39
CHAPTER 1 INTRODUCTION
CHAPTER 1
INTRODUCTION
1.1 IMPORTANCE OF LATTICE ENERGY
The study of cohesion in ionic solids is of fundamental
importance to understand the nature of interionic forces and their
effects on elastic, thermal and enharmonic properties. At present,
greater attention is being paid to the determination of the relation
between elastic and other physical and energy properties of
crystals. Lattice energy of ionic crystals is an important parameter
which is directly connected to the binding forces in ionic solids
and hence to the elastic constants. For this reason, various1theoretical approaches, starting with the Born-Mayer model have
been used to evaluate the lattice energy of ionic crystals apart from
the experimental method based on the Born-Haber cycle. The lattice
energy determined from the Born-Haber cycle is an experimental
lattice energy and is independent of the nature of assumptions made
about the bonding in the crystal. The classical theoretical
evaluations are based on the assumption of the ionic nature of the
bonding in the crystalline lattice.
Ionic crystals are made up of positive and negative ions.
When the crystal is formed, the ions arrange themselves with the
Coulomb interaction between ions of opposite sign being stronger
than the Coulomb repulsion between the ions of the same sign.
The ionic bond is thus the bond resulting from the electrostatic
interaction of oppositely charged ions. The common crystal
structures found in ionic crystals are the sodium chloride structure,
1
2
the caesium chloride structure and the fluorite type structure.
Among the physical properties of crystals, the elastic
constants are more important, since they allow us to judge
immediately the strength of the interatomic coupling and to calculate
a number of the most important parameters of solids, such as the
Debye temperature, the lattice energy and infrared dispersion
2frequency. The Born model has been found to be very much
successful in predicting the binding energies and moderately successful
in explaining the elastic properties. Hence, elastic constant data
can be utilised to evaluate the lattice energies of ionic crystals.
A brief review of the different theoretical approaches for
the evaluation of lattice energies of ionic crystals is presented in
the following section.
3
1.2 BRIEF REVIEW OF THE DIFFERENT THEORIES EMPLOYED FOR
THE EVALUATION OF LATTICE ENERGIES OF IONIC CRYSTALS
2Born devised formulae which permit the calculations of the
lattice energy of an ionic crystal. The basic assumption in the
theory of the cohesive energy of an ionic crystal is that the solid
may be considered as a system of positive and negative ions. The
lattice energy of such a crystal is defined as the increase in
internal energy at absolute zero accompanying the separation of the
constituent ions, to positions where they are infinitely removed from
one another. In order to account for the stability of crystals known
to be composed of ions, it is necessary to introduce forces between
the ions that are non-Coulombic. This is because no stable
equilibrium is possible in an electrostatic system of charges unless
other forces are present.
In Born's theory of the lattice energy of ionic crystals, it
is assumed that the charge distribution in the positive and negative
ions is spherically symmetric and hence the force between two such
ions is dependent only on their separation and is independent of
direction. As an example, considering a lattice of the NaCT-type
structure, a given sodium ion is surrounded by 6 Cl ions at a
+ — distance r, 12 Na ions at a distance r 8 Cl ions at a
distance r /? etc., where r represents the shortest interionic
distance. The Coulomb energy of the sodium ion in the field of
all other ions is given by
4
e c£r
6/T
12
/IT8
T *T}
... (1.2.1)where e is the charge per ion.
The coefficient e^/r is a pure number determined only by the
crystal structure. For the NaCl type structure, the result is
e c(1.2.2)
with A = 1.747558.
The constant A is called the Madelung constant. For other
crystal structures composed of positive and negative ions of the
same valency, the values of the Madelung constants are different.
In equation (1.2.2) e represents in general the electronic
charge times the valency of the ions under consideration. The minus
sign in equation (1.2.2) indicates that the average influence of all
other ions on the one under consideration is of an attractive nature.
To prevent the lattice from collapsing there must also be repulsive
forces between the ions. These repulsive forces become noticeable
when the electron shells of neighbouring ions begin to overlap and
they increase strongly in this region with increasing values of r.
These forces, can be understood best on the basis of wave mechanics
like all other types of overlap forces, because these are of a non-2classical nature. Born has simply assumed that the repulsive
energy between two ions as a function of their separation could be
expressed by a power law of the type B'/r11 where B1 and n are
as yet undetermined constants characteristic of the ions under
5
consideration. Thus, the repulsive energy of one ion due to the
presence of all other ions may be written as
£ rep*Bnr
... (1.2.3)
where B is related to B* by a numerical factor. Because of the
fact that the repulsive forces depend so strongly on the distance
between the ions, the repulsive energy given by equation (1.2.3)
is mainly determined by the nearest neighbours of the central ion.
The total energy of one ion due to the presence of all other ions
is obtained by adding the equations (1.2.2) and (1.2.3)
eAe2 A B--- + --r n
... (1.2.4)r
Assuming that the two types of forces just discussed are the only
ones that have to be taken into account and neglecting surface
effects, the total binding energy of a crystal containing N positive
and N negative ions can be given as
U = E(r) = N[- ... (1.2.5)r
The variation of U with r is shown in Fig. 1.2.1.
Considering the crystal at absolute zero, the equilibrium
conditions require U to be a minimum. This will be the case for
the equilibrium value of r = rQ as shown in Fig. 1.2.1 where
rQ represents the smallest interionic distance in the crystal at
T = 0. For this minimum
(iM)v 3r r 0 (1.2.6)
Fig. 1.2.1. Variation of lattice energy (U) with
interionic distance (r).
Curve a - attraction as a function of
Curve b - repulsion as a function of
Resultant of a and b exhibits a minimum
at r at absolute zero.
