- Email: [email protected]
Ultrasonic velocity and attenuation in kcl and KBr containing Li+∗
.I. Phys. Chem. Solids
Pergamon Press 1968. Vol. 29, pp. 677-687.
ULTRASONIC
Printed in Great Britain.
VELOCITY AND A~EN~ATION and KBr CONTAINING Li+*
IN KC3
N. E. BYER and H. S. SACK Department of Engineering Physics, Cornell University, Ithaca, New York 14850, U.S.A. (Received 24 July 1967) Abstract-The temperature and stress dependence of the ultrasonic velocity and attenuation in KC1 and KBr containing Li+ was measured by the pulse-echo technique in the frequency range lo-150 Mclsec, and in the temperature range 1.6”-77°K. Only the Tza mode interacts with the Li+ in KC1 indicating that the Li+ is displaced from the lattice site in the ( 11I) directions. In KBr the Li+ appears lo be on the lattice site since no acoustic interaction was observed. The experimental results agree well with the classical elastic dipole theory. The ‘shape factor’ (A, -A,) of the elastic dipole ellipsoid is (6,6~?0*6) x IO-’ as compared with 6.3 x lO-2 calculated on the basis of the lattice distortion derived by Hatcher et ul. Annealing and aging experiments indicate that Li’ precipitates even at .low concentrations and at room temperature. With increasingconcentration N interactions between the defects are observed, but there is no evidence that this is a cooperative phenomenon. The elastic losses are proportional to l/T and N; a good theoretical model is still missing,
1. IN~ODUCTION
account for the large electric dipole moment (2.5D) observed in KC1 : LiCl mixed crystals, it was proposed that the equilibrium position of the Li+ is displaced from the lattice site [ 1, 21. Ultrasonic velocity measurements have shown that eight equiiibrium positions exist for the Li+ along the ( 11 I ) directions [3]. Tunneling between potential minima results in a splitting of the ground vibrational state of 0.8 cm-’ as determined by microwave [4] and low frequency 151 dielectric, and by thermal conductivity [4] measurements. Calculations of the potential of a displaced Li+ ion agree with the model of ( 111) off-center defect [7-91. This paper isconcerned with the dependence of the acoustic properties of KC1 : LiCl on Li+ concentration, acoustic mode, static uniaxial stress, and heat treatment. The results are analyzed in terms of the IN
ORDER
to
*Part of a thesis submitted to Cornell University in partial fulfillment of the requirements for the Ph.D. degree. This research was mainly supported by the U.S. Atomic Energy Commission. Additional support was obtained from The Advanced Projects Agency through the use of space and facilities provided by the Materials Science Center.
classical ‘elastic dipole’ theory in the form developed by Nowick and Heller[lO]. The elastic dipole moment is compared with the atomic displacements calculated by Hatcher et af.[8]. The effects of interaction between defects are also discussed. 2. APPARATUS AND TECHNIQUES
All the velocity and attenuation measurements were made by the pulse-echo technique[ I 13. The attenuation was determined by a Sperry Ultrasonic Attenuation Comparator (frequency range 1O-ZOO Mclsec). The velocity was obtained by measurements of the time delay between echoes [12]. Considerable simplification was possible because only the relative change in velocity was needed, e.g. the per cent change due to a change in temperature. A change in transit time of about 10 ppm could be resolved. The low temperature apparatus and techniques and the sample preparation have been described elsewhere [ 11, 131. However, some modifications in the bonding of the quartz transducer to the sample were made. In order to minimize the thermal strains in the bond caused by the difference between the expan-
N. E. BYER and H. S. SACK
678
sion coefficients of quartz and KCI, bonds were made at low temperature. 1-pentene (m.p.[ 141 108°K and b-p. 29°C) proved to be more reliable than any one of the bonding agents mentioned in the literature [ 151. The bonding agent was applied at 120°K and then frozen. Measurements were made below 77°K where the bond was completely rigid. The crystals were grown in a Cl2 atmosphere in the Crystal Growing Facility of the Cornell Materials Science Center from Cl, treated analytic reagent-grade material to which controlled amounts of LiCl had been added. Ix. and U.V.absorption measurements indicate that the concentration of impurities such as CN-, OH- and NO,- were reduced to less than one ppm. The ionic conductivity of KC1 is unaltered by the addition of LiCl indicating that the Li+ is not interstitialr I]. Recent measurements have shown that the Li+ diffusion is appreciable even at room temperature [6, 161. Inorder to minimze precipitation, the acoustic measurements were made immediately after the crystals had been annealed. The effect of heat treatment and aging will be discussed later in more detail. Use was
made of the high diffusion coe~cient of Li+ to prepare highly doped samples (2-7 x 10’” cmP3Li). An analytic reagent-grade Isomet KC1 crystal coated with a layer of LiCl powder was annealed in a Cl, atmosphere at 700°C for a week. The impurity concentrations in all samples were determined by the method of flame emission spectrophotometry[ 171. 3. EXPERIMENTAL
A.
RESULTS
Pure ~~~ta~s
The transit time of an acoustic pulse is a function of the compliance, density, and length of the sample, all of which are temperature dependent. The per, cent change in transit time for a pure KC1 crystal in the region 1.6-77°K is shown in Fig. 1 for two modes. In the doped sample, the compliance consists of a component due to the impurity in addition to that of the host lattice. A small amount of impurity changes very little the temperature dependence of the density, length and compliance of the host lattice, so that the observed change in transit time entirely reflects the impurity dependent compliance com-
0
50
Tomporoturr
(‘to
Fig. 1. Temperature dependence of the transit time in pure KCI. The T,, acoustic wave propagated in the [I lo] direction with [OOlJpolarization, while the & wave propagated in the [ 1 lo] direction with [I IO] polarization. Measurements were made at 30 mcisec
ULTRASONIC
VELOCITY
AND
ATTENUATION
ponent. The compliance curves &r/s, presented later were al1 obtained from the difference between the transit times in the impure and pure crystal. Below 15°K there is negligible change in the pure crystal transit time with temperature and thus only the doped crystal need be considered. As will be discussed later, of the three different modes in a cubic crystal, only the EQ and the Tz9 modes will produce ‘stressinduced ordering’. For this reason the A,, hydrostatic mode is not shown in Fig. 1. The compliance for these two modes at low T are[l8]: sO(Tz8) = s4., = 15 x 1O-12cm2/dyn &I(&!) =
s11-
B. Temperature
s12 =
2.3 X lO-12 cm2/dyn.
dependence&he
compliance
The compliance changes as/s in KCl:LiCl crystals for three Li+ concentrations are plotted as a function of IIT in Fig. 2. In all three crystals, the compliance for the &mode is unchanged from that of the pure crystal
IN KC1 AND
KBr
while the Tzo compliance shows a marked contribution from the impurity. This confirms the results reported earlier[3]. The EB and Tzg modes propagating in the [ 1001 direction (not reproduced here) also show the same anisotropy. This striking difference between the interaction of the Eg and the T,modes with the defect will be used later to determine the symmetry of the Li +potential. The &z/s for the T, mode is proportional to l/T with a slope that scales with concentration The error in measu~ng the concentration of free Li+ is large (&20 per cent) because of errors in chemical analysis and concentration gradients. For the sample with the highest concentration (2.7 x lo’* cm+) as/s is linear in l/T only at high temperatures with a slope too small for this high concentration. This crystal is near the solubility limit for Li + and some regions are cloudy because of precipitation. The data can be fit to a cruve proportional to lI(T+@) where 8= 04S5”is0~05°K. From dielectric measurements [5] where the same concentration and T dependence is
t (‘K)
0
0.2
0.4 l/f
679
Ob
PK -1)
Fig. 2. Dependence of the compliance on acoustic mode in KCI with various concentrations Lit. Curves A, 8, and C are for T,,acoustic waves propagating in the [ 1101direction with [OOf1 polarization while curve D is for an E,wave in the [f IO] direction with [l IO] polarization. The Li+ concentrations are A-2-7x 101~cm-s, B-2.3 x lOiR cmm3, C-5.3 X 10”cm-3 and D-2*7X lO**~rn-~. Curve A is the function Gsls = 2.2 X 10-sl (T -k O-65).f = 30 mcjsec.
