import math
#Given Data
V0 = 9.1*10**-5; # Atomic volume of Pb, metre cube per kg
K = 2.3*10**-11; # Compressibility of Pb, metre square per newton
alpha = 86*10**-6; # Coefficient of thermal expansion, per K
Cv = 1.4*10**2; # Specific heat at constant volume, J/kg
gama = alpha*V0/(K*Cv); # Grunesien parameter for Pb
print"The Grunesien parameter for Pb = ",round(gama,3);
import math
#Given Data
V0 = 11*10**-5; # Atomic volume of Cu, metre cube per kg
K = 0.75*10**-11; # Compressibility of Cu, metre square per newton
alpha = 49*10**-6; # Coefficient of thermal expansion, per K
gama = 1.9; # The Grunesien parameter for Cu = 2.4
Cv = alpha*V0/(K*gama); # Specific heat of Cu at constant volume, J/kg
print"The specific heat capacity of Cu = ",round(Cv,3),"J/kg";
import math
#Given Data
N = 6.02*10**26; # Avogadro's number, per kmole
C_t = 6.32*10**3; # Velocity of transverse wave, m/s
C_l = 3.1*10**3; # Velocity of longitudinal wave, m/s
rho = 2.7*10**3; # Density of Al, kg per metre cube
M = 26.97; # Atomic weight of Al, gram per mol
V = M/rho; # Atomic volume of Al, metre cube
f_c = (9*N/(4*math.pi*V*(1.0/C_t**3+2.0/C_l**3)))**(1.0/3);
print"The Debye cut-off frequency of Al = ","{0:.3e}".format(f_c),"per sec";
import math
#Given Data
N = 6.02*10**23; # Avogadro's number, per mole
k = 1.38*10**-23; # Boltzmann constant, J/K
R = N*k; # Molar gas constant, J/mol/K
theta_D = 2230; # Debye temperature for diamond, K
T = 300.0; # Room temperature, K
C_v = 12.0/5*(math.pi**4*R)*(T/theta_D)**3; # Specific heat capacity per unit volume of diamond, J/mol-K
print"The heat capacity per unit volume of diamond = ",round(C_v,3),"J/mol-K";
import math
#Given Data
k = 1.38*10**-23; # Boltzmann constant, J/K
theta_D = 1440.0; # Debye temperature for Be, K
h = 6.626*10**-34; # Planck's constant, Js
f_D = k*theta_D/h; # Debye cut off frequency of Be, Hz
print"The Debye cut off frequency of Be = ","{0:.3e}".format(f_D),"sec";
import math
#Given Data
N = 6.023*10**23; # Avogadro's number, per kmol
e = 1.6*10**-19; # Energy equivalent of 1 eV, J/eV
k = 1.38*10**-23; # Boltzmann constant, J/K
R = N*k; # Molar gas constant, J/kmol/K
E_F = 7; # Fermi energy of Cu, eV
theta_D = 348.0; # Debye temperature of Cu, K
T = 300.0; # Room temperature, K
T_F = E_F/k; # Fermi temperature of Cu, K
C_e = math.pi**2/2*R*10**3*(T/(T_F*e)); # Electronic heat capacity of Cu, J/kmol/K
C_l = 12.0/5*(math.pi**4*R)*(T/theta_D)**3; # Lattice heat capacity of Cu, J/kmol/K
print"The electronic heat capacity of Cu = ",round(C_e,3),"J/kmol/K";
print"The lattice heat capacity of Cu = ",round(C_l,3),"J/mol/K";
import math
#Given Data
N = 6.023*10**23; # Avogadro's number, per kmol
e = 1.602*10**-19; # Energy equivalent of 1 eV, J/eV
k = 1.38*10**-23; # Boltzmann constant, J/K
R = N*k; # Molar gas constant, J/kmol/K
E_F = 7.0; # Fermi energy of Cu, eV
theta_D = 348.0; # Debye temperature of Cu, K
