A PERTURBATION THEORY FOR THE POPULATION OF ATOMIC ENERGY LEVELS IN NON-LTE PLASMAS

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A PERTURBATION THEORY FOR THE

POPULATION OF ATOMIC ENERGY LEVELS IN NON-LTE PLASMAS

M. Klapisch

To cite this version:

M. Klapisch. A PERTURBATION THEORY FOR THE POPULATION OF ATOMIC ENERGY

LEVELS IN NON-LTE PLASMAS. Journal de Physique Colloques, 1988, 49 (C1), pp.C1-343-C1-

347. �10.1051/jphyscol:1988174�. �jpa-00227588�

J O U R N A L D E PHYSIQUE

Colloque C1, Supplbment au n 0 3 , Tome 49, Mars 1988

A PERTURBATION THEORY FOR THE POPULATION OF ATOMIC ENERGY LEVELS IN NON-LTE PLASMAS

M. KLAPXSCH

Racah Institute of Physics, Hebrew University.

IL-91904 Jerusalem, Israel

RCsumC: On dkcrit un developpement formel du ModCle Collisionnel Radiatif en puissance d'un petit paramCtre. Les terrnes du developpement sont des produits de matrices. Les inverses de matrices sont interpret& comrne effet des cascades. On montre que c e schtma permet la separation des diffkrentes contributions, donnant ainsi une classification naturelle des processus atomiques dans les plasmas. En employant des rCgles de somme, on montre que dans des cas simples, les populations des niveaux excitCs sont l i k s aux nombres quantiques de maniCre transparente.

Abstract: A formal expansion of the CRM in powers of a small parameter is presented. The terms of the expansion are products of matrices. Inverses are interpreted as effects of cascades.It will be shown that this allows for the separation of the different contributions to the populations, thus providing a natural classification scheme for processes involving atoms in plasmas. Sum mles can be formulated, ailowing the population of the levels, in some simple cases, to be related in a transparent way to the quantum numbers.

1.Background

The well known Collisional -Radiative model (CRM) [1,2,3] has been extensively applied, mainly to simple atoms and ions-e.g.H or He-like -for which the conceDt of "collisional-radiative ratesWproved to be meaningful. For Gghly ionized heavy ions of interest &laser produced plasmas, or for X ray laser schemes, the amount of atomic data necessary is so large that using CRM has become synonymous with numerically solving the rate equations [4]. For an optically thin ,time independent, plasma, these can be written:

M is the number of all possible bound states of all ionization stages of the atom , ordered in increasine enemies .The N's are the ~ o ~ u l a t i o n densities of the excited states. A;& are the soontuneow tran~ition"~roba%iliries, including auioiGnization, and Q.. are the collisionally iZced rate coefficients, for excitations. ionizations - s i m ~ l e and multiole-

.

afirect recombination (radiative and resonant capture), and 3-body recombination. Note that'wheh all the states are inclided in a complete and detailed model, only one-step processes should be explicited, so dielectronic recombination @R) is not mentionned at this point. That part of DR which can be considered as a resonant radiative recombination[8] willbe discussed later

.

For complex ions, M may be of the order of several thousands or more, and an enormous number of rate coefficients for collisions have to be calculated. For this reason, the set of eqs.(l) is usually truncated, yielding an "atomic model", i.e. a specific choice of states for some ionization stages. Moreover, for the same reason, approximations for the cross sections are essential. However, checking the effect of neglecting so many states , and of making these approximations is practically impossible sir.;e this would usually mean unacceptable computing time. Consequently, different codes and models would yield widely differing results, especially for the ionization balance[5].

The present work is an attempt to address these questions before performing the computations of the so-called "atomic data base". The aim is not necessarily to replace the full computation procedure, but at least to provide a guide for the analysis and the choice of proper approximations.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988174

Cl-344 JOURNAL DE PHYSIQUE

II.The Matrix Ex~ansion.

1.Rate Eauations in ~ a t r i x Form.

Let

A

be the

M*M

mamx of spontaneous decay, Q that of induced transitions, and

N

the vector of unknowns. According to eq.(l), the elements of

A

and ($ are

k<i k t i

A

is a triangular matrix.Let us define A =max iil ,Q = max I iil, where i refers to the states of the ionization stage on which interest is focused. Its ground state is used for the normalization of the populations, i.e. No =I,and we obtain:

(A+ qQ) N= [-qQdl

(2)

where A= / A ,

Q=

^{/ Q} are now (M-I)*(M-I) dimensionless matrices,

N

=

N/NO,

^q = ne Q / A

,

^and

Q0

is a vector describing excitations from the reference ground state. Using the Van Regernorter formula [6], one obtains: q = ne10-161&,v1R.

