PHYSICAL REVIEW 8 VOLUME 47, NUMBER 2 1 JANUARY 1993-II

Low-frequency excitations in sodium P-alumina: An NMR study

H. Sieranski, O. Kanert, and M. BackensInstitute ofPhysics, Uniuersity ofDortmund, 4600 Dortmund 50, Germany

U. Strom and K. L. Ngainaval Research Laboratory, Washington, D.C. 20375

(Received 30 June 1992)

The temperature and frequency dependence of Al and Na nuclear-spin relaxation (NSR) has beenstudied in Na p-alumina between about 1.5 and 100 K. Below 50 K the NSR is shown to be caused bylow-frequency excitations of disorder modes, which are intrinsic to the glasslike state of p aluminas. Thedata can be interpreted consistently in the framework of thermally activated excitations of modes thatare represented by asymmetric double-well-potential configurations with suitably chosen densities ofstates. Further, the observed NSR rates are shown to be related to corresponding ac-conductivity datavia the Auctuation-dissipation theorem.

INTRODUCTION

Sodium P-alumina is a two-dimensional superionic con-ductor composed of slabs of alumina separa, ted by planescontaining Na ions which become mobile at elevated tem-peratures. Generally the crystal contains a large(15—35 Po) excess of Na so that the planes are highlydisordered even when the crystal is cooled to low temper-atures. ' A corresponding number of excess oxygen isalso present for charge compensation. It has beendemonstrated that, very similar to dielectric glasses, thestructural disorder gives rise to a broad spectrum of lo-calized low-frequency excitations (LFE) which dominatethe dynamics of p-alumina at low temperatures. The mi-croscopic nature, however, of the disorder modes is stilllacking and therefore a subject of current interest.

While the ionic transport properties in P alurninas havebeen widely studied in the past by a variety of tech-niques, ' rather few investigations exist at present intothe low-temperature properties of P aluminas connectedwith dynamics of the disorder modes. Dobbs, Anderson,and Hayes have shown by dielectric measurements per-formed between 0.07 and 2 K that the density of LFEseems to be independent of the Na excess over the range7—35%. Kurtz and Stapleton have measured the elec-tron spin relaxation rate of electron irradiation-inducedcolor centers in various P aluminas between about 1 and25 K. They were able to interpret the findings by as-suming a coupling mechanism of a color center with dis-order modes excited by single-phonon-assisted tunneling.Some Al and Na NSR measureInents are reported inthe low-temperature range by Greenbaum, Strom, andRubinstein. Within some scatter, the data have beenshown to exhibit a simple power-law dependence on tem-perature ( T, ~ T "+ ') which was attributed to thepresence of LFE. As the NMR data were taken over anarrow frequency range, the dependence of the NSR onfrequency could not be established with certainty. Re-cently, Strom, Ngai, and Kanert have related the NSRdata to electrical ac-conductivity data published widely

in the literature by means of the fluctuation-dissipationtheorem. They have demonstrated that the same LFEare responsible for both the observed NSR rate and theac conductivity. The interpretation, however, suffersfrom a lack of sufficient NSR data taken in the low-temperature range at different frequencies. It was theaim of the present investigation to obtain such a set ofdata, to interpret these data in the framework of LFE ofthe disorder modes due to the structural disorder of Nain crystalline f3-alumina, and to renew the comparison ofthe data with corresponding ac-conductivity data usingthe fluctuation-dissipation theorem.

EXPERIMENT

The sample consisted of a melt-grown single crystalwith the approximate composition 1.25 Na20 11 A1203and size of about 10X 10X 1 mm . Infrared (IR) absorp-tion measurements showed that the sample was nearlyfree of absorbed H20. The sample came from the sameoriginal boule used for the microwave loss studies de-scribed in Ref. 2. The NSR measurements were carriedout with a modified Bruker SXP 4-100 coherent pulsedNMR spectrometer including an on-line data acquisitionsystem. The temperature of the sample was varied bymeans of a commercial He bath —gas-fiow system (Konti-Cryostat IT, Cryovac). The sample orientation was fixedwith the c axis parallel to the direction of the externalfield Bo. The NSR rates 1/T, were measured with a sat-uration technique using a saturation pulse comb at time v.

