PH Y SICAL RE VIE% B VOLUME 15, 5 UMBER 7 1 APRIL 19'? 7

2sNa nuclear relaxation in Na P-alumina: Barrier-height distributions

and the diffusion process

R. E. %alstedt, R. Dupree, ~ J. P. Rerneika, and A. RodriguezBell Laboratories, Murray Hill, New Jersey 07974

(Received 27 August 1976)

Measurements of the "Na nuclear-relaxation time T, are reported for two types of Na P-alumina specimen,i.e., flux-grown crystals from our laboratory and melt-grown crystals from Union Carbide Corp. A

phenomenological calculation of T, due to activated hopping motion, taking account of local variations in

energy-barrier heights, is presented. This theory is used to interpret our own T, data, plus data on T, in theliterature for specimens prePared by a third method, in terms of barrier-height distributions. It gives a goodaccount of both the temperature and frequency variation of these three contrasting sets of T, data, where thedifferences are attributed to variations in the structure and/or arrangement of charge defects. Thecharacteristics of the barrier-height distributions obtained are discussed in terms of preparation methods andother available data. The rather larger conductivity for melt-grown (as opposed to flux-grown) materialreported by Barker et al. is found to be associated with a displacement of activation energies to lower values.Values of the quadrupolar coupbng constants from the T, data fits as well as from second-order quadrupolarbroadening are found to be essentially independent of sample source, suggesting that this coupling isdetermined primarily by the host structure. Finally, the T, fits yield values of the local vibration or "attempt"frequency vo for activated hopping which are -10" Hz for all three cases considered in sharp contrast withthe vibration frequency (-2X 10" Hz) reported from infrared spectra and Raman scattering. Our value,which is derived directly from the hopping motion itself, does not yield the good agreement with experimentnoted earlier when used in available theories of the diffusion constant.

I. INTRODUCTION

By now it is well established that the p -aluminacompounds form with a substantial excess (- 25/g)of mobile ions beyond the stoichiometric compo-sition (e.g. , Na, O 11AI,O,). While this excess isthought to be primarily responsible for the highionic conductivity of these materials, ' it is alsoknown to result in a similarly large number ofcharge-compensating defects in the host struc-ture. The precise nature of these defects is notknown with certainty, but they are thought to haveimportant effects on various measured properties.For example, the presence of charge defects in ornear the conducting planes in p-alumina implies adistribution of local vibration frequencies andbarrier heights for hopping motion. Such distri-butions of vibration frequencies are present ininfrared spectra' and have been invoked to explainspecific-heat data4 on p -alumina compounds.

The defect-related distribution of barrier heightsfor hopping motion has important consequencesfor the microscopic motion of the conducting ions.Nuclear quadrupolar spin-lattice relaxation is asensitive probe of ionic motion, and the mainthrust of this paper is to develop the interpreta-tion of such relaxation data in terms of diffusionalbarrier-height distributions. We report relaxa-tion time (T,) data for "Na in Na P -Al,O, speci-

mens taken from two different sources: melt-grown crystals from Union Carbide Corp. , andflux-grown samples made at our laboratory.Microwave conductivity data on these two kindsof material' show the same temperature depen-dence, with, however, the melt-grown materialmore conductive by a factor -4. Correspondingly,our T, data analysis reveals substantial differ-ences in the position and shape of the barrierheight distribution for these two cases, leading toan interesting conjecture as to the origin of thedifference in conductivities (see Sec. IV). A use-ful by-product yielded by this analysis is the pre-factor for the mean hopping frequency in thesesamples. This number, when compared with in-frared frequencies identified as "attempt frequen-cies,"' casts doubt on the validity of simple theo-ries of diffusion in these materials.

The paper is organized as follows. In Sec. IIwe discuss the theory of nuclear relaxation due toactivated hopping. A rate-equation model is de-veloped for this kind of motion from which nu-clear-relaxation-rate expressions are deduced.Simplified versions of these results are employedin Sec. IV to interpret experimental data. Forreaders interested only in the results and con-clusions, this section can be bypassed withoutloss of continuity. In Sec. III we discuss brieflythe specimens and experimental details. T, data

15

15 23 Na NUCLEAR RELAXATION IN Na P-ALUMINA:. . .

and its interpretation are given in Sec. IV, andin Sec. V we summarize our conclusions.

II. MODEL FOR CALCULATION OF T

and

A, (I()= [eQ/4I(2I —1)](I),igg+ Iggig, ),

A, (I,)= [eQ/4I(2I —1)]P„

V„=V,„,(t)+iV„.(t),V, 2= g[V, (t) —V( (t)]+i V„y(t)

(2a)

(2b)

(3a)

(3b)

In Eqs. (1)-(3) the subscript i stands for the nu-clear spin of ion i and the V, e(t)'s are the time-varying EFG tensor components at the ith nucleus.The time variation of V, ~ arises both becauseion i is moving through the crystal, and becausecharged ions in the vicinity of ion i are moving.

There are two fundamental rates of transitionamong the m levels of a particular nucleus causedby the fluctuating Rc, namely, W, (s& =+&) andW, (s-', =+ -', ) corresponding to

~&m~ = 1 and 2,

respectively. These give rise to two eigenmodesin the approach of energy-level populations towardequilibrium, with associated characteristic rateswhich are for this case simply R, ,= 2S', , Undercertain assumptions spelled out below, the timevariation of the nuclear signal amplitude S may beparametrized as

S(t) = S +S,e "&'+S,e "2~. (4)

While normal phonon contributions to T, maybe present, nuclear relaxation in Na p -aluminais clearly dominated by the diffusive motion, i.e. ,by the interaction of fluctuating electric-field-gradient (EFG) contributions with the nuclear quad-rupole moments. In this section we formu1ate amodel for the calculation of T, for the case ofactivated hopping between allowed sites when thereis a distribution of barrier heights present. Noattempt will be made to solve this model for ageneral case, but the solution for a simplifiedspecial case will be discussed in the Appendix.

We specialize our discussion of quadrupolarrelaxation to the case of I= 2 ("Na). Extensionto other spin values is straightforward. In anapplied field H, one has four energy levels Ewith energies given to zero order by E Q

= —~Qm, m=+&, + 2, where (dQ=yHQ. Nuclear quad-rupolar coupling gives rise to average shifts inthe energies' E as well as transitions among thelevels caused by the fluctuating part of the inter-action term

2

Xc = Q [A„(I()V(e(t)+c.c.],

The relative magnitudes of S, and S, depend on thequadrupolar splittings of the energy levels andthe pulse conditions employed in a particular ex-periment.

