UCRL-JC-125S42PREPRINT

Systematic Investigation of NLTE Phenomenathe Limit of SmalI Departures from LTE

S. B. LibbyF. R. Graziani

R. M. MoreT. Kato

This paper was prepared for submittal to the

in

13th Intem-at~onalConference on Laser Interactions andRelated Plasma Phenomena

Monterey, CAApril 13-18,1997

July 1,1997

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Systematic Investigation of NLTE Phenomena inthe Limit of Small Departures from LTE

S. B. Libby, F. R. Graziani, R. M. More

Lawrence Livermore National LaboratoryUniversity of California

Livermore, California 94550USA

T. Kato

National Institute for Fusion ScienceNagoya 464-01

Japan

Abstract. In this paper, we begin a systematic study of Non-Local ThermalEquilibrium (NLTE) phenomena in near equilibrium (LTE) high energy density,highly radiative plasmas. It is shown that the principle of minimum entropyproduction rate characterizes NLTE steady states for average atom rateequations in the case of small departures form LTE. With the aid of a novelhohlraum-reaction box thought experiment, we use the principles of minimumentropy production and detailed balance to derive Onsager reciprocity relationsfor the NLTE responses of a near equilibrium sample to non-Planckianperturbations in different frequency groups. This result is a significantsymmetry constraint on the linear comections to Kirchoff’s law. We envisageapplying our strategy to a number of test problems which include: the NLTEcorrections to the ionization state of an ion located near the edge of an otherwiseLTE medium; the effect of a monochromatic radiation field perturbation on anLTE medium; the deviation of Rydberg state populations from LTE inrecombining or ionizing plasmas; multi-electron temperature models such asthat of Busquet; and finally, the effect of NLTE populations shifts on opacitymodels.

This paper is aimed at the question: is there a simple way to ---extract the consequences of Non-Local Thermal Equilibrium, (NLTE), atleast for plasmas which are near Local Thermal Equilibrium, (LTE)? Theneed and ca ability to perform detailed NLTE calculations for ICF’,

?laser-plasma , and astrophysical problems3 have existed for many years.However, the computational effort required to solve the full NLTE rateequations (collisional-radiative modeling), for realistic problems involvinghydrodynamics, radiation, and charged particle transport has beenprohibitively expensive. The strategy of this effort is to add NLTE effectsin a perturbatively complete fashion to the LTE calculations at a fractionof the computational cost of detailed accounting methods. Among theimmediate consequences of our perturbative treatment are symmetryrelations (Onsager relations)4 that must be obeyed by any NLTE treatment

taken to the linear response limit. These relations are based on the conceptof minimum entropy production rate near equilibrium, and thesimultaneous requirement of detailed balance for all the underlyingkinetics. These relations thus provide a significant computationalconsistency check for any NLTE model in the near equilibrium limit.

There is precedence in the literature6’7 for the treatment of nearequilibrium plasma processes via a Kubo-like linear response analysis.Since the high Rydberg states of an otherwise NLTE ion distribution arebrought close to equilibrium by collisional coupling with the freeelectrons, one may consider the flow of electrons through the Rydbergstates in a recombining or ionizing plasma to be a linear response problem.Indeed, Pitaevskii6, and subsequently, Gurevich, Beigman, Syrkin, andothers6’7presented a Fokker-Planck treatment of this flow in principalquantum number space, where the conductivity was a suitable averageover otherwise complex atomic rate data.

In our present work, we focus on problems where radiation-mattercoupling is significant. At this point, it is appropriate to point out thedifference between two distinct manifestations of NLTE physics. Thefirst, and most obvious form, occurs when significant phenomena occur ona time scale shorter than the necessary transient relaxation times for LTEequilibration. The second form occurs when a sample system is in aquasi-steady state NLTE population distribution that is maintained by twoor more temperature reservoirs (for either photons and/or electrons) thatare at different temperatures. The main thrust of this paper is analysis ofthe latter type of case, though our analysis yields insights into the former.It should be emphasized that the steady state case has direct application toa variety of laser-plasma situations as the following examples show (fastion transition rates -picoseconds, in conjunction with a slow x-ray drivetime scale -nanoseconds).

In figure 1, we show three examples of experimental arrangementswhere we want to compute the NLTE response to a perturbation on anotherwise LTE medium. In the first example, consider an ion buried in anLTE medium with electron and radiation temperature set to the reservoirvalues of T~=TWWe then consider the response of the ion to an externallyimposed weak radiation beam (“X-Ray Laser or XRL”) at a givenfrequency v. In the second example, consider an ion near the edge of anotherwise LTE medium. The ion sees a Planckian radiation field BV(T)minus the intensity ~ of lost photons. This arrangement is applicable tothe NLTE corrections to shocks and Marshak waves. Finally, we considerthe NLTE variation in the behaviors of ions in a tamped quasi-LTE sampleinside of a hohlraum whose driving output is some B,(T)+(gold M band). -- –

Each of these three cases requires that we compute the response ofthe ions to both its thermal reservoirs and to a possibly multigroupradiation perturbation. If we could compute the response of our test ion infrequency group v to an effective temperature perturbation ~v, in an

arbitrary group v’, we could compute the response for such cases as an

integral over v’. Thus, if the LTE/NLTE populations are ~“and ~

respectively, we seek to tabulate the linear response matrix Mjv in:

$ -c” = ~Mjv~v. It is important to note that the matrix Mjv can be

tabulated o;ce and for all for any given LTE state. Mjv can be computed

using a Kubo type of linear response theory with a model Hamiltonian, orwith more conventional rate equations. Here, to illustrate the ideas, wediscuss the physics in the context of simple two and three level atoms.

Ion plus photon besm

EEEEEl

Ion near boundaryof medium

EzElIon in hohlraum

● Opacity experiment

FIGURE 1. Three classes of experiments that maybe treated by the methods ofthis paper. In the third case, a tamped, constant density, plasma sample isdriven by a gold hohlraum Planckian with a small M band perturbation. Theeventual object is to compute the deviation of the sample opacity in eachfrequency group from its LTE value. Such deviations will be averages over theM band perturbations convolved with the atomic data.

In analyzing the quasi-steady state response due to theperturbations of conflicting reservoir temperatures, it is useful to considerthe rate of entropy production in the entire system.4,5 In the case of linearresponse, the local ion NLTE population distribution is equivalentlyunderstood in terms of the entropy back reaction in the reservoirs.sSpecifically, for linear response, the total rate of entropy change for theentire system is minimized by NLTE populations that satisfy the steady ---state NLTE rate equations.s. 8

These ideas are illustrated by the behavior of a two level ioninteracting in quasi-steady state with a free electron reservoir oftemperature T. and a photon reservoir of temperature TR#Te. Denoting

the upper and lower state populations as P2 and P, (P2+P1=1,E2- E,=hv),equation (1) gives the rate of the lower state change in terms of thecollisional excitation rate a to go from level 1 to 2 and the corresponding

photon absorption rate bn, (where ~ is the number of photons per modecoming from the photon reservoir). The reverse rates are given bydetailed balance and the Einstein relations.

d~E

E=–a~+ae ‘P2– bnv~ + b(nv + 1)P2x

In steady state we get:

E E— —

‘lC + bnvekTRFy’l’e= aeE

E

(1)

(2). .

a(l+e~)+bn, (l+ez)

Near equilibrium, we put T~=T,v+6T and T~ = T,v-i5T to get the linear

response limit:a - bnv m

Pnh’l = P’tel– P“’1(1– P~’l);: + 0((7)2)a + bnv av (w (JV (3)

The same result can be obtained by minimizing the rate of entropygeneration for the entire system, including the reservoirs:

c

s=–kPIIn(;)+ ~(–a~ + ae=~) + $(-bn,? + b(n. + l)g)2 e R

(4)as

~=o

Note that the heat flow to and from the reservoirs is crucial to maintaining

(3sthe steady state ~ = O populations. The key point is that setting —=0

6!!

(but allowing ~ to vary) also yields equation (3) to order 5Tfl~v.However, it is not true that the variational principle and the full kinetics

solution agree at higher order -(?iT/T8v)2.The multigroup NLTE entropy flow is analyzed in the same spirit

as the two reservoir case. Consider the thought experiment shown in

figure 2. Here the sample of density p and thickness d is subjected to anelectron reservoir temperature Tc and is driven by radiation from the . . .different hohlraums. Each hohlraum is held in thermal equilibrium at a setof reservoir temperatures T~, all of which obey T~-Teee Te. The sampleviews and retransmits to each hohlraum at a single frequency groupselected by the spectrometer. Thus, each hohlraum directs an intensityinto the sample equal to the Planck function BV(TJ at the group andtemperature of the kth hohlraum. Since the sample ions are responding toconflicting group effective temperatures set by the disparate reservoirs, itis in quasi-steady state NLTE. If we imagine placing an absolutelyreflective x-ray mirror behind the sample, we find that the sample returnsthe non-Planckian intensity 1,, which we can express in terms of the

group emission and absorption opacities &end K (in the optically thin limit

of (pd)”’>>s, K):

1. – Bv = 2@(&v – Kv)Bv

The rate of entropy change is given by (5), reflecting the energyconservation between the electron bath at the sample and the energy flowin and out of the aggregate hohlraum array:

(5)

In the limit of near LTE we have: Te=TR,end TR=T~-8T~.Defining

Iv, - Bv(~) = ~MjkWkk

T1

@

.

@

T2X-ray mirror

Plasma with: T3

fixed electron

d

y,,/” // ,/

ternperat ure - Te + /’

@

@/ //’ ,“ . . ....

.%4nple .,“>.. +!’,(/ PEP,. / /

,/~ /“0‘> --<

‘.‘\ Hohlraum array

T4

Ideal spectrometw IFIGURE 2. The thermodynamics of a steady near LTE plasma is clarified by the“NLTE Reaction Box” thought experiment. Each hohlraum has a controlledtemperature and intaracts with the sample through a single frequency group. Iftha hohlraum temperatures are all the same, tha plasma is in LTE. Focusing onthe rate of change of the entropy due to perturbing a set of hohlraumtemperatures yields the correct radiativa deviations from LTE that obey Onsagersymmetry relations.

we find, upon expanding: S = ~ Mj,~wj~~ +... . Thus, we discover an

Onsager symmetry MW,= Mv,v that must hold for any pair ofgroups v and v’. Explicitly, this requires, for equal group widths andequal group temperature shifts, that the resultant Planck weighted

deviation from Kirchoff’s law in group v is equal to the appropriatelyweighted deviation we would get in the perturbing group v’ upon

switching to perturbing in group v: (S, - lcv)Bv= (E,, - Kv, )Bv,.

The simplest model illustrating the symmetry of MW, is a threelevel ion with populations P,, P2, and P~, and three radiation groups given

by hva=E2-E1,hv~=E~-Ez, and hvl~=Es- El. The kinetics are given by:& Ek3

~= –%24 + q2e TP2– %3< + a,3eF< - b12q2~

Plus similar equations for Pz and P3.

Working in the steady state, linear response limit, we compare the~

effect of a perturbation c$T~on (E. - ~~)Ba ochvaAa(3P2e T - 6.) to aEb

perturbation ~~ on (&~- JC~)B~= hv~A~(/i~e7 – 6?).Though the individual population shifts 6? ,6.., and 6E are

complicated, we find the two expressions equal and proportional to therate combination a13+ b13q3.

Turning briefly to an example of a model multigroup NLTEresponse in a more realistic situation, we present in figure (3) an exampleof a crude multigroup calculation of the response of an optically thinsample of aluminum at .001 gr/cc. and T,=50 eV. In this case, simpleestimatesl’ suggest insufficient collisionality to maintain LTE in theabsence of a Phmckian drive. The two curves represent the emission andabsorption opacities in response to a 12% effective temperature increase inthe group centered at 422 eV. It is interesting to note that this rathermodest perturbation redistributes significantly to the lower energy groups.AS our Onsager relation requires, calculations show a similarredistribution upward in energy for a perturbation centered around 150 eV.Precise Onsager symmetry, however, requires that all kinetics processes, ---including line formation, obey detailed balance.

Future work on this program will involve more realisticcomputations of the response matrix MW,(p, T) that gives the unique

NLTE, quasi-steady state deviation from LTE at a given p and T. Asstated earlier, we envisage the computation of small NLTE effects usingMW.(p, T) in an off-line tabular manner. We intend to apply these resultsto problems such as those shown in figure (l). Also, It would beinterestin to test other NLTE ionization models, such as that of

1?Busquet , in the linear response regime. Finally, since LTE opacitymodels such as the Super Transition Array model 13involve a convolution

of a Saha distribution with spectral arrays, one can build a NLTE versionbased on linear response.

6000 ~ I

5500

5000

4500

‘K,e4000

3500

cm21g3000

2500

2000

1500

1000

500

II

1

I

10-2

I10-1 100

keV

FIGURE 3. A multigroup linear r sponse calculation of 50eV Aluminum at .001gr/cc. The calculation was done ith LASNEX9 and XSNQ’O. In this case, theperturbation was in the group centered at 422eV. Curves 1 and 2 arerespectively the absorption and e ission opacities.

IAc owledgments

The authors would like t thank John Castor and George Zimmerman forstimulating conversations and help th the simulations. This work was performed underthe auspices of the U. S. Depart ent of Energy by Lawrence Livermore NationalLaboratory under Contract No. W 405-ENG-48. The contribution of T. Kato wassupported in part by the US-Japan Sc ence and Technology Cooperation Program.

References

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