Chapter 5 : Electron Theory of Metals

Example 5.1 Page No : 169

In [13]:
import math 


# Variables
#The probability that a particular quantum state at energy E is filled, is given by
#f(E)  = 1/(1+exp(E-E_f)/kT)
e = 1.6*10**(-19);			#charge on the electron
dE = 0.5*e;			#E-E_f in joule

# Calculation
#0.01 = 1/(1+exp(x))
#1+exp(x) = 100
x = math.log(99);
k = 1.38*10**(-23);			#consmath.tant
T = dE/(x*k);			#temperature

# Results
print 'temperature at which there is one per cent probability that a state with an energy\
 0.5 eV above the Fermi energy will be \noccupied by an electron in = %.0f K'%round(T)

# rounding off error

temperature at which there is one per cent probability that a state with an energy 0.5 eV above the Fermi energy will be 
occupied by an electron in = 1262 K

Example 5.2 Page No : 169

In [2]:
			
import math 

# Variables
n = 10**19;			#electrons per m**3
V = 0.017;			#applied voltage 
d = 0.27*10**-2;			#dismath.tance with material
e = 1.602*10**-19;			#in coulomb
m = 9.1*10**-31;			#mass of an electron(in kg)

# Calculation
conductivity = 0.01;			#in mho.m**-1)
E = V/d;			#Electric field(in V/m)
v = (conductivity*E/(n*e))*10**2;			#drift velocity of carriers(in meter/sec)

# Results
print 'drift velocity of carriers in = %.2f m/s'%v

drift velocity of carriers in = 3.93 m/s

Example 5.3 Page No : 170

In [3]:
			
import math 

# Variables
T = 300;			#Temperature(in Kelevin)
t = 2*10**-14;			#time(in sec)
V_c = 8.9;			#volume of 63.54gm of copper(in cc)
Aw_c = 63.54;			#Atomic weight of copper(in a.m.u)
e = 1.6*10**(-19);
m = 9.1*10**-31;
N_a = 6.023*10**23;			#avogadro's number

# Calculation
n = (N_a/(Aw_c/V_c))*10**6;			#Number of electrons per m**3
conductivity = (e**2)*n*t/m;			#conductivity of copper at 300K(in mho/m)

# Results
print 'conductivity of copper at 300K in = %.2e mho/m'%conductivity

# note : answer in book is wrong.

conductivity of copper at 300K in = 4.75e+07 mho/m

Example 5.4 Page No : 170

In [4]:
			
import math 

# Variables
t = 10**(-14);			#mean free time between the collisions(in second)
e = 1.6*10**-19;
m = 9.1*10**-31;

# Calculation
Mobility = e*t/m;			#in m**2/V-s

# Results
print 'mobility of condution electron in = %.2e m**2/V-s'%Mobility

mobility of condution electron in = 1.76e-03 m**2/V-s

Example 5.5 Page No : 170

In [14]:
			
import math 

# Variables
n = 6.*10**23;			#conduction electron per m**3
conductivity = 6.5*10**7;			#in mho/m
E = 1.;			#electric field intensity (in V/m)
e = 1.602*10**-19;
m = 9.1*10**-31;

# Calculation
Mobility = conductivity/(n*e);			#in m**2/V-s
v = Mobility*E;			#drift velocity(in m/sec)

# Results
print 'mobility of condution electron in = %.2e m**2/V-s'%Mobility
print 'drift velocity in = %.2e m/sec'%v

mobility of condution electron in = 6.76e+02 m**2/V-s
drift velocity in = 6.76e+02 m/sec

Example 5.6 Page No : 171

In [9]:
			

import math 

# Variables
d = 10.5;			#density of silver(in gm/cc)
At_w = 107.9;       #atomic weight
e = 1.6*10**-19;
conductivity = 6.8*10**5;			#in mho/centimeter

# Calculation
N = 6.023*10**23;
n = N*d/At_w;			#number of free electrons
Mobility = conductivity/(n*e);			#mobility of electrons(in cm**2/V-s);

# Results
print 'number of free electrons = %.2e'%n
print 'mobility of electrons in = %.2f cm**2/V-s'%Mobility

# rounding off error

number of free electrons = 5.86e+22
mobility of electrons in = 72.51 cm**2/V-s

Example 5.7 Page No : 172

In [4]:
import math 

# Variables
E_f = 3.75;			#Fermi energy(in eV)
e = 1.602*10**-19;
W_f = e*E_f;			#fermi energy in joules
t = 10**-14;			#mean free time between the collisions(in second)

# Calculation
m = 9.1*10**-31;			#mass of electron
v_f = ((2*W_f)/m)**(1./2);			#maximum velocity of an electron in a metal(in m/s)
mobility = e*t/m;			#mobility of electrons(in m**2/V-s)

# Results
print 'maximum velocity of an electron in a metal in = %.2e m/s'%v_f
print 'mobility of electrons in = %.2e m**2/V-s'%mobility

# incorrect answer in the textbook

maximum velocity of an electron in a metal in = 1.15e+06 m/s
mobility of electrons in = 1.76e-03 m**2/V-s

Example 5.8 Page No : 172

In [12]:
			

import math 

# Variables
E_f = 2.1;			#fermi energy(in eV)
e = 1.602*10**-19;
m = 9.1*10**-31;

# Calculation
W_f = e*E_f;			#fermi energy in joules
v_f = (2*W_f/m)**(1./2);			#velocity of an electrons at fermi level(in m/sec)

# Results
print 'velocity of an electrons at fermi level in = %.1e m/sec'%v_f

velocity of an electrons at fermi level in = 8.6e+05 m/sec

Example 5.9 Page No : 172

In [16]:
			
import math 

# Variables
t = 10**-9;			#collision time(in seconds)
E_f = 7;			#fermi energy(in eV)
e = 1.6*10**-19;
m = 9.1*10**-31;

# Calculation
W_f = E_f*e;			#fermi energy(in joules)
v_f = (2*W_f/m)**(1./2);			#velocity of an electrons at fermi level(in m/sec)
P = v_f*t;			#Mean free path(in meter)

# Results
print 'Mean free path in = %.2e m'%P

Mean free path in = 1.57e-03 meter

Example 5.10 Page No : 173

In [17]:
			
import math 

# Variables
N_a = 6.023*10**23;
V_c = 8.9;			#volume of 63.54gm of copper(in cc)
Aw_c = 63.54;			#Atomic weight of copper(in a.m.u)

# Calculation
n = (N_a/(Aw_c/V_c))*10**6;			#Number of electrons per m**3
e = 1.6*10**-19;
m = 9.1*10**-31;
t = 2*10**-14;			#collision time
conductivity = n*(e**2)*t/m;			#conductivity of copper

# Results
print 'conductivity of copper in = %.1e ohm**-1/m'%conductivity

conductivity of copper in = 4.7e+07 ohm**-1/m