Crystallography

Example number 6.1, Page number 185

In [3]:
#importing modules
import math

#Variable declaration
r=0.071;      #radius in nm
N=6.022*10**26;    

#Calculation
r=r*10**-9;        #converting r from nm to m
#mass of carbon atom m = 12/N
m=12/N;
#mass of diamond M = 8*mass of one carbon atom
M=8*m;
#volume of diamond V = (8*r/sqrt(3))^3
V=(8*r/math.sqrt(3))**3;
d=M/V;    #density in kg/m^3
d=math.ceil(d*100)/100;   #rounding off to 2 decimals

#Result
print("density of diamond in kg/m^3 is",d);

('density of diamond in kg/m^3 is', 4520.31)

Example number 6.2, Page number 185

In [4]:
#importing modules
import math

#Variable declaration
aBCC=0.332;      #lattice constant in nm
aHCP=0.296;      #lattice constant in nm
c=0.468;      #c in nm

#Calculation
aBCC=aBCC*10**-9;     #converting nm to m
Vbcc=aBCC**3;
aHCP=aHCP*10**-9;     #converting nm to m
c=c*10**-9;     #converting nm to m
Vhcp=6*(math.sqrt(3)/4)*aHCP**2*c;
V=Vhcp-Vbcc;
Vch=(V*100)/Vbcc;
Vch=math.ceil(Vch*100)/100;   #rounding off to 2 decimals

#Result
print("percentage change in volume is",Vch);

#answer given in the book is wrong

('percentage change in volume is', 191.12)

Example number 6.3, Page number 186

In [6]:
#importing modules
import math

#Variable declaration
r=1.278;     #atomic radius of Cu in Angstrom
A=63.54;     #atomic weight of Cu
n=4;     #for FCC n=4
Na=6.022*10**26;

#Calculation
r=r*10**-10;    #converting atomic radius from Angstrom to m
a=2*math.sqrt(2)*r;     
rho=(n*A)/(Na*a**3);
rho=math.ceil(rho*100)/100;   #rounding off to 2 decimals

#Result
print("density of Cu in kg/m^3 is",rho);

#answer given in the book is wrong

('density of Cu in kg/m^3 is', 8935.92)

Example number 6.4, Page number 186

In [7]:
#importing modules
import math
import numpy as np

#Variable declaration
rho=2180;     #density of NaCl in kg/m^3
wNa=23;      #atomic weight of Na
wCl=35.5;      #atomic weight of Cl
n=4;   #for FCC n=4
Na=6.022*10**26;

#Calculation
A=wNa+wCl;     #molecular weight of NaCl
x=np.reciprocal(3.);
a=((n*A)/(Na*rho))**x;

#Result
print("interatomic distance in NaCl in m is",a); 

('interatomic distance in NaCl in m is', 5.6278114346454509e-10)

Example number 6.5, Page number 187

In [8]:
#importing modules
import math

#Variable declaration
a=0.42;    #lattice constant in nm
h1=1;
k1=0;
l1=1;    #indices of the plane (101)
h2=2;
k2=2;
l2=1;    #indices of the plane (221)

#Calculation
a=a*10**-9;    #converting from nm to m
d1=a/math.sqrt((h1**2)+(k1**2)+(l1**2));     #interplanar spacing for plane (101)
d1=d1*10**9;    #converting from m to nm
d1=math.ceil(d1*10**5)/10**5;   #rounding off to 5 decimals
d2=a/math.sqrt((h2**2)+(k2**2)+(l2**2));     #interplanar spacing for plane (221)
d2=d2*10**9;    #converting from m to nm

#Result
print("interplanar spacing for (101) in nm is",d1);
print("interplanar spacing for (221) in nm is",d2);

('interplanar spacing for (101) in nm is', 0.29699)
('interplanar spacing for (221) in nm is', 0.14)

Example number 6.6, Page number 187

In [10]:
#Variable declaration
h1=1;
k1=0;
l1=2;    #indices for plane (102)
h2=2;
k2=3;
l2=1;    #indices for plane (231)
h3=3;
k3=-1;
l3=2;    #indices for plane (31'2)

#Calculation
#intercepts made by the plane is a/h, b/k, c/l
#for plane (102) intercepts are a/1=a, b/0=infinite, c/2
#for plane (231) intercepts are a/2, b/3, c/1=c
#for plane (31'2) intercepts are a/3=a, b/-1=-b, c/2

#Result
print("for plane (102) intercepts are a/1=a, b/0=infinite, c/2");
print("for plane (231) intercepts are a/2, b/3, c/1=c");
print("for plane (312) intercepts are a/3=a, b/-1=-b, c/2");

for plane (102) intercepts are a/1=a, b/0=infinite, c/2
for plane (231) intercepts are a/2, b/3, c/1=c
for plane (312) intercepts are a/3=a, b/-1=-b, c/2

Example number 6.7, Page number 188

In [13]:
#importing modules
import math

#Variable declaration
u1=1;
v1=1;
w1=1;    #indices for plane (111)
u2=2;
v2=1;
w2=2;    #indices for plane (212)

#Calculation
A=u1*u2+v1*v2+w1*w2;    
B1=math.sqrt((u1**2)+(v1**2)+(w1**2));
B2=math.sqrt((u2**2)+(v2**2)+(w2**2));
B=A/(B1*B2);
B=math.ceil(B*10**4)/10**4;   #rounding off to 4 decimals
theta=math.acos(B);    #angle in radian
theta=theta*57.2957795;     #converting radian to degrees
theeta=math.ceil(theta*10**3)/10**3;   #rounding off to 3 decimals
deg=int(theta);     #converting to degrees
t=60*(theta-deg);
mi=int(t);        #converting to minutes
sec=60*(t-mi);      #converting to seconds
sec=math.ceil(sec*10**2)/10**2;   #rounding off to 2 decimals

#Result
print("angle between the planes in degrees is",theeta);
print("angle between the planes is",deg,"degrees",mi,"minutes",sec,"seconds");

#answer given in the book is wrong

('angle between the planes in degrees is', 15.783)
('angle between the planes is', 15, 'degrees', 46, 'minutes', 57.85, 'seconds')

Example number 6.8, Page number 188

In [14]:

Example number 6.9, Page number 189

In [15]:
#importing modules
import math

#Variable declaration
d=0.2338;     #interplanar distance in nm
h=-1;
k=1;
l=1;   #indices of the plane (1'11)

#Calculation
d=d*10**-9;       #converting from nm to m
a=d*math.sqrt((h**2)+(k**2)+(l**2));
a=a*10**9;      #converting lattice constant from m to nm
a=math.ceil(a*10**5)/10**5;   #rounding off to 5 decimals

#Result
print("lattice constant in nm is",a);

('lattice constant in nm is', 0.40496)

Example number 6.10, Page number 189

In [16]:
#importing modules
import math

#variable declaration
h1=1;
k1=0;
l1=0;    #indices for plane (100)
h2=1;
k2=1;
l2=0;    #indices for plane (110)
h3=1;
k3=1;
l3=1;    #indices for plane (111)

#Calculation
#d=a/math.sqrt((h**2)+(k**2)+(l**2))
#d100=a/math.sqrt((h1**2)+(k1**2)+(l1**2))
x1=math.sqrt((h1**2)+(k1**2)+(l1**2));
#d100=a/x1 = a/1 = a
#d110=a/math.sqrt((h2**2)+(k2**2)+(l2**2))
x2=math.sqrt((h2**2)+(k2**2)+(l2**2));
x2=math.ceil(x2*10**4)/10**4;   #rounding off to 4 decimals
#d110=a/x2 = a/sqrt(2)
#d111=a/math.sqrt((h3**2)+(k3**2)+(l3**2))
x3=math.sqrt((h3**2)+(k3**2)+(l3**2));
x3=math.ceil(x3*10**4)/10**4;   #rounding off to 4 decimals
#d111=a/x3 = a/sqrt(3)
#hence d100:d110:d111=a:a/sqrt(2):a/sqrt(3)
#multiplying RHS by sqrt(6) we get d100:d110:d111=sqrt(6):sqrt(3):sqrt(2)

#Result
print("value of x1 is",x1);
print("value of x2 is",x2);
print("value of x3 is",x3);
print("d100:d110:d111=sqrt(6):sqrt(3):sqrt(2)");

('value of x1 is', 1.0)
('value of x2 is', 1.4143)
('value of x3 is', 1.7321)
d100:d110:d111=sqrt(6):sqrt(3):sqrt(2)

Example number 6.11, Page number 190

In [17]:
#variable declaration
h=2;
k=3;
l=1;    #indices for plane (231)

#Calculation
#intercepts made by the plane is a/h, b/k, c/l
#for a cubic unit cell, a=b=c
#for plane (231) intercepts are a/2, a/3, a/1 = a
#ratio of the intercepts is 1/2:1/3:1
#LCM is 6. multiplying by LCM, we get ratio l1:l2:l3 = 3:2:6

#Result
print("l1:l2:l3 = 3:2:6");

l1:l2:l3 = 3:2:6

Example number 6.12, Page number 190

In [18]:
#variable declaration
h=1;
k=2;
l=3;    #indices for plane (123)
l1=0.8;       #l1 in armstrong
a=0.8;       #a in armstrong
b=1.2;       #b in armstrong
c=1.5;       #c in armstrong

#Calculation
#intercepts made by the plane is a/h, b/k, c/l
#for plane (123) intercepts are a/1 = a, b/2, c/3
#ratio of the intercepts l1:l2:l3 = a:b/2:c/3
#thus 0.8:l2:l3 = 0.8:1.2/2:1.5/3
l2=1.2/2;     #l2 in armstrong
l3=1.5/3;     #l3 in armstrong

#Result
print("value of l2 in armstrong is",l2);
print("value of l3 in armstrong is",l3);

('value of l2 in armstrong is', 0.6)
('value of l3 in armstrong is', 0.5)

Example number 6.13, Page number 191

In [19]:
#Result
print("in simple cubic unit cell nearest neighbour distance is a");
print("in body centered cubic unit cell nearest neighbour distance is sqrt(3)*a/2");
print("in face centered cubic unit cell nearest neighbour distance is a/sqrt(2)");

in simple cubic unit cell nearest neighbour distance is a
in body centered cubic unit cell nearest neighbour distance is sqrt(3)*a/2
in face centered cubic unit cell nearest neighbour distance is a/sqrt(2)

Example number 6.14, Page number 191

In [20]:
#importing modules
import math

#variable declaration
a=2.04;     #lattice parameter in armstrong
h=2;
k=1;
l=2;    #indices for plane (212)

#Calculation
a=a*10**-10;    #converting from armstrong to m
d=a/math.sqrt((h**2)+(k**2)+(l**2));
d=d*10**10;    #converting from m to armstrong
d=math.ceil(d*10**3)/10**3;   #rounding off to 3 decimals

#Result
print("interplanar distance in armstrong is",d);

('interplanar distance in armstrong is', 0.681)

Example number 6.15, Page number 191

In [21]:
#importing modules
import math

#variable declaration
r=1.278;      #radius of Cu in armstrong
M=63.54;     #atomic weight of Cu
rho=8980;    #density in kg/m^3
Na=6.022*10**26;

#Calculation
r=r*10**-10;    #radius in m
a=math.sqrt(8)*r;
n=(rho*Na*a**3)/M;

#Result
print("interatomic distance in m is",a);
print("number of atoms per Cu unit cell is",int(n));

('interatomic distance in m is', 3.6147298654256317e-10)
('number of atoms per Cu unit cell is', 4)

Example number 6.16, Page number 192

In [12]:
#variable declaration
a=0.429;
b=1;
c=0.379;    #intercepts of an orthorhombic crystal

#Calculation
#ratio of intercepts are 0.214:1:0.188 = (a/0.429)*0.214:1:(c/0.379)*0.188 = a/2:b:c/2
#thus the coefficients are 1/2:1:1/2. inverses are 2,1,2.
#thus miller indices for the first plane are (212)
#ratio of intercepts are 0.858:1:0.754 = (a/0.429)*0.0.858:1:(c/0.379)*0.754 = 2a:b:2c
#thus the coefficients are 2:1:2. inverses are 1/2,1,1/2. LCM is 2. multiplying with LCM we get 1,2,1
#thus miller indices for the second plane are (121)
#ratio of intercepts are 0.429:infinite:0.126 = (a/0.429)*0.429:infinite:(c/0.379)*0.126 = a:infiniteb:c/3
#thus the coefficients are 1:infinte:1/3. inverses are 1,0,3.
#thus miller indices for the third plane are (103)

#Result
print("miller indices for the first plane are (212)");
print("miller indices for the second plane are (121)");
print("miller indices for the third plane are (103)");

miller indices for the first plane are (212)
miller indices for the second plane are (121)
miller indices for the third plane are (103)

Example number 6.17, Page number 193

In [22]:
#importing modules
import math
import numpy as np

#variable declaration
h1=1;
k1=0;
l1=0;    #indices of the first plane (100)
h2=1;
k2=1;
l2=0;    #indices of the second plane (110)
h3=1;
k3=1;
l3=1;    #indices of the third plane (111)

#Calculation
n_1=np.reciprocal(4.);
n_2=np.reciprocal(2.);
n_3=np.reciprocal(6.);
n1=(n_1*4)+1;    #number of atoms per unit cell in (100)
#number of atoms per m^2 is 2/a**2. but a=sqrt(8)*r.
#hence number of atoms per m^2 is 1/(4*r**2)
n2=(n_1*4)+(2*n_2);   #number of atoms per unit cell in (110)
#number of atoms per m^2 is 1/a*sqrt(2)*a. but a=sqrt(8)*r.
#hence number of atoms per m^2 is 1/(8*sqrt(2)*r**2)
n3=(n_3*3)+(3*n_2);   #number of atoms per unit cell in (111)
#number of atoms per m^2 is 2/(sqrt(3)/4)*a**2. but a=4*r.
#hence number of atoms per m^2 is 1/(2*sqrt(3)*r**2)

#Result
print("number of atoms per unit cell in (100)",n1);
print("number of atoms per m^2 is 1/(4*r**2)");
print("number of atoms per unit cell in (110)",n2);
print("number of atoms per m^2 is 1/(8*sqrt(2)*r**2)");
print("number of atoms per unit cell in (111)",n3);
print("number of atoms per m^2 is 1/(2*sqrt(3)*r**2)");

('number of atoms per unit cell in (100)', 2.0)
number of atoms per m^2 is 1/(4*r**2)
('number of atoms per unit cell in (110)', 2.0)
number of atoms per m^2 is 1/(8*sqrt(2)*r**2)
('number of atoms per unit cell in (111)', 2.0)
number of atoms per m^2 is 1/(2*sqrt(3)*r**2)

Example number 6.18, Page number 194

In [24]:
#importing modules
import math

#variable declaration
r=0.97;    #radius of Na+ ion in armstrong
R=1.81;    #radius of Cl- ion in armstrong

#Calculation
#atomic packing factor=packing density PD
#PD=Volume of atoms/Volume of unit cell
#volume of unit cell=a**3
#volume of atoms=number of atoms*volume of 1 atom = 4*(4/3)*math.pi*r**3
#but r=a/sqrt(8). hence PD = 4*(4/3)*math.pi*(a/(2*sqrt(2)))**3*(1/a**3) = 0.74
#atomic packing factor = 0.74
r=r*10**-10;   #radius of Na+ ion in m
R=R*10**-10;   #radius of Cl- ion in m
Vna = (4*4*math.pi*r**3)/3;    #volume of Na atoms
Vcl = (4*4*math.pi*R**3)/3;    #volume of Cl atoms 
V=(2*(r+R))**3;      #volume of unit cell
IPF=(Vna+Vcl)/V;     #ionic packing factor
IPF=math.ceil(IPF*10**4)/10**4;   #rounding off to 4 decimals

#Result
print("atomic packing factor = 0.74");
print("ionic packing factor of NaCl crystal is",IPF);

atomic packing factor = 0.74
('ionic packing factor of NaCl crystal is', 0.6671)
In [ ]: