Chapter 4:Defects in Solids¶

Example 4.1, Page number 4.6¶

In :
import math

#Variable declaration
k = 1.38*10**-23   #Boltzmann constant(eV/K)
e = 1.6*10**-19    #Electronic charge(C)
T1 = 500           #First temperature for metal(K)
T2 = 1000          #Second temperature for metal(K)
Ev = 1             #Average energy required to create a vacancy in metal(eV)

#Calculations
x = k/e
#n_500 = N*exp(-Ev/T1*k)  ---(1)
#n_1000 = N*exp(-Ev/T2*k) ---(2)
#Dividing (1) by (2), we get the following expression
n = math.exp(Ev/(T2*x))

#Result
print "Ratio of vacancies=",round((n/1E+5),3),"*10^5"
Ratio of vacancies= 1.085 *10^5

Example 4.2, Page number 4.7¶

In :
import math

#Variable declaration
n1_by_N = 1.*10**-10  #frequency of vacancy sites at 500 C
T1 = 500.+273.        #K
T2 = 1000.+273.       #K

#Calculations
x = math.exp((T1/T2)*math.log(n1_by_N))

#Result
print "Frequency of vacancy sites at 1000 C =",round((x/1E-7),4),"*10^-7"
Frequency of vacancy sites at 1000 C = 8.467 *10^-7

Example 4.3, Page number 4.9¶

In :
#Variable declaration
r = 2.82*10**-10  #interionic distance(m)
n = 5*10**11      #density of Schottky defect(per m^3)
T = 25+273        #temperature(K)
k = 8.625*10**-5  #Boltzmann constant(/K)

#Calculations
v = (2*r)**3       #volume of one unit cell(m^3)
N = 4/v           #density of ion pairs
Es = 2*k*T*2.303*math.log10(N/n)

#Result
print "The average energy required for creation of one Schottky defect is",round(Es,3),"eV"
The average energy required for creation of one Schottky defect is 1.971 eV

Example 4.4, Page number 4.11¶

In :
#Variable declaration
T1 = 20+273  #K
T2 = 300+273 #K
Ef = 1.4     #average energy for creating a Freknel defect(eV)
k = 8.625*10**-5  #Boltzmann constant(J/K)
N = 1             #For simplicity assume total number of metal ions to be unity
Ni = 1            #For simplicity assume total number of metal ions to be unity

#Calculations
n1 = (N*Ni)**0.5*math.exp(-Ef/(2*k*T1))
n2 = (N*Ni)**0.5*math.exp(-Ef/(2*k*T2))
x = n1/n2

#Result
print "The ratio of the number of Frenkel defects is",round((x/1E-6),2),"*10^-6 or",round(((1/x)/1E+5),2),"*10^5"
The ratio of the number of Frenkel defects is 1.32 *10^-6 or 7.56 *10^5