2.Ex~ansion for Small q

Now, if q <<I, one could formally invert the 1.h.s of eq.(2), and expand in powers of q:

N=(I

-T~A-'Q

+

q 2 ~ - ' ~ ~ - 1 ~

+...I

(-~A-'Q) (3)

However, eq.(3) , is incorrect,

A.

being singular, due to the existence of metastable and ground states in the various ionization stages. In order to overcome this difficulty, we split the set of the M possible states in two subsets with the aid of projection operators: PC selecting the ground and metastable states, and Px the ordinary excited states. Different expansions are then obtained:

2 - 1

N,

=

- q ~ k ' & , ~ , +q

A=

ax A&~&,N, - ...

⁽⁴⁾

with obvious notations

QXG =pXQ

PG etc

...,

and

Eqs. (4) and (5) show that instead of the usual "decoupliig'; between ionization balance and excited states populatior~ justified by time scales, a formal decoupling is obtained between ground (metastable) and excited states , yielding a ratio of q between the two sets, in the lowest order.

However, eq.(5) shows that , even in the limit q+ 0, the excited states cannot be ignored in the ground state balance, since they intervene , at least in the mamx elements. Also, note that metastable states have the same status as ground states, and are obtained from eq.(5). Actually, the distinction between PG and PX is in a sense flexible and PG can refer to all the levels for which the radiative lifetime is much smaller Lhan the collisionnal lifetime.

3.Expansion f o m

n(

near LTEl Let us divide eq. (2) by q (now>> 1).

Now,it is well known that, when radiative processes are negligible, the solution of eq.(2) is the Saha-BoItzmann population of

- -

LTE.

Q N

LTE = - Q (7)

Here autoionization is included in

Q

to satisfy the micro-detailed balance. One obtains :

A + -Q-'AQ-'A- 1 ....p

⁽⁸⁾

q 2

Remark that

Q

is not singular, since there are no states which cannot be depopulated by collisions, and there is no need to use the projection operators. This corresponds to the fact that, near LTE, ground states are not essentially different from other states[3].However, it is easy to use the same projection operators as above,in order to make the connection between the two expansions.

1II.Inverse Matrices and Cascades.

1.Ex~ansion of Inverse Matrices.

Let M represent a square non-singular matrix,i.e.either

AXX

or

Q.

It is always possible to write:

M= D+(M- D)

(9)

where

D

is an arbitrary non- singular diagonal matrix.Then,

ml= 1 I + D- '(M- D)I-'D-'

(10)

= {

I + ~ 1 - l D - l

the last equation defining the

T

matrix. Let ki be the eigenvalues of

T

and p = maxl %I its spectral radius. Then, if p<l, the following expansion

M-'=

{

I - T + ++ ...I D-

¹ (1 1)

converges with a rate of linear convergence p.

D .

being arbitrary, can be choosen to satisfy this requirement.

The fact that the inverse matrix involves successive powers of

M

prompts us to interpret the appearance of inverses in eq. (4),(5) and(8) as the effect of cascades.

2.Converrrence of the Cascade Series.

Two possible choices of

D

will be considered:

(i) D

=DI ,

D constant. It will appear below that this simplifies greatly the evaluation of the erms 'n the expansion (4),(5), etc

....

Minimization of p is achieved for D= l/Zmaxlhi I, with Pmin =\ 1 - 2

1.

where r is the ratio of the smallest eigenvalue to the largest. For the A mamx, r is very small, because the radiative lifetimes in one spectrum can differ by many orders of magnitude, so the convergence of expansion (1 1) with this choice of

D

would be poor. However for the

Q

mamx, it can be checked numerically that the eigenvalues, related to the total inelastic cross sections, do not vary much from one level to another, and this choice of D is well adapted.

(ii) It can be shown that if

T

is triangular with zero diagonal, that is, for the A mamx with the choice Dii=Aii , then it is nilpotent. Convergence is achieved in a finite number of steps.The highest non vanishing power is the maximum number of spontaneous cascades in the "Atomic Model". Note that in the

T

mamx, the transition probabilities are now normalized by the branching ratios 11 Aii

.

1V.Evaluation of Matrix elements.

1.Interwretation of the matrix exwansions,

Eqs.(4), (5),(8) and (1 1) provide for a systematic classification of atomic processes in plasmas, according to the number of "steps", or "elementary events" involved in the processes. There are clearly two kinds of events:

(i) some involve the perturbing particles, -e.g.electron collisions for low n, , escaping photons for high ne

-

and their magnitudes are proportionnal to that power of the small parameter (q or ,,l/q resp.), equal to the number of steps.

(ii) some involve the kind of particles connected to the zero-th order population matrix (e.g.photons for low n, ) and the convergence is probably slower (cascades).

The composite events appear as products of matrices. Dielectronic recombination appears naturally in eq.(4) in the lowest order of q

,

in the product A - I Q , alongside with excitation-autoionization, etc..Rad'ative cascades effects on dielectronic recombination [81 ,are

1

included naturally. An example of q dependance is the effect of collision transfer from the upper level of a dielectronic recombination satellite, by ionization or deexcitation

.

2.Atomic wavefunctions perturbation expansion.

