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+{

+ "cells": [

+  {

+   "cell_type": "markdown",

+   "metadata": {},

+   "source": [

+    "#9: Energy bands in solids"

+   ]

+  },

+  {

+   "cell_type": "markdown",

+   "metadata": {},

+   "source": [

+    "##Example number 9.1, Page number 240"

+   ]

+  },

+  {

+   "cell_type": "code",

+   "execution_count": 29,

+   "metadata": {

+    "collapsed": false

+   },

+   "outputs": [

+    {

+     "name": "stdout",

+     "output_type": "stream",

+     "text": [

+      "The fermi function for an energy kt above fermi energy is 0.269\n"

+     ]

+    }

+   ],

+   "source": [

+    "#importing modules\n",

+    "import math\n",

+    "from __future__ import division\n",

+    "\n",

+    "#Variable declaration\n",

+    "#E-EF=KT\n",

+    "#K=KB is the boltzmann constant in m^2 Kg s^-2 k^-1\n",

+    "\n",

+    "#Calculation\n",

+    "f=1/(1+math.exp(1));    #The fermi function for an energy kt above fermi energy\n",

+    "\n",

+    "#Result\n",

+    "print \"The fermi function for an energy kt above fermi energy is\",round(f,3)"

+   ]

+  },

+  {

+   "cell_type": "markdown",

+   "metadata": {},

+   "source": [

+    "##Example number 9.2, Page number 240"

+   ]

+  },

+  {

+   "cell_type": "code",

+   "execution_count": 32,

+   "metadata": {

+    "collapsed": false

+   },

+   "outputs": [

+    {

+     "name": "stdout",

+     "output_type": "stream",

+     "text": [

+      "The fermi function is 0.358999\n",

+      "answer given in the book is wrong\n"

+     ]

+    }

+   ],

+   "source": [

+    "#importing modules\n",

+    "import math\n",

+    "from __future__ import division\n",

+    "\n",

+    "#Variable declaration\n",

+    "X=0.01*1.6*10**-19;    #difference between energy and fermi energy(J)\n",

+    "T=200;    #temperature(K)\n",

+    "KB=1.38*10**-23;    #Boltzmann's Constant(J/K)\n",

+    "\n",

+    "#Calculation\n",

+    "f=1/(1+math.exp(X/(KB*T)));     #The fermi function\n",

+    "\n",

+    "#Result\n",

+    "print \"The fermi function is\",round(f,6)\n",

+    "print \"answer given in the book is wrong\""

+   ]

+  },

+  {

+   "cell_type": "markdown",

+   "metadata": {},

+   "source": [

+    "##Example number 9.3, Page number 241"

+   ]

+  },

+  {

+   "cell_type": "code",

+   "execution_count": 37,

+   "metadata": {

+    "collapsed": false

+   },

+   "outputs": [

+    {

+     "name": "stdout",

+     "output_type": "stream",

+     "text": [

+      "The fermi velocity fo conducting electron in aluminium is 2.02118 *10**6 ms^-1\n",

+      "The mean free path for conducting electron of aluminium is 14.7546 nm\n"

+     ]

+    }

+   ],

+   "source": [

+    "#importing modules\n",

+    "import math\n",

+    "from __future__ import division\n",

+    "\n",

+    "#Variable declaration\n",

+    "EF=11.63*1.6*10**-19;    #fermi energy of conducting electron in aluminium(J)\n",

+    "t=7.3*10**-15;       #relaxation time for electron(sec)\n",

+    "m=9.11*10**-31;      #mass of electon(Kg)\n",

+    "\n",

+    "#Calculation\n",

+    "Vf=math.sqrt(2*EF/m);     #The fermi velocity fo conducting electron in aluminium(ms^-1)\n",

+    "x=t*Vf*10**9;     #mean free path for conducting electron of aluminium(nm)\n",

+    "\n",

+    "#Result\n",

+    "print \"The fermi velocity fo conducting electron in aluminium is\",round(Vf/10**6,5),\"*10**6 ms^-1\"\n",

+    "print \"The mean free path for conducting electron of aluminium is\",round(x,4),\"nm\""

+   ]

+  },

+  {

+   "cell_type": "markdown",

+   "metadata": {},

+   "source": [

+    "##Example number 9.4, Page number 241"

+   ]