The calculation of the rates 8', , has been dis-cussed by Torrey' and others. ' The followingresults are essentially a paraphrasing of thisearlier work. By standard time-dependent per-turbation methods one obtains

Cl

x dv[e "o (V(,(t)V)~(t —r)), + c.c.],0

w, = f&+-,'/A f+-,*)f'(5a)

dr[e2'"0' (V&2(t) V&*,(t —r)), + c.c.],Q

(5b)

where ( ), denotes an average over all possibleenvironments in whig/. ion i finds itself during therelaxation process. A critical assumption is thatthis average is the same for all particles, where-upon the rates 8"» become unique quantities forthe entire system. This is closely linked to thefurther assumption that the correlation times 7,of Vy and V, are short compared with W, ' andrespectively, i.e., that the ions make many jumpsduring the relaxation process. %'e shall see inSec. IV that this assumption breaks down at lowtemperatures.

Modeling the relaxation process consists of cal-culating the correlation functions of V, and V, inEq. (5) in some approximation. We now discussa conceptual scheme which leads to the generalform of the answer for the case of uncorrelatedmotion of the diffusing ions. As justification forthis approximation we note that static correlationlengths have been found to be short (-1 latticespacing) in P -Al,O, materials, ' and that the localvibration mode as seen in neutron scattering isnearly dispersionless" and thus consistent withessentially independent ionic motion.

To study the time variation of the EFG tensorcomponents we introduce a set of variables p, ~~(t),which give the probability that ion i is at site jin "configuration" k at time t. By configuration,we mean local arrangement of mobile ions andcharge defects which give rise to a distinct set ofvalues for V„and V„while ion i is at site j. Forexample, a given site may be singly or doublyoccupied, "and this consideration alone leads toseven distinct configurations (see Appendix).

It is implicit in the above picture that ions areassumed to occupy a set of equilibrium sites atall points in time, i.e., the "transit time" as

3444 %ALSTEDT, DUPREE, REMEIKA, AND RODRIGUEZ

ions jump between sites is neglected. This ap-proximation is probably a reasonable one at allbut the highest temperatures, where the motionbecomes liquidlike.

In this framework one can write down a set ofrate equations for the time variation of the

p, »(t)'s. In accord with the physical picture ofhopping motion in t3-A1,0,developed by Wang,Gaffari, and Choi, ' we adopt as ground rules that

(a) only ions on doubly occupied sites can jump and

(b) their only possible destination is a singly occu-pied site, so that sites always have either one ortwo ions. The rate equations then have the generalform

(S) ~ (2)Pj» ~f fk ~ fk fk' ~ fk f'k'k' k'

With the above solution of the rate equation inmind, we can deduce the form of the correlationfunctions in Eq. (5).

First, we introduce the quantities v» and v»which are the EFG tensor components Vy and V2for an ion in configuration k. Next, we defineP(j ', k';j, k, t) to be the probability that an ion isat site j in configuration k at time t given that itwas at (j',k') at t=0. The time average ( ), inEq. (5) is then replaced by an average over j' andk' with their average occupation probabilityqf, k. . One then has immediately that

(Vn(t)V(~(t — )),= Q v,kvkk.kk k'

x Q qpk. p(j ', k'; j,k, r) (7}.

(3) (4)

++Pl�/k'

kU»', gk ~ Pgf'k' I'k', »'k' f'(8f)

In Eq. (6), w"' and w'" are rates for transitionsin which either ion i moves to another midoxygenposition at site j or ion i remains fixed and otherions move. Bates w'" and w"' are for processesin which ion i jumps to another site. In the gen-eral case the rates w"' '" are functions of theother p, , »'s (k' xi) because of the site availabilityfactor and because of the possibility of multiplejump processes. The set of Eq. (6) is thereforenonlinear and one cannot discuss a general solutionby any simple means.

To consider this model further, we retain onlyterms linear in the P,»'s. This has the effect ofneglecting correlations in the ionic motion. Asecond simplification we introduce is to replacethe quantities Q, .„,&P, .» which occur in thew" "s by (Z, P, ») which is the mean occupationprobability of configuration k at site j. In theresulting set of first-order linear differentialequations the motion of different ions is uncou-pled and the ion index i is totally redundant. How-ever, the site index j is not redundant because ofthe symmetry-breaking effect of charge defectsin or near the conducting plane. We therefore ex-pect an essentially continuous distribution of solu-tions to these simplified equations of motion.

The solution of the linearized rate equations isreduced to the problem of diagonalizing a matrixwith the substitution P, »(t) = a»e "'. If the ther-modynamic weight factors are absorbed into thea»'s, then the resulting matrix is symmetric bydetailed balance. We simplify this aspect of theproblem by assuming all sites and configurationsto be of equal energy. The eigenvalues of the rateequation matrix are assumed to be real and posi-tive, i.e. , no oscillatory solutions are considered.

Further, it is straightforward to show that

P(~', k;j, k, r)= g ep, (I)~»(l)e-" ',

where e»(I) is the j,k element of the Ith eigen-vector and X, the lth eigenvalue of the matrix ofthe linearised set of Eq. (6). Equation (I) thenbecomes

(V, (t)V, (t —r))

Z "ik"xa' ~&&'k' ~ &'k'(I)»( }e (6)

with a similar relation for V„. Combining Eqs.(6) and (5a) one finds for the measured relaxationrate R, = 2W„

Xg1 1& y2 2

l

where

(9)

ki= I&'I"kl'&I'Z kk Xk Z&rk Pk(} gk(}k', k fk, f(IO}

Equation (9) forms the basis for our interpreta-tion of experimental T, data in Sec. IV. However,in order to put this result in an appropriate formfor comparison with experiment, we must intro-duce additional phenomenologically motivatedassumptions concerning how the X, 's vary withtemperature. The X, 's are basically combinationsof the ur&„'& k

's from. E. q. (6) which, in turn, havethe form ve . Here v is a local vibrationalfrequency, modified by site availability factors,and U is a barrier height. We therefore expectthe X, 's to be combinations (not necessarily linear)of such terms. However, we shall make themathematically convenient assumption that theX, 's can be represented as a continuous distribu-