T = 0.01; # Room temperature, K
T_F = E_F/k; # Fermi temperature of Cu, K
C_e = math.pi**2/2*R*(T/(T_F*e)); # Electronic heat capacity of Cu, J/mol/K
C_l = 12.0/5*(math.pi**4*R)*(T/theta_D)**3; # Lattice heat capacity of Cu, J/kmol/K
print"The electronic heat capacity of Cu = ","{0:.3e}".format(C_e),"J/mol/K";
print"The lattice heat capacity of Cu = ","{0:.3e}".format(C_l),"J/mol/K";
import math
#Given Data
N = 6.023*10**23; # Avogadro's number, per kmol
e = 1.602*10**-19; # Energy equivalent of 1 eV, J/eV
k = 1.38*10**-23; # Boltzmann constant, J/K
R = N*k; # Molar gas constant, J/kmol/K
E_F = 3.2; # Fermi energy of Cu, eV
theta_D = 150.0; # Debye temperature of Cu, K
T = 20.0; # Given temperature, K
T_F = E_F/k; # Fermi temperature of Cu, K
C_e = math.pi**2/2*R*(T/(T_F*e)); # Electronic heat capacity of Cu, J/mol/K
C_l = 12.0/5*(math.pi**4*R)*(T/theta_D)**3; # Lattice heat capacity of Cu, J/kmol/K
print"The electronic heat capacity of Na = ","{0:.3e}".format(C_e),"J/mol/K";
print"The lattice heat capacity of Na = ",round(C_l,4),"J/mol/K";
import math
#Given Data
N = 6.023*10**23; # Avogadro's number, per kmol
e = 1.602*10**-19; # Energy equivalent of 1 eV, J/eV
k = 1.38*10**-23; # Boltzmann constant, J/K
R = N*k; # Molar gas constant, J/kmol/K
E_F = 3.2; # Fermi energy of Hf, eV
theta_D = 242.0; # Debye temperature of Hf, K
T_F = E_F/k; # Fermi temperature of Hf, K
T = [300.0, 200.0, 100.0, 10.0, 5.0]; # Declare a vector of 5 temperature values, K
print"________________________";
print"T(K) C_l (J/kmol/K)";
print"________________________";
for i in xrange(len(T)):
C_l = 12.0/5*(math.pi**4*R)*(T[i]/theta_D)**3; # Lattice heat capacity of Hf, J/kmol/K
print"",T[i]," ",round(C_l,3);
print"________________________"
import math
#Given Data
N = 6.023*10**23; # Avogadro's number, per kmol
e = 1.602*10**-19; # Energy equivalent of 1 eV, J/eV
k = 1.38*10**-23; # Boltzmann constant, J/K
R = N*k; # Molar gas constant, J/kmol/K
E_F = 7.0; # Fermi energy of Hf, eV
theta_D = 343.0; # Debye temperature of Hf, K
T_F = E_F/k; # Fermi temperature of Hf, K
# As C_l = 12/5*(pi**4*R)*(T/theta_D)**3 and C_e = pi**2/2*R*(T/(T_F*e)) so that
# For C_l = C_e, we have
T = math.sqrt((math.pi**2/2*R*1/(T_F*e))/(12.0/5*math.pi**4*R)*theta_D**3); # Required temperature when C_l = C_e, K
print"The temperature at which lattice specific heat equals electronic specific heat for Cu =",round(T,3),"K";
import math
#Given Data
C11 = 1.08*10**12; C12 = 0.62*10**12; C44 = 0.28*10**12; # Elastic constants of Al, dynes/cm square
a = 4.05*10**-8; # Lattice constant for Al cubic structure, cm
rho = 2.70; # g/cm cube
k = 1.38*10**-23; # Boltzmann constant, J/K
h = 6.626*10**-34; # Planck's constant, Js
s = 4.0; # Number of atoms in Al unit cell
Va = a**3; # Volume of unit cell, cm cube
theta_D =(3.15/(8*math.pi)*(h/k)**3*s/(rho**(3.0/2)*Va)*(C11-C12)**(1.0/2)*(C11+C12+2*C44)**(1.0/2)*C44**(1.0/2))**(1.0/3);
print"The Debye temperature of Al =",round(theta_D,3),"K";
import math
#Given Data
k = 1.38*10**-23; # Boltzmann constant, J/K
h = 6.626*10**-34; # Planck's constant, Js
A =[[1,2,3,4,5,6,7,8],[9,10,11,12,13,14,15,16]]; # Declare a matrix of 2X8
A[0][0] = 'Cu';
A[0][1] = 1.684*10**12;
A[0][2] = 1.214*10**12;
A[0][3] = 0.754*10**12;
A[0][4] = 4;
A[0][5] = 3.61*10**-8;
A[0][6] = 8.96;
A[1][0] = 'Na';
A[1][1] = 0.055*10**12;
A[1][2] = 0.047*10**12;
A[1][3] = 0.049*10**12;
A[1][4] = 2;
A[1][5] = 4.225*10**-8;
A[1][6] = 0.971;
# For Cu
Va = A[0][5]**3; # Volume of unit cell, cm cube
A[0][7] = (3.15/(8*math.pi)*(h/k)**3*A[0][4]/(A[0][6]**(3.0/2)*Va)*(A[0][1]-A[0][2])**(1.0/2)*(A[0][1]+A[0][2]+2*A[0][3])**(1.0/2)*A[0][3]**(1.0/2))**(1.0/3);
# For Na
Va =A[1][5]**3; # Volume of unit cell, cm cube
A[1][7] = (3.15/(8*math.pi)*(h/k)**3*A[1][4]/(A[1][6]**(3.0/2)*Va)*(A[1][1]-A[1][2])**(1.0/2)*(A[1][1]+A[1][2]+2*A[1][3])**(1.0/2)*A[1][3]**(1.0/2))**(1.0/3);
print"________________________________________";
print"Metal C11 C12 C44 thetaD";
print"________________________________________";
for i in range (0,2) :
print"",A[i][0]," ",A[i][1]/10**12," ",A[i][2]/10**12," ",A[i][3]/10**12," ",round(A[i][7],2);
print"________________________________________";
import math
#Given Data
k = 1.38*10**-23; # Boltzmann constant, J/K
h = 6.626*10**-34; # Planck's constant, Js
A =[[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15],[16,17,18,19,20]]; # Declare a matrix of 4X5
A[0][0] = 300;
A[0][1] = 0.878*10**10;
A[0][2] = 0.483*10**10;
A[0][3] = 0.448*10**10;
A[1][0] = 200;
A[1][1] = 0.968*10**10;
A[1][2] = 0.508*10**10;
A[1][3] = 0.512*10**10;
A[2][0] = 100;
A[2][1] = 1.050*10**10;
A[2][2] = 0.540*10**10;
A[2][3] = 0.579*10**10;
A[3][0] = 20;
A[3][1] = 1.101*10**10;
A[3][2] = 0.551*10**10;
A[3][3] = 0.624*10**10;
s = 2; # Number of atoms in a unit cell
a = 4.225*10**-10; # Lattice parameter of Na, m
rho = 0.971*10**3; # Density of Na, kg/metre-cube
Va = a**3; # Volume of unit cell, metre cube
print"________________________________________";
print"T C11 C12 C44 thetaD"
print"________________________________________";
for i in range (0,4) :
A[i][4] = (3.15/(8*math.pi)*(h/k)**3*s/(rho**(3.0/2)*Va)*(A[i][1]-A[i][2])**(1.0/2)*(A[i][1]+A[i][2]+2*A[i][3])**(1.0/2)*A[i][3]**(1.0/2))**(1.0/3);
print"",A[i][0]," ",A[i][1]/10**10," ",A[i][2]/10**10," ",A[i][3]/10**10," ",round(A[i][4],2);
print"________________________________________";
import math
from scipy.integrate import quad
#Given Data
Lu =[[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15],[16,17,18,19,20],[21,22,23,24,25],[26,27,28,29,30]]; # Declare a matrix of 6X5
Lu[0][0] = 0;
Lu[0][1] = 5.58;
Lu[0][2] = 3.517;
Lu[0][4] = 0.750;
Lu[1][0] = 36;
Lu[1][1] = 5.409;
Lu[1][2] = 3.440;
Lu[1][4] = 0.560;
Lu[2][0] = 103;
Lu[2][1] = 5.213;
Lu[2][2] = 3.341;
Lu[2][4] = 0.492;
Lu[3][0] = 157;
Lu[3][1] = 5.067;
Lu[3][2] = 3.259;