680
N. E. BYER and H. S. SACK
observed, it appears that 8 is proportional to the concentration. This is consistent with the straight lines for as/s observed for lower concentrations. It seems, then, that at high concentrations (> 3 x lOta cm+) Li+ ions tend to become inactive because of interactions between defects or precipitation. In contrast with these results for KCI, L.i+ in KBr shows no interaction with any acoustic mode. The compliance in a KBr sample with 2 X fOIRcm-” Li+ was identical with that of pure KBr to within +-2 x IO-” per cent.
in order to determine the effect of a large uniaxial stress, a static load of 2-2.5 kg/cm” was applied to the transducer face of the sample while the velocity was being measured. In comparison the stress produced by the sound wave was about 0.1 kg/cm”. Although in pure KC1 at room temperature the yield point occurs at about 10 kg/cm2, much larger elastic stresses are possible in impure samples at low temperatures; in fact no plastic deformation was observed with stresses up to 25 kg/cm”. The compliance change with static stress consists of contributions by the impurity and by
the host lattice. The latter component, determined by expe~ments on pure crystals, was deducted from the total compliance changes in the impure crystals, to yield the contribution &s/s from the impurity. Figures 3, 4, and 5 show the effect of uniaxial static stress in a KCI sample doped with Li+. The bias stress and the sound wave were both in the [lOO] direction with the wave polarized in the [OOI] direction. The wave produced a T,,stress white the static stress had T,,. E,, and A,, components. A marked decrease in compliance results from the bias stress (Fig. 3). For 25 kg/cm? the impurity dependent component ofcompliancedecreases by 14 per cent at 1.7%. Figure 4 is a plot of &s/s due to the static stress (the difference between the two curves of Fig. 3). A curve proportional to l/T” fits the data rather well. For this same sample, the curves of &Y/Svs. static stress are shown in Fig. 5 for two temperatures; &s/s is a linear function of stress. Thus &s/s due to a bias stress, CT?is proportional to a/T”. Experiments on crystals of various concentrations show that the proportionality constant varies linearly with concentration.
T (OK)
Fig. 3. Dependence of the T,, mode compliance on a uniaxial hias \trt‘\\ in ICC.1 containing 1.7 X 10 ix cm-:’ Li‘ The sound propagated in the [I IO] dir~ctiti)n IXith [OOl] polarization while the’static stress was applied acrt~ the (I IO] 1;tw. j’= 30mcisec.
ULTRASONIC
VELOCITY
5020
AND ATTENUATION
IO
f I’K) 3.0
5.0
IN KCI AND KBr
2.0 I
0.’
I
I.6 A
--
I 4 0
*
0 Z
2.0
-40
0
0.4
0.2 I/f
0.6
(OK-‘)
Fig. 4. The difference between the compliance in the stressed and unstressed vs. l/T at 30 mclsec. The curve Ss/s = I.25 X IO-W’? is fit to the dataof(
Prop. A [IlO]
Sound wave Polar.
Mode
w?t
Direct.
Static stress Mag. (kg/cm3
e:::;
20 25
Tzo
3 [II01 c rm
Mode. Tw =E,.A ts
[loo]
17
&gr GA,, &.A ,z,
-_(
i 1.7x iO’8c~-3KCl:Ll
Acoustic Mode Tzg Static
Stress fjl0-j&
I-
I IO
O
I 20
Static Bias Stress fKg/cnt2) Fig. 5. Compliance
vs. static stress for various temperatures in KCI with 1.7 x 1W cm-zELi+ at 30 mclsec. The acoustic and static stress fields are as in Fig. 3.
crystal
Concentration KC1 : Li+ f 1(YRcme3) 1.7 27 13
681
N. E. BYER and H. S. SACK
682
In order to determine which of the stress components affect the T,, acoustic stress, a static stress (Al,, Q,) and a shear wave (T,) were hot.: direct along the [ 1001 axis (Fig. 4-C). The stress had no effect on the wave although a variation of about -+l per cent of the impurity dependent velocity component could be discerned. Thus only a T,, static stress can affect a trigonal wave. Moreover, Fig. 4-B shows that an E, wave, which is unaffected by the Li+ (cf. Fig. 2), is still unaffected when a static bias stress is apphed. The expe~menta1 results shown in Fig. 2-5 can be summarized by the following empirical formulafor the fractional change in compliance:
is zero for A,, and Eg bias stresses. For the stress and acoustic fields shown in Fig. 3, B = (9.2 Z!Z 0.7) x lo-“OK cm”/dyn. However, since the temperature range is limited and the deviation from a l/T law is small, the first factor can also be written as NA/T ( 1- 8/T). D. Effect of heut treatment and aging Figure 6 shows the effect of heat treatment and aging on &s/s for a low Li+ concentration. The label ‘untreated’ refers to measurements made after the sample had been poli,‘*ed, usually a month after growth. The crystal was then annealed at 450°C for a day in a Cl, atmosphere, slowly cooled in about a day, and measured. It was measured again after aging 6 days at room temperature. The heat treatment and aging did not change the l/T dependence; however, the anneal did increase the slope, while aging decreased it. No change in the slope was observed when the sample was aged at 77°K. The slope ‘decreases with room temperature aging but at a diminished rate. Similar heat treatment effects were found for It appears then higher Li+ concentrations.