Na NUCLEAR RELAXATION IN Na P-ALUMINA:. . .

tion of ve ~~, centered at U, :

trFi U Ui~U

In accord with simple models' ' we take v(U) ~ U'i'throughout. In Eq. (11}, U, is the key variable;accordingly, we identify C» in Eq. (10) as a mean-squared variation in the quadrupolar couplingconstant" times a distribution density for U, .Thus, C„=0+G(U, ), where G is normalized,whereupon Eq. (9) becomes

(12)

Equations (11) and (12) are employed in Sec. IVto interpret nuclear relaxation data. Evidently,one must obtain further information about the dis-tributions F, and G in order to use this model.The only further theoretical input comes from amodel calculation by Varma' which gives the over-all distribution of barrier heights, representedapproximately by the convolution of F, and G, asa symmetric function of prescribed form. Beyondthis we employ these equations in a strictly phe-nomenological fashion to deduce the propertiesof F, and G from experiment.

III. SAMPLE MATERIALS AND EXPERIMENTALTECHNIQUE

Two sources of Na P -alumina sample materialwere used in this study, melt-grown crystals fromthe Union Carbide Corp. and flux-grown crystalsmade at our laboratory. There arp importantdifferences in the properties of these two kindsof material which are apparently a consequenceof the constrasting methods of crystal prepara-tion. The melt-grown crystals are, in effect,quenched from a high temperature (- 2000 'C),whereas those made in a flux form over manyhours at T-1250 C. Powder x-ray photographsshow these two types of material to be essentiallyidentical in structure. Chemical analyses, whichare not highly reliable in this ease, and ion-ex-change weight-change data indicate roughly thesame excess Na concentration (-25%) in bothtypes. X-ray diffuse scattering results, "how-ever, show the melt-grown crystals to be morehighly disordered. This observation is furthersupported by the fact that the "Al NMR line issubstantially weaker relative to the "Na in themelt-grown crystals, suggesting greater crystal-line disorder in the spinel block regions.

Other important experimental differences havebeen reported. While there are subtle differencesin their respective infrared spectra, "the micro-wave conductivity of melt-grown material has been

found' to be - 4 times that of flux grown with verynearly the same temperature dependence. Final-ly, of course, striking differences in the nuclearrelaxation properties of these two specimen typesare reported and discussed in Sec. IV.

For the melt-grown material, specimens werecut from single-crystal boules with a string saw.In contrast, the flux method tended to produceroughly hexagonal platelets a few mm across and1-2 mm thick. A dozen or so of these were as-sembled to form a specimen with a common c axis.(Whereas T, is strikingly anisotropic with respectto the angle between the applied field H, and thec axis, it is shown in the Appendix by symmetryarguments that T, is azimuthally isotropic. ) Bothtypes of specimen were sealed off inside quartztubes after being pumped to -10~ Torr overnightat 800 'C to eliminate water contamination.

T, measurements were made on "Na with apulsed NMR spectrometer of standard design.For temperatures up to -200'C the sample tem-perature was regulated by means of a Varian mod-el V-4540 temperature controller and measuredusing a Cu-constantan thermocouple. Above200'C a water-cooled furnace using a thermo-couple sensor and current-regulation loop wasemployed. In either case thermal homogeneityover the sample was estimated to be a 2 'C and ab-solute temperature accurate to + 2 'C below 200 'Cand +5'C above 200'C. A Fabritek model 1062Instrument Computer was used to improve thesignal-to-noise ratio on T, decay curves.

A, Second-order quadrupolar shift

The second-order quadrupolar shift of the(-, = -k} transition has been measured as a func-tion of angle 8 between H, and the c axis using amelt-grown (Union Carbide Corp. ) single-crystalspecimen. These data were taken at room tem-perature at a frequency of 25.8 MHz, and areplotted in Fig. 1. The solid line shown was deter-mined by least-squares fitting the data to the ex-pression'

&H = (sv+l8y(oo}tl(I+ 1) —~]

&& (1 —cos28) (9 cos28 —1), (13)

IV. EXPERIMENTAL RESULTS AND DISCUSSION

The model theory of Sec. II is now employedto interpret experimental T, data on the specimensdescribed above. T, has been studied as a functionof frequency, temperature, and angle 8 betweenHp and the c axis of the crystal . In addition to ourown data, we also discuss briefly the T, -vs-T dataon "Na in Na p -alumina reported by Jerome andBoilot. '4

3446 %AI STKDT, DUPRKE, RKMKIKA, AND RODRIGUEZ

12

10—

cil 2C/l

0C9 -2—XCI

I I I I I j I I I I I I I l

160 180 200 220 240 260 280 3008

FIG. 1. Second-order quadrupolar shift of the (~—~) transition of Na in a Union Carbide Corp. (melt)

crystal of Na P-alumina as a function of the angle 8between the applied field and the c axis. These datawere taken at 25.8 MHz and room temperature. Thesolid line is a fit of Eq. (13) to the data points.

where vo = 3e'qQ/2hI(2I —1), from which the quad-rupolar coupling parameter is determined to bev~ = 1.017+0.024 MHz. This value agrees closelywith those reported in the literature, "includingthat given for a crystal prepared in a ratherdifferent fashion" and having strongly contrastingT, properties (see below). Conjecturing that thesedifferences are caused by changes in the natureand/or arrangement of charge defects, one wouldthen conclude that the average nuclear quadrupolarcoupling is only weakly affected by the defectstructure.

B. Anisotropy of T&

C. Temperature variation of T&

Measurements of spin-lattice relaxation rateswere carried out on both types of sample materialdescribed in Sec. III at two NMB frequencies near

12

10

MELT CRYSTAL

(14) and (15) and are seen to give a reasonablerepresentation of the data except, perhaps, nearthe "crossover" regions. The data for R, areseen to be rather more scattered than for R„re-sulting from the small coefficient of the R, termin the decay curves.

The five-parameter fitting procedure is a verydemanding one in terms of data quality requiredto obtain reproducible results. Although the inputdata were excellent, deduced values of R, and R,near the crossover points must be regarded asquestionable. We cannot say whether the apparent"avoidance" of the intersection of R, and R, curvesby the data points is a real effect or an artifactof the fitting procedure.

