3: X-ray Diffraction

Example number 3.1, Page number 80

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

#Variable declaration
d=2.82*10**-10;    #lattice spacing(m)
theta=10;   #glancing angle(degree)
n=1;   #order

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

#Result
print "wavelength is",round(lamda*10**10,5),"angstrom"

wavelength is 0.97938 angstrom

Example number 3.2, Page number 80

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

#Variable declaration
d=3.035*10**-10;    #lattice spacing(m)
theta=12;   #glancing angle(degree)
n=1;   #order

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

#Result
print "wavelength is",round(lamda*10**10,3),"angstrom"

wavelength is 1.262 angstrom

Example number 3.3, Page number 81

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

#Variable declaration
d=2.81;    #lattice spacing(angstrom)
theta1=15.1;   #glancing angle(degree)
theta2=17.1;   #glancing angle(degree)

#Calculation
theta1=theta1*math.pi/180;    #angle(radian)
lamda1=2*d*math.sin(theta1);    #wavelength(angstrom)
theta2=theta2*math.pi/180;    #angle(radian)
lamda2=2*d*math.sin(theta2);    #wavelength(angstrom)

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

wavelengths are 1.464 angstrom and 1.6525 angstrom
answer varies due to rounding off errors

Example number 3.4, Page number 81

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

#Variable declaration
lamda=1.54;      #wavelength(angstrom)
theta=11;   #glancing angle(degree)

#Calculation
theta=theta*math.pi/180;    #angle(radian)
d=lamda/(2*math.sin(theta));   #separation between lattice planes(angstrom)

#Result
print "separation between lattice planes is",round(d,3),"angstrom"

separation between lattice planes is 4.035 angstrom

Example number 3.5, Page number 81

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

#Variable declaration
lamdaB=0.92;    #wavelength(angstrom)
theta1=30;   #glancing angle(degree)
theta2=60;   #glancing angle(degree)

#Calculation
theta1=theta1*math.pi/180;    #angle(radian)
theta2=theta2*math.pi/180;    #angle(radian)
lamdaA=2*lamdaB*math.sin(theta1)/math.sin(theta1);    #wavelength of line A(angstrom)

#Result
print "wavelength is",lamdaA,"angstrom"
print "answer in the book is wrong"

wavelength is 1.84 angstrom
answer in the book is wrong

Example number 3.6, Page number 81

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

#Variable declaration
d=0.4086*10**-10;    #lattice spacing(m)
theta=65;   #glancing angle(degree)
h=6.6*10**-34;    #plank's constant(Js)
m=9.1*10**-31;    #mass(kg)
n=1;

#Calculation
theta=theta*math.pi/180;    #angle(radian)
lamda=2*d*math.sin(theta)/n;    #debroglie wavelength(m)
v=h/(m*lamda);        #velocity(m/sec)

#Result
print "debroglie wavelength is",round(lamda*10**10,4),"*10**-10 metre"
print "velocity is",round(v/10**6,3),"*10**6 m/sec"
print "answer in the book is wrong"

debroglie wavelength is 0.7406 *10**-10 metre
velocity is 9.793 *10**6 m/sec
answer in the book is wrong

Example number 3.7, Page number 82

In [6]:
#importing modules
import math
from __future__ import division

#Variable declaration
d=2.82*10**-10;    #lattice spacing(m)
sintheta=1;   
n=1;

#Calculation
lamda_max=2*d*sintheta/n;    #longest wavelength(m)

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

longest wavelength is 5.64 angstrom

Example number 3.8, Page number 82

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

#Variable declaration
d=0.842*10**-10;    #lattice spacing(m)
theta1=8+(35/60);   #glancing angle(degree)
n1=1;   #order
n2=3;   #order

#Calculation
theta1=theta1*math.pi/180;    #angle(radian)
theta3=math.asin(n2*math.sin(theta1));      #glancing angle(radian)
theta3=theta3*180/math.pi;       #glancing angle(degree)

#Result
print "glancing angle is",round(theta3,3),"degree"

glancing angle is 26.599 degree

Example number 3.9, Page number 82

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

#Variable declaration
lamda=0.58;      #wavelength(angstrom)
theta1=6+(45/60);   #glancing angle(degree)
theta2=9+(15/60);   #glancing angle(degree)
theta3=13;   #glancing angle(degree)

#Calculation
theta1=theta1*math.pi/180;    #angle(radian)
theta2=theta2*math.pi/180;    #angle(radian)
theta3=theta3*math.pi/180;    #angle(radian)
x1=lamda/(2*math.sin(theta1));
x2=lamda/(2*math.sin(theta2));

