179

10 EPR Spectroscopy

10.1 Introduction to EPR Spectroscopy

Electron paramagnetic resonance (EPR) shares the theoretical description and many

experimental concepts with NMR spectroscopy. From a methodological point of view, the

main difference is the much larger magnetic moment of electron spins, which exceeds the one

of protons by a factor of 660. Therefore EPR has a higher sensitivity at given spin

concentration, can observe spin-spin interactions over longer distances, and is sensitive to

molecular motion at shorter time scales. Resonance frequencies are in the microwave (mw)

range (see Table 10.1). Typical pulse lengths and signal observation times are also shorter by

about a factor of 500. Spectrometer technology is thus more complicated and may differ

considerably between the different mw bands. In this lecture course, we shall refer to the most

common band, the X band, unless noted otherwise.

Table 10.1: Microwave frequency bands used in EPR spectroscopy (band definition by Radio Society

of Great Britain)..

From an application point of view almost all pure substances contain magnetic nuclei

and are thus accessible to NMR spectroscopy, while only few pure substances contain

unpaired electrons and are thus accessible to EPR spectroscopy. This is because chemical

binding is based on electron pair formation with spin cancellation. Most stable compounds are

Band Frequency rangeTypical EPRFrequency

[GHz]

Typicalwavelength

[mm]

Typicala EPRField

[mT]

a. Assuming = 2.002319

LSXKu

QVWD

1-22-48-1212-1830-5050-7575-110110-170

1.53.09.517367095140

20010030178432

541103406001280250033905000

g ge

B0

180

thus diamagnetic. The occurence of an EPR signal is thus often related to enhanced reactivity,

as for instance in transition metal complexes and free radicals. Therefore, EPR is particularly

well suited for studies on synthetic and biological catalysts and for studies of radical-induced

degradation processes in organisms and materials.

As an example consider the enzyme methyl-coenzyme M reductase (MCR), which

catalyzes methane formation from methyl-coenzyme M (methyl-CoM) and coenzyme B.

Biochemical studies do not reveal whether MCR catalyzes the reaction by attacking the

thioether sulfur atom of methyl-CoM or the carbon atom of the CH3S group. This question was

addressed by studying the reaction of MCR with the inhibitor 3-bromopropane sulfonate

(BPS), which results in a stable paramagnetic species with a single electron in a orbital

on the nickel ion. By using BPS 13C labeled at the methyl group and the two-dimensional

HYSCORE experiment at X band (Section 10.4.4.4), it was possible to detect hyperfine

coupling between the unpaired electron and the methyl carbon atom (Figure 10.1). The

couplings to the protons of BPS were obtained from Q-band HYSCORE and electron nuclear

double resonance (ENDOR) spectra. From this information a model for the binding mode

could be proposed.

Figure 10.1: Determination of the binding mode of the inhibitor 3-bromopropane sulfonate (BPS) tothe active center F430 of methyl-coenzyme M reductase (MCR). A) Structure of the active center. B)Structure of the inhibitor. C) X-band HYSCORE spectrum for the product obtained from F430 and BPS13C-labelled at the -position. The left quadrant contains the 13C peaks. D) Simulated HYSCOREspectrum. E) Structural model for BPS bound to F430 (from Hinderberger D. et al, Angew. Chem. Int.Ed. 2006, 45, 3602-3607).

Some defects in inorganic materials, such as semiconductors, are also paramagnetic.

These defects determine the electronic and optical properties of these materials. EPR

dx2 y2–

-30 -20 -10 0 10 20 300

10

20

30

-30 -20 -10 0 10 20 300

10

20

30

�2 /MHz

�2 /MHz

�1 / MHz

�1 / MHz

gz

Ni

OGln� ‘147

H�

H�1

H� H�2

SO

SO

3

3

-

-

N

N

N

NNi

*N

N

NN

Nin+

HN

H2NOC

H3C

–OOC

O

CH3

COO–

COO–

COO–

–OOC

O

H

H

O

H2NGln

�'147F430

��

� BPSBr

A C

D

E

B

181

spectroscopy provides more detailed information on their electronic and spatial structure than

electrical or optical measurements. Furthermore, species with unpaired electrons are often

observed as reaction intermediates, in particular in one-electron transfer processes. Such

processes are the basis of energetics of living cells and occur in many redox reactions. Short-

lived intermediates can be studied with special transient EPR experiments (photosynthetic

reaction centers and light-sensitive proteins) or after spin trapping (Section 10.5.5). For

instance, neurodegeneration during Alzheimers disease is related to an inflammatory process

that is in turn characterized by abnormally high levels of free radicals. EPR spectra recorded in

the presence of the spin trap N-tert-butyl--phenylnitrone (PBN) showed that toxic forms of

amyloid- peptide lead to a formation of radicals. Such radical formation is not observed in the

absence of these peptides or in the presence of a peptide that is rendered nontoxic by

replacement of the methionine residue at position 35 by norleucine (Figure 10.2). This

established the role of residue Met35 in the neurodegenerative process, the probable

involvement of the sulfur atom of this residue in the chemistry of radical formation, and a

potential protective effect of spin traps against Alzheimer’s disease. Meanwhile PBN and

similar compounds are discussed as potential drugs.

Figure 10.2: Demonstration of radical formation by amyloid- peptide using spin trap EPR. A) Spintrap N-tert-butyl--phenylnitrone (PBN). B-E) EPR spectra for B) a control solution withoutA-peptide, C) with A 1-42 added, D) as before, but with residue Met35 replaced by anorleucine residue, E) with A 25-35 added. Residues 1-24 and 36-42 appear to be notessential for radical formation (from: S. Varadarajan et al., J. Struct. Biol. 2000, 130, 184-208.

For such applications EPR is better suited than NMR, mainly because of the low

concentrations involved and the higher sensitivity of EPR. Unless paramagnetic centers relax

very fast, they also cause strong broadening of NMR lines of nuclei in their vicinity. Active

centers of paramagnetic metalloproteins thus cannot be characterized by NMR. On the other

hand, the structure of domains remote from the active center are not accessible to EPR, but can

be studied by NMR. The two spectroscopies are thus often complementary.

B

C

D

E

A

182

Figure 10.3: Structural model of the dimer of the Na+/H+ antiporter of Escherichia coli determinedfrom the x-ray structure of the monomer by EPR distance measurements between spin-labeled residues.A) A selected residue is mutated to a cystein and the methanethiosulfonate spin label (MTSSL) isattached. B) For a homodimer with a C2 symmetry axis, four degrees of freedom (x, y, , ) determinethe relative position of the two moieties. C) Nine distances were measured between spin labels in thetwo moieties of the dimer were measured (red lines). The dashed line marks a distance that was toolong for the applied technique. The eight known distances overdetermine the structure. D) Interactionbetween -sheets in the two moieties contributes most to binding in the dimer. E) Another contributionto binding results from contacts between two helices (original publication: Hilger D. et al, Biophys. J.2007, 93, 3675-3683).

EPR spectroscopy can also complement NMR spectroscopy on diamagnetic systems by

providing access to larger distances between sites and to dynamics on shorter time scales. For

this purpose, stable free radicals have to be introduced in the system. EPR is then used as a

probe technique. Usually, nitroxides (Section 10.5.1) are introduced as non-covalently

attached spin probes or covalently bound spin labels. For proteins site-directed spin labeling

(SDSL) have been developed. This technique is based on mutation of an amino acid near the

site of interest to a cystein and covalent attachment of a thiol-specific spin marker. As is the

case with NMR, structure determination of biomacromolecules by EPR does not require

crystallization.

For instance, the dimer of the Na+/H+ antiporter NhaA of Escherichia coli is weakly

bound and does not survive crystallization. A structural model could be determined from the x-

ray structure of the monomer and measurements of nine distances between spin-labeled

residues in the two moieties of the dimer (Figure 10.3). If the two monomers in a homodimer

are considered as rigid bodies, their relative position is fixed by only four degrees of freedom.

Hence, the problem is overdetermined. The structural model shows that, to a large extent,

binding between the monomers is due to interactions of two -sheets and to a lesser extent by

N OS

S

O

O

N OS

MTSSL

C -SH�

C -S�

C2

x

y180°

�

�Z

X

Y

A B C

D E

R49

Q47

VIIA

IXB

R204

V254

L210

W258

183

hydrophobic interactions between two helices. The -sheet binding motif is common for

soluble proteins, but was found here for the first time for a membrane protein. Later

biochemical experiments confirmed the importance of hydrogen-bonding residues in the -

sheets for dimer formation.

10.2 Differences between EPR and NMR spectroscopy

The basics of the description of spin systems and spin dynamics are the same for EPR

and NMR. Many of the experimental concepts from NMR can also be applied. In the

remaining lectures we mostly discuss concepts that are particular to EPR. Such concepts result

from the following differences. First, the larger magnetic moment implies frequencies in the

mw range (Table 10.1) that pose more difficult problems for spectrometer technology. Larger

excitation bandwidths are required, which implies pulses with lengths in the nanosecond rather

than microsecond range. For these reasons pulse techniques were developed later in EPR than

in NMR. The typical number of pulses in an experiment is smaller in EPR and pulse shape,

phase, and frequency cannot be varied so easily. Experimental schemes are thus less complex

in EPR compared to NMR.

Due to the larger magnetic moment of the electron spin, typical hyperfine couplings

between an electron and a nuclear spin are by at least two orders of magnitude larger than

typical couplings between two nuclear spins. At the same time, nuclear Zeeman frequencies

are smaller than in NMR as external magnetic fields are usually smaller. Thus, hyperfine

couplings are often of the same order of magnitude as the nuclear Zeeman interaction and lead

to mixing of nuclear spin states. The usual spectroscopic selection rules do no longer apply.

Formally forbidden transitions can be excited by a single pulse or even by continuous

irradiation. This allows for detection of nuclear frequencies in electron spin echo envelope

modulation (ESEEM) experiments (Section 10.4.4). The pulse sequences consist of only mw

pulses that are far off-resonant for the nuclear spins. Such experiments have no equivalent in

NMR. Generally, the spin dynamics is more complex in EPR compared to NMR.

In addition, the larger ratio between couplings and resonance frequencies than in NMR

makes relaxation rates faster compared to the resonance frequencies. As line widths are

proportional to the transversal relaxation rate, this leads to a larger ratio between line widths

184

and resonance frequencies, i.e. to lower resolution. To some extent this lower resolution is

compensated by a larger spread of magnetic parameters. The equivalent to the chemical shift in

NMR is the relative deviation (g-ge)/ge of the electron g value from the ge value for the free

electron. This deviation is typically three orders of magnitude larger than chemical shifts (ppt

instead of ppm).

Because of the larger frequeny shifts and the larger couplings, EPR spectra extend over

a broader frequency range relative to the mean resonance frequency. This requires a larger

relative excitation bandwidth. However, the relative excitation bandwidths of pulses in the

radio frequency (rf) and mw range are almost the same, if the same pulse power is available. It

is thus not usually possible in EPR spectroscopy to excite the whole spectrum by a single

pulse. This eliminates the sensitivity advantage of pulse Fourier transform experiments, so that

continuous-wave (cw) EPR is still the method of choice for recording basic EPR spectra

(Section 10.7.1). Furthermore, cw techniques are applicable even if the transverse relaxation

time T2 is shorter than the shortest available pulses. Therefore, cw EPR is applicable in a

broader temperature range than pulsed EPR.

NMR spectroscopy relies strongly on the concept of separation of interactions by

multipulse sequences and 2D spectroscopy. These concepts can be transferred to pulse EPR.

By using them, resolution can be improved tremendously and more information can be

obtained. Therefore both cw and pulse EPR are commonly applied. Even in pulsed EPR the

external magnetic field is varied to overcome the limitations in excitation bandwidth.

Another difference arises from the much larger change of the EPR resonance frequency

associated with a change of the orientation of the molecule in the magnetic field. Anisotropies

of EPR frequencies are on the order of tens of Megahertz to several Gigahertz. This matches

the range of inverse rotational correlations times of molecules in fluid solution or soft matter.

Hence, orientations of the molecule exchange on the time scale of cw EPR experiments. The

effect on the spectral line shape can be treated in the same way as chemical-exchange

phenomena in NMR were treated in Chapter 3. The rotational correlation time of the molecule

can then be inferered from the line shape (Section 10.5.2).

Like NMR, EPR spectroscopy can be applied to fluid or solid samples. As the coupling

between two electrons at the same distance is by a factor (e/H)2 = 436’000 larger than

185

between two protons, the concentration of paramagnetic centers needs to be smaller to avoid

excessive line broadening by spin-spin interaction. Best resolution and sensitivity for liquid

samples is achieved at concentrations of of about 10-3-10-5 M. In crystalline samples the

paramagnetic centers are often diluted by doping the paramagnetic compound into an

isomorphous diamagnetic compound. Typical dilutions are 1:100 to 1:10000. If isomorphous

diamagnetic substitution is impossible, solid-state measurements are performed in frozen

solution. Aggregation of the dissolved paramagnetic compound on freezing is avoided by

using glass-forming solvents. A selection of pure glas-forming solvents is given in Table 10.2.

For amphiphilic compounds mixtures of polar and apolar solvents have to be used. Such glass-

forming solvents are, for instance, isopentane/isopropanol in a 8/2 mixture, toluene/acetone in

a 1/1 mixture or with an excess of toluene, and toluene/methanol in a 1/1 mixture or with an

excess of toluene. For biomacromolecules, water/glycerol mixtures are often used.

Table 10.2: Glass-forming solvents with their melting temperatures Tm and

glass transition temperatures Tg.

Solvent Tm/K Tg/K

2-methylbutane 113.3 68.2

2-methyltetrahydrofuran 137 91

ethanol 155.7 97.2

methanol 175.2 102.6

1-propanol 146.6 109

toluene 178 117.2

ethylene glycol 255.6 154.2

di-1,2-n-butyl phthalate 238 179

glycerol 291.2 190.9

o-terphenyl 329.3 246

poly(styrene) 513 373

186

Another important difference between NMR and EPR is the appearance of the spectra.

For technical reasons (see Section 10.7.1), best sensitivity and resolution in EPR is achieved

with cw experiments at fixed mw frequency. The magnetic field is swept and at the same time

modulated with a typical frequency of 100 kHz. Phase-sensitive detection of the signal

modulation at this frequency yields the derivative of the absorption spectrum (Fig. 10.4).

Although the shape of a derivative Lorentzian absorption line looks like the shape of a

dispersion line, the derivative absorption line has a smaller linewidth.

Figure 10.4: Detection of the derivative of the absorption spectrum by field modulation withmodulation amplitude B0. The amplitude of the reflected mw V is measured (A) providing aderivative lineshape (B). To avoid artificial line broadening the condition has to befulfilled. The peak-to-peak linewidth Bpp = in field-swept spectra scales as 1/g (roughly with B0) ifthe transversal relaxation time T2 does not depend on g (C).

10.3 Interactions and EPR Hamiltonians

10.3.1 Classification of electron spin systems

To avoid excessive relaxational line broadening, EPR spectra are usually measured in a

concentration regime where interactions between paramagnetic centers are negligible. An

exception are systems where the distance between two (or more) paramagnetic centers is

relatively well defined and is measured by EPR (Section 10.3.6). An isolated paramagnetic

center can contain a single unpaired electron (electron spin S=1/2, Section 10.3.3) or several

tightly coupled electron spins (group spin S>1/2, Section 10.3.4). The latter situation is

�B0

�Bpp

�V

�V

A

B

C

= 2 h

T2 g µB3

B0

B0 3Bpp

187

encountered in high-spin states of transition metal complexes or triplet states of organic

molecules. If the zero-field splitting in such a high-spin system exceeds the mw frequency,

transitions can be induced only with pairs of levels with magnetic quantum numbers of

the electron group spin S. Such a pair of levels is a Kramers doublet, meaning that the levels

are degenerate in the absence of an external magnetic field and are split by the field. The

Kramers doublet can be treated as an effective spin S’=1/2 (Section 10.3.5).

Figure 10.5: Topology of spin systems. A) System consisting of seven nuclear spins. All magneticmoments and hence all 21 spin-spin couplings are comparable. B) System consisting of one electronspin with a large magnetic moment and six nuclear spins with much smaller moments. Only the sixhyperfine couplings between the electron spin and the nuclear spins are significant.

In most paramagnetic centers the electron spin is coupled to nuclear spins. Because of

the presence of one spin that has a much higher magnetic moment than all the other spins, such

electron-nuclear spin systems have a different topology compared to the nuclear spin systems

observed in NMR. In typical nuclear spin systems all spins exhibit significant couplings to

several other spins (Fig. 10.5A), while in typical electron-nuclear spin systems the nuclear-

nuclear couplings are negligible. Hence, the electron spin couples to many nuclei, but each

nucleus couples significantly only to the one electron spin (Fig. 10.5B). The spectrum of the

nuclear spins is thus easier to analyze and better resolved. It can be observed by ENDOR

(Section 10.4.3) or ESEEM (Section 10.4.4) experiments. Furthermore, spin evolution can be

described by considering only subsystems consisting of the electron spin and one nuclear spin,

since the matrix representation of the spin Hamiltonian factorizes into Hamiltonians of two-

spin systems. This is an important simplification, since two-body problems can be solved

exactly, while many-body problems cannot.

mS

A B

188

10.3.2 Units of magnetic parameters in EPR spectroscopy

In NMR spectroscopy magnetic parameters are generally given in frequency units or

are dimensionless. In EPR spectroscopy, magnetic field units are also used. Furthermore,

wavenumbers are sometimes given for zero-field splittings and large hyperfine couplings that

may be resolved in optical spectra.

