Chapter 6: Electron theory of Metals

Example 6.1, Page number 6.5

#Variable declaration
rho_s = 10.5*10**3   #density of silver(kg/m^3)
Na = 6.02*10**26     #Avogadro's number
Ma = 107.9           #atomic weight of silver
sigma = 6.8*10**7    #conductivity(/ohm-m)
e = 1.6*10**-19      #charge of an electron(C)

#Calculations
n = (rho_s*Na)/Ma
u = sigma/(n*e)

#Results
print "Density of electrons =",round((n/1E+28),2),"*10^28"
print "Mobility of electrons =",round((u/1E-2),3),"*10^-2 m^2/V-s"

Density of electrons = 5.86 *10^28
Mobility of electrons = 0.725 *10^-2 m^2/V-s

Example 6.2, Page number 6.6

#Variable declaration
den = 8.92*10**3  #density(kg/m^3)
rho = 1.73*10**-8 #resistivity of copper(ohm-m)
Ma = 63.5         #atomic weight
e = 1.6*10**-19   #charge of an electron(C)
Na = 6.02*10**26  #Avogadro's number
m = 9.1*10**-31   #mass of an electron(kg)

#Calculations
n = (den*Na)/Ma
u = 1/(rho*n*e)
tou = m/(n*e**2*rho)

#Results
print "Mobility of electrons =",round((u/1E-2),3),"*10^-2 m/V-s"
print "Average time of collision of electrons =",round((tou/1E-14),2),"*10^-14 s"

Mobility of electrons = 0.427 *10^-2 m/V-s
Average time of collision of electrons = 2.43 *10^-14 s

Example 6.3, Page number 6.7

#Variable declaration
P = 1.54*10**-8    #resistivity(ohm-m)
n = 5.8*10**28     #no. of electrons per m^3
m = 9.108*10**-31  #mass of an elecron(kg)
e = 1.602*10**-19  #charge of an electron(C)

#Calculations
tou = m/(n*e**2*P)

#Result
print "The relaxation time of conducton of electrons is",round((tou/1E-14),2),"*10^-14 s"

The relaxation time of conducton of electrons is 3.97 *10^-14 s

Example 6.4, Page number 6.8

#Varaible declaration
R = 0.06             #resistance(ohms)
D = 5                #length of wire(m)
I = 15               #current(A)
p = 2.7*10**-8       #resistivity of aluminium(ohm-m)
Ma = 26.98           #atomic weight
Na = 6.025*10**26    #Avogadro's number
rho_s = 2.7*10**3    #sensity(kg/m^3)

#Calculations
#Since each free atom atom contains 3 electrons, therefore,
n = (3*rho_s*Na)/Ma

#For mobility
u = 1/(n*e*p)

#For drift velocity
E = (I*R)/D
vd = u*E

#Results
print "Free electron concentration =",round((n/1E+29),4),"*10^29 electrons/m^2"
print "Mobility of electrons =",round((u/1E-3),3),"*10^-3 m/V-s"
print "Drift velocity of electrons =",round((vd/1E-3),3),"*10^-3 m/s"

Free electron concentration = 1.8088 *10^29 electrons/m^2
Mobility of electrons = 1.278 *10^-3 m/V-s
Drift velocity of electrons = 0.23 *10^-3 m/s

Example 6.5, Page number 6.13

#Variable declaration
L = 0.1*10**-9    #length of each side of box(m)
h = 6.62*10**-34  #Planck's constant(J-s)
m = 9.1*10**-31   #mass of electron(kg)
#For lowest energy,
nx = 1
ny = 1
nz = 1

#Calculations
E1 = (((h**2)*(nx**2+ny**2+nz**2))/(8*m*L**2))//(1.6*10**-19)

#Result
print "The lowest energy of electron is",round(E1,2),"eV" 

The lowest energy of electron is 112.0 eV

Exmple 6.6, Page number 6.13

import math

#Variable declaration
'''Fermi equation
F(E) =       1
          ---------------
          1+exp((E-Ef)/kT)
Given, E-Ef = kT
therefore,
F(E) =    1
      --------
      1+exp(1)
'''

#Calculation
Fe = 1./(1.+math.exp(1.))

#Result
print "F(E) =",round(Fe,3)

F(E) = 0.269

Example 6.7, Page number 6.13

import math 

#Variable declaration
Fe = 10./100.   #probability
Ef = 5.5        #Fermi energy(eV)
k = 1.38*10**-23
 
#Calculations
'''Fermi equation
F(E) =       1
          ---------------
          1+exp((E-Ef)/kT)
'''
E = Ef+(Ef/100)
E_Ef = (E - Ef)*1.6*10**-19  #(J)

#Let x be E-Ef/k
x = E_Ef/k
T = x/math.log(-(1-(1/Fe)))

#Result
print "Temperature =",round(T,2),"K"

Temperature = 290.22 K

Example 6.8, Page number 6.16

import math 

#Variable declaration
Fe = 1./100.   #probability
Ef = 0.5       #Fermi energy(eV)
k = 1.38*10**-23
 
#Calculations
'''Fermi equation
F(E) =       1
          ---------------
          1+exp((E-Ef)/kT)
'''

E = Ef+0.5
E_Ef = (E - Ef)*1.6*10**-19  #(J)

#Let x be E-Ef/k
x = E_Ef/k
T = x/math.log(-(1-(1/Fe)))

#Result
print "Temperature =",round(T),"K"

Temperature = 1262.0 K