#Result
print "interplanar spacing is",round(x2,3),"angstrom"
print "answer varies due to rounding off errors"

interplanar spacing is 1.804 angstrom
answer varies due to rounding off errors

Example number 3.10, Page number 83

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

#Variable declaration
lamda=1.3922;      #wavelength(angstrom)
n=1;
theta=14+(27/60)+(26/(60*60));   #glancing angle(degree)
M=58.454;          #molecular weight
rho=2163;          #density(kg/m**3)

#Calculation
theta=theta*math.pi/180;    #angle(radian)
d=n*lamda/(2*math.sin(theta));   #lattice spacing(angstrom)
d_m=d*10**-10;        #lattice spacing(m)
N=M/(2*rho*d_m**3);     #avagadro number(mol/k-mole)

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

lattice spacing is 2.7882 angstrom
avagadro number is 6.2337 *10**26 mol/k-mole
answer varies due to rounding off errors

Example number 3.11, Page number 84

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

#Variable declaration
lamda=0.586*10**-10;      #wavelength(m)
theta1=5+(58/60);   #glancing angle(degree)
theta2=12+(10/60);   #glancing angle(degree)
theta3=18+(12/60);   #glancing angle(degree)

#Calculation
theta1=theta1*math.pi/180;    #angle(radian)
theta2=theta2*math.pi/180;    #angle(radian)
theta3=theta3*math.pi/180;    #angle(radian)
x1=math.sin(theta1);
x2=math.sin(theta2);
x3=math.sin(theta3);
d1=lamda/(2*math.sin(theta1));     #spacing for 1st order(m)
d2=2*lamda/(2*math.sin(theta2));     #spacing for 2nd order(m)
d3=3*lamda/(2*math.sin(theta3));     #spacing for 3rd order(m)
d=(d1+d2+d3)/3;      #spacing(m)

#Result
print "ratio of angles of incidence are",round(x1,3),":",round(x2,4),":",round(x3,4),"which is nothing but",round(x1,1)*10,":",round(x2,1)*10,":",round(x3,1)*10
print "angles of incidence should be 1st, 2nd and 3rd orders"
print "spacing is",round(d*10**10,3),"*10**-10 m"
print "answer varies due to rounding off errors"

ratio of angles of incidence are 0.104 : 0.2108 : 0.3123 which is nothing but 1.0 : 2.0 : 3.0
angles of incidence should be 1st, 2nd and 3rd orders
spacing is 2.804 *10**-10 m
answer varies due to rounding off errors

Example number 3.12, Page number 84

In [11]:
#importing modules
import math
from __future__ import division

#Variable declaration
theta1=5+(23/60);   #glancing angle(degree)
theta2=7+(37/60);   #glancing angle(degree)
theta3=9+(25/60);   #glancing angle(degree)

#Calculation
theta1=theta1*math.pi/180;    #angle(radian)
theta2=theta2*math.pi/180;    #angle(radian)
theta3=theta3*math.pi/180;    #angle(radian)
x1=math.sin(theta1);
X1=1/(10*x1);
x2=math.sin(theta2)/x1;
x3=math.sin(theta3)/x1;

#Result
print "ratio of angles of incidence are",int(x1),":",round(x2,3),":",round(x3,3)
print "the crystal is a simple cubic crystal"

ratio of angles of incidence are 0 : 1.413 : 1.744
the crystal is a simple cubic crystal

Example number 3.13, Page number 85

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

#Variable declaration
h=6.62*10**-34;      #planck's constant(J sec)
e=1.6*10**-19;       #charge(coulomb)
m=9*10**-31;        #mass(kg) 
E=344;     #energy(volts)
n=1;
theta=60;   #angle(degrees)

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

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

spacing of crystal is 0.38 angstrom

Example number 3.14, Page number 85

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

#Variable declaration
h=2;
k=2;
l=0;
n=1;
theta=32;   #angle(degrees)
lamda=1.54*10**-10;      #wavelength(m)

#Calculation
theta=theta*math.pi/180;    #angle(radian)
d=n*lamda*10**10/(2*math.sin(theta));           #spacing of crystal(angstrom)
a=d*math.sqrt(h**2+k**2+l**2);     #lattice parameter(angstrom)
r=a/(2*math.sqrt(2));          #radius of atom(angstrom)

#Result
print "lattice parameter is",round(a,1),"angstrom"
print "radius of atom is",round(r,2),"angstrom"

lattice parameter is 4.1 angstrom
radius of atom is 1.45 angstrom