- Email: [email protected]
Unified study of lattice mechanics of alkali cyanides
Physica B 150 (1988) 385-396 North-Holland. Amsterdam
UNIFIED STUDY OF LATTICE MECHANICS OF ALKALI CYANIDES R.K. SINGH and N.K. GAUR School of Physics, Bhopal University, MAPCOST, Bhopal-462 016, India
Bhopal-462026
Received 23 March 1987 Revised manuscript received 28 September
and Centre for Science
and Technology
Development
Studies,
1987
An extended three-body force shell model (ETSM) for alkali cyanides has been developed by including the effects of coupling between the translational and rotational motions of cyanide (CN-) molecules in the framework of original TSM devised by Singh and Sanyal [Physica 132B (1985) 201). This ETSM has been applied to describe the static, dynamic. anharmonic elastic and photoelastic properties of the alkali cyanide crystals. This model has satisfactorily reproduced their experimental phonon dispersion curves and cohesive energy values. However, the agreements between the experimental and our theoretical values on the third order elastic constants are not so good. These agreements have been found to improve significantly with the exclusion of three-body interaction (TBI) effects, which play a vital role in the prediction of harmonic and anharmonic elastic properties of alkali halides. This might be so due to the overestimated effects of TBl from the Cauchy violations (C,,-C,,) which are much larger for alkali cyanides than those for alkali halides. It has been concluded that the ETSM, which has reproduced the lattice statics and dynamics of alkali cyanides so well, is incapable to explain their anharmonic elastic and thermal properties. However, this situation is expected to improve by using a more appropriate method for estimating TBI contributions and incorporating the effects of covalent and anharmonic interactions.
1. Introduction The alkali cyanides are polymorphic molecular crystals of the type M’XY- in which the cyanide (CN-) molecules attain the required effective spherical symmetry by either rotation [l] or random distribution of their orientations [2] in the unit cell along [RIO] and [ill] symmetry directions. These orientation distributions are different for various crystals and also they vary with temperature [3] within the cubic phase. At room temperature and pressure, the alkali cyanides (LiCN, NaCN, KCN and RbCN) are plastic crystals with the rocksalt structure (phase 1, spacegroup Fm3m) [4]. These crystals undergo a first-order phase transition at low temperatures to orthorhombic (for NaCN and KCN) and to monoclinic for (RbCN) structures [5-71. The additional striking features exhibited by the alkali cyanides seem to be due to strong Cauchy violations (C,,-C,,) whose magnitudes are several times larger with reverse sign as compared to those exhibited by the alkali halides (81.
During this decade, the lattice statics and dynamics of alkali cyanides have been the subject of many theoretical [l, 3,9-171 and experimental [ 17-241 investigations. The phonon dispersion curves (PDCs) of NaCN [ 181, KCN [18], RbCN [17] and CsCN [23] have been measured by making use of the most powerful technique of inelastic neutron scattering. There have been several attempts to explain the unusual features of alkali cyanides using phenomenological and microscopic approaches. The shell model calculation [lo] of the PDCs of NaCN and KCN has considered the cyanide (CN-) molecules to be rigid with dumb-bell shape and rotational degree of freedom. This model, however, failed to describe the gross features exhibited by PDCs of both NaCN and KCN. The reproduction of their PDCs obtained by Jex and Maetz [14] is although satisfactory, but they have employed the fitting procedure. Michel and Naudts [ll, 15,251 have employed molecular dynamics simulation to predict the PDCs of these crystals by incorporating the ef-
03784363/88/ $03.50 0 Elsevier Science Publishers B .V. (North-Holland Physics Publishing Division)
386
R.K.
Singh and N. K. Gaur
I Lattice mechanics
fects of coupling of the translational modes (phonons) to the re-orientations of the dumbbell shaped (CN-) molecules. The shell model calculations [15] have although explained the acoustic branches of PDCs satisfactorily, but it has surprisingly failed to predict the optical branches of alkali cyanides. Moreover, it has exhibited the unusual feature of cross-over of longitudinal optic (LO) and transverse optic (TO) branches, which is not consistent with the experimental results. Later on, the PDCs of RbCN [17] have been predicted by a model which combines the shell model and the rotational-translation coupling [14] introduced due to the dumbbell character of cyanide (CN-) molecules. This model has satisfactorily explained the acoustic branches with spoiled agreement for the TO and LO branches along (qO0) and (qq0) directions, respectively. Also, these models have, generally, reproduced the PDCs of NaCN, KCN, and RbCN but they have failed to explain their static properties. However, the analysis from them has not been carried out to predict their interesting anharmonic elastic properties [21] which differ strongly from those exhibited by the alkali halides [26] and divalent metal oxides [28]. Motivated with these remarks, we thought it pertinent to describe their lattice dynamic and static properties simultaneously by means of introducing modifications in the framework of the three-body force shell model (TSM) [26], which has been adequately successful in predicting the static, dynamic, dielectric, optic and anharmonic elastic properties of various ionic, covalent and semiconducting materials [26-281. Its wide ranging applicability is chiefly due to the realistic and appealing approach of representing the long range three body interactions [28], which are essentially required to explain the Cauchy violations in alkali halides [29]. Moreover, the framework of this TSM is appropriate for a unified description of the latice static and dynamic properties of ionic crystals [28,30]. The primary aim of the present paper is to introduce the translational-rotational coupling effects in the framework of TSM [28] and use the resultant model, called the extended three-body force shell model (ETSM), for a comprehensive
of alkali cyanides
and unified description of lattice statics and dynamics of alkali cyanide crystals. The appropriateness of the present ETSM may also be emphasised from the fact that there is a large ionic size difference between M’ and CN- ions of cyanide crystals and consequently there should be a pronounced effect of three-body interactions (TBI) [26,29] as the Cauchy discrepancy in these crystals is several times larger than that in alkali halides. We have applied the present ETSM to all the members of the alkali cyanide family of rocksalt structure for investigating their static, dynamic, anharmonic elastic and photoelastic properties. Our results have predicted the experimental PDCs quite satisfactorily. However, many of the tabulated values for the third-order elastic constants and pressure and temperature derivatives of second-order elastic constants differ considerably from their available experimental values. These discrepancies have been found to be largly due to the overestimated effects of the three-body interactions due to very strong Cauchy violations (C,?C,,) which have opposite sign as compared to those for alkali halides. The essential theory of the present model is given in section 2. Its applications are presented in section 3. A general discussion of the results along with the conclusion is given in the last section.
2. Theory of ETSM The interaction expressed as [28]
potential
rk +
b c kk’
Pkk’
exp
+
rk’
used in ETSM
-
rkk’ +
P
>
is
c#JTR . (1)
The first two terms represent the long-range Coulomb and three-body interactions (TBI) [28]. The third and fourth terms are the vdW contri-
R. K. Singh and N. K. Gaur
I Lattice mechanics of alkali qanides
butions due to dipole-dipole (d-d) and dipolequadrupole (d-q) interactions [31]. The fifth term is the Hafemeister and Flygare (HF) type short-range (SR) overlap repulsion [28] extended upto the second neighbor ions with Pkk8 as the Pauling coefficients PLk’ =
1+ (.2,/n,)
+ (Z/(./n,.).
(2)
Here, Z,(Z,,) and nk(nk,) are the valence and number of electrons in the outermost orbit of cation (anion). b and p are the hardness and range parameters determined from the equilibrium condition:
and they are determined from the method suggested by Michel and Naudts [ 11,251. Here, A 12 and B,, are the short-range force constants (equal to C and D as referred in [15]) between the nearest neighbours (nn). The dynamical matrix corresponding to the ETSM can be derived from eq. (1) and written as [28]: D(q) = (R(q) +Z,C’(q)z,) -(R(q)
+ Z,D’(q)Y,)
x (R’(q) + Y,$‘(q)Y,J-‘(K + Y,C’( 0,)
W(r) dr
o
(3)
T=ro=
and the bulk modulus: B = (9&J-’
[
d244r) 1r=ro,
dr2
The last term, 4’9 is the contribution due to the translational-rotational (TR) coupling effects [ll, 251. The second-order elastic constants (SOEC) derived from eq. (1) for the alkali cyanides can be expressed as cij = CYM + c;”
(ij = 11.12.44)
(
(5)
where the first and second terms represent the contributions from TSM [28] and TR coupling [11,25] given by TR
Cl1 = -8X,,A,2Ir,
2
CT. = 8X,,Ai2/3r, CT: = -2X,,B:,Ir,
,
(64
,
(64
Here, Xii are the elements of the rotation-rotation response matrix [ll]
x=
I
+ R=(q) (8)
+ DTR( 4) 9
where the various terms have their usual meanings [15,28]. The last term is contributed by the TR coupling effects and has non-zero elements only in alkali ion submatrix Dy( q), which is due to the approximation that the cyanide molecules are treated as elastic dipoles [ll]. Its relevant expression is given by [15]
D:R(q) 0 *) DTRW =( o
3
(9)
3
(10)
with D?(q)
= -V(#o+(q)
where V(q) is the rotational-vibrational matrix defined as [15]
V(q)= ’
(6b) .