Frequency units and wavenumbers are energy units that can be interconverted without

further knowledge (1 MHz = 3.335641·10-5 cm-1). In contrast, the conversion between

magnetic field units and frequencies is not trivial, as it depends on the g value, and the g value

often cannot directly be read off the spectrum. The conversion may thus require a full analysis

of the spectrum. If the g value is known with sufficient precision, we have

. [10.1]

In particular, for (organic radicals), 1 MHz corresponds to 0.0356828 mT or 1

mT to about 28 MHz. The cgs unit Gauss (G) is often used to specify the magnetic field. In fact

it is a unit of magnetic induction (SI unit Tesla (T)), with 10000 G = 1 T. In both NMR and

EPR the magnetic induction is usually referred to as the magnetic field. More precisely, the

magnetic field unit is A m-1 in SI units or Oersted in cgs units (1 A m-1 = 4·10-3 Oersted). If

the relative permeability of a material is unity, the magnetic induction in Gauss is the same as

the magnetic field in Oersted. This is a good approximation for diamagnetic and paramagnetic

substances, but not for ferromagnetic materials. Finally, when discussing spin Hamiltonians,

we use angular frequency units for energies. Angular frequencies are related to frequencies

by =2, and to energies E by .

10.3.3 One electron spin S=1/2 coupled to nuclear spins

The spin Hamiltonian in the absence of mw or rf irradiation (static Hamiltonian) is

given by

, [10.2]

B0 h gB =

g ge

E h=

H0 HEZ HHF HNZ HNQ+ + +=

189

where the terms are the electron Zeeman interaction , the hyperfine interaction , the

nuclear Zeeman interaction , and the nuclear quadrupole interaction . In the

following they are discussed in turn.

10.3.3.1 The electron Zeeman interaction

This interaction is formally equivalent to the (nuclear) Zeeman interaction treated in

Sections 5.1 and 5.4. It is given by

, [10.3]

where J·T-1 is the Bohr magneton,

J·s Planck’s quantum of action, the transpose (T)

of the magnetic field vector , the g tensor, and the electron spin

vector operator . In the PAS of the g tensor, the magnetic field vector can be

written as

, [10.4]

where and are polar angles that characterize the direction of this vector in the PAS.

Expansion of Eq. [10.3] in this coordinate system yields

. [10.5]

For an anisotropic g tensor,1 the direction of the effective field depends on the orientation of

the molecule in the magnetic field. In other words, the quantization axis of the electron spin is

not necessarily parallel to the magnetic field and can have different directions for different

orientations of the molecule. The effective g value is thus given by

. [10.6]

1 Strictly speaking, does not have the transformation properties of a tensor, as it connects two independent

spaces (spin space and laboratory space). Therefore it is sometimes called an interaction matrix. In this lec-

ture, we call all interaction matrices tensors, as most magnetic resonance literature does.

HEZ HHF

HNZ HNQ

HEZB

h------B0

TgS=

B e 9.27400899 37 24–10= =

h h 2 1.0545715968234–10= = B0

T

B0 B0x B0y B0z = g S

Sx Sy Sz

B0 B0 cossinsincossin =

HEZB

h------B0 gx Sx gy Sysinsin gz Szcos+ +cossin =

g

geff gx2 sin2 cos2 gy

2 sin2 sin2 gz2 cos2+ +=

190

In the high-field approximation, where the electron Zeeman interaction dominates over all the

other interactions, its contribution to the energy of the spin states is proportional to ,

where mS is the magnetic quantum number of the electron spin. The resonance field at this

orientation of the molecule with respect to the magnetic field is then

. [10.7]

The deviation of the g value of bound electrons from the ge value of the free electron is

mostly due to spin-orbit coupling. As orbital angular momentum is quenched for non-

degenerate ground states, spin-orbit coupling is due to admixture of excited states to the

ground state by the orbital angluar momentum operator . This admixture is usually small and

can thus be treated by perturbation theory. To second order the g tensor is given by

, [10.8]

where indices i and j run over the Cartesian directions x, y, and z and is the Kronecker

delta. The outer sum with index k runs over all atoms in the molecules, where is the spin-

orbit coupling constant for the kth atom and and are Cartesian components of the orbital

angular momentum operator of this atom. As spin-orbit coupling is a relativistic effect,

increases with the mass of the atomic nucleus. The inner sum with index m runs over the

molecular orbitals with index 0 designating the singly occupied molecular orbital (SOMO),

where the unpaired electron resides in the ground state. Note that this index can be both

negative (occupied orbitals) and positive (empty orbitals). Accordingly, the energy difference

in the denominator can also be positive or negative. Spin-orbit coupling with empty

p-, d-, or f-orbitals thus leads to negative deviations of g from ge (high-field shifts), while spin-

orbit coupling with occupied orbitals leads to positive deviations (low-field shifts). The latter

case is more often encountered. Spin density in s orbitals does not contribute to deviations of g

from ge.

The g tensor is thus a complicated function of the frontier orbitals of the molecule. It is

used as fingerprint information for the class of paramagnetic centers. Since light elements (first

and second period) have a small spin-orbit coupling constant, deviations of g from ge are only

mSgeff

B0 reshmw

geffB---------------=

l

gij geij 2 k

m lki 0 0 lkj m E0 Em–

----------------------------------------------m 0

k+ geij 2 k k ij

k+= =

ij

k

lki lkj

k

E0 Em–

191

of the order of for organic radicals, unless there exist near degenerate frontier

orbitals ( ). Typical cases of orbital degeneracy are the hydroxyl radical ·OH,

alkoxy radicals RO·, and thiyl radicals RS· in the gas phase. In condensed phase, degeneracy is

lifted by interaction with neighboring molecules. For these radicals, the g tensor thus strongly

depends on intermolecular interactions. For first row transition metal ions deviations of g from

ge are of the order of . They are dominated by contributions from orbitals on the

transition metal ion, so that we find

, [10.9]

where is the spin-orbit coupling constant of the transition metal.

10.3.3.2 The hyperfine interaction

More detailed information on the electronic and spatial structure of the paramagnetic

center can be obtained from the hyperfine couplings. The hyperfine interaction term of the spin

Hamiltonian is given by

, [10.10]

where k runs over all magnetic nuclei (Ik >0) in the molecule that have hyperfine

couplings larger than the linewidths in the spectrum under discussion. The are nuclear spin

vector operators for these nuclei and the are hyperfine tensors. When analyzing liquid-state

EPR spectra, hyperfine couplings exceeding 3 MHz have to be considered. In solid-state EPR

spectra, such couplings are usually resolved only when they exceed 15 MHz. When analyzing

ENDOR, ESEEM, or hyperfine sublevel correlation (HYSCORE) spectra the resolution limit

is at 0.2-0.5 MHz. Within the high-field approximation the contribution of the hyperfine

coupling to the energies of the spin states is , where Aeff is an effective hyperfine

coupling.

Hyperfine coupling of the electron spin to a nuclear spin comes about by through-space

dipole-dipole coupling of the two magnetic moments and by the Fermi contact interaction.

The Fermi contact interaction is due to the non-zero probability to find the electron spin at the

103– 10

2–

E0 Em– 0

102– 10

1–

g ge1 2m lki 0 0 lkj m

E0 Em–----------------------------------------------

m 0+ ge1 2+= =

HHF ST

AkIk

k=

Ik

Ak

mSmIAeff

192

same point in space as the nuclear spin. This happens only when the unpaired electron is in an

s orbital. The Fermi contact interaction leads to a purely isotropic coupling with

, [10.11]

where s is the spin density in the s orbital under consideration, gn is the nuclear g value and

J·T-1 the nuclear magneton ( ). The factor

is the probability to find the electron at this nucleus in the ground state with

wavefunction .1 The magnetic moment of the electron is fully characterized by the ge value

of the free electron, as there is no orbital angular momentum in s orbitals.

Unpaired electrons in p-, d, and f- orbitals do not contribute to Fermi contact

interaction, as these orbitals have a node at the nucleus. However, due to the non-spherical

symmetry of these orbitals, dipole-dipole coupling between the magnetic moments of the

electron and nuclear spin does not average. Spin density in such orbitals thus gives rise to

purely anisotropic couplings.2 In general, the matrix elements Tij of the total anisotropic

hyperfine coupling tensor of a given nucleus are computed from the ground state

wavefunction by

. [10.12]

Quantum chemistry programs such as ORCA, ADF, or Gaussian can compute as well as

aiso.

A special situation applies to protons, alkali metals and earth alkaline metals, which

have no significant spin densities in p-, d-, or f-orbitals. In this case, the anisotropic

contribution can only arise from through-space dipole-dipole coupling to other centers of spin

density. In a point-dipole approximation the hyperfine tensor is then given by

1 Tabulated in: J. R. Morton, K. F. Preston, J. Magn. Reson. 1978, 30, 577.

2 For given spin densities these couplings can be computed from parameters given in: J. R. Morton, K. F.

Preston, J. Magn. Reson. 1978, 30, 577.

aisoST

I

aiso s23---0

h-----geBgnn 0 0 2=

n n 5.05078317 20 27–10= = gnn n h=

0 0 2

0

T

Tij

0

4h----------geBgnn 0

3rirj ijr2

–

r5

--------------------------- 0 =

T

193

, [10.13]

where the sum runs over all centers with significant spin density j (summed over all orbitals at

this center), the Rj are distances between the nucleus under consideration and the centers of

spin density, and the are unit vectors along the direction from the considered nucleus to the

center of spin density. For protons in transition metal complexes it is often a good

approximation to consider spin density only at the central ion. The distance R from the proton

to the central ion can then be directly inferred from the anisotropy of the hyperfine coupling.

Admixed orbital angular momentum also contributes to dipole-dipole coupling. This

can be considered by a simple correction, giving the dipole-dipole hyperfine tensor

. [10.14]

We thus obtain for the total hyperfine tensor in Eq. [10.10]

. [10.15]

Note that the product may have an isotropic part, although is purely anisotropic. This

pseudocontact contribution depends on the relative orientation of the g tensor and the spin-

only dipole-dipole hyperfine tensor .

The hyperfine couplings can be used to map the SOMO in terms of a linear

combination of atomic orbitals. For catalytic species this can provide insight into reactivity and

thus into the mechanism of catalysis. For organic compounds that form reasonably stable

radical anions or cations the SOMO of the radical anion corresponds to the LUMO of the

parent compound and the SOMO of the radical cation to the HOMO of the parent compound.

The contributions discussed so far can be understood in a single-electron

representation. A further contribution arises from correlation between electrons in the

molecule. Assume that the pz orbital on a carbon atom contributes to the SOMO, so that the

spin state of the electron is preferred in that orbital (Fig. 10.6). Electrons in other orbitals on

the same atom will then also have a slight preference for the state, as electrons with the same

Tk

0

4h----------geBgnn j

3njnjT

1–

Rj3

-----------------------j k=

nj

Add

gT

ge-------=

Ak

Ak aiso k 1gTk

ge---------+=

gTk Tk

Tk

194

spin “tend to avoid each other” and thus have less electrostatic repulsion.1 In particular, this

means that the spin configuration in Fig. 10.6A is slightly more preferable than the one in Fig.

10.6B. According to the Pauli principle the two electrons that share the bond orbital of the C-

H bond must have antiparallel spin. Thus, the electron in the s orbital of the hydrogen atom

that is bound to the spin-carrying carbon atom has a slight preference for the state. This

corresponds to a negative isotropic hyperfine coupling of the directly bound proton, which is

induced by the positive hyperfine coupling of the adjacent carbon atom. The effect is termed

‘spin polarization’, although it is entirely different from the polarization of electron spin

transitions in an external magnetic field.

Figure 10.6: Spin polarization on an adjacent hydrogen atom ( proton) induced by spin density in a pzorbital on carbon. The spin configuration in A) is slightly favored with respect to the one in B).

Figure 10.7: Isotropic hyperfine coupling of a next-neighbor hydrogen atom ( proton) induced byhyperconjugation. The overlap of the carbon pz orbital and the hydrogen s orbital depends on dihedralangle .

Spin polarization is observed both in radicals, where the spin density is confined to a

single pz orbital, and in radicals, where the spin density is delocalized in a system. In

1 This preference for electrons on the same atom to have parallel spin is also the basis of

Hund’s rule.

C CH H� �

pz pz

A B

C

H

H�A B

195

radicals spin polarization is the only contribution to the isotropic hyperfine coupling of a

proton that is directly bound to a carbon atom of the conjugated system. The isotropic

hyperfine coupling of such an proton can thus be predicted by the McConnell equation

, [10.16]

where is the spin density at the adjacent carbon atom and QH is a parameter of the order of

mT, which slightly depends on the structure of the system.

The isotropic hyperfine coupling of protons in radicals is caused by spin

delocalization from the spin-carrying pz orbital to the s orbitals on the hydrogen atoms.

These orbitals are sufficiently close in space to allow for overlap. The extent of this

hyperconjugation depends on the dihedral angle between the pz orbital lobes and the C-H

bond. When the C group rotates freely with respect to the spin-carrying orbital, an average

hyperfine coupling is observed. Unlike spin polarization, hyperconjugation does not depend on

electron correlation, but can rather be seen as spin density transfer.

10.3.3.3 Nuclear Zeeman interaction

The nuclear Zeeman interaction can be neglected in the analysis of EPR spectra, unless

it is of the same order of magnitude as a resolved hyperfine coupling of the same nucleus. In

analysis of ENDOR, ESEEM, and HYSCORE spectra, the nuclear Zeeman interaction is used

to assign hyperfine couplings to elements (isotopes). This information is missing in EPR

spectra.

Chemical shift information cannot be obtained from EPR, ENDOR, ESEEM, or

HYSCORE spectra, as chemical shifts are smaller than the linewidths for nuclear spins with

significant hyperfine couplings. Chemical shift is thus neglected in the nuclear Zeeman

Hamiltonian, which is given by

. [10.17]

In the laboratory frame this simplifies to

, [10.18]

Aiso H QH=

2.5–

HNZgnn

h-----------B0

TI–=

HNZ IIz=

196

with I = -nB0. Within the high-field approximation the nuclear Zeeman interaction

contributes an energy mI I to the spin states.

At S-band to Q-band frequencies the nuclear Zeeman frequency is often comparable to

hyperfine couplings or smaller. This leads to state mixing by anisotropic hyperfine

interactions, which is the basis of ESEEM and HYSCORE experiments. At W-band

frequencies and above this phenomenon is observed only for a few low- nuclei with large

hyperfine couplings (large spin densities), such as for nitrogen atoms that are directly

coordinated to a transition metal ion.

10.3.3.4 Nuclear quadrupole interaction

The nuclear quadrupole interaction (Section 5.9) does not contribute to EPR spectra

unless it is of the same order of magnitude as the hyperfine coupling of the same nucleus and

the hyperfine coupling of this nucleus is resolved. Usually this happens only for heavy

elements from the third row of the periodic table onwards. In analyzing ENDOR, ESEEM, and

HYSCORE spectra of nuclei with spin Ik>1/2 the nuclear quadrupole interaction has to be

considered. In an EPR context, the nuclear quadrupole Hamiltonian is often written as

, [10.19]

where the principal values of the nuclear quadrupole tensor are given by

, [10.20]

, [10.21]

and

. [10.22]

The principal values are thus directly related to the field gradient components Vxx, Vyy, and Vzz

defined in Section 5.9. Within the high-field approximation the contribution of the nuclear

quadrupole interactions to the energies of the spin states is proportional to , where Peff

is an effective nuclear quadrupole interaction. To first order, the transition energy for a pair of

HNQ IT

PI=

Pze

2qQ

2I 2I 1– h----------------------------=

Px Py– Pz=

Px Py Pz+ + 0=

mI2Peff

197

levels thus does not depend on the nuclear quadrupole interaction. In particular, this

applies to the allowed transition .

10.3.4 One electron group spin S>1/2 coupled to nuclear spins

If m electrons are distributed indegenerate orbitals, each orbital is first singly occupied

(Fig. 10.8A, Hund’s rule). This is because electron pairing in the same orbital would lead to

increased Coulomb repulsion. In the presence of a ligand field that removes orbital

degeneracy, pairing is preferred if the energy difference between the upper and lower set of

orbitals exceeds electron pairing energy (Fig. 10.8B,C). Depending on the strength of the

ligand field, several electrons may thus be unpaired (high-spin state) or paired as far as

possible (low-spin state).

Figure 10.8: High-spin and low-spin configurations for a d5 transition ion, such as Fe3+. A) In theabsnece of a ligand field all unpaired electrons in the degenerate d orbitals have parallel spin (high spinS= 5/2). B) In a weak ligand field, where the splitting between the eg and t2g levels is smaller than theelectron pairing energy, all spins are parallel (high spin, S= 5/2). C) In a strong ligand field, where thelevel splitting exceeds the electron pairing energy, all electrons occupy the set of lowest-lying orbitals,where they pair (low spin, S= 1/2).

In high spin states of transition metal ions all the unpaired electrons reside mainly in

orbitals on the same atom. They are so close in space that they couple strongly and cannot be

excited separately from each other. Hence, it is appropriate to describe them as a single group

spin S = m/2. For instance, Fe3+ with a 3d5 configuration in weak ligand fields assumes an S =

5/2 high-spin state, while in strong ligand fields it assumes an S = 1/2 low-spin state.

On first sight one might expect that the distribution of five unpaired electrons on all

five 3d orbitals is spherically symmetric so that no equivalent to the nuclear quadrupole

mI

mI 1 2 1 2– =

A B C

d

eg

t2g

t2g

eg

198

interaction would result. However, spin-orbit coupling breaks this symmetry. Electron-group

spins S > 1/2 thus have an additional interaction that is formally analogous to the nuclear

quadrupole interaction. This zero-field splitting1 contribution to the spin Hamiltonian is

written as

, [10.23]

where is the traceless zero-field splitting tensor. In the principal axes frame of this tensor,

the Hamiltonian can be written as

. [10.24]

with D = 3Dz/2 and E = (Dx-Dy)/2. By convention, Dz is the principal value with the largest

magnitude, so that E cannot exceed D/3. For axial symmetry, E = 0. In cubic symmetry,

quadrupolar zero-field splitting vanishes (D= E = 0). In that case and for S > 3/2, the much

smaller hexadecapolar contribution may become observable.

For transition metal ions, g shifts with respect to ge (Eq. [10.9]) and zero-field splitting

are related,

. [10.25]

Within the high-field approximation the contribution of the zero-field splitting to the energy

levels is proportional to . Thus the transition that is allowed in

EPR spectra of half-integer group spins S is not affected to first order by zero-field splitting.

10.3.5 Effective spin S’=1/2 in a high-spin system S>1/2

If the zero-field splitting is much larger than the mw quantum and the electron Zeeman

interaction at accessible magnetic fields, only part of the transitions of a spin S > 1/2 are

accessible for EPR. In conventional EPR at X band, the limit is at about 20 GHz. For integer

group spins S, none of the transitions is accessible unless the zero-field splitting tensor has at

1 The term of the Hamiltonian is sometimes called fine structure term instead of zero-field splitting.

HZF ST

DS=

D

HZFPAS

DxSx2

DySy2

DzSz2

+ +=

D Sz2 1

3---S S 1+ – E Sx

2Sy

2– +=

D 2=

mS2Deff mS 1 2 1 2– =

199

least axial symmetry. Such integer-spin systems are termed EPR silent, although they are

paramagnetic and can be studied at higher fields and frequencies. For half-integer group spins

there is at least one pair of low-lying levels that are degenerate in the absence of a magnetic

field and are split in its presence. Such a Kramers doublet can be described as an effective spin

S’ = 1/2. For instance, for high-spin Fe3+ (S = 5/2) in non-cubic symmetry, EPR transitions can

be observed only within the three Kramers doublets, but not between them (Fig. 10.9). The

spin Hamiltonian for each Kramers doublet is equivalent to the spin Hamiltonian of a spin S =

1/2 (Section 10.3.3), except that the effective g values depend on the zero-field splitting

parameters D and E and are not strictly field-independent.

Figure 10.9: Kramers doublets with effective spin S’=1/2 for high-spin Fe3+ (3d5) with E/D = 1/3. Themw quantum (vertical bars) is too small to excite transitions between different Kramers doublets. Thethree Kramers doublets have different effective g values.

10.3.6 A pair of weakly coupled electron spins

10.3.6.1 Exchange coupling

Consider two electron spins S1 = 1/2 and S2 = 1/2 as individual paramagnetic centers. At

distances up to at least 15 Å there is still significant overlap of the two SOMOs, so that the two

electrons can exchange. This exchange interaction leads to a splitting between the singlet state

(group spin S = 0) and the triplet state (group spin S = 1) of the coupled system. If this splitting

B00

g = 9.67

g = 4.3

g = 0.6

200

is so small that transitions between the singlet and triplet state can be excited in an EPR

experiment, it is more convenient to treat the system in terms of two individual spins that are

coupled to each other. Typically at distances larger than 5-10 Å the individual spin treatment is

applied.

The energy difference between the singlet and triplet state is the exchange integral

, [10.26]

where 1 and 2 are the wavefunctions of the two unpaired electrons.1 For positive J, the

singlet state is lower in energy, i.e., the orbital overlap is bonding and the interaction is

antiferromagnetic. Negative J correspond to a lower-lying triplet state, i.e., antibonding orbital

overlap and a ferromagnetic interaction.

The exchange term of the spin Hamiltonian is given by

. [10.27]

This term describes a scalar (purely isotropic) coupling. Anisotropic contributions to the

exchange coupling may occur for species involving heavy elements, but can be neglected for

organic radicals. Within the high-field approximation the contribution of the exchange

interaction to energy levels is proportional to mS,1mS,2J.

To a good approximation the exchange interaction decays exponentially with distance r

between the paramagnetic centers. The decay rate depends on the conductivity of the medium

between the centers and on the presence or absence of a conjugated network of bonds between

them. If conjugation is weak and the matrix isolating, exchange coupling is much smaller than

the dipole-dipole coupling through space at distances longer than 15 Å.

1 There exist different conventions for the sign of J and the factor 2 may be missing. One should always

ascertain which convention a certain author is adhering to.

J 2e2 1

r1 2 r2 1 r2 2 r1

r1 r2–------------------------------------------------------------------------ r1d r2d–=

HEX JS1T

S2 J S1xS2x S1yS2y S1zS2z+ + = =

201

10.3.6.2 Dipole-dipole coupling

The dipole-dipole coupling between two electron spins is analogous to that between

two nuclear spins (Section 5.8). For transition metal and rare earth ions the g anisotropy is so

large that the two magnetic moments are not parallel to the magnetic field. The interaction

energy according to Eq. [5.74] is then parametrized by three angles 1, 2, and (Fig. 10.10)

and is given by

. [10.28]

The dipole-dipole coupling term of the spin Hamiltonian assumes the form

. [10.29]

Figure 10.10: Coupling between two magnetic moments and in a general orientation withrespect to the external magnetic field and definition of the angles 1, 2, and that parametrize thisinteraction. Note that these angles implicitly depend on the orientation of the molecule in the magneticfield.

In the special case of two parallel magnetic moments, which is a good approximation

for two organic radicals, the dipole-dipole coupling tensor has the principal values -dd, -

dd, 2dd with

. [10.30]

The orientation of the molecule can then be characterized by a single angle between the

magnetic field axis and the spin-spin vector. The orientation-dependent dipolar splitting thus

assumes the form

. [10.31]

E0

4------ 12

1

r3

---- 2 1 2coscos 1 2 cossinsin– –=

Hdd S1T

DS21

r3

----0

4h---------- g1g2B

2S1S2

3

r2

---- S1r S2r – = =

r

0B

µ1

µ2

��

���

1 2B0

D

dd1r3----

0

4h---------- g1g2B

2 =

d 3 cos2 1– dd=

202

For organic radicals, is a good approximation, so that the dipole-dipole

coupling dd=dd/2 in frequency units is given by

. [10.32]

With typical transversal relaxation times T2 of electron spins of a few microseconds,

dipole-dipole couplings can be measured down to about 100 kHz in favourable cases,

corresponding to distances of 8 nm. For soluble proteins and membrane proteins reconstituted

into detergent micelles this upper distance limit may reduce to about 6 nm and for membrane

proteins reconstituted into liposomes to about 5 nm. These limits are comparable to the

diameter of protein molecules.

Within the high-field approximation the contribution of the dipole-dipole interaction to

energy levels is proportional to mS,1mS,2 dd,eff. This is the same dependence on magnetic

quantum numbers as for the exchange coupling. The two contributions to electron-electron

coupling thus cannot be separated by spin manipulation.

Figure 10.11: Transversal and longitudinal relaxation of the electron spin of a nitroxide molecule by acollsion with triplet dioxygen. (a) The molecules are separated and diffuse towards each other. Forexample, both unpaired electrons of the oxygen molecule are in the state, while the unpaired electronof the nitroxide molecule is in the state. (b) The molceules collide, their orbitals overlap, and the threeunpaired electrons are no longer distinguishable. (c) The molecules have separated again. With aprobability of 2/3 the nitroxide molecule is now left with one of the unpaired electrons that originallybelonged to the oxygen molecule (and vice versa). The electron spin of the nitroxide thus has lost phasememory and has changed its spin state.

10.4 Measurement of hyperfine couplings

10.4.1 cw and echo-detected EPR spectroscopy

Hyperfine couplings manifest in the spectra of both electron and nuclear spins. Large

hyperfine couplings (>20...50 MHz or 0.7...2 mT at g = ge) can often be determined with

g1 g2 ge

dd52.04 MHz

r3

nm3–

-----------------------------=

N NNO O

OO=O O=OO=O� ��� ��

a b c

203

sufficient precision from EPR spectra even in the solid state. For small radicals in solutions

with low viscosity, precision of an EPR measurement may even be sufficient for hyperfine

couplings as small as 3 MHz. In these situations, cw EPR or field-swept echo-detected EPR is

the method of choice, as it is more sensitive and technically easier than measurement of the

nuclear spectrum. To obtain utmost hyperfine resolution, line broadening due to couplings

between electron spins has to be avoided. In the solid state this requires concentrations of 1

mmol L-1 or less. .

Figure 10.12: Line broadening due to collisonal exchange. 1,8-Dimethyl naphthaline radical anion at aconcentration of a) 10-3 M, b) 10-5 M. c) Wurster’s blue without (top) and with (bottom) oxygen in thesolution.

In the liquid state linewidths are smaller. Furthermore, exchange broadening can also

arise due to collisions between paramagnetic molecules. During such a collision the orbitals

overlap, the unpaired electrons of both molecules become indistinguishable and may be

exchanged when the molecules separate again (Fig. 10.11). This leads to sudden changes in

resonance frequency and thus to phase relaxation. The transversal relaxation time T2 is

shortened and lines are broadened. In exceptional cases, concentrations down to 200 mol L-1

may be required to avoid such broadening (Fig. 10.12A).

Exchange broadening is also caused by dissolved oxygen, as dioxygen has a triplet

ground state and is thus paramagnetic. The effect is often tolerable for measurements in

aqueous solution, but highly detrimental for measurements in unpolar solvents, where oxygen

solubility is much higher (Fig. 10.12B).

c)

204

.

Figure 10.13: Simulated EPR spectrum of the phenyl radical (bottom) and schematic drawing of howthe splitting pattern arises (top).

The hyperfine splitting pattern of radicals in solution is generated in the same way as

discussed in Section 7.1 for J-coupled spectra of weakly-coupled nuclear spins in NMR, except

that only the spectrum of one of the spins (the electron spin is observed). Furthermore, the

number of nuclear spins with significant hyperfine coupling to the electron spin is usually

larger than the corresponding number in J-coupled NMR spectra. As a particularly simple case,

the EPR spectrum of the phenyl radical is shown in Fig. 10.13. This radical can be generated

for instance by photolysis of phenylbromide in solution. Due to the low natural abundance of13C, carbon hyperfine splittings are apparent only in weak satellite lines that do not concern us

here. The splitting pattern is thus entirely due to proton hyperfine couplings. Among the

protons, the ones in positions 2 and 6 are magnetically equivalent with a coupling of 1.743 mT,

the ones in positions 3 and 5 with a coupling of 0.625 mT and the proton in position 4 has a

2H(1.743 mT)

2H(0.625 mT)

1H(0.204 mT)

375 380 385

B0 (mT)

HH

H

H

H

26

35

4

205

coupling of 0.204 mT. The resulting spectrum, a triplet of triplets of doublets has 18 lines. As

the pattern is clearly apparent, the hyperfine couplings can be read off directly.

Figure 10.14: Experimental EPR spectrum of the paracyclophan free radical.

Such a simple analysis may no longer be possible if the subpatterns overlap. This

happens almost with certainty when the number of hyperfine coupled nuclei increases, as the

number of lines in the EPR spectrum increases multiplicatively

, [10.33]

where index i runs over the groups of equivalent nuclei, the ki are the numbers of nuclei within

each group, and the Ii their nuclear spin quantum numbers. For instance, for the free cyclophan

radical in solution with 4 equivalent aromatic protons, 9 equivalent exo, and 9 equivalent endo

protons, lines arise (Fig. 10.14). Coordination with a K+ ion (nuclear spin I =

3/2 for 39K, quadruplet splitting) removes the equivalence between the two moieties of the

molecule, so that each group of equivalent protons splits into two inequivalent subgroups. The

number of lines increases to .

Spectra like this are difficult to analyse. Due to the special topology of electron-nuclear

spin systems (Fig. 10.5B), the nuclear spectrum in the liquid state has a much smaller number

of lines

nEPR 2kiIi 1+ i=

5 9 9 405=

3 3 5 5 5 5 4 22500=

206

, [10.34]

where i again runs over the groups of equivalent nuclei. For instance, the nuclear spectra of

phenyl radicals, uncoordinated paracyclophan radicals, and K+-coordinated paracyclophane

radicals have only 6, 6, and 14 lines, respectively. In the solid state, additional splitting due to

nuclear quadrupole couplings leads to a number of lines

. [10.35]

Among our examples, this would change the number of lines only for K+-coordinated

paracyclophane radicals to 18.

Figure 10.15: Typical S = 1/2 EPR spectra in the absence of hyperfine couplings. The frequencydispersion is caused only by a g tensor with axial symmetry (A,B) or orthorhombic symmetry (C,D).The low-field and high-field edges of the absorption spectra (A,C) correspond to selection of a smallsubset of orientations near the x or z axis of the g tensor PAS. The first derivative lineshape, as detectedin cw EPR spectroscopy, is dominated by the singularities that arise at principal axis directions.

nNMR liq 2S 1+i=

nNMR sol 2Ii 2S 1+ i=

300 320 340

h g� �mw B/ ||

h g� �mw B/ z

h g� �mw B/ ||

h g� �mw B/ z

h g� �mw B/ x

h g� �mw B/ � h g� �mw B/ y

h g� �mw B/ �

h g� �mw B/ x

h g� �mw B/ y

300 320 340

B0 (mT) B0 (mT)

� = 0°

� = 90°

300 320 340

A C

B D

300 320 340

207

The nuclear spin spectra cannot be measured with an NMR spectrometer, as the

hyperfine splittings exceed the bandwidth of NMR probeheads by orders of magnitude.

Furthermore, detection at the low NMR frequency insted of the high EPR frequency would

lead to a drastic loss in sensitivity. For these reasons spectra of hyperfine coupled nuclei are

detected by EPR-based methods such as ENDOR or ESEEM.

The dispersion of electron spin resonance frequencies due to g anisotropy for powders

or frozen solutions causes a substantial drop in hyperfine resolution compared to the liquid

state. Even for transtion metal ions with only moderate g anisotropy, such as V(IV), Cu(II), or

Co(II) the spectrum in the absence of any hyperfine couplings extends over tens of mT (Fig.

10.15).

Except for the singularities corresponding to principal axis directions, the first

derivative of the absorption lineshape is barely detectable. If hyperfine anisotropy is much

smaller than g ansiotropy or if the PAS of the g and hyperfine tensors coincide, observable

hyperfine splittings thus correspond to the principal axis directions of the g tensor. If g and

hyperfine anisotropy are of the same order of magnitude and the PAS are non-coincident,

spectrum analysis requires lineshape simulations.

Figure 10.16: Hyperfine splittings in solid state EPR spectra of powders, glasses or frozen solutions.(simulations for a square planar copper(II) complex with four equal ligands L). A) Each nuclear spinstate gives rise to a separate axial powder pattern. B) The absorption spectrum, as detected by field-swept echo-detected EPR is the sum of the four separate patterns. C) The derivative of the absorptionspectrum, as detected by cw EPR, exhibits hyperfine splitting along g||. The inset shows a stereoview ofthe complex with the unique axis of the g and copper hyperfine tensor.

0.3 0.32 0.34 0.3 0.32 0.34

A

B

C

mI ��������

mI ��������

mI ��������

mI ��������

B0 (T) B0 (T)

h g� �mw B/ ||

hfi at g||

A||h g� �mw B/ �

hfi at g�

Cu

L

L L

L

g||,A||

208

A simple case is illustrated in Fig. 10.16 on the example of a square planar copper

complex with four equal ligands L. This species has a C4 symmetry axis. A Cn symmetry axis

with implies an axial g tensor with its unique axis coinciding with the symmetry axis. As

the SOMO also has C4 point symmetry at the Cu2+ ion, the copper hyperfine tensor is also

axial and has the same unique axis. For each magnetic quantum number mI = -3/2, -1/2, +1/2,

and +3/2 the two tensors for the nuclear Zeeman and copper hyperfine interaction add to a total

axial tensor that describes the anisotropy of the resonance frequency.1 Each nuclear spin state

thus gives rise to a separate axial powder pattern (Fig. 10.16A). The edges of the patterns are

shifted with respect to each other by multiples of the hyperfine coupling A|| (low-field edge)

and (high-field edge). The small splitting is usually unresolved in the EPR spectrum

(Fig. 10.16B). The parameters g||, , and A|| can be directly read off the cw EPR spectrum

(Fig. 10.16C).

