydrophobic or hydro-philic regions in biological systems].Avery recent and “hot” label is thegreenfluorescent protein (238 amino acids, 26.9 kDa) first isolated from the jellyfish

Sample cell

Excitationmonochromator

emissionmonochromator

Photo detector

Xe lamp

FIGURE 11.27

Schematic diagram of a fluorome-ter with a Xe light source and a 90�

measuring geometry.

60Gregorio Weber (1916–1997).

684 11 INSTRUMENTS

Aequorea Victoria (lmax¼ 509 nm) and developed for analytical use byChalfie,61 Shimomura,62 and Tsuen.63

11.13 MICROWAVE SPECTROSCOPY

Microwave spectroscopy is a very sensitive technique for detecting therotational spectra of gas-phase molecules that possess a permanent dipolemoment (or for which the center of mass is not coincident with the center ofcharge).All that is needed is amicrowave source andawaveguide suitable forthe microwave frequency range selected, along with a microwave detector(aided by a transfer oscillator and frequency counter). Given the tremendousprecision of frequency measurements (1 part in 1010 or better), the rotationalspectra ofmolecules and their isotopic variations can bemeasured to fantasticprecision. After corrections for departures from the rigid-rotor model de-scribed in Section 3.5, a normal-mode analysiswill yield themoment of inertiaandwill determine bond lengths and bond angles to precisions that exceed byseveral orders of magnitude those available from least-squares structuredeterminations based on X-ray or neutron diffraction data. Several differentwaveguide dimensions and microwave sources are needed for versatility;this has been partially compensated by Fourier transform microwave spec-troscopy, developed by Flygare64.

11.14 SURFACE PLASMON RESONANCE

Surface plasmons, or surface plasmon polaritons, are surface electromag-netic waves that propagate inside ametal along ametal/dielectric (or metal/vacuum) interface; their excitation by light is surface plasmon resonance(SPR) for planar surfaces or localized surface plasmon resonance (LSPR) fornanometer-sized metal particles.

These electromagnetic waves are very sensitive to any change in theboundary—for example, to the adsorption of molecules onto the metalsurface. SPR has measured the absorption of material onto planar metalsurfaces (typically Au, Ag, Cu, Ti, or Cr) or onto metal nanoparticles and isused in many color-based biosensor applications and lab-on-a-chip sensors.To observe SPR, the complex dielectric constants e1 of the metal and e2 of thedielectric (glass or air) must satisfy the conditions Re(e1)< 0 and |e1|> |e2|,which aremet for visible or IR light; the light frequencymust equal that of themetal plasmon and must be p-polarized.

Two configurations are used: (1) In the Otto65 configuration (Fig. 11.28)the light impinges on the wall of a glass substrate, typically a prism, and istotally reflected. A thin metal (for example Au) film is positioned close

61Martin Chalfie (1947– ).62Osamu Shimomura (1928– ).63 Roger Y. Tsien (1952– ).64Willis H. Flygare (1936–1981).65Andreas Otto (1936– ).

11.14 SURFACE PLASMON RESONANCE 685

enough that the evanescent waves can interact with the plasma waves. (2)In the more popularKretschmann66 configuration (Fig. 11.29), the metal filmis evaporated directly onto the glass prism; the light again impinges from theglass side, and the evanescent wave penetrates into the metal film. Theplasmons are excited at the outer side of the film. The angle of the reflectancemaximum is usually quoted: It changes about 0.1� with nanometer-thickadsorbates; otherwise the wavelength of the absorption maximum shiftsslightly. Thedata (Fig. 11.30) are usually analyzedusing the Fresnel formulas.

11.15 ELECTRIC SUSCEPTIBILITY

One canmeasure thedielectric constant e of gases, liquids and solids byplacingthe sample in a capacitance cell. Frommeasuring e as a functionof temperature,one routinely gets the scalar first-order electric dipole susceptibility w(1).

FIGURE 11.28

Otto configuration.

FIGURE 11.29

Kretschmann configuration.

66 E. Kretschmann (ca. 1938– ).

686 11 INSTRUMENTS

By applying the Mossotti67–Clausius,68 Lorentz–Lorenz,69 or Debye orOnsager70 equations (Section 5.10), one can extract the dipole moments ofpolar molecules and the polarizability of any solute molecule. One needs acapacitancecellwhoseelectrodesareasclosetoeachotheraspractical(forhighercapacitances) and reasonable solubilities. If the shape of the solute is verydifferent from the sphere used in the Debye model, then the ellipsoidal cavityhas been treated theoretically [13] and applied to hypsochromism [14].

Very polar molecules tend to not dissolve in nonpolar solvents; fewtheories deal with the electrical effects of polar solvents on the electricalparameters for polar solutes.

11.16 NONLINEAR OPTICAL PROPERTIES

The technological interest in nonlinear susceptibilities is greatest when theseare not enhanced by resonant interactions with an allowed optical transitionof themolecule or crystal. As shown in Section 2.7, one can define the variousnonlinear susceptibilities in terms of the angular frequencies of input andoutput light. The convention is that a negative sign (indicating negative, oroutput, momentum) is used for output light. The general second-order

45444342Internal angle (degrees)

Nor

mal

ized

ref

lect

ivity

41400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FIGURE 11.30

SPR normalized reflectivity for“glass/Cr/Au” (squares) and for“glass/Cr/Au Langmuir–Blodgettmonolayer of hexadecylquinoli-nium tricyano-quinodimethanideg-C16H33Q-3CNQ” (triangles) mea-sured at 632.8 nm using theKretschmann configuration: theorganic LB monolayer thicknesswasdeterminedtobe2� 0.2nm[12](structure is shown in Fig. 11.31).

C C

CNC16H33

N

N

N

+-

FIGURE 11.31

Chemical structure of hexadecyl-quinolinium tricyanoquinodi-methanide g-C16H33Q-3CNQ [12].

67Ottaviano Fabrizio Mossotti (1791–1863).68 Rudolf Julius Emanuel Clausius¼Rudolf Gottlieb (1822–1888).69 Ludvig Lorenz (1829–1891).70 Lars Onsager (1903–1976).

11.16 NONLINEAR OPTICAL PROPERTIES 687

nonlinear susceptibility tensor is w(2)(�o3; o1, o2) (several special cases aredescribed in Fig. 11.32), and the third-order tensor is w(3)(-o4; o1, o2, o3).

For second harmonic generation (SHG), the tensor is w(2)(�2o; o, o)(useful for frequency doubling and parametric down-conversion) while forthe linear electrooptic or Pockels71 effect the tensor is w(2)(�o; o, 0) (usefulfor Q-switching of lasers, for phase or amplitude modulators, and for beamdeflectors); for optical rectification the tensor is w(2)(0; o, �o); for frequencymixing the tensor is w(2)(�o3; o1, o2) (useful for frequency up-converters,optical parametric oscillators, and spectroscopy).

For third harmonic generation the tensor is w(3)(�3o;o,o,o) (useful forspectroscopy and for deep UV conversion). For degenerate four-wavemixing it is w(3)(�o; o, �o, o). For four-wave mixing it is w(3)(�o4; o1, �o2,o3), useful for the generation of new frequencies). For electric-field-inducedsecond-harmonic (EFISH) generation, the tensor is w(3)(�2o;o,o, 0), usefulfor measuring molecular hyperpolarizabilities. For the DC Kerr72 effectthe tensor is w(3)(�o; o, 0, 0), used to study macromolecules and biopoly-mers. For the AC Kerr effect the tensor is w(3)(�o; o, o, �o), useful for fastswitching, time-resolved (gating) experiments, optically bistable systems,optical limiters, and phase conjugation. For Raman scattering the tensor is

ω1ω3

ω2

χ(2)IJK(-ω3; ω1; ω2)

SUM GENERATION

ω3 = ω1 + ω2

ω1ω2

ω3

χ(2)IJK(-ω1; ω3; ω2)

DIFFERENCE GENERATION

ω1 = ω3 - ω2

ω1

ω3

ω2

χ(2)IJK(-ω2; ω3; ω1)

DIFFERENCE GENERATION

ω2 = ω3 - ω1

ω1

ω3

ω2

χ(2)IJK(-ω1; -ω2; ω3)

PARAMETRIC FLUORESCENCE

ω1 + ω2 = ω3

IN IN

IN IN

IN

IN

INFIGURE 11.32

Three-way interactions that pro-duce w(2) radiation.

71 Friedrich Carl Alwin Pockels (1865–1913).72 John Kerr (1824–1907).

688 11 INSTRUMENTS

w(3)(�os; op, �op, os). For Brillouin73 scattering the tensor is w(3)(�os; op,�op, �os). These tensors are all different and require separate measure-ments, since different physical processes are at work.

In general, there are 9¼ 32 tensor components for the second-rank tensorsa and w(1), 27¼ 33 tensor components for both b and w(2), and 81¼ 34 compo-nents for both g and w(3). In practice, however, crystal symmetry will reducethe number of unique tensor elements.

Thus, for instance, of the nine second-rank elements of the tensors for aand for w(1), three are related by

aij ¼ aji and wij1ð Þ ¼ wji

1ð Þ ð11:16:1Þ

leaving only six unique elements for the general triclinic crystal; they reduceto one for a cubic crystal. As mentioned above, for SHG the experimen-talists use the 27 dijk coefficients of tensors d, each of which is half the size ofthe wijk

(2):

wijk2ð Þ �2o; o; oð Þ ¼ 2dijk �2o; o; oð Þ ð11:16:2Þ

These 27 general SHG coefficients are similarly reducible, by symmetry, to18 unique ones by the relationship

dijk ¼ dikj; or wijk2ð Þ ¼ wikj

2ð Þ; or bijk ¼ bikj ð11:16:3Þ

By contracting the last two suffixes, these 18 unique d coefficients are re-labeled as follows:

di11 di1; di12 di2; di13 di3; di23 di4; di31 di5; di12 d16; i ¼ 1; 2; 3ð Þf gð11:16:4Þ

When one is far from resonance, then a further simplification is possible.Kleinman74 showed [15] that energy is simply exchanged among the fields Ei,Ej, and Ek along the three axes i, j, and k, so that the suffixes can be inter-changed freely; this means that the three fields act independently and can beapplied in arbitrary order (at resonance, one field will distort the electroniccloud in such a way that a second field will act on a severely perturbedelectron configuration). Thus, when this is not a problem (i.e., when one is farfrom resonance) for SHG the 27 (or 18) coefficients reduce to 10:

wijk2ð Þ ¼ wikj

2ð Þ ¼ wjki2ð Þ ¼ wjik

2ð Þ ¼ wkji2ð Þ ¼ wkij

2ð Þ ð11:16:5Þ

while for THG the 81 coefficients reduce to 15.For SHG, we leave alone the 12 centrosymmetric crystal classes and

consider [16] only the 20 acentric ones [one triclinic: 1, two monoclinic: m, 2,two orthorhombic: mm2, 222, five tetragonal 4, �4, 4mm, �42m, 422, threetrigonal: 3, 3m, 32, four hexagonal: �6, �6m2, 6mm, 622, and three cubic: 23, 4�3m,

73 L�eon Nicolas Brillouin (1889–1969).74D. A. Kleinman (���).

11.16 NONLINEAR OPTICAL PROPERTIES 689

and 432]. By using the crystal symmetry, the number of unique d componentsreduces further; all components vanish for point group 432.

To reduce w(3) measurements in dilute solution or polymer dispersions(esu: these are “bulk” values) to values of g (esu per active solute molecule),the formula is

g ¼ w 3ð ÞL�1N�1 cgs-esu� �

ð11:16:6Þ

where N is the concentration (molecules of solute cm�3) and L is the Lorentzfactor:

L ¼ no2 þ 2

� �=3

� �3n3o

2 þ 2� �

=3 ð11:16:7Þ

where no is the linear index of refraction of the solution (index of refractionof the pure solvent), and n3o is the index of refraction at frequency 3o. Ifn3o¼ no, then this factor reduces to L¼ [(no

2þ 2)/3]4 Typically, L�1¼ 0.242for pyridine, 0.289 for chloroform, 0.1715 for CS2.

In particular, if the molecule has no center of symmetry and the crystal isin an acentric space group, then only the even-order susceptibilities x(2), x(4)

(and the corresponding molecular dipole moment m0 and the even-orderhyperpolarizabilities b, d, etc.) are nonzero. For all materials, regardless ofsymmetry, the odd-order molecular moments (a, g, etc.) and susceptibilities(x(1), x(3), . . .) can be nonzero.

This was of technological interest in the 1980s because, while“GaAs|Ga1�xAlx|As” diode lasers that resonate in the near-IR region(> 850 nm)weremass-produced at lowcost, there are nopractical lightweightlasers that operate in the 200- to 500-nm (UV to visible) region of theelectromagnetic spectrum. There was a great need for higher densities ofinformation storage and retrieval. Very expensive materials with high x(2)

[e.g., potassium dihydrogen phosphate (KDP), lithium niobate (LiNbO3), orthe organic crystals 2-methyl-4-nitroaniline (MNA)ormethyl-2-(2,4-dinitroa-nilino)-propanoate (MAP)] can efficiently double the frequency of an inputlaser beam, say from 1064 nm to 532 nm, thanks to their high x(2). Somerelevant data are collected in Tables 11.8 and 11.9. However, the advent ofreasonably priced green and blue diode lasers in the mid-1990s(GaN|GaAlN) demolished the need for organic x(2) materials.

At present, materials that have high x(3) are being studied, but theirtechnological use is more remote, while x(2) crystals are used routinely inresearch and are fast approaching commercialization.

The requirement of a high x(2) is a necessary but not sufficient conditionfor technological usefulness in the frequency-doubling regime. The otherrequirements are (a) the production of large single crystals with smooth faces

Table 11.8 Crystallographic Parameters for Two Inorganic and Two Organic Crystals [17–20]

Crystal a/A�

b/A�

c/A�

a b g Space Group Class

LiNbO3 5.14829 5.14829 13.8631 90� 90� 120� R3c (#161) 3mKH2PO4 7.4527 7.4527 6.9751 90� 90� 90� I�42d (#122) �42mMAP 6.829 8.116 11.121 90� 90� 95.59� P21 (#4) 2MNA 11.57 11.62 8.22 90� 139.2� 90� Cc (#9) m

690 11 INSTRUMENTS

and (b) the ease of coupling light energy into and out of the crystal (phasematching). Although d11 is large forMNA (40 times larger than for LiNbO3), itis not phase-matcheable, because the largest optical nonlinear axis is parallelto the fiber axis.

Another way to evaluate the suitability of frequency-doubling crystals isto quote afigure ofmerit, suchas [w(2)]2/n3,wheren is the indexof refraction atthe exciting wavelength. For lithium niobate, w(2)¼ 1.4 �10�8 esu; for MAP(methyl-2-(2,4-dinitroanilino)-propanoate), w(2) is almost 3 times larger thanfor LiNbO3, but [w

(2)]2/n3 is 15 times larger. However, inorganic crystals canbe grown as large as 0.3m� 0.3m� 1m, whereas organic crystal growthtechniques have rarely been perfected to grow crystals that size! Polingguest–host polymers can reorient functional groups and raise the NLOcharacteristics, but frequency-doubling use will heat the polymers andgradually undo the effects of poling.

Much less is understood about frequency tripling. Given a molecule orpolymerwith conjugation lengthL (L is the effective lengthof the conjugatedpelectron framework), two important issues are (i) what is the exact interde-pendence between g and L, and (ii) will symmetrical or unsymmetricalsubstituent electron donor groups D and/or electron acceptor groups A aidin increasing g?

Most experiments can measure either w(3)(�3o; o, o, o) (third harmonicgeneration) or w(3) (�o; o, -o, o) (degenerate four-wave mixing).

A simple but slow measurement is self-trapping (Fig. 11.33), due tosolution heating, to nonlinear processes aided by resonance (usingmolecular

Table 11.9 Experimental Nonlinear Optical Coefficients xijk(2) (2v; v, v) (pm/V) [21–23]

Crystal w111(2) w122

(2) w133(2) w123

(2) w131(2) w121

(2) w222(2) w233

(2) w223(2) w333

(2)

2d12 2d13 2d14 2d15 2d16 2d22 2d23 2d24 2d33 2d11

LiNbO3 0 0 0 0 11.9 �0 �0 0 �0 68.8KH2PO4 0 0 0 1.26 0 0 0 0 0 0MAP 0 0 0 0 0 226 510 54 0 0MNA 500 �0.5 75 0 �0 0 0 0 0 0

Ar+ laser, CW514.5 nmor other fixed frequency

Screen

Lens

A'

F'

A

F

FIGURE 11.33

Laser self-trapping. Using a lens,the Arþ ion beam is focused intoa quartz cuvette containing thesample. The image of the beam onthe screen (diffused light withoutprecise boundaries, AA’ here) isnoted, then the beam intensity issteadily increased, until, at a criticalpower Pc, the beam shrinks sud-denly, to a smaller ring (FF’) on thescreen. Then w(3)¼no 1.86cl2/48p3Pc, where no is the linear re-fractive index, c is the speed oflight, and l is the laser wavelength.

11.16 NONLINEAR OPTICAL PROPERTIES 691

excited states), and to the nonresonant “pure” w(3), all of which contribute tothe self-focusing of the laser beam (assumed to be Gaussian75).

Moredifficult is theTHGmeasurement (Fig. 11.34),whichusually requiresa way to generate an infrared primary beam from the Nd:YAG source, since anear-IR Nd:YAG source (l¼ 1064nm), if frequency-tripled, yields l/3¼ 354.7nm, which is usually in themiddle of the UV absorption band ofmost organicsystems. The IR shifting of the primary beam is done by an H2 cell.

Another difficult experiment is degenerate four-wave mixing (DFWM),which measures w(3) (�o; o, �o, o) (Fig. 11.35).

Much work was done on w(3) of polymerizable diacetylenes, along withsystematics on oligothienyls. Theoretical calculations (by either finite field orsum-of-states methods) give results that depend too much on the theoreticalmethod and on the basis set employed, but they do indicate that, as themolecule gets longer, w(3) should increasewith somepower (3 or higher) of theconjugation length. Also, heteroatom substitution should help greatly inincreasing w(3). At present, the observed w(3) values are said to be about twoorders of magnitude smaller than what is needed for technological use.

frequency shifter

monochromator

filter

Samplecuvette

Nd-YAG laser, λ = 1064 nm

Detector

lens

FIGURE 11.34

Measurement of w(3) (�3o; o, o, o) by third harmonic generation (THG).The laser wavelength is either (i) used as is (no frequency shifter), in whichcase l/3¼ 351nm (i.e., within the absorption spectrum of most molecules), orelse (with “frequency shifter”) (ii) Raman-shifted to�1800nmbyusing aH2 gascell, or else (iii) frequency-doubled to l¼ 532nmby using aKH2PO4 crystal, andthen this wavelength is red-shifted by using a dye laser, and mixed with thel¼ 1064nm in a LiNbO3 crystal, to generate a difference frequency.

75Karl Friedrich Gauss (1777–1855).

692 11 INSTRUMENTS

11.17 ELLIPSOMETRY

Ellipsometry is the study of the change of polarization of a light waveimpinging on a solid surface and partially reflected from it, which ismodifiedby the wavefunction of the substrate or of the molecules adsorbed on it(Fig. 11.36) (see Section 2.15 for the mathematics). It is a nondestructivemethod to study theoptical properties of thinfilmsand is sensitive to even justamonolayer of inorganic, organic, or biochemical adsorbates on a flat surface.

The green line of an Hg lamp or the red light of an HeNe laser beam(632.8 nm) are the usual sources for single-frequency ellipsometry. Thewide-band light of incandescent lamp is used for spectroscopic ellipsometry.The beam of light comes out of the laser or lamp, then a first polarizer selectsan angle for linear polarization, and then an optional 1/4 wave-plate com-pensator converts it to elliptically polarized light, such that after reflection offthe sample surface at an angle F, it will be linearly polarized, and then theanalyzer (also anoptical polarizer) is adjusted to crosswith that angle to finda

I3

I1

I4

I4

I4BS3

M4

M3M2

M1

Detector

Nd-YAG laser, λ = 1064 nm90 MW/pulse, 9 ns/pulse,

10 pulses per second

BS1

BS2

Samplecuvette

FIGURE 11.35

Measurements of w(3)(�o;o,o,�o)by backward-wave degeneratefour-wave mixing (DFWM). Twopump beams of equal intensity I1and I2 and opposite phase meet atthe sample and, with a weakerprobebeam I3, cause the formationof an emitted beam I4. M1, M2, M3and M4 are mirrors; BS1, BS2, andBS3 are beam splitters.

Sample base

Sample

Φ

Light source

Polarizer

Compensator(optional)

Compensator(optional)

Analyzer

Detector

FIGURE 11.36

Simplified diagram of an ellipso-meter.

11.17 ELLIPSOMETRY 693

null (Fig. 11.36). The image of a sample is detected and recorded by aphotodiode matrix.

Ellipsometers are usually set at a fixed angles of incidence and reflection(typically 45�), which can be mechanically changed to other angles (e.g., 60�)if desired. Single-frequency ellipsometers then measure two parameters,D and C:

r ¼ tanCexpðiDÞ ðð2:15:61ÞÞ

fromwhich the application of Fresnel’s76 equationswill yield two of the threescalar parameters of interest for a thin film: (1) the real part of the scalarrefractive index n, (2) the imaginary component of the scalar refractive index,k, and (3) the film thickness t. Usually k is assumed to be zero at the observingfrequency, unless the film absorbs the light and its absorbance is known, sothe measurement of D and C will yield n and t.

Spectroscopic ellipsometry repeats all measurements at several sourcewavelengths, and aCauchy77 fit to the datawill yield values for n, k, and t for athin film. For crystals, the dielectric tensor can also be obtained.

11.18 OPTICAL AND ELECTRON MICROSCOPY

Although the magnifying lens was first mentioned in 1021 by Ibn al-Haytham,78 and eyeglasses may have been invented in 1284 by d’Armato79

in Florence, the first compoundopticalmicroscope is due toGalileo80 in 1624,and itwas first brought to the attention of biologists by van Leeuwenhoek81 inthe 1670s.

At its simplest, the compound microscope consists of two magnifyinglenses: the objective lens, held close to the sample and controlled by coarseand fine focusing adjustments, and the eyepiece lens, held close to theobserver’s eye. Improvements are the methods of illuminating the sample,moving the sample platform laterally, and correcting the objective lens fortypical optical defects (spherical aberration, chromatic aberration, etc.) byusing a compound lens system. The optical microscope, using visible light,can achieve magnifications of between 50 and 1000 diameters.

All detection phenomena that use interference or diffraction have anatural limit, known as the Rayleigh criterion: The maximum resolutionobtainable with light of wavelength l is at best:

R ¼ l=w ð11:18:1Þ

76Augustin–Jean Fresnel (1788–1827).77Augustin-Louis Cauchy (1789–1857).78Abu Ali al-Hassan Ibn al-Haytham (965–ca.1039).79 Salvino d’Armato (1258–1312).80Galileo Galilei (1564–1642).81Anton van Leeuwenhoek (1632–1723).

694 11 INSTRUMENTS

where w is the width of the slit used. This Rayleigh criterion is usuallyquoted as

R � l=2 ð11:18:2Þ

Therefore visible light (350–750 nm) cannot resolve objects smaller than175 nm. This Rayleigh limit applies to all optical and electron microscopes,diffraction methods, and so on.

The resolution of the microscope was dramatically improved in 1931 byRuska82 and Knoll83 by using an electron beam instead of visible light, with40- to 400-keV electrons: this is the electron microscope, which by nowprovides magnification factors as large as 106. There are several varieties:

1. Transmission electronmicroscopes (TEM)use a 100-keV electron gunand can study �50-nm-thick specimens, provide electron diffractionfrom them, and in thehigh-resolutionTEM (HRTEM) version achievea resolution below 0.05 nm and magnifications up to 5� 107.One drawback of TEM is that the sample has to be thinned to notover 50 nm, to enable the electron beam to traverse the sample.

2. Scanning electron microscopes (SEMs) study surfaces by rastering—that is, slowly scanning across the surfaces and imaging the secondaryelectrons ejected from the surface; because the depth of field is large,SEMs give good three-dimensional representations, but have resolu-tions about 10 times worse than TEM; the lateral resolution of SEMs istypically 10 nm.One of drawbacks of SEMs is that surface chargingof anonconducting substrate is avoided by “shadowing” the surface withconducting graphite or OsO4; the eternal question is whether theshadowing had not distorted the surface morphology.

3. Reflection electron microscopes (REMs) use an input beam ofelastically scattered electrons and measure the resultant reflectedbeam; REM is usually coupled with RHEED (reflection high-energyelectron diffraction) or with RHELS (reflection high-energy lossspectroscopy). Another variant of REM is spin-polarized low-energyelectron microscopy (SPLEM).

4. LEED. Electron diffraction started with Davisson84 and Germer’s85

1927finding that electrons candiffract likewaves,with thedeBroglie86

wavelength [Eq. (3.1.2)]. Since electrons are easily scattered by otherelectrons, an electron beam cannot penetrate more than 5 nm into asolid without losing coherence, and also suffers scattering by gasatoms and molecules. In the 1960s, thanks in part to ultra-high vacua(10�12 bar), low-energy electron diffraction (LEED) became a practicaltool with which to study solid surface structure by its diffractionprofile: LEED gives readily the point symmetry of the solid surface.An accelerating voltage of 20 to 200V is used. The experimental setup

82 Ernst Ruska (1906–1988).83Max Knoll (1897–1969).84 Clinton Joseph Davisson (1888–1958).85 Lester Halbert Germer (1896–1971).86 Louis Victor Pierre Raymond, seventh duke de Broglie (1892–1987).

11.18 OPTICAL AND ELECTRON MICROSCOPY 695

is shown in Fig. 11.37. The diffracted beam ismeasured in a hemispher-ical screen held atþ5 kV, with some grids set at retarding potentials tocapture the inelastically scattered electrons due to electron–plasmon,electron–phonon, and electron–electron scattering events within thefirst 0.5 to 1.0 nmdepthwithin the sample.Modern CCD cameras focuson the screen and transfer the image to a computer for analysis.

5. If the electron beam is held at higher energies (10–30 kV) and impingeson the surface of a solid at glancing angles (very small scatteringangles), then the technique becomes RHEED: again the surface sym-metry of the solid is probed.

6. Scanning transmission electron microscopes (STEMs) combine thethin samples used in TEM with the rastering capability of SEM.

7. Brewster87 angle microscopy is a technique that exploits total reflec-tion of monochromatic visible light at Brewster’s angle, Eq. (2.14.31),from an interface: Any small adsorbate at the interface—for example,the presence of parts of an adventitious monolayer of different refrac-tive index—can be detected with great sensitivity as a change ofreflectivity.

As we shall see, scanned probe microscopies are exempt from theRayleigh criterion, because the images are not obtained by interference ordiffraction, but are scanned piezoelectrically, that is, mechanically.

11.19 SCANNED PROBE MICROSCOPIES: STM, AFM,MFM, LFM

Scanning tunneling microscopy (STM) was discovered in 1981 by Binnig88

(who had the idea) and Rohrer (who backed it):89 Instead of using photonsand lenses, or electrons andmagnetic fields, the ideawas to use an atomicallysharp tip, bringing it down gradually (using a piezoelectric drive) onto an

Tube

Screen, +5kv

Filament

Sample

Grids

FIGURE 11.37

Schematic of LEED [24].

87 Sir David Brewster (1781–1868).88Gerd K. Binnig (1947– ).89Heinrich Rohrer (1933– ).

696 11 INSTRUMENTS

electrically conducting surface and monitoring the current between surfaceand tip; this current will be zero at large distances, and is small (of the orderof nA) when the tip atom is within a quantummechanical tunneling distancefrom the surface, but very large if the tip touches (“crashes” into) thesurface [25] (Fig. 11.38).

Piezoelectric drives (electrical leads painted onto a piezoelectric leadzirconium tantalate ceramic) are used (i) to allow the up-down (z-) motion ofthe tip with 0.10 nm precision, and (ii) to allow the tip (or the substrate) alimited rasteredmotion in the plane of the substrate (x-motion and y-motion).The idea reminds one of the Edison90 phonograph! The DC current betweenthe tip and the substrate is monitored by an ultra-high-gain low-noise circuit,whose feedback controls the Z-drive to keep the current within the tunnelingregime. The atomically sharp tips had beendevelopedpreviously byM€uller91

forfield-emissionmicroscopy. These tips are trivially fabricated by (i) simplycutting a 0.010-in. Pt/Ir wire with a wire cutter or (ii) sharpening aWwire inan alkaline solution while passing a current through it. [Sideline: An STMproject (“topografiner”) had started elsewhere (1965–1971) [26], but an idiotmanager “killed” the project prematurely!]

