{
"cells": [
 {
		   "cell_type": "markdown",
	   "metadata": {},
	   "source": [
       "# Chapter 3: Crystal Planes and Defects"
	   ]
	},
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 3.10: Average_energy_required_to_create_Schottky_defect.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// Scilab Code Ex3.10 :Page-3.16 (2004)\n",
"T = 273+25; // Temperature , K\n",
"r = 2.82e-10;   // Interionic distance, m\n",
"N = 4/((2*r)^3);    // Density of ion pairs, ion pairs\n",
"k = 8.625e-5;  // Boltzmann constant, J/K\n",
"n = 5e+11;  // Density od Schottky effects, per unit volume\n",
"E_s = 2*k*T*2.303*log10(N/n);   // Average energy required to creat Schottky defect\n",
"printf('\nAverage energy required to create Schottky defect = %5.3f eV', E_s);\n",
"\n",
"// Result \n",
"// Average energy required to create Schottky defect = 1.971 eV "
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 3.11: Ratio_of_vacancies_in_metal_to_create_Frenkel_defect.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// Scilab Code Ex3.11 : Ratio of vacancies in metal to create Frenkel defect:Page-3.18 (2004)\n",
"N = 1;  // For simplicity assume total number of metal ions to be unity\n",
"Ni = 1; // For simplicity assume total number of metal ions to be unity\n",
"k = 8.625e-5;  // Boltzmann constant, J/K\n",
"T1 = 273+20;   // First temperature for metal, K\n",
"T2 = 300+273; // Second temperature for metal, K\n",
"E_v = 1.4;   // Average energy required to create a vacancy in metal, eV \n",
"n_293 = N*exp(-E_v/(2*k*T1));  // Number of vacancies at 500 K\n",
"n_573 = N*exp(-E_v/(2*k*T2));  // Number of vacancies at 500 K\n",
"n_ratio1 = n_573/n_293;     // Ratio of vacancies in metal \n",
"n_ratio2 = n_293/n_573;     // Ratio of vacancies in metal \n",
"\n",
"printf('\nThe ratio 1 of vacancies in metal to create Frenkel defect = %5.3e', n_ratio1);\n",
"printf('\nThe ratio 2 of vacancies in metal to create Frenkel defect = %5.3e', n_ratio2);\n",
"\n",
"// Result \n",
"//  The ratio 1 of vacancies in metal to create Frenkel defect = 7.558e+05\n",
"// The ratio 2 of vacancies in metal to create Frenkel defect = 1.323e-06 "
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 3.1: Number_of_atoms_per_square_mm_in_SC.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// Scilab Code Ex3.1 : Page-3.4 (2004)\n",
" // In a SC structure number of planes are having three arrangement (100),(110) and (111)\n",
" clc;clear;\n",
"a =  1;     // For simplicity lattice constant is taken to be unity\n",
"A_100 = a^2;    // Area of the plane (100), mm^2\n",
"N_100 = 1/A_100;    //   Number of atoms along (100) plane, atoms per square mm\n",
"A_110 = sqrt(2)*a^2;    // Area of the plane (110), mm^2\n",
"N_110 = 1/A_110;    // //   Number of atoms along (110) plane, atoms per square mm\n",
"A_111 = 1/2*a*sqrt(2)*sqrt(2)*a^2*cosd(30);    // Area of the plane (110), mm^2\n",
"A_111t = 0.5;   //   Total no of atoms in (111) plane\n",
"N_111 = A_111t/A_111;    // //   Number of atoms along (110) plane, atoms per square mm\n",
"printf('\nNumber of atoms along (100) plane= %d /a^2 atoms per square mm', N_100);\n",
"printf('\nNumber of atoms along (110) plane= %f  atoms per square mm', N_110);\n",
"printf('\nNumber of atoms along (111) plane= %5.3f /a^2 atoms per square mm', N_111);\n",
"// Result \n",
"// Number of atoms along (100) plane= 1 /a^2 atoms per square mm\n",
