-
4
Physico ' CkEMicAl Properties of AUcaU hAlidES
4.1 Introduction
The alkali halides have special significance in chemistry. Since they are the simplest
known ‘ionic’ molecules1'3. Ionic crystals have played very important role in the development of
solid state physics. Ionic crystals are made up of positive and negative ions. When the crystal is
formed, the ions arranged themselves with the coulomb interaction between ions of opposite sign
being stronger than the coulomb repulsion between the same sign. It has been well established in
*
the literature that they are easy to produce as large and pure single crystals suitable for
experimental investigations. Because of their simple structures, they are relatively easy to
understand. The alkali halides are of great interest both from theoretical and experimental
importance. The ionic bonding of these crystals play a dominent role in crystal field. Many
physico-chemical parameters of these crystals are directly connected to the binding forces in ionic
solids Though all alkali halides are cubic in nature having a common structural pattern, they
nevertheless exhibit a wide range of physical behaviour. The fluorides have distinctive properties
than those of chlorides, bromides and iodides. The common crystal structure found in ionic
crystals are the sodium chloride structures, the caesium chloride structure and the fluorite type
structure. Exhaustive study has been available in the literature on the properties of alkali halides.
The crystal structure nature can be known with the help of the ionicity. The magnitude of the
electronegativity difference between two atoms indicates the degree of ionicity.
151
Larger
electronegativity differences involve higher
icnicity in the bonding. According to
Pauling4, the percentage of ionic bonding in a co. ipouiri increases from 1 to 2 to 3, where the
values of electronegativity is not sufficient by accurate in comparing the relative degree of ionicity
in different compounds. According to the equation proposed by Pauling the bonds between atoms
with electronegativity difference 1.7 are have 50 percent of ionic and covalent characteristic. In
recent years Villars and Phillips5 have succesfully used the
more accurate values of
electronegativity estimated by Martynov and Bastsanov6 to study the crystal structures of some
300 compounds of different type. According to them electronegativity of an atom is proportional to
the square root of the average ionization potential of the valence electrons. A detailed knowledge
and understanding of band structure has become increasingly important in the design and
development of many materials. The two most interesting and fundamental properties of some
materials are the absorption edge or optical energy gap and the refractive index. The elucidation of
the physical, chemical and optical properties of the alkali halides utilizing physico-chemical
parameters, such as electronegativity and refractive index is the subject of the present work. The
various physico-chemical properties were reviewed comprehensively by different authors on the
alkali halides It is therefore, not possible to mention all the works carried out by different
investigators. A brief review on the properties of alkali halides is given below. A detailed knowledge
of the ionization potentials of atoms and molecules is very important if their electron structure is to
be determined. It is also noted that both ionization potentials and electron affinities are used as
important parameters in which semi-empirical theories from which molecular quantities are
calculated. Garbato ei al.7,8 deduced an empirical relation between average nuclear effective
charge and ionization potential of the compound. Koopman’s theorem8 provides an opportunity to
estimate ionization potentials of the molecules. Gaydon19 has given a procedure to evaluate
ionization potentials through cyclic process. Electron impact photoionization, photoelectron and
spectroscopic experimental methods11 19 are used to evaluate ionization potentials of the
molecules. Reed
evaluated ionization energies for several molecules and concluded that the
predicted ionization energies are too low for bonding electrons and two high for antibonding
152
elections. Ionization potentials are of gteat iii’pe.L
o
the interpretation of molecular spectra
and structures, and in studies of the interaction. L>,wreen molecules and between molecules and
atoms. Reddy et al.21 have proposed sm.
'
rnulus relating electronegativity differences,
ionization potentials and dissociation energ!"-
The estimated ionization potentials are in good
agreement with values cited in the literature. Martynov and Batsanov6 have revised the values of
electronegativity for the elements by assuming the concept that the electronegativity of an atom is
the square root of the average ionization potential of the valence electrons. These electronegativity
values of the elements have been used in the well known Pauling’s equation22 and estimated the
energy of formation of condensed substances. The energy of formation of Schottky defects in
alkali halides is usually estimated from the experimental data on ionic conductivity of alkali halide
crystals. The linear relationship between the Debye temperature Bo and the energy of formation of
Schottky defects has been demonstrated by Sastry and Mulinani23 in alkali halides.
Niwas and co-workers24 have revealed the effect of many-body potential on the
volume dependence of static dielectric constant, static polarizability, ionic distortions and effective
charges with in the framework of Lundquist's expression25 for ionic cohesion, The importance of
the inclusion of ionic distortions and their variation with change in the lattice parameter in the
studies of volume dependence of static dielectric properties, have been emphasized by evaluating
the transverse optic mode Grijneisen parameter for alkali halides crystallizing with the NaCl
structure
. Considerable success has been achieved in calculating the various bulk crystal
properties based on an understanding of the Interatomic forces. Elastic constants of a crystal are
helpful in predicting the behaviour of a crystal under certain special conditions. The relations
connecting the elastic constants with the interatomic forces are outlined by Kachhava and
Saxena
, with particular reference to two molecular models due to kellermann27 and krishnan and
Roy28, respectively. The authors have compared their results with the Krishnan and Roy s model28
for the specific case of alkali halide crystals.
153
Thakur and Sinha29 have sugn^-l- 1 a no'./ equation for the calculation of lattice
energy 'which utilizes crystal structure and hit'. Ionic distance values
By using these lattice
energies the authors have been evaluated the heat of atomization values of 28 diatomic ionic
crystals of alkali halides, hydrides and deuterides. Many workers30,31 have used Born-Mayer
equation of exponential form for the repulsive part of crystal energy, logarithmic form for the same
has been used by some workers31'33 in the calculation of lattice energy of alkali halides. Binding
energy, rotational and vibrational constants of alkali halide and hydride molecules were calculated
by Dass and Saxena34, using a potential function consisting of eletrostatlc, van der Waal’s,
polarization and Gaussian type of overlap, and compared with the available experimental data.
Usha Puri35 suggested a new type of repulsive term in the interaction potential and used to
calculate the cohesive energy and atomization energy of alkali halides.
Several theoretical expressions reported already in literature have been used by
Kachhava and Saxena36 and compute the Reststrahlen frequency of alkali halide crystals.
Regarding the constituent ions of crystals as non-polarizable point masses and assuming no
♦
explicit form for the overiap part of the interaction potential, Ketlermann27 deduced an expression
for Reststrahlen frequency of NaCS type crystals. Following the same approach Sharan and
Tiwari37 derived an expression for CsCl type crystals. Krishnan and Roy38 developed a theory for
the Reststrahlen phenomenon by considering the polarization of the medium resulting from the
oscillation of the interpenetrating lattices of the alkali and the halogen ions with respect to each
other. Woods et al.39 have worked out the lattice dynamics of alkali halide crystals on the
assumption that only the negative ion is polarizable. Blackman40 suggested a simple relation
between Debye temperature and circular Reststrahlen frequency.
On the basis of theory proposed by Hanlon and Lawson41, Kachhava42 has derived a
new relation for the estimation of dipolemoments of the alkali halide crystals. In his derivation he
adopted a Rittners s expression
. These estimated dipolemoments are used to determine the
effective charge in molecules. Gohe!44 and Gohel and Trivedi45 have suggested new potentials and
154
evaluated We, WeXi* and De values for alkali lc!ld.
Durstein48 introduced the concept of localized
effective charge parameter which has been wide!> use d to discuss the nature of the chemical bond
and dielectric properties of ionic crystals47,48. The localized effective charge parameter and its
strain dependence in ionic crystals has been studied and analysed by Shanker and Gupta49.
Behaviour of the szigeti effective charge parameter e* in alkali halides has been discussed by
Lowndes and Martin50. Since the effective charge is related to many different phenomena in
crystals, es has been studied within several different theoretical approaches. For example,
Lawaetz51 has found a simple empirical relationship between es and electronic ionicity in the
framework of the Philiips-Van Vehten theory. Ab initio calculations of es have been performed with
the use of the local density-approximation by -.Uian52 In alkali halides an interesting correlation
between effective ionic charge and interatomic separation has also been reported53. Kucharczyk54
has estimated the pressure dependance of the szigeti effective ionic charge in alkali halides. In his
method the empirical connection between electronic and lattice ionicity proposed by Lawaetz51 is
taken into accout.
