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MICROSCOPIC THEORY OF FERROELASTIC

PHASE TRANSITIONS IN ALKALI CYANIDES

By

Devarai Sahu

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics

1983

ABSTRACT
MICROSCOPIC THEORY OF FERROELASTIC

PHASE TRANSITIONS IN ALKALI CYANIDES

BY

Devarai Sahu

Alkali cyanides are classic examples of ionic
molecular solids which show anomalous elastic and phonon
softening in their high temperature pseudo-cubic phases. As
the temperature is decreased, there is a critical
temperature at which they order ferroelastically via a first
order phase transition. In this work we have developed a
microsc0pic theory based on a translation-rotation (TR)
model, originally proposed by Michel and Naudts, to explain
the above mentioned prOperties of both the disordered and
the ordered phases.

We use Zubarev's Green's function technique and an
RPA-type approximation to obtain the phonon frequencies in
the disordered phase. We find important competing mechanisms
in these solids which had been missed in previous works.
These are (i) competition between short range repulsion and
quadrupole electric field gradient contribution to TR

coupling, (ii) competition between short range single site

potential and direct quadrupolar interaction, and (iii)
competition between direct quadrupolar interactions and
lattice mediated interactions. To explain the observed
elastic properties we have to assume that the quadrupole
moment of the anion in the crystal environment is consider-
ably reduced from its free ion value. This conclusion
agrees with the results of molecular dynamics simulations on
orientational correlation functions hi KCN. However, in
contrast to Imolecular dynamics shnulations for CsCN our
calculation predicts a strong dependence of elastic
properties on the quadrupole moment of the anion.

To study the ordering in the ferroelasic phase, we
use a canonical transformation to eliminate the linear TR
coupling and derive an effective orientational Hamiltonian.
We use this Hamiltonian to obtain the free energy in terms
Iof five orientational order parameters. We find that the
form of this free energy has all the necessary third order
terms to account for the first order nature of the
transitions and the symmetries of the ordered phases. As a
concrete example we apply our theory to CsCN. We point out
the difficulties associated with explaining simultaneously
the properties of both the disordered and ordered phases in
terms of a single microscopic Hamiltoninan. Finally we

suggest improvements which can give better results.

To My Parents

ACKNOWLEDGEMENTS

It is a great pleasure to acknowledge my debt to Dr.
S.D. Mahanti for suggesting the problem in this dissertation
to me and for his guidance throughout the course of this
work. His intuition and nmticulous attention to details
have greatly contributed to my learning process. I am par-
ticularly thankful to him for allowing me to interrupt his
work at my convenience and for listening patiently to my

questions.

I would like to express my sincere thanks to Dr.
T.A.Kaplan for his easy accessibility and for clarifying my
doubts at great length and depth. I am thankful to Dr.

M.F.Thorpe for allowing me to take part in his Tuesday
seminars. I am quite grateful to Dr. I.S.Kovacs for his
help and cordiality. My warmest thanks are also due to Dr.
P.K.Misra for his constant encouragement. It is a delight
to express my gratitude to Dr. Rasik H. Raythatha for his
extremely friendly interaction with me and for his genuine
concern for me.

I am quite grateful to Mr. Balan Chenera, an

unselfish friend, but for whoninw stay in East Lansing would

have been a whole lot tougher.

I would like to thank my friends in the solid state
theory group, Mr. Hong Xing He, Mr. Edward I. Garboczi and
Mr. J. Marshall Thomsen for their cordiality in discussing
physics with me. I am grateful to Dr. R.F.Stein for letting
me use his editor and printer and to Dr. I.B.Hoffer for his
assistance in using the editor.

It is my greatest pleasure to thank my wife Urmila
for her love, patience and understanding during my graduate
studies and for bearing ungrudgingly the pains of long hours
of absence from home. I would like to thank my son Ashutosh
whose smiles have meant so much for our happiness. Words
cannot convey my gratitude to my father Mr. Narasimha Sahu
and my mother Mrs. Radha Devi for the love and support they
have given me in such abundance. I would also like to thank
Mrs. onasree (Ranoo) Mahanti for her help during my stay in
East Lansing.

Finally i am grateful to Michigan State University
and the National Science Foundation, through grant number

DMR 81-17297, for financial assistance.

iv

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

LIST OF ABBREVIATIONS

CHAPTER

1.

2.

3.

4.

INTRODUCTION

SUMMARY OF EXPERIMENTAL DATA

(a)The Free (CN)' Ion

(b)Crystal Structure and Orientational Probability
in the Disordered Phase

(c)Some Physical Properties

REVIEW OF EARLIER THEORIES ELASTIC AND PHONON

SOFTENINC IN ALKALI CYANIDES

(a)Purer Phenomenological Theories

(b)Semi-phenomenological Theory

(c)Microscopic Theories

(d)Suggestions for Improvement

MICROSCOPIC THEORY OF ELASTIC AND PHONON SOFTENINC

IN IONIC MOLECULAR SOLIDS

(a)Mode| and Crystal Field (CF) Potential

page

vii

ix

xi

18

19

22

23

29

32

33

(b)Hamiltonian 33

(c)Direct Interaction 36
(d)Renormalization of Phonon Frequencies 41
(e)Rotational Susceptibility 43
(f)Rotation Translation Coupling Coefficients 46
(g)Elastic Constants 52
5. RESULTS AND DISCUSSIONS ON DISORDERED PHASES 57
(a)Calculation of Parameters 57
(b)lsotherma| Rotational Susceptibility 63
(c)E|astic Softening and Transition Temperatures 74
(d)Phonon Softening in KCN 82
6. LANDAU THEORY OF ORIENTATIONAL ORDER 86
(a)Ear|ier Theory 86
(b)Effective Hamiltonian 88
(c)VariationaI Free Energy 89
(d)lndirect Interaction 94
(e)ResuIts 96
7. SUMMARY 99
APPENDICES
A.EQUIVALENCE WITH AN EARLIER THEORY 104
B.REPR|NT FROM Phys. Rev. B (Sah 82) 106
LIST OF REFERENCES 107

vi

LIST OF TABLES

TABLE

2.1 Symmetry changes in I-II transition in alkali
cyanides

2.2 Transition temperature, entropy, enthalpy
and dielectric constant changes in I-II
transition in alkali cyanides

4.1 Strengths of direct interactions between
a pair of molecules

4.2 Euler angles of nn intermolecular axis
for NaCl structure

4.3 Euler angles of nn intermolecular axis
for CsCl structure

4.4 The transpose of the coupling matrix vR(a,a,a)

for CsCl structure. p=AR/4, q=BR/4, r=CR/4.

page

16

17

38

4O

41

49

4.5 The transpose of the coupling matrix 1/8vR(k) for CsCl

structure, with si=Sin(kia),ci=Cos(kia), i=x,y,z.

4.6 The correspondence for obtaining Viaef8(K,k) for

CsCl structure from that of NaCl structure,

where p=1/16,q=1/64/2 and r=8/27/3

50

53

Summary table

Parameters for alkali cyanides (see also Table 4.1)
Repulsion & quadrupolar contribution to TR coupling
Bare elastic constants, Tfit and anharmonicity
parameters in units of 1011dyn/cm2, K and
109dyn/cm2/K respectively.

'Curie-Weiss' temperatures for KCN (f=.6)

and CsCN (f=.15) for k=0.

First order FE transition temperature T'(expt),
extrapolated and theoretical C44 softening

temperatures Tc and T44

viii

56

59

64

69

83

LIST OF FIGURES

FIGURE page
2.1 Elastic constant for KCI (in units of 1011dyn/cm2);
see(Nor 58) 6
2.2 Crystal structure of KCN 10
2.3 Crystal structure of CsCN 11
2.4 Single site short range repulsion potential for KCN
Vmax=9914K,Vmin=8350K,Vs.p.=8687K 12

Single site short range repulsion potential for CsCN
Single site susceptibilities Txiioand
enhancement factors Ri=ini(T)/Xiio for KCN.

(a)i=1 (e8) (b)i=4 (tzg)

Single site susceptibilities of eg and tzg symmetries
for CsCN.
Direct interaction strength D(00k) (in units of 100K)

as a function of the reduced wave vector x.
KCN (f=.6), CsCN (f=.4)
Direct interaction strength D(kkO) (in units of 100K)

as a function of the reduced wave vector x.

13

66

67

71

KCN (f=.6), CsCN (f=.4)

Direct interaction strength D(kkk) (in units of 100K)
as a function of the reduced wave vector x.

KCN (f=.6), CsCN (f=.4)

T- and f-dependence of C44. (a) KCN (1010dyn/cm2) and
(b)CsCN (1011dyn/cm2).

T dependence of C11 and C44 (1011dyn/cm2) for NaCN,
with f=0.6; 1:expt, 2:theory with anharmonicity,
3:theory without anharmonicity.

T dependence of C11 and C44 (1011dyn/cm2) for KCN,
with f=0.6; 1:expt, 2:theory with anharmonicity,
3:theory without anharmonicity.

T dependence'of C11 and C44 (1011dyn/cm2) for RbCN,
with f=0.6; 1:expt, 2:theory with anharmonicity,

3:theory without anharmonicity.

5.10 T-dependence of elastic constants for CsCN

(1011dyn/cm2) including anharmonicity.

72

73

77

78

79

80

81

AD

BKM

Bl

CF

MD

ME

MN

nn

TR

LIST OF ABBREVIATIONS

anistropic dispersion
Bounds, Klein and McDonald
Brillouin Zone

crystal field

center of mass

dimension

deRaedt, Binder and Michel
electric field gradient
ferro elastic

molecular dynamics
molecular field

Michel and Naudts

nearest neighbor
quadrupole moment

short range repulsion
random phase approximation
saddle point

short range

temperature

translation-rotation

CHAPTER 1

INTRODUCTION

The theory of phase transitions has been a very
active area of research in recent years. Understanding the
nature of phase transitions in a variety of physical systems
such as fluids, magnets, superconductors, molecular solids
and so forth has been quite challenging. In the study of
phase transitions in these systenw one can ask several
interesting questions, for example: what drives the phase
transitions? Is there any universal behavior in the phase
transitions in spite of the diversity of the systems? What
are the origins of the observed macroscopic properties?
Eventhough we do not attempt to answer all these questions
in this thesis, we would like to point out two main
directions iri which theories of phase transitions have
proceeded to answer some of these questions.

The first approach has been to concentrate on the
universal aspects of the phase transitions. In this
approach one uses the fact that a physical system is charac-
terized by its spatial dimensionality (D), the dimensional-
ity of its order parameter (d), and the syrrmetry of its
order parameter. Systems otherwise distinct, but having the

same D, d and symmetry of its order parameter are said to

belong to the same universality class. Such systems exhibit
identical critical behavior in spite of the fact that the
interactions which give rise to this behavior are entirely
distinct. In particular, one can add terms to the
Hamiltonian consistent with a given universality class and
yet leave the critical behavior unchanged. The details of
the interactions are relatively unimportant hi such an
approach (Fis 79, Fle 81).

On the other hand, in a different approach, one
identifies the physical Inechanisnu that drive the phase
transitions and studies the relative importance of various
competing processes that contribute to the phase
transitions. In particular one likes to know whether the
parameters that enter the theory have a microscopic justifi-
cation and if not, whether they are even plausible from a
microscopic point of view. In this approach the emphasis is
on the details of interactions, their sources and
microscopic origin.

The work in this thesis is based on the latter
approach. The motivation for pursuing such an approach is
due to the fact that not nmch enmhasis has been given in
recent years to develop microscopic theories of structural
phase transitions. On the other hand, phenomenological
theories of structural phase transitions have been

reasonably successful in explaining soft mode behavior,

order-disorder transitions etc. in terms of adjustable
parameters. One can ask several pertinent questions
regarding such phenomenological theories. Do these theories
convey any information as to the details of the interac-
tions? Are there any competing processes on which the
observed properties depend sensitively? A proper microscopic
theory can address and shed light on some of these
questions.

R.A.Cowley has formulated stringent criteria (Cow
81) to find out whether a theory is microscopic or not.
According to Cowley a proper microscopic theory of phase
transition should be able to predict parameters such as
transition temperature, soft mode wave vector, the symmetry
of the soft mode etc. If one has to introduce extra
parameters to explain the above properties, then one should
be also able to predict other properties such as phonon
dispersion relation, optical reflectivity of a metal and so
on. On the basis of these criteria Cowley points out that
there has been little progress in so far as the microscopic
theory of structural phase transitions is concerned. It is
therefore clear that developing a proper microscopic theory
has been a challenge to theorists. Hence we have chosen such
an approach.

The outline of this thesis is as follows. lri

chapter 2 we summarize the experimental observations

pertinent to the systems under investigation, viz. the
alkali cyanides. In chapter 3 *we review the earlier
theoretical works in the literature that attempt to explain
soft mode behavior in these systems. In chapter 4 we
present our theory of phonon softening and discuss the
different sources that contribute to translation-rotation
(TR) coupling iri these solids: short range repulsion and
anisotropic electrostatic interaction. The latter has not
been taken into account in previous works (except in recent
molecular dynamics simulations, see Bou 81, Kle 81,82) and
is shmwn to play a significant role in understanding the
physics of these solids. In chapter 5 we present the results
of our elastic and phonon softening calculations and show
that the two effects mentioned above compete against each
other. In chapter 6 we review an earlier theory (deR 81)
which tried to explain the ferroelastic order in the
cyanides in a semi-phenomenological way. We point out some
of the inadequacies of this work and present our microscopic
theory starting from Bogolyubov's variational principle for
the free energy. As a concrete example we apply the theory
to understand the ferroelastic order in CsCN. In the last
chapter we summarize our results and point out some of the
unresolved problmns which should be taken in) in a future

investigation.

CHAPTER 2

SUMMARY OF EXPERIMENTAL DATA

In this chapter we summarize some of the experimen-
tal data on the alkali cyanides MT(CN)', where M=Na,K,Rb and
Cs. The high temperature (high T) solid phase in these
compounds is denoted as phase I (pseudo-cubic phase) while
the low T non-cubic phases are denoted as phases II and Ill
respectively. Many of the phase transformation properties of
the alkai cyanides can be traced to properties of the
cyanide ion. For example, eventhough the alkali cyanides
are structural kins of the alkali halides, the two families
show remarkably different physical properties. In the
latter, phases II and III are absent (Nor 58), “mile in
phase I the isothermal elastic constants C44 and C11 harden
with decrease of T (e.g. Figure 2.1). Over a range of T from
300K to 4K, C44 and C11 increase by 5% and 19% respectively
for KCI. This is due to the fact that at low T, absence of
lattice anharmonicities renders the alkali halide lattice
hard. On the other hand, in alkali cyanides precisely the
opposite effect happens: C44 and C11 dramatically soften (in
phase I) with decrease of T. It is clear that this softening

is due to the interactions of the CN' molecular ions (Hau

73) with the lattice and with themselves. Thus it is

 

 

 

 

 

 

 

K at
5.0—
C”
l
4.0-
?
0.7 -
k k
0.6 -
J 1
' Tuoz K) 2 3

FIGURE 2.1 Elastic constants for KCI (in units of 1011
dyn/cmz) see (Nor 58).

apprOpriate to discuss some of the properties of the CN‘ ion

before summarizing the properties of the cyanide crystals.

(a)The Free CN' Ion

 

The free CN' ion has a nonspherical shape. Its
properties are characterized by the following parameters
(Bou 81): total charge q0=-eo (e0 is the charge on a
proton), dipole moment pz=0.318 Debye, quadrupole moment
Qo=-4.51 DebyexA. Taking a typical lattice distance r, the
energy scale of the direct dipole-dipole (d-d) interaction
Edd=p2/r3=10K while the energy scale of direct quadrupole-
quadrupole (Q-Q) interaction is ten times larger i.e.
EQQ=QZ/r5=100K. Thus to a first approximation Q-Q interac-
tions dominate over d-d interactions. The» two nuclear
centers of the molecule are separated by a distance of
2d=1.19A. Bound et al. (Bou 81, also referred to as BKM)
have assumed a dumbbell model of the CN' ion for their
molecular dynamics (MD) calculations with the following
charge distribution: qN=-1.28eo, 9C=‘1-3790aqc.m.=1-5590-
Recently LeSar and Gordon (LeS 82) have constucted a seven
charge model of the anion which takes into account the
higher multipole moments of the anion. Another quantity of

interest is the rotational frequency of the free CN'

molecule: 10'1=3x1010 1/sec (Lut 81,83). In phase II of the
solid, imaginary part of the dielectric response measure-
ments (Lut 81) show that 1'1(T)=a.exp(-b/kT)<Io'1, whereas

in phase ': T-1lleT0'1. This suggests that in phase I the

CN' ions undergo quick molecular reorientations whereas in
the ferroelastically ordered phase, the reorientation rate

is thermally activated.

(b) Crystal Structure and Orientational Probability in the

Disordered Phase

 

In phase I the alkali cyanides have pseudo-cubic
structure (see Figures 2.2 and 2.3). Powder X-ray
diffraction measurements (Ver 38, Bii 40) show that NaCN,KCN
and RbCN have NaCI-type structure whereas CsCN has CsCl-type
structure (Lel 42). In this phase the CN' ions are randomly
oriented so that the crystal symmetry is effectively cubic.
However the probabilities of orientations are not the same
for all directions and they change with T. Single crystal
neutron scattering data (Row 75) in NaCN and KCN suggest
that the CN' orientation probability is large along (111)
and (100) directions. We have made plots of the T-
independent single site steric potential Vo(n) to find out
the maximum and the minimum of the potentials, on the basis

of a simple Born-Mayer form of the potential. For KCN (e.g.

Figure 2.4) it is found that the orientation probability,
P=exp[-BVO(n)] is a maximum for the eight (111) directions,
minimum for the six (100) directions and saddle points
(s.p.) for the twelve (110) directions. The discrepancy
between experiment and theory as regards the orientational
probability distribution could be due to the fact that
experimentally it is hard to isolate the single site
effects, whereas the theoretical plot is a plot of the
single site potential. In contrast, for CsCN (e.g. Figure
2.5) the maxima and minima are along (100) and (111) respec-
tively whereas (110) is still the saddle point. The form of
the single site potential plays a very important role in
determining the bare orientational susceptibilities in the
paraelastic phase. In the lowest order, there are five such
susceptibilities for the pseudo-cubic phase, two of “Mich
have Eg symmetry ( for example <(3cosZB-1)2>), and the rest
have T2g symmetry (for example<(sin6cos¢sinesin¢)2>), where
the angular brackets indicate thermal averaging with respect

to the single site potential.

(c)Some Physical Properties

 

Single crstals of Na,K and Rb cyanides are
transparent in phase I (Hau 73,77; Kon 79). As the
temperature is lowered, there is a first order ferroelastic

phase transition at T=T. to a non-cubic phase (phase II).

10

 

 

 

 

 

 

 

 

 

 

KCN
D
a, o =K*
.0 =(CNI'
0——
o 5 fl
<———— 2Q——->
PHASE I

 

 

PHASE II

FIGURE 2.2 Crystal structure of KCN

11

CsCN

 

 

 

 

 

 

 

 

 

 

<——2a——s

PHASE I

 

 

 

 

 

PHASE II

FIGURE 2.3 Crystal structure of CsCN

        
  

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\\\\\\ \‘\\ “\ “\\\\\\‘\\\\\
\\MMMMM“W“ \“

‘“‘\§\1\\\\\\
\\\\\\\\\\\\\\\\
l \\\l

    

       
   
     
 

Ill/Ill I]
""1:Z,%,,,

 

 

 
         

“
3“”411115/44,’
“I! IIII“;’II {I

 

FIG
Kc URE
N. 2
I V .4
m 5.
99 "g
1,, le
K s.
I V lte
min=83ghort
0
K angavnge
l'
5 ep
'P =8ul
7K n
. Dot
en
ti
aI
f0
I’

13

 
  
  
      

    

\‘\\\
' 1'0" w. . \\\\ \.
. MO ‘0‘
:;\§\ .3 ::::.& ..”. ’:.:.’: MO. [7”] 7” 4““ ....

”4,101? \‘MRR 1“:\\\:“‘ W. 4“.
9'33.” ‘:\\\\\\\ 04:?” , ‘8“\
\‘\\‘ o‘“ ”’47?" {:‘\ g“ \\ “33$“ \\\“

$\\\\!

 
    

V0 (9.4»)

GC>

9:“
¢=O

FIGURE 2.5 Single site short range repulsion potential for
CsCN; Vmax=6082K, Vmin=5578K, Vs.p.=5919K.