6
From equations (1.2.5) and (1.2.6) the relation between the
unknown parameters B and n can be written as
B = (As!,nn-1
ro(1.2.7)
Substituting equation (1.2.7) into equation (1.2.5), we obtain the
following expression for the lattice energy f
2UL = U(ro} = “NA ~ U ~ •** (1.2.8)
o
where UT = U(r )• The interionic distance can be obtained fromL O
X-ray diffraction, the charge per ion is known, and thus the lattice
energy can be calculated if the repulsive exponent n is known.
2Born obtained the value of n from the measurements on the
compressibility of the crystals and using the following expression
49C r
n = 1 + ------- %— ... (1.2.9)K e A
o
where C is a constant determined by the type of the lattice, KQ
is the compressibility at absolute zero and the other terms have
their usual meaning.
According to equation (1.2.8) and in view of the relatively
large value of n which varies from crystal to crystal, most of the
lattice energy is due to the Coulomb interaction, and the repulsion
contributes only a relatively small fraction. On the other hand,
the repulsive and attractive forces acting on any one ion just
balance for r = rQ and thus are equal in magnitude.
7
The development of the theory of lattice energies is largely
on account of the development of the ideas about the non-Coulombic
3forces between the ions. Born and Lande represented these forcestilas varying as the inverse n power of the distance and treated
the potential energy of the crystal as the sum of the two terms,
given by the equation
Uo
N A Zj Z2 e2
rNB
nr(1.2.10)
where N, A, Zj, Z2, e and r represent the Avogadro number,
Madelung constant, valencies of the ions, charge of the electron
and distance between unlike ions respectively. B is a constant.
The value of n is usually determined empirically from the
compressibility.
The constant B can be eliminated by utilising the fact that
at the equilibrium separation of the ions in the crystal, say atr = r , (4r) = 0, so that
o dr r = ro2
N A Z-i Z? eU = --------------------- |1.- —I ... (1.2.11)
o r no
The development of wave mechanics led to the realisation
that ions with completed subgroups are spherically symmetric and
that, for the outermost shell, the electron density falls off
exponentially with distance. Utilising this fact and considering
only the Coulombic and repulsive forces, Born and Mayer^ obtained
the following expression for the lattice energy
8
N A Zj eNB' -r / p
... (1.2.12)
In the above expression B' is another constant and has the
3Udimensions of length. Making use of the condition (r—) _ = 0,dr r — ro
we get
N A Z. Z e2U = ---------- -—------- [1 - -£-} ... (1.2.13)
o r r o
These two expressions are called the simple Born-Mayer^ expressions
for the lattice energy of an ionic crystal.
The above simple Born-Mayer equation was further modified
by introducing terms to allow for the induced dipole-dipole and
dipole-quadrupole interactions in the lattice and for the zero point
energy of the lattice. The extended form of the Born-Mayer
equation is given by the expression
N A Z, Z. e2 . . a
U = -----------—------ - NB' e~r/p + Ne/rb + -j n h Vor 4 max
... (1.2.14)
where h is Planck's constant and u is the Debye maximum fre-max J
quency.4Using the values of 'basic radii1, Huggins recalculated the
lattice energies. These 'basic radii', which were determined using
the condition
(r ro
= 0,
9
represent the equilibrium internuclear distance when put into the4equation for the lattice energy. The treatment given by Huggins is
similar to the extended Born-Mayer^ equation except that adjusted
crystal 'basic radii' are used to obtain B' and optical data on the
crystal itself to obtain C.
Owing to the difficulty of assigning 'basic radii' in many
5salts, Ladd and Lee have extended the simple Born-Mayer equation.
They have eliminated B' and hence the 'basic radii' and included
9Udispersion energy terms. Using the condition, (-r—) _ = 0, theo
expression obtained for the lattice energy is given below.
N A Zj Z2 e[1 NC
6 [1 6_r
ND 8 ,8 r J
r oo+ I N hv
max
(1.2.15)
The theoretical methods of extended calculation of lattice
energies described previously are restricted to salts for which the
crystal structures are known exactly. For less simple structures
than those of the alkali halides, the extended calculation of lattice
energies becomes increasingly laborious as the symmetry decreases.
Considering that the expression for the lattice energy contains
basically the first two terms namely attractive and repulsive and
adopting either an inverse power law or an exponential law for the
repulsive term, Uq may be written as either N A 2. Z0 e2
••• (1.2,11)
10
or
Uo
N A Zx Z2 e2
ro[i ... (1.2.13)
If the number of ions in the chemical molecule is X then
the total number of ions in a mole is NX. The above equations
may then be written as
Uo
NX2
01 Z1 Z2 e] [1 - (1.2.16)
and
NX r “Z1 Z2 6 2 1 *o ] [1 (1.2.17)
where a = ZA/X is a constant. Although ot is not identical for6,7different lattice types, Kapustinskii found empirically that in
passing from one lattice type to another, the change in the
constant a was proportional to the change in the interionic distance. 6 7Kapustinskii's ’ equation for the lattice energy is given as
287’2 V Z1 Z2
+[1 0.345 .
y *f rrc rA... (1.2.18)
where r0 + r, is the sum of the ionic radii for the coordination C Anumber 6 and taking P = 0.345 A°, a = 1 .745 and Ne2 = 329.7 K cal/A°.
6 7Kapustinskii's ’ formula gives approximate lattice energies,
especially in those cases where the structure is not known. The
values of lattice energies given by the above equation fall on the
lower side of the correct lattice energy.
11
An altogether different approach to the calculation of lattice
genergies was given by Hylleras . He applied a general quantum
gmechanical treatment to the evaluation of lattice energies. Hylleras
used one-electron wave functions of the hydrogenic type with nuclear
screening so that the entire calculation could be performed
analytically. The wave functions are given by
= exp. (— (Z— fj>r/ahl ••• (1-2.19)
where a^ is the Bohr unit distance (0.58 A°).
Generalising the above discussion, the lattice energy of a
simple ionic crystal can be evaulated from the equation
U = UE - UR + UD • • • (1.2.20)
where Ug is the electrostatic energy, % isthe repulsive part
of the lattice energy and is the dispersion energy. Ug may
further be expressed as + USE where U..M
and lfeE are the
Madelung energy and the self-energy respectively of the crystal.