STM requires extensive isolation from ambient, thermal, or instrumentalvibrations. The height resolution of STM is an amazing� 0.01 nm; the hor-izontal resolution is� 0.1 nm: One could finally see and control atoms andmolecules on a conducting surface! STM avoids the Rayleigh criterion,because the image is controlled by mechanical (piezoelectric) movements.The early STMs all functioned in high vacuum, but by now STMs can be usedat room temperature in air (as long as the vibration isolationproblem is solvedby bungee cords). Figure 11.39 shows the image of anHOPG (highly orientedpyrolitic graphite) sample in air.

The image can be quite different, depending on theuser-selected constant“set-point” bias (within a range of about� 2 V) between tip and substrate: thebias will move the tip up and down relative to the substrate. There are twoways of collecting an STM image: (a) the constant-currentmode,where the tip

Z-drive X-drive Y-drive

TunnelingCurrentMonitor

TIP

PIEZO

SUBSTRATE

FIGURE 11.38

Schematic diagram of STM. Notshown is the microcomputer(whose program controls andmonitors the X-, Y-, and Z-drivesand the tunneling current).

90 Thomas Alva Edison (1847–1931).91 Edwin Wilhelm M€uller (1911–1977).

11.19 SCANNED PROBE MICROSCOPIES: STM, AFM, MFM, LFM 697

travels up and down under constant set-point bias and computer control,maintaining a constant tunneling current, and (b) the constant-height mode,whereby, again with user-selected set-point bias, the tip current is allowed tovary, but the tip is not allowed to move.

The STM image is not amere “photograph” of the conducting surface, butis rather a convolution of the wavefunctions of the surface and of theatomically sharp tip: If the tip is deformed, or if there are multiple atomson the tip, strange images can be obtained! The STM micrograph can becalculated if there is a good formalism for (i) the surface wavefunction and(ii) the wavefunction in the immediate neighborhood of the nanotip.

Thus it is easier to see a change in the periodicity of the surface due, forexample, to an ordered adsorbed layer on the surface (Fig. 11.40) than it is todetect a single molecule on the surface.

FIGURE 11.39

Room-temperature image of anHOPG substrate in air. The repeatdistance is 0.24 nm, the distancebetween second-nearest-neighborC atoms in graphite. In a singlegraphene sheet the C–C bond dis-tance should be the aromatic0.14nm, but the surface wavefunc-tion of graphite consists of twosheets of graphene longitudinallydisplaced so that the overlap Catoms are 21/2 0.14¼ 0.24nmapart.

FIGURE 11.40

STM image of Langmuir–Blodgettmonolayer of g-C16H33-Q-3CNQ(structure is shown in Fig 11.31) onHOPG: The 0.6� 1.2-nm shape isconsistent with the moleculesviewed “end-on” from the dicya-nomethylene end [27].

698 11 INSTRUMENTS

Single molecules (and not surface artifacts or specs of dust) are easier toseewhen the surface can be cleaned in ultra-high vacuumat low temperature,imaged to make sure no artifacts are present, and then volatile adsorbatesare injected onto the cold surface without breaking vacuum. A nice tour deforce was when at 4K under ultra-high vacuum, Xe atomswere picked up bythe STM tip and deposited onto a cold Ni substrate to “write” “IBM” onNi [28]. It is not too clear exactly how far above the surface the atomically tipfloats; a good guess is� 0.1 nm.

1. A very useful ancillary technique to STM is scanning tunnelingspectroscopy (STS): The XY scan is briefly interrupted (hoping thatthe tip does not move!), the tip voltage V is scanned (� 2V), and thecurrent I is measured (typically 1 to 10 nA or even 50pA), thusproducing an IV plot (Fig. 11.41).

2. Adding a very small solution sample cell, coveringmost of the STM tipwith lacquer (to limit unwanted conductivity from the shaft of thenanotip), and adding a third “reference” electrode permits scanningelectrochemical microscopy (SECM); this is electrochemistry prac-ticed on a nanoscale.

3. One can also use the atomically sharp tip to “drill” holes in substrates;this has been called “nano-dozing” (i.e., a nanoscopic Bulldozer�).

4. Spin-polarized scanned tunneling microscopy (SPSTM). Replacinga nonmagneticW or Pt or Au nanotip by a ferromagnetic Fe, Co, Ni, orGd nanotip permits studying magnetic surfaces and measuring theeffects of spin polarization:

P ¼ ðn"�n#Þ=ðn" þ n#Þ ð11:19:1Þ

where n" and n# are the density of “spin-up” and spin-downelectrons, respectively. However, progress for this technique hasbeen slow. There are tunneling magnetoresistance effects. The fieldreversals in samples studied with ferromagnetic tips has led to theuse of antiferromagnetic tips.

FIGURE 11.41

STS IV curve of 15 Z-typeLangmuir–Blodgett monolayers ofg-C16H33-Q-3CNQ (structure isshown in Fig. 11.31) on HOPG [27].

11.19 SCANNED PROBE MICROSCOPIES: STM, AFM, MFM, LFM 699

The next radical improvement in scanned probe microscopies was theinvention of the atomic force microscope (AFM) in 1986 by Binnig, Quate,92

and Gerber93 [29]. The X- and Y-piezoelectric scanners were kept, but theatomically sharp conducting tip was replaced by a sharp (but not atomicallysharp!) Si cantilever (Fig. 11.42), with a mirror glued to its back.

Now the substrate does not have to be electrically conducting, becausewhat is measured in the Z-direction is the change in the natural vibrations ofthe cantilever caused by van der Waals forces between it and the surface.The Z-movements of the cantilever are monitored by a visible laser, whosebeam is reflected by themirror on the back of the cantilever andmeasured byfour semiconductorphotoelectric detectors arranged in a quadrant tomonitorthe laser reflection and thus the sample position (Fig. 11.43).

In theAFM theXY resolution is nowno longer “atomic”: It is now limitedby the sharpness of the AFM cantilever (to typically 2–3 nm). However,liberating scanned probe microscopy from needing a conducting surfacevastly extended its usefulness. AFM has become very popular and useful,

FIGURE 11.42

AFM Si cantilever.

FIGURE 11.43

AFM and microcomputercontroller.

92Calvin F. Quate (1923– ).93Christoph Gerber (1942– ).

700 11 INSTRUMENTS

particularly for structures above 50 nm in size. Figure 11.44 shows an AFMmicrograph of the surface ofAl after acid-anodizing it and thendissolving theamorphous Al2O3 pores away; the hexagonal pattern of the interface is seenvery clearly.

“Children” of AFM are

1. Lateral force microscopy (LFM), where the longitudinal vibrations ofthe cantilever are monitored.

2. Conducting-tip atomic force microscopy (CTAFM), where the AFMtip is coated by a layer of metal, thus allowing the study of conductingsurfaces, with a reduced lateral sensitivity (�5 nm) but with a much“gentler” van der Waals force applied to the substrate (thanks to theAFM control), so that very soft materials can be studied with minimaldamage to the surface.

3. Magnetic force microscopy (MFM), where the AFM tip is coated by alayer of magnetized metal, thus allowing the study of magneticsurfaces, with a reduced lateral sensitivity (�5 nm).

Initially it was hoped that STMorAFM could allow huge information storagedensities: If each “nano” bit had size 5� 5 nm2 on a planar storage area of10 cm� 10 cm, this would imply a bit density of 4 Tbit cm�2. However, theX,Y, or Z piezoelectric controllers canmove, at best, only at the speed of sound,which is too slow for addressing and retrieving all these nanobits. IBMCorporation devised an array of 32� 32 AFM tips, or 1024 tips, and calledthis multibit programmable multiplexed detector the “millipede,” but it wasa tour de force, not a practical data storage technology.

Near-field scanning optical microscopy (NSOM) also studies nanos-tructures much smaller than the far-field Rayleigh limit [Eq. (11.18.2)], but byexploiting evanescent waves. These fast-decaying electromagnetic waves areformedverynear to an interface between twomediawhen an electromagneticwave ofwavelength lundergoes “almost’ total internal reflectionwithin onemedium (at or beyond its Brewster’s angle); continuity requires that a smalland exponentially decaying wave (the evanescent wave) penetrate at leastpartially into the second medium. NSOM is done by placing the detector

FIGURE 11.44

Room-temperature AFM micro-graph in air (1.5 mm� 1.5mm) of anelectropolished Al surface after24h of growing acid-anodizedAl2O3 pores nanopores on top ofit, then dissolving away the Al2O3

pores [30].

11.19 SCANNED PROBE MICROSCOPIES: STM, AFM, MFM, LFM 701

above the interface by a distance that is but a small fraction of the light’swavelength l. Lateral and vertical resolutions of the order of 20 nm and2–5 nm, respectively, have been found.

11.20 MAGNETIC MEASUREMENTS

Detecting magnetic materials involves measuring a force between the mag-neticmaterial and apermanentmagnet.Manymethods of detectingmagneticfields with varying sensitivities and precision have been developed(Fig. 11.45).

A simple instrument to measure paramagnetic materials is the Gouy94

balance, which suspends the magnetic sample inside the pole gaps of a largepermanent magnet or electromagnet (typically 0.2 to 5 T) and measures thedownward force (for a paramagnet) or the upwards force (for a diamagnet)using a laboratory balance (difference in mass between the sample residinginside and outside the magnetic field) (Fig. 11.46):

ðmB�m0Þg ¼ ð1=2Þm0wAB2 ðSIÞ; ðmB�m0Þg ¼ ð1=2ÞwAB2 ðcgsÞð11:20:1Þ

where B is themagnetic induction, m0 is themagnetic permittivity of vacuum,A is the cross-sectional area of sample, g is the acceleration due to gravity, wis the volumemagnetic susceptibility,mB is sample mass with B on, andm0 isthe sample mass at B¼ 0 [32].

The sensitivity of the Gouy balance (typically 10�8 cgs-emu cm�3) isacceptable forparamagnetic substances; diamagnetic samples give very small

FIGURE 11.45

Precision versus sensitivity of mag-netometers versus source [31].

94 Louis Georges Gouy (1854–1926).

702 11 INSTRUMENTS

apparent mass differences; ferromagnetic samples are strongly attracted toone of the poles, and thus cannot be studied by this method.

The Evans95 balance, a small modification of the Gouy balance, movesthe magnet instead of the sample. The Sucksmith96 ring balance usesoptical detection to measure the rotation of a ring bearing the specimen inthe pole gap [33]. The Faraday balance resembles the Gouy balance,except that the pole faces are not parallel but are so configured that B(dB/dz) is constant over z¼ several centimeters (Fig. 11.47). The static fieldis typically 1 T; the detectable magnetic moment range is 2� 10�9 to6� 10�5 A-turn m2 (this upper limit is the moment for a 0.025-mm3

sample of Fe) [34].

SN

Sample

FIGURE 11.46

The Gouy balance.

30º

0 1 2 in

20016012080400,0,0

6

4

3

2

1

cm

Z

A BC

D

E

0.5″B2 x 10–6

19x103AMP-TURNS

40 x 103AMP-TURNS

3 cm210

FIGURE 11.47

Faraday balance: Pole caps of elec-tromagnet.

95Dennis F. Evans (1928–1990).96W. Sucksmith (1896–1981).

11.20 MAGNETIC MEASUREMENTS 703

The vibrating sample magnetometer (VSM: 50 memu to 10 emu;�0.5 memu) is a standard instrument for measuring MH loops (Fig. 5.12C) byvibrating the sample in aDC external field at a frequencyo, andmeasuring ato the change in inductance of thewire loops surrounding the sample. Amoresensitive instrument is Flanders’97alternating gradient magnetometer(AGM: 1 memu to 3 emu;� 0.01 memu [35].

The Hall98 probe gaussmeter measures fields in magnetic materials (mTto 20-T range) by the Hall effect: The magnetic field B interacts with aperpendicular “control” current I sent through a semiconductor probe (thesedays, inside an integrated circuit) and produces a voltage VH given byEq. (8.1.10) (Fig. 11.48). At high enough fields and low enough temperatures,the Hall effect is quantized: The Hall electrical conductivity s is given by

s ¼ ne2=h ¼ n ð3:874� 10�5 SÞ ¼ n ð38; 740 nSÞ ð11:20:2Þ

where e is the electronic charge, h is Planck’s constant, and n can be either aninteger (integer Hall effect: Landau99 quantization) or a rational fraction(fractional Hall effect, n¼ 1/3, 1/5, 2/5, 12/5, etc.: electron–electron inter-actions). The reciprocal of (e2/h¼ 25,812.8O) is called the von Klitzing100

constant or the Landauer101 quantum of conductance.TheNMRgaussmeter (or teslameter, to be “modern”) detects the nuclear

magnetic resonance frequency nof protons in awater sample in the field beingmeasured:

B ¼ ðhn=gNbNÞ ¼ 4:2577� 107 Hz=Tesla ð11:20:3Þ

The precision is very high, but the sensitivity is low (20mT to 9T,� 0.5 mT).The fluxgate magnetometer, invented by Vacquier102 in the 1930s,

consists of a small, magnetically susceptible core wrapped by two coils ofwire. An AC electrical current passing through one coil drives a permeable

MagneticFm = magnetic

Fm

Fe = magnetic force from charge buildup

Fe

++

++

++

––

––

–

–

field B

VH

d

Direction of conventionalelectric currentI

I

force onnegative chargecarriers.

FIGURE 11.48

Hall voltage.

97 Philip J. Flanders (ca. 1929– ).98 Edwin Herbert Hall (1855–1938).99 Lev Davidovich Landau (1908–1968).100Klaus von Klitzing (1943– ).101 Rolf William Landauer (1927–1999).102Victor Vacquier, Sr. (1907–2009).

704 11 INSTRUMENTS

(“soft”) magnet through cycles of magnetic saturation (i.e., magnetized, thenunmagnetized, then inversely magnetized, then unmagnetized, then remag-netized, etc.); an induced electrical current in the second coil is measured bya detector. In an external magnetic field B, the magnet will saturate whenaligned with that field and will be less easily saturated in opposition to it.Fluxgate magnetometers, paired in a gradiometer configuration, measureboth direction and magnitude of B, and thus are used for detecting enemysubmarines and for archaeological prospecting. Phase-sensitive detection isoften used.

The optically pumped Cs vapor magnetometer (OPM) is a highlysensitive (0.004 nT/

ffiffiffiffiffiffiffiHz

p) and accurate device, with a Cs lamp source,

an absorption chamber with Cs vapor plus a “buffer” gas, and a photondetector. It detects weak magnetic fields in the range of 15 mT to 100 mT.Its development was pushed by a military need to detect enemy submar-ines from flying aircraft. Of nine states and energy levels of Cs atoms, threecan absorb the light emitted by the Cs lamp, until the levels are saturatedand will absorb no further energy; then the system is “polarized.” A verysmall AC magnetic field at frequency n is then applied to the absorptioncell, to force induced emission from the excited Cs atoms, and the detectorwill see a drop in light intensity because of renewed absorption at v.This magnetometer is used for geomagnetism. The Bell103–Bloom104 Csmagnetometer modulates the light applied to the cell at a frequency n, andit detects optically the changes at the frequency corresponding to theearth’s field.

Spin-exchange relaxation-free (SERF) atomic magnetometers containK,Cs, orRbvapor andhave sensitivities below1 fT/

ffiffiffiffiffiffiffiHz

p; they operate only in

small magnetic fields (< 0.5 mT), but have greater sensitivity per unit volumethan SQUID detectors.

11.21 MAGNETIC RESONANCE

The Family of Magnetic Resonance Techniques. Magnetic resonancespectroscopy is a sub-branch of radio-frequency (RF) spectroscopy: it iscalled a “resonant” technique because, as the frequency of the instrumentis systematically varied, or swept, or “tuned”, a quantum transition in thesample occurs (Fig. 11.49), caused by a coupling between themagnetic dipolemoment of the sample and themagnetic component of theRF electromagneticfield, which induces the absorption of a photon that bridges the energydifference between two magnetic states (Fig. 11.50). When all thishappens, the RF energy absorbed by the sample is detected electrically:The sample “resonates” (absorbs) at the same frequency as the source.Much effort is dedicated to build electrical circuits that are sensitiveto very small signals (in EPR, a microwave cavity of high-quality factor Qis used).

103William Earl Bell (1921–1991).104Arnold L. Bloom (1923– ).

11.21 MAGNETIC RESONANCE 705

Magnetic resonance spectroscopy started in 1938whenRabi105 improvedthe Stern106–Gerlach107 experiment: He sent a molecular beam bearingindividual magnetic moments through an inhomogeneous magnetic field,as before, but measured a resonant absorption of RF energy by the beam: thisresonance is caused by a magnetic-dipole-allowed quantum transition [37].

Magnetic resonance is a series of related techniques that include twoprincipal methods:

(i) Nuclear magnetic resonance (NMR) (which could be called nucleardiamagnetic resonance) discovered almost simultaneously in 1946 bythe groups of Bloch108 and Purcell109 [38–40].

(ii) Electron paramagnetic resonance (EPR), discovered in 1944 byZavoisky110 [41]; it is also called, less frequently, electron spin res-onance (ESR); its application to ferromagnets is called ferromagneticresonance (FMR).

There are many “children” of NMR and EPR:

(iii) Spin-echo and multiple-pulse techniques, such as nuclear spin-echo(NSE), electron spin-echo (ESE), electron–nucleus double resonance(ENDOR), electron–electron double resonance (ELDOR), andelectron–electron–nucleus triple resonance (TRIPLE),

(iv) Nuclear quadrupole resonance (NQR),

(v) Anoptical off-shoot:optically detectedmagnetic resonance (ODMR),and

| β > mI = -1/2

| α > mI = 1/2

spin down

spin up

h ν = gN βN H

0 Magnetic field H (Tesla)

Energy

| β > mS = 1/2

| α > mS = -1/2

spin up

spin down

h ν = ge βe H

0 Magnetic field H (Tesla)

Energy

Nuclear Spin transitions Electron spin transitions

| β>

| α>

Pαβ=P=Pβα Wβα < Wαβ

Stimulatedtransitionshave equal prob. up and down

Spontaneous transitions in absorption are more likely up than down

FIGURE 11.49

Magnetic transitions in externalmagnetic field H [(top left) NMR:I¼ 1/2; (top right) EPR: S¼ 1/2)] anddepiction of transitions for sponta-neous and stimulated EPR transi-tions (bottom). Adapted fromCarrington and McLachlan [36].

105 Isidor Isaac Rabi (1898–1988).106Otto Stern (1888–1969).107Walther Gerlach (1889–1979).108 Felix Bloch (1905–1983).109 Edward Mills Purcell (1912–1997).110 Yegeny Konstantinovich Zavoisky (1907–1976).

706 11 INSTRUMENTS

y

x

z

B0

β-state: ms = -1/2 for electronsα-state: mI = +1/2 for nuclei

ω0

μi

μf

ω0

ABSORPTIONOF

PHOTON

Larmor precessiondirection for electrons

Larmor precessiondirection for electrons

Larmor precessiondirection for nuclei

α-state: ms = +1/2 for electronsβ-state: mI = -1/2 for nuclei

Larmor precessiondirection for nuclei

Circularlypolarized

field for

electron transition

FIGURE 11.50

EPRorNMRtransitioninexternalmagneticfieldH0, showingthecircularlypolarizedelectromagnetic field at the frequencyo0 whichmatches the Larmor 111 precessionv0¼H0/g of the (EPR: electron) (NMR: nucleus) and, for EPR, causes the transitionfrom the spin-down (b-state, Sz¼�(1/2)h) (bottom vector) to the spin-up (a-state,Sz¼ (1/2)h) (upper vector) (for NMR the spin assignments are reversed). Initially, theelectron[nucleus] is inthe lower-energy statemS¼�1/2 (b spinstate) [NMR:mI¼ 1/2(a spin state)], and it precesses around the static field H0 at the Larmor frequencyv0¼H0 /g: The precessional motion can be drawn as a cone or “coolie hat.” Themagnitudeofthespinis |S |¼ [(1/2) (1/2þ1)]1/2h [NMRfornuclear I¼ 1/2: | I |¼ [(1/2)(1/2þ1)]1/2h]. The initialmoment ismi¼ (|e|h/2me)S; itsz-component ismi,z¼ (|e|h/8pme).The circularly polarized exciting field in the XY plane, when it attains the exactfrequencyv0, can cause the absorption of a photon, and the electron goes into thehigher-energy state mS¼þ1/2 (a spin state) and will now precess around thestatic field H0 at the same Larmor frequency v0¼H0/g. The magnitude of the spinisunchanged,butthefinalmomentismf¼ (|e|h/2me)S; itsz-componentismi,z¼þ(1/2)(|e|h/4pme). [For NMR, since nuclei are positively charged, the relative positions ofspin-up and spin-down, aswell as the directions of Larmor precession, are reversed:the nuclear b spin state (for protons, Iz¼�(1/2)h) is higher in energy, while thenuclear a spin state (for protons, Iz¼þ(1/2)h) is lower in energy.]

111 Sir Joseph Larmor (1857–1942).

11.21 MAGNETIC RESONANCE 707

(vi) Last, but by no means least, the hugely successful medical offshoot,(nuclear) magnetic resonance imaging (MRI). [Note: The adjective“nuclear” was dropped from it to allay the public’s possible fearsabout a “nuclear’ technique in NMRI!]

The absorption of radiation (for NMR, in the radio wave range,10–80MHz, or in the radar range, up to 1GHz; for EPR, in the microwaverange, 3–100GHz) depends on the relative population of the ground, or lower(l) state and first excited, or upper (u) state, that is, it depends on a Boltzmannfactor of the type Nu/Nl¼ exp (�DE/kBT), where DE is the relevant energydifference.

The Hamiltonian112 H for an electron of spin S and a nucleus of spin I inan external magnetic field H0 is given by

H ¼ beH0 ge Sþ bNH0 gN IþP

ijJijIi Ij ð11:21:1Þ

where ge is the anisotropic second-rank g-tensor for the electron, and gN is theanisotropic second-rank g-tensor for the nucleus. The g-tensor for the nucleusis usually approximated by its scalar value. If ge is indeed anisotropic, then Sin Eq. (11.21.1) is a fictitious spin, not the “true spin,” because of theinteraction between H0 and the orbital angular momentum L.

We first introduce the simplest NMR and EPR instruments, then presentthe relevant theory, and finally discuss the more specialized and advancedtechniques.

Nuclear Magnetic Resonance (NMR) [42]. When a sample (solid or liquid)consisting of nuclei of spin I andmagnetizationM is placed in an external DCmagnetic field H0, the Larmor frequency n0 (Hz, or cycles per second) fornuclear spin projection transitions DmI¼ � 1 is

n0 o0=2p¼DE=h¼ h�1H0 M¼ h�1gNH0 mN ¼ h�1gNbNH0 I¼ðgN=2pÞH0 Ið11:21:2Þ

where h is Planck’s113 constant, e is the electronic change, and (as discussed inSection 3.20) gN is the nuclear gyromagnetic or magnetogyric ratio, gN is the“g-value” for the nucleus (a pure number betweenþ4 and�1 for most nucleiof interest),mN is the nuclearmoment, and bNeh/2mp¼ 5.051� 10�27 J T�1 isthe nuclearmagneton,wheremp is themass of the proton. Exciting the sampleby applying an AC electromagnetic field of frequency n0 (Hz; or o0 radianss�1) induces a change in orientation of the nuclear spin in the DC magneticfield. The transition is thus a magnetic-dipole transition, which can bedetected by NMR: RF energy is absorbed by the sample, as either the fieldH0 or (more usually) the RF frequency n is “swept” (varied) across theresonance frequency n0. It can be detected by nuclear magnetic induction:A signal is induced in a “pickup coil” as either the field or the frequency isvaried (Fig. 11.51).

112 Sir William Rowan Hamilton (1805–1865).113Max Planck (1858–1947).

708 11 INSTRUMENTS

It is known from nuclear structure theory that nuclei with I� 1 also havean electric quadrupole moment Q, which is a measure of the asphericity ofthe nucleus. Nuclear quadrupole moments affect the rate of magnetic dipolerelaxation, and nuclei with Q $ 0 are candidates for NQR measurements(see Table 3.3).

For chemically interesting nuclei, Table 11.10 lists values of the nuclearspin quantum number I, the nuclear gyromagnetic ratio gN, the nuclearelectric quadrupole momentQ, and the nuclear magnetic resonance frequen-cy v (Hz, for H0¼ 1 tesla).

At 1.4 Tesla114 and 60MHz (for H1), typically 1015 nuclear spins can bedetected by continuous-wave (CW) NMR spectroscopy. Usually the NMRfield is fixed (and made very homogeneous by “shimming” coils for electro-magnets), and the RF frequency is swept. For increased sensitivity, Fouriertransform (FT) techniques are used, which nowmake C13 spectroscopy fairlyroutine. Typical NMR instruments in the 1960s were based on conventionaliron-core electromagnets; the numbering refers to the H1 frequency, evenwhen other frequencies were used (for the same fixedmagnetic field): VarianA60 and JEOL FX60Q (H0¼1.42 T, n¼60MHz forH1), VarianHA100 (H0¼2.37T, n¼100MHz). In the 1980s a transition to superconducting magnets oc-curred:NicoletA200, Bruker EM360 (H0¼8.52 T, n¼ 360MHz), Bruker EM500

Sample

Northpole ofelectro-megnetor super-conductingmagnet

South pole ofelectro-megnetor super-conductingmagnet

Electromagnet power supply (for 11,000-20,000Gauss) and also for shimming coilsor flux induction supply (for superconductingmagnet)

60-900 MHz RF source

Computer and display

Z sweep coils

Samplespinner

(compressed-air-driven)

Detectorfor y-pickup coils or Power supplyfor y-pulse coils

Detectorfor z-pickup coils or Power supplyfor z-pulse coils

Y-coils

Z-coilsZ sweep coils

A/D converters,FT hardware, signal processors

FIGURE 11.51

Schematic of NMR spectrometer.

114Nikola Tesla (1856–1943).

11.21 MAGNETIC RESONANCE 709

(H0¼ 11.8 T, n¼ 500MHz). In the late 1990s, 800-Mz instruments have arrived(H0¼ 18.9 T).

Electron Paramagnetic Resonance (EPR) Spectrometer [43–46]. When aparamagnetic or ferromagnetic sample (solid or liquid) of molecules orions with total spin S and magnetization M is placed in an external DCmagnetic field H0, the frequency n (Hz, or cycles per second) for spintransitions DmS¼ � 1 is

n0 o0=2p ¼ DE=h ¼ geH0 M ¼ ðgebe=hÞH0 S ¼ ðge=2pÞH0 S ð11:21:3Þ

where h is Planck’s constant, e is the electronic change, me is the mass of theelectron, ge is the electronic gyromagnetic ratio, ge is the electronic “g-value”(a pure number between þ1 and þ6 for most paramagnetic molecules orions, but 2.00232 for the free electron in vacuo), and be is the Bohr magneton,

Table 11.10 Table of Nuclei (plus the Free Electron, the Electron in the Crystal Diphenyl Picryl Hydrazyl(DPPH),andtheFreeNeutron),TheirNaturalAbundanceonEarth,TheirMass(1amu¼ 1.6605402� 10�27 kg),Their Electrical Charge, Their Nuclear Spin I (� the Electron Spin S for the Electron), Their Magnetic Moment(in Units of the Nuclear Magneton for the Nuclei and Neutron (mN¼eh/4pM, whereM¼proton mass¼1.672631� 10�27 kg, so mN¼ 5.0507866� 10�27 J T�1), and in Units of the Electronic Bohr 115 Magneton(be¼eh/4pm¼ 9.2740154� 10�24 J T�1) for the Electron

ChargeSpin Gyromagnetic

ElectricQuadrudopleMoment Q

ResonanceFrequency

NucleusNatural

Abundance (%)Half-Life

Mass(amu) e I Ratio (gN) Ratio (ge) (10�24 cm2) MHz@1T

Electron 100 1 1/1831 �1 1/2� 2.00232 0 2.8027E5DPPH electron Small 1 1/1831 �1 1/2� 2.0036 0 2.8045E5Neutron 0 12min 1.008665 0 1/2 �1.9131 0 29.165

1H1proton 99.985 1 1.007825 1 1/2 2.79284 0 42.577

1H2 0.015 1 2.0140 1 1 0.85743 0.002738 6.636

1H3 0 12.26 y 3.01605 1 1/2 2.97896 0 45.414

3Li6 7.5 1 6.015121 3 1 0.822056 0.045 6.265

3Li7 92.5 1 7.016003 3 3/2 3.25644 þ0.02 16.547

6C12 98.90 1 12.00000 6 0 0 0 0 0

6C13 1.10 1 12.003355 6 1/2 0.70241 0 10.705

6C14 0 5730 y 14.003241 6 0 0 0 0

7N14 99.63 1 14.003074 7 1 0.40376 0.02 3.076

7N15 0.37 1 15.000108 7 1/2 �0.28319 0 4.315

9F19 100.0 1 18.998403 9 1/2 2.62887 0 40.055

11Na23 100.0 1 23.989767 11 3/2 2.2161 0.1 11.262

13Al27 100.0 1 26.98154 13 5/2 3.64150 0.149 11.094

14Si29 4.67 1 28.976495 14 1/2 �0.5553 0 8.460

15P31 100.0 1 30.973762 15 1/2 1.13160 0 17.235

16S33 0.74 1 32.971456 16 3/2 0.64382 �0.064 3.266

16S35 0 35 d 34.969031 16 3/2 1.00 0.045 5.08

17Cl35 75.77 1 34.968852 17 3/2 0.82187 �0.07894 4.172

17Cl37 24.23 1 36.965903 17 3/2 0.68412 �0.06213 3.472

115Niels Hendrik David Bohr (1885–1962).