+  },

+  {

+   "cell_type": "code",

+   "execution_count": 46,

+   "metadata": {

+    "collapsed": false

+   },

+   "outputs": [

+    {

+     "name": "stdout",

+     "output_type": "stream",

+     "text": [

+      "The fermi energy in a metal is 3.36888 *10**-19 J or 2.1055 eV\n",

+      "The fermi temperature in a metal is 24.41 *10**3 K\n"

+     ]

+    }

+   ],

+   "source": [

+    "#importing modules\n",

+    "import math\n",

+    "from __future__ import division\n",

+    "\n",

+    "#Variable declaration\n",

+    "Vf=0.86*10**6;    #The femi energy of electons in the metal(m/sec)\n",

+    "m=9.11*10**-31;    #mass of electon(Kg)\n",

+    "KB=1.38*10**-23;   #Boltzmann's Constant(m^2 Kg s^-2 k^-1)\n",

+    "\n",

+    "#Calculation\n",

+    "Ef=(1/2)*m*Vf**2;   #The fermi energy in a metal(J)\n",

+    "Tf=Ef/KB;    #The fermi temperature in a metal(K)\n",

+    "\n",

+    "#Result\n",

+    "print \"The fermi energy in a metal is\",round(Ef/10**-19,5),\"*10**-19 J or\",round(Ef/(1.6*10**-19),4),\"eV\"\n",

+    "print \"The fermi temperature in a metal is\",round(Tf/10**3,2),\"*10**3 K\""

+   ]

+  },

+  {

+   "cell_type": "markdown",

+   "metadata": {},

+   "source": [

+    "##Example number 9.5, Page number 242"

+   ]

+  },

+  {

+   "cell_type": "code",

+   "execution_count": 3,

+   "metadata": {

+    "collapsed": false

+   },

+   "outputs": [

+    {

+     "name": "stdout",

+     "output_type": "stream",

+     "text": [

+      "The fermi temparature for sodium is 37.1 *10**3 K\n",

+      "The fermi velocity fo conducting electron in aluminium is 1.0602 *10**6 ms^-1\n"

+     ]

+    }

+   ],

+   "source": [

+    "#importing modules\n",

+    "import math\n",

+    "from __future__ import division\n",

+    "\n",

+    "#Variable declaration\n",

+    "Ef=3.2*1.6*10**-19;    #The fermi energy in a metal(J)\n",

+    "m=9.11*10**-31;   #mass of electon(Kg)\n",

+    "KB=1.38*10**-23;    #Boltzmann's Constant(m^2 Kg s^-2 k^-1)\n",

+    "\n",

+    "#Calculation\n",

+    "Tf=Ef/KB;     #The fermi temparature for sodium(K)\n",

+    "Vf=math.sqrt(2*Ef/m);     #The fermi velocity fo conducting electron in aluminium(ms^-1)\n",

+    "\n",

+    "#Result\n",

+    "print \"The fermi temparature for sodium is\",round(Tf/10**3,2),\"*10**3 K\"\n",

+    "print \"The fermi velocity fo conducting electron in aluminium is\",round(Vf/10**6,4),\"*10**6 ms^-1\""

+   ]

+  },

+  {

+   "cell_type": "markdown",

+   "metadata": {},

+   "source": [

+    "##Example number 9.6, Page number 242"

+   ]

+  },

+  {

+   "cell_type": "code",

+   "execution_count": 7,

+   "metadata": {

+    "collapsed": false

+   },

+   "outputs": [

+    {

+     "name": "stdout",

+     "output_type": "stream",

+     "text": [

+      "The temperature at which there is 1% probability that an electron in a solid is 1.26158 *10**3 K\n"

+     ]

+    }

+   ],

+   "source": [

+    "#importing modules\n",

+    "import math\n",

+    "from __future__ import division\n",

+    "\n",

+    "#Variable declaration\n",

+    "E=5.5*1.6*10**-19;     #energy level(J)\n",

+    "Ef=5*1.6*10**-19;     #fermi energy level(J)\n",

+    "x=0.5*1.6*10**-19;    #Difference between energy and fermi energy(J)\n",

+    "f=0.01;   #fermi function at which there is 1% probability that an electron in a solid\n",

+    "KB=1.38*10**-23;    #Boltzmann's Constant(m^2 Kg s^-2 k^-1)\n",

+    "\n",

+    "#Calculation\n",

+    "T=x/(KB*(math.log(1-f)-math.log(f)));      #The temperature at which there is 1% probability that an electron in a solid(K)\n",

+    "\n",

+    "#Result\n",

+    "print \"The temperature at which there is 1% probability that an electron in a solid is\",round(T/10**3,5),\"*10**3 K\""

+   ]

+  },

+  {

+   "cell_type": "markdown",

+   "metadata": {},

+   "source": [

+    "##Example number 9.7, Page number 244"

+   ]

+  },

+  {

+   "cell_type": "code",

+   "execution_count": 9,

+   "metadata": {

+    "collapsed": false

+   },

+   "outputs": [

+    {

+     "name": "stdout",

+     "output_type": "stream",

+     "text": [

+      "The energy for probability of occupancy at 1st is 1.98 eV\n",

+      "The energy for probability of occupancy at 2nd is 2.219 eV\n",

+      "The energy for probability of occupancy at 3rd is 2.1 eV\n"

+     ]

+    }

+   ],

+   "source": [

+    "#importing modules\n",

+    "import math\n",

+    "from __future__ import division\n",

+    "\n",

+    "#Variable declaration\n",

+    "Ef=2.1*1.6*10**-19;    #fermi energy level in potassium(J)\n",

+    "f1=0.99;    #fermi factor for 1st\n",

+    "f2=0.01;    #fermi factor for 2nd\n",

+    "f3=0.5;     #fermi factor for 3rd\n",

+    "T=300;     #temperature(K)\n",

+    "e=1.6*10**-19;    #charge of electron(C)\n",

+    "KB=1.38*10**-23;     #Boltzmann's Constant(m^2 Kg s^-2 k^-1)\n",

+    "\n",

+    "#Calculation\n",

+    "E1=(Ef+((KB*T)*(math.log(1-f1)-math.log(f1))))/e;    #The energy for probability of occupancy at 1st(eV)\n",

+    "E2=(Ef+((KB*T)*(math.log(1-f2)-math.log(f2))))/e;    #The energy for 1st at which the probability of occupancy(eV)\n",

+    "E3=(Ef+((KB*T)*(math.log(1-f3)-math.log(f3))))/e;    #The energy for 1st at which the probability of occupancy(eV)\n",

+    "\n",

+    "#Result\n",

+    "print \"The energy for probability of occupancy at 1st is\",round(E1,2),\"eV\"\n",

+    "print \"The energy for probability of occupancy at 2nd is\",round(E2,3),\"eV\"\n",

+    "print \"The energy for probability of occupancy at 3rd is\",E3,\"eV\""

+   ]

+  },

+  {

+   "cell_type": "markdown",

+   "metadata": {},

+   "source": [

+    "##Example number 9.8, Page number 245"

+   ]

+  },

+  {

+   "cell_type": "code",

+   "execution_count": 11,

+   "metadata": {

+    "collapsed": false

+   },

+   "outputs": [

+    {

+     "name": "stdout",

+     "output_type": "stream",

+     "text": [

+      "The probability of unoccupancy by an electron at room temperature is 0.97946\n"

+     ]

+    }

+   ],

+   "source": [

+    "#importing modules\n",

+    "import math\n",

+    "from __future__ import division\n",

+    "\n",

+    "#Variable declaration\n",

+    "X=0.1*1.6*10**-19;     #difference between energy and fermi energy(J)\n",

+    "T=300;    #temperature(K)\n",

+    "KB=1.38*10**-23;    #Boltzmann's Constant(m^2 Kg s^-2 k^-1)\n",

+    "\n",

+    "#Calculation\n",

+    "f=1-1/(1+math.exp(X/(KB*T)));     #The probability of unoccupancy by an electron at room temperature \n",

+    "\n",

+    "#Result\n",

+    "print \"The probability of unoccupancy by an electron at room temperature is\",round(f,5)"

+   ]

+  },

+  {

+   "cell_type": "markdown",

+   "metadata": {},

+   "source": [

+    "##Example number 9.9, Page number 252"

+   ]

+  },

+  {

+   "cell_type": "code",

+   "execution_count": 18,

+   "metadata": {

+    "collapsed": false

+   },

+   "outputs": [

+    {

+     "name": "stdout",

+     "output_type": "stream",

+     "text": [

+      "The fermi energy for the metal is 11.66 eV\n",

+      "The fermi factor is 0.0205\n"

+     ]

+    }

+   ],

+   "source": [

+    "#importing modules\n",

+    "import math\n",

+    "from __future__ import division\n",

+    "\n",

+    "#Variable declaration\n",

+    "n=4;     #number of atoms/unit cell in Al\n",

+    "a=4.05*10**-10;    #lattice constant of Aluminium which is FCC crystal(m)\n",

+    "nf=3;   #number of free electrons per atom in Al\n",

+    "T=300;   #ambient temperature(K)\n",

+    "x=0.1*1.6*10**-19;   #The same difference energy and fermi energy(J)\n",

+    "m=9.11*10**-31;    #mass of electon(kg)\n",

+    "h=6.625*10**-34;   #plank's constant(m^2 Kg/sec)\n",

+    "KB=1.38*10**-23;   #Boltzmann's Constant(m^2 Kg s^-2 k^-1)\n",

+    "\n",

+    "#Calculation\n",

+    "nc=n*nf/(a**3);   #number of electrons per unit volume\n",

+    "Ef=h**2/(8*m)*((3*nc)/math.pi)**(2/3);   #The fermi energy for the metal(eV)\n",

+    "f=1/(1+math.exp(x/(KB*T)));    #he fermi factor\n",

+    "\n",

+    "#Result\n",

+    "print \"The fermi energy for the metal is\",round(Ef/(1.6*10**-19),2),\"eV\"\n",

+    "print \"The fermi factor is\",round(f,4)"

+   ]

+  },

+  {

+   "cell_type": "markdown",

+   "metadata": {},

+   "source": [

+    "##Example number 9.10, Page number 253"

+   ]

+  },

+  {

+   "cell_type": "code",

+   "execution_count": 20,

+   "metadata": {

+    "collapsed": false

+   },

+   "outputs": [

+    {

+     "name": "stdout",

+     "output_type": "stream",

+     "text": [

+      "The fermi energy for cesium is 1.537 eV\n"

+     ]

+    }

+   ],

+   "source": [

+    "#importing modules\n",

+    "import math\n",

+    "from __future__ import division\n",

+    "\n",

+    "#Variable declaration\n",

+    "n=2;   #number of atoms/unit cell in cesium which is Bcc\n",

+    "a=6.14*10**-10;   #lattice constant of cesium which is BCC crystal(m)\n",

+    "nf=1;   #number of free electrons per atom in cesium\n",

+    "m=9.11*10**-31;   #mass of electon(kg)\n",

+    "h=6.625*10**-34;    #plank's constant(m^2 Kg/sec)\n",

+    "KB=1.38*10**-23;   #Boltzmann's Constant(m^2 Kg s^-2 k^-1)\n",

+    "e=1.6*10**-19;    #charge of electron(C)\n",

+    "\n",

+    "#Calculation\n",

+    "nc=n*nf/(a**3);   #number of electrons per unit volume\n",

+    "Ef=(h**2/(8*m)*((3*nc)/math.pi)**(2/3))/e;    #The fermi energy for the metal(eV)\n",

+    "\n",

+    "#Result\n",

+    "print \"The fermi energy for cesium is\",round(Ef,3),\"eV\""

+   ]

+  },

+  {

+   "cell_type": "markdown",

+   "metadata": {},

+   "source": [

+    "##Example number 9.11, Page number 254"

+   ]

+  },

+  {

+   "cell_type": "code",

+   "execution_count": 22,

+   "metadata": {

+    "collapsed": false

+   },

+   "outputs": [

+    {

+     "name": "stdout",

+     "output_type": "stream",

+     "text": [

+      "The number of free electrons per unit volume in potassium is 1.38 *10**28 electrons/m^3\n"

+     ]

+    }

+   ],

+   "source": [

+    "#importing modules\n",

+    "import math\n",

+    "from __future__ import division\n",

+    "\n",

+    "#Variable declaration\n",

+    "Ef=2.1*1.6*10**-19;    #The fermi energy level in potassium at a particular temperature(J)\n",

+    "m=9.11*10**-31;     #mass of electron(kg)\n",

+    "h=6.625*10**-34;    #plank's constant(m^2 Kg/sec)\n",

+    "\n",

+    "#Calculation\n",

+    "nc=(8*m/(h**2)*Ef)**(3/2)*(math.pi/3);   #ThE Number of free electrons per unit volume in potassium(electrons/m^3)\n",

+    "\n",

+    "#Result\n",

+    "print \"The number of free electrons per unit volume in potassium is\",round(nc/10**28,2),\"*10**28 electrons/m^3\""

+   ]

+  },

+  {

+   "cell_type": "markdown",

+   "metadata": {},

+   "source": [

+    "##Example number 9.12, Page number 254"

+   ]

+  },

+  {

+   "cell_type": "code",

+   "execution_count": 24,

+   "metadata": {

+    "collapsed": false

+   },

+   "outputs": [

+    {

+     "name": "stdout",

+     "output_type": "stream",

+     "text": [

+      "The fermi energy for the sodium is 3.155 eV\n"

+     ]

+    }

+   ],

+   "source": [

+    "#importing modules\n",

+    "import math\n",

+    "from __future__ import division\n",

+    "\n",

+    "#Variable declaration\n",

+    "AW=23;    #atomic weight of sodium(gm/mole)\n",

+    "d=0.971*10**6;   #density of sodium(gm/m^3)\n",

+    "m=9.11*10**-31;   #mass of electon(kg)\n",

+    "h=6.625*10**-34;  #plank's constant(m^2 Kg/sec)\n",

+    "AV=6.02*10**23;   #Avagadro number(mole^-1)\n",

+    "e=1.6*10**-19;    #charge of electron(C)\n",

+    "\n",

+    "#Calculation\n",

+    "nc=AV*d/AW;    #number of electrons per unit volume\n",

+    "Ef=(h**2/(8*m)*((3*nc)/math.pi)**(2/3))/e;    #The fermi energy for the sodium(eV)\n",

+    "\n",

+    "#Result\n",

+    "print \"The fermi energy for the sodium is\",round(Ef,3),\"eV\""

+   ]

+  },

+  {

+   "cell_type": "markdown",

+   "metadata": {},

+   "source": [

+    "##Example number 9.13, Page number 255"

+   ]

+  },

+  {

+   "cell_type": "code",

+   "execution_count": 26,

+   "metadata": {

+    "collapsed": false

+   },

+   "outputs": [

+    {

+     "name": "stdout",

+     "output_type": "stream",

+     "text": [

+      "The fermi energy for the sodium is 7.046 eV\n",

+      "answer varies due to rounding off errors\n"

+     ]

+    }

+   ],

+   "source": [

+    "#importing modules\n",

+    "import math\n",

+    "from __future__ import division\n",

+    "\n",

+    "#Variable declaration\n",

+    "AW=63.5;    #atomic weight of copper(u)\n",

+    "M=63.5*1.66*10**-27;   #mass of one copper atom(kg)\n",

+    "d=8.94*10**3;    #density of sodium(Kg/m^3)\n",

+    "m=9.11*10**-31;  #mass of electon(Kg)\n",

+    "h=6.625*10**-34;   #plank's constant(m^2 Kg/sec)\n",

+    "e=1.6*10**-19;   #charge of electron(C)\n",

+    "\n",

+    "#Calculation\n",

+    "nc=d/M;   #number of electrons per unit volume(electrons/m^3)\n",

+    "Ef=h**2/(8*m)*((3*nc)/math.pi)**(2/3)/e;   #The fermi energy for the sodium(eV)\n",

+    "\n",

+    "#Result\n",

+    "print \"The fermi energy for the sodium is\",round(Ef,3),\"eV\"\n",

+    "print \"answer varies due to rounding off errors\""

+   ]

+  }

+ ],

+ "metadata": {

+  "kernelspec": {

+   "display_name": "Python 2",

+   "language": "python",

+   "name": "python2"

+  },

+  "language_info": {

+   "codemirror_mode": {

+    "name": "ipython",

+    "version": 2

+   },

+   "file_extension": ".py",

+   "mimetype": "text/x-python",

+   "name": "python",

+   "nbconvert_exporter": "python",

+   "pygments_lexer": "ipython2",

+   "version": "2.7.9"

+  }

+ },

+ "nbformat": 4,

+ "nbformat_minor": 0

+}