There have been numerous attempts to study the cohesive, elastic, lattice dynamical,
thermal and dielectric behaviour of the alkali halides55-58. The Rydberg potential has been
analysed by many workers58'60 in the field. Varshni and Shukla58 used this function to explain a
few properties of alkali halides but the results were not found to be very satisfactory. Gupta and
Agrawal59, while reviewing this potential, extended its application to ionic crystals crystallizing in
the NaCI and CsCI structures to calculate some crystal properties. However, Gupta and Agrawal59
in their work ignored the van der Waals interactions, zero-point energy, and many-body
interactions which contribute significantly to the lattice dynamical properties of the ionic solids.
Gupta and Sipani60 have successfully employed this potential to explain the first and Second order
pressure derivatives of the bulk modulus, Gruneisen parameter, Anderson- Griineisen parameter
and the coefficent of thermal expansion using Slater, Dugdale-MacDonald and free volume
theories.
155
Pauling’s procedure61 gives fairly accurate estimate of bond energy between two
uni-univalent atoms held together by a single bond. However, this approach emphasize the ionicity
and type of bonding through electronegativities. Degree of ionicity in a chemical bond can be
understood through bond energy. Matcha62 has proposed a simple quantum mechanical theory to
the interaction of potential between two atoms and obtained an appropriate form of bond energy,
electronegativity relationship which is valid for both ionic as well as covalent bond. Manca63 has
attempted to calculate the energy of a single bond for different semiconductors crystalizing in the
diamond and blende structures by means of Pauling relation. Thermal and mass spectroscopic
methods are the most important experimental methods used to measure bond energy
values10,61,64. Wilkinson65 has pointed out that a knowledge of ionization potentials and
dissociation energies is important in astrophysical as well as in many physical and chemical
problems. Ramani and Ghadgaonkar66 have developed different methods for the evaluation of
bond energies. Matcha62 has pointed out that the Paulings empirical relation fares badly when
applied to ionic bonds. Different correlations have been proposed by Reddy et al.67 between band
gaps, electronegativities and bond energies of alkali halides. A simplified formula relating bond
orders to spectroscopic constants has also suggested by the same authors. Reddy et al.68,69 have
estimated the bond energies of alkali halides based on the derived relations. Linear correlations
have been found between the magnetic shielding constants of the nuclei of the alkali metals,
thallium and the halogens or the 19p NMR chemical shift and the dissociation energy of the
corresponding halides of the alkali and alkaline earth metals and thallium by Chizhervskii and
Kharitonov70.
In the chemical approaches to the approximate prediction of band gaps of
semiconductors, correlations have been proposed between band gaps and various other quatities,
e.g. single bond energies63, heats of formation per mole71,72, heats of formation per equivalent73,
atomic numbers74, electronegativites75 etc. Some of these approaches, especially those seeking
relationships between binding and band gaps have also been presented in the form of reviews,
156
either detailed76 or synoptic63. Vijh77 has been derived a relation between bond energies and band
gaps for the case of ionic compounds. Rincon78 has discussed the relation between Debye
temperature, melting point and compressibility of ternary chalcopyrite compounds which has
previously developed by Oshcherin79.
Fluorine exhibits a number of anomalous properties, both as an atom and as a
molecule80,81,82. Politzer80 has enlighted the formation of fluoride ion includes an unusual
destabilizing effect of same short, with which is associated an energy of about 26 K cal/mol and
mentioned that this effect will manifest itself in various situations involving this ion. The electron
affinity of the fluorine deviates very markedly from the general trend, however-it is less than that of
chlorine. Atomic radius, bond length of the molecule, dissociation energy of the molecule of the
diatomic halides are increasing from fluorine to iodine. Several explanations have been given for
the deviation of fluoride from other halides. Normally the first member of the periodic group differ
from the remaining members of the group. In the case of fluoride, it is more pronounced. Politzer80
has exposed the odd behaviour by observing the weaking of bonding by fluorine to other elements
compared with expecting on the basis of extrapolations from the heavier halogens. It is particularly
interesting to consider the case of fluorine molecule. The dissociation energy of F2 is very low,
relative to the tend among the other hologen molecules. Several interesting explanations for this an
anomaly have been proposed in the literature80,81'82 The effect of the fluorine as an atom in the
case of alkali halides has also been explained clearly by several workers. In view of the attractive
and special properties of the alkali halides mentioned above, the author has chosen to investigate
different properties of alkali halides like ionization potentials, bond energies, plasmon energies,
energy gaps, interionic distances, heats of atomization and heats of formation and melting points
of alkali halides using electronegativities83 and refractive indices.
157
4.2 Physico-chemical parameters and methods of analysis
4.2.1 Ionization potentials:
The ionization potential of an atom or molecule is defined as the energy required to
remove an electron from the highest occupied atomic or molecuier orbital of the species in its
ground electronic state. A detailed knowledge of the ionization potentials of atoms and molecules
is very important if their electron structure are to be determined. It is also noted that both ionization
potentials and electron affinities are used as important parameters in the semi-empiricai theories
from which molecular quantities are calculated. Ionization potential plays a vital role in the field of
astrophysics and many physical and chemical problems. The ionization potentials and dissociation
energies are of great importance in the interpretation of molecular spectra and structure in the
studies of interaction between molecules and molecules and atoms10,65.
Different experimental methods are available in the literature84 for the determination of
ionization potentials of the atoms and molecules. The followings are some useful experimental and
theoretical methods for the determination of ionization potentials.
Experimental Method:
There are two main processes which lead to the removal of electron or electrons from
atoms or molecules to produce the corresponding ions. Impinge electrons with sufficient energy
can ionize atoms or molecules by the following process:
AB + e = AB+ + 2e
......................................................................(4.1)
Ionization can also be caused by interaction of quanta of sufficiently high energy to
enable the process
AB + hy = AB+ + e
......................................................................(4,2)
to occur. This latter process is known as photoionization.
158
Koopman’s theorem:
Koopman in his theorem drawn ;
-
-r : that there is a link between the position
of bands in photo electron spectra end tb~ e.k
Hartree-Fock limit is the negative of the ieniz_.it
orbital. The theorem is valid strictly only to mol
i
,
olar orbitals. The orbital energy in the
_n;s t .t r the removal of an eletron from the
.n which all orbitals are either filled with two
electrons or empty, but, this covers the majoiity of the stable molecules. It also rests on the
assumption that on ionization the remaining electrons do not adjust their positions to the new
potential in an anion. In other words, the molecular orbitals for the -t-ve ion are the same as those
for the neutral molecule. The last assumption has specified that inherent errors in it are not usually
sufficiently large to invalidate the association between molecular orbital energies and ionization
potentials expressed through Koopman’s theorem.
A corollary of Koopman’s theorem is that distribution of the + ve charge in the ion may
be determined from the wave function of the molecular orbital from which the electron have been
removed. If /?(r) is the total electron density function for the ground state of the neutral mqlecule,
then the electron density in the ion obtained by removing an electron
is [p(r) - M'2k(r)]. If this
total electron density function is broken down to coritibutions on individual atoms, then, after
combining those with the positive nuclear charges, then one can obtain the distribution of the net
positive charge on the individual atoms.
Cyclic process:
An ionized molecule AB+ In its lowest electronic state may be expected normally to
dissociate to an ionized atom A+ and a neutral atom B, both in their lowest states. If the
dissociation energies of the neutral molecule, D(AB) and of the ionization potential of the molecule,
D(AB+) and the ionization potential of the atom A, then the ionization potential l(AB) of the
molecule are clearly related by the expression10
i{AB) = D(AB) + I (A) - D(A8 + )
(4,3)
The atomic ionization potential 1(A), D(AB) and D(AB + ) are known accurately from
spectroscopic data and other experimental methods1005,86. In few cases the molecular ionization
potentials can be determined from the limli of Rydberg series in a band spectra, or from the
maximum wavelength of photo-ionization, In other cases ionization values are available from the
measurement of ionization thesholds by electron beams; this method tends to give vertical
ionization potentials. Known values of I (A), D(AB) and D(AB+) are substituted in equation (4.3),
then the ionization potential of a molecule i.e., I(AB) can be evaluated.
