INTRODUCTION, CLASSICAL FREE ELECTRON THEORY OF METALS, DRAWBACKS OF CLASSICAL THEORY, RELAXATION TIME, COLLISION TIME AND MEAN FREE PATH etc.

1. Electrical Properties of Metals
1. INTRODUCTION: In solids the carriers of electric current are
electrons alone, and their interaction with the relatively stable rigid
ionic lattice can be approximately to some extent by viewing the
interaction as a collision between small electrons and individual large
ions. We begin by applying this concept to metallic conductors and
then extend it to semiconductors. Following are some of the
outstanding physical properties of metals:
(i) Metallic conductors obey Ohm’s law, which states that the
current in the steady state is proportional to the electric field
strength.
(ii) Metals have high electric and thermal conductivities.
(iii) At low temps, the resistivity is proportional to the fifth power of
absolute temp, i.e.   T 5
(iv) The resistivity of metals at room temp is of the order of 10-7 ohm
meter and above Debye’s temp varies linearly with temp, i.e.
  T
(v) For most metals, resistivity is inversely proportional to the
pressure, i.e.   1/P
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2. (vi) The resistivity of an impure specimen is given by Mathiessen’s
rule.  = 0 + (T), where 0 is a constant for the impure specimen
and (T) is the temp dependent resistivity of the pure specimen.
(vii) Near absolute zero, the resistivity of certain metals tends towards
zero, i.e. exhibit the phenomena of superconductivity.
(viii) The conductivity varies in the presence of magnetic field. This
effect is known as magneto resistance.
(ix) The ratio of thermal to electrical conductivities is directly
proportional to the absolute temp. This is known as Wiedemann-
Franz Law.
2. CLASSICAL FREE ELECTRON THEORY OF METALS: Metals
achieve structural stability by letting their valance electrons roam
freely through the crystal lattice. These valance electrons are the
equivalents of the molecules of an ordinary gas. Since electrons are
negatively charged particles, their motion corresponds to a flow of
electricity or electric current. It is assumed that the electrons are
moving about at random and colliding frequently with the residual
ions. Hence, the laws of classical kinetic theory of gases can be
applied to a free electron gas also. Thus, the electrons can be
assigned a mean free path , a mean collision time  and an average
speed c.
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3. In the absence of an externally applied p.d., there are on an average
as many electrons wandering through a given cross-section of the
conductor in one direction as there are in the opposite direction.
Hence the net current is zero. In between two collisions, the electron
may move with uniform velocity. During every collision both direction
and magnitude of the velocity get changed in general. The zigzag
motion is the thermal motion of the electron.
P. Drude made use of the electron gas model to explain theoretically
electrical conduction in metals. According to this theory the kinetic
velocities of the electrons are assumed to have a root mean square
velocity given by kinetic theory of gases in conjunction with the law
of equipartition of energy. is obtained as follows:
For unit volume of the metal, P = (1/3) = (1/3) mn
where, m is the mass of the electron and n is the no. of free electrons
in unit volume and is density  of the electron gas.
For molar volume of the metal,
P = (1/3) = (1/3) mNA /Vm
where, Vm is molar volume and NA is Avagadro’s number.
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4. Now PVm = (1/3) mNA = Ru T
Thus, m = [3 RuT /NA ] = 3 kBT
where, Ru and kB are universal gas constant and Boltzmann’s
constant respectively.
The kinetic energy of the electron is, (1/2)m = (3/2) kBT … (1)
and = 3kBT/m … (2)
At 20oC,
= [3 x 1.38 x10-23x293/9.11x10-31]1/2
= 1.154 x 105 m/s
Eq (2) indicates that the root mean square velocity of the electron is
directly proportional to the square root of the absolute temp of the
metal. At room temp, the drift velocity imparted to the electrons by
an applied electrical field is very much smaller than the average
thermal velocity. The time  taken by the electrons in traversing the
distance  will thus be decided not by the drift velocity due to the
field but by much greater velocity due to the kinetic thermal
motion. Now,
 = / =  [m/3kBT … (2a)
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5. A vital test of the validity of any theory of electrical conductivity is
whether or not it can account for Ohm’s law, which is V = IR, where, I
is the current flowing through a conductor as the result of applying a
p. d. V across its ends, at a given temp, R the resistance, is a
constant for a given conductor. In order to simplify the analysis let
us rewrite the law in terms of the current density, J = I/A and field
strength along the conductor, E = - dV/dx.
