Chapter 2:Band Theory of Solids

Example 2.1,Page No:2.2

In [1]:
import math

#variable declaration
h   = 6.63*10**-34;          # plancks constant in J.s
m   = 9.1*10**-31;           # mass of electron in kg
a   = 2.5*10**-10;           # width of infinite square well
e   = 1.6*10**-19;           # charge of electron coulombs
n2  = 2;                     #number of  permiissable quantum
n3  = 3;                     #number of  permiissable quantum

# Calculations
E1  = (h**2)/float(8*m*a**2*e);      # first lowest permissable quantum energy  in eV
E2  = n2**2 *E1;                     # second lowest permissable quantum energy in eV
E3  = n3**2 *E1;                     # second lowest permissable quantum energy in eV

# Result
print'Lowest three permissable quantum energies are  E1 = %d'%E1,'eV';
print' E2 = %d'%E2,'eV';
print' E3 = %d'%E3,'eV';

Lowest three permissable quantum energies are  E1 = 6 eV
 E2 = 24 eV
 E3 = 54 eV

Example 2.2,Page No:2.4

In [2]:
import math

#variable declaration
h   = 6.63*10**-34;          # plancks constant in J.s
m   = 9.1*10**-31;           # mass of electron in kg
a   = 10**-10;               # width of infinite square well in m
e   = 1.6*10**-19;           # charge of electron in coulombs
n1  = 1;                     #energy level constant
n2  = 2;                     #energy level constant

# calculations
E1  = ((n1**2)*(h**2))/float(8*m*(a**2)*e);      # ground state energy in eV
E2  = ((n2**2)*(h**2))/float(8*m*(a**2)*e);      # first excited state in energy in eV
dE  = E2-E1                                      # difference between first excited and ground state(E2 - E1)

#Result
print'Energy Difference = %3.2f '%dE,'eV';

 

Energy Difference = 113.21  eV

Example 2.3,Page No:2.5

In [3]:
import math

# Variable declaration
h   = 6.63*10**-34;           # plancks constant in J.s
m   = 9.1*10**-31;            # mass of electron in kg
a   = 5*10**-10;              # width of infinite potential well in m
e   = 1.6*10**-19;            # charge of electron in coulombs
n1  = 1;                      # energy level constant
n2  = 2;                      # energy level constant
n3  = 3;                      # energy level constant

#Calculations
E1  = ((n1**2)*(h**2))/(8*m*(a**2)*e);      # first energy level in eV
E2  = ((n2**2)*(h**2))/(8*m*(a**2)*e);      # second energy level in eV
E3  = ((n3**2)*(h**2))/(8*m*(a**2)*e);      # third energy level in eV

# Result
print'First Three Energy levels are \n E1 = %3.2f'%E1,'eV';
print' E2 = %d'%E2,'eV';
print' E3 = %3.2f'%E3,'eV';
print'\n Above calculation shows that the energy of the bound electron cannot be continuous';

First Three Energy levels are 
 E1 = 1.51 eV
 E2 = 6 eV
 E3 = 13.59 eV

 Above calculation shows that the energy of the bound electron cannot be continuous

Example 2.4,Page No:2.5

In [4]:
import math

#variable declaration
h   = 1.054*10**-34;          #plancks constant in J.s
m   = 9.1*10**-31;            #mass of electron in kg
a   = 5*10**-10;              #width of infinite potential well in m
e   = 1.6*10**-19;            # charge of electron coulombs

# Calculations
#cos(ka) = ((Psin(alpha*a))/(alpha*a)) + cos(alpha*a)
#to find the lowest allowed energy bandwidth,we have to find the difference in αa values, as ka changes from 0 to π
# for ka = 0 in above eq becomes
# 1 = 10*sin(αa))/(αa)) + cos(αa)
# This gives αa = 2.628 rad
# ka = π , αa = π
# sqrt((2*m*E2)/h**2)*a = π

E2   = ((math.pi*math.pi)*h**2)/(2*m*a**2*e);        #energy in eV
E1   = ((2.628**2)*h**2)/(2*m*a**2*e);               #for αa = 2.628 rad energy in eV
dE   = E2 - E1;                                      #lowest energy bandwidth in eV

# Result
print'Lowest energy bandwidth = %3.3f'%dE,'eV';

Lowest energy bandwidth = 0.452 eV

Example 2.5,Page No:2.8

In [5]:
import math

# Variable declaration
a   = 3*10**-10;             # side of 2d square lattice in m
h   = 6.63*10**-34;          # plancks constant in J.s
e   = 1.6*10**-19            # charge of electron in coulombs
m   = 9.1*10**-31;           # mass of electron in kg

# calculations
#p   = h*k                                              # momentum of the electron
k   = math.pi/float(a);                                 # first Brillouin zone
p   = (h/float(2*math.pi))*(math.pi/float(a));          # momentum of electron
E   = (p**2)/float(2*m*e)                               # Energyin eV

#Result
print'Electron Momentum for first Brillouin zone appearance = %g'%p,'eV';
print'\n Energy of free electron with this momentum = %4.1f'%E,'eV';
print'\n Note: in Textbook Momentum value is wrongly printed as 1.1*10**-10';

Electron Momentum for first Brillouin zone appearance = 1.105e-24 eV

 Energy of free electron with this momentum =  4.2 eV

 Note: in Textbook Momentum value is wrongly printed as 1.1*10**-10