# Chapter 1 : Elementary Materials Science Concepts¶

## Example 1.1 Page No : 4¶

In [1]:
# Given
A = 8.0*10**-77         # in J m**6
B = 1.12*10**-133       # in J m**12

# lennard-Jones 6-12 potential Energy (PE)curve is E(r)=-A*r**-6+B*r**-12
# For bonding to occur PE should be minimum, hence differentiating the PE equation and setting it to Zero at r=ro we get

# Calculations and Results
ro = (2.0*B/A)**(1.0/6)
print("Bond length in meters is {0:.5g}".format(ro))
E_bond = -A*ro**-6+(B*ro**-12)        # in J
E_bond = abs(E_bond/(1.6*10**-19))
print("Bond Energy for solid argon in ev is {0:.4f}".format(E_bond))

Bond length in meters is 3.7543e-10
Bond Energy for solid argon in ev is 0.0893


## Example 1.2 Page No : 5¶

In [2]:
import math

# Given
R = 8.314        # in J/mol/K
T = 27.0         # in degree celcius
T += 273.0       # in Kelvin
M_at = 14.0      # in g/mol
# From Kinetic Theory

# Calculations and Results
V_rms = math.sqrt((3.0*R*T)/(2.0*M_at*10**-3))
print("rms velocity of Nitrogen molecule in atmosphere at 300K in m/s is {0:.4f}".format(V_rms))
V_rmsx = V_rms/math.sqrt(3.0)
print("rms velocity in one direction in m/s is {0:.4f}".format(V_rmsx))

rms velocity of Nitrogen molecule in atmosphere at 300K in m/s is 516.9485
rms velocity in one direction in m/s is 298.4603


## Example 1.3 Page No : 10¶

In [3]:
# Given
R = 8.314        # in J/mol/K
M_at = 63.6      # in g/mol
# Acc. to Dulong -Petit rule Cm=3R for NA atoms

# Calculations and Results
C_gram = 3.0*R/M_at
print("Heat Capacity of copper per unit gram in J/g/K is {0:.4f}".format(C_gram))

Heat Capacity of copper per unit gram in J/g/K is 0.3922


## Example 1.4 Page No : 12¶

In [4]:
import math
# Given
k = 1.38*10**-23        # in J/K
m = 9.1*10**-31         # in Kg
T = 300.0               # in Kelvin

# Calculations and Results
v_av = math.sqrt(8.0*k*T/(math.pi*m))
print("Mean speed for a gas of non interacting electrons in Km is {0:.4f}".format(v_av*10**-3))
v = math.sqrt(2.0*k*T/m)
print("Most probable speed for a gas of non interacting electrons in Km is {0:.4f}".format(v*10**-3))
v_rms = math.sqrt(3.0*k*T/m)
print("rms velocity for a gas of non interacting electrons in Km is {0:.4f}".format(v_rms*10**-3))

Mean speed for a gas of non interacting electrons in Km is 107.6340
Most probable speed for a gas of non interacting electrons in Km is 95.3882
rms velocity for a gas of non interacting electrons in Km is 116.8262


## Example 1.5 Page No : 13¶

In [5]:
import math
# Given
L = 100*10**-6          # in Henry
C = 100*10**-12         # in Farad
T = 300.0               # in Kelvin
R = 200*10**3           # in ohms
k = 1.38*10**-23        # in J/K

# Calculations and Results
fo = 1.0/(2.0*math.pi*math.sqrt(L*C))       # resonant frequency
Q = 2.0*math.pi*fo*C*R                      # quality factor
B = fo/Q                                    # Bandwidth of tuned RLC
# Acc. to Johnson resistor noise equation

Vrms = math.sqrt(4*k*T*R*B)  # in volts
Vrms /= 10**-6  # in micro volts
print("Minimum rms radio signal that can be detected in micro volts is {0:.4f}".format(Vrms))

Minimum rms radio signal that can be detected in micro volts is 5.1338


## Example 1.7 Page No : 18¶

In [6]:
import math

# Given
n = 4.0
M_at = 63.55*10**-3         # Kg/mol
NA = 6.022*10**23           # mol**-1
R = 0.128                   # in nm
c = 8.0                     # no.of corners of unit cells
f = 6.0                       # no.of faces of unit cells

# Calculations and Results
# a
N = c*(1.0/8)+f*(1.0/2)
print("No. of atoms per unit cells is {0:.4f}".format(N))
# b
# Lattice parameter
a = R*2*2**(1.0/2)
print("Lattice Parameter in nm is {0:.4f}".format(a))
a *= 10**-9  # in m
# c
# APF=(No.of atoms in unit cell)*(Vol. of atom)/(Vol. of unit cell)
APF = 4**2*math.pi/(3*(2*math.sqrt(2.0))**3)
print("Atomic Packing Factor is {0:.4f}".format(APF))
# d
p = n*M_at/(a**3*NA)  # density
print("density of Copper in Kg/m3 is {0:.4f}".format(p))

No. of atoms per unit cells is 4.0000
Lattice Parameter in nm is 0.3620
Atomic Packing Factor is 0.7405
density of Copper in Kg/m3 is 8895.4906


## Example 1.8 Page No : 23¶

In [9]:
import math

# lcm function
def lcm(*args):
return cal_lcm(args[0][0], cal_lcm(args[0][1], args[0][2]))

def cal_lcm(x, y):
if x > y:
greater = x
else:
greater = y
while True:
if(greater % x == 0) and (greater % y == 0):
result = greater
break
greater += 1
return result

# Given
a = 1.0/float("inf")
b = -1.0/1
c = 2.0/1
p = (1.0, 1.0, 1.0)
# 1/%inf = 0  ; (0/1 -1/1 2/1) hence lcm is taken for [1 1 1]

# Calculations and Results
LCM = lcm(p)
h = a*float(LCM)
k = b*float(LCM)
l = c*float(LCM)
print("miller indices = {0:.4f} {1:.4f} {2:.4f}".format(h, k, l))

miller indices = 0.0000 -1.0000 2.0000


## Example 1.9 Page No : 28¶

In [10]:
import math

# Given
k = 1.38*10**-23        # J/K
T = 300.0               # kelvin
Ev = 0.75               # eV/atom
Ev = Ev*1.6*10**-19     # in J
T1 = 660.0              # degree celcius
T1 += 273.0             # in kelvin
# at room temperature
# let nv/N=nv_N for convenience

# Calculations and Results
nv_N = math.exp(-Ev/(k*T))
print("Fractional concentration of vacancies in the aluminium crystal at room temperature is {0:.4g}".format(nv_N))
# at melting temperature
# let nv/N=nv_N for convenience
nv_N = math.exp(-Ev/(k*T1))
print("Fractional concentration of vacancies in the aluminium crystal at melting temperature is {0:.4g}".format(nv_N))

Fractional concentration of vacancies in the aluminium crystal at room temperature is 2.581e-13
Fractional concentration of vacancies in the aluminium crystal at melting temperature is 8.961e-05


## Example 1.10 Page No : 33¶

In [11]:
import math

# Given
NA = 6.023*10**23       # mol**-1
d = 2.33                # density of Si in g/cm3
Mat = 28.09             # g/mol
Ev = 2.4                # ev/atom
Ev = 2.4*1.6*10**-19    # J/atom
k = 1.38*10**-23        # J/K
T = 300.0                 # kelvin
T1 = 1000.0               # degree celsius
T1 += 273.0               # in kelvin

# Calculations and Results
N = (NA*d)/Mat
# at room temperature
nv = N*math.exp(-(Ev/(k*T)))
print("concentration of vacancies in a Si crystal at room temperature in cm**-3 is {0:.4g}".format(nv))
# at 1000 degree celsius
nv = N*math.exp(-(Ev/(k*T1)))
print("concentration of vacancies in a Si crystal at 1000 degree celsius in cm**-3 is {0:.4g}".format(nv))

concentration of vacancies in a Si crystal at room temperature in cm**-3 is 2.608e-18
concentration of vacancies in a Si crystal at 1000 degree celsius in cm**-3 is 1.605e+13


## Example 1.11 Page No : 34¶

In [12]:
# Given
# from fig 7.1
# at 210 degree celsius
print("At 210 degree celsius")
C_L = 50.0        # CL=50% Sn
C_alpha = 18.0    # C_alpha=18% Sn
Co = 40.0         # solidification of alloy

# Calculations and Results
# lever rule
W_alpha = (C_L-Co)/(C_L-C_alpha)
print("weight fraction of alpha in the alloy is {0:.4f}".format(W_alpha*100))
W_L = 1-W_alpha
print("weight fraction of liquid phase in the alloy is {0:.4f}".format(W_L*100))

# Given
# at 183.5 degree celsius
print("At 183.5 degree celsius")
C_L = 61.9        # CL=50% Sn
C_alpha = 19.2    # C_alpha=18% Sn
Co = 40.0         # solidification of alloy

# Calculation and Results
# lever rule
W_alpha = (C_L-Co)/(C_L-C_alpha)
print("weight fraction of alpha in the alloy is {0:.4f}".format(W_alpha*100))
W_L = 1-W_alpha
print("weight fraction of liquid phase in the alloy is {0:.4f}".format(W_L*100))

# Given
# at 182.5 degree celsius
print("At 182.5 degree celsius")
C_beta = 97.5     # CL=50% Sn
C_alpha = 19.2    # C_alpha=18% Sn
Co = 40.0         # solidification of alloy

# Calculations and Results
# lever rule
W_alpha = (C_beta-Co)/(C_beta-C_alpha)
print("weight fraction of alpha in the alloy is {0:.4f}".format(W_alpha*100))
W_beta = 1-W_alpha
print("weight fraction of beta phase in the alloy is {0:.4f}".format(W_beta*100))

At 210 degree celsius
weight fraction of alpha in the alloy is 31.2500
weight fraction of liquid phase in the alloy is 68.7500
At 183.5 degree celsius
weight fraction of alpha in the alloy is 51.2881
weight fraction of liquid phase in the alloy is 48.7119
At 182.5 degree celsius
weight fraction of alpha in the alloy is 73.4355
weight fraction of beta phase in the alloy is 26.5645