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An analytical description of populations of highly excited levels
J. Quant. Spectrosc. Radiat. Transfer Vol. 28, No. 5, pp. 441-454, 1982
0022~.073182[110~1-14503.0010
Printed in Great Britain.
© 1982 Pergamon Press Ltd.
AN ANALYTICAL DESCRIPTION OF HIGHLY EXCITED
OF POPULATIONS LEVELS
I. L. BEIGMANand I. M. GAISINSKY LebedevPhysicalInstitute,USSRAcademyof Sciences,Moscow,U.S.S.R. (Received 15 May 1981)
Abstract--Analyticalmethodsare developedto describepopulationsof highlyexcitedlevels.The Kramers approximationwas used for radiative transitions and the Born approximationfor transitions induced by electroncollisions. An exact solution is derived for the limiting case of small densities. In the limiting case of large densities, an analyticalsolution is obtainedwith constants determinedfrom a comparisonwith computer solutionsof the balanceequations.A generalinterpolationformulais presented,whichincludesthe limiting cases and describesresultsfor intermediatedensitieswith 20-30%. 1. I N T R O D U C T I O N
Populations of atomic levels in a plasma are described by an infinite system of balance equations derived from particle conservation conditions. Usually, this system is solved numerically for each specific case, taking into account the restricted number of the equations. Since levels with large principal quantum number are always hydrogen-like, the problem for an arbitrary, highly excited atom can be solved using the hydrogen-atom approach. Systematic computer calculations for populations of highly excited hydrogen levels in low density plasma were made by Brocklehurst, ~-3 Brocklehurst and Seaton4 and Salem and Brocklehurst.5 Populations of highly excited levels (unlike lower levels where the structure depends strongly on the specific atom) are comparatively smooth functions of the principal quantum number n. Therefore, it is of interest to find unified analytical descriptions for the solution. When charged particle collisions are of greatest importance for the levels under study, transitions between nearby levels are most probable. This effect permits a diffusion approximation of the Focker-Planck type to be used for the populations. This approach was first suggested by Belyaev and Budker 6 and further developed for the recombination process by Abramov and Smirnovs and Gurevich and Pitaevsky7 (for detailed references, see also the review paper by Biberman et al.9). Radiative transitions can not be described by the Focker-Planck approximation. Beigman and Mikhalchi 1° showed that, for high levels, the system of balance equations for radiative transitions is equivalent to an integral equation. This integral equation admits an exact analytical solution if the Kramers approximation is used for radiative transition probabilities. An attempt to combine both methods was made by Beigman." The present paper is aimed at derivation of an analytical description of a general solution for the kinetics of highly excited levels in a low-density plasma. For this purpose, it is first necessary to derive a solution for the case without an externally pumped flux (hereafter referred to as a free solution) and, second, to obtain the Green function of the problem, i.e. a solution for the case of excitation to an arbitrary single level. We shall first write general kinetic equations for the limiting case of large quantum numbers. Two circumstances are important for collisional transitions within the Focker-Planck approximation: (1) the diffusion approximation is valid with logarithmic accuracy only; (2) an analytical expression for the flux depends on the specific situation. The 'free' solution is derived by approximating the computational solution for the specified kinetic equations. The resultant formula is compared with a direct solution of the discrete system of the balance equations. The Green function is derived as follows. First, the Green function is considered for the limiting case low electron density. The levels below the pumped level are described by the Green function derived from general equations (Beigman and Mikhalchil°), which allow for radiative transitions only. For the levels above the pumped level, the coronal approximation is 441 JQSRT Vol. 28, No. 5--F
442
I.L. BEIGMANand I. M. GAIS1NSKY
valid. Then the limiting case of high electron density is considered. From the general kinetic equations, the main functional dependences can be obtained. The constants in these dependences result from a comparison with a direct solution of the discrete system of the balance equations. The general interpolation formula is derived on the basis of the two limiting cases and the solutions of the discrete system. In conclusion, we consider applicability and possible errors of the derived formulae. 2. KINETIC EQUATIONS Let N(3`) be the population of atoms at a sublevel with a set of quantum numbers 3`, so that population at a level 3` is g(3`)N(3`), where g(y) is the statistical weight of the level 3`. We assume that charged particle collisions and radiative decays play a leading role in kinetics. The probabilities of transitions per unit time from the level 3` to the level 3`' due to charged particle collisions and spontaneous photon emission are denoted, respectively, by W ( y ' ~ y) and A(y' ~- y). The e.c. W and A are, to some extent, similar to the averaged transition strengths. The usual y-3`' transition probabilities are W(y'~-y)/g(3`) and A(y'*--y)/g(y), respectively. The functions W and A are not only highly symmetric, but they also allow the balance equations to be written without statistical weights. The descriptions of populations for highly excited and low atomic levels are different and thus we write an infinite system of the balance equations as N(3`){w(3`) + ~. [W(3`'~ 3,)+ A(3`'~ 3`)1} 7' ~ Y
-~
N(y')[ W(3`' ~ 3`')+ A(y ~ 3`')] -- F~(y); E~, e¢> El
(1)
N(y)w(y) + (£c + £ , ) N ( y ) = F(y); e,. < E,; £,,N(3`) -= y'#), ~ [W(3`' ~ 3`)N(3`)- W(3` ~ y')N(y')],
£~N(y) =- ~
y'~ y
[A(3`' *-- y)N(y) - A(3` , - 3`')N(y')].
(2)
e~, = E~]Z2Ry, 0 = kTe[ZeRy. Here, Ev is the ionization energy of the level 3`, T~ the electron temperature, Z the spectroscopic symbol (for a hydrogen atom Z = 1, • = l]n2), n the principal quantum number, w(y) the rate of level decay due to collisional processes omitted in the sum in Eq. (1). The spontaneous radiative decay of the level y is allowed for by the operator £~ and the appropriate sum in Eq. (1), F(3`) is the rate of upper level population by external processes (relative to these levels), F~(3`) is a similar function for low levels. In the general case the function F(y) depends on the population distribution over levels with •, > •t. The value E~ is a conditional boundary between two spectral regions. It is essentially below this level that processes between •7 < ~r levels may be described analytically (for a hydrogen atom, E~<0.1); for E-E~, radiative processes should prevail over collisions. The latter condition permits us to disregard levels described by Eq. (1) when we consider the system (2).t For the existence of ~ it is necessary that the plasma electron density N~ not too large [N~<(10 4-10-3) x / T x 1016cm ~ for hydrogen-like ions]. We shall consider only the system (2). This solution is independent of e~. First, we consider the radiative operator £~ We e.c. that, for levels •~ < •~, N(y), is a continuous function of •~ The radiative transition probabilities then follow the Kramers formula (• > •3: (E~'P) 3/2
A(e~e')=2Ao
E_Er,
" Ao=0.810 I°Z 4s i
(3)
tThe functionF(y) is thendeterminedonlyby the populationof the groundstateand the concentrationof recombingions.
An analyticaldescriptionof populationsof highlyexcitedlevels
443
Replacement of the summation in Eq. (2) by integration and the use of Eq. (3) for radiative transition probabilities with integration by parts, permits us to reduce the operator £¢ to (see the detailed transformations in Beigman and Mikhalchit°) /-~,Ar(e) = Aoe'/2[ N(e)ln(~) - fo" N ( ? - - : ( e ) d e ' ] .
(4)
Here, ~o refers to the lowest level for a radiative transition. For hydrogen in an optically thin plasma ~o= 1. We now e.c. the collisional operator/~c. Probabilities of transitions between highly excited levels induced by electron collisions have been considered by many authors. 12"t4'15 Both the Born and Kramers quasiclassical approximations show that transitions with small An(An = In n'l) are most probable and the basic dependence of the function W(~', E) on quantum numbers is given by (Ee')3/2/(e - e')3. Thus, W(e', E) contains a third-order pole with respect to the difference E - E' or An. Replacement of summation by integration and subsequent inte~ation by parts for the operator/_~ shows that, at high n, the main contribution to the operator Lc is made by the pole vicinity. In this situation, however, the replacement of summation by integration is not valid since, in the vicinity of the pole, variations of W(d, e) between neighbouring levels are of the order of the function itself. We therefore derive an expression for Lc in the limit n -->~ by means of a flux in the space of quantum numbers. We define the flux ] between the levels n + 1, n as
y(n + I, n) = 5'. X
(W~,~.N~.- W..o,N.,).
(5)
We assume the flux to be positive if it is directed towards the ground state, The flux difference j(n, n + 1 ) - j ( n - 1 , n) should be equal to the externally pumped flux at the level n. This difference is i(n, n + l)-j(n- I, n) = ~ (W..,N.,- W.,,,N,,)
(6)
n'~tl
according to Eqs. (2). Unlike the functions W(e', e) ] is usually a smooth function of ~. Taking account of Eqs. (5, 6), we may therefore write /~N(e) = 2e 3/2d./ de"
(7)
We continue to assume that N(e) is a sufficiently smooth function so that Nn,--Nn - dNn/de x (n' - n)8, where 8 = 2e 3/2 is the distance between neighbouring levels. Using also a comparatively smooth dependence of W on quantum numbers (but not on their difference), W(n - k', n + k') = W(n, n + k' + k") and the detailed balance conditions for the case 8/0 ~ 1 (0 = k T e [ f Ry, kT~ is the electron temperature) becomes
W(n+k", n-k')= W(n-k',
n+ k")exp( - ~
=W(n-k',n+k")x
8)
k' + k" \
1-~8).
Equation (5) now reduces to
](n,n+l)=
N e)
~
8 ~
~
k'>0 k">l
W(n,n+k'+k'~x(k'+k").
(8)
444
I.L. BEIGMANand I. M. GAISINSKY
Inserting Eqs. (4) and (7) into Eq. (2), we obtain 2E3/2 dJ + A o E 3 / 2 [ N ( E ) I n ( ~ ) - f " N ( E ' ) - N ( E ) d e ' ] = F(E), de I E - E'
Nz(E)---U~--~ No =- SN~N~+~: S = (4~r)3/2
0 3/~.
(9) tl0t
The Eq. (9) is the main tool to derive a general analytical solution of Eqs, (2), The e.c. for •--~0 follows from the physical condition that sufficiently high levels must be in thermodynamic equilibrium with the continuum. For su~ciently large E, radiative processes are dominant; hence, Eq. (9) may be transformed to an integral equation without the collisional operator £,.. Since the solution of Eq. (4) decreases with E ~ ~, the solution of Eq. (9) decreases with E--, ~c as well. Below we show that this conditions is sufficient. In Eq. (9), we omit the direct ionization [w(E) in Eqs. (1, 2)]. To do this, it is necessary to extend the summation over the continuous spectrum as well. Within the approximation considered key us, this results in no significant change of the expression for the flux. 3. FORMULAE FOR TRANSITION PROBABILITIES INDUCED BY ELECTRON COLLISIONS: AN ANALYTICAL EXPRESSION FOR THE FLUX In the Born approximation, the transition rate coefficient for highly excited levels is given by ~2 2
~xexp(_Ad0)
+(1--~-)
(e"3/2x{4 x--EU \~
x8{(l
0.25](Ed)3/2 . . . . .
' + l~/x[1-(dO) ]
x ¢(d8)]1,
¢(x) = - e x p ( x ) E i ( - x),
ae = lE- E'l, an = ln - n' I, e = l/n:.
(ll)
where Vo= 2.18 x 108 cm/s. When k ~ n , E'~< 0, Eqs. (11) reduce to
W ( n , n + k ) = N e g ( n + k ) < v c r ...... k)~-N~
x [rao°J × vo
E30 ,/2 [ ( 0.25]¢,8[, 0.611] ( 0 ) x ---~y-- x 1-Tj~tg~l k / k 3 J ' ¢ ~ - l n I.-~-~E"
(12)
It is easy to see in Eq. (8) that, if the function W(n, n + k) decreases with increasing k faster than 1/k 3, the series in Eq. (8) is convergent. This convergence provides asymptotic accuracy for Eq. (9). It is, however, clear from Eq. (12) that, in collisional transition probabilities, the term proportional to Ilk 2 is important and this term gives a logarithmic divergence. This means that the sum in Eq. (8) is a logarithmic function of the summation limits and, in particular, that the diffusion approximation of Eqs. (7, 8) for the operator £c is only valid for a logarithmic accuracy. We fellow use Eq. (12) to derive an explicit expression for the flux,? viz. j=--N~K(e)E 3/2(~N
K(e) = ~
~
N(E)),
g(n + k' + k")(v8 .... k,+z,) × (k' + k")& 3/2.
(13)
k'~-O k ' ~ l
tWe dealwithdiscretesums;the replacementof summationby integrationis incorrectdue to a strongk-dependenceforthe function W.
An analyticaldescriptionof populationsof highlyexcitedlevels It
445
tt
This relation is symmetric with respect to k" and k,,. For k,, ~ k', the summation index k = k' + k" is used and explicit separation of the first sum gives 2 2 ) 1 K(e) = ~---~['n'ao]v0 x ~ ( 0 . 7 5 q ~ + l . 0 6 ) x [ l + A ( ~ ) ] , *~ *~+*~( a ~ . a~ + ~a3) + k ~ . ( a2 ~_a_a_~3 A ( , ) = ~k'~2 k>k' ~ kk2.k3 k~2 ~ + k 3 " k4] ' 2.66 at=0.75y+l.06,a2
(14)
~ - 1.6 - 0.25 3' 0.75y+l.06;a3=0.75y+l.06"
t O0 , we find If we change the order of summation and take km~
a3)
A(e)= k;;,+l ~ ~,--~--r~---~ (a, . a2+~__~ + ~ k~2
k>~k~n+2
( k " + l ) x (ala-a2~-a3~ kk2- k3" k4)"
(15)
The limit k ' ~ ~ is justified by convergence of the appropriate sums. For k" = 0 and k" ,> 1, we obtain from Eq. (15) A(E) = al
k~>2
~-~+ a2
+ a3
E
k~2
4
~- 0.64a~ + 0.2a2, k~, = O; A ( , ) = a t ( ~ . Z+ k~2
a,
2,
~J+a2
k>_k~n+2 a. /
+a3 k>~2 a,
(16)
~- a~ In [1.78(k" + 1)] + 0.64a2 + 0.2a3.
Derivation of an interpolation formula requires also an expression for the total probability of collisional transitions W,. According to Refs. 12 and 15, W, = N~K,; K. = ~ g(n)(vtr,+k.,) k
6
= 2.18 x 10-8 x ~
{0.82~o+ 1.4711 - (d0)~o(d0)]}.
(17)
4. A FREE SOLUTION We consider first a solution of Eq. (9) with the boundary condition (10) when the function F is zero. This case is of methodological interest since it permits us to introduce the main concept required to derive the general solution. We use the reduced population /q = NINo. The dimensionless quantity N has a limit ~ r o 1 when c o 0 . If we assume F = 0 and neglect (N/O) in the flux expression, then, from Eqs. (9), (10), (13) ~--~+ ~- [N(~) In
\~/
Jo
e-
e'
j = - N , r ( m - ~ -e , N ~--;--~--~1.
= (18)
K(~) in Eq. (18) is a weak (logarithmic) function of 6. The assumption ~(~)= r(~c)= const. and the substitutions x = (d~c)TM, ~c = [49/8 x N,K(Ec)/Aoln (~o/Ec)]217, allow us to reduce the integro-differential equation to the form
x,t7~_~(x_,17~x)
- 4 1 " I" lnx ] ~ + N - ~ t Lln (~o/~)J
+
In 1 fo~'
_ x"
dx'} = O.
(19)
446
I.L. BEIGMANand 1. M. GAISINSKY
From the asymptotic expression ~o-~ In (0/1.78~c) and the assumption k " + 1 = ~c[2~ 312 for E~, we have 0.82•
"~2/7 [
N~
= 7
×
0.5 in (110) + in (1.6eo)]2/7 1-
j
,
20)
Since k~ is not uniquely defined, we have introduced a scale factor q~ ~ 1. An exact analytical solution of Eq. (19) presents considerable difficulties. Therefore we consider first the bebaviour of .N for small and large x, and use a local approximation valid for intermediate values; and finally, we obtain a general approximation from a comparison with the computed solution. Equation (19) describes the transformation of the collisional flux into the radiative flux as the electron moves from the ionization limit to the ground state. The collisional flux j is of a differential nature, while radiative transitions are described by integral expressions. At small x ( e ~ ~,.), transitions induced by collisions are most important (the expression in square brackets) and the solution is that for a diffusion problem with constant j, viz. 5 = 1 - L x '°/7, j = "~ K(e~)N~e~ m L .
The value of the flux j is determined by the behaviour of N at large x, i.e. by the radiative part of the solution. At large x(e ~> ec), collisional transitions become insignificant and we have the usual radiative equation considered by Beigman and Mikhalchi/° The integral form of Eq. (19) is essential here, since levels with x >> 1 (e t> e,) are primarily populated by radiative transitions from levels with x ~ 1. Since the total intensity of radiative transitions is equal to the flux j, the asymptotic solution ~° is
2i ]V(x)--, A0(e - Ec)ln2[(e - ec)/eo y] 40 In (eo/ec) = 4-9 + (x TM - 1) in 2 [(X 4/7 --
1 ) e c / ~ 0 ~ ]' ')/ =
1.78.
(21)
At x - 1, the collisional flux j transforms into radiative transitions to levels with large x. In this region, we may neglect population due to radiative transitions and consider only radiative decay, which is described by a local expression. This approach was first suggested in Abramov and Smirnov. s If we assume the logarithmic functions to be constant, then, the local approximation corresponds to the neglect of the expression in curly brackets in Eq. (19). The solution with boundary conditions N - ~ 1 at x-~0 and N - ~ 0 at x--,~ ist
Ne(X) = /~e(X) ~
2 sin m, F(I - v
q/"
K , ( x ) , v -- 5/7:
F(1 - v) 1 F(1 + /j) (2) 2u , F(1 -
v)
(2) 2
'x'"
''~
= 1 - 1.28x l°n + 0.875x2; 4(1) = 1.197x T M exp ( - x).
(22)
K,, is Macdonald function. tThe index v differs from that obtained by Abramovand S m i r n o v 7 since there authors used the Bethe approximationfor radiative transitions.
An analyticaldescriptionof populationsof highlyexcitedlevels
447
Equation (22) is described by the following interpolation formula with an error of less than 3%: 1 + X+ 1.5X2+6x 3 /Q,(X) = 1 + 1.28X1°/7+ 5x 39m exp (- x).
(23)
For the limiting cases of large and small x, Eqs. (22) and (23) become identical. The function tie(X) decreases exponentially at large x, whereas the solution of Eq. (19) decreases much slower. However, the asymptotic expression (21) and the curly brackets neglected in Eq. (19) are proportional to l/ln (l/e~), i.e., when ec~0, the solution of Eq. (19) transforms to a local solution/VtIn the general case ec > 0, an exact analytical solution is impossible. Computational results for ]Q at ec =0.05, 0.01, 0.0001 are shown in Fig. 1. It is seen that the function /V at e , # 0 decreases much slower than N,. The convergence of /V to /Qe at e e l 0 is logarithmic. The general aproximation formula for/V is a combination of Eq. (21) and the local solution (23), i.e.
,
JQf= ]Qe +O'81n
c
ln(~3,/~)
,
}
ln[,o3,1(e-ec(X311+ x3))] "
(24)
The accuracy of approximation is illustrated in Fig. 1. Equation (24) is compared with the solutions of the discrete system of equations in Fig. 2. The relation between ec and the electron density Ne is given by Eq. (20). It is seen that populations calculated with the Born and quasi-classical cross sections are close to each other (see also Beigmant6). Populations derived from the solution of the discrete system are satisfactorily described by Eq. (24) with qc = 1 for large and small values of dec. In the intermediate values e - ec, the error is about 20%. At q~ = 0.5, the description is somewhat better on the average, but worse for e ~ ec. The proceeding analysis shows that ec is a boundary between the regions of collisional transitions (e < e~) and radiative transitions (e > ec). 5. A GENERAL SOLUTION AND THE GREEN FUNCTION We write a general solution of the kinetic equations as f " F(eo)
N(e) = Ny(6)+jo 2ep3/2G(¢p, E)dep,
G%, E ) ~
0, G(~., ~) .--~, 0,
(25)
where NF(e) is the solution of Eq. (18) with appropriate boundary conditions, G(ep, ¢) is the Green function with homogeneous boundary conditions (the subscript p denotes the pumped level for the case when the external pumped flux is delta-shaped). The expression
10
O5"5
A
q~ 0~25
~c = 0 . 0 , 5 ,ec = 0 . 0 1 ,~c = 0 . 0 O O I
E,.
=o_..
Fig. 1. Comparison of the approximation formula for a free solution with the computer solution of the
kineticequation(thesolidlineis the computersolution,the dottedlineis calculatedfromthe approximation formula).
448
1. L. BEIGMANand I. M. GAISINSKY
'o
v
i i
f
~
001
,,I
Fig. 2. Comparison of the approximation formula for a free solution with the solution of the discrete system (the solid line is the calculated from the approximation formula); the followingdesignations are used here: NE=NJlO~6; A:q,.=l, NE=0.88×I0 r (transitions are in the Born approximation): 0:qc=0.5, NE~ 1.76x 10 7 (Born approximation); + : q~ - 1, NE = 0.88 x 10 7 (Transitions with ~n = 1 arc quasiclassical, the rest are in the Born approximation); × : q~ = 0.5, NE = 1.7 x 10 7 (Transitions with ~n = 1 are quasiclassical, the other are in the Born approximationl. F(Ep)/2Ep3/2 represents the pumping density over the energy interval. The Green function under consideration has a discontinuity of the first kind at ~ =Ep and our analytical methods are inappropriate for steep changes of the function F in the interval comparable with the distance b e t w e e n neighbouring levels, We consider separately the regions E > Ep (the lower wing), e < Ep (the'upper wing), and the vicinity E = Ep. There are no methods to derive analytically an exact function G(~p, E). We consider first extremely small densities Ne-~0 (the radiative case) and then the case Ne-->Go (the collisional or diffusion case). For these limiting cases, we shall derive a general interpolation formula. (a) Small densities. For the lower wing when Ne->0 the collisional transitions can be neglected and the solution of the problem is 1° I r'p
N(e)
F(e')
de'(e_e,)
A-~ J ' p ~" ~ +
2Fe ( £ Z _ ~ ) '
(26)
a,,(e - e,,)In z
(e - e ' ) i n 2 \
Eoy
I
\
Eoy
I
where Fp is the pumped flux to the level p, 6p = 2Ep~/:. A comparison of Eq. (26) with the solutions of the discrete system of equations is shown for np = 17.35 in Fig. 3.
°
13
t
q =
/
//
// //
o (% z'1
_ _
/ /
/VE = 1O
/
I
i
i: i I I ~-~
,,-'~/E = 10"2/
_ o 6or
-
-
,
r,~= 17; h E = IO' /VE = tO ~
,-'
fT~ = 3 5 1 N E
~ v
. . . . .
= 10";
i?~
-- -
Fig. 3. The Green function in the limitingcase for small densities. (NE = NJI0J6t . - represents the radiative Green function, "A" is the value for E = ~p, x is the calculation for np = 17: 0- holds for n = 35, the dotted I ne shows the so ut on of the k net c equat ons for E, = I).8x 10 ', i.e. for VE =- 10 ~').
An analyticaldescriptionof populationsof highlyexcitedlevels
449
The population of the pumped level Np is determined by the total probability of decay A~ of the level p, i.e.,
Np = FplAp, where Ap = Aoep3121n(~o/2ep m) in the Kramers approximation and Ao = Aoep3[21n(eo/1.3~p3/z) according to a more exact semiempirical Seaton formula, u For the upper wing ~ < ~p, the situation is more complicated. The solution of Eq. (19) in this region is proportional to exp [(de~) v4- (~p/e~)TM] near e,. This difference is at variance with the solutions for the discrete system (see Fig. 3). In this region, the approximation (19) is not applicable since direct transitions from ep to e are more effective than diffusion from ~p by small jumps. This situation described by a coronal approximation with N(~)= N p W ( ~ ) / A ( e ) . Comparison of the coronal relation with the solutions of the discrete system (see Fig. 3) confirms this assumption. (b) Large densities. We assume Ne to be sufficiently large at ~ >>e,. We first consider the upper wing e ~
(27)
K(e.) in Eq. (27) depends on ep. This dependence arises because the function N(e) is nonanalytic at the point ep and the summation in Eqs. (14, 17) is, therefore, limited by ep. If the functions in K(t.) are assumed to be constant, Eq. (27) becomes
K0= 2.18 x 10-8
+ 1.o6
Z3X/(8)
,
(28)
Ao= al In (yet,]2ep3/2)+ 0.64a2 + 0.2a3, where logarithmic functions are truncates at ~ = ep. The value of ;to determines a flux distant from the pumped level [ ~ ~p, k~,>> 1 in Eq. (15)]. In the immediate vicinity of the pumped level, A(e) = ;t(~p) is given by Eq. (16) (k" = 0). Since ;t(e)-o ;t0 when ~ 0 , the second integral decreases at small ~ faster than the first integral, and the solution N is given by the well-known 5/2-law: F_._F_z -/2
E5/2
The second integral cannot be calculated analytically: below we derive an approximation for it. We consider first the full integral over the whole region ~ < ~p. The function A(~) is actually a function of km- (~p - e)/Sp and, hence, A(~) differs from ;to in the vicinity of ~ ~ ~p - qSp. Thus, the second integral is proportional to (~,)3j2 ; t % ) - ;to [1 + A(~p)](1 + Aoj d~'--- const+ ~p3, q - 1 (const).
f [ ~-qsp
Consequently, we write Eq. (28) may be approximated by formula N(~)
2 Fp ,512 [1 + cx/(~p) ] = 5 N, Ko(1+ ;t0) 1 + c'(~p - e)~p-m].
(29)
Equation (29) confirms the fact that the correction to the 5/2-law decreases with the increasing (~p- E)/Sp. Comparison with the solutions for the discrete system shows that satisfactory
450
I. L. BE/GMANand I. M. GAIS1NSKY
n:= 17
2
n
o ~
:35
o
--
o
b~
~r
Fig. 4. Comparison of the upper wing of the Green function with the 5/2-law (solid line refers to the 5/2-law). agreement is achieved at c = 12; c ' = 0.5 (see Fig. 4). Since, in the limiting case of high densities, the upper wing is described by a diffusion approximation, the Green function is continuous and the value at • = ~ is obtained from Eq. (29), viz.' N ( e ) = Fp] NeLp,
Lp =
2"18×10-8 (~ eP'/2 '~ Z3~/(O ) × ( 0 . 7 5 ¢ + 1.06)(I +A) 1 + c~/(ep)]'
(30)
where Lp is the effective probability of a diffusion transition from the level p. For E,/> % (the lower wing), a free solution of Eq. (24) is inappropriate. In fact, Eqs. (13-17) for the flux because of a discontinuity of the first derivative at • = e , neglect transitions between the regions e < ep and E > ep. These transitions are especially important in the vicinity of the point %. Near ep in the region e,. ~> • > Ep, we may neglect radiative transitions and write the kinetic equation as 2~ 3/2 ~ + F(~) = 0,
F(e) =- F(e,)= ~ (Wp+k. ,-IV, - W,.p+kNp+k).
(31)
k>O
where the subscript p refers to the pumped level. We repeat for F(e) transformations used to derive an analytical expression for the flux and assume iV, ~ N t, + (p - n ) d N /dep × 6, Np+, ~Np - kdN+/dep × 6 (the indices - and + refer to the regions e > Ep and e < %, respectively). We assume all functions near ~p (except for those which depend on the difference e - ep) to be constant and note that dN-/dep .~ dN+/dep. Then integration of Eq. (31) gives const N(e) = const + 211 +,~(ep)]
dN* dEp
x - -
x
" 6p-+ 3p)' ( E - ep
(32)
The unknown constant in Eq. (32) is determined by the condition that N(E) transforms to a free solution for e - ~p >> 8p. The expression for the lower wing of the Green function for high densities is
(1 + cV(~p)/2)
(33)
Comparison with solutions for the discrete system (see Fig. 5) allows us to find the constant c =8.5.
An analytical description of populations of highly excited levels
~
ID-
~
=
~
~
2
o
OsF
451
o
8=0.1
np=17
S
o ;,
0
o ;2
IO
'~
05
-
I
np:17
o
o
bl
ob~
t
E
Fig. 5. The normalized Green function G(~p, ~) in the limiting case for high density (1 is the calculated from the approximation formula, 2 is the free solution, 0 is the solution for the discrete system).
Interpolation formula. We consider first the case N, ~ 0. For small densities, the population Np is determined only by the radiative lifetime. As the density increases, the decay due to electron collisions is added to the radiative decay. The population is determined as before by the lifetime of the level Np = FJ(Ap + N,.Np), where Kp is given by Eq. (17). For large
I
~, : n p = 1 7
08 07 06
o : np = 3 5
\..\ "'\'~
05 04
• : np = I 0 0
\\ "o,\
03
•
\ \
=,oo
02
0.1
•
008 006 005 004 003 002
I0 ,c,
i0 "9
~
i
10 .8
107 ns
~\~'\
106
N e "1
[ Z 7 i0,o =
Fig. 6. Comparison of the approximation formula with the solution of the discrete system for the region -~p. When the Green function is used (solid, dashed and dotted lines represent calculations by the approximation formula).
452
I.L. BEIGMANand I. M. GAISINSKY
densities, the population of the level p is determined by diffusion processes [see Eqs. (29) and (30)]. Therefore, G(%,
Ep) Go(~p)
1
(34)
Ap + N~Kp(Ap + clN~Lp)/(Ap + ctNeKp)"
Computed results are satisfactorily described at c~ = 0.25 (see Fig. 6). The general interpolation formula, including limiting cases for small and large densities [see the solutions of Eqs. (26, 29-30, 30--34)] is G(%, e)= Go(%)O(%, e);
12V'(e) ( ~ ) '/21+ A(E)%)+(I - Y)3 1 +(% ~ E)/Se, e < ~ : l+12V(e)
V3 W(,~---
¢
(351
1, e=~;
d(E., E)=
1+8.5
+
% E-~+
•~-t(x)
ap + + I ( E -
+0.4~ In -a~+6p
,
e.,
&)
e>ep;
(361
here
A.
Y = Ap + N~Kp ' x = \ zL ] 1 + x + 1.5x 2 + 6x 3
/~(X) = 1 + 1.28X1°/7+ { II(e-~p,A~)=~c
5X 39/14 exp
I in[y~o/(e_G) ]
Ac
- I+5",/(E,,IEc) ,
1 + 3X/(G/ec) (- x),/~c = 1 + 6(eJec) 3/2 Ec' l
},
ln[y~o/(e_~p_Ac(X3/l+x3))]
=2,p3j2. 6p
The general Green function is depicted in Fig. 7, which shows a steep radiative function with a first-order discontinuity at the pumped level. As the electron density increases, this function transforms to a smooth diffusion function with a discontinuous derivative.
,~ -,o
,,..c~ ~
Fig. 7. Contourmap of the Green function.
n~.
=17
An analytical description of populations of highly excited levels
453
50
I
2.0 o
,0
~o.~1
,0
~j"
20
o// I
30
I
,So
40
L
60
_,
?0
_,_
80
_L
90
L_
,00
"
n
Fig. 8. The relative populations N~/N4 of atomic levels in a hydrogen plasma at N, = liP cm -3, T, = 8000°K [1 is calculated by the method presented in this paper, 2 is the computer calculation of Brocklehyrst t'2 and Salem and Brocklehurst, 5 is calculated from (5*); discrete points are observational results for relative intensities of optical lines in the planetary nebula NGC-7009].
CONCLUSIONS
Here we discuss the application of the derived formulae. The analytical expression for the general solution consists of two parts: the expression for a free solution and the expression for the Green function. Formulae for the free solution are based on Eq. (18) derived in the limit n ~ o ; in the expression for the coilisional operator Lc, we assumed NIO < dNlde. Collisional transitions are mostly important in the region e < ec. Since in this region dNlde ~ N/e, our approximation is justified if ec < O. This condition imposes the following limit on the electron density: Ne < 1016 × Z 7 × 0 4. Comparison with the results of a direct solution of the discrete system shows that, if n > 5, the error of Eq. (24) is within 20-30%. The case of the Green function is more complicated. The approximations used are listed in Table 1. Table 1. Approximation used to solve problem involving a Green function e> ~c
Corona
~p ~ ~c
Interpolation formulae
e=ep
e>ep
Kinetic Eq. (18) in the limit N,-*0
Kinetic Eq. (19) with expressions (13-16) for the flux limited on one side.
Interpolation formula based on a qualitative solution of Eq (33) for Ec "> ~ > ep and a free solution for
- Ep/~p,> 1
In this case, no new limitations arise and the density condition is sufficient. At the same time, comparison with the results of a direct solution of the discrete system shows that in the region np > 10, n > 5, the error is within 20-30%. Acknowledgements--The authors are grateful to L. A. Vainstein and L. P. Presnyakov for discussions. REFERENCES M. Brocklehurst, Mon. Not. R. Astr. Soc. 148, 417 (1970). M. Brocklehurst, Mon. Not. R. Astr. Soc. 153, 471 (1971). M. Brocklehurst and M. J. Seaton, Mort. Not. R. Astr. Soc. 157, 179 (1972). M. Brocklehurst, Mon. Not. R. Astr. Soc. 157, 211 (1972). M. Salem and M. Brocklehurst, Astrop& J. Suppl. Set., 39, 633 (1979). S. T. Belyaev and G. I. Budker, Plasma Physics and the Problem o/ Controllable Thermonuclear Reactions. 3, 41 (1956). 7. L. V. Gurevich and L. P. Pitaevsky, JETP 46, 1281 (1964). 8. V. A. Abramov and B. M. Smirnov, Opt. i Spectr. 21, 19 (1966).
1. 2. 3. 4. 5. 6.
I. L. BEIGMANand I. M. GAISINSKY
454 9. 10. 11. 12. 13. 14. 15. 16. 17.
L. M. Biberman, V. S. Vorobyov, and I. T. Yakubo~,, UFN 107, 353 (1972). 1. L. Beigman and E. D. Mikhalchi, JQSRT 9, 1365 (1969). I. L. Beigman, Preprint No. 167, Lebedev Phys. Inst., Moscow (1975). L. A. Vainstein, I. I. Sobelman, and E. A Yukov, Excitation of Atoms and Broadening of Spectral Lines. Springer Verlag, Heidelberg (1981). I. L. Beigman, A. M. Urnov, and V. P. Shevelko, JETP 58, 1825 (1970). I. C. Perciva'l and D. Richards, Adv. At. Molec. Phys. 11, 2 (1975). I. L. Beigman, JETP 73, 1729 (1977). I. L. Beigman, ICAP-VI, 302, Riga (1978). J. B. Kaler, Astrophys. J. 143, 722 (1966).
APPENDIX Approximation Formulae for the Photorecombination Mechanism of Atomic Level Population From the practical point of view, the case of a purely photorecombination excitation to highly excited levels of great interest. Then, with the Kramers approximation for photorecombination, Eq. (9) may be rewritten as 2•312dd~ ~ - Ao•S/2{N(O)ln• - N(•)ln•o
+
fo'[N,¢•,)In
I•-,pl]d•p}= f(•),
,l*)
j = - N~r• 3/2d_N_N;[(~) = AoNo•3/2 exp (•/O)lEi(- •/0)i. At low density, when collisional transitions are neglected, the solution of the problem is determined by the radiative Green function
- f" exp(•t'JO)jEi(-•P]O)l " Nu(•) = Nob~n(•); NR(•) - Jo (e - •p)In 2 [(• - •p)/EoY]oep
(2*)
In the two limiting cases •,~ 0 and • ~ 0, asymptotic estimates of the solution follow from Eq. (2'), i.e.
I In(coy2/0)
e < O;
In (•oy/•) ' MR(El ~
(3'~ In (•y/0) E~>0. (e/O) In2 (~oy/E)'
It is seen that the approximation formula MR(e) = In (1 + 0/y,)[ln (~o34•) + y In (I + •7/0)] × In2 (dO) + 1.25 In2 (•o"fl•) In2 (dO) + 1
14'~
describes both limiting cases. The accuracy of the approximation for intermediate • is illustrated in Table 2. Table 2. Data showing the accuracy of the approximation used
O=O,1
0.5.10 3 0.2.10 3 0.8.10 -2 0.032 0.128
0=0.02
Approximation
Computer solution
Approximation
Computer solution
0.59 0.51 0.41 0.31 0.27
0.61 0.54 0.45 0.35 0.27
0.39 0.30 0.20 0.13 0.08
0.42 0.32 0.23 0.14 0.09
For E ,> ec determined by Eq. (20) collisional transitions are insignificant; hence, Eq. (4*) is applicable. For • ~ •,., collisional transitions dominate and photorecombination, as well as other radiative transitions, may be neglected. In this region, the local approximation (23) is valid. Therefore, the following approximation formula is suggested: N(•) = MRtE÷ ~(~/~,) × •c) + [1 - MR(a(0)•c)J x M,(d~,),
(5*)
a(x) = 3/2[2 exp (- 1.4x13) - I], which is asymptotically correct in the two limiting cases • ,> •c and •-> e, and describes intermediate cases within about 5-10%. As an illustration we compare in Fig. 8 the calculations for atomic level populations in a hydrogen plasma with calculatons of Brocklehurst 1'2 and Salem and Brocldehurst. 5 The plasma parameters Ne = llP cm -3, T, = 8000°K are characteristic of planetary nebulae. Figure 8 also shows observational results for relative intensities of optical lines in the nebula NGC-7009 (Kaler~7).
0022~.073182[110~1-14503.0010
Printed in Great Britain.
© 1982 Pergamon Press Ltd.
AN ANALYTICAL DESCRIPTION OF HIGHLY EXCITED
OF POPULATIONS LEVELS
I. L. BEIGMANand I. M. GAISINSKY LebedevPhysicalInstitute,USSRAcademyof Sciences,Moscow,U.S.S.R. (Received 15 May 1981)
Abstract--Analyticalmethodsare developedto describepopulationsof highlyexcitedlevels.The Kramers approximationwas used for radiative transitions and the Born approximationfor transitions induced by electroncollisions. An exact solution is derived for the limiting case of small densities. In the limiting case of large densities, an analyticalsolution is obtainedwith constants determinedfrom a comparisonwith computer solutionsof the balanceequations.A generalinterpolationformulais presented,whichincludesthe limiting cases and describesresultsfor intermediatedensitieswith 20-30%. 1. I N T R O D U C T I O N
Populations of atomic levels in a plasma are described by an infinite system of balance equations derived from particle conservation conditions. Usually, this system is solved numerically for each specific case, taking into account the restricted number of the equations. Since levels with large principal quantum number are always hydrogen-like, the problem for an arbitrary, highly excited atom can be solved using the hydrogen-atom approach. Systematic computer calculations for populations of highly excited hydrogen levels in low density plasma were made by Brocklehurst, ~-3 Brocklehurst and Seaton4 and Salem and Brocklehurst.5 Populations of highly excited levels (unlike lower levels where the structure depends strongly on the specific atom) are comparatively smooth functions of the principal quantum number n. Therefore, it is of interest to find unified analytical descriptions for the solution. When charged particle collisions are of greatest importance for the levels under study, transitions between nearby levels are most probable. This effect permits a diffusion approximation of the Focker-Planck type to be used for the populations. This approach was first suggested by Belyaev and Budker 6 and further developed for the recombination process by Abramov and Smirnovs and Gurevich and Pitaevsky7 (for detailed references, see also the review paper by Biberman et al.9). Radiative transitions can not be described by the Focker-Planck approximation. Beigman and Mikhalchi 1° showed that, for high levels, the system of balance equations for radiative transitions is equivalent to an integral equation. This integral equation admits an exact analytical solution if the Kramers approximation is used for radiative transition probabilities. An attempt to combine both methods was made by Beigman." The present paper is aimed at derivation of an analytical description of a general solution for the kinetics of highly excited levels in a low-density plasma. For this purpose, it is first necessary to derive a solution for the case without an externally pumped flux (hereafter referred to as a free solution) and, second, to obtain the Green function of the problem, i.e. a solution for the case of excitation to an arbitrary single level. We shall first write general kinetic equations for the limiting case of large quantum numbers. Two circumstances are important for collisional transitions within the Focker-Planck approximation: (1) the diffusion approximation is valid with logarithmic accuracy only; (2) an analytical expression for the flux depends on the specific situation. The 'free' solution is derived by approximating the computational solution for the specified kinetic equations. The resultant formula is compared with a direct solution of the discrete system of the balance equations. The Green function is derived as follows. First, the Green function is considered for the limiting case low electron density. The levels below the pumped level are described by the Green function derived from general equations (Beigman and Mikhalchil°), which allow for radiative transitions only. For the levels above the pumped level, the coronal approximation is 441 JQSRT Vol. 28, No. 5--F
442
I.L. BEIGMANand I. M. GAIS1NSKY
valid. Then the limiting case of high electron density is considered. From the general kinetic equations, the main functional dependences can be obtained. The constants in these dependences result from a comparison with a direct solution of the discrete system of the balance equations. The general interpolation formula is derived on the basis of the two limiting cases and the solutions of the discrete system. In conclusion, we consider applicability and possible errors of the derived formulae. 2. KINETIC EQUATIONS Let N(3`) be the population of atoms at a sublevel with a set of quantum numbers 3`, so that population at a level 3` is g(3`)N(3`), where g(y) is the statistical weight of the level 3`. We assume that charged particle collisions and radiative decays play a leading role in kinetics. The probabilities of transitions per unit time from the level 3` to the level 3`' due to charged particle collisions and spontaneous photon emission are denoted, respectively, by W ( y ' ~ y) and A(y' ~- y). The e.c. W and A are, to some extent, similar to the averaged transition strengths. The usual y-3`' transition probabilities are W(y'~-y)/g(3`) and A(y'*--y)/g(y), respectively. The functions W and A are not only highly symmetric, but they also allow the balance equations to be written without statistical weights. The descriptions of populations for highly excited and low atomic levels are different and thus we write an infinite system of the balance equations as N(3`){w(3`) + ~. [W(3`'~ 3,)+ A(3`'~ 3`)1} 7' ~ Y
-~
N(y')[ W(3`' ~ 3`')+ A(y ~ 3`')] -- F~(y); E~, e¢> El
(1)
N(y)w(y) + (£c + £ , ) N ( y ) = F(y); e,. < E,; £,,N(3`) -= y'#), ~ [W(3`' ~ 3`)N(3`)- W(3` ~ y')N(y')],
£~N(y) =- ~
y'~ y
[A(3`' *-- y)N(y) - A(3` , - 3`')N(y')].
(2)
e~, = E~]Z2Ry, 0 = kTe[ZeRy. Here, Ev is the ionization energy of the level 3`, T~ the electron temperature, Z the spectroscopic symbol (for a hydrogen atom Z = 1, • = l]n2), n the principal quantum number, w(y) the rate of level decay due to collisional processes omitted in the sum in Eq. (1). The spontaneous radiative decay of the level y is allowed for by the operator £~ and the appropriate sum in Eq. (1), F(3`) is the rate of upper level population by external processes (relative to these levels), F~(3`) is a similar function for low levels. In the general case the function F(y) depends on the population distribution over levels with •, > •t. The value E~ is a conditional boundary between two spectral regions. It is essentially below this level that processes between •7 < ~r levels may be described analytically (for a hydrogen atom, E~<0.1); for E-E~, radiative processes should prevail over collisions. The latter condition permits us to disregard levels described by Eq. (1) when we consider the system (2).t For the existence of ~ it is necessary that the plasma electron density N~ not too large [N~<(10 4-10-3) x / T x 1016cm ~ for hydrogen-like ions]. We shall consider only the system (2). This solution is independent of e~. First, we consider the radiative operator £~ We e.c. that, for levels •~ < •~, N(y), is a continuous function of •~ The radiative transition probabilities then follow the Kramers formula (• > •3: (E~'P) 3/2
A(e~e')=2Ao
E_Er,
" Ao=0.810 I°Z 4s i
(3)
tThe functionF(y) is thendeterminedonlyby the populationof the groundstateand the concentrationof recombingions.
An analyticaldescriptionof populationsof highlyexcitedlevels
443
Replacement of the summation in Eq. (2) by integration and the use of Eq. (3) for radiative transition probabilities with integration by parts, permits us to reduce the operator £¢ to (see the detailed transformations in Beigman and Mikhalchit°) /-~,Ar(e) = Aoe'/2[ N(e)ln(~) - fo" N ( ? - - : ( e ) d e ' ] .
(4)
Here, ~o refers to the lowest level for a radiative transition. For hydrogen in an optically thin plasma ~o= 1. We now e.c. the collisional operator/~c. Probabilities of transitions between highly excited levels induced by electron collisions have been considered by many authors. 12"t4'15 Both the Born and Kramers quasiclassical approximations show that transitions with small An(An = In n'l) are most probable and the basic dependence of the function W(~', E) on quantum numbers is given by (Ee')3/2/(e - e')3. Thus, W(e', E) contains a third-order pole with respect to the difference E - E' or An. Replacement of summation by integration and subsequent inte~ation by parts for the operator/_~ shows that, at high n, the main contribution to the operator Lc is made by the pole vicinity. In this situation, however, the replacement of summation by integration is not valid since, in the vicinity of the pole, variations of W(d, e) between neighbouring levels are of the order of the function itself. We therefore derive an expression for Lc in the limit n -->~ by means of a flux in the space of quantum numbers. We define the flux ] between the levels n + 1, n as
y(n + I, n) = 5'. X
(W~,~.N~.- W..o,N.,).
(5)
We assume the flux to be positive if it is directed towards the ground state, The flux difference j(n, n + 1 ) - j ( n - 1 , n) should be equal to the externally pumped flux at the level n. This difference is i(n, n + l)-j(n- I, n) = ~ (W..,N.,- W.,,,N,,)
(6)
n'~tl
according to Eqs. (2). Unlike the functions W(e', e) ] is usually a smooth function of ~. Taking account of Eqs. (5, 6), we may therefore write /~N(e) = 2e 3/2d./ de"
(7)
We continue to assume that N(e) is a sufficiently smooth function so that Nn,--Nn - dNn/de x (n' - n)8, where 8 = 2e 3/2 is the distance between neighbouring levels. Using also a comparatively smooth dependence of W on quantum numbers (but not on their difference), W(n - k', n + k') = W(n, n + k' + k") and the detailed balance conditions for the case 8/0 ~ 1 (0 = k T e [ f Ry, kT~ is the electron temperature) becomes
W(n+k", n-k')= W(n-k',
n+ k")exp( - ~
=W(n-k',n+k")x
8)
k' + k" \
1-~8).
Equation (5) now reduces to
](n,n+l)=
N e)
~
8 ~
~
k'>0 k">l
W(n,n+k'+k'~x(k'+k").
(8)
444
I.L. BEIGMANand I. M. GAISINSKY
Inserting Eqs. (4) and (7) into Eq. (2), we obtain 2E3/2 dJ + A o E 3 / 2 [ N ( E ) I n ( ~ ) - f " N ( E ' ) - N ( E ) d e ' ] = F(E), de I E - E'
Nz(E)---U~--~ No =- SN~N~+~: S = (4~r)3/2
0 3/~.
(9) tl0t
The Eq. (9) is the main tool to derive a general analytical solution of Eqs, (2), The e.c. for •--~0 follows from the physical condition that sufficiently high levels must be in thermodynamic equilibrium with the continuum. For su~ciently large E, radiative processes are dominant; hence, Eq. (9) may be transformed to an integral equation without the collisional operator £,.. Since the solution of Eq. (4) decreases with E ~ ~, the solution of Eq. (9) decreases with E--, ~c as well. Below we show that this conditions is sufficient. In Eq. (9), we omit the direct ionization [w(E) in Eqs. (1, 2)]. To do this, it is necessary to extend the summation over the continuous spectrum as well. Within the approximation considered key us, this results in no significant change of the expression for the flux. 3. FORMULAE FOR TRANSITION PROBABILITIES INDUCED BY ELECTRON COLLISIONS: AN ANALYTICAL EXPRESSION FOR THE FLUX In the Born approximation, the transition rate coefficient for highly excited levels is given by ~2 2
~xexp(_Ad0)
+(1--~-)
(e"3/2x{4 x--EU \~
x8{(l
0.25](Ed)3/2 . . . . .
' + l~/x[1-(dO) ]
x ¢(d8)]1,
¢(x) = - e x p ( x ) E i ( - x),
ae = lE- E'l, an = ln - n' I, e = l/n:.
(ll)
where Vo= 2.18 x 108 cm/s. When k ~ n , E'~< 0, Eqs. (11) reduce to
W ( n , n + k ) = N e g ( n + k ) < v c r ...... k)~-N~
x [rao°J × vo
E30 ,/2 [ ( 0.25]¢,8[, 0.611] ( 0 ) x ---~y-- x 1-Tj~tg~l k / k 3 J ' ¢ ~ - l n I.-~-~E"
(12)
It is easy to see in Eq. (8) that, if the function W(n, n + k) decreases with increasing k faster than 1/k 3, the series in Eq. (8) is convergent. This convergence provides asymptotic accuracy for Eq. (9). It is, however, clear from Eq. (12) that, in collisional transition probabilities, the term proportional to Ilk 2 is important and this term gives a logarithmic divergence. This means that the sum in Eq. (8) is a logarithmic function of the summation limits and, in particular, that the diffusion approximation of Eqs. (7, 8) for the operator £c is only valid for a logarithmic accuracy. We fellow use Eq. (12) to derive an explicit expression for the flux,? viz. j=--N~K(e)E 3/2(~N
K(e) = ~
~
N(E)),
g(n + k' + k")(v8 .... k,+z,) × (k' + k")& 3/2.
(13)
k'~-O k ' ~ l
tWe dealwithdiscretesums;the replacementof summationby integrationis incorrectdue to a strongk-dependenceforthe function W.
An analyticaldescriptionof populationsof highlyexcitedlevels It
445
tt
This relation is symmetric with respect to k" and k,,. For k,, ~ k', the summation index k = k' + k" is used and explicit separation of the first sum gives 2 2 ) 1 K(e) = ~---~['n'ao]v0 x ~ ( 0 . 7 5 q ~ + l . 0 6 ) x [ l + A ( ~ ) ] , *~ *~+*~( a ~ . a~ + ~a3) + k ~ . ( a2 ~_a_a_~3 A ( , ) = ~k'~2 k>k' ~ kk2.k3 k~2 ~ + k 3 " k4] ' 2.66 at=0.75y+l.06,a2
(14)
~ - 1.6 - 0.25 3' 0.75y+l.06;a3=0.75y+l.06"
t O0 , we find If we change the order of summation and take km~
a3)
A(e)= k;;,+l ~ ~,--~--r~---~ (a, . a2+~__~ + ~ k~2
k>~k~n+2
( k " + l ) x (ala-a2~-a3~ kk2- k3" k4)"
(15)
The limit k ' ~ ~ is justified by convergence of the appropriate sums. For k" = 0 and k" ,> 1, we obtain from Eq. (15) A(E) = al
k~>2
~-~+ a2
+ a3
E
k~2
4
~- 0.64a~ + 0.2a2, k~, = O; A ( , ) = a t ( ~ . Z+ k~2
a,
2,
~J+a2
k>_k~n+2 a. /
+a3 k>~2 a,
(16)
~- a~ In [1.78(k" + 1)] + 0.64a2 + 0.2a3.
Derivation of an interpolation formula requires also an expression for the total probability of collisional transitions W,. According to Refs. 12 and 15, W, = N~K,; K. = ~ g(n)(vtr,+k.,) k
6
= 2.18 x 10-8 x ~
{0.82~o+ 1.4711 - (d0)~o(d0)]}.
(17)
4. A FREE SOLUTION We consider first a solution of Eq. (9) with the boundary condition (10) when the function F is zero. This case is of methodological interest since it permits us to introduce the main concept required to derive the general solution. We use the reduced population /q = NINo. The dimensionless quantity N has a limit ~ r o 1 when c o 0 . If we assume F = 0 and neglect (N/O) in the flux expression, then, from Eqs. (9), (10), (13) ~--~+ ~- [N(~) In
\~/
Jo
e-
e'
j = - N , r ( m - ~ -e , N ~--;--~--~1.
= (18)
K(~) in Eq. (18) is a weak (logarithmic) function of 6. The assumption ~(~)= r(~c)= const. and the substitutions x = (d~c)TM, ~c = [49/8 x N,K(Ec)/Aoln (~o/Ec)]217, allow us to reduce the integro-differential equation to the form
x,t7~_~(x_,17~x)
- 4 1 " I" lnx ] ~ + N - ~ t Lln (~o/~)J
+
In 1 fo~'
_ x"
dx'} = O.
(19)
446
I.L. BEIGMANand 1. M. GAISINSKY
From the asymptotic expression ~o-~ In (0/1.78~c) and the assumption k " + 1 = ~c[2~ 312 for E~, we have 0.82•
"~2/7 [
N~
= 7
×
0.5 in (110) + in (1.6eo)]2/7 1-
j
,
20)
Since k~ is not uniquely defined, we have introduced a scale factor q~ ~ 1. An exact analytical solution of Eq. (19) presents considerable difficulties. Therefore we consider first the bebaviour of .N for small and large x, and use a local approximation valid for intermediate values; and finally, we obtain a general approximation from a comparison with the computed solution. Equation (19) describes the transformation of the collisional flux into the radiative flux as the electron moves from the ionization limit to the ground state. The collisional flux j is of a differential nature, while radiative transitions are described by integral expressions. At small x ( e ~ ~,.), transitions induced by collisions are most important (the expression in square brackets) and the solution is that for a diffusion problem with constant j, viz. 5 = 1 - L x '°/7, j = "~ K(e~)N~e~ m L .
The value of the flux j is determined by the behaviour of N at large x, i.e. by the radiative part of the solution. At large x(e ~> ec), collisional transitions become insignificant and we have the usual radiative equation considered by Beigman and Mikhalchi/° The integral form of Eq. (19) is essential here, since levels with x >> 1 (e t> e,) are primarily populated by radiative transitions from levels with x ~ 1. Since the total intensity of radiative transitions is equal to the flux j, the asymptotic solution ~° is
2i ]V(x)--, A0(e - Ec)ln2[(e - ec)/eo y] 40 In (eo/ec) = 4-9 + (x TM - 1) in 2 [(X 4/7 --
1 ) e c / ~ 0 ~ ]' ')/ =
1.78.
(21)
At x - 1, the collisional flux j transforms into radiative transitions to levels with large x. In this region, we may neglect population due to radiative transitions and consider only radiative decay, which is described by a local expression. This approach was first suggested in Abramov and Smirnov. s If we assume the logarithmic functions to be constant, then, the local approximation corresponds to the neglect of the expression in curly brackets in Eq. (19). The solution with boundary conditions N - ~ 1 at x-~0 and N - ~ 0 at x--,~ ist
Ne(X) = /~e(X) ~
2 sin m, F(I - v
q/"
K , ( x ) , v -- 5/7:
F(1 - v) 1 F(1 + /j) (2) 2u , F(1 -
v)
(2) 2
'x'"
''~
= 1 - 1.28x l°n + 0.875x2; 4(1) = 1.197x T M exp ( - x).
(22)
K,, is Macdonald function. tThe index v differs from that obtained by Abramovand S m i r n o v 7 since there authors used the Bethe approximationfor radiative transitions.
An analyticaldescriptionof populationsof highlyexcitedlevels
447
Equation (22) is described by the following interpolation formula with an error of less than 3%: 1 + X+ 1.5X2+6x 3 /Q,(X) = 1 + 1.28X1°/7+ 5x 39m exp (- x).
(23)
For the limiting cases of large and small x, Eqs. (22) and (23) become identical. The function tie(X) decreases exponentially at large x, whereas the solution of Eq. (19) decreases much slower. However, the asymptotic expression (21) and the curly brackets neglected in Eq. (19) are proportional to l/ln (l/e~), i.e., when ec~0, the solution of Eq. (19) transforms to a local solution/VtIn the general case ec > 0, an exact analytical solution is impossible. Computational results for ]Q at ec =0.05, 0.01, 0.0001 are shown in Fig. 1. It is seen that the function /V at e , # 0 decreases much slower than N,. The convergence of /V to /Qe at e e l 0 is logarithmic. The general aproximation formula for/V is a combination of Eq. (21) and the local solution (23), i.e.
,
JQf= ]Qe +O'81n
c
ln(~3,/~)
,
}
ln[,o3,1(e-ec(X311+ x3))] "
(24)
The accuracy of approximation is illustrated in Fig. 1. Equation (24) is compared with the solutions of the discrete system of equations in Fig. 2. The relation between ec and the electron density Ne is given by Eq. (20). It is seen that populations calculated with the Born and quasi-classical cross sections are close to each other (see also Beigmant6). Populations derived from the solution of the discrete system are satisfactorily described by Eq. (24) with qc = 1 for large and small values of dec. In the intermediate values e - ec, the error is about 20%. At q~ = 0.5, the description is somewhat better on the average, but worse for e ~ ec. The proceeding analysis shows that ec is a boundary between the regions of collisional transitions (e < e~) and radiative transitions (e > ec). 5. A GENERAL SOLUTION AND THE GREEN FUNCTION We write a general solution of the kinetic equations as f " F(eo)
N(e) = Ny(6)+jo 2ep3/2G(¢p, E)dep,
G%, E ) ~
0, G(~., ~) .--~, 0,
(25)
where NF(e) is the solution of Eq. (18) with appropriate boundary conditions, G(ep, ¢) is the Green function with homogeneous boundary conditions (the subscript p denotes the pumped level for the case when the external pumped flux is delta-shaped). The expression
10
O5"5
A
q~ 0~25
~c = 0 . 0 , 5 ,ec = 0 . 0 1 ,~c = 0 . 0 O O I
E,.
=o_..
Fig. 1. Comparison of the approximation formula for a free solution with the computer solution of the
kineticequation(thesolidlineis the computersolution,the dottedlineis calculatedfromthe approximation formula).
448
1. L. BEIGMANand I. M. GAISINSKY
'o
v
i i
f
~
001
,,I
Fig. 2. Comparison of the approximation formula for a free solution with the solution of the discrete system (the solid line is the calculated from the approximation formula); the followingdesignations are used here: NE=NJlO~6; A:q,.=l, NE=0.88×I0 r (transitions are in the Born approximation): 0:qc=0.5, NE~ 1.76x 10 7 (Born approximation); + : q~ - 1, NE = 0.88 x 10 7 (Transitions with ~n = 1 arc quasiclassical, the rest are in the Born approximation); × : q~ = 0.5, NE = 1.7 x 10 7 (Transitions with ~n = 1 are quasiclassical, the other are in the Born approximationl. F(Ep)/2Ep3/2 represents the pumping density over the energy interval. The Green function under consideration has a discontinuity of the first kind at ~ =Ep and our analytical methods are inappropriate for steep changes of the function F in the interval comparable with the distance b e t w e e n neighbouring levels, We consider separately the regions E > Ep (the lower wing), e < Ep (the'upper wing), and the vicinity E = Ep. There are no methods to derive analytically an exact function G(~p, E). We consider first extremely small densities Ne-~0 (the radiative case) and then the case Ne-->Go (the collisional or diffusion case). For these limiting cases, we shall derive a general interpolation formula. (a) Small densities. For the lower wing when Ne->0 the collisional transitions can be neglected and the solution of the problem is 1° I r'p
N(e)
F(e')
de'(e_e,)
A-~ J ' p ~" ~ +
2Fe ( £ Z _ ~ ) '
(26)
a,,(e - e,,)In z
(e - e ' ) i n 2 \
Eoy
I
\
Eoy
I
where Fp is the pumped flux to the level p, 6p = 2Ep~/:. A comparison of Eq. (26) with the solutions of the discrete system of equations is shown for np = 17.35 in Fig. 3.
°
13
t
q =
/
//
// //
o (% z'1
_ _
/ /
/VE = 1O
/
I
i
i: i I I ~-~
,,-'~/E = 10"2/
_ o 6or
-
-
,
r,~= 17; h E = IO' /VE = tO ~
,-'
fT~ = 3 5 1 N E
~ v
. . . . .
= 10";
i?~
-- -
Fig. 3. The Green function in the limitingcase for small densities. (NE = NJI0J6t . - represents the radiative Green function, "A" is the value for E = ~p, x is the calculation for np = 17: 0- holds for n = 35, the dotted I ne shows the so ut on of the k net c equat ons for E, = I).8x 10 ', i.e. for VE =- 10 ~').
An analyticaldescriptionof populationsof highlyexcitedlevels
449
The population of the pumped level Np is determined by the total probability of decay A~ of the level p, i.e.,
Np = FplAp, where Ap = Aoep3121n(~o/2ep m) in the Kramers approximation and Ao = Aoep3[21n(eo/1.3~p3/z) according to a more exact semiempirical Seaton formula, u For the upper wing ~ < ~p, the situation is more complicated. The solution of Eq. (19) in this region is proportional to exp [(de~) v4- (~p/e~)TM] near e,. This difference is at variance with the solutions for the discrete system (see Fig. 3). In this region, the approximation (19) is not applicable since direct transitions from ep to e are more effective than diffusion from ~p by small jumps. This situation described by a coronal approximation with N(~)= N p W ( ~ ) / A ( e ) . Comparison of the coronal relation with the solutions of the discrete system (see Fig. 3) confirms this assumption. (b) Large densities. We assume Ne to be sufficiently large at ~ >>e,. We first consider the upper wing e ~
(27)
K(e.) in Eq. (27) depends on ep. This dependence arises because the function N(e) is nonanalytic at the point ep and the summation in Eqs. (14, 17) is, therefore, limited by ep. If the functions in K(t.) are assumed to be constant, Eq. (27) becomes
K0= 2.18 x 10-8
+ 1.o6
Z3X/(8)
,
(28)
Ao= al In (yet,]2ep3/2)+ 0.64a2 + 0.2a3, where logarithmic functions are truncates at ~ = ep. The value of ;to determines a flux distant from the pumped level [ ~ ~p, k~,>> 1 in Eq. (15)]. In the immediate vicinity of the pumped level, A(e) = ;t(~p) is given by Eq. (16) (k" = 0). Since ;t(e)-o ;t0 when ~ 0 , the second integral decreases at small ~ faster than the first integral, and the solution N is given by the well-known 5/2-law: F_._F_z -/2
E5/2
The second integral cannot be calculated analytically: below we derive an approximation for it. We consider first the full integral over the whole region ~ < ~p. The function A(~) is actually a function of km- (~p - e)/Sp and, hence, A(~) differs from ;to in the vicinity of ~ ~ ~p - qSp. Thus, the second integral is proportional to (~,)3j2 ; t % ) - ;to [1 + A(~p)](1 + Aoj d~'--- const+ ~p3, q - 1 (const).
f [ ~-qsp
Consequently, we write Eq. (28) may be approximated by formula N(~)
2 Fp ,512 [1 + cx/(~p) ] = 5 N, Ko(1+ ;t0) 1 + c'(~p - e)~p-m].
(29)
Equation (29) confirms the fact that the correction to the 5/2-law decreases with the increasing (~p- E)/Sp. Comparison with the solutions for the discrete system shows that satisfactory
450
I. L. BE/GMANand I. M. GAIS1NSKY
n:= 17
2
n
o ~
:35
o
--
o
b~
~r
Fig. 4. Comparison of the upper wing of the Green function with the 5/2-law (solid line refers to the 5/2-law). agreement is achieved at c = 12; c ' = 0.5 (see Fig. 4). Since, in the limiting case of high densities, the upper wing is described by a diffusion approximation, the Green function is continuous and the value at • = ~ is obtained from Eq. (29), viz.' N ( e ) = Fp] NeLp,
Lp =
2"18×10-8 (~ eP'/2 '~ Z3~/(O ) × ( 0 . 7 5 ¢ + 1.06)(I +A) 1 + c~/(ep)]'
(30)
where Lp is the effective probability of a diffusion transition from the level p. For E,/> % (the lower wing), a free solution of Eq. (24) is inappropriate. In fact, Eqs. (13-17) for the flux because of a discontinuity of the first derivative at • = e , neglect transitions between the regions e < ep and E > ep. These transitions are especially important in the vicinity of the point %. Near ep in the region e,. ~> • > Ep, we may neglect radiative transitions and write the kinetic equation as 2~ 3/2 ~ + F(~) = 0,
F(e) =- F(e,)= ~ (Wp+k. ,-IV, - W,.p+kNp+k).
(31)
k>O
where the subscript p refers to the pumped level. We repeat for F(e) transformations used to derive an analytical expression for the flux and assume iV, ~ N t, + (p - n ) d N /dep × 6, Np+, ~Np - kdN+/dep × 6 (the indices - and + refer to the regions e > Ep and e < %, respectively). We assume all functions near ~p (except for those which depend on the difference e - ep) to be constant and note that dN-/dep .~ dN+/dep. Then integration of Eq. (31) gives const N(e) = const + 211 +,~(ep)]
dN* dEp
x - -
x
" 6p-+ 3p)' ( E - ep
(32)
The unknown constant in Eq. (32) is determined by the condition that N(E) transforms to a free solution for e - ~p >> 8p. The expression for the lower wing of the Green function for high densities is
(1 + cV(~p)/2)
(33)
Comparison with solutions for the discrete system (see Fig. 5) allows us to find the constant c =8.5.
