45
3-
APPROXIMATIONS USED IN THE THEORY OF
LATTICE DYNAMICS
3.1
INTRODUCTION:The theoretical and experimental study of lattice dynamical
problem has been of interest to physicists for understanding the
thermodynamic, elastic, optical, electrical and numerous other physical
properties of solids.1,2 The dynamical theory of metallic crystal is still
one of the most interesting area in the theory of lattice-dynamics,
despite the effort which has been devoted to its study. The metallic
crystal is supposed to consist an array of positive bare ions forming a
periodic lattice which is embedded in a uniform sea of negative charge
of conduction electrons. It is basically a many body problem involving
the interaction of ions in the field of many moving electrons. The exact
solution of equation of motion generated for this system becomes
formidable and therefore, it is necessary to make some approximations
in order to solve the problem.
3.2 THE ADIABATIC APPROXIMATION:When a thermal wave disturbs the lattice, the atoms execute
small oscillation about their mean position. It is
assumed that the
core-electrons belonging to the ions move rigidly with the nuclei and
can not be excited at the energies available. The conduction electrons
respond easily to screen out the local charge fluctuations generated by
the vibrations of positive ions. A rigorous approach to predict the
possible frequencies for the normal modes of vibration of lattice is to
46
express the basic Hamiltonian of the lattice and then to determine its
eigen values.
The basic Hamiltonian operator3,4 H, for the whole system
consisting of ions and electrons is written as
H
= Hii + Hee + Hie
…..
(3.1)
Where
∑
∑
∑
(
∑
)
|
|
and
∑
(
)
Where m is the mass of the electron and M is the mass of the ion,
ri represents the position of ith electron and
position of the
th
denotes the displaced
ion. The indices and ’ run over all the ions where
as the indices i and j extend over all the valence electrons. The first
term Hii includes the (i) the kinetic energy operator of ions and the (ii)
potential energy UI (Rl - Rl’ ) of direct interaction between ions which
includes the coulomb repulsion between the ions and the core
exchange interaction. The second term Hee corresponds to the valence
electrons which interact through coulomb potential. This also includes
exchange interaction between conduction and core electrons. The
47
ionic cores and valence electrons interaction is represented by H ie. This
interaction takes place through a potential VI ( ri - Rl).
The singularities originating on account of the coulomb
interaction present in equation (3.1) are avoided by taking the spatial
average of the potential to be zero and by excluding the interaction of
the particle with itself5. The problem of determining the eigen-values
of
Hamiltonian
(3.1)
is
a
complicated
task.
The
adiabatic
approximation6 introduce tremendous simplifications in the problem
by separating the dynamic aspect of electron and ionic motion.
The total Hamiltonian (3.1) may be expressed as
H= HI + He
…..
(3.2)
…..
(3.3)
Where
∑
∑
(
)
and
∑
∑
(
)
∑
|
|
…..
(3.4)
…..
(3.5)
Let the wave function for Hamiltonian H be
(
)
(
)
48
The crystal wave function ψq satisfies the Schrödinger equation for
the electrons in a static lattice as given below.
(
)
(
)
(
)
(
)
…..
(3.6)
Now applying the operator H to the crystal wave function ψq and
using (3.6) we get.
(
)[ ∑
∑
(
)
(
(
[
)]
)]
(
)
…..
(3.7)
…..
(3.8)
Where
G(R)
∑
(
)
If the second line of (3.7) is ignored, we can solve our complete eigenvalue problem, H ψq = Eq
( ) to satisfy a
ψq , by making
schrodinger type equation
[ ∑
(
)
(
)]
(
)=
(
)
…..
(3.9)
49
This is an equation for a wave function of the ions alone, which
shows that the motion of ions is governed by the effective potential,
G( r ) + Ee( R ) , i.e. electrons contribute adiabatically to the lattice
energy.
Justification for dropping the terms is (3-7) can be shown in the
following way. The second live of (3.7) contains two terms which
contribute almost nothing to the expection value of the energy of the
system in the state ψq. The first term vanishes because it produces
integrals like.
∫
∫
…..
(3.10)
Where ne is the total number of electrons. The second term is
small because, at worst, the electrons would be tightly bound to their
ions.
Ψ( r, R ) = ψ ( ri - Rl )
Which would give a contribution like
∫
∫
…..
(3.11)
50
This is just m/M times the kinetic energy of the electrons. Since
m/M is of the order of 10-4 or 10-5, which is negligible small in
comparison to the ordinary thermal energies.
In this way, adiabatic principle allows us to separate the ionic
motion from the electronic motion, leaving only a residual interaction
between the electrons and the phonons. So we can treat the electrons
and lattice waves as nearly independent entities and allot the electrons
the same coordinate as that of ions.
3.3 LATTICE VIBRATION IN HARMONIC APPROXIMATION
AND SECULAR EQUATION :In addition to the adiabatic approximation already discussed the
theory of lattice vibration is based on the so called harmonic
approximation viz; when the displacement of the nuclei are small this
assumption postulates that the potential energy of the crystal is
dependent only on second order terms in the displacement of atoms
from their equilibrium position.
As a result of thermal fluctuations, each ion in the lattice is
displaced from its equilibrium position. The total potential energy of
the crystal is assumed to be some function of the instantaneous
positions of all the atoms. However the potential energy of a crystal is
expanded in power of the amplitude of the atomic vibrations and all
the terms higher than those which are quadratic in amplitude are
51
neglected,
the
remaining
expansion
is
called
the
harmonic
approximation.
Let us confine ourselves to the crystal lattice with a basis and
having no impurities, vacancies, interstitials or dislocations. The
equilibrium position of the kth ion in the lth cell is given by
R0 (l,K) = R0 (l) + R0 (K)
…..
(3.12)
The constituent ions in a crystalline solid execute small
oscillations about their equilibrium positions as a result of thermal
fluctuation at finite temperature. Let the displacement of the (l,K)th ion
from the mean position be u (l,K), the displaced position of ion is given
by
R (l,k)= R0 (l, k) + u (l, k)
Distance between the kth ion in the
…..
lth
(3.13)
cell and k’th ion in the l’th
cell at any instant is given by
R (l,K) -
R (l’, k’)= R *
+
…..
(3.14)
As the two ions vibrate about their equilibrium positions, the
potential energy V of the whole lattice will be a function of R *
The total kinetic energy of the lattice is expressed by
+.
52
T =
∑
̇ (
…..
)
(3.15)
Where Mk is the mass of the kth atom and ux (l,k), uy(l,k) and uz (l,k)
are the components of u (l,k) along x, y, z axes respectively. This
approximation gives the expression for potential energy as follows.
V( R ) = V0+ V1+ V2
…..
(3.16)
Where V0 is the equilibrium potential energy of the crystal and
V1= ∑
Vα ( l, k ) Uα ( l, k)
=∑
(
uα ( l, k )
)
…..
(3.17)
and
V2=
∑
∑
(
)
(
),
(
)
….. (3.18 a)
where
(
)=
((
)
(
)
..… (3.18 b)
The subscript zero means that the derivative are
to be
evaluated for the equilibrium configuration. The coefficient Vα(l,k)
53
represents the negative of the force acting on (l,k)th ion in the α
(=x,y,z) direction in the equilibrium configuration. The coefficient Vαβ(l
k,l’k’) is the force constant acting on the ion at R (l, k) in the
direction when the ion at R (l’,k’) is displaced a unit distance along β
direction. As in equilibrium the force on any particle must be zero as
Vα( l, k )=0
The equation of motion for (l,k) atom, using above argument is,
Mk ̈ α (l ,k) =
=
∑
( )
(
)
(
{
)
(
)}
) …..
(
(3.19)
This forms an infinite set of simultaneous linear differential
equitation. Their solution is simplified by the periodicity of the lattice.
Let us consider a running wave solution of (3.19) in the form
uα (l,k) =
√
( )
* ( )
(
)+
…..