Garbato et a!., process:
Garbato et al.
proposed a relation between nuclear effective charge and ionization
potential of a compound AB as given below:
A
Zab
1/2
/ Iab
ab \
= n(----VR
R
............................................................................ (4-4)
/
where n is the arithmetic mean of nA and ns. Here
ra
and ne are the principal quantum numbers of
the outermost electrons of the constituent atoms. Iab and R are the ionization potential and
Rydberg constant, respectively. Further Garbato et a!7’8 suggested an empirical relation to
estimate the nuclear effective charge by means of the arithmetic mean of the atomic nuclear
effective charges and it is as follows:
*
—
Zab = aZ + b
where a, b are constants and these values are different for different groups of materials.
These methods are very expensive and time-consuming methods. In view of this and
also to verify the empirical studies made earlier, the author has proposed some simple and straight
empirical relations for the evaluation of ionization potentials of alkali halides by utilising the
refractive index (n) and Martynov and Batsanov electronegativity values alone. The suggested83
empirical relations are as follows.
160
Iab = 10.26- 1.34 n
..................... ................................................ (4.5)
lAB = 3,76 (^A^b)-0.b>
........................................................(4.6)
where Iab (eV), n and *a *b are the ionizer ,i ;
tential, refractive indox and Martynov and
Batsanov6 electronegativities, respectively. Thu > lumorical constants in the above expressions are
the result of a fit of the experimental data.
4.2.2 Bond energy:
Bond energies of the molecules are of great interest in thermochemistry, combustion
physics and astrophysics. Reliable bond energies are essential to estimate moleculer abundances
and interpret dissociation equilibria and ionization potentials in Stellar and Planetary atmospheres.
If a molecule is more than diatomic, two kinds of bond energies may be considered64. One is
represented by the energy required to break a particular bond, isolating the two neutral molecular
fragments. This is called ‘bond dissociation energy’. The other is the ‘average bond energy’, which
is the total heat of atomization divided by the number of bonds. Wilkinson65 has pointed out that a
knowledge of ionization potentials and dissociation energies is important in astrophysics! as well
as in many physical and chemical problems. Since the dissociation energy is a prime factor in such
phenomena, astrophysicists, chemists and spectroscopists are concerned with the determination
of reliable values of dissociation energies for diatomic molecules65. Also it is of use in testing
different theories of chemical bonding and electronic structure of molecule. Calculations of
theoretical parameters like heats of chemical reaction, heats of sublimation and atomic heats of
formation of polyatomic molecules as well as of statistical equilibria require a knowledge of
dissociation energies of the molecules involved in the reactions. Thermal and mass spectroscopic
methods are the most important experimental methods used to evaluate bond energy values87’89.
In view of the above importance the author has taken up this investigation to find the
bond energies of alkali halides.
161
Thermo chemical process:
In this process the bond energy of ’ iaC can be obtained by the equation
D (Na-CI) = A Hs + 1/2 A Hdiss + A Hf............. ...........................
(4.7)
where D (Na-CI) is the required bond energy; A Hs is the heat of sublimation of Na metal per mole,
A Haiss is the heat of dissociation of CI2 molecule into Cl atoms per mole, A Hf is the heat of
formation of NaCI per mole in Its standard state. The bond energy in equation (4.7) is nothing but
heat of atomization of NaCI. In the case of NaCI, only one shared electron pair is involved in the
bonding. In the case of complex compounds like TaaOs more than one bond (ie., shared electron
pair) is involved and hence a suitable normalizing factor must be used to obtain from equation
(4.7) , the bond energy per bond and not tfie total energy for all the bonds in TaaOs. This may be
carried out by calculating first from equation (4.7) the heat of atomization per equivalent (ie., the
bond energy ) for Ta20s is 1/10 of the heat of atomization per mole as calculated from equation
(4.7) .
The bond energies (ie., heat of atomization per equivalent) thus obtained after second
order corrections (ex: spin correlation stabilization energies, coordinate valence) as discussed by
Howald90 are the actual values of the average bond energies as obtained from experimental,
theoretical data. The accuracy of these energies is limited only by the accuracy of the
thermodynamic data used in computing them. Vijh77 concluded that this procedure is strictly valid
for alkali halides only because alkali halides do not involve any coordinate bonds.
Pauling process:
Pauling4 stated that the energy of an actual bond between unlike atoms is greater than
the energy of a normal covalent bond between these atoms. This additional energy is due to the
additional ionic resonance energy or ionic character ie.,
D(A-B) >. (1/2) (D(A-A) + D(B-B)] ................................................ (4.8)
162
If the additional ionic resonance encna !; I. c’catod by A, the equation (4.8) can be
written as
A = D(A-B) - (1/2) [D(A-A) + D(B-E)]............................................................ (4-9)
This postulate of additive law is not valid for alkali hydrides. When treated quantum
mechanically electron bonds, the arithmetic mean is replaced by the geometric mean. The values
for energy of normal covalent bonds where equation of geometric mean is involved is more
satisfactory than the arithmetic mean equation (4.9).
A (A-B) = A(A-B) - [A(A-A)A(B-B)|,/4........................................................... (4.10)
A (A-B) is obtained directly from heats of reaction using the relation
A (A-B) - 30(XA-J£6r.................................................................................(4.11)
where
xa
and %b are Pauling electronegativities of the atoms. The bond energy (K cai/mol) is
evaluated through Pauling’s relation4
D(A-B) = [(Daa.Dbb)]1/2 + 30 Ax2........................................................... (4.12)
where Dm and Dbb are the single bond energies of atoms A and B, respectively. Ax =
xa^b
(Pauling’s electronegativity difference).
The Pauling’s empirical relation [Equn.(4.12)J fares badly when applied to ionic bonds.
The error in estimated ionic bond energies is as much as 190 percent. So Reddy et a|.69*91,92
worked out a relation, using which they calculated bond energies of both ionic and covalent
molecules. The relation developed by them is simple and straight forward, which is as follows:
Dab = Dab + 32.058 Ax.............................................................. (4.13)
where Dab, Dab = (Dab.Dab)
and
t\% =
are the bond energy of a molecule, single bond
energies of constituent atoms and Pauling’s electronegativity difference, respectively.
Matcha Process:
Matcha62 used simple quantum mechanical theory to the interaction of potential
between two atoms and obtained an appropriate form of bond energy, electronegativity
relatibnship which is valid for both ionic as well as covalent bond. The final equation be obtained is
as follows:
f f
Re Ect*
Dab = (1-0 Dab + — Rc Dion.............
RoL
f
(4.14)
Ect is the energy associated with partial charge transfer, Dab is the average covalent bond energy,
f is the effective charge transferred in the curve cussing region during bond formation and Re is
the internuclear distance of the molecule. Substituting the constants Ect, Re and Dion appropriate
to LiF
Dab =•• (1-f) Dab t 252 (f/Re)...................................................... (4.15)
A comparison between (f/Re) and i (ionic character defined by Pauling) suggests a
strong correlation between these quantities. This suggests that Dab can be related to A* by
replacing (f/Re) by (2i/3) where i = [1 - exp(- A*2/4)].
Matcha
gave a relation for the construction electronegativity scale as
Dab = Dab + (f/FU) [252 - r»DabJ
............................................(4.16)
The term in the square brackets is approximately constant (169) for a series of bonds.
Thus the above equation can be written as
DabU Dab + 169 (f/R*).............................. .............................. (4.17)
replacing (f/Re) by (2i/3)
DabU Dab + 113 i
.................................................................. (4.18)
164
To increase the accuracy of this equation, Met..-;
aJtic.l hvo adjustable parameters K & i,
thus
Dab
Dab + Ki
........................................................................ (4.19)
for smaller values of A%, Ki becomes
K (1 - exp(-« A,v)2]
Matcha found that K = 103 and a = 0.29
,inai,y
Dab = Dab + K [1 - exp(-« A*)2]
................................................. (4.20)
here Dab = (Daa.Dbb)1/2 and A* — xk ~7,B are ttie arithmetic mean Gf the single bond energies of
constituent atoms and electronegativity difference respectively.
