Chapter 9: Superconductivity

Example 9.1,Page number 278

In [1]:
import math

#Given Data

H_c0 = 0.0803;    # Critical field at absolute zero, Tesla
T_c = 7.19;    # Transition temperature of specimen  lead, Kelvin
T = 5.0;         # Temperature at which destruction of superconductivity is to be found, Kelvin
H_c = H_c0*(1.0-(T/T_c)**2);    # Critical field required to destroy superconductivity, Tesla
print"Critical field required to destroy superconductivity = ",round(H_c,4),"T";
Critical field required to destroy superconductivity =  0.0415 T

Example 9.2,Page number 278

In [2]:
import math

#Given Data
H0 = 1970.0;    # Critical field at absolute zero, Oe
T_c = 9.25;    # Transition temperature of specimen  Nb, Kelvin
T = 4.0;    # Temperature at which destruction of superconductivity is to be found, Kelvin
H_c = H0*(1.0-(T/T_c)**2);    # Limiting magnetic field, Oe
print"Limiting magnetic field of Nb to serve as superconductor = ",round(H_c),"Oe";
Limiting magnetic field of Nb to serve as superconductor =  1602.0 Oe

Example 9.3,Page number 278

In [4]:
import math

#Given Data

T_1 = 14.0;    # Temperature, K
T_2 = 13.0;    # Temperature, K
H_c1 = 1.4*10**5;    # Critical field at T_1, K
H_c2 = 4.2*10**5;    # Critical field at T_2, K#As H_c1/H_c2 = (T_c**2-T_1**2)/(T_c**2-T_2**2), solving for T_c
T_c = math.sqrt((H_c2/H_c1*T_1**2 - T_2**2)/2);  # The superconducting transition temperature of a specimen, K
print"Transition temperature of a specimen = ",round(T_c,4),"K";
Transition temperature of a specimen =  14.4741 K

Example 9.4,Page number 280

In [5]:
import math

#Given Data
e = 1.6*10**-19;   # Energy equivalent of 1 eV, J/eV
E_g = 3.4*10**-4;    # Energy gap of aluminium, eV
v_F = 2.02*10**8;    # Fermi velocity of aluminium, cm/sec
h_bar = 1.05*10**-34;    # Planck's constant
L = h_bar*v_F/(2*E_g*e);    # Coherence Length of aluminium, cm

print"The coherence length of aluminium = ","{0:.3e}".format(L),"cm";
The coherence length of aluminium =  1.949e-04 cm

Example 9.6,Page number 284

In [6]:
import math

#Given Data
h = 6.6*10**-34;     # Planck's constant, Js
e = 1.6*10**-19;       # Energy eqivalent of 1 eV, eV/J
k = 0.86*10**-4;  # Boltzmann constant, eV/K
T_c = 0.56;     # Critical temperature for superconducting Zr, K
E_g = 3.52*k*T_c;    # Energy gap of aluminium, J
c = 3*10**8;      # Speed of light, m/s
lamda = h*c/(E_g*e);       # Wavelength of photon required to break a Cooper pair, m

print"The wavelength of photon required to break a Cooper pair = ","{0:.3e}".format(lamda),"m";
The wavelength of photon required to break a Cooper pair =  7.300e-03 m

Example 9.7,Page number 285

In [7]:
import math

#Given Data

Lamda_0 = 390.0;    # Penetration depth at absolute zero, angstorm
T_c = 7.0;    # Transition temperature of Pb, K
T = 2.0;      # Givn temperature, K
Lamda = Lamda_0*(1.0-(T/T_c)**2)**(-1.0/2);    # London penetration depth in Pb at 2K, angstorm 
print"The London penetration depth in Pb at 2K = ",round(Lamda,4),"angstorm";
print"The London penetration depth at T = T_c becomes Inf";
The London penetration depth in Pb at 2K =  406.9644 angstorm
The London penetration depth at T = T_c becomes Inf

Example 9.8,Page number 286

In [17]:
import math
import numpy  

#Given data

M = (199.5, 200.7, 202.0, 203.3);      # Isotopic mass of Hg, amu
T_c = (4.185, 4.173, 4.159, 4.146);    # Critical temperature of Hg, kelvin
alpha = 0.5;    # Trial value of Isotopic exponent
# Accroding to isotopic effect, T_c = K*M**(-alpha), solving for K
K = T_c[1]/M[1]**(-alpha);  # Isoptopic coefficent 
Tc = numpy.zeros(3);  
for i in xrange(len(Tc)):
    Tc[i] = K*M[i+1]**(-alpha)
    print"Tc[",i,"] = ",round(Tc[i],4);
if T_c[1]-Tc[0]<0.001 and T_c[2]-Tc[1]<0.001 and T_c[3]-Tc[2]<0.001 :
    print"The isotopic exponent in Isotopic effect of Hg =",alpha;
Tc[ 0 ] =  4.173
Tc[ 1 ] =  4.1596
Tc[ 2 ] =  4.1462
The isotopic exponent in Isotopic effect of Hg = 0.5

Example 9.9,Page number 286

In [8]:
import math


#GIven Data
M_1 = 202.0;    # Mass of first isotope of mercury, amu
M_2 = 199.0;    # Mass of second isotope of mercury, amu
T_c1 = 4.153;    # Transition temperature of first isotope of mercury, K 
#As T_c1/T_c2 = (M_2/M_1)**1/2, solving for T_c2
T_c2 = math.sqrt(M_1/M_2)*T_c1; 
print"The transition temperature of isotope of Hg whose mass number is ",M_2,"=",round(T_c2,4),"K";
The transition temperature of isotope of Hg whose mass number is  199.0 = 4.1842 K

