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{
"cells": [
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"# Chapter 2:Band Theory of Solids"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": true
},
"source": [
"## Example 2.1,Page No:2.2"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Lowest three permissable quantum energies are E1 = 6 eV\n",
" E2 = 24 eV\n",
" E3 = 54 eV\n"
]
}
],
"source": [
"import math\n",
"\n",
"#variable declaration\n",
"h = 6.63*10**-34; # plancks constant in J.s\n",
"m = 9.1*10**-31; # mass of electron in kg\n",
"a = 2.5*10**-10; # width of infinite square well\n",
"e = 1.6*10**-19; # charge of electron coulombs\n",
"n2 = 2; #number of permiissable quantum\n",
"n3 = 3; #number of permiissable quantum\n",
"\n",
"# Calculations\n",
"E1 = (h**2)/float(8*m*a**2*e); # first lowest permissable quantum energy in eV\n",
"E2 = n2**2 *E1; # second lowest permissable quantum energy in eV\n",
"E3 = n3**2 *E1; # second lowest permissable quantum energy in eV\n",
"\n",
"# Result\n",
"print'Lowest three permissable quantum energies are E1 = %d'%E1,'eV';\n",
"print' E2 = %d'%E2,'eV';\n",
"print' E3 = %d'%E3,'eV';"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 2.2,Page No:2.4"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Energy Difference = 113.21 eV\n"
]
}
],
"source": [
"import math\n",
"\n",
"#variable declaration\n",
"h = 6.63*10**-34; # plancks constant in J.s\n",
"m = 9.1*10**-31; # mass of electron in kg\n",
"a = 10**-10; # width of infinite square well in m\n",
"e = 1.6*10**-19; # charge of electron in coulombs\n",
"n1 = 1; #energy level constant\n",
"n2 = 2; #energy level constant\n",
"\n",
"# calculations\n",
"E1 = ((n1**2)*(h**2))/float(8*m*(a**2)*e); # ground state energy in eV\n",
"E2 = ((n2**2)*(h**2))/float(8*m*(a**2)*e); # first excited state in energy in eV\n",
"dE = E2-E1 # difference between first excited and ground state(E2 - E1)\n",
"\n",
"#Result\n",
"print'Energy Difference = %3.2f '%dE,'eV';\n",
"\n",
" \n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 2.3,Page No:2.5"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"First Three Energy levels are \n",
" E1 = 1.51 eV\n",
" E2 = 6 eV\n",
" E3 = 13.59 eV\n",
"\n",
" Above calculation shows that the energy of the bound electron cannot be continuous\n"
]
}
],
"source": [
"import math\n",
"\n",
"# Variable declaration\n",
"h = 6.63*10**-34; # plancks constant in J.s\n",
"m = 9.1*10**-31; # mass of electron in kg\n",
"a = 5*10**-10; # width of infinite potential well in m\n",
"e = 1.6*10**-19; # charge of electron in coulombs\n",
"n1 = 1; # energy level constant\n",
"n2 = 2; # energy level constant\n",
"n3 = 3; # energy level constant\n",
"\n",
"#Calculations\n",
"E1 = ((n1**2)*(h**2))/(8*m*(a**2)*e); # first energy level in eV\n",
"E2 = ((n2**2)*(h**2))/(8*m*(a**2)*e); # second energy level in eV\n",
"E3 = ((n3**2)*(h**2))/(8*m*(a**2)*e); # third energy level in eV\n",
"\n",
"# Result\n",
"print'First Three Energy levels are \\n E1 = %3.2f'%E1,'eV';\n",
"print' E2 = %d'%E2,'eV';\n",
"print' E3 = %3.2f'%E3,'eV';\n",
"print'\\n Above calculation shows that the energy of the bound electron cannot be continuous';\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 2.4,Page No:2.5"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Lowest energy bandwidth = 0.452 eV\n"
]
}
],
"source": [
"import math\n",
"\n",
"#variable declaration\n",
"h = 1.054*10**-34; #plancks constant in J.s\n",
"m = 9.1*10**-31; #mass of electron in kg\n",
"a = 5*10**-10; #width of infinite potential well in m\n",
"e = 1.6*10**-19; # charge of electron coulombs\n",
"\n",
"# Calculations\n",
"#cos(ka) = ((Psin(alpha*a))/(alpha*a)) + cos(alpha*a)\n",
"#to find the lowest allowed energy bandwidth,we have to find the difference in αa values, as ka changes from 0 to π\n",
"# for ka = 0 in above eq becomes\n",
"# 1 = 10*sin(αa))/(αa)) + cos(αa)\n",
"# This gives αa = 2.628 rad\n",
"# ka = π , αa = π\n",
"# sqrt((2*m*E2)/h**2)*a = π\n",
"\n",
"E2 = ((math.pi*math.pi)*h**2)/(2*m*a**2*e); #energy in eV\n",
"E1 = ((2.628**2)*h**2)/(2*m*a**2*e); #for αa = 2.628 rad energy in eV\n",
"dE = E2 - E1; #lowest energy bandwidth in eV\n",
"\n",
"# Result\n",
"print'Lowest energy bandwidth = %3.3f'%dE,'eV';\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 2.5,Page No:2.8"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Electron Momentum for first Brillouin zone appearance = 1.105e-24 eV\n",
"\n",
" Energy of free electron with this momentum = 4.2 eV\n",
"\n",
" Note: in Textbook Momentum value is wrongly printed as 1.1*10**-10\n"
]
}
],
"source": [
"import math\n",
"\n",
"# Variable declaration\n",
"a = 3*10**-10; # side of 2d square lattice in m\n",
"h = 6.63*10**-34; # plancks constant in J.s\n",
"e = 1.6*10**-19 # charge of electron in coulombs\n",
"m = 9.1*10**-31; # mass of electron in kg\n",
"\n",
"# calculations\n",
"#p = h*k # momentum of the electron\n",
"k = math.pi/float(a); # first Brillouin zone\n",
"p = (h/float(2*math.pi))*(math.pi/float(a)); # momentum of electron\n",
"E = (p**2)/float(2*m*e) # Energyin eV\n",
"\n",
"#Result\n",
"print'Electron Momentum for first Brillouin zone appearance = %g'%p,'eV';\n",
"print'\\n Energy of free electron with this momentum = %4.1f'%E,'eV';\n",
"print'\\n Note: in Textbook Momentum value is wrongly printed as 1.1*10**-10';"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 2",
"language": "python",
"name": "python2"
},
"language_info": {
"codemirror_mode": {
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"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
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|