The following relations which are proposed in the present study are found to yield
good results for alkali halides83 for the estimation of bond energy (Es):
Es = 219.82-75.82 n
.......................................................... (421)
Es = 55.18 (}fA£B) - 48.93
................................................................ (4 22)
where Es, n and^B are the bond energy (K cal/mol), refractive index and Martynov and Batsanov
electronegativity6, respectively.
4.2.3 Valence electron plasmon energy:
From the classical oscillator theory the density of oscillators, the oscillator strength,
the angular frequency at the resonance and the dielectric constant of free space is well correlated.
This simple classical oscillator model would give a good quantitative explanation of the refractive
index and which was pointed out by Moss93 in 1952. Penn94 gave a modified expression for the
refractive index, valence electron plasmon frequency and the energy of (JV resonance. Jackson95
165
has given a following relation between valence electron plasmon energy (fiwp), the total number of
valence electrons (2), density ip) and the molecular weight (M) of a material:
'fiWp
pP
= 28.8 /-----
V
........................................................................(4.23)
M
The significance of valence electron plasmon energy in describing fermi energy, Penn
gap, dielectric constant and electronic plarizability has been given by different workers96'101. The
above
parameters
are highly useful
in
studying the
structural
properties
of different
semiconductors. For a model semiconductor, the high frequency dielectric constant is explicitly
dependent on the valence electron plasmon energy, an average gap referred to as the Penn gap
and the fermi energy. The Penn gap is determined by fitting the dielectric constant with the
plasmon energy. In view of the above importance, the author has proposed83 the following relation
for the estimation of high frequency plasmon energy:
"ftWp
where-fiwp,
- 8.407 (jp%b) - 7.466
..........................................................(4.24)
and %b are the valence electron plasmon energy (eV), Martynov and Batsanov
electronegativities of the constituent atom A and B, respectively.
4.2.4 Energy gap and Refractive index:
The most interesting and fundamental properties of a material are the optical energy
gap or absorption edge and the refractive index. The magnitude of the refractive index can serve
as a guide in indicating the nature of the bonding because it is closely related to the electronic
polarizability of ions and the local field inside the material. Polarization is also the nature of its
degree of ionicity. Many attempts93,102'106 have been made to find more relationships between
these parameters both from the point of view of fundamental interest and also as a technological
aid. Moss
has given an excellent review on the various correlations suggested between optical
refractive index and energy gap of a material. Several workers63'67,103,104 have been proposed
different relationships between energy gap, simv..
’ -mergy, atomic numbers, refractive indices
and electronegativities of various groups of material::.
The first proposal was made by Moss93 on the very general grounds that all energy
levels in a solid are scaled down by a factor 1/fopi , where t0pt = n2 is the optical dielectic
constant. In terms of energy gap Eg, the Moss relation is given by
n4 Eg = 95 eV ................... ................................................................. (4.25)
For zincblende and diamond structures, Moss93 has given another relation
n4 Eg * 173 eV ..................................................................................(4.26)
Ravindra and Srivastava105 have proposed another relation with a revised value of the
constant.
n4 Eg = 108 eV
(4.27)
Duffy107 relation reads as
Eg = 3.72 Ax
where &x
................................. .................................................. (4.28)
= X (anion)^ (cation); x (anion) and /(cation) are the optical electronegativities of the
anion and the cation, respectively. By using the above relation [Eqn.(4.2S)] Reddy et al.108 have
been proposed a relation between refractive index and optical electronegativity which is as follows:
‘9.76"
where n is the refractive index and x
= X (anion)-/(cation); x’(anion) and X*(cation) are the
optical electronegativities of the anion and the cation, respectively.
In the present study, the author83 assuming the following linear relationships belween
the energy gap, refractive index and Martynov and Babanov electronegativity6 for aikafi halides:
167
(4.30)
Eg = 22.55 - 8.94 n
and
Eg = 4.786 (*A*B) + 4.244
........................................................... (4.31)
n = 2.286 - 0.271 feA xb)
........................................................... (4-32)
here Eg (eV), n and *a *b are the energy gap, refractive index and the product of Martynov and
Batsanov electronegativities6 of constituent atoms, respectively.
4.2.5 Interionic distance:
In the beglning of X-ray crystallographic Investigations109,110 the ionic radii based on
the concept of hard atomic or ionic spheres have played a very significant role in determining the
crystal structure, interionic forces, and the nature of the chemical bond. In traditional methods
used by earlier workers111'113 reproduced ionic radii which differ largely from the corresponding
values estimated from the X-ray diffraction and electron density measurements in crystals.
Singh114 has been obtained ionic radii from the knowledge of electron density measurements for
NaCI crystal and use of the additivity rule. The method of analysis is as follows:
The lattice parameters or nearest-neighbour Interlonlc separations can be divided Into
two parts by introducing the concept of ionic radii such that
R = r+ + r.......................................................................................... (4.33)
where R is the nearest-neighbour interionic separation and r+ (r-) the radiii of cation (anion).
Pauling111 has been suggested a relation connecting the ionic radii and effective
nuclear charges as follows:
Cn
-
(4.34)
--------(Z-S)A
168
where Cn is a constant depending on the number of electrons in the ion. Cn is same for a
sequence of isoelectronic ions. S is the size-screersiny parameter. Z is the nuclear charge. (Z - S) is
then taken as effective nuclear charge. + and - subscripts for cation and anion, respectively.
Keeping in view of the above the author has been proposed83 the following empirical
relations
between
interionic
distance,
refractive
index
and
Martynov
and
Batsanov
electronegativity:
r0 = 0.75 + 1.46 n
........................................................................ (4.35)
r0 = 6.038-1.15 CfA ZB) .................................................................... (4.36)
here r0 (A), n and
xa xb
are the interionic distance, refractive index and the product of Martynov
and Batsanov6 electronegativities, respectively.
4.2.6 Heat of atomization:
The heat of atomization of a crystal gives the crystal stability condition than cohesive
energy. It plays a significant role to understand the nature of the force. In the case of
non-molecular solids each atom is surrounded by several others in an interconnected aggregate
extending the boundaries of the crystal. In terms of the energy of atomization one can easily
determine the total forces holding the atoms together64. The total energy of atomization is
obtained simply by the difference between sum of the separate heats of atomization of its
component elements and the standard heat of formation. A more realistic indication of the stability
of a compound is readily available from thermodynamical measurements. This inturn gives the
concept of the heat of atomization or atomization energy. Sanderson64 has dearly demonstrated
various influencing factors of atomization energy. In alkali halides, four pairs of electrons per atom
pair are available for the formation of six bonds per atom pair is achieved by utilizing atomization
energy. Using a special repulsive term in the interaction potential, Usha Puri35 has made an
attempt to evaluate atomization energy for alkali halide crystals. The potential function employed
by the above author for the repulsive part of the energy is given by
«ZiZ2e2
Zb
<P (r) ................... + -- e
r
where
<p(r)
(4.37)
r
is the potential energy per unit cell, a is the Madelung constant, Zie and Z2e are the
charges on the ions, r is the ionic separation, Z is the number of nearest neighbours of any ion.
The constants A and b are the potential parameters which are evaluated by applying the following
conditions;
(i) the condition of equilibrium
/d.n
=o
\ dr /r = r0
(ii) the compressibility expression is
/ d2<£ \
9 C r0
V dr2 / r = r0
where
(3o
<p is the potential energy of the substance, Bo is the compressibility, C is the structure
dependent parameter, defined by molar volume V = N C r2; r0 is the equilibrium nearest neighbour
distance. The cohesive energy per mole (W) is related to <p (r0) by the expression
W = -N
ip (r0)....................................................................................... (4.38)
where N is the Avagadro number.
