2: Crystallography and Crystal Structures

Example number 2.3, Page number 2.9

In [5]:
#importing modules
import math
from __future__ import division
from sympy import Symbol
import numpy as np

#Variable declaration
a=Symbol('a');      #lattice constant(mm)
x1=4;
x2=math.sqrt(2);
b=a*math.sqrt(2);
theta=30;   #angle(degrees)

#Calculation
theta=theta*math.pi/180;    #angle(radian)
na1=x1*1/(x1*a**2);     #number of atoms in (100)(per mm**2)
na2=1/(x2*a**2);      #number of atoms in (110)(per mm**2)
A3=(1/2)*b*b*math.cos(theta);  
t=60/360*3;
na3=t/A3;     #number of atoms in (111)(per mm**2)

#Result
print "number of atoms in (100) is",na1,"atoms/mm**2"
print "number of atoms in (110) is",na2,"atoms/mm**2"
print "number of atoms in (111) is",na3,"atoms/mm**2"

number of atoms in (100) is a**(-2) atoms/mm**2
number of atoms in (110) is 0.707106781186547/a**2 atoms/mm**2
number of atoms in (111) is 0.577350269189626/a**2 atoms/mm**2

Example number 2.4, Page number 2.11

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

#Variable declaration
r=0.1278;     #atomic radius(m)
h1=1;
k1=1;
l1=0;
h2=2;
k2=1;
l2=2;

#Calculation
a=round(4*r/math.sqrt(2),4);
d110=a/math.sqrt(h1**2+k1**2+l1**2);    #interplanar spacing for (110)(nm)
d212=a/math.sqrt(h2**2+k2**2+l2**2);    #interplanar spacing for (212)(nm)

#Result
print "interplanar spacing for (110) is",round(d110,4),"nm"
print "interplanar spacing for (212) is",d212,"nm"

interplanar spacing for (110) is 0.2556 nm
interplanar spacing for (212) is 0.1205 nm

Example number 2.5, Page number 2.11

In [3]:
#importing modules
import math
from __future__ import division

#Variable declaration
h1=1;
k1=0;
l1=0;
h2=1;
k2=1;
l2=0;
h3=1;
k3=1;
l3=1;

#Calculation
d100=1/math.sqrt(h1**2+k1**2+l1**2);    #interplanar spacing for (110)
d110=1/math.sqrt(h2**2+k2**2+l2**2);    #interplanar spacing for (110)
d111=1/math.sqrt(h3**2+k3**2+l3**2);    #interplanar spacing for (111)

#Result
print "seperation between successive lattice planes is",int(d100),":",round(d110,2),":",round(d111,2)

seperation between successive lattice planes is 1 : 0.71 : 0.58

Example number 2.6, Page number 2.12

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

#Variable declaration
a=1;
b=1/2;
c=3;

#Calculation
A=1/a;
B=1/b;
C=1/c;
h=A*c;
k=B*c;
l=C*c;      #miller indices of plane

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

miller indices of plane is ( 3.0 6.0 1.0 )

Example number 2.7, Page number 2.22

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

#Variable declaration
r=1;    #assume

#Calculation
a=4/math.sqrt(3);
R=(a-(2*r))/2;      #radius of interstitial sphere(r)

#Result
print "radius of interstitial sphere is",round(R,3),"r"

radius of interstitial sphere is 0.155 r

Example number 2.8, Page number 2.23

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

#Variable declaration
r1=1.258;     #atomic radius(angstrom)
r2=1.292;     #atomic radius(angstrom)

#Calculation
a1=4*r1/math.sqrt(3);       #spacing(angstrom)
n1=((1/8)*8)+1;      #number of atoms per unit cell
v1=a1**3/n1;           #volume occupied by 1 atom(m**3)
n2=(1/2*6)+(1/8*8);    #number of atoms per unit cell
a2=2*math.sqrt(2)*r2;   #spacing(angstrom)
v2=a2**3/n2;           #volume occupied by 1 atom(m**3)
dc=(v1-v2)*100/v1;     #change in volume(%)