10.4.2 Nuclear spin spectra

Allowed transitions in nuclear spin spectra involve a change of the magnetic quantum

number of the nuclear spin mI by unity while the magnetic quantum number of the electron

spin mS remains constant, i.e., they are of the type . If the high-field

approximation applies to both the electron and nuclear spin, the angular frequencies of such

transitions are given by

. [10.36]

For simplicity, the following considerations are restricted to a system consisting of a nuclear

spin I=1/2 coupled to an electron spin S=1/2. This system has two nuclear transitions with the

angular frequencies

. [10.37]

Depending on the sign of the gyromagnetic ratio of the nuclear spin, the sign of the hyperfine

coupling, and the relative magnitudes of the two interactions, the transition frequencies can be

1 The argument applies strictly only in frequency domain, but the qualitative conclusions are true also in

field domain.

n 3

A A

g

mI mI 1+

I mSAeff 2mI 1+ Peff+ +=

I

Aeff

2----------=

209

either negative or positive and they can have either the same or a different sign. The absolute

sign of the frequency can only be detected if the nuclear transitions are directly excited with

circularly polarized radiofrequency (rf) irradiation. As linearly polarized irradiation is used

this information is lost. The relative sign can be detected in experiments that correlate the two

transitions, such as in the hyperfine sublevel correlation (HYSCORE) experiment (Section

10.4.2.3).

The sign uncertainty complicates assignment and interpretation of the spectra. This is

first discussed for the case of isotropic hyperfine coupling. The same considerations apply to

spectra of single crystals.

10.4.2.1 Isotropic case

Consider the case of the phenyl radical whose EPR spectrum is shown in Fig. 10.13. At

X-band frequencies the nuclear Zeeman frequency I=I/2 is about -15 MHz. The hyperfine

couplings are A1/2 = 5.7 MHz for the para proton H4, A2/2 = 17.5 MHz for the meta protons

H3 and H5, and A3/2 = 48.8 MHz for the ortho protons H2 and H6. The couplings of the para

and meta protons are in the weak coupling regime

(weak coupling) [10.38]

while the couplings of the ortho protons are in the strong coupling regime

(strong coupling) . [10.39]

In the weak coupling case the doublet is centered at |I| and split by |A|/2(Fig. 10.17A,B)1 In

the strong coupling case, one of the frequencies has opposite sign, corresponding to precession

of the magnetization with an opposite sense of rotation. As the sign is lost, the corresponding

line is “mirrored” to positive frequencies. The doublet is thus centered at |A|/4|A|/2 on an

angular frequency scale) and split by 2|I|. When different nuclear isotopes contribute to the

spectrum, transitions of strongly coupled low- nuclei, e.g. 14N, may thus overlap with

transitions of weakly coupled high- nuclei, e.g. 1H. This problem is common at X-band

1 In the literature, A is often used as an angular frequency in equations but as a frequency in spectra or

tables. Beware!

A 2 I

A 2 I

210

frequencies and below. The overlap may be eliminated by going to higher frequencies or by

HYSCORE.

10.4.2.2 Anisotropic case

In the regime of very weak coupling , the high-field approximation is valid

for the nuclear spins. The nuclear spins are then quantized along the direction of the external

magnetic field. The hyperfine interaction perpendicular to this direction is truncated, and Eq.

[10.37] applies with Aeff = Az being the hyperfine coupling along the external field direction.

Figure 10.17: Schematic nuclear spectrum of the phenyl radical. A) Subspectrum of the para proton(weak coupling). B) Subspectrum of the meta protons (weak coupling). C) Subspectrum of the orthoprotons (strong coupling). The line at (relative) negative frequencies is “mirrored” to positivefrequencies. D) Total spectrum.

If the nuclear Zeeman and hyperfine interaction are of the same order of magnitude, the

hyperfine field at the nucleus perpendicular to the external field (terms and

of the Hamiltonian) can no longer be neglected. Instead of being non-secular, as in the high-

field approximation, these terms are now pseudosecular. Together they contribute an effective

interaction in the xy plane of the laboratory frame. The corresponding field

A I«

�( H)1

��( H)1

�( H)1

�( H)1

�( H)1

0

0

0

0

C

B

A

D

A3

A1

A2

A3/2

�nuclear

�nuclear

�nuclear

�nuclear

AzxSzIx AzySzIy

B Azx2

Azy2

+=

211

at the electron spin (terms and of the Hamiltonian) is usually still negligible

as are the remaining off-diagonal terms, since the high-field approximation still applies to the

electron spin with its much larger Zeeman interaction. Thus, the truncated Hamiltonian for the

S=1/2, I=1/2 spin system becomes

. [10.40]

A new x direction was chosen for the nuclear spin frame to simplify the Hamiltonian.1 The

Hamiltonian is written in a singly rotating frame, where the electron spin space rotates with the

mw frequency while the nuclear spin space is fixed with respect to the laboratory frame.

Hence, S is a resonance offset, while I is the total nuclear Zeeman frequency.

Figure 10.18: Local fields on the nucleus in the mS=1/2 () and mS=-1/2 () states. The effective fielddirections for the two states differ; they are tilted by angles and with respect to the external fielddirection z.

When divided by the gyromagnetic ratio n, the Hamiltonian terms containing I spin

operators can be interpreted as local fields at the nuclear spin (Fig. 10.12). The pseudosecular

contribution causes a tilt of the effective field away from the external field direction z. As the

pseudosecular contribution has different signs for the and states of the electron spin, the

effective field axis ist tilted by angles and in different directions. As the effective field

component along z is once the sum and once the difference of the nuclear Zeeman field and the

secular hyperfine field, the magnitudes of and also differ. The quantization axis of the

nuclear spin thus is no longer well defined and mI is no longer a good quantum number.

1 This is convenient, unless the nuclear spins are directly irradiated by rf.

AxzSxIz AyzSyIz

H0 SSz IIz ASzIz BSzIx+ + +=

z

x

A/2

-A/2

- /2B B/2

�I

2

���

�

�

�

��

��

�

� �

212

This has two important consequences. First, an mw pulse that excites the electron spin

will also have an influence on nuclear spin state, as the direction of the local field at the

nucleus is changed. To some extent the pulse thus excites forbidden transitions in which both

mS and mI change. This is the basis of ESEEM experiments (Section 10.4.4). Second, Eq.

[10.37] for the nuclear frequencies does no longer apply. The two nuclear frequencies are now

given by

[10.41]

and

, [10.42]

as can be inferred by vector addition (Fig. 10.18). The doublet is now centered at a frequency

that is slightly higher than I while the splitting is slightly smaller than A.

For the hyperfine and nuclear Zeeman field along z cancel exactly for one

of the two electron spin states, either or . In this situation of exact cancellation the effective

field is within the xy plane, the nuclear frequency is B/2, and the transition moments of

‘allowed’ and ‘forbidden’ transitions are nearly the same. For quadrupolar nuclei (I>1/2) with

small hyperfine anisotropy the nucleus at exact cancellation experiences a near zero-field

situation. Narrow lines at the zero-field nuclear quadrupole resonance (NQR) frequencies are

then observed.

In macroscopically disordered systems, such as powders, glasses, and frozen solutions,

A and B depend on the relative orientation of the hyperfine tensor PAS and the magnetic field.

For a hyperfine tensor with axial symmetry and principal values (aiso+2T, aiso-T, aiso-T), the

orientation dependence is fully described by the angle between the unique axis of the

hyperfine tensor and the magnetic field,

, [10.43]

. [10.44]

IA2----+

2 B2

4------+=

IA2----–

2 B2

4------+=

A 2 I=

A aiso T 3 1–cos2 +=

B 3T cossin=

213

Powder patterns computed from Eq. [10.41-10.44] are shown for weak coupling, exact

cancellation, and strong coupling in Fig. A, B, and C, respectively.

Figure 10.19: Simulated anisotropic nuclear frequency spectra for an S=1/2, I=1/2 system with weakcoupling (A), at exact cancellation (B), and with strong coupling (C).The same sign is assumed for aisoand I.

10.4.2.3 Hyperfine sublevel correlation (HYSCORE)

The frequencies of a pair of transitions of the same nucleus in the electron spin and

manifold can be correlated in a 2D experiment by using a mw pulse for mixing. To keep

with usual notation in EPR literature flip angles are given in radians from now on. A

pulse is thus a pulse and a pulse a /2 pulse. The correlation peaks created by mixing

with the pulse appear in the quadrant ( , ) for weakly coupled nuclei, while they

appear in the quadrant ( , ) for strongly coupled nuclei. A schematic spectrum for

the phenyl radical is shown in Fig. 10.20 and can be compared to the EPR spectrum in Fig.

10.13 and the one-dimensional nuclear spectrum in Fig. 10.17. For macroscopically disordered

systems with anisotropic hyperfine coupling, curved ridges result. The anisotropy of the

hyperfine coupling can be extracted from their curvature and their shift with respect to the

antidiagonal at I, even if only part of the correlation ridge can be observed (Fig. 10.21).

A

B

C

�I

�I

0

0

0

aiso/2

B/2

��

��

��

��

180

180

90

1 0 2 0

1 0 2 0

214

Figure 10.20: Schematic HYSCORE spectrum for the phenyl radical. The two doublets of the weaklycoupled para and meta protons are separated from the doublet of the strongly coupled ortho protons.

Figure 10.21: Schematic HYSCORE spectrum for a macroscopically disordered system withansiotropic hyperfine coupling.The dashed line is the I antidiagonal.

�( H)1

��( H)1

0 �2

�2

��(H)

1

A3/2

A1

A2

2+Taiso

Ta+iso

9T32I

I�I

2

�I antidiagonal

215

10.4.3 ENDOR spectroscopy

10.4.3.1 Davies ENDOR

The polarization of electron spin transitions is at least 660 times larger than the one of

nuclear spin transitions. In ENDOR experiments, part of this polarization is transferred to the

nuclear transitions, so that these transitions can be detected with much higher sensitivity. The

detection is also performed on electron spin transitions. Because of the higher energy of mw

photons compared to rf photons, this results in an additional sensitivity gain.

For liquid samples, ENDOR is best performed as a cw experiment. As cw ENDOR

involves simulataneous detection and strong mw and rf irradiation, it is a technically

demanding experiment. It is often easier to analyze the more complicated EPR spectra or to

perform a pulsed ENDOR experiment on a frozen solution, which yields additional

information on the anisotropy of the hyperfine couplings. For solid samples, pulsed ENDOR

experiments have replaced cw ENDOR almost completely, as they are technically less

demanding and work over a broader range of temperatures. Therefore, only pulsed ENDOR is

discussed in this lecture course.

Figure 10.22: Pulse sequence of the Davies ENDOR experiment. The influence of a rf pulse withvariable frequency on echo inversion is observed. The echo is formed and inverted by selective mwpulses with an excitation bandwidth that is smaller than the hyperfine coupling. Level populations atpoints 0 (thermal equilibrium), 1 (after inversion), and 2 (after the rf pulse) are shown in Fig. 10.23.

The conceptually most simple ENDOR experiment is Davies ENDOR (Fig. 10.22). In

this experiment one of the two electron spin transitions of a hyperfine multiplet is selectively

inverted by an mw pulse. The influence of this inversion pulse on level populations for the S

= 1/2, I = 1/2 system is shown in Fig. A,B. The pulse creates a state of two-spin order. An

�

�/2 �

�

� �T

m.w.

r.f.

0 1

2

216

ideally selective mw pulse creates polarization with opposite sign on the two individual

nuclear spin transitions. This polarization is just as large as the thermal equilibrium

polarization of the electron spin transitions.

Figure 10.23: Polarization transfers in the Davies ENDOR experiment for an S = 1/2, I = 1/2 system.A) Thermal equilibrium populations corresponding to point 0 in the pulse sequence shown in Fig.10.22. Only the electron transitions are significantly polarized. B) The selective mw pulse hasgenerated a state of two-spin order, where both nuclear transitions are oppositely polarized (point 1 inthe pulse sequence). C-E) Situations corresponding to point 2 in the pulse sequence for differentfrequencies of the selective rf pulse. C) Situation after an rf pulse that is resonant with the

transition. Both electron transitions are saturated, i.e. devoid of polarization. D)Situation after a non-resonant rf pulse. The originally excited electron transition is still inverted. E)Situation after an rf pulse that is resonant with the transition. Both electron transitionsare saturated.

The rf pulse is also transition-selective. It inverts either the transition

(Fig. 10.23C) or the transition (Fig. 10.23E) or it is off-resonant (Fig. 10.23D). In

the cases where the rf pulse is resonant with one of the nuclear transitions, both electron

transitions become saturated. In this situation the subsequent detection echo subsequence,

consisting of selective mw /2 and pulses on transition , does not give rise to an

echo (dotted zero line in Fig. 10.22). For an off-resonant rf pulse the transition is

still inverted and a negative echo is observed (solid signal line).

A Davies ENDOR spectrum of a single crystal of a copper(II) complex diluted into the

corresponding nickel(II) complex is shown in Fig. 10.24A. The lines are due to protons with

| ���

| ���

| ���

| ���

MW

RF

RF

A B

C

D

E

217

small hyperfine couplings (arrows) and directly coordinated nitrogens with large hyperfine

couplings. Note that unlike integral NMR line intensities, ENDOR line intensities are not

proportional to the number of observed nuclei. This is because the rf field amplitude is not

constant over such a broad frequency range. Furthermore the rf field at the nucleus is enhanced

by the hyperfine coupling. This hyperfine enhancement results from the non-resonant adiabatic

motion of the magnetic moment of the electron spin induced by the rf field.

Figure 10.24: Davies ENDOR spectra of [(2,2’-bipyridylamine)(diethylenetriamine)Cu]2+ diluted intoa single crystal of the corresponding isomorphous Ni(II) complex obtained at a temperature of 10 K. A)Lengths of the mw pulses t

(1) = 400 ns, t= 200 ns, t(2) = 400 ns, both proton and nitrogen

transitions are observed, arrows indicate proton transitions; B) Lengths of the mw pulses t(1) = 20 ns,

t= 100 ns, t(2) = 200 ns, most of the proton transitions are suppressed. C) Structure of the complex

(adapted from: C. Gemperle, A. Schweiger, Chem. Rev. 1991, 91, 1481-1505).

Polarization transfer in Davies ENDOR crucially depends on the selective excitation of

one electron spin transition. The excitation bandwidth of the mw pulses thus has to be smaller

than the hyperfine coupling. With decreasing hyperfine coupling excitation thus has to be more

and more selective, so that the experiment looses sensitivity. Davies ENDOR is thus best

suited to detect nuclei with relatively large hyperfine couplings (5-50 MHz).

The suppression of signals with small hyperfine couplings can be described by a

selectivity parameter

, [10.45]

A

C

B

15105 20�rf (MHz)

Cu

N

HN NH

NH

NHN

S

Aefft1

2------------------=

218

where Aeff is the hyperfine splitting and the length of the first mw pulse. Maximum

absolute ENDOR intensity Vmax is obtained for . As a function of S, the absolute

ENDOR intensity is given by

. [10.46]

Eq. [10.46] describes the hyperfine contrast selectivity that can be used to edit spectra. By

shortening the mw pulse length from 400 to 20 ns in the example shown in Fig. 10.24

ENDOR lines from the weakly coupled protons are strongly suppressed.

Figure 10.25: Effect of selective irradiation of an inhomogeneously broadened spectral line. A) Theinhomogeneously broadened line is a superposition of lines of spin packets with a smallerhomogeneous linewidth determined by T2 and different resonance offsets. B) The spin packets can beindividually excited, so that a hole is burnt into the spectral line. The minimum width of the hole forvery weak irradiation is the homogeneous linewidth 2/T2. For pulsed irradiation, the width of the wholeis determined by the maximum of 2/T2 and the excitation bandwidth of the pulse.

The suppression effect can be understood by considering the polarization changes in an

inhomogeneously broadened EPR line. Such a line with width inh consists of many

superimposed lines of spin packets with different resonance frequency (Fig. 10.25). Each of

these lines has a homogeneous linewidth . With weak continuous mw

irradiation it is possible to saturate just one spin packet and burn a hole with width hom into

the line. An mw pulse affects spin packets within its excitation bandwidth. Sensitivity of

pulsed EPR experiments increases with the number of excited spin packets.

A resonant rf pulse shifts half of the hole (half of the electron spin polarization) to a

transition whose resonance frequency differs by the hyperfine splitting Aeff. This shift occurs

towards higher or lower frequencies, depending on which of the transitions or

was inverted by the mw pulse. Thus half of the whole remains at zero

t1

S 2 2=

V S Vmax

2S

S2

1 2+----------------------

=

t1

inh

hom

> 2/T2

�mw

0

A B

hom 2 T2=

219

resonance offset, while a quarter is shifted by Aeff and another quarter by -Aeff (Fig. 10.26).

This shift of the hole can only occur if Aeff is larger than the width of the hole.

Figure 10.26: An on-resonant rf pulse in Davies ENDOR shifts intensity from a hole burnt into theinhomogeneous EPR line to side holes. A) Situation before the rf pulse (point 1 in Fig. 10.22). B)Situation after the rf pulse (point 2).

10.4.3.2 Mims ENDOR

The inversion of spin packets can be performed in a more controlled way by splitting

the mw pulse in two /2 pulses that are separated by an interpulse delay . This can be seen

by computing the effect of such a subsequence (/2)x--(/2)x on an inhomogeneously

broadened line with product operator formalism. For brevity, only the electron spin S is

considered so that the rotating-frame Hamiltonian is . The experiment starts at

thermal equlibrium with density operator .1 Application of an ideal mw /2 pulse

along x,

, [10.47]

results in

. [10.48]

After subsequent evolution for time under the density operator is given by

. [10.49]

1 The negative sign is a consequence of the negative electron charge that leads to preferential population of

states of electron spins with their spin antiparallel to the external field.