14

In the ferroelastic (FE) phase all the CN' ions orient along
some definite fixed direction in the crystal. It has been
pointed out in the case of KCN (Sto 81) that there is
evidence of some elastic disorder in phase II, which however
averages out to zero. In the FE phase there is still a head
to tail randomness of the anions, so that there is no
electric dipole order. We list below some of the changes in
the properties of the crystal in the l-ll transition. (i)
The changes in the crystal symmetry and the directions of
the FE ordering of the anions are given in Table 2.1. (ii)
The single crystals of phase I give rise to multiple domains
upon transition to phase II. Typically the domains have a
mean size of 80 microns and because of strong light
scattering from the domains the crystal becomes opaque upon
transition. (iii) The phase transition is first order and
accompanied by changes in the configurational entropy (AS),
the dielectric constant (As) and enthalpy (AH) (see Table
2.2). Experimental measurement of specific heat (Lut 81) in
the neighborhood of the FE phase transition indicates the
existence of hysteresis effects. (iv) The most dramatic
consequence of the transition is a strong pre-transitional
softening of the (001) TA phonons and to a less remarkable
extent that of the (001) LA phonons. Ultrasonic (Hau
73,77,79), Brillouin (Kra 79; Boi 78,80;Sat 77;Reh 77) and

inelastic neutron scattering (Loi 80a,b;82) experiments show

15

that the elastic constants C44 and C11 decrease with
decrease of 'L. The extrapolated temperatures T44 and 111
where the respective elastic constants vanish are such that
T44>T11. Since: the actual transition is ‘first order, the
experimental transition temperature T. is higher than the
extrapolated temperature TC=T44 where the elasic constant
C44 vanishes. The numerical values of the actual and the
extrapolated transition temperatures are given in Table 2.2.

(v) A plot of ln(T') vs. the lattice constant 2a shows that

T = a"3 (Lut 81). Since there is no electric dipole order in
phase II, this has lead to suggest (LUt 81) that the FE
ordering in phase II is due to elastic dipole interaction.

(vi) Apart from the changes in the static properties, there
are changes in the dynamic pr0perties. Eventhough we will
not be concerned directly with these properties, we mention

two of these properties briefly. (a) In the high-T phase the

Raman spectrum is liquid like and structureless, whereas in
the low-T phase, the Raman spectrum shows well defined
lines. In stress aligned samples in the low-T phase, there

is a prominent Raman scattering peak due to the CN'
stretching vibration. (b) In the high-T phase, the
dielectric response is loss free, whereas in the low-T phase

there is a frequency dependent dielectric loss.

16

 

 

 

TABLE 2.1 Symmetry changes H1 l-II transition in
alkali cyanides.

System NaCN & KCN RbCN CsCN

Phase I fcc fcc sc

Phase II body-centerd monoclinic trigonal

orthorhombic

 

 

FE order (110) dir of (111) (111)
phase I
Reference Kon 79 Kon 79 deR 81
In addition to phases l and II, there is a phase

III which is realised in some systems as the temperature is
further lowered. In this phase there is evidence for
electric dipole ordering (Lut 81). In NaCN and KCN this

ordering is antiferroelectric in nature while in RbCN the

TABLE 2.2

Transition temperatures,entropy,enthalpy,

 

 

 

 

 

and dielectric constant changes in I-lI transition in

alkali cyanides.

System NaCN KCN RbCN CsCN

TC(K) 255.4 156 130 150

T'(K) 288 168 132 193

AS(erg/K/mole) Rln4 RIn2.7 R|n2 Rln3.7

AH(ca|/mo|) 783.7 339 177 461

Ae/e 15% 5% 3% -

Reference Kon 79 Kon 79 Kon 79 deR 81,

Sug 68

electric ordering is perhaps that of a! dipolar glass. In
CsCN on the otherhand, no electric ordering has been

reported upto 14K.
with the

cyanides.

In this work we shall

low-T electrically ordered phases

of

not be concerned

the alkali

CHAPTER 3
REVIEW OF EARLIER THEORIES OF ELASTIC AND PHONON

SOFTENINC IN ALKALl CYANIDES

In this chapter we review the earlier theories on
elastic and phonon softening in the alkali cyanides. The
soft phonon frequencies, in which we are interested in,
depend on two quantities: (a) the phonon frequencies of the
bare lattice and (b) the corrections arising from the
translation-rotation (TR) coupling. Since the calculation of
bare phonon frequencies for ionic solids is. by itself a

complex problem and has been exhaustively studied (Har 79),

we will not deal with such theories in the present chapter.
We will only review the earlier theories which take into
account the corrections to bare phonon frequencies. We

categorize these theories as (a) purely phenomenological,
(b) semi-phenomenological, and (c) microscopic. In a purely
phenomenological theory the coupling of the lattice strains
to the other degrees of freedom is written down in a form
that is consistent with the symmetry of the high-T phase,
but the coupling constants themselves are obtained by
fitting the theory to experimental data. In a microscopic
theory the coupling constants are calculated from interac-

tions which have been obtained independently. A semi-pheno-

18

19

menological theory is a combination of the former and the

latter. In the following we review these theories.

(a)Purely Phenomenological Theories:

 

(i) The phenomenology of elastic softening and coupled
order- parameter dynamics was given by Courtens (Cou 76) in
a work relating to succinonitrile. In this compound the
molecule has an anisotropic polarizability Pi which gives
rise to a polarizability-polarizability interaction of the
forni (1/2)BiiPiPi and a strain(ei)-polarizability
interaction of the form Diipiei' In addition, the lattice
has the usual harmonic strain energy terms. IA canonical
transformation leads to the appearance of an extra strain-
strain term which in the low frequency limit leads to the

renormalization of the elastic constants.

(ii) Boissier et al. (Boi 78,80) explain the elastic
softening in KCN and NaCN, the phonon line shapes, and
address the question of owientational order “1 phase II.

They expand the free energy upto second order in the orien-
tational order parameters n1,n2 and n3 (having T2g
symmetry). These second order terms correspond to a pseudo-
spin exchange energy. They also include a linear coupling
between the order parameters and the strains. For TA
phonons with wave vectors along (110) direction and “Nth

polarization vector e=(001) the free energy is

20

F1=(1/2)C44°(652+862)+s<e5n2+e6n3)+(1/2>(T-To)1(n22+n32)

where ITO is the pseudo-spin exchange energy and e's are the
strains. Note that n12 has been dropped in the above
equation because it does not lead to any softening. Assuming
further that the spin dynamics is diffusive, Boissier et al.
obtain the equations for softening of the elastic constant
C44 and the line shapes for the corresponding phonons. For
LA phonons, however, the coupling of strains can be through
order parameters of E8 symmetry only (due to symmetry
reasons) and these strains cannot couple directly to the
order parameters of T28 symmetry. But the LA phonons can
couple to the squares of the order parameters of T2g
symmetry. This fact gives rise to the softening and line
broadening of the LA phonons in the theory of Boissier et
al.

This phenomenological theory of Boissier et al.
gives good fits for the elastic constants and for phonon
life times (in the small wave vector regime) for KCN.
However it fails to explain the C44 softening in NaCN
because for this system the experimental softening of C44
goes linearly with (T-To). Moreover there is usually a wide
range of values of To over which C11 and phonon line widths
could be fitted equally well. Besides, one serious drawback

of the theory is that these authors neglect the fact that

21

the strains corresponding to LA phonons can couple linearly

to the order parameter of £8 symmetry.

(iii) Rehwald et al. (Reh 77) expand the free energy
of phase I upto second order in terms of a different set of
order parameters which are assumed to be the six discrete
<110> orientations of the cyanide ions. Linear combinations
of these six orientations are taken to obtain orientations
of following symnetry: one of A18 synmetry, two of £8
symmetry (£1,52) and three of T2g symmetry (n1.n2 and n3).
The strains c's couple linearly to the order parameters so

that
Fc=8t(“164*nzestn3eeltfiel51(81*52'263ltfi2lez'63ll
This gives rise to
C44=C44°(T'728'l/(T+To)

and c=co(T-T.g')/(T+To)

Here To is positive and the quantities T28. and Tea. are

related to the coupling strengths 8t and Be respectively,
and C=(1/2)(C11-C12).

One of the drawbacks of this theory is that in the
high T phase, all orientations are possible, not just the
six discrete ones chosen in the model. In fact the nmst
probable orientation is not along (110), but along <111>.

In addition in the Landau expansion third and fourth order

22

terms have been neglected in the works of both Boissier et
al. and Rehwald et al. Since the actual transition is first
order, these terms are extremely important and should be
included to have a proper understanding of the first order

nature of the transition.

(b)Semi-phenomenological Theory:

 

Loidl et al. (Loi 808,b;83) apply the theory of
magneto-elastic coupling in rare-earth solids to the
molecular solids and map the electronic degrees of freedom
into the rotational degrees of freedom of the molecular
species. In presence of TR coupling and three phonon
anharmonic processes, the frequencies of phonons are given

by the equation:
w(k)2=w0(k)2+2wo(k)w(k,m)-2A2w02(k)x(k,w)

where mg is the bare phonon frequency; w is an anharmonicity
parameter, A is a TR coupling constant and x is the
rotational susceptibility in presence of direct anion-anion
interaction. By taking the long wave length limit of the
above equation Loidl et al. obtain the renormalization of
the elastic constants, similar in form to that obtained by
(Reh 77).

The following limitations of the equation used by
Loidl et al. should be noted. First, there is as single

coupling constant A for both LA and TA phonons. Second, it

23

is not clear from the above equation, the role, for any
general k, of the off-diagonal elements of the susceptibil-
ity matrix in phonon softening. Moreover, the parameters of
the phonon Green's function which give rise to the
excitation frequencies given above, have well-defined
microscopic origin. However Loidl et al. donot obtain these
parameters from independent sources, instead they fit their
theory with experimental data in cyanides to obtain these
numbers. In addition this theory does not explore the
interesting competition between various physical effects,
which we show are crucial to have a proper understanding of

phase transition in these systems.

(c) Microscopic Theories:

 

(1) Elastic Softening:

A nficroscopic theowy ‘har anomalous thermoelastic
behavior in the alkali cyanides with NaCl type structure in
phase I was first given by Michel and Naudts (Mic 77a,77b).
We will review their theory in some detail because the
present work is similar in spirit to theirs. They start
from a nmdel in which the anion is taken to be a dumbbell
with two repulsion centers at the positions of the two
nuclei. Each molecule is surrounded by an octahedral cage of
six alkali ions. The single site repulsive potential on a

given molecule due to its nearest neighbors (nn) is given by

24

6

V0(i)=i§1V0(Riirni) (3.1a)

with V0(Rii,ni)= X C1exp[-C2|Rii+sdni|] (3.1b)
5=+'-

and "i is a unit vector specifying the orientation of the

ith molecule. The lowest order term in the crystal field
potential Ms a Devonshire potential involving spherical
harmonics of order l=4. However as the cations move from
their average positions, the potential does not have cubic
symmetry. This instantaneous noncubic potential as given by
(3.1b) is then expanded upto terms linear in the displace-

ments “ii
VO(R+uIn)=V0(R'n)-P(R,n)ou+o00oo

Here R=Rii0, and “="ii- The coefficient of the linear term
P(R,n) hs again expanded iri spherical harmonics. If one
retains only l=2 terms, one obtains the leading order con-

tribution to the TR Hamiltonian:
HTRU)=iZaYa(ni)vaR<Ri,0).u, (3.2)

Here and in the following two chapters the five symmetry
adapted spherical harmonics (not all normalized) of order

l=2 are defined by:

Y1(n)=(5/4n)1/2(1/2)(3ct2-1)

25
Y2(n)=(1.5)(5/4n)1/25t2(cp2-sp2)
Y3(n)=-c.st2.sp.cp (3-3)
Y4(n)=-c.st.ct.cp

Y5(n)=-c.st.ct.sp

where c=(15/2fl)1/2, ct=cosB, st=sin6, cp=cos¢, sp=sin¢, 9
and ¢ being the angles associated with the unit vector n.
It is well known that Y1 and Y2 have Eg symmetry whereas Y3,
Y4, and Y5 have T2g symmetry.The TR coupling matrix due to

repulsion, Vua: are calculated from:
vua(l)=ca2pr(l,Q)Ya(9)dQ ; (l=nn vector)
where ca'2=fYa2dQ (3.4)

For NaCl type structures the 3x5 coupling matrix v has only
two independent coefficients which we will denote as AR and
BR, where the subscript R stands for repulsion contribution.
The non-spherical shape of the cyanide ion, therefore leads
to a TR coupling which leads to an effective indirect

orientation-orientation interaction of the form

(1/2) X Ya'(k)la8(k)Y3(k)
aBk

where the indirect interaction matrix I is given by

la5(k)=-i% vai(k)Rii(k)viB(k)

26
and R'1(k)=MO is the bare dynamical Inatrix. Ya(k) is

defined by

va(k)=(1/N)1/2Zexp(ik.ki)Ya(ni)
I

The TR coupling not only introduces an effective orienta-
tional interaction, but also changes the dynamical matrix

from M0 to M such that
M=[1+F.R]'1M0 (3.5a)
where F=vxvt (3.5b)

In presence of the indirect interaction I, the rotational
susceptibility x=x(k) is given in the molecular field

approximation by the solution to the matrix equation

x=x°-x°|(k)x (3.6)
where xaBo=(1/kT)fexp(-8V0)YQ(Q)YB(Q)dQ/Z (3.7a)
with z=fexp(-8vo)dn (3.7b)

It should be pointed out that the TR coupling matrices enter
not only in equation (3.5) but also in equation (3.6)
through indirect interaction I. Such a treatment leads to
the correct equations for the elastic and phonon softening.
In contrast, in our theory (see section (e) of Chapter 4)
when the renormalized phonon frequencies are calculated in

RPA (Randmn Phase Approximation), it: is not necessary to

27

include the indirect interaction in the expression for the
susceptibility x, because it will lead to overcounting of
interactions. The equivalence between the approach of this
thesis and that of MN (Mic 77a) is proved in Appendix A.
Taking the elastic limit in equation (3.5) and (3.6)
Michel and Naudts (MN) obtain the softening of the elastic

constants:
C44=C440[1-(ZBR2xt0)/(aC440)] (3.83)
C11=C110[1-(8AR2xe0)/(aC110)] (3.8b)

where 2a is the lattice constant (see Figures 2.2 and 2.3),
xo's are susceptibilities of appropriate symmetries.

The parameters AR and BR in equation (3.8) depend
on: the strength C1 and inverse range C2 of the Born-Mayer
potential; on a and on the bond length 2d. They are given in
Appendix B of (Sah 82,enclosed with this thesis) and were
first derived by MN (Mic 77a).

For KCN the above theory for elastic softening gave
good agreement with experiment. However while checking the
orientational susceptibility curves of E8 and T28 symmetry
given in Figures 1 and 2 of (Mic 77a,b) we were unable to
reproduce these curves. We also noted that a different
choice of the repulsion strength C1, viz. C1=2CMN could
easily reproduce their curves. However since AR and BR are

proportional to C1 it is clear that doubling C1 should also

28

double the coupling constants AR and BR and as such will
spoil the agreement between theory and experiment. In other
words a simple rescaling of the parameters would not give
the reported (Mic 77a) agreement between theory and
experiment. Therefore it is clear that in order to obtain
agreement of the theory with experimental data one has to
invoke additional physical processes. We outline such a

physical process in section (d).

(2)Phonon Softening Over the Entire Brillouin Zone (Bl):

The original microscopic theory of MN was primarily
focused on calculating elastic properies. However there have

been several attempts to apply the theory to calculate

phonon dispersion relations. We review these theories
below.
(i) Strauch et al. (Str 79) start from the basic TR

coupling matrix discussed above and write the dynamical
matrix as M=M0+5M, where GM is :3 6x6 matrix which can be
partitioned into sub-matrices of order 3x3. The upper left
hand side sub-matrix in 6M is then given in terms of the TR
coupling matrix v and the bare susceptibilities x0. Strauch
et al. find that the use of v as given by MN does not give
good agreement between theory and experiment for the systems
NaCN and KCN. Hence they chose different constants that

give good fits to the experiment. It is clear that there is

29

no microscopic justification for the choice of parameters

used in this work [but see (Sah 82)].

(ii) Ehrhardt et al. (Ehr 80,83) obtain the phonon

dispersion relations for RbCN basing their theory again on

the microscopic model of MN. Their dynamical matrix is
identical to that given in equation (3.5) where the
rotational susceptibilities are calculated in the presence

of lattice mediated interaction, i.e. according to equation
(3.6). Note that the dynamical matrices of (i) above and
that of (Ehr 80) are different from each other. Whereas in
(i) the upper right hand side submatrix in GM is a 3x3 null
matrix, in (Ehr 80) the corresponding quantity is a non-null

matrix.

It should be pointed out that in both the works (i)
and in (ii) the rotational susceptibilities have been
assumed to be frequency independent. This certainly breaks
dowm when the time scales of orientational motion are
comparable to phonon time scales. For a proper dynamical
calculation involving projection operators, we refer the

reader to the later work of MN (Mic 78).

(d) Suggestions for Improvement:

 

As mentioned earlier, the model due to MN (Mic 77a)
which had only SR contribution to TR coupling, failed to

account har the observed thermoelastic softening hi KCN.

30

Moreover the theory of Loidl et al. (Loi 80a,b), in which
the phonon frequencies were calculated from the poles of
phonon (Breen's function, were ultimately fitted to the
experiments to obtain the parameters, thus revealing no
information at the microscopic level. Our aim in the next
chapter is to develop a microscopic theory similar in spirit
to the work of MN, but uses the Green's function formalism
to derive an equation similar to that used by Loidl et al.
(Loi 80a,b). The combination of the two approaches and the
inclusion of a physical mechanism mentioned below, leads to
a theory that brings out the essential physics of the
softening in the alkali cyanides in a straight forward
manner.

One of the major contribution of this work was to
realize that an improved agreement theory and experiment
could be obtained by incorporating TR coupling due to
quadrupole moment (Q) of the anion interacting with the
fluctuating electric field gradient (efg) at the site of the
anion. Since Q happens to be negative, the TR coupling con-
tributions due to SR repulsion and Q-efg interaction compete
against each other. To see the physics of this competition,
consider a (CN)' ion oriented along the z-axis and a
positive charge also on the positive zeaxis, at a distance a
from the c.m. of the anion. If one moves the positive charge
towards the anion along the z-axis, then the short range

repulsion energy increases, whereas the electrostatic energy

31

decreases because of the charge distribution associated with
the anion. Thus we see that there is a competition between
these two effects in their contribution to TR coupling which
is important in understanding the observed elastic softening
in the alkali cyanides. The details of this and other

competing processes are presented in the next chapter.

CHAPTER 4
MICROSCOPIC THEORY OF ELASTIC AND PHONON SOFTENING

IN IONIC MOLECULAR SOLIDS

In this chapter we present a microsc0pic theory of
elastic and phonon softening in ionic molecular solids in
general, with particular application to solids with NaCl and

CsCl structures. We show that our theory reproduces the

earlier microscopic theory of MN (Mic 77a) in the
appropriate limiting cases. Improvements over the earlier
microsconic theory which takes into account the non-

vanishing electric quadrupole moment Q of the molecular ion
are presented. These improvements are two fold: (i) there
is a non-zero contribution to TR coupling arising due to Q-
efg interaction, (ii) there is a modification of the
rotational susceptibility arising from direct Q-Q
interaction between the anions. We show that the presence
of (i) drastically modifies the conclusions of earlier
works. We find that in the contribution (i) it is
inadequate to retain just the nn contributions because con-
tributions from other neighbors may be of opposite in sign
to nn contributions. It is also found that going upto third
nn is usually adequate. However for direct Q-Q interaction

nn approximation is adequate because the contributions fall

32

33

off as 1/R5.
We divide this chapter into seven sections. Details
of some of the results given in this chapter can be found in

Appendix B of this thesis.

(a) Model and Crystal Field (CF) Potential

We treat the CN‘ ion as a rigid dumbbell which has
two identical centers separated by El distance 2d. The
alkali ion M4' on the other hand is represented by a
Spherically symmetric charge distribution. The short range
(SR) repulsion between a given (CN)' and an M+ ion is char-
acterized by two constants C1 and C2 which represent the
strength and the inverse range of the repulsion potential
respectively. Since we are treating the two centers of the
anion as identical, there is only one set of atom-atom
potential parameters C1 and C2 for each cyanide crystal.

Since the short range potential falls off exponentially with

distance, we will ignore the SR interaction beyond the first
nn. With these assumptions, the orientation dependent
single site CF potential is easily obtained from equation
(3.1).

(b) Hamiltonian

 

The Hamiltonian H consists of three parts H=HT + “R

+HTR. The translational part HT is obtained by treating the

34

ions as spherical charge distributions and keeping the

harmonic terms

HT=<1/2)kg"[(1/mK)pu'(r.k)pu(x.k)+

3;CHU(K,A,k)uu.(K,k)uu(A,k)] (4.1)

where mK is the mass of the Kth ion (+,-) in a unit cell, k
is the wave vector, 11 is the cartesian component x,y,z;
CuU(K,A,k) is the dynamical matrix (note that the matrix M0
introduced in Chapter 3 gives only the acoustic modes), and
u,p are the fourier transforms of the displacements (from
the appropriate cubic structure) and momentum respectively.
The rotational part of the Hamiltonian is obtained

by fixing the c.m. of the (CN)‘ ions at cubic lattice sites:

H = 2 1 2| L A,k 'L A,k +Ev - 2 v " 4.2
R kAl /( )l ( ) ( ) i 0(n.)*<ii> d(I]) ( )

where I is the moment of inertia of the dumbbell about each
of its two principal axes (i.e. A=1,2) and L(A,k) is the
fourier transform of the angular momentum L(A,i), VO(“i) is
the CF potential energy term already discussed in the
previous section. The tern: Vd(ij) represents the direct
interaction between two molecules at sites i and j and

oriented arbitrarily with respect to crystal axes.