By utilizing the cohesive energy in the below relation one can evaluate the atomization
energy:
AE = W + E -1
(4,39)
170
where AE, W, E and I are the atomization energy, cohesive energy, electron affinity of an anion and
the ionization energy of a cation.
The author in the present study, proposed83 following empirical equations connecting
the
atomization
energy,
refractive
index and
the
product
of
Martynov and
Batsanov
electronegativities6 of two atoms for alkali halides. The expressions are as follows:
AhUt = 315.54 - 104.12 n ...................................................................(4.40)
AHat = 50.198 (*a*b) + 21.07............................................................ (4.41)
where AHat and n are the heat of atomization (K cal/mol) and refractive index , respectively;
xa
and
XB are the Martynov and Batsanov electronegativities6 of the constituent atoms of alkali halides.
4.2.7 Heat of formation and Melting point temperature:
Heat of formation of any compound is the entholpy of the reaction which represents
the formation of a compound from its elements. In any chemical reaction the species participated
in it are in their standared state, the heat of formation AH? is called standard heat of formation64. In
determination of bond energies, heat of sublimation and heat of formation are play an important
role. Vijh73,77 has correlated the band gap and the heat of formation for a diatomic compound.
Garbato et at.7,8 have given an empirical relation between the standared heat of atomization and
the standared heat of formation for various semiconducting compounds and a linear dependence
between meting point, the arithmetic mean of the atomic nuclear effective charges. If heat of
formation of the reactants and the products of chemical reactions are known, heat of reaction can
be calculated from the relation:
AH° = Eni AHt(products) - Enj AHf(reactants)............................. (4.42)
*
J
where ni and nj are number of mole of product species and reactant species, respectively. AH? is
the molar heat of formation. In the calculation of bond energy, Vijh77 has used the thermochemical
procedure in which he used the following relation:
171
D(Na-CI)
AHS i (1/?) AHtiiss
t
AH?
(4.43)
where D(Na-CI) is the required bond energy; AHS is the heat of sublimation of Na metal per mole;
O
AHdiss is the heat of dissociation of CI2 molecule into Cl atoms, again, per mole; AHf is the heat of
formation of NaCI in its standard state, again, per mole.
From the first law of thermodynamics, it may be readily shown by means of a
Born-Haber cycle that
AH? = AHs + lm + (1/2) AHd + A, + U .................................... (4.44)
where AH? is the heat of formation, e.g of NaCI; AHS is the heat of sublimation of Na; lm is the
ionization potential of Na; AHd is the energy of dissociation of CI2 (g) into atoms; Ar is the electron
affinity of Cl; U is the lattice energy of NaCI. Garbato et al.7 have given a relation between standard
heat of atomization and standared heat of formation which is as follows;
AHat(AB) = A Hat (A) + AHat(B) + dHf(AB)............................... (4.45)
where AHat is the standard heat of atomization of the compound and its constituent atoms; dHf is
the standard heat of formation.
Keeping in view of the above importance, the author has been proposed83 straight
forward empirical relations between heat of formation, melting point temperature, refractive index
and Martynov and Batsanov electronegativities6. These equations are valid only for alkali halides
and given as:
AHf = 131.28 n-309............................................................................................. (4.46)
AHf" = 27.535 - 49.72 (7 Ay a)................................................................................ (4.47)
Tm = 1756.66 - 660.08 n........................................................................................ (4.48)
where AH?, n, y_A ya and Tm are heat of atomization (K cal/mol), refractive index, the product of
Martynov and Batsanov electronegativity and meting point temperature (C), respectively.
172
4.3 Results and discussion:
The present study demonstrates the importance of Martynov and
Batsanov
electronegativities6 and refractive index in evaluating various physico-chemical parameters of alkali
halides.
The
necessary
input
data
of
alkali
halides
namely
Martynov
and
Batsanov
electronegativities6 and refractive index103 have been given in relavant tables [4.1 to 4.8],
The ionization potentials of alkali halides have been calculated using equations (4.5) &
(4.6) and the results are presented in table 4.1 along with the available experimental data of Huber
and Herzberg85. A glance of the data on ionization potentials presented in table reveals that
agreement between the values of ionization potential evaluated from equations (4.5) and (4.6) and
those obtained from literature85 is excellent. The proposed83 equations relates Martynov and
Batsanov electronegativities, refractive index and ionization potential. In the present study the
average standard deviations estimated for the alkali halides are 5.6 and 6.56. Figure 4.1 reveals
that the values of Iab calculated from equations (4.5) and (4.6) are in good agreement with the
literature values85. The ionization potentials exhibit a gradual increase from iodine to fluorine. The
proposed correlations are linear in nature and the numerical constants involved in the equations
are unique in the sense that they represent the best fit with the experimental values85. The
adiabatic ionization potential values given by Berkowitz et al.17 for CsF, CsCI, CsBr and Csl are
8.80, 7.84, 7.46 and 7.10 eV respectively are in close ageement with the present study. Potts et
al.16 have demonstrated the photoelectron spectra of alkali halides. In the case of alkali fluorides
the estimated ionization potentials are high. Several explanations80,81 have been given for the
deviation of fluoride from other halides. Pitzer
and Potitzer
have given several explanations for
this odd behaviour. In view of the Mulliken61, eletroneyativity is mainly depends on the ionization
potential and
electron affinity of the concerned atom.
The proposed
relation involving
electronegativity to estimate ionization potential is more opt in the present study. This equation has
direct bearing on the concept of ionicity of a molecule.
Table 4 .1: Ionization potentials.
JAB (eV) nt alkali halide crystals
Tonizot i-: . potential.
Alkali
halides
Refractive
index, n
[Ref.103]
M-B electronegativity,
%
[Ref .6]
Present
study
Eqn.(4.t;
Present
study
Eqn.(4.6)
tab
<eV)
Known
[Ref.853
LiF
1.3915
3.402
8 *i 0
12 18
LiCl
1.6620
2.682
3 . uj
10.08
liBr
1.7840
2.547
7.8 7
8.97
Li I
1.9520
2.484
7.64
8.73
NaF
1.3360
3.364
8.45
12.04
NaCl
1.5400
2.652
8.20
9.36
8.93
NaBr
1.6412
2.519
8.10
8.86
8.31
Nal
1.7745
2.456
7.88
8.62
7.64
KF
1.3610
3.024
8.44
10.76
KC1
1.4900
2.384
8.26
8.35
8.44
KBr
1.5590
2.264
8.17
7.90
7.85
KI
1.6770
2.208
8.01
7.69
7.21
RbF
1.3980
3.024
8.39
10.76
RbCl
1.4930
2.384
8.25
8.35
8.26
RbBr
1.5530
2.264
8.17
7.90
7.75
Rbl
1.6474
2.208
8.05
7.69
7.12
CsF
1.4830
2.911
8.27
10.34
8.80
CsCl
1.6100
2.295
8.10
8.02
8.32
CsBr
1.6700
•2.179
8.02
7.58
7.72
Csl
1.7876
2.125
7.86
7.38
7.25
Average percentage deviation
6.56
5.60
174
10.00
. . .
rife
11
rife
rife
rife----------rife
*
rife
<0
rife----------
CO
3F
Calculated
o
*7
i ll______ HI-1
7
I____ Li
______ __ u__________________ ___________________
8
9
10
11
Known values
Fig 4.1 Plot showings the relationship between the known values of ionization potential of alkali
halides and those calculated from n [Eqn. (4.5)] and xa xb [Eqn. (4.6)] (data taken from
Table 4.1). Squares indicate Eqn. (4.5) and stars indicate Eqn. (4.6).