A B Aeff Aeff

! �2 /t�

H0 SSz=

eq Sz–=

eq 1 2 Sx

1 Sy=

H0

2 S Sycos S sin Sx–=

220

The following /2 pulse along x leaves the second term on the right-hand side of Eq. [10.49]

unaffected, while the first term is converted to negative polarization

. [10.50]

. The second term on the right-hand side of Eq. [10.50] decays by transversal relaxation or can

be removed by a phase cycle. The first term corresponds to a polarization grid in the

inhomogeneously broadened EPR line whose resolution depends on interpulse delay , but not

on the excitation bandwidth (Fig. 10.27). The excitation bandwidth determines the envelope of

the grating. It is thus possible to create a finely spaced grid to which many spin packets

contribute.

Figure 10.27: Polarization grating in an inhomogeneous line created by a (/2)x--(/2)x subsequence(numerical simulation). Unlike in the product operator treatment, the limited excitation bandwidth ofthe mw pulses is considered in this simulation.

The polarization grid can be detected by applying another mw /2 pulse. This creates

an FID signal of the oscillatory grating. As the Fourier transform of a cosine function is a Dirac

delta function, the FID signal corresponding to the first term on the right-hand side of Eq.

[10.50] is confined at time . It is a stimulated echo signal. The width of the echo is determined

by the excitation bandwidth of the /2 pulses and is thus comparable to the pulse length.

In the Mims ENDOR experiment the effect of an rf pulse with variable frequency on

the intensity of the stimulated echo is observed (Fig. 10.28). An on-resonant rf pulse shifts

two quarters of the polarization grating by Aeff and -Aeff, respectively, just as the rf pulse in

Davies ENDOR shifts two quarters of the burnt hole (compare Fig. 10.26). The unshifted half

of the grating and the two shifted quarters interfer. For with

interference is destructive. The stimulated echo is canceled. For an on-resonant rf pulse and

, the shifted gratings interfer constructively and the stimulated echo is nearly

3 S Szcos S sin Sx–=

�

1/�

Aeff 2k 1+ = k 0 1 =

Aeff 2k=

221

unaffected. This situation cannot be distinguished from the one for an off-resonant rf pulse.

The Mims ENDOR experiment thus features blind spots at ..

Figure 10.28: Pulse sequence for the Mims ENDOR experiment. The intensity of the stimulated echo

is observed as a function of the frequency of the rf pulse. The density operators are given in

Eq. [10.47-10.50] with .

The ENDOR efficiency

[10.51]

quantifies this blindspot behavior. The total intensity of the stimulated echo scales with

. By considering this and Eq. [10.51], it can be shown that very small hyperfine

couplings are detected with highest sensitivity at = T2. To safely detect all large couplings,

Mims ENDOR spectra have to be recorded at different and added. This averages the blind

spots.

10.4.4 ESEEM spectroscopy and the HYSCORE experiment

In NMR spectroscopy spin echo signals usually decay smoothly as a function of the

interpulse delays. In EPR spectroscopy, the decay is often superimposed by modulations with

nuclear frequencies. This electron spin echo envelope modulation (ESEEM) effect allows for a

detection of nuclear spectra without directly exciting the nuclear spins. The indirect excitation

of the nuclear spins arises from the tilt of the effective field at the nucleus with respect to the

external field (Fig. 10.18). Excitation of coherence on formally forbidden transitions can be

understood by transforming the mw Hamiltonian to the eigensystem of the static Hamiltonian

given in Eq. [10.40].

Aeff 2k=

�/2 �/2�/2

� �T

�m.w.

r.f.

�0^ �1

^ �2^ �3

^

13

0 eq=

FENDOR14--- 1 Aeff cos– =

e2– T2

222

10.4.4.1 Transition moments for allowed and forbidden transitions

In Eq. [10.40] the static Hamiltonian of the S=1/2, I=1/2 spin system is given in a

system where the z axes of both the electron and nuclear spin frame are parallel to the external

magnetic field and perpendicular to the linearly polarized mw field. In this frame the

oscillatory mw Hamiltonian, which describes the excitation, is given by

, [10.52]

with . The effective g value geff,x pertains to the x direction of the mw

field.

The spin transitions are, however, defined between eigenstates of the spin system. To

compute the excitation strengths for these transitions, Eq. [10.52] thus has to be transformed to

the eigenframe of . The necessary transformations can be read off directly from Fig. 10.18.

To bring the effective field axis to the z axis, the subspace of the electron spin S has to be

rotated clockwise about the y axis of the nuclear frame by angle . This is described by a

unitary transformation

, [10.53]

where is defined as

. [10.54]

The polarization operator is given by . The negative sign of the argument

of the arcustangens arises since a clockwise rotation is opposite to a rotation in mathematical

sense. A clockwise rotation is required since the y axis of a right-handed frame points into the

paper plane in Fig. 10.18. Likewise, the subspace corresponding to the state of S has to be

rotated anticlockwise about the y axis of the nuclear frame by angle . This is described by a

unitary transformation

[10.55]

with

H0

H1 1Sx=

1 geff x BB1 h=

H0

U iS

Iy– exp=

arcB–

2I A+------------------- tan=

S

S

E 2 Sz+=

U iSIy– exp=

223

, [10.56]

and the polarization operator . In this frame the static Hamiltonian for the

weak coupling case1 takes the form

. [10.57]

Since and commute, the transformation from Eq. [10.40] to Eq. [10.57] can be

described by a single unitary transformation

. [10.58]

The transformation of Eq. [10.52] to the eigensystem of is more simple with

Cartesian operators. By substituting the polarization operators, UEB takes the form

, [10.59]

with

, . [10.60]

Since , the second term in the argument on the right-hand side of Eq. [10.59] has

no effect on . In the eigenbasis, the oscillatory Hamiltonian takes the form

. [10.61]

The first term on the right hand side of Eq. [10.61] describes excitation of the allowed

transitions. The second term is more easily interpreted when the product operator is written in

terms of ladder operators,

. [10.62]

1 In the strong coupling case, the sign of either the or the term changes. Which sign changes depends

on the relative signs of I and A.

arcB

2I A–------------------- tan=

S

E 2 Sz–=

H0 SSz S

Iz SIz+ +=

S

Iy SIy

UEB U

U

UU

i S

Iy SIy+ – exp= = =

H0

UEB i 2SzIy Iy+ – exp=

–

2-------------------=

+

2-------------------=

Iy Sx[ , ] 0=

H1

H1EB

UEBH1UEB† 1Sxcos sin 1SyIy+= =

SyIy14--- S

+I-

S-I+

S+I+

– S-I-

–+ =

224

The terms on the right hand side of Eq. [10.62] drive the forbidden electron-nuclear transitions

(first two terms) and (last two terms) as indicated in Fig. 10.29A.

Figure 10.29: Level scheme and schematic EPR spectrum of the S=1/2, I=1/2 system. A) Level schemewith allowed electron (green), forbidden electron-nuclear (red), and nuclear (blue) transitions. B)Spectrum with allowe (green) and forbidden (red) transitions, where and

The transition moment of allowed transitions thus scales with cos, while the transition

moment of forbidden transitions scales with sin. Spectral intensities are proportional to the

transition probability, which is the square of the transition moment. This is because the

transition moment applies to both excitation and detection. The EPR spectrum of the S=1/2,

I=1/2 system with anisotropic hyperfine interaction thus has the appearance shown in Fig.

10.29B. The hyperfine splitting is given by the difference frequency

[10.63]

and the splitting of the forbidden transitions by the sum frequency

. [10.64]

10.4.4.2 Two-pulse ESEEM

Two-pulse ESEEM is observed by measuring the amplitude of a (/2)--()- echo as a

function of the interpulse delay . The first /2 pulse excites coherence with amplitude cos on

the two allowed transitions and coherence with amplitude sin on the two forbidden

transitions. The coherence evolves during the first interpulse delay , defocuses due to the

distribution of resonance offsets S, and decays with the transversal relaxation times T2a or T2f

of the respective transitions. The pulse inverts the phase of all coherences and thus leads to a

1

2

3

4

��

��

�24�13

�14 �23

�

�

s

+

��

��

��

��

A B

cos2�

sin2�

- –=+ +=

Aeff - –= =

+ +=

225

refocusing of magnetization after another delay . Those pathways, where the coherence

remains on the same transition during the transfer by the pulse, contribute two-pulse echoes

that smoothly decay with the respective relaxation rates.

Figure 10.30: Branching of magnetization (coherence transfers) during the pulse in the two-pulseESEEM experiment. A) Transfer of coherence on the allowed transition to the two allowedand two forbidden transitions combined with a phase inversion. B) Transfer of coherence on theforbidden transition to the two allowed and two forbidden transitions combined with a phaseinversion.

However, the pulse also changes the nuclear spin states with a probability sin2, and

this corresponds to a transfer of coherence between the different transitions. This branching of

magnetization is indicated in a density matrix representation in Fig. 10.30. For instance,

coherence on the allowed transition is merely phase-inverted with probability

cos2and transferred with probability sin()cos() to the forbidden transition (Fig.

10.30A). In contrast, coherence on the forbbiden transition is phase-inverted only

with the small probability sin2and transferred to the forbidden transition with the

larger probability cos2 (Fig. 10.30B).

All transferred coherences also experience phase inversion. Thus they also form

echoes. These coherence transfer echoes oscillate with the difference of the transition

frequencies before and after the pulse. From Fig. 10.29 and Eqs. [10.63] and [10.64] it can be

verified that the possible difference frequencies are , , , and . Each of the basic

nuclear frequencies and appears in two coherence transfer pathways, and each of the

combination frequencies and appears in only one pathway. Furthermore, the product of

the excitation probability (transition moment for excited coherence), transfer probability

(branching factor), and detection probability (transition moment for detected coherence) is the

A B

cos �

cos �cos �cos �

cos �-sin � sin �

cos �-sin �

sin � sin �

sin �

2 2

22

|���

"��|

"��|

"��|

"��|

|���|��� |���|��� |���|��� |���

226

same for each pathway, it takes the value sin2 cos2 = sin2(2)/4. The amplitude of all

nuclear modulations thus scales with the modulation depth parameter

. [10.65]

With proper bookeeping of the coherence transfer pathways, the two-pulse ESEEM formula

[10.66]

results. A schematic spectrum for the weak coupling case is shown in Fig. 10.31. Note that he

combination frequencies are close to A and 2I, but are slightly shifted towards lower and

higher frequencies, respectively. The phase information (positive basic frequency peaks and

negative combination peaks) is often hard to access, as spectrometer dead time prevents

observation of the signal at short values. Usually phase correction fails and magnitude

spectra are displayed, where all peaks are positive.

Figure 10.31: Schematic two-pulse ESEEM spectrum for a weakly coupled S=1/2, I=1/2 system.

10.4.4.3 Three-pulse ESEEM

Nuclear frequencies are measured by two-pulse ESEEM as differences between the

frequencies of electron spin transitions. Hence, the linewidth in the spectra is twice the

homogeneous EPR linewidth, which is much larger than the natural linewidth for nuclear

k 2sin2BI

--------------

2= =

V2p 1k4--- 2 2 cos– 2 cos - cos + cos+ +– –=

A

�I

2�I

227

transitions. The resolution can thus be drastically improved by observing the evolution of

coherence on nuclear transitions.

Figure 10.32: Magnetization branching during the second /2 pulse in a mw pulse subsequence. A) Coherence on the allowed electron spin transition is transferred

to nuclear coherences with probability sin. B) Coherence on the forbidden transition istransferred to nuclear coherences with probability cos.

A single, ideal mw pulse does not excite nuclear coherence. However, an mw pulse

subsequence (/2)--(/2) does generate such coherence due to magnetization branching

during the second /2 pulse (Fig. 10.32). This nuclear coherence evolves during delay T and is

then transferred to observable electron coherence by another /2 pulse. After another delay ,

the dispersion of electron spin resonance offsets S is refocused. The coherence transfer

pathways that involve nuclear coherence contribute signals that oscillate with frequencies and as a function of time T. Coherence transfer pathways that do not involve nuclear

coherence give rise to a smoothly decaying stimulated echo. The total echo signal as a function

of the fixed interpulse delay and the variable delay T, neglecting relaxation, is given by

. [10.67]

No oscillation with combination frequencies is observed. This simplifies spectra and is an

additional advantage of three-pulse ESEEM compared to two-pulse ESEEM. However, the

amplitude of the nuclear oscillation now depends on the nuclear frequency in the other electron

spin state and on the fixed delay . This leads to blind spots, which are a disadvantage of three-

A B|���

"��|

"��|

"��|

"��|

|���|��� |���|��� |���|��� |���

cos�

cos�

cos�

sin�

cos�i2

i2

sin�

sin�

i2

i2

i2

i2

-

i2

-

sin�i2

-

2 – 2 –

V3p T 1k4--- 1 cos– 1 T + cos– –=

1 cos– 1 T + cos– +

228

pulse ESEEM. To avoid suppression of peaks, the experiment thus has to be performed at

several values and the magnitude spectra have to be added.

Figure 10.33: Pulse sequence for three-pulse ESEEM. The echo amplitude is observed as a function ofinterpulse delay T for fixed delays .

10.4.4.4 HYSCORE

Hyperfine sublevel correlation spectra, as introduced in Section 10.4.2.3, can be

measured by a two-dimensional extension of the three-pulse ESEEM experiment. For that

purpose, an mw pulse is introduced at a variable delay t1 after the second /2 pulse (Fig.

10.34). This pulse transfers nuclear coherence between the two manifolds corresponding to

the electron spin states and . Thus it correlates frequency of a given nucleus with

frequency of the same nucleus and vice versa. The new frequency is measured by

introducing another variable delay t2 between the pulse and the final /2 pulse that

reconverts the nuclear coherence to observable electron coherence.

Figure 10.34: Pulse sequence of the two-dimensional HYSCORE experiment. Echo amplitude isobserved as a function of the two variable delays t1 and t2 for fixed delays .

Forbidden transitions during the pulse convert some nuclear coherence to electron

polarization and vice versa. This leads to axial peaks at 1 = 0 and 2 = 0. These peaks are

unwanted, as they do not contain correlation information. They can be removed by baseline

correction of the time-domain data before Fourier transformation. Because of its limited

excitation bandwidth, the nominal pulse does not fully invert the electron spin for large

resonance offsets S. This leads to diagonal peaks 1 =2 = and 1 = 2 = . These

�/2 �/2 �/2

� �T

�/2 �/2 �/2

�

� �t1 t2

229

diagonal peaks cannot reliably be removed by data processing and may obscure cross peaks for

small hyperfine couplings. Their contribution scales with the ratio between the excitation

bandwidths of the /2 pulses and the pulse. Therefore, it is advantageous to use the full mw

power for the pulse and less power for the /2 pulses, as indicated in Fig. 10.34.

The modulation in the HYSCORE experiment corresponding to the cross peaks is

described by

[10.68]

with

,[10.69]

.[10.70]

In this representation with unsigned nuclear frequencies, the weak coupling case corresponds

to and the strong coupling case to . In the former case, the cross

peaks are thus stronger in the quadrant (1>0, 2>0) and in the latter case in the quadrant

(1>0, 2<0).

The blind spot behavior of the HYSCORE experiment is described by the two factors

and in Eq. [10.68]. A cross peak is thus suppressed when one of the

two correlated nuclear peaks is suppressed in three-pulse ESEEM. Therefore it is even more

important in HYSCORE to add several magnitude spectra obtained at different delays .

10.5 Spin probes, spin labels, and spin traps

Structure and dynamics of complex biological systems or synthetic materials are often

difficult to characterize by techniques that derive their signal from the whole sample. The

problems are a lack of contrast between different components, insufficient resolution, and

ambiguities in signal assignment. The situation simplifies considerably with techniques that

detect signals only from a probe molecule that is attached to a certain site of interest in the

system. EPR spectroscopy can be used as such a probe technique for diamagnetic systems. The

V4p t1 t2 k2---

2

---------- sin

2

--------- V

t1 t2 V t1 t2 + sin=

V t1 t2 cos2 t1 t2 +

2---+ +

cos sin2 t1 t2 -2---+–

cos–=

V t1 t2 cos2 t1 t2 +

2---+ +

cos sin2 t1 t2 -2---+–

cos–=

cos2 sin2 sin2 cos2

2 sin 2 sin

230

spectroscopic response of the spin probe is interpreted in terms of structure and dynamics of

the system. Ideal probes do not perturb the system under investigation, can be directed at will

to sites of interest, and give a spectroscopic response that strongly depends on their

environment.

The EPR spin probe technique is particularly suitable for soft organic matter, as it does

not rely on long-range order and as the spectra of typical spin probes are very sensitive to

molecular dynamics on time scales between 10 ps and 1 s. This time range corresponds to

motion on length scales longer than a chemical bond, but smaller than colloidal dimensions

that start at about 100 nm. On these length scales supramolecular interactions determine the

self-organization, stability, and functionality of soft matter systems. Distances between spin

probes can be measured on length scales between about 0.8 and 8 nm.