The dispersion energy U ^ can be expressed as a sum of and
Uqd where and IT^ are the dipole-dipole and dipole-quadrupole
dispersion energies.
Neglecting the dispersion energy , equation (1.2.20) can be
written as
U = UE ~ Ur d-2.21)
Expressing the repulsive energy Ur as UR = B/r" where B is a
constant and minimising the equation with respect to rQ , we get
3the simple Born-Lande equation for the lattice energy
12
A Zj e NBn
(1.2.10)
The value of the repulsive exponent has to be assigned from the
compressibility measurements.
If the repulsive energy is expressed in an exponential form
as = B' exp. (-r/p), where B' is also a constant, upon
minimising the equation, we arrive at the simple Born-Mayer^
equation2
N A Z Z eUQ = ---------i— ----------NB' exp. (-r/p) ... (1.2.12)
Here also the value of the repulsion parameter p has to be
assigned.
6 TKapustinskii ’ assumed p to be a constant with a value
of 0.345 A° and put forward an equation for lattice energy which
is given by
U287.2 v z Z2
(rc + rA) U 0.345<rC+rA) (1.2.18)
To make use of this equation, the thermochemical radii of the consti
tuent ions are necessary.
On the other hand, retaining the dispersion energy term Up
in equation (1.2.20) in tact, expressing as B/rn, and minimising
the equation with respect to r^ , we get the extended Born-Lande
equation. In this equation also, the value of n is to be assigned.
Writing = B1 exp (-r/p) and performing minimisation with respect
5to rQ results in the extended Born-Mayer equation or Ladd and Lee
13
equation as given by equation (1.2.15). Here also the values for
the repulsion parameter p can be obtained from the compressibility data.
9 10For the case of complex ionic crystals, Jenkins and Waddingfon
have proposed a new approach. In the generalised equation (1.2.2)
for lattice energy, the repulsive component is expressed as
= B' exp. (-r/p), the electrostatic energy U is expressed as U„ =
Uy + Ugg. The new lattice energy equation is obtained upon
minimising the general equation. The Madelung and self-energy terms
are calculated by the Bertaut^ method. The repulsion energy
4is estimated by closely following the procedure given by Huggins and
12the dispersion energy is evaluated by employing Mayer's
method. In this equation also the values for p can be assigned
from the compressibility data. The important feature of this method
is the assignment of the charge distribution q in the complex ion.
13Jenkins and Pratt have suggested another alternative equation.
This new equation is arrived at by expressing as =
B £ exp. (-r/p) and minimising the general equation. Here, only
the value of the repulsion parameter p needs to be assigned. Apart
from these two procedures, there is also another method known as
the term by term approach^ ^ to evaluate the lattice energies
of complex crystals which needs no minisation of the general lattice
energy equation.
There are some disadvantages in both the approaches.
Although the compressibility data of the crystal can be used to
determine the repulsion parameter p, lack of compressiibility data
14
makes it difficult to do so. Moreover, if the assignment of the
value of p is done, it will be advantageous to use the minimisation
9 10procedure. The approach given by Jenkins and Waddington ’ and
14-16the term by term approach require the assignment of the charge
distribution in the complex ion. The latter approach^ ^ also re
quires the assignment of 'basic radii1 to the ions when using the
17Huggins and Mayer form of repulsion potential.
Apart from the generalised equation for the lattice energy
and its modifications discussed above, there are a number of new
approaches which have been applied to evaluate the lattice energies.
These new approaches differ from one another in the form of the
repulsion energy considered. A few of such studies are outlined
here. Jain and Shanker^’^ have used the following equation for
evaluating the lattice energies of alkali and silver halides.
U c_6
r
D8 + UR (1.2.22)
where is the Madelung constant, C and D are the dipole-dipole
and dipole-quadrupole coefficients respectively. is the repulsive
energy and is expressed in two forms proposed by Hafemeister and 20 21
Rlygare and Hafemeister and Zahrt respectively. These two
potential forms are given by
UE = c iAB and UR = C ( iAB/r)
where A and B correspond to the two types of ions and C and C'
are the constants of proportionality. The term A is the sum
of the squares of overlap integrals.
15
Considering short-range overlap repulsive interactions between
22nearest neighbours and next nearest neighbours, Shankar et al
have used the following equation to obtain the lattice energies of
alkali halides
u = - JL---------~6 - -8 + M <Z>+ _ + 1/2 M« (0++ + 0_Jr r
... (1.2.23)
where the first three terms have their usual meaning. The fourth
and fifth terms are the short-range overlap repulsive interactions
between nearest neighbours and next nearest neighbours given by
the Born-Mayer-Huggins exponential law and Pauling's inverse power 23law . An equation of the type
U Ll,m
lb6lm e*P'(rl + rm * rlm)/pl
... (1.2.24)
24has been used by Singh et al for evaluating the lattice energies
of some fluorite type crystals. In the above equation, apart from
the usual first three terms, the fourth term is the overlap repulsive
20 21energy due to first and second neighbour interactions ’ . Here, fs
is the Pauling coefficient, b and p are the repulsive potential
25parameters and r^ and r^ are the ionic radii. Sinha and Thakur
have made use of the relation
U = Up + Un + U_ - e + 2RT E D R o (1.2.25)
to evaluate the lattice energies of alkali halides and hydrides.
16
In the above equation, the first two terms have their usual signifi
cance. The repulsive energy is expressed in a logarithmic2 *“16 2 form as log (1 + p/r ) where p is a constant equal to 3 x 16 cm .