175
To evaluate the bond energies of alkali halides, the values of refractive index103 and
Martynov and Batsanov electronegativity6 are used. Table 4.2 lists the calculated value of single
bond energies using equations (4.21) and (4.22). The present values are compared with those of
Pauling61. A close look of the table reveals that the values of single bond energy calculated from
the above relations are in the excellent agreement with the experimental values61,69. It has been
pointed out by the Matcha62 that the Pauling’s equation61 donot work well in the case of
compounds with ionic bonds. The estimated bond energies can be in error by as much as 190
percent and the average percentage deviation observed with the Pauling’s equation is found to be
79.5. Equation (4.21) is based on the relation between the single bond energy and refractive Index,
while the other equation (4.22) is In between single bond energy and Martynov and Batsanov
electronegativities, respectively. The average percentage deviation at the end of the table reveals
that the proposed equations are quite suitable In estimating the single bond energy. Figure 4.2
illustrates the close relationship between the literature values and the calculated values from
equations (4.21) and (4.22). Matcha62 and Ramanl and Ghodgaonkar66 have given different
methods earlier to evaluate bond energies in diatomic molecules. The proposed relations by them
have their own drawbacks. Matcha62 has not given tne electronegativity values appropriate to his
relation sofar, where as Ramani and Ghodgaonkar66 relation mainly depends on the group
constant. Estimation of the group constant from the above equation is often involved with tedious
mathematical calculations. In view of the above difficulties, the author has given simple empirical
relations
to evaluate bond energies in alkali halides. Correlations between these parameters are
entirely new and simple, and give better understanding of the nature of the bonding. The relative
superiority of the simple relations can be gauged from the fact that they yeild theoretical values
which agree well with the experimental values.
The valence electron plasmon energy (hwp) for alkali halides have been calculated
using equation (4.24) and the results are presented in table 4.3 along with available literature
values
. A perusal of the table reveals that the values obtained from equation (4.24) are in good
176
Table 4.2: Bond energies, K
(K cal/n.'l; ut
ilk ill halide crystals.
s
I- nd energy, Eg (K cal/raol)
Alkali Refractive M-B electronegativity,
halides index, n
[Ref.103] %
[Ref.6]
Present
study
Eqn.<4.2l)
Present
study
Eqn.(4.22)
Pauling
Known
[Ref.61 [Ref.61
& 69
LiP
1.3915
3.402
114.3
138.8
136.2
300.7
LiCl
1.6620
2.682
93.8
99.1
111.6
158.5
LiBr
1.7840
2.547
84.0
91.6
99.8
131.1
Lil
1.9520
2.484
71.8
88.1
84.6
97.7
NaF
1.3360
3.364
118.5
136.7
122.9
313.6
NaCl
1.5400
2.652
10’ .0
97.4
97.5
163.9
NaBr
1.6412
2.519
95.4
90.0
86.2
136.3
Nal
1.7745
2.456
85.2
86.0
77.7
101.6
KF
1.3610
3.024
116.6
117.9
116.9
328.1
KC1
1.4900
2.384
106.8
82.6
100.1
171.3
KBr
1.5590
2.264
101.6
76.0
90.1
143.1
KI
1.6770
2.208
92.6
72.9
77.3
107.2
RbF
1.3980
3.024
113.8
117.9
115.3
327.2
RbCl
1.4930
2.384
106.6
82.6
100.1
170.2
RbBr
1.5530
2.264
102.0
76.0
89.9
142.2
Rbi
1.6474
.2.208
94.9
72.9
76.1
106.2
CsF
1.4830
2.911
107.4
111.7
118.2
346.3
CsCl
1.6100
2.295
97.7
77.7
105.6
183.3
CsBr
1.6700
2.179
93.2
71.3
96.2
154.0
Csl
1.7876
2.125
84.3
68.3
82.2
116.4
'
Average percentage deviation
9.98
177
10.3
79.5
Calculated
Known values
Fig 4.2 Plot showings the relationship between the known values of bond energy of alkali halides
and those calculated from n [Eqn. (4.2 i)],
/B(Eqn. (4.22)] and Pauling’s relation (Ref.
61] (data taken from Table 4.2). The line is of unit slope. Squares indicate Eqn. (4.21),
stars indicate Eqn. (4.22) and diamonds indicate Pauling's relation (Ref. 61],
Table 4.3: Valence electron plasron r ..
alkali halide crystals.
ilv
(eV) of
Valence elect.!.-:. ,. ;a..n..on energy, -Mw
Alkali
halides
M-B electronegativity,
[Ref.6]
Present
study
Eqn. (4.24}
LiF
3.402
21.13
25.96
LiCl
2.682
15.08
17.99
LiBr
2.547
13.94
10.87
Li I
2.484
13.41
13.16
NaF
3.364
20.82
20.11
NaCi
2.652
14.83
15.68
NaBr
2.519
13.71
14.37
Nal
2.456
13.16
12.74
KF
3.024
17.96
16.83
KC1
2.384
12.56
13.29
KBr
2.264
11.57
12.38
KI
2.208
11.10
11.19
RbF
3.024
17.96
15.03
RbCl
2.384
12.58
12.40
RbBr
2.264
11.58
11.59
Rbl
2.208
11.10
10.53
CsF
2.911
17.01
13.41
CsCl
2.295
11.83
12.54
CsBr
2.179
10.8.6
11.77
Csl
2.125
10.4 J
10.73
Vb
Average percentage deviation
=
8.5
179
Known
[Ref.95]
(eV)
30
28
-
26
-
10
12
14
16
18
20
22
24
26
28
30
Known values
Fig 4.3 Plot showings the relationship between the known values of valence electron plasmon
energy of alkali halides and those calculated from xa xb [Eqn. (4.24)] (data taken from
Table 4.3). The line is of unit slope. Squares indicate Eqn. (4.24).
1BD
agreement with the literature values95. Tisir is wei! illustrated in figure 4.3. The valance electron
piasmon energy is of particular importance in the estimation of dielectric constant, fermi energy
and Penn gap96'101. These parameters are highly helpful in explaining the band structure details of
semiconductors. For a model semiconductor, the high frequency dielectric constant is explicitly
dependent on the valence electron piasmon energy, it is evident from the analysis of the present
study that the proposed empirical relation, despite Its simplicity, is capable of providing reasonably
reliable, valence electron piasmon energy irt alkali halides. The concept of the electronegativity
proposed by Martynov and Batsanov6 involves the ionization potential. The parameters hwp and
Xk
xb
describing the behaviour of an electron. From the above one can conclude that there is
definite relationship between these two parameters. The valence electron piasmon energy is of
having special importance in solid state physic
Table 4.4 represents the vtluss
‘'J1.
/ energy gap for alkali halides computed from
equations (4.30) and (4.31). It is obsderved from the table that the present values are in resonable
agreement with the literature values86,115. Figure 4.4 illustrates the close relationship between the
measured values86,115 and the calculated values from equations (4.30) and (4.31). Equations (4.30)
and (4.31) used in the present study are valid foi alkali halides only. The proposed correlations83
are empirical in nature and the numerical constants involved in the equations are unique in the
sense that they represent the best fit with the experimental data. Equations (4.30) and (4.31) are
formulated on the basis of correlations earlier suggested by Moss93,99,102 and Bube74. Moss102
has first given correlation between the energy band gap and refractive index for photoconductors.
Bube74 has given the relation between the atom’s position in the periodic table and its
eietronegativity; and between the eletronegativity difference of the two atoms forming a compound
and tha type of chemical bonding to be expected ..
oirelation between the refractive Index and
the Martynov and Batsanov6 electronegativity vC,. XB) has been proposed [Eqn.(4.32)] in the
present study; and applied tne same for the evaluation of refractive indices for alkali halides and
the results are presented in table 4.5. An inspection of the figure 4.5 reveals that the refractive
Table 4.4: Energy gap.
halides
Eg (eV) of alkali bolide crystals.
