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| Keywords: Surface ionization of cesium.svg en Surface ionization effect in vaporized Cs atoms at 1500 kelvin calculated using a grand canonical ensemble <br/>Y-axis Average number of electrons on Cs atom X-axis negative of electron chemical potential plus elementary charge times electrostatic potential or in other words work function of surface <br/>See article http //dx doi org/10 1063/1 1139776 Characterization of a cesium surface ionization source with a porous tungsten ionizer for more information on the surface ionization effect 2014-01-20 12 21 24 own Nanite Source Created with Matplotlib Python source code Requires matplotlib Bonus features 1 Also makes entropy energy grand potential plots 2 Also makes plots for Cl and ionization of P B impurities in silicon <source lang python > Plot various quantities related to thermal ionization of an atom calculated from simple model using grand canonical ensemble from pylab import import matplotlib transforms as transforms plt rc 'text' usetex True plt rc 'font' family 'serif' plt rc 'font' serif 'Computer Modern' figtype ' svg' saveopts 'bbox_inches' 'tight' 'transparent' True 'frameon' True axsize 0 2 0 16 0 79 0 77 Thermodynamic functions def Omega x x_I x_A g_0 g_I g_A Grand potential in terms of dimensionless parameters x - \mu+e\phi /kT where \mu is chemical potential \phi is electrostatic potential of vacuum e is elementary charge x_I E_I/kT where E_I is ionization energy of atom x_A E_A/kT where E_A is electron affinity of atom g_0 g_I g_A degeneracies of neutral oxidized reduced states Returns grand potential with neutral offsets left off in units of kT Omega + mu N_0 - E_0 /kT return -log g_0 + g_I exp x-x_I + g_A exp x_A-x def navg x x_I x_A g_0 g_I g_A Average occupation number in terms of dimensionless parameters see Omega for parameters' meaning Returns <N> - N_0 The occupation number is given by <N> - d\Omega/d\mu return -g_I exp x-x_I + g_A exp x_A-x / g_0 + g_I exp x-x_I + g_A exp x_A-x def entropy x x_I x_A g_0 g_I g_A Entropy/k in terms of dimensionless parameters see Omega for parameters' meaning This function is calculated from the grand potential \Omega of this system and is given by S/k - d\Omega/d kT t1 log g_0 + g_I exp x-x_I + g_A exp x_A-x t2 - g_I x-x_I exp x-x_I + g_A x_A-x exp x_A-x / g_0 + g_I exp x-x_I + g_A exp x_A-x return t1+t2 Figure maker template def makefigs name W kT 1 N_0 1 E_I 1 E_A 1 g_0 1 g_I 1 g_A 1 bandgap None def makefig fig figure fig set_size_inches 3 3 fig patch set_alpha 0 ax axes axsize xlim amin W amax W trans ax get_xaxis_transform if amin W < E_A < amax W axvline E_A color 'gray' linestyle 'dotted' text E_A 1 01 r' \Delta E_ \rm A ' ha 'center' va 'bottom' transform trans if amin W < E_I < amax W axvline E_I color 'gray' linestyle 'dotted' text E_I 1 01 r' \Delta E_ \rm I ' ha 'center' va 'bottom' transform trans if bandgap is None free atom terminology xlabel r' W -\mu-e\phi ~ eV ' else semiconductor terminology xlabel r' \epsilon_ \rm C -\mu ~ eV ' axvline bandgap color 'gray' linestyle 'dashed' text bandgap 1 01 r' \Delta E_ \rm gap ' ha 'center' va 'bottom' transform trans return fig ax N navg W/kT E_I/kT E_A/kT g_0 g_I g_A S entropy W/kT E_I/kT