35

The TR coupling term HTR represents a coupling
between the owientational degrees of freedom Yh(ni) of a
molecular ion and the translational degrees of all the other

ions. Following MN we*write

HTR=iauXKkYQ'(k)vau(1<,k)uu(k,k) (4.3)

This tenn is obtained by displacing the c.m.s from their
equilibrium positions and keeping terms linear in the dis-
placements. Equation (4.3) is a fourier space representation
of equation (3.2) after summing over all lattice sites i.
The coupling matrix v will be discussed in detail in section
(f).

If we describe the translational degrees of freedom
in terms of phonons, then HT and “TR can be rewritten in
terms of phonon creation and destruction operators b. and b
respectively. Let k and j denote the wave vector and the

polarization index respectively. Then
HT=ikaik0(bjk.bik+1/2) (4-4)

where the bare phonon frequencies “jko are obtained by
solving a secular equation involving the dynamical matrix C
in equation (4.1). For detailed description about the
calculation of bare phonon frequencies in ionic solids with

long range interactions but without TR coupling we refer the

36

reader to the book by Hardy and Karo (Har 79). We can

express the displacements Uu(K,k) as a sum over the normal

modes of vibration:

uu(K,k)=%(ZwOmK)'1/2eu(K,kj)(bik+bi-k.) (4.5)

where eu(K,kj) is the polarization vector of a K type ion

for the phonon mode jk. One then obtains

HTR=iiEkYQ'(k)VQi(k)(bik+bi-k.) (4.6)
where Vaj (k)=(2wjk°)'1/2 Z (mK)‘1/2eu(1<kj)vua(r<,k) (4.7)
pK

(c) Direct Interaction

 

Direct interaction between two nmlecules at sites
Ri and RI can arise due to (a) 0-0 interaction, (b)
anistropic dispersion (AD) interaction and (c) short range

repulsion (R) interaction. One can write this interaction as

(Koh 60)
leill=r§1AmY2,m(‘”i)Y2,-m(wj) (4.8)

where the unit vectors mi and mi are the orientations of the
molecules i and j respectively with respect to the line
joining their c.m. taken as the polar axis. The coeffi-

cients Am can be written as

37
Am=A.n(QQ)+Am(AD)*Amm) (m=-2.2)

and A.m=Am

The individual contributions to the direct interaction from
the various sources are given in Table 4.1 (Coo 71,Dun
77,deB 42). As seen from this Table, the 00 interaction
falls off as 1/R5, while the AD interaction falls off as
1/R5. In Table 4.1, e and o are parameters of the Lenard-
lones potential and K is the anistropic polarizability. To
obtain the anionic repulsion interaction that is listed in
Table 4.1 a power law potential of the form 4A/R12 between
the anion centers was assumed. In this work we neglect the
effects C” IN) and R contributions. Because of the large
quadrupole moment of the (CN)' anion, and large intermolecu-
Iar separation, this assumption is justified. Estimates of
p,q and r give the following values (for nn anions):
q=1146K, p=-41K and r=1K where we have used the following
parameters e=118K,o=3.46A, K=.25 and A=8x107K(A)12.
Moreover since the dipole moment of the (CN)' ion is small,
we ignore the direct dipole-dipole interaction. The direct
interaction between the anions arising due to their

quadrupole moments can be written as (using 4.8):

Hd=1/2 X Da8(k)va(k)'Y5(k) (4.9)
oBk

38

TABLE 4.1 Strengths of direct interaction between a
pair of molecules

 

 

 

QQ AD R
A0 q p 8.41r
A1 (2/3)q (2/3lp (5/3r)
A2 (1/6)q (1/3)p 37.33r
where q=24an/5R5 p=-48n€06K2 r=2411336Ad4
/5R6 /5R16

 

 

The form of the interaction term Da8(k) is explicitly given
in (Sah 82), attached to this thesis as Appendix B. In
evaluating the direct interaction terms above one has to sum
over the Euler angles of the lattice vectors joining a given

anion to its nn anions (see Figures 2.2 and 2.3). These

39

Euler angles are given in Tables 4.2 and 4.3 for NaCl and
CsCI structures respectively.

The k-dependence of the direct interaction matrix
D(k) for various symmetry directions will be given in

chapter 5 and will be discussed in that chapter.

40

TABLE 4.2 Euler angles of nn

for NaCI structure

intermolecuar axis

 

 

position(units of a)

'B,-Y(units of u)

 

1/2,1/4
1/2,7/4
1/2,5/4
1/2,3/4
1/4,1/2
1/4,3/2
3/4,3/2
3/4,1/2
1/4,0

1/4,1

3/4,1

3/4,0

 

 

41

TABLE 4.3 Euler angles for nn intermolecular
axis for CsCI structure

 

 

 

position(units of 2a) -B,-Y(units of w)
1,0,0 1/2,0

-1,0,0 1/2,1

0,1,0 1/2,1/2

0,-1,0 1/2.3/2

0,0,1 0,0

0,0,-1 1,0

 

 

(d) Renormalization of Phonon Frequencies

 

In alkali cyanides the bare phonon frequencies of
the high-T phase are renormalized because of the TR coupling
given by equation (4.6). 'h: obtain these modified phonon

frequencies we use Zubarev's Green's function technique (Zub

42

60). We define the time- and temperature-dependent retarded

Green's function G by
Ciklt't')=(1/i)<[¢ik(t),¢jk.(t')]>6(t-t') (4.10)

For simplicity of»notation if we drop the jk's for the
moment, then ¢(t)=b(t)+b.(t) is the phonon field operator in
the Heisenberg representation. The square bracket is the
conmutator, < > stands for the average over a canonical
ensemble and 6(t-t') is the step function. The equation of

motion for C is
i(d/dt)c(t-t')=6(t-t')<I¢<t>.¢'<t')1>
+<1/i)e<t-t'><[[¢<t).H1,¢'<t')1>
The fourier transfornrof C(t-t') is given by

w<<¢;¢’>>w=<[¢,¢'l>+<<[¢.Hl;¢'>>w (4.11)

where "
<<¢;¢'>>w=C(w)= l<3(r)exp(imr)dr (4.12)

The Hamiltonian H appearing in the above equation is the sum
of rotational, translational and TR parts. Following the
treatment given in Appendix B, we obtain the renormalized

phonon frequencies in RPA as:

mik2=(wiko)2-2wik0£%vai(k)x08(kw)v8i(k) (4.13)

43

where the rotational susceptibility is defined as the time

fourier transform of the angle-angle Green's function:
xaB(k,w)=-<<Ya(k);YB'(k)>>w (4.14)

We would like to point out the assumptions under which
equation (4.13) has been derived. One assumption is that
the rotational dynamics is determined by HR alone, i.e. the
rotational dynamics has a faster time scale compared to
translational dynamics. N1 the elastic reginw this is a
reasonable approximation. For higher phonon frequencies,
one has to consider the retardation effects in an adequate
fashion. Another assumption in obtaining equation (4.13) has
been that a Green's function involving two orientational
Operators and two phonon operators has been replaced by the
average value of orientational Operators times the Green's

function of the phonon operators.

(e) Rotational Susceptibility

 

For calculation of elastic: constants it is
sufficient to consider the w=0 limit of the rotational sus-
ceptibility. We will further assume that xag(k), which we
define to be the zero frequency limit of the susceptibility,
can be replaced by the isothermal susceptibility xaBT(k).
We calculate this latter susceptibility in presence of

direct interaction between the molecules within a MF approx-

44

imation. We have also neglected fluctuation effects in our
calculations. As we have shown (Sah 82) this susceptibility
in the paraelastic phase is obtained by solving the matrix

equation
x<k)=[1+x°D(k)1'1x° (4.15)

where X0 is the bare susceptibility given by equation (3.7)
and is calculated by taking the thermal average over the

single site potential V0. Because of cubic symmetry xago is

diagonal in 0,8 and has only two independent elements xeo
and xtoz
xaa°=(xe°.3xe°.xr°.xt°.xt°) (4.16)
Similarly in the elastic limit (k+0), the direct interaction
matrix is diagonal with
Daa(k=0)=(f.f/3.8.8.8) (4-17)

where f=D110, g=D330. Note that the factors 3 and 1/3 in
equations (4.16) and (4.17) respectively appear because the
symmetry adapted spherical harmonic Y2 is unnormalized (see
equation 3.3). From equations (4.15) to (4.17) it follows

that for k=0, x is diagonal with

xaa=(m,3m,n,n,n) (4.18)

45

where num11, and n=x33. H1 the absence of single site
potential, i.e. for a free rotator, one has xeo=(1/kT)(1/4n)

and xt0=(1/kT)(1/2n). Clearly these values will be modified

when the single site potential is non-zero, the precise
nature of the modification will depend on the type of
lattice considered. The completeness relation of the

spherical harmonics allows one to write down a sum rule on

the bare susceptibilities:
2er+1.5xt0=(1/kr)(5/4r) (4.19)

Next we quote, for the sake of completeness, how
equation (4.15) is modified if we include self-interaction
(Geh 75) effects. Self-interactions can alter the single
site susceptibility in two ways: (1) by changing the single
site potential and (2) by changing the effective indirect
interaction between the molecular ions. In presence of self-

interactions the new single site potential is VT:
VT(9)=V0(9)+V5(Q)
where V0 is defined in equation (3.1) and

v 0 = Z I k Y 0 Y 0 4.20
s()a8ka3()a()3() ( )

with the indirect interaction between the molecules given by

'a8(k)='§2Vaj.(k)V8j(k)/(wjk0) (4.21)

46

One has then to calculate the bare susceptibility given by
equation (3.7) with respect to the total single site
potential VT(Q). In addition the nutrix equation for the

susceptibility (4.15) becomes

x(k)=l1+x°(D(k)-Elag(k))l'1x° (4.22)

Note that equation (4.22) has been obtained by defining the
self-interaction contribution to the single-site potential
through equation (4.20). One could as well define pVS
(where r) is an arbitrary number) as the self-interaction
contribution to single site potential with the consequence
that H1 equation (4422) (NH! obtains -plaB(k) instead of
-IaB(k). In an exact calculation such arbitrariness
(through the number p) is expected to have no effect.
However since we are using a meanfield approximation, the

nature of this arbitrariness might affect the final results.

(f) Rotation Translation Coupling Coefficients

 

In this section we discuss in detail the different
contributions 'N) the 'HQ coupling nmtrix that appears in
equation (4.6). These coefficients can be written as a sum
of two parts:

v(K,k)=vR(K,k)+vef3(K,k), (K=+,')
where R indicates SR repulsion contribution and efg

indicates the contribution coming from the interaction

47

between the Q of the anion and the fluctuating efg produced
at its site by all the other ions (taken to be point
charges). Here K indictes the sign of the charges and could
be + or - depending on the type of neighbor considered. For
SR repulsion, it is sufficient to retain contributions from
nn's only (K=+), while for efg terms we include contribu-

tions upto third nn's.

(1) Repulsion Contribution

 

The TR coupling matrix due to repulsion, in
coordinate space, is calculated from equation (3.4) with the

help of equation (3.2). We discuss two cases:

(i)NaCl structure:

This case has already been discussed by MN (Mic 77b)

and the corresponding results for the k space is also
discussed by them. It should be noted that there are only
two constants AR and In; in this coupling nntrix. These

constants depend on the parameters C1,C2,d and a which are
given in Section (a) of Chapter 5 which deals with the

calculation of parameters (see Table 5.1).

(ii)CsCl structure:

We will evaluate the coupling matrix for this case
in the coordinate space for the nn lattice vector I=(a,a,a)

(see Figure 2.3). Let N be an operator such that Nl=l and

48

Nn=n', where n is a unit vector specifying the direction of

the molecule. From equation (3.4) we obtain
vua(l)=ca2f[NPu(l,n)][NYa(n)]dn (4.23)

Choosing N=Sxy, an operator for reflection along the [110]
direction and also choosing N=R3, an operator for 3-fold
rotation about the [111] axis, we find that there are three
independent constants ARrBR and CR. In contrast, for NaCI
structure the SR coupling matrix has only two constants AR
and BR. The coupling matrix for CsCI structure in direct
space and in momentum space are given in Tables 4.4 and 4.5

respectively.

(2) Q-efg_Contribution

 

As the lattice vibrates, there is a deviation from
local cubic symmetry at the positions of the molecules and
the resulting efg couples to Q. To the lowest order in dis-
placements this leads tx: an additional coupling between
rotation and translation. Depending on the sign of Q the net
TR coupling will be either enhanced or suppressed.

The Q-efg interaction is given by (Coh 57)

H'=(1/6)i%iQ|i(i)U|i(i) (4.24)

where the Cartesian components of the Quadrupole and field

gradient tensors at site i are:

49
Q|j(i)=f[3riIrii-6|ir52]p(ri)dr (ri=r-Ri) (4.25)

U|j(i)=(8/3xl )(a/axi))'qn/IRi-Rnl, (xeRi) (4.26)
n

TABLE 4.4 The transpose of the coupling matrix
vR(a,a,a) for CsCI structure.
p=AR/4,q=BR/4,r=CR/4.

 

 

 

x y z
1 p p -20
2 -p p 0
3 q q r
4 q r q

 

 

50

TABLE 4.5 The transpose of the coupling matrix
(1/8)vR(k) for CsCI structure, where
si=Sin(kia),ci=Cos(kia),i=x,y,z.

 

 

 

x y z

1 p.sx.cy.cz p.cx.sy.cz -2p.cx.cy.sz
2 -p.sx.cy.cz p.cx.sy.cz o

3 q.cx.sy.cz q.sx.cy.cz -r.sx.sy.sz
4 q.cx.cy.sz -r.sx.sy.sz q.sx.cy.cz

5 -r.sx.sy.sz q.cx.cy.sz q.cx.sy.cz

 

 

Here p(ri) is the charge density at site i and U|i(i) is the
second derivative of the electrostatic potential at the ith
site due to all the other charges. We can express the
Cartesian quadrupole moment tensor as a linear combination

of spherical quadrupole moment tensors of rank 2:

02m(i)=frr2Y2m(i)p(ri)dr (4.27)

51

SO that H'=.Z sz(i)U2'm(i) (4028)
r,m

where Uz'm(i)'s are appropriate linear combinations of
U|i(i)'s. Now we can express 02m: which is defined with
respect to lab axes, in terms of Q which is measured with
respect to the nmlecular axes. Let (9i:¢i) be the polar
coordinates of the principal axis of the molecule i with
respect to the lab axes. Then an Euler angle transformation

gives:

szli)=Y2m(9i.¢i)Q (4-29)

Equations (4.28) and (4.29) can then be rewritten as

H'=Q.Z Ya(i)Ua(i) (4.24')

to
where the Ua(i)'s are appropriate linear combinations of the
second derivatives of the electric potential given in

equation (4.26), and are given in the reference (Sah 82, see
Appendix 8). Finally, one can take the fourier transform of

the Q-efg interaction term and express it as:

H'=ikEuYa'(k)vuaef8(K,k)uu(K,k) (4.30)

We now discuss the coupling constant matrix vuaef8(K,k) for

the two types of lattices that we are interested in.

52

The explicit expressions for the coupling matrices
coming from Q-efg interaction are given in Appendix A of
(Sah 82) for NaCI-structure.

The coupling matrix for Q-efg interaction for the
CsCI structure are easily obtained from those of the NaCl
structure by making use of the correspondence shown in Table
4.6. Note that the factors p, q and r defined in that Table
take care of the different signs and different distances of
the neighbors of the CsCI-structure as compared to that of

NaCI-structure.

(g) Elastic Constants

 

The elastic constants can be obtained from the k=0

limit of “jk' We define:
Aeff=AR+aAQ

Beff=BR+aBQ

a=1+1/(4/2)-8/(27/3)-1/16=0.943 (NaCl) (4.31)
e=8/(27/3)+1/16-1/(64/2)=.2225 (CsCI) (4.32)
AQ=i/(9n/4)Qeo/(a5), (+ NaCI, - CsCI) (4.33)

BQ=-AQ/(2/3) (4.34)

TABLE 4.6 The

correspondence

53

for obtaining

 

 

 

 

 

via9f8(K,k) for CsCI-structure from that of
NaCI-structure, where p=1/16, q=1/64/2 and
r=8/27/3.
NaCI CsCI
2a 23
1st nn 2nd nn
2nd nn 3rd nn
3rd nn 1st nn
A0 ‘PAQ
AQ qAQ
AQ 'fAQ
Case 1:

For a wave along

and kz=k and consider LA and TA phonon frequencies.

(001) direction we take kx=0, ky=0

The LA

54
frequency in the limit k=0 is
wLA2=(C11/D)k2
and the corresponding polarization vector is
C(K,k)=/(mK/m)(0,0,1)

where mK is the mass of the K-type ion, m=m++m- and p is the
mass density: p=m/(2a3) for NaCI structure and p=m/(8a3) for

CsCI structure. From equation (4.13) one then obtains
C11=C110'a11Aeff2Xe (T) (4'35)

where a11=8/a for NaCI structure and a11=2/a for CsCI

structure. Similarly for the TA phonon branch one has
wTA2=(C44/D)k2
e(K,k)=/(mK/m)(1,0,0)
and
C44=C44°'a443eff2thg(Tl (4°35)
where a44=2/a (NaCI) and a44=1/(2a) (CsCI)
Case 2:

For a wave propagating along (110) direction we take

kx=ky=k//2 and kz=0. For a TA wave in the xy plane one has

55
wTA2=(k2/2)(C11'C12)/D.
e(K,k)=/(mK/2m).(1,-1,0)
so that

C12=C120*a12Aeff2Xeg(T) (4.37)
where a12=4/a (NaCl) and a12=1/a (CsCI)

Equations (4.35) to (4.37) are in agreement with the results
of Michel and Naudts (Mic 77a) although they have been
derived in a completely different and straight forward way.
Note also that the notational susceptibilities “1 these
equations have to be calculated in presence of direct inter-
actions. The numerical results of our calculations will be
discussed in the next chapter.

The summary of the final results of elastic

softening are given in Table 4.7 (see next page).

56

TABLE 4.7 Summary table.

 

 

 

quantity NaCI CsCI
a11 B/a 2/a
344 2/a 1/28‘
a12 4/3 1/3

 

C11=C110'a11Aeff2Xeg(T)

C44=C440’a443effzxtzg(T)

C12=C12°*a12Aeff2Xeg(T)

 

 

CHAPTER 5

RESULTS AND DISCUSSIONS ON THE DISORDERED PHASES

In the previous chapter we developed a theory of
elastic and phonon softening for the orientationally
disordered phase of the alkali cyanides. In this chapter we
indicate how the various parameters appearing in the theory
are calculated and then discuss our results. We then make
comparision with experiment and comment on various aspects

of the theory.

(a)CalcuIation of Parameters

 

 

We calculate the atom-atom parameters C1 and C2 of
equation (3.1b) from the data of Fumi and Tosi (Fum 64) and

of Hirshfeld and Mirsky (Hir 79). The former work lists the

repulsion parameters between identical pairs of alkali ions
while the latter work lists the repulsion parameters between
C-C and N-N atoms. The parameters for the alkali cyanides

are then obtained by applying simple averaging schemes. For
example, for calculating C1 for a dissimilar pair of atoms
we take the geometric mean of the constituent pairs while
for calculating C2 we take the arithmatic mean of the values
of the pairs. For simplicity we further assume that C and N

ends of the anion are identical. For the alkali cyanides we

57

58

found that this turns out to be not too bad an assumption. We
indicate below the calculation of C1 and C2 for KCN as a
concrete example. The repulsion energy between a pair of K+

ions is
V(r)=C1(K‘-K*)exp[-C2(K*-K*)r]
so that following the notation of equation 3 of (Fum 64)
c1(K*-K*)=c..b.2 and c2(K*-K*)=1/p

Here b+2=(46.92)2 erg/molecule and p=0.3394 A, and the
Pauling constant c++=1.25. In a similar fashion we obtain

the atom-atom parameters for C-C and N-N repulsion. Then we

use
C1(K-CN)=[C1(K-K)2.C1(C-C).C1(N-N)]1/4
and C2(K-CN)=(1/4)[2C2(K-K)+C2(C-C)+C2(N-N)]
The parameters obtained in this way are listed in Table 5.1.

It should be noted that there are two sets of parameters for
C1 nd C2 given in Table 2 of (Fum 64). Our final results are
not very sensitive to choosing either of these two sets.
This agrees with the observation of Bound, Klein and
McDonald (BKM) (Bou 81) that modest changes of atom-atom
potentials do not affect their MD results in the disordered
phase to any appreciable extent. Moreover the parameters

for NaCN obtained by us agree with those listed by Klein and

60

adopted by Raich and Huller (Rai 79) in their work on the
azides. In the MD study of KCN, BKM (Bou 81) found that f
approximately equal to 0.5 gave good agreement between their
MD calculations on the orientational probability distribu-
tion functions and the experimental results. Our calcula-
tions also suggest that f<1. For cyanides having NaCI-type
structure we found that f=0.6 gave reasonable agreement of
our theory with experimental softening of the elastic
constants C11 and C44. For CsCN on the other hand we found
that f is less than 0.15. This should be contrasted with the
MD simulations for CsCN by Klein et al. (Kle 82) who found
that the cnfientational probability distribution functions
and phonon dispersion relations (for a few k points) are not
affected appreciably by choosing different f values. Our
calculations, on the other hand suggest that the phonon
dispersion relations are sensitively dependent on f for both
type of structures (for all k).

