12: X-ray Diffraction

Example number 1, Page number 323

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

#Variable declaration  
n=1;         #order
d=2.82*10**-10;          #spacing(m)
theta=10*math.pi/180;    #angle of diffraction(radian)

#Calculations
lamda=2*d*math.sin(theta)/n;    #wavelength of X-rays(m)

#Result
print "wavelength of X-rays is",round(lamda*10**10,5),"angstrom"

wavelength of X-rays is 0.97938 angstrom

Example number 2, Page number 323

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

#Variable declaration  
n=1;         #order
d=3.035*10**-10;          #spacing(m)
theta=12*math.pi/180;    #angle of diffraction(radian)

#Calculations
lamda=2*d*math.sin(theta)/n;    #wavelength of X-rays(m)

#Result
print "wavelength of X-rays is",round(lamda*10**10,3),"angstrom"

wavelength of X-rays is 1.262 angstrom

Example number 3, Page number 323

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

#Variable declaration  
n=1;         #order
d=2.81;          #spacing(angstrom)
theta1=15.1*math.pi/180;    #angle of diffraction(radian)
theta2=17.1*math.pi/180;    #angle of diffraction(radian)

#Calculations
lamda1=2*d*math.sin(theta1)/n;    #wavelength(angstrom)
lamda2=2*d*math.sin(theta2)/n;    #wavelength(angstrom)

#Result
print "their wavelengths are",round(lamda1,3),"angstrom and",round(lamda2,4),"angstrom"
print "answer in the book varies due to rounding off errors"

their wavelengths are 1.464 angstrom and 1.6525 angstrom
answer in the book varies due to rounding off errors

Example number 4, Page number 324

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

#Variable declaration  
n=1;         #order
lamda=1.54;          #wavelength of X-rays(angstrom)
theta=11*math.pi/180;    #angle of diffraction(radian)

#Calculations
d=lamda/(2*math.sin(theta));       #spacing(angstrom)

#Result
print "spacing is",round(d,3),"angstrom"

spacing is 4.035 angstrom

Example number 5, Page number 324

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

#Variable declaration  
n1=1;         #order
n2=3;         #order
theta1=30*math.pi/180;    #angle of diffraction(radian)
theta2=60*math.pi/180;    #angle of diffraction(radian)
lamdaB=0.92;    #wavelength(angstrom)

#Calculations
lamdaA=n2*lamdaB*math.sin(theta1)/math.sin(theta2);    #wavelength of line A(angstrom)

#Result
print "wavelength of line A is",round(lamdaA,3),"angstrom"
print "answer given in the book is wrong"

wavelength of line A is 1.593 angstrom
answer given in the book is wrong

Example number 6, Page number 324

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

#Variable declaration  
n=1;         #order
d=0.4086*10**-10;          #spacing(m)
theta=65*math.pi/180;      #angle of diffraction(radian)
m=9.1*10**-31;             #mass(kg)
h=6.6*10**-34;             #planks constant(Js)

#Calculations
lamda=2*d*math.sin(theta)/n;    #wavelength of electrons(m)
v=h/(m*lamda);                  #velocity of electrons(m/sec)

#Result
print "wavelength of electrons is",round(lamda*10**10,4),"*10**-10 m"
print "velocity of electrons is",round(v/10**6,3),"*10**6 m/sec"
print "answers given in the book are wrong"

wavelength of electrons is 0.7406 *10**-10 m
velocity of electrons is 9.793 *10**6 m/sec
answers given in the book are wrong

Example number 7, Page number 325

In [28]:
#importing modules
import math
from __future__ import division

#Variable declaration  
n=1;         #order
d=2.82*10**-10;          #spacing(m)
sintheta=1;              #angle of diffraction(radian)

#Calculations
lamdamax=2*d*sintheta/n;    #longest wavelength(m)

#Result
print "longest wavelength is",lamdamax*10**10,"angstrom"

longest wavelength is 5.64 angstrom

Example number 8, Page number 325

In [33]:
#importing modules
import math
from __future__ import division

#Variable declaration  
n1=1;         #order
n2=3;         #order
lamda=0.842*10**-10;     #wavelength(m)
theta1=(8+(35/60))*math.pi/180;      #angle of diffraction(radian)

#Calculations
theta3=math.asin(n2*math.sin(theta1));     #glancing order(radian)

#Result
print "glancing order is",round(theta3*180/math.pi,3),"degrees"

glancing order is 26.599 degrees

Example number 9, Page number 325

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

#Variable declaration  
n=1;         #order
lamda=0.58;          #wavelength of X-rays(angstrom)
theta1=6.45*math.pi/180;    #angle of diffraction(radian)
theta2=9.15*math.pi/180;    #angle of diffraction(radian)
theta1=13*math.pi/180;    #angle of diffraction(radian)

#Calculations
d1=lamda/(2*math.sin(theta1));       #spacing in 1st case(angstrom)
d2=lamda/(2*math.sin(theta2));       #spacing in 2nd case(angstrom)
d3=lamda/(2*math.sin(theta3));       #spacing in 3rd case(angstrom)

#Result"
print "spacing in 1st case is",round(d1,3),"angstrom"
print "spacing in 2nd case is",round(d2,3),"angstrom"
print "spacing in 3rd case is",round(d3,3),"angstrom"
print "answers given in the book are wrong"

spacing in 1st case is 1.289 angstrom
spacing in 2nd case is 1.824 angstrom
spacing in 3rd case is 0.648 angstrom
answers given in the book are wrong