Thus, we have,
V/R = I = AJ  1/J = AR/V = Al/AEl = /E
J =E/ = E … (2b)
where,  the resistivity and conductivity  of a conductor of length l
and area of cross-section A are related by,
 = 1/ = RA/l
Let us now assume that there are n free electrons/m3, and that in
the absence of any applied field, these are darting about in all
directions with no net velocity just like gas molecules in a container.
When a field Ex is applied in the x-dirn, all electrons are accelerated
in the x-dirn with the acceleration of the ith electron, aix , is given by,
aix = - (eEx/m)
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6. Alternatively, this eq may be written as,
dvix/dt = - (eEx/m) … (3)
Ex
where, vix is the x-component of velocity of the ith electron, the
subscript Ex means the acceleration arising from the applied field.
Since the rhs of this eqn is the same for all electrons, we may write
eqn (3) in another form:
d/dt vx  = - [eEx/m] … (4)
Ex
where, vx  is the average velocity of all n free electrons as given by,
vx  = 1/n vix
we have used vx  to denote the average value of v. The current
density J is the charge flowing through unit are in unit time, so that
x-component, Jx must be number of electrons times the charge on
each in a volume of unit area and length vx , i.e.
Jx = n vx  (-e) = - nevx … (5)
The minus sign, which is coupled to the charge on an electron,
underlines the fact that conventional direction of current (i.e. of Jx )
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7. is in the opposite dirn to vx .
On differentiating eq (5), we have,
d/dt [vx  ] = - (1/ne ) [dJx/dt ] … (6)
Comparing this with eq (4), we see that,
(1/ne) [dJx/dt ] = (e/m) Ex ,
[dJx/dt ] = constant x Ex … (7)
so that for a constant Ex , this free electron model leads to a current
which increases linearly with time. Nothing could be farther from
Ohm’s law.
It is clear from eqn (6) that law can only be satisfied (i.e. dJx/dt = 0) if
d/dt vx  = 0.
Since we have just written is actually wrong, the difficulty must arise
because we have omitted to include a decelerating term which, when
added to,
d/dt [vx  , gives a net d/dt [vx  = 0
Ex
Once again we conclude that the electrons cannot be completely free. A
conductor always warms up when a current is flowing, we deduce
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8. that the conduction electrons must collide and exchange energy with
ions – a process normally referred to as electron-lattice scattering.
The eqn governing the sum of the accelerations should therefore
contain two terms:
d/dt [vx  + d/dt [vx  = 0 … (8)
Ex el  la
In principle, we can feed details of applied electric and magnetic
fields as well as temp gradient into this type of eqn, and solve it in
order to obtain the resultant electric and thermal currents.
Unfortunately, in practice, this is an exceedingly difficult operation.
The main problem is in obtaining the scattering term, which can only
be obtained by quantum mechanical technique. Furthermore, this
method normally ends up with an integro-differential eqn which is
extremely difficult to solve.
We shall therefore, obtain an expression for d/dt [vx  by a method
el  la
which by-passes these difficulties. Suppose at an instant, t = 0, the
average velocity of an electron is vx0. Just at that instant we switch
the field off, so that vx  subsequently tends to zero.
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9. In other words, the velocity will be randomized by electron-lattice
collisions. Now let us assume that this process follows the simplest
law of decay:
vx  = vx 0 e - t/ … (9)
The factor  is a constant of the system known as the relaxation time,
because it gives a measure of the time that the system takes to relax
when a constraint (the electric field) is removed (Fig.1a).
It is, in fact, the time taken for the drift velocity to decay to 1/e of its
initial value. Since Ex = 0 during this relaxation process, we see that
eqn (9) can be used to obtain electron-lattice term in eqn (8), and this
is done in such a way to avoid having to consider any details of the
collision process. The result is,
d/dt [vx  = - vx 0 e - t/T /
el  la
= - vx / … (10).
If we assume that this expression is unchanged by the presence of
field, we have, from eqns (4), (8) and (10),
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10. 11/13/2023 DR VBP 10
Fig.1 Calculation of current density
Fig.1a. Relaxation time of a system
of electrons to equilibrium after
removing the electric field.

11. - (e/m)Ex - vx / = 0  vx  = - [eEx /m ] … (11)
This steady average velocity imposed by the field and proportional to
it, is called the drift velocity of the electrons. The constant of
proportionality, e /m, is known as their mobility,  and the drift
velocity for unit electric field is given by,
 = vx /Ex = e /m
The units of  are meter/sec/volt/meter or m2V -1s -1.
