Critical dynamics at the structural phase transitions in NaNO2 and NaNbO3 and quadrupole relaxation T1 and T1ϱ in high fields

JOURNAL OF MAGNETIC RESONANCE20,399-410 (1975)

Critical Dynamics at the Structural Phase Transitions in NaNO, and NaNbO, and Quadrupole Relaxation IV1and I’,, in High Fields A. AVOGADRO, G. BONERA, AND A. RIGAMONTI Istituto di Fisica dell’Universitri di Pavia, 27100 Pavia, Italy Presented at Third International Symposium on Nuclear Quadrupole Resonance, Tampa, Florida, April, 1975 Measurements of quadrupole spin-lattice relaxation in the laboratory and in the rotating frames for 23Na in NaNO, and for 23Na and 93Nb in NaNbO, are reported and discussed. From the analysis of the temperature dependence of the critical contribution to Ti and Tip, information is obtained on the dimensionality and the characteristic frequencies of fluctuations. For the ferroelectric order-disorder transition in NaNO*, the orientational fluctuations in adjacent chains of the NO; dipoles are uncorrelated and their frequencies remain greater than wL = 130 MHz even at the transition. For the typical displacive cubic-tetragonal transition in NaNbO, at T, = 641°C the fluctuations of the oxygen octahedra have a quasiplanar character and a strong slowing down occurs so that at T- T” + 4°C the characteristic frequency at the critical wave vector is about 270 MHz. An analysis of Tip near the phase transition is given, taking into account the fact that a further mechanism of relaxation due to vacancy diffusion is present, as revealed by the high-temperature measurements. INTRODUCTION

The critical dynamics occurring around the structural phase transitions in crystals can be detected and investigated through the shortening of the nuclear relaxation times due to the time-dependent quadrupole interaction. In fact the Hamiltonian &‘,Jt) = C,Q,V-,(t) in high magnetic field H (i.e., such that y#iH$ ZQ) induces transitions between the Zeeman levels, driving the relaxation process. The transition probabilities are related to the spectral densities J,(w) of the fluctuating functions V,,(t) of the efg components Vj, (in the 2” frame of reference with z/jH) at two frequencies: wL = yN H for the relaxation in the laboratory frame (TJ and w1 = y,H, for the relaxation in the rotating frame (T,,). Therefore, since Vjk(t) are related to the fluctuations in the atomic positions, by means of T1 and TIP measurements, in principle, one can investigate the low-frequency range (typically from lo3 to 10’ Hz) of the dynamic structure factor S(q, o), which spectralizes the collective fluctuations at wave vector q. The enhancement and slowing-down of the fluctuations on approaching T, cause a divergence in S(q) 3 & J”S(q, w)do for q 2 q, and an increase in the low-frequency components of S(q,, co) (qc, wave vector of the critical

excitation).

In this paper, we will present and discuss experimental results for T, and T,, near the crrtical temperature T, in two crystals: NaNO, and NaNbO,. The measurements were performed in high fields (HZ 20 kG) by conventional NMR pulse techniques. Copyright ci\ 1975by AcademicPress,Inc. All rights of reproduction in any form reserved. Printed in Great Brikn

399

400

AVOGADRO,

BONERA

AND

RIGAMONTI

Sodium nitrite can be considered as a typical order-disorder Ising system: The ferroelectric transition at 163°C is associated with the ordering of the fluctuating NO; dipoles. The critical dynamics can be considered as superimposed on the ordinary phonon excitations and can cause a relaxational-type central component in S(q, o), For the width zpl of this central component (which is a measure of the mean time of the life of the polarization fluctuations), one can expect a divergence for q = qc, i.e., sqc + JCfor T-t T,. The central component was evidenced as a quasi-elastic contribution in the energy distribution of scattered neutrons (I), while the temperature behavior of rqzO was obtained by dielectric measurements (2). In sodium niobate the cubic-tetragonal phase transition occurring at 641°C is the typical displacive structural transition; it is associated with the rotational fluctuations of the NbO, octahedra. The slowing-down of the fluctuations in this case involves both the softening in a branch of the lattice phonon modes and the occurrence of an additional contribution to S(q, o) centered at zero frequency (central peak). The simultaneous presence of the soft mode doublet and of the central peak can be justified by introducing a frequency-dependent damping factor into the lattice phonon equations. The central peak was observed (3) by inelastic neutron scattering in some perovskites (SrTiO,, KMnF,, LaAlO,) undergoing phase transitions similar to the one in NaNbO,. However, lack of suitable resolution in q and o inhibits, in general, determination of the width r4 of the central peak and study of its temperature behavior. No dielectric investigations of the critical fluctuations can be performed for these nonpolar crystals. QUADRUPOLE

RELAXATION

RATES

AND

CRITICAL

CONTRIBUTIONS

In the weak-collisions limit, and assuming the existence of a spin temperature, both in the laboratory and in the rotating frame, the relaxation process is described by an exponential law with time constants T, and TIP given by

~1 = 40

[.wd

Dal

+ 4~mh)i

T;,’= A(r) 4 [~J,(w

+ 1wd

+ 4~,wdi

[lb1

where A(I) = (2Zf 3) e* Q2/40Z2(21 - 1)h2 and J,(o) = 1 exp(-iwt)

< Vfi(0) V...,(t) > dt.

PI

To obtain the critical contribution to the relaxation rates, the P’jk components must be related to the critical variable &(t) whose anomalous temperature behavior causes the structural phase transition (1, index of cell). By writing,

and by introducing

the collective critical variables

one can directly relate, in the random phase approximation, dynamic structure factor S(q, o) = j exp(-iot)

< 4,(O) 4-,(t)

the relaxation rates to the > dt.

[31

PHASE TRANSITIONS AND QUADRUPOLE RELAXATION

401

One has (4,5) : [41 The presencein Eq. [4] of the q-dependentfactor ZX!;reflectsthe fact that the relaxation process is sensitive to the auto (I = I’) and to the pair correlation (I # 1’) for the local critical variable. For a given q, &; dependson the charge distribution in the crystal and on the orientation of the crystalline axes in x”. It must be observedthat in a polycrystalline sample, in practice, one observesan exponential relaxation process with a time constant given by the averagesover the Euler angles. In this case,from the symmetry properties of the efg tensor, one can show that

PI consequently, if S(q, 2w,) = S(q, oL) = S(q, 20,) (“white” spectrum) one has T, = T 10'

The above results apply directly to the 23Na and g3Nb relaxation processesin the cubic phase(T > 641“C) of NaNbO,. For polycrystalline samplesthe &t factor for the 23Na nucleus, in the point charge model, can be evaluated easily (4) for the critical wave vectors at the zone boundary (qs = ~/a, Z/U, qsZ); one has : where E = 144e’(1 - Y~)~/& (a lattice constant) (for the R,, zone boundary mode qsz = n/a, while for the M3 mode qBz = 0). Since ~52:are slowly varying functions of q and S(q, o) exhibits an enhancementonly for q around qs, in performing the q-summation in Eq. [4], one can assume-01;= d,,.a On the other hand, for the g3Nb nucleus,the -OZ&factors are zero for all values of qeZ.This requires that the exact q-dependencebe taken into account. One has (4) : J&t:= + 2d: = +f cd: = $+ E[ l-

- qY)a].

[71 From Eqs. [l] and [4] we can write for the relaxation rates by neglectingthe difference in the values of S(q, w) at oL and 20, : g3Nb:

T;l = 32A(Z)E 1 [l-cos(q,

P

COS(q,

- qy) a]S(q, oL)

T;; = a T;l + &,32A(Z)E 2 [l - cos(q, - q,,) a] S(q, 204)

Ml

T;’ = 2A(Z)E(l + cosq&

Pal

P

Z3Na:

PaI

x S(q,w,)

9

T;; = &i T;l + & 2/i(Z) E( 1 + cosq,,a) 2 S(q, 20,).

P

WI

For the analysis of the experimental results it is convenient to introduce certain assumptions about S(q, 0). These assumptions are suggestedby general theories of phasetransitions and critical phenomenaand are in agreementwith a semiphenomenological treatment of the critical dynamics of the oxygen octahedra in SrTiO, (6). We can write for the correlation function of the collective rotational fluctuations of the NbO, octahedra at long times : (4,(O) b,(t)> = +$,l’>e-‘q’.

DOI

402

AVOGADRO,

BONERA

AND

RIGAMONTI

For the q-dependenceof the static structure factor S(q) = (I@,\“>,one can assume(6) the anisotropic Orstein-Zernike expression:

44,1’> = k2/[q’ + (1 - 4q:+ k21

[[II

where A is the anisotropy parameter: For A = 1 the rotational fluctuations are threedimensionally correlated, while for the limiting case A = 0 the fluctuations are correlated in a plane but not in adjacent planes. A = 1 correspondsto the softening of a pure R,, or MS mode at the zone boundary, while A = 0 correspondsto the softening of the whole branch from R to M. For the temperature-dependencewe assume

and k = k.

k.

EY =

Wbl

Eyi2

where E= (T- 7:)/T,. Finally, for rcl in Eq. [lo], which in the above picture representsthe width of the central peak, we assumethe thermodynamical slowing-down and therefore

rm= $$Lg. ro 0

[ 1

From the aboverelations, by performing the q-summation, the completetemperature and frequency dependenceof the relaxation rates both for 23Na and g3Nb can be obtained (7). The general expressionsare quite complex and therefore, in the following section we will recall only the more significant features. For 23Na in the temperature range in which the width of the central peak r,= is greater than 201, (fast-motion regime) one has : T;‘=T;,‘=$(I)E$$

0

ki (T) x 2 (1 + cosq,,a) Aeli c

EC”

tan-l

[I31

In the neighborhood of the transition temperature this equation reduces, for A = 1, T;l ALE-’ while for A < 1 one has T;l a~-” (provided E”2 A; for E < A one again has T;’ NE-“). However,when the slowing-down reachesthe r.f. range so that rnC ,$ oL, Eq. [13] is no longer valid and the complete expression of the relaxation rates (7) predicts that a flattening in the temperature behavior of T-’ occurs (as shown in Fig. 3); moreover T,, # T, and for rPe 9 2110, one has: T;; = &T;l

+

2

(1 +

cos qsZa)ki A-‘/’

EC’;

[I41

while closeto T,, T;‘reaches a practically constant value, in T;,’ one still has a contribution that exhibits a divergenceas E-’ (see Fig. 3). For g3Nb in the fast-motion regime one has :

PHASE TRANSITIONS

AND

QUADRUPOLE

RELAXATION

403

where a2 = (4/x) + fl” and p = (&C&C) Ey/z. For T-t T,, Eq. [I 51 does not exhibit a divergence. This lack of a divergence is due to the fact that the J&“& factors are zero. In particular, for A $ 1, T;’ = T;,’ is practically temperature independent (for E 5 5 x lo-’ (see Fig. 3), while for A = 1 the relaxation rates are characterized by a cuspedshape behavior. In regard to NaNO,, because the crystalline structure, for T> T,, is body-centered orthorhombic, a static quadrupole interaction is present and the Zeeman levels are not equally spaced. Therefore, Eqs. [I], which stand on the assumption of a common spintemperature are no longer valid. For a single crystal with the ferroelectric c-axis perpendicular to H, only the central line is irradiated. In this case, the recovery law for 23Na (I= +) is given, both in the laboratory and in the rotating frame, by two exponentials with time constants T, and T,; however, due to the small difference between the two time contants, the experimental recovery plots for t 5 Ta,b are practically exponential, with a time constant given by (+)(T;l + T;‘). One can then write for the experimental relaxation rates : C-l = 3 4Z)[J,(d

+52@Ml

T;; = & A(Z) [9&(2w,) + 16J1(oL) + lOJ,(2w,)].

P6al D6bl

The ~2; have been evaluated by taking into account only the contribution from the NO; dipoles, for fluctuation waves propagating along the a-axis (q = 0, 0, q). One has for the three-dimensional correlation, forH/ja: forH/jb:

Lie;=0 at;=0

d; = 3.60( I - u) L&q=0

While assuming a complete uncorrelation different a-chains of dipoles one obtains : forH//a:

sd;=o cd; = 3.60(1 - u).

iI71

between the waves propagating along the

-ae; = D(1 - 0.392.4)2 22: = D(1.49 - 0.54U - 0.95U’)

forHllb:

df = D(2.72 f0.43~ + 0.027~~) 22; = D(1 + 0.17U)*

D81

Lzz:= 0(0.47+0.4724) ~4; = 0(3.74-

0.65~ -0.9~~)

where u = cos (qa), D = 18,~*(1 - yD)*/(c/2)’ and p is the effective dipole moment of NO- (a and c lattice parameters). Regarding the assumptions for S(q, o) for the dipole reorientational fluctuations in NaNO,, from the theory (8) for the time-dependent statistics of the Ising system, one can still use a correlation function as in Eq. [IO], where the correlation time zq = r-t is : zq = 1 - Z&T

[I91

and r0 is the fluctuation time in the absence of interaction; moreover, the static structure factor results S(q) = z,/~,. In Eq. [19], Z(q) is the Fourier transform of the interaction. By taking into account the effect of the depolarization field for the electric dipolar

404

AVOGADRO,

BONERA

AND

RIGAMONTI

interaction one has (9,ZU) Z(q) = Z(0) [ 1 - c$ - pcos’$1 and Eq. [ 191can be written as : 7q k2 Zq=(q2+k2+&os29)

PO1

where 9 is the angle between the q vector and the ferroelectric c-axis: zq,.= zO(T’T,-,)c ’ and k = k0 P; since the ferroelectric transition in NaNO, cannot be assumed of second order, here E means (T- T,)/T, and according to dielectric measurements(2) T, = T, - 1.27”C. In performing the q-summation in Eq. [4] we will assumefor -0”; the q-dependence obtained for the excitation waves propagating along the a-axis (seeEqs. [17] and 181). For the singular term in the 23Na relaxation rates, in the fast motions regime zq < (2wJ’ one obtains :

wheref(u) = [&: + &z]q=q,; an analogous expression holds for T;,1 where instead of f(u) one has g(u) = [(9&i + 16~2: + 106:)/16]q=qe As already pointed out (IO), the logarithmic divergence in the relaxation rate is a characteristic feature of the order-disorder ferroelectric phase transition driven by the long-range dipolar forces. If, for T-tTc, rq, becomes greater than w;l, while T;’ flattens, a logarithmic raising in T;,’ still occurs, until zq, - w;l. EXPERIMENTAL

RESULTS AND

DISCUSSION

In Figs. 1 and 2 the experimental results for the relaxation rates in the laboratory and in the rotating frames are presented for 23Na in NaNO, and for 93Nb and 23Na in NaNbO,, in a temperature range including the phasetransitions of interest. (i) NaNO,. The measurementsfor sodium nitrite were performed on a single crystal with the ferroelectric axis c perpendicular to H and with irradiation and detection of the central line only. The data shown in Fig. 1 refer to two orientations of the a-axis; only a slight difference is present for the TIP data. The measurementsin the rotating frame were performed with an r.f. field of amplitude H, = 20 gauss. No dependenceof T,, from ZZl was observed in the range from 5 to about 30 gauss; as expected for low fields (ZZ15 3 gauss), the time constant for the decay of the magnetization in the rotating frame was observed to decreasetowards the value of the dipolar relaxation time T,, 21 T,/2. In light of the theoretical results presented in the preceding section, the following information concerning the critical reorientational dynamics associated to the ferroelectric phase transition can be obtained. From Eq. [21], by taking into account that the critical wave vector is qc = (0, 0,O) for the ferroelectric transition [qc G:(0, 0, rc/lOa) for the transition to the sinusoidal phase (II) occurring some degrees above Tc] the existence of a critical contribution to the relaxation rates allows us to conclude that the reorientational fluctuations of the NO; dipoles are correlated along the a-axis; no correlation exists among fluctuations of dipoles belonging to two adjacent a-chains. In fact, for three-dimensionally correlated fluctuations (see Eq. [17]), one has for

PHASE TRANSITIONS

AND

QUADRUPOLE

/

405

RELAXATION

I

I

I 120

100

I 160

1 180

I 160

I 200

I 220

'C

FIG. 1. 23Na spin-lattice relaxation rates in the laboratory and in the rotating frames in a single crystal of NaNOz for the irradition and the detection of the central line only (seeEqs. (16)).

T,;

1

see

i i 'i c -1 1; i' .I* I . .-

j I

a) T, -. -. -*.

I 900

850

900

I I

* . . . ..

I 950

I 1000

950

1000

.

. --

. .

1050

T 'K

FIG. 2. Spin-lattice relaxation rates in the laboratory and in the rotating frames in a polycrystalline sample of NaNbO,. (a); 23Na; (b), 93Nb.

406

AVOGADRO,

BONERA

AND

RIGAMONTI

f(u) in Eq. [21] and for g(u) in T;,l, f(u) = g(u) = 0; instead, for one-dimensionally correlated fluctuations (seeEq. [IS]), one has: Hi[a: H/lb:

f(u) = 3.170

f(u) = 3.140

g(u) = 2.20 g(u) = 3.08 D.

In addition, the ratio TJT,, for Hllb and [Tlp](nla) /[Tlp]~H,,b~ can be accounted foi better by the hypothesis of one-dimensionallycorrelated fluctuations. Regarding the detailed temperature dependenceof T,, we will only remark that according to earlier measurements(10, 12) the logarithmic divergencetheoretically obtained in Eq. [21] is well supported by the experimental results. In addition, one can observe from Fig. 1 that the very small temperature gradients achieved in the measurements(~lO-Z”C/cm) allowed us to place in evidencethe dip in Tl at about 4°C above T,, whose existencewas already suggested(12). In regard to the temperature behavior of TIP, one can observe that Eq. [21] explains the experimental results; becauseTl p z Tl even at T, and no flattening in T;’ is observed,one can conclude that S(q,, 20,) 2: S(q,, 2w,); in other words, the slowing-down of the fluctuations is such that the width of the central peak is greater than 130 MHz even for Tz T,. This indication can be compared with the value of r;;,’N 140 MHz at T, obtained by dielectric measurements(2). (ii) NaNb03. The measurementsfor sodium niobate were performed in a polycrystalline sample.No dependenceof T,, on r.f. field amplitude (for Lfl 2 3 +- 4 gauss) was observedin the entire temperature range explored, neither for 23Na nor for y3Nb; the data reported in Fig. 2 refer to H1 = 15 gauss. As appears from the figure, T,, shows a temperature behavior strongly different from the one of T,, even far from the transition temperature. This suggeststhat a further mechanismof relaxation is present, which is effective only for the relaxation in the rotating frame. We will discuss the temperature behavior of T,, later, after somecommentsconcerninginformation on the critical dynamics that can be obtained from the analysis of the temperature behavior of Tl. A marked increase in the relaxation rate near T, is present for 23Na and not for g3Nb. An inspection of Eq. [13] (or Eqs. [6]) allows one to conclude that the cubictetragonal phase transition must involve the softening of the M3 mode (qez = 0) or of a large part of the M-R branch and not of a quasi-pureRzs mode (qsz = n/a), as happens instead in SrTiO,. In Fig. 3, the critical contribution to the 23Na relaxation rate (obtained by subtraction of a background-contribution of 1 see-‘) is reported versus E in a log-log plot. The flattening of T;’ for E 5 6 x 10s3indicates that the slowing-down of the rotational fluctuations reaches frequencies less than 20,. In fact, Eq. [13], based on the assumption of the fast-motion regime, can fit the experimentaldata only for E 2 6 x 10M3(seedotted line). On the other hand, a satisfactory fitting of the data in the entire range of E can be obtained by using the complete expression(7) for the relaxation rate (seesolid line, Fig. 3). The best-fit gives a value v = 0.6 for the critical index, and a value d = & for the asymmetry parameter. These numerical values are only indicative, since one has two adjustable parameters; however, one can conclude firmly that d < 1, sincefor A = 1 one would have a straight line in the log-log plot, for s 2 6 x 10m2.A small value of the asymmetry parameter allows us to rule out the occurrence of a softening of the pure M3 mode; therefore, we conclude that the

