The Internal Field Ei Can Be Written As Difference of Two Fields
Introduction
When an electric field is applied to a medium made up of a large number of atoms and molecules,the charges bound in each molecule will respond to the applied field and execute perturbed motion.The molecular charge density will be distorted. The multipole moment of each molecule will be different from what they were in the absence of the field. In simple substances, when there is no applied field, the multipole moments averaged over all molecules is zero. The dominant molecular multipole with the applied field is the dipole. There is thus produced in the medium an electric polarization P, which is the dipole moment per unit volume and is given by
(1)
where pi is the dipole moment of the ith type of molecule in the medium, the average is taken over a small volume centered at x and Ni is the average number per unit volume of the ith type of molecule at a point x.
In dense media with closely packed molecules, the polarization of neighboring molecules give rise to an internal field Ei experienced by a molecule in addition to the average macroscopic field E. Thus total field experienced by a molecule is E+Ei.
The internal field Ei can be written as difference of two fields
(2)Ei=Enear-Ep
where,Enear is the actual contribution of the molecules close to the given molecule and Ep is the contribution from those molecules treated in an average continuum approximation described be the polarization P.
figure no. 1
If we consider a small spherical volume of radius R , then the total dipole moment inside is given by
(3)
provided V is so small that P is essentially constant throughout the molecule.Hence the average electric field inside the sphere is
(4)
The internal field can thus be written as
(5)
It is a good working assumption that Enear= 0 for most materials.
The polarisation vector can be defined as
(6) P=N<pmol
Where <pmol> is the average dipole the molecules. This dipole moment is approximately proportional to the electric field acting on the molecule.To exhibit this dependence we define the molecular polarizabilityas a ratio of the average molecular dipole moment to times the applied field at the molecule.Thus,
(7)
is in principle , afunction of the electric field , but for a wide range of field strengths is a constant that characterizes the response of the molecules to an applied field.
Clausius Mossotti Equation
One of the earliest model of electric polarizability is that by Clausius in 1879 and Mossotti in 1850 . Here molecular polarizability is established in terms of dielectric constant.
Now ,using eqs. 6, 7 and 5 , and assumingEnear= 0 ,we get;
(8) P=Nmol(0E+P/3),
Now the polarization vector is related to the applied electric field as,
P=0eE, where as e is defined as electric susceptibility of the substance.
We have,
(9) e=
since the dielectric constant is 0 = 1+e , so the molecular polarizability can be expressed in terms of the dielectric constant ;
(10)
This is called the Clausius Mossotti Equation.
This relation holds best for dilute substances such as gases . For liquids and gases and solids the relation is only approximately valid , especially if the dielectric constant is large .
Polarizability
The total polarizability can usually be separated into three parts : electrical , ionic and dipolar , as shown in the figure
The applied field distorts the charge distribution and so produces an induced dipole moment in each molecule.
The ionic contribution comes from the displacement of a charged ion with respect to other ions .
The dipolar polarizability arises from molecules with a permanent electric dipole moment that can change the orientation in an applied electric field .
figure no.2
Here we are going to deal with the electric contribution only.
Atomic polarization
In this section, we consider what happens to a neutral atom when it is placed in an electric field E. Although the atom as a whole is electrically neutral, there is a positively charged core (the nucleus) and a negatively charged electron cloud surrounding it. These
two regions of charge within the atom are influenced by the field :
the nucleus is pushed in the direction of the field
and the electron is pushed in the opposite direction
In principle if the field is large enough, it can pull the atom apart completely. With less extreme fields, however, an equilibrium is soon established because the positive and negative charges attract one another and this holds the atom together. The two opposing forces – E pulling the electrons and nucleus apart, and their mutual attraction drawing them together – reach a balance, leaving the atom polarized. When an atom is polarized the plus charge is shifted slightly in one way, and the minus charge in the opposite direction. Thus the atom now has a dipole moment p which points in the same direction as E. This induced dipole moment is approximately proportional to the field :
(11)p = aE
The constant of proportionality ais called the atomic polarizability.
The values of atomic polarizability depend on the detailed structure of the atom concerned. We list below the atomic polarizability of some elements :
Element : H He Li Be C Ne Na Ar K
0.66 0.21 12 9.3 1.5 0.4 27 1.6 34
This table lists 1/(4πεo ) a in units of 10-30
A Model for Atomic Polarizability :
A primitive model for an atom consists of a point charge (+q) surrounded by a uniformly spherical electron cloud (-q) of radius “ a”. The atomic polarizability of such an atom is calculated as follows:
figure no.3
In the presence of an external field E, the nucleus will be slightly shifted to the right and the electron cloud to the left. Since the actual displacements are very small, it is reasonable to assume that the electron cloud retains its spherical shape. At equilibrium, let the nucleus be displaced by a distance d from the centre of the sphere. When the external field pushing the nucleus to the right exactly balances the internal field to the left :
E = Ee where Ee is the field produced by the electron cloud.
The field at a distance d from the centre of a uniformly charged sphere is :
(12) IEeI = qd / (4πε0 a3)
Thus at equilibrium, we have E = qd / (4πε0 a3) , or p = qd = (4πε0 a3)E .
The atomic polarizability is therefore given by,
(13)α = (4πε0 a3) = 3 ε0 v
Although this atomic model is a crude one, the result is surprisingly good – it is accurate to a factor of four for most simple atoms.
Models for the Molecular Polarizability
We deal with molecular polarizabilty by considering the following two cases:-
Case 1:The displaced charge is bound elastically to an equilibrium position.
Case 2:Charge has several equilibrium positions,each of which it occupies with a probability which depends on the strength of an external field.
Case 1:
The interpretation of this case is that on displacing a charge e, carried by a particle of mass m and radiusx, a restoring force proportional to x acts on the particle in a direction opposite to the displacement .Thus if a constant external field Eis applied
(14)
where 0/2denotes the frequency of oscillation and -m02x is the restoring force. The above equation can be written as
(15)
where
ie, dxo/dt=0.The charge e therefore carries out harmonic oscillations about the position xowhich thus represents time average of its displacements,ie, if C and are constants then
(16)x=xo+Ccos(0t+
The average electric moment is therefore:
(17) pmol=ex0=(e2/m02)E
This means that the polarizability is =e2/m02.If there is a set of charges ejwith masses mj and oscillation frequencies jin each molecule then the molecular polrizability is
(18)
Effects of thermal agitation
Suppose that our system is in temperature equilibrium with its surroundings, then the charge will make irregular oscillations, and the question is if these oscillations will change the induced moments. It can be seen at once that this is not the case, for since the restoring force is proportional to the displacement a diminutive of the moment by a certain amount will involve the same change in potential energy as an increase by the same amount .Therefore both corresponding position of the charge will be equally probable and in the average the electric moment will have the same value as given before, independent of the intensity of its heat motion. The same is illustrated by the following calculation:-
Assuming classical statistics to hold for the system,the probability distribution of particles in phase space is proportional to the Boltzmann factor exp(-H/kT) where H is the Hamiltonian.Assuming the applied field Eis in the z-direction the Hamiltonian for a harmonically bound charge is
(eqns 19 – 21)
which yields the average value of the dipole moment:
(22)p>=(e2/m02)E
and that of the polarizability:
(23)=(e2/m02)
thus the equation obtained shows that the thermal motion is not able to alter the induced polarization.
Case 2:
As an example of case 2, consider a particle with charge e, possessing two equilibrium positions A and B separated by a distance b.In the absence of an electric field the particle has the same energy in each position. Thus it may be assumed to move in a potential field of the type shown in the figure below:
figure no. 4
If in equilibrium with its surroundings it will oscillate with an energy of order kT about either of the equilibrium positions say A. Occasionally however through a fluctuation it will acquire sufficient energy to jump over the potential wall separating it from B.On a time average therefore it will stay in A as long as in B,ie the probability of finding the particle in either A or B is 0.5.
The presence of a field Ewill affect it in two ways.Firstly as in case 1 the equilibrium positions will be shifted by an amount xo which for simplicity will be assumed to be the same in A and B.Secondly the potential energies UA and UB of the particle in the two equilibrium positions will be altered because its interaction energy with the external field differs by ebE,ie
(25) UA - UB=ebE
The particle should then in average stay near B longer than near A.Actually since according to statistical mechanics the probability of finding a particle with energy U is proportional to exp(-U/kT),
(26) a.pA=exp(-UA/kT)/[exp(-UA/kT)+exp(-UB/kT)] b. pB=exp(-UB/kT)/[exp(-UA/kT)+exp(-UB/kT)]
are the probabilities for the positions A and B respectively.From the above three equations it is obvious that pB - pA>0.
