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Strain dependence of static and high frequency dielectric constants of some alkali halides
J. Phys.Chem. Solids, 1972,Vol.33, pp. 1079-1089. PergamonPress. Printedin Great Britain
STRAIN D E P E N D E N C E OF STATIC A N D H I G H F R E Q U E N C Y D I E L E C T R I C C O N S T A N T S OF SOME A L K A L I H A L I D E S R. SRINIVASAN and K. SRINIVASAN Department of Physics, Indian Institute of Technology, Madras-36, India
(Received 3 M a y 197 ! ; in revised form 13 October 1971 ) A b s t r a c t - T h e strain derivatives of the static and high frequency dielectric constants of some alkali halides with the NaCI structure have been computed on negative ion polarisable shell models. The shells are assumed to be undeformed during straining. Three models are considered Model I assumes a central overlap interaction of B o r n - M a y e r form to extend to the nearest neighbours. In Model II, besides the above interaction between nearest neighbours, a central overlap interaction of B o r n - M a y e r form is taken to act between second neighbour anions. Model III differs from Model II only in taking the first neighbour interactions to be non-central. The strain derivative WH,11c°)of the static dielectric constant is an order of magnitude larger than W11,2 2(0) and Wl2,12c0)in these alkali halides. Some general trends in the behaviour of the strain derivatives are brought out. In the calculation of the strain optical constants, the present models enable one to include the strain polarisability effect due to the third order parameter in the nearest neighbour interaction. It is shown that the inclusion of this effect brings the value of Pn.H closer to the experimental values. It is also concluded that a deformation of the ionic shell on straining the crystal will play a dominant role in the strain polarisability contributions topal.2v 1. INTRODUCTION
Cochran and Brockhouse have developed non-deformable shell models for the lattice dynamics of alkali halides. These shell models are superior to the rigid ion models in that they take into account the polarisability of the ions. The simplest of these shell models is one in which the negative ion alone is taken to be polarisable and this model has been able to reproduce the dispersion curves fairly well in some alkali halides. Recently Schr6der[1] has taken into account radial deformations of the electronic shells in his breathing shell model and has shown that the inclusion of the deformation improves the agreement between theory and experiment in the dispersion WOODS,
curves.
Srinivasan[2] has developed expressions for the third order elastic constants and the strain dependence of the dielectric constants in ionic crystals. These expressions allow us to calculate the strain derivatives of the static and high frequency dielectric constants of crystals using reasonable models like the shell
and breathing shell models. The values of the strain derivatives of the dielectric constants depend on the third order parameters in the overlap interactions between near neighbours. These third order parameters cause the polarisability of the ion to change on straining the crystal. The third order parameters can be obtained if a form like the B o r n - M a y e r expression is chosen for the overlap interaction. It can also be obtained as an empirical parameter from measured third order elastic constant data. The pressure and temperature dependence of the static dielectric constants of several cubic crystals have been accurately measured recently by Lowndes e t a/.[3] and Jones[4]. These measurements lead to an understanding of the deformation of electronic charge clouds when the lattice is subjected to a hydrostatic compression. We believe that with the available experimental techniques it should be possible to measure the changes in the static dielectric constant due to uniaxial stresses and such measurements will provide more
1079
1080
R. SRIN1VASAN and K. SR1NIVASAN
detailed information about the deformation of charge clouds. In the case of the high frequency dielectric constants such strain derivatives have been measured for a number of crystals and over a range of wavelengths. In this paper as a first step we have calculated the strain derivatives of the static and high frequency dielectric constants of several alkali halides using the negative ion polarisable shell model of Woods et al.[5]. Such calculations will give information about the relative magnitudes of the different strain derivatives of static dielectric constants which may be of help in planning experiments to measure these constants and in revealing certain general trends in the variation of these constants. For the high frequency dielectric constants these calculations are expected to reveal the effect of strain polarisability as taken into account in this simple model.
primitive unit cell. The shell is assumed to shift relative to the core on straining or during lattice vibrations; but it is not assumed to undergo any deformation. In other words the shell is taken to be rigid. Electrical neutrality requires that the total charge on the ion, which is (Z2+ Y2)lel,is-lel. The assumptions regarding the overlap interactions in the three models are as follows: Model 1 The overlap interaction acts only between the nearest neighbours; it is central and is expressed in the B o r n - M a y e r form. The nearest neighbour interaction is then characterised by two constants A' and B' given by e----~-~B ' = 1 d._._~_~ 2va r dr
(2.1)
and 2. NEGATIVE ION POLARISABLE SHELL MODELS FOR THE ALKALI HALIDES
We have considered three negative ion polarisable shell models which differ only in the nature and range of overlap interactions assumed. The Coulomb interaction in all these models arises due to (i) a point ion of charge ]el at the cation site (ii) a core of charge Z2]e] at the anion site and (iii) a spherical shell of charge Y2e surrounding the core at the anion site. The numbering and positions of the particles in the unit cell are given in Table 1. The shell is bound to the core by an isotropic spring of constant k2 = + (e2/2va)K2 where Va is the volume of the Table 1. Constituents of the Unit Cell of the alkali halide crystal on the basis o f the negative ion polarisable shell model and their positions Particle index
Description Mass
Charge
Position
1 2 4
Cation M+ Anioncore M_ Anion.shell 0
lel
Z2e Yze
a0(0,0,0) ao(l, 1, 1) ao(1, 1, i)
e---~--2A' = dz~b 2Va dr z" Here r is the nearest neighbour distance. The third order interaction parameter is given by
A'2( e~ )
e2a _daub 2vaao dr a = B'
~
(2.2)
since the potential is assumed to be of the B o r n - M a y e r form. 2a0 is the lattice constant. The equilibrium condition is 2 B' = ---~aM
(2.3)
where cxM is the Madelung constant, equal tO 1.7476. Model 2 Here the overlap forces are assumed to extend between s e c o n d neighbour anions also. The first and second neighbour interactions are both taken to be of the BornMayer type. So in addition to the second order constants A' and B' and third order parameter
DIELECTRIC CONSTANTS OF SOME ALKALI HALIDES
'a' describing the first neighbour interaction, we have the following parameters describing the second neighbour interaction. e2
10~b(r')
2vaB2 = r' e2
Or' 02~b
(2.4)
2v~ A2 = Or2
and e2
=[l
2vaao4 C2
0 ]3
[~0-Tr'] ~b(r').
As the second neighbour interaction is assumed to be of the B o r n - M a y e r form, we have 1,422 C2 -- 4 B2
3 (Az_B2). 4
(2.5)
The equilibrium condition now becomes B' + 2B2 =
2
---~OtM,
(2.6)
Model 3
Here the first neighbour interaction is taken to be noncentral. Such a noncentral interaction is defined by two force constants, the radial force constant defined by (e2/2va)A ' and the tangential force constant (e212va)B '. As no form of the potential can be defined A' and B' are two adjustable parameters chosen to fit the experimental data on the elastic constants. The third order interaction is defined in terms of two parameters (e2a/2vaao) and (e2b/2vaao). However, the application of rotational invariance conditions on the second and third order parameters yields the relation b=A'--B'. So we are left only with one unknown parameter 'a' to describe the third order interaction. This is an adjustable parameter which can be obtained from the measured value of C l l I . In addition to this noncentral first neighbour interaction, we include second neighbour anion-anion interaction. This is taken to be
1081
central and of the B o r n - M a y e r form and is represented by the three parameters A2, B2 and C2 defined in Model 2. The equilibrium condition (2.6) has no validity now as the first neighbour interaction is taken as noncentral. This model has to be invoked in the case of LiF since the second order elastic constants of this crystal do not satisfy the Cauchy relations even approximately. Models 1 and 2 where the interactions are purely central will lead to the Cauchy relations. T h e y are not therefore applicable to LiF. We have used model 3 also for NaCI and KC1 to see what is the effect on the strain derivatives of the dielectric constant of such a noncentral interaction. Since C m value has been measured only in LiF, NaCI and KCI and since the value of 'a' is to be determined from the measured value of Clxl, we could not apply model 3 to the other alkali halides. 3. RELEVANT EXPRESSIONS FOR THE SECOND AND THIRD ORDER ELASTIC CONSTANTS USED TO FIX THE PARAMETERS
The expressions for the second order elastic constants Cn, C12 and C44 and the third order elastic constant C m are given as follows: C n = ~
e2
[-- 2"5560 -~
A' + ~ + B 2 ] -"
(3.1)
CIS = v~e2aoL[0"1130 (B'+22B2) q_ (A2 4 B 2 ) ]
(3.2) C44 =
e2 [1-2780-+ ( B ' + 2 B 2 ) ~ (AshBY)I• v~ao L
2
,+
..I
(3.3) CI11 =
18.7811+~+C2---~A --3B2 . (3.4)
F r o m the measured static and high frequency dielectric constants and the transverse optic frequency at the zone centre we get two additional equations given by Woods et a/.[5].