710 11 INSTRUMENTS

be¼ eh/4pme¼ 9.274�10�24 J T�1.Given the ratio between theBohrmagnetonand the nuclear magneton, it is clear that EPR transitions involve energiesabout 2000 times larger than NMR transitions. Electron paramagneticresonance (EPR) spectroscopy studies organic and inorganic free-radicalsystems, with a thousand-fold greater sensitivity than NMR but asignificant restriction to free radicals which are stable for more than 1ms(because of the RF energy used to kick the system on and off-resonance). Thespectrometers use klystrons, magnetrons, or Gunn 116 diodes, as powersources (Fig. 11.52). Typical commercial EPR instruments in the 1960s

hybrid Tee

Resistive termination

Micro-wavecavity9.5 GHz

Detecting crystal diode(1N23F)

Preamplifier(detects μλpower level +100 kHz modul.)

OptionalMicrowavefreq.meter

phase shifter

Klystron(tunable from 9.2 to 9.7GHz)

Automaticfrequencycontrol (AFC)

Klystronpower supply(100-400 VDC)

Tuned Amplifier(for 100 kHz)

Isolator

optional NMR“gaussmeter”

Northpole ofelectro-magnet

Southpole ofelectro-magnet

Electromagnet power supply (for 3,500 Gauss)

Hall probegaussmeter forelectromagnetfield control

100 kHz Field ModulationOscillator andAmplifier

Phaseshifter

Phase-sensitivedetector

DC amplifier, andoutput device (CRTscreen, recorder, orcomputer)

Sample iscenteredin cavity

Iriscoupler

Modu-lationcoils

FIGURE 11.52

SchematicofX-bandEPR spectrom-eter (Varian E-series).

116 John Battiscombe Gunn (1928–2008).

11.21 MAGNETIC RESONANCE 711

were the Varian V-4500 and E-series, the JEOL JES-ME-1X, and the BrukerESP. Today JEOL and Bruker are still manufacturing EPR spectrometers.

The Bloch Equations for Magnetic Resonance. At thermal equilibrium, themagnetizationM0, due to individual electron or nuclear magnetizations mi, isassumed to be proportional to the static field H0 along the z axis:

M0 ¼ w0H0 ¼X

imi ð11:21:4Þ

For both NMR and EPR, the phenomenological Bloch equations [47] canbe used to track the time-dependence of the magnetization of the sample Min the total field H:

dMx=dt ¼ gðH�MÞ ex�Mx=T2

dMy=dt ¼ gðH�MÞ ey�My=T2

dMz=dt ¼ gðH�MÞ ez þ ðM0�MzÞ=T1

ð11:21:5Þ

(To discuss nuclei and electrons together, we assume g > 0 for electrons,and g< 0 for nuclei). The excited spin state (nuclei or electrons) will tend torelax and reorient, over time, by spin–spin interactions (sample-averagecharacteristic spin–spin or transverse relaxation time T2, which estimateshow long it takes for the relative phases between spins to be randomized)and interactions with the lattice (sample-average spin-lattice or longitu-dinal relaxation time T1, which estimates how long it takes for the spindistribution to come into agreement with the relative population distri-bution dictated by temperature and by the Boltzmann factor). The relax-ation times are characteristic of the material being studied. The equationscan be solved under either adiabatic rapid-passage conditions or slow-passage (steady-state) conditions. If there is no relaxation (i.e., if T1¼T2

¼1), then the surviving dM/dt¼ g (H�M) implies that the Larmorprecession will go on forever, unchanged.

For a solid, the spin-lattice relaxation will be related to interactions withspins at fixed lattice positions: This tends to be slow (T1 is large);the interactions between neighboring spin orientations occur very rapidly,soT2 tends to bemuch shorter,T2�T1. In liquids,wherediffusiondominates,T1 will be shorter, and T1T2. In typical H1 NMR of solutions, a typicalT2 1 s, so that linewidths of 10�5 T are usual.

The spin relaxation time for paramagnetic Mn2þ in aqueous solution isabout 3� 10�9 s; the rotational correlation time is about 10�11 s; the mean freetime for a proton to reside in a hydration sphere is about 2� 10�8 s.

At thermal equilibrium we have

dMz=dt ¼ 0 ð11:21:6Þ

We seek a solution to the Bloch equations in the case of an electromagneticfield H1, circularly polarized in the xy plane and rotating counterclockwisewith direction �v (as seen from the þz axis):

H1 ¼ H1excos ðotÞ�H1eysinðotÞ ð11:21:7Þ

712 11 INSTRUMENTS

Using H¼H0þH1 in Eq (11.21.5) yields

dMx=dt ¼ g½MyH0 þMzH1sin ðotÞ��Mx=T2

dMy=dt ¼ g½MzH1cos ðotÞ�MzH0��My=T2

dMz=dt ¼ g½�MzH1sin ðotÞ�MyH1cos ðotÞ��ðMz�M0Þ=T1

ð11:21:8Þ

These equations become simpler in a coordinate frame (u, v, z) that rotates inthe xy plane with the frequency o; this coordinate system is defined by

Mx ucosðotÞ�vsinðotÞMy �usinðotÞ�vcosðotÞ

ð11:21:9Þ

and its inverse:

u ¼ Mxcos ðotÞ�Mysin ðotÞv ¼ �Mxsin ðotÞ�Mycos ðotÞ

ð11:21:10Þ

whence

du=dtþ u=T2 þ ðo�o0Þv ¼ 0

dv=dtþ u=T2 þ ðo�o0Þuþ gH1 Mx ¼ 0

dMz=dtþ ðMz�M0Þ=T1�gH1v ¼ 0

ð11:21:11Þ

Slow Passage or Equilibrium or Steady-State Solution. We seek the “slow-passage” or “equilibrium” or “steady-state” solution, where all three timederivatives in Eq. (11.21.11) are set equal to zero; the answers are

u ¼M0gH1T22ðo0�oÞ ½1þ T2

2 ðo0�oÞ2 þ g2H21T1T2��1

v ¼ �M0gH1T2 ½1þ T22 ðo0�oÞ2 þ g2H2

1T1T2��1

Mz ¼M0 ½1þ T22 ðo0�oÞ2� ½1þ T2

2 ðo0�oÞ2 þ g2H21T1T2��1

ð11:21:12Þ

which for the magnetizations along the three axes yield:

Mx ¼M0gT2½H1T2ðo0�oÞcosðotÞþH1sinðotÞ� ½1þT22ðo0�oÞ2þg2H2

1T1T2��1

My ¼M0gT2½H1cosðotÞ�H1T2ðo0�oÞsinðotÞ� ½1þT22ðo0�oÞ2þg2H2

1T1T2��1

Mz ¼M0½1þT22ðo0�oÞ2� ½1þT2

2ðo0�oÞ2þg2H21T1T2��1

ð11:21:13Þ

The “properly” rotatingmagnetic fieldH1¼H1excosðotÞ�H1eysinðotÞ can beconsidered as the sum of two fields: an oscillating field

H1x¼2H1cosðotÞ ð11:21:14Þ

11.21 MAGNETIC RESONANCE 713

for which the counter-rotating part is H1excosðotÞþH1eysinðotÞ. In otherwords,H1¼H1expðiotÞþH1expð�iotÞ¼2H1cosðotÞ. Therefore Eq. (11.18.13)can be rewritten in terms of a complex susceptibility (sometimes called Blochsusceptibility):

ww0 þ iw00 ð11:21:15Þ

where the oscillating field 2H1cos(ot) causes an in-phase real susceptibility w0

with magnetization 2w0H1cos(ot) and an out-of-phase imaginary suscepti-bility w00 with magnetization 2w00 H1sin(ot):

w0 ¼ ð1=2Þ w0o0½T22ðo0�oÞ� ½1þ T2

2 ðo0�oÞ2 þ g2H21T1T2��1

w00¼ ð1=2Þ w0o0T2 ½1þ T22 ðo0�oÞ2 þ g2H2

1T1T2��1ð11:21:16Þ

The rate of absorption of energy dE/dt (Wm�3) from the RFmagnetic fieldHx

is�Mx(dH1x/dt), which in Eq. (11.21.13) contains (i) a termproportional to thetrigonometric sin(ot)cos(ot), which, integrated over one cycle, averages tozero and (ii) a term proportional to sin2(ot) which averages to 0.5, yielding anet absorption A per cycle given by

A ¼ ðo=2pÞÐ t¼2p=ot¼0 dt H ðdM=dtÞ ¼ 2ow00 H2

1

¼ ð1=2Þoo0w0H21T2 ½1þ T2

2ðo0�oÞ2 þ g2H21T1T2��1

ð11:21:17Þ

The rate of absorption of RF energy is

dE=dt ¼ n0 DE=½1=Pþ 2T1� ð11:21:18Þ

where n0 is the population difference at thermal equilibrium,DE is the energydifference, T1 is the spin-lattice relaxation time, and P, the rate for stimulatedtransitions, is given by Fermi’s golden rule formula for the transition betweenthe lower nuclear spin state |ai and the upper nuclear spin state |bi:

Pab ¼ ð2p=hÞjhbjVjaij2 gðoÞ ð11:21:19Þ

P is the same for absorption |ai ! |bi as for emission {|bi ! |ai in NMR}:

Pab ¼ Pab ¼ P ð11:21:20Þ

For EPR (discussed below), because electrons carry a negative, not a positivechange, the electron spin state |bi is lower in energy, and the state |ai is higherin energy.

The sensitivity of the NMR experiment depends on the frequency n; itfollows, to some extent, the 1/n law of decreasing noise, and therefore highersensitivity, as the Larmor frequency n and themagnetic fieldH0 are increased.

PROBLEM 11.21.1. Hydrogen is the most abundant element in space. The“song of hydrogen” at 1.42GHz is the emission of radiation by excited H1

atoms in interstellar gas clouds. Assuming that it is due to the coupling

714 11 INSTRUMENTS

between the electron spin S¼ 1/2 and the nuclear spin I¼ 1/2, evaluate thepossible transitions and energies.

PROBLEM 11.21.2. Derive Eq. (3.22.1) for electric dipoles interacting withelectric dipoles from Eqs. (2.7.63) and (2.7.67). This dipole-dipole interactionenergy is equally valid for the interactions between magnetic dipoles.

m ¼ðrðrÞr dvðrÞ ðð2:7:67ÞÞ

and from Eqs. (2.7.63) and (3.22.1).

PROBLEM 11.21.3. Derive Eq. (11.21.18) from Eq. (11.21.17).

Small Magnetic Field: Measurement of T2. If the oscillating magnetic field issmall (H1 � g�1T

�1=21 T

�1=22 ), the third term in the denominator of

Eq. (11.21.16) can be neglected, and

w00 ð1=2Þ w0o0T2 ½1þ T22 ðo0�oÞ2��1 ðif g2H2

1T1T2 � 1Þw0 ð1=2Þ w0o0 ½T2

2ðo0�oÞ� ½1þ T22ðo0�oÞ2��1 ðif g2H2

1T1T2 � 1ÞA oM0H

21T2 ½1þ T2

2ðo0�oÞ2��1 ðif g2H21T1T2 � 1Þ

ð11:21:21Þ

The first and third of Eq. (11.21.21) yield a Lorentzian line-shape. Figure 11.53shows a plot of w00 (proportional to absorption) and w0 (proportional todispersion).

The total area under this Lorentzian line isÐ x¼1x¼�1 dx ½1þ x2��1 ¼

½tan�1x�x¼1x¼�1 ¼ p. Most NMR and EPR lines are Lorentzian, if they are the

expression of a single magnetization with characteristic spin–spin relaxation

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

420-2-4

Lorentzian

χ" o

r χ'

x=(ω – ω0)T2

χ"(x)=[1+x2]-1

χ'(x)=-x[1+x2]-1

FIGURE 11.53

Unnormalized Lorentzian line-shapes w00(x) [1þx2]�1 and w0 �x[1þx2]�2, where x T2(o�o0).

11.21 MAGNETIC RESONANCE 715

time T2. Experimentally, the lineshapes can also be Gaussian, that is, withfunctional form exp(�x2), if they are due to a distribution of magnetizationsthat follow a normal curve of error (i.e., a Gaussian). To compare the twolineshapes, we define the normalized forms of the Lorentzian and Gaussianlineshapes for the imaginary susceptibility:

w00Lðo�o0Þ p�1T2 ½1þ T22 ðo�o0Þ2��1 ð11:21:22Þ

w00Gðo�o0Þ 2�1p�1=2T2 exp ½�2�1T22ðo�o0Þ2�: ð11:21:23Þ

These two functions are plotted in Fig. 11.54.The full width at half-maximum (FWHM) of the Lorentzian curve of

Fig. 11.54 is FWHML¼ 3.1905; for theGaussian of equal peak height, FWHMG

¼ 2.3549: at equal peak heights, a Gaussian lineshape is narrower, while theLorentzian is broader—that is, has more intensity far from the peak. Inpractice, NMR or EPR lineshapes can also be intermediate between Lorent-zian and Gaussian.

Measurement of T2. It is assumed for Fig. 11.54 that g2H21T1T2

� 1; that is; H21 � gT1

�1T2�12. If increasing H1 causes the signal to

decrease, then “saturation” has set in; if, instead, there is no change, thenthe condition H2

1 � gT�11 T�1

2 is still satisfied; and from FWHML or FWHMG

evaluated in frequency units (o�o0) instead of T2(o�o0), one can obtain T2

from T2¼ 1 /Do1/2, where Do1/2 is the half-width at half-maximum(Bloembergen convention).

Measurement of T1. If the circularly polarized or oscillating magnetic fieldH1 is increased toward infinity, then the maximum rate of absorption of

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

6420-2-4-6

χ"L

or 1

.128

4 χ"

G

T2(ω-ω0)

χ"L = 3.1415-1 [ 1 + T

2

2(ω-ω0)2]-1

1.1284 * χ"G = 0.5642 š -1/2 exp[-0.5*T

2

2(ω-ω0)2]

FWHMFIGURE 11.54

Comparison of normalized Lorent-zian w00L(o�o0) p�1T2[1þ T2

2(o�o0)

2]�1 lineshape and Gaussianw00G(o � o0) 1.1284�2�1p�1/2T2exp[�2�1T2

2(o � o0)2] lineshape

(premultiplied by 1.1284 to matchthe central peak of the Lorentzian).The abscissa is x¼ T2(o � o0).

716 11 INSTRUMENTS

power, consistentwith the ability of the spin-lattice interactions todissipate it,is given from Eq. (11.12.16) as

LimH1!1

f2oH21w

00g

¼ LimH1!1

f2oH212

�1w0o0T2½1þ T22ðo0�oÞ2 þ g2H2

1T1T2��1g ¼ w0H20T

�11

ð11:21:24Þ

This is proportional to the reciprocal of the spin-lattice relaxation time T1.Therefore, while the spin–spin relaxation time T2 is estimated directly fromthe half-width of the signal before “saturation,” T1 is obtained from asaturation experiment: For example, when increasing H1 leads, forinstance, to a halving of the signal intensity, then at this half intensity thecondition g2H1

2T1T2¼ 1 is satisfied, and the spin-lattice relaxation time T1 isestimated from the known H1 and T2.

Rapid Passage. During the magnetic resonance event, the important thingto realize is that the time-varying fieldH1, as it is swept in frequency throughthe resonance condition, will force the magnetization from one orientation(precessing about H0) to another. Under the conditions of Ehrenfestadiabaticity, H�1|dH/dt| �g|H|, when applied to H¼H0þH1, yields thecondition H�1

1 jðd=dtÞ ðH0�vÞ=gÞj � gH1. Under those conditions, the totalfield H moves from the condition of Fig. 11.55(a) before resonance, to

x

z

B0

ω B1

-ω / γ

x'

θ

Y

Beff

x

z

B0

ω

B1= Beff

- ω / γ

Beff = B0-ω/γ+ B1

x'Y

x

z

B0

ω B1

- ω / γ

x'

θ

Y

Beff

(a): ω < ω0 (b): ω = ω0 (c): ω > ω0

FIGURE 11.55

Successive orientations of Heff (orBeff) and therefore M: (A) Beforeresonance (o< o0) the magnetiza-tionM, if initially (H1¼ 0 at t¼ 0) itis along H0 (except for Larmor pre-cession); then, after the circularlypolarized field H1 is turned on, Mwill follow Heff, which is not farfrom H0. (B) At resonance (o¼o0)M is along the (small) H1 field inthe xy plane. (C) Beyond resonance(o > o0).

11.21 MAGNETIC RESONANCE 717

Fig. 11.55(b) at resonance (o¼o0), and to Fig. 11.55(c) after resonance; themagnetizationM and the individual spins followH and are in the horizontalplane at resonance for a very short time.

Free Induction Decay. If one subjects the sample to a so-called “90-degreepulse,” or “p/2 (radians) pulse”—that is, if the RFmagnetic field is kept at (orclose to) the resonance frequency o0 for a time long enough for the magneticmoments to turn from the condition of Fig. 11.55(A) (vertical, M along Z) tothat of Fig. 11.55(B) (horizontal,M inXY plane) [for NMR this time, similar tothe spin–spin relaxation time T2, is typically a few microseconds]—then adetecting coil placed with its axis in the XY plane will see “ringing,” or free-induction decay, with the functional formMx(t)¼M0 cos(ot) exp(�t/T2): thissignal decayswith time as the individual spins of the samplemove out of stepfrom each other (Fig. 11.56).

Spin-Echo NMR. A macroscopic set of spins undergoing Larmorprecession at the same frequency will feel spin–spin interactions and willlose phase coherence as time passes; this is seen in Fig. 11.57, which shows asequence of two RF impulses: (1) a p/2 (90�) pulse, followed by a wait time t,within which the spins, if they were initially aligned and in phase with eachother,arestartingtorepeleachother(T2process)andfanoutintheXYplane; (2)a p (180�) pulse turns their gradual divergence into a gradual convergence at180� from the initial pulse; after another equal (short)wait time t, the spins aretogetheragain.Thisphase reconstitutingcanbedoneboth inNMRandinEPR.

Selective Saturation. If there are several spin types, then one set ofchemically equivalent spins (say spins A) can be selectively excited tosaturation; a second set of spins (B) can then be seen clearly, andsaturating A can enhance the signals from spins B. This is the basis for “2-D” NMR, where two frequencies are plotted, and the histograms of signalscan be interpreted more clearly than if no saturation had occurred.

NMR Spectrum of Ethanol. Figure 11.58 shows the H1 NMR spectrum ofethanol (after a drop of HCl was added, which causes rapid spin exchangebetween hydroxyl protons of neighboring molecules, thus simplifying thespectrum).

Two very important pieces of information are immediately discerniblein the ethanol spectrum of Fig. 11.58: First of all, the OH proton peak is

Z

X

YM at ω = ω0

B0

(a)

90° pulse

0 2 4 6 8 10

Mag

netiz

atio

n M

(t)

= c

os(π

t)ex

p(-t

/10)

Time t (μs)

(b)

FIGURE 11.56

Free induction decay after a 90�

pulse. (A) A 90� pulse rotates Mfrom (close) to the Z axis to the Yaxis, where a pick-up coil detectsthe signal. (B) Signal detected de-cays with time; this is the inducedsignal in the Y-axis coils.

718 11 INSTRUMENTS

“chemically shifted” from the methylene protons and from the methylprotons (this is due to the different electron density that these protons findthemselves in); next, there is a multiplet structure.

Chemical Shifts in NMR. The first effect is very useful in chemical analysis:The nuclear spin transitions are affected by the “hyperfine” I S couplingbetween electron spin S (for any single electron in the molecule that hasdensity at the nuclear position) and nuclear spin I; this is due to the isotropicFermi contact term:

DE ¼ a I S ¼ ð8p=3Þh2gegnjc1sð0Þj2 I S ð11:21:25Þ

where a is the hyperfine splitting constant. Equation (11.21.25) uses theexperimental fact that the probability density |c1s(0)|

2 that 1s electronsreside in (actually, transit through) the nucleus is nonzero. Through thishyperfine interaction, the transition between nuclear spin states undergoes a“chemical shift” a, either directly with the 1s electron or, indirectly, with thevalence electrons in the atomormolecule. This shift is quite small andwas notseen in the firstNMRexperiments in the 1940s, due to the lack of homogeneityin the DCmagnetic field. With better magnet homogeneity, distinct chemicalshifts were detected, in the parts per million range, andmade NMR instantlyuseful as a tool for chemical analysis. Decades of work has resulted in

Z

X

Y

B0

(a)

90° pulse

B0

wait time τ

"fast" spins

"slow" spins

π /2 pulse

B0 π pulseB0

wait time τ

"slow" spins

"fast" spins

B0

Focusedmagnetization

Refocusedmagnetization

refocusing

defocusing

Time

π /2

šτ τ

(A)

(b)

(c)(d)

(e)

(f)

FIGURE 11.57

Spin-echo pulse sequence: (a) A p/2pulse is administered by coils along(i.e., with cylinder axis) along X, toa focused set of spins initially alongZ; these spins are rotated onto theY axis, but (b) after an interval t,they are somewhat defocused, so(c) a p pulse is administered by coilsalong Y, and (d) after an interval tthey (e) are refocused along Y; astrong “echo” signal is then re-corded along Y from all the refo-cused spins. Inset diagram (f) showsa conventional way of represent-ing the pulses and the wait times.

11.21 MAGNETIC RESONANCE 719

experimental estimates of the chemical shift, expressed as a chemical shield-ing constant s: It reduces the external fieldH at the nucleus by a small amountsH because of the diamagnetism of the effective electron density at thenucleus (but there is also a paramagnetic component!). This s is a smallquantity, maddeningly difficult to calculate, but experimentally easy totabulate and use, as a part per million chemical shift d:

d 1:00� 106 ðn� nrefÞ=nref 1:00� 106ðs� srefÞ=ð1� srefÞ ð11:21:26Þ

A reference standard compound is traditionally used [tetramethylsilane(TMS) Si(CH3)4 for both H1 and C13, 85% H3PO4 for P

31], and the chemicalshift d, relatable to s, is usually a down-shift of the resonance from thereference compound, a small amount of which is added to each sample as aninternal standard: Table 11.11 lists some typical H1 NMR chemical shifts d.Table 11.12 shows some C13 chemical shifts. Sometimes rare-earth salts areadded to the solution, to deliberately shift resonances by a Coulomb inter-action; these are called lanthanide shift reagents.

Multiplets inNMR. The second effect yields a spin–spin coupling constantJ (usually quoted in hertz), which it generates amultiplet structure that is dueto nuclear spin–nuclear spin interactions between equivalent or inequivalentprotons (in H1NMR). The spin interaction is actually a tensor quantity due to

0.511.522.533.544.5

0

0.2

0.4

0.6

0.8

1

Inte

nsi

ty (

arb

. un

its)

δ (ppm from TMS1H)

CH3CH2OH

CH3CH2OH

CH3CH2OH

Numericalintegration

of quartet: 2.0

Numericalintegration

of singlet: 1.0

Numericalintegrationof triplet: 3.0

FIGURE 11.58

H1NMR spectrumof ethanol (þ1dropofHCl),with an indication inboldface ofwhichprotons are responsible. TheOHproton (CH3CH2OH) (d¼ 4.0 ppm) of relative integrated intensity 1.0, is a singlet, because it exchanges rapidly withneighboringethanolmolecules anddoesnot “see” thehyperfine structuredue to theneighboringmethyleneprotonsor themore distantmethyl protons. Themethylene signal (CH3CH2OH) (d¼ 3.0ppm) is a quartet of relative integratedintensity 2.0,with intra-quartet intensity ratios 1:2:2:1, due to the interactionwith the threemethyl groupprotonsbutnotwith the hydroxyl proton. Themethyl group signal (CH3CH2OH) (d¼ 1.0ppm) is a tripletwith relative intensity 3.0,with intra-triplet intensity ratios 1:2:1, due to interactionwith the twomethyleneprotons. In pure ethanol (not shownhere), thehydroxyl signal is a triplet, due to interactionwith themethylenegroup; themethylene signal is amess of 24peaks, and the methyl signal is almost the same as in the acidified sample.

720 11 INSTRUMENTS

orientational dependence of the interaction energy between nuclearmagneticdipole moments mi and mj spaced a distance rij apart:

DEij ¼ mi mjr�3ij � 3 mi rijmj rij r �5

ij ð11:21:27Þ

This energy,when using the explicit formulas for the nuclearmagnetic dipolemoments, becomes

DEij ¼ gNibNigNjbNj ½Ii Ij r �3ij � 3 Ii rij Ij rij r �5

ij � ð11:21:28Þ

but again, sinceNMR isusuallymeasured in solutions,with intense rotationalaveraging, a scalar average coupling constant J is obtained from experiment.If there are N equivalent nuclei, there will be Nþ1 NMR lines, with intensityratios dictated by binomial coefficients. N equivalent nuclei will generateNþ1 lines with relative intensities given by the binomial coefficientsNm

� �¼ ðNÞ!

ðN � mÞ!m! (these intensity ratios are known as Pascal’s117 or

Tartaglia’s118 triangle: The elements in any row are the sum of the twonearest-neighbor elements above that element) (Table 11.13).

Table 11.11 NMR Chemical Shifts d (pmm) for H1 (Boldface) in SomeOrganic Groups

Si(CH3)4 0.0 (by Definition) �COOCH3 3.8–4.1RCH3 0.5–5.4 ROH 3.0–4.3–CH2� 0.5–2.2 �C¼CH� 5.0–7.2–CH� 2.0–3.2 ArOH 5.0–7.3RCOCH3 3.5–4.0 ArH 5.0–8.0ArOCH3 3.7–4.0 �CHO 9.0–10.0

�COOH 10.8–13

Table 11.12 NMR Chemical Shifts d (ppm) for C13 (Boldface) in SomeOrganic Groups

Si(CH3)4 0.0 (by definition)R-C-H 0–75R3C� 25–110>C¼C<X 60–175�CC� 70–110�C¼C< 110–170

C-X in Ar-X 110–150R-COOH 160–190R-CHO 190–220R2C¼O 200–230R¼C¼R 190–235R3C

þ 200–30

117 Blaise Pascal (1623–1662).118Niccolo Fontana “Il Tartaglia” (1500–1577).

11.21 MAGNETIC RESONANCE 721

The rationale for this distribution of signal intensities can be understoodfrom Fig. 11.59, which shows how five nuclear spins interact:

The analysis of complicated NMR multiplet spectra (AB, A2B, where Aand B have relatively close chemical shifts; and AX, A2X, etc., where the Aand B nuclei are well separated by chemical shift) is done by perturbationtheory, involving nuclear spin eigenfunctions.

Paramagnetism Kills the NMR Spectrum by Broadening. NMR is not usuallystudied in paramagnetic samples, because the intense local magnetic fieldproduced by amolecule with S> 0 broadens the NMR signal greatly, makingthe experiment rather difficult, but not impossible.

Magnitude of Relaxation Times. The relaxation times are such that veryshort-lived systems (e.g., transition states in chemical reactions) cannot beseen in NMR. NMR can detect species whose lifetime exceeds 1 ms. Atremendous advantage of work in solutions is that the rotationalrelaxation times (typically 1 ms to ns) average in three dimensions the

Table 11.13 For N Equivalent Nuclei, Tartaglia’s (or Pascal’s) Table of

Binomial Coefficients can be Generated From Nm

� �¼ ðNÞ!

ðN�mÞ!m!

N ¼ 0, m¼ 0 1N ¼ 1, m¼ 0! 1 1 1N ¼ 2, m¼ 0! 2 1 2 1N ¼ 3, m¼ 0! 3 1 3 3 1N ¼ 4, m¼ 0! 4 1 4 6 4 1N ¼ 5, m¼ 0! 5 1 5 10 10 5 1N ¼ 6, m¼ 0! 6 1 6 15 20 15 6 1N ¼ 7, m¼ 0! 7 1 7 21 35 35 21 7 1

N = 0

N = 1

N = 2

N = 3

N = 4

N = 5

Singlet

Doublet, 1:1

Triplet, 1:2:1

Quartet, 1:3:3:1

Quintet, 1:4:6:4:1

Sextet, 1:5:10:10:5:1FIGURE 11.59

Multiplet structure for N equiva-lent spins.

