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Summary

Description
English: Surface ionization effect in vaporized Cs atoms at 1500 kelvin, calculated using a grand canonical ensemble.
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.
See article "Characterization of a cesium surface ionization source with a porous tungsten ionizer." for more information on the surface ionization effect.
Date
Source Own work
Author Nanite
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Source code
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Python code

#Python source code. Requires [[:wikipedia:matplotlib|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.

<syntaxhighlight 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,
    )

Licensing

I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

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20 January 2014

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Date/TimeThumbnailDimensionsUserComment
current15:50, 20 January 2014Thumbnail for version as of 15:50, 20 January 2014270 × 270 (32 KB)Nanitetaking into account double degeneracy of neutral state
11:34, 20 January 2014Thumbnail for version as of 11:34, 20 January 2014270 × 270 (26 KB)NaniteChanged to 1500 K (was 295 K)
11:27, 20 January 2014Thumbnail for version as of 11:27, 20 January 2014270 × 270 (25 KB)NaniteUser created page with UploadWizard

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