(3.20)
Where uα(k) is independent of l and q is wave – vector of the
disturbance. Substitution of (3.20) in (3. 19) gives (for monatomic
crystal)
∑
[
( )
(
)
( (
)
(
)]
…..
(3.21)
54
Where δαβ and δkk’ are Kroneeker delta’s and dynamical matrix
elements Dαβ (q, kk’) are,
(
)
(
∑
)
* (
)
(
…..
)+
(3.22)
Here Vαβ (lk, l’k’) do not depend on l or l’ due to the translational
symmetry, but only on their difference (l - l’), Because of general
properties of invariance under arbitrary translation, the
force
constants must satisfy the relation7.
(
∑
)
…..
(3.23)
Equation (3.22) includes the terms for which l - I’=0 and k=k’. such
arise due to the interactions of atoms with themselves and obtained
their meaning from (3.23) and are, as a matter of fact the reaction of an
atom to the force exerted upon it by the crystal as a whole. Because of
mathematical difficulties, it would be desirable to avoid such terms,
and this can be obtained by subtracting (3.23) from (3.22). we also
set l’=-0 for the sake by simplicity , to obtain
(
)
∑
(
)*
()
+
55
(
∑
)
…..
(3.24)
While for k≠ k’, we retain the form of (3.22) with l=o. The
equation (3.24) is more convenient through less simple than (3.22),
since it avoids the term Vαβ (Ok, ok).
Therefore the secular determinant for a monatomic hexagonal
crystal can be written as
det|
(
)
|
( )
…..
(3.25)
Dαβ (q, kk’) is a element of the dynamical matrix of order 6×6. The
solution of the secular determinant along the symmetry direction of a
crystal becomes easy as the secular determinantial equation in these
directions are factorisable in three 2×2 equations.
If Q is a general vector in reciprocal space, we can take a Fourier
expansion of the interaction potential U(R)
between two ions
separated by distance R.
U(R)= ∑
( )
(
)
…..
(3.26)
The lattice potential energy V to be the sum of two body
potential
U { R(l,k)-R(l’,k’)} given as
Vαβ(lk, l’k’) = -
( )
R = R (l, k) – R (l’, K’)
…..
(3.27)
56
Substituting Vαβ (lk, l’k’) with l’=0, in (3.24) with the help of (3.27) and
using the standard relation.
(
N-1∑
(
))
∑
We have
(
(
)
) (
)
∑ (
(
) (
)
(
)
)
…..
(3.28)
Here N is the total number of unit cells in the crystal and summations
extend overall the reciprocal lattice vector
.
In general a three dimension crystal with n atoms per unit cell
has 3n frequencies for each wave vector. The” acoustic modes” are 3 in
number and for them ω tends to zero as q goes to zero. The remaining
[(n-1).3] frequencies are called as “optical modes” and they are
different from zero when q = 0.
For
a
quantum
mechanical
description,
the
harmonic
Hamiltonian for the crystal lattice can be quantized in the fashion of
second quantization in field theory8.
This leads to a description of lattice vibration in terms of
phonons, which can be thought of as travelling quanta of vibrational
energy.
57
3.4
THE SELF-CONSISTENT FIELD APPROXIMATION:In the present theory, the most decisive approximation is the
self consistent field approximation. The electrons in a metal form, a
kind of gas inside the metal, which is different from a perfect gas on
account of strong interaction of electrons among themselves and also
with positive ions. Because of this large interaction, a gas of this kind
must be too complicated for simple mathematical treatment. However,
in quantum-mechanics, the effect of one electron on all other to a
large extent can be averaged; one can treat each electron as moving in
the field of the other electrons. The average potential depends on the
distribution of electrons and upon the states which are occupied by
them. These states in turn depend upon the potential, thus we must
compute the potential self- consistently. Ultimately the only important
interaction between electrons is the coulomb repulsion, but this is
conveniently divided into three distinct contributions. First is the
Hartee potential9, obtained by computing
the time average of the
electrons distribution and then using Poisson’s equation to determine
the corresponding potential. Second is the correction for the potential
seen by an electron due to the Pauli principle. If an electron with given
spin is at a position
r, no other electron of the same spin can lie at
that point, simply because of the antisymmetry of the wave functions.