Lu[3][4] = 0.388;
Lu[4][0] = 191;
Lu[4][1] = 4.987;
Lu[4][2] = 3.217;
Lu[4][4] = 0.357;
Lu[5][0] = 236;
Lu[5][1] = 4.921;
Lu[5][2] = 3.179;
Lu[5][4] = 0.331;
V0 = 3*math.sqrt(3)/2*Lu[0][2]**2*Lu[0][1];
V = [0,0,0,0,0,0]; # Declare volume array
print"______________________________________________________________";
print"P(kbar) c(angstrom) a(angstrom) gamma_G nu_G ";
print"______________________________________________________________";
for i in range (0,6) :
V[i] = 3*math.sqrt(3)/2*Lu[i][2]**2*Lu[i][1];
Lu[i][3] = Lu[i][4]*V[i]/V0+2.0/3*(1-V[i]/V0)**(1.0/2);
print"",Lu[i][0]," ",Lu[i][1]," ",Lu[i][2]," ",round(Lu[i][3],3)," ",Lu[i][4];
print"______________________________________________________________";
cnt = 0;
print"________________________";
print"P(kbar) Theta_D(K)";
print"________________________";
for i in range (0,6) :
def integrand(x, a, b):
return (-1*Lu[i][4]*(math.exp(x)/V0)-(2.0/3)*(1-math.exp(x)/V0)**(1.0/2))
a=1;
b=1;
I = quad(integrand,-0.8+cnt,math.log(V[i]/1000000), args=(a,b));
theta_D = math.exp(I[0]);
cnt = cnt + 0.01;
print"",Lu[i][0]," ",round(theta_D,0);
print"________________________";
import math
#GIven Data
T_M = 1356.0; # Melting temperature of Cu, K
V = 7.114; # Atomic volume of Cu, cm cube per g-atom
M = 63.5; # atomic weight of Cu, g/mole
K = 138.5; # Lindemann constant
theta_M = K*(T_M/M)**(1.0/2)*(1/V)**(1.0/3); # Debye temperature by Lindemann method, K
print"The Debye temperature by Lindemann method =",round(theta_M,3),"K";
print"The values obtained from other methods are:";
print"theta_s = 342 K; theta_R = 336 K; theta_E = 345 K";
import math
#Given Data
N_A = 6.023*10**23; # Avogadro's number
c = 3.0*10**8; # Speed of light, m/s
epsilon_0 = 15.0; # Dielectric constant of the medium
m = 2.0*10**-22; # Mass of ion, g
e = 4.8*10**-10; # Charge on the ion, C
rho = 7.0; # Average density of solid, g/cc
A = 120.0; # Average atomic weight of solid, g
N = rho/A*N_A; # Number of ions per cc, per cm cube
f_P = 1/(2*math.pi)*math.sqrt(4*math.pi*N*e**2/(m*epsilon_0)); # Plasma frequency of vibrating ions in the crystal, Hz
lamda_P = c/f_P; # Plasma wavelength of vibrating ions in the crystal, cm
print"The plasma frequency of vibrating ions in InSb crystal = ","{0:.3e}".format(f_P),"Hz";
print"The plasma wavelength of vibrating ions in InSb crystal =",round(lamda_P/10**-6,3),"micron";
print"The calculated frequency lies in the infrared region.";
import math
#Given Data
h = 6.624*10**-34; # Planck's constant, Js
k = 1.38*10**-23; # Boltzmann constant, J/mol/K
q = 1.486*10**11; # Young's modulus of diamond, N/metre-square
rho = 3500; # Density of diamond, kg/metre-cube
c = math.sqrt(q/rho); # Speed of transverse wave through diamond, m/s
m = 12*1.66*10**-27; # Atomic weight of carbon, kg
theta_D = (h/k)*c*(3*rho/(4*math.pi*m))**(1.0/3); # Debye temperature for diamond, K
print"The Debye temperature for diamond =",round(theta_D,3),"K";