prior to the m /2 reading pulse to measure the time evolu-tion of the nuclear magnetization. Above 50 K, the in-version recovery technique (pulse sequence rr r rr/2) was--used to observe the magnetization decay. As depicted fortwo examples in Fig. 1, the magnetization of both nuclearprobes Al and Na was found to decay nonexponential-ly in the entire temperature region in agreement with ear-lier observations. Because of the noncubic structure ofp-alumina, the NMR spectra of 2 Al as well as of Naare strongly split by quadrupole interactions. Hence,

47 681 1993 The American Physical Society

682 SIERANSKI, KANERT, BACKENS, STROM, AND NGAI 47

10

10-

10

0. 110

10

100 200 300time (s)

400 500 1010

T (K)100

FIG. 1. Examples of nonexponential time evolution of nu-

clear magnetization in Na P-alumina for 'Na at 4 K and 95.2MHz (~ ) and for Al at 2 K and 46.9 MHz (0 ). Solid lines arefits to the data by means of Eq. (1) using the parameters given inthe text.

FIG. 2. Temperature dependence of 'Na NSR rates 1/T&for the different frequencies: 47.6 MHz (~ ), 95.2 MHz (o).The figure also exhibits data of Greenbaum, Strom, and Rubin-stein (Ref. 7) measured at 10.4 MHz (0). The solid linesrepresent a fit to the data by means of the approach discussed inthe text. Corresponding best-fit parameters are listed in Table I.

only the almost unperturbed central transition m=

—,' —+ —

—,' is selectively saturated by the pulse comb. In

that case, the nuclear magnetization relaxes according tothe theoretical expression'

Mo —M(t) = A exp( —W, t)+B exp( —W2t) .0

Equation (1) represents the solution of the correspondingset of rate equations of transitions among the m Zeemanlevels where 8'& 2 denote combinations of rates for

Ib m

I

= 1 and 2.The solid lines in Fig. 1 are fits to the data by means of

Eq. (1) using A =0.4, B =0.6, and Wt =6.25X10 s8'2=22. 7X10 s ' for Na, and 8'& =6.8X10 s8'2 =1.54X 10 s ' for Al, respectively, as best-fit pa-rameters. As expected theoretically, within a fairly largescatter the ratios 3 /B and 8'& /W2 were found to be in-dependent of frequency and of temperature over the en-tire temperature range for the two nuclei. The corre-sponding NSR rates were determined by means of the re-lation

A 8'i +BR'2

T) A+B

RESULTS AND DISCUSSION

The observed NSR rate dependence on temperature forAl at 15.6 and 46.9 MHz and for Na at 47.6 and 95.2

MHz is shown in Figs. 2 and 3, respectively. Further, thefigures exhibit the data points as measured by Green-baum, Strom, and Rubinstein ( Na at 10.4 MHz; Al at5.2 and 10.4 MHz). Two regions of the temperaturedependence of the NSR rates can be distinguished:Below about 50 K the LFE of disorder modes are shownto be responsible for the NSR rates whereas above 50 Kthe NSR rates start to increase exponentially due tothermally activated diffusive motions of the Na+ ions.Hence, the total NSR rate can be expressed as

10

10I

10

10

10

T (K)100

FIG. 3. Al NSR rates vs temperature for two different fre-quencies: 15.6 MHz ( o ), 46.9 MHz (0). The figure also exhibitsdata of Greenbaum, Strom, and Rubinstein (Ref. 7) taken at 5.2MHz (Q') and 10.4 MHz (+). Solid lines are best fits to thedata analogous to Fig. 2.