#Result
print "decrease of volume is",round(dc,1),"%"                                    

decrease of volume is 0.5 %

Example number 2.9, Page number 2.24

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

#Variable declaration
a=0.27*10**-9;     #spacing(m)
c=0.494*10**-9;
n=6;    #number of atoms
M=65.37;    #atomic weight
N=6.023*10**26;    #avagadro number

#Calculation
V=3*math.sqrt(3)*a**2*c/2;     #volume of unit cell(m**3)
rho=n*M/(N*V);     #density of zinc(kg/m**3)

#Result
print "volume of unit cell is",round(V*10**29,3),"*10**-29 m**3"
print "density of zinc is",int(rho),"kg/m**3"
print "answer varies due to rounding off errors"

volume of unit cell is 9.356 *10**-29 m**3
density of zinc is 6960 kg/m**3
answer varies due to rounding off errors

Example number 2.10, Page number 2.24

In [14]:
#importing modules
import math
from __future__ import division

#Variable declaration
r=1;    #assume

#Calculation
a=4*r/math.sqrt(2);
R=(a/2)-r;      #radius of interstitial sphere(r)

#Result
print "radius of interstitial sphere is",round(R,3),"r"

radius of interstitial sphere is 0.414 r

Example number 2.11, Page number 2.25

In [19]:
#importing modules
import math
from __future__ import division

#Variable declaration
a=0.356*10**-9;       #cube edge(m)
m=12.01;    #atomic weight of carbon
N=6.023*10**26;    #avagadro number

#Calculation
n=8/a**3;     #number of atoms per m**3
M=m/N;
d=M*n;       #density of diamond(kg/m**3)

#Result
print "number of atoms per m**3 is",round(n/10**29,2),"*10**29"
print "density of diamond is",round(d,1),"kg/m**3"
print "answer varies due to rounding off errors"

number of atoms per m**3 is 1.77 *10**29
density of diamond is 3535.7 kg/m**3
answer varies due to rounding off errors

Example number 2.12, Page number 2.26

In [22]:
#importing modules
import math
from __future__ import division

#Variable declaration
mw=23+35.5;     #molecular weight of NaCl(gm/mol)
N=6.023*10**23;    #avagadro number(per mol)
d=2.18;       #mass of unit volume

#Calculation
M=mw/N;     #mass of NaCl molecule(gm)
n=2*d/M;      #number of atoms per unit volume(atoms/cm**3)
a=(1/n)**(1/3);     #distance between 2 adjacent atoms(cm)

#Result
print "distance between 2 adjacent atoms is",round(a*10**8,2),"angstrom"

distance between 2 adjacent atoms is 2.81 angstrom

Example number 2.13, Page number 2.26

In [27]:
#importing modules
import math
from __future__ import division

#Variable declaration
M=63.5;    #atomic weight
N=6.023*10**23;    #avagadro number
r=1.278*10**-8;    #radius(m)
n=4;

#Calculation
m=M/N;     #mass of copper atom(gm)
a=4*r/math.sqrt(2);
Mu=n*m;     #mass of unit cell
d=Mu/a**3;    #density of copper crystal(gm/cm**3)

#Result
print "density of copper crystal is",round(d,3),"gm/cm**3"

density of copper crystal is 8.929 gm/cm**3

Example number 2.14, Page number 2.27

In [30]:
#importing modules
import math
from __future__ import division

#Variable declaration
r=0.1249*10**-9;     #radius(m)
pf=0.68;      #packing factor

#Calculation
a=4*r/math.sqrt(3);    #lattice constant(m)
v=a**3;     #volume of unit cell(m**3)
Fv=(1-pf)*v;     #free volume per unit cell(m**3)

#Result
print "free volume per unit cell is",round(Fv*10**30,4),"*10**-30 m**3"
print "answer varies due to rounding off errors"

free volume per unit cell is 7.6795 *10**-30 m**3
answer varies due to rounding off errors