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+{
+"cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Chapter 6: Beta Decay"
+ ]
+ },
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 6.10: Half_life_of_tritium.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// Scilab code Exa6.10: : Page-244 (2011)\n",
+"clc; clear;\n",
+"tau_0 = 7000; // Time constant, sec\n",
+"M_mod_sqr = 3; // Nuclear matrix\n",
+"E_0 = 0.018; // Energy of beta spectrum, MeV \n",
+"ft = 0.693*tau_0/M_mod_sqr; // Comparative half life\n",
+"fb = 10^(4.0*log10(E_0)+0.78+0.02); //\n",
+"t = 10^(log10(ft)-log10(fb)); // Half life of H3, sec\n",
+"printf('\nThe half life of H3 = %4.2e sec', t);\n",
+"\n",
+"// Result\n",
+"// The half life of H3 = 2.44e+009 sec "
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 6.11: Degree_of_forbiddenness_of_transition.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// Scilab code Exa6.11: : Page-244 (2011)\n",
+"clc; clear;\n",
+"t_p = 33/0.92*365*84800; // Partial half life for beta emission, sec\n",
+"E_0 = 0.51; // Kinetic energy\n",
+"Z = 55; // Atomic number of cesium\n",
+"log_fb = 4.0*log10(E_0)+0.78+0.02*Z-0.005*(Z-1)*log10(E_0); // Comparitive half life\n",
+"log_ft1 = log_fb+log10(t_p); // Forbidden tansition\n",
+"// For 8 percent beta minus emission\n",
+"t_p = 33/0.08*365*84800; // Partial half life, sec\n",
+"E_0 = 1.17; // Kinetic energy\n",
+"Z = 55; // Atomic energy\n",
+"log_fb = 4.0*log10(E_0)+0.78+0.02*Z-0.005*(Z-1)*log10(E_0); // Comparitive half life\n",
+"log_ft2 = log_fb+log10(t_p); // Forbidden transition\n",
+"// Check the degree of forbiddenness !!!!!\n",
+"if log_ft1 <= 10 then\n",
+" printf('\nFor 92 percent beta emission :')\n",
+" printf('\n\tTransition is once forbidden and parity change');\n",
+"end\n",
+"if log_ft2 >= 10 then\n",
+" printf('\nFor 8 percent beta emission :')\n",
+" printf('\n\t ransition is twice forbidden and no parity change');\n",
+"end\n",
+"\n",
+"// Result\n",
+"// For 92 percent beta emission :\n",
+"// Transition is once forbidden and parity change\n",
+"// For 8 percent beta emission :\n",
+"// Transition is twice forbidden and no parity change\n",
+" "
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 6.12: Coupling_constant_and_ratio_of_coupling_strengths_for_beta_transitons.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// Scilab code Exa6.12: : Page-244(2011)\n",
+"clc; clear;\n",
+"h_kt = 1.05457e-34; // Reduced planck's constant, joule sec\n",
+"c = 3e+08; // velocity of light, metre per sec\n",
+"m_e = 9.1e-31; // Mass of the electron, Kg\n",
+"ft_O = 3162.28; // Comparative half life for oxygen\n",
+"ft_n = 1174.90; // Comparative half life for neutron\n",
+"M_f_sqr = 2 // Matrix element\n",
+"g_f = sqrt(2*%pi^3*h_kt^7*log(2)/(m_e^5*c^4*ft_O*M_f_sqr)); // Coupling constant, joule cubic metre\n",
+"C_ratio = (2*ft_O/(ft_n)-1)/3; // Ratio of coupling strength\n",
+"printf('\nThe value of coupling constant = %6.4e joule cubic metre\nThe ratio of coupling constant = %5.3f', g_f, C_ratio);\n",
+"\n",
+"// Result\n",
+"// The value of coupling constant = 1.3965e-062 joule cubic metre\n",
+"// The ratio of coupling constant = 1.461 "
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 6.13: Relative_capture_rate_in_holmium_for_3p_to_3s_sublevels.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// Scilab code Exa6.13: : Page-245 (2011)\n",
+"clc; clear;\n",
+"Q_EC = 850; // Q value for holmium 161, keV\n",
+"B_p = 2.0; // Binding energy for p-orbital electron, keV\n",
+"B_s = 1.8; // Binding energy for s-orbital electron, keV\n",
+"M_ratio = 0.05*(Q_EC-B_p)^2/(Q_EC-B_s)^2; // Matrix ratio\n",
+"Q_EC = 2.5; // Q value for holmium 163, keV\n",
+"C_rate = M_ratio*(Q_EC-B_s)^2/(Q_EC-B_p)^2*100; // The relative capture rate in holmium, percent\n",
