3: Crystal Physics

Example number 3.3, Page number 60

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

#Variable declaration
h=3;     #miller indices with respect to x axis
k=1;     #miller indices with respect to y axis
l=1;     #miller indices with respect to z axis
a=2.109*10**-10;    #lattice constant of plane in a simple cubic lattice(m)

#Calculation
d=(a/(math.sqrt(h**2+k**2+l**2)));      #The interplanar distance(m)

#Result
print "The interplanar distance is",round(d*10**11,4),"*10**-11 m"

The interplanar distance is 6.3589 *10**-11 m

Example number 3.4, Page number 60

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

#Variable declaration
h=1;    #miller indices with respect to x axis
k=1;    #miller indices with respect to y axis
l=0;    #miller indices with respect to z axis
d=2.86*10**-10;    #the distance between miller indices(m)

#Calculation
a=(d*(math.sqrt(h**2+k**2+l**2)));     #The lattice constant(m)

#Result
print "The lattice constant is",round(a*10**10,4),"*10**-10 m"

The lattice constant is 4.0447 *10**-10 m

Example number 3.6, Page number 61

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

#Variable declaration
h=1;    #miller indices of x-axis
k=1;    #miller indices of y-axis
l=1;    #miller indices of z-axis

#Calculation
p=1/h;    #intercept on x-axis
q=1/k;    #intercept on y-axis
r=1/l;    #intercept on z-axis

#Result
print "The ratio of intercepts on the three axis by (",h,k,l,") plane is",p,":",q,":",r

The ratio of intercepts on the three axis by ( 1 1 1 ) plane is 1.0 : 1.0 : 1.0

Example number 3.7, Page number 61

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

#Variable declaration
r=1.246*10**-10;     #atomic radius of Fcc crystal(m)
h1=1;     #miller indices with respect to x axis in 1st plane
k1=1;     #miller indices with respect to y axis in 1st plane
l1=1;     #miller indices with respect to z axis in 1st plane
h2=2;     #miller indices with respect to x axis in 2nd plane
k2=0;     #miller indices with respect to y axis in 2nd plane
l2=0;     #miller indices with respect to z axis in 2nd plane
h3=2;     #miller indices with respect to x axis in 3rd plane
k3=2;     #miller indices with respect to y axis in 3rd plane
l3=0;     #miller indices with respect to z axis in 3rd plane

#Calculation
a=(4*r)/math.sqrt(2);     #The lattice constant in a FCC crystal(m)
d1=(a/(math.sqrt(h1**2+k1**2+l1**2)));     #inter planar spacing distance in 1st plane(m)
d2=(a/(math.sqrt(h2**2+k2**2+l2**2)));     #inter planar spacing distance in 2nd plane(m)
d3=(a/(math.sqrt(h3**2+k3**2+l3**2)));     #inter planar spacing distance in 3rd plane(m)

#Result
print "The inter planar spacing distance in 1st plane is",round(d1*10**10,4),"*10**-10 m"
print "The inter planar spacing distance in 2nd plane is",round(d2*10**10,4),"*10**-10 m"
print "The inter planar spacing distance in 3rd plane is",d3,"m"

The inter planar spacing distance in 1st plane is 2.0347 *10**-10 m
The inter planar spacing distance in 2nd plane is 1.7621 *10**-10 m
The inter planar spacing distance in 3rd plane is 1.246e-10 m

Example number 3.8, Page number 62

In [21]:
#importing modules
import math
from __future__ import division

#Variable declaration
a=1;    #assume
h1=1;   #miller indices with respect to x axis in 1st plane
k1=0;   #miller indices with respect to y axis in 1st plane
l1=0;   #miller indices with respect to z axis in 1st plane
h2=1;   #miller indices with respect to x axis in 2nd plane
k2=1;   #miller indices with respect to y axis in 2nd plane
l2=0;   #miller indices with respect to z axis in 2nd plane
h3=1;   #miller indices with respect to x axis in 3rd plane
k3=1;   #miller indices with respect to y axis in 3rd plane
l3=1;   #miller indices with respect to z axis in 3rd plane

