Chapter 16: Crystal Physics

Example 16.1, Page 820

In [4]:
#Variable declaration
p = 1; q = 2; r = 3; # Coefficients of intercepts along three axes

#Calculations
p_inv = 1./p;        # Reciprocate the first coefficient
q_inv = 1./q;        # Reciprocate the second coefficient
r_inv = 1./r;        # Reciprocate the third coefficient
mul_fact = p*q*r; # Find l.c.m. of m,n and p
m1 = p_inv*mul_fact;    # Clear the first fraction
m2 = q_inv*mul_fact;    # Clear the second fraction
m3 = r_inv*mul_fact;    # Clear the third fraction

#Result
print "The required miller indices are : (%d %d %d) "%(m1,m2,m3)

The required miller indices are : (6 3 2)

Example 16.2, Page 820

In [5]:
#Variable declaration
p = 2; q = 3; r = -4; # Coefficients of intercepts along three axes

#Calculations
p_inv = 1./p;        # Reciprocate the first coefficient
q_inv = 1./q;        # Reciprocate the second coefficient
r_inv = 1./r;        # Reciprocate the third coefficient
mul_fact = p*q*abs(r); # Find l.c.m. of m,n and p
m1 = p_inv*mul_fact;    # Clear the first fraction
m2 = q_inv*mul_fact;    # Clear the second fraction
m3 = r_inv*mul_fact;    # Clear the third fraction

#Result
print "The miller indices of laticce plane are : (%d %d %d) "%(m1,m2,m3)

The miller indices of laticce plane are : (12 8 -6)

Example 16.3, Page 821

In [14]:
import numpy

#Variable declaration
p = 3; q = 4; r = numpy.inf; # Coefficients of intercepts along three axes

#Calculations
p_inv = 1./p;        # Reciprocate the first coefficient
q_inv = 1./q;        # Reciprocate the second coefficient
r_inv = 1./r;        # Reciprocate the third coefficient
mul_fact = p*q; # Find l.c.m. of m,n and p
m1 = p_inv*mul_fact;    # Clear the first fraction
m2 = q_inv*mul_fact;    # Clear the second fraction
m3 = r_inv*mul_fact;    # Clear the third fraction

#Result
print "The miller indices of the given planes are : (%d %d %d) "%(m1,m2,m3)

The miller indices of the given planes are : (4 3 0)

Example 16.4, Page 822

In [18]:
#Variable declaration
p = 1.2;  # First coefficient of intercept along X-axis, angstrom
a = 1.2
b = 1.8
c = 2.0;    # Lattice parameters along three axes, angstrom
h = 2.
k = 3.
l = 1.;    # Miller indices of lattice plane

#Calculations
# As p:q:r = a/h:b/k:c/l, solving for q and r
q = p*(b/k)/(a/h); # Second coefficient of intercept along X-axis, angstrom   
r = p*(c/l)/(a/h); # Third coefficient of intercept along X-axis, angstrom   

#Result
print "The lengths of the intercepts on Y and Z axes are %3.1f angstrom and %3.1f angstrom respectively"%(q, r)

The lengths of the intercepts on Y and Z axes are 1.2 angstrom and 4.0 angstrom respectively

Example 16.5, Page 822

In [20]:
#Variable declaration
M = 58.5;        # Molecular weight of NaCl, g-mole
rho = 2.198e+03;    # Density of Nacl, kg per metre cube
n = 4;    # No. of atoms per unit cell for an fcc lattice of NaCl crystal
NA = 6.023e+26;    # Avogadro's No., atoms/k-mol

#Calculations
# Volume of the unit cell is given by
# a^3 = M*n/(N*d)
# Solving for a
a = (n*M/(rho*NA))**(1./3);    # Lattice constant of unit cell of NaCl

#Result
print "Lattice constant for the NaCl crystal = %4.2f angstrom"%(a/1e-010)

Lattice constant for the NaCl crystal = 5.61 angstrom

Example 16.6, Page 823

In [21]:
#Variable declaration
M = 119;        # Molecular weight of KBr, g-mole
rho = 2.7;    # Density of KBr, g per cm-cube
n = 4;    # No. of atoms per unit cell for an fcc lattice of KBr crystal
NA = 6.023e+23;    # Avogadro's No., atoms/mol