PHASE TRANSITIONS

AND

QUADRUPOLE

RELAXATION

407

rotational fluctuations of the NbOs octahedra are two-dimensionally correlated, while low correlation exists in the fluctuations occurring in adjacent planes. This result is also confirmed by the temperature behavior of the g3Nb relaxation rate; in fact, the theoretical expression given by Eq. [ 151 with v = 0.6 and A = & (see solid line for 93Nb in Fig. 3) ,fits the experimental results adequately. Quasi-two-dimensional correlations are also indicated by the strongly anisotropic X ray diffuse scattering (13).

w,: 2n.22 MHz

FIG. 3. Critical contribution to the relaxation rate T;’ for Z3Na and g3Nb in the cubic phase of NaNbO, versus E in a log-log plot. The solid lines are the best-fits of the experimental results according to the theoretical expressions and corresponding to the values v = 0.6 and A = 1/SO. The dashed line represents the theoretical behavior of the relaxation rate if the width r,, of the central component would remain greater than 2110~ (seeEq. (13)), and the solid line corresponds to l-q, = 2 x 10” P. The dotted-dashed line represents the theoretical behavior of T,, (Eq. (14) corresponding to the same values of v, A, and fq, as found in TI.

The occurrence of the flattening in the 23Na relaxation allows one to obtain an evaluation of the width r4C of the central component in NaNbO, (see Eq. [lo]). Since the departure of the relaxation rate from the temperature behavior given by Eq. [ 131 is expected where Ta, = 2w,, from Fig. 3 one can immediately deduce that Ta, = 270 MHz for E = 4 x 10e3 f 2 x 10e3. (If one assumes FQC= To s2’, one can obviously obtain rUC as a function of temperature; in Fig. 3 the solid line is derived from the complete expression for the relaxation rate with rq, = 2 x 10” ~r,~.) A slowing-down of the rotational fluctuations of the oxygen octahedra of this order of magnitude has also been evaluated in SrTiO, from the critical broadening of the EPR line of the Fe3+- V, center (14). The critical contribution to T,, from the rotational fluctuations should obey Eq. [ 141. The plot of T;,’ according to Eq. [14] and using the same values of v and A obtained from Tl data is reported in Fig. 3 (dotted-dashed line). As already mentioned, a further mechanism of relaxation is present in the rotating frame (which is practically ineffective for T,). From the analysis of T,, at high temperatures for 23Na and g3Nb, we can 15

408

AVOGADRO,

BONERA

AND

RIGAMONTI

conclude that this relaxation is due to the diffusion of vacancies. A discussion of the contribution to 2”,, from the diffusion of vacancies and the main conclusions that can be obtained are presented in the Appendix. The vacancy-diffusion contribution to the relaxation rate in the rotating frame is reported in Fig. 4. The data reported have been obtained by subtracting from the experimental results the values of T;: for the critical contribution, in the temperature range in which one expects r,, = T,(E 2 IO-‘). By extrapolating the behavior of (T,,), due to the vacancy-diffusion mechanism for c 5 lo-‘, one can try to obtain the critical contribution to T,, in the region where a

NaNbO,

IO 0.9

0.95

I 'K-1

m

1.05

L

T

FIG. 4. Contribution from the vacancy-diffusion rotating frame in NaNbO,.

mechanisms to the 23Na relaxation rate in the

difference between r,, and r, could be observed. Unfortunately, a sizable difference between TIP and Tl in NaNbO, should occur only very close to T, (E 5 10s3), where the uncertainty in the temperature for T,, measurements due to the r.f. heating does not allow a meaningful analysis (it must be observed that the poor signal-to-noise ratio requires the use of averaging techniques, with many repetitions of the pulse sequency for measuring T,,). However, the data so obtained seem to indicate the presence of a critical contribution to T,p which does indeed follow Eq. [ 141. Therefore, the T,, data can be fitted adequately by a contribution related to an activated diffusion process and by a critical contribution in agreement with the results on the rotational fluctuations of the oxygen octahedra obtained from Tl measurements. In conclusion, the hypothesis regarding the dynamic structure factor, the existence of a central component and its width, together with the values of v and d, are supported even from the measurements in the rotating frame. APPENDIX.

