4: Electron Theory of Metals

Example number 4.1, Page number 4.5

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

#Variable declaration
rho_s = 10.5*10**3    #density of silver(kg/m^3)
NA = 6.02*10**26     #avagadro number(per k-mol)
MA = 107.9     #atomic weight of silver
sigma = 6.8*10**7;     #conductivity of silver(ohm-1 m-1)
e = 1.6*10**-19

#Calculation
n = rho_s*NA/MA      #molar volume of silver
mew = sigma/(n*e)     #mobility of electrons(m^2/Vs)
mew = mew*10**2
mew = math.ceil(mew*10**4)/10**4;   #rounding off to 4 decimals

#Result
print "density of electrons in silver is",round(n/1e28,2),"*10^28"
print "mobility of electrons is",mew,"*10**-2 m**2/Vs"

density of electrons in silver is 5.86 *10^28
mobility of electrons is 0.7255 *10**-2 m**2/Vs

Example number 4.2, Page number 4.6

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

#Variable declaration
d = 8.92*10**3      #density(kg/m^3)
e = 1.6*10**-19
m = 9.1*10**-31      #mass of electron(kg)
N = 6.02*10**26      #avagadro's number(per k-mol)
AW = 63.5     #atomic weight
rho = 1.73*10**-8     #resistivity of copper, ohm-m

#Calculation
n = d*N/AW       #number of cu atoms(per m^3)
mew = 1/(rho*n*e)      #mobility of electrons(m/Vs)
mew = mew*10**2
mew = math.ceil(mew*10**4)/10**4;   #rounding off to 4 decimals
tow = m/(n*e**2*rho)    #relaxation time(s)

#Result
print "mobility of electrons is",round(mew,3),"*10**-2 m/Vs"
print "relaxation time is",round(tow/1e-14,3),"*10^-14 sec"

mobility of electrons is 0.427 *10**-2 m/Vs
relaxation time is 2.43 *10^-14 sec

Example number 4.3, Page number 4.7

In [8]:
#importing modules
import math
from __future__ import division

#Variable declaration
rho = 1.54*10**-8      #resistivity(ohm-m)
n = 5.8*10**28      #conduction electrons(per m^3)
m = 9.108*10**-31    #mass of electron(kg)
e = 1.602*10**-19

#Calculation
tow = m/(n*(e**2)*rho)    #relaxation time(sec)

#Result
print "relaxation time of conduction electrons is",round(tow/1e-14,2),"*10^-14 sec" 

relaxation time of conduction electrons is 3.97 *10^-14 sec

Example number 4.4, Page number 4.8

In [17]:
#importing modules
import math
from __future__ import division

#Variable declaration
R = 0.06      #resistance(ohm)
D = 5      #length of Al wire(m)
e = 1.602*10**-19
rho = 2.7*10**-8     #resistivity of Al(ohm-m)
MA = 26.98     #atomic weight
NA = 6.025*10**26     #avagadro number(k/mol)
rho_s = 2.7*10**3     #density(kg/m^3)
I = 15     #current(A)

#Calculation
n = 3*rho_s*NA/MA     #free electron concentration(electrons/m^3)
mew = 1/(n*e*rho)     #mobility(m/Vs)
E = I*R/D      #electric field(V/m)
vd = mew*E     #drift velocity(m/s)
vd = vd*10**3
mew = mew*10**3
mew = math.ceil(mew*10**4)/10**4;   #rounding off to 4 decimals
vd = math.ceil(vd*10**4)/10**4;   #rounding off to 4 decimals

#Result
print "free electron concentration is",round(n/1e29,4),"*10^29 electrons/m^2"
print "mobility is",round(mew,3),"*10^-3 m/Vs"
print "drift velocity is",round(vd,2),"*10^-3 m/s"

free electron concentration is 1.8088 *10^29 electrons/m^2
mobility is 1.278 *10^-3 m/Vs
drift velocity is 0.23 *10^-3 m/s

Example number 4.5, Page number 4.13

In [18]:
#importing modules
import math
from __future__ import division

#Variable declaration
n1 = 1
n2 = 1
n3 = 1      #for lowest energy
h = 6.62*10**-34     #planck's constant(Js)
e = 1.6*10**-19
m = 9.1*10**-31      #mass of electron(kg)
L = 0.1     #side of box(nm)

#Calculation
L = L*10**-9    #side of box(m)
E1 = (h**2)*(n1**2+n2**2+n3**2)/(8*m*L**2)      #lowest energy(J)
E1 = E1/e        #lowest energy(eV)
E1 = math.ceil(E1*10)/10   #rounding off to 1 decimal

#Result
print "lowest energy of electron is",E1,"eV"

lowest energy of electron is 112.9 eV

Example number 4.6, Page number 4.14

In [19]:
#importing modules
import math

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

#Result
print "fermi function is",F_E

fermi function is 0.269

Example number 4.7, Page number 4.14

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

#Variable declaration
F_E = 10      #probability in percent
k = 1.38*10**-23
EF = 5.5      #fermi energy(eV)

#Calculation
E = EF+(EF/100)     #energy(eV)
X = E-EF     #E-EF(eV)
X = X*e      #E-EF(J)
T = X/(k*math.log(F_E-1))      #temperature(K)
T = math.ceil(T*10**2)/10**2   #rounding off to 2 decimals

#Result
print "temperature is",round(T,1),"K"

temperature is 290.2 K

Example number 4.8, Page number 4.17

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

#Variable declaration
F_E = 0.01      #probability in percent
k = 1.38*10**-23
e = 1.6*10**-19
#let E-EF be X
X = 0.5      #E-EF(eV)

#Calculation
kT = X/(2.303*math.log10((1-F_E)*100))    #value of kT(eV)
T = kT*e/k        #temperature(K)
T = math.ceil(T*10)/10   #rounding off to 1 decimal

#Result
print "temperature is",T,"K"
print "answer given in the book is wrong by a decimal point"

temperature is 1261.4 K
answer given in the book is wrong by a decimal point