File:Surface ionization of cesium.svg
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Summary
DescriptionSurface ionization of cesium.svg |
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 |
SVG development InfoField | ![]() This plot was created with Matplotlib. |
Source code InfoField | 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,
)
|
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20 January 2014
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 15:50, 20 January 2014 | ![]() | 270 × 270 (32 KB) | Nanite | taking into account double degeneracy of neutral state |
11:34, 20 January 2014 | ![]() | 270 × 270 (26 KB) | Nanite | Changed to 1500 K (was 295 K) | |
11:27, 20 January 2014 | ![]() | 270 × 270 (25 KB) | Nanite | User created page with UploadWizard |
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