1 1 1

Ti Ti LFE T& diff

The LFE-induced contribution 1/T,~ L„E does not follow

a simple power law of the form 1/T, ~L„E~ T'+ as pro-posed by Greenbaum, Strom, and Rubinstein, based ontheir data for T & 5 K. However, the observed tempera-ture dependence as well as the frequency dependence of1/T, ~„„Ecan be interpreted well by means of a more de-tailed LFE model as introduced for vitreous materials. "In the framework of the model, the NSR rate is due to anuclear-spin-Rip process induced by LFE which are cou-pled to the nuclear spin via magnetic or quadrupolar in-teractions. Although the microscopic origin of themodes is mostly unknown, generally atoms or groups ofatoms possessing two or more configurations of nearlyequal energy are believed to form these modes. Accord-ing to Phillips' and Anderson, Halparin, and Varma'

47 LOW-FREQUENCY EXCITATIONS IN SODIUM P-ALUMINA: 683

the modes are commonly described phenomenologicallyby an asymmetric double-well potential (ADWP) with abroad distribution f (6, V) of the barrier height V be-tween the two wells and of their energy difference A. Theresulting NSR rate can be written as"

=(~ )™vmd~dV

T, L„E'

o o cosh (5/2kT) I+cooed

(4)

parameters in Eq. (4).As shown by Walstedt et al. for Na NMR (Ref. 19)

and later by Villa and Bjorkstam for Al, the diffusion-induced part of the NSR rate 1/T, ~d;ff leads to an asym-metric maximum at about 250 K. An explanation of theasymmetric maximum in terms of the coupling modelwas first proposed by Ngai. According to the investiga-tions, the NSR rate at the low-temperature site(coor d ff & 1 ) of the maximum can be expressed as

where the distribution function f (b, , V) is normalized asfollows:

1cdiff~o exp( —

ENMR /kT),di8'

(10)

I J dhdVf(b„V)=1 . (5)

Here (co; ) denotes the strength of the ADWP probe-nucleus coupling (magnetic dipole or nuclear quadrupole)and 6 and V are the maximum energy difference andmaximum barrier height, respectively, of the ADWP.According to Phillips' the correlation rate 1/w of theLFE is given by two terms:

—= A b oE coth1

2kT

1 —V+ — cosh expTO 2 kT (6)

f ( &, V) =p ( & )g ( V)

is used where p (b, ) is assumed to be uniform or increas-ing weakly with 6:

p(a)= a'+'a~ (0&a&a1

y+1 (8)

with 0 (y & 2. Using a high-resolution spectral-hole-burning technique, Kohler and Friedrich obtained for thedistribution of the barrier heights V a relationg ( V) ~ I /& V for a number of polymers' while Phillipsproposed a distribution g ( V) =sech( V/ Vo ). In theanalysis presented here a uniform distribution for g(V)was chosen, i.e.,

The first term describes one-phonon-assisted tunnelingwith b,o the ground-state tunnel splitting, E =Qb, o+b, ,and 3 the strength of the phonon-tunneling coupling,while the second term represents the classical rate pro-cess. Experiments show that typically below about 1 K,tunneling processes become important while above 1 Kthe excitations are found to be thermally activated. '

These observations agree with the present results on NaP-alumina.

Different suggestions exist in the literature concerningthe distribution function f (6, V). ' ' Mostly, a separa-tion of the form

where P&2 ["non-BPP (Bloembergen-Purcell-Pound) be-havior"] and ENMR was found to be about 0.04 eV forboth nuclear probes. A microscopic interpretation of the0.04-eV activation energy in terms of the cation escapefrom "associated clusters" was given by Wolf. ' Anotherinterpretation in terms of a true activation energy of iontransport and its relation to dc-conductivity activationenergy were given in Refs. 20, 22, and 23.

The solid lines in Figs. 2 and 3 are best fits to the datausing Eqs. (3), (4), (8), and (9) and assuming a classicalrate process for the LFE determined by the second termof Eq. (6). The set of parameters is listed in Table I. It isimportant to note that just one single set of parametersfor V, 5, and y describing the distribution of theADWP configurations is used for both nuclear probes inorder to fit all the temperature and frequency depen-dences of the experimental data below 50 K. Further, theapproach also depicts the diffusion-induced increase ofthe NSR rates above 50 K in agreement with experimen-tal findings published in the literature. '

Surprisingly, the relative strength of the coupling con-stant (co; ) of the two nuclei indicates that the interactionbetween the nuclear probes and the ADWPconfigurations is obviously of magnetic type. The experi-mentally observed ratio (co; [ Na] ) /(co; [ Al] ) =0.6agrees approximately with that expected for magnetic di-pole interaction, i.e.,

y I(I+1)[ Na]/y I(I+1)[ Al] =0.45 .