Figure 10.35: Structures of the most common spin probes and labels. R is a functional group used fordierctiong the probe or attaching the label to the site of interest. 1 4-oxo-2,2,6,6-tetramethyl-piperidin.2 2,2,6,6-tetramethyl-piperidin-1-oxyls, TEMPO derivatives. 3 2,2,5,5-tetramethyl-pyrrolydine-1-oxyls, PROXYL derivatives. 4 2,2,5,5-tetramethyl-pyrroline-1-oxyls. 5 4,4,-dimethyl-3-oxazolidinyloxy probes, DOXYL derivatives. 6 Methanethiosulfonate spin label, MTSSL.

The most widely applied class of spin probes are nitroxides (Fig. 10.35). They are

originally derived from compound 1, 4-oxo-2,2,6,6-tetramethyl-piperidin, which is the basis

compound of hindered amine light stabilizers. This compound is formed in a one-pot reaction from

acetone and ammonia and is easily oxidized to TEMPON, 4-oxo-2,2,6,6-tetramethyl-piperidin-1-oxyl

(compound 2 with -R being =O). The carbonyl group can be converted to other functional groups (e.g. -

OH, -NH2, -N+(CH3)3). Such groups direct site attachment via hydrogen bonds or electrostatic

interactions (spin probes in a strict sense). Reactive groups R can be used for covalent attachment to the

site of interest (spin labels). Somewhat smaller probes or labels are obtained by reducing the ring size to

N

O

R

N

H

O

N

O

R

N

O

R

O N O

N OS

S

O

O

H3C

1

5 6

2 3 4

231

PROXYL derivatives 3 or 3,4-dehydro-PROXYL derivatives 4. The most important label is the

methanethiosulfonate spin label (MTSSL) 6 that attaches with high selectivity under very mild

conditions to cysteine residues in proteins. The size of the MTSSL-derived sidechain is similar to the

one of an aromatic amino acid residue. The SDSL technique for proteins involves mutation of an amino

acid at the site of interest to cystein and subsequent reaction with a thiol-selective spin labels such as

MTSSL, iodoacetamido-PROXYL, or maleimido-PROXYL. For distance measurements two selected

sites are labeled.

Synthesis of DOXYL derivatives 5 is more difficult and proceeds with lower yields.

Their advantage is a very rigid attachment to alkyl chains that allows to probe motion and

liquid crystalline ordering of surfactant molecules, lipids, and steroids in more detail.

The unpaired electron in nitroxides is stabilized by delocalization over the N-O bond

and by the steric hindrance due to the four methyl groups that prevents dimerization.

Derivatives of TEMPO, PROXYL, and 3,4-dehydro-PROXYL are usually stable up to

temperatures of 130 C. They are degraded by strong acids (pH 1 and below) and slowly

decompose in solution at pH 2 and below. Nitroxides are reduced to the corresponding

hydroxylamines by ascorbic acid and may be reduced by some thiols. PROXYL derivatives

are less susceptible to this reduction than TEMPO derivatives. The hydroxylamines are

reoxidized by air.

Figure 10.36: Molecular frame (PAS of the g tensor) for nitroxides and dependence of the 14Nhyperfine coupling on orientation of the magnetic field with respect to this frame.

H

N

R

O.

z

y

x

A( N)14

A( N)14

232

10.5.1 Nitroxide spectra with and without orientational averaging

The unpaired electron in nitroxides is distributed mainly in a orbital along the N-O

bond that is made up of the pz orbitals of the nitrogen and oxygen atom. There is also some

spin density in the s orbitals of these two atoms, but few delocalization to the neighboring

carbon atoms. The deviation of the g value from ge is due to spin orbit coupling in excited

states that involve the lone pair orbitals on oxygen. This deviation is maximum in the x

direction of the molecular frame along the N-O bond (Fig. 10.36), intermediate in the y

direction, and almost negligible in the z direction. Typical principal g values are gxx = 2.0090,

gyy=2.0060, and gzz= 2.0024. In contrast, the hyperfine coupling to 14N is strongest if the

external field is along the z direction, i.e. parallel to the lobes of the pz orbital on nitrogen.

Differences between the hyperfine coupling in the x and y direction are minute, so that an

axially symmetric hyperfine tensor can be assumed. Typical hyperfine couplings are Axx= Ayy

= 18 MHz, Azz= 96 MHz. All magnetic parameters differ slightly between the different classes

of nitroxides. The z axes of the hyperfine and g tensor are nearly coincident. Hyperfine

couplings to the protons of the methyl groups in the 2- and 6-positions may be resolved in the

limit of fast motion (see Section 10.5.2) in the absence of oxygen. In other cases the proton

couplings are unresolved and lead to line broadening. Narrower lines are then obtained with

deuterated nitroxides.

Figure 10.37: Schematic energy level scheme and cw EPR spectrum of a nitroxide radical. Only theelectron Zeeman and 14N hyperfine interactions are considered.

The EPR spectrum of a nitroxide in a single orientation or in the regime of fast

orientational averaging is schematically shown in Fig. 10.37. The energy levels of the unpaired

electron result from electron Zeeman splitting in the external field (magnetic quantum number

E

B0

+1/2

-1/2

h�mw

-10

+1

+1 0 -1+1

0-1

mS mI

233

mS) and hyperfine coupling to the 14N nuclear spin (product of quantum numbers mS and mI).

For the allowed transitions mI is constant, so that a hyperfine triplet results. As the hyperfine

coupling is positive, the transition with the highest frequency corresponds to mI= +1. In a field

sweep this transition is detected at lowest field. When Fig. 10.37 is interpreted as a spectrum in

the regime of fast orientational averaging, the center line is at a field corresponding to the

isotropic g value giso= (gxx+gyy+gzz)/3 and the splitting between two outer lines is 2Aiso=

2(Axx+Ayy+Azz)/3. If the Figure is interpreted as a single-crystal spectrum, the center line is

positioned at the g value along the magnetic field direction in the molecular frame and the

splitting is determined by the hyperfine coupling along this direction.

The powder spectrum is the superposition of the spectra at all orientations with proper

weighting. Each of the lines with different 14N magnetic quantum number mI can be

considered separately (Fig. 10.38A). The variation of the resonance field is minimum for the

mI= 0 transition, where the hyperfine contribution vanishes. It is maximum for the mI= -1

transition, where the hyperfine contribution and electron Zeeman contribution have the same

sign, and intermediate for the mI= +1 transition, where the two contributions have different

sign.

Figure 10.38: Nitroxide spectrum for a macroscopically disordered system at X band frequencies. A)Absorption subspectra of the three transitions with different 14N nuclear magnetic quantum number mI.B) Total absorption spectrum, resulting from the sum of the three subspectra (top) and its derivative(bottom) that corresponds to the cw EPR spectrum.

The total absorption spectrum is the sum of the two subspectra (Fig. 10.38B). It can be

measured by echo-detected field-swept EPR. In cw EPR the derivative of this absorption

spectrum is observed. It has a positive peak at the low-field edge that corresponds to the edge

of the mI= +1 line and a negative peak at the high-field edge that corresponds to the edge of the

mI= -1 line. The outer extrema splitting between these two peaks is twice the hyperfine

0.34 0.340.345 0.3450.35 0.35

mI=-1

mI=0

mI=+1

B0 (T) B0 (T)

A B 2Azz

234

coupling along the molecular z orientation 2Azz. The central line with a quasi-dispersive shape

results mainly from the mI= 0 transition.

10.5.2 Nitroxide spectra for incomplete orientational averaging

The effect of rotational motion of the nitroxide on the spectrum can be understood in

terms of a multi-site exchange between different orientations. The exchange rate is related to

the rotational diffusion rate R, which in turn is given by R= 1/6r, where r is the rotational

correlation time. The basic concepts are the same as for two-site chemical exchange (Sections

3.1 and 3.2). In the limit of slow exchange, the spectrum is lifetime broadened (Figure 3.8).

This effect is unresolved in nitroxide spectra. If the motion becomes faster, exchange between

orientations in addition to the line broadening leads to partial averaging of the two resonance

frequencies (Fig. 3.6). As a consequence the outer extrema splitting 2A’zz reduces (Fig. 10.39).

This effect becomes observable at rotational correlation times shorter than about 1 s.

Figure 10.39: Dependence of the nitroxide cw EPR spectrum on the rotational correlation time r.

Fast regime Slow tumbling

2 3.0 mTA 'zz !

2 6.8 mTA 'zz !

�r �r

10 ps 4 ns

32 ps 10 ns

100 ps 32 ns

316 ps 200 ps

1 ns 316 ns

3 ns 1 µs

235

For a symmetric two-site exchange a normalized exchange rate constant of

corresponds to coalescence, i.e. a collapse of the doublet into a single line, and maximum

exchange broadening (Fig. 3.6). For the multi-site orientational exchange in nitroxides such

coalescence is observed at a rotational correlation time of about 3.5 ns, where the appearance

of the spectrum changes from a powder like lineshape (right column in Fig. 10.39, slow

tumbling) to a triplet of Lorentzian broadened lines (left column, fast regime). In the fast

regime the outer extrema splitting 2A’zz is only slightly larger than twice the isotropic

hyperfine coupling (Fig. 10.40). The outer extrema splitting depends most strongly on r at

coalescence, where it is mT. Dynamic processes are often studied as a function of

temperature. They can then be characterized by the temperature T5mT (or T50G), where

mT is attained.

Figure 10.40: Dependence of the outer extrema splitting 2A’zz on the rotational correlation time (semi-logarithmic plot).

For thermally activated processes, 1/r is expected to have an Arrhenius dependence on

temperature,

, [10.71]

where EA is the activation energy. The rotational correlation times at different temperatures

can be determined by lineshape fitting.

These considerations assume isotropic Brownian rotational diffusion, which is often a

good approximation for small, almost spherical spin probes like TEMPO (compound 2, R=H).

Different lineshapes are observed for more complicated dynamic processes. However, only in

simple cases it is possible to derive a model for the motion from the spectral lineshape.

1 2 2

2A'zz 5

2A'zz 5=

-10 -9 -8 -7 -63

4

5

6

7

2'(m

T)

Azz

log( /s)�r

slow

tumbling

fast

regime

1 r ln A EA RT +=

236

Lineshapes predicted from molecular dynamics simulations are often in good agreement with

experimental lineshapes.

In the fast regime, the widths of the three lines are different. This can be understood

from Fig. 3.9. The mean square difference of the resonance frequencies of

exchanging orientations is smallest for the narrow central line with mI= 0 and largest for the

broad high-field line with mI= -1. Therefore, the high-field line is broader also in the case of

fast orientational exchange. Only in the fast limit, where other contributions to line broadening

dominate, equal widths and amplitudes are observed for the three lines. This limit is attained at

rotational correlation times of 10 ps or shorter.

In the fast regime r can be obtained from analyzing the linewidths B according to

Kivelson theory.1 According to this theory, the ratio of the line width of one of the outer lines

to the line width of the central line is given by

, [10.72]

where

[10.73]

and

, [10.74]

with the hyperfine anisotropy parameter

[10.75]

and the electron Zeeman anisotropy parameter

. [10.76]

1 D. Kivelson, J. Chem. Phys. 1960, 33, 1094-1106. Theory of ESR Linewidths of Free Radicals.

1 2– 2

T21– mI

T21– 0

-------------------- 1 BmI CmI2+ +=

B415------– bB0T2 0 r=

C18---b2T2 0 r=

b43

------ Azz

Axx Ayy+

2-------------------------–=

2B

h------------- gzz

gxx gyy+

2----------------------–=

237

The relaxation time T2(0) for the central line can be computed from the corresponding peak-to-

peak linewidth in field domain as

. [10.77]

Thus, Eqs. [10.72-10.74] can be solved for the only remaining unknown r. In practice, ratios

of peak-to-peak line amplitudes I(mI) are analyzed rather than linewidth ratios, as they can be

measured with higher precision. The linewidth ratio is related to the amplitude ratio by

, [10.78]

since the integral intensity of the absorption line is the same for each of the three transitions.

The rotational correlation time can thus be determined by, e.g.,

, [10.79]

where B0 is the peak-to-peak linewidth of the central line. Similar formulas can be obtained

by using different pairs of line amplitudes or all three line amplitudes. It is good practice to test

whether several of these formulas give similar values for r. If this is not the case, the motion

cannot be characterized by a single rotational correlation time.

10.5.3 Dependence of magnetic parameters on polarity and hydrogen bonds

The g and hyperfine tensor of a nitroxide depend on the distribution of the unpaired

electron between the orbitals on the nitrogen and oxygen atom. This distribution in turn is

influenced by the polarity of the environment, as can be understood from the mesomeric

structures shown in Fig. 10.41. Localization of the unpaired electron at the nitrogen atom

corresponds to a charge-separated structure that is stabilized by a polar environment. Therefore

more spin density N on the nitrogen and less spin density O on the oxygen atom is found in

such a polar environment.

Bpp 0

T2 0 h

3gisoBBpp 0 ------------------------------------------------=

T21– mI

T21– 0

--------------------I mI I 0

-------------=

r 3

b----------

b8---

4B0

15----------------–

1– gisoB

h----------------Bpp 0 I 0

I 1– ------------- 1–=

N NO O+ -

238

Figure 10.41: Mesomeric structures of a nitroxide with the unpaired electron localized on the nitrogenatom (left) and the oxygen atom (right).

The isotropic 14N hyperfine coupling Aiso and the coupling along the lobes of the pz

orbital (Azz) increase with increasing N and thus with increasing polarity. The deviation of the

g value from ge changes in the opposite way, as it results mainly from spin-orbit coupling in an

excited state that involves alone pair localized on oxygen. The strongest change is observed for

gxx, as is expected when the lone pair orbitals have mainly py character.1 The g value shift

gxx= gxx-ge is influenced not only by changes in the spin density distribution, but also by

changes in the energy difference between the SOMO and the lone pair orbital. Hydrogen

bonding lowers the energy of the lone pair and thus increases this energy difference. This

corresponds to a blue shift of the n-* transition in optical spectra and to an increase of the

denominator in Eq. [10.8]. Thus hydrogen bonding leads to a smaller g shift gxx at same

polarity. Hydrogen bonding also leads to some delocalization of the lone pair orbitals, which

leads to a minor further decrease of gxx. By measuring both Azz and gxx with high precision

with high-field EPR, it is possible to separate the contributions due to polarity and hydrogen

bonding.

10.5.4 Accessibility measurements

The influence of Heisenberg exchange of unpaired electrons between paramagnetic

species on relaxation times (see Section 10.4.1, Fig. 10.12) can be used to estimate the local

concentration of a paramagnetic quencher near a nitroxide spin label. An ubiquitous

paramagnetic quencher is triplet oxygen, which is much better soluble in apolar environments,

such as lipid bilayers, than in polar environments. Water-soluble quenchers are transition metal

complexes such as the electroneutral Ni(II) complex of ethylendiaminediacetic acid (EDDA)

or the negatively charged Cr(III) complex of oxalate [Cr(C2O4)3]3-. At very high local

concentrations such quenchers cause line broadening. At lower concentration their influence

on relaxation times can be quantified easily and precisely by measuring saturation curves (see

Section 2.7.2).

1 Fo a full discussion, see: T. Kawamura, S. Matsunami, T. Yonezawa, Bull. Chem. Soc. Japan 1967, 40,

1111-1115. Solvent Effects on the g-Value of Di-t-butyl Nitric Oxide.

239

The change in relaxtion rates is purely through collisions (Fig. 10.11) if the

longitudinal relaxation time of the quencher is much shorter than the lifetime of the collisional

encounter complex. In this regime it is equal to the exchange rate Wex,

, [10.80]

where kex ist the exchange rate constant and C the local quencher concentration.1 In the

following we assume the case of insignificant exchange broadening, . Up to a

constant factor , Wex can then be determined from saturation curves. For this, the peak-to-

peak amplitude A of the first derivative central line of the nitroxide spectrum (mI= 0) is

measured as a function of mw power P. The data are described by the function

[10.81]

with the power-independent amplitude factor I, the power at half saturation P1/2, and the

inhomogeneity parameter as adjustable parameters. The inhomogeneity parameter takes the

value = 1.5 in the homogeneous limit, which is assumed from here on, and = 0.5 in the

inhomogeneous limit. More precisely, P1/2 is the incident mw power where A is reduced to

half of its unsaturated value. This is given by

, [10.82]

where is the conversion efficiency of the mw resonator. For small quencher

concentrations exchange broadening is insignificant and T2 is the same in the presence and

absence of the quencher. The change in P1/2 due to addition of the quencher is then given by

. [10.83]

1 The term ‘local’ refers to the characteristic diffusion length of the quencher on the time scale given by Wex.

The factorization into kex and C may be ill-defined when comparing labels attached at different sites of a

protein, but Wex remains a meaningful parameter.

1T1------ 1

T2------ Wex kexC= = =

Wex 1 T2«

A P I P

1 21

1– P P1 2+ -----------------------------------------------------------=

P1 243 1–

e22T1T2

-------------------------=

B1 P=

P1 243 1– Wex

e22T2

--------------------------------=

240

To eliminate the dependencies on T2 and , a dimensionless accessibility parameter is

defined as1

, [10.84]

where the denominator is obtained from measurements on a reference substance such as dilute

diphenylpicrylhydrazyl powder in KCl.