The most significant result is the minimum foundin R, (8) near 8= 0. The peak-to-minimum rationR, /R, „-100 demonstrates the mirror-planesymmetry of the EFG tensor even in this morestrongly disordered of the two sample materials.The discussion of Sec. IV A suggested that per-turbation of the EFG tensor by the defects wassmall. The null point in R, is consistent with thisconclusion and further demonstrates that thedefect perturbation of the EFG tensor retains themirror symmetry of the host crystal.

The variation of spin-lattice relaxation ratesR, and R, with angle 8 has been measured for theUnion Carbide crystal at room temperature.While the relaxation curve obtained with a g-&gpulse sequence was found in general to exhibitboth rates, R, was almost always found to be thepredominant term with R, only very weakly rep-resented in some instances. The two rates wereextracted from S(t) using Eq. (4) in a least-squares-fitting procedure. The results are plottedin Fig. 2.

In the Appendix it is shown that symmetry con-siderations lead to the forms

8

6F

CV

00 60 90

R, (8) =A sin'8+ 8 sin'28,

R,(8) = D+ E cos28+ F cos'28,(14)

(15)where the parametrization has been chosen sothat the constants A-F are positive. The solidcurves in Fig. 2 has been calculated using Eqs.

FIG. 2. Spin-lattice relaxation rates R&

and R2 ina Union Carbide Corp. (melt) crystal taken at roomtemperature at a frequency of 25.8 MHz, as a functionof angle 8 between the applied field and the c axis. Thesolid curves are plots of Eqs. (14) and (15), hand fittedto the data, whereA =2.3, B=6.2, D=0.9, E=1.5, andE=7.6, all in msec '.

Na N UC LEAR RE LAXATION IN Na P-ALUMINA:. . .

10

RYSTAL

ORE:NTZIAN DISTRIBUTION

GAUSSIAN DISTRIBUTION

by itself would lead to T, cc&', below the minimum.In Fig. 3 we find a change with frequency of 1.6at low T in contrast with the ratio 2.2 of u~~ val-ues. We therefore assume throughout, for sim-plicity, that F, (U —U, )=6(U —U, ), deferring fur-ther discussion of this point to Sec. IVD." Equa-tion (12) then becomes simply

i i I

0 1 2

—w 0

I i I

4 51000/T (K )

o 17.2 MHzI

6 7 8

FIG. 3. Plot of lnT i vs 1000/T for the Union CarbideCorp. (melt) crystal at two resonance frequencies asshown. The solid and dashed lines are fits of Eq. (16)to the data using a Lorentzian and a Gaussian shape forG (U), respectively.

17 and 25 MHz, respectively, and at temperaturesranging from 125 to 1125 K. We denote T, —=8,'throughout the discussion. " All measurementswere taken with the magnetic field at 45' to thec axis, which gives the peak value of R, (Fig. 2).

Plotting the data as 1nT~ vs 1/T one expects, inthe simple case of a unique barrier height, to findthe symmetrical V-shaped curve of the Bloem-bergen-Purcell-Pound model theory. As we seefrom the Union Carbide Corp. (melt) crystal dataplotted in Fig. 3, the experimental curve is highlyasymmetric. This reflects the presence of abroad distribution of barrier energies U as anti-cipated in the model theory of Sec. II. Moreover,the diminished frequency variation of T, foundat low temperatures leads us to surmise that thedominant distribution effect in the model is G(U)[Eq. (12)] rather than F,(U- U, ) [Eq. (11)), which

(16)1 0

where 1/&=v(U)e r. Simple model theories"give v(U) = vo(U/Uo)' for sinusoidal barriershapes. We also adopt this relation throughoutthe discussion.

T, data are interpreted with Eq. (16) by modelingthe distribution function G(U}. We employ Gaus-sian and Lorentzian distributions as extremecases of rapid and slow decay in the wings. Thedata fits were carried out by trial-and-error ad-justment of parameters. Gaussian and Lorentzianfits are shown in Fig. 3 by the dashed and solidlines, respectively. One notes that the tempera-ture extremes are well represented by eithershape for G(U), the primary difference appearingin the region of the minimum. Furthermore, forthis crystal the data are seen to lie between thetwo curves, suggesting that some shape inter-mediate between Gaussian and Lorentzian is theappropriate one. This leads to the interestingconjecture that the model distribution calculatedin Ref. 4 is very nearly correct, since it is inter-mediate in the sense required. " One interestingfeature of these fits is that the two rather differentshapes tried for G(U} give frequency dependencesat low T which are nearly indistinguishable andare in good agreement with the data. We empha-size that this frequency dependence is primarilya function of the width of G(U) and is in no sensefitted.

The parameters from the fitted curves in Fig. 3are listed in Table I. Deferring a detailed dis-cussion of these to the end of this section, weonly note that the peak energies of the fitted

TABLE I. Fitting parameters for Ti vs T. Thevation energy &0=0.17 eV.

v values quoted correspond to the bulk acti-

Crystal G(U) Up (eV) 4Uiy2 (eV) & 0(Hz) & (sec i)

meltmeltfluxflux

Jerome-Boilot(polycrystal)

LorentzianGaussianLorentz ianGaussian

bimodal

Gaussian

0.1340.1500.2080.2110.1800.3010.189

0.0470.0640.0740.089

0.056

0.045

2.01x 10«2.76 x 102.77x10"2.46x 10"

1.92 x 10ii

1 79x ]0ii

3.6 x 10~

2.91x 106

3.46 x 106

3.15x 10

3.22 x 106

2 45x 10

%ALSTEDT, DUPREE, REMEIKA, AND RODRIGUEZ 15

1.0—

RIBUTION

BUTION

VNE

I-

O.I— . I MHz

5.1 MHz

I i I i I i I i I i I i I

2 3 4 5 6 7IOOO/T (K-1)

FIG. 4. Plot of lnT& vs 1000/T for flux-grown crys-tals from our laboratory at two resonance frequenciesas shown. The solid and dashed lines are fits of Eq.P6) to the data using a Lorentzian and a Gaussianshape, respectively, for G(U).

curves are somewhat below the bulk activationenergy, ' and that the widths are -0.4 times thepeak values. As expected, the Lorentzian half-width is less than the Gaussian one because of itsbroader wings.