£q is the zero-point energy and the term 2RT is added for making
a PV-correction applied to the gaseous ionic species from 0 to
298.15 K, and is equal to 5.02 kJ mole A generalised logarithmic
form of repulsion energy
UR = A l0« <! *
where A and B are potential parameters and n is a pure number26 27 28
has been used by Prakash and Behari , Thakur , Uma Rani Pant29
and Jha and Thakur to evaluate the lattice energies of alkali
halides, heavy metal halides and alkaline earth oxides. Mishra et al° 31 32 33 34Thakur ’ and Thakur and Pandey ' have also made use of the
35logarithmic functions of the repulsive energy. Usha Puri and
36Hasan et al have employed an exponential form of repulsive
energy given by the relation = p/r exp (-A r) where A and p
are the potential parameters which are evaluated using crystal
37stability and compressibility conditions. Overhauser and Dick ,
38 39 40Karo and Hardy ’ and Dixit and Sharma have also applied
exponential type functions to describe the repulsive energy. Fumi
41and Tosi have included the van der Waals interactions estimated
by Mayer. A number of similar potentials have been developed
42-49and employed by many workers for the prediction of various
properties of ionic crystals.
17
50Thakur et al have proposed an empirical expression for
the lattice energy based on the logarithmic form of repulsive term
and estimated the lattice energies of a few ionic crystals. Their
expression is given by
U = UElec.[1 ~ log10*a + Pr02)] ••• d‘2.26)
2where a = 1.015 and p = 1.60416 A . Following the theory
51 5 described by Tosi , the modified Ladd and Lee equation with ex
ponential law for repulsive interactions with a term appropriate52to the Hildebrand equation of state is given as
U = D (1 8P 3V TBP Kr
... (1.2.27)
where the first three terms have their usual meaning. In the
fourth term, V is the volume per ion pair, 3 is the thermal
expansion coefficient and K is the compressibility at room
temperature T. The values of the hardness repulsive parameter P
are determined by the Hildebrand equation of state. The above
equation was used 53 54by Cantor , Ladd , Bakshi 55et al Sharma + ,56et al
to evaluate the lattice energies of alkaline earth oxides, alkali
halides and rutile type compounds and heavy metal halides respecti
vely. The same equation without the last term 3VT £p/Kr was used 57by Shanker et al to estimate the energies of rutile type
compounds.
Taking into account the many body interaction effects within
18
the frame work of Hafemeister-Zahrt potential , the following equation
58 59was used by Gupta and Goyal ’ to calculate the lattice energies
of chalcogenides with NaCl structure and some alkaline earth
fluorides.
U = 1/2 li.j
z. z,..,L....2r. . il
2e +
i.j.k if j fk
Z. Z.——e f(r.,) + r. . lk
il
Nb li»j
litrij
exp [r. + r. i_____1
Pij-ii J-1/2 l
i.l1/2 l D_
i.j 8 J r
... (1.2.28)
The first term in the above equation is the Coulomb energy. The
second term arises as a result of charge transfer between adjacent
ions and the three body interactions. The function f(r) is a
parameter dependent on the overlap integrals calculated from 60
Cochran's formula. The third term represents the Hafemeister- 21
Zahrt form of short range repulsive interaction. The fourth and
fifth terms represent the van der Waals dipole-dipole and dipole- quadrupole interactions. Singh and Pandey^ have also used the
above equation to evaluate the lattice energies of lead, barium and
strontium nitrates.
A modified form of the Kapustinskii-Yatsimirskii equation 62
using logarithmic form of lattice potential energy was proposed by
Thakur and Sinha63 and is given as
1201.61 l n Z. ZuL = - - - - - -—..... -. . . . . ^ n - iogl0 (i ♦
(r + Oc a(r + r ) c a7
;)] +
10.5 2 n Zj Z2 • * • (1.2.29)
19
where p is a constant equal to 2A° when (r + r ) < 4A° and 10A°c s
when (r + r ) > 4A°, for complex crystals. In is the totalC 3.
number of ions present in the molecular formula of the compound,
63and Zj and are the ionic charges. Thakur and Sinha have cal
culated the lattice energies of alkali halides and alkaline earth
64chalcogenides using the above equation. Later, Thakur and Sandwar ,
65 66 67Sharmma et al , Thakur and Sandwar and Thakur have estimated
the lattice energies of first row transition metal fluorides, divalent
fluoride and hydride cyrstals, second row transition metal oxides
and fluorides and some metal complexes respectively, applying the
above equation.
Sharma^ made use of the Rydberg potential energy function^to
calculate the lattice energies of alkali halides. The Rydberg
potential energy function (r) of the ionic crystal per pair of ions
can be expressed in the form
$ (r)2 ?a a 7 —hr -hr—-—+ ( A e Dr - pr e ) + e ... (1.2.30)
where a is the Madelung constant, e is the electronic charge, z
is the ionic valency, e is the zero-point energy and A , |J and b
are the constants of the potential function respectively. The same
70 71equation was applied by Misra and Sharma and Sharma to evaluate
the lattice energies of some lighter and heavier ionic crystals.
71Sharma had also calculated the lattice energies from the equation
3f + ( (b2 - 2)/r2 - (2b/r)J «e2/r 4- (r) = ------------------------ 5------------------------------------ ... (1.2.31)
20
where f is the force constant which can be computed from the72 73 71
expressions given by Krishnan and Roy and Born and Huang . Sharma
had found good agreement with experimental values for the case of
some heavier ionic crystals.
An equation was derived for the calculation of polarisation
energy and crystal lattice energy of ionic crystals in terms of
effective nuclear charge and rate of cation polarisation by Wen and
Sho74 The equation was used to calculate the polarisation and
75lattice energies of many ionic compounds. Jenkins and Morris have
assumed a linear relationship between the viscosity B-coefficients
76of the Jones-Dole equation for aqueous solutions of ammonium salts
and the enthalpy of hydration of the gaseous monoatomic constituent
ions. These enthalpies of solution of NH^ (g) + X (g)have been used
77to estimate the lattice energies of ammonium halides. Ivanova et al
have obtained an equation relating the longitudinal and transverse
components of the sound velocity to the lattice energy on the basis
of the general expression for compressibility of ionic crystals. They
78have evaluated lattice energies of alkali halides. Belomestnykh had
79used the Nemilov's relation for lattice energy, U = a GV where G is
the shear modulus, V is the molar volume and a is a constant whose
78value depends upon the structure and type of the crystal. Belomestnykh
had evaluated the lattice energies of a number of ionic crystals.
80 81Pandey and Singh and Sharma have employed an overlap
2potential of the form P/r exp.(-kr ) where P and k are the potential
82parameters. Eggenhoffner et al have commented upon some recent potentials for alkali halidies, proposed by Woodcock*^, Romano et al®**
21
85 “3r/oand Narayan and Ramaseshan . By postulating an exponential form Ae
for the compression energy of individual ions where A and p are the
repulsion parameters, the lattice energies of a number of peroviskite
type crystals of the form ABC^ have been evaluated by Narayan and 86Ramaseshan . Their equation apart from the usual Madelung and
dispersion energy terms, consists of compression energy terms for every
contact an ion makes with its neighbours and is given by
■ e2 C D “'»» ,P» ~rAB/Pk" - - ----- 6 ' *8 + “a e + 8AA e
r r
. "rAC / PA ~rBA/pB , ~rBC / PB+ 12Aa e + 8Ag e + 6Ag e
~rCA 1 PC , ~rCB /pC .. -rCC 1 P° + 12AC e + 6AC e + 24AC e
+ 12A^ e•rCC 1 Pc
... (1.2.32)
where Q is the Madelung constant and C and D are the van der Waals
coefficients. A. and p . are the repulsion parameters of the A ion A Aetc., rAg is the radius of an A ion in the direction of a neighbouring
B ion etc.
The dispersion component of the lattice energy and the
compressibility of the ionic crystals were calculated based on87statistical theory by Dedkov and Temrokov . An additivity rule for
the repulsive potential softness parameter in alkali halides, derived88from overlap integrals was proposed by Shanker et al . Shanker and
89Agarwal have also used the same method to evaluate the lattice
22
energies of silver and alkaline earth fluorides. Lister had putforth
a method of calculating lattice energies from pressure volume data91(P-V data). Mackrodt and Stewart have used the electron-gas
92approximation for calculating the interaction potentials. Islam had
proposed an exponential potential function of the type UD = I exp.(-sr 0 )K
rwhere a - z + (d/z ) where rQ and d are the equilibrium ionic
separation and the difference of the ionic radii respectively, z is
the largest common factor of the valencies of the cations and anions
and I and s are the two potential parameters.
A quantum mechanical analytical potential form following the93tight binding theory was applied by Narayanan et al . Ghadgaonkar
94and Ramani have used a modified Morse-Kratzer potential and
estimated the lattice energies of alkali hydrides, silver and alkaline
earth fluorides. The expression used by them to evaluate lattice
energies is
? r - r 2U = N o D [ 1 - exp ,l~3(r-r)l] +D [ ----------- J
o e r e e r
... (1.2.33)
where D is the dissociation energy, r is the equilibrium distance 6 6
17 1/2and fb is the Morse parameter given by fs = 1.2277 x 10 (uA/u>eXe)
where pA and ujgXe are the reduced mass and spectroscopic constant
respectively. A new repulsive term of the form A r n e r^was pro-
95posed by Yadav for alkali halides where the constant A has the
dimensions of energy, X is the range of interaction and n is a pure
number.
23
An empirical method based on the effective charges on atoms
in molecules and ions was used to estimate the lattice energies of
ionic crystals by Kazin et al^, The equation used by them is
given by
U = 332 A* q [ W^L— + ] ... (1.2.34)
cat cat.Y cat.X
*where A is the Madelung constant, q is the charge on the cation,
catq and q^ are the effective charges on the atoms of the anion YX^”1.
96Kazin et al have estimated the lattice energies of several
compounds of the Ca(Fe02)2> BaSiO^, etc., type. More recently,
97Reddy et al have assumed a linear relationship between the lattice
energy and interionic distance and suggested the following relations
to estimate the lattice energies.
u = -49.51 (rQ) + 324.72 (alkali halides) ... (1.2.35)
u = -144.30 <ro> + 901.00 (divalent halides)
• • • (1.2.36)
u = -47.75 <r0) + 286.54 (Ga, In and T1 halides)
(1.2.37)
The temperature dependence of lattice energy of ionic crystals
was studied by very few workers. A relation between the lattice
energy and temperature derivative of Young's modulus E given by
the following expression
U = ((-6.70 x 10~3 ||) + (5.20 x 105)] ... (1.2.38)
24
was proposed by Botaki to "evauate the lattice energies of alkali
99halides. Fermor and Kjekshus have taken into account the
dispersion and multipolar forces and calculated the temperature
dependence of the lattice energies of univalent nitrates in the calcite-
like modifications. The temperature dependence of lattice energy
of MgO and L^O was calculated by Mackrodt^^.
There have been a few studies on the estimation of lattice
energies of mixed ionic crystals. For mixed crystals of alkali
101halides, Tobolsky used the Born equation (equation 1.2.4).
102Wallace employed equation (1.2.12) but included the van der Waal
103terms also. Botaki made use of equation (1.2.28) to calculate
the lattice energies of NaCl-NaBr and KCl-KBr mixed crystals.
