{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#3: Crystal planes, X-ray diffraction and defects in solids"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"##Example number 3.1, Page number 3.19"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"glancing angle is 21 degrees\n"
]
}
],
"source": [
"#importing modules\n",
"import math\n",
"from __future__ import division\n",
"\n",
"#Variable declaration\n",
"lamda=0.071*10**-9; #wavelength(m)\n",
"a=0.28*10**-9; #lattice constant(m)\n",
"h=1;\n",
"k=1;\n",
"l=0;\n",
"n=2; #order of diffraction\n",
"\n",
"#Calculation\n",
"d=a/math.sqrt(h**2+k**2+l**2);\n",
"x=n*lamda/(2*d); \n",
"theta=math.asin(x); #angle(radian)\n",
"theta=theta*180/math.pi; #glancing angle(degrees)\n",
"\n",
"#Result\n",
"print \"glancing angle is\",int(theta),\"degrees\""
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"##Example number 3.2, Page number 3.19"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"wavelength is 0.0842 nm\n",
"maximum order of diffraction is 7.0\n"
]
}
],
"source": [
"#importing modules\n",
"import math\n",
"from __future__ import division\n",
"\n",
"#Variable declaration\n",
"n=1; #order of diffraction\n",
"theta1=8+(35/60); #angle(degrees)\n",
"d=0.282; #spacing(nm)\n",
"theta2=90;\n",
"\n",
"#Calculation\n",
"theta1=theta1*math.pi/180; #angle(radian)\n",
"lamda=2*d*math.sin(theta1)/n; #wavelength(nm)\n",
"theta2=theta2*math.pi/180; #angle(radian)\n",
"nmax=2*d/lamda; #maximum order of diffraction\n",
"\n",
"#Result\n",
"print \"wavelength is\",round(lamda,4),\"nm\"\n",
"print \"maximum order of diffraction is\",round(nmax)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"##Example number 3.3, Page number 3.20"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"fraction of vacancy sites is 8.466 *10**-7\n"
]
}
],
"source": [
"#importing modules\n",
"import math\n",
"from __future__ import division\n",
"\n",
"#Variable declaration\n",
"T1=500+273; #temperature(K)\n",
"T2=1000+273; #temperature(K)\n",
"f=1*10**-10; #fraction\n",
"\n",
"#Calculation\n",
"x=round(T1/T2,5);\n",
"y=round(math.log(f),3);\n",
"w=round(x*y,3);\n",
"F=math.exp(w); #fraction of vacancy sites\n",
"\n",
"#Result\n",
"print \"fraction of vacancy sites is\",round(F*10**7,3),\"*10**-7\""
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"##Example number 3.4, Page number 3.21"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ratio is math.sqrt( 6.0 ): math.sqrt( 3.0 ): math.sqrt( 2.0 )\n"
]
}
],
"source": [
"#importing modules\n",
"import math\n",
"from __future__ import division\n",
"\n",
"#Variable declaration\n",
"a=1; #assume\n",
"h1=1;\n",
"k1=0;\n",
"l1=0;\n",
"h2=1;\n",
"k2=1;\n",
"l2=0;\n",
"h3=1;\n",
"k3=1;\n",
"l3=1;\n",
"\n",
"#Calculation\n",
"d100=a*6/(h1**2+k1**2+l1**2);\n",
"d110=a*6/(h2**2+k2**2+l2**2);\n",
"d111=a*(6)/(h3**2+k3**2+l3**2);\n",
"\n",
"#Result\n",
"print \"ratio is math.sqrt(\",d100,\"): math.sqrt(\",d110,\"): math.sqrt(\",d111,\")\""
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"##Example number 3.5, Page number 3.21"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"lattice parameter of nickel is 3.522 angstrom\n"
]
}
],
"source": [
"#importing modules\n",
"import math\n",
"from __future__ import division\n",
"\n",
"#Variable declaration\n",
"n=1; #order of diffraction\n",
"theta=38.2; #angle(degrees)\n",
"lamda=1.54; #wavelength(angstrom)\n",
"h=2;\n",
"k=2;\n",
"l=0;\n",
"\n",
"#Calculation\n",
"theta=theta*math.pi/180; #angle(radian)\n",
"d=n*lamda/(2*math.sin(theta));\n",
"a=d*math.sqrt(h**2+k**2+l**2); #lattice parameter of nickel(angstrom)\n",
"\n",
"#Result\n",
"print \"lattice parameter of nickel is\",round(a,3),\"angstrom\""
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"##Example number 3.6, Page number 3.22"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"order of diffraction is 2\n"
]
}
],
"source": [
"#importing modules\n",
"import math\n",
"from __future__ import division\n",
"\n",
"#Variable declaration\n",
"theta=90; #angle(degrees)\n",
"lamda=1.5; #wavelength(angstrom)\n",
"d=1.6; #spacing(angstrom)\n",
"\n",
"#Calculation\n",
"theta=theta*math.pi/180; #angle(radian)\n",
"n=2*d*math.sin(theta)/lamda; #order of diffraction\n",
"\n",
"#Result\n",
"print \"order of diffraction is\",int(n)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"##Example number 3.7, Page number 3.22"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"length of unit cell is 0.287 *10**-9 m\n",
"volume of unit cell is 0.02366 *10**-27 m**3\n",
"radius of the atom is 0.1243 *10**-9 m\n"
]
}
],
"source": [
"#importing modules\n",
"import math\n",
"from __future__ import division\n",
"\n",
"#Variable declaration\n",
"h=1;\n",
"k=1;\n",
"l=0;\n",
"d=0.203*10**-9; #spacing(m)\n",
"\n",
"#Calculation\n",
"a=d*math.sqrt(h**2+k**2+l**2); #length of unit cell(m)\n",
"V=a**3; #volume of unit cell(m**3)\n",
"r=math.sqrt(3)*a/4; #radius of the atom(m)\n",
"\n",
"#Result\n",
"print \"length of unit cell is\",round(a*10**9,3),\"*10**-9 m\"\n",
"print \"volume of unit cell is\",round(V*10**27,5),\"*10**-27 m**3\"\n",
"print \"radius of the atom is\",round(r*10**9,4),\"*10**-9 m\""
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"##Example number 3.8, Page number 3.22"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"order of diffraction is 2\n"
]
}
],
"source": [
"#importing modules\n",
"import math\n",
"from __future__ import division\n",
"\n",
"#Variable declaration\n",
"theta=90; #angle(degrees)\n",
"lamda=1.5; #wavelength(angstrom)\n",
"d=1.6; #spacing(angstrom)\n",
"\n",
"#Calculation\n",
"theta=theta*math.pi/180; #angle(radian)\n",
"n=2*d*math.sin(theta)/lamda; #order of diffraction\n",
"\n",
"#Result\n",
"print \"order of diffraction is\",int(n)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"##Example number 3.9, Page number 3.23"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"glancing angle is 20 degrees 42 minutes 17 seconds\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",
"lamda=0.065; #wavelength(nm)\n",
"a=0.26; #edge length(nm)\n",
"h=1;\n",
"k=1;\n",
"l=0;\n",
"n=2;\n",
"\n",
"#Calculation\n",
"d=a/math.sqrt(h**2+k**2+l**2); \n",
"x=n*lamda/(2*d); \n",
"theta=math.asin(x); #glancing angle(radian)\n",
"theta=theta*180/math.pi; #glancing angle(degrees)\n",
"theta_d=int(theta); \n",
"theta_m=(theta-theta_d)*60;\n",
"theta_s=(theta_m-int(theta_m))*60;\n",
"\n",
"#Result\n",
"print \"glancing angle is\",theta_d,\"degrees\",int(theta_m),\"minutes\",int(theta_s),\"seconds\"\n",
"print \"answer varies due to rounding off errors\""
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"##Example number 3.10, Page number 3.23"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"cube edge of unit cell is 4.055 angstrom\n"
]
}
],
"source": [
"#importing modules\n",
"import math\n",
"from __future__ import division\n",
"\n",
"#Variable declaration\n",
"lamda=1.54; #wavelength(angstrom)\n",
"h=1;\n",
"k=1;\n",
"l=1;\n",
"n=1;\n",
"theta=19.2; #angle(degrees)\n",
"\n",
"#Calculation\n",
"theta=theta*math.pi/180; #angle(radian)\n",
"d=n*lamda/(2*math.sin(theta)); \n",
"a=d*math.sqrt(h**2+k**2+l**2); #cube edge of unit cell(angstrom)\n",
"\n",
"#Result\n",
"print \"cube edge of unit cell is\",round(a,3),\"angstrom\""
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"##Example number 3.11, Page number 3.24"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"lattice parameter of nickel is 3.522 angstrom\n"
]
}
],
"source": [
"#importing modules\n",
"import math\n",
"from __future__ import division\n",
"\n",
"#Variable declaration\n",
"lamda=1.54; #wavelength(angstrom)\n",
"h=2;\n",
"k=2;\n",
"l=0;\n",
"n=1;\n",
"theta=38.2; #angle(degrees)\n",
"\n",
"#Calculation\n",
"theta=theta*math.pi/180; #angle(radian)\n",
"d=n*lamda/(2*math.sin(theta)); \n",
"a=d*math.sqrt(h**2+k**2+l**2); #lattice parameter of nickel(angstrom)\n",
"\n",
"#Result\n",
"print \"lattice parameter of nickel is\",round(a,3),\"angstrom\""
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"##Example number 3.12, Page number 3.24"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"interplanar spacing for (111) is 0.208 nm\n",
"interplanar spacing for (321) is 0.096 nm\n"
]
}
],
"source": [
"#importing modules\n",
"import math\n",
"from __future__ import division\n",
"\n",
"#Variable declaration\n",
"a=0.36; #edge length(nm)\n",
"h1=1;\n",
"k1=1;\n",
"l1=1;\n",
"h2=3;\n",
"k2=2;\n",
"l2=1;\n",
"\n",
"#Calculation\n",
"d1=a/math.sqrt(h1**2+k1**2+l1**2); #interplanar spacing for (111)(nm)\n",
"d2=a/math.sqrt(h2**2+k2**2+l2**2); #interplanar spacing for (321)(nm)\n",
"\n",
"#Result\n",
"print \"interplanar spacing for (111) is\",round(d1,3),\"nm\"\n",
"print \"interplanar spacing for (321) is\",round(d2,3),\"nm\""
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"##Example number 3.13, Page number 3.25"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"glancing angle is 16 degrees 27 minutes\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",
"lamda=0.675; #wavelength(angstrom)\n",
"n=3; #order of diffraction\n",
"theta=5+(25/60); #angle(degrees)\n",
"\n",
"#Calculation\n",
"theta=theta*math.pi/180; #angle(radian)\n",
"d=lamda/(2*math.sin(theta)); \n",
"theta3=math.asin(3*lamda/(2*d)); #glancing angle(radian)\n",
"theta3=theta3*180/math.pi; #glancing angle(degrees)\n",
"theta_d=int(theta3); \n",
"theta_m=(theta3-theta_d)*60;\n",
"\n",
"#Result\n",
"print \"glancing angle is\",theta_d,\"degrees\",int(theta_m),\"minutes\"\n",
"print \"answer varies due to rounding off errors\""
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"##Example number 3.14, Page number 3.25"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"glancing angle is 22 degrees 56 minutes 31 seconds\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",
"lamda=0.79; #wavelength(angstrom)\n",
"n=3; #order of diffraction\n",
"d=3.04; #spacing(angstrom)\n",
"\n",
"#Calculation\n",
"x=round(n*lamda/(2*d),4);\n",
"theta=math.asin(x); #glancing angle(radian)\n",
"theta=theta*180/math.pi; #glancing angle(degrees)\n",
"theta_d=int(theta); \n",
"theta_m=(theta-theta_d)*60;\n",
"theta_s=(theta_m-int(theta_m))*60;\n",
"\n",
"#Result\n",
"print \"glancing angle is\",theta_d,\"degrees\",int(theta_m),\"minutes\",int(theta_s),\"seconds\"\n",
"print \"answer given in the book is wrong\""
]
}
],
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