Chapter 8.: Statics and Band theory of Solids

Example 8.1, page no-208

In [9]:
# Fermi Energy of metals

import math
# variable declaration
d_cu=8.96*10**3                         # density of cu
a_cu=63.55                              # Atomic weight of cu
d_z=7.14*10**3                          # density of Zn     
a_z=65.38                               # Atomic weight of Zn
d_al=2700                               # density of Al
a_al=27                                 # Atomic weight of Al     
avg=6.022*10**26                        # Avogadro's number    
h=6.626*10**-34                         # Planck's constant
m=9.1*10**-31                           # mass of an electrons
e=1.6*10**-19                           # charge of an electron



#(i)

# Calculations
n_cu=d_cu*avg/a_cu
e_cu=(h**2/(8*m))*(3*n_cu/math.pi)**(2.0/3.0)
e_cu=e_cu/e

#Result
print("\n(i)For Cu\nThe electron concentration in Cu is %.4f*10^28 per m^3\nFermi energy at 0 k =%.4f eV "%(n_cu*10**-28,e_cu))

#(ii)

# calculations
n_z=d_z*avg*2/a_z
e_z=(h**2/(8*m))*(3*n_z/math.pi)**(2.0/3.0)
e_z=e_z/e

# Result
print("\n(ii)For Zn\nThe electron concentration in Zn is %.5f*10^28 per m^3\nFermi energy at 0 k =%.2f eV "%(n_z*10**-28,e_z))

#(iii)

# Calculations
n_al=d_al*avg*3/a_al
e_al=(h**2/(8*m))*(3*n_al/math.pi)**(2.0/3.0)
e_al=e_al/e

#Result
print("\n(iii)For Al\nThe electron concentration in Al is %.3f*10^28 per m^3\nFermi energy at 0 k =%.2f eV "%(n_al*10**-28,e_al))

(i)For Cu
The electron concentration in Cu is 8.4905*10^28 per m^3
Fermi energy at 0 k =7.0608 eV 

(ii)For Zn
The electron concentration in Zn is 13.15298*10^28 per m^3
Fermi energy at 0 k =9.45 eV 

(iii)For Al
The electron concentration in Al is 18.066*10^28 per m^3
Fermi energy at 0 k =11.68 eV

Example 8.2, page no-210

In [11]:
# Density of states for Cu

import math
# variable declaration
avg=6.023*10**26                            # avogadro's number
h=6.626*10**-34                             # Planck's constant 
m=9.1*10**-31                               # mass of an electron
e=1.6*10**-19                               # charge of an electron
n=8.4905*10**28                             # sphere of radius
gam=6.82*10**27                             # gamma

# Calculations
ef=(h**2/(8*m))*(3*n/math.pi)**(2.0/3.0)
ef=ef/e
x=(gam*math.sqrt(ef))/2

#Result
print("The density of states for Cu at the Fermi level for T = 0 K is %.0f*10^27 m^-3"%(x*10**-27))

The density of states for Cu at the Fermi level for T = 0 K is 9*10^27 m^-3

Example 8.3, page no-210

In [13]:
# Nordheims coeeficient

import math
#Variable declaration
rni=63                            # Resistivity of Ni
rcr=129                           # Resistivity of Cr
k=1120                            # Resistivity of 80% Ni + 20% Cr

#Calculations
c=(k*10**-9)/(0.8*(1-0.8))

#Result
print("The Nordheims coeeficient is %.0f *10^-6 Ohm-m"%(c*10**6))

The Nordheims coeeficient is 7 *10^-6 Ohm-m

Example 8.4, page no-211

In [15]:
# Conductivity of Al

import math
#Variable declaaration
d=2700                            # Density of Al
awt=27                            # Atomic weight
t=10**-14                         # Relaxation time
e=1.6*10**-19                     # charge of an electron
m=9.1*10**-31                     # mass of an electron
avg=6.022*10**26                  # Avogadros number

# calculation
n=avg*d*3/awt
sig=(n*t*e**2)/m

#Result
print("The conductivity of Al is %.4f*10^7 ohm-m."%(sig*10**-7))

The conductivity of Al is 5.0823*10^7 ohm-m.

Example 8.5, page no-211

In [18]:
#Fermi distribution function

import math
#variable declaration
e1=0.01                   # difference between energy level to fermi level in eV
e=1.6*10**-19             # charge of an electron
ed=e*e1                   # difference between energy level to fermi level in J
T=200                     # Temperature
k=1.38*10**-23            # Boltzmann's constant

# Calculations
E=1/(1+math.e**(ed/(T*k)))
print("The Fermi distribution function for energy E is %.4f"%E)

The Fermi distribution function for energy E is 0.3590

Example 8.6, page no-212

In [25]:
# Fermi energy and fermi temperature

import math
#variable declaration
v=0.86*10**6                      # velocity of electron
m=9.11*10**-31                     # mass of electron
e=1.6*10**-19                     # electronic charge   
k=1.38*10**-23                    # Boltzmann's constant 

#calculations
E=(m*v**2)/2
E= math.floor(E*10**22)/10**22
T=E/k

#Result
print("\nThe fermi energy is %.3f*10^-19 J\nThe Fermi Temperature Tf is %.2f*10^4 K"%(E*10**19,T*10**-4))
# answer in the book for Temperature id 2.43 x 10^4

The fermi energy is 3.368*10^-19 J
The Fermi Temperature Tf is 2.44*10^4 K

Example 8.7, page no-212

In [27]:
# No of states lying between energy levels

import math
# variable declaration
m=9.1*10**-31                            # mass of electron
dE=0.01                                  # energy interval
h=6.63*10**-34                           # planck's constant
eF=3.0                                   # Fermi energy
e=1.6*10**-19                            # electronic charge

#Calculations
E1=eF*e
E2=E1+e*dE
n=(4*math.pi*(2*m)**(1.5))/h**3
k=((2*0.3523/3)*((E2**(1.5)-(E1**(1.5)))))
n=n*k

#Result
print("The number of states lying between the energy level is %.2f*10^25"%(n*10**-25))

The number of states lying between the energy level is 4.14*10^25

Example 8.8, page no-214

In [29]:
#Fermi Velocity

import math
#Variable declaration
Tf=24600                       # Fermi temperature of the metal
m=9.11*10**-31                 # mass of electron
k=1.38*10**-23                 # Boltzmann's constant

#Calculations
vf=math.sqrt(2*k*Tf/m)

#Result
print("The Fermi Velocity is %.4f *10^6 m/s"%(vf*10**-6))

The Fermi Velocity is 0.8633 *10^6 m/s

Example 8.9, page no-214

In [40]:
#Fermi energy

import math
#variable declaration
n=18.1*10**28                   # elecron density of electron
h=6.62*10**-34                  # Planck's constant
m=9.1*10**-31                   # mass of an electron
e=1.6*10**-19                   # electronic charge

#calculations
ef=((3*n/(8*math.pi))**(2.0/3.0))*((h**2)/(2*m))
ef=ef/e
ef=math.ceil(ef*100)/100

#Result
print("The Fermi energy at 0 K is %.2f eV "%(ef))

The Fermi energy at 0 K is 11.68 eV

Example 8.10, page no-215

In [45]:
#Fermi energy 

import math
#variable declaration
n=18.1*10**28                # elecron density of electron
h=6.62*10**-34               # Planck's constant
m=9.1*10**-31                # mass of an electron
e=1.6*10**-19                # electronic charge

#calculations
ef=((3*n/(8*math.pi))**(2.0/3.0))*((h**2)/(2*m))
ef=ef/e
ef=math.ceil(ef*100)/100

#result
print("The Fermi energy at 0 K is %.2f eV "%ef)

The Fermi energy at 0 K is 11.68 eV

Example 8.11, page no-215

In [49]:
# Temperature calculation

import math
#variable declaration
e=1.6*10**-19             # electronic charge
Ed=0.5*e                  # difference between energy level to fermi level
k=1.38*10**-23            # Boltzmann's constant
x=0.01                    # probability

#Calculaations
T=Ed/(k*math.log((1/x)-1))

#Result
print("Temperature at which there is 1%% probability that a state with 0.5 eV energy occupied above the Fermi energy level is %.1f K"%T)
#answer is not matching with the answer given in the book

Temperature at which there is 1% probability that a state with 0.5 eV energy occupied above the Fermi energy level is 1261.6 K

Example 8.14, page no-218

In [53]:
#energies for the occupying of electrons
import math

#variable declaration
ef=2.1                      # Fermi energy
k=1.38*10**-23              # Boltzmann's constant
T=300                       # Temperature
e=1.6*10**-19               # Electronic charge

#calculations

#(i)
p1=0.99                     # probability
E1=ef+(k*T*math.log(-1+1/p1))/e

#(ii)
p2=0.01                     # probability
E2=ef+(k*T*math.log(-1+1/p2))/e

#(iii)
p3=0.5                     # probability
E3=ef+(k*T*math.log(-1+1/p3))/e

#Result

print("\nThe energies for the occupying of electrons at %d K for the probability of %.2f are %.2f eV"%(T,p1,E1))

print("\nThe energies for the occupying of electrons at %d K for the probability of %.2f are %.2f eV"%(T,p2,E2))

print("\nThe energies for the occupying of electrons at %d K for the probability of %.2f are %.2f eV"%(T,p3,E3))

The energies for the occupying of electrons at 300 K for the probability of 0.99 are 1.98 eV