Next we discuss about the constants AR, BR, CR, AQ
and BQ which play an important role in elastic and phonon
softening. These constants are calculated from a knowledge
of C1, C2 and Q about which we have already mentioned. For
an NaCI type structure with short range repulsion between
the nearest neighbors CR=0 and for a CsCI-type crystal
eventhough CRaéO, it does not appear in the calculation of

elasic constants. We have listed values of these constants

in Table 5.2.

have been calculated by taking

factor f that is

indicated

61

in the table.

The quadrupolar contributions

into account

the

in this

TABLE 5.2 Repulsion and quadrupolar contributions
to TR coupling

 

 

 

 

 

table

reduction

quantity NaCN KCN RbCN CsCN

f 0.6 0.6 0.6 0.15

AR(K/A) S578 4379 3323 -706.5
BRl') -1390 -988 -713 3261.7
CRl') x x x 4127.2
AQ(') -3065 -2064 -1693 2822

BQ(') 2503 1685 1382 -2304
BR/AR -0.249 -0.226 -0.215 -4.617
BQ/AQ -0.816 -0.816 -0.816 -0.816

Another set of parameters that enter our calcula-

tions are the bare elastic constants C110, C440, and C120.

MN

(Mic

77a)

obtained the bare elastic constant C440 by

62

fitting the experimental value of the extrapolated
transition temperature to their theoretical value. Another
procedure is to use the high T elastic constants of the
alkali bromides as the bare elastic constants because Br'
and (CN)' ions are roughly of the same size (deR 81). Both
these fitting schemes have some limitations. For example it
is well known that near but above the transition, fluctua-
tions become very large and can in cases reduce the
transition temperature significantly. Since the present
theory does not incorporate fluctuation effects the first
fitting procedure is unreasonable. The second procedure,
though less unsatisfactory, has the limitation that it
refers to a crystal other than the one under investigation.
We have, therfore, followed a different procedure. We have
obtained the bare elastic constants by fitting to the exper-
imental values of the elastic constants of the cyanides at a
temperature Tfit far above the transition temperature. In
fact we choose Tfit as the highest temperature at which the
elastic constants have been nmasured. The values of the
bare elastic constants and Tfit are given in Table 5.3.

In our calculations we have incorporated the
effects of anharmonicity on elastic constants by supposing-
ing that there is an additional renormalization of elastic

constants which decrease linearly with T, at the tempera-

63

tures of interest, i.e. 6Ciian=-yiT . The reason for this
assumption is that experimentally anharmonicity effects in
the alkali halides show a similar variation. To find 7; we
have further assumed that Yi(MCN)=yi(MBr). These values are
given in Table 5.3 and have been obtained from (Nor 58, Cha
70, Ove S1, Rei 61). The elastic constant can now be

expressed as
Cij(T)=Cij°+5Ciia"(T)*5CijTR(T) (i,j=1, 2,0r 4) (5.1)

where the TR coupling contributions to the renormalization
of the elastic constants are given by equations (4.35) to
(4.37). The importance of the anharmonic terms is discussed

at the end of section (c).

(b)Isothermal Rotational Susceptibility

 

The isothermal rotational susceptibility x08(k) can
be calculated with a knowledge of the bare susceptibilities
and the direct interaction between the anions (see section
(e) of Chapter 4). The bare susceptibilities were calculated
using equation (3.7) by numerical techniques. We used the

16-point Gaussian quadrature formula of Fehlner and Vosko

(Feh 76) to calculate the bare susceptibilities. For temper-
atures T>100K, this interpolation formula gives the
paraelastic susceptibilities quite accurately. For lower

temperatures, though, one should use an interpolation

64

TABLE 5.3 Bare elastic constants, Tfit and anhar-
monicity parameters in units of 1011 dyn/cmz, K and
109 dyn/cmZ/K respectively

 

 

 

 

 

quantity NaCN KCN RbCN CsCN

C110 5.749 5.115 4.022 2.548

c440 0.752 0.470 0.411 1.190

Tfit 473 453 380 300

Y4 0.019 0.013 0.008 0.106
formula of still greater accuracy. The susceptibilities of

the paraelastic phase are plotted in Figures 5.1 and 5.2
over the temperature range of about 150 to 500K. In the long

wave length limit'we have
Xii(k=0)=XiiO/l1+Dii(k=0)xii°I (i=1 or 4) (5.2)

where X110=xego and X440=Xt230° The TVdependence of the
bare susceptibilities can be written as XiiolT)=5ii(T)/Tr so

that the total susceptibility can be expressed in a Curie-

65

Weiss forn1with a T-dependent 'Curie Constant',i.e.
xii(k=0)=Sii(T)/[T+Tcwi] (5.3)

From equation (5.3) it is clear that the direct interac-
tions, depending on their sign, can either enhance or
suppress the susceptibilities from their bare values. We
have plotted, for the sake of illustration, the enhancement
factors R1=Xeg/Xeg0 and R4=Xt2g/Xt2gQ for KCN in Figure 5.1.
Note that in our theory the Curie-Weiss temperatures are not
only functions of T but also different for IE8 and T28
symmetries. We list these quantities for two temperatures
in Table 5.4 and compare than with other works. We would
like to point out that in our work the Curie-Weiss tempera-
tures for susceptibilities of E8 and T28 symmetries are of
opposite sign and (H‘ different magnitude, whereas in the
other works they are of the same sign and of equal
magnitude. Note also that the Curie-Weiss temperatures for
CsCN are very close to zero which is a consequence of the
fact that quadrupole Imoment of the anion in the solid
environment is reduced by about 85% of its free ion value.

The short range potential strongly affects the bare

rotational susceptibilities. In the limit T approaching e,
the anions tend to be more like free rotators so that $1 is
about 0.08 and S4 is about 0.16. In the presence of a CF

potential, though, the anions prefer to orient along some

66

KCN

 

1.13

(a)

1.12

1.09 1.10 1.11 v
i
l l L
0.26 0.36 0.46
Txgxio

1.08

0.16

 

 

 

p
u)-
).

TilO’K)

KCN

 

(b)

0.25

0 23
1x3.

 

A A

 

 

 

3
1002 x)

FIGURE 5.1 Single site susceptibilities TXiiO and
enhancement factors Ri=Xii(T)/Xiio(T) for KCN. (a)i=1 (eg)

(b)l=4 (t2g)

67

 

 

 

 

CsCN
:- TXO
.14 t29
.13 -
.12 r-
TX
.11 -
.10 - 7X0
99
09 i 1 '
' 1 2 :3 4 5

1(1 02 Kl

FIGURE 5.2 Single site susceptibilities of e8 and t2g
symmetries for CsCN.

68

specific directions of the crystal. For NaCI structure,
there are strong repulsion centers along the principal
crystal axes, whereas no such centers exist along the body
diagonal directions. Hence for NaCI structure S4>S1. In
fact from Figure 5.1 it is found that 54/51 is about 10 for
the temperature range of 100 to 500 K. However as T becomes
very large, this ratio tends to the free rotator value of 2.
On the other hand for CsCI structure, it is found from
Figure 5.2 that S4/S1 is of the order of 1 in the same
temperature range. For CsCI structure S1 is large in the

low-T regime because orientations with IE8 synmetry reduce

the repulsion energy. With increase of temperature,
however, S1 decreases and 54 increases. In contrast in the
NaCl structure the (opposite happens. These differences

strongly affect the nature of elastic and phonon softenings

in the alkali cyanides with different crystal structures.

Next we examine the effect of direct interactions on
the rotational susceptibilities and hence on the elastic
softening. We discuss two cases:(i) KCN and (ii) CsCN. For
KCN, taking f=0.6, we find that D11(O)=-704K and
D44(O)=235K. From equation (5.2) it follows that R1>1 and

R4<1 which indicate that direct intermolecular interactions

69

TABLE 5.4 'Curie-Weiss' temperatures for
KCN(f=.6) and CsCN(f=.15) for k=0

 

 

 

this work 100 -13 212

' 500 -34 47

KCN Boi 78 x 86 86
Reh 77 x 231 230
Loi 80 200 -42 -42

this work 150 17 -4

CsCN ' 500 13 -6
Loi 83 x 0. 0.

 

 

enhance CH1 softening and suppress C44 softening. Since
54/51 is about 10 in the temperature range of interest we

have 1.08<R1<1.13 and 0.6<R4<0.9. For CsCN, the suscepti-

70

bilities and enhancement factors show a completely different
behavior. Taking, for example, a value of f=0.4 we find
that D11(O)=947K and D44(O)=-316K. Hence over a temperature
range of 100-500K, 0.42<R1<0.85 and 1.28>R4>1.09, so that
044(0) tends to enhance C44 softening. Since 54/51 is about
1, direct interactions are important for both types of sus-
ceptibilities. In the limit f=1 and Yi=0, direct interaction
produces most dramatic effect on elastic softening. In this
limit BR is approximately -aBQ, and therefore Beff=0. 50 C44
shows a very gradual softening in the range 300-500K.
However since Tcw4<0, the denominator in equation (5.2)
enhances C44 softening. Hence as T+chw4l, the rotational
susceptibility of T2g symnetry diverges. From equation
(4.13) we find that there is an abrupt softening of the
elastic constant C44 (see Figure 5.6b) in the neighborhood
of Tcw4-

For the sake of completeness we have plotted the
k-dependence of DaB(k) for the three symnetry directions
(001), (110) and (111) in figures (5.3) through (5.5). Some
of the direct interaction strengths show quite a bit of
dispersion for some of the directions considered. Also
notice that the direct interaction strengths change sign in
some cases as one goes from the zone center to the zone
boundary. We will discuss more about phonon softening at

non-zero k for KCN later in section (d) of this chapter.

71

 

 

 

 

 

 

 

 

 

l5 CICNIOOkI
H II
.. D---O 22
(25 om_°33
um»- 44
IO-
1
5
0
“a, (r)

 

    
 
 

..-... ........ .m...
..,.¢.‘"
a."

-15 _.....- ..----°"' ....
Y: «m-0—--0~-m<#w—~0~—~«»--4—--0~-"
J 1 J l l 1 l l

O. I 0.4 0.7 1.0

 

FIGURE 5.3 Direct interaction strengths D(00k) (in units of
100K) as a function of the reduced wave vector x; KCN
(f=.6), CsCN (f=.4)

72
2 O mutual

 

 

 

 

 

 

 

l 2 CsClek O)

 

H11
IO' D---O 22

 

 

 

 

 

FIGURE 5.4 Direct interaction strengths D(kkO) (in units of

100K) as a function of the reduced ‘wave vector x; KCN
(f=.6), CsCN (f=.4)

73

8 xcmm)

 

 

 

 

 

 

 

 

 

0.2 0.4 0,6 0.8 l
X
4% CsCNikkkl

7.5

5

5

0

Eh“ “I

 

 

 

 

FIGURE 5.5 Direct interaction strengths D(kkk) (in units of
100K) as a function of the reduced wave vector x; KCN

(f=.6), CsCN (f=.4)

74

(c)E|astic Softening and Transition Temperatures
As seen from equations (4.35) to (4.37), the elastic

softening due to TR coupling depends on A3 and Bi (i=R.Q)

and the rotational susceptibilities of E8 and T28
symetries. First we discuss the effect of the coupling
constants. The relevant quantities that deternfine the

elastic softening are the squares of Aeff and Beff.
Eventhough the signs of the coupling constants are not
relevant for elastic softening, diffuse inelastic neutron
scattering can probe the signs of these quantities. For KCN
Rowe has found that our sign of Beff (which is positive)

agrees with his experimental findings (Row 82). The second

POI"t to notice is that AR and A0 are opposite in sign as are
BR and 80. Thus there is a cancellation effect in Aeff and
Beff and this arises due to the fact that Q is negative. The
nature of this cancellation is quite different for the two
types of structures that we have studied. For the NaCl
structure, the 2nd and 3rd neighbors of a given anion have
opposite charges and it turns out that they very nearly
cancel out each others contributions to Q-efg coupling thus
making a about 1 (equation 4.31). In contrast for the CsCI
structure, in spite of the opposite signs of the charges of

the second and third neighbors, they donot cancel each

75

other's contribution appreciably, so that a is about 0.2225
(equation 4.32). Thus the exact nature of the cancellation
depends not only on the sign and value of Q, but also
sensitively on the structure.

Next we discuss the results of our calculations in

various limiting cases.

(i)Anharmonicity Absent
In the lhnit Q=0, for land Beff/Aeff=-0.226. The
large value of Aeff leads to a very strong C11 softening and
a relatively weak C44 softening. In this limit one expects
to recover the results of MN (Mic 77a) who only considered
SR repulsion contribution to TR coupling. However we donot
recover their numerical results because of a factor-of-two
error in their calculation of rotational susceptibilities.
For a brief discussion on this point we refer the reader to
the papers (Mah 82 ,page 938 reference 10; Sah 82 ,page
2993). In the other limit where repulsion is small (or T is
large) and f=1, Beff/Aeff is about -0.8 and S4/S1 is about
2, so that the TA phonons soften appreciably leading to a
vanishing of C44. ln realistic cases both repulsion and
quadrupolar effects are important and vwe have found that
f=0.6 gives a good fit to the experiment (see Figure 5.6a

and Figures 5.7 to 5.9).
For CsCN the situation is completely different. In

the limit Q=0, Beff/Aeff='4.62 so that there is a strong C44

76

softening (Figure 5.6b) and a weak C11 softening. In the
limit f=1, BR and aBQ are comparable in magnitude but
Opposite in sign. Hence Beff is small. Thus for large T ,
C44 doesnot soften appreciably whereas C11 shows a
pronounced softening due to the large difference in the
values of AR=-706.5K/A and aAQ=4186K/A. With decrease of T,
though, the susceptibility Of ng symmetry becomes hnportant
and shows a very sharp drop Off (Figure 5.6b). To Obtain a
good fit to the experiments (Loi 83) f=0.4 seems a good

choice provided anharmonicity effects are neglected.

(ii)Anharmonicity Present

The combined effects of TR coupling and anharmoni-
city' are given ir1 equation (5.1) and the anharmonicity
parameters in Table 5.3. lt is clear from that table that
for NaCI structure 74/11 is about .1 so that anharmonicities
affect C11; more strongly than C44. For NaCN and KCN the
overall agreement of C11 with experiment (Figures 5.7 and
5.8) seems to be very good. In particular the peak in C11(T)
is understood in terms of a competition between the TR
coupling and anharmonicity affect. For RbCN inclusion of
anharmonicity gives a peak, but the agreement (Figure 5.9)
is not as good. This might be due to the mean field nature

of the theory. The temperatures T44 where the elastic

constants C44 would extrapolate to zero are given in Table

5.5 . For NaCI structure the affect of including anharmoni-

77

 

 

 

 

 

 

 

 

 

KChl la]
1=expi
3. 2 o=0.5oo
3 0=0.600
4 O=0.750°
2
2 .
Q“ 3
4
I r-
2 3Tl102KI ‘ -'
CsCN lbl
o=roo
1I=L0
21:08
31:06
41:04
51:02
“0‘ 64:00
08»
C44
1
06- 2
04~
0.2 '- l 6
7 1 4
1 2 3 4 s
T(102K)

FIGURE 5.6 T- and f-degendence of C44 (a)KCN (1010dyn/cm2)
and (b)CsCN (1011dyn/cm )

78

 

 

 

 

 

 

 

 

NaCN lal
26 I- ‘
2
Cu 3
L8?
1 1 1
2 3 mo’K) ‘1 5
NaCN “’1
0J2-
C44
0" )-
1 2 3‘
1 .1 l
2 4 5
TIIOzKl
FIGURE 5.7 T-dependence Of C11 and C44
1011dyn/cm2) for NaCN; with f=0.6; 1:expt,

anharmonicity 3:theory without anharmonicity

2:

(in units of
theory with

79

 

 

 

 

 

KCN lal
2 r- 2
1
3
Cu
‘r—
L l 1
4 S
2 100210
KCN

lbl

02~

 

 

 

 

1 (102K)

FIGURE 5.8 T-dependence of C11 and C44 (in
1011dyn/cm2) for KCN, with f=0.6; 1:expt, 2:

units of
anharmonicity 3:theory without anharmonicity

theory with

80

 

 

 

 

 

 

 

 

 

 

RbCN [a]
24)»
Cu
LS?
J 1 J
l 2 132 4 S
1410 K)
RbCN lb]
(12‘
C44 ‘
0.1 '-
1 L _L
1 2 32 4 5
1110 K)
FIGURE 5.9 T-dependence C11 and C44 (in units of

1011dyn/cm2) for

RbCN with f=0.6;

1:expt, 2: theory wfith

anharmonicity 3:theory without anharmonicity

0.5

0.4

0.3

0.2

0.l

0.0

81

 

 

CsCN

---f'0J5

—f=0.0

circleszExpt

--
g-
C -
’ ~~
".
~‘
g-
5
.

‘======‘======='====-'-——£l-

<-C

 

0.5

 

FIGURE 5.10
(1011dyn/cm2) including anharmonicity.

T-dependence

Of

elastic

 

constants

for

CsCN

82

city is to reduce T44 by about 10K. For CsCN anharmonicity
effects are not only important for C11, but also for C44
because y4/y1 is about 0.9 (Table 5.3). We have calculated
the elastic softening in CsCN using equation (5.1) for f=0
and f=0.15 which are plotted in Figure 5.10. For f=0, the
agreement between our theory and experiment is excellent for
C11 and C=(C11-C12)/2 , whereas the agreement is reasonable
for C44, if we note that ours. is a rnean field theory.
Choosing f=0.15 tends to improve the agreement of our theory
with experiment for C44 but the agreement for C11 and C is
not as good. This suggests that for CsCN the bare quadrupole
moment of the anion has to be reduced by more than 85% to
obtain fits to the elastic constant data. The T-dependence
of the elastic constants are plotted in Figures 5.6 to 5.10
We conclude this Chapter by giving a brief
discussion of the softening of phonons over the entire
Brillouin Zone (Bl). For brevity we choose KCN, although

the arguments are general.

(d)Phonon Softening in KCN

 

In chapter 3 welhentioned the two theories Of phonon
softening (Ehr 80,Str 79) in alkali cyanides which incorpo-
rated the TR coupling model. Our phonon calculations are
improvements over both the above works since we have incor-
porated the effects of ionic quadrupole moment on both the

TR coupling and the rotational susceptibility. We compare

83

our calculated changes ir1 phonon frequencies due to TR
coupling in KCN with the work of (Str 79) who in turn

Obtained good fit with experiment (Sah 82).

TABLE 5.5 First order FE transition temperature

T°(expt), extrapolated and theoretical C44
softening temperatures Tc and T44

 

 

 

system T.(K) TclK) T44(K)
NaCN 288 255.4 337.5
KCN 168 '156 190
RbCN 132 130 179

CsCN 193 150 179

 

 

84

It should be pointed out that Strauch et al.
considered only 1H1 contribution to V4Q(K,k). Our
calculation provides a microscopic justification for the
validity' of this assumption because we showed that the
secomd and third nn contributions to the coupling matrix
very nearly cancel each other out for the NaCl structure.
However Strauch et al. considered only SR repulsion contri-
bution whereas our analysis also brings out the importance
of quadrupolar effects.

We define rjk=lek02'wjk2)1/2/1o13 as a measure of
phonon renormalization. We have calculated rjk for the
acoustic phonons propagating along the (001) direction. Our
results for phonon renormalization and those of Strauch et
al. are given in Table VI Of (Sah 82). It should be pointed
out that the calculation of Fig involves the calculation of
polarization vectors Of phonons. In the limit k=0 one can
analytically calculate these polarization vectors. But for
arbitrary k the calculation (H: polarization vectors is
nontrivial. We have calculated these polarization vectors
numerically by taking a rigid ion model for an fcc crystal.
While it is well known that rigid ion model is inadequate
for describing the lattice dynamics of an ionic solid, we
have nevertheless used this model tO calculate the eigenvec-

tors, mainly for simplicity. We find that at T=300K

85

inclusion (H: direct interaction affects LA softening by
about 7% whereas the TA phonons are affected to a maximum of
25%. It is clear from Figure (5.3) that D11(00k) shows a
wider variation in its range of values than D44(00k). In
spite of this, the TA phonons are influenced more strongly
by direct interaction. This is due to the fact that S4/S1 is
about 10 in the temperature range 100-500K. As can be seen
from Table VI of the attached reprint (Appendix B), for LA
phonons our calculated values are about 15-30% higher than
experiment while for TA phonons our calculated values are
about 7-32% unaller than experiment. These differences most
likely arise due to the fact that we have used a static sus-
ceptibility ir1 our calculathm1 of the renormalization of
phonon frequencies. For a proper phonon calculation one
should use the dynamical susceptibility xa3(k,w). As
discussed in (Sah 82) the inclusion of prOper dynamical sus-

ceptibility should improve the agreement with experiment.