#Calculation
x1=math.sqrt(h1**2+k1**2+l1**2);
d100=a/x1;    #inter planar spacing distance in 1st plane(m)
x2=math.sqrt(h2**2+k2**2+l2**2);
d110=a/x2;    #inter planar spacing distance in 2nd plane(m)
x3=math.sqrt(h3**2+k3**2+l3**2);
d111=a/x3;    #inter planar spacing distance in 3rd plane(m)

#Result
print "The inter planar spacing distance in 1st plane is a*",d100,"m"
print "The inter planar spacing distance in 2nd plane is a*",round(d110,3),"m"
print "The inter planar spacing distance in 3rd plane is a*",round(d111,3),"fm"
print "Ratio of interplanar distance of three planes d100:d110:d111=",(1/x1),":",round((1/x2),3),":",round((1/x3),3)

The inter planar spacing distance in 1st plane is a* 1.0 m
The inter planar spacing distance in 2nd plane is a* 0.707 m
The inter planar spacing distance in 3rd plane is a* 0.577 fm
Ratio of interplanar distance of three planes d100:d110:d111= 1.0 : 0.707 : 0.577

Example number 3.9, Page number 62

In [23]:
#importing modules
import math
from __future__ import division

#Variable declaration
p=1;   #x-intercept of the plane
q=1/2;  #y-intercept of the plane
r=3;    #z-intercept of the plane

#Calculation
h=(1/p)*3;     #miller indices with respect to x axis
k=(1/q)*3;     #miller indices with respect to y axis
l=(1/r)*3;     #miller indices with respect to z axis

#Result
print "The miller indices of the plane is (h k l)=(",h,k,l,")"

The miller indices of the plane is (h k l)=( 3.0 6.0 1.0 )

Example number 3.10, Page number 63

In [24]:
#importing modules
import math
from __future__ import division

#Variable declaration
a=2.814;     #the lattice constant of a simple cubic system(angstrom)
h1=1;        #miller indices with respect to x axis
k1=0;        #miller indices with respect to y axis
l1=0;        #miller indices with respect to z axis

#Calculation
d=a/math.sqrt(h1**2+k1**2+l1**2);     #inter planar d spacing distance(angstrom)

#Result
print "The inter planar d-spacing distance is",d,"angstrom"

The inter planar d-spacing distance is 2.814 angstrom

Example number 3.11, Page number 63

In [26]:
#importing modules
import math
from __future__ import division

#Variable declaration
OA=0.025;    #The unit cell makes intercepts on a(nm)
OB=0.02;     #The unit cell makes intercepts on b(nm)
OC=0.01;     #The unit cell makes intercepts on c(nm)
a=0.05;      #The unit cell edge of an orthorhombic crystal(nm)
b=0.04;      #The unit cell edge of an orthorhombic crystal(nm)
c=0.03;    #The unit cell edge of an orthorhombic crystal(nm)

#Calculation
p=a/OA;    #miller indices with respect to x axis
q=b/OB;    #miller indices with respect to y axis
r=c/OC;    #miller indices with respect to z axis

#Result
print "The miller indices of the set of parallel lines is (",p,q,r,")"

The miller indices of the set of parallel lines is ( 2.0 2.0 3.0 )

Example number 3.12, Page number 63

In [36]:
#importing modules
import math
from __future__ import division

#Variable declaration
a=0.424;    #value of one axial unit
b=1;       #value of second axial unit
c=0.367;    #value of third axial unit
i1=0.212;   #value at x-intercept
j1=1;      #value at y-intercept
k1=0.183;    #value at z-intercept
i2=0.848;    #value at x-intercept
j2=1;       #value at y-intercept
k2=0.732;    #value at z-intercept
i3=0.424;    #value at x-intercept
k3=0.123;    #value at z-intercept