VACANCY-DIFFUSION

CONTRIBUTION

TO TIP IN NaNbO,

A self-diffusion process via a single vacancy mechanism of Nat ions or 02- ions can produce a sizable contribution to the spin-lattice relaxation in the high temperature region. For the self-diffusion of Na+ the contribution to the 23Na or 93Nb relaxation mechanism can be both of quadrupole or magnetic origin, while for the self-diffusion of 02- the relaxation can be due only to the electric quadrupole interaction. For the magnetic mechanism, the relaxation is induced by the change in the dipolar local field when an ion jumps (15). Taking into account the ineffectiveness of the

PHASE TRANSITIONS

AND

QUADRUPOLE

RELAXATION

409

diffusion process on T,, one can assume the condition of slow diffusion with respect to w,(i.e., or- ~~$1) and therefore write

T;; = y2M,r,[3G(k,2w,z,/2)]

[AlI

z, is the mean diffusion jump time of the Na+ ion (2, = v-l = a2/6D, where D is the macroscopic self-diffusion coefficient) and M2 is the rigid lattice dipolar second moment (in NaNbQ, we have (4) for ‘3Na y&M2 =4x2.2.45 x IO6 Hz’, for 93Nb, &,,MZ = 47?.0.33 x IO6 Hz’). For wlr, 2 10 (very slow diffusion) Eq. [Al] takes the simple for ni

T;; = 0.42y2M,/'w~~,

[A21

while for w,~.,g 1 (and always wL r0 $ I. i.e., fast diffusion only with respect to CO,) one has: T;,' = 0.96;12M2r,. [A31 A maximum in the relaxation rate occurs when q~, = 0.869. The evaluation of the relaxation rate due to the quadrupole mechanism is more complex. Following an old suggestion by Reif (16) we will consider the efg produced by a vacancy as due to a point charge Ze and its fluctuations as due to the diffusion of the vacancy itself. For the efg by the ith vacancy at r, we assume an exponential-type isotropic correlation function with a correlation time T(Y) = (Y/u)~T~, where T" = (NV/N,) ?a = c,, TV is the mean jumping time of the vacancy and N, the number of vacancies. In the assumption T;,j 2: ($) A(/)J,(2w,) (see Eq. [I b]) we can write:

T;,l =$A(/) 1 f exp(-2 iw,t)(V6” i .

c

For

Q,T~

(O)V~‘(t))dt

2T(Y) - Y2 dYdQ. 1+4W : T2(Y)

[A41

>> 1, Eq. [A41 becomes:

[A51 while for (11~T" < 1 one has: T;j

= 7$Z2e2(l

- yr,)‘,4(z)~-‘d-’

NV?,

where nis the minimum distance ofthe approach ofthe diffusing vacancy to the resonant nucleus. It can be observed from Eqs. [A2]-[A51 that if the relaxation is due to the magnetic mechanism, from the temperature behavior of r,, one can evaluate the activation energy for T, = TV c;’ and therefore the sum (E, + E2) ofthe activation energies for the vacancy mean frequency of jump E, and for the concentration of the intrinsic vacancies E2. On the other hand, if the relaxation is due to the quadrupole mechanism from the temperature behavior of TIP one should obtain (El + E2) in the low-temperature range (T,, > w;‘) and (E, - I?,) in the high-temperature range (zV < 0;‘). As mentioned previously, the experimental results for NaNb03 indicate the presence, both in the 13Na and 93Nb relaxation rates on the rotating frame, of a vacancy-diffusion contribution (TY,‘)~ which increases with the increase in temperature. Measure-