1082
R. S R 1 N I V A S A N and K. S R I N I V A S A N
3Va ~® -- 1 4Ir e ® + 2
Y2Zva K2+A'+2B'
(3.5)
and
(,0-,o),,,
3
1--O')T'O'~T] (e®+2)t--~-] =--Y2(A'+2B')/(K2+A'+2B').
(3.6)
Here eo is the static dielectric constant, ~® is the high frequency dielectric constant, OJT.O.is the zone centre T.O. mode frequency and /z is the reduced mass of the ions. N o w we explain how the parameters were obtained in various models.
Model 1 In this model we have only three parameters to determine, namely A', Yz and K2. These three parameters are obtained from (3.1), (3.5) and (3.6). The value o f B ' = -1-165 from (2.3). The value of 'a' can be calculated from (2.2), andZ2 = -- (1 + 172). Model 2 In this model we have six parameters to determine namely Yz, K2, A', B ' and A2, B2. The values of A' and B ' , A~ and B2 were obtained from (3.1) and (3.2), the equilibrium condition (2.6) and the value of the hardness parameter P for the second neighbour BornMayer interaction as given by Tosi [6]. Yz and K2 are then obtained from (3.5) and (3.6). The third order parameters 'a' and C2 are then obtained using (2.2) and (2.5). Model 3 Here we have seven parameters ]I2, K 2 , A', B', A2, B~ and 'a' to be determined. We cannot use the equilibrium condition (2.6). The values o f A ' , B', A~ and B~ are now determined from Ca1, Ca2 and C~ and the value of the hardness parameter for the second neighbour interaction as given by Tosi. The third order parameter 'a' for first neighbour interaction is determined from C m (3.4) and C2 is determined as in Model 2 from (2.5). Since
the value of C m is available only for LiF, NaCI and KCI this model was applied only to these three crystals. In NaCI and KCI, the second order elastic constants have been measured as a function of temperature[7, 8]. In these crystals the data were extrapolated linearly to absolute zero to get the constants used in obtaining the parameters A', B', A2 and B2. Leibfried and Ludwig[9] have stated this to be the proper procedure to follow. In these crystals the values of OJr.o., eo and E~ at 4.2°K were taken from the work of Lowndes et a/.[10]. The values of C m used in Model 3 are the extrapolated values at 0°K obtained from the measured values at room temperature using the theoretically calculated temperature coefficients [11]. In NaI as no data were available on the temperature variation of the elastic constants, values of the second order parameters in Models 1 and 2 are taken from the work of Cowley et a/.[5] who fitted the measured dispersion curves at 100°K. F o r N a F , NaBr, KBr and KI the input data are at room temperature. For LiF the second order parameters in Model 3 were taken from the work of Dolling[12]. In Table 2 are collected the input data used in these calculations for the different alkali halides. 4. RELEVANT EXPRESSIONS FOR THE CALCULATION OF THE STRAIN DERIVATIVES OF STATIC AND HIGH FREQUENCY DIELECTRIC CONSTANTS
The notation followed is the same as in Srinivasan [2]. F o r want of space the notation is not explained here. The static dielectric constant is eo = 1 + 47rCg°1 where e2
f~, = ~
[Zz2{2x, 2x} + 2Z2Y~{2x, 4x}
+ Yz2{4x, 4x}].
(4.1)
DIELECTRIC
CONSTANTS
OF SOME ALKALI
HALIDES
1083
Table 2. Input data used in the calculation of strain derivatives of the static and high frequency dielectric constants of the alkali halides C12 C44 in 10~ dynes/cm 2
Cll Crystal LiF*t~ NaF NaCI NaBr NalCe~ KCI KBr KI
Cl1| ~0
~®
9-69~ 5.76 ~° 4.00 ~
2.45 1.26 1.00
2-80 1.41 1.00
142.3 cb) -5.08 --91.2 cb~ 5.45 -6.27
1.74 2-35 2.60
4-94 tr~ 3.98 t~ 2-677 c~
0.51 0-56 0.405
0-671 0-52 0-369
- 7 9 - 6 ¢~ 4.49 -' 4-90 -5"09
2-20 2"36 2.646
ao in A U
tOT.O. in 10x3 /sec
- -
2-31 2.79 2.98 3"21 3-116 3"29 3"526
4.64 (d) 3-36ta~ 2.54 c~ 2"85 ta~ 2"15 ta~ 1 "9"2¢~)
Remarks Room temperature Room temperature 0°K Room temperature 100°K 0°K R o o m temperature Room temperature
CarRel. [12] tb)DRAI~BLE J. R. and S T R A T H E N R. E. N., Proc. R. Soc. Lond. 92, 1090 (1967). C°LEWiS J. F., Phys. Rev. 161,877 (1967). Cd~Ref. [10] ~Ref. [5] cr~NORWOOD M. H. and B R I S C O E C. V., Phys. Rev. 112, 45 (1958). CU)REDDY J. and R U O F F A. L., Physics of Solids at High Press, Academic Press, New York (1965). *All the parameters were taken from Ref. [ 12]. The data for ~0 and ~® were taken from Ref. [ 10].
The high frequency dielectric constant is
constants are defined for the alkali halides by
eo~= 1 + 4zrfYll
w{o.). Oe---~°~= 47rf~Pu'kzl "" ~"". = a~kt
o0
(4.3)
where =
fen
e2
y2 2
v~2 [4x, 4x]"
The elements {hx, Ixx} are obtained from the elements [ hx, tzx] by omitting the row X = 1 and the column /~ = 1, inverting the 2 × 2 matrix so obtained and bordering it with zeros to make a 3 x 3 matrix. [ hx, tzx] is defined in equations (3.29a), (3.40) of Srinivasan [2]. Expressions for [ hx, I~x] are given below: [2x, 2x] =Va'-"~[e2 ---~Z22--I-K2]
where eo is the dielectric constant tensor and "0kt are Lagrangian strain components. In the original paper of Srinivasan[2] there is an error in the expression for f~tkz,,,,,]. The correct expressions are ~ [ k l , toni ~ P[kl, mn] - - ~ k t ~ m n
where P[kl, m n ] -
1 Z Z Z e(k){hk, aa}e(p)
Va 2 kP eta Bb
x {fib, pl}{[aa, fib, mn] - ~, [eta, fib, vl]A (vl, ran) }
[2x, 4x] = ~a2 [ - - ~ Z2Y2 -- K Q
vl
(4.2)
[4x, 4x] = Va 2e---~[-~Y22+K2+A'+2B'] where (X, tz) = 2, 4. The strain derivatives of the static dielectric
- ½ [ f~k,,~l,. + fY,,,Skm + ffz,.8~. + f~kmB~,]
(4.4)
The factor ½ before the last term is omitted in Srinivasan's paper. The correct expressions ~ k l , mn] for the alkali halide s are
1084
R. S R I N I V A S A N and K. S R I N I V A S A N e
fC0 lX,l,
[ 2x, 2x, xx] = Va e--~-2 [--23"7782Z2 + 13"3576Z22 2
Z Z kO ~O
-- 2K2 ]
x [~xx, fix, xx] {fix, px}e (p) -- 3f~°11
[2x, 4x, xx] = -e2 - [ 13-3576Z2112 + K2] Va 2
1
o _ '~[11,221-
Va 2 Z Xp Z aO
e(~.){kx, otx}
X [otx, fix, yy] {fix, px}e (p) -- f~°a, 1 ~ 2 , 1 2 ] = - - Va'-"2Z
Z
[ ax, fly, xy] {fix, px} e (p ) -- f~°l. (4.5) In the present models, k, p, ~ and fl take the values 2 and 4, e ( 2 ) = / 2 1 e l and e ( 4 ) = I"21el. These are the three independent components of the tensor o[kl, toni" The corresponding expressions for the strain derivatives of the high frequency dielectric constants are
Va2
~[~1,22]
~
-
co ~[12.121
-~" - -
[4-4x,4x
~**
e2(4) ([ 4x, 4x, yy]~ ~ va2 \ [4x, 4x]' /--f~1%
(4.6)
-
e~(4) ([4x,4y, xy]~~ - - f~® " ~ 11. Va 2 k [4x, 4x] /
The W~,~,t are related to the strain optical constants P~,kt by P~,kz =
_ W
+a--2K2--2A'--4B'] e___~2_ [2x, 2x, yy] = va2 [ 11-8991Z2 + 3"7932Z22 ]
e (X) {Xx, ax}
kP ~O
e2(4) ([4x, 4x, xx]~
[ 4x, 4x, xx] = va e--~-~ 2 [--23"7782Y2 + 13"3576Y22
2 t~.ku/ E ®. ~
The quantities [ai, flj, kl] are defined in equation (3.29d) of Srinivasan[2]. Given below are the expressions for these quantities for the alkali halides with the sodium chloride structure in the negative ion polarisable shell models considered here.
e___z2_ [4x, 4x, yy] = v"2 [ 11.8991 Yz + 3.7932 Yz2 +A'--B'] [2x, 4x, yy]
= e__2. 2 Va2 [3"7932Z2Y2]
[kx, vy, xy] = [kx, vx, yy]--[hy, vy]
(4.7)
5. RESULTS AND DISCUSSION
(a) Strain derivatives of co Table 3 gives the values of wto~ Wt°~ and "" 1 1 , 1 1 ' "" 1 1 , 2 2 WCO~ for the different alkali halides on the 12,12 various models. We observe the following general trends. (i) In all the alkali halides, the value of WCO~ is about an order of magnitude II,II larger than the values of W c°~ and "" 11,22 W~O~ All these alkali halides have the 12,12" sodium chloride structure. The sign of WCO~ is negative in all these crystals. 11,22 (ii) When the overlap interaction is extended to the second neighbours in Model 2, the value UA,, I ,1~t0~ I'V 1 1 , 1 1 decreases in all these crystals. (iii) In NaCl and KCI, the use ofnoncentral first neighbour interaction in Model 3 as opposed to Central first neighbour interaction in Model 2 causes a further reduction in the value of W~°~ ll,n" (iv) Comparison of the corresponding sodium and potassium halides reveals a decrease in W ~°) as sodium is replaced "" 1 1 , 1 1 by potassium.
DIELECTRIC CONSTANTS OF SOME ALKALI HALIDES
(v) T h e r e is a t e n d e n c y for the value o f WOO) to go up as we go f r o m the 11,11 chloride to the iodide. (vi) Srinivasan[13] had previously calculated the w~0~ values for the alkaline " i3,kl earth fluorides, CaF2, BaF2 and SrF2 using A x e ' s shell model. But his results have to be c o r r e c t e d for the e r r o r in the expressions for Ptk/,,,,,j. It is instructive to c o m p a r e the vaIues o f WC0~ O,kl in the alkaline earth fluorides with the values in the rock salt structure. In the fluorite structure the cation is s u r r o u n d e d by eight anions at the corners o f a cube while in the rocksalt structure each cation is s u r r o u n d e d by
1085
six anions at the corners of an octahedron. T a b l e 4 gives the values o f W t°~ 11,11' WtO~ and w t°~ for the alkaline earth 11,22 "" 1 2 , 1 2 fluorides so that they may be c o m p a r e d with the values for the alkali halides in T a b l e 3. In the alkaline earth fluorides the magnitudes of ""W c°~ ~o) W ~°~ are nearly of 1 1 , 1 1 ' W 1 1 , 2 2 and " ' 1 2 , 1 2 the same order unlike in the alkali halides having N a C ! structure where wco~ is one '' 11,11 o r d e r of magnitude larger than W ~°) and "" 1 1 , 2 2 WOO) This difference is clearly due to the 12,1'2- " difference o f the a r r a n g e m e n t of the near neighbours in the two structures. (vii) T h e pressure derivatives of the static dielectric constant can be c o m p u t e d from the
Table 3. Strain derivatives and the pressure derivatives o f the static dielectric constant o f the alkali halides Crystal
W,°,.1,
W°L22
W12,12 °
LiF
52"4
-- 5"3
1"2
NaF
NaCI
NaBr
11~o(a~Jap)X 1011dyne/cm 2 Calculated Observed -- 0"27
-- 0"48~ taJ -0"50J --0"38 ,el
72-3
--6-0
- - 1-5
--0.74
--0.50
,a~
1
-6"8
--2"1
--0"59
--0"53
te~
2
64.3
--5.2
--0.3
-
59"9
--5.1
-0.3
- - 1.03
55"0
-- 4-2
- - 0" 1
-- 0"95
1"12
I - - 1 - 0 1 ; - - 1 - 0 6 ' ~ 1 ; - - 0 " 9 2 °:1
--5-5
0.3
- - 1.54
-- 5.4
0-3
- - 1-56
Nal
71-8 67.3
--5-0 --4.9
0.71 0"84
- - 1.77 - - 1.65
_ 1.5 p c j . _ '
KCI
45-8 38"6
--2-9 --4.1
--0"6 --0-8
- 1.49 - - 1.18
-- 1-06;-- l'09tal; -
28"7
- - 3"8
- - 0"6
- - 0"83
48.1
- - 4.1
-- 0.45
- - 1-66
43"7
-- 4.2
- - 1.0
-- 1.50
- - 1-29~a~; -
1.19 ~c~
1 2
1.6yal
1 2
1"00 t~
1 2 3
- - 1" 1 8 ; - - l ' 2 0 t a ~ ; -
1' 15 ~
1
2
58.3
--4-6
-0"05
-2"47
- - 1 " 2 6 ; - - 1 " 0 9 ; - - I ' I Y "~
1
52-5
-- 4-7
-- 0-20
-- 2-25
- - 1-37 ~)
2
C~Ref. [4]. Cb~Ref.[13]. 'CIRef. [3]. CaIGIBBSD. F. and JARMAN M., Phil. Mag. 7,663 (1962).
J P C S Vol. 33 No. 5 - - 1
2 3
74.3
KI
3
60"8
75-3
KBr
Model
1086
R.
SRINIVASAN
and
eo Op
1 W(o~ "" 11,11 +2W
~o
SRINIVASAN
for M o d e l I in NaCI and using s~ = 23-8 × 10 - u and s~2 = - 5 " 1 7 × 10 -11 cm2/dyne and taking T to be 100 kg/cm z we obtain
relation, 1 0Co_
K.
•
Caa + 2C12
A e o = 1"58 × 10 -z
T h e c o m p u t e d values for the various alkali halides are given in T a b l e 3 and for the alkaline earth fluorides in T a b l e 4. T h e m e a s u r e d values o f these quantities are also given in these tables. In general the calculated values follow the trend o f the m e a s u r e d values. In some cases the agreement between calculated and m e a s u r e d values is good and in some o t h e r cases like KI and L i F the agreement is bad. W e will not speculate on the reasons for the disagreement at this stage o f the work. Finally we would like to consider briefly the possibility o f measuring the strain derivatives o f the static dielectric constant. One must subject plates having different crystallographic orientations to uniaxial stresses and measure the dielectric constant. In uniaxial stress m e a s u r e m e n t one cannot reach as high a level o f stress as in hydrostatic stress measurements for fear o f plastically deforming the crystal. O n e could go up to a m a x i m u m stress level of 1 0 0 k g / c m z. Considering a plate o f NaCI cut normal to the [100] direction and subjecting it to a uniaxial stress T along the [100] direction the change in dielectric constant, AE0, is given by (0) (0) AEO = [ W11,11Sll + 2 W11,22Slz ]T
w h e r e s~ are the elastic compliance coefficients. T a k i n g the values of W ~°~ and WCO~ "" 11,I1 ""
II,22
or Aeo/eo ~ 2.8 × 10-3. With the present day techniques it is possible to determine Aeo/Eo to an a c c u r a c y o f 10-L So o n e could determine the strain derivatives o f the dielectric constants to within a few per cent. (b) Strain derivatives o f e~ Mueller[14] has w o r k e d out the t h e o r y o f the strain optical constants in the alkali halides. In his notation Pz, Px andp~, stand for P u , u , Pn,22 and P12,12 respectively. Mueller has s h o w n that there are f o u r distinct contributions to the photoelastic constants. T h e y are as follows: (i) T h e contribution arising from the change in the density o f the crystal. This contribution is present only for pz and p:~ and is indicated by p a and pz a respectively. (ii) T h e L o r e n t z force at an ion b e c o m e s anisotropic on straining a crystal. This anisot r o p y makes contributions p ~ , p ~ and p~, to Pz, Px and p~, respectively. (iii) Considering the ions to have oscillating electrons, the coulomb f o r c e at the site o f the ion b e c o m e s anisotropic and will t h e r e f o r e cause an anisotropic shift in the f r e q u e n c y o f the electron. This contribution to p~, px and p~, is d e n o t e d b y p ~ , px ~ and p~, respectively. (iv) T h e .above three contributions can be e v a l u a t e d f r o m the structure o f the alkali
Table 4. Strain derivatives and the pressure derivatives of the static dielectric constant o f the Alkaline Earth Fluorides Crystal
o 11 W.,
o Wnm
Won, ,2
ll¢o(a¢olap) x 10n dyne/cm2 Calculated Observed
Model
CaF~ 24"66tb) 22.13 17"63 --0-392 --0"349; --0.348Ca~ SrF2 24.10tb~ 19-29 14-73 -- 0.372 -BaFz 23"48tm 15.60 14-54 --0.49 --0.424;--0.428 ~a~ -~a~ Ref. [4]. ~b~ Srinivasan, Ref. [13]. The values of W~ 11etc. given in the above paper have been corrected for the error in the expression -
Pkhmn.