#Calculation
p1=1/(i1/a);     #miller indices at x-intercept
q1=1/(j1/b);     #miller indices at y-intercept
r1=1/(k1/c);     #miller indices at z-intercept
p2=1/(i2/a)*2;    #miller indices at x-intercept
q2=1/(j2/b)*2;    #miller indices at y-intercept
r2=1/(k2/c)*2;    #miller indices at z-intercept
p3=1/(i3/a);      #miller indices at x-intercept
q3=0;             #miller indices at y-intercept
r3=1/(k3/c);      #miller indices at z-intercept

#Result
print "The miller indices are",int(p1),int(q1),int(r1)
print "The miller indices are",int(p2),int(q2),int(r2)
print "The miller indices are",int(p3),int(q3),round(r3)

The miller indices are 2 1 2
The miller indices are 1 2 1
The miller indices are 1 0 3.0

Example number 3.13, Page number 65

In [39]:
#importing modules
import math
from __future__ import division

#Variable declaration
OB=2;    #The intercept made by the parrell line ,OB=2b
OC=7;    #The intercept made by the parrell line ,OC=2c
#OA=infinite  The intercept made by the parrell line ,OB=2b

#Calculation
A=0;     #miller indice along x-axis
B=1/OB;     #miller indice along y-axis
C=1/OC;     #miller indice along z-axis
X=(B*(OC*OB));   #taking L.C.M
Y=(C*(OC*OB));   #taking L.C.M

#Result
print "Miller indices are (1/infinite 1/",OB,"1/",OC,")=",A,int(X),int(Y)

Miller indices are (1/infinite 1/ 2 1/ 7 )= 0 7 2

Example number 3.14, Page number 75

In [41]:
#importing modules
import math
from __future__ import division

#Variable declaration
a=3.6;     #lattice parameter of copper(angstrom)

#Calculation
r=(a*math.sqrt(2))/4;     #The atomic radius of copper(angstrom)

#Result
print "The atomic radius of copper is",round(r,3),"angstrom"

The atomic radius of copper is 1.273 angstrom

Example number 3.15, Page number 76

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

#Variable declaration
a=4.12;     #the lattice constant of a simple cubic system(angstrom)
h1=3;     #miller indices with respect to x axis
k1=2;    #miller indices with respect to y axis
l1=1;    #miller indices with respect to z axis

#Calculation
d=a/math.sqrt(h1**2+k1**2+l1**2);    #inter planar d spacing distance(angstrom)

#Result
print "The inter planar d-spacing distance is",round(d,4),"angstrom"

The inter planar d-spacing distance is 1.1011 angstrom

Example number 3.16, Page number 76

In [40]:
#importing modules
import math
from __future__ import division

#Variable declaration
n=4;    #no.of atoms in FCC structure
A=63.54;    #Atomic weight of copper
r=1.278*10**-10;   #atomic radius(m)
N=6.023*10**26;    #Avogadro's Number(per Kg mol)

#Calculation
a=(4*r/math.sqrt(2));    #The lattice constant(m)
d=A*n/(N*a**3);    #The density of copper(Kg/m^3)

#Result
print "The density of copper is",int(d),"Kg/m^3"
print "answer varies due to rounding off errors"

The density of copper is 8934 Kg/m^3
answer varies due to rounding off errors

Example number 3.17, Page number 76

In [41]:
#importing modules
import math
from __future__ import division

#Variable declaration
h1=1;    #miller indices with respect to x axis in 1st plane
k1=0;    #miller indices with respect to y axis in 1st plane
l1=0;    #miller indices with respect to z axis in 1st plane
h2=1;    #miller indices with respect to x axis in 2nd plane
k2=1;    #miller indices with respect to y axis in 2nd plane
l2=0;    #miller indices with respect to z axis in 2nd plane
h3=1;    #miller indices with respect to x axis in 3rd plane
k3=1;    #miller indices with respect to y axis in 3rd plane
l3=1;    #miller indices with respect to z axis in 3rd plane
a=1;     #The lattice constant in a in a simple cubic lattice(m)

#Calculation
d100=a/math.sqrt(h1**2+k1**2+l1**2);    #inter planar spacing distance in 1st plane(m)
d110=a/math.sqrt(h2**2+k2**2+l2**2);    #inter planar spacing distance in 2nd plane(m)
d111=a/math.sqrt(h3**2+k3**2+l3**2);    #inter planar spacing distance in 3rd plane(m)

#Result
print "The ratio of interplanar distance between successive lattice planes in a simple cubic lattice is d100:d110:d111=",int(d100),":",round(d110,3),":",round(d111,3)

The ratio of interplanar distance between successive lattice planes in a simple cubic lattice is d100:d110:d111= 1 : 0.707 : 0.577

Example number 3.18, Page number 76

In [42]:
#importing modules
import math
from __future__ import division

#Variable declaration
x=23;     #atomic weight of sodium
y=35.45;   #atomic weight of chloride
AW=58.45;    #atomic weight of sodium chloride(NaCl)
n=4;    #no.of atoms in FCC structure
d=2.18*10**6;   #density of NaCl crystal of FCC structure(kg/m^3)
N=6.023*10**23;    #Avogadro's Number(per Kg mol)

#Calculation
a=(n*AW/(d*N))**(1/3);   #The lattice constant(m)
r=a/2;    #The distance between two adjacent atoms(m)

#Result
print "The distance between two adjacent atoms is",round(r*10**10,2),"*10**-10 m"

The distance between two adjacent atoms is 2.81 *10**-10 m

Example number 3.19, Page number 77

In [43]:
#importing modules
import math
from __future__ import division

#Variable declaration
n=2;    #no.of atoms in BCC structure
d=7.86*10**6;    #density of iron of FCC structure(kg/m^3)
AW=55.85;      #atomic weight of Fe
N=6.023*10**23;   #Avogadro's Number(per Kg mol)

#Calculation
a=(n*AW/(d*N))**(1/3);    #The lattice constant(m)
r=a*math.sqrt(3)*10**10/4;     #The atomic radius of Fe which has BCC structure(angstrom)

#Result
print "The atomic radius of Fe which has BCC structure is",round(r,3),"angstrom"

The atomic radius of Fe which has BCC structure is 1.242 angstrom

Example number 3.20, Page number 77

In [44]:
#importing modules
import math
from __future__ import division

#Variable declaration
n=4;    #no.of atoms in FCC structure
d=2.7*10**3;   #density of potassium bromide(Kg/m^3)
AW=119;    #molecular weight of KBr
N=6.023*10**26;   #Avagadro's number(Kg mol)

#Calculation
a=((n*AW/(d*N))**(1/3))*10**10;      #The lattice constant(angstrom)

#Result
print "The lattice constant is",round(a,1),"angstrom"

The lattice constant is 6.6 angstrom

Example number 3.21, Page number 78

In [45]:
#importing modules
import math
from __future__ import division

#Variable declaration
d=9.6*10**2;    #density of crystal(Kg/m^3)
AW=23;     #molecular weight of the crystal
N=6.023*10**26;   #Avagadro's number(per Kg mol)
a=4.3*10**-10;    #lattice constant(m)

#Calculation
n=d*N*a**3/AW;    #Number of atoms per unit cell of a crystal

#Result
print "Number of atoms per unit cell of a crystal is",round(n)
print "If n=2,the crystal system is body centered cubic"

Number of atoms per unit cell of a crystal is 2.0
If n=2,the crystal system is body centered cubic

Example number 3.22, Page number 78

In [46]:
#importing modules
import math
from __future__ import division

#Variable declaration
r=1.2*10**-10;    #atomic radius of crystal of BCC structure(m)

#Calculation
a=4*r/math.sqrt(3);     #lattice constant of BCC structure(m)
V=a**3;     #The volume of cell(m^3)