CHAPTER 6

LANDAU THEORY OF ORIENTATIONAL ORDER

To study the orientational order in the FE phase of
the alkali cyanides we start from a variational form of the
free energy due to Bogolyubov. Then we make a Landau
expansion Of that free energy in terms of the tensor order
parameters "i (i=1,5). The coefficients of the free energy
expansion are obtained in terms of the parameters Of the
microscOpic Hamiltonian introduced in Chapter 4. The free
energy is then minimized with respect to the orientational
order parameters to obtain the ordering. Thus we would like
to understand not only the high-T elastic properties of
these molecular solids, but also the nature of low-T orien-
tational order with the help Of the model that we have
introduced. As a concrete example we examine the orienta-
tional order in CsCN which undergoes a first order FE phase
transition frmn a pseudo-cubic to a trigonal structure at

Tc=193K.

(a)Ear|ier Theony

 

There has been only one serious attempt (deR 81) to
explain the orientational order in the cyanides of Na,K and

Rb. We will refer to this as the work by dBM. They

86

87

eliminate the TR coupling by means of a canonical transfor-
mation and write down an effective orientational Hamiltonian
which is then approximated by its mean field value Hmf. The

free energy is then calculated from

Fd3M=-lenTr[exp(-8Hmf)]-

One should then self-consistently solve the AM: equations
na=<Ya>mf and obtain the free energy. However dBM donot
perform such a self-consistent calculation. Instead, they
expand FdBM in powers of "i and minimize the free energy
with respect to "i° Since the above procedure does not
necessarily give the true minimum (Sah 83b), we have
followed a different procedure and Obtained the free energy
from Bogolyubov's variational theorem.

In the work of dBM the interaction terms TQB=DQB+IGB
appear in all orders in the expansion (see section (c) Of
Chapter 4 for discussion on direct interaction and equation
(4.21) for an expression for indirect interaction). In
contrast, in our treatment T08 appears only in the second
order terms. This is particularly useful since the non-
analytic terms involving the indirect interactions as k+0
(the indirect interaction depends on the direction in which
k+0; see Geh 75) appear only once in our expression. The
Landau coefficients in our theory depend on the single site

susceptibilities and hence on the nature Of the short range

88

CF potential. It should be pointed out that for simplicity
dBM take the CF potential to be a constant, which is rather
unrealistic (Figures 2.4 and 2.5). Our calculation is free
from this unrealistic assumption.

Furthermore dBM assunm» that the ordering fields
have T28 synnmtry. Before making such an assumption one
should examine whether the ordering energies associated with
Eg-type symmetry are indeed high enough to be ignored from
the minimization process. For NaCN,KCN and RbCN we find
that the assumption of dBM about the ordering fields is not
quite correct whereas for CsCN ordering energies associated
with Eg-type symmetry are considerably higher up in energy

than those of ng-type symmetry.

(b)Effective Hamiltonian

 

We start lnr defining normalized symmetry adapted

N

spherical harnmnics ‘Ya=caYa’ Ca(-2)=IYa2d9o where the un-
normalized spherical harmonics have been defined through
equation (3.3). Then the effective rotational Hamiltonian

is
Heff=§lvoli)*vs(i)

~ ~ ~

+(1/2)Zi'T08(ii)Y0(i)YB(l)] (6.1)

89

For the sake of convenience we will drop the tilde (~) in the
following. In equation (6.1) the total interaction matrix T
is the sum of the direct quadrupolar interaction matrix D
and the indirect interaction matrix I. Matrices D and I can
be obtained from equations (4.9) and (4.21) by multiplying
these equations by proper normalization constants. In
equation (6.1), Vs(i) is the self-interaction part of the
indirect interaction which contributes to the single site
potential, defined through equation (4.20), and the prime in
the second summation indicates that the term i=j should be

excluded .

(c)Variational Free Energy

 

Let us consider the Hamiltonian H=Heff given by
equation (6.1). Let Fex be the exact free energy for this
Hamiltonian. Then the Bogolyubov variational free energy

theorem (Hub 68) states that Fex‘Fvar with
Fvar=<H'Ht>t'le"Tl[€XD('BHt)] , (6.2)

where the subscript t indicates that the trace has to be

performed with respect tO a trial density matrix

pt=Ilpit (6.3a)
i

with pit=exp(-3Hit)/Trexp(-BHit) (6.3b)

90

and Hir=Vo(i)+2ha(i)Ya(i) (6.4)
01

Here ha(i) is a variational parameter at site i with
reference to the trial density matrix. Note that we have
neglected the term Vs(i) in equation (6.4) for the sake of

simplicity. We define the tensor order parameters by
nal=tr[Ya(i)exp(-BHit]/tr[exp(-8Hit)] (6.5)

Since we are dealing with FE ordering we choose nai=na for
all i. In addition we define the following susceptibilities
with respect to the single site SR CF potential ( for
simplicity Of notation we will drop the superscript 0 in the

susceptibilities):
XaB=(1/kT)<Y0YB>
xaBY=(1/kT)2<YaYBYY> (6.6)
XaBy6=(1/kT)3<YaYBYYY5>

The order parameters can be expressed as a function Of the
ordering fields , i.e. na=<Ya>t=f(ha) which can be formally

inverted to give:

ha=¢a5n5+¢a55in5n51+...... (6.7)

Consistency then demands that

91
¢aé='(x-1)ad
9688'='(1/2)¢au¢cu¢8'exuue (6-3)

We can now expand the free energy in powers of her and
eliminate the ordering fields in favor of the order

parameters through equation (6.7) to Obtain

fvar=vaar'F0)/N=12*f3*14 (6-9)
f2=(1/2)[Ta8(k+0)+(1/pa)5ag]nan3 (6.10a)
f3=('1/6)(1/papoy)XaBynqn8"y (6-10b)

f4=1/24(1/papgpypal

xl'XaBY5i3BpadeaBGY5

+(3/pp)GpoxasprGOlnaanYn5 (6.10c)
and pa=B<Ya2>
Define g=p1
r=p3
Imposition of cubic symmetry on the susceptibilities further

simplifies the third and fouth order terms in the free
energy. The nonvanishing susceptibilities M1 the third

order are

CA=X111=-X122=82e3(1/2)(54A6+9A4-5)

where

and

Here (x,y,z)

92

CB=x345=82t3A6

CC=x331=82(e.t)(1/6)(18A6+A4-1)

CD=X441=xss1=(-1/2)X331

CE=x442=-x552=‘[({3)/2]X331

e=/(5/16n)
t=/(15/4n)
A4=<x4+y4+z4>
A6=<x2y222>
A4s=<(x4+y4+z4)2>

specify the orientation of the molecule.

Similarly the non-vanishing suceptibilities in the fourth

order are

CF=x1111=x2222=3x1122=B3e4(3/2)(9445'5A4ti)

CG=x3333=X4444=xsss5=83t4(1/12)(A4s'ZA4'8A6tl)

CH=X3344=X4455=X5533=53t4A6/3

C|=X1133=B3(et)2(1/6)(1-A4-18A6)

CJ=X1144=X1155=B3(et)2(1/12)(13A4'9A4s+18A6-4)

93
CK=x2233=B3(et)2(1/6)(3A4-2A4s+6A5-1)
CL=X2244=X2255=B33let)2(1/12)(A4'A4s'646)
CM=X1244=-X1255=B3/3(et)2(1/12)(3A4s-5A4-18A5+2)
We can now express f3 and f4 as:
f3=(-1/6)[(CA/g3)n1(n12-3n22)+6(CB/r3)n3n4n5
+(1.5CC/gr2)( n1(2n32-n42-n52)+31/2n22(n52-n42))j
f4=(1/24)[ A41(n12+n22)2+A42(n34+n44+n54)
+A43(03204ztn420523‘n52032)
+841012032+B42012M42+nszP343022032
+B44022("42*0521’34501nzln42‘nszl l
where A41=-CF/g4+3/(T.g2)+3CA2/85
A42=-cc/r4+3/(T.r2)+3cc2/(g.r4)
A43=-6CH/r4+6/(T.r2)+(3/r4)(4CB2/r-CC2/g)
B41=-6CI/(g.r)2+6/(T.g.r)+(3/g2r2)(2CA.CC/g+4CC2/r)
B42=-6CJ/(g.r)2+6/(T.g.r)+<3/gzr2)(CCZ/r-cc.CA/g)
B43=-6CK/(gr)2+6/(T.g.r)-(6/g2r2)CA.CC/g

B44=-6CL/(gr)2+6/(T.g.r)+(3/g2r2)(CA.CC/g+3CC2/r)

94

B45=-12CM/(g.r)2+6[(3)1/2/(g.r)2](CA.CC/g+CC2/r)

The B-coefficients are run: all independent. In fact a
symmetry analysis Of the fourth order terms using polynomial

invariants (syzygies) of rank 2 gives (Sah 83b)
B41/B43=(4R-1)/3; 842/843=(R+2)/3
B45/B43=2(R-1)/(3)1/2
R=B44/B43

These relations provide a check (”I the B-coefficients in
equation (6.10c') that we Obtain by using the parameters of

equation (6.1). Before presenting our results Of orienta-

tional order in CsCN, we would like to discuss about the
indirect interaction rnatrix l which appears in equation
(6.10a).

(d)lndirect Interaction

 

It is well known (Geh 75) that the indirect
interaction matrix I is non-analytic as k+0. As a result,
the eigen values of the matrix I depend on the direction in

which k approaches zero. We examine the eigen values of I
along the three symmetry directions (001), (110) and (111)
in the limit of long wavelengths. For CsCN we find that the
lowest eigen values are realized for both (001) and (110)

directions. We would like to remind the reader that in order

95

to fit our theory to the elastic softening data, we had to
choose a value of the quadrupole moment of the anion in the
which was considerably reduced from its free ion value,
namely f=0.15. With this choice of f, the non-vanishing
elements Of the indirect interaction matrix I for the (001)

direction in k-space are:
l11=-2Aeff2/(C110a)=-3.6K,
'44='55='Beff2/(2C44°a)=-2924K

For (110) cHrection the l nutrix is complicated because
there is some mixing between E8 and T28 symmetries. However
since this mixing is less than 2%, we ignore the mixing and
confine ourselves to T28 symmetry for which all elements of
I have lower energy compared to those of E8 symmetry. We

have
l33='23eff2/I8(C11°+C12°+2C440)]=-1447K
and l44=|55=l45=-Beff2/[2aC440]='1462K.

Eventhough both the directions give the lowest energy in the
(n4, n5) space, we have preferred the (110) direction over
the (001) direction for our analysis because the former
gives the possibility of obtaining a lower free energy

through a non-vanishing n3 order.

96

(e)Results

We discuss our results for CsCN in two limiting
cases of the total interaction: (i) in which the direct
interaction is dominant and we assume that I=0 and, (ii) in
which the indirect interaction is dominant (i.e. Q=.15Q0 or
Q=0). It should be reminded that in the latter case alone we
were able 1x) explain the elastic softening U1 CsCN (Sah

83a).

(i) In the quadrupole dominated direct interaction
regime, i.e. when f=1, we find that D11(O)=D22(0)=5916K and
D33(O)=D44(0)=055(0)=-3942K. Thus the ordering is in a
manifold which has T2g symmetry. Hence putting n1=n2=0 and

n3=n4=n5=n in the free energy we Obtain
fvar=(3/2)(033(0)*1/r)nz-(1/r3).CB.n3+(1/8)(A424A43)n4

Minimizing with respect to n we find that there is a first
order phase transition at Tc=255K; n=.3no at T=Tc-e where
no=(1/3)/(15/4n) and e+0(+). When the strength of the single
site potential is reduced to zero (free rotator limit), we
Obtain Tc=335K and n=.3no. Thus we see that the effect of
the single site potential is to lower the transition
temperature. Physically this makes sense because the single
site potential has minima along (001) directions and hence

disfavors ordering with T2g symmetry.

97

(ii) Eventhough (i) provides an explanation for the
first order nature Of the transition and gives rise to an
(111) ordering as seen in the experiment for CsCN, the
direct quadrupole dominated interaction is inconsistent with
elastic softening. Hence we take the reduced values of Q
(with f=.15 and 0) to investigate nature of the order. For
the k-vector along the (110) direction we find that the
transition is second order with Tc=175.2K and n3=.0005no,
n4=n5=.02n0 at T=Tc-e. For Q=0, the results are not quali-
tatively different, although the transition temperature is
increased to 209K because Beff is increased.

It should be noted that if one arbitrarily increased
I33 by a factor of 2, one could get a first order phase
transition. (Mu: has this freedom ir1 a phenomenological
theory, but not in a microscopic theory. Hence although in
principle we know how to Obtain a first order transition in
CsCN having a (111) ordering, such a transition is not
compatible with values of the parameters used in the theory
to explain the experimentally observed elastic softening in
the high-T disordered phase.

Thus ("1 the basis Of the nficrosc0pic Hamiltonian
used by us, we can understand the origin of (111) orienta-
tional order in CsCN, but the order of the transition
Obtained by us is different from that of experiment. We

Obtain a second order I-lI phase transition, whereas experi-

98

mentally the transition is first order. “1 our work we
investigated the effect Of quadrupole moment of the anion on
TR coupling and the direct interaction between anions and
also on the order of the transition. Since the cyanide ion
has higher electric multipole moments (LeS 82), it would be
interesting to investigate the effect of these moments on
the couplings and interactions mentioned above and to the

order of the transition.

CHAPTER 7

SUMMARY

In this thesis we have investigated the orienta-
tional order-disorder phase transitions in the alkali
cyanides with the help of a nflcroscopic Hamiltonian. we
have tried to understand both the elastic properties of the
disordered phase and the nature of orientational order in
the FE phase in terms of a given microscopic Hamiltonian. We
have proposed a physical mechanism, e.g. the quadrupole
moment of the anion interacting with the fluctuating efg
which contributes to TR coupling in the cyanides. We have
showed that in the absence of this mechanism previous expla-
nations Of acoustic phonon softening become incorrect.
Evidence in support of the contribution Of Q of the anion to
the properties of KCN crystal has come from MD simulations
(Bou 81). For CsCN, the MD simulations (Kle 82) do not
predict a strong dependence of the properties Of phase I on
the quadrupole moment of the anion. In contrast, we find
that the elastic prOperties of CsCN depend on the quadrupole
moment of the anion. Furthermore diffuse inelastic neutron
scattering data for KCN and NaCN (Row 82) has established
the fact that sign Of the coupling constant Beff is

positive. This is in agreement with our theory, but does

99

100

not agree with the earlier work of MN. It would be
intersting to see if our prediction of a positive Beff for
CsCN is borne out by diffuse inelastic neutron scattering
studies in this system.

This work has focused attention, for the first time,
on several important competition effects in the alkali
cyanides . These are: (i) competition between SR repulsion
and Q-efg contribution to TR coupling, (ii) competition
between SR CF potential and ordering via direct quadrupolar
interaction and (iii) competition between direct and lattice
mediated interactions. We have also found that the nature of
this competition depends sensitively on the crystal
structure. Exploring these competing mechanisms has been
one of the major contributions of this thesis.

We have also studied the orientational order in the
alkali cyanides by (deriving, an expression for the free
energy based on Bogolyubov's variational theorem. We found
that the third order terms in the free energy involve terms
which not only have E8 and T28 symmetries, but also have
ngng mixed symnetry. This finding holds promise for
explaining Observed (110) order in KCN because such an order
can be expressed as a combination of the two types Of
symmetries involed. As a concrete example of the
application of our variational theory, we have investigated

the cubic to trigonal distortion in CKCN. Eventhough we

101

understand the circumstances in which our theory can give a
first order phase transition, the actual parameters used in
the calculation of elastic softening give a: second order
transition for CsCN. It would be intersting tO see how
inclusion of other interactions affect our results

Our study has shown that the anomalous thermoelas-
tic properties of the cyanides can be understood by
postulating a reduction in the value value of the free ion
Q, e.g. 40% for NaCI structure and more than 85% for CsCI
structure. One can therefore ask: *what is the physical
basis for such a reduction? This is a difficult question to
answer and perhaps electronic band structure calculations
might throw more light on this aspect of the problem. In a
recent density functional calculation (LeS 82) it was found
that there is I“) evidence for appreciable reduction of
quadrupole and higher order electric multipole moments of
the cyanide ions in the 'solid'. Therefore one can ask the
following questions: what are the effects of including
higher order (l>2) electric multipole moments on the TR
coupling? DO these higher moments always lead to a
softening, Of elastic constants or do they compete with
quadrupolar and SR coupling terms in the TR Hamiltonian? In
the former case one has to look for additional mechanisms
that suppress the softening. In the latter case, on the

other hand, it would be necessary to examine whether the

102

assumption of an effective quadrupole moment is equivalent
to full Q plus higher order multipole moments.

For the sake of simplicity we have assumed a very
simple model for anharmonicity and obtained the parameters
in a phenomenological way. One needs to investigate why
anharmonicity effects are important for LA phonons and not
so important for TA phonons for NaCI structure. One needs
also to understand why anharmonicity effects are important
for both LA and TA phonons for CsCI structure.

A source of difficulty in the study of orientational
order is the non-analyticity Of indirect interaction matrix
I(k) as k+0. We have looked at the symmetry directions of k
and took that direction as the ordering direction that gave
minimum eigen value. One needs to find out if there is a
better way of handling this non-analyticity of the lattice
mediated interactions.

Another intersting question is: does the
application of an external perturbation to the Hamiltonian
(6.1) lead to a cross over from first order to second order
behavior? If indeed such a cross over point exists then one
should study the nature of elastic softening near the
tricritical point. Moreover one can make a renormalization
group calculation and extract the critical exponents.

we would like to conclude by saying that our work on

the alkali cyanides has focused attention on the importance

103

of some of tflu: physical mechanisms involved ir1 the under-
standing of structural phase transitions in these systems.

Improvements over our work could be the starting point for

further future studies.

APPENDICES

APPENDIX A

EQUIVALENCE WITH AN EARLIER THEORY

In this appendix we prove that equations (3.5) along
with equation (3.6) [which were derived by MN (Mic 77a)]
reduces to equation (4.13) derived by us for arbitrary k if
we assume that direct intermolecular interactions are zero
in equation (4.13). Incidentally, it may be mentioned that
our method is less cumbersome because we donot have to
invert matrices, whereas this is the case in the work of MN.

From equation (3.5) and (3.6), the dynamical matrix

in presence of TR coupling is

M=(1+F.R)'1M0 (A-1)
F=vxvt (A.2)
where x, the rotational susceptibility in presence of

lattice mediated interaction is given from equation (3.6) by

x'1=xo'1"I
and I=-vth
Hence R=(M0)'1=-(vt)"1lv'1
Let F0=vavt

104

105

where )u) is the bare susceptibility in the absence of any

interaction. Then
F-1=Fo-1-(M0)'1 (A.3)
From equation (A.1) using equation (A.3) we Obtain
MOM'1=(FO-1-R)-1Fo-1
which gives M=Mo-F0=M0-vxovt (A.4)

Since the eigenvalues of the dynamical matrices M and M0 are
w2 and woz respectiveLy, it is easy to see that equation
(A.4) is the same equation (4.13) derived by us provided we
put the direct interaction to be zero in the latter
equation. This proves the equivalence of the two

approaches.

APPENDIX B

REPRINT FROM Phys. Rev. B (Sah 82)

106

PHYSICAL REVIEW B

VOLUME 26. NUMBER 6

15 SEPTEMBER 1982

Theory of elastic and phonon softening in ionic molecular solids.
Application to alkali cyanides

D. Sahu and S. D. Mahanti
Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824
(Received 18 February I982)

We have carried out a theoretical study of the effect of coupling between rotational and
translational degrees of freedom first proposed by Michel and Naudts on the elastic con-
stants and phonon frequencies of ionic molecular solids. We have applied our theory to
the high-temperature plastic phase of alkali cyanidu NaCN, KCN, and RbCN. We find
that the competition between short-range repulsion and the interaction of the electric
quadrupole moment of the CN’ ion with the fluctuating electric field gradient strongly
influences the elastic softening and ferroelastic instabilities in these systems. The effect
of direct intermolecular interaction and anharmonicity is found to be significant in some
cases. The ferroelastic transition temperatures for the above three compounds are found
to be 337.5, 190, and 179 K which compare favorably with the experimental values 255.4,
156, and I30 K if we note the mean-field nature of our theory. Within our model we can
understand the qualitative differences between the cyanides and the superoxides, a similar
class of compounds showing drastically different ferroelastic behavior. Our calculations
provide a microscopic justification for the use of certain phenomenological parameters by
Strauch er al. in their calculation of phonon frequencies in NaCN and KCN at 300 K.

I. INTRODUCTION

Ionic molecular solids undergo a series of struc-
tural phase transitions and show anomalous ther-
moelastic prOperties which are intimately connect-
ed with the orientational, spin, and orbital degrees
Of freedom of the ionic molecular species. Typical
examples are alkali cyanides"7 (MCN), superox-
ides"'° (MOz), azidas“ (MN 3), hydroxides”
(MOE), and nitrites" (M N02) where M is an al-
kali ion. In this class cyanides are the simplest,
the (CN)' molecular ion possessing only orienta-
tional degrees of freedom whereas the superoxides
are perhaps the most complex, the 02‘ ion pos-
sessing all three, i.e., orientational, orbital, and spin
degrees of freedom.

The structure of the highest-temperature solid
phase in almost all these systems is face-centered
pseudocubic, the molecules undergoing hindered
rotations between several equivalent directions of
minimum energy. This high-temperature solid
phase (referred to as phase I in the literature)
shows anomalous thermoelastic properties and the
systems behave like plastic crystals. In the case of
cyanides careful measurements"3'° of elastic con-
stants have been made and it is found that C n and
C“ decrease with temperature and Cu approaches
zero at a temperature T‘ where one expects a fer-
roelastic instability of the pseudocubic phase.

251

However, the transition to the ferroelastic phase is
usually"”"‘ first order, the transition tempera-
ture 7', being higher than 7“ (see Table I for
values of T, and T‘ in cyanides).

The symmetry of the low-temperature phase
(phase II) is different for different classes of these
systems. For example, in cyanides, the orientation
of the (CN)" molecular axis is along the original
[110] direction" of the phase I; the structure of
phase II is body-centered orthorhombic. In con-
trast, the average orientation of the superoxide
molecule is parallel to the z axis, and the structure
of phase II is body-centered tetragonal (CaC;
structure). There are, however, significant fluctua-
tions in the molecular orientations about the c axis
due to the Jahn-Teller (IT) splitting of the 02’ or-
bital degeneracy.u In this work we are primarily
concerned with the cyanides and superoxides al-

TABLE I. First-order ferroelastic transition tempera-
ture T, (experimental), extrapolated and theoretical C“
softening temperatures T', and T“.

 

 

System T‘ (K) T‘ (K) T“ (K)
NaCN 283.5 255.4 337.5
KCN I68 156 I90
RbCN I33 I30 I79

 

2981 @1982 111. American Physical Society

2982 D. SAHU AND S. D. MAHANTI 26

though our results should be applicable to other
ionic molecular solids as well.

In this paper we deveIOp a microscopic theory of
elastic softening and phonon renormalization in
these systems. We analyze the effect of the orien-
tational degrees of freedom on the elastic proper-
ties and phonons by extending the earlier work of
Michel and Naudts" (MN). In the evaluation of
the coupling between the translational and rota-
tional degrees of freedom, we include" the effects
of (i) short-range steric (repulsion) forces, (ii) aniso-
trOpic electrostatic forces,” and (iii) the effects as-
sociated with the splitting of the orbital degeneracy
of the molecular ions. In addition we include the
direct interaction between the molecules. We do
not consider here the coupling between the spin
and either the translational or the rotational de
grees of freedom. Alkali superoxides show low-T
structural phase transitions involving large-scale
molecular reorientations (referred to as magneto-
gyric phase transitions) which can be understood in
terms of spin-rotational coupling (see Ref. 19).
This interaction does not appreciably affect the
high-T ferroelastic phase transitions. One possible
exception is the ordered pyrite to marcasite transi-
tion in NaO; (see Ref. 20) which occurs at about
200 K.

For cyanides, (iii) is not present and only (i) was
. considered in the earlier work" on elastic soften-
ing. The importance of (ii) for the cubic phase of
cyanides was recently discussed by Bound er al.,2|
although for the noncubic phase of superoxides it
has been pointed out by Mahanti and Kemeny.20
Bound et al., in their molecular dynamics calcula-
tion of the rotational-translational dynamics of
KCN, NaCN, and RbCN, have found that the in-
clusion of the electric quadrupole moment Q of the
(CN)' ion was essential to understand the experi-
mental orientational probability distribution func-
tions (OPDF) and other low-frequency local
dynamic prOperties. A careful study of the inter-
play Of (i) and (ii) in the observcd"'3 anomalous
elastic softening, ferroelastic phase transitions, and
phonon softening in cyanides is the main subject of
this paper. In a separate paper we will report the
combined effects of (i), (ii), and (iii) on the ferro-
elastic instabilities and apply our theory to the case
of superoxides.

Our main results can be summarized as follows.
Because of the large electric quadrupole moment
(Q) of the (CN)’ molecular ion, there is an appre-
ciable contribution to the rotational-translational
coupling (I‘Q) arising from the interaction between

Q and the fluctuating electric field gradient (EFG)
present in the high-T orientationally disordered
pseudocubic phase. Because of the negative sign of
Q, this coupling has Opposite sign to that obtained
from considering short-range repulsive forces alone
(to be denoted as I“). We find that when 1‘, $0
and I‘Q=0, c..—40 at a temperature T“ which is
higher than T“ where C44-+0. On the other
hand, when 1‘. =0 and rgeeo. T“ > T“, i.e., C“
softens at a higher temperature than C 11 which is
observed experimentally in NaCN, KCN, and
RbCN. Actually 1‘9 and I“. are nonzero and ap-
preciable, with I'Q dominating the ferroelastic in-
stabilities in cyanides. In contrast, I“ is more im-
portant ir1 superoxides because of smaller value of
Q Of the 01’ ions. As a result C”-+0 at higher
temperature than C“ and since C” couples to the
order parameter (1’20 ), one expects the molecules
to orient parallel to the c axis, giving rise to a
CaC; structure. This structure is seen experimen-
tally.I However, for a quantitative understanding
of the ferroelastic transition temperature in the su-
peroxides one must incorporate the orbital degen-
eracy of the superoxide ion and go beyond simple
molecular field theory.22 Within a molecular field
treatment of translational-rotational coupling and
the intermolecular interaction, our theoretical tran-
sition temperature T“ compares favorably with
the experimental values for the three cyanides (see
Table I).

For a better agreement between theoretical and
experimental C n and C“ values, we find that
anharmonicity effects,23 particularly for C n are
very impo ant. As a measure of the anharmonici-
ty we take the values of dC” /dT (for Tz300 K)
appropriate for alkali-halide crystals and find that
the observed peak in CM?) for the cyanides can
be understood in terms of two canceling contribu-
tions to dC” /dT; one coming from anharmonicity
effects and the other from the rotational-transla-
tional coupling. The results for NaCN and KCN
are extremely good but for RbCN there are
discrepancies.

The outline of the paper is as follows. In Sec. II
we discuss the model and the Hamiltonian that we
have used to study the elastic properties and pho-
nons of ionic molecular solids. In Sec. III, a
Green‘s-function method is used to calculate pho-
non frequencies and elastic constants which are re-
normalized by the coupling between the transla-
tional and rotational degrees of freedom. Section
IV contains a brief discussion on the isothermal ro-
tational susceptibility which plays an important

26 THEORY OF ELASTIC AND PHONON SOFTENING IN IONIC . . . 2983

role in the T dependence of elastic softening. In
Sec. V, we discuss the different contributions to
the translational-rotational coupling. Finally in
Sec. VI we discuss our results and make compar-
ison with earlier theories and available experi-
ments.

II. HAMILTONIAN
A. Model

We treat the CN' ion as a rigid dumbbell con-
sisting of two identical centers separated by a dis-
tance 24. Each molecule sits in an octahedral cage
(in the highoT phase) of six nearest-neighbor (NN)
M * ions, the NN distance being a. The M 1’ ions
are represented by spherically symmetric charge
distributions.

In addition to the electrostatic forces, there are
short-range (SR) repulsive forces between the ions.
This repulsion can be expressed in a Born-Mayer
for'm,

V33<fl=tc.).,e"cz’°“?‘ , (2.1)

where a (or 5) stands for any one of the two atoms
of the anion or cation. The constants (C . )GB and
(C2)” represent the strength and the inverse of the
range of the repulsion potential, respectively. The
quantities (C 1 I“ and (C 2 la, are available in the
literature,"‘” and one can use the equations

(C1)”: V (C) )MIC1)”
and (2.2)
(C,).,,=-;- (C,),,+(c,),,]

to Obtain the values of C 1 and C; for appropriate
systems.

The short-range repulsive interaction between a
(CN)" ion whose center of mass (c.m.) is at R,- and
a M "’ ion at R, is given by a sum of atom-atom
potential,

VZR+(CN,-(ij)=C,‘ 2+ Ie’cz' “‘I*‘“" , (2.3)
where if,- is a unit vector specifying the orientation
of the (CN)’ ion with respect to the crystal axes.
We have also assumed that both C and N atoms
can be replaced by an average atom whose repul-
sion with M 1’ is characterized by the parameters

C I and C 2. Following MN we discuss the elastic
properties of these systems using a Hamiltonian H
that consists of three parts, i.e.,

H =11, +H,,, +19“, , (2.4)

where

 

f .0 —0
H"=iz.,, 2m, pu(x| k)p,,(x| k)

+%_. 2 Cw-(xx'l Itohrzlxl It.)u,,v(x'| k)

“"3"" (2.5)

represents the translational part of the Hamiltonian
in the harmonic approximation and is obtained by
treating the ions as spherical charge distributions.
For example, this part would be analogous to that
of a KBr crystal.” Here m‘js the mass of the xth
ion (+ or -) in a unit cell, k is the wave vector,
y is the Cartesian component x,y,z; CM-(xx'l k) is
the dynamical matrix and 11,3 are the Fourier
transforms of the displacements (from the fee
structure) and momentum, respectively. Here and
in the following we define Fourier transforms by
the equation

fir): ftir'k‘r'T. (2.6)

I
~47 ’5-
where N is the total number of unit cells. We will
use l/t/I—V- in the definition unless otherwise speci-
fied. The rotational part of H is Obtained by fix-
ing the c.m. of all the ions at fcc sites (a?) and is
given by

2 _, 4 N
3,0,:2 2 fiLXIkILLIkH-Z V001,.)
k 1-] [II

+ 2 Vd(ij) . (2.7)
(:1)
Here I is the moment of inertia of the dumbell
about each of the two principal axes and L is the
angular momentum. Vo(fi,-) is the orientation-
dependent single-site potential which is given by

6 -°0 ..
Volfin=q 2 2 e-C2| “‘I*""" . (2.8)
I I I S I 11
Only the repulsion contributes to VOW) because
the electric field and electric field gradient at the
lattice sites vanish because of cubic symmetry. In
Eq. (2.8) only NN contributions are retained be-
cause Of the short-range nature of the repulsive
potential. The other contribution to the cubic
single-site potential will come from the anisotropic
dispersion interaction between (CNI‘ molecules.
This contribution was evaluated20 for sodium su-
peroxide (NaOzl and was found to be about 6% of
the short-range repulsion contribution. We expect
a similar behavior for the cyanides.
We have included the term le1’]) in Eq. (2.7)
which represents the direct interaction between two
(CN)' ions at sites R? and R). We write

2984 D. SAHU AND S. D. MAI-[ANTI _2_§

2
V,(ij)= 2 .4... ma, )Y;"(o,) , (2.9)
n --2
A... =AEQ+A:'+A.’,‘° . (2.10)

Here A... is a measure of the strength of the direct
interaction which has three main sources”:
quadrupole-quadrupole interaction, short-range
repulsion, and anisotropic dispersion, the last one
arising from the fluctuating dipole moments of the
(CN)’ ion. In addition, for cyanides there is a
direct electric dipole-dipole contribution to V4117)
and its effect will be discussed later. The unit vec-
tors 0, and (’0‘,- are the orientations of the molecules
i and j with respect to the intermolecular axis tak-

_I

 

 

4 2
V,(ij)=2t/4rr(21+1) 2 A...

[.0 III—2 m

 

 

en as polar axis. The coefficients 4,?“ are explicit-
ly given as follows:
489:124775102/(1/‘211 )5 .

AWeAggs-i-AEQ , (2.11)
499=4€i=iAF°.

where Q is the quadrupole moment of the mole-
cule. For the rest of the coefficients we refer the
reader to the literature.”

In order to go from a system of reference where
the intermolecular axis is the z axis to the crystal
axis system one makes a transformation involving
the Euler angles” (01113111711) associated with the
vector Ru to obtain

2 I

M) m; -—(m1+m;)

X YLRI+M1(BU’YU)YZM (fiiyhfifij) . (2.12)

where the quantities in the square brackets are the Clebsch-Gordan (CG) coefficients. Using the prOperties
of CG coefficients one finds that only even I terms contribute in Eq. (2.12). Further if one considers only
the quadrupole contribution to A,” then only the 1= -4 term survives.

Next we introduce the five symmetry-adapted spherical harmonics I’. (see Ref. 16) through

5
rh(£,)= 2 c,.Y,(fi,-) ,

 

 

(2.13)
«In!
where the 5x5 matrix (cu) is given in Table II, and obtain
2 my): ~23 2 0,,(E’)r§(i€)r,(it'), (2.14)
((1) It c.5-l
where j
.. - - 2 2 2 2 4
D (1t)= “P" War
68 Rae ”£2 "1' n1; —("1)+M2) §Am "I —m 0
X Ym'+mz(Bfi,Yfi )cmacmzp . (2.15)

Here the sum R is over (CN)’ ions surrounding
the central (CN)" ion at R=0.

Finally the last term in Eq. (2.4) represents a
coupling between the orientational degrees of free-
dom Y. of the molecular ion and the translational
degrees of freedom of the anions and cations. We
write
H.,.,,=i 2 2 Y;(i£)v.,,(x I if»), (x I it‘).

kuach‘F'

(2.16)
This is obtained by displacing the center of mass

 

I
from the equilibrium position, i.e., i, =§?+ii,
and calculating terms in the Hamiltonian which
are linear in the displacements 11,. The elements
of the coupling constant matrix 124,, form a 3x5
matrix. An explicit form of 0,,“ including dif-
ferent contributions will be given in Sec. V.

B. Phonon description

If we describe the translational degrees of free-
dom in terms of phonons, then H" and H" m, can
be rewritten in terms of creation (b 7;) and destruc-

26 THEORY OF ELASTIC AND PHONON SOFTENING IN IONIC . . . 2985

TABLE 11. Coefficients of expansion of unnormal-
ized real order parameters Y. in terms of Y;,,,‘s.

 

 

 

N 1 2 3 4 s

—2 o \/l—/_6 i/2 o o

—1 o 0 0 —% in
o 1 o o o o
1 o o 0 § :72
2 o m —1'/2 o o

 

 

tion (bi?) Operators of phonons of wave vector I;
and polarization index j. We have

11,, = 2 fingflbI-gbj; + -;-) . (2.17)

The bare phonon frequencies (of? are obtained by
solvingthe secular equation

[Cw-(xx'l k.)—w%2‘5w'8“o| =0. (2.18)

In terms of phonon creation and destruction Opera-
tors, we have

H,,,,,,,=i 2 r;(E)Va,(E)(bj-,o+bj_;),
1.411:

(2.19)

 

V,,(iE)=(1/2w,°-;)"’2

\/1— e”
(2.20)

In Eq. (2. 20), eu(x| 11' j) gives the pin component of
polarization vector for x-type ion for the mode jk.

V,,-( k) is determined from a knowledge of bare
phonon frequencies, the masses, polarization vec-
tors, and the coefficients 1),“.

III. RENORMALIZATION OF PHONON
FREQUENCIES AND ELASTIC CONSTANTS

The rotation-phonon coupling [Eq. (2.19)] renor-
malizes the phonon frequencies from their bare
value (09;. We use the Zubarev’s Green’s-function
method to obtain the renormalized phonon fre-
quencies “jif- We define the time- and tempera-
ture-dependent retarded Green’s function G by

I l + I I
Gjrtr—r )=7([¢j;(1),¢j;(t )])9(t—r ) , (3.1)

where ¢j;(t)=bj;~(t)+b;_;-(t) is the phonon field
operator in the Heisenberg representation. The

e(x | kj)vau(x l k) .

square bracket is the commutator, ( ) stands for
the average over a grand canonical ensemble, and
9(1—1') is the step function. The equation of
motion for G is

. d ,
IEijU-I )

=5“—r')([d,;-(r),¢;;(r')])

+1.90 -t')<[[¢,;~(:),H],¢};(i')]> . (3.2)

For simplicity of notation let wo=w°;° and

d,‘ . =42). 11 (r =0). The Fourier transform of
G,-,~(: —r') is given by the equation

w(<¢,;~;¢};->>.=Media-l)

+ « [¢,;.H1;¢};))., . (3.3)
where
((¢,r;¢;r)).,='_-Gj;(w)= f; G,;-(:)e‘°'d:.
(3 4.)

Since b}; and b 3; satisfy the usual boson commu-
tation relations, we obtain

[¢j" k oHtrl=w0¢j~ k s

I ¢i?'”ml=l¢jrflmml=0. (3.5)
[d’jrsutg]=wo¢j‘r ,
where
1
"OF jr- ;r. (3.6)

Using the above equalities it follows that
(mi—eg)((¢,;;¢};»,
=22)o 1+2 V:,<< hm]; »., . (3.7)
a

and

w«Y¢;¢:-;°»..=«Y1.a;¢;1‘»., . (3.3)
where

Y,,,(E)E[Y,(E),Hm] . (3.9)

It is clear that to obtain the Green's functions on
the right-hand side (RHS) of Eq. (3.7) one needs
the Green's function on the RHS of Eq. (3.8) and
the hierarchy of equations extends to infinity. We
further write

Yr.a(E)=[Yr—l.a(E)oHrot]
(f=2,3,...,oc). (3.10)

2986 D. SAHU AND S. D. MAI-IANT I 26

 

Note that the operators Y, ”1’2. m. . .do not con- ‘ with d and H". Hence one can write. using Eq.
tain the operators b and b and hence commute (2.19),
1
a)« Y.,.;¢};~»,z (< 1'2,.;¢};~»,.+2 V,,-(E)([Y,,,(E).1',(i’)])«(iflmj-r», . (3.11)
a?

In obtaining Eq. (3.11) we have replaced the commutator [Yup 11,] by its average value. This is equivalent
to a random-phase approximation. We can now generalize Eq. (3.11) to higher-order Green’s functions and

obtain

[w2—w3—2,r(w)l((¢,;;¢]; ))..=2wo . (3.12)
where the phonon self-energy 2,,(w) is given by
2 ;(a))=2wo2 V;,(k)V,,(k)2w ——+T,<[Y,,,(i€), Y;(k)]). _ (3.13)

We will now relate the phonon self-energy to the frequency and wave-vector-dependent rotational suscepti-

bility rag k pm)

The rotational susceptibility is defined by a related orientational Green’s function,

 

x,,( i,:—:')= —%([Ya(k°,r),}';( E,:')])9(1 —t'). (3.14)
The time Fourier transform of Xa5( kit—1') is given by the equation
rad Em): _ (( 1'.( E); min)”: - % f_" :11 ew'eu -1')( [ m E1), Y;( 16,131) , (3.15)
w((Ya(k.);Y5(k.)»u=((Y1.5(E),Y;(k.)». (3.16)
1
Let us assume that the rotational response is deter- 2 o2 0 -° , '° "
.-o.-. 2.20,... V‘ k , V ~ k .
mined by Hm, alone. With this approximation, 0’ k a” k I " B “1‘ ”(‘5‘ k w) B’( )

which is equivalent to the assumption that rota-
tional dynamics has a faster time scale compared
to translation,31 we obtain

e(( 13,211}, ».,=(1r.,.,.Y;1)
+ << Y”; Y} )>., . (3.17)

Generalizing this procedure to higher-order
Green‘s functions, we obtain

«11.3%»=2([Y,,(E),Yg(i£)]>—

r-I

(3.18)
Using Eqs. (3.18) and (3.15) in Eq. (3.13) we get

217(w)=—2¢ooEbfij(k)V,J-(k)xap(k,w) .
a.

(3.19)

Thus from (3.11) and (3.18) and noting that the
renormalized phonon frequencies ((1).; are obtained
from the poles of the Green’ 5 function

«blindly )).,. we get

(3.20)

The above equation ignores vertex corrections
and is not adequate when the time scales of rota-
tional and translational dynamics are comparable.
However, we are primarily interested in the elastic
softening (“1): —>0) at relatively high temperatures
where the rotational motion is rapid and the above
approximation is quite reasonable. For the calcula-
tion of phonon frequencies at finite k particularly
when ((1,), ~01”, where aim, is a characteristic rota-
tional frequency, one has to consider the frequency
dependence3| of lad k ,w) and also include vertex
corrections.

From Eq. (3.20) one can easily obtain the effect
of rotational-translatiogal coupling on the elastic
constants by choosing k along several symmetry
directions and studying the frequencies of longitu-
dinal and transverse phonons 1n the li_r_nit k, (1)—.0.
The details of the calculation of X0,“ k ,0) are dis-
cussed in Sec. IV and in Sec. V, we will give expli-
cit expressions for the renormalized elastic con-
stants.

26 THEORY OF ELASTIC AND PHONON SOFTENING IN IONIC . . . 2987

IV. ROTATIONA}. SUSCEPTIBILITY
Id k,m-=0)

For the calculation of elastic constants and pho-
non frequencies we replace X.p( k,w) by its static
values Xadk )=X.g(k, (11:0). This 1s adequate for
the elastic constants and the limitations for phonon
frequency calculation will be discussed in Sec.
VIE. I'd k) ts the static susceptibility of an iso-
lated system subjected to an adiabatic perturbation.
Following the commonly made approximation for
large systems we rgalace lad k) by the isothermal

tibility 14 k), 1 e we assume that
I‘d k )zl’dk) even though the differences be-
tween the two need not be zero in general. 32 Next
we calculate 115 k) 1n the presence of only the
direct intermolecular interaction Dap( k) using a
molecular field approximation.