-
DIELECTRIC
CONSTANTS OF SOME ALKALI HALIDES
halides. H o w e v e r a calculation of the photoelastic constants taking only these three contributions still leads to a serious discrepancy with experiment. Mueller therefore concluded that there must be a fourth contribution which arises because the polarisability of the ions may be directly changed due to strain due to the change in the overlap of the neighbours. This contribution is called the strain polarisability effect and is denoted by pz A, px a and pa,. From the measured photoelastic constants, Mueller subtracted the first three contributions and obtained the magniA tudes ofpz a, px A a n d Px," Szigeti and Aggarwal[15] starting from the Lorentz-Lorenz relation for the alkali halides have computed again the first two contributions pa and pL. They then calculated the difference between the measured constants and the sum of the two contributions pa and pL. The difference obviously arises due to the total change in polarisability of the ion. If we picture the ions to be oscillators then this total change arises due to anisotropy in the coulomb force pC and the anisotropy in the short range force pA. H o w e v e r Szigeti and Aggarwal did not adopt any such model for the ion. They calculated empirically the change in the polarisability of the ions with changes in first and second neighbour distances so that a fit could be obtained to the measured photoelastic constants. The reason why the present work is carried out is the following. On this simple model one could obtain the anisotropic change in the overlap forces. The strain polarisability effect can therefore be calculated on this model in which the change in polarisability arises only through the parameter 'a' in the overlap interaction. We do not envisage any deformation of the shell here. At least such a calculation will give us an independent evaluation of the strain polarisability effect. In the actual lattice the strain polarisability will also have a contribution from the deformatiofi of the charge distribution from a spherical shape. The present calculation may yield a clue as to
1087
the importance of the latter effect. First we should show that the present expressions for P~J,kZ agree with Mueller's as regards the first three contributions. This is done in the appendix IA. We calculated P~J,k~for the alkali halides on the various models. We found that the calculated values ofpu, k~ were sensibly the same for any crystal on the different models. The values only differed in the third decimal place. So unlike the static dielectric constant, the strain derivatives of the high frequency dielectric constant are not much sensitive to the details of the overlap interaction. In view of this fact, we have given the values of Pu, ut in Table 5 for all the alkali halides except LiF on Model 2. F o r LiF only Model 3 is applicable and the values of P~,kl quoted are on this model. In the same table are quoted the values of Szigeti and Aggarwal which represent the sum of the Lorentz-Lorenz and density contributions. Szigeti and Aggarwal claim that the addition of the coulomb contribution only increases the discrepancy between theory and experiment. One striking fact emerges on studying the Table 5. The inclusion of the strain polarisability effect through the parameter 'a' in the present model brings about a large reduction in the value of P~I. This reduces the discrepancy between the calculated and measured values of P11. For example in KCI, the inclusion of the strain polarisability on the simple model described here reduces Pll from the value 0.61 given by Szigeti and Agarwal to 0.251 The measured value is 0.19. H o w e v e r the present calculation does not bring about any such drastic .change in the values of Pn,z2 and PI2,1z. The reason for this becomes apparent on looking at the expressions for [ax, fix, xx] and [ax, fix, yy]. While the anharmonic third order parameter enters the former expressions it does not enter the latter. This means that on applying a uniaxial stress along [ 100], the short range anharmonic interactions of the ion with its two first neighbours along [ 100] changes the polarisability of the ion in this direction.
1088
R. S R I N I V A S A N and K. S R I N I V A S A N
Table 5. Strain optical constants of the alkali halides Pn,n Crystal
Model
LiF NaF NaCI NaBr Nal KCI KBr KI
3 2 2 2 2 2 2 2
P11,22 P12,12 Present work
0.247 0-213 0-241 0.238 0.239 0-248 0-253 0.241
0-243 0-255 0-273 0.267 0.259 0.288 0.288 0.280
Pn,n Pn,m P12,12 Aggarwal and Szigeti*
Pn, n
0-38
0-30
0-052
0.02
0-13
--0-045
0-51
0-28
0.033
0.11
0.153
-0.010
0.61 0.62 0.61
0.22 0.23 0-24
--0.03 --0.016 0.010
0.19 0.216 0.210
0.14 0.168 0-169
--0-03 --0.024 --
0-005 0-006 0-030 0.029 0-044 0-040 0.042 0-047
P11.22 P12.12 Experiment*
*Ref. [15].
This result will be true even if we take a model in which both ions are polafisable. So we do not think that a significant improvement in the agreement of P12 or P44 can be brought about merely by including the polarisation of the cation also. On the other hand if a compression along [100] brings about a distortion of the shell, then one would also obtain a change in the polarisability in the [010] direction. Thus deformation of the ion will be the major effect deciding the strain polarisability contribution to Pn,2z. On the other hand the anharmonic third order parameter in the nearest neighbour overlap interaction is of importance in the strain polarisability contribution to Pn, 11. These conclusions we believe, should be valid in the alkali halide crystals with rock salt structure. Acknowledgement--The zuthors are thankful to Miss G. Lakshmi for her help in the computations. REFERENCES 1. S C H R O D E R U., Solid State Commun. 4, 347 (1966). 2. S R I N I V A S A N R., Phys. Rev. 165, 1041 (1968). 3. L O W N D E S R. P. and M A R T I N D. H., Proc. R. Soc. Lond. A316, 351 (1970). 4. J O N E S B. W.,Phil. Mag. 16, 1085 (1967). 5. C O W L E ' ~ , R. A., C O C H R A N W. and BROCKH O U S E B. N. and W O O D S A. D., Phys. Rev. 131, 1030 (1963). 6. TOSI M. P., Solid State Physics, Vol. 16, p. 1, Academic Press, New York (1964). 7. LEWIS J. F.,Phys. Rev. 161,877 (1967). 8. N O R W O O D M. H. and B R I S C O E C. V., Phys. Rev. 112, 45 (1968). 9. L E I B F R I E D G. and L U D W I G W., Solid State
10. 1 I. 12. 13. 14. 15.
Physics, Vol. 12, p. 276, Academic Press, New York (1961). L O W N D E S R. P. and M A R T I N D. H., Proc. R. Soc. Lond. 308, 473 (1969). G H A T E P. B.,Phys. Rev. 139, A1666 (1965). D O L L I N G G., S M I T H H. G., N I C K L O W L. M., V I J A Y A R A G H A V A N P. R, and W I L K I N S O N M. K., Phys. Rev. 168, 970 (1968). S R I N I V A S A N R.,Phys. Rev. 165, 1054 (1968). M U E L L E R H., Phys. Rev.47,947 (1935). A G G A R W A L K. G. and S Z I G E T I , J. Phys. (Part C ) 3 , 1097(1970).
APPENDIX IA Mueller[14] gives the following expressions for the strain optical constants. In his notation:
Pn.ll = Pz; Pn.22 = px and P12.12 = -'-fWritingpl (i = z o r x , o r x ' ) as
Pt=P~e+PlL+p~C+pi A
(IA.I)
as mentioned in the text we have p d=
~= + 21) unless i = x', p~, = 0 3(~,--
pl L = -- ~ sjk~NjNkRIRe/[ E
NkRk]
2
Jk
p c=_~j
NjRj2 fJ ~" N,Zks~** / [~ NkRk] 2
(IA.2)
pt a is the contribution arising from the anharmonic short range interaction and distortion of the electronic cloud of the ion on straining the lattice. Here Nk = = Zk = R~ =
number of ions per cnP of the k th type. I/va in our notation. is the charge of the ion k, 4~rNoc9/3 where 09 is the polarisability and No is the Avogadro number. f j = is the oscillator strength o f t h e j th ion.
sj~ are lattice sums whose values are summarised in
DIELECTRIC
CONSTANTS
T a b l e I A . j = 1 refers to the cation a n d j = 2 to the anion.
Table I A . The lattice sums sjkt f o r the sodium chloride lattice Sn,z = s22.z = (L~a:) st2,z = (LlalB) Sll.x = s22,z = (Lla~~) s12,x = (L~) s=,x, = sz2,~, = (L~4a) s12,.~, = (L4a~)
= --0"0630 = -- 1"9566 = 0"0315 = 0"9783 = - 0"6037 = 1"2901
T h e notation o f A g g a r w a l a n d Szigeti is indicated in the brackets. In o u r model
OF SOME
ALKALI
HALIDES
a l = 0, RI = 0; a2 =
1089
Y22v" [K2+A'+2B'I
f j = 0, w h e n j = 1 a n d f 2 = -- Y2
(IA.3)
(Y2 is negative as it is the charge on the shell). S u b s t i t u t i o n for the v a r i o u s quantities in (IA.2) we c a n get the e x p r e s s i o n s for p j d + p L + p ~ a n d h e n c e the c o n t r i b u t i o n s from t h e s e t e r m s to Pu.k~- In o u r model PtJ, kt "~
E2
Substituting for various quantities from (4.2) a n d (4.6) a n d (4.7) a n d omitting t h e t e r m s u a n d b in [4i, 4j, kl] ( t h e y r e p r e s e n t t h e effect o f s h o r t range a n h a r m o n i c interaction), we obtain e x p r e s s i o n s identical with t h o s e derived f r o m M u e l l e r ' s theory.
