3: Crystal planes, X-ray diffraction and defects in solids

Example number 3.1, Page number 3.19

In [13]:
#importing modules
import math
from __future__ import division

#Variable declaration
lamda=0.071*10**-9;     #wavelength(m)
a=0.28*10**-9;    #lattice constant(m)
h=1;
k=1;
l=0;
n=2;    #order of diffraction

#Calculation
d=a/math.sqrt(h**2+k**2+l**2);
x=n*lamda/(2*d);     
theta=math.asin(x);     #angle(radian)
theta=theta*180/math.pi;    #glancing angle(degrees)

#Result
print "glancing angle is",int(theta),"degrees"

glancing angle is 21 degrees

Example number 3.2, Page number 3.19

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

#Variable declaration
n=1;    #order of diffraction
theta1=8+(35/60);    #angle(degrees)
d=0.282;     #spacing(nm)
theta2=90;

#Calculation
theta1=theta1*math.pi/180;    #angle(radian)
lamda=2*d*math.sin(theta1)/n;    #wavelength(nm)
theta2=theta2*math.pi/180;    #angle(radian)
nmax=2*d/lamda;     #maximum order of diffraction

#Result
print "wavelength is",round(lamda,4),"nm"
print "maximum order of diffraction is",round(nmax)

wavelength is 0.0842 nm
maximum order of diffraction is 7.0

Example number 3.3, Page number 3.20

In [15]:
#importing modules
import math
from __future__ import division

#Variable declaration
T1=500+273;    #temperature(K)
T2=1000+273;   #temperature(K)
f=1*10**-10;   #fraction

#Calculation
x=round(T1/T2,5);
y=round(math.log(f),3);
w=round(x*y,3);
F=math.exp(w);    #fraction of vacancy sites

#Result
print "fraction of vacancy sites is",round(F*10**7,3),"*10**-7"

fraction of vacancy sites is 8.466 *10**-7

Example number 3.4, Page number 3.21

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

#Variable declaration
a=1;    #assume
h1=1;
k1=0;
l1=0;
h2=1;
k2=1;
l2=0;
h3=1;
k3=1;
l3=1;

#Calculation
d100=a*6/(h1**2+k1**2+l1**2);
d110=a*6/(h2**2+k2**2+l2**2);
d111=a*(6)/(h3**2+k3**2+l3**2);

#Result
print "ratio is math.sqrt(",d100,"): math.sqrt(",d110,"): math.sqrt(",d111,")"

ratio is math.sqrt( 6.0 ): math.sqrt( 3.0 ): math.sqrt( 2.0 )

Example number 3.5, Page number 3.21

In [17]:
#importing modules
import math
from __future__ import division

#Variable declaration
n=1;    #order of diffraction
theta=38.2;    #angle(degrees)
lamda=1.54;    #wavelength(angstrom)
h=2;
k=2;
l=0;

#Calculation
theta=theta*math.pi/180;    #angle(radian)
d=n*lamda/(2*math.sin(theta));
a=d*math.sqrt(h**2+k**2+l**2);    #lattice parameter of nickel(angstrom)

#Result
print "lattice parameter of nickel is",round(a,3),"angstrom"

lattice parameter of nickel is 3.522 angstrom

Example number 3.6, Page number 3.22

In [18]:
#importing modules
import math
from __future__ import division

#Variable declaration
theta=90;    #angle(degrees)
lamda=1.5;    #wavelength(angstrom)
d=1.6;    #spacing(angstrom)

#Calculation
theta=theta*math.pi/180;    #angle(radian)
n=2*d*math.sin(theta)/lamda;    #order of diffraction

#Result
print "order of diffraction is",int(n)

order of diffraction is 2

Example number 3.7, Page number 3.22

In [19]:
#importing modules
import math
from __future__ import division

#Variable declaration
h=1;
k=1;
l=0;
d=0.203*10**-9;    #spacing(m)

#Calculation
a=d*math.sqrt(h**2+k**2+l**2);    #length of unit cell(m)
V=a**3;    #volume of unit cell(m**3)
r=math.sqrt(3)*a/4;    #radius of the atom(m)

#Result
print "length of unit cell is",round(a*10**9,3),"*10**-9 m"
print "volume of unit cell is",round(V*10**27,5),"*10**-27 m**3"
print "radius of the atom is",round(r*10**9,4),"*10**-9 m"

length of unit cell is 0.287 *10**-9 m
volume of unit cell is 0.02366 *10**-27 m**3
radius of the atom is 0.1243 *10**-9 m

Example number 3.8, Page number 3.22

In [20]:
#importing modules
import math
from __future__ import division

#Variable declaration
theta=90;    #angle(degrees)
lamda=1.5;    #wavelength(angstrom)
d=1.6;    #spacing(angstrom)