410

AVOGADRO, BONERA AND RIGAMONTI

ments at various amplitudes of the r.f. field have shown that the diffusion contribution does not depend on or. In the light of Eqs. [A.2]-[A.51 one can obtain the following conclusions: (i) the effective relaxation mechanism is due to the quadrupole interaction: (ii) ty is smaller than o;‘, i.e., T” 4 IO-“; (iii) the activation energy E, for the concentration of vacancies is greater than the activation energy E, for the mean frequency of the jump of the vacancies 7;‘. A confirmation of the quadrupole origin of the vacancy-diffusion mechanism of relaxation can be obtained by comparing the experimental value of the ratio of the (TY,‘)~ for 23Na and ‘jNb, R ~0.5, with the order of magnitude which can be estimated theoretically. For magnetic mechanisms (see Eq. [A.2]) or [A.3]) one should have R N (Y~M~)~~/(Y~M&,, v 7, while for the yuadrupole mechanism (see Eq. [A.51 or [A.6]) R N [(I - ys)2 A(I)],,/[(l - yz)’ A(Z)],, z 0.4 f 0.25 (where the large uncertainty is mainly related to the range of values for the quadrupole moment of 93Nb present in the literature). From the semilog plot of (T;,‘)d versus l/T(see Fig. 4) one can obtain for (EL - E,) the value 0.25 eV/atom. For Tgreater than about 1020 K a marked changeover in the temperature behavior of (TY,‘)~ occurs. Since for T 2 1000 K thermogravimetric measurements reveal a sizable loss of weight of NaNbO,, one can infer that the marked increase in the relaxation rate is due to a strong increase in the number of vacancies. Note that the temperature at which this large increase in the number of vacancies starts seems to depend on the nature (powder or polycrystalline) and on the handling of the sample. ACKNOWLEDGMENTS Thanks

are due to M. Mali

for earlier

measurements

of Z3Na T,, in NaNbO,.

REFERENCES I. J. SAKURAI, R. A. COWLEY, AND G. DOLLING, J. Phys. Sot. Jup. 28,1426 (1970). 2. I. HATTA, J. Phys. Sot. Jup. 28, 1266 (1970). 3. S. M. SHAPIRO, J. D. AXE, G. SHIRANE, AND T. RISTE, Phys. Rev. B, 6,4332 (1972); J. K. KIEMS, G. SHIRANE, K. A. MULLER, AND H. J. SCHEEL, Phys. Rev. B. &I119 (1973). 4. A. AVO~ADRO, G. BONERA, F. BORSA, AND A. RIGAMONTI, Phys. Rev. B. 9,3905, (1974). 5. G. BONERA, M. MALI, AND A. RIGAMONTI, in “Magnetic Resonance and Related Phenomena” (P. S. Allen, E. R. Andrew, and C. A. Bates, Eds.), p. 301, Nottingham University Press, Nottingham, 1975. 6. F. SCHWABL, Z. Physik 254,57 (1972). 7. A. RIGAMONTI, in “Local Properties at Phase Transitions” (K. A. Muller and A. Rigamonti, Eds.), S.I.F., Bologna, 1975. 8. M. SUZUKI AND R. KUBO, J. Phys. Sot. Jap. 24, 51 (1968). 9. M. H. COHEN AND F. KEFFER, Phys. Rev. 99, 1128 (1955). 10. G. BONERA, F. B~RSA, AND A. RIGAMONTI, Phys. Rev. B. 2,2784 (1970). I I. T. YAGI, I. TATSAZAKI, AND I. TODO, J. Phys. Sot. Jap. 28,311 (1970) and references therein. 12. A. AVOGADRO, E. CAVELIUS, D. MULLER, AND J. PETERSSON, Phys. Status Solidi (b) 44,639 (1971). 13. F. DENOYER, R. COMES, AND M. LAMBERT, Solid State Comm. 8, 1969 (1970); Acta Crystallogr. 27A, 414 (1971); K. ISHIDA AND G. HONJO, J. Phys. Sot. Jap. 34,1279 (1973). 14. K. A. MOLLER, W. BERLINGER, C. H. WEST, AND P. HELLER, Phys. Rev. Letters 32, 160 (1974); see also K. A. MULLER in the text quoted in Ref. (7). 15. H. C. TORREY, Phys. Rev. 12,962 (1953); H. A. RESING AND H. C. TORREY, Phys. Rev. 131, 1102

(1963). 16. F. REIF,

Phys. Reo. 100,

1597 (1955).