104Zavadovskaya et al have calculated the lattice energies of mixed
105crystals on the basis of the Durham and Hawkins method. Belomestnykh
106 107and Sukhushin ’ have utilised both the equation (1.2.11) and
79Nemilov's equation and evaluated the lattice energies of alkali
halide mixed crystals. Lattice energies of mixed alkali halides,108have also been calculated by Zavadovskaya et al
109Based on the
pseudo unit cell model of Chang and Mitra*-", and accounting for
van der Waals dipole-dipole and dipole-quadrupole terms and
including short range repulsive interactions effective up to second
110neighbours, Shanker and Jain have evaluated the lattice energies
of NaCl-NaBr, KCl-KBr and KBr-KI mixed crystals. Singh and
Sanyal^^, Singh et a]_H2,113 and Rana and Daud^^ have applied
equation (1.2.28) with slight modifications to satisfy the conditions
115of a mixed crystal. Jain and Shanker used an overlap repulsive
25
potential model in equation (1.2,20). Peressi and Baldereschi gen
eralised the Born-Mayer model and included ionic polarisation to
estimate the cohesive energies in KBr-RbBr, RbBr-RbI and KCl-KBr
crystals.
A brief review of the lattice energy calculations carried out
on various crystal systems is presented in the following section.
26
1.3 BRIEF REVIEW OF THE LATTICE ENERGY CALCULATIONS
PERFORMED ON VARIOUS IONIC CRYSTALS
During the last few decades the studies on the cohesive
energy and static properties of ionic crystals have been of much
interest to the investigators. The attractive interactions in these
crystals are quite well understood so that much of the effort has
been directed towards computing the repulsive interactions, as
discussed in the preceeding section. Of all the ionic crystals,
much attention has been paid to the alkali halides because of their
relatively simple crystal structure. A detailed account of the
individual lattice energy calculations made on various ionic crystals,
starting from the alkali halides is presented in this section.
1.3.1 ALKALI HALIDES
The lattice energies of alkali metal halides have been
calculated theoretically by a large number of workers. The
principal calculations by the classical ionic theory have been made117 6 118
by Mayer and Helmholtz , Kapustinskii , Verwey and de Boer ,
Landshoff , Huggins^ and by Kapustinskii^. Lowdin*^*’has
estimated the lattice energies of LiCl, NaCl, KC1 and NaF. Ladd
5and Lee have used a method to evaluate lattice energies, eliminat-
123 124ing the need for basic radii. Cubbiccotti , Saxena and Kachhava ,
Pandey^^ and Prakash and Behari^ have also evaluated the lattice
energies of alkali halides. Basing on an exponential form of
repulsive energy, Dixit and Sharma^ and Pandey^ have estimated
the lattice enerigies of alkali halides. Sharma^ had used the
27
Rydberg potential function^. Misra et al^ have employed a logari-
126 Sarkar andthmic form of repulsive potential. Goyal and Sankar
Sengupta*^, Sangster^^* , Lister^^’^^ have calculated the lattice
132energies of alkali halides. Pandey had used the Born-Mayer
133potential. Cohen and Gardon used a modified electron gas
134treatment to obtain the lattice energies. Calais had made a
74review of the calculations. Wen and Sho have made calculations on
the basis of the effective nuclear charge and rate of cation polari-
135sation. Shorczyk had made use of a semi-empirical force model
136with atomic constants as potential parameters. Thakur and Ivanova
77et al have also evaluated the lattice energies of alkali halides.
137Zhadanov and Polyakov have used a parameterless calcul
ation to obtain the lattice energies of NaBr, KBr, RbCl and RbBr.
138Ghosh and Basu have used a deformable shell model for CsCl
139 79and Csl crystals. Belomestnykh utilised the Nemilov relation
0 . , .140 . , . , D. , 141 c 142 _ . ,143Puri et ai , Andzeim and Piela , Saxena , Garg et al ,144 25 35Zhadanov and Polyakov , Sinha and Thakur , Usha Puri , Thakur
and Sinha^, Andzeim and Piela^^, Kaganyuk^^, Singh et al^^ and
148Shanker et al have all evaluated the lattice energies of alkali
149halides. Alonso and Iniguez have applied density functional
90formalism to obtain the lattice energies. Lister , Singh and
Shanker*^, Dedkov and Temrokov^, Nirwal and Singh‘S, Shanker and
Agrawal^^, Islam^, Shanker et al^^ and Nirwal and Singhhave
154obtained the lattice energies using different methods. Boswara had
used the generalised Huggins-Mayer form of the Born potential.
28
The lattice energies of alkali halides have also been29 95 155
estimated by Jha and Thakur , Yadav , Kaur et al (from dis
sociation energy studies), Singh et al156
(from relation between
the lattice energy U and thermal expansion coefficient a )t
RehmanYamashita and Asano157, Shukla et al158, Kaur et al159
and Shams*89 and more recently by Reddy et al97,
1.3.2 ALKALI HYDRIDES
gHylleras had evaluated the lattice energy of LiH by the
quantum mechanical calculations. Bichowskii and Rossini*8* have also
estimated the lattice energy of LiH by Born-Lande expression.
Lindquist had also applied quantum mechanical model for LiH. 163Waddington had obtained the values for the lattice energies of
all the alkali metal hydrides using a simple Born-Mayer expression
and ignoring van der Waals terms. Using the Born-Mayer equation
with second neighbour interactions and including van der Waals164
terms, Bowman Jr., , had estimated the lattice energies of the
alkali metal hydrides.
165Dass and Kachhava have made calculations based on the
molecular behaviour. Calculations for the determination of the
lattice energies of alkali metal hydrides have also been made in
148 25recent times by Zhadanov and Polyakov , Sinha and Thakur , Singh
and Sharma8*, Pandey and Uma Rani Pant*88, Thakur and Thakur*87,
89 168 36Shanker and Agrawal , Singh and Tiwary , Hasan et al and by
Pandey and Uma Rani Pant 169
29
1.3.3 UNIVALENT HEAVY METAL HALIDES
The lattice energies of the argentous, thallous and cuprous
170halides were evaluated by Sherman . The lattice energies of the
171above ionic crystals were recalculated by Mayer and by Mayer and
172 5Levy . Later, Ladd and Lee have recalculated the lattice
energies of the silver and thallous halides by a method avoiding
the use of Huggin's basic radii which are difficult to fix for these
173salts. Recently, Saxena et al have evaluated the lattice energies
of these heavy metal halides. Calculations on these halides have
also been made by Gohel and Trivedi174, Pandey176 for cuprous,
176silver and thallous chlorides, bromides and iodides and by Sharma
177for the same compounds. Later, Ladd had evaluated the lattice
178 179energies of thallous halides. Murthy and Murti , Gupta and Sharma,
180 71Kumari Jha et al and Sharma have obtained the lattice energies
181of these halides. Thakur had used logarithmic potential and
182Ladd had employed van der Waals potential to estimate the lattice
28energies of these univalent halides. Uma Rani Pant used different
interaction potentials including the logarithmic potentials to calculate
the lattice energies of these crystals.