The rotational response is determined by H“,
which is replaced by its man-field value H “F:

11.5 I!
(4.1)

where
m,(i€)=<1',(i£1> . (4.2)

We apply a staggered external field ha( 1:) which
adds a term H“, to the Hamiltonian H ”F,

11...: —2 h,(i€)Y,(iE) , (4.3)
B

and calculate the susceptibility X in the limit when
the external field vanishes. In the presence of

hp( k)

m,( k )= Tre-‘Vk'nmm+”“‘) . (4.4)
The generalized susceptibility matrix X.g( link"),
defined by the equation

 

m.(i£') , (4.5)

 

a
X k' ,k)= lim
d hr“, 8h,(k)

is found to satisfy the matrix integral equation,
1

‘ f9 -voto.Te)/k. Y‘(0¢)Y5(91 ¢)sin6d94¢.

x.,( in?)
=x°.,(i",i° )— x°....(it",i°")
«'5 "f"
xnd,.(i’")x,.,( it"s?) .
(4.5)
where

1:,(i‘,i")=-‘—-[(r:(i°')r,(i‘))
1,1

-<Y:(i"))(1',(i'))l. (4.7)

the thermal averages being taken 1n the absence of
H“, .For the disordered phase (Y+(k))= -0and
Hur- =2 VOW) and we have

x°.,(i(", E1=x°., 5;. 1... ‘ (4.8)

where X2, 1s the k-independent single-site suscepti-
bility. '6 In this case I adk, k' ) 1s diagonal in the k
index and defining XJk):-— X.g(k, k), we obtain

x.,1i£1=x°.,— 2 1°...Dd,-(i'1x,.,(i‘). (4.9)
a’fi'
From Eq. (4.9) one obtains,
mal solution
X=(1+X°D)"X° . (4.10)

As shown in Ref. 16 because of cubic symmetry 1°
is a diagonal matrix with

r°..=(x?..3x?..x23.x‘3..x231 . (4.11)

where the first two quantities have e, symmetry
and the last three have 1,, symmetry. Similarly 1n
the k-eO limit D(k) is diagonal with

D "(k—+0)=(D?.,;D”,D§’,,D§,,D§3).
(4.12)

From (4.8)-(4.10) 1: follows that in the iii-.0 lim-
it I is diagonal with

xm=(X|1,3X11.XJ3.X33,X33) °

The k-independent susceptibility X24 T) is calcu-
lated from

symbolically, the for-

 

x°,,,(r)= kT

-V(0,T‘)/k
[9 o a

sm6d6d¢

(4.13)

V. ROTATIONAL-TRANSLATIONAL COUPLING COEFFICIENTS

As discussed 1n the Introduction, there are two main physical sources that contribute to the rotational-
translational coupling matrix vm(x| k) and we write down this as a sum of two parts,

2988 1). sum AND 8. D. m g

“(x1i1+u§§°(x1i') (x=_t),
(51)

vuhrl kl=vk

where 1:2,,(xl k) denotes the short- -range repulsion
contribution and 113° (xl k) denotes the contribu-
tion coming from the interaction between electric
quadrupole moment of the (CN)' ion and the flue-
tuating EFG produced at its site by all the other
ions (taken to be point charges). We denote by
l,2,3,. . . the contributions from first, second,
third, etc., . . . neighbors, respectively, which have
charges +, -, +, and soon. Then

L,(+|i€)= 6,,(+|i’).+v:,,,(+|k),

+ . .. 11:11, rare), (5.2a)
vg,(— 1 E1=v:.,,(— | if),+v:.,.(— If).
+ - -- (1' =12, EFG). (5.2b)

Following Michel and Naudts" we will take
2,,(xl k)=v:,,(+ l k), because the short-range
repulsion falls off rapidly with distance. In calcu-
lating 05:00:] k) we include contributions up to

fourth neighbors only, further neighbors making

 

A. Short-range repulsion contribution
:J + I k )1

The coefficients 0' in k space can be obtained
from their F-space values through the relation
.. .. ‘30
5.1+ 11:),= 2651-1112319 0. (5.3)
I.

where the _sum is to be carried over all the NN
positions R3 given by (10.9.0.0), (0,169.0).
(0,0,:45‘). From Ref. 16 we have

)1. -4. 0 o 0

6:,(+1a£).= 0 0 a. 0 0 ,
0 0 0 a. o
'0 0 a. 0 0
65,,(+|ay).= A. A. 0 0 0 , (5.4)

L0 0 008.

l 0 0 0 a. 0

62,,(+ |a£),= 0 0 0 0 3,.

-2A,. 0 0 0 O

The quantities AR and 8,; depend on C1, C2, (1,
and a. They are explicitly given in Eqs. (A13) and
(A15) of Ref. 16. The values of C1, C2, (1, and a

for the cyanides are given in Table III and the
values of A, and B. are given ink Table IV. The

 

 

insignificant contributions since the interaction Fourier-transformed quantities 0‘ ”H- I k )1 are
falls off as l/r‘. then
1
4,5, —A.s, 3,5, 3,3, 0 ’
65,,(4- | i).=2 11,5, 4,5, 8,5, 0 13.5, , (5.5)
—ZARS, O O 8.5, 3.8,
1
where net rotation translation coupling is either enhanced
. or suppressed.
Sg=flnk‘a, Cl=005kla (I =X,y,Z) . (5.6)

3. Calculation of 83°(x1i')

As the lattice vibrates there is deviation from
the local cubic symmetry at the positions of the
molecules and the resulting EFG couples to Q. To
the lowest order in displacement this leads to an
additional coupling between rotation and transla-
tion with strength given by vffiGMI k). Depend-
ing on the sign of the quadrupole moment Q the

TABLE III. Repulsion parameters ( C ..C;), lattice
constant (a), molecular size (21!), and free-ion quadru-
pole moment value (Q0). See Eq. (2.11) for .4 8°.

 

 

NaCN KCN RbCN
c, (10’ K) 1.013 2.347 3.421
C2 (A"l 3.3382 3.3382 3.3382
0 (A) 2.944 3.250 3.415
d (A) . 0.615 0.600 0.575
Q0 (10-'° esu A2) —4.64 -—4.64 —4.64
.189 (K) 1870 1143 892

 

 

TABLE IV. Repulsion (A.,B.) and quadrupolar
(240,89) contributions to the translation-rotation cou-

pling.

 

 

 

 

NaCN KCN RbCN
Q= 0-6Qo 06% 0696
A. (K/A) 5578 4379 3323
a. (it/A) -1390 —988 ..713
.19 (it/A) -3065 -2064 —1693
so (K/A) 2503 1685 1382
B. M. —0.249 —O.226 —0.215
some -0.816 -0.816 —0.816

 

 

The quadrupole EFG interaction can be written
33
as

2 Q;,U;',, , (5.7)

i um

where the Cartesian components of the quadrupole
moment and the field gradient tensor at site 1' are

QL.= f13x,,,x,,-8,,,;})p(r,)dr

 

 

(5.8)
(fl,V=X,y,Z;fi=?—R;) ,
:- .__§_ 3 ~
U" — 6X” 8X, U(X)
6 a (I; ~
=— ' (XER,)
6X, 6X, 5;: 111,—le
(5.9)

Here p( r,- ) is the charge density at the site 1' and
U (R ) is the electrostatic potential at the ith site
due to all other charges q, at the R. We can ex-
press Q“, as a linear combination of spherical
quadrupole moment tensors of rank 2 defined by

Q5... = f r,’i',,.,(?,1p(r,)dr (5.10)
so that Eq. (5.7) can be rewritten as
2
=2 2 Q§,U{”(i), (5.11)
i ll! .1 -2
where U I ""s are appropriate linear combinations

of Uzv’s. Since Q is measured with respect to
molecular axes (MA) one should transform from
the lab axes to the molecular axes, i.e.,

ng .—_ Yum, )Q , (5.12)I

 

26 THEORY OF ELASTIC AND PHONON SOFTENING IN-IONIC . . . 2989

 

 

 

 

     

2
z " pmdr (5.13)
2 MA
Using (5.11) and (5.12) we can write
, .
H'=QZ 2 743.102.. (5.14)

l 0.1
where (dropping the superscript i in UL)
U, =1/iT/‘5tl.
u,=1/u743(u,.-U,,).
U,= —1/2';r/—15U., , (5.15)
U,= - VziT/Tw.
U, = nan—71511,.

The components 0,,' are obtained from Eq. (5.9),
i. e. .,

523'” A: [(Xiivulu'l'xiiuulv)

"(sji'jjifi—Oflyliu’fijl , (5J6)

WhCl’C iij=§?-fitog £7 =21) /Xu, 8116! X0, is the
vth component of le' In obtaining Eq. (5.16), be-
cause of inversion symmetry the Ti, terms drop out
when the sum over j is carried out. Finally one
obtains the coefficients ”er-‘0‘“ | X11) and its
Fourier components from” the identity

H=2nmwg
l

= 2 ms, 16555132111141.

Ma.»

El; Y; (klvg,

k.“

The quantities vEFthl k) have been explicitly cal-
culated in Appendix A.

F0(1:1'1't')ii,,(x|i€). (5.17)

C. Elastic limit

To extract information regarding the elastic con-
stants C11” we take the long-wavelength limit
( k-sO) and retain the leading terms in sines and
cosines and obtain

Aefka -Aefi’kx Beffky Bell": 0

u,p(+ 1E)+u,,,(—|'1E'1=2a A,,,k, A,,,k,

—2.(,,,k, 0

Bmkx 0 Beflkz , (5.18)

0 Bmk, Bah,

2990 D. SAHUANDS. D.M.AHANTI 2_6

where
Aefl=AR +GAQ , (5.19.)
BdT=BR +089 , (5.19.2)
and
AQ=V9«/5flf—L= —\/3/239 .
a
(5.20)

a=1+1/(41/2)—8/(271/§1-%+~-

I. Care!

For a wave along [001] direction we take 1:,
=k,=0 and k,==k and consider the frequencies of
LA and TA branches.’4 The longitudinal acoustic
(LA) frequency in the limit k—~0 is given by

mi,=(c,,/p)k2 (5.21a)
and the polarization of the vibration is
0
e,,(x| k)=\/m./m 0 , (5.21b)

1

where m, is the mass of the x-type ion, m =m .,_
+m _, and p is the mass density (m /Za 3l. Substi-
tuting (5.18) and (5.21) in (2.20) and using (3.20)
we obtain

Cu=C?1-%Agflx“(n . (5.22)
Similarly for the transverse acoustic branch (TA),
using

mi-A=(C« /p))c2
and

1
("(KI kl=v al./m 0 ,

0
we obtain
c“ =c2, — %B§,x.,( T) . (523)
2. Case 2

For a wave propagating along the [110] direc-
tion, we take 1:, =k,=k/t/2, k,=0. For a TA

wave in the xy plane, using
612,,=k’(c,,—c,,1/(2p1
and

1
2.11:1 i)=\/——m./m 1M2 -1 ,
0

we obtain
c.,=c?,+-:-A;,x,.m . (5.24)

Equations (522)-1524) are in agreement with
the results of Ref. 16 when Q20, although they
have been derived in a completely different way.
These equations will be used in the next section to
study the T dependence of the elastic constants for
the cyanides.

VI. RESULTS AND DISCUSSION

The temperature dependence of elastic constants
C", C“, and Cu given in Eqs. (5.22)-(5.24) de-
pend upon (i) the short-range repulsion (243.83)
and quadrupole contributions 049,89) to the
translational rotational coupling, and (ii) the rota-
tional susceptibility 1.51 k) obtained in the pres-
ence of direct interaction D.p( k ). The T depen-
dence of Xdk) comes from that of X25, the
single-site susceptibility. From Eq. (4.13) we see
that apart from the l/k3T factor, the T depen-
dence of m k) is determined by the single-site po-
tential V001} ). In the cubic phase, the electric field
and the electric field gradient vanish at the lattice
sites and therefore the only contribution to V0033)
comes from the repulsive (steric) forces and be-
cause of the short-range nature of the latter only
the neighboring cations contribute to 170(6)). In
Fig. l we give the (0,¢) dependence of Vo(fi,l for
KCN which shows the four minima along the
[111] and equivalent bodyodiagonal directions. The
maxima are along the [i100], [0:10], [0011], and
the saddle points are along the [110] and its
equivalent directions. The values and the T depen-
dence of the different components of the single-site
susceptibility X3 are determined by the strength of
the repulsion and will be discussed in detail later.
For the superoxides, there is an additional contri-
bution to V008,) which comes from the splitting of
the orbital degeneracy of the 02" ion as the mole-
cule orients away from the symmetry directions.20

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_ THEORY OF ELASTIC AND PHONON SOFTENING IN IONIC . . .

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2991

  

 
 
             
   
   
   
   

  

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FIG. l. Single-site potential Vo(0.d) for KCN. Vo(0,0)=99l4 K (max); Vo(99,1r/4)=8350 K (min), where 00: 54';
Vo(1r/2.rr/4)=8387 K (saddle point). 9 is measured from the [00l] and 4: from the [100] axis.

A. Repulsion parameters and quadrupole moment

In the present and all earlier"2| calculations, C
and N atoms of the (CN)‘ molecule have been as-
sumed to be equivalent so far as the strength of
atom-atom repulsion is concerned. In other words,
the repulsion parameter between a positive ion M
and any one atom of the CN molecule is given by

C1 = V (Cl )M-C(Cl )M-N v

I (6.1)
2="[(C2IM.c+(C2IM it]-

Using Tosi- Fumi" parameters for metal ions and
the parameters of Mirsky el al. 25 for CC and NN
we have calculated C l and C 2 and the values are
given in Table III. Bound et al.2! have used the
same values (excepting for a few minor differences)
in their molecular dynamic (MD) study. Actually
there are two sets of parameters given in the paper
by Tosi and Fumi; our final results are not very
sensitive to the choice of these two different sets.
The value21 of the quadrupole moment (Q0) of the
free (CNI' ion is given in Table III. In their MD
study involving KCN molecules confined to a fin-
ite cube, Bound et al.“ found that in order to ex-
plain the on'entational probability distribution

function (OPDF) they had to reduce the value of Q
by a factor f=0.5, i.e., Q=fQo. They argued that
such a reduction could arise from the charge redis-
tribution of (CN)" ion when it is placed in a solid
environment. While this is an important physical
effect. the precise value off depends on the nature
of the approximations made in obtaining the
OPDF. In particular, Bound et al. considered a
small system and did not allow for volume fluctua-
tions. Instead of using their value of f, we have
used a slightly different value, i.e., f =0.6 to fit the
long-wavelength elastic constant data for the three
compounds NaCN, KCN, and RbCN.

For the superoxides we find that the repulsive
forces are slightly stronger compared to cyanides
whereas the free-ion quadrupole moment20 is about
a factor of 2 smaller. These differences along with
the orbital degeneracy of the 02" ion are primarily
responsible for the experimentally observed” qual-
itative differences between the cyanides and super-
oxides as regards the nature of structural phase
transition in these systems is concerned. In Table
III we also give the values of a quantity
A8m=(2411'/5)Q(‘~;(\/2_a)s which measures the
strength of the direct intermolecular quadrupole
interaction. The repulsion and anisotropic disper-

2992 D. SAHU AND S. D. MAHANTI 2_6

sion contribution to the direct interaction which
was found20 to be important for NaOz are not so
important for the cyanides because of large Q and
for the other superoxides because of a large value
of a. lnthe present calculation, repulsion and an-
isotrOpic dispersion contributions to the direct in-
termolecular interaction are ignored.

We would like to make a few comments about
the effect of the nonzero electric dipole moment
(9 ) of the (CN)‘ ion on the elastic softening at
high temperatures. Taking the calculated value2|
of la | =.O 30, we find that the strength of the
dipole-dipole interaction measured by the quantity
e.=92/tax/2')3~ ID x which IS a factor of 10
smaller than the quadrupole-quadrupole interaction
energy measured by 69: =Qo/(a\/2)5= 100 K.
Therefore the dipole-dipole interaction can be
neglected while calculating Xap. The coupling be-
tween .9 and the fluctuating electric field E can
soften the phonons just like the coupling between
Q and the fluctuating EFG. But we have found

that for symmetry reasons the former does not
J

a
5,:—
R

and depends upon the parameters (d /a) and (C20).
For the cyanides 532—025 (see Table IV) and it
turns out that if one considers repulsion alone, C ,1
goes to zero at a higher temperature than C“.

The parameters Ag and Hg depend on the values
of the quadrupole moment of the (CN)" ion in the
solid. As has been pointed out before, there is
some evidence from the molecular dynamic stud-

2' of rotational autocorrelation functions in the
cubic phase of KCN that Quid < Q0, the free-ion
value. For the entries in the Table IV, we have
used the value Q=O.6Qo; this value was obtained
by making a reasonable fit to the experimental
values of elastic constants C u and C“ over a large
temperature range. With Q=0.5Qo (a value sug-
gested by Bound et al.) we were unable to obtain a
decent fit over the entire experimental temperature
range. Of course it is possible to change the value
of repulsion parameters slightly and obtain a dif-
ferent value for Q which gives an equally good fit.
However, our main purpose is to point out the im-
portant role of the quadrupole-fluctuating EFG in-
teraction on the elastic softening rather than to ob-
tain a very good fit to the experiment. It will be
pointed out later that anharmonicity and non-
mean-field effects are important and should be tak-

contribute to the softening of the elastic constants
because in the presence of this coupling only
(02 —woc: k‘ for k—vO.

B. Rotational-translational coupling
A]. Bl: A09 ’0

Knowing C l and C; for the various compounds
we have calculated A, and BR using the expres-
sions of Michel and Naudts (MN)."’ For the sake
of completeness we have reproduced their expres-
sions for A, and B, in Appendix B. Our values
of A, and B, (see Table IV) for KCN are about a
factor of 2 larger than that given by MN and this
difference is due to the large value of C; given by
Tosi-Fumi parameters. An important quantity
that determines the nature of the ferroelastic insta-
bility, i.e., which of the two elastic constants C“
or C“ softens first, is the ratio 8,; MR. This ratio
(see Appendix B) is given by

=-‘/élad I[(ft-f3)/[3f2—fo+‘;(ft-3f3)JJ (6-2)

 

l
en into account for a better quantitative under-

standing of the experiment.

It turns out that the dominant contribution to
Ag and Bo comes from the nearest-neighbor posi-
tive ion. This is because the second and third
neighbors [see Eqs. (5.1%) and (5.1%)] make con-
tribution! of opposite sign and almost equal
(within 0.1%) magnitude to A9 and 39- The
fourth neighbor‘s contribution is 57% of that of
the nearest neighbor. For the quadrupolar contri-
bution we find that
SQ =BQ( l + ' ° '

)/AQ(1+ ' =BQ/AQ

= —V 2/3 . (6.3)

The large value of [SQ I compared to [5,. I causes
C44-+0 at a higher temperature than C I. if we
consider the quadrupole contribution to the trans-
lational-rotational coupling alone. The above
analysis concerning the competition between qua-
drupolar and repulsive forces in determining the
effective rotation-translation coupling suggests that
there should be interesting pressure effects because
of the different volume dependences of AR,BR and
AQ,BQ. Such a study is under present investiga-
tron.

g THEORY OF ELASTIC AND PHONON SOFTENING IN IONIC . . . 2993

C. Isothermal rotational susceptibility 1.;

The temperature dependences of C u and C“ are
determined by 1¢p( k). For the symmetry direc-
tion k= -[0 O k] that we are interested 1n, 1.3(k)
=6a51m ( k) and furthermore 1n the limit k—cO,
which is relevant for the calculation of elastic con-
stants

xa101=xg/[1+Du(it‘=o1xfil . (6.4)

where 1° is defined in Eq. (4.13). As has been dis-
cussed in the first agraph of this subsection, the
T dependence of 1581’s determined by the single-site
potential Vo(fi,). Because of cubic symmetry 1°
=3x22=x and x‘,’,= x2.=x‘,’,=x, and the
quantities T10, and T10 are given in Figs. 2(a)
and 2(b). Our results differ from that of MN even
if we use their C 1 and C2 parameters and this
difference is due to a factor of 2 error in the calcu-
lation of their VOUI‘, ). In fact, their numerical
values of rx‘.’. and 7x2, given in Figs. 1 and 2 are
apprOpriate for a system for which C 1 22C?", a
value much closer to that obtained from Tosi-Fumi
parameters. However, for this stronger repulsion,
AR and Hg should be increased by a factor of 2.
This will change their results on T dependence of
C“ and C“ drastically and spoil the agreement
with the experiment that they found.

In Figs. 2(a) and 2(b) we also give R1=1n/X?1
and R,=x,,/x2,, x” and x“ being the c, and) t2,
susceptibilities obtained in the presence of direct
quadrupole-quadrupole interaction. Using the
value of Q=0. 6Qo, we find that Du(k= 0)
= —704 K and D«(k= 0)=235 K. From Eq.

(6. 4) we immediately see that 1"/1?| > 1 and

 

Ara/xi, <1 which indicates that direct intermolec-
ular interaction enhances C" softening and
suppresses C“ softening. Over the temperature
range of interest 1(1) 5 1000 K,
1 man/1°. <1.13, and o.sgx“/x£’,<o..9 Thus
effect of the direct interaction on elastic softening
is quite important for C“ and not so for C 11
which is essentially due to the fact that
124/13": 10 in the temperature range of interest.
From Eqs. (5.22) and (5.23) we see that the effect
of direct molecular interaction is to decrease T“
and increase T". the actual amount of decrease
will depend upon other parameters like C“,C 11
and values of 10.,124. The maximum effect of
direct interaction on T. in cyanides is found to be
~20%.

