single-crystal nmr studies of low-concentration hydrous ... · ed by standard multiple-pulse...

American Mineralogist, Volume 78, pages 1149-1164, 1993

Single-crystal NMR studies of low-concentration hydrous species inminerals: Grossular garnet

Hnnnn,q.N CHoxA. A. Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, California 91 125, U.S.A.

Gnoncn R. RossNr,qsDivision of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California 91 125, U.S.A.

Ansrnlcr

A detailed solid-state proton NMR study has been performed on garnets containing lowlevels of OH (0.2-0.3 wto/o as HrO), including a colorless and a pale orange grossularsample from Asbestos, Quebec, and a colorless grossular sample from klatema Hills,Tanzania. Synthetic hydrogrossular powder, CarAlr(OoHo)r, has been investigated as well.The NMR spectra of the three natural crystals have a broad line (40 kHz) with a weaksharper feature superimposed. Multiple-quantum spectra indicate that in the Tanzaniangarnet the dominant cluster size is two protons; in the colorless grossular from Asbestosa mixture of two proton and four proton clusters is in evidence; in the hydrogrossularpowder, the cluster size is four protons.

Solid echo envelope decay experiments show that for all the natural samples a two-stepdecay process occurs, implying the existence of two distinct proton environments. Thearea ratios indicate that the spectrum of each of the natural crystals is dominated by abroad line with a Gaussian shape and a line width that differs from the line width of thesynthetic hydrogrossular. In the case of the Tanzanian sample, the broad band constitutes-850/o of the area in the NMR spectrum.

Moment analysis shows that although the synthetic hydrogrossular has an average in-terproton separation of 2.16 A in the interacting cluster, it is 1.69 A in the glossular.

INrnonucrroN

In crystalline silicates, the hydrogarnet substitutionSiOX- * 4OH has been presumed to be a common andfavored form of OH- substitution, its existence well es-tablished in the series grossular-katoite [CarAl,(SiOo)3-CarAlr(HoOo)rl. Recent infrared spectral studies of nat-urally occurring garnets indicate that minor amounts ofOH- commonly occur in macroscopic garnets, but thatmodes of substitution other than the hydrogarnet substi-tution are prevalent (Rossman and Aines, l99l). Thispossibility is reinforced by the NMR studies of Kalini-chenko et al. (1987) of polycrystalline hydrogarnet with3.44 wto/o HrO, which proved that H occurs in the sixfold-coordinated Al site. On the basis of crystallographic stud-ies, others have also concluded that H may possibly oc-cupy sites other than the Si site. (Basso and Cabella, 1990;Birkett and Trzcienski, 1984).

A combination of analytical methods is required to ob-tain a more complete picture of the identity and structureof hydrous defects in anhydrous minerals. Along with IRabsorption spectroscopy, NMR spectroscopy is perhaps

* Present address: Molecular Science Research Center MSINK2-20, Pacific Northwest Laboratory, Richland, Washington99352, U.S.A.

one of the most informative options. The principal chal-lenge in the investigation of silicate materials with protonNMR spectroscopy is primarily a technical one, namely,overcoming or accommodating its intrinsically low sen-sitivity. Many of the most interesting phenomena asso-ciated with hydration of silicate minerals are observed atconcentrations that test the sensitivity thresholds of themost advanced NMR spectrometers available. Conse-quently, few NMR studies of hydrous species in mineralshave been reported (Yesinowski et al., 1988).

We have attempted to address in this work the dualproblems oflow H concentration and interference ofpro-ton background signals by a number of experimentalstrategies, and we report the results of NMR investiga-tions on the incorporation ofhydrous species at low con-centrations on three single-crystal grossular garnet sam-ples and one synthetic hydrogarnet powder. This paperis organized in three parts. First, we present results ofholeburning experiments on the three single-crystal gros-sular samples, which indicate that H nuclei are clustered.Next, we characterize these presumptive clusters throughthe use of recently developed multiple-quantum NMRmethods. We test the reliability of this technique in thecontext of silicate mineral systems and provide evidencefrom experiments on a synthetic hydrogrossular powderthat this technique can be applied to obtain reliable counts

0003-004x/93 / | r t2-n 49$02.00 lt49

l 1 5 0

TABLE 1. Descriptions of garnet NMR samples

Material Hydro- Grossulargrossutar

Locality Asbestos,Quebec,Canada

CHO AND ROSSMAN: HYDROUS SPECIES IN GROSSULAR

Color colorless colorless colorless pale orangeCatalog no. GRR1386 cRR1537 GRR53Mass for NMR 189.3 mg 193.6 mg 110.6 mg 264.1 mgH,O content' 0.17 wt% O.27 wt"h 0.31 wt%

Compositions(1) Ca3At'(O4H4)3 (Lager et at., 1gg7)(2) Car.uMgo *Mno 01 Feo o, Al, eTTio orsi2 eso,, oo (Gr2 in Shan-

non and Rossman, 1992)(3) Car."Mgo *Mno 01 Feo o1Al1 eTTio oosi, sO,, @ (Gr3 in Shan-

non and Rossman, 1992)(4) Car rrMgo @Mno o4Feo,4Al2 ooTio oosi3 @O2 oo (Gr1 in Shan-

non and Rossman, 1992). OH content, expressed as wty" H2O, was obtained from the integrated

absorbance in the region of 3800-3400 cm ,. The HrO contents wereestimated using a linear fit to the calibration data from Table 3 of Rossmanand Aines (1991), which apply to grossular with <0.5 wt% H.O.

of the number of protons in closely spaced groups of Hnuclei. We then proceed to apply this method to the nat-ural garnets to determine the number of H nuclei consti-tuting the clusters at the hydrated sites.

The final part of this paper is an analysis of the NMRline shape of the grossular garnets and a discussion of theinformation that can be deduced from the spectrum, basedon the knowledge gained from the previous experiments.

ExpnnrnanNr,lL

Sample selection and preparation

Descriptions of the samples studied in this work arecompiled in Table l. A primary consideration governingthe choice of these samples was the HrO concentration:for the specific experiments and samples investigated here,we estimate the proton detection threshold to be approx-imately l0'7 H nuclei in the solid state, with a maximumsample volume of 0.5 cc.

The single crystals were judged by several other crite-ria: freedom from cracks, twinning, and visible inclusions(as determined by examination at 40 x magnification) andconcentration of paramagnetic and ferromagnetic impu-rities. Specimens with the fewest crystal defects and low-est concentrations of paramagnetic and ferromagrreticimpurities were deemed the most desirable for investi-gation by NMR.

Single-crystal specimens were prepared for the NMRexperiments by immersing in acetone, wiping with papertissue, and drying with a heat gun. The crystals werecleaned two or more times afterward by a similar pro-cedure, but with spectral grade carbon tetrachloride asthe solvent. After the final cleaning, the crystals werestored under 0. I -torr vacuum at I 00 "C for 20 h and thenwrapped in small amounts of Teflon tape to preventmovement of the sample while positioned inside the NMRcoil. The Teflon tape was checked for H content by scan-

Synthetic Lelatema Jeffrey mine,powder Hills, Asbestos,

Tanzania Ouebec,Canada

ning the proton NMR resonance of several strips ran-domly taken from different rolls of tape. None was foundto contain amounts of H detectable by NMR at roomtemperature, even after several thousand scans.

All cleaned samples were handled with latex gloves andclean metal instruments only, to protect against contam-ination of the crystal surface by H-containing residue.The Teflon-wrapped samples were periodically un-wrapped and rewrapped with new tape to ensure thecleanliness of the outer surface for experiments lastingseveral days.