Recently, calculations for the determination of the lattice
energies of these univalent heavy metal halides have been made
183 55 56 50by Thakur , Bakshi et al , Sharma et al , Thakur et al , Jain
19 184 89and Shanker , Shanker et al , Shanker and Agrawal and Singh
185and Khare
30
1.3.4 AMMONIUM HALIDES
The lattice energies of ammonium halides have been calculated
, „ . 186 . , 187 .. 188 _ , . , . 189 , T , .177 . .by Grimm , Bleick , May , Ladd and Lee and Ladd , treat
ing the ammonium ion as a point charge. Later, Murthy and Murti 178
179 190Gupta and Sharma and Jenkins and Waddington have also
191estimated the lattice energies of ammonium halides. Goodlife et al
have calculated the lattice energies of these halides basing on the
point charge model and the distributed charge model. Both the
models were satisfactory for the ammonium ion. In an altogether
75new approach, Jenkins and Morris have estimated the lattice
energies of the ammonium halides by relating the viscosity B-coeffici-
76ent of the Jones-Dole equation with the enthalpy of hydration of
the gaseous monoatomic constituent ion. Apart from the above
works, the estimation of lattice energies of ammonium halides has
51 192been carried out by Thakur et al , Singh and Rana , Agarwal
193 194 195et al , Raghuram and Narayan , Satyanarayana and Rana and
Varshney^^, in recent times.
1.3.5 FLUORITE TYPE AB^ CRYSTALS
The lattice energies of divalent metal halide fluorite type
170AB2 crystals have been evaluated by Sherman on the basis of the
197Born-Lande equation. Morris has extended the theoretical
calculation, again using the Born-Lande equation and has recalculated
198the thermochemical data. Reitz et al have evaluated the lattice
energy of CaF^ using the Born-Mayer potential and incorporating
199van der Waal interactions. Benson and Dempsey have also repeated
31
the same equation to evaluate the lattice energies. Benson et al^°°
have estimated the lattice energies of UO^ and Th02. Later, Brackett
and Brackett^, Shanker and Agrawal^, Shanker et al^^ , Singh
24 59 203et al , Gupta and Goyal and Dutt et al have all evaluated
the lattice energies of divalent metal halides of the fluorite type.
1.3.6 ALKALINE EARTH OXIDES
170Sherman used the simple Born formula for the evaluation
204of lattice energies of alkaline earth oxides. Mayer and Maltbie used
the Born-Mayer expression for the lattice energies, calculating the
London dispersion energies from the polarisabilities of the free ions.
205Later, de Boer and Verwey recalculated the lattice energies of
204the oxides because the Mayer and Maltbie calculations of the inter
atomic distances differed significantly from those obtained from
206X-ray data. Kapustinskii and Yatsimirskii have recalculated the
lattice energies for a large number of crystals. Huggins and
207Sakamoto have recalculated the lattice energies of all the alkaline
earth chalcogenides.
The lattice energies of some alkaline earth oxides have been
173 180calculated by Saxena et al , Kumari Jha et al and Pandey and
208 209Pant . Son and Bartels have applied a simple Born model for
the cohesive energy of MgO, CaO and SrO and found it less
satisfactory. They have also considered the ionic charge on the
oxygen ion to be varying between two values. The lattice energies53 27of these oxides have also been estimated by Cantor , Thakur ,
03A Oil / o
Upadhyaya and Singh , Thakur , Thakur and Sinha ,
32
50 150 91Thakur et al , Singh and Shanker and Mackrodt and Steward .
92 212 213More recently, Islam , Shanker et al and Singh et al have also
evaluated the lattice energies of the alkaline earth oxides.
1.3.7 UNIVALENT METAL CHALCOGENIDES
The lattice energies of a few compounds of this category
170 214have been calculated by Sherman . Recently Agarwal et al , Jain
215 216and Shanker and Agnihotri et al have evaluated the lattice
170 217energy of Na£S. Sherman and Mamulov have estimated the
lattice energy of C^O. Mackrodt and Steward have calculated
the lattice energy of Li^O.
1.3.8 OXIDES OF TRIVALENT METALS
170Sherman has calculated the lattice energies of A^O^ and
62C^Og. Yatsimirskii has obtained the lattice energies of A^O^,
217and In2®3 * Mamulov has evaluated the lattice energies
of ^2^3’ ^r2^3’ ^e2^3 anc* T*2®3* The lattice energies of A^O^ ,
218^r2^3’ a ~^e2^3 an<^ ^2^3 ^ave been calculated by Van Gool and Picken.
219Recently, Morss has evaluated the lattice energies of yttrium,
lanthanum and lanthanide oxides.
1.3.9 ALKALI METAL CHLORATES AND BROMATES
The lattice energy of NaClO^ has been calculated byYatsimirskii^^using Kapustinskii1 s^ equation. Morris^* has also
evaluated the lattice energy of NaClO^ from hydration heat value.
62Yatsimirskii has estimated the lattice energy of NaBrO^. Finch
222and Gardner have obtained the lattice energy of NaBrO^ by
33
hydration heat value and also using Kapustinskii - Yatsimirskii
206 223equation . Herzig et al have obtained the lattice energies of
NaClO^ and NaBrO^ making use of different methods.