"//  Number of atoms along (110) plane= 0.707107  atoms per square mm\n",
"//  Number of atoms along (111) plane= 0.577 /a^2 atoms per square mm "
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 3.2: Maximum_radius_of_sphere_in_BCC_lattice.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// Scilab Code Ex3.2 :  Page-3.5(2004)\n",
"clc;clear;\n",
"r = 1;  // For simplicity assume radius of atom to be unity, unit\n",
"a = 4*r/sqrt(3);  // Lattice constant, unit\n",
"R = (a/2)-r;      // R be the radius of interstitial sphere that can fit into void, unit\n",
"printf ('\nMaximum Radius of sphere that can fit into BCC = %5.3fr', R);\n",
"\n",
"//  Result\n",
"//  Maximum Radius of sphere that can fit into BCC = 0.155r "
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 3.3: Volume_change_during_BCC_to_FCC.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// Scilab Code Ex3.3 :  Page-3.6 (2010)\n",
"clc;clear;\n",
"r1 = 1.258e-10;  // Atomic radius in BCC, metre\n",
"a1 = 4*r1/sqrt(3);  // Lattice constant for BCC,  metre \n",
"V1 = a1^3;   // Volume of unit cell in BCC, metre cube\n",
"Vpa = V1/2; // Volume occupied by one atom in BCC, metre cube\n",
"r2 = 1.292e-10;  // Atomic radius in FCC, metre \n",
"a2 = 2*r2*sqrt(2);  // Lattice constant for F    CC, cube\n",
"V2 = a2^3;   // Volume of unit cell in FCC, meter cube\n",
"Vpa1 = V2/4; // Volume occupied by one atom in FCC, metre cube\n",
"dV = (Vpa-Vpa1)/Vpa*100;    // Change in volume, percentage\n",
"printf('\nChange in volume in percentage = %4.3f percentage', dV);\n",
"\n",
"// Result \n",
"// Change in volume in percentage = 0.493 percentage"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 3.4: Volume_and_density_of_unit_cell_in_HCP_Zn_structure.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// Scilab Code Ex3.4 : Page-3.7 (2010)\n",
"clc;clear;\n",
"a = 0.27e-9;  // Lattice constant for BCC,  metre \n",
"c = 0.494e-9;  //  Height of the unit cell, metre\n",
"M = 65.37;  // Atomic weight of zn, kg\n",
"N = 6.02e+26;   // Avogadro number per k mol\n",
"m = 6*M/N;  // Mass per unit cell in HCP structure, kg\n",
"V = 3*sqrt(3)*a^2*c/2;   // Volume of unit cell in HCP, metre cube\n",
"rho = m/V;    // Density of HCP Zn structure, kg per metrecube\n",
"\n",
"printf('\nVolume of HCP Zn structure = %4.3e metrecube', V);\n",
"printf('\nDensity of HCP Zn structure = %4.0f kg per metrecube', rho);\n",
"\n",
"// Result \n",
"//  Volume of HCP Zn structure = 9.356e-29 metrecube\n",
"//  Density of HCP Zn structure = 6963 kg per metrecube "
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 3.5: Interplanar_spacing_in_110_and_212_planes_in_FCC_lattice.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"/// Scilab Code Ex3.5 : Page-3.5 (2004)\n",
"clc;clear;\n",
"r = 0.1278; // Atomic radius, nm\n",
"a = 4*r/sqrt(2);  // Lattice constant, nm\n",
"h1 = 1, k1 = 1, l1 = 0;    // Miller Indices of (110) planes\n",
"d_110 = a/sqrt(h1^2 + k1^2 + l1^2);    // Interplanar spacing for (110) planes, nm\n",
"h2 = 2, k2 = 1, l2 = 2;    // Indices of third set of parallel planes\n",
"d_212 = a/sqrt(h2^2 + k2^2 + l2^2);    // Interplanar spacing for (111) planes, nm\n",
"printf('\nInterplanar spacing for (110) planes = %6.4f nm', d_110);\n",
"printf('\nInterplanar spacing for (212) planes = %6.4f nm', d_212);\n",
"\n",
"// Result \n",
"// Interplanar spacing for (110) planes = 0.2556 nm\n",
"//  Interplanar spacing for (212) planes = 0.1205 nm "
