Atom-atom excitation and ionization in shock waves of the noble gases

26 March 1973

PHYSICS LETTERS

Volume 43A, number 4

ATOM-ATOM EXCITATION AND IONIZATION IN SHOCK WAVES OF THE NOBLE GASES H.W. DRAWIN and F. EMARD Dkpartement de la Physique du Plasma et de la Fusion ContrGlke, Centre d’Etudes Nuclkaires, 92 Fontenay-aux-Roses, France

Received 9 February 1973 Ratecoefficients for electron production in the relaxation zone behind the shock front in the noble gases have been calculated on the basis of a simplified model using generalized cross sections for atom-atom excitation and ionization of noble gas atoms.

The relaxation region behind an ionizing shock front in monatomic gases is usually divided into three regimes [ 1, 21: in regime I the ionization is built up from a negligible value. Since ionization starts with no electrons present, the electron production must begin with atom-atom collisions [3]. When a sufficient number of electrons, N,, has been produced ionization by electron-atom impact takes over as the dominant process. This is regime II. Regime III begins with the onset of radiation followed by the relaxation phase towards a quasi-steady (equilibrium) state. In the following we consider only regime I. It is generally assumed that the ionization process is a two-step process. First, excitation of the lowest resonance (or metastable) level occurs according to the reaction A+A+A*+A

(1)

followed by the ionization

process

A*+A+A’+e+A

A*+A-tAi+e.

or

For this phase the rate equation

tion takes the simple form aNelat = SA(TA)Ni

(3)

where SA is some effective rate coefficient which depends only on the atom temperature TA and the kind of chemical species considered and is independent of the numbeYdGGty frA of atom_. DiftZent authors [4-81 have determined SA from measured values of aN,/at and NA. When TA is known the mean reaction energy can additionally be determined from an Arrhenius plot. At present no theoretical predictions for the rate coefficient SA are available. In order to calculate S, we decompose the rate coefficient into two terms according to the relation SA&)

= Lo(TA) + 2Gor(TA)

(4)

where Lo is the rate coefficient for direct ionization from the ground level due to the reaction Ai-AdA’+e+A.

(2)

for electron produc-

(5)

The second term of (4) has been put equal to two times the excitation coefficient for excitation from

Table 1 Atomic constants to be used for the noble gases Element

Eo(eV)

50

Eor(eV)

f or

mAlmH

mA(10-23 gl

He Ne Ar Kr Xe

24.58 21.56 15.76 14.01 12.13

2 6 6 6 6

21.2 16.5 11.6 10.2 9.03

0.28 0.18 0.32 0.28 0.35

4 20.2 40 83.8 131.3

0.669 3.38 6.69 14.2 22.0

333

Volume 43A, number 4

PHYSICS LETTERS

26 March 1973

Table 2 Numerical values of the rate coefficient SA(cm3 see-‘) 7-A(103 OK)

He-He

Ne-Ne

Ar-Ar

Kr-Kr

5

_

_

_

_

I .2, -23

6 7 8 9 10

_ _ _ _ _

_ _ _ 4.6, --23

8.9, 1.4, 1.2, 7.2,

_ -23 -21 -20 -20

3.7, 9.3, 1.1, 7.7, 3.8,

-23 -22 -20 -20 -19

6.3, 1.1, 1.0, 6.1, 2.6,

-22 -20 -19 -19 -18

Xe-Xe

12 14 16 18 20

7.8, 1.0, 7.8, 4.1,

-23 -21 -21 -20

1.8, 2.6, 2.1, 1.1, 4.2,

--21 -20 -19 -18 -18

1.1, 8.3, 4.0, 1.4, 3.9,

-18 -18 -17 -16 -16

4.4, 2.7, 1.1, 3.6, 9.3,

-18 -17 -16 -16 -16

2.4, 1.3, 4.6, 1.3, 3.2,

-17 -16 -16 -15 -15

22 24 26 28 30

1.7, 5.4, 1.5, 3.7, 8.1,

-19 -19 -18 -18 -18

1.3, 3.5, 8.1, 1.7, 3.3,

-17 -17 -17 -16 -16

9.5, 2.0, 3.9, 6.9, 1.2,

-16 -15 -15 -15 -14

2.1, 4.2, 7.6, 1.3, 2.1,

-15 -15 -15 -14 -14

6.7, 1.3, 2.4, 3.7, 5.8,

-15 -14 -14 -14 -14

32 34 36 38 40

1.6, 3.1, 5.4, 9.1, 1.5,

-17 -17 --17 -17 -16

5.9, 9.9, 1.6, 2.5, 3.7,

-16 -16 -15 -15 -15

1.8, -14 2.8, -14 4.1, -14 5.8, -14 8.1, -14 _

3.2, 4.8, 6.8, 9.5, 1.3,

-14 -14 -14 -14 -13

8.6, 1.2, 1.7, 2.3, 3.1,

-14 -13 -13 -13 -13

1.8, -14 9.6, -13

3.0, -13 7.5, -12

4.4, -13 9.0, -12

50 100 Read

9.6, -16 8.4, -14

9.7, -13 1.6, -11

: 7.8, -23 = 7.8 X 10mz3, etc.

the ground into the first resonance level Yaccording to reaction (1). The factor two shall account for the fact that excitation is not only into the first excited state but into all excited levels. This increases the rate by approximately a factor two. In (4) we made the tacit assumption as in [ 1] that all particles which have been excited are ionized according to reaction (2) before excitation energy in form of photons escapes from the plasma. We use for process (5) the recently derived cross section formula [9]

--+-KELLY -a.--MC

(6)

9

9

(1666)

ELWAK

rt al. mocolculatiin

-Present

ti’?

334

-_

10

12x103 TA [~K]-D

Fig. 1. Comparison between measured and calculated rate coefficients for atom-atom ionization in argon gas.

Volume

43A, number

PHYSICS

4

LETTERS

where IV, is the free kinetic energy E available for an inelastic collision, in units of the ionization energy E,, i.e., W, = E/E,. mA is the mass of the colliding noble gas atoms, to the number of equivalent electrons in the outer (ground state) shell. Ey, mH and a, are the ionization energy (13.58 eV), the mass and the first Bohr radius of atomic hydrogen respectively. me is the electron mass. We note that compared to [9] to has been replaced by ti. This yields a much better agreement with measured cross sections. Cross sections for process (1) are not yet known, neither experimentally nor theoretically. Since in noble gas atoms the energy interval between ground and first resonance level is comparable with the ionization energy, one can expect that excitation and ionization cross sections will approximately show the same relative energy dependence. In analogy to eq. (6) we therefore write for the excitation cross section

(Ware

1 -2

2me + m A

(7)

I>

where for is the absorption oscillator strength for transition from ground to first resonance level, W,, is the free collision energy in units of the excitation energy Ear, i.e., W,, = E/E,, . Multiplication of the cross section formulas by the relative velocity and integration over a Maxwellian velocity distribution at temperature TA yields the rate coefficients

26 March

1973

With the atomic constants listed in table 1, eqs. (4, 8, 9) give rate coefficients SA as listed in table 2. Measured values are only available for relatively low temperatures. Fig. 1 shows a comparison between measured and calculated rate coefficients for argon. The calculated values deviate considerably from the older measurements of Kelly [S] but rather good agreement is obtained with recent values of McElwain et al. [7] especially when one goes to higher temperatures. A similar behavior as for argon is obtained for the other noble gases which confirms that the order of magnitude as given by the formulas (8, 9) for the rate coefficients and, thus, by the cross section formulas (6,7) is correct. In all cases, however., the discrepancy between measured and calculated SA-values increases when one goes to low temperatures where the calculated values are always smaller than the measured ones. Since at low temperatures Lo in (4) is negligible compared to 2G,,, SA is mainly determined by the slope of the excitation cross section with collision energy E near threshold energy E,,. When the cross section increases near threshold linearly with E as it is generally assumed, the rate coefficient should for a Maxwellian velocity distribution have the TA-dependence Ti12 exp (-Eo,/kTA) if kTA
‘kA(wo>

Lo =

(8)

References

[ll

G,r = 64na; (E”r(%J2 +OK

x

52f o or

memA

mH(m,+mA)

%(Wor)

with w. = Eo/kTA, war = Eo,/kTA _ The function @A(X) can be approximated by the relation *A(x) =

1 + 2/x 1 + [2me/(m,+mA)

00) xl 2

H. Wong and D. Bershader, J. Fluid Mech. 26 part 3 (1966) 459. 121 R.M. Hobson, Proc. Tenth Intern. Conf. on Phenomena in ionized gases, Oxford 1971. 131 K.E. Harwell and R.G. Jahn, Phys. Fluids 7 (1964) 214. 141 E.J. Morgan and R.D. Morrison, Phys. Fluids 7 (1965) 214. [51 A.J. Kelly, J. Chem. Phys. 45 (1966) 1723. [61 T.I. McLaren and R.M. Hobson, Phys. Fluids 11 (1968) 1262. Phys. 171 D.L.S. McElwain, L. Wagschal and H.O. Pritchard, Fluids 13 (1970) 2200. [81 S.P. Kalra and R.M. Measures, Phys. Fluids 14 (1971) 2544. [91 H.W. Drawin, Z. Physik 211 (1968) 404. 335