1.3.10 PEROVISKITE TYPE AMX3 CRYSTALS
The lattice energies of peroviskite type AMX^ crystals where A
is a monovalent atom, M is a divalent metal and X is a halogen
or oxygen have been evaluated by very few workers. Rousseau
224et al have estimated the lattice energies of KMnF^, RbMnFj ,
KCoFj and RbCoF^ crystals using Born-Mayer type interactions.
86Narayan and Ramaseshan have estimated the lattice energies of
a number of peroviskite type crystals by using the compressible91
ion approach to repulsion. Maekrodt and Steward have estimated the
lattice energies of CaTiO^ and BaTiO^*
1.3.11 HEX A HA LOMETA LLATE A2MX6 TYPE CRYSTALS
Compounds having the general formula A2MX^, where A is an
alkali metal, thallium or ammonium, M is a main-group or transition
element of oxidation state (IV) and X is a halogen, and possessing
the antifluorite structure have received much attention. The lattice
energies of these crystals have been calculated by a large number
225of workers. Jenkins and Pratt have presented a review of the
calculations made and have also reported the lattice energy values
of a number of these compounds. They have used a new minimi
sation procedure to obtain the lattice energies. The lattice energies
34
of compounds possessing non-cubic structure have been226
estimated by Jenkins and Pratt
1.3.12 ORTHORHOMBIC SULPHATES
227Ladd and Lee have calculated the lattice energies of BeSO^,
228MgSO^, SrSO^, Al^SO^ and T^SO^. Selivanova and Karapet'yants
have evaluated the lattice energies of a number of sulphates MSO^
(M = Ba, Sr, Mn, Fe, Co, Ni, Cu, Zn and Pb) and SCL (M =b *1
229Tl, Cu, Ag and Hg). Later, Ladd and Lee making use of the
Born model, have computed the lattice energies of Ga, Sr, Ba and230
Pb sulphates. Jenkins has reported the lattice energy value
of K2S04. The lattice energies of alkali metal sulphates231
(M = Li, K, Rb and Cs) have been evaluated by Jenkins . Jenkins 232and Smith have reported the lattice energy value of (NH^^SO^.
233-244Jenkins and his co-workers have evaluated the lattice
energies of a number of complex ionic crystals.
In the following section, the aim and scope of the present
work is highlighted.
35
1.4 AIM AND SCOPE OF THE PRESENT STUDY
The evaluation of cohesive energy of ionic crystals has been
a subject of extensive study for the past six decades. Various
theoretical as well as experimental methods have been developed
in course of time with varying degrees of accuracy. The detailed
calculation of lattice energy according to the theoretical methods
requires large input data like crystal structure, compressibility,
interionic distance etc., which are not always accurately known for
all the ionic crystals. In the absence of any of these parameters,
the lattice energy values cannot be computed by these methods.
For better results, a knowledge of van der Waal's energy, zero
point energy and other interaction terms are required.
In the course of time, a large number of interaction potential
functions have been suggested by different workers. Their repulsive
parts are either inverse power functions or exponential functions.
31It has been found recently by Thakur that such potential functions
are not capable of yielding an acceptable behaviour of the potential
energy and force curves of ionic solids. Regarding the exponential
245form of the repulsion energy, Dobbs and Jones remark that the
calculation of the lattice properties using the exponential form
becomes complicated. They also comment that in the region near
the minimum of the total potential, the exponential form is perhaps
not valid. The logarithmic form of the repulsive potential has also
some limitations.
Lattice energies of complex ionic crystals are not accessible
36
by direct experimentation. Theoretical calculation involving quantum
246mechanical procedures though accurate, has met with only limited
success due to complexity in the spin correlation effects. Conse
quently one has to resort to classical, sem-empirical or empirical
methods to evaluate the lattice. energies of ionic crystals and related
thermodynamic properties. The classical treatment assumes ions
9to be point charges. Jenkins and Waddington have replaced the
older point charge calculations by a term by term calculation,
assuming ions having extended, charge distributions in any crystal.
These procedures require a knowledge of the internal structure of
the lattice and of the complex ion, which become difficult to
ascertain.
An examination of the review presented in the preceeding
2sections indicates the usefulness of the Born's theory for the lattice
energy of ionic crystals. An exhaustive survey of the literature
shows that in majority of the cases, the lattice energies have been
2calculated only from theories based on the Born's theory and its
refinements. It can also be inferred that a large number of workers
have applied this theory considering some modifications to the Born-
Mayer^ equation in the repulsive term and also incorporating terms
involving the van der Waal's energy, the dispersion energy and
the zero point energy. From the above discussion, it has been
observed that a single equation is unable to account for the crystal
lattice energies of all types of ionic compounds maintaining the
simplicity and accuracy in any procedure.
37
Measurements of the elastic properties of ionic crystals have
been very much useful providing information about the short range
interatomic forces, anharmonic properties and equations of state.
The dynamical theory of crystal lattice dynamics is highly
developed for ionic crystals. Therefore, it is possible to use
experimentally determined elastic constants and other parameters
in the calculation of such important characteristics of a solid as
the crystal lattice energy, Debye temperature and infrared
dispersion frequency.
Motivated by the situation outlined above, the author thought
it pertinent to evaluate the lattice energies of ionic and mixed ionic
crystals and their temperature dependences using Kudriavtsev's
247theory which relates the lattice energy with sound velocity in
247the crystal. The details of Kudriavtsev's theory are outlined in
248Chapter 2. The Voigt-Reuss-Hill approximation , which is used
to obtain sound velocity in the polycrystal from single crystal
elastic constant data is also presented in Chapter 2. The results
on the calculation of the lattice energies of ionic crystals are
presented and discussed in Chapter 3. The results on the
temperature dependence of lattice energy are discussed in Chapter 4.
In Chapter 5, results on the evaluation of lattice energies of mixed
ionic crystals are discussed.
