# Chapter 4 : Bonding the Band Theory of Solids and Statistics¶

## Example 4.6 Page No : 154¶

In :
import math

# Given
E_FO = 7.0          # in eV
e = 1.6*10**-19     # in coulombs
E_FO *= e           # in Joules
me = 9.1*10**-31    # in Kg

# Calculations and Results
v_f = math.sqrt(2*E_FO/me)
print("Speed of the conduction electrons in m/s is {0:.4g}".format(v_f))

Speed of the conduction electrons in m/s is 1.569e+06


## Example 4.7 Page No : 159¶

In :
# Given
e = 1.6 * 10 ** -19     # in coulombs
Eg = 1.1                # in eV
Eg *= e                 # in Joules
h = 6.6 * 10 ** -34     # in Js
c = 3 * 10 ** 8         # in m/s

# Calculations and Results
lambda_val = h * c / Eg
print("Wavelength of light that can be absorbed by an Si photo-detector"
" at Eg=1.1 eV in micro meter is {0:.4g}".format(lambda_val * 10 ** 6))
print("Hence the light of wavelength 1.31 micro meter and 1.55 micro meter will not "
"be absorbed by Si and thus cannot be detected by detector")

Wavelength of light that can be absorbed by an Si photo-detector at Eg=1.1 eV in micro meter is 1.125
Hence the light of wavelength 1.31 micro meter and 1.55 micro meter will not be absorbed by Si and thus cannot be detected by detector


## Example 4.8 Page No : 162¶

In :
import math

# Given
e = 1.6*10**-19     # in coulombs
h = 6.626*10**-34   # in Js
me = 9.1*10**-31    # in Kg
# let x=k*T
x = 0.026           # in eV
E = 5.0               # in ev
E *= e              # in Joules

# Calculations and Results
g_E = (8*math.pi*math.sqrt(2))*(me/h**2)**(3./2)*math.sqrt(E)  # in J**-1*m**-3
# conversion of units
g_E = g_E*10**-6*e  # in eV**-1 cm**-3
print("density of states at the center of the band in cm**-3*J**-1 is {0:.4g}".format(g_E))
# part(b)
n_E = g_E*x         # in cm**-3
print("No.of states per unit volume within kT about the center in cm**-3 is {0:.4g}".format(n_E))
# part(c)
# Given
E = 0.026           # in eV
E *= e              # in joules

# Calculations and Results
g_E = (8*math.pi*math.sqrt(2))*(me/h**2)**(3./2)*math.sqrt(E)  # in J**-1*m**-3
# conversion of units
g_E = g_E*10**-6*e  # in eV**-1 cm**-3
print("density of states at at kT above the band in cm**-3*J**-1 is {0:.4g}".format(g_E))
# part(d)
n_E = g_E*x         # in cm**-3
print(" No.of states per unit volume within kT about the center in cm**-3 is {0:.4g}".format(n_E))
# solved using the values taken from the solution of textbook

density of states at the center of the band in cm**-3*J**-1 is 1.518e+22
No.of states per unit volume within kT about the center in cm**-3 is 3.946e+20
density of states at at kT above the band in cm**-3*J**-1 is 1.095e+21
No.of states per unit volume within kT about the center in cm**-3 is 2.846e+19


## Example 4.9 Page No : 165¶

In :
import math

# Given
e = 1.6*10**-19     # in coulombs
h = 6.626*10**-34   # in Js
me = 9.1*10**-31    # in Kg
d = 10.5            # in g/cm
Mat = 107.9         # g/mol
NA = 6.023*10**23   # mol**-1
E_ctr = 5.0           # in ev
E_ctr *= e          # in Joules

# Calculations and Results
S_band = 2*(16*math.pi*math.sqrt(2)/3)*(me/h**2)**(3./2)*E_ctr**(3./2)  # in states m**-3
# conversion of units
S_band *= 10**-6    # in states cm**-3
print("No. of states in the band in states cm**-3 is {0:.4g}".format(S_band))
n_Ag = d*NA/Mat
print("No.of atoms per unit volume in silver in atoms per cm3 is {0:.4g}".format(n_Ag))

No. of states in the band in states cm**-3 is 1.012e+23
No.of atoms per unit volume in silver in atoms per cm3 is 5.861e+22


## Example 4.10 Page No : 169¶

In :
import math
# Given
e = 1.6*10**-19     # in coulombs
h = 6.626*10**-34   # in Js
me = 9.1*10**-31    # in Kg
d = 8.96            # in g/cm
Mat = 63.5          # g/ mol
NA = 6.023*10**23   # mol**-1
n = d*NA/Mat        # in cm**-3
n *= 10**6          # in m**-3

# Calculations and Results
E_FO = (h**2/(8*me))*(3*n/math.pi)**(2./3)   # in J
E_FO /= e           # in eV
print("Fermi energy at 0 Kelvin in eV is {0:.4f}".format(E_FO))
E_FO = (h**2/(8*me))*(3*n/math.pi)**(2./3)   # in J
v_e = math.sqrt(6*E_FO/(5*me))
print("Average speed of conduction electrons in m/s is {0:.4g}".format(v_e))

Fermi energy at 0 Kelvin in eV is 7.0653
Average speed of conduction electrons in m/s is 1.221e+06


## Example 4.11 Page No : 174¶

In :
# Given
e = 1.6*10**-19     # in coulombs
me = 9.1*10**-31    # in Kg
u_d = 43*10**-4     # in cm2/V/s
v_e = 1.22*10**6    # in m/s

# Calculations and Results
T = u_d*me/e
l_e = v_e*T
print("Mean free path of electrons in meters is {0:.4g}".format(l_e))

Mean free path of electrons in meters is 2.984e-08


## Example 4.13 Page No : 178¶

In :
import math
# Given
e = 1.6*10**-19   # in coulombs
T = 373.0         # in kelvin
To = 273.0        # in kelvin
k = 1.38*10**-23  # in m2 kg /k/s2
# from table 4.3
E_FAO = 11.6      # in eV
E_FAO *= e        # in J
x_A = 2.78
E_FBO = 7.01      # in eV
E_FBO *= e        # in J
x_B = -1.79

# Calculations and Results
# Mott jones Equation
V_AB = (-math.pi**2*k**2/(6*e))*((x_A/E_FAO)-(x_B/E_FBO))*(T**2-To**2)
print("EMF in micro volts available from Al and Cu thermocouple with the given respective"
" temperatures at the junctions is {0:.4f}".format(V_AB*10**6))

EMF in micro volts available from Al and Cu thermocouple with the given respective temperatures at the junctions is -391.2988


## Example 4.14 Page No : 182¶

In :
import math
# Given
phi = 2.6           # in eV
e = 1.6*10**-19     # in coulombs
phi *= e            # in Joules
Be = 3*10**4        # schottky coefficient in A/m2/K2
T = 1600.0            # in degree celsius
T += 273.0            # in Kelvin
k = 1.38*10**-23    # m2 kg s-2 K-1
d = 2*10**-3        # in m
l = 4*10**-2        # in in m

# Calculations and Results
# Richardson-Dushman Equation
J = Be*T**2*math.exp(-phi/(k*T))
A = math.pi*d*l
I = J*A
print("Saturation current in Amperes if the tube is operated at 1873 kelvin is {0:.4g}".format(I))

Saturation current in Amperes if the tube is operated at 1873 kelvin is 2.708