STRAIN D E P E N D E N C E OF STATIC A N D H I G H F R E Q U E N C Y D I E L E C T R I C C O N S T A N T S OF SOME A L K A L I H A L I D E S R. SRINIVASAN and K. SRINIVASAN Department of Physics, Indian Institute of Technology, Madras-36, India
(Received 3 M a y 197 ! ; in revised form 13 October 1971 ) A b s t r a c t - T h e strain derivatives of the static and high frequency dielectric constants of some alkali halides with the NaCI structure have been computed on negative ion polarisable shell models. The shells are assumed to be undeformed during straining. Three models are considered Model I assumes a central overlap interaction of B o r n - M a y e r form to extend to the nearest neighbours. In Model II, besides the above interaction between nearest neighbours, a central overlap interaction of B o r n - M a y e r form is taken to act between second neighbour anions. Model III differs from Model II only in taking the first neighbour interactions to be non-central. The strain derivative WH,11c°)of the static dielectric constant is an order of magnitude larger than W11,2 2(0) and Wl2,12c0)in these alkali halides. Some general trends in the behaviour of the strain derivatives are brought out. In the calculation of the strain optical constants, the present models enable one to include the strain polarisability effect due to the third order parameter in the nearest neighbour interaction. It is shown that the inclusion of this effect brings the value of Pn.H closer to the experimental values. It is also concluded that a deformation of the ionic shell on straining the crystal will play a dominant role in the strain polarisability contributions topal.2v 1. INTRODUCTION
Cochran and Brockhouse have developed non-deformable shell models for the lattice dynamics of alkali halides. These shell models are superior to the rigid ion models in that they take into account the polarisability of the ions. The simplest of these shell models is one in which the negative ion alone is taken to be polarisable and this model has been able to reproduce the dispersion curves fairly well in some alkali halides. Recently Schr6der[1] has taken into account radial deformations of the electronic shells in his breathing shell model and has shown that the inclusion of the deformation improves the agreement between theory and experiment in the dispersion WOODS,
curves.
Srinivasan[2] has developed expressions for the third order elastic constants and the strain dependence of the dielectric constants in ionic crystals. These expressions allow us to calculate the strain derivatives of the static and high frequency dielectric constants of crystals using reasonable models like the shell
and breathing shell models. The values of the strain derivatives of the dielectric constants depend on the third order parameters in the overlap interactions between near neighbours. These third order parameters cause the polarisability of the ion to change on straining the crystal. The third order parameters can be obtained if a form like the B o r n - M a y e r expression is chosen for the overlap interaction. It can also be obtained as an empirical parameter from measured third order elastic constant data. The pressure and temperature dependence of the static dielectric constants of several cubic crystals have been accurately measured recently by Lowndes e t a/.[3] and Jones[4]. These measurements lead to an understanding of the deformation of electronic charge clouds when the lattice is subjected to a hydrostatic compression. We believe that with the available experimental techniques it should be possible to measure the changes in the static dielectric constant due to uniaxial stresses and such measurements will provide more
1079
1080
R. SRIN1VASAN and K. SR1NIVASAN
detailed information about the deformation of charge clouds. In the case of the high frequency dielectric constants such strain derivatives have been measured for a number of crystals and over a range of wavelengths. In this paper as a first step we have calculated the strain derivatives of the static and high frequency dielectric constants of several alkali halides using the negative ion polarisable shell model of Woods et al.[5]. Such calculations will give information about the relative magnitudes of the different strain derivatives of static dielectric constants which may be of help in planning experiments to measure these constants and in revealing certain general trends in the variation of these constants. For the high frequency dielectric constants these calculations are expected to reveal the effect of strain polarisability as taken into account in this simple model.
primitive unit cell. The shell is assumed to shift relative to the core on straining or during lattice vibrations; but it is not assumed to undergo any deformation. In other words the shell is taken to be rigid. Electrical neutrality requires that the total charge on the ion, which is (Z2+ Y2)lel,is-lel. The assumptions regarding the overlap interactions in the three models are as follows: Model 1 The overlap interaction acts only between the nearest neighbours; it is central and is expressed in the B o r n - M a y e r form. The nearest neighbour interaction is then characterised by two constants A' and B' given by e----~-~B ' = 1 d._._~_~ 2va r dr
(2.1)
and 2. NEGATIVE ION POLARISABLE SHELL MODELS FOR THE ALKALI HALIDES
We have considered three negative ion polarisable shell models which differ only in the nature and range of overlap interactions assumed. The Coulomb interaction in all these models arises due to (i) a point ion of charge ]el at the cation site (ii) a core of charge Z2]e] at the anion site and (iii) a spherical shell of charge Y2e surrounding the core at the anion site. The numbering and positions of the particles in the unit cell are given in Table 1. The shell is bound to the core by an isotropic spring of constant k2 = + (e2/2va)K2 where Va is the volume of the Table 1. Constituents of the Unit Cell of the alkali halide crystal on the basis o f the negative ion polarisable shell model and their positions Particle index
Description Mass
Charge
Position
1 2 4
Cation M+ Anioncore M_ Anion.shell 0
lel
Z2e Yze
a0(0,0,0) ao(l, 1, 1) ao(1, 1, i)
e---~--2A' = dz~b 2Va dr z" Here r is the nearest neighbour distance. The third order interaction parameter is given by
A'2( e~ )
e2a _daub 2vaao dr a = B'
~
(2.2)
since the potential is assumed to be of the B o r n - M a y e r form. 2a0 is the lattice constant. The equilibrium condition is 2 B' = ---~aM
(2.3)
where cxM is the Madelung constant, equal tO 1.7476. Model 2 Here the overlap forces are assumed to extend between s e c o n d neighbour anions also. The first and second neighbour interactions are both taken to be of the BornMayer type. So in addition to the second order constants A' and B' and third order parameter
DIELECTRIC CONSTANTS OF SOME ALKALI HALIDES
'a' describing the first neighbour interaction, we have the following parameters describing the second neighbour interaction. e2
10~b(r')
2vaB2 = r' e2
Or' 02~b
(2.4)
2v~ A2 = Or2
and e2
=[l
2vaao4 C2
0 ]3
[~0-Tr'] ~b(r').
As the second neighbour interaction is assumed to be of the B o r n - M a y e r form, we have 1,422 C2 -- 4 B2
3 (Az_B2). 4
(2.5)
The equilibrium condition now becomes B' + 2B2 =
2
---~OtM,
(2.6)
Model 3
Here the first neighbour interaction is taken to be noncentral. Such a noncentral interaction is defined by two force constants, the radial force constant defined by (e2/2va)A ' and the tangential force constant (e212va)B '. As no form of the potential can be defined A' and B' are two adjustable parameters chosen to fit the experimental data on the elastic constants. The third order interaction is defined in terms of two parameters (e2a/2vaao) and (e2b/2vaao). However, the application of rotational invariance conditions on the second and third order parameters yields the relation b=A'--B'. So we are left only with one unknown parameter 'a' to describe the third order interaction. This is an adjustable parameter which can be obtained from the measured value of C l l I . In addition to this noncentral first neighbour interaction, we include second neighbour anion-anion interaction. This is taken to be
1081
central and of the B o r n - M a y e r form and is represented by the three parameters A2, B2 and C2 defined in Model 2. The equilibrium condition (2.6) has no validity now as the first neighbour interaction is taken as noncentral. This model has to be invoked in the case of LiF since the second order elastic constants of this crystal do not satisfy the Cauchy relations even approximately. Models 1 and 2 where the interactions are purely central will lead to the Cauchy relations. T h e y are not therefore applicable to LiF. We have used model 3 also for NaCI and KC1 to see what is the effect on the strain derivatives of the dielectric constant of such a noncentral interaction. Since C m value has been measured only in LiF, NaCI and KCI and since the value of 'a' is to be determined from the measured value of Clxl, we could not apply model 3 to the other alkali halides. 3. RELEVANT EXPRESSIONS FOR THE SECOND AND THIRD ORDER ELASTIC CONSTANTS USED TO FIX THE PARAMETERS
The expressions for the second order elastic constants Cn, C12 and C44 and the third order elastic constant C m are given as follows: C n = ~
e2
[-- 2"5560 -~
A' + ~ + B 2 ] -"
(3.1)
CIS = v~e2aoL[0"1130 (B'+22B2) q_ (A2 4 B 2 ) ]
(3.2) C44 =
e2 [1-2780-+ ( B ' + 2 B 2 ) ~ (AshBY)I• v~ao L
2
,+
..I
(3.3) CI11 =
18.7811+~+C2---~A --3B2 . (3.4)
F r o m the measured static and high frequency dielectric constants and the transverse optic frequency at the zone centre we get two additional equations given by Woods et a/.[5].