This effectively gives a hole in the electron distribution and gives rise
to the exchange interaction. The third contribution arises from the
correlated motion of the electron, which is known as correlation
energy. These two corrections fall under the well known Hartree-Fock
approximation.10 In the Hartree-Fock approximation the effect of
correlation has not been properly considered.
58
However, on account of the use of determinantal wave function
the correlation of the parallel spins only is taken into account while the
antiparallel spin
is still lacking. Such type of correlation has been
considered at the first time by Seitz11. Bohm and Pines12 “based on
collective vibrational approach”, have shown that the long rang part of
coulomb interaction leads to a coherent motion of the electrons which
can be described in terms of plasma oscillations. Toya 13 has
incorporated the correlation energy of both the spins in an
approximate way using the reduction of density of states near the
Fermi-surface to meet the defects of the H-F approximations arising
from the neglect of proper correlation on the screening.
3.5
PERTURBATION THEORY:The displacement of the ions in the crystal caused by lattice
vibrations provide perturbation
in the electronic Hamiltonian (3.2).
This theory can be used to calculate the conduction band state. We can
then write the perturbed Hamiltonian, total electronic wave function,
and the total energy occurring in equation (3.9) as.
….. (3.29 a)
(
)
….. (3.29 b)
( )
….. (3.29 c)
Where the superscript refers to the order to which u occurs in
the expansion terms. The contribution to the lattice vibrational
potential energy made by ion-electron-ion interaction will be
.
59
The total electronic wave function ψ(r, R)
in the Hartree
approximation is written as a product of one-electron wave function
(r, R) . The
satisfies the self-consistent Schrödinger equation.
H1 ψk ( r, R ) = E ( K ) ψk (r, R )
…..
(3.30)
…..
(3.31)
Where
H1= (
(
)
)
(
)
The bare ion potential VI is the energy of interaction with the
ions, and the self consistent Hartree potential Ve( r ) is due to
interaction with all the other electrons. They can be written as
VI (r, R ) =∑
(
(
))
….. (3.32 a)
and
Ve(r, R )=
∑ ∫
|
(
|
)|
|
r
….. (3.32 b)
Here K includes the spin quantum number of the electron when it is
needed. The displacement of the nuclei are so small that the core
electrons are not distorted and core electrons plus nucleus can still be
considered as a single entity producing a one – particle potential UI for
conduction electrons.
The total energy Ee is given in terms of one electron Hartree
energies E( k ) by the following expression,
60
( )* ( )
Ee=∑
| |
+
…..
(3.33)
Where nF (k) is the probability that the state k is occupied at T=
O0 K, we know that nF(k)( has a step function behaviour with nF(k)=1
for k < kf and nF( k)=0 for k > kf , here kf is Fermi wave vector for
electrons. If we make the perturbation expansion corresponding to
(3.29) for the one-electron Hamiltonian and for the wave function
as
(
∑
)
(
(
The first order term
))
….. (3.34 a)
is the sum of two interaction potentials,
due to the motion of ions and
originating from the response of
conduction electrons
=
=∑
(
)
(
(
))
+
0
(
= ∑
)
(
(
))
….. (3.34 b)
The second order terms are
=
= ∑
(
)
(
(
))
(
)
(
)
….. (3.34 c)
We also write
61
ψ k=
….. (3.34 d)
We have used the following expressions for
(
)
(
∑
)∫
,
( )
{
and
as
( ) |
|}
….. (3.35 a)
(
)
( )
{(
(
∑
)∫
|
( )
|
( )
( )}
….. (3.35b)
and
(
)
∑
( )
*(
( )
Where
(
)∫
|
( )
|
( )
( )
( )+
….. (3.35 c)
represents a set of one particle Block functions for
electrons which apply to a crystal for electrons with ions fixed in
equilibrium positions.
We can establish the following relation between the secondorder matrix elements of Ve,
|
|
=
|
|
|
|
|
|
….. (3.36)
The second-order perturbation theory together with (3.33) can
be used to obtain the lattice vibrational ion-electron ion potential
62
energy for the system in terms of one electron wave function, Hartree
potential, and their perturbations due to thermal motion as.