T, data for the flux-grown crystals are shownin Fig. 4. These data exhibit similar features tothe melt-crystal data with roughly the same ratioof high- to low-temperature slopes and a similarfrequency variation at low temperatures. Thereare also striking differences. The high-tempera-ture slope is markedly higher than in Fig. 3,suggesting the presence of higher energy barriers.Also, the data do not extend to nearly as low atemperature, a consequence of the onset of whatmight be termed "ion trapping" in this specimen.By this we mean that below -200 K a significantand increasing number of Na' ions become largelyimmobilized by high-energy' barriers. As a re-sult, their T, becomes very long and their con-tribution to the signal saturates to zero. One en-counters a distribution of T, values as a conse-quence of the breakdown of the averaging assump-tion introduced with Eq. (5). The T, decay be-comes nonexponential and no longer subject tostraightforward interpretation.

The signal loss from the trapping effect isshown in Fig. 5, where the repetition period ofthe (v, —,'v) pulse sequence was at least ten timesthe minimum T, in the system. Starting at T-200 K, the trapping effect reduces signal ampli-tude to -10% of its normal value by T = 135 K.

1.6

1.4—

1.2—

1.0

0.8—

0.6—

0.4—

0.2—

FLUX-GROWN CRYSTALS

p 00

0

00

I

100I

200T(K)

400

FIG. 5. NMR signal amplitude $ in flux-grown crys-tals at 25.1 MHz plotted vs temperature. Data are nor-malized to unity at room temperature and correctedfor Curie-law variation. The repetition frequency of thex-~ 7I pulse sequence was maintained less than 10% ofthe maximum relaxation rate present.

Moreover, the T, of trapped nuclei is apparentlylengthened by some orders of magnitude, sinceincreasing the repetition period of the pulse se-quence by a factor of 2 does not appreciablychange the numbers in Fig. 5. This effect wasnot an important one in the melt crystal over thetemperature range studied (Fig. 3).

Another interesting contrast with the melt crys-tal results appears when we fit the data to Lo-rentzian and Gaussian barrier-height distributionsas shown in Fig. 4. Again, one finds that a goodfit is possible in the temperature extremes and thelow-temperature frequency dependence is wellrepresented. In the region of the minimum, how-ever, there is a marked discrepancy with bothdistribution functions such that the predicted T,minimum is not attained. This suggests that theactual distribution in this material may havestructure which is not resolved by the T, curve.

In particular, the disparity seen in Fig. 4 sug-gests that the actual distribution G(U) may havemore than one mell-defined peak, with a gap be-tween which causes the rate R, to fall below themaximum value which would be exhibited by asingle-peak distribution. Accordingly, a fit tothese data has been attempted using a bimodalshape for G(U). In order to avoid an unwieldyproliferation of parameters, some simplifyingassumptions were introduced. The shape wastaken to be the sum of two Gaussians of equalwidth. Because of the extremely steep high-tem-perature slope it was surmised that in additionto a peak similar to the Gaussian curve fitted tothe melt crystal data (Fig. 3), one may also havea somewhat smaller peak at rather higher energy.The amplitude of the higher peak was arbitrarilychose to be 25/0 of the lower one. The fitting

15 N a N UC LEAR RELAXATION IN Na P-A LUNIINA:. . .

1,0

E

0.1—MHz

1 MHz

s ! s I s I i I s I i I i I

0 1 2 3 4 5 6 71000' T (K-')

FIG. 6. T j data on flux-grown crystals from Fig. 4,with the fit of Eq. (16) using a bimodal distribution(see text) shown as a solid line.

D. Discussion of results

The a,ssumption that G(U) is the main distribu-tion effect for barrier heights, i.e., that

parameters in this case include the centerpositionsof both peaks as well as the Gaussian width param-eter, the attempt frequency v, and the scale fac-tor 0~2.

The "best fit" to the T, data of Fig. 4 using abimodal distribution as set forth above is shownin Fig. 6. There one observes a marked improve-ment over the fits obtained with single-peak dis-tributions. In Table I we see that the fit presentedcorresponds to peak energy values split by nearlya factor of 2. The quality of fit obtained is quitesensitive to the splitting between these peaks,deteriorating very noticeably if this splitting isvaried by + 20%. Thus, while the bimodal fit isby no means unique, it is nonetheless quite sensi-tive to the details of G(U). This suggests that thebimodal fit contains at least the gross featuresof the true distribution.

Lastly we analyze the T, data reported by Jer-ome and Boilot,"which are plotted in Fig. V. Thespecimens were prepared by a method of slowcooling near the melting point, "which is differentfrom either method employed to prepare the crys-tals studied here. Likewise, the T, data are alsorather different from Figs. 3 and 4, with a rela-tively small ratio of high- to low-temperatureslopes. The solid line in Fig. 7 corresponds to aGaussian shape for G(U) and gives a reasonablefit to the data. As expected, the width parameteris the smallest of any found in our study (Table 1).

E,(U —U, ) is relatively much narrower, hasyielded good agreement with observed frequencydependences in all cases tried. On close examina-tion one finds a barely resolvable tendency forthis procedure to underestimate dT, /d~, at low

temperatures, an effect which could be accountedfor by giving F, a finite width. Further data overa wider frequency range will be necessary to re-solve this point. In the present approximation,then, one has simply a superposition of Bloem-bergen-Purcell-Pound rate curves, a result whichwould have been difficult to guess a priori fromthe formulation of Sec. II or from the simplifiedsolution in the Appendix.

The major uncertainty in the distributions G(U)obtained in Sec. IVC stems from the question ofuniqueness. It is not easy to rule out the possibi-lity that some rather different forms for G(U)would also lead to a reasonable representation ofT,. Barring this seemingly unlikely possibility,we feel that our results represent the major fea-tures of the energy barrier distributions. It isalso important to note that the energy resolutionof these experiments is not high, so that a wealthof finer structure may have escaped detection.

It is interesting to correlate the distributionparameters with crystal preparation techniquesand other known data. All fitting parameters arelisted in Table I with the best-fitting distributions(i.e., Gaussian for melt crystals and Jerome-Boilot data and bimodal for flux-grown crystals)plotted in Fig. 8. The melt crystal is known to bestrongly disordered from x-ray diffuse scatteringstudies, and this probably accounts for its ex-tremely wide profile for G(U). Also, its peak is

10

01 I I I I i I i ) j I

0 1 2 3 4 5 6 7 S1000/T (K-1 )

FEG. 7. Plot of lnT& vs 1000/T using data from Ref.14, showing fit of Eq. (16) with Gaussian shape forG (U).

3450 %ALSTEDT, DUPREE, REMEIKA, AND RODRIGUEZ 15

12—

0.8

cn 06

04

0.2

00 Oi 0.2

U(eV)

0.3 04

FIG. 8. Plot of Gaussian distribution G (U) determinedfor the melt-grown and Jerome-Boilot specimens andthe bimodal distribution determined for the flux-grownspecimen. The distributions are normalized to con-stant area and plotted in arbitrary units.

well below the bulk activation energy (in contrastwith the others which peak above it) giving rise toa great deal of low-barrier motion. Again, thiseffect is undoubtedly connected with disorder.One must be cautious in drawing conclusions aboutdiffusional motion from these barrier height dis-tributions, because there may be modes of motionpresent which contribute to nuclear relaxation butnot to diffusion (e.g. , the circulatory mode ana-lyzed in the Appendix). Nonetheless, the abun-dance of low-barrier motion in the melt crystalseems to be clearly connected with its superiorconductivity.

Likewise, the cause of relatively lower con-ductivity in flux-grown crystals is possibly re-vealed by the bimodal G(U) fitted to the T, curve.In Fig. 8 we see, first, a subsidiary peak at highenergies which is very probably connected withthe ion trapping effect described in Sec. IV C. Theoccurrence of such a peak is no doubt an inhibitingfactor for diffusional motion. In addition the mainpeak is also centered somewhat above the bulkactivation energy, leading to a relative paucityof low-barrier motion in strong contrast with themelt crystals.

The occurrence of a high-energy satellite peakfor flux crystals is probably associated with theirslow low-temperature method of preparation.This method might lead to a partial orderingamong charge-compensating defects or, alterna-tively, to a different species of defect from thosefound in crystals formed near the melting point.The only evidence on the ordering hypothesis isthe relatively strong tendency for superlatticeformation in flux-grown crystals revealed bydiffuse x-ray studies. " A detailed knowledge ofdefect structure is only now beginning to emergefor these compounds, so that a detailed accounting

for these properties is not yet possible.The crystal used in the Orsay work has the

narrowest distribution of U values of the threeanalyzed. In the absence of further data, one

might conjecture that the lengthy annealing nearthe melting point employed in preparing thesecrystals results in a lower concentration of Na'

as well as charge defects. The narrowness of

G(U) would then be simply a defect concentrationeffect. The absence of any high-energy contribu-tion in G(U) suggests that whatever gives rise tothis does not occur near the melting point.

Examining the EFG coefficients + and jumpfrequency prefactors v0 in Table I, one finds anoteworthy consistency among the various fits and

crystals used. Considering 0 first, if one sup-poses this parameter for the melt crystal to bebracketed by the Lorentzian and Gaussian fits(which bracket the data), then an excellent agree-ment is found with the bimodal value of Oz forthe flux crystals. In contrast, the Jerome-Boilotdata have a coupling smaller by -25/z, a disparitywhich undoubtedly stems from the polycrystallinecharacter of the sample. In such a sample T, willbe dominated by its value near 0™&m because ofthe preponderance of solid angle near that value.Thus, from Fig. 2 we see that R, (-,'m) is rathersmaller than R, (4w), which was measured in thepresent work. It is not possible to make quantita-tive arguments in this case, because the R, aniso-tropy may be different in the Orsay specimen.The A~ value is consistent with the expected aniso-tropy, however, and the 0 values in Table Icombined with the second-order quadrupolar shiftresults of Sec. IVA give a clear impression thatthe EFG tensor components are very nearly inde-pendent of crystal preparation methods.

The values of v0 in Table I are remarkably con-sistent in view of the sensitivity of this parameterto errors in U. In interpreting v0 one must becautious about the effects of a distributionE, (U- U, ). For example, suppose E, ~ e 'U' ~~'.

Then, taking v(U') = v(U, ) in Eq. (11) one finds X, tobe multiplied by a correction factor e+For large values of 4 this factor can make v(U, )look significantly smaller than it really is. Inthe present case we estimate 4~ 300 K, giving acorrection & 25% in the region of the T, minimum,which is rather small.

The prefactors v0 found here are an order ofmagnitude smaller than those estimated from in-frared spectra and have the advantage of beingderived from the actual jumping motion of diffusingions. From the discussion in the Appendix, weconclude that v0-2v,«, where v,« is the vibra-tion or "attempt" frequency of a diffusing ion ex-clusive of site availability factors. Thus we find

Na NUCLEAR RELAXATION IN Na P-ALUMINA:. . . 8451

v,« -10"Hz in contrast with the peak frequencyof the infrared resonance v„-2 x 10"Hz. Ac-cording to the model of Wang et al. ' the vibrationfrequency of doubly occupied cells is lower by-& than the singly occupied ones which dominatethe spectrum. Even taking this into account thereremains a large discrepancy which suggests thatsimple theories of the diffusion constant may notbe correct. '

V. SUMMARY AND CONCLUSIONS

The NMR T, is found to reflect the presence ofa distribution of barrier heights G(U} for ionicmotion in Na P -alumina and to provide a roughbut effective probe of its profile when interpretedwith the phenomenological theory of Sec. II.While T, is sensitive to local as well as diffusivemotion, the strong correlation observed betweendiffusion properties and G(U) suggests that diffu-sive motion plays a major role.

The breadth of G(U) may be said to arise fromtwo effects, (a) the interaction between mobileions, which would depend mainly on their concen-tration, and (b) the spatial fluctuations in poten-tial caused by charge defects, which will dependon their structure and distribution. We attributethe variations in G(U) among the three crystalsanalyzed to the latter effect, concluding that thedefect structure is strongly influenced by crystalpreparation method. The width of G(U) is foundto be -0.4 times peak value, in accord with theo-retical estimates. ' The Czochralski (melt) tech-nique results in highly disordered material due tothe rapid quenching of temperature. The conse-quent G(U) is broad and centered below the bulkactivation energy tJ~,„=0.17 eV." In contrast,the flux method leads to only slight disorder withhigher barrier energies in general as well as asubsidiary peak at - twice the bulk value. Thisbarrier configuration results in a significant andprogressive immobilization of ions below T-200K, found only in this specimen. Qualitatively,these results are in accord with the substantiallyhigher conductivity found for the melt crystals. 'The actual conductivity process would undoubtedlyinvolve percolative effects of great complexitybecause of the barrier height variations; to ourknowledge, this problem has not yet even beenclearly formulated.