It follows from the definitions of the probabilities pB and pA that if the condition of the system over a long time t1 the particle spends time
(27) pAt1=[0.5-0.5(pB - pA)]t1
in position A and a time
(28) pBt1=[0.5+0.5(pB - pA)]t1
in a position B.Thus it has been displaced by a distance b from A to B during the fraction 0.5(pB - pA) of t1.The average moment induced is thus 0.5eb(pB - pA).
Hence is the angle between E and b,the projection of the induced moment into the field direction is given by
(29)
In most cases we can assume ebEkT. Developing in terms of ebE/kT the average induced moment in the field direction is found to be
(30) (0.5ebcos)2E/kT+eIxoI
where eIxoI is a term similar to those considered in case1 which has been added to account for the elastic displacement .
Often two charges +e and –e are strongly bound forming an electric dipole p=ed,where d is the distance between the two charges.The above case 2 then leads to the same result as that of a dipole p having two equilibrium positions with opposite dipole directions,with equal energy in the absence of a field.In a field Ethe energy of interaction between the field and the dipole is given by -p.Eso that 2p.E is the energy difference between the two positions.This is equivalent to equation (25) if p=0.5eb.
Actually putting an immobile charge –e halfway between A and B turns case 2 into the present case. Clearly the induced moment must be the same in both cases because the charge e is immobile and its distance from A and B is 0.5bleading to the dipole moment p.Introducing this into (30) will give for the induced moment in the field direction
(31)
In contrast to case1 the electric moment is now temperature dependent. Therefore a substance consisting of large number of such dipoles will have temperature dependent dielectric constant ,in contrast to substances in which all charges are bound elastically
This means that the entropy of the substance is decreased by the field [,
Here,S0is the entropy in the absence of the field].This is evident because the field causes the fraction pB of the dipoles with components in the field diretion to be larger than the fraction pA of dipoles with components in the opposite direction,thus leading to state which is more ordered(having lower entropy)than the state of complete disorder.
The difference between cases 1 and 2 is as follows:
In case1 the field exerts a force on the elastically bound charge thus shifting its equilibrium position.In case 2 this force of the field on the charge leads to contributions of type 1denoted byeIx0Iin equations (30) and (31).It would be wrong however to assume that the field by this force turns a dipole from one equilibrium position to another.It should also be realized that though every charge is displaced elastically (case 1) the fraction of dipoles turned by a field of reasonable strength(case 2)is very small.This fraction is given by 0.5(pB - pA) which si of the order of 10-4 for a field of 30000 volts per metre.
Demerits of the models for molecular polarizabilty:
A model such as we have used,in which the electronic motions are represented by harmonic oscillators,is not compatible with the modern knowledge of atomic structure.We know that actually the electrons are subject to inverse square rather than linear restoring forces,and move in approximately Keplerian orbits instead of executing simple harmonic vibrations about positions of static equilibrium. In fact Earnshaw’s theorem in electrostatics tells us that there are no such positions for all the charges.In actual molecules, to be sure , the motions of the nuclei, in distinction from the electrons,can be regarded as approximately simple harmonic motions about equilibrium, as the nuclei are sluggish because of their large masses, but for this very reason the amplitudes of their vibrations are so small that the contributions of these oscillations to the susceptibility is usually smallthough not always negligible.Hence the part of the molecular motion which is really simple harmonic is of secondary importance for susceptibilities.
Many of these demerits are removed when we consider the Quantum Mechanical model.
REFERENCES :
1) Classical Electrodynamics - J.D. Jackson(chapter 4)
2) Introduction to Electrodynamics - David J. Griffiths (chapter 4)
3) Theory of Dielectrics-H. Frohlich (chapter2)
4) Classical Electrodynamics -S.P . Puri (section 2.2)
5) Foundations of Electromagnetic Theory –John Reitz,
Frederick Milford,
Robert Christy
6)The Theory of Electric and Magnetic
Susceptibilities-Van Vleck(chapter 2)
7)Introduction to Solid State PhysicsCharles Kittel (chapter 13)
(7th edition)
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