PHYSICAL SCIENCES
05
Lowering of the Ionization Potential in Dense
Aluminum Plasmas t
R. E. BRUCE and F. C. TODD
Physles Department, Oklahoma State University, Stillwater
From the hypervelocity impact of microparticles on a massive target,
the high concentration of energy in the material at the impact region
produces a dense plasma. As the energy content and the per cent of
ionization of a dense plasma increases, the interaction between the electric
microfields from the ions and electrons results in a decrease in the ionizatJon potential of the atoms and ions iIi the plasma. The effective ionization potential, I,·, for each ionic species, i, is given in Fig. 1 by equation 1,
where 61, is the reduction in the ionization potential for the i'~ species
of the atom and I, is the ionization potential of the isolated ionIc specIes.
The reduction in the Ionization potentlal was found by experiment and
was reported (Elenbaas, 1951) and it has been recognized in several theoretical studies (Ecker et al., 1956; Margenau et al., 1959). The mlcrofields result in the disappearance of the upper energy levels ot the atoma
and ions as they merge into the continuum.
The purpose of this study is to ascertain the most accurate value for
the density of ionIzation from an application of the corrected Saba equation, Equation 2 in Fig. 1, to dense plasmas. Solutions are obtained wIth
a computer by a cyclic calculation of the convergence on the electron
density. The results in this paper indicate that the amount of ionization
for dense plasmas at low temperatures Is substantially higher than preViously predicted. As a consequence, the effective temperature is much
lower than was expected from earUer studies.
Low
DENSITY PLASMAS
There are essentially two methods tor determining the reduction in
ionization potential, 6I,. One becomes a direct calculation of 61 u 8uch
as on the basis of thermodynamics (Ecker et al., 1956), or on the broadening of bound energy levels (Margenau et al., 1959). The second is a
calculation of the maximum quantum number, g ,", which is used to obtain
61 and is illustrated by the Rouse paper which is considered below. The
81mplest way to estimate 61, from the maximum quantum number is to
assume that no bound state may have a radius greater than the Debye
radius.
j
In one form or another, the preceding calculations usually employ the
Debye radius. This quantity is an effective radius for the action ot the
electric field from an ion. Beyond this radius, the electric field from an
ion is assumed to decrease abruptly from the Coulomb value to zero. Th1a
assumption was postulated during early work on electrolytes (Debye et a.l.,
1923; Fowler et al., 1939). The density range tor an accurate application
of the Debye equation was established by Kirkwood and Poirier (Kirkwood et al., 19M) and will be compared with the results which wlll be
given later in this paper. According to Equation " Fig. 1, the numerical
value of the Debye radius, D, varies as the square root of the temperature,
or the square root of the effective kinetic energy ot the atoms and iou,
and inversely as the square root of the maximum Ionic chargM in the
pluma. The term, ZI1 in this equation is the dimensionle8s charge parameter expressed in number ot electron charges.
ISapported b~ Natlo_) Aeroaautie. and Spue AclJDtDbtraUoa Coatnlet No. NASr.7
IIdJaIDJaterecl throu..h a._reb Foundation. Oklahoma ShUe UnfY.r8ft~.
PROC. OJ' mE OKLA. ACAD. OJ' SCI. FOa 1963
• Reduction of loal&atlon Pot,ntl,l
2.
I.h. IquaUon:
~.
ft
l
n
R~b.r Den.lty of lth Sp.cl••
•
1
2(2na,kT)3/ 2Q1 + ,-11*/kT
l
h 3 Q1
n, • Huab.r D,n.lty of Electronl
Q. • El.ctronlc Partition Function.
).
El.ctronlc Partition 'unction (Q.):
QI.~ Wlge-Elg/kT
~
W
• Degeneracy of 8 th Level
K*
•
I1
Principal Qu.ntum Number
H.xl.~
4.
I. • Ionic Core Charg. P.ra.eter (Numb.r of Electron
Chug.. )
~
.
K.ll.y'. Equation:
11 *2.
~
•
o
1 1*
• Maxl.um Prlnclp.l Quantua "~btr for l th Species
'0
• ladlua of l·t aOhr Orbit
6.
11 • l.ol.ted
7.
L1ait of Deb,. t1leory:
IlU
At~'. (or
100'.) lonl&.t10n Pot.ntl.l
·(tn)[(.I~)ij3
Ilcr • tot.l Partlcl. DeIl.lt7
'1 ... • Kaal_ IODlc Chari' (ill •••• u.) 1n the Ph.
l'Ic.l. SquatloDa U8ed lD article
(Me
text).
PHYSICAL SCIENCES
91
Solutions of the 8aha equation on the basis of the g,. method were
reported in a series of papers for a plasma in ionization equtUbrium.