From measurements of T, and second-orderquadrupolar shifts, the nuclear quadrupolar cou-pling parameters do not appear to vary signifi-cantly between the specimens analyzed. We con-clude that this coupling is determined primarilyby the host structure and is not strongly pertur-bed by variations in the defect structure. The

deep null of rate R, for applied field along thec axis corroborates the expected mirror-planesymmetry of the EFG tensor.

Perhaps the most puzzling result we find is theconsistently low value of jump frequency prefactorv,(-10"Hz) for all specimens. A similar figurewas recently quoted in the literature. " Curiously,however, it is not given in the references cited,and its origin is therefore not entirely clear. Wealso note that data consistent with this value ofattempt frequency as well as the Lorentzian widthparameter for melt crystals (Table I) has recentlybeen obtained by means of acoustic loss measure-ments. " In any ease, such a value for v, is a fullorder of magnitude smaller than estimates basedon infrared spectral frequencies, where the lattergive a reasonable estimate of the diffusion con-stant on the Sato-Kikuchi" model. We see nosimple explanation for this discrepancy. On thecontrary, it suggests that ion-ion correlationeffects may play an important role in the diffusionof Na' ions in this material.

ACKNOWLEDGMENTS

We wish to acknowledge informative discussionswith D. B. McWhan and S. J. Allen, Jr. , and help-ful correspondence with Mlle. J. Theory of theEcole Nationale Supbrieure de Chimie de Paris.We also thank C. M. Varma for helpful sugges-tions regarding the nature of the barrier-height dis-tributions and L. R. Walker for advice on the theory.

APPENDIX

We illustrate the analysis of Sec. II with thesolution of a simplified case of the linearizedequation (6), namely that of a defect-free latticeso that the site index j becomes redundant. Wefurthermore take account of the EFG contributionfrom other mobile ions only when they are nearestneighbors occupying the same site. The EFGtensor has two contributions: (a) that of the hostlattice (with other mobile ion sites given theiraverage occupation) and (b) that of the nearest-neighbor mobile ion on a doubly occupied site.Naturally, the latter contribution is absent forsingly occupied sites. These conditions describethe simplest model which is not totally unrealistic.

It then follows that there are seven distinguish-able configurations in the sense of Sec. II. Six ofthese are illustrated in Fig. 9, where we show thedoubly occupied configurations. The seventh isthe state of single occupation. In the figure weshow only the immediate vicinity of an allowed(Beevers-Ross} site for Na' ions. These sitesoccupy alternate positions on a planar hexagonallattice in the p -alumina structure.

3452 %ALSTEDT, DUFREE, REMEIKA, AND RODRIGUEZ 15

FIG. 9. Six distinguishable doubly occupied configura-tions used in the model calculation, where the ions areshown in midoxygen sites neighboring a Beevers-Rosssite at the vertex of the solid lines. The filled circleis the ion under consideration.

Considering first the equations of motion, wewrite rate equations involving two types of single-particle hopping processes. First, in the con-figurations of Fig. 9, one of the two particles canjump to the unoccupied midoxygen site. In thisway one can make transitions, for example, fromconfiguration 1 to 5 and 6, 2 to 4 and 6, 3 to 4 and

5, etc. We assign the rate r, to the transitioncoupling any allowed pair of states. In the secondprocess, one particle of configuration 1-6 canjump to a neighboring singly occupied site. Wea.ssume that such a jump is made to one of thetwo nearest-neighbor Beevers-Ross sites on thecondition that one or both of them is singly occu-pied, in which case the jump is assumed to pro-ceed at rate r2. If both neighbor sites are singlyoccupied, the ion moves to each with equal proba-bility and in any case takes up the closestrnidoxy-gen position at its new site. If the Na' concentra-tion exceeds the stoichiornetric value by a frac-tion f, then the mean occupational parameter qz~is f/3(1+ f) for configurations 1-6 and (1 -f)/(1+f) for the seventh.

Using the above conditions it is a straightfor-ward matter to determine the total transition ratesbetween all pairs of configurations, and thus de-termine the following set of rate equations: dP/dt=AP, where P is a vector with components P,and the matrix A is given by

—(Sa+ 2p)

—(Sa+ 2p)

—(Sa+ 2p)

(a+ p)

(a+p)

—(Sa+ yp)

- (Sa+ 2'p)

(Al)

&i.a=4 ra(I f')+2ri+ lr +-4r.~(l-f')I

(doubly degenerate), (A2)

r~(21 f)+4r„-X,=r,(1+f)',

(AS)

(A4)

where &= e"' '. In discussing these decay modes

where a~ 4r, (1-f'), p =r„and-q=, r,f(I+f).These seven equations are not independent becauseof the condition+, P, = 1. If this condition is usedto eliminate P, from the first six equations, thenthe matrix of coefficients for these six becomessymmetric and an inhomogeneous term appearswhich gives the equilibrium values of P,~. Theseventh equation is then simply the sum of thefirst six. Diagonalizing the resulting 6 x 6 matrixgives the six nonzero eigenvalues of the problem.

This diagonalization is carried out straightfor-wardly, yielding the following set of eigenvalues:

we note that r1 and r2 are of the form v1, 2e-'1 2A

where by the nature of the problem v, and v,must be very nearly the same, though U, andU, may differ somewhat. X, 3 represent mixedmodes whereas X4 is a purely diffusive mode.For x2»x„ the coefficients of x2 range from1.56 to 2.44 for X, 4, whereas for r1»12 X1 3range from 1-4 times x, with X,4 being a relativelyslow mode by comparison. It is possible to inter-pret the data of Sec. IV using the mixed modes of~1 3 but one does not find reasonable values forthe parameters. Thus the defects must play amajor role in determining the actual mode struc-ture found in these materials.

It is useful to discuss the EFG tensor compo-nents in this model, because this enables one tointerpret the anisotropy of T, in Na p -aluminadiscussed in Sec. IVA. To find the form of thev»'s and vs, 's [i.e. , Eqs. (3a) and (3b) for specificconfigurations] for a general orientation of the

Na NUCLEAR RELAXATION IN Na P-ALUMINA:. . .