Finally, we see that by combining eqns (2b), (5) and (11), we have,
 = ne2 /m or  = m /ne2 … (12)
and  = ne or  = 1/ne … (13)
This last eqn is of great important in the theory of conductivity, and
embodies the difference between the electrical properties of metals
and semiconductors. In metals, where all the valence electrons are
free at all temps, n is a constant and the temp variation of  is
essentially the temp variation of . In semiconductors, however, n
may also vary with temp and it is usually the most important factor.
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12. Alternate Method: Materials in solid form exhibit quite a lot of
interesting properties very useful for devices in science and
engineering. In ionic materials, the movement of ions is mainly
responsible for electric conduction. The charge can be carried by both
positive or negative ions, which become more mobile as the temp
increases and thus more free to wander through the lattice when an
external electric field is applied across the crystal. Such a charge
transfer process is termed ionic conduction. However, in metals the
atoms no longer hold their outer electrons because of their low
ionization energy (or loosely bound to the nucleus), held within the
metal, but otherwise essentially free. The metal is supposed to be
held together by the electrostatic forces between the positively
charged ions and the free electrons.
The two types of internal energy in metals are (i) the vibrational
energy of the metallic atoms (ions) about their mean lattice positions
and (ii) the free energy (kinetic energy) of free electrons. If thermal
energy is supplied from any external source to the metal under
consideration, its temp rises and hence the internal energy increases.
The well known thermal properties of solids such as thermal
expansion, thermal conductivity and heat capacity depend totally
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13. upon the changes in the energy of the lattices and free electrons.
Once an electric field is established across the metallic solid the free
electrons are accelerated. Their kinetic energy increases, a part of the
kinetic energy is of course lost by collisions with atoms attached to
the lattices. The resulting flow of charge or current is directly
proportional to the velocity of electrons. This velocity is determined
by the applied electric field and also the collision frequency.
It is reasonable to suppose that the electron gas in metals could be
considered as a classical gas obeying the classical laws including
laws of kinetic theory of gases. Thus, in the absence of electric field,
the electrons can move from place to place randomly in crystal,
without any change in their energy and collide occasionally with the
atoms. In between two collisions the electron may move with a
uniform velocity, but during every collision both the direction and
magnitude of the velocity gets altered in general. This zigzag motion
of electron due to thermal energy is shown in Fig.1(b). The average
speed of this thermal motion say depends on absolute temp and is
calculated using eqn (2).
The thermal velocities calculated using eqn (2) may not bring any net
transport of electric charges, since on the average, for every electron
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14. moving in one direction there
will be another moving in the
opposite direction.
Fig.1b. Zigzag motion of the electron
due to frequent collisions with
atoms at the lattice.
Classical Theory of Electric Conduction: Because of random
motion of electrons in a free field, the resultant motion is zero, hence
no current flows. When an electric field E is applied to a conductor
an electric current begins to flow and the current density by Ohm’s
law is, J = E.
The constant  is the electrical conductivity of specimen and its
reciprocal is electrical resistivity . This indicates that the electrons
move in a specific direction under the influence of the field.
Naturally, the distribution function of electrons in the conductor
undergoes a change. The directional motion of free electron is called
a drift. The average velocity gained during this drift motion is termed
as drift velocity.
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15. 11/13/2023 DR VBP 15

16. As electron is being a negatively charged particle, the force acting on
it under the electric intensity E is, F = - eE. The electron drift is in a
direction opposite to that of the applied field. During the accelerated
motion, the electron collides with the defects in the lattice. As a
result of the consequent scattering, the electron loses the velocity it
gained from the field. The effect of the lattice may be reduced
considerably due to a retarding force (may be due to damping). This
force is proportional to the velocity v and mass m of the electron.
Thus the retarding force is represented as –mv, where  is a
constant. Now the eqn of motion of electron is,
mdv/dt = - eE - mv …(13a)
On application of electric field, the velocity rises till the retarding
force which is proportional to the velocity equals the force due to the
applied field.
When these forces become equal the acceleration ceases. Thereafter
the electron moves with the drift vd .