387
X
coupling
&
A,,&
--A,,&
42%
Ad,
A
\-2A,,S,
125
0
42Sz
42Sx
42%
0
0
BJ,
B,,S,
0
,
(11) where
(7)
Si = sin qiro ,
i = x, y, 2
(12)
with qi = ( q,, q, , q,) as the phonon wavevector,
R. K. Singh and N. K. Gaur
388
/ Lattice mechanics o.f alkali cyanides
rO the equilibrium interionic distance, and m, as the alkali ion mass. The formalism of ETSM derived from the present crystal potential differs slightly from that of TSM [28] due to the inclusion of TR coupling [31] effects. There are eight potential parameters (four parameters (b, p, f(r,) and r,f’(rO)) and four shell model parameters (d,, d2, Y, and Y2) in the ETSM framework. Here d, and d2 are the distortion polarizabilities and Y, and Y, are the shell charge parameters. Using the expressions corresponding to ETSM, the values of these model parameters
have been evaluated from the experimental data on elastic constants (C,, . C,,). bulk modulus lattice constant (2r,), electronic (BL polarizabilities (cy_ , a_) and zone-centre vibration frequencies ( vLo. +o). The higher-order derivatives of f(r) have been obtained from the analytical expression. f(r,,) = f;, exp(- r,,/p). 3s done by others [28]. The values of input data and vdW coefficients are presented, respectively, in tables I and II and used to compute the model parameters listed in table III. The ETSM has been applied for the following descriptions.
Table I Input data for alkali cyanide crystals Properties
LiCN
Ref.
NaCN
Ref.
KCN
Ref.
RbCN
Ref.
r,(A) C,,(lO” dyn.cm-*) C,Z(lO1’ dyn.cm-*) C,,( 10” dyn cm:‘) GTHz) a+@‘) cI_(A’) lO e,
2.55 3.15 1.75 0.02 4.50 0.03 1.80 8.95 2.18
a a a a b c d e f
2.95 2.53 1.44 0.03 4.38 0.28 1.80 7.55 2.11
g h h h i
3.26 1.92 1.20 0.14 4.11 1.30 1.80 5.72 1.99
! h h i c d I k
3.42 1.75 1.04 0.17 4.23 2.10 1.80 4.50 1.77
g m m
Zl j k
Y d d e f
a - Extrapolated values obtained from the plot of r,, versus (r + + r_) and C,, from C,, versus r0 for alkali cyanides. b - Extrapolated values obtained from the plot of r,, versus vTo for alkali cyanidrs. c - [38], d - [17], e - Extrapolated values obtained from r,, versus e0 for alkali cyanides. f - Obtained from the Lorentz-Lorenz relation. g- [l], h-[39], i- [18,20], j - [40]. k- [23]. I - [41]. m- [42]
Table II Van der Waals coefficients for alkali cyanide crystals (C,,. and d,,. are in the units of 10mhoerg. cm6 and 1Om76 erg. cm’. respectively)
Table III Model parameters
for alkali cyanide crystals
Properties
LiCN
NaCN
KCN
RbCN
Parameters
LiCN
cc*
3.43 0.14 105.25 1.55 0.03 61.56 117.81 34.16
22.86 5.89 105.25 10.46 1.49 61.56 251.19 89.52
79.79 63.22 105.25 39.56 24.26 61.56 678.41 277.49
137.28 181.60 105.25 72.36 83.76 61.56 1164.52 502.86
b(10mL2 erg)
0.270 0.232 -0.005 0.049 0.020 0.537 -9.354 -0.857
c** ck ii d,, d d::,
C D
P(A)
f(r,J
r,,f’(rl,) d, d: y, Y,
NaCN 0.391 0.372 -0.009 0.073 0.094 0.318 0.664 1.253
KCN
RbCN
0.389 0.345 --0.009 0.089 0.197 0.287 - I .258 - 1.372
0.364 0.350 -0.007 0.069 0.126 0.330 -2.970 -0.970
R.K.
Singh and N. K. Gaur
I Lattice mechanics of alkali cyanides
3. Application of ETSM
III, we have computed the phonon dispersion curves (PDCs) for LiCN, NaCN, KCN and RbCN along the symmetry directions and displayed them in figs. l-4. We have compared our results with the neutron scattering data [16,18201 available for NaCN, KCN and RbCN. An inspection of these figures reveals that the overall agreement between experimental and our theoretical results is quite satisfactory. Both the optic and acoustic branches have shown reasonably good agreement with their experimental data. This is not surprising as the present model is capable to take proper account of both the elastic (Cauchy violation) as well as the dielectric (polarizabilities) properties on which these branches depend. In an earlier shell model calculation of PDCs
We have explored the extent to which ETSM is capable to explain simultaneously both the lattice dynamics and the statics of alkali cyanides. In the evaluation of the model parameters, we have used the values of 6C,, ( = - 1.40, -1.35, -1.30 and -1.24) and SC,, (= -0.64, -0.44, -0.20 and -0.19) (in units of 10” dyn *cm-‘) for LiCN, NaCN, KCN and RbCN, respectively, along the lines suggested by Strauch et al. [15]. The method of these computations and the results are presented below. 3.1. Lattice dynamic properties Using the model parameters I
I
reported
in table
I
I
I
(cl001
(qqo) LiCN(300K)
REDUCED
389
WAVE
VECTOR(q)
-
Fig. 1. Phonon dispersion curves of LiCN obtained from ETSM.
R. K. Singh and N. K. Gaur
1 Lattice mechanics of alkali cyanides 1
I
((4 00)
I
(990)
1
I
1
(wl)
-le
NaCN (3wK) /l./L____ .
0
0 0 0 _:;
0
00
1 0
REDUCED Fig. 2. Phonon
dispersion
curves
of NaCN
obtained
0.5
0.0
from
of alkali cyanides, Bill et al. [lo] have introduced the rotational degree of (CN - ) molecule. phenomenologically. This has although given satisfactory prediction of PDCs, but failed to explain the Cauchy discrepancy as they themselves have concluded that the librational coupling to the transverse branches alone cannot explain the Cauchy violation exhibited by C,z and C,, in alkali cyanides. Jex and Maetz [14] have also explained the PDCs of alkali cyanides (NaCN and KCN) using a shell model with second neighbour SR forces besides the translational-rotational and rotational-rotational part of the dynamical matrix. This model has although reproduced a satisfactory agreement with experiment for KCN, but it
WAVE
ETSM.
VECTOR
The circles
(‘2 )
represent
the experimental
data
(181.
has a large number of model parameters and gives poor prediction of the static properties. The PDCs of NaCN and KCN have recently been analysed by Strauch et al. [15] by incorporating the microscopic translational-rotational coupling effect [ll, 151 in the framework of the rigid shell model (RSM). This model has given a good description of their PDCs. However, it has revealed an unusual feature of cross-over of LO and TO branches along (qO0) direction. This limitation might be due to the constraint on taking cation and anion shell charges equal and the exclusion of the van der Waals and three body interaction (TBI) effects. Recently, Ehrhardt et al. [17] have predicted the PDCs of RbCN using a shell model modified
R. K. Singh and N. K. Gaur I Lattice mechanics of alkali cyanides
(4 00)
(9 40) KCN
(‘JOOK)
6
0
0
1.0
REDUCED
WAVE
VECTOR
(9)
0.5
w
Fie. 3. Phonon dispersion curves of KCN obtained from ETSM. The circles and open triangles are the experimental data [g. 201; full triangles are from [19].
to include the TR coupling effects. This model has reproduced the PDCs fairly well at the cost of static properties. It is interesting to note that our PDCs for LiCN have followed the trends more or less similar to those revealed by the measured PDCs available for NaCN [18], KCN [18-1201 and RbCN [17]. Our decisive comments on the results on LiCN will be restricted until the report of their measured data on them. A critical c.omparison shows that our results on LA and TA modes are almost similar to their experimental values. Our results on optical modes are in closer agreement with their experimental data as compared to those obtained by earlier workers [14-171. Also, our results have not shown any unusual cross-over between
LO and TO branches as revealed by the results of Strauch et al. [15]. 3.2. Lattice static properties In order to assess the relative merit of the present potential and the logarithmic potential (LP) of Thakur [12], we have calculated the compressibility ( p), molecular force constant (f), infrared absorption frequency (I+,), Debye temperature (On), Griineisen parameter ( y ), ratio of volume expansion coefficient (cz,) to specific heat (cV) at constant volume and the Moelwyn Hughes constant (F1 ) which are directly derived from the cohesive energy, 4(r), expressed by eq. (1). Their expressions [12] are given below for ready reference.
R. K. Singh and N. K. Gaur
I Lattice mechanics of alkali cyanides
Rb CN 13GOK)
Reduced Fig. 4. Phonon
dispersion
The compressibility
curves
of RbCN
obtained
Wave Vector from
is well known to be given
ETSM.
(q) The circles
and triangles
are the experimental
data
116)
Debye temperature
by
(16)
p=- 3Kr,, f in terms of molecular
(13) force constants
with h and k as the Planck and Boltzmann constants, respectively. The values of the Griineisen parameter (y) have been calculated from the relation rlr
with 4:,“.(r) as the short-range nearest neighbour (k # k’) part of 4(r) given by the last three terms in eq. (1). This force constant (f) gives the infrared absorption frequency
’ = -6
L4”‘(r) I 4”(r)
r=I,,
.
We have calculated the ratio of the volume expansion coefficient (+) to the volume specific heat (c,) from its well known expression
(15) with the knowledge of the reduced mass (CL) of the cyanide crystals. This frequency gives us the
(17)
(18) The Moelwyn Hughes constant (F,) calculated from the relation [12]
has been
393
R. K. Singh and N. K. Gaur I Lattice mechanics of alkali cyanides
4 = 1-
&
(19)
M"(~L,,
which was earlier known as the Rao-Keer constant . The values of the cohesive energy listed in table IV show that our values are in better agreement with their experimental data than those revealed by other models [l, 121. It is interesting to note that our results from the ETSM for compressibility (j3) and infrared absorption frequency (~~‘0)are much closer to the experimental data than those obtained by Thakur [12] from the logarithmic potential. Also, our values of the Griineisen parameter (y) seem to be more realistic as they are closer in magnitude to the measured values for alkali halides [26]. The other thermodynamic quantities listed in table IV could not be compared due to the lack of experimental data on them. 3.3. Anharmonic
ties, like thermal expansion, thermoelastic constants and thermal conductivity. The expressions for these elastic constants and pressure derivatives of second order elastic constants (SOECs) have been derived by us corresponding to the three-body lattice energy given by eq. (1) and they have been found to be similar to those derived by Garg et al. [32] for the rocksalt structure crystals. The short-range parameters (A;, B;, C,, and D,; i = 1,2) involved in our expressions differ from those of Garg et al. [32] in respect of the additional contributions from the van der Waals (d-d) and (d-q) interactions. The parameters involved in these expressions are redefined as
A, =
L[d2~r:‘r’]r-,k,, (20)
elastic properties
We have studied the anharmonic elastic properties by calculating third and fourth-order elastic constants (TOEC and FOEC) as they provide physical insight into the nature of the binding forces between the constituents of a crystal and are strongly related to other anharmonic properTable IV Cohesive and thermodynamic
(21) B
=
2
1
WzkR(r) + W?bW
_L r [
Ci = AfIBi
dr ,
dr
(22)
r=krkk.
and Di = AjlBf,
(23)
properties of alkali cyanide crystals Y
cr,,lc,. (10’ J)
F,
Ref.
470
2.32
4.33
5.68
Present Expt. Others
6.54
314
1.69
5.70
4.38
1.22’
585’
2.46’
-
5.91’
Present Expt. Others
5.47 4.118 1.03’
262 197’ 474’
1.86 2.68’
5.73 -
4.74 6.34’
Present Expt. Others
271 203’ 450’
1.95 2.58’
5.74 -
4.91 6.17’
Present Expt. Others
P
f
;kJ . moII’)
(lo-“dyn.cm-‘)
(lOJdyn.cm-‘)
&-Iz)
LiCN
-855 - 849” -864b
4.45 4.51’
3.45 3.40’
9.80 -
NaCN
-729 -738” -755’, -743’
5.28 5.56’ 3.61’
3.41 3.18’ 4.91’
5.63 4.23h 0.94’
KCN
-689 -674” -6824 -678’
6.39 6.95’ 5.08’
3.06 2.82’ 3.85’
RbCN
-653 -646” -648: -63Sd
7.01 7.83’ 5.90’
2.91 2.62’ 3.47’
a - (22). b - [l]. c - [12]. d - [9]. e-Value calculated from fl = 3/(C,, + 2C,?). f-Obtained g - [la, 201. h - [17]. i - Obtained from eq. (16) using experimental data.
from eq. 13 using experimental data.
394
R. K. Singh and N. K. Gaur
where the SR potential
energies are
4:,“,(r) = b/3,,. exp ” $- ‘;! -d
,,,r,-,“.
“‘0
(k, k’ =
I Lattice mechanics of alkali cyanides
in terms of TOEC and SOEC. The present results on TOEC and pressure and temperature derivatives of SOEC and FOEC and pressure derivatives of TOEC have been presented in tables V and VI. The values of TOEC and pressure and temperature derivatives of SOEC on NaCN, KCN and RbCN are closer to the experimental data [21.24] except for a few cases. However, it is interesting to note that our values for them show a close resemblance in the trend of their magnitude with the available experimental data for alkali halides [34-361. In order to present a visual comparison of the individual contributions from various energy terms, we have evaluated the values of TOECs and the pressure and temperature derivatives of SOEC without the inclusion of TBI, vdW and TR Coupling effects, separately. It has been found that the agreements improve quite significantly in all the cases the exclusion of TBI effects. This indicates that the contributions from the TBI are overestimated and this might be due to the strong Cauchy violations, which are several times larger than those for the alkali halides. This fact leads to the conclusion that ETSM,
- C,,.ril,
1.2)
(24)
with rkk = rkeks = kr,,.
.
(25)
where L = 2V/e* and K = 1.4142 for NaCl structure. Here V (=2ri) is the unit cell volume for the NaCl structure and e is the absolute value of the electronic charge. Besides them, we have calculated the temperature derivatives of SOEC expressed as [33] dC,, dT
= [C,,, + 2C,,, + 2C,, +
2Gzl’yv7
dC,, dT = [Cm + 2G,z + CIZb” ’ dc,,
dT
(10” dvn .crn-‘I C III
Properties LiCN
NaCN
N&l KCN
KCI RbCN
RbCl
(27)
= [C,,, + 2C,,, + 3C,, + 3C,,Ja,
Table V Values of TOECs
Expt. Present Without Without Without
and oressure
C II?
_
(26)
c ,:i
3
(28)
and temDerature C 114
_
C 100
-5.24 -3.90 -7.14 -4.71
0.4b 0.4s 0.4b 0%
0.38 0.38 0.38 0.35
-4.7X -3 65 -b.Yl - 1.33
Expt. 121,241 Present Without TBI Without vdW Without TR Expt. 1341
-1.62 -9.79 -4.53 -8.82 -17.90 -8.80
-0.97 -1.31 -1.07 -0.7x -1.95 -0.57
-0.b7 0.28 0.21 0.2X 0.40 (I.28
0.24 0.22 U 72 0.22 0.73 0.26
-0.27 -0.71) -0.31 -0.30 -0.67
Expt. 121,241 Present Without TBI Without vdW Without TR Expt. [35]
-1.29 -11.34 -5.51 -11.80 -23.76 -7.30
-0.5s -0.75 -0.45 -0.67 -1.37 -0.24
- 1 .Ol n.20 0.14 0.19 0.36 0. I1
U.lh 0 I5 0.15 0.15 0.20 0.23
0.15 0.15 0.15 0.24 0.05
_ 0.12 0.12 0.12 0.14 0.11
Expt. [24] Present Without TBI Without vdW Without TR Expt. [36]
_ -9.48 -6.39 -9.48 -17.32 -6.71
-0.52 -0.77 -0.46 -0.91 -0.18
C,%,
_
-21.55 - 16.31 -22.90 - 19.36
TBI vdW TR
derivatives
U.35 0.24 0.34
of SOEC
dC,,/dP
dC,JdP
_
_
Y.88 8X 9.60
0.26 5.06
(dimensionless dC,,ldP
dC,,idT
_
_
4.34 5.77 4.21 2.X8
-0.56 -0.53 -0.S8 -0.20
and
-0.71 I -0.451 -0.631 -0.300
in 10” dyn ‘cm dC,JdT
dC,,idT
-0.306 -0.252 -0.477 -0.100
-0.1 17 -0.221 -0.041 -0.040
-U.OlX
-0.01x
-(I.030 -0.023 -0.050
o.Oot 0.001 -0.010
5.55 Y.68 8.01 Y.21 8.M) 1 I.85
S.9H
-0.36
-(l.bl
-0. l-l 0.23 0.12 0.22 0.2’) 0.27
2.5x 3.751 7.17 2.42 2.06
-0.41 -0.33 -0.46 -0.59 -0.37
_
-0.77 -0.30 -0.33 -0.30 -0.0’) -0.26
-0.08 O.Ih 0.07 O.Ih 0.25 ll.lh
4.46 11.35 10.32 11.3.5 10.71 12.93
5.0
-0.11
2.1’) 3.62 2.05 7.01 1.59
-0.12 -0.11 -0.12 -0.12 -0.39
-Il.‘02 -0.094 -0.2Ob -0.290 --
-0.022 -0.01 I -0.018 -0.030
_
-0.003 -0.003 -0.002 -0.010
-0.22 -0.56 -0.21 -0.04 -0.17
0.10 0.11 0.13 0.08 0.14
4.46 12.06 10.52 11.95 11.17 12.50
5.38 2.06 3.81 1.91 1.97 1.48
-0.13 -0.13 -0.13 -0.14 -0.14 -0.64
0.083 -0.s59 -0.137 -0.522 -0.241 _
-0.1x1 -0.047 -0.076 -0.040 -0.020 _
-0.053 -0.001 -0.039 -0.002 -0.010 _
-0.701 -0 II3 -0. IYO -0.290 _ _
