{
"metadata": {
"name": "",
"signature": "sha256:201c0242c8a2063b9a355e5a7deec7582a5a21b0e1187464be6b9fa32a384889"
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"worksheets": [
{
"cells": [
{
"cell_type": "heading",
"level": 1,
"metadata": {},
"source": [
"Chapter 3:Crystal planes and X-ray diffraction"
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 3.2, Page number 3.5"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"import math\n",
"from sympy import *\n",
"\n",
"#Variable declaration\n",
"'''In a simple cubic structure, there are three types of atomic arrangement\n",
"(i)(100)\n",
"(ii)(110)\n",
"(iii)(111)'''\n",
"\n",
"#Calculations and results\n",
"\n",
"#Consider (100) plane\n",
"n = (1./4.)*4 #no. of atoms in this plane\n",
"#Let a be the lattice constant in mm\n",
"a = Symbol('a')\n",
"A1 = a**2\n",
"nm = n/A1 #no. of atoms per mm^2\n",
"print \"The number of atoms per square millimeter for (100) plane is\",nm\n",
"\n",
"#Consider (110) plane\n",
"n2 = 1 \n",
"A2 = math.sqrt(2)*a**2\n",
"nm2 = n2/A2\n",
"print \"The number of atoms per square millimeter for (110) plane is\",nm2\n",
"\n",
"#Consider (111) plane\n",
"n3 = (1./360.)*60*3\n",
"EO = a*math.sqrt(2)*math.cos(math.pi/6)\n",
"A3 = (a*math.sqrt(2)*EO)/2\n",
"nm3 = n3/A3\n",
"print \"The number of atoms per square millimeter for (111) plane is\",nm3"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The number of atoms per square millimeter for (100) plane is 1.0/a**2\n",
"The number of atoms per square millimeter for (110) plane is 0.707106781186547/a**2\n",
"The number of atoms per square millimeter for (111) plane is 0.577350269189626/a**2\n"
]
}
],
"prompt_number": 2
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 3.3, Page number 3.6"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"from math import sqrt\n",
"\n",
"#Variable declartion\n",
"r = 0.1278*10**-9 #atomic radius(m)\n",
"\n",
"#Calculations\n",
"#For FCC structure,\n",
"a = (4*r)/sqrt(2)\n",
"\n",
"#For (110) plane,\n",
"h1 = 1\n",
"k1 = 1\n",
"l1 = 0\n",
"d1 = a/((h1**2+k1**2+l1**2)**0.5) \n",
"\n",
"#For (212) plane,\n",
"h2 = 2\n",
"k2 = 1\n",
"l2 = 2\n",
"d2 = a/((h2**2+k2**2+l2**2)**0.5)\n",
"\n",
"#Results\n",
"print \"Interplanar spacing for (110) plane =\",d1/1E-9,\"nm\"\n",
"print \"Interplanar spacing for (212) plane =\",round((d2/1E-9),4),\"nm\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Interplanar spacing for (110) plane = 0.2556 nm\n",
"Interplanar spacing for (212) plane = 0.1205 nm\n"
]
}
],
"prompt_number": 3
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 3.4, Page number 3.7"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"from sympy import *\n",
"\n",
"#Variable declaration\n",
"#Let a be the lattice constant\n",
"#For calculations, let us assume a = 1\n",
"a = 1\n",
"\n",
"#Calculations\n",
"\n",
"#For (100) plane,\n",
"h1 = 1\n",
"k1 = 0\n",
"l1 = 0\n",
"d1 = 1/((h1**2+k1**2+l1**2)**0.5) \n",
"\n",
"#For (110) plane,\n",
"h2 = 1\n",
"k2 = 1\n",
"l2 = 0\n",
"d2 = 1/((h2**2+k2**2+l2**2)**0.5)\n",
"\n",
"#For (111) plane,\n",
"h3 = 1\n",
"k3 = 1\n",
"l3 = 1\n",
"d3 = 1/((h3**2+k3**2+l3**2)**0.5)\n",
"\n",
"#Result\n",
"print \"d100:d110:d111 =\",d1,\":\",d2,\":\",d3"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"d100:d110:d111 = 1.0 : 0.707106781187 : 0.57735026919\n"
]
}
],
"prompt_number": 4
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 3.5, Page number 3.7"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"#Variable declaration\n",
"#Coefficients of intercepts along three axes\n",
"m = 1.\n",
"n = 1./2.\n",
"p = 3.\n",
"\n",
"#Calculations\n",
"m_inv = 1/m \n",
"n_inv = 1/n\n",
"p_inv = 1/p\n",
"\n",
"def gcd(a, b):\n",
" while b: \n",
" a, b = b, a % b\n",
" return a\n",
"\n",
"def lcm(a, b):\n",
" return (a * b)/ gcd(a, b)\n",
"\n",
"def lcmm(*args): \n",
" return reduce(lcm, args)\n",
"\n",
"mul_fact = lcmm(1,1,3)\n",
"m1 = m_inv*mul_fact #Clear the first fraction\n",
"m2 = n_inv*mul_fact #Clear the second fraction\n",
"m3 = p_inv*mul_fact #Clear the third fraction\n",
"print \"The required miller indices are\", m1,m2,m3\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The required miller indices are 3.0 6.0 1.0\n"
]
}
],
"prompt_number": 5
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 3.6, Page number 3.14"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"import math\n",
"\n",
"#Variable declaration\n",
"d = 0.282*10**-9 #lattice spacing(m)\n",
"n = 1 #first order\n",
"theta = 8.35 #glancing angle(degrees)\n",
"\n",
"#Calculations\n",
"lamda = (2*d*math.sin(math.radians(theta)))/n\n",
"\n",
"#For maximum value possible,\n",
"theta = 90\n",
"n = (2*d)/lamda\n",
"\n",
"#Results\n",
"print \"Wavelength of X-rays =\",round((lamda/1E-9),4),\"nm\"\n",
"print \"Maximum order of diffraction possible =\",round(n,2)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Wavelength of X-rays = 0.0819 nm\n",
"Maximum order of diffraction possible = 6.89\n"
]
}
],
"prompt_number": 8
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 3.8, Page number 3.15"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"import math\n",
"\n",
"#Variable declaration\n",
"lamda = 1.5418*10**-10 #wavelength(m)\n",
"theta = 30 #angle(degrees)\n",
"n = 1 #first order\n",
"#For (111) plane\n",
"h = 1\n",
"k = 1\n",
"l = 1\n",
"\n",
"#Calculations\n",
"d = (n*lamda)/(2*math.sin(math.radians(theta)))\n",
"a = d*((h**2+k**2+l**2)**0.5)\n",
"\n",
"#Result\n",
"print \"Interatomic spacing =\",round((a/1E-10),3),\"A\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Interatomic spacing = 2.67 A\n"
]
}
],
"prompt_number": 9
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 3.9, Page number 3.15"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"from math import sqrt, pi \n",
"\n",
"#Variable declaration\n",
"d = 0.28 #lattice spacing\n",
"lamda = 0.074*10**-9 #Wavelength(m)\n",
"n = 2 #2nd order\n",
"\n",
"#Calculations\n",
"d110 = d/sqrt(2)\n",
"theta = math.asin((n*lamda)/(2*d110))*180/pi\n",
"\n",
"#Result\n",
"print \"Glancing angle =\",round(theta/1E-9),\"degrees\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Glancing angle = 21.0 degrees\n"
]
}
],
"prompt_number": 10
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 3.10, Page number 3.16"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"#Variable declaration\n",
"a = 0.38*10**-9 #lattice constant\n",
"h = 1\n",
"k = 1\n",
"l = 0\n",
"\n",
"#Calculations\n",
"d = a/math.sqrt(h**2+k**2+l**2)\n",
"\n",
"#Result\n",
"print \"Distance =\",round((d/1E-9),2),\"nm\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Distance = 0.27 nm\n"
]
}
],
"prompt_number": 11
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 3.11, Page number 3.16"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"import math\n",
"from sympy import *\n",
"\n",
"#Variable declaration\n",
"a = Symbol('a')\n",
"\n",
"#Calculations\n",
"#For (110) plane\n",
"a1 = math.sqrt(2)*a**2 #area of plane\n",
"n1 = (1./4 )*4 #no. of atoms in this plane\n",
"rho1 = n1/a1\n",
"\n",
"#For (111) plane\n",
"EO = (a*math.sqrt(3))/math.sqrt(2)\n",
"a2 = (a*EO)/math.sqrt(2)\n",
"n2 = 3*(1./6.)\n",
"rho2 = n2/a2\n",
"\n",
"#Result\n",
"print \"Density of lattice points (111) plane:density of lattice points (110) plane =\",rho2,\":\",rho1"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Density of lattice points (111) plane:density of lattice points (110) plane = 0.577350269189626/a**2 : 0.707106781186547/a**2\n"
]
}
],
"prompt_number": 12
}
],
"metadata": {}
}
]
}