722 11 INSTRUMENTS

spin–spin splittings (due to the nuclei of the solvent adjacent to the moleculebeing studied). This minimizes, particularly with solvents of low viscosity orat high temperature, the effects of the solvent and “sharpens” the signal. Asthe temperature is lowered, or the solvent becomes more viscous, theseaveraging mechanisms will fail, and anisotropies in the signal will emerge.

NMR inSolids. In solids, these spin–spin interactions are not averaged:AnH1 NMR signal that is 1Hz wide in solution will be broadened to about100 kHz in a solid, decreasing signal-to-noise ratios and losing muchchemically useful information. Two techniques have evolved in tandemto combat this broadening: magic-angle spinning and multiple-pulsenarrowing.

Magic-Angle Spinning. One technique is to spin the sample at the so-called“magic angle” of 54.74�, whichminimizes the spin–spin interaction effects: Inthe dipolar expansion rewritten in terms of the dipole–dipole interactionbetween two dipoles mi and mj, with mutual angle of orientation yij:

Edd ¼X

i

Xiðmimj � 3mi rij mj rij r �2

ij Þ r �3ij ð11:21:29Þ

¼X

i

Ximimjr

�3ij ½1� 3 cos2 ðyijÞ� ð11:21:30Þ

the numerator will vanish, and the dipole–dipole forces will cancel when1� 3 cos2 yij¼ 0, that is, when yij¼ 54.73561032�. In practice, the sample, fittedwith a plastic propeller, is spun at 54.74� by a compressed-air jet at 3–5 kHz;this spinning cancels a large part of the dipolar broadening.

Multiple-Pulse Narrowing. The other technique is to artificially realign thenuclear spins in a solid; several mutually orthogonal RF coils are mountedaround the sample area; these coils receive RF energy at the frequency n ofinterest, but for varying times, in a precise sequence of impulses, firstintroduced by Hahn,119 Purcell, and Waugh:120 these multiple pulses arecalculated to combat, and even exploit, thermal relaxation. The net effect is tonarrow the NMR resonances by “kicking” the Boltzmann population ofthermally varied orientations into a single orientation, watching as the freeinduction decay lets these excited nuclei slowly dephase, and then kickingthem again at 90�, etc., forcing the dephasing back toward sharpeningthe signal.

2DNMR. Suchmultiple-pulse sequencesnot only canhelp todetect solid-state NMR spectra, but also are applied to decouple spectra of moleculesin solution, where certain chemical groups can be studied, by applyingcombinations of two or more NMR frequencies. Names, such asOverhauser121, “Underhauser,” COSY, MAGIC, and so on, have beengiven to these pulse sequences. By varying two saturating frequencies, so-called “two-dimensional NMR” plots become possible: Plotting the signal

119 Erwin L. Hahn (1921– ).120 John S. Waugh (1929– ).121Albert W. Overhauser (1925– ).

11.21 MAGNETIC RESONANCE 723

intensities as contour diagrams in a two-dimensional plot of varyingfrequencies may isolate the NMR transition of interest within verycomplicated biomolecules, by first saturating one resonance, then theother, thus allowing for the decipherment of local structure.

Derivative Detection of EPR Transition. The EPR spectrum is usuallydisplayed as the first derivative of the absorption w00(H), becausethe nonresonant low-frequency and low-amplitude RF modulation (o1/2p¼ typically 100 kHz) applied to the coils near the magnet is detected bya rectifier in addition to the drop in microwave power level due to the RFresonant absorption (typically o0/2p¼ 9.1 GHz if H0¼ 0.34 T): The signalis processed by a phase-sensitive circuit, which detects a back-and-forthsweep across resonance in small magnetic field increments (relative to theDC field and to the width of the measured spectrum), thus generating aresponse dw00/dH (see Fig. 11.60).

A 9.5-GHz 0.34-T EPR spectrometer can detect 1011 spins if the linewidthis 1 gauss; that is, an EPR spectrometer is four orders of magnitude moresensitive than an NMR spectrometer. However, paramagnetic samples areless prevalent than diamagnetic ones, so NMR has proven to be much moreuseful than EPR. EPR spectrometers now can reach 95GHz, with a 10-foldincrease in sensitivity over the 9.5-GHz instrument.

A typical EPR spectrum of electrochemically or chemically generatedbenzene radical anion C6H6

� in solution is displayed in Fig. 11.61. Thespectrum is centered at g¼ 2.003, and consists of 7 lines, due to the hyperfinesplitting of the electron resonance by 6 chemically equivalent protons with ahyperfine splitting constant a¼ 0.375mT. If there are M chemically or topo-logically inequivalent nuclei, each spectrum is splitM times, with a hyperfinesplitting constant a; all the splittings (and splittings of splittings) are centeredaround the Larmor frequency; this can make a very complex spectrum. Forinstance, the naphthalene negative ion radical has 25 lines, due to fourequivalent protons at positions 1, 4, 5, and 8, which generates a quintet with

–1

–0.5

0

0.5

1

1.5

0.3430.3420.3410.340.3390.3380.337

Ab

s (B

) o

r (d

Ab

s / d

B)

B (Tesla)

Abs (B)

(d Abs / d B)

Modulationsweepwidth

FIGURE 11.60

Normalized Lorentzian absorptionlineshapefunctionAbs(H)¼ w00L(o�o0) p�1T2[1þT2

2(o� o0)2]�1 (dot-

ted line) and its rescaled derivative(d Abs(H)/d H)¼dw00L/d(o � o0)�2p�1 (o�o0)T2[1þ T2

2(o�o0)2]�2

(solid line) as a function of the DCmagneticfieldH.ThepeakofAbs(H)is at T2/p; The horizontal line indi-catesFWHM¼2T2

�1;at thecenterofthe absorbance (w00L(o � o0)¼max)the derivative vanishes; the deri-vative peaks are separated by2� 3�1/2T2

�1.

724 11 INSTRUMENTS

relative intensities 1:4:6:4:1, with a¼ 0.490mT, and four other protons atpositions 2, 3, 6, and 7, with a¼ 0.183mT, which generates a smaller quintetwith relative intensities 1:4:6:4:1. The anthracene negative ion has a 1:2:1triplet of splitting 0.43mT, split into a 1:4:6:4:1 quintet of splitting 1:4:6:4:1,which is split again into a 1:4:6:4:1 quintet with splitting 0.11mT. All of thismakes sense in the McConnell122 equation:

a ¼ Qr ð11:21:31Þ

whereQ 2.3mT and r is the spin density of the C atom closest to the proton.Actually, thisQ not too constant:Q¼ 2.304mT for *CH3,�2.99mT for *C5H5,�2.25mT forC6H6

�,�2.74mT for *C7H7, and 2.57mT forC8H8�. Figure 11.62

shows the spin densities calculated from the spectra for several cyclichydrocarbons. These spin densities can also be obtained from open-shelltheoretical calculations of the spin densities (¼density at atom of spin-up(alpha) electrons minus spin-down (beta) electrons $ charge densities).

Stable Free Radicals. Stable free radicals are a small minority of the morethan 6 million chemical compounds known by 2005. The oxygen molecule isparamagnetic (S¼ 1). In 1896, Ostwald stated that “free radicals cannot beisolated.”Only four years later,Gomberg123made triphenylmethyl (Fig. 11.63),the first proven stable and persistent free radical [48]! An infinitely stable freeradical used as a reference in EPR is diphenyl-picryl hydrazyl (DPPH). Otherpersistent free radicals are Fremy’s124 salt (dipotassium nitrosodisulfonateKþ�O3S-NO-SO3

� Kþ) 2,2-diphenyl-1-picrylhydrazy (DPPH)l, Galvinoxyl(2,6-di-tert-butyl-a-(3,5-di-tert-butyl-4-oxo-2,5-cyclohexadien-1-ylidene)-p-

-60

-40

-20

0

20

40

60

0.34250.3420.34150.3410.34050.340.3395

d χ"

(d

B)

Field B (Tesla)

FIGURE 11.61

EPR spectrum of electrochemicallygenerated benzene radical anion,C6H6

�. The hyperfine interactionbetween the free spin and the sixH1 nuclei generates a seven-linespectrumofnominal relative inten-sities 1:6:15:20:15:6:1. The hyper-fine splitting constant is 0.375mT.

122Harden Marsden McConnell (1927– ).123Moses Gomberg (1866–1947).124 Edmond Fr�emy (1814–1894).

11.21 MAGNETIC RESONANCE 725

tolyl-oxy), named after Galvin Coppinger125, and nitroxides R-NO, such as2,2,6,6-tetramethyl-1-piperidinyloxy (TEMPO), with the spin concentrated inthe terminal NO group and protected sterically from chemical attack byadjacent methyl groups.

g-Tensor. So far, the g-value has been presented as an isotropic quantity; itactually is a tensor, so that the spin Hamiltonian HEZ for the electronicZeeman126 effect should be written as

HEZ ¼ beH0 ge S ð11:21:32Þ

In organic radicals in solution, the g-factor anisotropy cannot be detected; oneneeds oriented samples. In crystals of free radicals, this anisotropy is easilymeasured—for example, in crystals of sodium formate (Naþ HCOO�) theprincipal-axis components are gxx¼ 2.0032, gyy¼ 1.9975, and gzz¼ 2.0014.If there is some spin–orbit interaction in an organic molecule (e.g., if acompound contains S or Cl), then g-values as high as 2.0080 are encountered.In disordered powders with narrow EPR lineshapes, the g-factor anisotropycan produce considerable distortion in the overall signal, due to averaging ofthe g-tensor.

0.166

0.22

0.08

0.193 0.097

0.048

C H

(a)

(c)

0.200

0.1429

0.333

0.125

(b)

H

H

H

C1.000

FIGURE 11.62

(a) Experimental spin densities oncyclic hydrocarbon radicals (b) De-piction of spin density of 1.000 oncarbon atom of methyl radical,*

CH3. (c) Qualitative explanation ofthe McConnell equation: An elec-tron spin in 2pz orbital onC inducesan antiparallel nuclear spin orien-tation on the adjacent H1 nucleus,by polarizing the C–H electron pairbond.

125Galvin M. Coppinger (1923– ).126 Pieter Zeeman (1865–1943).

726 11 INSTRUMENTS

Fine-Structure Splittings in ESR Spectra of Triplet States. Consider thehamiltonian H for spin–spin (fine-structure) and Zeeman interactions oftwo spins S1 and S2 a mutual distance r12 apart, in an external magneticfield H0:

H ¼ beH0 ge ðS1 þ S2Þ þ g 2e b

2e ½S1 S2 r

�312 � 3 ðS1 r12Þ ðS2 r12Þ r �5

12 �ð11:21:33Þ

Assume (1) an isotropic g-tensor for simplicity, (2) that the two spins couple:S¼ S1þ S2, to form a singlet state 2�1/2{|aebe > �|beae>} and three tripletstates |aeae>, 2�1/2{|aebe>þ|beae>}, and |bebe>|. Then the Hamiltoniansimplifies to

H ¼ gebeH0*Sþ S*D*S ð11:21:34Þ

where the symmetric fine-structure tensor D has diagonal and off-diagonalcomponents (in an arbitrary coordinate system, e.g. when H0 is along the zaxis) of the type

Dxx ¼ ð1=2Þg 2e b 2

e < r �312 � 3x 2

12 r�5

12 > ð11:21:35Þ

Dxy ¼ ð1=2Þg 2e b 2

e <� 3x12y12r�5

12 > ð11:21:36Þ

CN

N

O2N

NO2

NO2

NH3C

H3C CH3

CH3

O O

C(CH3)3(H3C)3C

O

C(CH3)3

C(CH3)3

FIGURE 11.63

Some stable free radicals: (top left)triphenylmethyl; (top right) 2,2-diphenyl-1-picrylhydrazyl (DPPH);(bottom left) TEMPO; (bottomright) Galvinoxyl.

11.21 MAGNETIC RESONANCE 727

After a transformation into a principal-axis system (X,Y,Z) the fine-structuretensor becomes a traceless symmetric diagonal tensor:

S D S ¼ �XS 2X � YS 2

Y � ZS 2Z ¼ D S 2

z � 3�1S 2z

� �þ E S 2

X � S 2Y

� �ð11:21:37Þ

Experimentally, this means that, between measurements, a crystal must berotated along two mutually orthogonal axes, until extrema in the signals(usually symmetrical about the “impurity” signal at g¼ 2) are detected atcertain angles. It is important and interesting to correlate the “principal axes”with the crystallographic axes and determine how they relate to axes in themolecules that exhibit the triplet signal. The D value gives the size of theinteractions (typically, a fraction of 1 cm�1), while Emeasures the asymmetryof the triplet state charge distribution (usually E is smaller than D). At highmagnetic fields (H0 > 0.3 T) the transitions in the principal-axis plane are|D� 3E|and (3/2)|D|; at zero external field, the transitions are|DþE|and|D�E|; the absolute signs of D and E cannot be determined from an EPRspectrum. Even the EPR “powder” spectrum of randomly oriented crystal-lites can sometimes yield D and E values at the “turning points” of thedistribution of spins.

Spin Labeling. The EPR of Fremy’s salt in water (or TEMPO in low-viscosity organic solvents) shows a 1:1:1 triplet with hyperfine splitting1.3mT, due to the I¼ 1 spin of N14, centered around g¼ 2.002. McConnellshowed that TEMPO and similar nitroxides, appropriately synthesized to bebiocompatiblewith the target region, can be incorporated as “spin labels” intobiologically interesting regions:mitochondria andother cellular components,phospholipid bilayers, nerve cells, and active sites of enzymes; if themediumis viscous, then the symmetrical pattern of Fig. 11.61 becomes unsymmetricalanddistorted; this probes the relaxation timeswithin the biological system. Inbiophysical chemistry, the spin label method competes with the fluorescentlabelmethod, but both labels tend tobe largemolecules,which are intrusive inthe very region they are probing.

Nuclear Resonance in Paramagnetic Systems: Knight127 shift. If there is aparamagnetic specieswith z-component of spin Sz andanuclear z-componentof spin Iz in an external magnetic field H0 along z, then the interactionHamiltonian is

H ¼ gebeH0Sz � gNbNH0Iz þ aIzSz ¼ �gNbNH0Iz H0 � aSz=gNbNð Þ þ gebeH0Sz

ð11:21:38Þ

By collecting the terms in Iz we see that there is an effective fieldHeff acting onthe nucleus:

Heff H0 � aSz=gNbN ð11:21:39Þ

which can be very large: For instance, when H0¼ 1T, then the H1 Larmorfrequency is 42MHz; an admittedly large hyperfine splitting constant

127Walter David Knight (1919–2000).

728 11 INSTRUMENTS

a¼ 84MHz will then causeHeff to be either 0 T or 2 T, depending on the spinorientation! Therefore a nuclear resonance will shift upfield or downfield byan amount DH:

DH=H0 ahSzi=gNbNH0 ð11:21:40Þ

which strongly depends on T. Using the magnetic susceptibility of electronsof spin S:

w ¼ Ng 2e b

2e SðSþ 1Þ=3kBT ðð5:9:12ÞÞ

and the Fermi contact term for the hyperfine interaction:

a ð8p=3ÞgebegNbNjc1sð0Þj2 ðð11:21:25ÞÞ

this becomes the Knight shift DH/H0:

DH=H0 ¼ ð8p=3NÞwjc1sð0Þj2 ð11:21:41Þ

The Knight shift was first measured in metals, but is also appreciable forsolutions containing paramagnetic ions. La3þ salts have been used to shift H1

resonances, although the spin-lattice relaxation times lengthen considerably,and thus the signals become harder to detect.

Overhauser Effect. If one measures a nuclear transition in a paramagneticsystem, while saturating the electron spin resonance, then the nucleartransition can be enhanced 100-fold, or a nuclear absorption can mutateinto a nuclear emission. The reason is that one is playing with coupledBoltzmann populations of spins (electronic or nuclear): The relaxation ratefor one is affected by the relaxation rate of the other. Consider thesimultaneous change of electron spin (e) and nuclear spin (N),—forexample, aebN K beaN or aeaN K bebN (see Fig. 11.64). Assume that thecontact hyperfine interaction term a(t) I*S has a time-average value ofzero, wiping out the hyperfine splittings: thus in the external magneticfield H0 the Hamiltonian is

H ¼ gebeH0Sz � gNbNH0Iz ð11:21:42Þ

At thermal equilibrium, when the transition rates between upper andlower states become equal, the ratio of the population N0

þ of nuclear “up”spins bN (Iz¼þ1/2) to the population N0

� of nuclear “down”-spinsaN (Iz¼�1/2) is given by a Boltzmann factor:

N þ0 =N �

0 ðNaa þNbaÞ ðNab þNbbÞ � 1 ¼ expðgNbNH0=kTBÞ ðat equilibriumÞ

ð11:21:43Þ

If, however, the electron spin transition is saturated (this is shownby thewidearrows in Fig. 11.64A), then the populations of the electron spin-up and spin-down states are forced to become equal: Naa¼Nba, and Nab¼Nbb. Underthese conditions, the spin populations will depend only on the rate of

11.21 MAGNETIC RESONANCE 729

simultaneousflips of both electron spin and nuclear spin (aebNK beaN, arrowX in Fig. 11.64A); this is permitted by contributions of the type IþS� and I�Sþ

in the expansion of the contact term a(t) I*S. Then the population ratiosbecome

Nþ=N� ¼ 2Naa=2Nab

¼ exp½ðgNbN þ gebeÞH0=kBT� ðat electron spin saturationÞ

ð11:21:44Þ

Thus, the nuclear spin population difference should increase by a factorof (1þgebe/gNbN)¼ 639, which is not always reached in practice. Whendipole–dipole interactions dominate (Fig. 11.64B), then the nuclei will emitenergy, and the enhancement factor becomes negative.

Electron–Nuclear Double Resonance (ENDOR) Spectroscopy. This observesa spin resonance transition after a nuclear resonance transition has beensaturated by a radio-frequency pulse (Fig. 11.65) so as to invert the relativepopulations of the |aeaN> and |aebN> spin states; this forces the populationsof the |aeaN> and|beaN> states to bedifferent and thus offers the opportunityto measure hyperfine splittings much more carefully, with better resolutionthan in standard EPR.

There are many other specialized methods: electron–electron doubleresonance (ELDOR), TRIPLE, HYSCORE (hyperfine sublevel correlationspectroscopy, which is similar to 2D-EPR), electron spin-echo, and so on;these methods are not discussed here.

Optically Detected Magnetic Resonance (ODMR). The first opticallydetected magnetic resonance experiment was done using the 3P1 state of

αeβN

βeβN

αeαN

βeαN

Pe Pe

X

αeβN

βeβN

αeαN

βeαN

Pe Pe

X

(a)(b)

Y

PN

PN

SATU-RATE

SATU-RATE

SATU-RATE

SATU-RATE

exp [(geβe-gNβN)H/2kT]

exp [(geβe+gNβN)H/2kT]

exp [(-geβe-gNβN)H/2kT]

exp [(-geβe+gNβN)H/2kT]

Y

FIGURE 11.64

Energy levelswithOverhauser effect: (a) Relaxation due to a time-dependent isotropic contact electron-spin–nuclear-spin hyperfine interaction a(t)I Swhich has a zero time-average, but allows processes X and Y and enhances nuclearspin transitionswhen the electronpopulations aremadeequal by saturation. (b) Relaxation is due to all dipole–dipoleinteractions, which allow processes X,Y, and PN; nuclear spin transitions are forced into emission by the Overhausereffect. In (a) the relative Boltzmann populations before saturation are shown.

730 11 INSTRUMENTS

gas-phase mercury atoms [49]. While the sensitivity for a CW X-Band EPRexperiment is typically 1011 spins (for a 1-gauss linewidth), ODMR can detectoptically 106 to 108 spins and has even been improved to detect the spin on asingle molecule (pentacene radical anion or cation embedded in terphenyl).ODMR is a double-resonance technique, in which transitions between spinsublevels aredetectedbyopticalmeans.Usually these are sublevels of a tripletstate, and the transitions are inducedbymicrowaves. For thedifferent types ofoptical detection the following abbreviations are used: ADMR (absorption-detected magnetic resonance), DEDMR (delayed-emission, non-specified-detected magnetic resonance), DFDMR (delayed-fluorescence-detectedmagnetic resonance), FDMR (fluorescence-detected magnetic resonance),and PDMR (phosphorescence-detected magnetic resonance). If a reactionyield is followed, the expression RYDMR (reaction-yield-detected magneticresonance) is used. A “spinmicroscope” has been proposed, based onODMRand using a Si cantilever similar to AFM.

Nuclear Quadrupole Resonance (NQR) [50–54]. Nuclear (electric)quadrupole resonance (NQR) was invented in 1950 [55] and is applicableto nuclei with nonzero nuclear electric quadrupoles eQ, which are 3� 3tensors, whose significant components are the quadrupole couplingconstant QCC:

QCC e2Qqzz=h ð11:21:45Þ

and the electric field gradient asymmetry parameter:

Z qxx � qyy� �

=qzz0 ð11:21:46Þ

Here qij is defined as the gradient of the electrical field at the nucleusundergoing NQR, due to the local electron density and the nearby nuclearcharges:

qij @E=@xi ¼ @2V=@xij2 ð11:21:47Þ

βe

αe

αN

βN

αN

βN

αeαN

αeβN

βeβN

βeαN

gNβNH-a/2

gNβNH+a/2

geβeH

Population beforeαeαN<->αeβN is saturated

Population afterαeαN<->αeβN is saturated

1 + q

1 - p + r - q

1+p-r-q

1+q

SATURATE

1 - p + r - q

1 + q

1+p-r-q

1+q

OBSERVE

FIGURE 11.65

ENDORexperiment. pgebe H/2 kBT; q gNbNH/2 kB T; r a/4 kBT;Hyperfine interaction present; butOverhauser effect is absent; thepopulation expressions above arevalid at T high enough to have p, q,r� 1: then the Boltzmann factor isexp(�x) 1 �x.

11.21 MAGNETIC RESONANCE 731

The potential energy due to the electrical quadrupole in a local principal-axis system, where the eQ tensor is diagonal and |qzz| � |qyy| � |qxx| ischosen, is

�jejq¼ � ðjej=2ÞðrnðrÞdvðrÞ x2 @2V=@x2

� �þ y2 @2V=@y2

� �þ z2 @2V=@z2

� �� �¼ � jej

ðrnðrÞ 3z2 � r2

� �dvðrÞ

¼ �jej �ðc rð Þ* 3 cos2y� 1

� �r � 2c rð ÞdvðrÞ þ

XiZiRi

� 2 3 cos2yi � 1� �

ð11:21:48Þ

where �|e| is the electronic charge, rn(r) is the charge density, c(r)� is theelectronic wavefunction, and Zi and Ri refer to the nuclear charge and thedistance of nucleus i from the nucleus undergoing NQR, respectively.

The energy levels for nuclear spin I and its z-componentm for the axiallysymmetric crystal are

Em ¼ e2Qq 4 I 2 I� 1ð Þ½ � � 1 3m2 � I I þ 1ð Þ� �

ð11:21:49Þ

As mentioned earlier, nuclei have nonzero quadrupole momentQ if andonly if their nuclear spin quantum number I is �1; such nuclei, if stable, arelisted in Table 11.14. The formal expressions for the transition frequencies as afunction of I and Z are shown in Table 11.15. A few 17Cl

35NQR frequencies arelisted inTable 11.16; a fewhalogenNQR frequencies are shown inTable 11.17.

The NQR signal is measured by coupling an oscillating radio-frequencymagnetic field H with the magnetic dipole moment of the nucleus (as in

Table 11.14 Stable Nuclei and Their Quadrupole Moments[50]

Nucleus I Q (barns) Nucleus I Q (barns) Nucleus I Q (barns) Nucleus I Q (barns)

1H2 1 0.0027965 27Co

59 7/2 0.40440 51Sb123 7/2 �0.7 67Ho165 7/2 2.82

3Li6 1 �0.000741 29Cu

63 3/2 �0.163 53I127 5/2 �0.785 68Er

167 7/2 2.83

3Li7 3/2 �0.039 30Zn

67 5/2 0.15 55Cs133 7/2 �0.003 70Yb

173 5/2 2.8

4Be9 3/2 þ0.029 31Ga69 3/2 0.178 56Ba

135 3/2 þ0.182 71Lu175 7/2 5.68

5B10 3 0.074 31Ga71 3/2 0.112 56Ba

137 3/2 þ0.283 73Ta181 7/2 3

5B11 3/2 0.036 33As75 3/2 þ0.32 57La

139 7/2 0.21 75Re185 5/2 þ2.8

7N14 1 0.0166 33Br

79 3/2 þ0.332 59Pr141 5/2 �0.059 76Os189 3/2 0.8

8O17 5/2 �0.0301 33Br

81 3/2 þ0.282 60Nd143 7/2 �0.25 77Ir191 3/2 1.5

11Na23 3/2 0.14�0.15 37Rb85 5/2 0.27 62Sm

147 7/2 �0.208 77Ir193 3/2 1.5

13Al27 5/2 þ0.151 37Rb87 3/2 0.13 62Sm

149 7/2 �0.060 79Au197 3/2 þ0.606

16S33 3/2 �0.064 38Sr

87 9/2 0.2 63Eu151 5/2 1.16 80Hg201 3/2 0.50

17Cl35 3/2 �0.0802 41Nb93 9/2 �0.2 63Eu

152 5/2 2.9

17Cl37 3/2 �0.0632 42Mo95 5/2 0.12 64Gd155 3/2 1.6

17K39 3/2 0.11 42Mo97 5/2 1.1 64Gd157 3/2 2

21Sc45 7/2 �0.22 49In

113 9/2 þ1.145 65Tb159 3/2 1.3

23V51 7/2 �0.04 49In

115 9/2 þ1.165 66Dy161 5/2 1.4

25Mn55 5/2 0.355 51Sb121 5/2 �0.5310 66Dy163 5/2 1.6

aOne barn equals 10�24 cm2.

732 11 INSTRUMENTS

Table 11.15 Transition Frequencies inUnits of theQCC (e2Qq/h) and as aFunction of the Asymmetry Parameter h [50]

I Transition Frequency

1 �(þ1! �1) (3/4)(1� Z/3)3/2 �(3/2! 1/2) (1/2)(1þ Z2/3)1/2

5/2 �(5/2! 3/2) (3/10)(1� 0.2037 Z2þ 0.1622 Z4)�(3/2! 1/2) (3/20)(1þ 1.0926 Z2� 0.6340 Z4)

7/2 �(7/2! 5/2) (3/14)(1� 0.1001 Z2� 0.0180 Z4)�(5/2! 3/2) (2/14)(1� 0.5667 Z2þ 1.8595 Z4)�(3/2! 1/2) (1/14)(1þ 3.6333 Z2� 7.2607 Z4)

9/2 �(9/2! 7/2) (4/24)(1� 0.0809 Z2� 0.0043 Z4)�(7/2! 5/2) (3/24)(1� 0.1875 Z2� 0.1233 Z4)�(5/2! 3/2) (2/24)(1� 1.3381 Z2þ 11.7224 Z4)�(3/2! 1/2) (1/24)(1þ 9.0333 Z2� 45.6910 Z4)

Table 11.16 Cl35 NQR Frequencies for Several Chlorine-ContainingCompounds [50]

Compound Frequencies (MHz)

Cl2(s) 108.9CHCl3 38.254 & 38.308 @ 77KHgCl2 22.251 & 22.0964 @ 296K, 22.240 & 22.058 @ 300KGaCl2 20.302 & 19.204 @ 300K, 22.23 & 19.08 @ 305KGeCl2 24.449 & 25.451 @ 77KBiCl3 15.952 & 19.173 @ 291KK2TeCl6 15.13 & 14.99 @ 298KK2SnCl6 15.06 @ 298KK2PtCl6 25.82 @ 298KK2ReCl6 13.89 @ 298KRb2TeCl6 15.14 @ 298KRb2SnCl6 15.60 @ 298KRb2PtCl6 26.29 @ 298KRb2ReCl6 14.28 @ 298KCs2TeCl6 15.60 @ 298KRb2SnCl6 16.05 @ 298KRb2PtCl6 26.60 @ 298KRb2ReCl6 14.61 @ 298KNaClO3 30.62 @ 77KNaClO3 29.92 @ 296KBa(ClO3)2H2O 29.923 @ 77KBa(ClO3)2H2O 29.322 @ 299K

Table 11.17 Typical Values of e2Qq (MHz) for Halogen Nuclei inCovalently Bonded Crystals [55]

17Cl35: 80 33Br

79: 500 53I127: 2,000

11.21 MAGNETIC RESONANCE 733

NMR), but the NQR signal is due to the interaction of the nuclear electricquadrupole moment eQwith the local electric field gradient. Large samples,preferably single crystals (typically � 5 g, or � 1 cm3), are placed in anRF pickup coil, and an adsorption is registered, using (i) marginal oscillators(< 10MHz), (ii) regenerative or marginal oscillators (< 100MHz), (iii) super-regenerative oscillators (20–300MHz) [55], (iv) microwave cavity oscillators(100–380MHz), (v) microstrip oscillators (0.250–1GHz), and (vi) pulsed(quadrupole spin-echo) methods. As in NMR, the width of the NQR linehas contributions from the crystal inhomogeneityDB and from the reciprocalsof the spin-lattice relaxation time T1, and the spin-spin or “spin memory”relaxation time T2. NQR is sometimes referred to as “zero-field NMR.”Figure 11.66 shows an old NQR signal for KClO3.