NMR experiments

General remarks. NMR experiments were performedat room temperature rn a 4.7-T magnetic field, corre-sponding to a proton Larmor frequency of 200. I 35 MHz.The radio frequency pulse times and phases were adjust-ed by standard multiple-pulse calibration techniques ona large HrO sample sealed inside a glass tube (Burum etal., l98l). The > 130 kHz radio frequency field generatedby our NMR probe sufficed to ensure the uniform exci-tation of the entire spectral range of all samples in Ta-ble l .

The near ubiquity of H was a significant complicationattending our efforts to observe proton NMR signals fromsamples with low H content. To accomplish the isolationof the sample from H-containing contaminants, we haveconstructed a customized NMR probe, with its coil sus-pended on top of two Cu rods approximately 2 cm abovea Teflon base. The Teflon base, although not free of H,did not produce a measurable background signal in theNMR spectrum. This base and the Teflon dielectric jack-eting the coaxial cable conducting radio frequency currentto the coil were the only nonmetallic parts within 6 cmof the sample. The NMR coil, which was wound from astrip cut from an unvarnished Cu sheet, was washed byrepeated immersion in concentrated HCI or carbon tet-rachloride, followed by drying with a heat gun. The coilwas attached to the Cu rods by minimal amounts of sol-der and then sprayed and wiped with solder flux removerand carbon tetrachloride.

To reduce the signal originating from humidity in theatmosphere, the bore of the superconducting magnet wassealed at both ends, except for a small outlet at the bot-tom, and the air within purged by directing high-pressure,dried, filtered air through a transfer tube directly onto thesample and out the exhaust port.

In addition to the above measures, every NMR exper-iment on the grossular garnets was performed twice underidentical conditions, once with the sample loaded in theprobe and once without. The NMR signal of the samplewas thus obtained by forming the diference of the twoexperimental signals.

Multiple.quantum NMR experiments. Procedures forgenerating and detecting multiple-quantum spin coher-ences in a two-dimensional NMR experiment have beenrecently documented (Weitekamp, 1983; Munowitz andPines. 1987). We have utilized versions of the two-D

multiple-quantum experiment wherein the mixing periodwas a phase-shifted, time-reversed counterpart of thepreparation period and the evolution period was omitted.The different multiple-quantum orders were resolved byincrementing the phase of the preparation period fromacquisition to acquisition and keeping the evolution pe-riod equal to zero throughout, in the manner describedby Shykind et al. (1988). The incremented phase shifts ofthe preparation period were implemented by inserting acomposite z pulse (Freeman et al., 1981) between thepreparation period and mixing periods of the multiple-quantum pulse sequence, as illustrated in Figure la andlb. The problems arising from the sensitivity of thismethod to radio frequency field heterogeneity were min-imized by employing NMR coils with inner volumes sev-eral times larger than the volume of the samples. We haveverified that the radio frequency homogeneity was ade-quate for these experiments by monitoring the Rabi os-cillations (Abragam, 196l) of an adamantane powdersample over several periods and by performing multiple-quantum measurements on solid samples having knownhigh maximum multiple-quantum transition orders.

Nonselective multiple-quantum spectra were acquiredwith the use ofthe one-quantum pulse sequence ofSuteret al. (1987), whereas even-order selective spectra wereacquired with the even-quantum pulse sequence devel-oped by Warren et al. (1980). A typical time for one cycleofthese excitation sequences was 60 ps.

A delay of l0 ms was inserted after the mixing periodto permit the irreversible decay of spin order not alignedalong the axis of quantization. This time was short com-pared with spin-lattice relaxation times, but long relativeto the time constant of the sample's free induction decay(FID) transient. The remaining longitudinal magnetiza-tion was sampled by means of a pulsed spinlock train(Ostroffand Waugh, 1966), with acquisition of data oc-curring between spinlock pulses. Radio frequency pulsescorresponding to an approximately % flip angle were uti-lized in the spinlock train (Rhim et al., 1976), spaced byl9-ps intervals, and 4096 complex spinlock points wereacquired in these experiments. An eight-step phase cycleof the detection-period pulses was implemented to reduceartifacts induced by the receiver and flip angle.

Detector and filter ringing due to pulse breakthroughduring the spinlock detection period was attenuatedthrough the placement of a blanking switch on the inputof the NMR receiver. Because of the reduction of pulsebreakthrough, the signal-to-noise ratio could be furtherimproved by using narrower filter band widths than wouldotherwise be practical in a multiplo-pulse NMR experi-ment. In these experiments, we have set the filter widthto be approximately the inverse of the separation betweenspinlock pulses.

Two complete periods of the multiple-quantum inter-ferograms were acquired in all multiple-quantum NMRexperiments recounted here. The dala were analyzed byadding the 4096 complex points in the summed spinlocktransient for each value of the phase increment individ-

I l 5 l

(a)

(b)

(d)

(e)

Fig. l. Pulse sequences of the NMR experiments discussedin this paper. (a) Generic multiple-quantum NMR experiment,composed of preparation (P), evolution (E), mixing (M), anddetection (D) intervals; (b) composite z pulse constructed froma sandwich of three pulses with standard quadrature phases; (c)two-pulse solid echo sequence; (d) pulse sequence for measuringsolid echo envelope decays by pulsed spinlock detection; (e) hole-burning experiment with two-pulse solid echo detection.

ually, Fourier transforming the complex interferogramwith respect to the phase variable, and then computingthe magnitude spectrum. The positive and negative orderintensities of spectra computed in this way represent un-correlated measurements and were both used in the least-squares fits to be discussed later.

Spin echo and solid echo envelope decay experiments.The garnet NMR spectra were recorded with the standardsolid echo sequence shown in Figure lc @owles andStrange, 1963). The phases ofthe pulses and receiverwereiterated through a l6-step phase cycle designed to elim-inate receiver and pulse artifacts. The refocusing time forthese experiments was chosen to be 20 ps, although dif-ferent values were explored to verifo that the areas andline widths of the garnet spectra were independent of r.


tt52

Frequency (kHz)Fig. 2. Proton NMR spectra of the Tanzanian glossular gar-

net both (top) without and (bottom) with a hole-burning field.The carrier frequency ofthe hole-burning field is marked by thearrow on the horizontal axis. Both spectra are the averages of2000 scans with a 4-s delay between scans and are plotted withequal vertical scaling.

The z independence of our results was affirmed for r >12 ' " '

The solid echo envelope experiment depicted in Figureld is a version of the standard experiment, modified byus to increase its sensitivity. The modification consists ofmeasuring the echo amplitude with a pulsed spinlock train,as described in the previous section, rather than by itsrefocused free induction decay. The enhancement of thesignal-to-noise ratio depends on the 2,, of the sample andthe recovery time of the detector and tank circuit, but infavorable cases it can be several orders ofmagnitude.

Rnsur,rs AND DrscussroN

NMR spectra: Homogeneous vs. heterogeneousbroadening

Spectra were recorded for at least three random ori-entations of each of the single-crystal grossular samples,using the two-pulse solid echo sequence diagrammed inFigure lc. Representative spectra for all three samplesare pictured in Figures 2a, 3a, and 4a. To within theresolution of the experiment, the appearance of the NMRspectrum for each crystal was not altered by changes inthe orientation of the crystal with respect to the direction


+50 -50o +'

Frequency (kHz)Fig. 3. Same as Fig. 2, but for the colorless grossular from

the Jeffrey quarry, Asbestos, Quebec.

of the static magnetic field. More detailed analyses ofthese spectra (discussed below) confirm this assessmentquantitatively. As a result of this apparent orientationindependence, no measures were taken to position thecrystals in special, known orientations for any of the NMRexperiments described below.