#Calculation
theta=theta*math.pi/180;    #angle(radian)
n=2*d*math.sin(theta)/lamda;    #order of diffraction

#Result
print "order of diffraction is",int(n)

order of diffraction is 2

Example number 3.9, Page number 3.23

In [21]:
#importing modules
import math
from __future__ import division

#Variable declaration
lamda=0.065;    #wavelength(nm)
a=0.26;      #edge length(nm)
h=1;
k=1;
l=0;
n=2;

#Calculation
d=a/math.sqrt(h**2+k**2+l**2);      
x=n*lamda/(2*d);     
theta=math.asin(x);        #glancing angle(radian)
theta=theta*180/math.pi;    #glancing angle(degrees)
theta_d=int(theta);       
theta_m=(theta-theta_d)*60;
theta_s=(theta_m-int(theta_m))*60;

#Result
print "glancing angle is",theta_d,"degrees",int(theta_m),"minutes",int(theta_s),"seconds"
print "answer varies due to rounding off errors"

glancing angle is 20 degrees 42 minutes 17 seconds
answer varies due to rounding off errors

Example number 3.10, Page number 3.23

In [22]:
#importing modules
import math
from __future__ import division

#Variable declaration
lamda=1.54;    #wavelength(angstrom)
h=1;
k=1;
l=1;
n=1;
theta=19.2;    #angle(degrees)

#Calculation
theta=theta*math.pi/180;    #angle(radian)
d=n*lamda/(2*math.sin(theta));     
a=d*math.sqrt(h**2+k**2+l**2);      #cube edge of unit cell(angstrom)

#Result
print "cube edge of unit cell is",round(a,3),"angstrom"

cube edge of unit cell is 4.055 angstrom

Example number 3.11, Page number 3.24

In [23]:
#importing modules
import math
from __future__ import division

#Variable declaration
lamda=1.54;    #wavelength(angstrom)
h=2;
k=2;
l=0;
n=1;
theta=38.2;    #angle(degrees)

#Calculation
theta=theta*math.pi/180;    #angle(radian)
d=n*lamda/(2*math.sin(theta));     
a=d*math.sqrt(h**2+k**2+l**2);    #lattice parameter of nickel(angstrom)

#Result
print "lattice parameter of nickel is",round(a,3),"angstrom"

lattice parameter of nickel is 3.522 angstrom

Example number 3.12, Page number 3.24

In [24]:
#importing modules
import math
from __future__ import division

#Variable declaration
a=0.36;      #edge length(nm)
h1=1;
k1=1;
l1=1;
h2=3;
k2=2;
l2=1;

#Calculation
d1=a/math.sqrt(h1**2+k1**2+l1**2);    #interplanar spacing for (111)(nm)
d2=a/math.sqrt(h2**2+k2**2+l2**2);    #interplanar spacing for (321)(nm)

#Result
print "interplanar spacing for (111) is",round(d1,3),"nm"
print "interplanar spacing for (321) is",round(d2,3),"nm"

interplanar spacing for (111) is 0.208 nm
interplanar spacing for (321) is 0.096 nm

Example number 3.13, Page number 3.25

In [25]:
#importing modules
import math
from __future__ import division

#Variable declaration
lamda=0.675;    #wavelength(angstrom)
n=3;    #order of diffraction
theta=5+(25/60);    #angle(degrees)

#Calculation
theta=theta*math.pi/180;    #angle(radian)
d=lamda/(2*math.sin(theta));   
theta3=math.asin(3*lamda/(2*d));    #glancing angle(radian)
theta3=theta3*180/math.pi;    #glancing angle(degrees)
theta_d=int(theta3);       
theta_m=(theta3-theta_d)*60;

#Result
print "glancing angle is",theta_d,"degrees",int(theta_m),"minutes"
print "answer varies due to rounding off errors"

glancing angle is 16 degrees 27 minutes
answer varies due to rounding off errors

Example number 3.14, Page number 3.25

In [26]:
#importing modules
import math
from __future__ import division

#Variable declaration
lamda=0.79;    #wavelength(angstrom)
n=3;    #order of diffraction
d=3.04;    #spacing(angstrom)

#Calculation
x=round(n*lamda/(2*d),4);
theta=math.asin(x);         #glancing angle(radian)
theta=theta*180/math.pi;    #glancing angle(degrees)
theta_d=int(theta);       
theta_m=(theta-theta_d)*60;
theta_s=(theta_m-int(theta_m))*60;

#Result
print "glancing angle is",theta_d,"degrees",int(theta_m),"minutes",int(theta_s),"seconds"
print "answer given in the book is wrong"

glancing angle is 22 degrees 56 minutes 31 seconds
answer given in the book is wrong