+"printf('\nThe relative capture rate in holmium 161 = %3.1f percent', C_rate);\n",
+"\n",
+"// Result\n",
+"// The relative capture rate in holmium 161 = 9.8 percent "
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 6.14: Tritium_isotope_undergoing_beta_decay.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// Scilab code Exa6.14: : Page-246 (2011)\n",
+"clc; clear;\n",
+"t_half = 12.5*365*24; // Half life of hydrogen 3, hour\n",
+"lambda = log(2)/t_half; // Decay constant, per hour\n",
+"N_0 = 6.023e+26; // Avogadro's number, per mole\n",
+"m = 0.1e-03; // Mass of tritium, Kg\n",
+"dN_by_dt = lambda*m*N_0/3; // Decay rate, per hour\n",
+"H = 21*4.18; // Heat produed, joule \n",
+"E = H/dN_by_dt; // The average energy of the beta particle, joule\n",
+"printf('\nThe average energy of beta particles = %4.2e joule = %3.1f keV', E, E/1.6e-016);\n",
+"\n",
+"// Result\n",
+"// The average energy of beta particles = 6.91e-016 joule = 4.3 keV \n",
+" "
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 6.15: Fermi_and_Gamow_Teller_selection_rule_for_allowed_beta_transitions.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// Scilab code Exa6.15: : Page-246 (2011)\n",
+"clc; clear;\n",
+"S = string(rand(2,1))\n",
+"S(1,1) = 'antiparallel spin'\n",
+"S(2,1) = 'parallel spin'\n",
+"\n",
+"for i = 1:2\n",
+" if S(i,1) == 'antiparallel spin' then\n",
+" printf('\nFor Fermi types :')\n",
+" printf('\n\n The selection rules for allowed transitions are : \n\tdelta I is zero \n\tdelta pi is plus \nThe emited neutrino and electron have %s',S(i,1))\n",
+" elseif S(i,1) == 'parallel spin' then\n",
+" printf('\nFor Gamow-Teller types :')\n",
+" printf('\nThe selection rules for allowed transitions are : \n\tdelta I is zero,plus one and minus one\n\tdelta pi is plus\nThe emited neutrino and electron have %s',S(i,1)) \n",
+" end\n",
+" end\n",
+"// Calculation of ratio of transition probability\n",
+"M_F = 1; // Matrix for Fermi particles\n",
+"g_F = 1; // Coupling constant of fermi particles\n",
+"M_GT = 5/3; // Matrix for Gamow Teller\n",
+"g_GT = 1.24; // Coupling constant of Gamow Teller\n",
+"T_prob = g_F^2*M_F/(g_GT^2*M_GT); // Ratio of transition probability\n",
+"// Calculation of Space phase factor\n",
+"e = 1.6e-19; // Charge of an electron, coulomb\n",
+"c = 3e+08; // Velocity of light, metre per sec\n",
+"K = 8.99e+9; // Coulomb constant\n",
+"R_0 = 1.2e-15; // Distance of closest approach, metre\n",
+"A = 57; // Mass number\n",
+"Z = 28; // Atomic number \n",
+"m_n = 1.6749e-27; // Mass of neutron, Kg\n",
+"m_p = 1.6726e-27; // Mass of proton, Kg\n",
+"m_e = 9.1e-31; // Mass of electron. Kg\n",
+"E_1 = 0.76; // First excited state of nickel\n",
+"delta_E = ((3*e^2*K/(5*R_0*A^(1/3))*((Z+1)^2-Z^2))-(m_n-m_p)*c^2)/1.6e-13; // Mass difference, mega electron volts\n",
+"E_0 = delta_E-(2*m_e*c^2)/1.6e-13; // End point energy, mega electron volts\n",
+"P_factor = (E_0-E_1)^5/E_0^5; // Space phase factor \n",
+" printf('\nThe ratio of transition probability = %4.2f\nThe space phase factor = %4.2f', T_prob, P_factor);\n",
+" \n",
+"// Result\n",
+"// The emited neutrino and electron have antiparallel spin\n",
+"// For Gamow-Teller types :\n",
+"// The selection rules for allowed transitions are : \n",
+"// delta I is zero,plus one and minus one\n",
+"// delta pi is plus\n",
+"// The emited neutrino and electron have parallel spin\n",
+"// The ratio of transition probability = 0.39\n",
+"// The space phase factor = 0.62 "
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 6.1: Disintegration_of_the_beta_particles_by_Bi210.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// Scilab code Exa6.1: : Page- 240 (2011)\n",
+"clc; clear;\n",
+"T = 5*24*60*60; // Half life of the substance, sec\n",
+"N = 6.023e+026*4e-06/210; // Number of atoms\n",
+"lambda = 0.693/T; // Disintegration constant, per sec\n",
+"K = lambda*N; // Rate of disintegration, \n",
+"E = 0.34*1.60218e-013; // Energy of the beta particle, joule\n",
+"P = E*K; // Rate at which energy is emitted, watt\n",
+"printf('\nThe rate at which energy is emitted = %d watt', P);\n",
+"\n",
+"// Result\n",
+"// The rate at which energy is emitted = 1 watt "
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 6.2: Beta_particle_placed_in_the_magnetic_field.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// Scilab code Exa6.2 : : Page-241 (2011)\n",
+"clc; clear;\n",
+"M_0 = 9.10939e-031; // Rest mass of the electron, Kg\n",
+"C = 2.92e+08; // Velocity of the light, metre per sec\n",
+"E = 1.71*1.60218e-013; // Energy of the beta particle, joule\n",
+"e = 1.60218e-019; // Charge of the electron, C \n",
+"R = 0.1; // Radius of the orbit, metre\n",
+"B = M_0*C*(E/(M_0*C^2)+1)*1/(R*e); // Magnetic field perpendicular to the beam of the particle, weber per square metre\n",
+"\n",
+"printf('\nThe magnetic field perpendicular to the beam of the particle = %5.3f Wb/square-metre', B);\n",
+"\n",
+"// Result\n",
+"// The magnetic field perpendicular to the beam of the particle = 0.075 Wb/square-metre "
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 6.3: K_conversion.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// Scilab code Exa6.3 : : Page-241 (2011)\n",
+"clc; clear;\n",
+"m_0 = 9.10963e-031; // Rest mass of the electron, Kg\n",
+"e = 1.60218e-019; // Charge of the electron, C\n",
+"c = 2.9979e+08; // Velocity of the light, metre per sec\n",
+"BR = 3381e-006; // Field-radius product, tesla-m\n",
+"E_k = 37.44; // Binding energy of k-electron\n",
+"v = 1/sqrt((m_0/(BR*e))^2+1/c^2); // Velocity of the converson electron, m/s\n",
+"E = m_0*c^2*(1/sqrt(1-v^2/c^2)-1)/(e*1e+003); // Energy of the electron, keV \n",
+"E_C = E+E_k; // Energy of the converted gamma ray photon, KeV\n",
+"printf('\nThe energy of the electron = %6.2f keV \nThe energy of the converted gamma ray photon = %6.2f keV', E, E_C);\n",
+"\n",
+"// Result\n",
+"// The energy of the electron = 624.11 keV \n",
+"// The energy of the converted gamma ray photon = 661.55 keV "
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 6.4: Average_energy_carried_away_by_neutrino_during_beta_decay_process.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// Scilab code Exa6.4 : : Page-241 (2011)\n",
+"clc; clear;\n",
+"E = 18.1; // Energy carried by beta particle, keV \n",
+"E_av = E/3; // Average energy carried away by beta particle, keV\n",
+"E_r = E-E_av; // The rest energy carried out by the neutrino, keV\n",
+"\n",
+"printf('\nThe rest energy carried out by the neutrino : %5.3f KeV', E_r);\n",
+"\n",
+"// Result\n",
+"// The rest energy carried out by the neutrino : 12.067 KeV "
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 6.5: Maximum_energy_available_to_the_electrons_in_the_beta_decay_of_Na24.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// Scilab code Exa6.5: : Page-242(2011)\n",
+"clc; clear;\n",
+"M_Na = -8420.40; // Mass of sodium 24, keV\n",
+"M_Mg = -13933.567; // Mass of magnesium 24, keV\n",
+"E = (M_Na-M_Mg)/1000; // Energy of the electron, MeV\n",
+"printf('\nThe maximum energy available to the electrons in the beta decay = %5.3f MeV', E);\n",
+"\n",
+"// Result\n",
+"// The maximum energy available to the electrons in the beta decay = 5.513 MeV "
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 6.6: Linear_momenta_of_particles_during_beta_decay_process.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// Scilab code Exa6.6: : Page-242 (2011)\n",
+"clc; clear;\n",
+"c = 1; // For simplicity assume speed of light to be unity, m/s\n",
+"E_0 = 0.155; // End point energy, mega electron volts\n",
+"E_beta = 0.025; // Energy of beta particle, mega electron volts\n",
+"E_v = E_0-E_beta; // Energy of the neutrino, mega electron volts\n",