Alkali
Refractive M-B
negativity,
index, n
'X^xB
[Ref.103]
[Ref .6]
Energy yap,
electroPresent
study
Eqn. (4.30)
Eg SeV)
—_
Present
study
Eqn. (4.31)
Known
[Ref.86
& 115]
I,i F
1.3915
3.402
10.11
12.02
12.0
LiCl
1.6620
2.682
7.70
8.58
10.0
LiBr
1.7840
2,547
6.60
7.95
8.5
Lil
1.9520
2.484
5.10
7.63
5.9
NaF
1.3360
3.364
10.60
11.84
10.5
NaCl
1.5400
2.652
8.70
8.44
8.6
NaBr
1.6412
2.519
7.0 0
7.80
7.7
Nal
1 . 774 5
2.456
'' .70
7.50
5.8
KF
1.3610
3.024
10.4 0
10.21
10.9
FC3
1.4900
2.384
9.70
7.16
8.5
KBr
1.5590
2.264
8.60
6.58
7.8
KX
1.6770
2.208
7.50
6.31
6.2
RbF
1.3980
3.024
10.10
10.21
10.4
RbCl
1.4930
2.384
9.20
7.16
8.2
RbBr
1.5530
2.264
8.60
6.58
7.7
Rbl
1.6474
2.208
7.80
6.31
6.1
CsF
1.4830
2.911
9.30
9.67
10.0
CsCl
1.6100
2.295
8.10
6.73
8.0
CsBr
1.6700
2.179
7.60
6.18
7.5
Csl
1.7876
2.125
6.50
5.92
6.3
Average percentage deviation
10.25
10.50
182
13
CM
-«
ji,
<D
00
.
si/
5
6
s i/
-ar
CD
=*= ijjfc
sls
-*
.... .
=si/%
Calculated
O
If
JL
g _______ mm
i
7
11
i
8
n
>_________ ___ JJ___ LL.....................................
9
10
11
12
13
Known values
Fig 4 4 Plot showings the relationship between the known values of energy gap of alkali halides
and those calculated from n [Eqn. (4.30;] and jA /B [Eqn. (4.31)] (data taken from Table
4.4). Squares indicate Eqn. (4.30) and diamonds indicate Eqn. (4.31).
Table 4 .b: Refractive index, i) of a. ' . ' ; 1
lid:: crystals.
Rotra<:*!; i vc index. n
Alkali
halides
M-B electronegativity,
[Ref.6]
Present
study
Eqn. (4.32)
Known
(Ref.103]
Kumar et al.
[Ref.103]
LiF
3.402
1.365
1 . 391
1.442
LiCl
2.682
1.559
1.662
1.510
LiBr
2.547
1.596
1.78 4
1.572
Lil
2.484
1.613
1 . 952
1.723
NaF
3.364
1.374
1.336
1.491
NaCl
2.652
1.567
1.540
1.598
NaBr
2.519
1.603
1.641
1.612
Nal
2.456
1.620
1.774
1.730
KF
3.024
1.466
1.361
1.477
KC1
2.384
1.640
1.490
1.572
KBr
2.264
1.672
1.559
1.606
KI
2.208
1.688
1.677
1.701
RbF
3.024
1.466
1.398
1.495
RbCl
2.384
1.640
1.493
1.586
RbBr
2.264
1.672
1.553
1.612
•
Rbl
2.208
1.688
1.649
1.708
CsF
2.911
1.497
1.483
1.510
CsCl
2.295
1.664
1.610
1.596
CsBr
2.179
1.695
1 .670
1.650
Csl
2.125
1.710
1.787
1.695
Average percentage
deviation
5.59
5.10
8 i
2
-
1.8
-
alculated
1.9
Known values
Fig 4.5 Plot showings the relationship between the known values of refractive index of alkali
halides and those calculated from yJs ic lEqn. (4.32)] and Kumar et al., [Ref. 103] (data
taken from Table 4.5). Squares indicate
(4.32) and stars indicate Kumar et al.,
[Ref. 103],
indices obtained83 from the Martynov and Batsanc
P c'cc.tronegativities in the present study are in
satisfactory agreement with the experimental values'03. However, some deviation is noticed in the
case-of alkali fluorides. The refractive index of ‘.he fluorides is invariably lower than that of the other
halides, and it is note worthy that this feature appears despite the high density of the fluorides. The
dispersive power of the fluorides as indicated by the difference of the refractive indices in the
visible and ultraviolet is also small. In contrast, it is large for the other halides and particularly so in
the case of the bromides and iodides. The fluorides, in particular, have distinctive properties. Their
melting points are notably higher than those of the coresponding chlorides, bromides and iodides,
thereby indicating a stronger binding between the metal and halogen atoms. The peculiar
behaviour of alkali fluorides may be attributed to the extremely high reactivity and high
electronegativity of fluorine atom. Politzer80 has been mentioned that fluorine exhibits a number of
anomalous properties both as an atom and as a molecule. The good agreement, found between
the calculated and the literature values
in the present study
may be taken as an indication that
the proposed relation between the regractive index and (*a %b) is valid for alkali halides. The
estimated average standared deviation for alkalis halides is also given at the end of the table the
minimum average percentage deveiation indicates the soundness of the present study.
In the investigation of the structures of molecules and crystals there have been
developed empirical or semi-empirical systems of standard bond lengths or of atomic or ionic radii
which together with other structural information provide a useful basis for predictions of
approximate structures of molecules and crystals. Moreover, if the significance of such system is
know, say in terms of a theory of electronic structure, then it may be used as a basis for discussing
differences between observed bond lengths and those predicted by the system in terms of special
features of the electronic structures. Pauling and Huggins120 suggested that molecular bond
lengths could be adequately estimated by simple addition of the covalent radii. Schomaker and
Stevenson121 have given the continuous relationship between bond length and electronegativity
for homo and hetronuclear bonds. In viev~ of the above importance the author has proposed a
186
simple relation for the estimation of interionic distance in alkali halides. For the sake of comparison
the values of interionic distance (ro) given by Slater116 c-c also reported in table 4.6. At the glance
of the table 4.6 it is known that the values of r0 calculated from equations (4.35) and (4.36) are in
excellent agreement with the literature values [Fig. 4.6]. The most important point which deserves
to be emphasized here is that the linear relationship between the interionic distance and the
electronegativities of the constituent atoms hold good in the case of alkali halides. These interionic
distances have been employed in the estimation of the crystal stability and compressibility
condition.
The calculation of atomization energy using a repulsive term in the interaction potential
is always a difficult job and this procedure requires the data on electron affinity, ionization
potentials and cohesive energy. A brief analysis of the data on atomization energies presented in
table 4.7, reveals that the agreement between the values of atomization energy estimated from
equations (4.40) and (4.41) and those obtained from experiment64 Is excellent. The values reported
by Kittel117 are also given in the same table for comparison. Figure 4.7 shows the graphical
representation of the results listed In table 4.7. In the present study, the atomization energy of alkali
halides are evaluated from equations (4.40) and (4.41) using the values of refractive index103 and
Martynov and Batsanov electronegativities6 of the constituent atoms in alkali halides. Equations
(4.40) and (4.41) are entirely different in nature. The proposed83 relations not only yield most
satisfactory results but also their comparisons with literature data64 provide a direct and precise
check of the appropriateness and suitability of the relations.
Pauling4,22 has established good
correlations between bond energies and
electronegativities. Bond energy may also be estimated from the thermochemical procedure
involving heat of formation. In the determinatiop of bond energies, heat of sublimation and heat of
formation play an important role. In view of the above the author has formulatd empirical relations
between the heat of formation, the Martynov and Batsanov electronegativities6 and refractive index
and results are presented in table 4.8. Recently, Martynov and Batsanov6 have given an empirical
187
'table 4.6: Tnterionie distance, r
cl alkali halide crystals.
