Onsager symmetry of the average ion near equilibrium

Journal of Quantitative Spectroscopy &Radiative Transfer 71 (2001) 295–303

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Onsager symmetry of the average ion near equilibrium

Alain DecosterCommissariat a l’Energie Atomique, Bruyeres-le-Chatel BP 12, 91680 Bruyeres-le-Chatel, France

Abstract

The linear response of the average-ion model near local thermodynamical equilibrium (LTE) is writtenin Onsager’s symmetric form. The symmetry shows up through a “spectral signature” of one-electronlevels, which describes both the sensitivity of the level population to non-LTE radiation, and the e4ectof a non-LTE population on the spectral response of the plasma. In particular, we prove analytically thesymmetry and negativity properties of the “response matrix” de7ned by More and Kato. More generally,we expect to speed up non-LTE atomic physics computations by using these linear response equations.? 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Plasma; Atomic physics; Non-LTE; Onsager

Recent studies on the linear response of a plasma near thermodynamical equilibrium (LTE)[1–3] have dealt mainly with detailed atomic physics and discrete spectra; we consider here theaverage-ion model as it is coupled to radiative hydrodynamical codes. Two such models areXSN [4,5] and Nohel [6]. We show that the time-evolution of an average-ion coupled to theradiation 7eld near LTE can be cast into the symmetric form described by Onsager, giving newinsights on the way radiation and matter interact.

The set of equations of hydrodynamics, radiative transfer and atomic physics is in general acoupled set of equations. To simplify this paper, we do not write here the transport terms (nospace variables), and the mass density � is assumed constant. What remains is the temporalevolution of the atomic populations, of the radiation 7eld, and of the electron temperature. Asmost present atomic physics models assume, the free electrons are in local thermodynamicalequilibrium with a temperature Te, and their chemical potential �e is determined by the plasmaneutrality.

A con7guration of the average ion is de7ned by the vector P of the average populationsPi of the one-electron “levels” i, which are continuous variables 06Pi6Di. Bound levels iactually are atomic shells (with principal quantum number n) or subshells (quantum numbers n

E-mail address: [email protected] (A. Decoster).

0022-4073/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved.PII: S 0022-4073(01)00076-0

296 A. Decoster / Journal of Quantitative Spectroscopy & Radiative Transfer 71 (2001) 295–303

and ‘). The level degeneracy Di (maximum vacancy in level i) may be a function of density.The time-evolution of populations is written as

dPi

dt= Tci − Tic +

∑j

(Tji − Tij);

where T � is the transition rate from level to level � (including the continuum c). Thetime evolution of the spectral radiative energy E� (radiative intensity integrated over photondirections) is

dE�

dt= �c(j� − k�E�); (1)

where � is the mass density, j� is the spectral spontaneous emissivity, and k� is the spectralopacity corrected for stimulated emission.

Only one-electron transitions are written here, as we omit auto-ionization and dielectroniccapture. The total rates T are made up of radiative rates TR and collisional rates TC. The onlycollisions considered in practice are those between ions and free electrons. The radiative ratesTR are functions of the frequency of the absorbed, TR↑(�), or the emitted, TR↓(�), photon.These radiative rates depend on the photon number, n� ≡ c2E�=2h�3; the absorption terms areproportional to TR↑(�)n�; the emission terms are proportional to TR↓(�)(1 + n�). This yields

T ≡ TC + TR ≡ TC +∫ ∞

0[TR↓(�)(1 + n�) + TR↑(�)n�] d�:

Of course, for a given pair of levels ij, one of the two bound–bound transition radiative termsTR↓ij (�) (emission) or TR↑

ij (�) (absorption) vanishes, and the other one is peaked around the meanenergy of the transition. Further, the bound-free radiative rates have a threshold at the meanionization energy, but the precise functional forms are not necessary here. The time-evolutionof the average-ion populations is 7nally

dPi

dt= TC

ci − TCic +

∑j

(TCji − TC

ij ) +∫ ∞

0[TR↓

ci (�)(1 + n�)− TR↑ic (�)n�] d�

+∑j

∫ ∞

0[TR↓

ji (�)(1 + n�) + TR↑ji (�)n� − TR↓

ij (�)(1 + n�)− TR↑ij (�)n�] d�: (2)