_
‘)
R. K. Singh and N. K. Gaur
Table VI Values of FOECs (in 10” dyn . cm-‘), pressure derivatives of TOEC and photoelastic constants for alkali cyanides LiCN
NaCN
KCN
RbCN
c 1112 c 1166 c llZZ C 1266 C4444 C 1123 C *,44 C ,244 C 1456 C4456
235.56 26.35 21.96 23.28 22.88 22.69 -0.96 -0.70 -0.87 -0.74 -1.03
129.37 8.59 7.76 11.03 8.55 8.01 -0.80 -0.53 -0.48 -0.37 -1.04
167.80 7.23 4.20 7.22 4.84 4.32 -0.63 -0.45 -0.32 -0.24 -0.85
144.95 4.99 2.97 5.00 3.44 3.09 -0.47 -0.35 -0.27 -0.21 -0.63
dC,,,ldP dC,,,ldP dC,,ldP dC,,,ldP dC,,,ldP dC,,,IdP
-28.52 -5.64 0.17 1.22 0.82 1.17
-19.69 -2.98 -0.49 0.37 0.05 0.20
-40.59 -3.32 -0.62 0.54 0.06 -0.18
-29.46 -1.97 -0.32 0.32 0.02 0.18
c 1111
PI, P,*
-0.380 0.167 0.032
P 44
0.050 0.104 0.003
0.138 0.107 0.039
-0.004 0.135 0.018
adequately suitable for the descriptions of lattice statics and dynamics of alkali cyanides, is incapable to explain their nonlinear elastic and thermal properties. The values of FOECs and the pressure derivatives of TOEC are being reported by us, probably for the first time and as such our comments on their reliability are restricted until the report of their experimental data. 3.4. Photoelastic properties Earlier, the expressions for photoelastic constants (PEC) have been derived from the rigid shell model (RSM) [37] and TSM [28] and used for the calculation of PEC for the ionic crystals [26,28]. We have used the expressions derived by Singh and Sanyal [28] to calculate the PEC of the present molecular crystals of rocksalt structure. Their relevant expressions are given below: P,, = 9
[[3x, 3x, xx]Q2 + [4x, 4x, xx]Q’
+“z[3x, 4x9 =lQ,Q,
+ 3QJ
,
395
I Lattice mechanics of alkali cyanides
(29)
P,, = 5
[[3x, 3x, I
+2[3~, 4x,
P.+,= y
yy]Q’ + [4x,
4x3 YYIQ’
YYIQ,Q,+ Q,l 7
(30)
[[3x, 3y, xy]Q’ + [4x, 4y, xy]Q’
+;3x, 4~9 xylQ,Qz + Q,l
9
(31)
where the bracket and other symbols involved in them are the same as those defined by Singh and Sanyal [28]. The calculated values of PEC for alkali cyanides have been listed in table VI. Probably, these are the first results on them and hence no comparison could be possible due to the lack of experimental data.
4. General discussion and conclusion A comprehensive and unified description of the lattice dynamic, static, anharmonic and photoelastic properties of alkali cyanides has been presented in this paper. It is interesting to note from earlier discussions that the agreement between the experimental and our theoretical results on the acoustic and optic branches of the phonon dispersion curves is reasonably good. This is not surprising as the present model is capable of taking proper account of the Cauchy violation (elastic properties), the polarizabilities (dielectric properties) and the TR coupling in molecular crystals. Also, the ETSM has predicted the lattice statics of alkali cyanides with the same success (see table IV) as their lattice dynamics (see figs. l-4). These descriptions can be further improved by incorporating the effects of covalent and anharmonic interactions and by eliminating the restriction imposed on the hardness parameter (p) to have the same value for different ion-pairs for representing the first and second neighbour interactions. Further, it is surprising to note that the discrepancies of the TOEC and the pressure and temperature derivatives of SOEC (tables V and
396
R.K.
Singh and N.K.
Gaur
I Lattice mechanics
VI) have improved with the exclusion of TBI effects. This is probably due to the overestimation of TBI from the extremely large Cauchy discrepancy (C,,-C,,). This limitation of ETSM can be eliminated by adopting a more appropriate method for estimating the TBI contributions to the elastic constants, which have an altogether different trend as compared to the alkali halides for which the TSM is an adequately suitable model.
Acknowledgements The authors are thankful to Prof. T.S. Murty, Director General, MAPCOST for providing encouragements and to Dr. S.P. Sanyal for many useful discussions. The authors are also thankful to the University Grants Commission (UGC), New Delhi for providing the financial support to this project. One of us (NKG) is thankful to the Council of Scientific and Industrial Research (CSIR). New Delhi for the award of a Senior Research Fellowship.
References Jenkins and D.F.C. Morris. Mol. Phys. 33 111 H.D.B. (1978) 663. 121D.L. Price. J.M. Rowe, J.J. Rush. E. Price. D.G. Hinks and S. Susman, J. Chem. Phys. 56 (1972) 3697. I31 P.F. Fracassi and M.L. Klein, Canad. J. Phys. 62 (1984) 725. and L.A.K. Stavely. Disorder in Crystals I41 N. Personage (Clarendon. Oxford, 1978). Universitat Regenburg [51 W. Dultz. Habilitationsschrift, (1975). 14 (1961) bl N. Ellio and J. Hastings, J. Acta Crystallogr. 1018. I71 M. Sngisaka. T. Mastuo and T. Seki. Bull. Chem. Sot. (Jpn) 41 (1968) 1947. PI P.G. Lowdin. Ark Fys. 35A (1947) 30. Jenkins and K.F. Pratt. J. Inorp. Nuclear 191 H.D.B. Chem. 38 (1976) 1775. Phys. Lett. A 56 [lOI C.J. Bill. H. Jex and M. Mullner. (1976) 320. 1111 K.H. Michel and J. Naudts, Phys. Rev. Lett. 39 (1977) 212.
of alkali
cyanides
[12] K.P. Thakur. J. Chimie Physique 75 (1978) 300 [13j J. Shanker. G.G. Agrawai and R.P. Singh. J. Chem. Phys. 69 (1978) 670. 1141 H. Jex and C.J. Maetz. Phys. Stat. Sol. (b) 85 (197X) 511. U. Schroder and W. Bauernfeind. Solid 1151 D. Strauch. State Commun. 30 (1979) 559. 1161 J.P.S. Rana. S.P. Sanyal and G.B. Varshney. Acta Phys. Polon. A 66 (1984) 209. W. Press. J. Lefebvrae and S. Haussuhl. [I71 K.D. Ehrhardt. Solid State Commun. 34 (1980) 580. D.L. Price. D.G. [W J.M. Rowe, J.J. Rush, N. Vegelatos. Hinks and S. Susman. J. Chem. Phys. 62 (1975) 4551. K. Knorr. W. Dultz and R. Currant. J. I191 J. Daubert, Phys. C 9 (1976) L389. J.J. Rush. E. Prince and J.J. Chesser. PO1 J.M. Rowe, Ferroelectrics 16 (1977) 107. Acta Cryst. A 35 (1979) WI S. Haussuhl and W. Michaelis. 240. Weast. ed.. C.R.C. Handbook of Physics 6i 1-w R.C. Chemistry (CRC Press, 1981). D89. and K.J. Kjems. 2. Phys. B 50 1231 A. Loidl. S. Haussuhl (1983) 187. Solid State Commun. 32 (1979) 1X1. P41 S. Haussuhl. [251 K.H. Michel and J. Naudts. J. Chem. Phys. 67 (1977) 547. WI R.K. Singh. Phys. Rep. 85 (1982) 259. 1271 R.K. Singh and H.N. Gupta. Proc. Roy. Sot. (Lond.) A 349 ( 1976) 289. 1281 R.K. Singh and S.P. Sanyal. Physica 132B (1985) 201. Ark Fys. 9 (1955) 435. 1291 S.O. Londqvist. 1301 R.K. Singh and V.V.S. Nirwal. Phys. Rev. B 20 (1979) 5379. J. Phys. Chem. Solids 38 (1977) 1097. [311 R. Narayan. I321 V.K. Garg. D.S. Puri and M.P. Verma. Phys. Stat. Sol. (b) 82 (1977) 325. Phys. Stat. Sol. (b) 100 (1980) 641. [331 U.C. Shrivastava. I341 K.D. Swartz, J. Acoust. Sot. Amer. 41 (1967) 1083. and R.E.B. Strathen. Proc. Phys. Sot. I351 J.R. Drabble London 92 (1967) 1090. D.P. Dandekar and A.L. Rouff. J. 1361 M. Ghafelehbashi. Appl. Phys. 41 (1970) 652. and K. Srinivasan. Phys. Stat. Sol. (b) 57 1371 R. Srinivasan (1973) 757. A.H. Khan, W. Schockley. Phys. Rev. 1381 J.R. Tessmann. B 92 (1953) 890. Solid State Commun. 13 (1973) 147; Acta [391 H. Haussuhl. Cryst. A 30 (1974) 455. Institute of Physics Handbook, [401 D.E. Gray. American 3rd ed. (McGraw-Hill, New York, 1972). Naturw. 44 (1957) 325, Solid State Com[411 S. Haussuhl. mun. 13 (1957) 147. and H. Haussuhl. [421 W. Krasser, B. Janik. K.D. Ehrhardt Solid State Commun. 32 (1979) 181.