An analytical description of populations of highly excited levels
~
ID-
~
=
~
~
2
o
OsF
451
o
8=0.1
np=17
S
o ;,
0
o ;2
IO
'~
05
-
I
np:17
o
o
bl
ob~
t
E
Fig. 5. The normalized Green function G(~p, ~) in the limiting case for high density (1 is the calculated from the approximation formula, 2 is the free solution, 0 is the solution for the discrete system).
Interpolation formula. We consider first the case N, ~ 0. For small densities, the population Np is determined only by the radiative lifetime. As the density increases, the decay due to electron collisions is added to the radiative decay. The population is determined as before by the lifetime of the level Np = FJ(Ap + N,.Np), where Kp is given by Eq. (17). For large
I
~, : n p = 1 7
08 07 06
o : np = 3 5
\..\ "'\'~
05 04
• : np = I 0 0
\\ "o,\
03
•
\ \
=,oo
02
0.1
•
008 006 005 004 003 002
I0 ,c,
i0 "9
~
i
10 .8
107 ns
~\~'\
106
N e "1
[ Z 7 i0,o =
Fig. 6. Comparison of the approximation formula with the solution of the discrete system for the region -~p. When the Green function is used (solid, dashed and dotted lines represent calculations by the approximation formula).
452
I.L. BEIGMANand I. M. GAISINSKY
densities, the population of the level p is determined by diffusion processes [see Eqs. (29) and (30)]. Therefore, G(%,
Ep) Go(~p)
1
(34)
Ap + N~Kp(Ap + clN~Lp)/(Ap + ctNeKp)"
Computed results are satisfactorily described at c~ = 0.25 (see Fig. 6). The general interpolation formula, including limiting cases for small and large densities [see the solutions of Eqs. (26, 29-30, 30--34)] is G(%, e)= Go(%)O(%, e);
12V'(e) ( ~ ) '/21+ A(E)%)+(I - Y)3 1 +(% ~ E)/Se, e < ~ : l+12V(e)
V3 W(,~---
¢
(351
1, e=~;
d(E., E)=
1+8.5
+
% E-~+
•~-t(x)
ap + + I ( E -
+0.4~ In -a~+6p
,
e.,
&)
e>ep;
(361
here
A.
Y = Ap + N~Kp ' x = \ zL ] 1 + x + 1.5x 2 + 6x 3
/~(X) = 1 + 1.28X1°/7+ { II(e-~p,A~)=~c
5X 39/14 exp
I in[y~o/(e_G) ]
Ac
- I+5",/(E,,IEc) ,
1 + 3X/(G/ec) (- x),/~c = 1 + 6(eJec) 3/2 Ec' l
},
ln[y~o/(e_~p_Ac(X3/l+x3))]
=2,p3j2. 6p
The general Green function is depicted in Fig. 7, which shows a steep radiative function with a first-order discontinuity at the pumped level. As the electron density increases, this function transforms to a smooth diffusion function with a discontinuous derivative.
,~ -,o
,,..c~ ~
Fig. 7. Contourmap of the Green function.
n~.
=17
An analytical description of populations of highly excited levels
453
50
I
2.0 o
,0
~o.~1
,0
~j"
20
o// I
30
I
,So
40
L
60
_,
?0
_,_
80
_L
90
L_
,00
"
n
Fig. 8. The relative populations N~/N4 of atomic levels in a hydrogen plasma at N, = liP cm -3, T, = 8000°K [1 is calculated by the method presented in this paper, 2 is the computer calculation of Brocklehyrst t'2 and Salem and Brocklehurst, 5 is calculated from (5*); discrete points are observational results for relative intensities of optical lines in the planetary nebula NGC-7009].
CONCLUSIONS
Here we discuss the application of the derived formulae. The analytical expression for the general solution consists of two parts: the expression for a free solution and the expression for the Green function. Formulae for the free solution are based on Eq. (18) derived in the limit n ~ o ; in the expression for the coilisional operator Lc, we assumed NIO < dNlde. Collisional transitions are mostly important in the region e < ec. Since in this region dNlde ~ N/e, our approximation is justified if ec < O. This condition imposes the following limit on the electron density: Ne < 1016 × Z 7 × 0 4. Comparison with the results of a direct solution of the discrete system shows that, if n > 5, the error of Eq. (24) is within 20-30%. The case of the Green function is more complicated. The approximations used are listed in Table 1. Table 1. Approximation used to solve problem involving a Green function e
Corona
~p ~ ~c
Interpolation formulae
e=ep
e>ep
Kinetic Eq. (18) in the limit N,-*0
Kinetic Eq. (19) with expressions (13-16) for the flux limited on one side.
Interpolation formula based on a qualitative solution of Eq (33) for Ec "> ~ > ep and a free solution for
- Ep/~p,> 1
In this case, no new limitations arise and the density condition is sufficient. At the same time, comparison with the results of a direct solution of the discrete system shows that in the region np > 10, n > 5, the error is within 20-30%. Acknowledgements--The authors are grateful to L. A. Vainstein and L. P. Presnyakov for discussions. REFERENCES M. Brocklehurst, Mon. Not. R. Astr. Soc. 148, 417 (1970). M. Brocklehurst, Mon. Not. R. Astr. Soc. 153, 471 (1971). M. Brocklehurst and M. J. Seaton, Mort. Not. R. Astr. Soc. 157, 179 (1972). M. Brocklehurst, Mon. Not. R. Astr. Soc. 157, 211 (1972). M. Salem and M. Brocklehurst, Astrop& J. Suppl. Set., 39, 633 (1979). S. T. Belyaev and G. I. Budker, Plasma Physics and the Problem o/ Controllable Thermonuclear Reactions. 3, 41 (1956). 7. L. V. Gurevich and L. P. Pitaevsky, JETP 46, 1281 (1964). 8. V. A. Abramov and B. M. Smirnov, Opt. i Spectr. 21, 19 (1966).
1. 2. 3. 4. 5. 6.
I. L. BEIGMANand I. M. GAISINSKY
454 9. 10. 11. 12. 13. 14. 15. 16. 17.
L. M. Biberman, V. S. Vorobyov, and I. T. Yakubo~,, UFN 107, 353 (1972). 1. L. Beigman and E. D. Mikhalchi, JQSRT 9, 1365 (1969). I. L. Beigman, Preprint No. 167, Lebedev Phys. Inst., Moscow (1975). L. A. Vainstein, I. I. Sobelman, and E. A Yukov, Excitation of Atoms and Broadening of Spectral Lines. Springer Verlag, Heidelberg (1981). I. L. Beigman, A. M. Urnov, and V. P. Shevelko, JETP 58, 1825 (1970). I. C. Perciva'l and D. Richards, Adv. At. Molec. Phys. 11, 2 (1975). I. L. Beigman, JETP 73, 1729 (1977). I. L. Beigman, ICAP-VI, 302, Riga (1978). J. B. Kaler, Astrophys. J. 143, 722 (1966).
APPENDIX Approximation Formulae for the Photorecombination Mechanism of Atomic Level Population From the practical point of view, the case of a purely photorecombination excitation to highly excited levels of great interest. Then, with the Kramers approximation for photorecombination, Eq. (9) may be rewritten as 2•312dd~ ~ - Ao•S/2{N(O)ln• - N(•)ln•o
+
fo'[N,¢•,)In
I•-,pl]d•p}= f(•),
,l*)
j = - N~r• 3/2d_N_N;[(~) = AoNo•3/2 exp (•/O)lEi(- •/0)i. At low density, when collisional transitions are neglected, the solution of the problem is determined by the radiative Green function
- f" exp(•t'JO)jEi(-•P]O)l " Nu(•) = Nob~n(•); NR(•) - Jo (e - •p)In 2 [(• - •p)/EoY]oep
(2*)
In the two limiting cases •,~ 0 and • ~ 0, asymptotic estimates of the solution follow from Eq. (2'), i.e.
I In(coy2/0)
e < O;
In (•oy/•) ' MR(El ~
(3'~ In (•y/0) E~>0. (e/O) In2 (~oy/E)'
It is seen that the approximation formula MR(e) = In (1 + 0/y,)[ln (~o34•) + y In (I + •7/0)] × In2 (dO) + 1.25 In2 (•o"fl•) In2 (dO) + 1
14'~
describes both limiting cases. The accuracy of the approximation for intermediate • is illustrated in Table 2. Table 2. Data showing the accuracy of the approximation used
O=O,1
0.5.10 3 0.2.10 3 0.8.10 -2 0.032 0.128
0=0.02
Approximation
Computer solution
Approximation
Computer solution
0.59 0.51 0.41 0.31 0.27
0.61 0.54 0.45 0.35 0.27
0.39 0.30 0.20 0.13 0.08
0.42 0.32 0.23 0.14 0.09
For E ,> ec determined by Eq. (20) collisional transitions are insignificant; hence, Eq. (4*) is applicable. For • ~ •,., collisional transitions dominate and photorecombination, as well as other radiative transitions, may be neglected. In this region, the local approximation (23) is valid. Therefore, the following approximation formula is suggested: N(•) = MRtE÷ ~(~/~,) × •c) + [1 - MR(a(0)•c)J x M,(d~,),
(5*)
a(x) = 3/2[2 exp (- 1.4x13) - I], which is asymptotically correct in the two limiting cases • ,> •c and •-> e, and describes intermediate cases within about 5-10%. As an illustration we compare in Fig. 8 the calculations for atomic level populations in a hydrogen plasma with calculatons of Brocklehurst 1'2 and Salem and Brocldehurst. 5 The plasma parameters Ne = llP cm -3, T, = 8000°K are characteristic of planetary nebulae. Figure 8 also shows observational results for relative intensities of optical lines in the nebula NGC-7009 (Kaler~7).