The resonance frequency nNQR is very sensitive to the squaredwavefunc-tion at the nucleus |c1s(r¼ 0)|2, to the local crystal electric field, and also totemperature changes (Table 11.16).

The NQR data, combined with crystallographic information, can probestructure and bonding in the vicinity of the NQR nucleus. NQR has aregrettable appetite for large samples (grams), but applications have beenproposed for explosives detection (e.g., 7N

15 NQR at 700–900 kHz for thechemical RDX, as long as the sample is not encased in metal!).

PROBLEM 11.21.4. If for HCl the 17Cl35 signal is found at (e2Qq/h)¼ 67.9

MHz, and Q¼�0.0789 barns, then estimate the electric field gradient |e| q(whose cgs units are |e| cm�3).

11.22 ELECTROCHEMICAL METHODS

After earlier experiments with static electricity and with Franklin128 fishingfor thunderbolts, electrochemistry was born with Galvani’s129 electrostaticstimulation of deceased frog muscles in 1791 and Volta’s130 development ofthe “voltaic pile.” Themethodical and routine study of current versus voltage

RESONANCEABSORPTION

LINE

TRANSIENT FROMSWEEP SAWTOTH

VOLTAGE

28.2 Mc/s

APPROX 15 kc/s

FIGURE 11.66

Oscilloscope tracing of 17Cl35 NQR

signal from KClO3 [54].

128 Benjamin Franklin (1706–1790).129 Luigi Galvani (1737–1798).130 Count Alessandro Giuseppe Antonio Anastasio Volta (1745–1827).

734 11 INSTRUMENTS

characteristics started in 1922 with Heyrovsky’s131 invention of thepolarograph.

Modern electrochemical techniques can be divided into four groups: (1)In potentiometry, the electrical potential (voltage) is measured at almostzero or very low current with an unpolarized working electrode. (2) Involtammetry (of which polarography is one technique) a significant currentis measured as a function of voltage, and the working electrode is polarized.(3) In amperometry the current at apolarizedworking electrode, proportionalto the analyte concentration, is measured at fixed potential. (4) In coulometrythe complete conversion of the analyte to a product is determined bymeasuring the total charge consumed.

As a reminder, galvanic cells are spontaneous (Ecell > 0 and DGcell < 0),while electrolytic cells are driven by an external voltage supply (Ecell< 0 andDGcell > 0).

Primary reference electrodes, with their reduction potentials in H2O at298.15K, are:

1. Thestandardornormalhydrogenelectrode(SHEorNHE)“Pt|H2(g)|Hþ (aq, 1M)” at 0.000V by definition.

2. The saturated calomel electrode (SCE) “Hg | Hg2Cl2, KCl (aq, sat’d)”electrode at 0.2412V vs. SHE (which can also be used in nonaqueoussolvents).

3. The silver/silver chloride electrode “Ag | AgCl, KCl (aq, sat’d)”electrode at 0.22V vs. SHE.

4. The “Hg | Hg2SO4, K2SO4 (aq, sat’d)” electrode at 0.64V vs SHE.

5. The “Hg | HgO, NaOH (aq, 0.1M)” electrode at 0.926V vs SHE.

The solvent “windows”—that is, the potential ranges within whichelectrochemical measurements are possible, because within them the elec-trolyte does not undergo an unwanted side-reaction—are shown in Fig. 11.67.Electrochemical measurements require elaborately cleaned electrodes (po-lished metal surface, Hg drop, glassy carbon, etc.) and a “supporting electro-lyte” (often at 0.1M to 1.0M concentration) which transmits the potentialacross the cell. If the reaction at the anode and the cathodemust be “shielded”from each other, then a salt bridge is placed between the two solutions: Thesalt bridge typically consists of 4M aqueous KCl in gelatinous agar agar; theKþ andCl� ions have comparable sizes and hence almost equal ionmobilities(4% difference), so a small flow of these ions in and out of the salt bridgetransmits electrical potential differences between the two solutions, at anacceptable cost of a few millivolts of junction potential.

Empirical formulas exist to correct for the temperature dependence of thereferencepotentials in aqueous solution.Whenonemustwork innonaqueoussolvents, because of their conveniently large “window,” onemust add a 0.1Mto 1.0M salt (see Fig. 11.67) to help conduct current, but there can be aproblem with referencing the working electrode potential to a standardelectrode. SCE can be used in many nonaqueous solvents, but in some casessuch a direct experiment does not work; one must use the Ag|Agþ ion

131 Jaroslav Heyrovsky (1890–1967).

11.22 ELECTROCHEMICAL METHODS 735

electrode as a reference instead and must also use experiments with mixedcells that will allow a numerical change of reference to SCE.

In an ideal cell there are two half-reactions at the “left” and “right”electrodes, and most often there is also a finite internal cell resistance R:

E ¼ Eleft � Eright � IR ð11:22:1Þ

An ideal unpolarized cell would haveR¼ 0 and infinite current; an idealpolarized cell would have a fixed R independent of E and thus a constantcurrent. Reality is somewhere in between: There are several sources of“polarization” that can be considered as finite contributions to the overallresistance R > 0 (or better, the impedance Z). The IR drop, from whateversource, is also called the overpotential Z (i.e., IR> 0), which always decreasesthe overall E; remember that R is always a function of time and E. The causesof polarization are (1) diffusion-limited mass transfer of ions from bulk toelectrode (2) chemical side reactions (if any), and (3) slow electron transfer atthe electrode between the adsorbed species to be oxidized and the adsorbedspecies to be reduced.

In potentiometry, where little current is passed, the emphasis is onelectrodes that measure pH (pH electrode) or permit the chemical

+3.0 +2.0 +1.0 0.0 -1.0 -2.0 -3.0

+3.0 +2.0 +1.0 0.0 -1.0 -2.0 -3.0

1 M H2SO4(aq) | Pt

pH 7 buffer(aq) | Pt

1 M NaOH(aq) | Pt

1 M KCl(aq) | Hg

1 M H2SO4(aq) | Hg

1 M NaOH(aq) | Hg

0.1 M Et4NOH(aq) | Hg

1 M HClO4(aq) | graphite

1 M KCl(aq) | graphite

MeCN, 0.1 M TBABF4 | Pt

DMF, 0.1 M TBABF 4 | Pt

C6H5CN, 0.1 M TBABF4 | Pt

THF, 0.1 M TBAP | Pt

PC, 0.1 M TEAP | Pt

CH2Cl2, 0.1 M TBAP | Pt

SO2, 0.1 M TBAP | Pt

NH3, 0.1 M Kl | Pt

FIGURE 11.67

Practical limits or “windows” orpotential ranges for electrochemi-cal measurements in aqueous solu-tion or in nonaqueous solvents. PC,propylene carbonate. The electro-lytes are: TBAPF4, tetrabutylammo-nium tetrafluoroborate; TBAP, tet-rabutylammoniumphosphate, andTEAP, tetraethylammonium phos-phate.

736 11 INSTRUMENTS

identification of the analyte (ion-sensitive electrodes). The pH meter, in-vented by Beckmann132 in 1934 and by the Radiometer Co. in Denmark in1936, is a high-impedance voltmeter that uses a pH electrode, consisting of asmall Agwire connected to an “Ag|AgCl |KCl (sat’d)” electrode, immersedin a small 0.1M HCl solution, that is separated from bulk solution by a thin,H3O

þ-permeable thin glass membrane; a potential of 0.0592V per pH unit isdetected, amplified, corrected for temperature dependence, and converted todisplay pH units directly.

Indicator electrodes can be metallic, conductive, or membrane-based.Metallic indicator electrodes may use a metal and its cation—for example,Ag|Agþ or Hg|Hg2

2þ in neutral solutions, or Zn|Zn2þ, Cd|Cd2þ, Bi|Bi3þ,Tl|Tlþ, or Pb|Pb2þ in de-areated solutions (other metals cannot be usedbecause they are not too selective to specific cations or are too easily oxidizedor too refractory). For instance, for the reduction Cd2þþ 2e� ! Cd, theNernst133equationreads:Eind¼ECdy� (0.0592/2)log10(1/aCd

2þ).Othermetallicindicator electrodesuse ametal and avery stable salt of thatmetal; for example,for the reduction AgCl(s)þ e� ! Ag(s)þCl�, the Nernst equation yieldsEind¼ 0.222�0.0592log10aCl�;anothersuchapplicationusesScharzenbach’s134

ethylenetetracarboxylic acid, EDTA or COOH)2C¼C(COOH)2, also called “Y”for short: since Y can form a very stable complex HgY2� with Hg, thereforeaddingaknownamountofHgY2� tothesolution, forthereductionHgY4�þ 2e�

! Hg(l)þY� the Nernst equation Eind¼ 0.21 – (0.0592/2) log10(aY4�/aHgY

2�)can be used to measure aY

4�.Solid electrodes, consisting of any of the relatively few inorganic salts that

are electrically conducting, will allow for the determination of certain ions.For instance, LaF3 (“doped”with ErF3) exhibits relativelymobile F� ions, so itcan be used as a fluoride-sensitive electrodes, although above pH 8OH� is an“interfering” ion, as is Hþ below pH 5; these interfering ions would bedetected by this electrode as if they were F�. Table 11.18 lists several suchmetal salt electrodes for the detection of specific anions.

Table 11.18 Crystalline Salt Electrodes as Specific Ion-Sensitive Electrodes in H2O[56]

Salt Analyte Ion Concentration Range Interfering Species

AgBr Br� 100 to 5� 10�6 CN�, I�, S2�

CdEDTA Cd2þ 10�1 to 1� 10�7 Fe2þ, Pb2þ, Hg2þ, Agþ, Cu2þ

AgCl Cl� 100 to 5� 10�5 CN�, I�, S2�, OH�, NH3

Ag3CuS2 Cu2þ 10�1 to 1� 10�8 Hg2þ, Agþ, Cd2þ

AgCN CN� 10�1 to 1� 10�6 I�, S2�

LaF3þEuF3 F� sat’d to 1� 10�6 OH� above pH 8, Hþ below pH 5AgI I� 100 to 5� 10�8 CN�

PbS Pb2þ 10�1 to 1� 10�6 Hg2þ, Agþ, Cu2þ

Ag2S Agþ/S2� Agþ: 100 to 1� 10�7 Hgþþ

S2�:100 to 1� 10�7 Hgþþ

AgSCN SCN� 100 to 5� 10�6 Br�, CN�, I�, S2�

132Arnold Orville Beckmann (1900–2004).133Walther Hermann Nernst (1864–1941).134Gerold Karl Schwarzenbach (1904–1978).

11.22 ELECTROCHEMICAL METHODS 737

Ion-sensitive electrodes can also bemade using (i) specific ions dissolvedin a nonpolar liquid, (ii) specific ions embedded in an ion-exchange polymeror liquid membrane matrix, or (iii) “hollow” molecules that can surroundspecific cations. The ion-sensitive layer must be separated from the bulksolution and also from an indicating electrode by ion-permeable polymers,such as poly-tetrafluoroethylene (Teflon�), Nafion�, or polyvinyl chloride(PVC). Rugged ion-sensitive field-effect transistor s (ISFETs)may replace anindicating electrode, by measuring the current due to ions that penetrate aSi3N4 layer placed over the gate electrode (with the rest of the FET protectedfrom the solution by an impervious encapsulant polymer). Since the gateelectric field change is not specific to which ion enters the Si3N4 layer, ISFETsmust be designed with care and are coming to market rather slowly. Next,gas-sensitive membrane electrodes (GSME) are also in wide use. Finally,there are enzyme-based biosensors (EBB) that depend on specific enzyme–analyte interactions, which can be measured by various ion or molecule-sensitive electrodes. Table 11.19 reviews these various electrodes.

Electrochemical sensors that detect specific and preselected analytes arenow incorporated into convenient encapsulated hand-held packages and arein routine commercial use. A few multisensors for explosives or traceamounts of gases (e.g., the “Caltech nose”) also exist. However, the shelf lifeand re-usability of all these sensors have been a vexing problem.

Coulometry comes in several flavors: constant-potential or potentiostaticcoulometry, constant-current or amperostatic coulometry, coulometric titra-tions, and electrogravimetry.

Constant-potential coulometry is related to industrial electroplating:One wants to know how to completely deposit a certain metal ion onto anelectrode, without gas evolution (which may make the electrode surface notsmooth) or without depositing another ion that may be present in theelectrolyte; in practice, one may have to program the applied potentialelectronically to secure a complete deposition. Consider the equation

Eappl ¼ Eright � Eleft

� �þ Zconc;right � Zconc;left� �

þ Zkin;right � Zkin;left� �

� IR

ð11:22:2Þ

where the term in square brackets is the difference between standard reduc-tion potentials, corrected for the activitites of the relevant species andtherefore obtainable from theNernst equation. The next three terms representthe overpotential, due to either concentration effects, kinetic effects, or the“IR” drop due to the effective electrical resistance of the solution; alas, thesemust be obtained from experiment. If the left-hand electrode is the referenceelectrode, we may neglect both Zconc,left and Zkin,left leaving

Eappl Eright � Eleft

� �þ Zconc;right þ Zkin;right � IR ð11:22:3Þ

As the electrolysis proceeds, the Nernst potential,R, and the two Z for theright-hand electrode will change with time. For instance, consider the elec-troplating reduction of Cu2þ [56]:

Cu2þ aqð Þ þH2O lð Þ!Cu sð Þ þ 1

2

� �O2 g

� �þ 2Hþ aqð Þ ð11:22:4Þ

738 11 INSTRUMENTS

Table 11.19 Liquid Membrane Electrodes (LME), Gas-Sensing Electrodes (GSME), and Enzyme-BasedBiosensors (EBB) [56]

Analyte TypeConcentration

RangeMajor Interferences,or Active Enzyme Reaction

SensingElectrode

NH4þ LME 100 to 5� 10�7 <Hþ, <5� 10�1 Liþ, AgCl

<8� 10�2 Naþ,<6� 10�4 Kþ,<5� 10�3 Csþ,>1 (Mg2þ or Ca2þ orSr2þ), > 1� 10�2 Zn2þ

Cd2þ LME 100 to 5� 10�7 >5� 10�7 Hg2þ or Agþ, AgCl?<Fe3þ when Cd2þ>0.1,Pb2þ if [Pb2þ]>[Cd2þ],maybe Cu2þ

Ca2þ LME 100 to 5� 10�7 Hþ, 0.3 Liþ, 0.2 Naþ, 0.4 Kþ, AgCl?1.0 Mg2þ, 6� 10�3 Sr2þ,

0.7 Ba2þ,10�5 Pb2þ, 4� 10�3 Hg2þ,4� 10�2 Cu2þ, 1.0 Zn2þ,2� 10�2 Fe2þ, 5� 10�3 Ni2þ,0.2 NH3

Cl� LME 100 to 5� 10�6 < Max. ratio to [Cl�]: 80 OH�,3� 10�3 Br�, 5� 10�7 I�, AgCl2� 10�7 CN�, 10�6 S2�,0.01 S2O3

2�, 0.12 NH3,BF4

� LME 100 to 7� 10�6 5� 10�2 Cl�, 10�3 Br2�,5� 10�6 I�,

AgCl?

5� 10�7 ClO4�, 5� 10�5

ClO3�,

5� 10�4 CN�, 10�3 NO2�

5� 10�3 NO3�, 3� 10�3

HCO3�,

8� 10�2 H2PO4�, HPO4

2�,PO4

3�,0.2 OAc�, 0.6 F�, 1.0 SO4

3�,NO3

� LME 100 to 7� 10�6 0.6 F�, 5� 10�2 Cl�, 7� 10�4

Br�,AgCl

5� 10�6 I�,10�7 ClO4�,

5� 10�5 ClO3�,

10�3 HS�, 10�4 CN�, 10�3

NO2�,

10�2 HCO3�, 2� 10�2 CO3

2�,5� 10�2 H2PO4

�, HPO42�,

PO43�,

0.2 OAc�, 1.0 SO43�,

NO2� LME 1.4� 100 to 1� 10�1 F�, 1� 10�1 Cl�, 10�1

Br�,AgCl

3.6� 10�6 2� 10�3 I�, 3� 10�1 ClO3�,

1� 10�1 ClO4�,

10�4 CN�, 2� 10�1 NO3�,

10�3 HS�, 2� 10�1 SO42�,

2� 10�1 HCO3�, 2� 10�2

CO32�,

7� 10�1 salicylate, 2� 10�1

acetateClO4

� LME 100 to 7� 10�6 F�, Cl�, Br�, 2� 10�3 I�,2� 10�2, ClO3

�

(continued)

11.22 ELECTROCHEMICAL METHODS 739

for which the standard cell potential Eright – Eleft is 0.34 – 1.23¼ �0.94V. Alarge overvoltage forces us to operate the electrolytic cell at the much morenegative potential of �2.5V; the initial cathode potential is at Ecath¼þ0.34V(which is notmeasured directly). As the reaction proceeds, Cudeposits on thecathode, and oxygen gas is evolved at the anode, the [Cu2þ(aq)] decreases,and both I and R decrease. Therefore, at fixed Eappl¼�2.5V the potential atthe cathode will decrease steadily from Ecath¼þ0.34V; if some contaminantPb2þ (aq) is present, at Ecath¼�0.18V Pb will co-precipitate with Cu; atEcath¼�0.5V, the side reaction 2Hþ(aq) !H2(g) will produce bubbles ofhydrogen that will mix with the remaining Cu being plated, thus rougheningthe surface of the deposited Cu.

Using modern electronics and/or well-designed potentiostats, bothcontrolled-potential coulometry and controlled-current coulometry can beperformed, as can gravimetric determinations of analytes. Figure 11.68 showsa potentiostat that uses three operational amplifiers.

The electrodes can be Au, Pt, Al, graphite, glassy carbon, carbon nano-tubes, Sn, In2O3, and so on. Electrode surfaces are made as flat as possible

Table 11.19 (Continued )

Analyte TypeConcentration

RangeMajor Interferences,or Active Enzyme Reaction

SensingElectrode

1� 10�1 ClO4�, 5� 10�2 NO2

�,NO3

�

4� 10�2 CN�, 10�3 HS�,SO4

2�,2 HCO3

�, CO32�, H2PO4

�,HPO4

2�, PO43�

Kþ LME 100 to 1� 10�7 10�2 Hþ, 2.0 Liþ, Naþ, Tlþ, AgCl3� 10�4 Csþ, 6� 10�4 NH4

þ

1.0 Agþ

Kþ LME <?10�3Naþ, <?10�7 Ca2þ,<?10�7 Mg2þ, !valinomycin: Kþ

Kþ LME <? Rbþ,<? Csþ,<?Tlþ ! crown-8: Kþ

Ca2þ LME ! bis-thiourea: Ca2þ

Ca2þ LME ! (alkylphosphate)4-CaNH3 GSE — — !NH4þOH� pHCO2 GSE — — !HCO3

�þHþ pHHCN GSE — — !CN�þHþ pHHF GSE — — ! F�þHþ pH; LaF3H2S GSE — — ! S2�þ 2Hþ pH Ag2SSO2 GSE — — !HSO3

�þHþ pHNO2 GSE — — !NO2

�þ NO3�þ2Hþ pNO2

Urea EBB — Urease ! 2NH3þCO2 NH3

Creatinine EBB Creatininase !N-Me-hydantoinþNH3 NH4þ;NH3

L-amino acids EBB L-Amino acid oxidase !RCOCO2HþH2O2þNH3 NH4þ;NH3

D-Amino acids EBB D-Amino acid oxidase !RCOCO2HþH2O2þNH3 NH4þ;NH3

L-Glutamine EBB Glutaminase ! glutamic acidþNH3 NH4þ;NH3

Adenosine EBB Adenosine deoxidase ! inosineþNH3 NH4þ;NH3

L-Glutamate EBB Glutamate decarbonylase!GABAþCO2

CO2

Amigdalin EBB b-Glucosidase !Glucoseþ benzaldehydeþHCN

CN�

Glucose EBB Glucose oxidase ! gluconic acidþH2O2 pHPenicillin EBB Penicillinase !penicilloic acid pH

740 11 INSTRUMENTS

(using abrasives such as Al2O3 or diamond polish) and often as small aspossible (mm2 to mm2 to even nm2).

Hydrodynamic mixing, either by rotating the working electrode or bymechanical stirring of the solution, is preferred, since this makes the analyteconcentration as isotropic as possible; in the bulk solution the flow of analyteis chaotic, while close to the electrode laminar flow will dominate.

Patch-clamping was developed in 1970 by Neher135 and Sakmann136 tostudy the electrochemistry inside cells, single-ion channels, and neurons: Athin metal wire is inserted into a glass capillary, which is heated to shrink itaround the wire, thus producing electrodes as small as 1 mm in diameter.

Oligomer-modified or polymer-modified electrodes were studied byMurray137 and others in the 1980s; these oligomers and polymers were eitherspin-coated or covalently bonded by “self-assembly” to electrodes; one goalwas to study the electrical double layer by penetrating it. Alas, the doublelayer simply moved further toward the bulk solution!

Figure 11.69 shows a cyclic voltammogram (CV) for the reversibleNernstian redox involving the reduction of potassium hexacyanoferrate(III)to potassium hexayanoferrite(II):

Fe IIIð Þ CNð Þ63� þ e� ! Fe IIð Þ CNð Þ6

4� ð11:22:5Þ

CE

WE

RE

LINEAR SWEEPGENERATOR

DATAACQUISITION

SYSTEM

A

B

C

Potentiostaticcontrol circuit

Current-to-voltageconverter

EoutFIGURE 11.68

Potentiostat with three operation-al amplifiers A (adder control), B(voltage follower), and C (currentfollower). Adapted from Skooget al. [56].

135 Erwin Neher (1944– ).136 Bert Sakmann (1942– ).137 Royce Wilton Murray (1937– ).

11.22 ELECTROCHEMICAL METHODS 741

In a CV, the two criteria for Nernstian reversibility are (1) thatthe distance along the V axis between oxidation and reduction peaks be(0.0592/n) volts and (2) that themeasuredpeakheights for the oxidationwaveand the reduction wave be equal. If only one of two waves (oxidation orreduction) is present, then the reaction is clearly irreversible. If they both

cycle 1

b

c

forwardscan

reversescan

0 20

0.8

0.6

0.4

0.2

0

–0.2

40 60

TIME, s

PO

TE

NT

IAL,

V v

ersu

s S

CE

80

a

d EfinalEinitial

cycle 2

–0.200.20.4

POTENTIAL, V versus SCE

0.6

j

ik

a

b

h

g

f

e

tpc

d

c

ipc

Epa

ipa

0.8–20

–10

anod

icC

UR

EN

T, μ

Aca

thod

ic

0

10

20

(a)

(b)

FIGURE 11.69

(a) Applied potential versus timeand (b) resulting cyclic voltammo-gram of 6mM K3Fe(CN)6 in 1MKNO3. Scan was initiated at 0.8Vversus SCE in negative direction at50mV/s. Platinum electrode area¼ 2.54mm2. The labels a, b, c, d,e, f, etc., in (a) match the corre-sponding points in (b) [56,57].

742 11 INSTRUMENTS

exist, but one or both criteria are notmet, the term quasi-reversible is appliedloosely (its original definition [58] applied to electrochemical reactions thatshowed kinetic limitations due to the reverse reaction). The CV shape isdistorted if the potential sweep rate is too rapid (>200mV/s), because the iondiffusion rate cannot keep up. In water, dissolved O2 should be removedbybubblingN2gas through the solution for 20min (or else, at�0.22Vvs. SCE,O2 ! H2O2 in a first reduction, and, at �1.2V vs. SCE, H2O2 ! H2O in asecond reduction). The CV can get “fat” (i.e., the waves are still present, butthe currents on the forward and reverse cycles at intermediate voltages aresignificantly larger than those seen in Fig. 11.63), because the diffusion ofanalyte to the electrode is retarded by a partially permeable layer on theelectrodes. For conducting polymers, repeating the CV cycles several timesgrows one of the waves: the polymer is growing on the electrode!

The simple linear-sweep voltammetry (LSV) or linear potential sweepchronoamperometry (ofwhichpolarographywith adroppingHgelectrode isthe earliest example) can be understood simply if one looks at just the first riseto a peak in Fig. 11.70.

Other forms of voltammetry are as follows: (1) fast-scan cyclic voltam-metry: useful in neuroelectrochemistry; (2) nanosecond voltammetry: for a5-mm disk working microelectrode with RC < 1 ms, scan rates of 2.5MV/sallow for fast kinetics measurements; (3) differential-pulse voltammetry:with staircase pulses, potential resolutions of 0.04V and detection limits of10�8M can be attained; (4) anodic (cathodic) stripping voltammetry: traces

Voltage

Current

Time

Time

EXCITATION

RESPONSE

FIGURE 11.70

High-speed linear-sweep voltam-metry (LSV) or linear potentialsweep chronoamperometry: (top)potential waveform; (bottom) cur-rent response. The areas betweenthe solid lines and the dotted linesmeasure approximately the chargetransferred in the oxidation orreduction.

11.22 ELECTROCHEMICAL METHODS 743

of analyte pre-plated on an anode (cathode), and the current is measured asthe metal is removed by oxidation (reduction); (5) square-wave voltamme-try: allows for fast speeds and sensitity down to 10�8M; (6) Osteryoung138

square-wave stripping voltammetry: detection limits of 10�10M; (7) chron-oamperometry (Fig. 11.64); (8) scanning electrochemical microscopy: theworking electrode is an STMtip, protected byfingernail polish, except close toits atomically sharp tip.

11.23 X-RAYDIFFRACTIONOFORDEREDCRYSTALS, LIQUIDSAND DISORDERED SOLIDS

Crystals may not be too perfect: The condition for Bragg139 reflection,Eq. (8.3.2), is also the condition for total internal reflection. Thus, an absolutelyperfect millimeter-sized crystal will reflect internally almost all of the X-raybeam, even at the Bragg angles. However, each crystal contains crystallinedomains, 1–10 mm in size, which are slightly misaligned with each other (byseconds or a fewminutes of a degree); this is what permits the observation ofX-ray diffraction peaks. If the diffracted intensity is unacceptably low, a quickthermal shock to the crystalmayhelpmicro-shatter the crystal and form thosedomains.

In general, the linewidthDwof adiffraction line (for apowder or a crystal)is given by the Scherrer140 equation [see Eq. (8.3.5)]:

Dw ¼ l= 2L sin yð Þ ð11:23:1Þ

where y is the Bragg angle and L is the size of the coherence length normal tothe incident X-ray beam of wavelength l. When the crystallites extend toabout 100–1000 repeat units (i.e., from 3 to 3000 A

�), then, even for randomly

oriented crystallites, X-ray diffraction peaks obeying Bragg’s law can be seenquite easily, with angular widths of the order of 0.5� or so. The indexing ofthese peaks is, however, difficult in general.

The earliest diffraction photographs used a stationary nonmonochro-matic X-ray source, a stationary or almost-stationary crystal, and a stationaryplanar X-ray-sensitive film (masked from ambient light by black paper); thiswas the Laue141 camera. The first diffraction by NaCl was initially misinter-preted (the scattering power of Cl and Na is almost the same). The mostintense Ka radiation of a Coolidge142 water-cooled X-ray tube could beselected by thin filters, which absorbed Kb and a fair fraction of the contin-uous Bremsstrahlung white emission (Ni filter for Cu X-rays, Cr filter for Fe,etc.). Later, crystal monochromators (graphite, or LiF) replaced the filters toprovide an almost-monochromatic X-ray source beam.

If the compound does not crystallize, then diffraction by a powder is theonly recourse: The Scherrer powder camera has a strip of X-ray-sensitive film

138Robert Allen Osteryoung (1927–2004).139 Sir William Lawrence Bragg (1890–1971).140 Paul Scherrer (1890–1969).141Max Theodor Felix von Laue (1879–1960).142William David Coolidge (1873–1975).

744 11 INSTRUMENTS

wrapped around the inside of a cylindrical camera with a light-tight lid (thefilm is inserted in adark room), and a thin glass capillary of diameter 0.5-1mmis mounted along the axis of the cylinder. Each crystallite in the powder istypically 1–5 mm in size and is randomly oriented; nevertheless, the capillaryis rotated from the outside, to reduce any deviations from a random distri-bution. Modern two-circle y–2y powder diffractometers use a stationaryX-ray tube with Bragg–Brentano143 collimators, a photomultiplier detectormounted on the 2y axis, and the powder placed centrally on the y axis.

For crystalline samples, rather quickly, the problem of deciding whichX-ray spot was due to which set of crystal planes required that the crystaland/or the detector bemovedduring the film exposure; thus in 1924 came theWeissenberg144 camera (Fig. 11.71), with a cylindrically placed X-ray filmtranslated back and forth about a stationary cylindrical slit, as the crystal isrotated around the cylinder axis. A decent photograph resulted if the crystalwas so oriented that the reciprocal lattice axis a� orb� or c� was along the axialdirection of crystal rotation. To make the necessary adjustments, the singlecrystal, of typical dimensions (0.5mm)3 for organic compounds, is glued to ashort stub and mounted on a goniometer head, whose translations andshallow arcs permit manual adjustments to the crystal orientation relativeto the cylindrical axis of the Weissenberg camera. The Weissenberg photo-graph is a vast improvement over the Laue photograph but gives a distortedpicture of the reciprocal lattice.

Weissenberg film holder

Xray tube

X-raycollimator

Layerlinescreen

Layerlinescreen

RotatingCrystalgoniometerholder

Worm drive forfilm holdertranslation

Clutchforengagingfilm holdertranslation

X-raydirect beamstop

Crystalon

goniometerhead

Gearbox

FIGURE 11.71

Weissenberg camera.

143 J. C. M. Brentano (1888–1969).144K. Weissenberg (1893–1976).

11.23 X-RAY DIFFRACTION OF ORDERED CRYSTALS, LIQUIDS AND DISORDERED SOLIDS 745

In theBuerger145 precession camera (1944) themotions of the crystal andthe planar filmholderweremechanically coupled to provide a distortion-freephotograph of reciprocal space.

The crystallographer would develop his films and seek an indexingscheme for his photographs, assigning Miller146 h, k, and l integers for theobserved spots seen and noting the extinctions, or systematic absences,which would provide information about certain symmetry elements in thespace group of the crystal (Table 11.20).

Using Table 11.20, one can find a narrow range of between 1 and 6possible space groups for the crystal (the X-ray beam adds a center ofsymmetry); the correct space group is confirmed after least-squares refine-ments in each of the possible space groups and after a decision by statisticaltests about which is the “best” space group.

By comparison with a film strip with a relative intensity scale for somestrong hkl reflection exposed to X-rays for known increments of time, theintensity of each spot was read by hand (later by optical photometers), and aset of Miller indices hkl and intensities Ihkl were collected (this would take a

Table 11.20 Systematic Absences Used to Determine the Possible Space Groups of CrystalþX-RayBeama

Symmetry Element Affected Reflection Condition for Systematic Absence

Screw axis 21 or 42 or 63 along a h00 h¼ odd¼ 2 n þ1Screw axis 21 or 42 or 63 along b 0k0 k¼ odd¼ 2 n þ1Screw axis 21 or 42 or 63 along c 00l l¼ odd¼ 2 n þ1Screw axis 31 or 32 or 62 or 64 along c 00l l¼ 3 nþ 1 or 3 nþ2Screw axis 41 or 43 along a h00 h¼ 4 nþ 1, 2, or 3Screw axis 41 or 43 along b 0k0 k¼ 4 nþ 1, 2, or 3Screw axis 41 or 43 along c 00l l¼ 4 nþ 1, 2, or 3Screw axis 61 or 65 along c 00l l¼ 6 n þ1, 2, 3, 4, or 5Glide plane ?a, transl. b/2 (b-glide) 0kl k¼ 2 nþ 1Glide plane ?a, transl. c/2 (c-glide) 0kl l¼ 2 nþ 1Glide plane ?a, transl. (bþ c)/2 (n-glide) 0kl kþ l¼ 2 nþ 1Glide plane ?a, transl. (bþ c)/4 (d-glide) 0kl kþ l¼ 4 nþ 1, 2, or 3Glide plane ?b, transl. a/2 (a-glide) h0l h¼ 2 nþ 1Glide plane ?b, transl. c/2 (c-glide) h0l l¼ 2 nþ 1Glide plane ?b, transl. (aþ c)/2 (n-glide) h0l hþ l¼ 2 nþ 1Glide plane ?b, transl. (aþ c)/4 (d-glide) h0l hþ l¼ 4 nþ 1, 2, or 3Glide plane ?c, transl. a/2 (a-glide) hk0 h¼ 2 nþ 1Glide plane ?c, transl. b/2 (b-glide) hk0 k¼ 2 nþ 1Glide plane ?c, transl. (aþ b)/2 (n-glide) hk0 hþ k¼ 2 nþ 1Glide plane ?c, transl. (aþ b)/4 (d-glide) hk0 hþ k¼ 4 nþ 1, 2, or 3A-centered lattice (A) hkl kþ l¼ 2 nþ 1B-centered lattice (B) hkl hþ l¼ 2 nþ 1C-centered lattice (C) hkl hþ k¼ 2 nþ 1F-centered lattice (F) hkl hþ k¼ 2 nþ1 & kþ l¼ 2 nþ 1 & lþ h¼ 2 nþ1

(i.e., hkl not all even, and not all odd)Body-centered lattice (I) hkl hþ kþ l¼ 2 nþ 1

aThis always adds a center of Inversion Symmetry. Usually the resulting choices are 1 to 6 possible space groups [59].

145Martin Julian Buerger (1903–1986).146William Hallowes Miller (1801–1880).

746 11 INSTRUMENTS

month or two). After collecting many data (typically, and hopefully, 6 to8 times asmany as therewere atom coordinates to determine), the attempts to“solve” the structure would ensue. The advent of digital computers andcharge-coupled detectors have changed two things dramatically: (1) datacollection and (2) structure solution. Since the mid-1960s, computer-controlled four-circle diffractometers used a fixed X-ray source (plus mono-chromator), a narrow-angle photomultiplier detector mounted on one circle(2y), and the crystalmounted ona cradle of three concentric circles (o, w, andf(Fig. 11.72), or y, k, and f), so almost all of the Ewald147 sphere could beaccessed, one diffraction peak at a time.

Powerful algorithms cut the data collection time to a few days. Usingcrystal samples cryocooled to 77K sharpens the diffraction intensities bytypical factors of 5 or so, and it hastens data collection to about 1 day.

Finally, the advent of planar charge-coupled devices replaced films andphotomultipliers, and it enabled Arndt148 oscillation cameras (logically

X-ray detector(photomultiplier)on 2 θ axis

2θ circle

ω circle

LiquidHe or N2

source

Optionalcoolinggas stream

Crystalgoniometerhead

Crystal X-ray collimator

χ circle

φcircle

CoolidgeX-raytube

X-rayport

FIGURE 11.72

Four-circle X-ray diffractometer (cables, power supplies, computer controller connections, and direct X-ray beam stopare not shown). The four circles are (2y,o, w, and j). The Coolidge X-ray tube is stationary. The 2y ando axes lie in thevertical plane,while the w axis turns in the horizontal plane,where the Ewald diffraction condition is satisfied. The 2y-circle swings the photomultiplier detector in the horizontal plane; the o circle turns independently of 2y, but iscomputer-controlled so that o¼Bragg y¼half of 2y. The w-circle is anchored atop the o-circle. The j-circle racesaround the w-circle, and it turns on an axis along the long axis of the goniometer head. The crystal is placed on agoniometer headwith (atmost) two shallowangular races and twoplanar races, one longitudinal andone transverse;with thehelpofanoptical telescope (not shown), these races are initially adjustedmanually, to center the crystal at theintersection of the j, w,o, and 2y axes. Also shown is the optional container for liquid He or N2, which produces a gasstream to cool the crystal to 85 K or to 10 K.

147 Paul Peter Ewald (1888–1985).148Ulrich Wolfgang Arndt (1924–2006).

11.23 X-RAY DIFFRACTION OF ORDERED CRYSTALS, LIQUIDS AND DISORDERED SOLIDS 747

related to the Buerger precession camera) to collect many diffraction inten-sities at once, thus decreasing crystal exposure to X rays (which had become amajor problem in protein crystallography, where the Bremsstrahlung tendedto limit the endurance of a typical protein crystals to only a fewdays on a four-circle diffractometer). Thus, data collection is now highly automated andneed not last more than a day.

X-Ray Scattering and Diffraction Intensities. As first discussed inSection 10.6, the X rays are produced with lack of phase coherence withintensity I0; if they impinge on a single stationary electron at the origin,they will scatter (Thomson149 scattering) with intensity I at a distance Rfrom the electron, at an angle y from the direction of the incoming beam asfollows:

I R; yð Þ ¼ I0 8 � 1p � 1eo � 2e4m � 2c � 4R � 2� �

0:5ð1þ cos2 yÞ� �

ðð10:6:16ÞÞ

The diffraction of X rays in a crystal occurs even for a phase-incoherentX-ray source; one can write the theoretical scattered amplitude as the struc-ture factor:

Fhkl ¼ ZXAj

fj exp½2piðhxj þ kyj þ l zjÞ� ðð10:6:17ÞÞ

where fj is given by:

fj sð Þ ¼ðcini rð Þexp is rð Þcfin rð Þdv rð Þ ð11:23:1Þ

where s is the scattering vector with magnitude:

s 4p sinyhkl=l ð11:23:2Þ

yhkl is the Bragg angle, l is the X-ray wavelength, and the wavefunctionscini(r) and cfin(r) describe the initial state and the final state of the atom. Thestructure factor can be resolved into real and imaginary components, or into amagnitude times a complex phase factor:

Fhkl ¼ Ahkl þ iBhkl ¼ Fhklj jexp iahklð Þ ðð10:6:19ÞÞ

For a spherically symmetric atom with radial wavefunction R(r),Eq. (10.6.16) simplifies to

fj sð Þ ¼ 4pð10r2R2 rð Þ sin srð Þ=sr½ �dr ð11:23:3Þ

149 Sir Joseph John Thomson (1856–1940).

748 11 INSTRUMENTS

The atomic scattering factor fj(s) decays exponentially with increasingscattering angle 2yhkl.

The phase factor exp[2pi(hxjþ kyjþ lzj)] in Eq. (10.6.17) contains the all-important information of the positions (xj, yj, zj) of the j¼ 1, . . .,A atoms in thecrystallographic asymmetric unit, which are the essential goal of X-raystructure determination. The observed X-ray intensities (which are assumedto be already corrected for several effects discussed in Section 10.6) are givenby

Ihkl ¼ Fhklj j2 ðð10:6:18ÞÞ

and contain no direct phase information; this phase problem makes findingthe (xj, yj, zj) not straightforward, as explained below.

PROBLEM 11.23.1 Prove that, unless there is significant X-ray absorptionwithin the crystal because of “anomalous dispersion,” Friedel’s law holds:Ihkl ¼ I ��h�k�l.

PROBLEM 11.23.2 The Bragg reflections are symmetrical to reflection andtransmission. However, Bijvoet showed that if there is “anomalous dis-persion”—that is, a small amount of X-ray absorption (usually due to ele-ments with high atomic number (Z > 60)—then asymmetry occurs, and thisfacilitates structure solution.

The ElectronDensity Function and the “Phase Problem.” The electron densityfunction r(x, y, z) at the general point (x, y, z) in the crystal is the goal of everycrystal structure solution; formally, it is the Fourier transform of the stucturefactors:

r x; y; zð Þ ¼ 1=Vð ÞX

h

Xk

XlFhklexp � 2pi hxþ kyþ lzð Þ½ � ð11:23:4Þ

whereV is the unit cell volume and the sums over so-calledMiller indices h, k,l span over all the available intensities (usually 200 to 20,000 but not infinity).The absolute square of the structure factor in Eq. (10.6.18) denies us directknowledge of the desired atom positions {(xj, yj, zj), j¼ 1, 2, . . ., A}: Theobserved intensities Ihkl, when Fourier-inverted, yield, instead, thePatterson150 function P(R), which is the convolution of the electrondensity function:

PðRÞ ¼ðdvðrÞrðrÞrðrþRÞ ð11:23:5Þ

and it is difficult to “deconvolute the problem.” A start on the structuresolution can occur in two lucky cases: (a) There are heavy atoms present,which dominate the X-ray scattering, or (b) special regions of Pattersonspace, due to atoms in special symmetry positions in the space group

150Arthur Lindo Patterson (1902–1966).

11.23 X-RAY DIFFRACTION OF ORDERED CRYSTALS, LIQUIDS AND DISORDERED SOLIDS 749

(Harker151 regions) can yield some atom coordinates. The other atompositions can then be sought by standard but very tedious Fouriermethods.

If two heavy-atom derivatives can be crystallized which preserve thespace group and unit cell size of a large protein, then the structure can besolved directly; this method of multiple isomorphous replacement was usedby Perutz152 and Kendrew153 to solve the first two protein structures bylaborious, decade-long film methods: hemoglobin and myoglobin.

Direct Methods. X-ray or neutron diffraction crystallography candetermine the crystal symmetries (translational and often even local). ButX-ray beams or neutron beams are not phase-coherent, so much informationis lost. There are, however, many significant relationships between theintensities of the diffraction peaks.

The best approach to date to automatic structure solution is the directmethod (DM), introduced by Hauptmann,154 Jerome Karle,155 and IsabellaKarle,156 and made possible by efficient computer programs, such asMULTAN (Main,157 Germain,158 and Woolfson159) and SHELX(Sheldrick160).

DM can be applied to “small” structures (< 1000 atoms in the asym-metric unit). Since a crystal with, say, 10 C atoms requires finding only x, y,and z variables, but typically several thousand intensity data can be col-lected, then, statistically, this is a vastly overdetermined problem. There arerelationships between the contributions to the scattering intensities of twodiffraction peaks (with different Miller indices h, k, l, and h0, k0, l0), due to thesame atom at (xm, ym, zm). DM solves the phase problem by a bootstrapalgorithm, which guesses the phases of a few reflections and uses statisticaltools to find all other phases and, thus, all atom positions xm, ym, zm. Howto start?

DM is easiest to explain for centrosymmetric crystals, for which all phasefactors exp [2pi(hxjþ kyjþ lzj)]must be equal, individually, to eitherþ 1 or�1;that is, the only phase choice is which Fhkl has a positive sign andwhich has anegative sign. Since a typical crystal structure is determined from about 2000independent reflections (Ihkl)obs and there may be 30 atoms to be found [foreach, three positional parameters xj, yj, zj and the six unique components ofthe second-rank thermal vibration tensor (thermal parameters) b11, b22, b33,b12, b23, b31, i.e., a total of 30� (3þ6)¼ 270 parameters], the problem isoverdetermined, by a comfortable ratio of 2000/270. It also uses the physical

151David Harker (1906–1991).152Max Ferdinand Perutz (1914–2002).153 Sir John Cowdery Kendrew (1917–1997).154Herbert Aaron Hauptmann (1917–2011).155 Jerome Karle (1918– ).156 Isabella Lugoski Karle (1921– ).157 Peter Main (1938– ).158Gabriel Germain (ca. 1935–2011).159Michael Mark Woolfson (1927– ).160George Michael Sheldrick (1942– ).

750 11 INSTRUMENTS

fact that the electron density function is never negative, is close to zero farfrom atoms, and has approximately spherical peaks around the correct atompositions.

DM uses the normalized structure factors Ehkl obtained from the ob-served structure factors Fhkl and the individual theoretical atomic scatteringfactors fi:

Ehkl Fhkl=XAJ¼1

fj2

� �" #1=2

ð11:23:6Þ

These “E’s” compensate for the fall-off in X-ray scattering intensity athigh diffraction angles. DM identifies the strongest E’s (say the top 10%)and selects from them a “S2” list of “triples” of E’s whose indices add: E(hkl),E(h0k0l0) andE(hþh0, kþk0, lþl0). It can be shown that if s(h, k, l) is the sign ofEhkl,then

sðh; k; lÞsðh0; k0; l0Þ sðhþ h0; kþ k0; lþ l0Þ ð11:23:7Þ

The tangent-angle formula gives the probability P that the triple productis positive:

P ¼ 1

2þ

�1

2Nð�1=2Þ

�tanh ðEhkl Eh0k0l0 Eh � h0k � k0 l � l0 Þ ð11:23:8Þ

(where N is the number of atoms in the unit cell); this formula is used to“bootstrap” an ever-increasing set of phases that can be used to define thecorrect structure. An “E-map” based on all these guesses will usually revealthe chemically correct structure.

DMalso applies to acentric structures, where a phase angle a(hkl)must befound for each reflection. Typically, the possible phase angles are divided into15� increments, so that 24 possible phase angles must be considered perreflection. About 90% of all structures can be solved in a day or two by directmethods.

Patterson and Symmetry Superposition Methods. An older bootstrapmethod, based on searches of the Patterson function and variants thereof(vector superposition and symmetry superposition functions), shouldpresent significant advantages for noncentric structures. Much recentprogress has been made in such alternative algorithms, which should beused when direct methods fail.

Least-Squares Refinement. When a chemically plausible structure is found,its correctness must be proven by a nonlinear least-squares refinementprocedure—for example, ORXFLS (Busing161 and Levy162). Since thealgorithm neglects higher-order nonlinearities, the fit becomes better, andthe convergence faster, if the beginning structure found by direct or Patterson

161William R. Busing (1924– ).162Henri A. Levy (1913–2003).

11.23 X-RAY DIFFRACTION OF ORDERED CRYSTALS, LIQUIDS AND DISORDERED SOLIDS 751

methods is very “close” to being correct. Each cyclemoves slightly (“refines”)the three atom position coordinates and the six parameters for thermalvibration for each unique atom in the asymmetric unit, until no moreimprovement of the fit is possible. The reliability indices, or discrepancyindices, or R-factors are the unweighted factor R:

R ¼X

hklF obshkl

� F calchkl

� �=X

hklF obshkl

ð11:23:9Þ

(the sum extends over all the measured intensities) and the weightedR-factor Rw:

Rw ¼X

hklwhkl F obs

hkl

� F calshkl

� �=X

hklwhkl F

obshkl

ð11:23:10Þ

wheretheweightwhklcorrectsforerrors inmeasurementandreliability.TheRw

is used to “drive” the refinement to its best fit with the observed X-rayintensities |Fhkl

obs|2. The unweighted R-factor gauges the quality of thestructure determination, or of the original set of X-ray intensities. R¼ 40%can be obtained with a totally incorrect structure, or with atom coordinatespicked fromrandomnumbergenerators or soccerball coordinates. IfR¼ 12%,thestructureischemicallycorrect,butthebonddistancesandanglesarenot tooreliable. For a “good” structure, R should be between 2% (R¼ 0.02) and 5%(R¼ 0.05). Thesmall rangeofpossible spacegroups for agivencrystal requiresthat refinements be done in each space group, accepting the assignment thatgives the lowest R-factor. In modern computerized data collection andstructure determination, symmetries easily discerned by old-fashioned filmmethods are often missed; published crystal structures are sometimesattributed to a lower-symmetry space group than warranted. Opticallyactive molecules must crystallize in acentric space groups.

The “last step” is to use a plotting program to yield a nice picture of atompositions, bonding, thermal vibration ellipsoids, and packing; this is anORTEP plot (Johnson’s163 Oak Ridge Thermal Ellipsoid Program); seeFig. 11.73.

The estimated precision in bond lengths obtained by a least-squarerefinement of a data set measured by X-ray diffraction can be� 0.003 A

�

(�0.3 pm), for a structure with unweighted R-factor less than 3%. If the dataset is collected at low temperatures (20K or 80K), the decrease in thermalvibration can yield even better bond distances and angles. For H atomcoordinates, the precision is one or two orders of magnitude lower, sincethe electron density around an H atom is relatively low; in these cases aneutron diffraction study (which requires very large crystals) can yield betterH atom positions.

Low-temperature data sets have also been exploited to detect the effectsof valence bonding electron densities on the scattering factors and, thus, onthe quality of the structure factors and of the refined structure.

For powder diffractograms, which usually involve no more than 20 to 50diffraction peaks, the Rietveld164 procedure is a least-squares program thatassumes that even the many noise “data” between diffraction peaks are

163Carroll K. Johnson (1929– ).164Hugo Rietveld (1932– ).

752 11 INSTRUMENTS

statistically significant; it has been used to “solve” structures and has its ownR-factor. However, given the low number of diffraction peaks, the risk of“refining” erroneous structures is much greater in Rietveldt analysis than intraditional single-crystal structure determination.

Protein Crystallography. Proteins are molecules of between 2 and 100 kDa(kg/mol); they often containmuchwater of crystallization, and thediffractingpower of the crystal is limited to a Bragg angle of no more than about 15�

or 20�. This limits the possibility of applying direct methods, and itmakes traditional least-squares refinement impossible. But biochemistsand geneticists do not insist on such precision, and they are content with“1.5-A

�” or “2-A

�”maps,which roughly show the regions of helical folding and

afford a rough idea of the active site where the important chemical reactionoccurs. The programsused in protein crystallography are quite different fromthose used for “small-crystal” diffractometry.

Liquids, Gases and Disordered Solids. Liquids, disordered solids, gases,and single crystals can diffract X rays. For liquids and disordered solids,where there is no long-range order, and the short-range order extends from0 to maybe 1 or 2 nm, the diffraction consists of very broad maxima in theintensity function I(s), where s is the scattering vector defined inEq. (11.23.2). The one-dimensional Fourier transform of I(s) is the radialdistribution function R(r):

RðrÞ ¼ ð2r=pÞðs¼smax

s¼0IðsÞ½sinðs rÞ=s�expð � a s2Þ ds ð11:23:11Þ

where the integration is carried out numerically up to the maximum valueobserved; the Gaussian factor exp(�a s2) helps with the numerical conver-gence of the integral. I(s) is often “improved” by “sharpening” techniques,which correct for independent-atom scattering. R(r) has meaningful broadpeaks centered around the first few interatomic distances r in the sample.

N(32)

C(31)

C(30)113.05(25)

C(33)

C(34)

F(36) F(35)

C(25)

C(29)

C(24)

N(19)

C(20)

C(21)113.20(24)

C(22)

N(23)

F(37)F(38)

1.14

2(3)

1.41

7(4)

123.

10(2

7)

123.41(26)

1.405(3)

C(26)118.80(23)

1.153(4)

1.344(4)

1.345(3)

1.421(4)

1.409(4)

1.147(4)

176.21(34)

117.

68(2

3)

123.

47(2

3)

C(27)123.85(27)

116.90(24)

123.73(25)

123.

84(2

4)

175.78(33)

118.

93(2

2)1.

352(

3)

1.14

5(3)

118.82(22)C(28)

1.34

5(3)

176.73(36)

1.353(3)

1.421(3)

118.72(22)

1.347(3)

112.73(23) 112.75(23)

117.94(23)

123.50(24)117.66(23)

1.409(3)

176.39(33)

122.89(26)

123.

62(2

3)

1.42

1(4)

124.

23(2

4)

123.

74(2

6)

1.41

6(3)

123.13(25)

FIGURE 11.73

ORTEP plot of TCNQF�4 , 2,3,5,6-tet-rafluoro-7,7,8.8-tetracyanoquino-dimethan radical anion in the saltn-butylphenzinium TCNQF�4 : Thering is benzenoid, not quinonoid[the ring C–C bond lengths (withestimated standard deviations) areclose to equal]; this indicates thatwe have the radical anion, not theneutral molecule TCNQF4

0 [60].

11.23 X-RAY DIFFRACTION OF ORDERED CRYSTALS, LIQUIDS AND DISORDERED SOLIDS 753

Aswas the Patterson function, Eq. (11.23.5),R(r) is also the convolution of theelectron density function r(r):

RðrÞ ¼ðx¼0

x¼1rðrÞrðrþ xÞdx ¼ 4pr2rind þ RstrðrÞ ð11:23:12Þ

Here 4pr2rind is the structure-independent scattering, while Rstr(r) is thestructure-dependent part. For a disordered liquid or solid, as r increa-ses––say, beyond 10 or 15 A

�––the peaks in R(r) fade into the structureless

scattering from electrons 4pr2rind, as the short-range order is exceeded.The area under the first few peaks of R(r) will yield the coordination numberof the atoms in the peak.

The local structure in liquids can be measured by X-ray diffraction anddescribed by either a radial distribution function or the pair correlationfunction. In particular, the oxygen–oxygen pair correlation function orreduced radial distribution function for water, gOO(r) Fig. 11.74, can beobtained from

gOOðrÞ ¼ 1þ ½2p2r0jrj�� 1

ðs¼0

s¼smax

sinðsrÞs ds½IðsÞ � < F2 >�= < F > ð11:23:13Þ

where r0 is the average density, I(s) is the total integrated experimentalscattering intensity in electron units per molecule, and s is the magnitude ofthe scattering vector:

s 4p siny=l ðð11:23:4ÞÞ

l is theX-rayorneutronwavelength,and2y is the(Bragg) scatteringangle (e.g.,smax¼ 7.67A

� �1whenl¼ 1.54A�and2ymax¼ 140�).F is thetheoreticalscattering

amplitude per molecule, whose mean square is given by the Debye formula:

hFðsÞ2i ¼X

i¼j

i¼mXj¼1

j¼ifiðsÞfjðsÞ½sinðsrijÞ=srij� ð11:23:14Þ

10

1

2

g 00(

r)

3

3 5

r / Å

7 9

FIGURE 11.74

Reduced radial distribution func-tion, or oxygen–oxygen pair corre-lation function gOO(r), for H2O(l)at 300 K: X rays, solid line [61,62];X rays: dashed line [63]; neutron,dot-dashed line [64]; neutron, grayline [65]. The peaks indicate a first(nearest)-neighbor, a second, andathird O–O distance at approx 2.9 A

�,

4.3 A�, and 6.5 A

�, respectively.

754 11 INSTRUMENTS

where fi(s) is the theoretical scattering amplitude for atom i, rij is the distancebetween atoms i and j in the molecule, and the sum is over all them atoms inthe molecule.

Small-Angle Scattering. X-ray diffraction becomes rather difficult at Braggangles below 2�, because of intensity interference from the collimated andmonochromatized direct X-ray beam. However, if a double monochromatoris used, then angles from 0.001� to 2� become accessible in special small-anglecameras. The particle sizes can be calculated from a fit to Guinier’s165 law:

IðsÞ ¼ n2expð � ð4=3Þps2RG2Þ ð11:23:10Þ

where s is given by Eq. (11.23.4), n is the number of electrons in the particle,and RG is its radius of gyration. This approximate equation is valid (to �5%)up to (sRG) � 0.2.

Diffuse X-Ray Scattering. Polymers yield only a few diffuse X-ray peaksand streaks, which give an idea of the relative “crystallinity” of the polymerand of its growth axis. The structure of deoxyribonucleic acid (DNA) wasinferred in 1953 by Watson166 and Crick167 [66] from the fiber-axis X-rayphotographs of DNA salts by Franklin,168 one of which is Fig. 11.75.

If one looks between diffraction peaks at high resolution, one finds“streaks” due to lattice phonons, which sharpen gradually at low tempera-tures: this is called thermal diffuse scattering.

FIGURE 11.75

Fiber X-ray diffractogram of Na de-oxyribose nucleate [67].

165Andr�e Guinier (1911–2000).166 James Dewey Watson (1928– )167 Francis Harry Compton Crick (1916–2004).168 Rosalind Elsie Franklin (1920–1958).

11.23 X-RAY DIFFRACTION OF ORDERED CRYSTALS, LIQUIDS AND DISORDERED SOLIDS 755

The Peierls169 metal-to-semiconductor phase transition in TTFr TCNQ�r

was detected in an oscillation camera; these streaks became bona fide X-rayspots only below the phase transition temperature of 55 K; this transition isincommensurate with the room-temperature crystal structure, due to itspartial ionicity r 0.59, and the “freezing” of the concomitant itinerantcharge density waves (this effect was missed by four-circle diffractometerexperiments, which had been set to interrogate only the intense Bragg peaksof either the commensurate room-temperature metallic structure, or thecommensurate low-temperature semiconducting structure).

Neutron Diffraction. X-ray diffractometry uses scattering of X-rays byelectrons, but is not very sensitive for the determination of H atompositions in a crystal structure, except if the data collection is performedaround 10 K; for 300 K data collections the H atom positions are usuallyassigned at “reasonable” calculated distances from the atoms to which H isexpected to be chemically bonded. X-ray diffraction is also insensitive tomagnetic effects, even in ferromagnetic or antiferromagnetic samples. On theother hand, hydrogen nuclei andmagnetic nuclei have a very large scatteringcross section for neutrons. A collimated beam of thermalized neutrons,energy-selected to have wavelengths close to 0.01 nm, and possiblycollimated by a monochromator crystal, is available as a beam line atnuclear reactor sites; with this beam one can carry out neutrondiffractometry, locate H atom nuclear positions with great precision, andalso detect superlattice effects of local high magnetic moments in magneticsamples. Neutron diffraction was first done in 1945 by Wollan170 andperfected by Wollan and Shull.171 Neutron diffraction needs much largersamples than X-ray diffraction: crystals of typical size 10 mm rather than 0.1mm are needed, and so single-crystal neutron diffractometry is rarely done.However, atom positions in the unit cell (which are typically found withsingle-crystal diffractometry) can be detected with some precision by 4.2 Kneutron powder diffractometry coupled with Rietveld refinement.

EXAFS and XANES. The absorption of X-rays increase dramatically closeto the band edge, because of core-level energies; they will also show certainsmall oscillations (not shown in Fig. 10.10), known at first as “Kossel172 lines.”The energy-dependent linear X-ray absorption coefficient m(E) in theexpression It¼ I0 exp(�m(E)x) has oscillations near the absorption edge inthe region 0 to 1 keV from the absorption edge; Kossel had only studied theregion 0 to 5 eV from the edge [68].

The X-ray absorption has three regions: (i) the edge region; (ii) the X-raynear-edge structure (XANES) (0 to 0.1 keV) due to multiple scatteringresonances; (iii) the extended X-ray absorption fine structure (EXAFS) (0.1to 1 keV), known earlier as “Kronig173 structure” and due to single scatteringof the excited photoelectron by neighboring atoms. NEXAFS is synonymousto XANES. The XANES and EXAFS techniques require very high input beamintensities I0 at multiple wavelengths (in the range 0.2 to 35 keV), so most

169 Sir Rudolf Ernst Peierls (1907–1995).170 Ernest Omar Wollan (1902–1984).171 Clifford Glenwood Shull (1916–2001).172Walther Ludwig Julius Kossel (1888–1956).173 Ralph de Laer Kronig (1904–1995).

756 11 INSTRUMENTS

experiments are carried out at beam lines at electron synchrotron sources.The fitting of m(E) to theory reveals the element-specific nature of theabsorbing atom (which controls the absorption edge) and its oxidation state,plus the number and coordination and distance of nearest-neighbor atoms tothe absorbing atom. The sensitivity is in the ppm range, and its uses inmaterials science and biochemistry are many. However, EXAFS requires amodel inorganic compound of known local structure, whose EXAFS spec-trum can be used to calibrate the experiment.

Ever since the first X-ray photographs (or radiographs) obtained byR€ontgen in 1895, medical X-ray studies have used, not the element-charac-teristicKa radiation, but rather the broad-spectrum“white radiation” emittedby an X-ray tube in a transmission mode. However, since the 1970s thedifferent absorbances of X-rays by body tissues of different densities andchemical composition, as a function of the angle between the stationaryhuman body and a rotating X-ray tube and X-ray detector, were exploitedto give a “picture” of bones, soft tissues, and even cancers. The mathematicaltreatment of the data reveals tissue features with a resolution approaching afew cubic centimeters. The first instruments for this computerized axialtomography (CAT or CT) were made by Hounsfield174 and Cormack.175

Later, this technology “bred” (nuclear) magnetic resonance imaging (MRI),mentioned earlier, and positron emission tomography (PET).

Positron emission tomography (PET) exploits the difference inpositron–electron annihilation rates in the reaction:

eþ þ e� ! 2g ð11:23:21Þ

A small dose of a soluble fast-decay positron-emitting artificial radioiso-tope (produced as needed not too far from the PET instrument: 6C

11, 8O15,

9F18 or 37Rb

82) is put into human tissue (e.g., blood); the positron typicallytravels about 1mm,meets an electron fromwithin the human body, and thepair decays into two g photons of energy 0.51 MeV each, within micro-seconds to nanoseconds. Two spin states are possible for the positron–-electron ion pair before their annihilation: singlet and triplet. The annihi-lation rate for the triplet state depends sensitively on the electron density ofthe body tissue. Two g counters are set in coincidence mode, and severalhundred thousand coincidence events are used to provide valuable tissueinformation (in addition to a CT scan).

11.24 CALORIMETRY

This section reviews calorimetry [69–71]: the measurement for a “system”(¼sampleþcontainer) of (1) the “latent” enthalpy DH, (2) the internal energyDE, (3) the heat capacity at either constant pressure, CP dH/dT, or (4) theheat capacity at constant volumeCV dE/dT. All thesemeasurements requirecareful control of the initial and final states, along with reliable temperaturemeasurements for the system relative to its surroundings. Around room

174 Sir Godfrey Newbold Hounsfield (1919–2004).175Allan McLeod Cormack (1924–1998).

11.24 CALORIMETRY 757

temperature, precision in enthalpy measurements to �1% is easily attained,0.1% precision is a few hundred times more difficult, and 0.01% precisionrequires really elaborate efforts.

Assume that initially the sample is in intimate thermal contact with a“bucket” that is as small as possible and is at a uniform temperature Tb; this“bucket” is surrounded by a “jacket” at temperatureTj. There are fourways ofdoing the general calorimetric experiment:

(i) In the adiabatic vacuum calorimeter the sample is placed in a goodvacuum, so it transfers no heat to the surroundings.

(ii) In the (quasi)-adiabatic calorimeter, Tj in the water jacket is raised asfast as possible by infusion of hot water, to follow Tb during the rapid-temperature-rise regime.

(iii) In the aneroid calorimeter the system is inside a metal block of highheat conductivity (e.g., OHFC copper alloy), and its (hopefully uni-form) temperature is monitored.

(iv) In the isoperibol (isothermal jacket) calorimeter the sample at tem-perature Tb, which changes during the experiment, is inside a water“bucket” that is surrounded, across an air gap, by a very large “jacket”ofwater at constant andfixed temperatureTj, and a correction ismadefor the hopefully small heat transfer between the “bucket” and the“jacket” by using Newton’s176 law of heating/cooling:

dTb=dt ¼ KðTj � TbÞ ð11:24:1Þ

which determines the time dependence of Tb as a function of aconstant K, which is instrument-dependent but can be determinedby electrical measurements.

In the twin calorimeter, first developed by Joule177 [72], the sample isplaced in one calorimeter, while a reference compound of known thermalproperties is placed in a second calorimeter matched as closely as possible tothe sample calorimeter. This has been very useful in studying rapid reactions,or for measurements of very small heats or slow reactions.

The drop calorimeter starts a sample at a high temperature (TH¼ 300�C to1600�C) and then suddenly “drops” it into a bucket at room temperature; the(small)riseinthebuckettemperatureisrelatedtoseveralinitialvaluesofTH.Thisispracticalforhigh-temperaturemeasurementsbuthasrelativelylowprecision.

“Bomb” combustion calorimetry or constant-volume calorimetry is atechnique that dates back to Lavoisier178 (Fig. 11.76), is nowmostly relegatedto undergraduate teaching laboratories and is in bad need of a renaissance. Itmeasures the internal energy of combustion DEc, which is easily converted toDHc, and then converted to standard enthalpies of formation DHy

f,298.15. In atypical “macro” experiment, with commercially available equipment, a verycarefully measured mass m (–2.0 g) of a sample of molar massM g/mol and

176 Sir Isaac Newton (1643–1727).177 James Prescott Joule (1818–1889).178Antoine–Laurent Lavoisier (1743–1794).

758 11 INSTRUMENTS

containing the elements C, H, N, and O (CxHyNzOw) is placed inside a smallopen crucible inside a sturdy sealed steel-walled “bomb,” of internal volume0.3 L, together with a small cotton fuse and a fuse wire (Pt or Fe), andpressurizedwith a large excess of pureO2 gas (30 atm). The bomb is placed ina known volume V of water (typically 1 L, carefully weighed, but minimizedto increase sensitivity) in a “bucket.” The initial temperature of the bucketwater is carefully measured, since the water is stirred mechanically to ensureuniformity of temperature, the input of stirring energy DEstir will cause the“bucket” temperature to rise slowly and linearly with time. An initialtemperature Ti is selected (typically 1 K below 298 K), and a small amountof electrical energyDEign is used to rapidlymelt the fusewire, burn the cottonfuse, and ignite the sample: the conflagration is mostly contained in thecrucible (the “bomb” must not explode!) The temperatures (to �0.001 K orbetter) are measured by Pt resistance thermometers or by quartz crystalthermometers. The dominant chemical reactions are

xCþ xO2 ¼ xCO2 ð11:24:2Þ

yH þ ðy=2ÞO2 ¼ ðy=2ÞH2O ð11:24:3Þ

Most of the nitrogen is released as N2, but a small amount, determinedlater in an acid–base titration, is converted to HNO3. After ignition, the“bucket” temperature will at first rise rapidly, as it receives the internalenergy of combustion DEc, and then it will plateau to a final temperatureTf (typically Tf Tiþ 2�); this final temperature will also rise slowly andlinearly because of DEstir. In order to obtain the molar energy of combustionDEc,m of CxHyNzOw, the so-called calorimeter constant Q (sampleþ bombþ“bucket”)must be predetermined, either by electricalmeans (finding out howmuch electrical energy is needed to raise the temperature of the water by1.0 K) or else by burning a calorimetry standard (typically very pure benzoicacid) under the same conditions. Then:

DEc;m ¼ QðTf � TiÞðM=mÞ � DEign � DEnitr ð11:24:4Þ

FIGURE 11.76

(a) Lavoisier’s calorimeter; (b)mod-ern static bomb combustioncalorimeter

11.24 CALORIMETRY 759

where DEnitr is the small correction due to the partial conversion of N to nitricacid. The temperature must be measured accurately, preferably with a Ptresistance thermometer, or with a temperature-sensitive quartz crystal os-cillator. A typical “macro” static bomb calorimeter has Q 16 kJ/K. In acommercial “semi-micro” static-bomb calorimeter, m is reduced to 50 mg,the bomb volume to 25 mL, and Q 2000 J/K. A “micro” static-bombcalorimeter exists for 10 mg samples, with Q¼ 0.58 kJ/K [73].

If the compound contains S, Se, Te, or halogens (Cl, Br, I), the bombmust lined with Pt to prevent dissolution of the bomb walls by thechalcogen oxyacids and must also be rotated under water after combus-tion, since the oxidation products of S, Se, and Te are the oxyacids H2SO4,and so on, whose heats of solution in the water produced inside the bombare very concentration-dependent; macro rotating-bomb calorimetersexist in research laboratories with Q¼ 16.3 kJ/K [74], as does a semi-microrotating-bomb calorimeter with Q¼ 4.2 kJ/K [75]. If halogen compoundsare to be burned, care must be taken (using a catalyst) that the highestoxidation is reached in the product halogen oxyacids. If F2 gas is used asthe oxidizer instead of O2, fluorine combustion bombs with nickel wallsmust be used.

At the end of the combustion experiment, one should collect all gasesproduced in the bomb, capturing the CO2 gravimetrically in LiOH, deter-mining the sample purity (99.99% is desirable), and finally titrating the bombwater for nitric acid and chalcogen acid contents. If a visual inspection revealssoot in the bomb, the oxidation of Eq. (11.24.2) was incomplete, and the dataare rejected.

The conversion of rawexperimental data to amolar heat of formationDHyf

at a standard temperature (typically 298.15 K) for CxHyNzOw or CxHyNzOwSv,and, so on requires elaborate corrections [76] and uses DHy

f values for theexpected oxidation products, such as CO2 and H2O; Computer programs forthis data reduction exist.

Chemists always need to know bond energies, often for unusual combi-nations of elements, for which bomb combustion calorimetry experimentshave never been done, partly because the appetite of conventional bombcombustion calorimeters for large samples is not easily met for rare com-pounds. Thus there is a need for future micro rotating-bomb calorimeters.

Reaction Calorimeters. The previous discussion focused on oxidationreactions (oxygen and fluorine bomb calorimeters), but many othercalorimeters of specialized design are used to monitor chemical reactions:phase change, solution, and so on.

The Nernst calorimeter is used for low-temperature heat capacity mea-surements. The sample is contained in a small metal case equipped with aheater and thermometer and is placed in an isoperibol (isothermal) jacket oflarge heat capacity, which in turn is surrounded by an evacuated chambersurrounded by, for example, a liquidN2 or H2 chamber (Fig. 11.77). A variantis to use an adiabatic jacket. Of course, what is measured is not CP, but ahopefully reasonable approximation to it:

hCPi ¼ ðH2 �H1Þ=ðT2 � T1Þ ð11:24:5Þ

For liquids, heat capacities can bemeasuredwithmuch simpler calorimeters.

760 11 INSTRUMENTS

8

9

101112131415

1617

18

19

20

21

38

370

4

8

12

1640

30

20

10

0

incm

36353433

323130

29

28

27

26

25

242322

76

54321

414039

FIGURE 11.77

Cryostat for heat-capacitymeasurements from10K to300K, derived fromearlier instruments byNernst andGiauque179

and improved by Stout180 [77]. 1, Kovar–ceramic shields; 2, connection to safety valve; 3, connection toH2 gas holder; 4,connection to vacuumpump; 5, rubber hose outlet to room; 6, tube for optional gas inlet; 7,Monel cup; 8, union joint inblow-outtube;9, tubeforoptionalgas inlet;10, solder cupat topof innervacuumcan;11,heatstation in thermal contactwithrefrigerantbath;12, shield-supportingscrewswithsupportingstring;13,Cuwire leadsthroughtopofshield;14, topof shield;15. resistancethermometer–heater;16, calorimeter;17,bottomofshieldwithventholes;18, innervacuumcan;19, Dewar heater; 20, balsawoodDewar support; 21,Woodblock; 22, glass tube to high-vacuumpumping line; 23, waxseal;24,monelpumpingtube;25,blow-intube;26,woodtopofinsulatingcase;27,coilofblow-intubeimmersedinmonelcup;28,stainlesssteelball-veejoint;29,holes infillingtube;30,galvanized-ironsheet-metalformingoutsideof insulatingcase; 31. evacuated jacket; 32,filling tube; 33,Monel sheet-metal formingwall ofouter can; 34, brass reinforcement ringsolderedtooutercan;35,PyrexDewarvessel;36,Custudsfor thermalcontactbetweenheatstationandbath;37,pushingrods; 38, metal base of insulating case; 39, wax seal; 40, electrical leads and shield thermocouples; 41, Monel tube.

179William Francis Giauque (1895–1982).180 John Willard Stout (1912–1999).

11.24 CALORIMETRY 761

A continuous-flow calorimeter is used for measuring hCPi (J mol�1 K�1)for liquids, gases, and vapors, and even mixed gases; a flow of liquid, gas, orvapor is passed at a known constant flow rate F (mol s�1) over an electricalheater with input power W (watts); the temperature is measured just before(T1) and immediately after (T2) the sample has passed over the heater. Then atsteady state:

hCPi ¼ W=FðT2 � T1Þ ð11:24:6Þ

Adiabatic conditions are difficult to secure, so the dependence of hCPi on F ismeasured and is extrapolated to infinite F.

The Bunsen ice calorimetermeasuresDH at 273.15 K bymelting ice in anice–water mixture that is in contact with an Hg reservoir; the partial conver-sion of some ice intowater of higher density draws aweighable amount ofHginto the calorimeter.

Constant-Pressure Reaction Calorimeters. A constant-pressure calorimetermeasures the change in enthalpy DH for a chemical reaction occurring insolution under constant atmospheric pressure: a trivial example is the coffee-cup calorimeter, which is constructed from two nested polystyrene(Styrofoam�) cups having holes through which a thermometer and astirring rod can be inserted. The inner cup holds the solution in which thereaction occurs, while the outer cup provides insulation. (A fancier versionuses a Dewar181 vessel to approximate adiabatic conditions for the reaction.)Then

CP ¼ ðm=MÞðDH=DTÞ ð11:24:7Þ

wherem is the mass of the solute,M is the molar mass of the solute, DT is themeasured change of temperature, DH is the change of enthalpy, and CP is thespecific heat or heat capacity of the solute at constant pressure (assumed to beknown from other experiments).

Most calorimeters described above rely on a measurement of temper-ature (heat-Flow Calorimeters). The Tian182–Calvet183 calorimeters (somewith a twin calorimeter design) use a thermopile (instead of a thermocouple)to measure heat flow directly (Fig. 11.78).

Pulse calorimeters pass electrical current through an electrically con-ducting sample to force a temperature increase,which ismeasured alongwiththe voltage drop across the sample. If the heat loss from the sample is known(or estimated by calibration), the energy input divided by the temperatureincrease determines the true heat capacity, if the temperature change is small.Pulse calorimetry eliminatesmany of the drawbacks of drop calorimetry. It isfast, reproducible, and, with proper calibration, accurate. However, its use islimited to conductive materials.

Heat-capacity calorimeters measure the absorption of heat DH by asample and the change of temperature DT, and thus typically determine CP

from Eq. (11.24.7). Heat capacity measurements are performed in (i) drop

181 Sir James Dewar (1842–1923).182Albert Tian (1880–1972).183 Edouard Calvet (1895–1966).

762 11 INSTRUMENTS

calorimeters (at very high temperatures, up to 1600�C), (ii) Bunsen or icecalorimeters (where the mass of ice that is liquefied is used to measure DH at273.15K), (iii) Tian–Calvet solution calorimeters, (iv) (almost)-adiabatic ca-lorimeters (a reaction is allowed to run to completion), (v) pulse calorimeters,(vi) heat-flow calorimeters, and (vii) heat-balance calorimeters. Usually thesecalorimeters have a huge appetite for large samples (10–100 g).

In a heat-loss calorimeter (Fig. 11.79) the heat developed inside the cell iscollected by the inner Cu cup and then flows through the silicone-rubberwhich serves as a thermal resistor and the outer Cu cup, to the flowing watersurrounding the outer Cu cup.

High-energy particle calorimeter. In particle physics, a calorimeter is acomponent of a detector that measures the energy of entering particles thatenter the reaction chamber.

A thermogravimetric analyzer (TGA) monitors weight loss by a smallsample as a function of temperature, usually because of sample decompo-sition or because of alterations in the sample composition. This technique ismentioned here just because it is often combined with the techniques men-tioned next.

Differential thermal analysis (DTA) and differential scanning calorim-etry (DSC) are closely related thermoanalytic techniques that measure asample (S) and a reference compound (R, e.g. Al2O3) placed in identical Alor Au pans; the temperature of S and R are monitored by a thermocouple,while a temperature programmer increases the temperature linearly withtime. DTA is the older, more qualitative “fingerprint“ technique that recordsthe temperature of R (TR) and the difference in the two temperatures, TS – TR,

as a function of TR: this is called a thermogram: heat lags (endotherms) andheat leads (exotherms) of S (relative to R) indicate phase transitions in S (glasstransitions, crystallization, melting, sublimation, transitions between poly-morphs, except for any known phase transitions of R). The area under theDTA peak is a measure of the enthalpy change in the transition. Present DTAinstruments are often combined with TGA.

temperaturesens or (PT 100)

cover

vaccumpump

N2 - gas bottleN2 - exit

liquid Nitrogen

vaccumtightcalometric tankthermostat

thermo couplesin seriesreference cellsample cell

electric heatingunit

FIGURE 11.78

Commercial Tian-Calvet microcalo-rimeter [78].

11.24 CALORIMETRY 763

In contrast, DSC, designed in 1960 by Watson184 and O’Neill,185 is anewer, more quantitative technique that does measure TS and TR, but alsomeasures very precisely the electrical energy used by separate heaters undereither pan to make TS¼TR (this is power-compensated DSC, useable below650�C). Thepower input into Sminus the power input intoR is plotted againstTR.High-temperatureDSC (useful for TR> 1000�C)measures the heat fluxesby Tian–Calvet thermopiles rather than the electrical power, as a function ofTR. In a heat-flux DSC, both pans sit on a small slab of material with acalibrated heat resistance. The temperature of the calorimeter is raisedlinearly with time. A schematic DSC curve is shown in Fig. 11.80.

11.25 X-RAY PHOTOELECTRON SPECTROSCOPY (XPS)AND AUGER ELECTRON SPECTROSCOPY

X-ray photoelectron spectroscopy (XPS), with a defunct “propaganda” nameof electron spectroscopy for chemical analysis (ESCA), was developed bySiegbahn186 in 1954; it measures the elemental composition and valence stateof elements in solids (atomic number Z¼ 3 to Z¼ 92) to within about 5 to10 nmof their surface by impingingX-rays, typicallymonochromatizedAlKa(EX¼ 1.4867 keV and lX¼ 0.83386 nm) in a beam of 0.02- to 0.2-mm diameter,onto a sample surface in ultra-high vacuum andmeasures to within�0.25 eV

FIGURE 11.79

Cross-section of a heat-loss calori-meter [79].

184 Emmett S. Watson (1943– ).185Michael J. O’Neill (ca. 1930– ).186Kai Siegbahn (1918–2007).

764 11 INSTRUMENTS

the kinetic energy EKE EXPS of the emitted photoelectron from the “core”electron states. The basic equation, in two flavors, is

EBE ¼ EX � EKE �f ð11:25:1Þ

EXPS EKE ¼ EX � EBE � f ð11:25:2Þ

where f is the work function of the instrument (slightly larger than the workfunction of the sample). Equation (11.25.1) then yields the binding energyEBE, which is tabulated for chemical elements and model compounds inTable 11.21. The energy diagram and photoelectron emission mechanism areshown in Fig. 11.81A, which also shows a related technique, ultravioletphotoelectron spectroscopy (UPS, Fig. 11.81B), which involves valenceelectron states instead of core states. The two ways, or “channels,” in whichatoms excited by X rays (Fig. 11.81A) or by a high-energy electron beam canrelax following the ejection of a core photoelectron are emission of an X-rayphoton (X-ray fluorescence (XRF), Fig. 11.81C) and Auger187 electron emis-sion (Auger electron spectroscopy (AES), Fig. 11.81D). The Auger effect wasdiscovered independently by Auger and Meitner188 in the 1920s.

As the XPS photoelectron is emitted from a “core” state (e.g., 1s or K-level), a vacancy (“hole”) is created in that core level, and an electron from anupper level (say 2s or L1 level) “descends to fill the hole”; at this point, theatom has twoways to respond: One “channel” is emission of an X-ray photoncorresponding to the energy difference between the L1 and K levels; this isX-ray fluorescence (XRF: Fig. 11.81C):

EXRF ¼ EK � EL1 ð11:25:3Þ

Features of a DSC curve

Crystalisation

Glass transition

Melting

Temperature (°C)

Hea

tflow

(m

W)

FIGURE 11.80

Schematic DSC thermogram, show-ing a glass transition, a glass-to-crystal transition (exotherm), anda melting transition (endotherm)[80].

187 Pierre Victor Auger (1899–1993).188 Lise Meitner (1878–1968).

11.25 X-RAY PHOTOELECTRON SPECTROSCOPY (XPS) AND AUGER ELECTRON SPECTROSCOPY 765

Table 11.21 XPS Data for Pure Elements and Selected Chemical Compoundsa

Grp 1

/1A/

Grp 2

/2A/

Grp 3

/3B/

Grp 4

/4B/

Grp 5

/5B/

Grp 6

/6B/

Grp 7

/7B/

Grp 8

/B/

Grp 9

/8B/

Grp 10

/8B/

Grp 11

/1B/

1 H 1s

H2° LiH

3 Li 1s

LiOH Li2O

54.9 55.6

±1.65 ±1.6

285.0 285.0

531.8 530.8

±1.6 ±1.5

4 Be 1s

Be° BeO

111.8 113.8

±0.79 ±1.73

286.2 285.0

111.9 531.2

±0.69 ±1.71

11 Na 1s

Na° Na2O

1071.8 1072.6

±1.1 ±1.78

285.0

530.6

±1.44

12 Mg 2p

Mg° MgO

49.7 50.8

±0.58 ±1.25

286.5 285.0

49.77 529.9

±0.60 ±1.38

19 K 2p3

K° K2O

294.4 292.7

±0.9 ±1.37

285.0

531.0

±1.6

20 Ca 2p3

Ca° CaCO3

346.0

21 Sc 2p3

Sc° Sc2O

3

398.6 401.9

±0.9 ±1.27

285.8 285.0

398.46 530.0

±0.69 ±1.33

22 Ti 2p3

Ti° TiO2

453.8 458.7

±0.79 ±1.09

285.2 285.0

453.95 530.0

±0.62 ±1.18

23 V 2p3

V° V2O

5

512.2 517.3

±0.79 ±1.32

285.0 285.0

453.95 530.0

±0.75 ±1.33

24 Cr 2p3

Cr° Cr2O3

574.2 575.7

±1.05 ±1.20

284.6 285.0

574.37 530.1

±0.89 ±1.24

25 Mn 2p3

Mn° MnO2

638.7 641.5

±1.00 ±1.12

296.4 285.0

638.74 529.5

±0.89 ±1.02

26 Fe 2p3

Fe°αFe2O

3

706.6 709.8

±0.90 ±1.32

284.9 285.0

706.78 532.9

±0.89 ±1.02

27 Co 2p3

Co° Co3O

4

778.1 779.5

±0.99 ±1.39

284.9 285.0

778.26 530.1

±0.85 ±1.00

28 Ni 2p3

Ni° NiO

852.6 853.8

±1.14 ±1.42

284.8 285.0

852.65 529.4

±1.02 ±1.03

29 Cu 2p3

Cu° Cu2O

932.7 932.5

±1.22 ±1.10

284.6 285.0

932.68 530.5

±0.92 ±1.01

37 Rb 3d5

Rb° RbOAc

38 Sr 3d5

Sr° SrCO3

39 Y 3d5 Y° 40 Zr 3d5

Zr° ZrO2

41 Nb 3d5

Nb° Nb2O5

42 Mo 3d5

Mo° MoO3

43 Tc 3d5

Tc°________

44 Ru 3d5

Ru° RuO2

45 Rh 3d5

Rh° Rh2O3

46 Pd 3d5

Pd° PdO

47 Ag 3d5

Ag° Ag2O

111.5 109.7

±1.40

285.0

530.9

±1.6

134.3 133.7

±1.67

285.0

531.5

±1.9

Y2O3 155.9

156.6 ±0.80 ±

1.25 286.0

285.0 155.92

531.0 ±0.82

±1.30

179.0 182.4

±0.90 ±1.18

285.3 285.0

178.80 530.3

±0.63 ±1.39

202.1 207.4

±0.78 ±1.14

285.0 285.0

202.35 530.4

±0.57 ±1.35

227.8 233.1

±0.66 ±1.05

285.4 285.0

227.94 531.0

±0.57 ±1.20

Radioactive

280.0 281.1

±0.67 ±0.79

285.0

280.11 529.7

±0.59 ±0.95

307.2 338.9

±0.73 ±0.80

284.5 285.0

307.21 530.5

±0.69 ±1.05

335.1 337.0

±0.86 ±0.97

284.5 285.0

368.28 529.4

±0.62 ±0.97

368.2 367.5

±0.64 ±1.00

284.7 285.0

368.28 529.4

±0.62 ±0.97

55 Cs 3d5

Cs° CsCl

726.4 724.6

±2.08

285.0

199.2

56 Ba 3d5

Ba° BaOAc

780.6 780.0

±1.80

285.0

531.4

±1.93

57 La* 3d5

La° La2O3

835.8 834.7

±3.0 285.0

529.2 ±1.6

72 Hf 4f7

Hf° HfO2

14.4 17.1

±0.63 ±1.26

285.7 285.0

14.32 530.5

±0.62 ±1.68

73 Ta 4f7

Ta° Ta2O5

21.8 26.8

±0.80 ±1.12

285.0 285.0

21.78 531.0

±0.56 ±1.46

74 W 4f7

W° WO3

31.4 35.8

±0.58 ±1.01

285.3 285.0

31.38 530.6

±0.63 ±1.27

75 Re 4f7

Re° Re2O7

40.3 46.8

±0.67 ±1.64

285.3 285.0

40.30 532.1

±0.54 ±1.58

76 Os 4f7

Os° OsO2

50.7 52.7

77 Ir 4f7

Ir° IrO2

60.8 62.0

±0.80 ±0.98

284.4 285.0

60.88 530.2

±0.82 ±0.97

78 Pt 4f7

Pt° PtO2

71.0 75.1

±0.96 ±1.16

284.3 285.0

71.15 531.3

±0.88 ±1.74

79 Au 4f7

Au° Au2O3

84.1 88.1

±0.83 ±1.12

284.1 285.0

83.98 531.6

±0.68 ±1.13

87 Fr 4f7

Radioactive

88 Ra 4f7

Radioactive

89 Ac‡ 4f7

Radioactive

104 Rf

Radioactive

105 Db

Radioactive

106 Sg

Radioactive

107 Bh

Radioactive

108 Hs

Radioactive

109 Mt

Radioactive

110 Ds

Radioactive

111 Rg

Radioactive

*58 Ce 4d5

Ce° CeO2

883.8 882.1

±2.3 ±2.0

59 Pr 4d5

Pr° Pr2O5

929.4 928.8

±2.9 ±3.2

60 Nd 4d5

Nd° Nd2O3

980.8 983.2

±2.9 ±2.7

61 Pm 4d5

Pm° Pm2O3

Radioactive

62 Sm 4d5

Sm° Sm2O3

1081.1 1083.8

±4.3

63 Eu 4d5

Eu° Eu2O3

128.2 136.2

±3.6

64 Gd 4d5

Gd° Gd2O3

1187. 1189. 0

±5.4

65 Tb 4d5

Tb° Tb2O3

146.0 1241.5

766 11 INSTRUMENTS

Table 11.21 (Continued )