1082
R. S R 1 N I V A S A N and K. S R I N I V A S A N
3Va ~® -- 1 4Ir e ® + 2
Y2Zva K2+A'+2B'
(3.5)
and
(,0-,o),,,
3
1--O')T'O'~T] (e®+2)t--~-] =--Y2(A'+2B')/(K2+A'+2B').
(3.6)
Here eo is the static dielectric constant, ~® is the high frequency dielectric constant, OJT.O.is the zone centre T.O. mode frequency and /z is the reduced mass of the ions. N o w we explain how the parameters were obtained in various models.
Model 1 In this model we have only three parameters to determine, namely A', Yz and K2. These three parameters are obtained from (3.1), (3.5) and (3.6). The value o f B ' = -1-165 from (2.3). The value of 'a' can be calculated from (2.2), andZ2 = -- (1 + 172). Model 2 In this model we have six parameters to determine namely Yz, K2, A', B ' and A2, B2. The values of A' and B ' , A~ and B2 were obtained from (3.1) and (3.2), the equilibrium condition (2.6) and the value of the hardness parameter P for the second neighbour BornMayer interaction as given by Tosi [6]. Yz and K2 are then obtained from (3.5) and (3.6). The third order parameters 'a' and C2 are then obtained using (2.2) and (2.5). Model 3 Here we have seven parameters ]I2, K 2 , A', B', A2, B~ and 'a' to be determined. We cannot use the equilibrium condition (2.6). The values o f A ' , B', A~ and B~ are now determined from Ca1, Ca2 and C~ and the value of the hardness parameter for the second neighbour interaction as given by Tosi. The third order parameter 'a' for first neighbour interaction is determined from C m (3.4) and C2 is determined as in Model 2 from (2.5). Since
the value of C m is available only for LiF, NaCI and KCI this model was applied only to these three crystals. In NaCI and KCI, the second order elastic constants have been measured as a function of temperature[7, 8]. In these crystals the data were extrapolated linearly to absolute zero to get the constants used in obtaining the parameters A', B', A2 and B2. Leibfried and Ludwig[9] have stated this to be the proper procedure to follow. In these crystals the values of OJr.o., eo and E~ at 4.2°K were taken from the work of Lowndes et a/.[10]. The values of C m used in Model 3 are the extrapolated values at 0°K obtained from the measured values at room temperature using the theoretically calculated temperature coefficients [11]. In NaI as no data were available on the temperature variation of the elastic constants, values of the second order parameters in Models 1 and 2 are taken from the work of Cowley et a/.[5] who fitted the measured dispersion curves at 100°K. F o r N a F , NaBr, KBr and KI the input data are at room temperature. For LiF the second order parameters in Model 3 were taken from the work of Dolling[12]. In Table 2 are collected the input data used in these calculations for the different alkali halides. 4. RELEVANT EXPRESSIONS FOR THE CALCULATION OF THE STRAIN DERIVATIVES OF STATIC AND HIGH FREQUENCY DIELECTRIC CONSTANTS
The notation followed is the same as in Srinivasan [2]. F o r want of space the notation is not explained here. The static dielectric constant is eo = 1 + 47rCg°1 where e2
f~, = ~
[Zz2{2x, 2x} + 2Z2Y~{2x, 4x}
+ Yz2{4x, 4x}].
(4.1)
DIELECTRIC
CONSTANTS
OF SOME ALKALI
HALIDES
1083
Table 2. Input data used in the calculation of strain derivatives of the static and high frequency dielectric constants of the alkali halides C12 C44 in 10~ dynes/cm 2
Cll Crystal LiF*t~ NaF NaCI NaBr NalCe~ KCI KBr KI
Cl1| ~0
~®
9-69~ 5.76 ~° 4.00 ~
2.45 1.26 1.00
2-80 1.41 1.00
142.3 cb) -5.08 --91.2 cb~ 5.45 -6.27
1.74 2-35 2.60
4-94 tr~ 3.98 t~ 2-677 c~
0.51 0-56 0.405
0-671 0-52 0-369
- 7 9 - 6 ¢~ 4.49 -' 4-90 -5"09
2-20 2"36 2.646
ao in A U
tOT.O. in 10x3 /sec
- -
2-31 2.79 2.98 3"21 3-116 3"29 3"526
4.64 (d) 3-36ta~ 2.54 c~ 2"85 ta~ 2"15 ta~ 1 "9"2¢~)
Remarks Room temperature Room temperature 0°K Room temperature 100°K 0°K R o o m temperature Room temperature
CarRel. [12] tb)DRAI~BLE J. R. and S T R A T H E N R. E. N., Proc. R. Soc. Lond. 92, 1090 (1967). C°LEWiS J. F., Phys. Rev. 161,877 (1967). Cd~Ref. [10] ~Ref. [5] cr~NORWOOD M. H. and B R I S C O E C. V., Phys. Rev. 112, 45 (1958). CU)REDDY J. and R U O F F A. L., Physics of Solids at High Press, Academic Press, New York (1965). *All the parameters were taken from Ref. [ 12]. The data for ~0 and ~® were taken from Ref. [ 10].
The high frequency dielectric constant is
constants are defined for the alkali halides by
eo~= 1 + 4zrfYll
w{o.). Oe---~°~= 47rf~Pu'kzl "" ~"". = a~kt
o0
(4.3)
where =
fen
e2
y2 2
v~2 [4x, 4x]"
The elements {hx, Ixx} are obtained from the elements [ hx, tzx] by omitting the row X = 1 and the column /~ = 1, inverting the 2 × 2 matrix so obtained and bordering it with zeros to make a 3 x 3 matrix. [ hx, tzx] is defined in equations (3.29a), (3.40) of Srinivasan [2]. Expressions for [ hx, I~x] are given below: [2x, 2x] =Va'-"~[e2 ---~Z22--I-K2]
where eo is the dielectric constant tensor and "0kt are Lagrangian strain components. In the original paper of Srinivasan[2] there is an error in the expression for f~tkz,,,,,]. The correct expressions are ~ [ k l , toni ~ P[kl, mn] - - ~ k t ~ m n
where P[kl, m n ] -
1 Z Z Z e(k){hk, aa}e(p)
Va 2 kP eta Bb
x {fib, pl}{[aa, fib, mn] - ~, [eta, fib, vl]A (vl, ran) }
[2x, 4x] = ~a2 [ - - ~ Z2Y2 -- K Q
vl
(4.2)
[4x, 4x] = Va 2e---~[-~Y22+K2+A'+2B'] where (X, tz) = 2, 4. The strain derivatives of the static dielectric
- ½ [ f~k,,~l,. + fY,,,Skm + ffz,.8~. + f~kmB~,]
(4.4)
The factor ½ before the last term is omitted in Srinivasan's paper. The correct expressions ~ k l , mn] for the alkali halide s are
1084
R. S R I N I V A S A N and K. S R I N I V A S A N e
fC0 lX,l,
[ 2x, 2x, xx] = Va e--~-2 [--23"7782Z2 + 13"3576Z22 2
Z Z kO ~O
-- 2K2 ]
x [~xx, fix, xx] {fix, px}e (p) -- 3f~°11
[2x, 4x, xx] = -e2 - [ 13-3576Z2112 + K2] Va 2
1
o _ '~[11,221-
Va 2 Z Xp Z aO
e(~.){kx, otx}
X [otx, fix, yy] {fix, px}e (p) -- f~°a, 1 ~ 2 , 1 2 ] = - - Va'-"2Z
Z
[ ax, fly, xy] {fix, px} e (p ) -- f~°l. (4.5) In the present models, k, p, ~ and fl take the values 2 and 4, e ( 2 ) = / 2 1 e l and e ( 4 ) = I"21el. These are the three independent components of the tensor o[kl, toni" The corresponding expressions for the strain derivatives of the high frequency dielectric constants are
Va2
~[~1,22]
~
-
co ~[12.121
-~" - -
[4-4x,4x
~**
e2(4) ([ 4x, 4x, yy]~ ~ va2 \ [4x, 4x]' /--f~1%
(4.6)
-
e~(4) ([4x,4y, xy]~~ - - f~® " ~ 11. Va 2 k [4x, 4x] /
The W~,~,t are related to the strain optical constants P~,kt by P~,kz =
_ W
+a--2K2--2A'--4B'] e___~2_ [2x, 2x, yy] = va2 [ 11-8991Z2 + 3"7932Z22 ]
e (X) {Xx, ax}
kP ~O
e2(4) ([4x, 4x, xx]~
[ 4x, 4x, xx] = va e--~-~ 2 [--23"7782Y2 + 13"3576Y22
2 t~.ku/ E ®. ~
The quantities [ai, flj, kl] are defined in equation (3.29d) of Srinivasan[2]. Given below are the expressions for these quantities for the alkali halides with the sodium chloride structure in the negative ion polarisable shell models considered here.
e___z2_ [4x, 4x, yy] = v"2 [ 11.8991 Yz + 3.7932 Yz2 +A'--B'] [2x, 4x, yy]
= e__2. 2 Va2 [3"7932Z2Y2]
[kx, vy, xy] = [kx, vx, yy]--[hy, vy]
(4.7)
5. RESULTS AND DISCUSSION
(a) Strain derivatives of co Table 3 gives the values of wto~ Wt°~ and "" 1 1 , 1 1 ' "" 1 1 , 2 2 WCO~ for the different alkali halides on the 12,12 various models. We observe the following general trends. (i) In all the alkali halides, the value of WCO~ is about an order of magnitude II,II larger than the values of W c°~ and "" 11,22 W~O~ All these alkali halides have the 12,12" sodium chloride structure. The sign of WCO~ is negative in all these crystals. 11,22 (ii) When the overlap interaction is extended to the second neighbours in Model 2, the value UA,, I ,1~t0~ I'V 1 1 , 1 1 decreases in all these crystals. (iii) In NaCl and KCI, the use ofnoncentral first neighbour interaction in Model 3 as opposed to Central first neighbour interaction in Model 2 causes a further reduction in the value of W~°~ ll,n" (iv) Comparison of the corresponding sodium and potassium halides reveals a decrease in W ~°) as sodium is replaced "" 1 1 , 1 1 by potassium.