Vxx =& sin28(V, „cos'p —sin2$V, „+sin'pV„„—V,},

(A6)Vrz = ~ sin8[(V„- V „)sin2$ + V» cos2$]

and

Vxx = cos'8(V„„cos'Q —V sin2$

+ V„„sin'Q}+V„sin'8,

V»= V»sin'@+ V~ sin2$+ Vyy cos

V»= & cos 8[(V„—V„„)sin2$+ 2V cos2$],

(A7)

FIG. 10. Definition of coordinates used to expressEFG tensor components. The vectors B indicate neigh-boring midoxygen sites to a Beevers-Ross site . The(Y,y') axis lies in the x-y plane.

8 cos

ecosoc

—cose sing —sine 8

8X 8X

sing cosQ8

8$

8sine cos(IJ) —sine sing cos 88Z

8

8z

from which we obtain the relations

(A5)

field H„we place H, along the Z axis in Fig. 10,and calculate the EFG components in the XYZ co-ordinate system. Here the crystal is representedby coordinates xyz, where the hexagonal latticebonds meeting at a Beevers-Ross site are labeledB. The gradient operator transformation is givenby

where we have made use of the fact that V„,= V„,=O because of the mirror-plane symmetry.

To complete the modeling process one wouldneed to substitute appropriate expressions for theV~'s for each configuration. However, one maydeduce the angular dependence of R, and R, byconsidering Eqs. (A6) and (AV) in conjunctionwith Eqs. (3), (9), and (10) from the text. Forboth R, and ft, it is clear that Q-dependent termsmust vary as sin2$, cos2$, sin4$, or cos4$.However, the trigonal symmetry of the crystalis clearly incompatible with such terms and there-fore their coefficients must vanish. Consideringthe 8 dependence, from Eq. (A6} one concludesimmediately that R, consists of a sin'8 term anda sin'28 term. For R, one has in addition a con-stant term. The absence of P dependence wouldbe expected to hold even when defects break thelocal symmetry, as the average environment ex-perienced by the ions over a time T, would remainhighly symmetric. The 8-dependence results arealso quite general as they depend only on the re-flection symmetry of the mirror plane, whichestablishes the c axis as a principal axis of theEFG tensor.

*Work performed at Bell Laboratories while on leavefrom University of Warwick, England.

'See the review article by J. T. Kummer [Prog. SolidState Chem. 7, 141 (1972)] for a discussion of thesynthesis of these materials.J. C. Wang, M. Gaffari, and Sang-il Choi, J. Chem.Phys. 63, 772 (1975).

3S. J. Allen, Jr. and J. P. Remeika, Phys. Rev. Lett.33, 1478 (1974).

4D. B. McWhan, C. M. Varma, F. L. S. Hsu, and J. P.Remeika, Phys. Rev. B 15, 553 (1977).

5A. S. Barker, Jr., J. A. Ditzenberger, and J. P.Remeika, Phy~. Rev. B (to be published).

8M. H. Cohen and F. Reif, Solid State Physics, editedby F. Seitz and D. Turnbull (Academic, New York,1957), Vol'. 5.

H. C. Torrey, Phys. Rev. 92, 652 (1953).Dieter Wolf, J. Magn. Res. 17, 1 (1975).D. B. McWhan, S. J. Allen, Jr., J. P. Remeika, andP. D. Dernier, Phys. Rev. Lett. 35, 953 (1975).

S. Shapiro, D. B. McWhan, and G. Shirane (unpub-lished).The site index enumerates Beevers-Ross sites, whicheither contain a single ion or are doubly occupied inthe sense that two ions reside at neighboring mid-oxygen sites. The anti-Beevers-Ross sites are un-occupied for the alkali P-alumina compounds [C. R.Peters, M. Bettman, J. W. Moore, and M. D. Glick,Acta Crystallogr. B 27, 1826 (1971)].' It is easy to show that C& &

vanishes for constant v f g.Thus, C& &

reflects the mean squared variation of theEFG tensor over the configurations k.D. B. McWhan, P. D. Dernier, and J. P. Remeika (ab-stract only), International Conference on SuperionicConductors: Chemistry, Physics and Applications,Schenectady, N. Y., May, 1976 (unpublished); andS. J. Allen, Jr., L. C. Feldman, D. B. McWhan, J. P.Remeika, and R. K. Walstedt, ibid.D. Jerome and J. P. Boilot, J. Phys. (Paris) 35, L-129(1974).

3454 WALSTKDT, DUPREE, REMEIKA, AND RODRIGUEZ

' (a) I. Chung, H. Story, and W. Both, J. Chem. Phys.63, 4903 (1975); (b) J. P. Boilot, L. Zuppipoli,G. Delplanque, and D. Jerome, Philos. Mag. 32, 343(1975).

' The crystal preparation method for Ref. 14 and inRef. 15(b) is described in J.Antoine, D. Uivian, J.Liv-age, J. Theory, and R. Collongues, Mater. Res. Bull.10, 865 (1975).As in Ref. 14 the R2 component appeared to vanish fromthe recovery curve at higher temperatures. Since thequadrupolar splitting of the line persists, there is noobvious reason for $2 to vanish [Eq. (4)] or for R2

8 f above the T, minimum. This behavior is there-fore not understood.As long as G(U) dominates the broadening this approxi-mation should not materially affect our results.Whatever small width would normally be given to F

g

will appear in G, whereas the total width of interest isthat of some combination of F& and G.

' The model of Ref. 4 gives G(U) as the Fourier trans-form (FT) of exp[-(o. t) .], which is therefore anintermediate shape between a Gaussian (|FTexp[ (n t) t

Qand a Lorentzian [FT exp{—& t)].Y. Y. Yao and J. T. Kummer, J. Inorg. Nucl. Chem.29, 2453 (1967); M. S. Whittingham and R. A. Huggins,J. Chem. Phys. 54, 414 (1971).R. S. Title and G. V. Chandrashekhar, Solid StateCommun. 20, 405 (1976).M. Barmatz and R. Farrow, 1976 Ultrasonics Sym-Posilm Proceedings, edited by J. deIQerk andB. McAvoy, IEEE Cat. No. 76, CH 1120-5su, (IEEE,New York, 1976), p. 662.H. Sato and R. Kikuchi, J. Chem. Phys. 55, 677 (1971);55, 702 (1971).