∑
( )(
(
( )
∑
|
|
))
|
|
(
(
∑
|
)–
(
|
))
|
( )
|
∑
( )
|
|
∑
( )
|
|
∑
( )
|
|
∑
( )
|
|
….. (3.37)
If we use the relation (3.36) together with the following
expression for the first order perturbed wave function
∑ (
|
(
(
|
)–
(
))
, as
)
…..
(3.38)
Then the expression (3.37) may be written as
∑
(
(
|
∑
( )
)(
(
)–
(
(
|
|
))
|
))
|
|
|
|
|
|
…..
(3.39)
Here the first two terms on the R.H.S. are caused by repeated
one-phonon process and the last term is caused by intrinsic two
phonon processes. Instead several approximation methods have been
introduced to evaluate the electronic energy bands in solids based on
63
Hartree-Fock method. Complete discussion of these method may be
found else where14.
The Hartree-Fock approximation was used by Toya to calculate
the frequencies of normal modes in metals, in which the treatment is
analogous to the one given above for Hartree approximation, so it will
not be repeated again. The form of equation (3.39) remains unchanged
inspite of introducing exchange term in the Hartree-potential energy.
3.6
THE SMALL ION CORE APPROXIMATION:The small core approximation plays an important role in
expressions of the dynamical matrix element in the frame-work of the
Toya’s Theory. In this approximation we make three assumptions as
(i)
According to this assumption, the variation of the potential due
to the conduction electrons over the core-region is neglected.
We assume that this potential shifts the energy of the core states
but does not change their wave functions.
(ii)
In the second assumption the radial variation of the potential in
evaluating the matrix element of smooth function over the core
region is neglected.
(iii)
This is the most important assumption which enters in the
calculation of dynamical matrix elements of electrons-phonon
interaction is that the overlap repulsive interaction between ion
–cores is neglected due to small core-radii i.e. the spatialvariation of neighbouring core-potentials over the core – region
is ignored.
64
However, the effective potential in the present theory is
expressible as a sum of pair potentials, which will not be the case
if
core-repulsion interactions remain present or three body
forces are included in the theory. Estimates of ionic radii15,16, in
the metals to be investigated, are considerably smaller than half
of the interionic spacing, therefore it is believed that the
assumption is justified.
65
REFERENCES
1-
Debye P.; Ann. Phys., 4,39, 789 (1912).
2-
Born, M. and Karman, V.Z., Physik, 13,297 (1912)
3-
Pines, D.; Elementary Excitations in Solids, Ch. I (W.A. Benzamin,
Inc., N.Y.), (1963).
4-
Ziman, J.M. Electrons and Phonon , Ch. II(Oxford clarandon
press), (1962).
5-
Bardeen. J and pines, D., Phys, Rev., 99, 140(1955).
6-
Born, M. and oppenheimer, R, Ann. Physik, 84, 457 (1927).
7-
Maradudin, A.A., Montroll, E.W. and Weiss, G.H.; In solid state
physics, edited by seitz, F. and Turnbull. D. (Acad. Press, Inc. New
York), (1963), (Suppl.3).
8-
Kettel, C., Quantum theory of solids (wiley, New York), (1963)
9-
Hartree, P., Phys. Rev., 36, 57 (1930).
10-
Hartree, Fock, Hand buch der, Physic, 24/1, 349 (1933).
11-
Seitz, F. phys. Rev. 47, 400 (1935).
12-
Bohm, D. and Pines D, Phys. Rev, 82, 625 (1951).
13-
Toya, T,J. Res. Inst. Cata. Hokkaido, 6, 161(1958).
66
14-
Callaway, J., “Energy Band Theory”. (Academic press, New York),
(1964).
15-
Pauling, L., J. Am. Chem. Soc. 49, 765 (1927).
16-
Hand book of chemistry and physics, (the chemical Rubber
Published co., Cleveland), 48th ed. Page F-143, 1967/68.
**********