Now, - eE - mvd = 0 or vd = - eE/m … (13b)
The drift velocity produced for unit electric field is called carrier
mobility,. i.e.  = e/m
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17. Suppose that as soon as the velocity of the directional motion of
electrons attains its steady value, the field is cut off. On account of
collision of electrons with the lattice defects the velocity starts
decreasing. After some time the electron gas resumes its equilibrium
condition. Such a process which leads to the establishment of
equilibrium in a system for which it was previously disturbed is
called the relaxation process. When the applied field is cut off, the
eqn of motion of the electron becomes,
mdv/dt = - mv  dv/v = -  dt
On integrating both sides, we get,
log v = - t + C … (13c)
When t = 0, v = vd and C = log vd
Thus eqn (13c) becomes, log v = - t + log vd
i.e. v = vd e (-t ) … (13d)
If we define the time taken by the electron to attain a directional
velocity which is 1/e of the drift velocity as the relaxation time
symbolized as , then
vd /e = vd e (-   )
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18. This yields  = 1/. Substituting this value of in eqn (13b), one gets,
vd = - eE  /m … (13e)
and vd /E = e /m is called the mobility of the electron, .
It is assumed in Fig.1c that the distance between the layers A and B
is nvd with n as the no. of free electrons in unit volume (one meter3).
Thus the charge flowing through unit area for unit time at the layer
B is nevd. This is nothing but the current density.
i.e. J = - nevd with vd = - e E /m or J = ne2E /m or
 = J/E = ne2/m
The electrical resistivity,  = m/ne2 … (13f)
This is the microscopic expression for the resistivity of the metal.
Here the measurable resistivity of metals is related to the density of
electrons, the charge of the electrons, the mass of the electron and
the relaxation time as expressed by eqn (12). Mobility of electron in
the metal is defined as the steady state drift velocity per unit electric
field,
i.e.  = vx/Ex = e /m … (14)
and hence,  = ne … (15)
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19. The unit of mobility is m2v-1s-1. Mobility is an important term in the
study of semiconductors. The macroscopic definition for resistivity is
given by,
R =  l/A ;  = AR/l … (15a)
Also,  = l/ and E = V/l … (15b)
Again  = J/E or J = E
Substituting for  and E = V/l from (15a) and (15b), we get,
J = E/ = V/ l
Substituting for , we get,
J = (V/l ) (l/AR) , JA = V/R
i.e. V = IR. This is Ohm’s law.
Temperature Dependence of Electrical Resistivity: In the absence
of electric field, the free electrons in a metal will be moving about at
random in all directions and will be in temp equilibrium with it. The
kinetic energy associated with the electron is,
m ( )2/2 = 3kBT/2
When an electric field is applied, the electron will require a drift
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20. velocity and the resulting acceleration is, a = eE/m. The drift velocity
is small compared to the random velocity . Further, the drift
velocity is not retained after a collision with an atom because of the
relatively large mass of the atom. Hence just after a collision the drift
velocity is zero. If the mean free path is , then the time that elapses
before the next collision takes place is / . Hence the drift velocity
acquired before the next collision takes place is,
u = accn x time interval = (eE/m)(/ )
Thus the average drift velocity is, u/2 = (eE /2m )
If n is the no. of electrons/unit volume, then the current flowing
through unit area for unit time is,
Jx = neu/2 = ne2E/2m or
 = Jx /E = ne2/2m , or  = 2m /ne2 … (16)
i.e.  = 2 x 3mkBT /ne2 ,  = ne2/3mkBT … (17)
It was assumed by Drude and Lorentz that  is independent of temp
and that is of the order of interatomic distance. Hence,  T . This
means that the specific resistance of an electric conductor is directly
proportional to the square root of the absolute temp. This is not in
agreement with exptal observation that  T. Apart from this
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21. discrepancy, it is also not correct to assume that the mean free path
is independent of temp and hence this classical theory is almost an
unacceptable one. However, the Ohm’s law is derived, since the
conductivity in eqn (16) is independent of the field.