On the contrary, an electric quadrupole interactionwould result in a remarkably larger coupling constant for

Na than for Al. From experimental data one obtains(co;[ Na])/(co;[ Al]) =10. Moreover, the small mag-nitude of the coupling constants (coupling frequencyv; = I/2rr+co; =300 Hz) suggests a low density cL„E ofthe LFE analogous to observations in dielectricglasses. ' ' Using the relation (co;) =cL„END (Ref. 26)and assuming 10 kHz as a typical value for the strengthof the dipolar interaction, i.e., co& =4X 10+ s, one ob-

g(V)= 1

V(0&V&V ) . (9)

TABLE I. Summary of the fitting parameters for NSR re-sults in Na P-alumina shown in Figs. 2 and 3.

As expected, however, from the integral form of Eq. (4)the results do not depend very sensitively on the exactform of f (b, , V), and the widths of the distribution func-tions and the cutoff energies 6 and V are the leading

(~'; &

Nucleus (s )

Tp

(s)V 6 y(K) (K)

Cdiff—( +p))f +NMR

(eV)

Al 4.6X10 1.0X10 32 64 1.1 1.5X10 0.7 0.038N 2.8X 10 0.4X 10 32 64 1.1 5.0X 10 0.7 0.038

684 SIERANSKI, KANERT, BACKENS, STROM, AND NGAI 47

tains cLFE-—10 in accord with data published by Hun-klinger for various dielectric glasses, ' The values of Vand 6, listed in Table I, show that the underlyingADWP configurations responsible for the LFE are highlyasymmetric compared to those in inorganic glasses: Here

/V =2, whereas in inorganic glasses typically0 1

11 2sm m

Trying to extract power laws from the low-temperatureNSR data, one obtains approximately the following rela-tions:

10

10

10

1 ~ T /co (1.5 K T 6 K; region I),LFE

10 10 10 10Frequency (Hz)

1 010

1 ~ T/cg ' (6 K & T & 50 K; region II) .LFE

On the other hand, as remarked in the Introduction, theNSR rate can be related to the electrical ac conductivityo.„by means of the Auctuation-dissipation theorem.The NSR rate 1/T1 is proportional to the spectral densi-ty J(co) of the random perturbation due to transitions ofthe ADWP. Also, it is well known from the celebratedfluctuation-dissipation theorem that the imaginary partof the generalized susceptibility y"(co) is proportional tothe corresponding spectral density. In the classical lim-it, the theorem states that y"(co)=(co/2kT) J(co). On theother hand, y"(co) is proportional to the dielectric lossE"(co), which is supposed to be caused by the transitionsof the ADWP. From Maxwell's relations the corre-sponding ac conductivity o„is given by coo"(co). Hence,the NSR rate can be related to the ac conductivity as

1 To„(co,T) .

T1 co

Using the above given power laws derived from the ob-served NSR rates, Eq. (11)predicts, for the ac conductivi-ty at comparable frequencies and temperatures,

0 „~nT2 (1.5 K & T & 6 K; region I),cr„~co' independent of T (6 K T& 50 K; region II).

The predictions are qualitatively in accord with ac-conductivity data for Na P-alumina available over the in-teresting temperature range between about 10 and 10Hz. First, as proposed, (o „),exhibits a fiat plateau (re-gion I) which occurs between about 4 and 40 K when ex-trapolating the ac frequency to the actual Larmor fre-quencies ( —10 —10 Hz). The frequency dependence ofthe ac conductivity, however, is proportional to co"within the plateau area. Secondly, below about 4 K,(o.„)„increases with rising temperature and frequency(region II), and seems to obey a power law o„~co'Tb.The exponents a, b are somewhat smaller than those de-rived from the NSR data: a =0.7, b = 1.5.