10.5.5 Spin trapping

In many processes that involve free radicals, these radicals react very fast, so that their

lifetimes are extremely short and their steady-state concentrations extremely low. Such

radicals can be detected indirectly by trapping them with a diamagnetic compound. The

reaction product is a stable free radical whose magnetic parameters provide some information

on the original, short-living radical. The most important class of diamagnetic spin traps are

nitrones, such as -phenyl-t-butyl nitrone (PBN) or 5,5-dimethyl-1-pyrroline N-oxide

(DMPO). Free radicals add to the carbon of nitrones and a nitroxide is formed (Fig. 10.42).

The hydrogen bound to the carbon makes nitroxides generated by such reactions less stable

than the nitroxides used as spin probes. The lifetime of spin trap products in solution typically

ranges between a few seconds and a few hours. This is usually sufficient to accumulate steady-

state concentrations that are detectable by cw EPR

Figure 10.42: Reaction of the spin traps -phenyl-t-butyl nitrone (PBN) or 5,5-dimethyl-1-pyrroline N-oxide(DMPO) with free radicals R·

1 C. Altenbach, W. Froncisz, R. Hemker, H. Mchaourab, W. L. Hubbell, Biophys. J. 2005, 89, 2103-2112.

Accessibility of Nitroxide Side Chains: Absolute Heisenberg Exchange Rates from Power Saturation EPR.

P1 2 Bpp 0 P1 2 Bpp 0 ref

-------------------------------------------------- Wex= =

N N

O O

NN

OO

PBN

DMPO

�

�

+R�

�

�

R

R

+R�

H

H

241

.

Figure 10.43: cw EPR spectrum observed by UV irradiation ( < 380 nm) of a dispersion of ZnO inheptane in the presence of oxygen and the spin trap DMPO (from C. Cheng, R. P. Veregin, J. R.Harbour, M. I. Hair, S. L. Issler, J. Tromp, J. Phys. Chem. 1989, 93, 2607-2609. Photochemistry of ZnOin Heptane: Detection by Oxygen Uptake and Spin Trapping.

The hyperfine coupling to the -proton is usually resolved, so that a six-line spectrum

results. In some cases, further couplings may be resolved. For instance, the spectrum of the

DMPO adduct of the superoxide anion radical O2 (Fig. ) has twelve lines with couplings AN

= 12.9 G, AH= 6.3 G, A

H= 1.6 G. The coupling to one of the two -protons is not fully

resolved. Spin trapping has its main field of application in studies on degeneration processes in

living cells. Such processes involve reactive oxygen species, most of which are radicals.

Furthermore spin-trapping is used in studies on hetrerogeneous catalysts that promote radical

formation. The radical adducts of spin traps are mainly identified via the 14N and -1H

hyperfine couplings. The isotropic g value of the adduct also depends on the type of original

radical, but is less characteristic. An internet database for identification of radical adducts of

several widely used spin traps is maintained by the NIH (http://tools.niehs.nih.gov/stdb/

index.cfm).

242

10.6 Distance measurements by EPR techniques

The high sensitivity of EPR spectroscopy and the larger magnetic moment of the

electron spin allow for the measurement of longer distances between spins than is possible

with NMR. In conjunction with SDSL this allows for studying structure and structural

dynamics of biomacromolecules and their complexes, even if they cannot be crystallized and

have molecular weights larger than 100 kDa. The main current application field are large

membrane proteins. Similar techniques can also be applied to synthetic polymers or to soft

matter that forms self-organized structures based on supramolecular interactions.

Distance measurements are based on the inverse cube dependence of the magnetic

dipole interaction on distance (Section 5.8). The data can be analyzed with high precison

( Å for distances up to 40 Å) if the spin pairs are sufficiently diluted and exchange

coupling (10.3.6.1) is negligible. On length scales between 0.8 and 8 nm, where such

measurements are possible, spins are sufficiently diluted at concentrations of 200 M or less.

Exchange couplings are usually negligble at distances beyond 1.5 nm. Shorter distances than

1.5 nm can be determined with a precision of about Å.

10.6.1 cw EPR

10.6.1.1 Analysis of dipolar line broadening

The peaks in a well resolved first derivative absorption spectrum of a nitroxide in the

rigid limit have a typical width of 0.4 mT corresponding to 11.2 MHz. An additional

broadening with about half this width can be safely detected. According to Eq. [10.32] this

corresponds to a spin-spin distance of up to 21 Å that still leads to recognizable dipolar

broadening. If the spin label environment is homogeneous and very similar for both labels, the

peak width in cw EPR spectra is determined by unresolved hyperfine couplings of methyl

protons in the label. The limit can then be shifted to distances of about 25 Å by using a

deuterated label. If the environment is heterogeneous or different for the two labels,

differences in hydrogen bonding and polarity may cause additional broadening due to a

distribution of the 14N hyperfine coupling and g value (Section 10.5.3). The upper limit for

0.5

2

243

elucidating accurate distances then decreases to about 16 Å and cannot be extended by

deuteration.

The distance can be determined with the highest precision if the cw EPR spectra both in

the absence and presence of the dipole-dipole interaction are available. Hence, the spectrum of

the doubly-labeled macromolecule and the two spectra of the corresponding singly-labeled

macromolecules have to be measured. The spectrum of the doubly labeled molecule can then

be fitted by convolution of the sum of the spectra of the isolated paramagnetic centers with a

dipolar broadening function (Fig. 10.44). The dipolar broadening function in turn is simulated

by superimposing Pake patterns for a Gaussian distribution of distances. Fit parameters are the

mean distance and width of the Gaussian peak.

Figure 10.44: Distance determination from a dipolar broadened cw EPR spectrum by fitting of aconvolution of the spectrum of isolated paramagnetic centers with a dipolar broadening function.

The convolution method assumes that the macromolecule is completely doubly

labelled. This assumption can be relaxed by fitting a superposition of the dipolar broadened

and unbroadened lineshapes. If the spectrum of the isolated centers is unknown, a simulated

spectrum can be used instead. Both these variations of the method cause a loss in accuracy and

a decrease in the upper distance limit.

10.6.1.2 Intensity of half-field transitions

The convolution technique neglects the influence of exchange coupling between the

electron spins on the line shape. This is a good approximation at the upper end of the distance

range (1.5... 2 nm), but a poor approximation at the lower end (0.8... 1.2 nm). The influence of

the isotropic exchange coupling is eliminated in a technique that relies on mixing of the states

r r

isolatedcenters

trialdistancedistribution

exp. dipolarbroadened

exp. dipolarbroadened

sim. dipolarbroadened

IntegrateFT

inverse FT

FTSimulate/ fitdipolarspectrum

Multiplyderivative

Convolution

B0

B0

B0

r

B0

" �r

�r

244

of the two spins by the anisotropic dipole-dipole coupling. The two types of electron-electron

coupling can be distinguished, since the anisotropic coupling contributes off-diagonal terms

that connect the and states of the S1= 1/2, S2= 1/2 two-spin system ( and terms

in the dipolar alphabet, see Eq. [5.80] and Fig. ), while the exchange coupling does not.

Therefore, only the dipole-dipole coupling mixes theses levels and leads to a non-zero

transition probability of the double quantum transition. At constant field, this

transition would be observed at the sum of the two electron Zeeman frequencies, i.e., at twice

the electron Zeeman frequency of the nitroxide radical. Since cw EPR experiments are

performed at constant mw frequency and variable field, the double-quantum transition is

observed at half the field of the usual single-quantum spectrum.

Figure 10.45: Energy level scheme for a system consisting of two electron spins S1= 1/2 and S2= 1/2with assignement of the terms of the dipolar alphabet and of zero-quantum (ZQ), single-quantum (SQ),and doubl;e-quantum (DQ) transitions.

The intensity of the half-field transition depends on the ratio between the dipole-dipole

coupling and the electron Zeeman interaction. The distance can thus be determined from the

integral intensities V(DQ) of the double-quantum (DQ) transition and V(SQ) the four single

quantum (SQ) transitions.1 Numerical computations provide the approximate relation2

[10.85]

that is valid up to X band frequencies. The approach neglects anisotropic contributions of the

exchange coupling, as is appropriate for distances larger than about 6 Å. For distances larger

1 S. S. Eaton, G. R. Eaton, J. Am. Chem. Soc. 1982, 104, 5002-5003. Measurement of Spin-Spin Distances

friom the Intensity of the EPR Half-Field Transition.

2 R. E. Coffman, A. Pezeshk, J. Magn. Reson. 1986, 70, 21-33. Analytical Considerations of Eaton’s For-

mula for the Interpsin Distance between Unpaired Electrons in ESR.

E F

����

��������

����

B^

E F^ ^

C,D

C,D

C,D

C,D^

^

^

^^

^

^

^SQ

SQ

SQ

SQ

DQ

ZQ

V DQ V SQ ------------------ 19.5 10.9g+ 9.1 GHz

mw-------------------- 2 Å6

r6------=

245

than about 12 Å the intensity of the half-field transition at X-band frequencies is too weak for

this kind of analysis.

10.6.2 DEER

At distances longer than 18... 20 Å the dipole-dipole coupling needs to be separated

from other interactions that lead to broadening of EPR lines. The experimentally most robust

technique for such separation is the double electron electron resonance (DEER) experiment,

which is also known als pulse electron electron double resonance (PELDOR). By employing

two mw frequencies this experiment excites each of the two coupled spins separately. It is thus

akin to a heteronuclear rather than a homonuclear NMR experiment. The separate excitation of

two electron spins is possible even if they both belong to nitroxide radicals, as the excitation

bandwidth of mw pulses is significantly smaller than the width of the nitroxide spectrum. The

two frequencies thus select different orientations or different spin states of the 14N nucleus in

the two radicals (Fig. 10.46).

Figure 10.46: Selective excitation of two radicals by observer and pump pulses in a doubly nitroxidelabeled macromolecule. The radicals have different orientations with respect to the external magneticfield or different 14N nuclear spin states and thus different resonance frequncies.

The subsequence for the observer spin A at frequency 1 is a refocused echo sequence

(Fig. 10.47). It consists of a part (/2)-1-()-1 that creates a first echo and a part 2-()-2 that

refocuses the signal once again and creates a second echo. This second echo is observed. As

the observer pulses excite only the A spin, they refocus all interactions that are linear in

operators of this spin. These are the electron Zeeman interaction, the hyperfine interaction of

the A spin, and also the dipole-dipole interaction between the A spin and the pumped B spin.

pump ( )�2

observer ( )�1

B0

N NO

O

..

B0

excitation bands

schematic spectra

A

B

246

Only the dipole-dipole interaction is reintroduced by the pump pulse at frequency 2,

which exclusively excites the B spin. This pump pulse inverts the state of the B spin and thus

leads to a change of the resonance frequency of the observer spin A by the dipole-dipole

splitting d() given in Eq. [10.31]. This frequency change is induced at a variable delay t with

respect to the first observer echo, while delays 1 and 2 are fixed. The echo signal is thus

modulated with dd as a function of time t. From dd the distance can be determined using Eq.

[10.30] or [10.32]. The modulation is not damped by relaxation, as the total duration of the

experiment is constant. Nevertheless distance resolution and the upper distance limit depend

on electron spin relaxation, as the maximum observation time tmax of the dipolar modulation

cannot be much longer than T2. Therefore, the experiment is performed at temperatures of

about 50 K, where transversal electron spin relaxation of nitroxides asymptotically approaches

its low-temperature maximum. At this temperature transversal relaxation is mainly driven by

fluctuating hyperfine fields from weakly coupled protons. A decrease of proton concentration

by deuterium exchange extends the upper distance limit and improves resolution.

Figure 10.47: Pulse sequence of the DEER experiment consisting of the refocused echo subsequence atfrequency 1 that excites exclusively observer spins A and a pump pulse at frequency 2 that excitesexclusively pumped spins B. Delays 1 and 2 are fixed, while delay t is varied. The echo amplitudeoscillates with the dipolar splitting d() as a function of delay t.

As the pump pulse has limited excitation bandwidth and inverts only a fraction of the

B spins, only a fraction of the echo intensity is modulated. The remaining fraction 1- of the

echo amplitude is independent of t for an isolated pair. The DEER signal F(t,) for an isolated

spin pair is thus given by

. [10.86]

The form factor F(t) for a macroscopically disordered sample is obtained by powder averaging

�/2

�

�

�

t

� �1 2

�1

�2

F t F 0 1 1 d t cos– – =

247

. [10.87]

Figure 10.48: DEER distance measurement on a rigid biradical. A) Structure of the biradical. B)Normalized echo amplitude as a function of delay t. The read line is a fit of the background functionB(t) according to Eq. [10.88]. C) Form factor F(t) obtained by dividing the normalized echo amplitudeby the background function. The red line is the theoretical form factor corresponding to the extracteddistance distribution P(r). D) Dipolar spectrum (Pake pattern) obtained by subtracting the constant partfrom the form factor F(t), multiplication with a Hamming window, zero filling, and Fouriertransformation. The red line is the theoretical spectrum corresponding to the extracted distancedistribution P(r). E) Distance distribution P(r) obtained by Tikhonov regularization.

Interaction of the A spin with spins in neighboring molecules modifies the signal, as a

fraction of theses spins is also inverted by the pump pulse. If the neighboring spins are

homogeneously distributed in space, the background factor B(t) assumes the form

F t V t sin d

0

2

=

N N

O

O

O

O

N NO

(CH2)5CH3

H3C(H2C)5

(CH2)5CH3

H3C(H2C)5

(CH2)5OCH3

H3CO(H2C)5

O

0 5 10

0.6

0.7

0.8

0.9

1

t (µs)0 5 10

0.8

0.9

1

t (µs)

3 4 5 6

r (nm)-1 0 1

� (MHz)

A

B

D

C

E

background

Tikhonov regularizationFo

urier

correction

transfo

rmat

ion

Vt

V(

)/(0

)

Pr()

I()�

�dd

#

248

, [10.88]

where the average modulation depth is the fraction of spins excited by the pump pulse, g is

an average g value, and c is the total concentration of spins. The total DEER signal

[10.89]

for a well-defined distance between the spins has an appearance as shown in Fig. 10.48B.

The form factor for the isolated pair can thus be obtained by fitting the background

function and dividing the DEER signal V(t) by this function (Fig. 10.48B). The dipolar

spectrum is computed by Fourier transformation after removing the constant component 1- of

the form factor (Fig. 10.48C). From the singularity of the Pake patern, dd can be determined

and the distance calculated according to Eq. [10.32].

Due to flexibility of the macromolecule and conformational degerees of freedom of the

label, the spin-spin distance is not sharply defined but rather distributed. Extraction of this

distribution of distances P(r) from the form factor F(t) is an ill-posed problem. In ill-posed

problems small errors in the input data F(t) may result in large errors of the output data P(r).

Errors in the input data come from noise or incomplete background correction and are

unavoidable. Therefore the solution has to be stabilized by additional constraints. Constraints

can be derived from the properties of the distribution to be non-negative (P(r)>0 for all

distances r) and smooth. Smoothness corresponds to a small square norm of the second

derivative

. [10.90]

Furthermore, the mean square deviation of the simulated form factor S(t) from the

experimental form factor F(t),

, [10.91]

has to be small (goodness of the fit). The relative weight of these two criteria for a good

solution of the problem is experessed by a regularization parameter . The best solution for the

distance distribution is obtained by minimizing

B t 2g2B

2 0NA

9 3h----------------------------------ct–

exp=

V t F t B t =

r

2

dd

P r 2

=

S t F t – 2=

249

[10.92]

This solution can be directly computed by matrix algebra with the Tikhonov regularization

algorithm.

The result of Tikhonov regularization depends on the choice of the regularization

parameter . A good compromise between undersmoothing (too small ) and oversmoothing

(too large ) can be found by plotting log vs. log . Because of its typical shape this

plot is termed L curve (Fig. 10.49). In the steep branch of the curve at small an increase in

leads to a strong decrease in the square norm (much smoother distribution), but to only a small

increase in the mean square deviation of the simulated from the experimental data. In the flat

branch at large , a further increase of causes a strong increase in the mean square deviation

(deterioration of the fit), but to only a small improvement in smoothness. The best compromise

corresponds to the corner of the L (arrow in Fig. 10.49).

Figure 10.49: L curve for the distance distribution of the biradical in Fig. 10.48. The optimumregularization parameter = 0.1 corresponds to the corner of the L (red). The distance distributionobtained with this regularization parameter is shown in Fig. 10.48E.

Determination of the distance distribution rather than only the mean distance is

particularly important for systems that simultaneously attain two or more conformational

states. This situation is encountered in studies of protein function or folding.