3. DRAWBACKS OF CLASSICAL THEORY:
(i) Heat Capacity of the Electron Gas: Classical free electron theory
assumes that all the valance electrons in a metal can absorb thermal
energy. According to the law of equi-partition of energy, every free
electron in a metal has an average kinetic energy 3kBT/2. Thus one
kmol of a metal which has NA atoms will therefore have NA free
electrons, assuming that each atom contributes one valance electron
to the electron gas. Now, the energy associated with one kmol of such
a metal is,
U = 3 NA kBT/2 … (18)
If heat is supplied to the metal, these free electrons also absorb part
of the heat, and the molar electronic specific heat is obtained as
follows:
[Cv]el = [dU/dT] = 3NA kB/2 = 1.5 Ru … (19)
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22. where Ru is the universal gas constant.
i.e. [Cv]el =1.5 x1.38 x 10-23 x 6.02 x 1026 = 12.5 x 103 J/kmol/K
That is, molar electronic specific heat = 12.5 x 103 J/kmol/K. The
value of 1.5 Ru , due to free electrons is about hundred times greater
than the exptally predicted value. Since the heat capacity of a solid
due to atomic vibrations in 3Ru , free electrons should make a
significant contribution to the total specific heat of a metal. It is,
however, that at least at high temps, Dulong and Petit law (lattice
specific heat) holds good and the total specific heat of a solid is 3Ru ,
this means that the free electrons do not contribute significantly to
the heat capacity of a metal. It is, therefore, concluded that, the law of
equi-partition and hence classical Maxwell-Boltzmann statistics must
not be applied to evaluate the electronic specific heat in metals.
(ii) Computation of Mean Free Path: The microscopic expression for
the resistivity of a metal is given by the eqn (12),  = m/ne2 . The
resistivity of the most useful metal copper at 20oC is 1.69 x 10-8 ohm-
m and the concentration of free electrons in copper,
n = 8.5 x 1028/m3. Thus,  = m/ne2
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23.  = (9.11x10-31)/8.5 x1028 x(1.6x10-19)2 x 1.69 x10-8
 = 2.47 x 10-14 sec , But  = / … (20)
  =  = 2.47 x 10-14 x 1.154 x 105   = 2.85 nm.
The exptally found value for  is about ten times above this value.
Classical theory could not explain the large variation in  values.
(iii) Relation between Electrical Conductivity and Thermal
Conductivity (Wiedemann-Franz Law):
Heat conduction in solids may take place either through the
mechanism of the atoms or through that of the free electrons. In
metals, electrons are the principal carriers of heat energy. Free
electrons do this by being excited by energetic scattering centers and
carrying the extra energy to another scattering center in a cooler part
of the metal.
Consider a copper conductor in the form of a rod of uniform cross-
section (say 1 sq.m.). Let us assume that a thermal gradient be
established. Consider three layers at A, B and C normal to the dirn of
heat flow. Let each of them be separated by a distance , (mean free
path of electron gas in metal).
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24. 11/13/2023 DR VBP 24
Fig.2 Flow of heat through a conductor

25. T + (dT/dx)  , T and T - (dT/dx)  are thee steady temp of the
layers as shown in Fig.2.
Let the thermal energy of an electrons in these layers be,
E + (dE/dx)  , E and E - (dE/dx)  respectively. Thus the excess of
energy carried by an electron from A to B is dE/dx . It is well
known from kinetic theory of gases, the no. of electrons flowing in a
given direction through unit area for unit time is (1/6) n , where n
is the no. of free electrons per cubic meter and is the average
thermal velocity of the electron. Thus the excess of energy
transported by the process of conduction through unit area in unit
time at the middle layer B is, (n  /6) (dE/dx).
Similarly, deficit of energy transported through B is the opposite
direction is, (- n  /6) (dE/dx). Hence, net heat energy transported
through unit area in unit time from A to B is,
(n  /6) (dE/dx) - (-n  /6) (dE/dx) = (n  /3)(dE/dx)
… (21)
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26. This must be equal to the product of thermal conductivity and the
temp gradient.
i.e. (n  /3)(dE/dx) = T [ dT/dx] … (22)
From the general definition, Q = T At (dT/dx )
Now, n[dE/dx ] = n [dE/dT ] [dT/dx ] = n [Cv ]el [dT/dx]
Substituting the value of n[dE/dx ] in eqn (22), we get,
(n  /3) [Cv ]el [dT/dx] = T (dT/dx )
Thus, T = (n  /3) [Cv ]el … (23)
The electronic specific heat associated with each electron is,
[Cv ]el = 3kB/2 and = 3 kBT/m
Now, T = (n /3 ) (1.5 kB ) 3 kBT/m
T = (n kB/2 ) 3 kBT/m
The expression for electrical conductivity is,
 = (n e2 ) /3mkBT … (24)
Thus, T / = [nkB /2] 3kBT/m [3mkBT /ne2]
T / = 3T/2 [kB /e ]2 … (25)
This linear dependence of T / on absolute temp is known as the
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27. law of Wiedemann and Franz (classical one). The multiplying
constant is called Lorentz number L, that is,
L = 3/2 [kB/e]2 = 1.12 x 10-8 WK-2 … (26)
The validity of eqn (25) can be checked using exptally determined
values of  and T . Eqn (25) is ,
T /(T) = L
For copper at 20oC, the electrical resistivity and thermal conductivity
are 1.72 x 10-8 -m and 386 Wm-1K-1 respectively.
Now, L = T / x T = 386 / 5.81 x 107 x 293
or L = 2.26 x 10-8 WK-2
The value of Lorentz number does not agree with the value calculated
from the classical formula given in eqn (25). Thus, the classical
assumption that all the free electrons of a metal participate in
thermal conduction is not correct. Quantum theory with Fermi-Dirac
statistics reveals that only the electrons near the fermi level take part
in thermal and electrical conductions.
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28. 4. RELAXATION TIME, COLLISION TIME AND MEAN FREE PATH:
(i) Relaxation Time: Suppose at a given time t = 0, the average
velocity of the electron is  vx 0. Just at that instant, we switch the
field off, so that  vx  will subsequently tend to zero. In other words,
the velocity will be randomized by electron-lattice collisions. Now let
us assume that this process follows the simplest law of decay.
 vx  =  vx 0 e-t/ … (27)
The factor is a constant relaxation time, because it gives a measure
of the time that the system takes to relax when a constraint (the
electric filed) is removed. When t = , the eqn (27) becomes,
 vx  =  vx 0/e
Thus, relaxation time may be stated as the time taken for the drift
velocity to decay to 1/e of its initial value.
Differentiating eqn (27), we get,
d/dt  vx  elec – latt = -  vx 0 e-t// = -  vx / … (28)
(ii) Collision Time: Suppose that the probability of an electron
making a collision in time dt is dt/c , so that on an average there
are 1/c , collisions per second.
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29. Now our intension is to show that c =  .
Let us assume that all electron collisions are elastic, i.e. energy is
conserved, so that the speed of the electron is the same before and
after collision. This assumption is not strictly correct, because a
conductor warms up when current is passing, it implies that energy
is transferred to the lattice during electron-lattice collisions.
However, the actual energy change at each collision is minute, and
the fractional change in velocity in a metal is typically 1 in 106, so
that the assumption is entirely reasonable. If we assume that after
collision the velocities are completely random, we have,
 vx after = 0. Since  vx before =  vx , it follows that the change of
 vx  on collision = -  vx .
There are 1/c collisions/second, so that, the rate of change of,
 vx  = -  vx /c … (29)
However, we could equally write the LHS of this eqn is,
[d  vx /dt ]elec – latt , so that on comparing eqns (29) and (28)
we see that  = c and the relaxation time, therefore, equals the mean
time between collisions.
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30. 11/13/2023 DR VBP 30
Fig.3 Relaxation of a system of electrons to
equilibrium after removing the electric field
as given by eqn (27).
Fig.4a Scattering equally probable in all
directions, so that the average angle is 90o.
Fig.4b Scattering predominantly in the
forward direction at average angle .
Change in any vx is therefore, on an
average, vx (1-cos ) .
Fig.4b
Fig.4a

31. In deriving this result, we assumed that the velocities took up random
dirns after each collision (Fig.4a), this is equivalent to stating that
after each collision the electron had no memory of what went before.
However, it is possible that the scattering is very weak, in which case
such an assumption is unrealistic. If, on an average, the velocity dirn
changes by  on collision (Fig.4b), then the change of  vx  is on an
average, -  vx  (1-cos ). Consequently,
 = c /[1-  cos   ] … (30)
where,  is the relaxation time and c is the mean time between
collisions.
In other words, the relaxation time is then much longer than the
mean time between collisions, these electrons have a memory.
(iii) Mean Free Path: The mean free path  of an electron is defined
as below:
 = c
In case, the obstacles are hard spheres, the mean free path  is
determined by the concentration of these obstacles. Then for a given
concentration of obstacles, the collision time c becomes inversely
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32. proportional to the velocity of the electron. Then, with  cos  
independent of the velocity of the electrons, in accordance with eqn
(30), the relaxation time  becomes proportional to c and hence,
inversely proportional to velocity .
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