D. Elastic constants C“ and Cu
and transition temperature

Before presenting the results on the T depen-
dence of C"(T) and C44(T) and the temperature
where they approach zero, we would like to discuss
the importance of anharmonic effects on C 11 and
C“. It is well known23 that anharmonic effects
give rise to phonon-phonon interaction and renor-
malize the elastic constants, i.e.,

cgm=c3+sc5nm . (6.5)

At high temperatures 8C§“(T)= -r, T. For
alkali-halide crystals careful elastic constant mea-
surernents35’” have been made and it is found
that y, » 74. In order to incorporate anharmoni-
city effects in our calculation, we assume that

 

 

n
u-'- (l)
N
.- 0
“:1- R «1‘.
" ' _, 1°
.82
2. cg.
O .. .-
0. Jo
.-‘- f".
O
.- - ______ ._ -
2
.19
I 4 o.
I 2 4 £5

 

 

3
1110211)

 

 

(b)
an
1“!
’ ____"°
6 "T ‘ 3‘4»
n
.3 No:
0 a x
.-
J
K» '.
d .—
__ 5"!
0
ii

 

3
1110’“

FIG. 2. Single-site susceptibility 1?,T and enhancement factor R, =1..(T)/1..( T) for e, symmetry (KCN). (b)
Single-site susceptibility 1147' and enhancement factor R4=Xu( T)/12.( T) for 1;, symmetry (KCN).

2994 D.SAHUANDS.D.MAHANTI _2_§

cyanides are equivalent to bromides“ excepting for
the nonzero translational-rotation coupling in the
former and therefore use

n(MCN)=‘n(MBr), M=Na ,K ,Rb . (6.6)

The values of y. and y. are given in Table V.

In the presence of both translational-rotational
(tr-rot) coupling and anharmonicity effects, we
then have

cum=c3+scrm+5c3W1r1 , (6.7)

where SCE'MU') is given by Eqs. (5.22) and (5.23).

To calculate C"(T) and C“(T) we need to
know the values of bare elastic constants C0. and
c2, In their calculation" MN obtained c.I and
C24 by fitting to the experimental value of the
transition temperatures T“ and T“. Since the
present theory is of mean-field nature and there-
fore does not include fluctuation effects, we have
obtained 6}. and C2. by fitting to the experimental
values of C" and C“ at temperatures Tm far
above the transition temperature T,. The values of
T“, are given in Table V and those of Te in Table
I

As has been pointed out earlier, the value of the
quadrupole moment that we have used in our cal-
cuiation is 0.6Qo compared to 0.5Qo used by
Bound et al. in their MD study. In Fig. 3 we give
C“( T) vs T for three values of Q, i.e., Q=O.SQ0,
0.6Qo, 0.75Qo in the absence of anharmonicity ef-
fects. We find that the temperature at which
C“ -o0 are (1(1), 200, 300, respectively. In-
clusion of anharmonicity effects reduces the transi-
tion temperatures further although by only a few
percent. Since our theory of phonon softening is
of mean-field nature, we expect that inclusion of
fluctuation effects will reduce the transition tem-

perature still further for Q=O.5Qo (away from the
experimental value 7“: 156 K). The choice

Q=0.6Q0 gives TIM Z 7:,“ and a reasonable fit
over the entire range of the experiment. For
Q=O.75Qo the agreement between theory and ex-

TABLE V. Anharrnonicity parameters (71.74) bare
elastic constants (C‘h,C2.) and temperature Tm where .
theoretical and experimental values are fitted.

NaCN KCN RbCN

 

 

y, (10" dyn/cm’K) 0.26 0.27 0.28
y. (10’ dyn/ctn’K) 0.019 0.013 0.008
C2, (10" dyn/cm’) 5.749 5.115 4.022
C2. (10" dyn/em’) 0.752 0.470 0.411
r... (x) 473 453 380

 

 

KCN

1.0pr

p 2 030.500
3 o-o.a

4 080.75 0

U

 

 

 

 

fl 3mo’m i -'
FIG. 3. Tdependence of C... (in units of 10'
dyn/cm’) for KCN for different values of the (CN')

quadrupole moment Q =fQo, where Q, is the free-ion
value.

periment for T» T‘ is rather poor. For all the
cyanides, instead of choosing different values of
Q to obtain an optimum fit, we have chosen
Q=O.6Qo. The overall agreement between theory
and experiment (to be discussed shortly) using this
value is quite good.

Before discussing individual systems separately
we make a few general remarks. If we consider
only the repulsion contribution to the translation-
rotation coupling, i.e., Ag =Bg =0, then we find
that C ”—+0 at temperature T“ which is higher
than T“ where C44-90. Thus the repulsive forces
tend to soften C" much more than C“. Since C"
couples to order parameter Y., one expects that
C” —o0 will imply a nonzero (Y1) in the ferro-
elastic phase. This corresponds to molecules
orienting ‘along the z axis with a concomitant
tetragonal structure. Such a structure is not seen
in the cyanides but in the superoxides. If on the
other hand, we choose A )1 =33 =0 and Ag,Bg;£=O,
then we find that T“ > T” which is observed in
the cyanides. Of course as can be seen from the
Table IV, A 3, BR, Ag, and Hg are all important
for the cyanides. The fact that in these systems
T“ > T” is due to the dominance of Hg over 8,;
and a significant reduction in A R caused by nega-
tive Ag. Since the quadrupole moment of the 02’
ion is about a factor of 2 smaller than that of
(CN)" and since the short-range repulsion is
stronger in the superoxides, we believe that the
qualitative features of the ferroelastic instability in
superoxides is determined by the short-range repul-
sive forces. However, for a quantitative under-
standing of the transition temperatures in superox-
ides, one has to include anisotr0pic (quadrupolar)
electrostatic forces and the effect of orbital degen-

 

 

 

 

NaCN
2 6 L. l
2
1.8 '-
1 J J
2 3 4 5

111030
FlG.j4. Tdependence of C” (in units of 10"

dyn/cmz) for NaCN. 1: experiment, 2: theory with

anharmonicity, 3: theory without anharmonicity.

eracy of the superoxide ion.

In Figs. 4—9, we give the T dependence of C 11
and C“ with and without the inclusion of anhar-
monicity effect and compare with the experimental
result. For NaCN and KCN, the overall agree-
ment appears to be very good. In particular the
peak in C ”(T) is understood in terms of a com-
petition between the two contributions to the re-
normalization of the elastic constants 8C}? and
8C '1'; '°'. For a proper understanding of the T
dependence of C 11 it is important to include the
anharmonicity effect whereas for C“( T) this is not
so. For RbCN, inclusion of anharmonicity effects
in C 11 gives a peak but at a much lower tempera-
ture than that seen experimentally. Our feeling is
that although our calculations bring out the impor-
tance of various physical effects it is necessary to
go beyond a simple mean-field theory for a com-
plete understanding of the elastic softening in the
orientationally disordered phases of molecular crys-
tals. In this regard we propose to extend the work
of Naudts and Mahanti38 on spin-phonon systems

 

NaCN

0.2 -

 

 

p—

2

U)-

 

_l J.
4 5

THO’K;

FIG. 5. Tdependence of C... (in units of 10”
dyn/cm2 ) for NaCN. 1: experiment. 2: theory with
anharmonicity, 3: theory without anharmonicity.

26 THEORY OF ELASTIC AND PHONON SOFTENING IN IONIC . . . 299$

 

KCN

 

3 'mo'x)
FIG. 6. Tdependence of C 11 (in units of 10"

dyn/cm’) for KCN. 1: experiment. 2: theory with
anharmonicity. 3: theory without anharmonicity.

 

 

l j
2 4 E)

and apply to molecular crystals.

The value of T“ is given in Table I. The effect
of including anharmonicity is to reduce T... by
~10 K. Comparing the theoretical values of T“
with T' (see Table I), we see that the agreement is
reasonably good in view of the mean-field nature
of the present theory. Particularly remarkable is
the trend in T“ in going from NaCN to RbCN.
The T dependence of elastic constants in CsCN are
not available but they will provide an additional
test of the present microscopic theory.

E. Softening of phonons over the entire
Brillouin zone

Strauch e1 01.39 have used the translation-rota-
tion (tr-rot) coupling model to calculate the pho-
non frequencies of NaCN and KNC at 300 K for
the three symmetry directions [4‘00], [cg-g], and
[ggg]. For the bare phonon frequencies which are
determined by the dynamic matrix M °, they have
used a IO-parameter shell model. Translation-

 

 

 

 

KCN
0.2)-
Cu ‘
0.1.
1 2 3
3 A 5
2 mo’m

FIG. 7. Tdependence of Cu (in units of 10"
dyn/cmzl for KCN. 1: experiment. 2: theory with
anharmonicity, 3: theory without anharmonicity.

2996 D. SAHU AND S. D. MAI-{ANTI 26

 

RbCN
20 '-

l-S '-

 

 

 

b
b»

5:
THO K)
FIG. 8. Tdependence of C" (in units of 10"

dyn/cm’) for RBCN. 1: experiment, 2: theory with
anharmonicity, 3: theory without anharmonicity.

rotation coupling is incorporated by adding to M o
a contribution 6M given by

 

 

831+ 0
5M: 0 0 (6.8)
where
15114, = —ux°v* , (6.9)

where v and 1° are the tr-rot coupling and rota-
tional susceptibility matrices discussed in Sec. III
of this paper. Only the nearest-neighbor contribu-
tions to v, i.e., the interaction between a (CN)' ion
and its nearest-neighbor M + ion was considered in
Ref. 39 just as in Ref. 16. Strauch er al. did not
include the direct interaction between the (CN)"
molecules and therefore their renormalized phonon
frequencies are the same as those given in Eq.

(3. 20) of this paper with 1.3( it' ,w) replaced by 12,,
the single-site static susceptibility. Thus our re-
sults can be thought of as a generalization of their
work.

 

RbCN
0.2 '-

0“ 1 2 :1

l L

3
7110310

 

 

 

D)-

FIG. 9. Tdependence of C“ (in units of 10”
dyn/cm’) for RBCN. 1: experiment, 2: theory with
anharmonicity, 3: theory without anharmonicity.

Since in the limit k—o 0, 6M+ gives the renor-
malization of elastic constants, the latter complete-
ly determines 5M provided only nearest neigh-

bors contribute to v,“ k). Then knowing 6C 1. and

5C“ at T= 300 K, one can calculate “51"" (05";

for all values of jk at this temperature. Strauch
et al. had to use values of 5C 11 and 6C“ different
from those obtained by Michel and Naudts "’ to fit
to the experimental data. In our analysis of the
short- -range repulsion and quadrupole contribution
to 1:3,,(11) we found that because of near perfect
cancellation between second- and third-neighbor
contributions to 11,,(k). considering only the
nearest neighbor contribution to v.“( k), is an ex-
cellent approximation. However, both the above
mechanisms contribute to 11;,( k ). As we discuss
below our present calculations provide a micro-
scOpic justification of the values of 5C” and 5C“
chosen by Strauch et al. to fit to the experiment.
Since these authors, with their phenomenological
choice of 6C 11 and 5C“, found excellent agree-
ment with experiment we will use their calculated
values of wfy- r; as an experimental measure of
the phonon renormalization;

We define a quantity I" " =(a1fiz—wfi )"2/10l3
cps which is a measure_of phonon renormalization.
We have calculated I" " for phonons prOpagating
along the [00k] direction by using Eq. (3.20) and
noting that the tr-rot coupling matrix (including
both short-range and quadrupole contributions) has
typically a form like Eq. (5.5) with k, =k,=0 and
k, =k_.. In Table VI the results of our calculation
of I" " are given along with those obtained from
the Fig. l of Ref. 39 where the phonon frequency
v=w/21r is given in units of THz. Using the ap-
propriate polarization vectors of phonons given in
Sec. VC it is easy to see that I" k for the LA and
TA phonons are determined by the rotational sus-
ceptibilities of e, and 12‘ symmetries, respectively.
These susceptibilities in turn depend on the direct
Q- Q interaction Dag(k) through Eq. (4.10). We
find that at T=r300 K inclusion of direct interac-
tion affects LA phonon softening by about 7%
whereas the TA phonons are affected by 5—25 %.
The TA phonons are influenced more strongly by
direct interaction because at this temperature
191/12, le, in spite of the fact that D”(k) IS

—704 K at k =0 and 1955 K at k=kaz and that
044(k) is 235 K at k =0 and -652 K at k=kaz.
As can be seen from Table V1, for LA modes our
calculated values of I" " are about 15-30%
higher than experiment. This difference is due to
the fact that for a proper calculation of phonon

22 THEORY OF ELASTIC AND PHONON SOFTENING IN IONIC . . . 2997

TABLE VI. Renormalization of TA and LA phonons along [001) direction;

r‘[(a)3)’-(w‘)’]'”/10" cps; i=LA, TA.

 

 

 

k/Ir... 1‘M (present) 1‘“ (Ref. 39) r“ (present) 1‘“ (Ref. 39)

0 0 0 0 0

0.2 024 035(5) 0.71 . 061(5)

0.4 0.42 0.52 1.12 0.88

0.6 0.49 0.53 1.11 0.37

0.8 0.35 038(5) 071 0.56

1.0 0 0 0 0
softening one has to include the frequency depen- ‘3. Conclusion

dence of 1.,( law). If one is far away from the
phase transition temperature, i.e., T > T, (this may
not be true foLNaCN) then the frequency depen-
dence of 1.4 k,a)) is determined primarily by that
of 125m). The rotational dynamics at T5300 K
will be almost diffusive with a frequency scale
I‘M-05x 10'3 cps, i.e.,

‘ " it' 0) r2

1(k,a)) 1( , F2+wz .

On physical grounds‘O one expects that for high-
frequency phonons (a) >wm,l‘m) the effect of tr-
rot coupling will be reduced from the values given
in Table VI. This will improve agreement between
theoretical and experimental values of mud k ).

On the other hand, for TA phonons one finds.
from Table VI that our calculated values of I" “
are smaller than the experimental values by about
7—32%, the large discrepancy being in the low-k
region. However, because of the direct interaction,
the agreement with experiment (~ Strauch et al.’s
work) is fairly good for large values of k. There-
fore, for the low-frequency TA phonons, if one ap-
proximates the rotational dynamics by a resonant-
type behavior, i.e., by

r2

+(w—wm)2

 

1(k',a))~1(it',0) r2

with I‘~a),.,,, then one can improve the agreement
between experiment and present theory. There is
some evidence of the behavior of the form given
above from the MD calculation.2| For the high;
frequency TA phonons though, the quantity 1" k.
may depend sensitively on the values of 192‘,

Didi), and the frequency scales involved and the
above-mentioned quantitative agreement should be
reexamined carefully. A quantitative study of the
phonon softening including the prOper frequency
dependence of 1,5( 16,10) is beyond the scope of the
present work.

In summary, we believe that the anomalous ther-
moelastic properties and softening of phonons in
the orientationally disordered phase of the cyanides
can be adequately described by the tr-rot coupling
model."“7 The physics of these systems depends
sensitively on the commition between the short-
range repulsive and anisotropic electrostatic
(predominantly quadrupole-EFG interaction)
forces.” Furthermore, anharmonicity effects are
also important for a proper understanding of the T
dependence of C 11- For the phonons in general, it
is necessary to include the retardation effects by
considering the frequency. dependence of the rota-
tional susceptibility 1.4 k,a)). Fluctuation effects
not included in the present mean-field theory ap-
proach should be considered for a better quantita-
tive understanding. We propose to extend our
theory to CsCN which has a different high-T cubic
structure and see if we can understand the large
ferroelastic transtition temperature“ Tom 2200
K. Finally for the alkali superoxides (which will
be discussed in detail in a separate paper) short-
range repulsion dominates over the quadrupole
coupling and the orbital degeneracy of the superox-
ide ion plays an important role.

ACKNOWLEDGMENTS

We thank Dr. G. Kemeny and Dr. J. Naudts for
helpful discussions. This work was partially sup-
ported by NSF Grant No. DMR 81-17297.

APPENDIX A: ROTATIONAL-TRANSLATIONAL
COUPLING FROM QUADRUPOLE

EFG INTERACTIONS

In this appendix we evaluate the coupling con-
stant matrices arising from the contribution of
various NN‘s to the electric field gradient. We ex-
plicitly consider the cases 0 =1 and a =4. other
terms being readin obtainable from these by sym-
metry considerations.

2998
1. First NN contribution

From (5.16), choosing the origin at R?

3 e
U.(ar)= - U.( —a2‘)= — J‘s-Lu," ..

(AI)

We use (5.6), (5.15), (5.17), and (Al) and express
the displacements iii in terms of their Fourier
components and obtain for a=1 [see Eq. (5.6) for
the definition of 5,, C," etc.]. We have

 

D. SAHU AND S. D. MAI-[ANTI

g
Sx
ufifGH |1't’)..—.2A,2 3, (A4)
“ZS:
Similarly for a=4,
(was): —U.,(—ar)=3-l"71-u,,. .. , (A5)
a

2 giant, 1113:2139 2 Y1( it‘)[u,(it‘)s, +u,(1't')s, 1
t r

 

(A6)
2 91301.10: =2“,2 Z rl(it°)[u,('1?)s, +u,(it')s, where
l
k _. Bg = — t/2/3AQ , (A7)
—2u,(k)S,] ,
and therefore,
(A2)
where am .. S,
e.,, (+llt)|=23g 0 (A8)
Ana/$32.19. (A3) 5
a I
so that ‘ Hence we obtain
A95, —AQs, 199$, 199.9, 0
uEfGt+|it°).=2 A95, .495, rigs, 0 nos, (A9)
4.105, 0 0 199$, 119$,
l
. . — 1
2. Second NN contnbutton =——
40 4,540 (A12)
From (5.16) we again obtain so that
U,(a£+afi)= - U.( —a£—af) s,(3c, —2C,)
EFG '* _ - _
=_235,:a‘(up+ub)’ e.,, (4102—249 sec, 26‘.) (A13)
(A10) —s,(c,+c,)
Ua(a£—af)= - Uzi “fwd” Similarly with
- e (u u ) " ‘ (A14)
__ k- , =___
25004 1) Bg “/2 Bq .
and .
gamma: “’c 0m“
l'
. - _. .. S(C —4C)
.1124 1"(1t1u (2)5(30 —2C) .. - ’ ‘ ’

92,; ' [" _.‘ ‘ ’ u$G(—|1t),=%ag s,(4c,-C,) ,
+u,(1t)s,(3c,—2C,) ss,(C,—C,)
+u.(i°)S.(Ca+C,)] . s,(3C,—2C,)

EFG " "
0

where

g THEORY OF ELASTIC AND PHONON SOFTENING IN IONIC . . . 2999

s,(3c,—2c,)
v$G(—IE)2=2§Q 0 s
s.(3c,-2c,)

and
0
u§f°(— |'1'E),=2§Q s,(3c,—2c,)
' s,(3c,-2c,)

3. Third NN contribution

Defining
. f=_%ag, (A16)
59": _ 37873-39 , (A17)

we obtain the following terms for the coupling
constant matrix:

_ s,c,c,
DF:G(+ I E)3=ZIQ SnyC: 9
45.0.6, 1
ngq
"$G(+|E)3=ZXQ S,C,C. '
0
I s,c,c,
EFG " :
113,, (+ I k)3=2-BQ SnyCs ’ (A18)
k-%s,s,s,

 

 

' s,c,.c,
uf,”(+|it’),=2§,, —%s,s,s, ,
L s,c,c,
’ '_%st,s’
vf;,""(+|it’),=2§‘2 s,c,c,
i S,C,C,

APPENDIX B: ROTATIONAL-TRANSLATIONAL
COUPLING FROM SHORT-RANGE REPULSION

Equations (A13) and (A15) of Ref. 16 are
correct. There is an error ofa factor of l/t/i in
Eq. (A14) where B, BB is expressed in terms of
I“ ’(a,;0), but the other equations are correct.
The parameters A and B (which are A 3 and B, in
the present paper) are

A, 5.4 =t/Er'rc,c,(d’+a1)-m
x[a(3f,—fo)+d(f,-3f,)] . (BI)
19,, 53:: —\/3Trrc,c,(d’+a’)-md(/, -f,) .
(132)
where
1. =8"'“’ jg (1 —y’)"e-"’dy (B3)
and
h =C2(d2+a’)m , (34)
8=Zda /(d’+a’) . (as)

In the above equations C “C; are the repulsion
parameters discussed in Sec. VIA of the text , 241
is the internuclear separation, and a is the distance
between the (CN)‘ ion and its nearest M1“ ion.

 

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3000 D. SAHU AND S. D. MAHANTI 26

"In Na02, the transition is from an disordered to a
four-sublattice antiferroelastic phase and there are no
accompanying elastic anomalies.

u"I'here is some evidence that in RbCN, the molecular
orientations in the ordered phase is along [111] direc-
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3'When the time scales of rotational motion is fast com-
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carrying out a canonical transformation. This leads
to a correction to the direct intermolecular interaction
and to‘ the single-site susceptibility. However, in the
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