# 4: Defects in Crystals¶

## Example number 4.1, Page number 4.5¶

In [5]:
#importing modules
import math
from __future__ import division

#Variable declaration
Ev=1;
k=1.38*10**-23;     #boltzmann constant(J/K)
e=1.6*10**-19;      #charge(eV)

#Calculation
r=Ev/(2.303*1000*k/e);
n=10**r;                #ratio of n1000/n500

#Result
print "ratio of vacancies is",round(n/10**5,3),"*10**5"

ratio of vacancies is 1.082 *10**5


## Example number 4.2, Page number 4.5¶

In [9]:
#importing modules
import math
from __future__ import division

#Variable declaration
Ev=1.2;
k=1.38*10**-23;     #boltzmann constant(J/K)
e=1.6*10**-19;      #charge(eV)
T1=350;    #temperature(K)
T2=500;    #temperature(K)

#Calculation
x1=Ev/(2.303*k*T1/e);
n1=1/(10**x1);     #number of vacancies per atom at 350K
x2=Ev/(2.303*k*T2/e);
n2=1/(10**x2);     #number of vacancies per atom at 500K

#Result
print "number of vacancies per atom at 350K is",round(n1*10**17,4),"*10**-17"
print "number of vacancies per atom at 500K is",round(n2*10**12,3),"*10**-12"

number of vacancies per atom at 350K is 0.5486 *10**-17
number of vacancies per atom at 500K is 0.827 *10**-12


## Example number 4.3, Page number 4.7¶

In [14]:
#importing modules
import math
from __future__ import division

#Variable declaration
d=2.82*10**-10;    #distance(m)
k=1.38*10**-23;     #boltzmann constant(J/K)
e=1.6*10**-19;      #charge(eV)
T=273+25;    #temperature(K)
sd=5*10**11;    #schotky defects(per m**3)

#Calculation
V=(2*d)**3;    #volume of unit cell(m**3)
N=4/V;    #density of ion pairs
x=round(math.log10(N/sd),2);
Es=2*(k/e)*T*2.303*x;       #average energy required(eV)

#Result
print "average energy required is",round(Es,3),"eV"

average energy required is 1.971 eV


## Example number 4.4, Page number 4.8¶

In [10]:
#importing modules
import math
from __future__ import division

#Variable declaration
T1=273+25;      #temperature(K)
T2=273+350;     #temperature(K)
Ef=1.35;     #energy(eV)
k=8.625*10**-5;

#Calculation
x=(Ef/k)*((1/(2*T1))-(1/(2*T2)))/2.303;
r=1/(10**round(x,3));      #ratio of Frenkel defects

#Result
print "ratio of Frenkel defects is",round(r*10**6,3),"*10**-6"

ratio of Frenkel defects is 1.125 *10**-6