Electron Theory of Metals

Example number 2.1, Page number 69

In [5]:
#import module
import math

#Calculation
# given that E-Ef = kT
# fermi function FE = 1/(1+exp((E-Ef)/kT)
# therefore FE = 1/(1+exp(kT/kT));
# FE = 1/(1+exp(1))
FE=1/(1+math.exp(1));
FE=math.ceil(FE*10**2)/10**2;   #rounding off to 2 decimals

#Result
print("fermi function is",FE);

('fermi function is', 0.27)

Example number 2.2, Page number 69

In [6]:
 
#import module
import math

#Calculation
# given that E-Ef = kT
# fermi function FE = 1/(1+exp((E-Ef)/kT)
# therefore FE = 1/(1+exp(kT/kT));
# FE = 1/(1+exp(1))
FE=1/(1+math.exp(1));
FE=math.ceil(FE*10**3)/10**3;   #rounding off to 3 decimals

#Result
print("fermi function is",FE);

('fermi function is', 0.269)

Example number 2.3, Page number 69

In [8]:
 
#import module
import math
from __future__ import division

#Variable decleration
FE=10/100;    #fermi function is 10%
Ef=5.5;   #fermi energy of silver in eV
k=1.38*10**-23;

#Calculation
E=Ef+(Ef/100);
#FE=1/(1+math.exp((E-Ef)/(k*T)))
#therefore 1/FE = 1+math.exp((E-Ef)/(k*T))
#therefore (1/FE)-1 = math.exp((E-Ef)/(k*T))
#therefore log((1/FE)-1) = (E-Ef)/(k*T)
#therefore T = (E-Ef)/(k*math.log((1/FE)-1))
#let X=E-Ef;    
X=E-Ef;     #energy in eV
X=X*1.6*10**-19;    #energy in J
T = (X/(k*math.log((1/FE)-1)));
T=math.ceil(T*10**2)/10**2;   #rounding off to 2 decimals

#Result
print("temperature in K is",T);

('temperature in K is', 290.23)

Example number 2.4, Page number 70 **

In [14]:
 
#import module
import math
from __future__ import division

#Variable decleration
#let X=E-Ef
X=0.5;   #E-Ef=0.5 in eV

#Calculation
X=X*1.6*10**-19;   #X in J
FE=1/100;    #fermi function is 1% 
k=1.38*10**-23;
#FE=1/(1+exp(X/(k*T)))
#therefore 1/FE = 1+math.exp(X/(k*T))
#therefore (1/FE)-1 = math.exp(X/(k*T))
#therefore log((1/FE)-1) = X/(k*T)
#but log(x) = 2.303*math.log10(x)
#therefore T = X/(k*math.log((1/FE)-1))
#but log(x)=2.303*math.log10(x)
#therefore T = X/(k*2.303*math.log10((1/FE)-1))
T = X/(k*2.303*math.log10((1/FE)-1));

#Result
print("temperature in K is",T);

('temperature in K is', 1261.3505710887953)

Example number 2.5, Page number 71 *

In [16]:
 
#import module
import math
from __future__ import division

#Variable decleration
rho_s=10.5*10**3;   #density in kg/m^3
NA=6.02*10**26;    #avagadro number per kmol
MA=107.9; 

#Calculation
n=(rho_s*NA)/MA;
sigma=6.8*10**7;
e=1.6*10**-19;   #charge in coulomb
mew=sigma/(n*e);
mew=math.ceil(mew*10**6)/10**6;   #rounding off to 6 decimals

#Result
print("density of electrons is",n);
print("mobility of electrons in silver in m^2/Vs is",mew);

('density of electrons is', 5.85820203892493e+28)
('mobility of electrons in silver in m^2/Vs is', 0.007255)

Example number 2.6, Page number 71 *

In [18]:
 
#import module
import math
from __future__ import division

#Variable decleration
d=8.92*10**3;   #density in kg/m^3
rho=1.73*10**-8;  #resistivity in ohm-m
m=9.1*10**-31;    #mass in kg
w=63.5;    #atomic weight
e=1.6*10**-19;    #charge in coulomb
A=6.02*10**26;    #avagadro number

#Calculation
n=(d*A)/w;
mew=1/(rho*n*e);
tow=m/(n*(e**2)*rho);
mew=math.ceil(mew*10**6)/10**6;   #rounding off to 6 decimals

#Result
print("mobility of electrons in Copper in m/Vs is",mew);
print("average time of collision of electrons in copper in sec is",tow);

('mobility of electrons in Copper in m/Vs is', 0.004273)
('average time of collision of electrons in copper in sec is', 2.4297841992299697e-14)

Example number 2.7, Page number 72

In [19]:
 
#import module
import math
from __future__ import division

#Variable decleration
rho=1.54*10**-8;    #resistivity in ohm-m
n=5.8*10**28;       #electron/m^3
m=9.108*10**-31;   #mass in kg
e=1.602*10**-19;   #charge in coulomb

#Calculation
tow=m/(n*(e**2)*rho);

#Result
print("relaxation time of conduction electrons in sec is",tow);

('relaxation time of conduction electrons in sec is', 3.973281032516849e-14)

Example number 2.8, Page number 73

In [21]:
 
#import module
import math
from __future__ import division

#Variable decleration
FE=10/100;    #fermi function is 10%
Ef=5.5;   #fermi energy of silver in eV
k=1.38*10**-23;

#Calculation
E=Ef+(Ef/100);
#FE=1/(1+math.exp((E-Ef)/(k*T)))
#therefore 1/FE = 1+math.exp((E-Ef)/(k*T))
#therefore (1/FE)-1 = math.exp((E-Ef)/(k*T))
#therefore log((1/FE)-1) = (E-Ef)/(k*T)
#therefore T = (E-Ef)/(k*math.log((1/FE)-1))
#let X=E-Ef;    
X=E-Ef;     #energy in eV
X=X*1.6*10**-19;    #energy in J
T = (X/(k*math.log((1/FE)-1)));
T=math.ceil(T*10**2)/10**2;   #rounding off to 2 decimals

#Result
print("temperature in K is",T);

('temperature in K is', 290.23)

Example number 2.9, Page number 73

In [23]:
 
#import module
import math

#Calculation
# given that E-Ef = kT
# fermi function FpE = 1/(1+exp((E-Ef)/kT)
# therefore FpE = 1/(1+exp(kT/kT));
# FpE = 1/(1+exp(1))
FpE=1/(1+math.exp(1));
FpE=math.ceil(FpE*10**2)/10**2;   #rounding off to 2 decimals

#Result
print("fermi function is",FpE);
#the presence of electron at that energy level is not certain

('fermi function is', 0.27)

Example number 2.10, Page number 74 **

In [22]:
 
#import module
import math
from __future__ import division

#Variable decleration
m=9.1*10**-31;   #mass in kg
h=6.626*10**-34;
A=(8*m)**(3/2);

#Calculation
B=math.pi/(2*h**3);
EfeV=3.10;    #fermi energy in eV
Ef=EfeV*1.6*10**-19;    #fermi energy in J
EFeV=EfeV+0.02;    #energy after interval in eV
EF=EFeV*1.6*10**-19;    #energy after interval in J
def f(E):
    Q=A*B*math.sqrt(E)
    
I=(Ef,EF,f)

#Result
print("number of energy states per unit volume is",I);

('number of energy states per unit volume is', (4.960000000000001e-19, 4.992000000000001e-19, <function f at 0x7f1495202848>))

Example number 2.11, Page number 74

In [27]:
 
#import module
import math
from __future__ import division

#Variable decleration
T=300;   #temperature in K
n=8.5*10**28;   #density per m^3
rho=1.69*10**-8;   #resistivity in ohm/m^3
me=9.11*10**-31;    #mass of electron in kg
e=1.6*10**-19;    #charge in coulomb
KB=1.38*10**-23;    #boltzmann constant in J/k

#Calculation
lamda=math.sqrt(3*KB*me*T)/(n*(e**2)*rho);

#Result
print("mean free path of electron in m is",lamda);

#answer given in the book is wrong

('mean free path of electron in m is', 2.892506814374228e-09)

Example number 2.12, Page number 75

In [28]:
 
import math
from __future__ import division

#Variable decleration
rho=1.43*10**-8;    #resistivity in ohm-m
n=6.5*10**28;    #electron/m^3
m=9.11*10**-34;  #mass in kg
e=1.6*10**-19;   #charge in coulomb

