4: Defects in Crystals

Example number 1, Page number 4.14

In [36]:
#importing modules
import math
from __future__ import division

#Variable declaration
N=6.023*10**26;  #avagadro number
T1=1/float('inf');     #temperature 0K(K)
T2=300;
T3=900;         #temperature(K)
k=1.38*10**-23; #boltzmann constant 
deltaHv=120*10**3*10**3/N;    #enthalpy(J/vacancy)

#Calculation
#n1=N*math.exp(-deltaHv/(k*T1));    #equilibrium concentration of vacancy at 0K
#value of n1 cant be calculated in python, as the denominator is 0 and it shows float division error
n2=N*math.exp(-deltaHv/(k*T2));    #equilibrium concentration of vacancy at 300K 
n3=N*math.exp(-deltaHv/(k*T3));    #equilibrium concentration of vacancy at 900K 

#Result
#print "equilibrium concentration of vacancy at 0K is",n1
print "equilibrium concentration of vacancy at 300K is",round(n2/10**5,3),"*10**5"
print "equilibrium concentration of vacancy at 900K is",round(n3/10**19,3),"*10**19"

equilibrium concentration of vacancy at 300K is 7.577 *10**5
equilibrium concentration of vacancy at 900K is 6.502 *10**19

Example number 2, Page number 4.15

In [2]:
#importing modules
import math
from __future__ import division

#Variable declaration
nbyN1=1*10**-10;      #fraction of vacancies
T1=500+273;
T2=1000+273;

#Calculation
lnx=T1*math.log(nbyN1)/T2;
x=math.exp(lnx);      #fraction of vacancies at 1000

#Result
print "fraction of vacancies at 1000 is",round(x*10**7,1),"*10**-7"

fraction of vacancies at 1000 is 8.5 *10**-7

Example number 3, Page number 4.16

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

#Variable declaration
d=2.82*10**-10;    #interionic distance(m)
T=300;             #temperature(K)
k=1.38*10**-23;    #boltzmann constant 
e=1.6*10**-19;     #charge(coulomb)
n=4;               #number of molecules
deltaHs=1.971*e;   #enthalpy(J)

#Calculation
V=(2*d)**3;        #volume of unit cell(m**3)
N=n/V;             #number of ion pairs
x=deltaHs/(2*k*T);
n=N*math.exp(-x);    #concentration of schottky defects(per m**3)

#Result
print "concentration of schottky defects is",round(n*10**-11,2),"*10**11 per m**3"

concentration of schottky defects is 6.42 *10**11 per m**3

Example number 4, Page number 4.17

In [16]:
#importing modules
import math
from __future__ import division

#Variable declaration
N=6.026*10**23;    #avagadro number 
T=500;             #temperature(K)
k=1.38*10**-23;    #boltzmann constant 
deltaHv=1.6*10**-19;     #charge(coulomb)
V=5.55;            #molar volume(cm**3)
nv=5*10**7*10**6;  #number of vacancies

#Calculation
n=N*math.exp(-deltaHv/(k*T))/V;    #concentration of schottky defects(per m**3)
x=round(n/nv,4);            #amount of climb down by the dislocations(step)
xcm=2*x*10**-8;             #amount of climb down by the dislocations(cm)

#Result
print "concentration of schottky defects is",round(n/10**12,2),"*10**12 per cm**3"
print "amount of climb down by the dislocations is",x,"step or",xcm*10**8,"*10**-8 cm" 

concentration of schottky defects is 9.23 *10**12 per cm**3
amount of climb down by the dislocations is 0.1846 step or 0.3692 *10**-8 cm