E_A/kT g_0 g_I g_A Om Omega W/kT E_I/kT E_A/kT g_0 g_I g_A aveE Om + S kT This is <E>-E_0 for the case when \mu 0 fig ax makefig plot W N_0 + N linewidth 1 5 Nmin Nmax N_0 + amin N - 0 2 N_0 + amax N + 0 2 ylim Nmin Nmax axhline N_0 color 'green' linestyle 'solid' linewidth 0 5 ax yaxis set_ticks t for t in range N_0+10 if Nmin < t < Nmax ylabel r' \langle N \rangle -\frac d\Omega d\mu ' savefig 'ionize_'+name+'_navg'+figtype saveopts fig ax makefig ylabel r' S/k -\frac d\Omega d kT ' plot W S linewidth 1 5 ylim -0 1 1 7 savefig 'ionize_'+name+'_entropy'+figtype saveopts fig ax makefig plot W Om kT linewidth 1 5 ax autoscale False plot W -E_A + W - kT log g_A color 'gray' linewidth 0 5 plot W W 0 - kT log g_0 color 'green' linewidth 0 5 plot W E_I - W - kT log g_I color 'gray' linewidth 0 5 text 0 5 0 95 r' for \mu 0 ' ha 'center' va 'top' transform ax transAxes ax yaxis set_label_coords -0 17 0 5 ylabel r' \Omega - E_0 eV ' savefig 'ionize_'+name+'_grand'+figtype saveopts fig ax makefig plot W aveE color 'b' linewidth 1 5 ax autoscale False plot W -E_A + W color 'gray' linewidth 0 5 plot W W 0 color 'green' linewidth 0 5 plot W E_I - W color 'gray' linewidth 0 5 text 0 5 0 95 r' for \mu 0 ' ha 'center' va 'top' transform ax transAxes ylabel r' \langle E \rangle - E_0 ~ eV ' ax yaxis set_label_coords -0 17 0 5 savefig 'ionize_'+name+'_energy'+figtype saveopts Specific data makefigs 'Cs' free Cesium linspace -0 4 5 2 541 kT 8 61733238e-5 1500 eV 1500 K N_0 55 E_I 3 89390 eV from WP Ionization_energies_of_the_elements_ data_page E_A 0 47164 eV from WP Electron_affinity_ data_page g_0 2 unpaired 6s electron spin degeneracy g_I 1 filled shells g_A 1 filled shells makefigs 'Cl' free Chlorine linspace -0 4 5 2 541 kT 8 61733238e-5 1500 eV 1500 K N_0 17 E_I 12 96764 eV from WP Ionization_energies_of_the_elements_ data_page E_A 3 612724 eV from WP Electron_affinity_ data_page g_0 2 unpaired 3p hole spin degeneracy g_I 1 irrelevant placeholder value no visible effect g_A 1 filled shells Below we try some ionization of dopants in silicon The real behaviour is a bit more complicated than indicated here but this gives the conventional textbook model of dopant ionization See Theory of shallow acceptor states in Si and Ge by Schechter 1962 also The electronic structure of impurities and other point defects in semiconductors by Pantelides 1978 makefigs 'Si-P' Phosphorus in silicon dopant linspace -0 1 1 25 261 kT 8 61733238e-5 295 eV 295 K N_0 15 E_I 0 045 eV from web E_A -10 eV random large value to prevent ionization g_0 2 3sp 5 electron spin degeneracy S 1/2 in this case g_I 1 half-filled shell of 3sp electrons apparently nonmagnetic g_A 1 irrelevant placeholder value no visible effect bandgap 1 1 makefigs 'Si-B' Boron in silicon dopant linspace -0 1 1 25 261 kT 8 61733238e-5 295 eV 295 K N_0 5 E_I 10 eV random large value to prevent ionization E_A 1 1-0 045 eV from web g_0 4 3sp 3 hole spin degeneracy two possible orbital states from two valence bands each with S 1/2 g_I 1 irrelevant placeholder value no visible effect g_A 1 half-filled shell of 3sp electrons apparently nonmagnetic bandgap 1 1 </source> cc-zero Uploaded with UploadWizard Caesium Ion source Statistical mechanics Images with Python source code | ||||