G +=

-5 -4.5 -4

-20

-16

-12

-8

log�

log $

�= 10-5

�= 10-1

�= 103

250

10.7 Technical considerations

10.7.1 The cw EPR experiment

The general setup of an EPR spectrometer is similar to the one of an NMR

spectrometer discussed in Chapter 2.5 (Fig. 10.50). The mw source in modern spectrometers is

a Gunn diode whose frequency can be tuned over a range of typically 1 GHz. The NMR coil

and capacitance are substituted by a cavity resonator (Fig. 10.51A) or, for special experiments,

a loop-gap resonator. The dimensions of cavity resonators are comparable to the wavelength of

the mw (Table 10.1). The standing wave in the resonator has spatial regions with a large

magnetic and a small electric component of the alternating field (Fig. 10.51B). This is where

the sample has to be placed, as resonant absorption of the magnetic component provides the

signal, while non-resonant absorption of the electric component by the electric dipoles of

molecules in the sample leads to mw losses and sample heating. Therefore sample size must be

much smaller than the wavelength. Typical outer diameters of sample tubes for cw EPR are 4

mm at X band, 1.6 mm at Q band, and 0.9 mm at W band. Samples that contain highly polar

solvents, such as water, have high dielectric losses and thus need to be confined to a region

where alternating electric fields are very small. They are thus measured in capillaries or flat

cells. Typical sample amounts are 150 mg/150 l at X band and 1 mg/1 l at W band for non-

lossy samples and by about a factor 2 to 5 less for aqueuos samples. It is possible to routinely

detect micromolar concentrations with a microliter of sample at X band, corresponding to 1

pmol of or paramagnetic centers.

Figure 10.50: Schematic drawing of a cw EPR spectrometer.

61110

microwave

source

reference

arm

bias

attenuator

resonator

magnet

modulation

coils

1

23

�

m.w.

diode

PSD

modulation

generator

Sig

nalcirculator

phase

251

The mw from the source (power up to Pmw = 200 mW at X band) is attenuated to a

level that does not cause saturation of the electron spin transitions (see Section 2.7.2). With

typical attenuations between 20 and 40 dB the power incident on the sample is about 2 mW to

20 W. This power reaches the cavity resonator through a circulator, so that no power can

directly pass on to the mw diode that functions as the detector (Fig. 10.50). The mw is

transmitted via a waveguide that is coupled to the resonator by an iris with a tuning screw (Fig.

10.51). For best sensitivity the spectrometer has to be tuned and matched.

Figure 10.51: Cavity resonator used in cw EPR experiments. A) Rectangular cavity with iris tuner. Themw is fed from the left front side by a rectangular waveguide. B) Distribution of the mw field. Solidlines with arrows marked B1 correspopnd to the magnetic field component, while dots and crossescorrespond to the electric field component.The central plane of the resonator is a nodal plane of theelectric field.

Tuning means that the frequency of the Gunn diode is adjusted to the resonance

frequency of the cavity resonator. For that the mw absorption of the cavity is measured as a

function of frequency and displayed as a tuning picture (Fig. 10.52). The dip is moved to the

center of the picture by adjusting the center frequency of the sweep.

Maximum sensitivity is achieved if the impedance R0 of the transmission line (50 ) is

matched to the impedance R of the cavity resonator. Due to the non-resonant mw losses, R

depends on the dielectric properties of the sample. Tuning and matching thus have to be

repeated for each sample. A matched cavity does not reflect any mw power if the electron

spins are off resonance (Fig. 10.52). This means that zero mw power is transmitted through

B1 B1

A B

252

ports 2 and 3 of the circulator towards the detecting mw diode (Fig. 10.50). Matching is

achieved by adjusting the iris tuning screw. This changes the coupling n between the

transmission line and the resonator until

, [10.93]

a condition that is termed critical coupling. The loaded quality factor, which characterizes the

resonant enhancement of the mw field in the cavity, is then given by

, [10.94]

where L is the inductivity of the cavity. The amplitude of the magnetic field component B1

becomes

, [10.95]

where Vc is the effective volume of the resonator, which scales roughly as . For the

same QL, less mw power Pmw is thus required at higher frequencies to achieve the same

amplitude of the alternating magnetic field. More power is needed if the sample is lossy (large

R), corresponding to low QL. .

Figure 10.52: Idealized tuning picture of a matched (solid line) cavity resonator with a loaded qualityfactor QL = 10000 at X band frequencies. The dashed line corresponds to resonant mw absorption bythe electron spins.The EPR signal Vrefl is due to the power reflected in this situation

The bandwidth of the resonator (Fig. 10.52) is given by

R0n2

R=

QL

2resonatorL

R0n2

R+---------------------------------=

B1

2QLPmw

0Vcresonator-----------------------------------=

resonator3–

-2 -1 0 1 2

���resonator (MHz)

��

no reflection

total reflection

-3 dB

sig

nalvoltage

aft

er

m.w

.dio

de

�Vrefl

253

. [10.96]

In cw EPR the bandwith of a high-quality cavity resonator is always sufficient. For

pulsed EPR, bandwidths of 30 to 200 MHz are required. This is achieved by intentionally

spoiling resonator quality (increasing R) or by overcoupling (increasing n beyond critical

coupling).

With the spectrometer tuned and matched, the measurement is performed by keeping

the mw frequency constant and sweeping the magnetic field through resonance of the electron

spins. On resonance, the spin systems absorbs mw energy and dissipates the energy through

longitudinal relaxation. This corresponds to an increase in cavity impedance R and thus to a

violation of the critical coupling condition, Eq. [10.93]. The cavity is no longer matched and

mw power is reflected (Fig. 10.52). This reflected power is transmitted through ports 2 and 3

of the circulator to the detecting mw diode (Fig. 10.50). The change in the voltage incident at

the mw diode is given by

, [10.97]

where ’’(B0) is the field-dependent imaginary part of magnetic susceptibility arising from the

electron spins, is the filling factor of the resonator and C is an apparative constant. The

function ’’(B0) is the EPR absorption spectrum. The filling factor is the ratio of the integrals

of the mw magnetic field amplitude over the sample and the whole resonator,

. [10.98]

Sensitivity of the cw EPR experiment can thus be increased by optimizing either the quality

factor QL (cavity resonators) or the filling factor (loop-gap resonators, dielectric ring

resonators). The first strategy works better for samples with low dielectric losses that are

available in sufficient quantities (100 mg). The second strategy is preferable for lossy samples

or small sample quantities. Dielectric resonators combine high QL and high . They are only

used for special applications or pulse EPR as the dielectric rings used for concentrating the mw

field have background signals.

resonator

QL----------------------=

Vrefl C B0 QL=

B1 Vdsample

B1 Vdresonator

---------------------------------=

254

The reflected mw power is detected by a mw diode, which can manufactured with a

low intrinsic noise figure. However, mw diodes have two drawbacks. First, they are not

sensitive at very low incident powers (Fig. 10.53). Second, they detect mw power over a very

broad frequency range, thus also collecting thermal noise from this whole frequency range.

These problems require two additions to the spectrometer.

To achieve best sensitivity and linearity of the output signal with respect to incident

voltage, the diode has to be biased to its operating point (Fig. 10.53). This is done by

transmitting a small fraction of the mw power directly from the source to the diode via the

reference arm (Fig. 10.50). The operating point is attained by adjusting the bias attenuator. The

output current with the sample off resonance should correspond to the horizontal dotted line in

Fig. 10.53 (typically 200 A). The mw transmitted through the reference arms must have the

same phase as the mw reflected from the resonator to interfere constructively at the diode

input. This is achieved by adjusting the mw phase in the reference arm.

Figure 10.53: Characteristic curve of an mw diode for cw EPR detection (solid line). Highestsensitivity and linear behavior is obtained near the operating point (full circle). The diode is damaged attoo high incident power (full diamond). In modern spectrometers, the diode is protected from damageby limiting the input power. The characteristic curve of a limiter-protected diode is shown as a dashedline.

The detection band can be narrowed and thus much of the noise excluded by imposing

a modulation on the signal. This can be done most easily by modulating the external magnetic

field at a low frequency that can pass the mw. diode (typically 100 kHz). Such modulation

leads to an oscillation of the reflected mw with the same frequency (Fig. 10.4). For sufficiently

low field modulation amplitudes B0 the amplitude of the voltage oscillation is proportional to

the derivative , i.e., the derivative of the absorption spectrum. This amplitude is

incident voltage

ou

tpu

tcu

rre

nt

operatingpoint

d dB0

255

measured by a phase-sensitive detector (PSD). As a result cw EPR spectra are derivative

absorption spectra.

With increasing modulation amplitude the signal increases, while noise remains the

same. This applies until the modulation amplitude reaches the peak-to-peak linewidth Bpp.

With this modulation amplitude detection is most sensitive. However, if the modulation

amplitude exceeds Bpp/3, the line is artificially broadened. Therefore, the modulation

amplitude is adjusted to about one third of the width of the most narrow line in the spectrum. In

rare cases natural linewidths can be smaller than 100 kHz, corresponding to 36 mG or 3.6 T

on a magnetic field axis. In this case the modulation frequency has to be decreased to avoid

line broadening. Finally, saturation of the electron spin transitions (see Section 2.7.2) may also

cause line broadening. Such power broadening is avoided in the linear regime where the signal

amplitude increases with the square root of the mw power, . In this regime the

amplitude increases by a factor of two when decreasing attenuation by 6 dB

( ).

Figure 10.54: Schematic drawing of a pulse EPR spectrometer.

10.7.2 Pulsed EPR experiments

In a pulsed EPR spectrometer the pathways for excitation power, signal and reference

mw are similar to the ones in a cw spectrometer (Fig. 10.54). The mw pulses are created in an

mw pulse forming unit (MPFU), where the mw power is divided into several channels. In the

V Pmw

100.6

4 4 2= =

m.w.

source

2 m.w.

source

nd

r.f.

source

r.f.

PFU

reference arm

main

attenuator

magnet

1

23

�

MPFU*

*MPFU m.w. pulse forming unit

**S protection switch or power limiter

S**

power

amplifier

receiver

amplifierm

ixer

vid

eo

am

plifier

r.f. amplifier resonator video signal

(MHz range)

r.f. coils

ELD

OR

EN

DO

R

phase shifter

circulator

256

most flexible setup each channel has its own attenuator, phase shifter, and PIN diode switch

(Fig. 10.55A). This allows for independent adjustment of power, phase, and timing of the

pulses. The PIN diode switches have rise and fall times of about 3-4 ns, so that a pulse with a

length of 32 ns is not quite rectangular (Fig. 10.55B). The power from all channels is

recombined at the output of the MPFU. Modern commercial spectrometers have one preset

four-channel MPFU with fixed quadrature phases (+x, +y, -x, -y). The power of the four

channels cannot be adjusted independently. This MPFU provides best performance in phase

cycling and ease in experiment setup. For full flexibility, additional MPFUs with adjustable

phase and amplitude in each channel have to be installed

Figure 10.55: MW pulse formation. A) MW pulse forming unit (MPFU) with four channels whoseamplitudes and phases can be adjusted independently. B) Typical pulse shape in EPR with the rise andfall times of the pulse-forming switch being comparable to the pulse length.

Much larger B1 fields are required for pulse EPR than for cw EPR. Therefore, the mw

excitation power is amplified in a traveling wave tube ( kW) or solid-state power

amplifier ( W). A precision attenuator after this amplifier allows for adjusting pulse

power. The pulses enter the resonator via the circulator. Critical coupling of the resonator is

not required in pulsed EPR, since excitation and detection are separated in time. In fact,

resonators are usually overcoupled on purpose to increase bandwidth (see Eqs. [10.94,10.96]).

For instance, a pulse with a length tp= 12 ns has an excitation bandwidth of about 1/tp = 83

MHz and thus requires QL < 115 at a frequency of 9.6 GHz. Such overcoupling leads to

substantial power reflection. Reflected mw power in the range of tens to hundreds of Watt

would destroy the sensitive receiver amplifier. Therefeore, the amplifier is protected by a

switch or a power limiter.

During detection the protection switch is open. The signal is amplified by about 20 dB

by the mw receiver amplifier. This amplifier has a very low noise figure. The signal is then fed

to a mixer, where a video signal at the difference frequency between the reference mw and the

MPFU

S+ x" �

S+ y" �

S- x" �

S- y" �

%

%

%

%

attn. phase PIN diode t = 0

t�

A B

32 ns

Pmw 1

Pmw 10

257

signal mw is generated. This frequency subtraction corresponds to detection in the rotating

frame. Sign information on the difference frequency is lost, unless two mixers with a reference

phase difference of are used. Such quadrature detection with a doubly-balanced mixer is

optionally available in modern EPR spectrometers. It should be used only when necessary, as

the splitting of the signal to two mixers decreases signal-to-noise ratio compared to single-

channel detection.

The video signal is passed through a low-pass filter, amplified, and detected by a fast

digitizer with a typical time resolution between 1 and 8 ns. Modern digitizers allow for signal

accumulation at repetiton rates up to 1 MHz. In echo experiments, the digitized signal is

integrated over a certain time range. This improves signal-to-noise ratio. The width of the

integration gate puts an upper limit on the frequencies that can be detected in the signal.

Optimum sensitivity is achieved by matching detection bandwidth to excitation bandwidth.

This corresponds to an integration gate that is approximately as wide as the full width at half

height of the echo, which is in turn given by the length of the longest excitation pulse (window

with length tintg,SN in Fig. 10.56). For recording field-swept echo-detected EPR spectra the

detection bandwidth should be much smaller than the width of the most narrow peaks in the

absorption spectrum. This is achieved by integrating over the whole echo. Typical widths for

the integration gate in this mode of operation (tintg,res in Fig. 10.56) are 100 to 200 ns.

Figure 10.56: Signal trace in a two-pulse echo experiment starting immediately after the falling edge ofthe pulse (simulation for an infinetely broad inhomogeneous line). Pulse lengths of 16 and 32 ns wereassumed for the /2 and pulse, respectively.

90

0 100 200 300 400

t (ns)

FID

echotd

�'

tintg,SN

tintg,res

sig

nalvoltage

(a.u

.)

t�

258

The theoretically expected signal trace after a two-pulse sequence with an interpulse

delay ’ measured between the falling edge of the /2 pulse and the rising edge of the pulse is

shown in Fig. 10.56. The simulation assumes that the EPR line is inhomogeneously broadened

and is much broader than the excitation bandwidth, as is often the case. For such a line, the FID

length tFID is determined by the length of the pulse. The echo appears at a time slightly

longer than ’, as the effective evolution time is approximately the delay between the centers of

the two mw pulses. In practice, only the echo but not the FID can be detected, as the signal

with a power of a few microwatts is superimposed by stronger residual power from the high-

power pulses within dead time td. For a residual power of 100 W to decay to the level of 1 W

a time ln(108)tring has to pass, where tring is the ringing time of the system. This time cannot be

shorter than the ring down time of the resonator tring,res= QL/(mw). For QL= 100 at 9.6 GHz

this corresponds to a dead time of approximately 60 ns. Any reflection of power in the mw

bridge increases tring and proportionally td.

The decrease in QL that allows for a larger excitation bandwidth leads to a loss in

sensitivity compared to cw EPR. This loss is only partially compensated by the contribution of

more spin packets to the signal. Furthermore, according to Eq. [10.95] this decrease also leads

to a smaller mw field amplitude B1 at given power Pmw. These unwanted effects can be

compensated by a different resonator design than in cw EPR. While cw EPR resonators are

optimized by maximizing QL, pulse EPR resonators are optimized for high filling factors

(Eq. [10.98]) and for effective volumes Vc that match available sample volume as closely as

possible. Both criteria require a confinement of the magnetic component of the mw field to

dimensions that are much smaller than the wavelength in air. There exists the additional

constraint that the electric field component in the sample should be as small as possible to

avoid sample heating.

The dimensions of an mw resonators can be reduced by filling it with a material with

much larger dielectric constant than the one of air (= 1). The mw wavelength is inversely

proportional to . Furthermore, if a hole is drilled in the dielectric medium, the electric

component is mainly condined in the dielectric ring and the magnetic component in empty

space. Such a dielectric ring (Fig. 10.57A) within a cylindical cavity resonator combines a

large filling factor , small effective volume, and high loaded quality factor QL at critical

coupling. When critically coupled it allows for cw EPR experiments with good sensitivity and

259

when overcoupled for pulse EPR experiments with good power conversion. Compared to an

optimized cavity resonator without dielectric ring it has somewhat lower cw EPR sensitivity.

For some cw EPR applications and pulse EPR at very low temperatures it is not suitable, as

dielectric materials generally have some background signal due to transition metal impurities.

Usually sapphire ( ) is employed.

Figure 10.57: Resonator designs used for pulsed EPR. A) Dielectric ring resonator, consisting of a ringof material with a high dielectric constant that is placed inside an mw shield or outer cylindricalresonator. The sample is placed in the center hole. B) Loop-gap resonator consisting of a metal ringwith a vertical slit. The sample is placed in the center hole.

More flexibility in scaling of the critical volume is achieved with lumped circuit

resonators, which can be pictured as LC resonance circuits with the inductivity L and the

capacitance C in the same place. More strictly defined the travel time of the electric signal

between the elements of a lumped circuit is negligible. The most simple lumped circuit

resonator is the loop-gap resonator, which is a massive metal ring with a slit that has a constant

width (Fig. 10.57B). The slit (gap) can be considered as the capacitance C while the metal

piece is a one-turn coil, i.e., the inductivity L. Accordingly, the electric component of the mw

field is concentrated across the gap, while the magnetic component is particularly strong inside

the ring (loop), where the sample is placed. The dimension of a loop-gap resonator can be

adjusted without changing its frequency, as L and C can be varied independently by changing

the radius of the loop and the width of the gap, respectively.

An antenna is used to couple mw into such a loop-gap resonator. The coupling n can be

changed by shifting the antenna vertically (along the B1 direction) with respect to the

resonator. Such antenna coupling can also be employed for the dielectric ring resonator.

10

B1

B0

B1

gap

loop

A B

260