#Result
print "The volume of cell is",round(V*10**29,3),"*10**-29 m^3"

The volume of cell is 2.128 *10**-29 m^3

Example number 3.23, Page number 78

In [47]:
#importing modules
import math
from __future__ import division

#Variable declaration
a=4*10**-7;    #lattice constant of the crystal(mm)
h1=1;    #miller indices with respect to x axis in 1st plane
k1=0;    #miller indices with respect to y axis in 1st plane
l1=0;    #miller indices with respect to z axis in 1st plane

#Calculation
n=4*(1/4);     #Number of atoms contained in a plane per unit cell
A=a**2;     #Area of the plane(mm^2)
d=n/A;      #The planar atomic density(atoms/mm^2)

#Result
print "The planar atomic density is",d,"atoms/mm^2"

The planar atomic density is 6.25e+12 atoms/mm^2

Example number 3.24, Page number 79

In [48]:
#importing modules
import math
from __future__ import division

#Variable declaration
n=4;     #no.of atoms in Face centered cubic lattice
d=6250;    #density of potassium bromide(Kg/m^3)
AW=60.2;   #molecular weight of crysal with face centered cubic lattice
N=6.023*10**26;    #Avagadro's number(per Kg mol)

#Calculation
a=((n*AW/(d*N))**(1/3));     #The lattice constant(m)

#Result
print "The lattice constant is",round(a*10**10),"*10**-10 m"

The lattice constant is 4.0 *10**-10 m

Example number 3.25, Page number 79

In [50]:
#importing modules
import math
from __future__ import division

#Variable declaration
r1=0.1258*10**-9;     #atomic radii of the iron atom in BCC structure(m)
r2=0.1292*10**-9;    #atomic radii of the iron atom in FCC structure(m)
T=910;    #metallic iron changes from BCC to FCC(C)

#Calculation
a1=(4*r1/math.sqrt(3));     #lattice constant of BCC structure(m)
v1=a1**3/2;     #The volume occupied by one BCC atom(m^3)
a2=4*r2/math.sqrt(2);    #lattice constant of FCC structure(m)
v2=a2**3/4;     #The volume occupied by one FCC atom(m^3)
V=((v1-v2)/v1)*100;    #The change in volume percentage

#Result
print "The change in volume percentage is",round(V,5)
print "answer varies due to rounding off errors"

The change in volume percentage is 0.49326
answer varies due to rounding off errors

Example number 3.26, Page number 80

In [52]:
#importing modules
import math
from __future__ import division

#Variable declaration
a=0.405*10**-9;     #lattice constant of unit cell of aluminium which is face centered cubic(m)
s=25*10**-2;      #Side of aluminium foil(m)
t=0.005*10**-2;    #Thickness of aluminium foil(m)

#Calculation
ar=s**2;      #area of aluminium foil(m^2)
V=ar*t;       #volume of the aluminium foil(m^3)
v=a**3;       #volume of the unit cell(m^3)
n=(V/v);      #Number of unit cells

#Result
print "The Number of unit cells is",round(n/10**22,5),"*10**22"

The Number of unit cells is 4.70419 *10**22

Example number 3.27, Page number 81

In [59]:
#importing modules
import math
from __future__ import division

#Variable declaration
r=0.1605*10**-9;      #radius of magnesium atom which has HCP structure(m)

#Calculation
a=2*r;       #lattice constant of magnesium which has HCP structure(m)
c=a*math.sqrt(8/3);     #height of the HCP structure(m)
V=3*math.sqrt(3)*(a**2)*c/3;      #Volume of the unit cell of Magnesium which has HCP structure(m^3)

#Result
print "The Volume of the unit cell of Magnesium which has HCP structure is",round(V*10**28),"*10**-28 m^3"
print "answer given in the book is wrong"

The Volume of the unit cell of Magnesium which has HCP structure is 1.0 *10**-28 m^3
answer given in the book is wrong