DIELECTRIC CONSTANTS OF SOME ALKALI HALIDES
(v) T h e r e is a t e n d e n c y for the value o f WOO) to go up as we go f r o m the 11,11 chloride to the iodide. (vi) Srinivasan[13] had previously calculated the w~0~ values for the alkaline " i3,kl earth fluorides, CaF2, BaF2 and SrF2 using A x e ' s shell model. But his results have to be c o r r e c t e d for the e r r o r in the expressions for Ptk/,,,,,j. It is instructive to c o m p a r e the vaIues o f WC0~ O,kl in the alkaline earth fluorides with the values in the rock salt structure. In the fluorite structure the cation is s u r r o u n d e d by eight anions at the corners o f a cube while in the rocksalt structure each cation is s u r r o u n d e d by
1085
six anions at the corners of an octahedron. T a b l e 4 gives the values o f W t°~ 11,11' WtO~ and w t°~ for the alkaline earth 11,22 "" 1 2 , 1 2 fluorides so that they may be c o m p a r e d with the values for the alkali halides in T a b l e 3. In the alkaline earth fluorides the magnitudes of ""W c°~ ~o) W ~°~ are nearly of 1 1 , 1 1 ' W 1 1 , 2 2 and " ' 1 2 , 1 2 the same order unlike in the alkali halides having N a C ! structure where wco~ is one '' 11,11 o r d e r of magnitude larger than W ~°) and "" 1 1 , 2 2 WOO) This difference is clearly due to the 12,1'2- " difference o f the a r r a n g e m e n t of the near neighbours in the two structures. (vii) T h e pressure derivatives of the static dielectric constant can be c o m p u t e d from the
Table 3. Strain derivatives and the pressure derivatives o f the static dielectric constant o f the alkali halides Crystal
W,°,.1,
W°L22
W12,12 °
LiF
52"4
-- 5"3
1"2
NaF
NaCI
NaBr
11~o(a~Jap)X 1011dyne/cm 2 Calculated Observed -- 0"27
-- 0"48~ taJ -0"50J --0"38 ,el
72-3
--6-0
- - 1-5
--0.74
--0.50
,a~
1
-6"8
--2"1
--0"59
--0"53
te~
2
64.3
--5.2
--0.3
-
59"9
--5.1
-0.3
- - 1.03
55"0
-- 4-2
- - 0" 1
-- 0"95
1"12
I - - 1 - 0 1 ; - - 1 - 0 6 ' ~ 1 ; - - 0 " 9 2 °:1
--5-5
0.3
- - 1.54
-- 5.4
0-3
- - 1-56
Nal
71-8 67.3
--5-0 --4.9
0.71 0"84
- - 1.77 - - 1.65
_ 1.5 p c j . _ '
KCI
45-8 38"6
--2-9 --4.1
--0"6 --0-8
- 1.49 - - 1.18
-- 1-06;-- l'09tal; -
28"7
- - 3"8
- - 0"6
- - 0"83
48.1
- - 4.1
-- 0.45
- - 1-66
43"7
-- 4.2
- - 1.0
-- 1.50
- - 1-29~a~; -
1.19 ~c~
1 2
1.6yal
1 2
1"00 t~
1 2 3
- - 1" 1 8 ; - - l ' 2 0 t a ~ ; -
1' 15 ~
1
2
58.3
--4-6
-0"05
-2"47
- - 1 " 2 6 ; - - 1 " 0 9 ; - - I ' I Y "~
1
52-5
-- 4-7
-- 0-20
-- 2-25
- - 1-37 ~)
2
C~Ref. [4]. Cb~Ref.[13]. 'CIRef. [3]. CaIGIBBSD. F. and JARMAN M., Phil. Mag. 7,663 (1962).
J P C S Vol. 33 No. 5 - - 1
2 3
74.3
KI
3
60"8
75-3
KBr
Model
1086
R.
SRINIVASAN
and
eo Op
1 W(o~ "" 11,11 +2W
~o
SRINIVASAN
for M o d e l I in NaCI and using s~ = 23-8 × 10 - u and s~2 = - 5 " 1 7 × 10 -11 cm2/dyne and taking T to be 100 kg/cm z we obtain
relation, 1 0Co_
K.
•
Caa + 2C12
A e o = 1"58 × 10 -z
T h e c o m p u t e d values for the various alkali halides are given in T a b l e 3 and for the alkaline earth fluorides in T a b l e 4. T h e m e a s u r e d values o f these quantities are also given in these tables. In general the calculated values follow the trend o f the m e a s u r e d values. In some cases the agreement between calculated and m e a s u r e d values is good and in some o t h e r cases like KI and L i F the agreement is bad. W e will not speculate on the reasons for the disagreement at this stage o f the work. Finally we would like to consider briefly the possibility o f measuring the strain derivatives o f the static dielectric constant. One must subject plates having different crystallographic orientations to uniaxial stresses and measure the dielectric constant. In uniaxial stress m e a s u r e m e n t one cannot reach as high a level o f stress as in hydrostatic stress measurements for fear o f plastically deforming the crystal. O n e could go up to a m a x i m u m stress level of 1 0 0 k g / c m z. Considering a plate o f NaCI cut normal to the [100] direction and subjecting it to a uniaxial stress T along the [100] direction the change in dielectric constant, AE0, is given by (0) (0) AEO = [ W11,11Sll + 2 W11,22Slz ]T
w h e r e s~ are the elastic compliance coefficients. T a k i n g the values of W ~°~ and WCO~ "" 11,I1 ""
II,22
or Aeo/eo ~ 2.8 × 10-3. With the present day techniques it is possible to determine Aeo/Eo to an a c c u r a c y o f 10-L So o n e could determine the strain derivatives o f the dielectric constants to within a few per cent. (b) Strain derivatives o f e~ Mueller[14] has w o r k e d out the t h e o r y o f the strain optical constants in the alkali halides. In his notation Pz, Px andp~, stand for P u , u , Pn,22 and P12,12 respectively. Mueller has s h o w n that there are f o u r distinct contributions to the photoelastic constants. T h e y are as follows: (i) T h e contribution arising from the change in the density o f the crystal. This contribution is present only for pz and p:~ and is indicated by p a and pz a respectively. (ii) T h e L o r e n t z force at an ion b e c o m e s anisotropic on straining a crystal. This anisot r o p y makes contributions p ~ , p ~ and p~, to Pz, Px and p~, respectively. (iii) Considering the ions to have oscillating electrons, the coulomb f o r c e at the site o f the ion b e c o m e s anisotropic and will t h e r e f o r e cause an anisotropic shift in the f r e q u e n c y o f the electron. This contribution to p~, px and p~, is d e n o t e d b y p ~ , px ~ and p~, respectively. (iv) T h e .above three contributions can be e v a l u a t e d f r o m the structure o f the alkali
Table 4. Strain derivatives and the pressure derivatives of the static dielectric constant o f the Alkaline Earth Fluorides Crystal
o 11 W.,
o Wnm
Won, ,2
ll¢o(a¢olap) x 10n dyne/cm2 Calculated Observed
Model
CaF~ 24"66tb) 22.13 17"63 --0-392 --0"349; --0.348Ca~ SrF2 24.10tb~ 19-29 14-73 -- 0.372 -BaFz 23"48tm 15.60 14-54 --0.49 --0.424;--0.428 ~a~ -~a~ Ref. [4]. ~b~ Srinivasan, Ref. [13]. The values of W~ 11etc. given in the above paper have been corrected for the error in the expression -
Pkhmn.
