Chapter 5 : Miller Indices and X-Ray Crystallograph Techniques

Example 5.1 pageno : 96

In [1]:
# Variables
p = 1.;
q = 1./2;
r = 3.;

# Calculations
h = 1./p;
k = 1./q;
l = 1./r;
h1 = 3.*h;
k1 = 3.*k;
l1 = 3.*l;

# Results
print "MILLER INDICES OF THE PLANE are h  = ",h1
print "k  =  ",k1
print "l  =  ",l1

MILLER INDICES OF THE PLANE are h  =  3.0
k  =   6.0
l  =   1.0

Example 5.3 pageno : 97

In [2]:
# Variables
p = 2./4;			#intercepts
q = 3./3;
r = 4./2;

# Calculations
h = 1./p;
k = 1./q;
l = 1./r;
h1 = 2.*h;
k1 = 2.*k;
l1 = 2.*l;

# Results
print "MILLER INDICES ARE ",l1,k1,h1

MILLER INDICES ARE  1.0 2.0 4.0

Example 5.5 pageno : 105

In [4]:
import math 

# Variables
r = 1.246;			#radius in angstorm
h = 2.;
k = 0.;
l = 0.;
h1 = 2.;
k1 = 2.;
l1 = 0.;
h2 = 1.;
k2 = 1.;
l2 = 1.;

# Calculations
x = math.sqrt(h**2+k**2+l**2);
a = 2*math.sqrt(2)*r;			#in angstorm
d_200 = a/x;			        #interplanar spacing in angstorm
x1 = math.sqrt(h1**2+k1**2+l1**2);
d_220 = a/x1;       			#interplanar spacing in angstorm
x2 = math.sqrt(h2**2+k2**2+l2**2);
d_111 = a/x2;		        	#interplanar spacing in angstorm


print "Interplanar Spacing (200) (in Angstorm)  =  %.3f A"%d_200
print "Interplanar Spacing (220) (in Angstorm)  =  %.3f A"%d_220
print "Interplanar Spacing (111) (in Angstorm)  =  %.3f A"%d_111

Interplanar Spacing (200) (in Angstorm)  =  1.762 A
Interplanar Spacing (220) (in Angstorm)  =  1.246 A
Interplanar Spacing (111) (in Angstorm)  =  2.035 A

Example 5.6 pageno : 106

In [6]:
import math 

# Variables
a = 3.61*10**-10;			#unit cell in m

# Calculations
r_110 = 2/(math.sqrt(2)*a);			#in atoms/m
r_a = r_110/10**3;	        		#in atoms/mm
r_111 = 1/(math.sqrt(3)*a);			#in atoms/m
r_b = r_111/10**3;			        #in atoms/mm

# Results
print "Linear Density per unit length along direction [110] (in atoms/mm)  = %.2e atoms/mm"%r_a
print "Linear Density per unit length along direction [111] (in atoms/mm)  = %.2e atoms/mm"%r_b

Linear Density per unit length along direction [110] (in atoms/mm)  = 3.92e+06 atoms/mm
Linear Density per unit length along direction [111] (in atoms/mm)  = 1.60e+06 atoms/mm

Example 5.7 pageno : 110

In [11]:
import math 

# Variables
r_po = 1.7*10**-10;			    #radius of polonium in m
r_rh = 1.34*10**-10;			#radius of rhodium in m
r_cr = 1.25*10**-10;			#radius of chromium in m

# Calculations
a_po = 2*r_po;		        	#in m
a_rh = 2*math.sqrt(2)*r_rh;		#in m
a_cr = 4*r_cr/math.sqrt(3);
p_po = 1/a_po**2;			    # /sqm
p_rh = 1.414/a_rh**2;			# /sqm
p_cr = 1.732/a_cr**2;			# /sqm

# Results
print "Planar Density on [100] in Polonium (per sqm)  =  %.2e /m**2"%p_po
print "Planar Density on [110] in Rhodium (per sqm)  =  %.2e /m**2"%p_rh
print "Planar Density on [111] in Chromium (per sqm)  =  %.2e /m**2"%p_cr

# Note : To check answer , please calculate manually for p_rh

Planar Density on [100] in Polonium (per sqm)  =  8.65e+18 /m**2
Planar Density on [110] in Rhodium (per sqm)  =  9.84e+18 /m**2
Planar Density on [111] in Chromium (per sqm)  =  2.08e+19 /m**2

Example 5.8 pageno : 113

In [14]:
import math 

# Variables
w = 0.824;			#wavelength in angstorm
theta1 = 8.35;			#angle at n = 1 in degrees
n1 = 1.;
n3 = 3.;

# Calculations
d = w/(2*math.sin(math.radians(theta1)));			#in angstorm
theta3 = math.degrees(math.asin(3*math.sin(math.radians(theta1))))

# Results
print "Glancing angle for third order diffraction  =  %f degrees"%theta3
print "Interplanar spacing of the crystal (in Angstorm)  =  %.3f A"%d

Glancing angle for third order diffraction  =  25.827235 degrees
Interplanar spacing of the crystal (in Angstorm)  =  2.837 A

Example 5.9 pageno : 115

In [15]:
import math 

# Variables
a = 17.03;			#in degrees
w = 0.71;			#in angstorm
n = 1.;

# Calculations
d = n*w/(2*math.sin(math.radians(a)));			#interplanar spacing in angstorm
# given that h**2+k**2+l**2 = 8
a = math.sqrt(8)*d;                     			#in angstorm

# Results
print "Interplanar Spacing (in angstorm)  =  %.3f A"%d
print "Lattice parameter of the crystal (in Angstorm)  =  %.2f A"%a

Interplanar Spacing (in angstorm)  =  1.212 A
Lattice parameter of the crystal (in Angstorm)  =  3.43 A

Example 5.10 pageno : 117

In [16]:
# Variables
w = 0.0708;			        #wavelength in nm
h = 1.;
k = 0.;
l = 0.;
s = 0.0132;     			#a common divisor i.e.math.sin**2(theta) = 0.0132

# Calculations
a = math.sqrt((w**2*(h**2+k**2+l**2))/(4*s));			#in nm
a1 = 10.**3*a;			    #in pm

# Results
print "Dimension of unit cell (in Picometer)  =  %.1f pm"%a1

Dimension of unit cell (in Picometer)  =  308.1 pm