4: Dislocations and Crystal Structure Determination

Example number 4.1, Page number 66

In [1]:
#importing modules
import math
from __future__ import division

#Variable declaration
d=0.282*10**-9;    #lattice spacing(m)
theta=8+(35/60);   #glancing angle(degree)
n=1;   #order
Theta=90;    #angle(degree)

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

#Result
print "wavelength is",round(lamda*10**10,3),"angstrom"
print "answer varies due to rounding off errors"
print "maximum order of diffraction is",round(nmax)

wavelength is 0.842 angstrom
answer varies due to rounding off errors
maximum order of diffraction is 7.0

Example number 4.2, Page number 66

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

#Variable declaration
d=3.04*10**-10;    #lattice spacing(m)
n=3;   #order
lamda=0.79*10**-10;    #wavelength(m)

#Calculation
theta=math.asin(n*lamda/(2*d));     #glancing angle(radian)
theta=theta*180/math.pi;          #glancing angle(degrees)

#Result
print "glancing angle is",round(theta,3),"degrees"

glancing angle is 22.942 degrees

Example number 4.3, Page number 66

In [3]:
#importing modules
import math
from __future__ import division

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

#Calculation
d110=a/math.sqrt(h**2+k**2+l**2);     #spacing(m)
theta=math.asin(n*lamda/(2*d110));    #glancing angle(radian)
theta=theta*180/math.pi;          #glancing angle(degrees)

#Result
print "glancing angle is",round(theta,2),"degrees"
print "answer in the book is wrong"

glancing angle is 21.01 degrees
answer in the book is wrong

Example number 4.4, Page number 67

In [4]:
#importing modules
import math
from __future__ import division

#Variable declaration
n=1;   #order
lamda=3*10**-10;    #wavelength(m)
h=1;
k=0;
l=0;
theta=40;    #angle(degree)

#Calculation
theta=theta*math.pi/180;    #angle(radian)
d=n*lamda/(2*math.sin(theta));     #space of plane(m)
a=d*math.sqrt(h**2+k**2+l**2);     
V=a**3;      #volume of unit cell(m**3)

#Result
print "space of plane is",round(d*10**10,4),"angstrom"
print "volume of unit cell is",round(V*10**30,3),"*10**-30 m**3"
print "answer varies due to rounding off errors"

space of plane is 2.3336 angstrom
volume of unit cell is 12.708 *10**-30 m**3
answer varies due to rounding off errors

Example number 4.5, Page number 67

In [7]:
#importing modules
import math
from __future__ import division

#Variable declaration
a=3;    #lattice spacing(m)
n=1;   #order
lamda=0.82*10**-9;    #wavelength(m)
theta=75.86;    #angle(degree)

#Calculation
theta=theta*math.pi/180;    #angle(radian)
d=n*10**10*lamda/(2*math.sin(theta));    #spacing(angstrom)

#Result
print "spacing is",round(d,2),"angstrom"
print "answer in the book is wrong. hence the miller indices given in the book are also wrong and cannot be calculated"

spacing is 4.23 angstrom
answer in the book is wrong. hence the miller indices given in the book are also wrong and cannot be calculated

Example number 4.6, Page number 68

In [8]:
#importing modules
import math
from __future__ import division

#Variable declaration
e=1.6*10**-19;    #charge(c)
m=9.1*10**-31;    #mass(kg)
h=6.625*10**-34;   #plank constant
n=1;   #order
theta=9+(12/60)+(25/(60*60));    #angle(degree)
V=235.2;    #kinetic energy of electron(eV)

#Calculation
theta=theta*math.pi/180;    #angle(radian)
lamda=h*10**10/math.sqrt(2*m*e*V);   
d=n*lamda/(2*math.sin(theta));       #interplanar spacing(angstrom)

#Result
print "interplanar spacing is",round(d,3),"angstrom"
print "answer in the book is wrong"

interplanar spacing is 2.502 angstrom
answer in the book is wrong

Example number 4.7, Page number 68

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

#Variable declaration
n=1;   #order
h=1;
k=1;
l=1;
e=1.6*10**-19;    #charge(c)
theta=27.5;    #angle(degree)
H=6.625*10**-34;    #plancks constant
c=3*10**10;    #velocity of light(m)
a=5.63*10**-10;     #lattice constant(m)

#Calculation
theta=theta*math.pi/180;    #angle(radian)
d=a/math.sqrt(h**2+k**2+l**2);
lamda=2*d*math.sin(theta)/n;      #wavelength of Xray beam(m)
E=H*c/(e*lamda);           #energy of Xray beam(eV)         

#Result
print "wavelength of X-ray beam is",int(lamda*10**10),"angstrom"
print "energy of Xray beam is",round(E/10**5,2),"*10**5 eV"
print "answer in the book is wrong"

wavelength of X-ray beam is 3 angstrom
energy of Xray beam is 4.14 *10**5 eV
answer in the book is wrong

Example number 4.8, Page number 69

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

#Variable declaration
e=1.6*10**-19;    #charge(c)
theta=56;    #angle(degree)
V=854;    #voltage(V)
n=1;      #order of diffraction
m=9.1*10**-31;    #mass(kg)
h=6.625*10**-34;   #plank constant

#Calculation
theta=theta*math.pi/180;    #angle(radian)
lamda=h/math.sqrt(2*m*e*V);    #wavelength(m)
d=n*lamda/(2*math.sin(theta));     #spacing of crystal(m)

#Result
print "spacing of crystal is",round(d*10**10,3),"angstrom"

spacing of crystal is 0.253 angstrom

Example number 4.9, Page number 69

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

#Variable declaration
n=1;   #order
h=2;
k=0;
l=2;
theta=34;    #angle(degree)
lamda=1.5;   #wavelength(angstrom)

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

#Result
print "lattice parameter is",round(a,3),"angstrom"
print "answer in the book is wrong"

lattice parameter is 3.794 angstrom
answer in the book is wrong

Example number 4.10, Page number 70

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

#Variable declaration
n=1;   #order
h=1;
k=1;
l=1;
e=1.6*10**-19;    #charge(c)
V=5000;    #voltage(V)
m=9.1*10**-31;    #mass(kg)
H=6.625*10**-34;   #plank constant
d=0.204*10**-9;    #interplanar spacing(m)

#Calculation
lamda=H/math.sqrt(2*m*e*V);    #wavelength(m)
theta=math.asin(n*lamda/(2*d));    #bragg's angle(radian)
theta=theta*180/math.pi;    #bragg's angle(degree)

#Result
print "bragg's angle is",round(theta,4),"degrees"

bragg's angle is 2.4389 degrees