In Eq. (1), the interaction terms between radiation and free electrons are the bremsstrahlungemissivity j4� and the corresponding free–free opacity k4� . The other interaction terms of radiationand matter are present in the temporal evolution equation (2) for bound electron populations.For a consistent coupling of radiation transfer with the average ion model, the same radiativerates must be included in Eq. (1). Thus we obtain

dn�

dt= �c

(c2

2h�3j4� − k4� n�

)+

nac3

8��2

∑i

[TR↓ci (�)(1 + n�)− TR↑

ic (�)n�]

+∑ij

[TR↓ij (�)(1 + n�)− TR↑

ij (�)n�]

; (3)

A. Decoster / Journal of Quantitative Spectroscopy & Radiative Transfer 71 (2001) 295–303 297

where we use the assumption of one atomic species at a density na. Identifying terms inEqs. (1) and (3) yield the expressions for the emissivity j�,

j� = j4� +na

�h�4�

∑

i

TR↓ci (�) +

∑ij

TR↓ij (�)

;

and the opacity k�,

k� = k4� +na

�c2

8��2

∑i

[TR↑ic (�)− TR↓

ci (�)] +∑ij

[TR↑ij (�)− TR↓

ij (�)]

: (4)

In an average-ion model, the con7guration energy E(P) is a function of the population vectorP. To each level is attributed an energy Ei which is the partial derivative Ei(P) ≡ @E(P)=@Pi[7,8]. The hypothesis is made that, for physical con7gurations, the energy is a concave functionof populations, i.e. its second derivatives make a positive matrix: @2E=@Pi@Pj¿ 0 ( NNM is apositive matrix if v · NNM · v¿ 0 for any vector v �= 0).

The LTE for bound electrons is characterized by two parameters, the chemical potential �and the temperature T . In an average-ion model, the bound electron populations in equilibriumare given by a Fermi–Dirac formula

PLTEi =

Di

1 + exp[(1=T )Ei(PLTE)− �](5)

with one-electron energies Ei. This LTE formula is not exact: on the contrary, this is one ofthe approximations which characterizes an average-ion model. In LTE at temperature T , theradiative energy E� is the Planck function B�:

ELTE� ≡ B�(T ) ≡ 2h�3=c2

eh�=T − 1; nLTE� ≡ 1

eh�=T − 1: (6)

If a stationary equilibrium state is reached with external sources in LTE, it cannot be otherthan thermodynamical equilibrium. The micro-reversibility relations indicate that the transitionrates are then compatible with LTE, and the same holds for each kind of transition separately,which is called detailed balance principle. The population of the initial level, and the vacancyof the 7nal level can be factorized from the rates T , so that we obtained reduced rates �,

Tci ≡ (Di − Pi)�ci; Tic ≡ Pi�ic; Tij ≡ Pi(Dj − Pj)�ij;

which is a statement that there can be no transitions starting from an empty level, nor endingin a full level. Micro-reversibility relations are imposed between reduced rates �C or �R, foreach pair of levels, and for each photon frequency:

�Cci = �Cic exp(�e − Ei

Te

); �R↓ci (�) = �R↑ic (�) exp

(�e − Ei + h�

Te

);

�Cji = �Cij exp(Ej − Ei

Te

); �R↓ji (�) = �R↑ij (�) exp

(Ej − Ei − h�

Te

):

(7)

Eqs. (2) and (3) are part of the radiative hydrodynamic model of plasmas. Together with adetermination of �e and Te (e.g., conservation of total energy and neutrality), Eqs. (2) and (3)


are a closed system describing the local evolution of radiation and bound and free electrons.The entire set is now linearized near a LTE reference state characterized by values T 0

e and �0e ofthe temperature and the chemical potential, respectively. Quantities evaluated in this referencestate are denoted by a superscript 0. Near LTE,

Pi = P0i + �Pi; n� = n0� + �n�; �e = �0e + ��e; Te = T 0

e + �Te:

We introduce new variables xi and y�, which are functions of Pi and n�, and which vanishin LTE:

xi ≡ ln(

Pi

Di − Pi

)+

Ei(P)Te

− �e; y� ≡ ln(

n�

1 + n�

)+

h�Te

: (8)

These will be the conjugate variables that are essential to show the Onsager symmetry of theaverage-ion model near LTE. When an entropy function does exist, the conjugate variables areits derivatives with respect to state variables; but Onsager symmetry appears in more generalcases. Near LTE, the conjugate variables xi; y� are small. De7nitions (8) can be linearized as

xi =∑j

!0ij�Pj − ��e + E0

i �1Te

; y� =�n�

n0�(1 + n0�)+ h��

1Te

; (9)

where

!ij ≡ Di

Pi(Di − Pi)�ij +

1Te

@2E@Pi@Pj

is a symmetric positive matrix often encountered in average ion calculations. Further, its inverseis the usual estimation of the matrix of population correlations.