-
DIELECTRIC
CONSTANTS OF SOME ALKALI HALIDES
halides. H o w e v e r a calculation of the photoelastic constants taking only these three contributions still leads to a serious discrepancy with experiment. Mueller therefore concluded that there must be a fourth contribution which arises because the polarisability of the ions may be directly changed due to strain due to the change in the overlap of the neighbours. This contribution is called the strain polarisability effect and is denoted by pz A, px a and pa,. From the measured photoelastic constants, Mueller subtracted the first three contributions and obtained the magniA tudes ofpz a, px A a n d Px," Szigeti and Aggarwal[15] starting from the Lorentz-Lorenz relation for the alkali halides have computed again the first two contributions pa and pL. They then calculated the difference between the measured constants and the sum of the two contributions pa and pL. The difference obviously arises due to the total change in polarisability of the ion. If we picture the ions to be oscillators then this total change arises due to anisotropy in the coulomb force pC and the anisotropy in the short range force pA. H o w e v e r Szigeti and Aggarwal did not adopt any such model for the ion. They calculated empirically the change in the polarisability of the ions with changes in first and second neighbour distances so that a fit could be obtained to the measured photoelastic constants. The reason why the present work is carried out is the following. On this simple model one could obtain the anisotropic change in the overlap forces. The strain polarisability effect can therefore be calculated on this model in which the change in polarisability arises only through the parameter 'a' in the overlap interaction. We do not envisage any deformation of the shell here. At least such a calculation will give us an independent evaluation of the strain polarisability effect. In the actual lattice the strain polarisability will also have a contribution from the deformatiofi of the charge distribution from a spherical shape. The present calculation may yield a clue as to
1087
the importance of the latter effect. First we should show that the present expressions for P~J,kZ agree with Mueller's as regards the first three contributions. This is done in the appendix IA. We calculated P~J,k~for the alkali halides on the various models. We found that the calculated values ofpu, k~ were sensibly the same for any crystal on the different models. The values only differed in the third decimal place. So unlike the static dielectric constant, the strain derivatives of the high frequency dielectric constant are not much sensitive to the details of the overlap interaction. In view of this fact, we have given the values of Pu, ut in Table 5 for all the alkali halides except LiF on Model 2. F o r LiF only Model 3 is applicable and the values of P~,kl quoted are on this model. In the same table are quoted the values of Szigeti and Aggarwal which represent the sum of the Lorentz-Lorenz and density contributions. Szigeti and Aggarwal claim that the addition of the coulomb contribution only increases the discrepancy between theory and experiment. One striking fact emerges on studying the Table 5. The inclusion of the strain polarisability effect through the parameter 'a' in the present model brings about a large reduction in the value of P~I. This reduces the discrepancy between the calculated and measured values of P11. For example in KCI, the inclusion of the strain polarisability on the simple model described here reduces Pll from the value 0.61 given by Szigeti and Agarwal to 0.251 The measured value is 0.19. H o w e v e r the present calculation does not bring about any such drastic .change in the values of Pn,z2 and PI2,1z. The reason for this becomes apparent on looking at the expressions for [ax, fix, xx] and [ax, fix, yy]. While the anharmonic third order parameter enters the former expressions it does not enter the latter. This means that on applying a uniaxial stress along [ 100], the short range anharmonic interactions of the ion with its two first neighbours along [ 100] changes the polarisability of the ion in this direction.
1088
R. S R I N I V A S A N and K. S R I N I V A S A N
Table 5. Strain optical constants of the alkali halides Pn,n Crystal
Model
LiF NaF NaCI NaBr Nal KCI KBr KI
3 2 2 2 2 2 2 2
P11,22 P12,12 Present work
0.247 0-213 0-241 0.238 0.239 0-248 0-253 0.241
0-243 0-255 0-273 0.267 0.259 0.288 0.288 0.280
Pn,n Pn,m P12,12 Aggarwal and Szigeti*
Pn, n
0-38
0-30
0-052
0.02
0-13
--0-045
0-51
0-28
0.033
0.11
0.153
-0.010
0.61 0.62 0.61
0.22 0.23 0-24
--0.03 --0.016 0.010
0.19 0.216 0.210
0.14 0.168 0-169
--0-03 --0.024 --
0-005 0-006 0-030 0.029 0-044 0-040 0.042 0-047
P11.22 P12.12 Experiment*
*Ref. [15].
This result will be true even if we take a model in which both ions are polafisable. So we do not think that a significant improvement in the agreement of P12 or P44 can be brought about merely by including the polarisation of the cation also. On the other hand if a compression along [100] brings about a distortion of the shell, then one would also obtain a change in the polarisability in the [010] direction. Thus deformation of the ion will be the major effect deciding the strain polarisability contribution to Pn,2z. On the other hand the anharmonic third order parameter in the nearest neighbour overlap interaction is of importance in the strain polarisability contribution to Pn, 11. These conclusions we believe, should be valid in the alkali halide crystals with rock salt structure. Acknowledgement--The zuthors are thankful to Miss G. Lakshmi for her help in the computations. REFERENCES 1. S C H R O D E R U., Solid State Commun. 4, 347 (1966). 2. S R I N I V A S A N R., Phys. Rev. 165, 1041 (1968). 3. L O W N D E S R. P. and M A R T I N D. H., Proc. R. Soc. Lond. A316, 351 (1970). 4. J O N E S B. W.,Phil. Mag. 16, 1085 (1967). 5. C O W L E ' ~ , R. A., C O C H R A N W. and BROCKH O U S E B. N. and W O O D S A. D., Phys. Rev. 131, 1030 (1963). 6. TOSI M. P., Solid State Physics, Vol. 16, p. 1, Academic Press, New York (1964). 7. LEWIS J. F.,Phys. Rev. 161,877 (1967). 8. N O R W O O D M. H. and B R I S C O E C. V., Phys. Rev. 112, 45 (1968). 9. L E I B F R I E D G. and L U D W I G W., Solid State
10. 1 I. 12. 13. 14. 15.
Physics, Vol. 12, p. 276, Academic Press, New York (1961). L O W N D E S R. P. and M A R T I N D. H., Proc. R. Soc. Lond. 308, 473 (1969). G H A T E P. B.,Phys. Rev. 139, A1666 (1965). D O L L I N G G., S M I T H H. G., N I C K L O W L. M., V I J A Y A R A G H A V A N P. R, and W I L K I N S O N M. K., Phys. Rev. 168, 970 (1968). S R I N I V A S A N R.,Phys. Rev. 165, 1054 (1968). M U E L L E R H., Phys. Rev.47,947 (1935). A G G A R W A L K. G. and S Z I G E T I , J. Phys. (Part C ) 3 , 1097(1970).
APPENDIX IA Mueller[14] gives the following expressions for the strain optical constants. In his notation:
Pn.ll = Pz; Pn.22 = px and P12.12 = -'-fWritingpl (i = z o r x , o r x ' ) as
Pt=P~e+PlL+p~C+pi A
(IA.I)
as mentioned in the text we have p d=
~= + 21) unless i = x', p~, = 0 3(~,--
pl L = -- ~ sjk~NjNkRIRe/[ E
NkRk]
2
Jk
p c=_~j
NjRj2 fJ ~" N,Zks~** / [~ NkRk] 2
(IA.2)
pt a is the contribution arising from the anharmonic short range interaction and distortion of the electronic cloud of the ion on straining the lattice. Here Nk = = Zk = R~ =
number of ions per cnP of the k th type. I/va in our notation. is the charge of the ion k, 4~rNoc9/3 where 09 is the polarisability and No is the Avogadro number. f j = is the oscillator strength o f t h e j th ion.
sj~ are lattice sums whose values are summarised in
DIELECTRIC
CONSTANTS
T a b l e I A . j = 1 refers to the cation a n d j = 2 to the anion.
Table I A . The lattice sums sjkt f o r the sodium chloride lattice Sn,z = s22.z = (L~a:) st2,z = (LlalB) Sll.x = s22,z = (Lla~~) s12,x = (L~) s=,x, = sz2,~, = (L~4a) s12,.~, = (L4a~)
= --0"0630 = -- 1"9566 = 0"0315 = 0"9783 = - 0"6037 = 1"2901
T h e notation o f A g g a r w a l a n d Szigeti is indicated in the brackets. In o u r model
OF SOME
ALKALI
HALIDES
a l = 0, RI = 0; a2 =
1089
Y22v" [K2+A'+2B'I
f j = 0, w h e n j = 1 a n d f 2 = -- Y2
(IA.3)
(Y2 is negative as it is the charge on the shell). S u b s t i t u t i o n for the v a r i o u s quantities in (IA.2) we c a n get the e x p r e s s i o n s for p j d + p L + p ~ a n d h e n c e the c o n t r i b u t i o n s from t h e s e t e r m s to Pu.k~- In o u r model PtJ, kt "~
E2
Substituting for various quantities from (4.2) a n d (4.6) a n d (4.7) a n d omitting t h e t e r m s u a n d b in [4i, 4j, kl] ( t h e y r e p r e s e n t t h e effect o f s h o r t range a n h a r m o n i c interaction), we obtain e x p r e s s i o n s identical with t h o s e derived f r o m M u e l l e r ' s theory.