Interionic distance, rQ (A°)
Alkali
halides
Refractive
index, n
[Ref.103]
M-B electronegativity,
■*A*B
[Ref.6]
Present
study
Eqn.(4.35)
Present
study
Eqn.(4.36)
Known
[Ref.116
LiF
1.3915
3.402
2.79
2.12
2.01
LiCl
1.6620
2.682
3.19
2.95
2.57
LiBr
1.7840
2.547
3.37
3.11
2.75
Lil
1.9520
2.484
3.61
3.18
3.00
NaF
1.3360
3.364
2.71
2.17
2.31
NaCl
1.5400
2.652
3.01
2.99
2.81
NaBr
1.6412
2.519
3.15
3.14
2.98
Nal
1.7745
2.456
3.35
3.21
3.23
KF
1.3610
3.024
2.75
2.56
2.67
KC1
1.4900
2.384
2.94
3.29
3.14
KBr
1.5590
2.264
3.04
3.43
3.29
KI
1.6770
2.208
3.21
3.50
3.53
RbF
1.3980
3.024
2.80
2.56
2.82
RbCl
1.4930
2.384
2.94
3.29
3.27
RbBr
1.5530
2.264
3.03
3.43
3.43
Rbl
1.6474
2.208
3.16
3.50
3.66
CgF
1.4830
2,911
2.93
2.69
3.00
CsCl
1.6100
2.295
3.11
3.40
3.56
CsBr
1.6700
2.179
3.20
3.53
3.71
Csl
1.7876
2.125
3.37
3.59
3.95
12.25
5.70
Average percentage deviation
=
188
4
3.8
3.6
*
*
*
3.4
*
&
Calculated
*
3.2
n
2.8
A
2.6
2.4
2.2
-A
• e
2
2,2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
Known values
Fig 4,6 Plot showings the relationship between the known values of interionic distance of alkali
halides and those calculated from n [Eqn. (4.35)] and XAXsIEqn. (4.36)] (data taken from
Table 4.6). Squares indicate Eqn. (4.35) and stars indicate Eqn. (4.36).
Table 4.7: Heat of atomization,
AH°t (K c^. 1/mol) of
alkali halide crystals.
Heat of atomization. AH°t <K cal/mol)
Alkali Refractive M-B electrohalides index, n
negativity,
Present
Present
[Ref.103] -^B
Kittel
study
study
Known
[Ref.6]
Eqn.(4.40) Eqn. (4.41) [Ref.64] [Ref.117]
LiP
1.3915
3.402
170.7
191.8
203.8
196.0
LiCl
1.6620
2.682
142.5
155.7
165.2
152.9
LiBr
1.7840
2.547
129.8
148.9
149.0
127.2
Lil
1.9520
2.484
112.3
145.7
129.5
115:7
NaF
1.3360
3.364
176.4
189.9
181.8
174.8
NaCl
1.5400
2.652
155.2
154.2
153.2
144.4
NaBr
1.6412
2.519
144.6
147.5
138.6
120.9
Nal
1.7745
2.456
130.8
144.3
120.2
112.0
KF
1.3610
3.024
173.8
172.8
175.8
167.1
KC1
1.4900
2.384
160.4
140.7
154.6
145.9
KBr
1.5590
2.264
153.2
134.7
138.6
125.0
KI
1.6770
2.208
140.9
139.1
125.1
118.4
RbF
1.3980
3.024
169.9
172.8
169.8
162.4
RbCl
1.4930
2.384
160.1
140.7
151.6
143.4
RbBr
1.5530
2.264
153.8
134.7
139.3
122.3
Rbl
1.6474
2.208
144.0
131.9
123.6
117.3
CsF
1.4830
2.911
161.1
167.2
164.8
CsCl
1.6100
2.295
147.9
136.2
151.6
CsBr
1.6700
2.179
141.6
130.4
139.7
Csl
1.7876
129.4
127.7
124.7
7.2
6.0
2.125
Average percentage deviation
«
190
. . .
. . .
7.4
110
120
130
140
150
160
170
180
190
200
210
Known values
Fig 4.7 Plot showings the relationship between the known values of heat of atomization of alkali
halides and those calculated from n [Eqn. (4.40)], xa xb [Epn- (4.41)] and Kittel [Ref. 117]
(data taken from Table 4.7). The line is of unit slope. Squares indicate Eqn. (4.40),
stars indicate Eqn. (4.41) and diamonds indicate Kittel [Ref. 117].
I9i
Table 4.8: Heat of formation,
ZiH^ (K cal/mol) and Melting point temperature, T'm (K)
of alkali halide crystals.
Heat of formation,
(K cal/mol)
Alkali Refractive M-B electrohalides index, n
negativity,
Present
[Ref.103] ^B
study
[Ref.6]
Eqn. (4.46)
Present
study
Eqn. (4. ■17)
Known
[Ref.86]
Melting point
temperature, Tm (K)
Present
study
Eqn.(4.48)
Known
[Ref.86]
LiF
1.3915
3.402
-126.3
-141.6
-145.7
838
845
LiCl
1.6620
2.682
- 90.8
-105.8
- 97.7
659
605
LiBr
1.7840
2.547
- 74.8
- 99.1
- 83.7
579
550
Li I
1.9520
2.484
- 52.7
- 95.9
- 64.8
468
449
NaF
1.3360
3.364
-133.6
-139.7
-136.3
875
.993
NaCl
1.5400
2.652
-106.8
-104.3
- 98.2
740
801
NaBr
1.6412
2.519
- 93.5
97.7
- 86.0
673
747
NaT
1.7746
2.456
- 76.0
- 94.5
- 68.8
585
661
KF
1.3610
3.024
-130.3
-122.8
-134.5
858
858
KC1
1.4900
2.384
-113.3
- 91.0
-104.2
773
770
KBr
1.5590
2.264
-104.3
- 85.2
- 93.7
727
734
KI
1.6770
2.208
- 88.8
- 82.2
- 78.3
650
681
RbF
1.3980
3.024
-125.5
-122.8
-131.3
833
795
RbCl
1.4930
2.384
-112.9
-91.0
-102.9
771
718
RbBr
1.5530
2.264
-105.1
- 85.0
- 93.5
731
693
Rbl
1,6474
2.208
- 92.7
- 82.2
- 79.0
669
647
CsF
1.4830
2.911
-114.3
-117.2
-126.9
777
682
CsCl
1.6100
2.295
- 97.6
-
86.6
-103.5
694
645
CsBr
1.6700
2.179
- 89.7
- 80.8
- 94.3
654
636
Csl
1.7876
2.125
- 74.3
-78.1
- 80.5
576
626
9.4
12.2
Average percentage deviation
192
5.96
O
C*
o
<o
o
a<
o
a>
o
Calculated
o
rKnown values
Fig 4.8 Plot showings the relationship between the known values of heat of formation of alkali
halides and those calculated from n [Eqn. {4.46)] and xa XB[Eqn. (4.47)] (data taken from
Table 4.8). The line is of unit slope. Squares indicate Eqn. (4.46) and stars indicate Eqn.
(4.47).
193
1000
O
P
Calculated
O
2
500
400
400
500
600
700
800
900
1000
Known values
Fig 4.9 Plot showings the relationship between the known values of melting point temperature of
alkali halides and those calculated from n [Eqn. (4.48)] (data taken from Table 4.8). The
line is of unit slope. Squares indicate Eqn. (4.48).
194
equation which describles the energy of atomization and eletronegativities the magic formula
suggested by them has applied to several crystalline substances and estimated the energy of
atomization. Hence it is most appropriate to establish a relation between the heat of formation and
electronegativities. Reddy et al.119 have proposed some more relations between refractive index,
the average nuclear effective charge and the standard heat of atomization. Garbato et al.7,8 have
evaluated cohesive energies and heat of formation based on average nuclear effictive charge in
tetrahedral semiconductors. The empirical studies mentioned above proved their significance and
importance in evaluating several properties of semiconductors. Following the above development
in this field, the author has been proposed a relation between heat of formation and refractive
index [Eqn.(4.46)j. On the similar grounds the author has been proposed a relationship between
heat of formation and Martynov and Batsanov electronegativities [Eqn.(4.47)J. The melting point
temperature is a measure of crystal cohesion which is largely due to electron transfer and so It may
be taken as a direct measure of crystal binding energy which inturn depends upon the nature of
bonding. A relation connecting the refractive index with the melting point is proposed and the
results are presented in table 4.8. The estimated values of Hf and Tm for different alkali halides are
in fair agreement with the experimental values84. A glance at figures 4.8 and 4.9 reveals that there
is a close agreement between the experimental and calculated values. The advantage of the above
equations is that one can evaluate AH? and Tm values with the Martynov and Batsanov
electronegativities6 and refractive index103.
The calculated values of physico-chemical parameters of alkali halides are in fair
agreement with the availbale experimental data. Although the approach adopted in the present
work is over simplified, and its present form makes no claim to be more than speculative, the
derived empirical relations, which interrelate different properties by introducing few numerical
constants, will stimulate basic research in the field of physical characterization of alkali halides.
195
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