%(a, T)i =E(l-T) s 7.1,
(1) where N is the concentration, (r is the static compressive stress, A = (3.7 i: O-7) x 10-2’0K cm3, and 0/N = (2.4 If; 0.2) X 10 20”K cm3: B
PKf 2.0
3p 1’
8
I
1
I
I
0.4
0.5
1.6
5.3 x IO” cni3 Lit: KCI Ttg
0
Acourtlc
0.1
Mode
0.2
0.3 I/T
(“K-l
0.6
1
Fig. 6. Dependence of heat treatment and aging on the T, compiiance at 30 mckec KC1 with 5.3 x 1(Y7 crnes Li+. (A) The first measurement was made a month after growth; (8) the sample was then annealed at 450°C and slow cooled: (Cf it was again measured after 6 days of room tempe~ture aging. in
ULTRASONIC
VELOCITY
AND
ATTENUATION
that the effective free Lit concentration increases with annealing but progressively decreases with aging. In addition to decreasing the compliance of a KCI: LiCl crystal, aging also increases the internal strains in the crystal as can be seen from photomicrographs under crossed polaroids. The photographs show that annealing minimizes the internal strains while aging significantly increases them. This aging behavior of the internal strains was not found in pure crystals. The internat stresses were estimated at about 1 kg/cm”. Efastic stresses of this magnitude are too small to noticeably affect the velocity (cf. Fig. 5). It is suggested that the observed strains result from precipitation of the Li+, leaving fewer free Li+ ions which agrees with the decrease of the slope of the compliance vs. l/T curve with aging. A decrease in the effective Li+ concentration has also been observed in thermal conductivity experiments [6]. Effects such as these have also been seen in other mixed crystals of alkali halides such as NaCl: KC1 [IS]. In conclusion it appears that mixed crystals of KC1 :LiCI are quite unstable
50
20
IO
5.3
x id7
Tpp
5.0
cm-3 Lit:
Acoutik
IN KC1 AND
even at low Lit concentrations temperature,
and at room
The results will be discussed in terms of a cIassica1 theory [ IO] which, though not applicable at low temperatures where quantum effects become important, does give physical insight into the nature of the problem. In the case of electric dipoles in an electric field, the classical
2.0
3.0
1.6
KCI
Mods
.75 0 o_ *
s 50
x
s z % s
1.0 .25
D” r J
-: P .f! u
0.0
0.0 0
Fig. 7.
0.1
0.2
0.3
683
E. Acoustic hses The uItrasonic attenuation in KCI containing Li+ has the same concentration and mode dependence as the velocity: The Eg mode losses are the same as in the pure crystal, while the T,, mode losses differ from those in the pure crystal by an amount proportional to the Li+ concentration for low concentrations. As shown in Fig. 7, the losses have the same i/T dependence as the velocity. The ratio of the compliance to the toss angle slopes is 47 at 30 mclsec. However, while the velocity changes are frequency independent (between IO and 30 mclsec), the loss changes increase with frequency, slightly less than proportionally.
2.0
E 5 z
KBr
0.4
0.5
0.6
Temperature dependence of the acoustic attestation for the Te. mode with frequency as a parameter. Concentration 5.3 x I@’ crneR KC1 : Li+.
684
N.
E. BYER and H. S. SACK
theory is valid in the region[9] kT > 6 and hf 4 6, where f is the frequency of the field and 6 the tunnel splitting of the vibrational ground state of Li+ (2.4 x 1Olucps) [4]. Since the present acoustic measurements were made in this region, the classical acoustic wave impurity interaction theory should be able to explain the experimental results. The central element in the classical theory is the ‘elastic dipole moment’ tensor, A$ It is related to the strain produced in the matrix by the mismatch of the impurity and can best be defined by the relation (in complete analogy to the electric case): II, = - I/” x
hf'jflij
(2)
ij
where U, is the energy density introduced by one defect of type p in a stress field cij and V,, the volume of a primitive cell. In general the hij will depend on the orientation of the defect, but it will be the same for crystallographically equivalent positions. The strain ellipsoid for Li+ is an ellipsoid of revolution with one of the principle axes along the direction the Li+ is displaced from the lattice site, provided this direction is along one of the symmetry axes. The principle value associated with this axis is denoted by A, = A,,, while the other two principle values are he = AZ3 = AJP Since u, depends on the mutual orientation of the elastic dipole and the stress field, the latter will destroy the degeneracy of the n equivalent equilibrium positions, and cause their popula-
tions to differ. This in turn produces a strain which is observed as a change in the compliance. This phenomenon is known as ‘stressinduced ordering’. Whether or not stress-induced ordering will occur for a given applied stress depends on the defect symmetry. The selection rules[20] are shown in Table 1. Three defect symmetries are considered. These are denoted by the axes along which the Li+ is displaced. In a cubic crystal for a single defect species only the T,, and E, stress modes can produce stressinduced ordering. By experimentally observing which of these two modes produces such an effect, the defect symmetry can be determined. The experiments for Li+ in KC1 indicate which only T,, stress mode interaction, according to Table 1 implies that Li+ is a ( I 11) defect with eight potential minima along the (1ll)axes. Since the orientation of the principle axes of the strain ellipsoid for Li+ is now known, the compliance change can be written in terms of the principle values AA,An,[ 131 using a series development for the Boltzman factors: 6s ( T,,) = &s,, =
4/V l’,,”/A,, - A,; j“ 91i( T- ,I‘,.)
V,,(A,--,,)a 3kl 6s(E,,)
= 6(s,, -s,z)
r%(A ,<,) = 6(s,, + 2s,,)
+O(rr:‘)
1
= 0; = 0
Table 1. Interaction (indicated by ‘x’) c$vrrrious ultrasonic stresses with dcfkt of‘difl~~re~lt symmetries in a cubic host lattice. The velocity is expressed in terms of’ the appropriate mod&i cij and the density p. ‘Stress symmetry indicates the representatiotls of’the crlbic group according to Mlhich the components of the stress tensor ,for a Rivet1 acwrrstic. ulode transforms
Propagation Direction
[IO01 [loo1 [1101 [1101
Polarization Direction
[IOQl LO101 [OOll [ ITO]
Velocity
{[(C~,,ik,,)
+2(c,,-c,2)]/3p)“’ (c /p)1’2 (& “2
[Cc,,-c,,)/2p]“”
E,, A, T2<, 7‘,,
\
k
.v
.\ .\
\ v .Y .v
(3)
ULTRASONIC
VELOCITY
AND
where N is the defect concentration, (T the magnitude of the bias compressiona stress, and T, a Curie temperature. The acoustic is assumed to be small, i.e. stress, c’ (T’V&A/(kT) 4 1. The numerical factor in the second term depends on the relative orientation of the bias and the measuring stress, and equation (3) applies for the case of Fig. 3. Only (AA- A,), the shape factor of the strain by acoustic ellipsoid, can be determined measurements. The dependence on T, f+ and N in equation (3) is experimentally verified as shown by equation (1). It should be noted that the second term in equation (3) gives a value for the A~-hR independent of the concentration and thus provides a sensitive check for the correctness of the model. In fact the shape factor is (6.7 &O-6) x lo+ from the first term and (6.1 _t O-9) x lo-” from the second. The sign of the second term gives the additional information that AA> AB so that the major axis of the ellipsoid lies along the { 111) axes. It should also be mentioned that these values are compatible with the chemically determined concentration. The results for KBr: Li+ indicate that IhA- AH/ < 6 x lo-“. The potential minima for Li+ in KBr apparently is on the lattice site. This has also been inferred from dielectric [5] and i.r. [2 I] measurements. The experimental results can be compared with the theoretical calculations of Wilson and Hatcher [7,8], which indeed show that the potential minima for the Li+ in KC1 are displacedalong the ( 111) axes. The dispIacements 6Z?, of the six neighboring Cl- were also determined. In or-der to calculate the elastic dipole tensor, a relation is needed between the local atomic distortion and the macroscopic strain produced by a defect. At present such relations are only known precisely for hydrostatic distortions in an isotropic medium [22-241. Since a satisfactory theory is missing, a procedure analogous to that used by K&rzig[25], is applied. In the undisturbed lattice the nearest neighbors of a K+ are considered located on a sphere of radius u/2
ATTENUATION
IN KCI AND
KBr
685
(a, lattice constant). This corresponds to the ‘undisturbed site’ in Ktinzig’s case. When the K+ is replaced by the Li+ the nearest neighbors are displaced by 6Ri and can now be considered to lie on an ellipsoid (Klnzig’s ‘ellipsoid of inclusion’) whose principle axis RI, R, and R3 can be determined from 6RI. The energy u, can thus be calculated (Kanzig, equation (7)) and compared with equation (2), and A determined. The result can be written as Ai=~~~~~~i-~),
i= 1,2,3.