The NMR spectra for all three samples appear to con-sist of two distinct lines, one broad line approximately40kHz full width at halfheight (FWHH), and one narrowline superimposed on the broader peak and with approx-imately the same center frequency. The observation ofthe broad line indicates immediately the existence of fixedor slow-moving H clusters in the crystal, since broad linesin proton spectra are usually the consequence of short-range homonuclear dipolar couplings (Abragam, 196l).Nevertheless, other sources of line broadening not in-volving H clustering could plausibly account for the ob-served line shape as well. Chief among these is the pres-ence in all three samples of trace amounts of paramagneticions and an unknown number of crystal lattice defectsites. The heterogeneity of magretic environments for in-dividual protons can give rise to line broadening in thissituation by creating a heterogeneous distribution ofpro-ton resonances, with a concomitantly broad NMR lineshape.

To decide which of these two mechanisms is respon-

Frequency (kHz)

Fig. 4. Same as Fig. 2, but for the orange grossular fromAsbestos, Quebec.

sible for the shape of the proton line, the hole-burningexperiment sketched in Figure le was carried out on thegrossular single crystals (Mehring, 1983). The predictedresponses to a hole-burning field for the case of hetero-geneous broadening and the case ofhomogeneous broad-ening are depicted in Figure 5a and 5b, respectively.

The results of the hole-burning experiments are dis-played in Figures 2b,3b, and 4b. The rotating frame am-plitude of the saturating field in these experiments wasestimated to be approximately 4 kHz, a value that ade-quately fulfills the condition that the excitation range ofthe weak pulse be narrow, compared with the line widthofthe resonance. A dephasing interval l0 ms in durationwas inserted after the weak rf pulse to allow residualtransverse magnetization to decay before acquisition ofthe echo. This interval, although long compared with theZ, for the samples, was short compared with the ?", (> Is at room temperature). The duration of the weak rf pulsefor these experiments was l0 ms.

The spectra show that almost the entire broad featuredisappears as a result of the irradiation with the weakfield. We conclude on the basis of this result that the linewidth is not due to heterogeneous distribution ofnarrowspectral packets but is more likely attributable to homo-geneous interactions, such as homonuclear dipolar cou-plings of clustered protons.

I 153

Fig. 5. Possible responses of a wide NMR line to a weakhole-burning field for the cases where the line width is due to (a)heterogeneous line broadening interactions, and (b) homoge-neous line broadening mechanisms. The narrow line representsthe unperturbed line shape, the bold line the line shape afterirradiation with a hole-burning field.

Multiple-quantum NMR

Application to cluster size deterrnination. The results ofthe previous section indicate that H nuclei in the threegrossular samples probably exist in the form of closelyspaced groups of protons, with interproton separationswithin a group of no more than a few dngstrdms, as in-ferred from the roughly l-mT width of the broad protonline. To gain further insight into the size, arrangement,and identity of these clusters, we performed the multiple-quantum NMR counting experiment illustrated in Figurela (Baum and Pines, 1986; Baum et al., 1985). The pro-cedure for determining cluster sizes by multiple-quantumNMR methods consists of fitting multiple-quantum spec-tra to known equations relating cluster size to multiple-quantum line intensities. Ifwe adopt certain assumptionsabout multiple-quantum transition probabilities, the in-tegrated intensity of thie +Am quantum line can be de-rived by a simplified combinatorial argument (Hoffman,1970; Wokaun and Ernst, 1978). This formula is givenby the expression

I ( + m ) :(2r1.|-1

( N - m ) t ( N + m ) l

where Nis the putative number of coupled-spin Yz nuclei.The reliability of this formula has been empirically con-

CHO AND ROSSMAN:HYDROUS SPECIES IN GROSSULAR

( l )

I 154

Multiple{uantum transition order

Fig. 6. Multiple-quantum NMR magnitude spectra of thesynthetic hydrogarnet powder for preparation periods of dura-tion (a) 240 ps (four cycles); (b) 360 ps (six cycles); and (c) 600ps (ten cycles). The solid circles represent least-squares fits of themultiple-quantum line intensities to Eq. I and indicate a protoncluster size of four. Each phase increment point in the experi-ment corresponds to the sum of40 scans of4096 complex spin-lock points each, with a recycle delay between scans of4 s.

firmed in previous experiments (Baum and Pines, 1986;Baum et al., 1985).

For more general systems containing nonintoractingclusters of more than one size, Equation 1 can be modi-fied in the following way:

I (+ m) : Ao +; 1 ---l+?i- - (z)7-

'- '(N, - m)t(N, + my

where the,4, represent the relative contribution oftheTthcluster to the total intensity observed at a given m value,and Z is the number of cluster sizes present.

Although Equations I and 2 contain no reference tothe properties of the spin system other than the clustersize, or to the specific details of the multiple-quantumexperiment used to obtain the spectrum, the intensitiescan in fact depend very strongly on a number of otherparameters, most notably on the duration of the intervalknown as the preparation period. The multiple-quantum


experiments attempted in this work were therefore per-formed several times, with successively longer prepara-tion periods for each repetition. Convergence to the sta-tistically determined limit was decided when the fittedcluster size inferred from the spectrum ceased to increaseas the preparation time was lengthened.

Synthetic hydrogarnet powder multiple-quantum re-sults. Multiple-quantum spectra of the synthetic hydro-garnet powder for three preparation times are displayedin Figure 6, accompanied by solid circles denoting thenonlinear least-squares fit of the experimental intensitiesto the single cluster intensity formula in Equation l. Thenonselective pulse sequence described in the experimen-tal section was employed during the preparation and mix-ing periods to generate the multiple-quantum spin coher-ences. Because of the unique susceptibility of the zeroquantum line to errors from experimental artifacts andinstrumental instabilities, the intensity of this line onlywas excluded from the calculation of the fit. The com-puter program employed to fit these data incorporates atable containing information on the maximum coherenceorders admissible for spin clusters of a given size N, thusensuring that the physical limits on the largest multiple-quantum transitions observable in clusters of finite sizeare recognized by the fitting program. The value of Nthatbest satisfied the least-squares criterion was determinedto be four for every spectrum, signifying that the statis-tical limit has indeed been reached, and that the system'spredominant cluster consists of four H nuclei. In view ofprevious independent work showing that H nuclei aregrouped in clusters of (OH)o tetrahedra (Cohen-Addad etal., 1967), the determination of a H cluster size of fourvalidates the reliability of multiple-quantum NMR as amethod for counting clustered H nuclei in garnets.

Despite the goodness of fit of the experimental spectrato Equation I for N : 4, the flnding that the hydrogarnetcluster size is four is contradicted by the presence ofclear-ly visible 4- and 5-quantum lines in Figure 6c. As dem-onstrated in the appendix, all multiple-quantum transi-tions involving lLml > 4 are forbidden in clusters withN: 4 for this particular version of the multiple-quantumexperiment. The fact that these higher order transitionsare observed implies that the (OH)o tetrahedra cannot beregarded as truly isolated in the NMR sense, and that aresidual magnetic coupling does exist between at leastsome of the tetrahedra. Inasmuch as the dipolar couplingis a function ofthe interproton vector and the directionof the magnetic field (Abragam, 1961), the magnitude ofthe intercluster coupling depends both on the position ofthe tetrahedron urithin the structure and on the orienta-tion of the individual crystallites constituting the powder.In this and other cases, the least-squares fit becomes crit-ical for extracting the most probable cluster size from theexperimental intensities.

