Continuum intensity for electron-ion recombination in nonequilibrium plasm

Chen, C. J.

English

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~ Technical Report No. 32-707 
Valid Conditions for the Krarners-Unsold Continuum 
Theory in a Non- Equilibrium Plasma 
Che Jen Chen 
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E T  P R O P U L S I O N  L A B O R A T O R Y  
C A L I F O R N I A  1 N S T l T U T E  OF T E C H N O L O G Y  i 
I PA s A DE N A, CALI FOR N I A 
December 30, 1964 
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Technical Report No. 32- 70 7 
Valid Conditions for the Kramers-Unsold Continuum 
Theory in a Non-Equilibrium Plasma 
Che Jen Chen 
D. R. Bartz, Chdf 
Propulsion Research and Advanced Concepts Section 
J E T  P R O P U L S I O N  L A B O R A T O R Y  
PA S A D  EN A, CALIFORNIA 
C A L I F O R N I A  I N S T I T U T E  O F  TECHNOLOGY 
December 30, 1 964 
Copyright @ 1965 
Jet Propulsion Laboratory 
California Institute of Technology 
Prepared Under Contract No. NAS 7-100 
National Aeronautics & Space Administration 
0 
J P L  TECHNICAL REPORT NO.  32-707 
ABSTRACT 30 
The Kramers-Unsold theory of continuum intensity for electron-ion 
recombination is formulated quantum mechanically. The valid condi- 
tions for the theory in a non-equilibrium plasma are deduced by using 
Griem’s criterion for partial local thermodynamic equilibrium. The 
results indicate that the conditions are much less restricted than those 
required for the Saha equation to be valid. 
Kramers (Ref. 1) and Unsijld (Ref. 2) have derived the 
expression for the intensity of electron-ion recombination 
continuum radiation from a plasma by calculating the 
coefficient of absorption of a photon interacting with 
atoms and using the radiational equilibrium argument to 
get the intensity of radiation. In such an approach the 
number density of the atoms Nu will appear in the expres- 
sion for the radiation and the electron density N e  is 
introduced by eliminating Nu through the use of the 
Saha equation. Because of the nature of the mathematical 
derivation one is apt to tie the physical restraints asso- 
ciated with the Saha equation to the theory for the 
continuum radiation (Ref. 3). The conditions required 
for the Saha equation to be valid, stated in Griem’s paper 
(Ref. 4), are that the plasma be in complete local thermo- 
dynamic equilibrium. These conditions usually cannot be 
expected in most laboratory plasmas. However, the 
electron-ion recombination continuum intensity theory 
can be formulated by using the free-bound transition 
probability of an electron, integrating over all possible 
electron velocities, and summing over only the energy 
states contributing to the continuum radiation at a given 
frequency. The assumption made in such a calculation is 
that the energy of the free electrons has a Maxwellian 
distribution, and there are no excited electrons in the ions. 
The problem is then reduced to that of a hydrogenlike 
plasma. 
The transition probability for the transition from states 
i to f per unit time for the recombination process, accord- 
ing to first-order perturbation theory, can be expressed 
as (Ref. 5, 6) 
where rif  is the position matrix element for dipole radia- 
tion and pf is the density of final states. After summing 
over the final-state orbital and azimuthal quantum num- 
bers and the two directions of polarization for the 
emitting photon, and averaging over the initial electron 
spins, the transition probability for a transition to a state 
of principal quantum number n with the emission of a 
photon of energy hv is given by (Ref. 7) 
where Z is the ground-state energy, w the velocity of free 
electrons, v the frequency of the photon emitted, n the 
. 
J P L  TECHNICAL REPORT NO. 32-707 
principal quantum number, V the volume of the system, 
and 6 the &function. The Gaunt factor g is approximately 
unity and is set equal to unity in the present discussion. 
The constant K in Eq. (1) stands for 24he2z2/3s'2m2c3, 
where h is Planck's constant, c the velocity of light in 
vacuum, e and me the electronic charge and electronic 
mass, respectively, and z the effective charge seen by the 
bound electron ( z  = 1 for a hydrogen atom). 
The intensity of radiation el.,, per unit volume per unit 
frequency at  frequency Y due to a transition of free elec- 
trons to a particular state of quantum number n is 
obtained by multiplying Eq. (1) by hv,  the number den- 
sity of ions n, ,  the number of electrons per unit volume 
with velocity w, and integrating over all possible values 
of w (0- w )  in a Maxwellian electron gas. This yields 
The intensity of the continuum radiation E y  at frequency v 
for all possible transitions is given by summing Eq. ( 2 )  
over all n such that h Z Z / n 2 :  
where C stands for 21K/(2nmek)%, and n,,, is equal to 
(Z/hv)w. In obtaining Eq. (3), the summation over n is 
replaced by integration, for at reasonably large n the 
energy level is close enough to justify such a replace- 
ment. The last factor in Eq. (3) gives the reduction of 
intensity by induced emission. 
The inequality 
I 
11" 2 - 
11 - (4) 
allows one to calculate the minimum value of quantum 
number n, above which the transition to a quantum state 
will have a contribution of continuum intensity at fre- 
quency v .  The energy levels with quantum number smaller 
than nnL will contribute no continuum intensity at the 
particular frequency. 
The valid conditions for Eq. (3) arc that the energy of 
the free electrons has a Maxwellian distribution and all the 
electrons in the ions are in their ground states. To verify 
Eq. (3), an experimental measurement in the decaying 
plasma is performed. A column of argon plasma is pro- 
duced by discharging a capacitor bank of 16 pf capaci- 
tance energized with a 7500-V dc source through a 
discharge tube. After a period of about 50 psec after 
the cessation of the discharge, the free-electron energy 
distribution is thought to relax from the Dniyvesteyn 
distribution (Ref. 8) to the Maxwellian one (in the order 
of 5 psec) and the excited electrons in the ions to deplete 
to the ground states (in the order of sec). The further 
excitation of the electrons in the ions, after cessation of 
the discharge, due to electron-ion collisions and photo- 
excitation is unlikely to happen because the electron tem- 
perature is of the order of 1.5 eV at the moment the 
measurement is initiated, which is small in comparison 
with the first excitation energy of argon (13.4 eV), and 
the plasma is optically thin (330 pHg initial pressure). 
The electron temperature of the decaying plasma as a 
function of time is measured by the spectral-line intensity 
ratio techniques, and the electron density as a function 
of time is measured by the ion current probe and micro- 
wave techniques. The relative intensity of continuum 
radiation at 5530A is obtained by using a spectroscope 
and photomultiplier. The results of the measurements are 
shown in Fig. 1. The electron density obtained from the 
10 
lo5 
5 
L? 
W 
Lz 
3 
n W
+a 
5 104 + 
z 
0 n + 
V W J 
W 
1 0 3  
102 
0 1 l 1  
CAPACITANCE 16 pf 
VOLTAGE 7500 V 
PRESSURE 330 p Hg ARGON 
0 55308 CONTINUUM 
0 ION PROBE 
A SAHA YALUES 
I - ,  
TIME, msec I 
TIME, msec 
Fig. 1.  Electron temperature and electron density 
a s  a function of time 
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J P L  TECHNICAL REPORT NO. 32-707 
I continuum intensity is calculated by using Eq. (3) (con- 
tinuum due to free-free transition is negligible in the I 
temperature range of the present experiment). The con- 
stant C in Eq. (3) is evaluated from the known electron 
density (microwave data), known electron temperature 
(spectral-line ratio data) and relative continuum inten- 
sity at one particular instant of the discharge. This point 
is at the microwave cutoff point. The percentage differ- 
ence of the electron density obtained from the continuum 
I 
intensity, and from the microwave and ion probe data, is 
less than 30%, covering the electron density range from 
10" to 10'j ~ m - ~ ;  while the values of electron density 
calculated from the Saha equation differ by two orders 
of magnitude from the experimental data, as also indi- 
cated in Fig. 1. This fact reveals that the plasma is not in 
a complete local thermodynamic equilibrium, but that 
the two valid conditions mentioned above do exist for the 
continuum radiation measured. 
1. Kramers, H. A. "On the Theory of  X-Ray Absorption and of  the Continuous X-Ray 
Spectrum," Philosophical Magazine, 6th Series, Vol. 44, p. 836, 1923. 
2. Unsold, Von A., "On the Continuing Spectrum of a Mercury High-pressure Lamp, 
Underwater Sparking, and Similar Discharges," Annalen der Physik, Vol. 33, p. 607, 
1938. 
3. Alpher, R. A., and White, D. R., "Visible Continuum Emission From Shocked Noble 
Gases," Physics of Fluids, Vol. 7, p. 1239, 1964. 
4. Griem, H. R., "Validity of Local Thermal Equilibrium in Plasma Spectroscopy," 
Physical Review, Vol. 131, p. 1170, 1963. 
5. Bethe, H. A., and Salpeter, E. E., Handbuch der Physik, Vol. XXXV, Springer Verlag, 
Vienna, Austria, 1957. 
6. Berman, S. M., Electromagnetic Radiation From an Ionized Hydrogen Plasma, Report 
No. PRL 9-27, Physical Research Lab., Space Technology Labs, Inc., Los Angeles, 
Calif., September 24, 1959. 
7. Spitzer, L., Jr., Physics of Fully Ionized Gases, lnterscience Tracts on Physics and 
Astronomy, lnterscience Publishers, Inc., New York, 1956. 
8. Druyvesteyn, M. J., "Calculation of Townsend's (Y for NE," Physica, Vol. 3, p. 65, 
1936. 
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J P L  TECHNICAL REPORT NO. 32-707 
ACKNOWLEDGEMENT 
The author wishes to thank Gary Russell for his suggestion of this 
problem and for helpful discussions. 


Valid conditions for the kramers-unsold continuum theory in a non-equilibrium plasma

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