(4)
The principle values of the A-tensor obtained in this way are A, = AA= -0.038,
A2= A3= AR= -0.10
(A, - A,) = 0.063,
$trA = -0.08.
Considering the agreement in the ment and theory is related to the volume by
assumptions involved, the shape factor between experiis remarkable. The trace of h fractional change in crystal AV - = NV&+X. V
The term Qtr& called the ‘size factor’, cannot be determined from stress-induced ordering experiments. The theoretical results indicate a hydrostatic contraction of the lattice upon the introduction of Li+. The deviation from the l/T, law which becomes more pronounced with increasing concentration is caused by interaction between the defects. Nowick and Heller[ 101 have developed a Curie-Weiss type theory of interaction leading to a Curie temperature as expressed in equation (3). if only elastic interaction is considered T,. can be calculated making certain assumptions about the averaging over different lattice sites: T, = 4NV~21AA-AR1zC44/(9k) (for a ( 1I 1) defect).
(6)
For the present case T, = 3-7 X lo-“’ N which is an order of magnitude too small and of the
686
N. EL
BYER and I-I. S. SACK
wrong sign as compared with the experimental results. If interpreted in terms of a cooperative phenomenon, the experiments would indicate an ‘anti-ferroelastic’ behavior (and from the dielectric measurements an anti-ferroelectric one). Both sign and magnitude of T, are sensitive to the exact nature of the model, e.g. the angular dependence of the interaction energy, other than elastic interaction, etc., so that the discrepancies with Nowick and Heller’s theory are not surprising. In the present case the electrical dipole interaction energy is of the order kT, = iVpFL2/~ = 1O-2o Nk. However, there are other reasons which make it doubtful that one is dealing here with a cooperative process: One is that in dielectric [5] and thermal [6] measurements where temperatures below T, have been reached, no sharp transition at T, was observed. There are other types of interaction which will influence the compliance (or dielectric potarization). Lawless E26] has discussed one case, in which an increase or decrease is possible. One can also consider the situation where one defect creates a field at the site of another one and impedes the orientation of his second defect, in the applied field, resulting in a decrease of the compliance (or polarization). In the classical theory discussed above, the frequency of the applied stress field is small enough so that thermodynamic equilibrium can be assumed at any instant. However, in order to determine the acoustic energy loss, the approach to equilibrium must also be considered. Since the tunneling model implies a resonance system, a Lorentzian frequency dependency for the complex compliance can be assumed:
#*z
Gsfu, T, o) = 6s(a, T) CmOz-
(7)
ultrasonic frequency, approximated by, 8S(EF, T, 0) = &(a,
u/w()e
is the compliance given in equation (31, o0 is the characteristic resonant frequency, i.e. the tunnel frequency, and I? is the line width. Since the tunnel frequency is much greater than the
T) (I + iwr/w,2) 1.
(8)
x (l-&f.
(9)
From the experiments a T of lo-lo set results. However, whether or not one is dealing here with a relaxation phenomenon superimposed on a resonance process can not be decided without further experimentation or a more complete quantum mechanical treatment of this problem. AcknowledgementsHelpful discussions with PJOfeSSOJ J. A. Krumhansl, Mr. S. P. Bowen, Mr. M. Gomex, Dr. J, P. Harrison, DJ. W. 3orgardus, and DJ. A. Lakatos are greatfully acknowledged. Thanks are due to Professor R. 0. Pohl for providing the crystals and to MJS. Alison Waters for polishing them. REFERENCES
1. LOMBARD0 2.
6s((r, T) = liio s(u, T, co>
(7) may be
The frequency and temperature dependence of the real and imaginary component of the compliance agree with experiment (cf. Fig. 7). Microwave dielectric measurements [27] give for the resonance frequency and the linewidth o,, = 1.5 X 10”’ set-I, r = 6 X i09 set-I. Using these values, the ratio of the imaginary to the real part of the compliance change diccro2 = 104 at w/(2n) = 30 mclsec. The corresponding experimental value for the acoustic loss is O-02 i.e. two orders of magnitude larger than the dielectric value. Which would imply a linewidth greater than oO. This case approaches that of a relaxation process
w2)--iwr
where
equation
3. 4. 5. 6.
G. and POHL R. O., Phys. Rea Left. 15,291 (1965). SACK H. S. and MORIARTY M. G., Solid Stute Commun. 3,93 (1956). BYER N. E. and SACK H. S., Phys. Ret’. Left. 17, 72(1966). LAKATOS A. and SACK H. S., Solid State Commun. 4,3 15 (1966). BOGARDUS N., Private communication. BAUMANN F. C., HARRfSON J. P., POHL R. O.and SEWARD W. D., Phps. Ra. To be published.
ULTRASONIC
VELOCITY
AND
ATTENUATION
7. DIENES G. J., HATCHER R. D., SMOLUCHOWSKI R. and WILSON W., Phys. Rev. Lett. 16, 25 (1965). 8. WILSON W. D., HATCHER R. D., DIENES G. J. and SMOLUCHOWSKI R. To be published. We wish to thank the authors for showing us the manuscript. 9. GOMEZ M., BOWEN P. S. and KRUMHANSL J. A., Phys. Rev. 153, 1009 (1967). 10. NOWICK A. S. and HELLER W. R., Adv. Phys. 12,25 1 (1963). 11. MOOG R. A., Thesis, Cornell University (19653. 12. DANIELS W. B. and SMITH C. S.., Phvs. , Rev. 111. 713 (1958). (1967). 13. BYER N. E., Thesis, Cornell University S. S., OLIVER G. D. and HUFFMAN 14. TODD H. M., J. Am. them. Sot. 69,15 19 (1947). A., LEWIS J. T. and BRISCOE 15. LEHOCZKY C. V., Cryogenics, 154 (1966). 16. HANSON R. C., Private communication.
IN KC1 AND
KBr
681
of Li+ was measured in the Cornell 17. The concentration Material Science Center Analytical Laboratory under the supervision of Dr. R. K. Skogerboe. M. H. and BRISCOE C. V., Phys. 18. NORWOOD Rev. 112.45 (1958). R. G.. KIBES W. and FINE M. E.. 19. WOLFSON J. a&. Phys. 37,704 (1966). 20. NOWICK A. S. and HELLER W. R.,Adu. Phys. 14, 101 (1965). 21. SIEVERS A. J. and TAKEN0 S., Phys. Rev. 140, Al030(1965). J. D., SofidSt. Phys. 3,79 (1951). 22. ESHELBY A.. MANN E. and VAN JAN R.. J. 23. SEEGER Phys. Chem. solids 23,639 (1962). H. B. and JOHNSON R. A., 24. HUNTINGTON A+ Metall. 10,281 (1962). W., J. Phys. Chem. Solids 23,479 (1962). 25. KANZIG 26. LAWLESS W. N., Phys. kondens Materie. 5, 100 (1966). 27. LAKATOS A., Thesis, Cornell University (1967).