(·Rouse, 1961, 1962&, 1962b, 1962c). This series of papers is chosen for
special consideration because it is claimed that they are applicable to
liquid metals and constitute the only group which has been found in the
pUbllshed literature. In the first two papers, values of the ionization
were obtained by use of the uncorrected Baha equation, which is equivalent
to using the perfect gas law for the equation of state. In the third paper,
a correction for 61. was introduced but this method was modified for the
last paper. In the final paper, which was to apply to liquid metals, 61.
was obtained from computed values of the maximum quantum number,
g/, on the basis that no bound state may have a radius greater than the
Debye radius. Kelley's relation, Equation 5 in Fig. I, was used for g,.,
in which (I. is the radius of the first Bohr orbit, D is the Debye radius and
Z. is the dimensionless core-charge parameter. A Balmer type of equation was assumed for determining the maximum effective ionization potential. For this approximation, 61, is given in Fig. 1 by Equation 6. This
is essentially a Debye correction of the ionization potential and it would
be valid only in regions of high temperature and low density for which the
Debye-HUckel theory is valid.
The preceding discussion on the Rouse papers Is concerned with lowering of the ionization potential on the assumption that the energy levels
of the bound states are not affected by the microflelds. In addition to the
reduction of the ionization potential, the microfields change the positions
of the bound levels. This may be introduced into the Saba equation by
means of the electronic partition function, Q" which is gIven in Fig. 1 by
Equation 3. The partition function is expressed as a summation over
bound states for which tv •• and E,. are the degeneracy and the energy.
respectively, of the g'. level. Corresponding to the lowering of the ionization potential, the atom has a highest bound excitation state with a principal quantum number 9 ,.. The summation, in Equation 3, is limited by
this maximum quantum number. Since the Saha equation is affected
much more by the ionization potential than by this secondary consideration, the effect of the modified partition functions wu neglected.
DENSE PLASMAS
The major difficulty in applying the Debye-HUckel theory is its limited
region of validity. In addition to this, Duclos and C&mbel (1962) have
shown that in the density region for which the theory ls valid, the resulting
corrections are very small and may, in many cases, be neglected. It may
be shown that the upper density limit for a valid application of the
Debye-HUckel theory is given in Fig. 1 by Equation 7 (Ecker and KrlSll,
1963). In this equation, the critical density, ncr, is the maximum total
particle density for which the Debye theory la valid. It la dependent upon
the cube of the absolute temperature and inversely upon the slxth power
of the maximum ionic charge in the plasma.
RecenUy, equations were reported that may be used for the dlrect
calculation of the reduction of the ionization potential for plasma deD8itlu
above and below the critical density; see Fig. 2 and Equation 7 (Ecker
and KriSll, 1963). The upper Umlt of validity of these equations fa reported
by the authors to be the seml-classlcal Umlt which la shown in Fig. 2 by
Equation 8. For thia limit, the electron density is the llmltlng factor. A
eompa.rison of the regions of vaUdlty, in a 8lDgly ionized pluma, for the
Debye theory and for the Ecker and KrlSll equations la shown in Fig. 3.
The lines are the upper llmit of vaHdity. The region of interat for deue
plaamas lies above the Debye ~mlt and probably below the Mml-e....lcal
Umtt.
98
PROC. OF THE OKLA. ACAD. OF SCI. FOR 1963
.vOl' deDM plallnU, the reduction of the ionlZation potentJal is found
with tbe Ecker and Kr6U equati0118. wblch are Equations 9 through 13 in
...... 2. In UIeIe equaUoJUJ. D .. the Debye radius. 6, is the core charge in
....u.. epeUoK' la the dielectric constant (taken .. unity to calculate the
CU1'\'e8) and T la the temperature in degrees Kelvin. The summation in
EquaUon 11 la over the dt8trlbuUon of lone and electroJUJ at the critical
deMity. The ave,...e dlatance between particl.. 18 r.. Tb.ia dependence
baa & marked effect on the calculated degree of ionization at blgh densities
and loW pluma temperatures. For particle deJUJities. ",", which are less
than tim the ionization potenUal la Jiven on Fig. 2 by Equation 9. When
..,. la ~ter than tI", Equation 10 must be used. The general characterlItlCl of the new .olution .. Illustrated by the Ecker and KliSll calculation
01 /iI, lor a hydrogen pluma at various temperatures and electron densiUu wblch .... IIhown In Fig.•.
REsULTS
Th. d.gree ot ionization at equUtbrium tor a dense aluminum plasma
wu calculated for .evera) temperatures and densities by use ot the uncor-
rected iontzation potentIal and by two other methods which determine a
value tOT 61,. The first calculation used the uncorrected Saba equation
and gave the lowe.t curve on both Figs. 5 and 6. A second computation
used RouH'. correction. F.quations 5 and 6. for the reduction ot the ioniza8.
S._lcla.. lcel Ll_1t:
('or Electronl)
_
~I
..L.
1
Coni tent
"c"
2
1 • 2eD '-1+1 • -1
U I
11.
(l!l!!a.!!)
),~
h2
<
n
• ~
2e r
r 2
2
'-1+1 ·.1
o
2.
+ e J
2
+ e
2,
J
1n 10:
t n 8 • Sum Ov.r Dlatrlbutlon at Critical Den.lty
9 cr
12.
Cona~.nt
r
"r " In 10;
o
..
o
13. Total Partlcl. Den.lty
•
(....L)l/J
4TTny
(n-)'
-[.
n r • n _ + r[ n
1
• Di.l.ctric Cooltaat
D • Debye Radiua (Equation 4)
' i • Cor. Chara. of tb. i
t
tb
Specie. (la •••• u.)
• t..,.rature (ia OK)
·.a • ~rae of
8 Speci•• (ia e ••• u.)
(.C~)a • lu.ber Deaait, of • Specie. at Critical D.nalty
J'lc. 2. BquaUcma
U8ed in article (Me
text).
PHYSICAL
SCIENCF~
Dun TH!OllV VALID
IN TMIS "felON
Fig. 3.
Regions in which Debye theory and Ecker and Krijll's equations
are valid in a singly ionized plasma.
tion potential and thia calculation gave the curve which 18 just above the
lowest In both Figs. 5 and 6. The third calculation used Equations 9
through 13, the Ecker and Kr6U method, to determine 61,. A much higher
degree of Ionization 18 found at high densities according to the curves In
Fig. 5 and at low temperatures according to the curves in Fig. 6. As the
energy per particle, or the effective temperature, is increased and as the
density of the plasma 18 reduced, the results from the three methods converge to an identical degree of ionization. These results are to be expected.
All corrections reduce to zero as the temperature increases, or the density
decreases. When the corrections become zero, the uncorrected perfect gu
law fa an accurate equation of state.
100
PROC. OF THE OKLA. ACAD. OF SCI. FOR 1963
1
IOM'-:-
to'
FI,.~.
--JI-
.L..
•
'04
----. T
~
(eK]
•
Lowertnc of the IoniaUOD potenUal (AI) as a fuDcUon ot the
carrier dena1ty fl and temperature T. tl cr Is the crlUcal
deDalty; fl. gtvea the llmlt for the claulcal descrlpUon of the
eleetroNJ (Ecker and KrGU. 1968).
ebarce
101
PHYSICAL SCIENCES
10-
1~I'
to
it
-.
IE
o
II:
i
IOIe
:::)
Z
Z
o
II:
E
... 10·'
-.
.....
101",--_~ _ _---~_---_.....I -----~~----"""'10-41
Fig. 5.
10
10
10
10"
~'.-
Comparison ot ionization electron density calculated for HVeral
densities ot an aluminum plasma. The three curves are solutions
ot the three methods tor determining ionization. Rho 18 the soUd
density of aluminum.
Iff'
10
t
TlMPIItATUIII
FIg. 6.
•
.
..---.
Comparlacm of tile calc:ulated electron deu1U. in an alumtnum
pJuma with a heavy partlcle deD8lty of 6.02 x lOU/ern'. The three
curves repreaent 8OluUona obtaJDed WIlDa the tndIcated metbod to
correct the fcm'ption poteDtIa1.
102
PROC. OF THE OKLA. ACAD. OF SCI. FOR 1963
LlTFJlATURt: CI".:v
Deb)'e, P., aDd E. HUckel. 1923. On the theory ot electrolytes, 1. Freez.
IftI' point deprea10n and related phenomena. Physik. 24 : 185-206.
DuclCMI, D. P., and A. B. cambel. 1962. Equation ot state of an ionized
.... Progr. In Intem. Research on Thermodynamic and Transport
Propertt•• Academic Press, New York.
Ecker, G.• and W. Weizel. 19M. Partition function and effective ionization
potenUAI ot an atom in the interior of a plasma. Ann. Physik. 117 :
126-140.
Ecker, 0., and W. KriJIJ. 1963. Lowering ot the ionization energy for a
pluma In thermodynamic equillbrium. The Physics of Fluids 6: 62·
89.
Elenbaas, W. 1961. The High Pressure Mercury Vapor Discharge. NorthHolland Publishing Company, Amsterdam.
Fowler, R., and E. A. Guggenheim. 195ft
Cambridge Univ. Preu. London.
Statistical Thermodynamics.
Kirkwood, J. C., and J. C. Poirier. 1954. The statistical mechanical basis
ot the Debye-HUckel theory ot strong electrolytes. J. Phys. Chem.
68: 691-696.
Marcen&u, H., and M. Lewia. 1959. Structure ot spectral lines from
plaamu. Rev. Mod Phya. 31: 596-615.
1961.
184: 486.
RoWMl, C. A.
J.
Ioniution equUlbrium equation of state.
Astrophya.
Rou.., C. A. 1962&. Ionization equilibrium equation of state 11.
tures. Aatl'Ophya. J. 1M: 599-615.
Mix-
1962b. 10nlzation-equUibrium equation of state m. HelUlu with Debye-HUckel corrections and Planck's partition function.
A8trophyl. J. 188: 636-664.
RoUM, C. A.
Rouee, C. A. 1962c. Ion1zation-equtlibrium equation of state IV.
pluma IUld liqUid metala. Astrophys. J. 136: 665-670.
Dense