#Calculation
tow=m/(n*(e**2)*rho);

#Result
print("relaxation time of conduction electrons in sec is",tow);

('relaxation time of conduction electrons in sec is', 3.8285032275416887e-17)

Example number 2.13, Page number 75 **

In [31]:
 
import math
from __future__ import division

#Variable decleration
d=8.92*10**3;   #density in kg/m^3
rho=1.73*10**-8;  #resistivity in ohm-m
m=9.1*10**-31;    #mass in kg
M=63.5;    #atomic weight
e=1.6*10**-19;    #charge in coulomb
A=6.02*10**26;    #avagadro number

#Calculation
n=(d*A)/M;
mew=1/(rho*n*e);
tow=m/(n*(e**2)*rho);
mew=math.ceil(mew*10**6)/10**6;   #rounding off to 6 decimals

#Result
print("mobility of electrons in Copper in m/Vs is",mew);
print("average time of collision of electrons in copper in sec is",tow);

('mobility of electrons in Copper in m/Vs is', 0.004273)
('average time of collision of electrons in copper in sec is', 2.4297841992299697e-14)

Example number 2.14, Page number 76

In [33]:
 
import math
from __future__ import division

#Variable decleration
MH=1.008*2*1.67*10**-27;    #mass in kg
T=30;   #temperature in C

#Calculation
T=T+273;   #temperature in K
KB=1.38*10**-23;    #boltzmann constant in J/k
KE=(3/2)*KB*T;   #kinetic energy in J
KEeV=KE*6.24*10**18;   #kinetic energy in eV
cbar=math.sqrt((3*KB*T)/MH);

#Result
print("average kinetic energy in J is",KE);
print("average kinetic energy in eV is",KEeV);
print("velocity of molecules in m/s is",cbar);

#answers for average kinetic energy in eV and velocity of electrons given in the book are wrong

('average kinetic energy in J is', 6.2720999999999986e-21)
('average kinetic energy in eV is', 0.039137903999999994)
('velocity of molecules in m/s is', 1930.269663853336)

Example number 2.15, Page number 77 **

In [35]:
 
import math
from __future__ import division

#Variable decleration
Ee=10;   #electron kinetic energy in eV
Ep=10;   #proton kinetic energy in eV
me=9.1*10**-31;   #mass of electron in kg
mp=1.67*10**-27;   #mass of proton in kg

#Calculation
EeeV=Ee*1.6*10**-19;   #electron kinetic energy in J
EpeV=Ep*1.6*10**-19;   #proton kinetic energy in J
cebar=math.sqrt((2*EeeV)/me);
cpbar=math.sqrt((2*EpeV)/mp);

#Result
print("velocity of electron in m/s is",cebar);
print("velocity of proton in m/s is",cpbar);

#answers given in the book are wrong

('velocity of electron in m/s is', 1875228.9237539817)
('velocity of proton in m/s is', 43774.05241316662)

Example number 2.16, Page number 77

In [36]:
 
#import module
import math
from __future__ import division

#Variable decleration
A=10;   #area of cross section in mm^2
A=A*10**-6;   #area of cross section in m^2
i=100;   #current in amp
n=8.5*10**28;    #number of electrons per mm^3
e=1.6*10**-19;    #electron charge in coulumb

#Calculation
vd=1/(n*A*e);

#Result
print("drift velocity in m/s is",vd);

#answer given in the book is wrong

('drift velocity in m/s is', 7.3529411764705884e-06)

Example number 2.17, Page number 78

In [38]:
 
#import module
import math
from __future__ import division

#Variable decleration
tow=3*10**-14;    #relaxation time in sec
n=8*10**28;    #density of electrons per m^3
KB=1.38*10**-23;   #boltzmann constant in J/k
T=0;   #temperature in C

#Calculation
T=T+273;   #temperature in K
m=9.1*10**-31;   #mass of electron in kg
sigma_T=((3*n*tow*(KB**2)*T)/(2*m));
sigma_T=math.ceil(sigma_T*10**2)/10**2;   #rounding off to 2 decimals

#Result
print("thermal conductivity of copper in ohm-1 is",sigma_T);

('thermal conductivity of copper in ohm-1 is', 205.68)
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