Pergamon Press 1968. Vol. 29, pp. 677-687.
ULTRASONIC
Printed in Great Britain.
VELOCITY AND A~EN~ATION and KBr CONTAINING Li+*
IN KC3
N. E. BYER and H. S. SACK Department of Engineering Physics, Cornell University, Ithaca, New York 14850, U.S.A. (Received 24 July 1967) Abstract-The temperature and stress dependence of the ultrasonic velocity and attenuation in KC1 and KBr containing Li+ was measured by the pulse-echo technique in the frequency range lo-150 Mclsec, and in the temperature range 1.6”-77°K. Only the Tza mode interacts with the Li+ in KC1 indicating that the Li+ is displaced from the lattice site in the ( 11I) directions. In KBr the Li+ appears lo be on the lattice site since no acoustic interaction was observed. The experimental results agree well with the classical elastic dipole theory. The ‘shape factor’ (A, -A,) of the elastic dipole ellipsoid is (6,6~?0*6) x IO-’ as compared with 6.3 x lO-2 calculated on the basis of the lattice distortion derived by Hatcher et ul. Annealing and aging experiments indicate that Li’ precipitates even at .low concentrations and at room temperature. With increasingconcentration N interactions between the defects are observed, but there is no evidence that this is a cooperative phenomenon. The elastic losses are proportional to l/T and N; a good theoretical model is still missing,
1. IN~ODUCTION
account for the large electric dipole moment (2.5D) observed in KC1 : LiCl mixed crystals, it was proposed that the equilibrium position of the Li+ is displaced from the lattice site [ 1, 21. Ultrasonic velocity measurements have shown that eight equiiibrium positions exist for the Li+ along the ( 11 I ) directions [3]. Tunneling between potential minima results in a splitting of the ground vibrational state of 0.8 cm-’ as determined by microwave [4] and low frequency 151 dielectric, and by thermal conductivity [4] measurements. Calculations of the potential of a displaced Li+ ion agree with the model of ( 111) off-center defect [7-91. This paper isconcerned with the dependence of the acoustic properties of KC1 : LiCl on Li+ concentration, acoustic mode, static uniaxial stress, and heat treatment. The results are analyzed in terms of the IN
ORDER
to
*Part of a thesis submitted to Cornell University in partial fulfillment of the requirements for the Ph.D. degree. This research was mainly supported by the U.S. Atomic Energy Commission. Additional support was obtained from The Advanced Projects Agency through the use of space and facilities provided by the Materials Science Center.
classical ‘elastic dipole’ theory in the form developed by Nowick and Heller[lO]. The elastic dipole moment is compared with the atomic displacements calculated by Hatcher et af.[8]. The effects of interaction between defects are also discussed. 2. APPARATUS AND TECHNIQUES
All the velocity and attenuation measurements were made by the pulse-echo technique[ I 13. The attenuation was determined by a Sperry Ultrasonic Attenuation Comparator (frequency range 1O-ZOO Mclsec). The velocity was obtained by measurements of the time delay between echoes [12]. Considerable simplification was possible because only the relative change in velocity was needed, e.g. the per cent change due to a change in temperature. A change in transit time of about 10 ppm could be resolved. The low temperature apparatus and techniques and the sample preparation have been described elsewhere [ 11, 131. However, some modifications in the bonding of the quartz transducer to the sample were made. In order to minimize the thermal strains in the bond caused by the difference between the expan-
N. E. BYER and H. S. SACK
678
sion coefficients of quartz and KCI, bonds were made at low temperature. 1-pentene (m.p.[ 141 108°K and b-p. 29°C) proved to be more reliable than any one of the bonding agents mentioned in the literature [ 151. The bonding agent was applied at 120°K and then frozen. Measurements were made below 77°K where the bond was completely rigid. The crystals were grown in a Cl2 atmosphere in the Crystal Growing Facility of the Cornell Materials Science Center from Cl, treated analytic reagent-grade material to which controlled amounts of LiCl had been added. Ix. and U.V.absorption measurements indicate that the concentration of impurities such as CN-, OH- and NO,- were reduced to less than one ppm. The ionic conductivity of KC1 is unaltered by the addition of LiCl indicating that the Li+ is not interstitialr I]. Recent measurements have shown that the Li+ diffusion is appreciable even at room temperature [6, 161. Inorder to minimze precipitation, the acoustic measurements were made immediately after the crystals had been annealed. The effect of heat treatment and aging will be discussed later in more detail. Use was
made of the high diffusion coe~cient of Li+ to prepare highly doped samples (2-7 x 10’” cmP3Li). An analytic reagent-grade Isomet KC1 crystal coated with a layer of LiCl powder was annealed in a Cl, atmosphere at 700°C for a week. The impurity concentrations in all samples were determined by the method of flame emission spectrophotometry[ 171. 3. EXPERIMENTAL
A.
RESULTS
Pure ~~~ta~s
The transit time of an acoustic pulse is a function of the compliance, density, and length of the sample, all of which are temperature dependent. The per, cent change in transit time for a pure KC1 crystal in the region 1.6-77°K is shown in Fig. 1 for two modes. In the doped sample, the compliance consists of a component due to the impurity in addition to that of the host lattice. A small amount of impurity changes very little the temperature dependence of the density, length and compliance of the host lattice, so that the observed change in transit time entirely reflects the impurity dependent compliance com-
0
50
Tomporoturr
(‘to
Fig. 1. Temperature dependence of the transit time in pure KCI. The T,, acoustic wave propagated in the [I lo] direction with [OOlJpolarization, while the & wave propagated in the [ 1 lo] direction with [I IO] polarization. Measurements were made at 30 mcisec
ULTRASONIC
VELOCITY
AND
ATTENUATION
ponent. The compliance curves &r/s, presented later were al1 obtained from the difference between the transit times in the impure and pure crystal. Below 15°K there is negligible change in the pure crystal transit time with temperature and thus only the doped crystal need be considered. As will be discussed later, of the three different modes in a cubic crystal, only the EQ and the Tz9 modes will produce ‘stressinduced ordering’. For this reason the A,, hydrostatic mode is not shown in Fig. 1. The compliance for these two modes at low T are[l8]: sO(Tz8) = s4., = 15 x 1O-12cm2/dyn &I(&!) =
s11-
B. Temperature
s12 =
2.3 X lO-12 cm2/dyn.
dependence&he
compliance
The compliance changes as/s in KCl:LiCl crystals for three Li+ concentrations are plotted as a function of IIT in Fig. 2. In all three crystals, the compliance for the &mode is unchanged from that of the pure crystal
IN KC1 AND
KBr
while the Tzo compliance shows a marked contribution from the impurity. This confirms the results reported earlier[3]. The EB and Tzg modes propagating in the [ 1001 direction (not reproduced here) also show the same anisotropy. This striking difference between the interaction of the Eg and the T,modes with the defect will be used later to determine the symmetry of the Li +potential. The &z/s for the T, mode is proportional to l/T with a slope that scales with concentration The error in measu~ng the concentration of free Li+ is large (&20 per cent) because of errors in chemical analysis and concentration gradients. For the sample with the highest concentration (2.7 x lo’* cm+) as/s is linear in l/T only at high temperatures with a slope too small for this high concentration. This crystal is near the solubility limit for Li + and some regions are cloudy because of precipitation. The data can be fit to a cruve proportional to lI(T+@) where 8= 04S5”is0~05°K. From dielectric measurements [5] where the same concentration and T dependence is
t (‘K)
0
0.2
0.4 l/f
679
Ob
PK -1)
Fig. 2. Dependence of the compliance on acoustic mode in KCI with various concentrations Lit. Curves A, 8, and C are for T,,acoustic waves propagating in the [ 1101direction with [OOf1 polarization while curve D is for an E,wave in the [f IO] direction with [l IO] polarization. The Li+ concentrations are A-2-7x 101~cm-s, B-2.3 x lOiR cmm3, C-5.3 X 10”cm-3 and D-2*7X lO**~rn-~. Curve A is the function Gsls = 2.2 X 10-sl (T -k O-65).f = 30 mcjsec.
680
N. E. BYER and H. S. SACK
observed, it appears that 8 is proportional to the concentration. This is consistent with the straight lines for as/s observed for lower concentrations. It seems, then, that at high concentrations (> 3 x lOta cm+) Li+ ions tend to become inactive because of interactions between defects or precipitation. In contrast with these results for KCI, L.i+ in KBr shows no interaction with any acoustic mode. The compliance in a KBr sample with 2 X fOIRcm-” Li+ was identical with that of pure KBr to within +-2 x IO-” per cent.
in order to determine the effect of a large uniaxial stress, a static load of 2-2.5 kg/cm” was applied to the transducer face of the sample while the velocity was being measured. In comparison the stress produced by the sound wave was about 0.1 kg/cm”. Although in pure KC1 at room temperature the yield point occurs at about 10 kg/cm2, much larger elastic stresses are possible in impure samples at low temperatures; in fact no plastic deformation was observed with stresses up to 25 kg/cm”. The compliance change with static stress consists of contributions by the impurity and by
the host lattice. The latter component, determined by expe~ments on pure crystals, was deducted from the total compliance changes in the impure crystals, to yield the contribution &s/s from the impurity. Figures 3, 4, and 5 show the effect of uniaxial static stress in a KCI sample doped with Li+. The bias stress and the sound wave were both in the [lOO] direction with the wave polarized in the [OOI] direction. The wave produced a T,,stress white the static stress had T,,. E,, and A,, components. A marked decrease in compliance results from the bias stress (Fig. 3). For 25 kg/cm? the impurity dependent component ofcompliancedecreases by 14 per cent at 1.7%. Figure 4 is a plot of &s/s due to the static stress (the difference between the two curves of Fig. 3). A curve proportional to l/T” fits the data rather well. For this same sample, the curves of &Y/Svs. static stress are shown in Fig. 5 for two temperatures; &s/s is a linear function of stress. Thus &s/s due to a bias stress, CT?is proportional to a/T”. Experiments on crystals of various concentrations show that the proportionality constant varies linearly with concentration.
T (OK)
Fig. 3. Dependence of the T,, mode compliance on a uniaxial hias \trt‘\\ in ICC.1 containing 1.7 X 10 ix cm-:’ Li‘ The sound propagated in the [I IO] dir~ctiti)n IXith [OOl] polarization while the’static stress was applied acrt~ the (I IO] 1;tw. j’= 30mcisec.
ULTRASONIC
VELOCITY
5020
AND ATTENUATION
IO
f I’K) 3.0
5.0
IN KCI AND KBr
2.0 I
0.’
I
I.6 A
--
I 4 0
*
0 Z
2.0
-40
0
0.4
0.2 I/f
0.6
(OK-‘)
Fig. 4. The difference between the compliance in the stressed and unstressed vs. l/T at 30 mclsec. The curve Ss/s = I.25 X IO-W’? is fit to the dataof(
Prop. A [IlO]
Sound wave Polar.
Mode
w?t
Direct.
Static stress Mag. (kg/cm3
e:::;
20 25
Tzo
3 [II01 c rm
Mode. Tw =E,.A ts
[loo]
17
&gr GA,, &.A ,z,
-_(
i 1.7x iO’8c~-3KCl:Ll
Acoustic Mode Tzg Static
Stress fjl0-j&
I-
I IO
O
I 20
Static Bias Stress fKg/cnt2) Fig. 5. Compliance
vs. static stress for various temperatures in KCI with 1.7 x 1W cm-zELi+ at 30 mclsec. The acoustic and static stress fields are as in Fig. 3.
crystal
Concentration KC1 : Li+ f 1(YRcme3) 1.7 27 13
681
N. E. BYER and H. S. SACK
682
In order to determine which of the stress components affect the T,, acoustic stress, a static stress (Al,, Q,) and a shear wave (T,) were hot.: direct along the [ 1001 axis (Fig. 4-C). The stress had no effect on the wave although a variation of about -+l per cent of the impurity dependent velocity component could be discerned. Thus only a T,, static stress can affect a trigonal wave. Moreover, Fig. 4-B shows that an E, wave, which is unaffected by the Li+ (cf. Fig. 2), is still unaffected when a static bias stress is apphed. The expe~menta1 results shown in Fig. 2-5 can be summarized by the following empirical formulafor the fractional change in compliance:
is zero for A,, and Eg bias stresses. For the stress and acoustic fields shown in Fig. 3, B = (9.2 Z!Z 0.7) x lo-“OK cm”/dyn. However, since the temperature range is limited and the deviation from a l/T law is small, the first factor can also be written as NA/T ( 1- 8/T). D. Effect of heut treatment and aging Figure 6 shows the effect of heat treatment and aging on &s/s for a low Li+ concentration. The label ‘untreated’ refers to measurements made after the sample had been poli,‘*ed, usually a month after growth. The crystal was then annealed at 450°C for a day in a Cl, atmosphere, slowly cooled in about a day, and measured. It was measured again after aging 6 days at room temperature. The heat treatment and aging did not change the l/T dependence; however, the anneal did increase the slope, while aging decreased it. No change in the slope was observed when the sample was aged at 77°K. The slope ‘decreases with room temperature aging but at a diminished rate. Similar heat treatment effects were found for It appears then higher Li+ concentrations.
%(a, T)i =E(l-T) s 7.1,
(1) where N is the concentration, (r is the static compressive stress, A = (3.7 i: O-7) x 10-2’0K cm3, and 0/N = (2.4 If; 0.2) X 10 20”K cm3: B
PKf 2.0
3p 1’
8
I
1
I
I
0.4
0.5
1.6
5.3 x IO” cni3 Lit: KCI Ttg
0
Acourtlc
0.1
Mode
0.2
0.3 I/T
(“K-l
0.6
1
Fig. 6. Dependence of heat treatment and aging on the T, compiiance at 30 mckec KC1 with 5.3 x 1(Y7 crnes Li+. (A) The first measurement was made a month after growth; (8) the sample was then annealed at 450°C and slow cooled: (Cf it was again measured after 6 days of room tempe~ture aging. in
ULTRASONIC
VELOCITY
AND
ATTENUATION
that the effective free Lit concentration increases with annealing but progressively decreases with aging. In addition to decreasing the compliance of a KCI: LiCl crystal, aging also increases the internal strains in the crystal as can be seen from photomicrographs under crossed polaroids. The photographs show that annealing minimizes the internal strains while aging significantly increases them. This aging behavior of the internal strains was not found in pure crystals. The internat stresses were estimated at about 1 kg/cm”. Efastic stresses of this magnitude are too small to noticeably affect the velocity (cf. Fig. 5). It is suggested that the observed strains result from precipitation of the Li+, leaving fewer free Li+ ions which agrees with the decrease of the slope of the compliance vs. l/T curve with aging. A decrease in the effective Li+ concentration has also been observed in thermal conductivity experiments [6]. Effects such as these have also been seen in other mixed crystals of alkali halides such as NaCl: KC1 [IS]. In conclusion it appears that mixed crystals of KC1 :LiCI are quite unstable
50
20
IO
5.3
x id7
Tpp
5.0
cm-3 Lit:
Acoutik
IN KC1 AND
even at low Lit concentrations temperature,
and at room
The results will be discussed in terms of a cIassica1 theory [ IO] which, though not applicable at low temperatures where quantum effects become important, does give physical insight into the nature of the problem. In the case of electric dipoles in an electric field, the classical
2.0
3.0
1.6
KCI
Mods
.75 0 o_ *
s 50
x
s z % s
1.0 .25
D” r J
-: P .f! u
0.0
0.0 0
Fig. 7.
0.1
0.2
0.3
683
E. Acoustic hses The uItrasonic attenuation in KCI containing Li+ has the same concentration and mode dependence as the velocity: The Eg mode losses are the same as in the pure crystal, while the T,, mode losses differ from those in the pure crystal by an amount proportional to the Li+ concentration for low concentrations. As shown in Fig. 7, the losses have the same i/T dependence as the velocity. The ratio of the compliance to the toss angle slopes is 47 at 30 mclsec. However, while the velocity changes are frequency independent (between IO and 30 mclsec), the loss changes increase with frequency, slightly less than proportionally.
2.0
E 5 z
KBr
0.4
0.5
0.6
Temperature dependence of the acoustic attestation for the Te. mode with frequency as a parameter. Concentration 5.3 x I@’ crneR KC1 : Li+.
684
N.
E. BYER and H. S. SACK
theory is valid in the region[9] kT > 6 and hf 4 6, where f is the frequency of the field and 6 the tunnel splitting of the vibrational ground state of Li+ (2.4 x 1Olucps) [4]. Since the present acoustic measurements were made in this region, the classical acoustic wave impurity interaction theory should be able to explain the experimental results. The central element in the classical theory is the ‘elastic dipole moment’ tensor, A$ It is related to the strain produced in the matrix by the mismatch of the impurity and can best be defined by the relation (in complete analogy to the electric case): II, = - I/” x
hf'jflij
(2)
ij
where U, is the energy density introduced by one defect of type p in a stress field cij and V,, the volume of a primitive cell. In general the hij will depend on the orientation of the defect, but it will be the same for crystallographically equivalent positions. The strain ellipsoid for Li+ is an ellipsoid of revolution with one of the principle axes along the direction the Li+ is displaced from the lattice site, provided this direction is along one of the symmetry axes. The principle value associated with this axis is denoted by A, = A,,, while the other two principle values are he = AZ3 = AJP Since u, depends on the mutual orientation of the elastic dipole and the stress field, the latter will destroy the degeneracy of the n equivalent equilibrium positions, and cause their popula-
tions to differ. This in turn produces a strain which is observed as a change in the compliance. This phenomenon is known as ‘stressinduced ordering’. Whether or not stress-induced ordering will occur for a given applied stress depends on the defect symmetry. The selection rules[20] are shown in Table 1. Three defect symmetries are considered. These are denoted by the axes along which the Li+ is displaced. In a cubic crystal for a single defect species only the T,, and E, stress modes can produce stressinduced ordering. By experimentally observing which of these two modes produces such an effect, the defect symmetry can be determined. The experiments for Li+ in KC1 indicate which only T,, stress mode interaction, according to Table 1 implies that Li+ is a ( I 11) defect with eight potential minima along the (1ll)axes. Since the orientation of the principle axes of the strain ellipsoid for Li+ is now known, the compliance change can be written in terms of the principle values AA,An,[ 131 using a series development for the Boltzman factors: 6s ( T,,) = &s,, =
4/V l’,,”/A,, - A,; j“ 91i( T- ,I‘,.)
V,,(A,--,,)a 3kl 6s(E,,)
= 6(s,, -s,z)
r%(A ,<,) = 6(s,, + 2s,,)
+O(rr:‘)
1
= 0; = 0
Table 1. Interaction (indicated by ‘x’) c$vrrrious ultrasonic stresses with dcfkt of‘difl~~re~lt symmetries in a cubic host lattice. The velocity is expressed in terms of’ the appropriate mod&i cij and the density p. ‘Stress symmetry indicates the representatiotls of’the crlbic group according to Mlhich the components of the stress tensor ,for a Rivet1 acwrrstic. ulode transforms
Propagation Direction
[IO01 [loo1 [1101 [1101
Polarization Direction
[IOQl LO101 [OOll [ ITO]
Velocity
{[(C~,,ik,,)
+2(c,,-c,2)]/3p)“’ (c /p)1’2 (& “2
[Cc,,-c,,)/2p]“”
E,, A, T2<, 7‘,,
\
k
.v
.\ .\
\ v .Y .v
(3)
ULTRASONIC
VELOCITY
AND
where N is the defect concentration, (T the magnitude of the bias compressiona stress, and T, a Curie temperature. The acoustic is assumed to be small, i.e. stress, c’ (T’V&A/(kT) 4 1. The numerical factor in the second term depends on the relative orientation of the bias and the measuring stress, and equation (3) applies for the case of Fig. 3. Only (AA- A,), the shape factor of the strain by acoustic ellipsoid, can be determined measurements. The dependence on T, f+ and N in equation (3) is experimentally verified as shown by equation (1). It should be noted that the second term in equation (3) gives a value for the A~-hR independent of the concentration and thus provides a sensitive check for the correctness of the model. In fact the shape factor is (6.7 &O-6) x lo+ from the first term and (6.1 _t O-9) x lo-” from the second. The sign of the second term gives the additional information that AA> AB so that the major axis of the ellipsoid lies along the { 111) axes. It should also be mentioned that these values are compatible with the chemically determined concentration. The results for KBr: Li+ indicate that IhA- AH/ < 6 x lo-“. The potential minima for Li+ in KBr apparently is on the lattice site. This has also been inferred from dielectric [5] and i.r. [2 I] measurements. The experimental results can be compared with the theoretical calculations of Wilson and Hatcher [7,8], which indeed show that the potential minima for the Li+ in KC1 are displacedalong the ( 111) axes. The dispIacements 6Z?, of the six neighboring Cl- were also determined. In or-der to calculate the elastic dipole tensor, a relation is needed between the local atomic distortion and the macroscopic strain produced by a defect. At present such relations are only known precisely for hydrostatic distortions in an isotropic medium [22-241. Since a satisfactory theory is missing, a procedure analogous to that used by K&rzig[25], is applied. In the undisturbed lattice the nearest neighbors of a K+ are considered located on a sphere of radius u/2
ATTENUATION
IN KCI AND
KBr
685
(a, lattice constant). This corresponds to the ‘undisturbed site’ in Ktinzig’s case. When the K+ is replaced by the Li+ the nearest neighbors are displaced by 6Ri and can now be considered to lie on an ellipsoid (Klnzig’s ‘ellipsoid of inclusion’) whose principle axis RI, R, and R3 can be determined from 6RI. The energy u, can thus be calculated (Kanzig, equation (7)) and compared with equation (2), and A determined. The result can be written as Ai=~~~~~~i-~),
i= 1,2,3.
(4)
The principle values of the A-tensor obtained in this way are A, = AA= -0.038,
A2= A3= AR= -0.10
(A, - A,) = 0.063,
$trA = -0.08.
Considering the agreement in the ment and theory is related to the volume by
assumptions involved, the shape factor between experiis remarkable. The trace of h fractional change in crystal AV - = NV&+X. V
The term Qtr& called the ‘size factor’, cannot be determined from stress-induced ordering experiments. The theoretical results indicate a hydrostatic contraction of the lattice upon the introduction of Li+. The deviation from the l/T, law which becomes more pronounced with increasing concentration is caused by interaction between the defects. Nowick and Heller[ 101 have developed a Curie-Weiss type theory of interaction leading to a Curie temperature as expressed in equation (3). if only elastic interaction is considered T,. can be calculated making certain assumptions about the averaging over different lattice sites: T, = 4NV~21AA-AR1zC44/(9k) (for a ( 1I 1) defect).
(6)
For the present case T, = 3-7 X lo-“’ N which is an order of magnitude too small and of the
686
N. EL
BYER and I-I. S. SACK
wrong sign as compared with the experimental results. If interpreted in terms of a cooperative phenomenon, the experiments would indicate an ‘anti-ferroelastic’ behavior (and from the dielectric measurements an anti-ferroelectric one). Both sign and magnitude of T, are sensitive to the exact nature of the model, e.g. the angular dependence of the interaction energy, other than elastic interaction, etc., so that the discrepancies with Nowick and Heller’s theory are not surprising. In the present case the electrical dipole interaction energy is of the order kT, = iVpFL2/~ = 1O-2o Nk. However, there are other reasons which make it doubtful that one is dealing here with a cooperative process: One is that in dielectric [5] and thermal [6] measurements where temperatures below T, have been reached, no sharp transition at T, was observed. There are other types of interaction which will influence the compliance (or dielectric potarization). Lawless E26] has discussed one case, in which an increase or decrease is possible. One can also consider the situation where one defect creates a field at the site of another one and impedes the orientation of his second defect, in the applied field, resulting in a decrease of the compliance (or polarization). In the classical theory discussed above, the frequency of the applied stress field is small enough so that thermodynamic equilibrium can be assumed at any instant. However, in order to determine the acoustic energy loss, the approach to equilibrium must also be considered. Since the tunneling model implies a resonance system, a Lorentzian frequency dependency for the complex compliance can be assumed:
#*z
Gsfu, T, o) = 6s(a, T) CmOz-
(7)
ultrasonic frequency, approximated by, 8S(EF, T, 0) = &(a,
u/w()e
is the compliance given in equation (31, o0 is the characteristic resonant frequency, i.e. the tunnel frequency, and I? is the line width. Since the tunnel frequency is much greater than the
T) (I + iwr/w,2) 1.
(8)
x (l-&f.
(9)
From the experiments a T of lo-lo set results. However, whether or not one is dealing here with a relaxation phenomenon superimposed on a resonance process can not be decided without further experimentation or a more complete quantum mechanical treatment of this problem. AcknowledgementsHelpful discussions with PJOfeSSOJ J. A. Krumhansl, Mr. S. P. Bowen, Mr. M. Gomex, Dr. J, P. Harrison, DJ. W. 3orgardus, and DJ. A. Lakatos are greatfully acknowledged. Thanks are due to Professor R. 0. Pohl for providing the crystals and to MJS. Alison Waters for polishing them. REFERENCES
1. LOMBARD0 2.
6s((r, T) = liio s(u, T, co>
(7) may be
The frequency and temperature dependence of the real and imaginary component of the compliance agree with experiment (cf. Fig. 7). Microwave dielectric measurements [27] give for the resonance frequency and the linewidth o,, = 1.5 X 10”’ set-I, r = 6 X i09 set-I. Using these values, the ratio of the imaginary to the real part of the compliance change diccro2 = 104 at w/(2n) = 30 mclsec. The corresponding experimental value for the acoustic loss is O-02 i.e. two orders of magnitude larger than the dielectric value. Which would imply a linewidth greater than oO. This case approaches that of a relaxation process
w2)--iwr
where
equation
3. 4. 5. 6.
G. and POHL R. O., Phys. Rea Left. 15,291 (1965). SACK H. S. and MORIARTY M. G., Solid Stute Commun. 3,93 (1956). BYER N. E. and SACK H. S., Phys. Ret’. Left. 17, 72(1966). LAKATOS A. and SACK H. S., Solid State Commun. 4,3 15 (1966). BOGARDUS N., Private communication. BAUMANN F. C., HARRfSON J. P., POHL R. O.and SEWARD W. D., Phps. Ra. To be published.
ULTRASONIC
VELOCITY
AND
ATTENUATION
7. DIENES G. J., HATCHER R. D., SMOLUCHOWSKI R. and WILSON W., Phys. Rev. Lett. 16, 25 (1965). 8. WILSON W. D., HATCHER R. D., DIENES G. J. and SMOLUCHOWSKI R. To be published. We wish to thank the authors for showing us the manuscript. 9. GOMEZ M., BOWEN P. S. and KRUMHANSL J. A., Phys. Rev. 153, 1009 (1967). 10. NOWICK A. S. and HELLER W. R., Adv. Phys. 12,25 1 (1963). 11. MOOG R. A., Thesis, Cornell University (19653. 12. DANIELS W. B. and SMITH C. S.., Phvs. , Rev. 111. 713 (1958). (1967). 13. BYER N. E., Thesis, Cornell University S. S., OLIVER G. D. and HUFFMAN 14. TODD H. M., J. Am. them. Sot. 69,15 19 (1947). A., LEWIS J. T. and BRISCOE 15. LEHOCZKY C. V., Cryogenics, 154 (1966). 16. HANSON R. C., Private communication.
IN KC1 AND
KBr
681
of Li+ was measured in the Cornell 17. The concentration Material Science Center Analytical Laboratory under the supervision of Dr. R. K. Skogerboe. M. H. and BRISCOE C. V., Phys. 18. NORWOOD Rev. 112.45 (1958). R. G.. KIBES W. and FINE M. E.. 19. WOLFSON J. a&. Phys. 37,704 (1966). 20. NOWICK A. S. and HELLER W. R.,Adu. Phys. 14, 101 (1965). 21. SIEVERS A. J. and TAKEN0 S., Phys. Rev. 140, Al030(1965). J. D., SofidSt. Phys. 3,79 (1951). 22. ESHELBY A.. MANN E. and VAN JAN R.. J. 23. SEEGER Phys. Chem. solids 23,639 (1962). H. B. and JOHNSON R. A., 24. HUNTINGTON A+ Metall. 10,281 (1962). W., J. Phys. Chem. Solids 23,479 (1962). 25. KANZIG 26. LAWLESS W. N., Phys. kondens Materie. 5, 100 (1966). 27. LAKATOS A., Thesis, Cornell University (1967).