The energies for the occupying of electrons at 300 K for the probability of 0.01 are 2.22 eV

The energies for the occupying of electrons at 300 K for the probability of 0.50 are 2.10 eV

Example 8.15, page no-219

In [4]:
# Fermi distribution function

import math
# Variable declarations
e=1.6*10**-19                       # Electronic charge
ed=0.02*e                           # difference between energy level to fermi level
T1=200                              # Temperature 1
T2=400                              # Temperature 2
k=1.38*10**-23                      # Boltzmann's constant

#Calculations
fe1=1/(1+math.e**(ed/(k*T1)))
fe2=1/(1+math.e**(ed/(k*T2)))

#Result
print("\nThe Fermi distribution function for the given energy at %d K is %.5f"%(T1,fe1))
print("\nThe Fermi distribution function for the given energy at %d K is %.4f"%(T2,fe2))

The Fermi distribution function for the given energy at 200 K is 0.23877

The Fermi distribution function for the given energy at 400 K is 0.3590

Example 8.16, page no-220

In [7]:
# Fermi energy for given metal

import math
#Variaable declaration
d=10500                           # Density of the metal
avg=6.022*10**26                  # Avogadro's number
awt=107.9                         # Atomic weight of metal
h=6.62*10**-34                    # Planck's constant
m=9.1*10**-31                     # mass of an electron
e=1.6*10**-19                     # electronic charge

#Calculattions
n=d*avg/awt
ef=((3*n/(8*math.pi))**(2.0/3.0))*((h**2)/(2*m))
ef=ef/e

#Result
print("The Fermi energy for given metal is %.1f eV "%ef)

The Fermi energy for given metal is 5.5 eV

Example 8.17, page no-221

In [9]:
# Fermi distribution function 

import math
#Variable declaration
e=1.6*10**-19                         # electronic charge
ed=0.2*e                              # difference between energy level to Fermi level
T1=300                                # Temperature 1
T2=1000                               # Temperature 2
k=1.38*10**-23                        # Boltzmann's constant

#Calculations
fe1=1/(1+math.e**(ed/(k*T1)))
fe2=1/(1+math.e**(ed/(k*T2)))

#Result
print("\nThe Fermi distribution function for the given energy at %d K is %.7f"%(T1,fe1))
print("\nThe Fermi distribution function for the given energy at %d K is %.4f"%(T2,fe2))
# Answer for 300 K is wrong in the book

The Fermi distribution function for the given energy at 300 K is 0.0004395

The Fermi distribution function for the given energy at 1000 K is 0.0896

Example 8.18, page no-221

In [12]:
# Free electrons concentration

import math
#Variable declarations
h=6.62*10**-34                         # Planck's constant
m=9.1*10**-31                          # Mass of electron
e=1.6*10**-19                          # Charge of an electron
ef=3*e                                 # Fermi Energy

#Calculations
k=((3/(8*math.pi))**(2.0/3.0))*((h**2)/(2*m))
k=ef/k
n=k**(1.5)

#Result
print("The number of free electrons concentration in metal is %.2f *10^28 per cubic meter "%(n*10**-28))

The number of free electrons concentration in metal is 2.36 *10^28 per cubic meter

Example 8.19, page no-221

In [31]:
# Free electrons concentration in metal  

import math
#Variable declaration
h=6.626*10**-34                     # Planck's constant
m=9.1*10**-31                       # Mass of electron
e=1.6*10**-19                       # Charge of electron
ef=5.5*e                            # Fermi energy

# Calculation
k=((3/(8*math.pi))**(2.0/3.0))*((h**2)/(2*m))
k=ef/k
n=k**(1.5)

#Result
print("The number of free electrons concentration in metal is %.3f * 10^28 per cubic meter "%(n*10**-28))
#Answer is matching with the answer given in the book

The number of free electrons concentration in metal is 5.837 * 10^28 per cubic meter

Example 8.20, page no-221

In [41]:
# electrons concentration and termal velocity of electrons

import math
#variable declaration
h=6.626*10**-34                             # Planck's constant
m=9.1*10**-31                              # mass of electron
e=1.6*10**-19                              # charge of electron
ef=7*e                                     # Fermi energy

#calculations
k=((3/(8*math.pi))**(2.0/3.0))*((h**2)/(2*m))
k=ef/k
n=k**(1.5)
vth=math.sqrt(2*ef/m)

#Result
print("The number of free electrons concentration in metal is %.2f *10^28 per cubic meter "%(math.ceil(n*10**-28*10**2)/10**2))
print("\nThe termal velocity of electrons in copper is %.3f *10^6 m/s"%(math.floor(vth*10**-6*10**3)/10**3))

The number of free electrons concentration in metal is 8.39 *10^28 per cubic meter 

The termal velocity of electrons in copper is 1.568 *10^6 m/s