Tanzanian garnet multiple-quantum results. The samemultiple-quantum experiment performed on the hydro-garnet in the previous section was carried out on a sampleconsisting of two Tanzanian garnet single crystals with a

- 8 - 6 - 4 - 2 0 2 4 6 8

- 8 - 6 - 4 - 2 0 2 4 6 8

-

Multiple-quantum transition order

Fig. 7. Multiple-quantum NMR magnitude spectra of theTanzanian grossular garnet for preparation periods of duration(a) 240 trs (four cycles); (b) 480 ps (eight cycles); and (c) 720 ps(12 cycles). Each phase increment point in the experiment is thesum of560 scans of4096 complex spinlock points each, with arecycle delay per scan of4 s.

combined mass of 646.1 mg. Three spectra correspond-ing to three different preparation times were recorded inthis manner, with the results shown in Figure 7.

Unlike the spectra for the synthetic hydrogarnet pow-der, in the multiple-quantum spectra of the grossular sin-gle crystals only the Am: + I quantum lines have ap-preciable intensity. For reasons given in the appendix, weconclude that the number of clustered H nuclei revealedby these data must be greater than or equal to two. More-over, if we accept the validity of Equation l, then we findthat the ratio of the t2 quantum transition intensity tothe + I quantum transition intensity for cluster sizes N> 3 must satisfy the inequality

I (+ 2)/I (+ l) = ,/,. (3)

It is clear from the signal-to-noise ratio ofall three spectrathat +2 quantum lines 400/o as intense as the + 1 quantumlines would readily be detectable if they were present.Since these lines are evidentlv absent. we infer that Nmust be two.

I 1 5 5

M u l L i p l e - q u a n l u m L r a n s i l i o n o r d e r

Fig. 8. Even-order selective multiple-quantum NMR mag-nitude spectrum of the Tanzanian garnet for a preparation pe-riod of 240 ps. The points in the interferogram are the sum of400 scans of4096 complex spinlock points each, with a recycledelay of4 s per scan.

Additional support for this assertion is provided by themultiple-quantum spectrum shown in Figure 8. The even-order selective multiple-pulse sequence used in the prep-aration period of this experiment preferentially populatesthe even-quantum coherences in the spin system, and sointensities of even order transitions are enhanced in thespectrum and those of odd order transitions suppressed.For this experiment, the +2 quantum transitions of clus-ters consisting ofjust two H nuclei have nonzero inten-sities. The observation of the t2 quantum lines in thisspectrum therefore lends credence to the claim that Hnuclei exist primarily as discrete, closely spaced pairs inthe garnet crystal structure.

The NMR spectrum of isolated pairs of magneticallyequivalent protons has a readily identified form for rigidsolids (Pake, 1948). No such pattem can be discerned inthe spectrum of the Tanzanian grossular in Figure 2a.Two inferences may tentatively be drawn from this ob-servation. The first is that the two protons constitutinga garnet pair are probably not magnetically equivalentnor well isolated from interactions with other magneticisotopes such as 27Al or '?eSi. When the effects of chem-ical shift anisotropy and orientational averaging are alsotaken into account, the NMR spectnrm of the proton pairscould come to resemble more closely the spectrum ofFigure 2a.

The second inference to be made from these results isthat proton pairs may occur in a variety of magneticallyinequivalent orientations within the crystal. The com-plexity of the unit cell, with numerous locations whereproton pairs could conceivably be situated, supports thisconjecture (Novak and Gibbs, l97l). The orientation in-dependencb of the NMR spectrum provides additionalevidence for this supposition. The NMR spectrum for


U'-=

LNL

;LG

;'6C.o

c.

- 8 - 6 - 4 - 2 0 2 4 6 8

l 156 CHO AND ROSSMAN: HYDROUS SPECIES IN GROSSULAR

u't'=.=t-Nt-

ELG

;-6c.o

C.

-6 -4 - 2 0 2 4 6

l l u l t i p l e -quan tum L rans i t i on o rde r

Fig. 9. Multiple-quantum NMR magnitude spectrum of thecolorless grossular (GRR 1537) from the Jeffrey quarry for apreparation period of 240 pLs. The spectrum was symmetrizedby zeroing the imaginary buffer prior to Fourier transformation;560 scans of 4096 complex spinlock points each were acquiredand summed per phase increment, with a recycle delay of 4 sper scan. The triangles symbolize the least-squares fit ofa singlecluster model to the experimental intensities (cluster size : 2),and the circles the fit of a two cluster model to the experimentaldata (cluster sizes : 2, 4).

this case would be a superposition of appropriatelyweighted multiplets, with each multiplet having diflerentsplittings and center frequencies depending on the direc-tion of the interproton vector, the magnetic field, and theorientation of the crystal.

Asbestos garnet multiple-quantum results. The multi-ple-quantum spectrum for the garnet from the Jefreyquarry (GRR 1537) appears in Figure 9. The small num-ber of H nuclei in the crystal made observation of themultiple-quantum signal difficult in this sample, as man-ifested by the poor signal-to-noise ratio. Consequently,our deteffnination of the H cluster size from this spec-trum is extremely uncertain.

The only peaks in the spectrum that we can confidentlyidentify are the +1 quantum lines. However, weak +2quantum lines are visible above the noise as well. Wehave performed nonlinear least-squares fits ofthe exper-imental intensities to Equation 2 for L: I (one cluster),and L: 2 (two clusters). The best one-cluster model fit,represented by the triangles, is obtained for a cluster sizeof two. For the two-cluster model, the best least-squaresfit is obtained for clusters of size two and four. The fittedequation for the two-cluster model contains more ad-justable parameters than the corresponding equation forthe one-cluster model, and thus more accurately fits thedata. Statistical tests we have performed that account forthis bias, however, verify that the two-cluster equationdoes in fact provide a superior fit to the experimentalintensities.

I

(b )

+50 0 -50

Frequency (kHz)Fig. 10. (a) Proton NMR spectrum of the synthetic hydro-

garnet powder. The spectrum is the average of 64 scans with arecycle delay of 4 s per scan. The thick black line represents aleast-squares fit of a Gaussian function to the experimental lineshape. (b) Difference ofthe least-squares fit and the experimentalspectrum, plotted on the same vertical scale as (a).

NMR line shape analysis

Synthetic hydrogarnet. Detailed microscopic informa-tion can be extracted from relatively featureless NMRlines, such as the broad peak in Figure 10, by the methodof moments (van Vleck, 1948). Though normally astraightforward numerical exercise, the presence of theoverlapped nzurow peak complicates the calculation, sincethe moments of the unwanted signal are superimposedon the moments of the line of interest. i.e.. the broadpeak. Hence, we have attempted to fit the main peak tovarious line shapes, and to use the fitted line shape toinfer the moments instead of the experimental line shape.Because the second peak is considerably narrower thanthe main peak, it is feasible to use only regions of thespectrum that lie outside the band width of the secondpeak in our least-squares fit calculation. The source ofthesecond resonance was not identified but is presumably aH-containing contaminant mixed with the hydrogarnetpowder.

The function that was found empirically to give thebest fit to the spectrum was the normalized Gaussianfunction:

-8

F@ ::, "(+) (4)

CHO AND ROSSMAN: HYDROUS SPECIES IN GROSSULAR Lt57

The second moment of this function is

* , " ^ " , , , ^ : * . ( 5 )

The plot of the fitted spectrum and the experimentalspectrum reveals the Gaussian to be a close representa-tion of the broad line shape.

