# 6: Crystallography¶

## Example number 6.1, Page number 134¶

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

#Variable declaration
M = 23+35.5;        #Molecular weight of NaCl(kg/k-mole)
d = 2.18*10**3;     #Density of rock salt(kg/m**3)
n = 4;    #Number of atoms per unit cell for an fcc lattice of NaCl crystal
N = 6.02*10**26;    #Avogadro's No., atoms/k-mol

#Calculation
a = (n*M/(d*N))**(1/3);     #Lattice constant of unit cell of NaCl(m)
a = a*10**9;      ##Lattice constant of unit cell of NaCl(nm)
a = math.ceil(a*10**3)/10**3;     #rounding off the value of a to 3 decimals

#Result
print "Lattice parameter for the NaCl crystal is",a, "nm"

Lattice parameter for the NaCl crystal is 0.563 nm


## Example number 6.2, Page number 134¶

In [2]:
#importing modules
import math

#Variable declaration
m = 3;
n = 2;
p = 1;     #Coefficients of intercepts along three axes

#Calculation
#reciprocals of the intercepts are 1/m, 1/n, 1/p i.e 1/3, 1/2, 1
#multiplying by LCM the reciprocals become 2, 3, 6

#Result
print "The required miller indices are : (2, 3, 6)"

The required miller indices are : (2, 3, 6)


## Example number 6.3, Page number 135¶

In [3]:
#importing modules
import math

#Variable declaration
m = 2;  #Coefficient of intercept along x-axis
#n = infinite    Coefficient of intercept along y-axis
p = 3/2;    #Coefficient of intercept along z-axis

#Calculation
#reciprocals of the intercepts are 1/m, 1/n, 1/p i.e 1/2, 0, 2/3
#multiplying by LCM the reciprocals become 3, 0, 4

#Result
print "The required miller indices are : (3, 0, 4)"

The required miller indices are : (3, 0, 4)


## Example number 6.5, Page number 136¶

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

#Variable declaration
#For (110) planes
h1 = 1;
k1 = 1;
l1 = 0;    #Miller Indices for planes in a cubic crystal
a1 = 0.43;     #Interatomic spacing(nm)
#For (212) planes
h2 = 2;
k2 = 1;
l2 = 2;    #Miller Indices for planes in a cubic crystal
a2 = 0.43;     #Interatomic spacing(nm)

#Calculation
d1 = a1/(h1**2+k1**2+l1**2)**(1/2);  #The interplanar spacing for cubic crystals(nm)
d1 = math.ceil(d1*10**4)/10**4;     #rounding off the value of d1 to 4 decimals
d2 = a2/(h2**2+k2**2+l2**2)**(1/2);    #The interplanar spacing for cubic crystals(nm)
d2 = math.ceil(d2*10**4)/10**4;     #rounding off the value of d2 to 4 decimals

#Result
print "The interplanar spacing between consecutive (110) planes is",d1, "nm";
print "The interplanar spacing between consecutive (212) planes is",d2, "nm"

The interplanar spacing between consecutive (110) planes is 0.3041 nm
The interplanar spacing between consecutive (212) planes is 0.1434 nm


## Example number 6.6, Page number 136¶

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

#Variable declaration
h = 2;
k = 3;
l = 1;      #Miller Indices for planes in a cubic crystal
r = 0.175;    #Atomic radius of fcc lattice(nm)

#Calculation
a = 2*math.sqrt(2)*r;     #Interatomic spacing of fcc lattice(nm)
d = a/(h**2+k**2+l**2)**(1/2);    #The interplanar spacing for cubic crystals(nm)
d = math.ceil(d*10**4)/10**4;     #rounding off the value of d to 4 decimals

#Result
print "The interplanar spacing between consecutive (231) planes is",d, "nm"

The interplanar spacing between consecutive (231) planes is 0.1323 nm


## Example number 6.7, Page number 136¶

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

#Variable declaration
lamda = 1.44;      #Wavelength of X-rays(A)
d = 2.8;     #Interplanar spacing of rocksalt crystal(A)
n1 = 1;      #For 1st Order diffraction
n2 = 2;     #For 2nd Order diffraction

