Population inversion in optically thick, recombining hydrogen plasmas

J. Quant. Spectrosc. Radiat. Transfer Vol. 29, No. I, pp. 75-79, 1983

0022--4073/831010075--05503.0010 © 1983 Pergamon Press Ltd.

Printed in Great Britain.

P O P U L A T I O N I N V E R S I O N IN O P T I C A L L Y THICK, RECOMBINING HYDROGEN PLASMAS UTARO FURUKANEand TOSHIAKiYOKOTA Department of Physics, College of General Education, Ehime University, Matsuyama 790, Japan KEN KAWASAKI Faculty of Education, Okayama University, Okayama 700, Japan and TOSHIATSU ODA Faculty of Science, Hiroshima University, Hiroshima 730, Japan

(Received 29 December 1981) Abstract--The plasma condition is investigated theoretically for population inversion between the first two

excited states of hydrogen atoms in a recombiningplasma. The rate equation, includingatom-atom collision terms, is solved consistently with the optical escape factors. The upper bound of the ground level population density (n0ma~necessary for inversion in the optically thick plasma at specified electron density and temperature is nearly inversely proportional to the mean radius of the plasma rz. With a decrease in the atom temperature, the upper bounds increase in the optically thin plasma but decrease in the optically thick plasma. 1. INTRODUCTION

In a previous paper, ~we have discussed the regions of plasma parameters (electron temperature Te, electron density ne, and population density of the ground level nl) for population inversion between lower levels of the hydrogen atom in a recombining plasma. Recently, Sato et al. have observed population inversion between lower levels of singly ionized helium in a TPD-I plasma, which was cooled by interaction with neutral helium gas introduced into the plasma? In such a plasma containing a large number of neutral particles, atom-atom collisions play an important role. We have determined the conditions for population inversion in Ref. 3, where the atom-atom collision process is taken into account. In a plasma with high neutral density, absorption by the Lyman series prevents population inversion. Tallents investigated the effect of L~ self-absorption on population inversions.4 His result shows that the plasma parameters for population inversion between the first two excited levels are restricted by self-absorption within a considerably smaller region than corresponds to the optically thin case. In his calculations, fixed values less than unity for the escape factor At2 for L~ radiation were introduced parametrically into the rate equations, and all the other Ao were set equal to unity. In order to quantify the effect of self-absorption on population inversion, we solve simultaneously the rate equations containing the atom-atom collision terms and the equations determining the optical escape factors of the Lyman series and then investigate the conditions for population inversion between the first two excited levels of hydrogen atoms. The result of our calculations provide a useful guide for experiments on population inversions.

2. FORMULATION

The rate equation for the population density of level i, ni(i >-2), is a non-linear equation in n~ when atom-atom collisions are taken into account) The equation is 2O

E (aij + nla'ij)nj = -(ai! + nta[l)nl - (8i + 6]nl),

j=2

(2 -< i -< 20). 75

(1)

U, Ft ~tl,. xNI. el at.

76

The coefficients a~j and ,S~ are given by the following quantities: (i) the radiative transition probability (A~i, i < j), (ii) the optical escape factor (A~i, A~), (iii) the rate coefficients for electron-impact ionization (K..), (iv) the rate coefficients for electron-impact recombination (K,~t, iv) the rate coefficients for electron-impact excitation (C~j, i ~ j), (vi) the rate coefficients for" electron-impact deexcitation (F~, i < D, (vii) the rate coefficients for radiative recombination (/3,). ' The coefficients a',j and a',. are given by the rate coefficients for atomic collisional ionization (KC~,). recombination (K oJ , excitation (C~, i < j) and deexcitation (FTi, i < j)2 We confine the discussion to population inversion between the first two excited levels on the axis of a cylindrically symmetric plasma. The optical escape factors for lines with Doppler profile for the Lyman series A~i(j-> 2) is given by 7

% =~ f ~dx f'dt exp l-x~- - ~exp(-x-')].

(2~

Here. 7: is the optical depth at the line center expressed in cgs and K units and 7 :=1 . 1 6 1 1 x l 0 ( ' & z f ~ k ( 1

-

/3o21)~,/(,~_)

nlrl,,

(3)

where o2k is the statistical weight of level k, /3 = ndnt, f~k is the absorption oscillator strength (t--,k), &~ is the wavelength for the transition k-+ 1, T,, is the atomic temperature, A is the atomic weight, and ro =- (f~ n @ ) dp)/n~, where I is the plasma boundary radius. All of the other optical escape factors Aq(i -> 2, for all j) are set equal to unity, that is, the plasma is considered to be optically thin except for the Lyman series. The optical escape factors, A~i(j---2), on the axis of a cylindrically symmetric plasma with given values of r0 and T,, are determined by n~. Therefore, the population densities of excited levels on the axis of the plasma are quite generally given as functions of T,,, n,, and n~ by consistent calculations involving the rate equation and the equation for the optical escape factor. The upper bounds of the ground-level p o p u l a t i o n (nl)ma x necessary for population inversion are determined by the method discussed in Ref. 3 using Eqs. (1) and (2) simultaneously. 3. N U M E R I C A L

CALCULATIONS

AND DISCUSSION

The calculation has been carried out for a recombining plasma in the atom and electron temperature ranges 0.15 e V _< 7],, _<0.5eV and 0.5eV_< T,,-< 1.5 eV, respectively, and in the electron density interval 108 c m - 3 <- ne-< l016 c m 3. For the mean radius of the plasma ro, four cases are considered: ro = 0, 0.42, 0.84, and 1.68 cm. The case r0 = 0 corresponds to an optically thin plasma. When ro > 0 and n, is increased beyond a certain value, ? no longer becomes small compared to unity [see Eq. (3)], i.e., the plasma grows to be optically thick. For r0 > 0, we consider such a region of n,. If atom-atom collisions are negligible in comparison with electron-atom collisions, the population density ratio n3/n2 has the largest value for n, = 0 (purely recombining plasma). 89 An increase in n~ raises the rate of electron-impact excitation from the ground level to the first excited level. Therefore, population inversion is reduced. At large values of nl, the rate of photo-absorptions from the ground level also grows strongly and the contribution of radiative deexcitations to population inversion decreases. Thus, the upper bounds of electron temperatures necessary for population inversion between levels 3 and 2 are obtained by letting n, be 0. The results have been given as a function of ne in Fig. 1 of Ref. r. In Fig. 1, the solid curve shows (nO .... in an optically thin plasma (ro = 0 cm) as a function of n,, at T,, = T¢ = 0.5 eV. The dashed curve also shows (n0max for the same temperatures and r~ = 0.84 cm. The dot-dash curve is calculated in a manner similar to that used by Tallents, 4 i.e., (tz~)...... is obtained by using constant optical escape factors. We have assigned the following values: At2 = 0.43, Al~ = 0.85, AI4 = 0.94, AIs = 0.97, AI6 = 0.98, At7 = 0.99, A_I~= 0.99; all other X,, are unity. + +These escape factor Aii correspond to the optical depth on the axis of a cylindrically symmetric plasma, e.g., with 71, = (I.5 eV, r,~ = 0.84 cm, and n~ = 1.5 × I0 t~ cm ~.

Population inversion in opticflly thick, recombining hydrogen plasmas 1020

,

I

, ~

, )

, I

,

I

,

77

, i

i

1019 'E 1018 u

1017

r:0/

.5c~1016

i i I

1015 1014

........L... i..

1013 1012 1011 1010 107

/ 108

Z 109 10 i0

..... I ) '

)I )

1011 1012 1013

1014 1015

ne(Cm-3)

Fig. 1. Upper bounds of the ground level population density for optically thin and thick plasmas at Te = T~ = 0.5 eV. The dot-dash curve is a calculated result with fixed values of Aii (see the text for details).

In an optically thin plasma (r0 = 0), (nl)max takes a value of about 1011c m -3 at ne "~ 108 cm 3 and increases to about 1019cm -3 as ne increases (cf. the solid curve). Finally, inversion is destroyed at ne = 5 x 1014cm-3. We first consider the case of low electron densities (he-< 1012cm-3). At nl = 0, radiative processes, which play the most important role in both populating and depopulating levels 3 and 2, depopulate level 2 rather than level 3 and produce population inversion between levels 3 and 2. If nl is increased so that the rate of populating level 2 increases due to collisional excitations from the ground level, the inversion disappears; consequently, (nl)ma x remains small. Next, we consider the higher density, case with n,-> 1013cm -3. The rate of populating level 3 is raised by collisional deexcitation from the upper levels; population inversion becomes more intense than at the lower electron density. Therefore, even for large nl, population inversion is possible, i.e., (nl)max increases up to about 1019 cm -3 as Y/eis raised. In the higher nl region (nl = 1019 cm-3), atom-atom collisions depopulate level 3 significantly and (nl)max will not exceed a value of about 1019 c m -3. A further increase in n, causes collisional processes to be dominant in place of radiative processes. The rate of depopulating level 3 grows and exceeds that of level 2. Finally, the inversion disappears at ne = 5 x 10~4cm -3. In the optically thick situation, the rate of depopulating level 2 is greatly reduced by an increase of self-absorption caused by the rise in nj and (nt)ma x remains much smaller than for the optically thin case. We now consider the dot-dash curve in Fig. 1, where a rather small value of 0.43 is assigned to Al2. We consider the case of the purely-recombining plasma (nl = 0). This assignment for A12 leads to an overestimation of the effect of self-absorption of the Lyman a line. At low electron density where the radiative process is dominant, population inversion is not expected. In the higher density region (10~3cm -3 -< ne <- 1014cm-3), the rate of populating level 3 due to electronimpact deexcitation from the upper levels becomes comparable with that caused by the radiative process and population inversion is produced. With a further increase in ne, the rate of depopulating level 3 by electron impact becomes significant. When the rate of depopulating level 3 exceeds that from level 2, population inversion is destroyed. The upper bounds of n~ are lower than those obtained from our method because of overestimation of self-absorption. When nl is increased in the higher electron density region, (n,)max becomes comparable to the value for ro = 0 because the fixed values of Aij introduced into Eq. (1) result in underestimation of self-absorption in the higher n~ region. We conclude that Tallents' calculated results for (n0max do not apply to an actual plasma.

78

U. FURUKANEet al.

Figure 2 shows ( n l ) m a x VS nefor ro = 0.42, 0.84 and 1.68 cm at T,, - T< = 0.5 eV. The relation r0(n,)m~x "=-constant holds for a change in ro at each value of n<. This result and the relation .;- :c ron~ indicate that ( h i ) m a x is mainly affected by self-absorption of the Lyman a line in the optically thick plasma. Figure 3 shows ( n l ) m a x VS n e at the lower atom temperature T, = 0.15 eV at T,, = 0.5 eV for r0 = 0 and 0.84 cm. In the optically thin plasma, with a decrease in T,,(nOm,~ is increased mainly by depression of the rate coefficient for atomic impact excitation to level 2 from the ground level. In the optically thick plasma for ro = 0.84 cm, (m) .... is decreased by a rise in the optical depth with a decrease of T, (see Eq. (3)). In Fig. 4, the (nl)ma x for r0 = 0 and 0.84 cm are plotted as functions of n, at T,, = 1.5 eV and

1020

, I

, I

,I

i #

it'--

, I

7

, I

1019

I+

~q

1018

i

E v

10 i7 i016

1 I015 10 lq

r0=0,42 cm', X ro=O.84cm\ ~,

1013

.0 =, 68cm.\',,

........

............. ::::::::i):::::"<, ====================== .......

1012 ,l~ t''°

.."

i011

/

l0 I0 1(

i t

} !

""

108

i I

, I

,

i I

1019

1 0 1 0 101* 10 2

i

I t

1013 1014 IQI5

ne(Cm-3)

Fig. 2. Upper bounds of the ground level population vs n,> for r0 = 0.42, 0.84 and 1.68 cm at T,, = T, = 0.5 eV.

i020

, ,

, ,

1019

, i

,,

/

i41015 i01610

// //

:o13

// Z .

, i

/' //Ta=O,5

eV

rO=O Ta=0.5 eV %=0,84 cm _::..:.i;:..i. "~°""

f "

i010

, ,

]o=O, 15 ev

1017

loll

, I

,...•

%

r0=0,84 cm

/ , ,

108

' , 109

'

I , , , , i 1010 1011 ,1012 1013 ,10,14 1015 ne ( cm- 3 )

Fig. 3. Upper bounds of the ground level population vs tie at T, = 0.15 e'V and 1], = 0.5 eV for r. = 0 and 0.84 cm.

Population inversion in optically thick, recombining hydrogen plasmas 1014

,,

, i

1013

, i

, i

, i

, i

79

, *

r0-0

*E 1012 1011 1010 109 108 107 106 105 104107

"10:s

: 109

: : : : 1014 . 1010 1011 1012 1013 1015 ne(cm-3)

Fig. 4. Upper bounds of the ground level population vs ne for Te = 1.5 eV and Ta = 0.5 eV at ro = 0 and 0.84 cm.

Ta = 0.5 eV. The increase in Te reduces (nl)max considerably at ro = 0 because the rates of electron-impact excitations increase while the rate of three-body recombination (EKcjn2n+) is decreased. The apparent transition probability AoAij is not significantly influenced by a change in ro when nl is less than about 10~2cm -3. 4. C O N C L U S I O N

The plasma conditions for population inversion between the first two excited states on the axis of a cylindrically symmetric recombining plasma have been determined from the rate equation and the equation for the optical escape factor. There is a significant discrepancy between Tallents' results and ours. Since fixed value of A 0 are introduced in Tallents' calculations, changes in n~ in the plasma cannot influence the optical escape factor. Our numerical calculations show: (i) In the optically thick plasma, the product ro(nl)max remains nearly constant for a change in ro at given electron density and temperature. (ii) With a decrease in Ta, (nl)max is increased in an optically thin plasma but is decreased in an optically thick plasma. (iii) When T, is greater than 1.5 eV, (n0mx is reduced about 1012cm -3 and the plasma is optically thin. Acknowledgements--This work was carried out under the collaborating research program at the Institute of Plasma Physics, Nagoya University. The numerical calculations were performed at the Computer Center of the Institute of Plasma Physics, Nagoya University, and at the Computer Center of Kyushu University. REFERENCES 1. U. Furukane, T. Yokota, and T. Oda, JQSRT 22, 239 (1979). 2. K. Sato, M. Shiho, M. Hosokawa, H. Sugawara, T. Oda, and T. Sasaki, Phys. Rev. Lett. 39, 1074 (1977). 3. U. Furukawa and T. Yamamoto, Jpn. J. App. Phys. 19, L 285 (1980). 4. G. J. Tallents, J. Phys. Bll, L 157 (1978). 5. L. J. Johnson, Astrophys.J. 174, 227 (1972). 6. H. W. Drawin, Z. Phys. 225, 483 (1969). 7. M. Otsuka, R. Ikee, and K. Ishii, JQSRT 21, 41 (1979). 8. T. Fujimoto, J. Phys. Soc. Japan 49, 1569 (1980). 9. H. W. Drawin and F. Enrard, EUR-CEA-FC-534, Asociation Euratom-C.E.A. Fontenay-aux-Roses (France) (1970).