path: root/Solid_State_Physics_Principles_And_Applications_by_R._Asokamani/chapter4.ipynb

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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Chapter 4: Free Electron Theory of Metals"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.1,Page number 112"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "m = 9.1*10**-31;   # Mass of an electron, kg\n",
      "e = 1.6*10**-19;   # Charge on an electron, C\n",
      "n = 8.5*10**28;   # Concentration of electron in Cu, per metre cube\n",
      "rho = 1.7*10**-8;  # Resistivity of Cu, ohm-m\n",
      "t = m/(n*e**2*rho);  # Collision time for an electron in monovalent Cu, s\n",
      "print\"The collision time for an electron in monovalent Cu =\",\"{0:.3e}\".format(t),\"s\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The collision time for an electron in monovalent Cu = 2.460e-14 s\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.2,Page number 112"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "m = 9.1*10**-31;   # Mass of an electron, kg\n",
      "e = 1.6*10**-19;   # Charge on an electron, C\n",
      "n = 10**29;   # Concentration of electron in material, per metre cube\n",
      "rho = 27*10**-8;  # Resistivity of the material, ohm-m\n",
      "tau = m/(n*e**2*rho);  # Collision time for an electron in the material, s\n",
      "v_F = 1*10**8;    # Velocity of free electron, cm/s\n",
      "lamda = v_F*tau;   # Mean free path of electron in the material, cm\n",
      "print\"The collision time for an electron in monovalent Cu =\",\"{0:.3e}\".format(tau),\"s\";\n",
      "print\"The mean free path of electron at 0K =\",\"{0:.3e}\".format(lamda),\"cm\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The collision time for an electron in monovalent Cu = 1.317e-15 s\n",
        "The mean free path of electron at 0K = 1.317e-07 cm\n"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.3,Page number 112"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "m = 9.1*10**-31;   # Mass of an electron, kg\n",
      "e = 1.6*10**-19;   # Charge on an electron, C\n",
      "r = 1.28*10**-10;  # Atomic radius of cupper, m\n",
      "a = 4*r/sqrt(2);    # Lattice parameter of fcc structure of Cu, m\n",
      "V = a**3;    # Volume of unit cell of Cu, metre cube\n",
      "n = 4/V;    # Number of atoms per unit volume of Cu, per metre cube\n",
      "tau = 2.7*10**-4;  # Relaxation time for an electron in monovalent Cu, s\n",
      "sigma = n*e**2*tau/m;    # Electrical conductivity of Cu, mho per cm\n",
      "print\"The free electron density in monovalent Cu =\",\"{0:.3e}\".format(n),\"per metre cube\";\n",
      "print\"The electrical conductivity of monovalent Cu =\",\"{0:.3e}\".format(sigma),\"mho per cm\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The free electron density in monovalent Cu = 8.429e+28 per metre cube\n",
        "The electrical conductivity of monovalent Cu = 6.403e+17 mho per cm\n"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.4,Page number 118"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "m = 9.1*10**-31;   # Mass of an electron, kg\n",
      "e = 1.6*10**-19;   # Energy equivalent of 1 eV, J/eV\n",
      "h = 6.625*10**-34;     # Planck's constant, Js\n",
      "L = 10*10**-3;     # Length of side of the cube, m\n",
      "# For nth level\n",
      "nx = 1; ny = 1; nz = 1;     # Positive integers along three axis\n",
      "En = h**2/(8*m*L**2)*(nx**2+ny**2+nz**2)/e;      # Energy of nth level for electrons, eV\n",
      "# For (n+1)th level\n",
      "nx = 2; ny = 1; nz = 1;     # Positive integers along three axis\n",
      "En_plus_1 = h**2/(8*m*L**2)*(nx**2+ny**2+nz**2)/e;      # Energy of (n+1)th level for electrons, eV\n",
      "delta_E = En_plus_1 - En;       # Energy difference between two levels for the free electrons\n",
      "print\"The energy difference between two levels for the free electrons =\",\"{0:.3e}\".format( delta_E),\"eV\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The energy difference between two levels for the free electrons = 1.130e-14 eV\n"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.5,Page number 119"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "T = 300.0;    # Room temperature of tungsten, K\n",
      "k = 1.38*10**-23;  # Boltzmann constant, J/mol/K\n",
      "e = 1.6*10**-19;   # Energy equivalent of 1 eV, J/eV\n",
      "E_F = 4.5*e;    # Fermi energy of tungsten, J\n",
      "E = E_F-0.1*E_F; # 10% energy below Fermi energy, J\n",
      "f_T = 1.0/(1+exp((E-E_F)/(k*T)));     # Probability of the electron in tungsten at room temperature at an nergy 10% below the Fermi energy\n",
      "print\"The probability of the electron at an energy 10 percent below the Fermi energy in tungsten at 300 K =\",round(f_T,3);\n",
      "E = 2*k*T+E_F; # For energy equal to 2kT + E_F\n",
      "f_T = 1.0/(1+exp((E-E_F)/(k*T)));     # Probability of the electron in tungsten at an energy 2kT above the Fermi energy\n",
      "print\"The probability of the electron at an energy 2kT above the Fermi energy =\",round(f_T,4);\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The probability of the electron at an energy 10 percent below the Fermi energy in tungsten at 300 K = 1.0\n",
        "The probability of the electron at an energy 2kT above the Fermi energy = 0.1192\n"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.6,Page number 121"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "h = 6.625*10**-34; # Planck's constant, Js\n",
      "h_cross = h/(2*pi);    # Reduced Planck's constant, Js\n",
      "m = 9.1*10**-31;   # Mass of an electron, kg\n",
      "e = 1.6*10**-19;   # Energy equivalent of 1 eV, J/eV\n",
      "a = 5.34*10**-10;      # Lattice constant of monovalent bcc lattice, m\n",
      "V = a**3;    # Volume of bcc unit cell, metre cube\n",
      "n = 2.0/V;    # Number of atoms per metre cube\n",
      "E_F = h_cross**2.0/(2*m*e)*(3*pi**2*n)**(2.0/3);    # Fermi energy of monovalent bcc solid, eV\n",
      "\n",
      "print\"The Fermi energy of a monovalent bcc solid =\",round(E_F,4),\"eV\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The Fermi energy of a monovalent bcc solid = 2.0341 eV\n"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.7,Page number 121"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "h = 6.625*10**-34; # Planck's constant, Js\n",
      "h_cross = h/(2*pi);    # Reduced Planck's constant, Js\n",
      "m = 9.11*10**-31;   # Mass of an electron, kg\n",
      "e = 1.6*10**-19;   # Energy equivalent of 1 eV, J/eV\n",
      "V = 1*10**-5;    # Volume of cubical box, metre cube\n",
      "E_F = 5*e;  # Fermi energy, J \n",
      "D_EF = V/(2*pi**2)*(2*m/h_cross**2)**(3.0/2)*E_F**(1.0/2)*e;     # Density of states at Fermi energy, states/eV\n",
      "print\"The density of states at Fermi energy =\",\"{0:.3e}\".format( D_EF),\"states/eV\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The density of states at Fermi energy = 1.521e+23 states/eV\n"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.8,Page number 121"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "h = 6.626*10**-34; # Planck's constant, Js\n",
      "h_cross = h/(2*pi);    # Reduced Planck's constant, Js\n",
      "m = 9.1*10**-31;   # Mass of an electron, kg\n",
      "e = 1.6*10**-19;   # Energy equivalent of 1 eV, J/eV\n",
      "V = 1*10**-6;    # Volume of cubical box, metre cube\n",
      "E_F = 7.13*e;  # Fermi energy for Mg, J \n",
      "D_EF = V/(2*pi**2)*(2*m/h_cross**2)**(3.0/2)*E_F**(1.0/2);     # Density of states at Fermi energy for Cs, states/eV\n",
      "E_Mg = 1.0/D_EF;  # The energy separation between adjacent energy levels of Mg, J\n",
      "print\"The energy separation between adjacent energy levels of Mg =\",\"{0:.3e}\".format(E_Mg/e),\"eV\";\n",
      "E_F = 1.58*e;  # Fermi energy for Cs, J \n",
      "D_EF = V/(2*pi**2)*(2*m/h_cross**2)**(3.0/2)*E_F**(1.0/2);     # Density of states at Fermi energy for Mg, states/eV\n",
      "E_Mg = 1.0/D_EF;  # The energy separation between adjacent energy levels of Cs, J\n",
      "print\"The energy separation between adjacent energy levels of Cs =\",\"{0:.3e}\".format(E_Mg/e),\"eV\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The energy separation between adjacent energy levels of Mg = 5.517e-23 eV\n",
        "The energy separation between adjacent energy levels of Cs = 1.172e-22 eV\n"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.9,Page number 122"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "m = 9.1*10**-31;   # Mass of an electron, kg\n",
      "e = 1.6*10**-19;   # Energy equivalent of 1 eV, J/eV\n",
      "E_F = 3.2*e;    # Fermi energy of sodium, J\n",
      "P_F = sqrt(E_F*2*m);    # Fermi momentum of sodium, kg-m/s\n",
      "print\"The Fermi momentum of sodium =\",\"{0:.3e}\".format(P_F),\"kg-m/sec\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The Fermi momentum of sodium = 9.653e-25 kg-m/sec\n"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.10,Page number 122"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "k = 1.38*10**-23;  # Boltzmann constant, J/mol/K\n",
      "T = 500.0;    # Rise in temperature of Al, K\n",
      "EF_0 = 11.63;    # Fermi energy of Al, eV\n",
      "EF_T = EF_0*(1-pi**2.0/12*(k*T/EF_0)**2);  # Change in Fermi energy of Al with temperature, eV\n",
      "print\"The change in Fermi energy of Al with tempertaure rise of 500 degree celsius =\",round(EF_T,3),\"eV\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The change in Fermi energy of Al with tempertaure rise of 500 degree celsius = 11.63 eV\n"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.11,Page number 122"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "m = 9.18*10**-31;   # Mass of an electron, kg\n",
      "e = 1.6*10**-19;   # Charge on an electron, C\n",
      "lamda = 1.0*10**-9;   # Mean free path of electron in metal, m\n",
      "v = 1.11*10**5;    # Average velocity of the electron in metal, m/s\n",
      "\n",
      "# For Lead\n",
      "n = 13.2*10**28;  # Electronic concentration of Pb, per metre cube\n",
      "sigma = n*e**2*lamda/(m*v);     # Electrical conductivity of lead, mho per metre\n",
      "print\"The electrical conductivity of lead =\",\"{0:.3e}\".format(sigma),\"mho per metre\";\n",
      "\n",
      "# For Silver\n",
      "n = 5.85*10**28;  # Electronic concentration of Ag, per metre cube\n",
      "sigma = n*e**2*lamda/(m*v);     # Electrical conductivity of Ag, mho per metre\n",
      "print\"The electrical conductivity of silver =\",\"{0:.3e}\".format(sigma),\"mho per metre\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The electrical conductivity of lead = 3.316e+07 mho per metre\n",
        "The electrical conductivity of silver = 1.470e+07 mho per metre\n"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.12,Page number 125"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "k = 1.38*10**-23;  # Boltzmann constant, J/mol/K\n",
      "e = 1.6*10**-19;   # Charge on an electron, C\n",
      "L = pi**2.0/3*(k/e)**2;    # Lorentz number, watt-ohm/degree-square\n",
      "print\"The Lorentz number =\",\"{0:.3e}\".format(L),\"watt-ohm/degree-square\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The Lorentz number = 2.447e-08 watt-ohm/degree-square\n"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.13,Page number 125"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "A =[[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]];      # Declare a 4X4 cell\n",
      "A[0][0] = 'Mg';  \n",
      "A[0][1] = 2.54*10**-5;  \n",
      "A[0][2] = 1.5;  \n",
      "A[0][3] = 2.32*10**2;  \n",
      "A[1][0] = 'Cu';  \n",
      "A[1][1] = 6.45*10**-5;  \n",
      "A[1][2] = 3.85;  \n",
      "A[1][3] = 2.30*10**2;  \n",
      "A[2][0] = 'Al';  \n",
      "A[2][1] = 4.0*10**-5;  \n",
      "A[2][2] = 2.38;\n",
      "A[2][3] = 2.57*10**2;  \n",
      "A[3][0] = 'Pt';  \n",
      "A[3][1] = 1.02*10**-5;  \n",
      "A[3][2] = 0.69;\n",
      "A[3][3] = 2.56*10**2;  \n",
      "T1 = 273;   # First temperature, K\n",
      "T2 = 373;   # Second temperature, K\n",
      "print\"_________________________________________________________________\";\n",
      "print\"Metal     sigma x 10**-05   K(W/cm-K)   Lorentz number             \";\n",
      "print\"          (mho per cm)                (watt-ohm/deg-square)x10**-2\";\n",
      "print\"_________________________________________________________________\";\n",
      "for i in range (0,4) :\n",
      "    L1 = A[i][2]/(A[i][1]*T1); \n",
      "    L2 = A[i][3];\n",
      "    print\"\",A[i][0],\"        \",A[i][1]/10**-5,\"         \",A[i][2],\"      \",L2/10**2,\"         \",L2/10**2;\n",
      "print\"_________________________________________________________________\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "_________________________________________________________________\n",
        "Metal     sigma x 10**-05   K(W/cm-K)   Lorentz number             \n",
        "          (mho per cm)                (watt-ohm/deg-square)x10**-2\n",
        "_________________________________________________________________\n",
        " Mg          2.54           1.5        2.32           2.32\n",
        " Cu          6.45           3.85        2.3           2.3\n",
        " Al          4.0           2.38        2.57           2.57\n",
        " Pt          1.02           0.69        2.56           2.56\n",
        "_________________________________________________________________\n"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.14,Page number 125"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "A = [[1,2],[3,4]];      # Declare a 2X3 cell\n",
      "A[0][0] = 1.6*10**8;  # Electrcal conductivity of Au at 100 K, mho per metre\n",
      "A[0][1] = 2.0*10**-8;   # Lorentz number of Au at 100 K, volt/K-square\n",
      "A[1][0] = 5.0*10**8;  # Electrcal conductivity of Au at 273  K, mho per metre\n",
      "A[1][1] = 2.4*10**-8;   # Lorentz number of Au at 273 K, volt/K-square\n",
      "T1 = 100;       # First temperature, K\n",
      "T2 = 273;       # Second temperature, K\n",
      "\n",
      "print\"___________________________________________________________________________\";\n",
      "print\"          T = 100 K                               T = 273 K                \";\n",
      "print\"_________________________________       ___________________________________\";\n",
      "print\"Electrical conductivity)  L              Electrical conductivity)  L       \";\n",
      "print\"    mho per metre       V/K-square           mho per metre       V/K-square\";\n",
      "print\"___________________________________________________________________________\";\n",
      "K1 = A[0][0]*T1*A[0][1]; \n",
      "K2 = A[1][0]*T2*A[1][1];\n",
      "print\"{0:.3e}\".format(A[0][0]),\"            \",\"{0:.3e}\".format(A[0][1]),\"         \",\"{0:.3e}\".format(A[1][0]),\"           \",\"{0:.3e}\".format(A[1][1]) \n",
      "print\"K =\",K1,\"W/cm-K                           K =\",K2,\"W/cm-K\";\n",
      "print\"___________________________________________________________________________\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "___________________________________________________________________________\n",
        "          T = 100 K                               T = 273 K                \n",
        "_________________________________       ___________________________________\n",
        "Electrical conductivity)  L              Electrical conductivity)  L       \n",
        "    mho per metre       V/K-square           mho per metre       V/K-square\n",
        "___________________________________________________________________________\n",
        "1.600e+08              2.000e-08           5.000e+08             2.400e-08\n",
        "K = 320.0 W/cm-K                           K = 3276.0 W/cm-K\n",
        "___________________________________________________________________________\n"
       ]
      }
     ],
     "prompt_number": 29
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.15,Page number 131"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "e = 1.6*10**-19;   # Electronic charge, C\n",
      "a = 0.428*10**-9;  # Lattice constant of Na, m\n",
      "V = a**3;    # Volume of unit cell, metre cube\n",
      "N = 2;  # No. of atoms per unit cell of Na\n",
      "n = N/V;    # No. of electrons per metre cube, per metre cube\n",
      "R_H = -1.0/(n*e);  # Hall coeffcient of Na, metre cube per coulomb\n",
      "print\"The Hall coefficient of sodium =\",\"{0:.3e}\".format(R_H),\"metre cube per coulomb\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The Hall coefficient of sodium = -2.450e-10 metre cube per coulomb\n"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.16,Page number 131"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "e = 1.6*10**-19;   # Electronic charge, C\n",
      "n = 24.2*10**28;    # No. of electrons per metre cube, per metre cube\n",
      "R_H = -1.0/(n*e);  # Hall coeffcient of Be, metre cube per coulomb\n",
      "print\"The Hall coefficient of beryllium =\",\"{0:.3e}\".format(R_H),\"metre cube per coulomb\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The Hall coefficient of beryllium = -2.583e-11 metre cube per coulomb\n"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.17,Page number 131"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "e = 1.6*10**-19;   # Electronic charge, C\n",
      "R_H = -8.4*10**-11;  # Hall coeffcient of Ag, metre cube per coulomb\n",
      "n = -3*pi/(8*R_H*e);    # Electronic concentration of Ag, per metre cube\n",
      "print\"The electronic concentration of Ag =\",\"{0:.3e}\".format(n),\"per metre cube\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The electronic concentration of Ag = 8.766e+28 per metre cube\n"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.18,Page number 134"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "# We have from Mattheissen rule, rho = rho_0 + alpha*T1\n",
      "T1 = 300.0;   # Initial temperature, K\n",
      "T2 = 1000.0;  # Final temperature, K\n",
      "rho = 1*10**-6;    # Resistivity of the metal, ohm-m\n",
      "delta_rho = 0.07*rho;      # Increase in resistivity of metal, ohm-m\n",
      "alpha = delta_rho/(T2-T1);  # A constant, ohm-m/K\n",
      "rho_0 = rho - alpha*T1;     # Resistivity at room temperature, ohm-m\n",
      "print\"The resistivity at room temperature =\",\"{0:.3e}\".format(rho),\"ohm-m\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The resistivity at room temperature = 1.000e-06 ohm-m\n"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.19,Page number 134"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "# We have from Mattheissen rule, rho = rho_0 + alpha*T1\n",
      "e = 1.6*10**-19;   # Energy equivalent of 1 eV, J/eV\n",
      "k = 1.38*10**-23;  # Boltzmann constant, J/mol/K\n",
      "rho_40 = 0.2;   # Resistivity of Ge at 40 degree celsius, ohm-m\n",
      "E_g = 0.7;  # Bandgap for Ge, eV\n",
      "T1 = 20+273;    # Second temperature, K\n",
      "T2 = 40 + 273;  # First temperature, K\n",
      "rho_20 = rho_40*exp(E_g*e/(2*k)*(1.0/T1-1.0/T2));    # Resistivity of Ge at 20 degree celsius, ohm-m\n",
      "print\"The resistivity of Ge at 20 degree celsius =\",round(rho_20,1),\"ohm-m\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The resistivity of Ge at 20 degree celsius = 0.5 ohm-m\n"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.20,Page number 135"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "rs_a0_ratio = 3.25;     # Ratio of solid radius to the lattice parameter\n",
      "E_F = 50.1*(rs_a0_ratio)**(-2);  # Fermi level energy of Li, eV\n",
      "T_F = 58.2e+04*(rs_a0_ratio)**(-2);  # Fermi level temperature of Li, K\n",
      "V_F = 4.20e+08*(rs_a0_ratio)**(-1);  # Fermi level velocity of electron in Li, cm/sec\n",
      "K_F = 3.63e+08*(rs_a0_ratio)**(-1);   \n",
      "print\"E_F =\",round(E_F,2),\"eV\";\n",
      "print\"T_F =\",\"{0:.3e}\".format(T_F),\"K\";\n",
      "print\"V_F =\",\"{0:.3e}\".format(V_F),\"cm/sec\";\n",
      "print\"K_F =\",\"{0:.3e}\".format(K_F),\"per cm\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "E_F = 4.74 eV\n",
        "T_F = 5.510e+04 K\n",
        "V_F = 1.292e+08 cm/sec\n",
        "K_F = 1.117e+08 per cm\n"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.21,Page number 135"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "n = 6.04*10**22;  # Concentration of electrons in yittrium, per metre cube\n",
      "r_s = (3/(4*pi*n))**(1.0/3)/10**-8;    # Radius of the solid, angstrom\n",
      "a0 = 0.529; # Lattice parameter of yittrium, angstrom\n",
      "rs_a0_ratio = r_s/a0;   # Solid radius to lattice parameter ratio\n",
      "E_F = 50.1*(rs_a0_ratio)**(-2);  # Fermi level energy of Y, eV\n",
      "print\"The Fermi energy of yittrium =\",round(E_F,4),\"eV\";\n",
      "Ryd = 13.6;     # Rydberg energy constant, eV\n",
      "E_bs = 0.396*Ryd;   # Band structure energy value of Y, eV\n",
      "print\"The band structure value of E_F =\",round(E_bs,3),\"eV is in close agreement with the calculated value of\",round(E_F,4),\"eV\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The Fermi energy of yittrium = 5.6083 eV\n",
        "The band structure value of E_F = 5.386 eV is in close agreement with the calculated value of 5.6083 eV\n"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.22,Page number 137"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "rs_a0_ratio = 2.07;   # Solid radius to lattice parameter ratio for Al\n",
      "E_F = 50.1*(rs_a0_ratio)**(-2);  # Fermi level energy of Y, eV\n",
      "# According to Jellium model, h_cross*omega_P = E = 47.1 eV *(rs_a0_ratio)**(-3/2)\n",
      "E = 47.1*(rs_a0_ratio)**(-3.0/2);      # Plasmon energy of Al, eV\n",
      "print\"The plasmon energy of Al =\",round(E,4),\"eV\";\n",
      "print\"The experimental value is 15 eV\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The plasmon energy of Al = 15.8149 eV\n",
        "The experimental value is 15 eV\n"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.1a,Page number 137"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "E_F = 1;    # For simplicity assume Fermi energy to be unity, eV\n",
      "k = 1.38*10**-23;      # Boltzmann constant, J/mol/K\n",
      "e = 1.6*10**-19;   # Energy equivalent of 1 eV, J/eV\n",
      "dE = 0.1;        # Exces energy above Fermi level, eV\n",
      "T = 300;        # Room temperature, K\n",
      "E = E_F + dE;    # Energy of the level above Fermi level, eV\n",
      "f_E = 1.0/(exp((E-E_F)*e/(k*T))+1);    # Occupation probability of the electron at 0.1 eV above E_F\n",
      "print\"At 300 K:\";\n",
      "print\"=========\";\n",
      "print\"The occupation probability of electron at\",round(dE,3),\"eV above Fermi energy =\",round(f_E,3);\n",
      "E = E_F - dE;    # Energy of the level below Fermi level, eV\n",
      "f_E = 1.0/(exp((E-E_F)*e/(k*T))+1);    # Occupation probability of the electron at 0.1 eV below E_F\n",
      "print\"The occupation probability of electron at\",round(dE,3),\"below Fermi energy =\",round(f_E,3);\n",
      "\n",
      "T = 1000;        # New temperature, K\n",
      "print\"At 1000 K:\";\n",
      "print\"=========\";\n",
      "E = E_F + dE;    # Energy of the level above Fermi level, eV\n",
      "f_E = 1.0/(exp((E-E_F)*e/(k*T))+1);    # Occupation probability of the electron at 0.1 eV above E_F\n",
      "print\"The occupation probability of electron at\",round(dE,3),\"eV above Fermi energy =\",round(f_E,3);\n",
      "E = E_F - dE;    # Energy of the level below Fermi level, eV\n",
      "f_E = 1.0/(exp((E-E_F)*e/(k*T))+1);    # Occupation probability of the electron at 0.1 eV below E_F\n",
      "print\"The occupation probability of electron at\",round(dE,3),\"eV below Fermi energy =\",round(f_E,3);\n",
      "\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "At 300 K:\n",
        "=========\n",
        "The occupation probability of electron at 0.1 eV above Fermi energy = 0.021\n",
        "The occupation probability of electron at 0.1 below Fermi energy = 0.979\n",
        "At 1000 K:\n",
        "=========\n",
        "The occupation probability of electron at 0.1 eV above Fermi energy = 0.239\n",
        "The occupation probability of electron at 0.1 eV below Fermi energy = 0.761\n"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.2a,Page number 138"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "f_E = 0.01;     # Occupation probability of electron\n",
      "E_F = 1;    # For simplicity assume Fermi energy to be unity, eV\n",
      "k = 1.38*10**-23;      # Boltzmann constant, J/mol/K\n",
      "e = 1.6*10**-19;   # Energy equivalent of 1 eV, J/eV\n",
      "dE = 0.5;        # Exces energy above Fermi level, eV\n",
      "E = E_F + dE;    # Energy of the level above Fermi level, eV\n",
      "# We have, f_E = 1/(exp((E-E_F)*e/(k*T))+1), solving for T\n",
      "T = (E-E_F)*e/k*1.0/log(1.0/f_E-1);   # Temperature at which the electron will have energy 0.1 eV above the Fermi energy, K\n",
      "print\"The temperature at which the electron will have energy\",round(dE,3),\"eV above the Fermi energy =\",round(T,3),\"K\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The temperature at which the electron will have energy 0.5 eV above the Fermi energy = 1261.578 K\n"
       ]
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.3a,Page number 139"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "E_F = 10;   # Fermi energy of electron in metal, eV\n",
      "e = 1.6*10**-19;   # Energy equivalent of 1 eV, J/eV\n",
      "m = 9.1*10**-31;   # Mass of an electron, kg\n",
      "E_av = 3.0/5*E_F;     # Average energy of free electron in metal at 0 K, eV\n",
      "V_F = sqrt(2*E_av*e/m);   # Speed of free electron in metal at 0 K, eV\n",
      "print\"The average energy of free electron in metal at 0 K =\",round(E_av,3),\"eV\";\n",
      "print\"The speed of free electron in metal at 0 K =\",\"{0:.3e}\".format(V_F),\"m/s\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The average energy of free electron in metal at 0 K = 6.0 eV\n",
        "The speed of free electron in metal at 0 K = 1.453e+06 m/s\n"
       ]
      }
     ],
     "prompt_number": 26
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.4a,Page number 139"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "f_E = 0.1;     # Occupation probability of electron\n",
      "E_F = 5.5;    # Fermi energy of Cu, eV\n",
      "k = 1.38*10**-23;      # Boltzmann constant, J/mol/K\n",
      "e = 1.6*10**-19;   # Energy equivalent of 1 eV, J/eV\n",
      "dE = 0.05*E_F;        # Exces energy above Fermi level, eV\n",
      "E = E_F + dE;    # Energy of the level above Fermi level, eV\n",
      "# We have, f_E = 1/(exp((E-E_F)*e/(k*T))+1), solving for T\n",
      "T = (E-E_F)*e/k*1.0/log(1.0/f_E-1);   # Temperature at which the electron will have energy 0.1 eV above the Fermi energy, K\n",
      "print\"The temperature at which the electron will have energy\",round(dE/E_F*100,3),\"percent above the Fermi energy\",round(T,3),\"K\";\n",
      "\n",
      "#(The answer given in the textbook is wrong) "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The temperature at which the electron will have energy 5.0 percent above the Fermi energy 1451.106 K\n"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.5a,Page number 139"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "T_F = 24600;   # Fermi temperature of potassium, K\n",
      "k = 1.38*10**-23;  # Boltzmann constant, J/mol/K\n",
      "m = 9.1*10**-31;   # Mass of an electron, kg\n",
      "E_F = k*T_F;    # Fermi energy of potassium, eV\n",
      "v_F = sqrt(2*k*T_F/m);      # Fermi velocity of potassium, m/s\n",
      "print\"The Fermi velocity of potassium =\",\"{0:.3e}\".format(v_F),\"m/s\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The Fermi velocity of potassium = 8.638e+05 m/s\n"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.6a,Page number 139"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "e = 1.6*10**-19;   # Energy equivalent of 1 eV, J/eV\n",
      "E_F = 7.0;  # Fermi energy of Cu, eV\n",
      "f_E = 0.9;  # Occupation probability of Cu\n",
      "k = 1.38*10**-23;  # Boltzmann constant, J/mol/K\n",
      "T = 1000;   # Given temperature, K\n",
      "# We have, f_E = 1/(exp((E-E_F)*e/(k*T))+1), solving for E\n",
      "E = k*T*log(1.0/f_E-1) + E_F*e;     # Energy level of Cu for 10% occupation probability at 1000 K, J\n",
      "print\"The energy level of Cu for 10 percent occupation probability at 1000 K =\",round(E/e,3),\"eV\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The energy level of Cu for 10 percent occupation probability at 1000 K = 6.81 eV\n"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.7a,Page number 140"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "m = 9.1*10**-31;   # Mass of an electron, kg\n",
      "e = 1.6*10**-19;   # Electronic charge, C\n",
      "h = 6.626*10**-34; # Planck's constant, Js\n",
      "E_F = 1.55;  # Fermi energy of Cu, eV\n",
      "n = (pi/3)*(8*m/h**2)**(3.0/2)*(E_F*e)**(3.0/2);    # Electronic concentration in cesium, electrons/cc\n",
      "print\"The electronic concentration in cesium =\",\"{0:.3e}\".format(n),\"electrons/cc\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The electronic concentration in cesium = 8.733e+27 electrons/cc\n"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.8a,Page number 141"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "e = 1.6*10**-19;   # Energy equivalent of 1 eV, J/eV\n",
      "E_F = 7;  # Fermi energy, eV\n",
      "k = 1.38*10**-23;  # Boltzmann constant, J/mol/K\n",
      "T_F = E_F*e/k;    # Fermi temperature, K\n",
      "print\"The Fermi temperature corresponding to Fermi energy =\",\"{0:.3e}\".format(T_F),\"K\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The Fermi temperature corresponding to Fermi energy = 8.116e+04 K\n"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.9a,Page number 141"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "m = 9.1*10**-31;   # Mass of the electron, kg\n",
      "h = 6.626*10**-34; # Planck's constant, Js\n",
      "e = 1.6*10**-19;   # Energy equivalent of 1 eV, J/eV\n",
      "h_cross = h/(2*pi);    # Reduced Planck's constant, Js\n",
      "s = 0.01;   # Side of the box, m\n",
      "E = 2;  # Energy range of the electron in the box, eV\n",
      "V = s**3;    # Volume of the box, metre cube\n",
      "I = I = 2*E**(3.0/2)/3;    # Definite integral over E : I = 2*E**(3/2)/3\n",
      "D_E = V/(2*pi**2)*(2*m/h_cross**2)**(3.0/2)*I*e**(3.0/2);  # Density of states for the electron in a cubical box, states\n",
      "print\"The density of states for the electron in a cubical box =\",\"{0:.3e}\".format(D_E),\"states\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The density of states for the electron in a cubical box = 1.280e+22 states\n"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.10a,Page number 141"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "E_F = 1;    # For simplicity assume Fermi energy to be unity, eV\n",
      "k = 1.38*10**-23;      # Boltzmann constant, J/mol/K\n",
      "e = 1.6*10**-19;   # Energy equivalent of 1 eV, J/eV\n",
      "dE = 0.5;        # Exces energy above Fermi level, eV\n",
      "T = 300;        # Room temperature, K\n",
      "E = E_F + dE;    # Energy of the level above Fermi level, eV\n",
      "f_E = 1./(exp((E-E_F)*e/(k*T))+1);    # Occupation probability of the electron at 0.1 eV above E_F\n",
      "print\"At 300 K:\";\n",
      "print\"=========\";\n",
      "print\"The occupation probability of electron at\",dE,\"eV above Fermi energy =\",\"{0:.3e}\".format(f_E);\n",
      "E = E_F - dE;    # Energy of the level below Fermi level, eV\n",
      "f_E = 1.0/(exp((E-E_F)*e/(k*T))+1);    # Occupation probability of the electron at 0.1 eV below E_F\n",
      "print\"The occupation probability of electron at\",dE,\"eV above Fermi energy =\",\"{0:.3e}\".format(f_E);\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "At 300 K:\n",
        "=========\n",
        "The occupation probability of electron at 0.5 eV above Fermi energy = 4.054e-09\n",
        "The occupation probability of electron at 0.5 eV above Fermi energy = 1.000e+00\n"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.11a,Page number 141"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "E_F = 1;    # For simplicity assume Fermi energy to be unity, eV\n",
      "k = 1.38*10**-23;      # Boltzmann constant, J/mol/K\n",
      "e = 1.6*10**-19;   # Energy equivalent of 1 eV, J/eV\n",
      "dE = 0.2;        # Exces energy above Fermi level, eV\n",
      "T = 0+273;        # Room temperature, K\n",
      "E = E_F + dE;    # Energy of the level above Fermi level, eV\n",
      "f_E = 1.0/(exp((E-E_F)*e/(k*T))+1);    # Occupation probability of the electron at 0.1 eV above E_F\n",
      "print\"At 273 K:\";\n",
      "print\"=========\";\n",
      "print\"The occupation probability of electron at\",dE,\"eV above Fermi energy =\",\"{0:.3e}\".format(f_E);\n",
      "T = 100+273;    # Given temperature of 100 degree celsius, K\n",
      "f_E = 1.0/(exp((E-E_F)*e/(k*T))+1);    # Occupation probability of the electron at 0.1 eV below E_F\n",
      "print\"At 373 K:\";\n",
      "print\"=========\";\n",
      "print\"The occupation probability of electron at\",dE,\"eV above Fermi energy =\",\"{0:.3e}\".format(f_E);\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "At 273 K:\n",
        "=========\n",
        "The occupation probability of electron at 0.2 eV above Fermi energy = 2.047e-04\n",
        "At 373 K:\n",
        "=========\n",
        "The occupation probability of electron at 0.2 eV above Fermi energy = 1.992e-03\n"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.12a,Page number 142"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "m = 9.1*10**-31;   # Mass of the electron, kg\n",
      "e = 1.6*10**-19;   # Energy equivalent of 1 eV, J/eV\n",
      "r = 1.28*10**-10;   # Atomic radius of Cu, m\n",
      "a = 4*r/sqrt(2);    # Lattice constant of Cu, m\n",
      "tau = 2.7*10**-14;      # Relaxation time for the electron in Cu, s\n",
      "V = a**3;    # Volume of the cell, metre cube\n",
      "n = 4.0/V;    # Concentration of free electrons in monovalent copper, \n",
      "sigma = n*e**2*tau/m;    # Electrical conductivity of monovalent copper, mho per m\n",
      "print\"The electrical conductivity of monovalent copper =\",\"{0:.3e}\".format( sigma/100),\"mho per cm\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The electrical conductivity of monovalent copper = 6.403e+05 mho per cm\n"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.13a,Page number 142"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "n = 18.1*10**22;  # Number of electrons per unit volume, per cm cube\n",
      "N = n/2;    # Pauli's principle for number of energy levels, per cm cube\n",
      "E_F = 11.58;     # Fermi energy of Al, eV\n",
      "E = E_F/N;     # Interelectronic energy separation between bands of Al, eV\n",
      "print\"The interelectronic energy separation between bands of Al =\",\"{0:.3e}\".format(E),\"eV\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The interelectronic energy separation between bands of Al = 1.280e-22 eV\n"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.14a,Page number 142"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "m = 9.1*10**-31;   # Mass of the electron, kg\n",
      "h = 6.626*10**-34; # Planck's constant, Js\n",
      "e = 1.6*10**-19;   # Energy equivalent of 1 eV, J/eV\n",
      "h_cross = h/(2*pi);    # Reduced Planck's constant, Js\n",
      "E_F = 7;    # Fermi energy of Cu, eV\n",
      "V = 10**-6;  # Volume of the cubic metal, metre cube\n",
      "D_EF = V/(2*pi**2)*(2*m/h_cross**2)**(3.0/2)*(E_F)**(1.0/2)*e**(3.0/2);   # Density of states in Cu contained in cubic metal, states/eV\n",
      "print\"The density of states in Cu contained in cubic metal =\",\"{0:.3e}\".format(D_EF),\"states/eV\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The density of states in Cu contained in cubic metal = 1.796e+22 states/eV\n"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.15a,Page number 143"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "m = 9.1*10**-31;   # Mass of the electron, kg\n",
      "h = 6.626*10**-34; # Planck's constant, Js\n",
      "e = 1.6*10**-19;   # Energy equivalent of 1 eV, J/eV\n",
      "h_cross = h/(2*pi);    # Reduced Planck's constant, Js\n",
      "E_F = 7;    # Fermi energy of Cu, eV\n",
      "V = 10**-6;  # Volume of the cubic metal, metre cube\n",
      "D_EF = V/(2*pi**2)*(2*m/h_cross**2)**(3.0/2)*(E_F)**(1.0/2)*e**(3.0/2);   # Density of states in Cu contained in cubic metal, states/eV\n",
      "d = 1.0/(D_EF);     # Electronic energy level spacing between successive levels of Cu, eV\n",
      "print\"The electronic energy level spacing between successive levels of Cu =\",\"{0:.3e}\".format(d),\"eV\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The electronic energy level spacing between successive levels of Cu = 5.568e-23 eV\n"
       ]
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.16a,Page number 143"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "A = [[1,2],[3,4],[5,6],[7,8]];      # Declare a 4X2 matrix \n",
      "A[0][0] = 'Li';       \n",
      "A[0][1] = -0.4039;   # Energy of outermost atomic orbital of Li, Rydberg unit\n",
      "A[1][0] = 'Na';      # \n",
      "A[1][1] = -0.3777;   # Energy of outermost atomic orbital of Na, Rydberg unit\n",
      "A[2][0] = 'F';      # \n",
      "A[2][1] = -1.2502;   # Energy of outermost atomic orbital of F, Rydberg unit\n",
      "A[3][0] = 'Cl';      # \n",
      "A[3][1] = -0.9067;   # Energy of outermost atomic orbital of Cl, Rydberg unit\n",
      "cf = 13.6;  # Conversion factor for Rydberg to eV\n",
      "print\"________________________________________\";\n",
      "print\" Atom                          Energy gap\";\n",
      "print\"\",A[1][0],A[3][0],\"                         \",(A[1][1]-A[3][1])*cf,\"eV\";\n",
      "print\"\",A[1][0],A[2][0],\"                          \",(A[1][1]-A[2][1])*cf,\"eV\";\n",
      "print\"\",A[0][0],A[2][0],\"                          \",(A[0][1]-A[2][1])*cf,\"eV\";\n",
      "\n",
      "print\"________________________________________\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "________________________________________\n",
        " Atom                          Energy gap\n",
        " Na Cl                           7.1944 eV\n",
        " Na F                            11.866 eV\n",
        " Li F                            11.50968 eV\n",
        "________________________________________\n"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 4.18a,Page number 144"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Given Data\n",
      "# For Cu\n",
      "rs_a0_ratio = 2.67;     # Ratio of solid radius to the lattice parameter\n",
      "E_F = 50.1*(rs_a0_ratio)**(-2);  # Fermi level energy of Cu, eV\n",
      "T_F = 58.2*10**4*(rs_a0_ratio)**(-2);  # Fermi level temperature of Cu, K\n",
      "V_F = 4.20*10**8*(rs_a0_ratio)**(-1);  # Fermi level velocity of electron in Cu, cm/sec\n",
      "K_F = 3.63*10**8*(rs_a0_ratio)**(-1);   \n",
      "print\"For Cu :\";\n",
      "print\"========\";\n",
      "print\"E_F =\",round(E_F,3),\"eV\";\n",
      "print\"T_F =\",\"{0:.3e}\".format(T_F),\"K\";\n",
      "print\"V_F =\",\"{0:.3e}\".format(V_F),\"cm/sec\";\n",
      "print\"K_F =\",\"{0:.3e}\".format(K_F),\"per cm\";\n",
      "rs_a0_ratio = 3.07;     # Ratio of solid radius to the lattice parameter\n",
      "E_F = 50.1*(rs_a0_ratio)**(-2);  # Fermi level energy of Nb, eV\n",
      "T_F = 58.2*10**4*(rs_a0_ratio)**(-2);  # Fermi level temperature of Nb, K\n",
      "V_F = 4.20*10**8*(rs_a0_ratio)**(-1);  # Fermi level velocity of electron in Nb, cm/sec\n",
      "K_F = 3.63*10**8*(rs_a0_ratio)**(-1);   \n",
      "print\"For Nb :\";\n",
      "print\"========\";\n",
      "print\"E_F =\",round(E_F,3),\"eV\";\n",
      "print\"T_F =\",\"{0:.3e}\".format(T_F),\"K\";\n",
      "print\"V_F =\",\"{0:.3e}\".format(V_F),\"cm/sec\";\n",
      "print\"K_F =\",\"{0:.3e}\".format(K_F),\"per cm\";\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "For Cu :\n",
        "========\n",
        "E_F = 7.028 eV\n",
        "T_F = 8.164e+04 K\n",
        "V_F = 1.573e+08 cm/sec\n",
        "K_F = 1.360e+08 per cm\n",
        "For Nb :\n",
        "========\n",
        "E_F = 5.316 eV\n",
        "T_F = 6.175e+04 K\n",
        "V_F = 1.368e+08 cm/sec\n",
        "K_F = 1.182e+08 per cm\n"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}