UNIFIED STUDY OF LATTICE MECHANICS OF ALKALI CYANIDES R.K. SINGH and N.K. GAUR School of Physics, Bhopal University, MAPCOST, Bhopal-462 016, India
Bhopal-462026
Received 23 March 1987 Revised manuscript received 28 September
and Centre for Science
and Technology
Development
Studies,
1987
An extended three-body force shell model (ETSM) for alkali cyanides has been developed by including the effects of coupling between the translational and rotational motions of cyanide (CN-) molecules in the framework of original TSM devised by Singh and Sanyal [Physica 132B (1985) 201). This ETSM has been applied to describe the static, dynamic. anharmonic elastic and photoelastic properties of the alkali cyanide crystals. This model has satisfactorily reproduced their experimental phonon dispersion curves and cohesive energy values. However, the agreements between the experimental and our theoretical values on the third order elastic constants are not so good. These agreements have been found to improve significantly with the exclusion of three-body interaction (TBI) effects, which play a vital role in the prediction of harmonic and anharmonic elastic properties of alkali halides. This might be so due to the overestimated effects of TBl from the Cauchy violations (C,,-C,,) which are much larger for alkali cyanides than those for alkali halides. It has been concluded that the ETSM, which has reproduced the lattice statics and dynamics of alkali cyanides so well, is incapable to explain their anharmonic elastic and thermal properties. However, this situation is expected to improve by using a more appropriate method for estimating TBI contributions and incorporating the effects of covalent and anharmonic interactions.
1. Introduction The alkali cyanides are polymorphic molecular crystals of the type M’XY- in which the cyanide (CN-) molecules attain the required effective spherical symmetry by either rotation [l] or random distribution of their orientations [2] in the unit cell along [RIO] and [ill] symmetry directions. These orientation distributions are different for various crystals and also they vary with temperature [3] within the cubic phase. At room temperature and pressure, the alkali cyanides (LiCN, NaCN, KCN and RbCN) are plastic crystals with the rocksalt structure (phase 1, spacegroup Fm3m) [4]. These crystals undergo a first-order phase transition at low temperatures to orthorhombic (for NaCN and KCN) and to monoclinic for (RbCN) structures [5-71. The additional striking features exhibited by the alkali cyanides seem to be due to strong Cauchy violations (C,,-C,,) whose magnitudes are several times larger with reverse sign as compared to those exhibited by the alkali halides (81.
During this decade, the lattice statics and dynamics of alkali cyanides have been the subject of many theoretical [l, 3,9-171 and experimental [ 17-241 investigations. The phonon dispersion curves (PDCs) of NaCN [ 181, KCN [18], RbCN [17] and CsCN [23] have been measured by making use of the most powerful technique of inelastic neutron scattering. There have been several attempts to explain the unusual features of alkali cyanides using phenomenological and microscopic approaches. The shell model calculation [lo] of the PDCs of NaCN and KCN has considered the cyanide (CN-) molecules to be rigid with dumb-bell shape and rotational degree of freedom. This model, however, failed to describe the gross features exhibited by PDCs of both NaCN and KCN. The reproduction of their PDCs obtained by Jex and Maetz [14] is although satisfactory, but they have employed the fitting procedure. Michel and Naudts [ll, 15,251 have employed molecular dynamics simulation to predict the PDCs of these crystals by incorporating the ef-
03784363/88/ $03.50 0 Elsevier Science Publishers B .V. (North-Holland Physics Publishing Division)
386
R.K.
Singh and N. K. Gaur
I Lattice mechanics
fects of coupling of the translational modes (phonons) to the re-orientations of the dumbbell shaped (CN-) molecules. The shell model calculations [15] have although explained the acoustic branches of PDCs satisfactorily, but it has surprisingly failed to predict the optical branches of alkali cyanides. Moreover, it has exhibited the unusual feature of cross-over of longitudinal optic (LO) and transverse optic (TO) branches, which is not consistent with the experimental results. Later on, the PDCs of RbCN [17] have been predicted by a model which combines the shell model and the rotational-translation coupling [14] introduced due to the dumbbell character of cyanide (CN-) molecules. This model has satisfactorily explained the acoustic branches with spoiled agreement for the TO and LO branches along (qO0) and (qq0) directions, respectively. Also, these models have, generally, reproduced the PDCs of NaCN, KCN, and RbCN but they have failed to explain their static properties. However, the analysis from them has not been carried out to predict their interesting anharmonic elastic properties [21] which differ strongly from those exhibited by the alkali halides [26] and divalent metal oxides [28]. Motivated with these remarks, we thought it pertinent to describe their lattice dynamic and static properties simultaneously by means of introducing modifications in the framework of the three-body force shell model (TSM) [26], which has been adequately successful in predicting the static, dynamic, dielectric, optic and anharmonic elastic properties of various ionic, covalent and semiconducting materials [26-281. Its wide ranging applicability is chiefly due to the realistic and appealing approach of representing the long range three body interactions [28], which are essentially required to explain the Cauchy violations in alkali halides [29]. Moreover, the framework of this TSM is appropriate for a unified description of the latice static and dynamic properties of ionic crystals [28,30]. The primary aim of the present paper is to introduce the translational-rotational coupling effects in the framework of TSM [28] and use the resultant model, called the extended three-body force shell model (ETSM), for a comprehensive
of alkali cyanides
and unified description of lattice statics and dynamics of alkali cyanide crystals. The appropriateness of the present ETSM may also be emphasised from the fact that there is a large ionic size difference between M’ and CN- ions of cyanide crystals and consequently there should be a pronounced effect of three-body interactions (TBI) [26,29] as the Cauchy discrepancy in these crystals is several times larger than that in alkali halides. We have applied the present ETSM to all the members of the alkali cyanide family of rocksalt structure for investigating their static, dynamic, anharmonic elastic and photoelastic properties. Our results have predicted the experimental PDCs quite satisfactorily. However, many of the tabulated values for the third-order elastic constants and pressure and temperature derivatives of second-order elastic constants differ considerably from their available experimental values. These discrepancies have been found to be largly due to the overestimated effects of the three-body interactions due to very strong Cauchy violations (C,?C,,) which have opposite sign as compared to those for alkali halides. The essential theory of the present model is given in section 2. Its applications are presented in section 3. A general discussion of the results along with the conclusion is given in the last section.
2. Theory of ETSM The interaction expressed as [28]
potential
rk +
b c kk’
Pkk’
exp
+
rk’
used in ETSM
-
rkk’ +
P
>
is
c#JTR . (1)
The first two terms represent the long-range Coulomb and three-body interactions (TBI) [28]. The third and fourth terms are the vdW contri-
R. K. Singh and N. K. Gaur
I Lattice mechanics of alkali qanides
butions due to dipole-dipole (d-d) and dipolequadrupole (d-q) interactions [31]. The fifth term is the Hafemeister and Flygare (HF) type short-range (SR) overlap repulsion [28] extended upto the second neighbor ions with Pkk8 as the Pauling coefficients PLk’ =
1+ (.2,/n,)
+ (Z/(./n,.).
(2)
Here, Z,(Z,,) and nk(nk,) are the valence and number of electrons in the outermost orbit of cation (anion). b and p are the hardness and range parameters determined from the equilibrium condition:
and they are determined from the method suggested by Michel and Naudts [ 11,251. Here, A 12 and B,, are the short-range force constants (equal to C and D as referred in [15]) between the nearest neighbours (nn). The dynamical matrix corresponding to the ETSM can be derived from eq. (1) and written as [28]: D(q) = (R(q) +Z,C’(q)z,) -(R(q)
+ Z,D’(q)Y,)
x (R’(q) + Y,$‘(q)Y,J-‘(K + Y,C’( 0,)
W(r) dr
o
(3)
T=ro=
and the bulk modulus: B = (9&J-’
[
d244r) 1r=ro,
dr2
The last term, 4’9 is the contribution due to the translational-rotational (TR) coupling effects [ll, 251. The second-order elastic constants (SOEC) derived from eq. (1) for the alkali cyanides can be expressed as cij = CYM + c;”
(ij = 11.12.44)
(
(5)
where the first and second terms represent the contributions from TSM [28] and TR coupling [11,25] given by TR
Cl1 = -8X,,A,2Ir,
2
CT. = 8X,,Ai2/3r, CT: = -2X,,B:,Ir,
,
(64
,
(64
Here, Xii are the elements of the rotation-rotation response matrix [ll]
x=
I
+ R=(q) (8)
+ DTR( 4) 9
where the various terms have their usual meanings [15,28]. The last term is contributed by the TR coupling effects and has non-zero elements only in alkali ion submatrix Dy( q), which is due to the approximation that the cyanide molecules are treated as elastic dipoles [ll]. Its relevant expression is given by [15]
D:R(q) 0 *) DTRW =( o
3
(9)
3
(10)
with D?(q)
= -V(#o+(q)
where V(q) is the rotational-vibrational matrix defined as [15]
V(q)= ’
(6b) .
387
X
coupling
&
A,,&
--A,,&
42%
Ad,
A
\-2A,,S,
125
0
42Sz
42Sx
42%
0
0
BJ,
B,,S,
0
,
(11) where
(7)
Si = sin qiro ,
i = x, y, 2
(12)
with qi = ( q,, q, , q,) as the phonon wavevector,
R. K. Singh and N. K. Gaur
388
/ Lattice mechanics o.f alkali cyanides
rO the equilibrium interionic distance, and m, as the alkali ion mass. The formalism of ETSM derived from the present crystal potential differs slightly from that of TSM [28] due to the inclusion of TR coupling [31] effects. There are eight potential parameters (four parameters (b, p, f(r,) and r,f’(rO)) and four shell model parameters (d,, d2, Y, and Y2) in the ETSM framework. Here d, and d2 are the distortion polarizabilities and Y, and Y, are the shell charge parameters. Using the expressions corresponding to ETSM, the values of these model parameters
have been evaluated from the experimental data on elastic constants (C,, . C,,). bulk modulus lattice constant (2r,), electronic (BL polarizabilities (cy_ , a_) and zone-centre vibration frequencies ( vLo. +o). The higher-order derivatives of f(r) have been obtained from the analytical expression. f(r,,) = f;, exp(- r,,/p). 3s done by others [28]. The values of input data and vdW coefficients are presented, respectively, in tables I and II and used to compute the model parameters listed in table III. The ETSM has been applied for the following descriptions.
Table I Input data for alkali cyanide crystals Properties
LiCN
Ref.
NaCN
Ref.
KCN
Ref.
RbCN
Ref.
r,(A) C,,(lO” dyn.cm-*) C,Z(lO1’ dyn.cm-*) C,,( 10” dyn cm:‘) GTHz) a+@‘) cI_(A’) lO e,
2.55 3.15 1.75 0.02 4.50 0.03 1.80 8.95 2.18
a a a a b c d e f
2.95 2.53 1.44 0.03 4.38 0.28 1.80 7.55 2.11
g h h h i
3.26 1.92 1.20 0.14 4.11 1.30 1.80 5.72 1.99
! h h i c d I k
3.42 1.75 1.04 0.17 4.23 2.10 1.80 4.50 1.77
g m m
Zl j k
Y d d e f
a - Extrapolated values obtained from the plot of r,, versus (r + + r_) and C,, from C,, versus r0 for alkali cyanides. b - Extrapolated values obtained from the plot of r,, versus vTo for alkali cyanidrs. c - [38], d - [17], e - Extrapolated values obtained from r,, versus e0 for alkali cyanides. f - Obtained from the Lorentz-Lorenz relation. g- [l], h-[39], i- [18,20], j - [40]. k- [23]. I - [41]. m- [42]
Table II Van der Waals coefficients for alkali cyanide crystals (C,,. and d,,. are in the units of 10mhoerg. cm6 and 1Om76 erg. cm’. respectively)
Table III Model parameters
for alkali cyanide crystals
Properties
LiCN
NaCN
KCN
RbCN
Parameters
LiCN
cc*
3.43 0.14 105.25 1.55 0.03 61.56 117.81 34.16
22.86 5.89 105.25 10.46 1.49 61.56 251.19 89.52
79.79 63.22 105.25 39.56 24.26 61.56 678.41 277.49
137.28 181.60 105.25 72.36 83.76 61.56 1164.52 502.86
b(10mL2 erg)
0.270 0.232 -0.005 0.049 0.020 0.537 -9.354 -0.857
c** ck ii d,, d d::,
C D
P(A)
f(r,J
r,,f’(rl,) d, d: y, Y,
NaCN 0.391 0.372 -0.009 0.073 0.094 0.318 0.664 1.253
KCN
RbCN
0.389 0.345 --0.009 0.089 0.197 0.287 - I .258 - 1.372
0.364 0.350 -0.007 0.069 0.126 0.330 -2.970 -0.970
R.K.
Singh and N. K. Gaur
I Lattice mechanics of alkali cyanides
3. Application of ETSM
III, we have computed the phonon dispersion curves (PDCs) for LiCN, NaCN, KCN and RbCN along the symmetry directions and displayed them in figs. l-4. We have compared our results with the neutron scattering data [16,18201 available for NaCN, KCN and RbCN. An inspection of these figures reveals that the overall agreement between experimental and our theoretical results is quite satisfactory. Both the optic and acoustic branches have shown reasonably good agreement with their experimental data. This is not surprising as the present model is capable to take proper account of both the elastic (Cauchy violation) as well as the dielectric (polarizabilities) properties on which these branches depend. In an earlier shell model calculation of PDCs
We have explored the extent to which ETSM is capable to explain simultaneously both the lattice dynamics and the statics of alkali cyanides. In the evaluation of the model parameters, we have used the values of 6C,, ( = - 1.40, -1.35, -1.30 and -1.24) and SC,, (= -0.64, -0.44, -0.20 and -0.19) (in units of 10” dyn *cm-‘) for LiCN, NaCN, KCN and RbCN, respectively, along the lines suggested by Strauch et al. [15]. The method of these computations and the results are presented below. 3.1. Lattice dynamic properties Using the model parameters I
I
reported
in table
I
I
I
(cl001
(qqo) LiCN(300K)
REDUCED
389
WAVE
VECTOR(q)
-
Fig. 1. Phonon dispersion curves of LiCN obtained from ETSM.
R. K. Singh and N. K. Gaur
1 Lattice mechanics of alkali cyanides 1
I
((4 00)
I
(990)
1
I
1
(wl)
-le
NaCN (3wK) /l./L____ .
0
0 0 0 _:;
0
00
1 0
REDUCED Fig. 2. Phonon
dispersion
curves
of NaCN
obtained
0.5
0.0
from
of alkali cyanides, Bill et al. [lo] have introduced the rotational degree of (CN - ) molecule. phenomenologically. This has although given satisfactory prediction of PDCs, but failed to explain the Cauchy discrepancy as they themselves have concluded that the librational coupling to the transverse branches alone cannot explain the Cauchy violation exhibited by C,z and C,, in alkali cyanides. Jex and Maetz [14] have also explained the PDCs of alkali cyanides (NaCN and KCN) using a shell model with second neighbour SR forces besides the translational-rotational and rotational-rotational part of the dynamical matrix. This model has although reproduced a satisfactory agreement with experiment for KCN, but it
WAVE
ETSM.
VECTOR
The circles
(‘2 )
represent
the experimental
data
(181.
has a large number of model parameters and gives poor prediction of the static properties. The PDCs of NaCN and KCN have recently been analysed by Strauch et al. [15] by incorporating the microscopic translational-rotational coupling effect [ll, 151 in the framework of the rigid shell model (RSM). This model has given a good description of their PDCs. However, it has revealed an unusual feature of cross-over of LO and TO branches along (qO0) direction. This limitation might be due to the constraint on taking cation and anion shell charges equal and the exclusion of the van der Waals and three body interaction (TBI) effects. Recently, Ehrhardt et al. [17] have predicted the PDCs of RbCN using a shell model modified
R. K. Singh and N. K. Gaur I Lattice mechanics of alkali cyanides
(4 00)
(9 40) KCN
(‘JOOK)
6
0
0
1.0
REDUCED
WAVE
VECTOR
(9)
0.5
w
Fie. 3. Phonon dispersion curves of KCN obtained from ETSM. The circles and open triangles are the experimental data [g. 201; full triangles are from [19].
to include the TR coupling effects. This model has reproduced the PDCs fairly well at the cost of static properties. It is interesting to note that our PDCs for LiCN have followed the trends more or less similar to those revealed by the measured PDCs available for NaCN [18], KCN [18-1201 and RbCN [17]. Our decisive comments on the results on LiCN will be restricted until the report of their measured data on them. A critical c.omparison shows that our results on LA and TA modes are almost similar to their experimental values. Our results on optical modes are in closer agreement with their experimental data as compared to those obtained by earlier workers [14-171. Also, our results have not shown any unusual cross-over between
LO and TO branches as revealed by the results of Strauch et al. [15]. 3.2. Lattice static properties In order to assess the relative merit of the present potential and the logarithmic potential (LP) of Thakur [12], we have calculated the compressibility ( p), molecular force constant (f), infrared absorption frequency (I+,), Debye temperature (On), Griineisen parameter ( y ), ratio of volume expansion coefficient (cz,) to specific heat (cV) at constant volume and the Moelwyn Hughes constant (F1 ) which are directly derived from the cohesive energy, 4(r), expressed by eq. (1). Their expressions [12] are given below for ready reference.