285.0

529.8

±2.00

285.0

528.5

±1.2

285.6 285.0

530.8

±1.3

285.0

128.7 529.9

±2.5 ±1.6

284.3 285.0

128.18 530.5

±1.08 ±1.4

281.4 285.0

140.31 529.3

±1.05 ±1.7

285.0

145.95 529.7

±1.16 ±1.5

‡90 Th 4d5

Th° ThO2

675.3 334.5

±1.6

91 Pa 4d5

Radioactive

92 U 4f7

U° UO2

377.3 380.0

±1.4

93 Np

Radioactive

94 Pu

Radioactive

95 Am

Radioactive

96 Cm

Radioactive

97 Bk

Radioactive

Grp 12

/2B/

Grp 13

/3A/

Grp 14

/4A/

Grp 15

/5A/

Grp16

/6A/

Grp 17

/7A/

Grp 18

/8A/

2 He 1s

He+/Be He°/C

B 1s

B° B2O3

187.5 197.7

±0.92 ±1.30

285.2 285.0

187.8 533.7

±1.03 ±1.6

6 C 1s

HOPG Black

284.5 284.4

±0.42 ±1.06

7 N 1s

Kapton BN

400.9 398.9

±1.31 ±1.10

285.0 285.0

191.3

±1.03

8 O 1s

CuO SiO2

529.6 532.5

±0.98 ±1.40

284.9 285.0

934.0 103.0

±1.42 ±1.14

9 F 1s

Teflon CaF2

689.2 685.1

±1.8 ±1.53

285.0 285.0

292.0 348.1

±1.6 ±1.6

10 Ne 1s

Ne°/Be Ne°/C

13 Al 2p3

Al° Al2O3

72.9 74.4

±0.62 ±1.39

284.9 285.0

72.82 530.8

±0.41 ±1.60

14 Si 2p3

Si° SiO2

99.8 103.1

±0.57 ±1.19

258.8 285.0

99.35 532.4

±0.45 ±1.27

15 P 2p3

P° InP

130.1 128.8

±0.67 ±0.62

285.0 285.5

444.7

±0.78

16 S 2p3

S° MoS2

164.0 162.7

±0.72 ±0.98

285.0

229.7

±0.80

17 Cl 2p3

PVC NaCl

199.8 199.3

±1.7 ±1.19

285.0 285.0

1072.0

±1.40

18 Ar 2p3

Ar+/B Ar+/HOPG

241.8 241.8

±1.4 ±0.89

285.2 284.5

188.1

30 Zn 2p3

Zn° ZnO

1021.8 1021.7

±1.10 ±1.50

284.8 285.0

1021.76 530.5

±0.97 ±1.11

31 Ga 3d5

Ga° Ga2O3

18.7 20.7

±0.70 ±1.37

285.0 285.0

18.5 531.3

±0.60 ±1.51

32 Ge 3d5

Ge° GeO2

29.3 33.2

±0.68 ±1.49

285.0 285.0

29.28 532.2

±0.64 ±1.40

33 As 3d5

As° As2O3

41.8

45.11

±0.67 ±1.26

284.5 285.0

41.69 532.0

±0.67 ±1.41

34 Se 3d5

Se° SeOx

54.6 59.3

±0.76 ±1.09

284.2 285.0

54.90 532.6

±0.76 ***

35 Br 3d5

KBr

68.8

±0.92

285.0

293.2

±1.31

36 Kr 3d5

Kr+/Be Kr+/C

*** 86.9

*** ±0.88

*** 285.0

***

***

(continued)

11.25 X-RAY PHOTOELECTRON SPECTROSCOPY (XPS) AND AUGER ELECTRON SPECTROSCOPY 767

Table 11.21 (Continued )

48 Cd 3d5

Cd° CdO

405.0 404.0

±0.90 ±1.38

285.0 285.0

405.04 526.6

±0.61 ±1.28

49 In 3d5

In° In2O3

443.8 444.3

±1.08 ±1.26

284.9 285.0

443.87 529.9

±0.71 ±1.19

50 Sn 3d5

Sn° SnO2

484.9 487.3

±0.81 ±1.28

284.7 285.0

485.01 531.1

±0.68 ±1.29

51 Sb 3d5

Sb° Sb2O5

528.2 530.4

±1.0 ±1.10

284.7 285.0

485.01 531.1

±0.80 ±0.86

52 Te 3d5

Te° TeO2

572.8 576.5

±1.12 ±1.27

284.2 285.0

572.97 530.7

±0.83 ±0.86

53 I 3d5

KI

619.2

±1.30

285.0

293.2

±1.11

54 Xe 3d5

Xe+/Be Xe°/C

669.6

±1.13

285.0

80 Hg 4f7

Hg° HgO

99.8 100.7

*** ±1.06

285.0 285.0

99.81 532.9

±0.65 ±0.96

81 Tl 4f7

Tl° Tl2O3

117.8 118.2

±0.97 ±1.01

285.1 285.0

117.77 528.8

±0.66 ±1.10

82 Pb 4f7

Pb° PbO

136.9 137.5

±0.67 ±1.10

284.9 285.0

136.95 528.9

±0.63 ±1.07

83 Bi 4f7

Bi° Bi2O3

157.0 158.8

±0.73 ±1.11

284.6 285.0

157.05 529.6

±0.62 ±1.58

84 Po 4f7

Radioactive

85 At 4f7

Radioactive

86 Rn 4f7

Radioactive

112 Cn 4f7

Radioactive

113 Uut 4f7

Radioactive

114 Uuq 4f7

Radioactive

115 Uup 4f7

Radioactive

116 Uuh 4f7

Radioactive

117 Uus 4f7

Radioactive

118 Uuo 4f7

Radioactive

66 Dy 4d5

Dy° Dy2O3

152.4 153.5

*** ±1.9

285.0

152.34 529.4

±0.90 ±1.7

67 Ho 4d5

Ho° Ho2O3

159.8 160.9

*** ±2.0

285.0

159.58 529.3

±0.96 ±1.6

68 Er 4d5

Er° Er2O3

167.7 168.5

*** ±1.9

285.0

167.25 529.7

±1.00 ±1.5

69 Tm 4d5

Tm° Tm2O3

175.3 178.3

*** ±1.1

285.0

175.37 529.6

±1.08 ±1.6

70 Yb 4f7

Yb° Yb2O3

182.2 185.0

*** ±2.7

285.6 285.0

182.39 529.2

±0.61 ±1.5

71 Lu 4f7

Lu° Lu2O3

7.1 8.0

±0.46 ±2.5

284.6 285.0

7.10 529.5

±0.69 ±1.4

98 Cf

Radioactive

99 Es

Radioactive

100 Fm

Radioactive

101 Md

Radioactive

102 No

Radioactive

103 Lr

Radioactive

a For each element the electron configuration shown is for the main XPS peak. Nonconductive materials were referenced to adventitioushydrocarbon with C 1s BE (binding energy) at 285.0 eV. Energy resolution settings for pure oxide data gave FWHM <0.75 eV for Ag 3d5

of ion-etched Ag0. All nonconductors were analyzed with a flood-gun mesh screen 0.5 to 1.0 mm above the specimen. C 1s BEs for“hydrocarbons” were collected from the hydrocarbon peak that formed on ion-etched elements. Carbon was from the cryo-pumpedvacuum (3 � 10�9 torr) for Ag 3d5 of ion-etched Ag0. Calbration was Cu 2p3 at 932.67 � 0.05 eV, Cu 3s at 122.45 � 0.05 eV, and Au 4f7 at8398 eV. The FWHM and BE values in this table were all obtained by one scientist using two SSI XPS systems with a theoretical resolutionof �0.1 eV. Each system was equipped with a monochromatic Al X-ray source, which has a theoretical energy resolution limit of about0.16 eV. The first BE listed for each element is the one for the element “under its native oxide” if present. The second BE for the pureelement is for the element sample after ion-etching; this BE for the ion-etched element is a reliable secondary energy reference value, witha standard deviation of �0.055, as measured by NIST; the BEs for the chemical compounds should be accurate to �0.15 eV. The BEs ofnonconductive materials are referenced to the hydrocarbon C 1s BE, defined to be at 285.0 eV, to match current-day methods of chargereferencing, but this method of charge referencing is not absolute. The C 1s BE of the adventitious hydrocarbon components on variousnaturally formed, thin native oxides (metal signal visible) was measured to occur between 285.5 eV and 286.5 eV. The native oxides of Ag,Al, Be, Bi, Cd, Co, Ga, Ge, Hf, In, Lu, Mg, Ni, Pb, Pd, Sc, Si, Ta, Y, Zn, and Zr give C1s BEs that are 0.5–1.5 eV above 285.0 eV; surface dipolemoments are suspected to be the cause of this shift. The BEs of native oxides for these metals are more correct. In each element entry, thevalence state for the principal XPS signal is given; the data are in electronvolts; the datamarkedwith “�” are the FWHM in eV. The entriesaround 285.0 eV are for the adventitious hydrocarbon C 1s signal; the entries around 533 eV are for oxygen in the oxide; OAc stands foracetate [83].

768 11 INSTRUMENTS

The other “channel” is ejection of anAuger electron from a neighboring level:After a new “hole” is produced in the L1 level, an electron is emitted from anearby level (say the 2p or L2 level) with energy EAES (AES, Fig. 11.81D):

EAES ¼ EK � EL1 � EL2 ð11:25:4Þ

TheXPS photoelectron energy depends on the kinetic energy of themeasuredphotoelectron, but theAuger photoelectron does not. The twoprocesses (XRFand AES) compete with each other; the X-ray fluorescence (i.e., radiative)yield dominates for elements of high atomic number Z, while the Augerelectron (nonradiative) yield dominates at low Z (3� Z� 50) (Fig. 11.82); thehigh-Z elements, whose core and near-core electrons have higher speeds, areless likely to absorb the X-ray photon and start the Auger process. X-rays dopenetrate quite deeply into a sample, but photoelectrons or input electronbeams cannot penetrate or escape a sample, except from a layer that is only 5to 10 nm deep, because they get scattered within the sample (the mean freepath of electrons is short). In XRF the emittedX rays are energy-analyzed. XPScan also be used with ancillary ion-etching (e.g., using a beam of Arþ ionsfocused to a beamdiameter of�30 nm; this focused ionbeam (FIB)) cleans offthe surface layers or contamination. Depth profiling by ion-beam etchingremoves surface layers to probe “inside” the solid (but not too far); lineprofiling allows probing uniformity across a sample surface. Elementalanalysis by XPS in parts per thousand are routine; parts per million arepossible. Specialized instruments can also study samples at low temperaturesor even in the gaseous state.

1s K 1s K

1s K

2s L1 2s L1

2s L1

2p L22p L2

2p L2

3s M1 3s M1

3s M1

Valence Band

Valence Band

Valence Band

Vacuum level (KE=0 )

(A): XPS (B): UPS

(D): Auger

(1) X-ray in (2) e- out (1) X-ray or UV in (2) e- out

"Free electron level" (with kinetic energy>0) EKE

1s K

2s L12p L2

3s M1

Vacuum level (KE=0)

(C): XRF

(3) e- down

(4) X-ray out

(3) e- down

(4) e- out

Valence Band

EBE

"Free electron level" (with kinetic energy>0)(1) X-ray in

(1') Electron in(2) Electron out

(1) X-ray in

(1') Electron in

FIGURE 11.81

(A) XPS, (B) UPS, (C) XRF, and (D)Auger processes.

11.25 X-RAY PHOTOELECTRON SPECTROSCOPY (XPS) AND AUGER ELECTRON SPECTROSCOPY 769

Organic chemicals are rarely analyzed by XPS, because they are readilydegraded by either the energy of the X rays or the heat from nonmonochro-matic X-ray sources; however, for the zwitterionic organic Z-typeLangmuir–-Blodgett monolayer or multilayer film of g-C16H33Q-3CNQ (Fig. 11.31): XPSspectra are shown in Figs. 11.83 and 11.84. Older UPS instruments used anultraviolet lamp in place of an X-ray source, butmodern XPS instruments canalso perform a UPS-type valence-state scan (Fig. 11.85).

One limitation for XPS is that the ejection of photoelectrons is simple formetallic or electrically conducting samples, but more difficult for insulators:The sample acquires and retains a formal charge (surface charging); thisaffects the ejection efficiency of photoelectrons from the sample surface.The compounds labeled in Table 11.20 are all referenced to a C 1s XPS peakat 285.0 V; this is because, even in ultra-high vacua, a clean metallic orinsulating surface acquires a physisorbed monolayer of adventitious hydro-carbons from pump oil or residual atmospheric gases, which can be drivenoff by ion-beam etching. The efficiency of elemental analysis by XPS is onlyabout 1%. The difference between the XPS line in a compound minus thevalue for the corresponding pure element is an indirect measure of chemical

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00 5 10 15 20 25 30 35 40 45

Auger electron yield

X-ray yield

Yie

ld p

er s

hell

vale

ncy

Atomic Number

FIGURE 11.82

X-rayfluorescence (XRF) andAugerelectron (AES) yields as functions ofthe atomic number for K-shell va-cancies. Auger transitions (solidcurve) are more probable for ligh-ter elements, while the X-ray yield(dashed curve) becomes dominantat higher atomic numbers. Similarplots can be obtained for L and Mshell transitions. Intra-shell transi-tions are ignored in thisanalysis [81].

20000

15000

10000

5000

01000 800 600

N 1s, 400 eV

C 1s, 286 eV

400 200

Binding energy (eV)

Inte

nsity

(C

ount

s pe

r se

cond

)

0

FIGURE 11.83

Wide-scan XPS survey spectrum forall elements in Langmuir–Blodgettmultilayer of g-C16H33Q-3CNQ onSi: N and C are easily detected [14].The chemical structure is given inFig. 11.31.

770 11 INSTRUMENTS

bonding, like a chemical shift in nuclearmagnetic resonance, but the XPS shiftis less sensitive andcanbeusedonly to estimatewhether the species studied isan anion or cation. Typical XPS instrumentation is shown in Fig. 11.86; typicalAES instrumentation is shown in Fig. 11.87, and a sample Auger spectrum isgiven in Fig. 11.88.

Fast-ion bombardment (FIB) of a metal (or a very thin ordered surfaceadsorbate, e.g. Langmuir–Blodgett monolayer on metal) by a narrowlyfocused beam of Arþ ions can be used to penerate the metal, and drill holesas small as 50-mmdiameter andmany nanometers deep. Typical ion energiesare 5–50 keV, and the narrowest beam diameter is about 3 nm. The ionsimpinging on the substrate are charged, so, just as in electron microscopy, amechanism is needed to disperse the extra charge (i.e., the substrate must belargely a metal, or a metal coating) or else the buildup of static charge willseverely defocus the ion beam.

FIB can also be used to then follow the XPS spectrum, to obtain a “depthprofile”: The disappearance of some element-characteristic XPS or Auger

Count

378000

376000

A 401.60 eV 2.00 eV 170.635 ctsB 399.45 eV 2.00 eV 522.447 cts

374000

372000

370000

368000

366000

364000

362000

360000

358000

356000

354000

408 404 400

Binding Energy, eV

396 392

FIGURE. 11.84

N 1s core-level XPS spectrum forLangmuir–Blodgett monolayer ofg-C16H33Q-3CNQ on Au: the tworesolved peaks are at 401.6 eV (Nþ)and at 399.45 eV (N0) [82].The chemical structure is given inFig. 11.31.

-50

0

50

100

150

200

250

300

350

010203040

Inte

nsity

(co

unts

)

Binding energy (eV)

25.520.0

17.613.6

7.63.7

FIGURE 11.85

Valence-level (UPS-like) spectrumfor Langmuir–Blodgett multilayerof g-C16H33Q-3CNQ on Si, withGaussian line fits to the energy le-vels, adjusted to the vacuum levelby assuming Si f¼ 4.1 eV [14].The chemical structure is given inFig. 11.31.

11.25 X-RAY PHOTOELECTRON SPECTROSCOPY (XPS) AND AUGER ELECTRON SPECTROSCOPY 771

signal as a function of FIB penetration yields an estimate of how far down inthe absorbate that element resides.

11.26 M€OSSBAUER SPECTROSCOPY

M€ossbauer spectroscopy was born in the 1957 Ph.D. dissertation ofM€ossbauer,189 who measured and explained the resonant emission andabsorption of a g ray from the radioactive decay of 77Ir

191 [87,88].Manynuclides exhibit this effect, but themost used is the 14.413 keV g-ray

found in the radioactive decay of 26Fe57 from 27Co

57 (Fig. 11.89).TheM€ossbauer source is a thin Rh (or Pd or Pt) foil with small amounts of

the substitutionally introduced radioactive isotope 27Co57, which has a half-

life of 271.79 days and decays into a short-lived 27Fe57 isotope, which in turn

decays very quickly (half-life t1/2¼ 98.3 ns, natural “Heisenberg” linewidthh/t1/2¼ 42.2 neV), emitting a photon of energy EM¼ 14.413 keV. TheM€ossbauer source is mounted on a translation stage whose longitudinallinearmotion can be varied [for 26Fe

57 the speed is 0.194(2)mm/s at resonanceand 1mm/s¼ 48.075 neV]. If the emitting excited 26Fe

57 atomwere in the gasphase, its energy would be decreased by a recoil energy ER:

ER ¼ pg2=2m ¼ EM

2=2c2m ð11:26:1Þ

HemisphericalAnalyzer.

Detector Head

HT ElectronicsIon Gun

Stage

Host Computer

Sample Stub

Slits

Aperture

XrayGun

FIGURE 11.86

XPS instrumentation [84].

189 Rudolf Ludwig M€ossbauer (1929–2011).

772 11 INSTRUMENTS

wherem is the mass and c is the speed of light, preventing its reabsorption bya stationary 26Fe

57 atom. In the foil, the recoiling 26Fe57 atoms interact with

the latticevibrationsof the foil (phononsof frequencynwithquantizedenergiesnhn at the Debye temperature YD); some “recoil-free” fraction f of this14.413–keV photon flux interacts with n¼ 0, and thus can be resonantlyabsorbedbythestationarysample containingFe (which includes2.14%natural

100 200 300

Cu

Cu

Nkll

400 500 600 700 800 900 1000

Kinetic Energy (eV)

D{E

*N(E

)}/D

E (

Arb

. Uni

ts)

FIGURE 11.88

Auger electron spectrum (in deriv-ative mode) of a Cu3N2 film [86].

E = 137 keV

E = 14.413 keV

7/2-

3/2-

1/2-

5/2-

2757Co (half-life 271.79 d)

Electron capture26

57Fe

91%9%

E = 0 keV

Mössbauergammaradiation

FIGURE 11.89

14.4413-keV M€ossbauer line in

26Fe57.

Target

Optional lonSource

Shields

Electron Gun ElectronDetector

DataAcquisition

SweepSupply

FIGURE 11.87

AES setup using a cylindrical mirroranalyzer (CMA). An electron beamis focused onto a specimen, andemitted electrons are deflectedaround the electron gun and passthrough an aperture toward theback of the CMA. These electronsare then directed into an electronmultiplier for analysis. Varyingvoltage at the sweep supply allowsderivative-mode plotting of theAuger data [85].

11.26 M €OSSBAUER SPECTROSCOPY 773

abundance 26Fe57). With the linear motion, the full M€ossbauer spectrum is

scanned, with its chemical shifts and quadrupole and Zeeman splittings.A detector situated at the far side senses the decreased signal level when

some of the g photons are absorbed by the sample (Fig. 11.90).M€ossbauer experiments are also possible in reflection mode.The fraction of the photons that is recoil-free is given by the Lamb–M€oss-

bauer factor f:

f ¼ jhLijexpðik:XjLiij2 ð11:26:2Þ

where |Li> is the initial state of the lattice,k is thewavevector of the emitted gray, andX is the coordinate vector of the center ofmomentum of the decayingnucleus. A necessary condition for M€ossbauer resonance is roughly

ER2=2c2m < kBYD ð11:26:3Þ

where kB is Boltzmann’s constant. This condition severely limits candidatenuclei to those with relatively low-energy gamma energies. The chemicalelements which have M€ossbauer isotopes are given in Fig. 11.91.

PROBLEM 11.26.1 Compute the Doppler recoil energy ER for 26Fe57 and a

linear speed vr needed to counteract the Doppler shift. Calculate also thespeed of a free recoiling nucleus vr, and comment on why this calculatedspeed is too big.

PROBLEM11.26.2 IfEq. (11.26.1)gives for 26Fe57 a linear speedvr¼ 81.21m/s,

whereas the experimental linear speed for detecting theM€ossbauer resonanceis only 0.194 mm/s, what is the “effective mass”?

Using the Debye lattice model, Eq. (11.26.3) becomes

f ¼ expð � 2wÞ ð11:26:4Þ

w ¼ ð3ER=kBYDÞ½ð1=4Þ þ ðT=YDÞ2ðx¼YD=Tx

x¼0dx=ðexp ðxÞ � 1Þ� ð11:26:5Þ

This formula has great similarities with the Debye–Waller factor w:

w ¼ 2huz2ið4p2sin2y=l2Þ ð11:26:6Þ

Co:Rh source Fe foil sample detector

FIGURE 11.90

M€ossbauer experiment for Fe.

774 11 INSTRUMENTS

for mean-square atom harmonic displacements huz2i in a crystal undergoingalmost elastic X-ray diffraction of wavelength l at Bragg angle y. The X-rayscattering event is much faster (�10�15 s) than the M€ossbauer event, whichinvolves lattice times (�10�13 s); but if we assume a harmonic expression forthe lattice reaction to the M€ossbauer gamma emission, then f of Eq. (11.26.5)simplifies to

f ¼ expð � 4 p2hLijXk2jLii=lg2Þ ð11:26:7Þ

where Xk is the component of the coordinate vector X in the direction of theemitted g-ray photon of wavelength lg. A classical treatment of the problemyields a very similar result:

f ¼ expð � 4p2hx2i=lg2Þ ð11:26:8Þ

and suggests that an observable “recoil-free” M€ossbauer event requires amean-square displacement hx2i1/2 much smaller than lg. Furthermore, thehalf-life of the excited-state t1/2 should also be between 10�6 and 10�9 s.

Other popular M€ossbauer nuclides are 53I129, 50Sn

119, and 51Sb121

(Table 11.22). The M€ossbauer linewidth is very narrow (energy resolutionof 1 part in 1011), enabling very sensitive tests of several disparate physicalphenomena. M€ossbauer spectra are very sensitive to electron–nucleusinteractions: They exhibit (i) a chemical shift, also called an isomer shift,proportional to the ground-state electron probability density at the nucleus,|c1s(0)|

2, and similar to the chemical shift seen in nuclear magnetic reso-nance; (ii) a quadrupole splitting, seen as a doubling of the M€ossbauer peak,due to the interaction of the nuclear electric quadrupole moment Q with thesurrounding electric field gradient at the nucleus; and (iii) a hyperfine or

H1

0

IIIB IVB VB VIB VIIB VII

IIIA IVA VA VIA VIIA

IB IB

2

3

4

5

6

7

IA

IIA

1

1

1 1 2

2

3

3

2 2

2

7 1 16

6

6 6

6 6

4 2 211

11

1

1 1 1

11

1 1

111

1

1 1 1

1

1 1 1

1 1 1

1 1

1 1

1 1 1 1 15

5

5

1 1 1 1

1

14 4 1 1 12

2

2

2

2 2 2

2

2 2 2 2

22

24

4

4

9

4

Li

Na

K

Rb

Ra

Be

Mg

NUMBER OF ISOTOPES IN WHICHTHE MOSSBAUER EFFECTHAS BEEN OBSERVED

NUMBER OF OBSERVEDMOSSBAUER TRANSITIONS

Ca Sc Ti

Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te XeI

Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn

V Cr Mn Fe

Fe

Co Cu Ga Ge As Se Br Kr

B

Al Si P S Cl Ar

C N O F Ne

He

Ni Zn

Sr Y

LaBa

AcCe

Th Pa U Np Pu Am

Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu

Cs

Fr

FIGURE 11.91

M€ossbauer-active nuclei [89].

11.26 M €OSSBAUER SPECTROSCOPY 775

Table 11.22 Popular M€ossbauer Nuclides [89]

Stable Abundance Eg ER vr kBYD t1/2 HostNuclide (atom %) (keV) (meV) (mm/s) (eV) (ns) Lattice

19K40 0.0117 29.83 11.9 2.184 0.0077 4.25 KF

26Fe57 2.14 14.4129 1.959 0.194 0.040 98.3 Cr, Cu, Rh, Pd,Pt

36Kr83 11.5 9.405 0.573 0.1980 0.0061 154.4 Kr(s)

47Ag107 51.84 93.13 43.5 –– –– 44.3E9

50Sn119 8.58 23.879 2.574 0.64 0.017 18.03 CaSnO3

51Sb121 57.3 37.133 6.1 2.1 0.0155 3.46 InSb

52Te123 0.908 159.0 110 –– 0.013 0.196 ZnTe

53I129 t1/2¼ 1.57 � 107 y 27.80 3.218 0.58 0.0091 16.8 KI

54Xe129 26.4 39.578 6.523 6.843 0.0054 7.1 Na4XeO6

55Cs133 100 80.997 26.5 0.5361 0.0032 6.28 CsCl

62Sm149 13.8 22.51 1.83 1.708 0.012 7.33 SmB6

63Eu151 47.8 21.54 1.65 1.303 0.012 9.6 Cs2NaEuCl6

64Gd155 14.8 60.01 12.5 29.41 0.017 0.2 GdCo266Dy160 2.34 86.788 25.3 1.592 0.018 2.02 Dy0.4Sc0.6H2

66Dy161 18.9 25.65 2.19 0.3795 0.018 29.1 DyF368Er

166 33.6 80.57 21.0 1.866 0.019 1.9 ErH2

69Tm169 100 8.41 0.225 8.330 0.020 4.04 TmAl2

70Yb170 3.05 84.25 22.4 2.029 0.010 1.61 YbAl3

72Hf177 18.6 113.0 38.7 4.843 0.021 0.54 HfZn274W

182 26.4 100.1 29.6 1.995 0.034 0.14 W

74W183 14.3 46.48 6.34 32.16 0.034 0.19 W

75Re187 62.6 134.2 51.7 203.8 0.037 0.010 Re

77Ir191 37.3 82.42 19.1 0.8258 0.037 4.1 Ir

77Ir193 62.7 73.0 14.8 0.5946 0.037 6.1 Ir

78Pt195 33.8 98.88 26.9 16.28 0.020 0.17 Pt

79Au197 100 77.34 16.3 1.861 0.014 1.91 Au

–4

100

92

0

Velocity (mm s–1)

Rel

ativ

e Tr

ansm

issi

on

4

γ-austenite

α-FeFe IFe IIFe III

FIGURE 11.92

Transmission M€ossbauer spectrumand Gaussian waveform line-fitsfor a sample containing 15 at%Mn, reducedwithNH3:H2, and tem-pered at 200�C for 150 min, yield-ing a solidmixture of g-N-austenite(31 at%), a-Fe (32.7 at%), andmetastable a00-Fe16N2 (with itsthree crystallographic lattice sitesFe I (14.2 at%), Fe II (8.4 at%), andFe III (13.7 at%)) [90].

776 11 INSTRUMENTS

Zeeman splitting, due to the interaction of the nucleus with the surroundingmagnetic field (for 26Fe

57, this yields six lines).TheM€ossbauer spectrum thus yields important information about lattice

sites, internal magnetic fields, and chemical shifts, made possible by least-squares fits of the observed spectrum to linear and nonlinear combinations ofLorentzian waveforms (Fig. 11.92).

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