Up to now, no approximation was involved in the mamx elements themselves.The preceding oaramaoh cave us alreadv a clue of which coefficients are im~ortant. _{a r}

-

Now, in order to check the kffects of approximations: it suffices to refer to the usual, atoiic Perturbation Theory.[9,10] .This may require a formulation of operators for collisions , where only bound states are explicit. Such a formulation has recently been proposed, for instance, for collisional excitations[7]

.

If one starts from the usual zero-order wavefunctions , then configuration mixing will yield additionnal virtual processes that will add up to the real processes, with possibly some interference terms, that will also be additive. This is a consequence of, and one of the reasons for, the linearization of eq.(4,5,etc..). The effect of resonnances in excitation , for instance, could be studied in this way.

3.Use of Sum rules and effective operators.

Another important desired consequence of the linearization of the rate equations is that it enables to define effective operators for composite events, whose mamx elements precisely define and generalize the concept of "collisional radiative ratesM[3] for complex ions. As an example, consider excitation from the ground state (Jo of a two electron ion, to an intermediate state (J M), followed

evaluate the sum

"p'

by a radiative decay to a final state ( ,MI). If we suppose the choice(i) for D,(see~~I.Z)we have to

which is averaged over the the initial states, in the usual way for transition probabilities.

The radiative transition

and likewise, the collisional excitation rate[7]:

C 1-346 JOURNAL DE PHYSIQUE

~ ( 1 ) and u w are the dipole and the unitary multiple operator respectively. In equations (12) and (13).

there is a factor which is dependant on the energy of the intermediate state, and on the radial wavefunction of the intermediate configuration. As mentionned above, we deal at this point with atomic wavefunctions corresponding to first order in energy. In order to perform the sum let us do the following steps:

-a)Resmct temporarily the sum to the states of a given configuration.

-b)Expand inTaylor series the energy dependant factor around the average E,, of this configuration X(E~)"C&E;, ~ ~ ( n l j , nYl)Qk(no ldo,n fj,AEJJJ

Where

Cg

is a numerical constant whose value depend on the system of units.

-c)The first term in the sum S=S(Eav)X( E,, )+Sl(Ej)

+..

is now purely angular, and we have:

Now the last sum can be shown to be zero through the use of Wigner-Eckhart theorem and orthogonality of 3j coefficients. Using the closure relation in the middle sum (15). and recoupling the operators, we obtain a series of effective multipoles:

s ( E ~ ~ ) = ~ ( J ~ M ~

I

T:

I

J ~ M ~ ) ~ x ~ ( E ~ ~ ) (16) 4

where XX is now a "radial factor" depending on through a recoupling coefficient involving orbital quantum numbers, and on the radial wavefunctions only. A similar result can be obtained by graphical methods[lO] for more complex systems. A physical analogy of this result would be interference of two incoherent radiations. Now , we can lift the restriction (a) because all intermediate configurations differing only by the principal quantum numbers n will have exactly the same value of the recoupling coefficient

.

^so that the radial integral

XX

can be replaced by a sum of similar integrals over all values of n. The intermediate configurations differing by orbital quantum numbers will contribute in a similar manner, but with another recoupling coefficient.

-d) The contribution to S of the second term in the Taylor expansion (14) can be made to vanish with a proper choice of E,,. The quadratic term can be taken into account by generalizing eq.(15) to two-body operators, the result involving now one-,two-,three..- body operators, and the radial integrals involve combinations of Slater integrals of the intermediate configuration, describing the effect of the width of the intermediate configuration being non-zero. We suppose for the time being that the third power term is negligible.

V. Discussion.

1.Conseauences of the model.

Since no numerical results have been obtained yet, it is difficult to make a statement about the convergence of the whole expansion. However, it is clear, that there are going to be many terms, even taking into account the sums over intermediate states. So, it is our feeling that this model will be used more like a conceptual framework, than like an actual computational method. In this sense, it will allow to check the quality of an "Atomic Model".

2.Extension to more eeneral cases,

The mamx expansion,(eq.2) can be easily generalized to the case where the plasma includes external radiation. For this, it suffices to add a mamx

B

describing the effect of radiation absorption.

The small parameter would be the number of photons in the external field, normalized to that of a black body. The matrix expansion would then include additional terms connected to the radiation field, and crossed collisional-absorption terms, etc ....Ln a similar fashion, it is conceivable to add to eq.(2) any number of external particle collisions(e.g. protons, heavy ions,...). For the optical thickness, which is not a linear process, an escape factor model could be applied. Work is in progress on these problems. Another problem that is under consideration is the inclusion of branching ratios in the sum rules,i.e. choice (ii) in 111.2, which introduces in the sum S of IV.3 a division by an operator dependant on J. This would improve considerably the convergence, but sum rules for divisions are not known.

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m , 1 5 5 ( 1 9 6 2 ) .

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Stewart,Phys. Rev. Lett.% 2300, (1987)

5. S.R. Stone and J. C.Weisheit, J.Quant. Spectrosc. Radiat Transfer,X,67,(1986).

6. H. Van Regemorter, Astrophys. J. m . 9 0 6 (1962).

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