# Chapter 3: Atomic Packing¶

## Exa 3.1¶

In [1]:
from __future__ import division
import math
#  Python  Code  Ex3.1  Packing  of  equal  spheres  in
#two  dimensional  square  lattice:Page-88(2010)

#Variable declaration

a=1;         #  Length  of  the  side  of  the  square,  unit
R  =  a/2;                               #  Radius  of  the  sphere,  unit

#Calculations

# Radius of  the  sphere  introduced  within  the  void  produced  by
# the  packing  of  equal  spheres  on  square  lattice,  unit
r  =  ((2)**0.5-1)*R;
A  =  math.pi*R**2;      #  Area  associated  with  a  sphere,  square  units
FA  =  a**2-A;#Free area occupied  by  void  in  square  lattice,square  units
FA_per  =  FA*100;    #  Percentage  free  area  in  square  lattice

#Result

print"\nFree  area  in  square  lattice  is  :",round(FA_per,2),"percent"

Free  area  in  square  lattice  is  : 21.46 percent


## Exa 3.2¶

In [2]:
from __future__ import division
import math
# Python Code Ex3.2 Packing efficiency in diamond structure: Page-92 (2010)

#Variable declaration

# For simplicity we may take radius of the atom, R = 1 unit
R = 1;                           # Radius of the atom in bcc lattice, unit
nc = 8;                   # Number of corner atoms in diamond structure
nfcc = 6;                  # Number of face centred atoms in diamond structure
na = 4;                      # Number of atoms completely within the unit cell

#Calculations

n = 1/8*nc+1/2*nfcc+1*na  # Effective number of atoms in the diamond structure
V_atom = 8*4/3*math.pi*R**3; # Volume of atoms within the unit cell, unit cube
# Since for a diamond cubic crystal, the space lattice is fcc,
#with two atos per lattice point, such that 8*R = sqrt(3)*a, solving for a
a = 8*R/(3)**0.5;       # lattice parameter of diamond structure, unit
V_cell = a**3;                 # Volume of the unit cell, unit cube
eta = V_atom/V_cell*100;    # Packing efficiency in diamond structure

#Result

print"\nThe packing efficiency in diamond structure is :",round(eta),"percent"


The packing efficiency in diamond structure is : 34.0 percent


## Exa 3.3¶

In [3]:
from __future__ import division
import math
#  Python  Code  Ex3.3  Radius  of  largest  sphere
#that  can  be  placed  at  the  octahedral  void:  Page-100  (2010)

#Variable declaration

#  For  simplicity  we  may  take  radius  of  the  atom,  R  =  1  unit
R  =  1;                 #  Radius  of  the  atom  in  bcc  lattice,  unit

#Calculation

#  For  a  bcc  lattice,  4*R  =  a*sqrt(3),  solving  for  a
a  =  4*R/(3)**0.5;  #  lattice  parameter  of  bcc  crystal,  unit
#  Since  R  +  Rx  =  a/2,  solving  for  Rx
#Radius of  the largest sphere that will fit into the octahedral void, unit
Rx  =  a/2  -  R;

#Results

print"\nThe  radius  of  the  largest  sphere  that"
print " will  fit  into  the  octahedral  void  is  :",round(Rx,2),"R"


The  radius  of  the  largest  sphere  that
will  fit  into  the  octahedral  void  is  : 0.15 R


## Exa 3.4¶

In [4]:
from __future__ import division
import math
#  Python  Code  Ex3.4  Radius  of  largest  sphere  that
#can  be  placed  at  the  tetrahedral  void:  Page-100  (2010)

#Variable declaration

#  For  simplicity  we  may  take  radius  of  the  atom,  RL  =  1  unit
RL  =  1;       #  Radius  of  the  atom  in  bcc  lattice,  unit

#Calculation

#  For  a  bcc  lattice,  4*RL  =  a*sqrt(3),  solving  for  a
a  =  4*RL/(3)**0.5;    #  Lattice  parameter  of  bcc  crystal,  unit
#  Further  RL  +  Rx  =  sqrt(5)*a/4,  solving  for  Rx
#  Radius of the largest sphere  that will fit into the octahedral void, unit
Rx  =  (5)**0.5*a/4-RL;

#Result

print"\nThe  radius  of  the  largest  sphere "
print " that  will  fit  into  the  tetrahedral  void  is  :",round(Rx,2),"RL"

The  radius  of  the  largest  sphere
that  will  fit  into  the  tetrahedral  void  is  : 0.29 RL


## Exa 3.5¶

In [5]:
from __future__ import division
import math
#Python  Code  Ex3.5  Diameter  of  the  largest  atom  that
# would  fit  into  the  tetrahedral  void:5  Page-101  (2010)

#Variable declaration

a  =  3.52*10**-10;                #  Lattice  parameter  for  Ni,  m

#Calculations

#  For  an  fcc  lattice,  sqrt(2)*a  =  4*R,  solving  for  R
R  =  (2)**0.5*a/4;          #  Radius  of  the  atom  in  fcc  lattice,  m
R_oct  =  0.414*R; #Radius of the octahedral void  in  fcc  close  packing, m
D  =  2*R_oct#Diameter of the octahedral void in the fcc structure of nickel,m

#Result

print"The diameter of the octahedral void in the fcc structure of nickel,"
print" in angstrom, is : ",round(D/(1*10**-10),2)


The diameter of the octahedral void in the fcc structure of nickel,
in angstrom, is :  1.03


## Exa 3.6¶

In [6]:
from __future__ import division
import math
# Python Code Ex3.6 Void space in cubic close packing:  Page-101  (2010)

#Variable declaration

R  =  1;               #  For  simplicity,  radius  of  the  sphere,  m
n  =  4;                 #  Number  of  lattice  points  in  fcc  unit  cell

#Calculations

#  For  cubic  close  packing,  side  of  the  unit  cell
#and  the  radius  of  the  sphere  is  related  as
#        sqrt(2)*a  =  4*R,  solving  for  a
a  =  2*(2)**0.5*R;     #Lattice  parameter  for  cubic  close  packing,  m
V_cell  =  a**3;                         #  Volume  of  the  unit  cell
#  Volume  occupied  by  the  atoms,  metre  cube
V_occupied  =  4*4/3*math.pi*((1.000)**3+(0.414)**3+2*(0.225)**3);
void_space  =  V_cell  -  V_occupied; #  Void  space  in  the  close  packing
percent_void  =  void_space/V_cell*100;     #  Percentage  void  space

#Result

print"\nVoid  space  in  the  close packing is:",round(percent_void),"percent"

Void  space  in  the  close packing is: 19.0 percent


## Exa 3.7¶

In [7]:
from __future__ import division
import math
#Python CodeEx3.7 The minimum value of radius ratio in AX-compound:Page-104

#Variable declaration

# For simplicity we may assume a = 1
a = 1;                            # Lattice parameter of the crystal, unit

# Calculation

b = 2/3*a*math.sin(math.pi/3);        # Lattice parameter of the crystal, unit
# Here a = 2*Rx, where a is the lattice parameter and
# Rx is the radius of X-ions representing the bigger spheres, solving for Rx
Rx = 0.5*a;
# Also b = RA + Rx, solving for RA
RA = b - Rx;          # Radius of A-ion representing teh smaller sphere, unit

 Minimum value of radius ratio in AX compound is : 0.15