+"p_v = E_v/c; // Linear momentum of neutrino, mega electron volts per c\n",
+"m = 0.511; // Mass of an electron, Kg\n",
+"M = 14*1.66e-27; // Mass of carbon 14,Kg\n",
+"c = 3e+8; // Velocity of light, metre per sec\n",
+"e = 1.60218e-19; // Charge of an electron, coulomb\n",
+"p_beta = sqrt(2*m*E_beta); // Linear momentum of beta particle, MeV/c\n",
+"sin_theta = p_beta/p_v*sind(45); // Sine of angle theta\n",
+"p_R = p_beta*cosd(45)+p_v*sqrt(1-sin_theta^2); // Linear momemtum of recoil nucleus, MeV/c\n",
+"E_R = (p_R*1.6e-13/2.9979e+08)^2/(2*M*e); // Recoil energy of product nucleus, MeV\n",
+"printf('\nThe linear momentum of neutrino = %4.2f MeV/c \nThe linear momentum of beta particle = %6.4f MeV/c \nThe energy of the recoil nucleus = %4.2f eV', p_v, p_beta, E_R);\n",
+"\n",
+"// Result\n",
+"// The linear momentum of neutrino = 0.13 MeV/c \n",
+"// The linear momentum of beta particle = 0.1598 MeV/c \n",
+"// The energy of the recoil nucleus = 1.20 eV "
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 6.7: Energies_during_disintergation_of_Bi210.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// Scilab code Exa6.7: : Page-242 (2011)\n",
+"clc; clear;\n",
+"N = 3.7e+10*60; // Number of disintegration, per sec\n",
+"H = 0.0268*4.182; // Heat produced at the output, joule\n",
+"E = H/(N*1.6e-013); // Energy of the beta particle, joule\n",
+"M_Bi = -14.815; // Mass of Bismuth, MeV\n",
+"M_Po = -15.977; // Mass of polonium, MeV\n",
+"E_0 = M_Bi-M_Po; // End point energy, MeV\n",
+"E_ratio = E/E_0; // Ratio of beta particle energy with end point energy\n",
+"printf('\nThe energy of the beta particle = %5.3f MeV \nThe ratio of beta particle energy with end point energy = %5.3f ', E, E_ratio);\n",
+"\n",
+"// Result\n",
+"// The energy of the beta particle = 0.316 MeV \n",
+"// The ratio of beta particle energy with end point energy = 0.272 "
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 6.9: The_unstable_nucleus_in_the_nuclide_pair.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// Scilab code Exa6.9: : Page-243(2011)\n",
+"clc; clear;\n",
+"M = rand(4,2);\n",
+"M(1,1) = 7.0182*931.5; // Mass of lithium, MeV\n",
+"M(1,2) = 7.0192*931.5; // Mass of beryllium, MeV\n",
+"M(2,1) = 13.0076*931.5; // Mass of carbon, MeV\n",
+"M(2,2) = 13.0100*931.5; // Mass of nitrogen, MeV\n",
+"M(3,1) = 19.0045*931.5; // Mass of fluorine, MeV\n",
+"M(3,2) = 19.0080*931.5; // Mass of neon, MeV\n",
+"M(4,1) = 33.9983*931.5; // Mass of phosphorous, MeV\n",
+"M(4,2) = 33.9987*931.5; // Mass of sulphur, MeV\n",
+"j = 1; \n",
+"// Check the stability !!!!\n",
+"for i = 1:4\n",
+" if round (M(i,j+1)-M(i,j)) == 1 then\n",
+" printf('\n From pair a :')\n",
+" printf('\n Be(4,7) is unstable');\n",
+" elseif round (M(i,j+1)-M(i,j)) == 2 then\n",
+" printf('\n From pair b :')\n",
+" printf('\n N(7,13) is unstable');\n",
+" elseif round (M(i,j+1)-M(i,j)) == 3 then\n",
+" printf('\n From pair c :')\n",
+" printf('\n Ne(10,19) is unstable');\n",
+" elseif round (M(i,j+1)-M(i,j)) == 0 then\n",
+" printf('\n From pair d :')\n",
+" printf('\n P(15,34) is unstable');\n",
+" end \n",
+"end\n",
+"\n",
+"// Result\n",
+"// \n",
+"// From pair a :\n",
+"// Be(4,7) is unstable\n",
+"// From pair b :\n",
+"// N(7,13) is unstable\n",
+"// From pair c :\n",
+"// Ne(10,19) is unstable\n",
+"// From pair d :\n",
+"// P(15,34) is unstable "
+ ]
+ }
+],
+"metadata": {
+ "kernelspec": {
+ "display_name": "Scilab",
+ "language": "scilab",
+ "name": "scilab"
+ },
+ "language_info": {
+ "file_extension": ".sce",
+ "help_links": [
+ {
+ "text": "MetaKernel Magics",
+ "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
+ }
+ ],
+ "mimetype": "text/x-octave",
+ "name": "scilab",
+ "version": "0.7.1"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 0
+}
generated by cgit (git 2.34.1) at 2024-12-22 20:15:25 +0000