The limiting form of the kinetic equations (2) and (3) near equilibrium does not requiredi4erentiating the rates when Eqs. (7) and (8) are used. Consider, for example, the radiativeionization and recombination terms:

TR↓ci (�)(1 + n�)− TR↑

ic (�)n� = TR↑ic (�)n�(e−xi−y� − 1) → −[TR↑

ic (�)n�]0(xi + y�):

With each pair of rates being treated in a similar way, the kinetic equations near equilibriumread

− ddt

�Pi = (TCic )

0 xi +∑j

(TCij )

0(xi − xj) +∫ ∞

0[TR↑

ic (�) n�]0(xi + y�) d�

+∑j

∫ ∞

0{[TR↑

ij (�)n�]0(xi − xj + y�)− [TR↑ji (�)n�]0(xj − xi + y�)} d�;

− ddt

�n� = �c[k4� n�(1 + n�)]0y�

+nac3

8��2

∑

i

[TR↑ic (�)n�]0(xi + y�) +

∑ij

[TR↑ij (�)n�]0(xi − xj + y�)

: (10)


This explicitly shows collisional ionization rates TCic and radiative absorption rates TR↑(�)n�;

but, there is, of course, an alternative form with recombination rates TCci and emission rates

TR↓(�)(1 + n�).According to the general principles of Onsager [9–11], the linear kinetic equations near

equilibrium, Eq. (10), expressed in terms of the conjugate variables, are written as a set of linearequations with a symmetric and negative matrix. The negativity property, which is physicallynecessary to insure relaxation to equilibrium, is readily demonstrated from Eq. (10):∑

i

xiddt

�Pi +8�nac3

∫ ∞

0y�

(ddt

�n�

)�2 d�

=− �na

8�c2

∫ ∞

0[k4� n�(1 + n�)]0y2

� �2 d�−

∑i

(TCic )

0x2i −12

∑ij

(TCij )

0(xi − xj)2

−∑i

∫ ∞

0[TR↑

ic (�)n�]0(xi + y�)2 d�−∑ij

∫ ∞

0[TR↑

ij (�)n�]0(xi − xj + y�)2 d�6 0:

(11)

Eq. (11) is an obviously negative quadratic function of the departure from equilibrium. Notethat the same expression is negative far from LTE, which may lead to an H theorem, i.e., ageneralization of the property of increasing entropy. We now introduce new notation to explicitlyshow the Onsager symmetry. First, taking into account the micro-reversibility relations, Eq. (4)becomes in LTE

k0� = (k4� )0 +

na

�c2

8��21

1 + n0�

∑

i

TR↑ic (�) +

∑ij

TR↑ij (�)

0

:

Second, the symmetric and positive matrix

Bij ≡[Tci + Tic +

∑k

(Tik + Tki)

]�ij − Tij − Tij;

with total rates T ≡ TC + TR, was introduced in a previous work (Eq. (9) in Ref. [12]) as thesource term in the non-LTE time-evolution equation of the population correlations. Here, wewrite the LTE limit of this matrix Bij. We 7nally de7ne the quantities

Ci(�) ≡TR↑

ic (�) +∑j

[TR↑ij (�)− TR↑

ji (�)]

0

n0� ;

which combine the LTE radiative rates of interest for a given level and a given frequency.The kinetic equations near equilibrium, Eq. (10), can now be written in a compact form:

ddt

�Pi =−12

∑j

B0ijxj −

∫ ∞

0Ci(�)y� d�;

ddt

�n� =−�c[k�n�(1 + n�)]0y� − nac3

8��2∑i

Ci(�)xi:

(12)


Onsager’s “symmetry of kinetic coePcients” is now apparent: the matrix Bij is symmetric andthe same function Ci(�) appears in the two equations. For any state of matter near LTE, thee4ects of the non-LTE bound electrons in the radiative balance (j� − k�E�), see the second lineof Eq. (12), are always a linear combination of hvCi(�) terms, which we shall call spectralsignatures of one-electron levels. The proportions in this linear combination are the variables xiconjugate to the bound electron populations, that vanish in LTE. Due to Onsager’s symmetryof kinetic coePcients, these same spectral signatures determine how populations vary under theaction of radiation, see the 7rst line of (12). Note that the idea of spectral signatures is onlyinteresting for an average-ion model, since there is only one for each one-electron level, whilefor detailed atomic physics, there is one for each con7guration, which is not practical.

In the general case, one must solve the coupled set of equations for atomic physics, radiativetransfer including transport terms, hydrodynamics, and energy conservation including also spatialtransport terms. For simplicity, we now consider the time-evolution of bound electrons withgiven radiation 7eld and electron temperature conditions. To simplify further, we consider thestationary equilibrium of bound electrons in a constant radiation 7eld, and constant temperature.In this case, there are absorption and emission processes, but it is assumed that they do notalter the constant radiation 7eld. The stationary equilibrium near LTE is given by equating the7rst line of Eq. (12) to zero; the positive matrix Bij can be inverted, and the solution is

xi =−2∑j

(B−1)0ij

∫ ∞

0Cj(�)y� d�:

If needed, the variations of populations �Pi and of chemical potential ��e can be deduced fromEq. (9) and the requirement of charge neutrality. Eliminating xi from the second line of Eq. (12)yields the radiative response dn�=dt. More and Kato [2] de7ned the “response matrix” RM��′

as the stationary net radiative response to a radiative temperature change at a single frequency,which in radiative transfer notations is

dE�

dt= �c

∫ ∞

0RM��′

E�′ − B�′(Te)@B�′=@Te

d�′:

We obtain here its expression in analytic form:

RM��′ =−k0�@B�

@Te�(�− �′) +

na

�12�

∑ij

h�Te

Ci(�)(B−1)0ijh�′

TeCj(�′); (13)

where the two terms are the direct radiative response, given with the LTE opacity k0� , and a“matrix” reaction term

In the present case of stationary atomic physics, the negativity property, Eq. (11), gives∫∞0 y�((d=dt)�n�)�2 d� 6 0, which means that the symmetric matrix RM��′ is negative. This isphysically necessary to ensure that the radiation returns to LTE. The direct radiative responseis a damping whose coePcient, −k0� , is negative; the reaction term in Eq. (13) is a manifestlysymmetric and positive matrix, but the net response RM��′ remains negative. These three al-ternating signs are an illustration of Le Chatelier’s law in its complete version: “The actioninduces a reaction that opposes it, but cannot completely cancel it”.

The reaction matrix is obtained in the rather simple analytical form of Eq. (13), requiringthe inversion of a matrix of moderate size. Moreover, the reaction matrix is factorized as a


sum of products of functions of � and �′, which has minimum memory requirements, and hasease of numerical manipulation. The reaction matrix is a “frequency redistribution”, where thespectral signatures, h�Ci(�), of the bound levels maintain the two meanings implied by Onsager’ssymmetry of kinetic coePcients. Thus, a change in radiation induces variations of the levelpopulation according to their spectral signatures. These population changes are redistributedby the e4ect of all transitions, collisional as well as radiative. They 7nally appear in the netradiative balance with again the spectral signatures.

The LTE opacity k0� and the spectral signatures h�Ci(�) are constructed from radiative tran-sition rates. On the contrary, the Bij matrix depends on total rates, sums of collisional andradiative rates. At low temperatures and high densities, collisional rates are much larger thanradiative ones, and the non-LTE reaction term becomes negligible compared to the LTE opacity.In the opposite case of low densities and high temperatures, the reaction term is of the sameorder of magnitude as the LTE opacity, even very near LTE. One can then distinguish twoopacities: one, a matrix, for the isotropic part of radiation, and the other, the LTE opacity, fordamping the radiation anisotropy.

It is diPcult to draw pictures of a response matrix. To show the size of the reaction opposingthe direct response, we write the net integrated radiative response as

ddt

∫ ∞

0E� d�= �c

∫ ∞

0kF� [B�(Te)− E�] d�;

where kF� , which is essentially the integral of the response matrix over one of the frequency

variables, acts as an “e4ective” opacity. It is the net absorption coePcient of radiative energy atfrequency �, taking into account reemissions and reabsorptions, at the same frequency (inducedemission), and at all other frequencies (reaction matrix). Note that some values of kF

� may benegative, or larger than the LTE opacity k0� .In Table 1 we give the example of iron at temperature 100 eV, and di4erent densities. The

table shows the ionization degrees at LTE, and non-LTE with no radiation, as calculated by the(non-linear) average-ion model; the non-LTE ionization is only there to indicate if the plasmais globally collisional or not. Fig. 1 shows the LTE opacity (upper curves) and the e4ectiveopacity (lower curves). When the e4ective opacity is much smaller than the LTE opacity, theabsorbed radiative energy is mostly reemitted. When the two opacities are close, the collisionalprocesses redistribute the absorbed radiative energy. It is 7rst noticed that the absorption byK-shell photo-ionization, above 8 keV, is apparently not e4ectively redistributed in this model.We note that this is physically wrong, as the Ruoresence yield of iron is not that large in reality.This simply shows that our model lacks the Auger e4ect which is the main decay process foriron with a hole in the K-shell. At other photon energies, the e4ective opacity is near the LTE

Table 1Ionization degree of iron at 100 eV temperature

Density (g=cm3) 10−5 10−3 10−1 10

LTE ionization 19.2 16.5 14.3 9.4Non-LTE ionization without radiation 14.9 15.5 14.2 9.4


Fig. 1. LTE opacity (upper curve) and e4ective opacity (lower curve) of iron at 100 eV temperature, and a densityof (a) 10−5 g=cm3, (b) 10−3 g=cm3, (c) 0:1 g=cm3, (d) 10 g=cm3.

one when the density is high and the absorbed photon energy is small, i.e., when the collisionalprocesses dominate the radiative ones. At low densities, the absorption of radiation is seen tobe ine4ective. In these cases, the LTE opacity loses its meaning, as it does not force the systemto LTE, because a large part of the absorbed energy is reemitted.

The e4ective opacity carries less information than the response matrix, which would tell,for example, if the energy absorbed in a line is reemitted in the same line, or otherwise after


frequency redistribution. The response matrix is itself a particular solution of the time-dependentequations (12). These are much simpler to solve implicitly than the non-linear equations (2),so that we would like to use the former as much as possible. The next step in this study willbe to evaluate how far from LTE we can use the linear response equations (12).

References

[1] Libby SB, Graziani FR, More RM, Kato T. Systematic investigation of NLTE phenomena in the limit of smalldepartures from LTE. In: Miley GH, Campbell EM, editors. Proceedings of the 13th International Conferenceon Lser Interactions and Related Plasma Phenomena. New York: AIP, 1997. p. 637–44.

[2] More R, Kato T. Near-LTE linear response calculations with a collisional-radiative model for He-like Al ions.Phys Rev Lett 1998;81:814–7.

[3] Faussurier G, More RM. NLTE steady state response matrix method. JQSRT 2000;65:387–91.[4] Lokke WA, Grassberger WH. XSNQ-U—a non LTE Emission and absorption coePcient subroutine. Lawrence

Livermore Laboratory Report UCRL-52276, 1977. 53pp.[5] Pollak G. Detailed physics of XSN-U opacity package. Los Alamos National Laboratory Report

LA-UR-90-2423, 1990. 72pp.[6] Decoster A. ModWele d’ionisation hors-ETL: NOHEL. Rapport des activitXes laser. Commissariat Wa I’Energie

Atomique, Limeil-Valenton, 1994.[7] Zimmerman GB, More RM. Pressure ionization in laser-fusion target simulation. JQSRT 1980;23:517–22.[8] More RM. Electronic energy-levels in dense plasmas. JQSRT 1982;27:345–57.[9] Onsager L. Reciprocal relations in irreversible processes. Phys Rev 1931;37:405–26; 38:2265–79.[10] Casimir HBG. On Onsager’s principle of microscopic reversibility. Rev Mod Phys 1945;17:343–50.[11] Landau LD, Lifshitz EM. Course of theoretical physics. Statistical Physics, Part 1, vol. 5. Oxford: Pergamon

Press, 1980.[12] Dallot P, Faussurier G, Decoster A, Mirone A. Average-ion level population correlations in o4-equilibrium

plasmas. Phys Rev E 1998;57:1017–28.

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Onsager symmetry of the average ion near equilibrium