Example 9.10,Page number 287

In [9]:
import math

#Given Data

alpha = 0.5;    # Isotopic exponent of Osmium
T_c = 0.655;    # Transition temperature of Osmium, K 
M = 190.2;    # Mass of Osmium, amu
K = T_c*M**alpha;    # K is the constant of proportionality

print"The value of constant of proportionality =",round(K,4);
The value of constant of proportionality = 9.0333

Example 9.11,Page number 298

In [11]:
import math

#Given Data
k = 1.38*10**-23;   # Boltzmann constant, J/mol/K
e = 1.6*10**-19;      # Energy equivalent of 1 eV, eV/J
Theta_D = 96;    # Debye temperature, kelvin
N0 = 0.3678;    # Density of states at Fermi energy
V = 1.0;          # Volume of the material, metre cube
T_c = 1.14*Theta_D*math.exp(-1.0/(N0*V));    # Critical temperature of the material, K
Delta_0 = k*Theta_D/math.sinh(1.0/(N0*V)); # Energy gap at absolute zero, J
print"The transition temperature of a material = ",round(T_c,3),"K";
print"The energy gap of a material = ","{0:.3e}".format(Delta_0/e),"eV";
The transition temperature of a material =  7.217 K
The energy gap of a material =  1.097e-03 eV

Example 9.12,Page number 298

In [12]:
import math

#Given Data
Theta_D = 350.0;    # Debye temperature, kelvin
Lamda = 0.828;    # Electron-phonon coupling constant
mu_prime = 0.1373;    # Reduced mass of a superconductor, amu
T_c = Theta_D/1.45*math.exp(-1.04*(1+Lamda)/(Lamda-mu_prime*(1+0.62*Lamda))); #Transition temperature of superconductor using McMillan formula,K

print"The transition temperature of the superconductor using McMillan formula = ",round(T_c,3),"K";
The transition temperature of the superconductor using McMillan formula =  11.258 K

Example 9.13,Page number 298

In [13]:
import math

#Given Data
Theta_D = 350;    # Debye temperature, kelvin
Lamda = 0.641;    # Electron-phonon coupling constant
mu_prime = 0.143;    # Reduced mass of a superconductor, amu
T_c = Theta_D/1.45*math.exp(-1.04*(1+Lamda)/(Lamda-mu_prime*(1+0.62*Lamda)));#Transition temperature of superconductor using McMillan formula,K

print"The superconducting transition temperature of a superconductor using McMillan formula = ",round(T_c,4),"K";
The superconducting transition temperature of a superconductor using McMillan formula =  5.0426 K

Example 9.15,Page number 314

In [14]:
import math

#Given Data
Theta_D = 490;    # Debye temperature, Kelvin
Lamda = 0.8;    # wavelength of a superconductor, angstorm
mu_prime = 0.13;    # Reduced mass of a superconductor, amu
T_c = Theta_D/1.45*math.exp(-1.04*(1+Lamda)/(Lamda-mu_prime*(1+0.62*Lamda)));
print"The superconducting transition temperature of a borocarbide superconductor =",round(T_c,4),"K";
The superconducting transition temperature of a borocarbide superconductor = 15.3526 K

Example 9.16,Page number 314

In [15]:
import math

#Given Data
T_c = 16.5;    # Transition temperature of a superconductor, K
Lamda = [0.7, 0.8, 0.9, 1.0]    # Electron-phonon coupling constants at different Tc values 
Theta_D = 503.0;    # Debye temperature, kelvin
mu_prime = 0.13;    # Reduced mass of a superconductor, amu
Tc = [0.0, 0.0, 0.0, 0.0];
print"_____________________";
print"Lamda          Tc";
print"_____________________";
for i in xrange(len(Lamda)):
   Tc[i] = Theta_D/1.45*math.exp(-1.04*(1+Lamda[i])/(Lamda[i]-mu_prime*(1+0.62*Lamda[i]))); 
   if abs(Tc[i] - 16.5) < 1.0 :
       best_Lvalue = Lamda[i];
   print"",Lamda[i],"       ",round(Tc[i],3),"K";
print"_____________________";
print"The best electron-phonon coupling constant should be slightly above ", best_Lvalue;
_____________________
Lamda          Tc
_____________________
 0.7         11.095 K
 0.8         15.76 K
 0.9         20.407 K
 1.0         24.881 K
_____________________
The best electron-phonon coupling constant should be slightly above  0.8

Example 9.17,Page number 317

In [16]:
import math

#Given Data
T_c = 39.4;    # Transition temperature of a superconductor, K
Lamda = 1;    # Electron-phonon coupling constant for a superconductor
mu_prime= 0.15;    # Reduced mass of a superconductor, amu
# As T_c = Theta_D/1.45*exp(-1.04*(1+Lamda)/(Lamda-mu_prime*(1+0.62*Lamda))), solving for Theta_D
Theta_D = T_c*1.45*math.exp(1.04*(1+Lamda)/(Lamda-mu_prime*(1+0.62*Lamda)));

print"The Debye temperature of a BCS superconductor = ",round(Theta_D,3),"K";
The Debye temperature of a BCS superconductor =  891.6 K
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