Expressions for the second moment of dipolar coupled-spin Yz nuclei have been derived from first principles forcomparison with experimentally measured second mo-ments. For samples that contain proton clusters with anisotropic continuum oforientations, such as powders andcrystals with complex unit cells, the second moment ofthe NMR line shape in the absence of chemical shifts is,in units of Hertz squared (Abragam, l96l):

l i N lM , : = - l l t t , I ( I + l ) ) . ( 6 )' 4r t 5 ' ' '

f= , r lo

where the r,o are interproton distances, 1 is the spin quan-tum number (in this case, t/z),,y is the gyromagnetic ratioof the isotope in radians per second, and Nis the numberof magnetically coupled spins, which, in theory, couldinclude every H nucleus within the crystallite. The mul-tiple-quantum NMR results on this sample demonstrate,however, that it is valid to regard the (OH)o tetrahedra inthe hydrogarnet powder as very nearly isolated clustersof four H nuclei in a proton NMR measurement. Weshall assume, therefore, that the NMR spectrum of thehydrogarnet is an orientationally averaged superpositionof spectra of tetrahedral H clusters, and that N : 4. If wesuppose that the protons of the four spin (OH)o clusterlie at the vertices of a regular tetrahedron, M, reduces to

M': |

'27 ^Yalt2

4tr2 Zo rf Q)

where r, is the interproton separation.From the fit ofthe Gaussian to the hydrogarnet powder

NMR spectrum, a value for M, can be estimated; thisvalue was determined to be Mr,"*, : 1.917 x 108 Hz2.Taking this to be the value of M, in Equation 7 aboveand solving for r,, we discover r,: 2.16 A.

Inasmuch as the line width of the experimental spec-trum contains contributions from sources other than thehomonuclear dipolar coupling, such as paramagneticbroadening, chemical shift anisotropy, and heteronuclearcouplings to 27Al and 2eSi nuclei, the second moment ac-tually attributable to the H-H dipolar coupling will besomewhat less than Mr,.*r. For this reason, our value ofr, is best regarded as a lower limit for the true dimensionof the (OH)o tetrahedron.

The configuration of the H nuclei in the hydrogarnet(OH)" tetrahedron has been previously determined byX-ray methods (Foreman, 1968). According to these re-sults, the tetrahedron formed by the four nuclei is dis-torted, with two characteristic interproton distances of1.82 and 2.42 A instead of one. The single distance wehave found lies close to the mean of these two values.

0.0

=

Gis=oco

F

.goE o.oIIJ

10{

10s

10{

10 '3

0.0

10€ 10€ 1

Time, sFig. I 1. Decay of the solid echo amplitude as a function of

the logarithm ofthe refocusing time for the Tanzanian grossulargarnet (top), the colorless grossular garnet specimen from theJeffrey quarry, Asbestos, Quebec (middle), and the orange gros-sular specimen from Asbestos, Quebec (bottom). The circles areexperimental data, the line a least-squares fit ofEq. 8 to the data.Each experimental point was obtained from the sum of256 spin-lock transients of4096 complex points each, with a recycle delayper scan of 4 s.

Grossular garnet single crystals. An accurate determi-nation of pair interproton distances by a line shape mo-ment analysis requires that we be able to separate thecontribution to the line width of short-range homonu-clear dipolar couplings from contributions due to other

o

oo

I 158

Frequency (kHz)Fig. 12. (Top) Proton NMR spectrum of the Tanzanian gar-

net single crystal. The spectrum is the average of 2000 scanswith a 4-s delay between scans. The bold line is a least-squaresfit of a Gaussian line shape to the experimental spectrum. (Bot-tom) Difference ofthe least-squares fit spectrum and the exper-imental spectrum plotted on the same vertical scale as the topspectnrm.

line broadening sources. This separation can be approx-imately effected by means of a procedure developed byBoden et al. (1974,1975), Boden and Mortimer (1973),Boden and krrine (1975, 1978), and Boden and Kahol(1983), based on m€asurements of the solid echo enve-lope decay (SEED). This experiment, shown in Figure ld,was performed on all three grossular single crystals, withthe results summarized by Figure I l. These data werefitted to a sum ofone Gaussian function and one expo-nential function ofthe form

f(t): A" - A,s '(t/ 'a^'\2 - Are-'h'. ' ' (8)

Significantly worse correlations were obtained by at-

TABLE 2. Spin echo envelope decay time constants

Sample tc* (s) r",o (s)

Tanzanian grossularColorless grossular, Jeftrey quarry,

Asbestos, QuebecOrange grossular, Asbestos,

Quebec


+500

-50

Frequency (kHz)Fig. 13. Same as Fig. 12, but for the colorless grossular spec-

imen from the Jeffrey quarry, Asbestos, Quebec.

tempts to fit the data to two Gaussians or two exponen-tials.

Values for the decay time constants ro"" and rexp aregathered in Table 2. These data and the decay curves ofFigure I I show that the decays of the solid echo enve-lopes take place in two distinct steps. The possibility thatthe natural gamet samples contain two chemically dis-tinct types ofprotons has already been mentioned in theprevious discussion of the NMR line shapes. The obser-vation of the two step echo envelope decay appears toaffirm this hypothesis.

A second point ofinterest revealed by these data is thatthe three decay curves, although superficially very dis-similar, have very nearly the same decay time constantsfor both the exponential and the Gaussian componentsof f(t). The meaning of this parallel is not clear but couldindicate that two of the same tlpe of proton are indeedpresent in all three garnet samples, but in varying relativeamounts.

Thirdly, we note that zo., and r"^o are both considerablylarger than the Z, measured from the line widths of eachof the samples. The implications of this observation willbe explored later in this section, but for now we note thatthe discrepancy provides a measure of the size of line

0

4 . 9 0 4 x 1 0 5 1 . 1 5 6 x 1 0 - 3

5.068 x 10 s 1.474 x 10 3

5.307 x 10 5 1.332 x 10 3


Tmue 3. Proton NMR line widths and line area ratios

Line width,Sample FWHH (kHz)

I 1 5 9

Synthetic hydrogarnet powderTanzanian grossularColorless grossular, Jefirey quarry,

Asbestos, QuebecOrange grossular, Asbestos,

Ouebec

32.60 na39 56 + 0.89 0.155 + 0.052

37.06 + 0.34 0.124 + 0.034

37.32 + 0.61 0.172 + 0.015

Note.- errors are standard deviations computed from three measure-ments of each quantity.

calculation to excise the narrow peak. The function thusfitted to the broad resonance may then be subtracted fromthe experimental spectrum and the area of the narrowpeak evaluated numerically for comparison with the in-tegral ofthe fitted function.

The outcome of this procedure is shown in Figures 12,13, and 14. The line shape function that agrees best withthe data is the Gaussian function in Equation 4, with afull width at half height of

The flatness ofthedence of the fittedresonance.

base line proves the close correspon-Gaussian to the shape of the broad

The line widths of the broad peak (as calculated withthe Gaussian model) and the ratios of the area of thenanow peak to the broad peak are shown in Table 3. Thevalues presented are averages computed from three fittedspectra per sample, measured for three orientations ofeach crystal. The line widths of the three grossular spec-imens are very similar to one another, but differ signifi-cantly from the line width of the hydrogarnet powderspectrum. Moreover, the ratios of narrow to broad areasnearly coincide. The line shapes ofthe narrow peaks dif-fer markedly, however, as may be seen in Figure 15.

The chemical identity of the H nuclei corresponding tothe narrow peak cannot be ascertained on the basis oftheavailable NMR data. The narrow width of this line andthe echo envelope decay results both indicate that the Hnuclei are either far removed from other H nuclei or arepart of a mobile species within the sample.

We have proposed on the basis of the multiple-quan-tum NMR spectra that H nuclei are grouped pairwise inthe Tanzanian grossular. If we assume that the interpro-ton vector of the H pair can point in any direction in asingle crystal with equal probability, then the contribu-tion to the second moment due solely to the intrapairdipolar interaction is

u.: zol-+ '(,t)]" (e)

+50 -50

Frequency (kHz)Fig. 14. Same as Fig.12, but for the orange grossular speci-

men from Asbestos, Quebec.

broadening interactions exclusive of the intrapair homo-nuclear dipolar couplings.