Example number 10, Page number 326

In [39]:
#importing modules
import math
from __future__ import division

#Variable declaration  
M=58.5;           #molecular weight(kg/k-mole)
N=6.02*10**26;    #avagadro number(mol/k-mole)
rho=2.16*10**3;   #density(kg/m**3)

#Calculations
d=(M/(2*N*rho))**(1/3);      #seperation of adjacent atoms(m)

#Result
print "seperation of adjacent atoms is",round(d*10**10,3),"angstrom"

seperation of adjacent atoms is 2.823 angstrom

Example number 11, Page number 327

In [45]:
#importing modules
import math
from __future__ import division

#Variable declaration  
M=58.454;           #molecular weight(kg/k-mole)
n=1;                #order
rho=2163;           #density(kg/m**3)
lamda=1.3922*10**-10;     #wavelength(m)
theta=(14+(27/60)+(26/3600))*math.pi/180;    #angle of diffraction(radian)

#Calculations
d=n*lamda/(2*math.sin(theta));      #lattice spacing(m)
N=M/(2*rho*d**3);                   #avagadro number(mol/k-mole)

#Result
print "lattice spacing is",round(d*10**10,4),"angstrom"
print "avagadro number is",round(N/10**26,3),"*10**26 mol/k-mole"
print "answer in the book varies due to rounding off errors"

lattice spacing is 2.7882 angstrom
avagadro number is 6.234 *10**26 mol/k-mole
answer in the book varies due to rounding off errors

Example number 12, Page number 327

In [12]:
#importing modules
import math
from __future__ import division

#Variable declaration  
lamda=0.586*10**-10;      #wavelength(m)
theta1=(5+(58/60))*math.pi/180;      #angle of diffraction(radian)
theta2=(12+(10/60))*math.pi/180;      #angle of diffraction(radian)
theta3=(18+(12/60))*math.pi/180;      #angle of diffraction(radian)

#Calculations
a=int(10*math.sin(theta1));          
b=int(10*math.sin(theta2));          
c=int(10*math.sin(theta3));          
d1=a*lamda/(2*math.sin(theta1));     #spacing at first order(m)
d2=b*lamda/(2*math.sin(theta2));     #spacing at second order(m)
d3=c*lamda/(2*math.sin(theta3));     #spacing at third order(m)
d=(d1+d2+d3)/3;                      #mean value of crystal lattice spacing(m)

#Result
print "orders of reflection is",a,":",b,":",c
print "spacing at first order is",round(d1*10**10,4),"*10**-10 m"
print "spacing at second order is",round(d2*10**10,4),"*10**10 m"
print "spacing at third order is",round(d3*10**10,4),"*10**-10 m"
print "mean value of crystal lattice spacing is",round(d*10**10,3),"*10**-10 m"
print "answers given in the book vary due to rounding off errors"

orders of reflection is 1 : 2 : 3
spacing at first order is 2.8187 *10**-10 m
spacing at second order is 2.7805 *10**10 m
spacing at third order is 2.8143 *10**-10 m
mean value of crystal lattice spacing is 2.804 *10**-10 m
answers given in the book vary due to rounding off errors

Example number 13, Page number 328

In [27]:
#importing modules
import math
from __future__ import division

#Variable declaration  
theta1=(5+(23/60))*math.pi/180;      #angle of diffraction(radian)
theta2=(7+(37/60))*math.pi/180;      #angle of diffraction(radian)
theta3=(9+(25/60))*math.pi/180;      #angle of diffraction(radian)

#Calculations
d1=math.sin(theta1);     #spacing at first order(m)
d2=math.sin(theta2);     #spacing at second order(m)
d3=math.sin(theta3);     #spacing at third order(m)
x=d1/d1;
y=round(d2/(d1*math.sqrt(2)));
z=round(d3/(math.sqrt(3)*d1));

#Result
print "ratio of spacing is",int(x),":",int(y),"*math.sqrt(2) :",int(z),"*math.sqrt(3)"
print "the crystal is simple cubic crystal"

ratio of spacing is 1 : 1 *math.sqrt(2) : 1 *math.sqrt(3)
the crystal is simple cubic crystal

Example number 14, Page number 328

In [50]:
#importing modules
import math
from __future__ import division

#Variable declaration  
M=58.46;           #molecular weight(kg/k-mole)
N=6.02*10**26;     #avagadro number(mol/k-mole)
rho=2170;          #density(kg/m**3)
theta=6*math.pi/180;      #angle of diffraction(radian)
n=1;               #order

#Calculations
d=(M/(2*N*rho))**(1/3);      #seperation of adjacent atoms(m)
lamda=2*d*math.sin(theta)/n;     #wavelength of Kalpha(m)

#Result
print "wavelength of Kalpha is",round(lamda*10**10,3),"angstrom"
print "answer given in the book is wrong due to printing mistake"

wavelength of Kalpha is 0.589 angstrom
answer given in the book is wrong due to printing mistake

Example number 15, Page number 329

In [59]:
#importing modules
import math
from __future__ import division

#Variable declaration  
e=1.6*10**-19;             #charge(coulomb)
E=344;                     #energy(V) 
theta=60*math.pi/180;      #angle of diffraction(radian)
n=1;                       #order
m=9*10**-31;               #mass(kg)
h=6.62*10**-34;            #planks constant(Js)

#Calculations
lamda=h/math.sqrt(2*m*e*E);     #wavelength(m)
d=n*lamda/(2*math.sin(theta));  #spacing of crystal(m)

#Result
print "spacing of crystal is",round(d*10**10,2),"angstrom"

spacing of crystal is 0.38 angstrom