#Calculations
# Volume of the unit cell is given by
# a^3 = M*n/(N*d)
# Solving for a
a = (n*M/(rho*NA))**(1./3);    # Lattice constant of unit cell of KBr

#Result
print "Lattice constant for the KBr crystal = %4.2f angstrom"%(a/1e-008)

Lattice constant for the KBr crystal = 6.64 angstrom

Example 16.7, Page 823

In [23]:
from math import *

#Variable declaration
M = 63.5;        # Molecular weight of Cu, g-mole
rho = 8.96;    # Density of Cu, g per cm-cube
n = 4;    # No. of atoms per unit cell for an fcc lattice of Cu 
NA = 6.023e+23;    # Avogadro's No., atoms/mol

#Calculations
# Volume of the unit cell is given by
# a^3 = M*n/(N*d)
# Solving for a
a = (n*M/(rho*NA))**(1./3);    # Lattice constant of unit cell of Cu
d = a/sqrt(2);    # Distance between the two nearest Cu atoms, angstrom 

#Results
print "Lattice constant for the Cu crystal = %4.2f angstrom"%(a/1e-008)
print "The distance between the two nearest Cu atoms = %4.2f angstrom"%(d/1e-008)

Lattice constant for the Cu crystal = 3.61 angstrom
The distance between the two nearest Cu atoms = 2.55 angstrom

Example 16.8, Page 824

In [24]:
a = 1;    # For simplicity assume lattice parameter of cubic crystal to be unity, unit
# For (011) planes
h = 0; k = 1; l = 1; # Miller Indices for planes in a cubic crystal
d_011 = a/(h**2+k**2+l**2)**(1./2);  # The interplanar spacing for cubic crystals, m
print "The interplanar spacing between consecutive (011) planes = a/sqrt(%d)"%(1/d_011**2)

# For (101) planes
h = 1; k = 0; l = 1; # Miller Indices for planes in a cubic crystal
d_101 = a/(h**2+k**2+l**2)**(1./2);  # The interplanar spacing for cubic crystals, m
print "The interplanar spacing between consecutive (101) planes = a/sqrt(%d)"%(1/d_101**2)

# For (112) planes
h = 1; k = 1; l = 2; # Miller Indices for planes in a cubic crystal
d_112 = a/(h**2+k**2+l**2)**(1./2);  # The interplanar spacing for cubic crystals, m
print "The interplanar spacing between consecutive (112) planes = a/sqrt(%d)"%(1/d_112**2) #incorrect answer in textbook

The interplanar spacing between consecutive (011) planes = a/sqrt(2)
The interplanar spacing between consecutive (101) planes = a/sqrt(2)
The interplanar spacing between consecutive (112) planes = a/sqrt(5)

Example 16.9, Page 824

In [25]:
#Variable declaration
a = 4.2e-010;    # Lattice parameter of cubic crystal, m
h = 3; k = 2; l = 1; # Miller Indices for planes in a cubic crystal

#Calculations
d_321 = a/(h**2+k**2+l**2)**(1./2);  # The interplanar spacing for cubic crystals, m

#Result
print "The interplanar spacing between consecutive (321) planes = %4.2f angstrom"%(d_321/1e-010)

The interplanar spacing between consecutive (321) planes = 1.12 angstrom

Example 16.10, Page 825

In [27]:
from math import *

#Variable declaration
a = 2.5
b = 2.5
c = 1.8;    # Lattice parameter of tetragonal crystal, angstrom
h = 1
k = 1
l = 1; # Miller Indices for planes in a tetragonal crystal

#Calculations
d_hkl = 1/sqrt((h/a)**2+(k/b)**2+(l/c)**2);  # The interplanar spacing for tetragonal crystals, m

#Result
print "The interplanar spacing between consecutive (111) planes = %4.2f angstrom"%d_hkl

The interplanar spacing between consecutive (111) planes = 1.26 angstrom