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 3.6: Ratio_of_interplanar_spacing_in_SC_lattice.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// Scilab Code Ex3.6 : Page-3.8 (2004)\n",
"clc;clear;\n",
"a = 1;  // For simplicity we assume a to be unity, unit\n",
"h1 = 1, k1 = 0, l1 = 0;    // Indices of first set of parallel planes\n",
"d_100 = a/sqrt(h1^2 + k1^2 + l1^2);    // Interplanar spacing for (100) planes, unit\n",
"h2 = 1, k2 = 1, l2 = 0;    // Indices of second set of parallel planes\n",
"d_110 = a/sqrt(h2^2 + k2^2 + l2^2);    // Interplanar spacing for (110) planes, unit\n",
"h3 = 1, k3 = 1, l3 = 1;    // Indices of third set of parallel planes\n",
"d_111 = a/sqrt(h3^2 + k3^2 + l3^2);    // Interplanar spacing for (111) planes, unit\n",
"printf('\nd_100 : d_110 : d_111 = %1d : %4.2f : %4.2f', d_100, d_110, d_111);\n",
"\n",
"// Result \n",
"// d_100 : d_110 : d_111 = 1 : 0.71 : 0.58 "
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 3.7: Miller_indices_of_a_plane_in_SC_lattice.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// Scilab Code Ex3.7 : Page-3.8 (2004)\n",
"clc;clear;\n",
"m = 1; n = 1/2; p = 3; // Coefficients of intercepts along three axes\n",
"m_inv = 1/m;        // Reciprocate the first coefficient\n",
"n_inv = 1/n;        // Reciprocate the second coefficient\n",
"p_inv = 1/p;        // Reciprocate the third coefficient\n",
"mul_fact = double(lcm(int32([1, 1, 3]))); // Find l.c.m. of 1, 1 and 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",
"printf('\nThe required miller indices are : (%d %d %d) ', m1,m2,m3);\n",
"\n",
"// Result \n",
"// The required miller indices are : (3 6 1)"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 3.8: Ratio_of_vacancies_in_metal.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// Scilab Code Ex3.8 : Page-3.13 (2004)\n",
"clc;clear;\n",
"N = 1;  // For simplicity assume total number of metal ions to be unity\n",
"e = 1.6e-019;   // Electronic charge, C\n",
"k = 1.38e-023/e;  // Boltzmann constant, eV/K\n",
"T1 = 500;   // First temperature for metal, K\n",
"T2 = 1000; // Second temperature for metal, K\n",
"E_v = 1;   // Average energy required to create a vacancy in metal, eV \n",
"n_500 = N*exp(-E_v/(k*T1));  // Number of vacancies at 500 K\n",
"n_1000 = N*exp(-E_v/(k*T2));  // Number of vacancies at 500 K\n",
"n_ratio = n_1000/n_500;     // Ratio of vacancies in metal \n",
"printf('\nThe ratio of vacancies in metal = %5.3e', n_ratio);\n",
"\n",
"// Result \n",
"// The ratio of vacancies in metal = 1.085e+05 "
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 3.9: Fraction_of_vacancy_sites_in_metal.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// Scilab Code Ex3.9 : Page-3.14 (2004)\n",
"clc;clear;\n",
"T1 = 500+273;   // First temperature for metal, K\n",
"T2 = 1000+273; // Second temperature for metal, K\n",
"frac_vac = 1e-010;  // n1/N, the fraction of vacancy sites at 500 degree celsius\n",
"e = 1.6e-019;   // Electronic charge, C\n",
"k = 1.38e-023/e;  // Boltzmann constant, eV/K\n",
"// n1 = N*exp(-E_f/(k*T1));  // Number of vacancies at 500 K\n",
"// n2 = N*exp(-E_f/(k*T2));  // Number of vacancies at 500 K, solving for n2/N = x\n",
"x = exp((T1/T2)*log(frac_vac));\n",
"printf('\nThe fraction of vacancy sites in metal = %6.4e', x);\n",
"\n",
"// Result \n",
"// The fraction of vacancy sites in metal = 8.4670e-07 "
   ]
   }
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