The fourth interesting feature in these data is the ex-ponential behavior ofthe slower decay process. The Fou-rier transform ofan exponential decay curve is al-orentz-ian function, a line shape function that is oftensymptomatic of a highly mobile species or nuclear spinsthat are very weakly coupled by magnetic interactions toother spins around them (Abragam, 196 l). Further work,such as a variable temperature study, is necessary beforewe can make a more definite interpretation of this result.

The evidence presented thus far indicates that H nucleiin the grossular garnets are incorporated in the crystal inat least two distinct forms. The broad line and the narrowline in the spectra of Figures 2-4 presumably correspondto these two species. Assuming that to be the case, therelative areas of the two NMR lines provide a measureof the relative amounts of the two species. For the pur-poses of computing these areas, we are helped by the largedisparity in line widths, which enables us to fit the broadresonance to some model line shape function using partsof the spectrum that contain little or no intensity contri-bution from the narrow peak. Since the narrow peak con-stitutes such a small fraction of the total spectral width,very few points need actually be excluded from the fit

0

Mr, rn tu :#*# (10)

This equation contains, besides the assumption of equi-probable H pair orientations, the second significant as-sumption, that all paired H nuclei are separated fromtheir partners by the same distance, R.

I 160

Mz: Mz . , i noJ_ Mr . " r n , . ( l l )

In this equation, M, is the second moment computeddirectly from the observed NMR line, and Mr.,.,* is thequantity defined in Equation 10.

TABLE 4. Second moments

Sample M2..,e (Hz2) M2SEED (Hz2) M2.tu. (Hz2)

Tanzaniangrossular 2.823 x 1Oa 0.662 x 10s 2.161 x 1geColorless grossular, Jef-

frey quarry, Asbestos,Ouebec 2477 x 1ge 0.612 x jOe 1.965 x 1Os

Orange grossular, Asbes-tos, Quebec 2.511 x 190 0.565 x 1Os 1.946 x 10s


2.0

r .5

1.0

0.5

0.03800

(I)C)dcd

-oL{

A)I

Frequency (kHz)

Fig. 15. Diference spectra of Figs. 12, 13, and 14 on anexpanded frequency axis. (a) Tanzanian grossular, (b) colorlessgrossular from the Jetrrey quarry, Asbestos, euebec, (c) orangegrossular from Asbestos, Quebec. The position of the TMSchemical shift standard is designated by the arrows.

In addition to intrapair dipolar couplings, NMR linescan be broadened by interactions about which little maybe known, such as heteronuclear couplings, chemical shiftanisotropies, and interpair homonuclear dipolar cou-plings. The cumulative effect of these additional sourcesof line broadening can be represented mathematically asa quantity Mz.o*.,, which must be added to Mr.,"roto ob_tain the measured second moment:

3700 3600 3500 3400Wavenumbers (crnt )

Fig. 16. Infrared absorption spectra in the OH-stretching re-gion of single-crystal grossular garnets plotted as 1.00 mm inthickness. From top to bottom: pale orange from Asbestos, Que-bec; colorless from Jetrrey mine, Quebec; colorless from Lela-tema Hills, Kenya.

A method for accomplishing the separation of secondmoments has been developedby Boden et al. (1974, 1975),Boden and Mortimer (1973), Boden and Levine (1975,1978), and Boden and Kahol (1983), and demonstratedon model two-spin and heteronuclear systems. Strictlyspeaking, none ofthe special cases treated by them con-tains simultaneously the full range of magnetic interac-tions potentially present in the Tanzanian garnet. Nev-ertheless, their theory is readily generalized to handle othersituations, and no significant modifications of their anal-ysis become necessary when applying their procedures tothe Tanzanian garnet.

According to the generalized analysis of the echo en-velope decay, the line obtained by Fourier transformationof the spin echo envelope has a second moment Mr."r.o,which can be shown to equal

Mr,ru.o: Mr,ornu. (r2)The measurement of Mr,.uro, therefore, ofers the meansfor determininE Mz.i",,^ by Equation I L

We have analyzed echo envelope decays as prescribedby this generalized method to obtain estimates for thevalues of Mz,o,nu and Mr.,",,^. The echo envelope decaycurves pictured with fits in Figure I I all show two over-lapping lines after Fourier transformation, one a broadGaussian and one a narrower Lorentzian.

Z O

Second moments computed from both the NMR spec-tra and the echo envelope decay curves through Equation5 are exhibited in Table 4. M2.in,^ has been calculatedfrom Mr. r.ro and M, through the use of Equations I 1 and12. Inserting Mr.rn,u for the Tanzanian garnet into Equa-tion l0 and solving for R, we find R : 1.76 A.If insteadwe use M, to solve for R, we obtain the slightly smallervalue R : 1.69 A. For comparison, we observe that in-terproton distances in crystalline HrO molecules have beenmeasured by NMR for a number of inorganic hydratesand have been found to lie in the range 1.58 < R"_" S1.62 A eake, I 948; Weitekamp et al., I 983; Keller, I 988).

We note in closing that as a practical matter the Fouriertransform ofthe echo envelope decay curve need not beactually computed in order to oblain Mr.".ro, as suggestedby the previous discussion; with the presumption that theecho envelope decay follows a Gaussian function, the sec-ond moment may instead be calculated from the Gauss-ian fit parameters through Equation 5. The echo envelopedecay moments measured in this work were in fact de-rived in this way.

Comparison with infrared data

The three single-crystal grossular samples exhibit dif-ferent types of IR spectroscopic behavior in the OHstretching region. The spectrum of the Tanzanian gros-sular (1386), with a complex pattern of at least 12 OHstretches (Fig. 16), is similar to the spectrum of grossular771, suggesting that its modes of OH incorporation aremore complex than the hydrogrossular substitution(Rossman and Aines, 1991). The corresponding NMRresult indicating that protons are clustered in pairs is sup-portive of this conclusion.

The infrared spectrum of the colorless Canadian gros-sular (1537) is a class 4 spectrum in the scheme of Ross-man and Aines, which, among the samples suitable forthe NMR experiment, most closely resembles that of thesynthetic hydrogrossular and, by inference, is the one mostlikely to contain H in the hydrogarnet substitution. It waschosen because it is the garnet that has the fewest spec-troscopic components in addition to the dominant 3621cm-' band. The indication of protons clustered in groupsof four from the multiple-quantum spectra are consistentwith the interpretation of the IR spectra. Additional weakfeatures in the IR spectrum could represent the protonsin clusters of size two indicated by the NMR results.

The pale orange grossular from Canada (53) containsboth the features of the colorless sample (1537) plusprominent added features at 3631 and 3664 cm ' in thespectrum, suggesting that multiple OH sites could bepresent.

AcxNowr.nncMENTS

Samples used in these experiments were generously provided by P.Flusser (Overland Gems, Los Angeles), R.D. Shannon (E.I. DuPont), andG A Lager (University of Louisville). H.C. thanks D.P. Weitekamp andD.N Shykind for helpful discussions and their interest in this work. Pri-mary funding for this project was prowided by the White Rose Founda-

1 1 6 1

tion, whose assistance is greatly appreciated. Additional funding for spec-trometer time was obtained from NSF grant EAR-89-16064. PacificNorthwest Laboratory is operated for the U.S. Department of Energy byBattelle Memorial Institute under contract DE-AC06-76RLO 1830.

RnrnnnNcns crrEDAbragam, A. (1961) Principles of nuclear magnetism, p. 1-599. Claren-

don, Oxford, England.Basso, R., and Cabella, R. (1990) Crystal chemical study ofgarnets from

metarodingites in the Voltri Group metaophilites (Ligurian Alps, It-aly). Neues Jahrbuch fiir Mineralogie Monatshefte, 127-136.

Baum, J , and Pines, A. (1986) NMR studies of clustering in solids. Jour-nal ofthe American Chemical Society, 108, 7447-7454.

Baum, J., Munowitz, M., Garroway, A.N., and Pines, A. (1985) Multiple-quantum dynamics in solid state NMR. Journal of Chemical Physics,83,2015-2025

Birkett, T.C., and Trzcienski, W.E. (1984) Hydrogarnet: Multi-site hy-drogen occupancy in the garnet structure. Canadian Mineralogist, 22,675-680.

Boden, N., and Kahol, P.K. (1983) NMR solid echoes in systems of cou-pled pairs ofspin-Vz. I. Theoretical model. Molecular Physics, 50, 645-665.

Boden, N., and Levine, Y.K. (1975) Interpair dipolar interactions in sol-ids containing spin-Vz pairs. Molecular Physics, 29, 122l-1227.

- ( I 978) Calculation of NMR spin echo responses in solids. Joumalof Magnetic Resonance, 30, 327 -342.

Boden, N., and Mortimer, M. (1973) An NMR "solid" echo experimentfor the direct measurement of the dipolar interactions between spin-y2pairs in solids. Chemical Physics Letters, 21, 538-540.

Boden, N., Gibb, M, kvine, Y.K., and Mortimer, M. (1974) Spin-echoexperiments for determination of the homo- and heteronuclear contri-butions to the van Vleck moments of NMR absorption spectra in sol-ids. Journal of Magnetic Resonance, 16, 47 1-482.

Boden, N., Levine, Y.K., Lightowlers, D., and Squires, R.T. (1975) NMRdipolar echoes in solids containing spin-Y: pairs. Molecular Physics, 29,1 877- 1 89 1.

Burum, D.P., Linder, M., and Ernst, R.R. (1981) A new "tune-up" NMRpulse cycle for minimizing and characterizing phase transients. Journalof Magnetic Resonance, 43, 463-47 l.

Cohen-Addad, C., Ducros, P, and Bertaut, E.F (1967) Etude de la sub-stitution du goupement SiO4 par (OH), dans les composes Al'Car(OlI)':et AlrCar(SiO.), ,u(OH),,u de tlpe grenat. Acta Crystallogmphica, 23,220-230.

Foreman, Jr., D.W (1968) Neutron and x-ray diffraction study ofCa,Alr(OoDn),, a garnetoid. Joumal of Chemical Physics, 48, 3037-304 l .

Freeman, R., Frenkiel, T.A., and kvitt, M H. (1981) Composite z-pulses.Journal of Magnetic Resonance, 44, 409-412.

Hoffinan, R.A. (1970) Lineshapes in high resolution NMR. In J.S. Wauglr,Ed., Advances in magnetic resonance, vol. 4, p. 87-200. Academic,New York.

Kalinichenko, A.M, Proshko, Y.Ya, Matyash, LV., Pavlishin, V.L, andGamarnik, M.Ya. (1987) Geochemistry international, 24, 132-135(translated from Geokhimiya, 9, 1363-1366, 1986.)

Keller, A. (1988) Spin-l behaviour ofsystems ofdipolar coupled pairs ofspin-Vz nuclei In W.S. Warren, Ed., Advances in magrretic resonance,vol. 12, p. 183-246. Academic, New York.

Lager, G.A., Armbruster, T., and Faber, J. (1987) Neutron and x-raydiffraction study of hydrogarnet Ca3Al,(O4H"). American Mineralogist,12,7 56-765

Mehring, M. (1983) Principles of higlr resolution NMR in solids, p. 295-304. Springer-Verlag. Berlin.

Messiah, A. (1976) Quantum mechanics, p.664-679. Wiley, New York.Munowitz, M., and Pines, A. (1987) Principles and applications of mul-

tiple-quantum NMR. In I Prigogine and S.A Rice, Eds., Advances inchemical physics, vol. LXVI, p. 1-152. Wiley Interscience, New York.

Novak, G.A, and Gibbs, G.V. (1971) The crystal chemistry of silicategarnets American Mineralogist, 56, 791-825.

Ostroff, E.D., and Waugh, J.S. (1966) Multiple spin echoes and spin lock-ing in solids. Physical Review Irtters, 16, 1097-1098.


tt62

Pake, G.E. (l 948) Nuclear resonance absorption in hydrated crystals: Finestructure ofthe proton line. Joumal ofChemical physics, 16,327-3j6.

Powles, J.G., and Strange, J.H. (1963) Zero time resolution nuclear rnag-netrc resonance (NMR) transients in solids Proceedings ofthe physicalSociety, London, 82, 6-15.

Rhim, W.-K., Burum, D.P., and Elleman, D.D. (1976) Multiple-pulsespinJocking in dipolar solids. Physical Rewiew Irtters, 37, 1764-1769.

Rossman, G.R., and Aines, R.D. (1991) The hydrous components in gar-nets: Grossular-hydrogrossular American Mineralogist, 7 6, ll53-l 164

Shannon, R.D., and Rossman, G.R (1992) Dielectric constants of silicategarnets and the oxide additivity rule. American Mineralogist, 77, 94-100.

Skykind, D.N., Baum, J., Liu, S.-B., Pines, A , and Garroway, A.N. (1988)Phase-incremented multiple-quantum NMR experiments. Joumal ofMagnetic Resonance, 7 6, 1 49-1 54.

Suter, D., Liu, S.-B, Baum, J., and Pines, A. (1987) Multiple-quantumNMR excitation with a one-quantum Hamiltonian. Chemical physicsLet ters. l l4 .103-109

van Vleck, J.H (1948) The dipolar broadening of magnetic resonancelines in crystals. Journal ofChemical Physics, 74, I 168-l 183.

Warren, W S., Weitekamp, D P., and Pines, A. (1980) Theory of selectiveexcitation of multiple-quantum transitions. Journal of Chemical phys-ics, 73, 2084-2099.

Weitekamp, D.P. (1983) Time domain multiple-quantum NMR. In J.S.Waugh, Ed., Advances in magnetic resonance, vol. ll, p. lll-zj4.Academic, New York.

Weitekamp, D.P., Bielecki, A.,Zax,D.B., Zilm, K., and pines, A. (1983)Zero-field nuclear magnetic resonance. Physical Review Letters, 50,1807 -1810 .

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Melluscntr'r REcETvED Fssnueny 22, 1993M,c.Nuscnrpr AccEprED Jvtv 27, 1993

ApprNorx 1.

This appendix presents a derivation of an a priori limiton the maximum multiple-quantum transition order thatcan be observed in the nonselective multiple-quantumexperiment described in the experimental section. Oneupper limit can be deduced immediately: a cluster of Nspin 7z nuclei can absorb or emit no more than Nphotonsin a single multiple-quantum transition. The observedmaximum can often be less, however, depending on thedetails of the experiment and on physical propertiesunique to the spin cluster under investigation. Here weshall demonstrate that for an N spin Yz cluster, the +Nquantum lines are not observable in the nonselectivemultiple-quantum experiment of Suter et al. (1987) andconfirrn the assertion made earlier that a multiple-quan-tum spectrum manifesting only t I quantum lines is in-dicative ofa proton cluster ofsize N > 2.