The transition from region I to region II at fixed fre-quency occurs over a relatively narrow temperaturerange near T=T . In Fig. 4 are plotted values of Tdetermined from ac-conductivity and microwave lossdata. The acoustic loss data of Doussineau et al. havealso been included for comparison. The solid line drawn

FIG. 4. log-log plot of the transition temperature T vs fre-quency obtained by different experiments: A, ac conductivity(Ref. 27); 0, Na; ~, 'Al NSR (present work);, ultrasonic at-tenuation (Ref. 28); A, microwave conductivity (Ref. 2). Thesolid line represents Eq. (12).

in Fig. 4 represents the equation

T ct 1 /3m (12)

which is predicted by the relaxation of the LFE due toone-phonon-assisted tunneling. Although one-phonon-assisted processes are expected to dominate only for T ~ 1

K, it is apparent from Fig. 4 that the acoustic and mi-crowave data, for which T ~ 10 K, are also in goodagreement with the predictions of Eq. (12). This impliesthat thermally activated processes [second term in Eq.(6)] do not introduce large variations from the predictionsof Eq. (12), at least for the temperature and frequencyrange considered here.

The NSR related data shown in Fig. 4 point to some-what larger values of T, i.e., of 1/T, , than predicted.These results are consistent with our preceding compar-ison of the prediction of Eq. (11) and the observed 0.„.Aplausible explanation for the deviation of the NMR datatoward larger than expected T is that there are contri-butions to 1/T1 for T & 5 K which are not apparent inthe measured dielectric loss or ultrasonic attenuation.One possible interpretation is that Auctuations due to ion-ic motion somewhat removed from the conduction planesare contributing to the nuclear-spin relaxation. The ap-parently larger deviation of the Al 1/T, data from thepredicted curve in Fig. 4 is consistent with this interpre-tation, in as much as the Al ions closest to the conductionplanes are in the form of columnar Al-O-A1, where the Alions are displaced perpendicular to the conduction planesfrom the in-plane oxygen ions.

One has to note that a more precise analysis comparingNSR and ac conductivity cannot be carried out at presentbecause of the lack of relevant conductivity data. An im-proved evaluation requires ac-conductivity data mea-sured in the entire temperature range at frequencieswhich are equal to the dift'erent actual Larmor frequen-cies. Nevertheless, the present results allow the con-clusion that the low-temperature ac conductivity in NaP-alumina is caused by the same spectrum of LFE whichis responsible for the NSR.

47 LOW-FREQUENCY EXCITATIONS IN SODIUM P-ALUMINA: 685

SUMMARY

Measurements of the Na and Al NSR rate 1/T&

in-dicate that the dominant relaxation mechanism below 50K is caused by thermally activated low-frequency excita-tions of modes which are due to the disordered arrange-ment of the Na ions in single-crystal P-alumina. Themodes are coupled via magnetic dipole interactions toboth the nuclear probes. Temperature and frequencydependences of the observed NSR rates can be explainedconsistently by means of a model which describes themodes phenomenologically by ADWP configurationswith appropriate distributions of the barrier heights andof the asymmetry. The data suggest that the ADWPconfigurations are highly asymmetric. The NSR rate canbe related to the electrical ac conductivity via thefIuctuation-dissipation theorem. A corresponding com-

parison of the present NSR rate data with ac-conductivity data from the literature suggests that theLFE are responsible for the NSR as well as for the acconductivity.

Above 50 K the NSR rates start to increase exponen-tially with rising temperature due to diA'usive motions ofthe Na+ ions. The measured activation energy and theobserved deviation from the ~ law agree with previousresults published in the literature.

ACKNQWLKDGMKNTS

H.S., O.K., and M.B. acknowledge financial supportfrom the Deutsche Forschungsgemeinschaft. U.S. andK.L.N. are grateful for support by the Ofhce of NavalResearch. K.L.N. was also supported in part by ONRContract No. N0001492 WX.

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