R. K. Singh and N. K. Gaur
I Lattice mechanics of alkali cyanides
Rb CN 13GOK)
Reduced Fig. 4. Phonon
dispersion
The compressibility
curves
of RbCN
obtained
Wave Vector from
is well known to be given
ETSM.
(q) The circles
and triangles
are the experimental
data
116)
Debye temperature
by
(16)
p=- 3Kr,, f in terms of molecular
(13) force constants
with h and k as the Planck and Boltzmann constants, respectively. The values of the Griineisen parameter (y) have been calculated from the relation rlr
with 4:,“.(r) as the short-range nearest neighbour (k # k’) part of 4(r) given by the last three terms in eq. (1). This force constant (f) gives the infrared absorption frequency
’ = -6
L4”‘(r) I 4”(r)
r=I,,
.
We have calculated the ratio of the volume expansion coefficient (+) to the volume specific heat (c,) from its well known expression
(15) with the knowledge of the reduced mass (CL) of the cyanide crystals. This frequency gives us the
(17)
(18) The Moelwyn Hughes constant (F,) calculated from the relation [12]
has been
393
R. K. Singh and N. K. Gaur I Lattice mechanics of alkali cyanides
4 = 1-
&
(19)
M"(~L,,
which was earlier known as the Rao-Keer constant . The values of the cohesive energy listed in table IV show that our values are in better agreement with their experimental data than those revealed by other models [l, 121. It is interesting to note that our results from the ETSM for compressibility (j3) and infrared absorption frequency (~~‘0)are much closer to the experimental data than those obtained by Thakur [12] from the logarithmic potential. Also, our values of the Griineisen parameter (y) seem to be more realistic as they are closer in magnitude to the measured values for alkali halides [26]. The other thermodynamic quantities listed in table IV could not be compared due to the lack of experimental data on them. 3.3. Anharmonic
ties, like thermal expansion, thermoelastic constants and thermal conductivity. The expressions for these elastic constants and pressure derivatives of second order elastic constants (SOECs) have been derived by us corresponding to the three-body lattice energy given by eq. (1) and they have been found to be similar to those derived by Garg et al. [32] for the rocksalt structure crystals. The short-range parameters (A;, B;, C,, and D,; i = 1,2) involved in our expressions differ from those of Garg et al. [32] in respect of the additional contributions from the van der Waals (d-d) and (d-q) interactions. The parameters involved in these expressions are redefined as
A, =
L[d2~r:‘r’]r-,k,, (20)
elastic properties
We have studied the anharmonic elastic properties by calculating third and fourth-order elastic constants (TOEC and FOEC) as they provide physical insight into the nature of the binding forces between the constituents of a crystal and are strongly related to other anharmonic properTable IV Cohesive and thermodynamic
(21) B
=
2
1
WzkR(r) + W?bW
_L r [
Ci = AfIBi
dr ,
dr
(22)
r=krkk.
and Di = AjlBf,
(23)
properties of alkali cyanide crystals Y
cr,,lc,. (10’ J)
F,
Ref.
470
2.32
4.33
5.68
Present Expt. Others
6.54
314
1.69
5.70
4.38
1.22’
585’
2.46’
-
5.91’
Present Expt. Others
5.47 4.118 1.03’
262 197’ 474’
1.86 2.68’
5.73 -
4.74 6.34’
Present Expt. Others
271 203’ 450’
1.95 2.58’
5.74 -
4.91 6.17’
Present Expt. Others
P
f
;kJ . moII’)
(lo-“dyn.cm-‘)
(lOJdyn.cm-‘)
&-Iz)
LiCN
-855 - 849” -864b
4.45 4.51’
3.45 3.40’
9.80 -
NaCN
-729 -738” -755’, -743’
5.28 5.56’ 3.61’
3.41 3.18’ 4.91’
5.63 4.23h 0.94’
KCN
-689 -674” -6824 -678’
6.39 6.95’ 5.08’
3.06 2.82’ 3.85’
RbCN
-653 -646” -648: -63Sd
7.01 7.83’ 5.90’
2.91 2.62’ 3.47’
a - (22). b - [l]. c - [12]. d - [9]. e-Value calculated from fl = 3/(C,, + 2C,?). f-Obtained g - [la, 201. h - [17]. i - Obtained from eq. (16) using experimental data.
from eq. 13 using experimental data.
394
R. K. Singh and N. K. Gaur
where the SR potential
energies are
4:,“,(r) = b/3,,. exp ” $- ‘;! -d
,,,r,-,“.
“‘0
(k, k’ =
I Lattice mechanics of alkali cyanides
in terms of TOEC and SOEC. The present results on TOEC and pressure and temperature derivatives of SOEC and FOEC and pressure derivatives of TOEC have been presented in tables V and VI. The values of TOEC and pressure and temperature derivatives of SOEC on NaCN, KCN and RbCN are closer to the experimental data [21.24] except for a few cases. However, it is interesting to note that our values for them show a close resemblance in the trend of their magnitude with the available experimental data for alkali halides [34-361. In order to present a visual comparison of the individual contributions from various energy terms, we have evaluated the values of TOECs and the pressure and temperature derivatives of SOEC without the inclusion of TBI, vdW and TR Coupling effects, separately. It has been found that the agreements improve quite significantly in all the cases the exclusion of TBI effects. This indicates that the contributions from the TBI are overestimated and this might be due to the strong Cauchy violations, which are several times larger than those for the alkali halides. This fact leads to the conclusion that ETSM,
- C,,.ril,
1.2)
(24)
with rkk = rkeks = kr,,.
.
(25)
where L = 2V/e* and K = 1.4142 for NaCl structure. Here V (=2ri) is the unit cell volume for the NaCl structure and e is the absolute value of the electronic charge. Besides them, we have calculated the temperature derivatives of SOEC expressed as [33] dC,, dT
= [C,,, + 2C,,, + 2C,, +
2Gzl’yv7
dC,, dT = [Cm + 2G,z + CIZb” ’ dc,,
dT
(10” dvn .crn-‘I C III
Properties LiCN
NaCN
N&l KCN
KCI RbCN
RbCl
(27)
= [C,,, + 2C,,, + 3C,, + 3C,,Ja,
Table V Values of TOECs
Expt. Present Without Without Without
and oressure
C II?
_
(26)
c ,:i
3
(28)
and temDerature C 114
_
C 100
-5.24 -3.90 -7.14 -4.71
0.4b 0.4s 0.4b 0%
0.38 0.38 0.38 0.35
-4.7X -3 65 -b.Yl - 1.33
Expt. 121,241 Present Without TBI Without vdW Without TR Expt. 1341
-1.62 -9.79 -4.53 -8.82 -17.90 -8.80
-0.97 -1.31 -1.07 -0.7x -1.95 -0.57
-0.b7 0.28 0.21 0.2X 0.40 (I.28
0.24 0.22 U 72 0.22 0.73 0.26
-0.27 -0.71) -0.31 -0.30 -0.67
Expt. 121,241 Present Without TBI Without vdW Without TR Expt. [35]
-1.29 -11.34 -5.51 -11.80 -23.76 -7.30
-0.5s -0.75 -0.45 -0.67 -1.37 -0.24
- 1 .Ol n.20 0.14 0.19 0.36 0. I1
U.lh 0 I5 0.15 0.15 0.20 0.23
0.15 0.15 0.15 0.24 0.05
_ 0.12 0.12 0.12 0.14 0.11
Expt. [24] Present Without TBI Without vdW Without TR Expt. [36]
_ -9.48 -6.39 -9.48 -17.32 -6.71
-0.52 -0.77 -0.46 -0.91 -0.18
C,%,
_
-21.55 - 16.31 -22.90 - 19.36
TBI vdW TR
derivatives
U.35 0.24 0.34
of SOEC
dC,,/dP
dC,JdP
_
_
Y.88 8X 9.60
0.26 5.06
(dimensionless dC,,ldP
dC,,idT
_
_
4.34 5.77 4.21 2.X8
-0.56 -0.53 -0.S8 -0.20
and
-0.71 I -0.451 -0.631 -0.300
in 10” dyn ‘cm dC,JdT
dC,,idT
-0.306 -0.252 -0.477 -0.100
-0.1 17 -0.221 -0.041 -0.040
-U.OlX
-0.01x
-(I.030 -0.023 -0.050
o.Oot 0.001 -0.010
5.55 Y.68 8.01 Y.21 8.M) 1 I.85
S.9H
-0.36
-(l.bl
-0. l-l 0.23 0.12 0.22 0.2’) 0.27
2.5x 3.751 7.17 2.42 2.06
-0.41 -0.33 -0.46 -0.59 -0.37
_
-0.77 -0.30 -0.33 -0.30 -0.0’) -0.26
-0.08 O.Ih 0.07 O.Ih 0.25 ll.lh
4.46 11.35 10.32 11.3.5 10.71 12.93
5.0
-0.11
2.1’) 3.62 2.05 7.01 1.59
-0.12 -0.11 -0.12 -0.12 -0.39
-Il.‘02 -0.094 -0.2Ob -0.290 --
-0.022 -0.01 I -0.018 -0.030
_
-0.003 -0.003 -0.002 -0.010
-0.22 -0.56 -0.21 -0.04 -0.17
0.10 0.11 0.13 0.08 0.14
4.46 12.06 10.52 11.95 11.17 12.50
5.38 2.06 3.81 1.91 1.97 1.48
-0.13 -0.13 -0.13 -0.14 -0.14 -0.64
0.083 -0.s59 -0.137 -0.522 -0.241 _
-0.1x1 -0.047 -0.076 -0.040 -0.020 _
-0.053 -0.001 -0.039 -0.002 -0.010 _
-0.701 -0 II3 -0. IYO -0.290 _ _
_
‘)
R. K. Singh and N. K. Gaur
Table VI Values of FOECs (in 10” dyn . cm-‘), pressure derivatives of TOEC and photoelastic constants for alkali cyanides LiCN
NaCN
KCN
RbCN
c 1112 c 1166 c llZZ C 1266 C4444 C 1123 C *,44 C ,244 C 1456 C4456
235.56 26.35 21.96 23.28 22.88 22.69 -0.96 -0.70 -0.87 -0.74 -1.03
129.37 8.59 7.76 11.03 8.55 8.01 -0.80 -0.53 -0.48 -0.37 -1.04
167.80 7.23 4.20 7.22 4.84 4.32 -0.63 -0.45 -0.32 -0.24 -0.85
144.95 4.99 2.97 5.00 3.44 3.09 -0.47 -0.35 -0.27 -0.21 -0.63
dC,,,ldP dC,,,ldP dC,,ldP dC,,,ldP dC,,,ldP dC,,,IdP
-28.52 -5.64 0.17 1.22 0.82 1.17
-19.69 -2.98 -0.49 0.37 0.05 0.20
-40.59 -3.32 -0.62 0.54 0.06 -0.18
-29.46 -1.97 -0.32 0.32 0.02 0.18
c 1111
PI, P,*
-0.380 0.167 0.032
P 44
0.050 0.104 0.003
0.138 0.107 0.039
-0.004 0.135 0.018
adequately suitable for the descriptions of lattice statics and dynamics of alkali cyanides, is incapable to explain their nonlinear elastic and thermal properties. The values of FOECs and the pressure derivatives of TOEC are being reported by us, probably for the first time and as such our comments on their reliability are restricted until the report of their experimental data. 3.4. Photoelastic properties Earlier, the expressions for photoelastic constants (PEC) have been derived from the rigid shell model (RSM) [37] and TSM [28] and used for the calculation of PEC for the ionic crystals [26,28]. We have used the expressions derived by Singh and Sanyal [28] to calculate the PEC of the present molecular crystals of rocksalt structure. Their relevant expressions are given below: P,, = 9
[[3x, 3x, xx]Q2 + [4x, 4x, xx]Q’
+“z[3x, 4x9 =lQ,Q,
+ 3QJ
,
395
I Lattice mechanics of alkali cyanides
(29)
P,, = 5
[[3x, 3x, I
+2[3~, 4x,
P.+,= y
yy]Q’ + [4x,
4x3 YYIQ’
YYIQ,Q,+ Q,l 7
(30)
[[3x, 3y, xy]Q’ + [4x, 4y, xy]Q’
+;3x, 4~9 xylQ,Qz + Q,l
9
(31)
where the bracket and other symbols involved in them are the same as those defined by Singh and Sanyal [28]. The calculated values of PEC for alkali cyanides have been listed in table VI. Probably, these are the first results on them and hence no comparison could be possible due to the lack of experimental data.
4. General discussion and conclusion A comprehensive and unified description of the lattice dynamic, static, anharmonic and photoelastic properties of alkali cyanides has been presented in this paper. It is interesting to note from earlier discussions that the agreement between the experimental and our theoretical results on the acoustic and optic branches of the phonon dispersion curves is reasonably good. This is not surprising as the present model is capable of taking proper account of the Cauchy violation (elastic properties), the polarizabilities (dielectric properties) and the TR coupling in molecular crystals. Also, the ETSM has predicted the lattice statics of alkali cyanides with the same success (see table IV) as their lattice dynamics (see figs. l-4). These descriptions can be further improved by incorporating the effects of covalent and anharmonic interactions and by eliminating the restriction imposed on the hardness parameter (p) to have the same value for different ion-pairs for representing the first and second neighbour interactions. Further, it is surprising to note that the discrepancies of the TOEC and the pressure and temperature derivatives of SOEC (tables V and
396
R.K.
Singh and N.K.
Gaur
I Lattice mechanics
VI) have improved with the exclusion of TBI effects. This is probably due to the overestimation of TBI from the extremely large Cauchy discrepancy (C,,-C,,). This limitation of ETSM can be eliminated by adopting a more appropriate method for estimating the TBI contributions to the elastic constants, which have an altogether different trend as compared to the alkali halides for which the TSM is an adequately suitable model.
Acknowledgements The authors are thankful to Prof. T.S. Murty, Director General, MAPCOST for providing encouragements and to Dr. S.P. Sanyal for many useful discussions. The authors are also thankful to the University Grants Commission (UGC), New Delhi for providing the financial support to this project. One of us (NKG) is thankful to the Council of Scientific and Industrial Research (CSIR). New Delhi for the award of a Senior Research Fellowship.
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[12] K.P. Thakur. J. Chimie Physique 75 (1978) 300 [13j J. Shanker. G.G. Agrawai and R.P. Singh. J. Chem. Phys. 69 (1978) 670. 1141 H. Jex and C.J. Maetz. Phys. Stat. Sol. (b) 85 (197X) 511. U. Schroder and W. Bauernfeind. Solid 1151 D. Strauch. State Commun. 30 (1979) 559. 1161 J.P.S. Rana. S.P. Sanyal and G.B. Varshney. Acta Phys. Polon. A 66 (1984) 209. W. Press. J. Lefebvrae and S. Haussuhl. [I71 K.D. Ehrhardt. Solid State Commun. 34 (1980) 580. D.L. Price. D.G. [W J.M. Rowe, J.J. Rush, N. Vegelatos. Hinks and S. Susman. J. Chem. Phys. 62 (1975) 4551. K. Knorr. W. Dultz and R. Currant. J. I191 J. Daubert, Phys. C 9 (1976) L389. J.J. Rush. E. Prince and J.J. Chesser. PO1 J.M. Rowe, Ferroelectrics 16 (1977) 107. Acta Cryst. A 35 (1979) WI S. Haussuhl and W. Michaelis. 240. Weast. ed.. C.R.C. Handbook of Physics 6i 1-w R.C. Chemistry (CRC Press, 1981). D89. and K.J. Kjems. 2. Phys. B 50 1231 A. Loidl. S. Haussuhl (1983) 187. Solid State Commun. 32 (1979) 1X1. P41 S. Haussuhl. [251 K.H. Michel and J. Naudts. J. Chem. Phys. 67 (1977) 547. WI R.K. Singh. Phys. Rep. 85 (1982) 259. 1271 R.K. Singh and H.N. Gupta. Proc. Roy. Sot. (Lond.) A 349 ( 1976) 289. 1281 R.K. Singh and S.P. Sanyal. Physica 132B (1985) 201. Ark Fys. 9 (1955) 435. 1291 S.O. Londqvist. 1301 R.K. Singh and V.V.S. Nirwal. Phys. Rev. B 20 (1979) 5379. J. Phys. Chem. Solids 38 (1977) 1097. [311 R. Narayan. I321 V.K. Garg. D.S. Puri and M.P. Verma. Phys. Stat. Sol. (b) 82 (1977) 325. Phys. Stat. Sol. (b) 100 (1980) 641. [331 U.C. Shrivastava. I341 K.D. Swartz, J. Acoust. Sot. Amer. 41 (1967) 1083. and R.E.B. Strathen. Proc. Phys. Sot. I351 J.R. Drabble London 92 (1967) 1090. D.P. Dandekar and A.L. Rouff. J. 1361 M. Ghafelehbashi. Appl. Phys. 41 (1970) 652. and K. Srinivasan. Phys. Stat. Sol. (b) 57 1371 R. Srinivasan (1973) 757. A.H. Khan, W. Schockley. Phys. Rev. 1381 J.R. Tessmann. B 92 (1953) 890. Solid State Commun. 13 (1973) 147; Acta [391 H. Haussuhl. Cryst. A 30 (1974) 455. Institute of Physics Handbook, [401 D.E. Gray. American 3rd ed. (McGraw-Hill, New York, 1972). Naturw. 44 (1957) 325, Solid State Com[411 S. Haussuhl. mun. 13 (1957) 147. and H. Haussuhl. [421 W. Krasser, B. Janik. K.D. Ehrhardt Solid State Commun. 32 (1979) 181.