The multiple-quantum NMR experiment utilized in thiswork consists ofa sequence ofevents that can be dividedinto four distinct time intervals, viz., the preparation,evolution, mixing, and detection periods. The multiple-quantum interferogram recorded in such an experimentcan be calculated from the general expression

CHO AND ROSSMAN:HYDROUS SPECIES IN GROSSULAR

S(ro, r^, x,) : TrIDU-Q)Ur(x,)UoQ")poU[(ro)

UL@)ULQ)]. (Al )

(Weitekamp, 1983; Munowitz and Pines, 1987). In theabove equation, r- and a p are the durations of the mixingand preparation periods, respectively, xr is some quantitysuch as time or radio frequency phase that is incrementedfrom acquisition to acquisition during the experiment(usually, but not exclusively, in the evolution period), Dis the quantum-mechanical operator corresponding to thedetected observable, po is the nuclear spin density oper-ator at the start of the experiment, and U^(r*), Uu(xr\,and Uo(r) are the unitary evolution operators describingthe time development of the spin system during the mix-ing, evolution, and preparation periods, respectively. Ifwe cyclically permute the order of multiplication of thematrices on the right side of Equation Al (which leavesthe trace invariant) and expand the expression in somecomplete orthonormal basis, we arrive at the equation

S (r o, r -, x,) : T r {lU l(x,) U fi(r -) D U ̂ (r -) U "(x,)l

.fUo(r )psU[(r,)l)

:2 < j lO' ( r* , x , ) lk) (k lp ' " ( r , ) l j ) (A2)

where D'(r-, x,) and p'oQ) are defined by the Hermitianoperators

D' (r -, x 1) = UL@ )Uk(r ̂ )DU -(r -)U'(x,)

p'oG) = UoQ)poU[(r).

(A3)

(A4)

Since the answer will be independent of the basis inwhich the trace expression is evaluated, we can choosean orthonormal basis which has as two of its basis vectorsthe Zeeman eigenstates l+N/2), corresponding to theproduct states in which all N spins are in the *t/z and -t/z

eigenstates of the 1, spin operator, respectively. In thisbasis, the entire +Nquantum amplitude of the detectedmultiple-quantum signal is mathematically contained injust two terms in the expansion of Equation 42, specifi-cally:

S*r(2,, r-, xr) : <N/2lD'(r-, x1)l - N/2)

' ( -N /2 lp 'oG) lN/2) (As)

and

S-.,,(zo, r-, xt): FN/2ID'G^, x,)lN/2)

'\N/2lp(,(r,)l - N/2t. (A6)

Note that because of the hermiticity of D'(r-, x,) andpl(ro), we have

S*"(2, a^, xt): [S "(r,

r-, xr)|* (A7)

where the asterisk symbolizes the complex conjugate.The task of proving that the +N quantum transitions

are not observable in the nonselective multiple-quantum

experiment therefore reduces to a demonstration of theequality

S*"( r ' r^ , xr ) :0. (A8)

To prove this statement, we shall explicitly evaluate thematrix element \-N/2lptoQ)lN/2) and show that it isidentically equal to zero.

A proof of this equality based on the symmetry of timereversal (Messiah, 1976) may be formulated in the fol-lowing way. The quantum mechanical time reversal op-erator, which we desigrrate by the symbol I, can be re-garded as the product oftwo operators, as below:

T : } K (Ae)where K is the operator that transforms wave functionsinto their complex conjugates, and 7 is the rotation op-erator:

I 1 6 3

which allows us to write

lUoQ)Il: e aes'Y"rr'

- ^ + t U , t "

: Utr?). (Al7)

To complete this simplication of Equation Al2, werecall from our definition of I that the time-reversedwavefunctions will have the form


i : g - i ' t '

The inverse off has the form

T T : K A I .

(Al0)

(Al l )

We proceed to make use of this symmetry operation byevaluating the (-N/2, N/2) element of p'oQ):

<- N/21pi ( r , ) lN/2)

: < - N / 2 | U,(r ")

p oUj(r,) | N / 2)

: ( - N / 2 | {ltI U "(r,)IrI

p sII I U!(r )Jr J I N / 2) I

: \ - N / 2 | $rlJ U,G )ytlfI polt]lI UIQ )It]

. J lN /2 ) j (Al2)

where we have made use of the equality I : 7t9, with 1the identity operator.

The initial density operator po in all of the multiple-quantum NMR experiments of this work was the reducedhigh-field equilibrium density operator, i.e., an operatorthat is proportional to the spin angular momentum op-erator 1". This fact, combined with the definition of 9,gives us the equality

TpoJr : -po. (Al3)

In the multiple-quantum experiment that concerns ushere, the time development operator of the preparationperiod is well approximated by the unitary propagator:

and

The minus sign on tho right of Equation (Al8) is a con-sequence of the fact that the eigenfunctions of a spin 7zparticle are spinors.

Combining Equations Al3, Al7, Al8, and A19 andutilizing the known properties of the time reversal oper-ator, we obtain:

( - N/21p(: ( r , ) lN/2)

: (- l)N+'( - N/2llItUj(r,)poU,(, ")l

- N/2)l

: ( - l ; " * ' t t , - N /2 lT t l

' lU[(r,) psU "(r,)

| - N / 2)l]*

: (-1)n+'[(N/2lU[(r)psU"G)l - N/2)l*. (A20)

The hermiticity of po and the unitarity of U,(2,) implythat the last expression above can be written

<- N/2 lptoQ) lN/2)

: ( - l )N+1(- N/2 lU|( r , )p6UoQ)lN/2) . (A21)

An alternative expression for the matrix element onthe left side may be derived by writing the identity op-erator as

l : z r z

wfihZ defined as the rotation operator:

(422)

(A23)Z = g i " t ,

Inserting this form for I into (N/2lpitG)lN/2), we see

(-N/2 lpt"Q) lN/2)

: ( - N / 2 lzt lzu oQ )ZtllZpozrllzur k )ztl.zlN/2). (424)

This expression may be reduced by observing that, sincepo is proportional to 1., then

JIN/2) : (- l )"1 - N/2)

<-N/2 lJ t : (N /21 .

IZ, pol: o

zpozr : po.

(Al8)

(Ale)

(A25)

(426)

where

J , kj > k

Since the effective preparation Hamiltonian 1I,, is purelybilinear in spin operators, then

Uo(tr) : € iH"'p

11",: > dik!" jI,k + I, jI"k).

(Al4)

(Al5)

Jff,Jr : ff_" (Al6)

so that

l t64

Furthermore, we observe that

zf1",zr : -11",

from which it follows that

ZUpG)Zt: (4(r,).

Finally, the fact that lN/2) and, l-N/2) are, by construc-tion, eigenstates of t with eigenvalues N/2 and, -N/2,respectively, enables us to deduce


Comparing Equations A2l and A,3l, we discover

(427) ( -N/2 lp 'oQ) lN/2) : - ( -N/z lp 'oQ) lN/2)

: 0 .

(A28)

and

Substituting Equations A26, A28, A29, and A30 intoEquation A24 leads to the final result:

( -N /2 lp ' o | ) lN /2 t

: ( -1) /v( -Nl2 lU!( r " )poU,(r , ) lN/2) . (A3l )

(A32)

We conclude, therefore, that the N quantum transition isnot observable in an N spin Yz system if the effectiveHamiltonian during the preparation period is of the formff," and the initial density operator is proportional to 1,.It follows from this result that a multiple-quantum spec-trum measured under these conditions exhibiting onlythe +l-quantum transitions must originate from spinclusters of size N > 2.

In closing, we note that an analogous time reversalsymmetry argument can be constructed to establish morerestrictive theoretical limits on the maximum multiple-quantum transition observable for the even-quantum se-lective version of the multiple-quantum experiment butforgo presenting the proofhere.

Z IN /2 ) : , i u " t z lN /2 )

(- N/2lZl : (- N/21 e-iN*tz

(A2e)

(A30)

single-crystal nmr studies of low-concentration hydrous ... · ed by standard multiple-pulse...

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