#Calculation
theta1 = math.asin(n1*lamda/(2*d));    #Angle of diffraction(radians)
theeta1 = theta1*57.2957795;       #Angle of diffraction(degrees)
theeta1 = math.ceil(theeta1*10**2)/10**2;     #rounding off the value of theeta1 to 2 decimals
theta2 = math.asin(n2*lamda/(2*d));     #Angle of diffraction(radians)
theeta2 = theta2*57.2957795;       #Angle of diffraction(degrees)
theeta2 = math.ceil(theeta2*10**2)/10**2;     #rounding off the value of theeta2 to 2 decimals

#Result
print "The angle of diffraction for first order maxima is",theeta1, "degrees"
print "The angle of diffraction for second order maxima is",theeta2, "degrees"

The angle of diffraction for first order maxima is 14.91 degrees
The angle of diffraction for second order maxima is 30.95 degrees


## Example number 6.8, Page number 136¶

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

#Variable declaration
a = 1;     #For convenience, assume interatomic spacing to be unity(m)

#Calculation
N = 8*(1/8) + 6*(1/2);      #total number of spheres in a unit cell
r = a/(2*math.sqrt(2));    #The atomic radius(m)
V_atom = N*(4/3)*math.pi*r**3;    #Volume of atoms(m**3)
V_uc = a**3;       #Volume of unit cell(m**3)
PV = (V_atom/V_uc)*100;      #percentage of actual volume
PV = math.ceil(PV*10)/10;     #rounding off the value of PV to 1 decimal

#Result
print "The percentage of actual volume occupied by the spheres in fcc structure is",PV, "percent"

The percentage of actual volume occupied by the spheres in fcc structure is 74.1 percent


## Example number 6.9, Page number 137¶

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

#Variable declaration
#For (221) planes
h = 2;
k = 2;
l = 1;    #Miller Indices for planes in a cubic crystal
a = 2.68;    #Interatomic spacing(A)
n1 = 1;      #First Order of diffraction
n2 = 2;    #Second order of diffraction
theta1 = 8.5;    #Glancing angle at which Bragg's reflection occurs(degrees)

#Calculation
theta1 = theta1*0.0174532925;     #Glancing angle at which Bragg's reflection occurs(radians)
a = a*10**-10;      #Interatomic spacing(m)
d = a/(h**2+k**2+l**2)**(1/2);   #The interplanar spacing for cubic crystal(m)
lamda = 2*d*math.sin(theta1)/n1;     #Bragg's Law for wavelength of X-rays(m)
lamda_A = lamda*10**10;        #Bragg's Law for wavelength of X-rays(A)
lamda_A = math.ceil(lamda_A*10**4)/10**4;     #rounding off the value of lamda_A to 4 decimals
theta2 = math.asin(n2*lamda/(2*d));    #Angle at which second order Bragg reflection occurs(radians)
theta2 = theta2*57.2957795;       #Angle at which second order Bragg reflection occurs(degrees)
theta2 = math.ceil(theta2*10)/10;     #rounding off the value of theta2 to 1 decimal

#Result
print "The interplanar spacing between consecutive (221) planes is",d, "m"
print "The wavelength of X-rays is",lamda_A, "angstrom"
print "The angle at which second order Bragg reflection occurs is",theta2, "degrees"

The interplanar spacing between consecutive (221) planes is 8.93333333333e-11 m
The wavelength of X-rays is 0.2641 angstrom
The angle at which second order Bragg reflection occurs is 17.2 degrees


## Example number 6.10, Page number 137¶

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

#Variable declaration
h = 1;
k = 1;
l = 0;    #Miller Indices for planes in a cubic crystal
n = 1;    #First Order of diffraction
theta = 25;    #Glancing angle at which Bragg's reflection occurs(degrees)
lamda = 0.7;     #Wavelength of X-rays(A)

#Calculation
theta = theta*0.0174532925;     #Glancing angle at which Bragg's reflection occurs(radians)
d = n*lamda/(2*math.sin(theta));    #Interplanar spacing of cubic crystal(A)
a = d*(h**2+k**2+l**2)**(1/2);      #The lattice parameter for cubic crystal(A)
a = math.ceil(a*10**3)/10**3;     #rounding off the value of a to 3 decimals

#Result
print "The lattice parameter for cubic crystal is",a, "angstrom"

The lattice parameter for cubic crystal is 1.172 angstrom


## Example number 6.11, Page number 138¶

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

#Variable declaration
d = 0.31;      #Interplanar spacing(nm)
n = 1;    #First Order of diffraction
theta = 9.25;    #Glancing angle at which Bragg's reflection occurs(degrees)
theta_max = 90;     #Maximum possible angle at which reflection can occur(degrees)
theta_max = theta_max*0.0174532925;      #Maximum possible angle at which reflection can occur(radians)

#Calculation
theta = theta*0.0174532925;    #Glancing angle at which Bragg's reflection occurs(radians)
lamda = 2*d*math.sin(theta)/n;    #Wavelength of X-rays(nm) (Bragg's Law)
lamda = math.ceil(lamda*10**5)/10**5;     #rounding off the value of lamda to 5 decimals
n = 2*d*math.sin(theta_max)/lamda;    #Maximum possible order of diffraction

#Result
print "The wavelength of X-rays is",lamda, "nm"
print "The Maximum possible order of diffraction is",round(n)

The wavelength of X-rays is 0.09967 nm
The Maximum possible order of diffraction is 6.0


## Example number 6.12, Page number 138¶

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

#Variable declaration
#For (110) planes
h1 = 1;
k1 = 1;
l1 = 0;     #Miller indices for (110) planes
d_110 = 0.195;     #Interplanar spacing between (110) planes(nm)
#For (210) planes
h2 = 2;
k2 = 1;
l2 = 0;     #Miller indices for (110) planes
n = 2;     #Second Order of diffraction
lamda = 0.071;     #Wavelength of X-rays(nm)

#Calculation
a = d_110*(h1**2 + k1**2 + l1**2)**(1/2);     #Lattice parameter for bcc crystal(nm)
d_210 = a/(h2**2 + k2**2 + l2**2)**(1/2);     #Interplanar spacing between (210) planes(nm)
theta = math.asin(n*lamda/(2*d_210));      #Bragg reflection angle for the second order diffraction(radians)
theeta = theta*57.2957795;      #Bragg reflection angle for the second order diffraction(degrees)
theeta = math.ceil(theeta*10**3)/10**3;     #rounding off the value of theeta to 3 decimals

#Result
print "Bragg reflection angle for the second order diffraction is",theeta, "degrees"

Bragg reflection angle for the second order diffraction is 35.149 degrees


## Example number 6.13, Page number 138¶

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

#Variable declaration
d = 2182;       #Density of rock salt(kg/m**3)
n = 4;        #Number of atoms per unit cell for an fcc lattice of NaCl crystal

#Calculation
M = 23+35.5;         #Molecular weight of NaCl(kg/k-mole)
#V = a^3 = M*n/(N*d)
a = (n*M/(d*N))**(1/3);      #Lattice constant of unit cell of NaCl(m)
D = a/2;       #distance between nearest neighbours(m)
D = D*10**9;    #distance between nearest neighbours(nm)
D = math.ceil(D*10**4)/10**4;     #rounding off the value of D to 4 decimals

#Result
print "The distance between nearest neighbours of NaCl structure is",D, "nm"

The distance between nearest neighbours of NaCl structure is 0.2814 nm


## Example number 6.14, Page number 139¶

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

#Variable declaration
r1 = 1.258;     #Atomic radius of bcc structure of iron(A)
N1 = 2;     #Number of atoms per unit cell in bcc structure
#For fcc structure
r2 = 1.292;     #Atomic radius of fcc structure of iron(A)
N2 = 4;     #Number of atoms per unit cell in fcc structure

#Calculation
a1 = 4*r1/math.sqrt(3);    #Lattice parameter of bcc structure of iron(A)
V1 = a1**3;     #Volume of bcc unit cell(A)
V_atom_bcc = V1/N1;    #Volume occupied by one atom(A)
a2 = 2*math.sqrt(2)*r2;     #Lattice parameter of fcc structure of iron(A)
V2 = a2**3;     #Volume of fcc unit cell(A)
V_atom_fcc = V2/N2;     #Volume occupied by one atom(A)
delta_V = (V_atom_bcc-V_atom_fcc)/V_atom_bcc*100;    #Percentage change in volume due to structural change of iron
delta_V = math.ceil(delta_V*10**3)/10**3;     #rounding off the value of delta_V to 3 decimals

#Result
print "The percentage change in volume of iron is",delta_V, "percent"

The percentage change in volume of iron is 0.494 percent

In [ ]: