Chapter 5: Band Theory of Solids

Example 5.1,Page number 176

In [1]:
import math

#Given Data
h = 6.626*10**-34;  # Planck's constant, Js
h_bar = h/(2*math.pi);  # Reduced Planck's constant, Js
e = 1.6*10**-19;   # Energy equivalent of 1 eV, J/eV
m = 9.1*10**-31;   # Mass of an electron, kg

# For Na
n_Na = 2.65*10**28;    # electronic concentration of Na, per metre cube
k_F = (3*math.pi**2*n_Na)**(1.0/3);    # Fermi wave vector, per cm
E_F = h_bar**2*k_F**2/(2*m*e);  # Fermi energy of Na, eV
print"The fermi energy of Na = ",round(E_F,4),"eV";
print"The band structure value of Na = ",0.263*13.6,"eV";

# For K
n_K = 1.4e+28;  # electronic concentration of K, per metre cube
k_F = (3*math.pi**2*n_K)**(1.0/3);    # Fermi wave vector, per cm
E_F = h_bar**2*k_F**2/(2*m*e);  # Fermi energy of K, eV
print"The fermi energy of K = ",round(E_F,4),"eV";
print"The band structure value of K = ", 0.164*13.6,"eV";
print"The agreement between the free electron and band theoretical values are fairly good both for Na and K";

The fermi energy of Na =  3.2489 eV
The band structure value of Na =  3.5768 eV
The fermi energy of K =  2.1232 eV
The band structure value of K =  2.2304 eV
The agreement between the free electron and band theoretical values are fairly good both for Na and K

Example 5.3,Page number 177

In [2]:
import math

#Given Data

n_Na = 2.65*10**22;    # electronic concentration of Na, per cm cube
k_F = (3*math.pi**2*n_Na)**(1.0/3);    # Fermi wave vector, per cm
print"The fermi momentum of Na =","{0:.3e}".format(k_F),"per cm";

The fermi momentum of Na = 9.223e+07 per cm

Example 5.5,Page number 177

In [3]:
import math

#Given Data

h = 6.626*10**-34;  # Planck's constant, Js
h_bar = h/(2*math.pi);  # Reduced Planck's constant, Js
e = 1.6*10**-19;   # Energy equivalent of 1 eV, J/eV
m = 9.1*10**-31;   # Mass of an electron, kg
V = 1.0*10**-6; # Volume of unit cube of material, metre cube

# For Mg
E_F = 7.13*e;  # Fermi energy of Mg, J
s = 2*math.pi**2/(e*V)*(h_bar**2/(2*m))**(3.0/2)*(E_F)**(-1.0/2);  # Energy separation between levels for Mg, eV
print"The energy separation between adjacent levels for Mg = ","{0:.3e}".format(s),"eV";

# For Cs
E_F = 1.58*e;  # Fermi energy of Cs, J
s = 2*math.pi**2/(e*V)*(h_bar**2/(2*m))**(3.0/2)*(E_F)**(-1.0/2);  # Energy separation between levels for Cs, eV
print"The energy separation between adjacent levels for Cs =","{0:.3e}".format(s),"eV";

The energy separation between adjacent levels for Mg =  5.517e-23 eV
The energy separation between adjacent levels for Cs = 1.172e-22 eV

Example 5.9,Page number 180

In [4]:
import math

#Given Data

gamma_expt = 7.0*10**-4;   # Experimental value of electronic specific heat, cal/mol/K-square
gamma_theory = 3.6*10**-4;   # Theoretical value of electronic specific heat, cal/mol/K-square
L = (gamma_expt - gamma_theory)/gamma_theory;
print"The electron-phonon coupling constant of superconductor = ",round(L,2);

The electron-phonon coupling constant of superconductor =  0.94

Example 5.10,Page number 181

In [5]:
import math

#Given Data

N_Ef = 1.235;   # Density of states at fermi energy, electrons/atom-eV
N = 6.023*10**23;  # Avogadro's number
k = 1.38*10**-23;   # Boltzmann constant, J/mol/K
e = 1.6*10**-19;   # Charge on an electron, C
gama = math.pi**2*k**2/3*(N_Ef*N/e);   # Electronic specific heat coefficient, J/g-atom-kelvin square

print"The electronic specific heat coefficient of superconductor = ",round(gama*1000,4),"mJ/g-atom-kelvin square";

The electronic specific heat coefficient of superconductor =  2.9127 mJ/g-atom-kelvin square

Example 5.11,Page number 181

In [6]:
import math

#Given Data

gamma_expt = 4.84;   # Experimental value of electronic specific heat of metal, mJ/g-atom/K-square
gamma_theory = 2.991;   # Theoretical value of electronic specific heat of metal, mJ/g-atom/K-square
L = (gamma_expt-gamma_theory)/gamma_theory;
print"The electron-phonon coupling constant for metal = ",round(L,4);

The electron-phonon coupling constant for metal =  0.6182

Example 5.12,Page number 181

In [8]:
import math

#Given data
mu_B = 9.24*10**-27; # Bohr's magneton, J/T
N_Ef = 0.826;   # Density of states at fermi energy, electrons/atom-eV
N = 6.023*10**23;  # Avogadro's number
e = 1.6*10**-19;   # Energy equivalent of 1 eV, J
chi_Pauli = mu_B**2*N_Ef*N/e;
print"Pauli spin susceptibility of Mg = ","{0:.3e}".format( chi_Pauli*1000),"cgs units";

Pauli spin susceptibility of Mg =  2.655e-07 cgs units
In [ ]: