A new algorithm for the solution and analysis of the Collisional Radiative Model equations

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1 High Energy Density Physics 3 (2007) 143e148 A new algorithm for the solution and analysis of the Collisional Radiative Model equations Marcel Klapisch a, *, Michel Busquet a,b a ARTEP, Inc., 2922 Excelsior Springs Ct., Ellicott City, MD 21042, USA b Observatoire de Paris, Meudon, France Available online 7 February 2007 Abstract A new general algorithm for the solution of the Collisional Radiative Model (CRM) for large atomic models is presented. It is based on the separation of total population, and reduced populations for the individual states of each charge state. The sum of the latter is unity. This leads to a double linearized coupled iterative system, including cycling on all charge states. This algorithm is robust. It gives insight into the quality of the atomic model by allowing the follow up of the charge states populations, the global rates and the average charge Z* as a function of the iteration index. We show results on Xe with 39,683 configuration states. Ó 2007 Elsevier B.V. All rights reserved. PACS: Hv; Os Keywords: Plasma spectroscopy; Non-LTE; Collisional Radiative Model 1. Introduction Most of hot and dense plasmas of interest for fusion research are too dense to be in the coronal regime and may radiate too much to be in local thermodynamic equilibrium (LTE). Therefore, they have to be modeled with the Collisional Radiative Model (CRM), where all transition rates between all the levels are explicitly included. The HULLAC code [1] is able to compute efficiently all the necessary rates, but has lacked an efficient CRM solver. The purpose of this paper is to present such a solver, which also has the property of giving information on the quality of the atomic model. It is now recognized [2,3] that in order to obtain the correct charge distribution, one has to include many charge states and in each of these, numerous singly and doubly excited states. This yields a rate matrix e the order of which is equal to the * Corresponding author. address: marcel.klapisch@nrl.navy.mil (M. Klapisch). number of levels included in the model e often in the tens of thousandse large enough to require solution by iterative procedure. One widely used method is the bi-conjugate gradient method [4,5]. However, it acts like a black box, giving little information about which rate processes are critical for the accuracy of the results. In addition, in cases where it does not converge, one cannot extract any useful information. This is a real concern because the lifetimes of resonant states, on the one hand, and ground and metastable states, on the other hand, are by nature extremely different. Thus, the condition of the rate matrix can be in excess of Other methods [6], more adapted to low density plasmas, involve separating the states into two groups, where only the ground and metastable states are used to obtain the charge distribution, while the excited states are obtained afterwards. An alternative method [7], which is also a two step procedure, splits the large matrix in smaller matrices, one per charge state. Inversion of each small matrix gives the population of the excited levels of one charge state relative to its ground state and to the next higher ground state. However, these decompositions are correct and significant only when ions are connected only through their /$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi: /j.hedp

2 144 M. Klapisch, M. Busquet / High Energy Density Physics 3 (2007) 143e148 ground states and there are no transitions between excited levels of different ions. A classical example of the failure of this approach occurs when neglecting the influence of metastable levels on charge state distribution in X-ray laser plasmas [2,8]. If excited-to-excited transitions are not accounted for e and such transitions happen to be numerous and non-negligible when looking at L- or M-shell ions, and more generally when the plasma is near LTE e this procedure becomes inexact, because it neglects populating levels by ionization from excited states of the lower ion. We describe in this paper a new method, taking into account excited-to-excited transitions on the same footing as transitions to the ground states. The populations of the levels (ground and excited) are split into reduced populations within each ion, normalized to unity, times the total population of that ion. Hereafter, we shall call total population of each ion the proportion of this ion in the total ion density of the plasma. We present in Section 2 the iterative procedure for solving the full time-dependent CRM. This procedure has three steps. In the first step, the variations of the reduced levels populations are linearized with respect to the variations of the two adjacent ion total populations. In the second step, the global rates between charge states are computed, using the actual reduced populations of the excited levels. The third step is determining each ion s stage s proportion by solving the global rate matrix equation. In Section 3 we present results on Xe at T e ¼ 50 ev and various densities. Section 4 is a conclusion. 2. Description of the algorithm 2.1. Definitions For generality, we consider a time-dependent model. We shall denote by a superscript o, the population of the previous time step, and we choose to replace the time derivative by a finite element difference: ðv=vtþn p ¼ðN p Np oþð1=dtþ. We shall denote by a tilde the population estimated at the previous iteration. Subscripts p 0 and p 00 will stand for levels of charge state Q 0 ¼ Q þ 1, Q 00 ¼ Q 1. The rate from a level q to a level p will be denoted indifferently W q,p or W q/p when needed for clarity. The time evolution of the population N Q,p of a level (Q, p) of ion Q is given by: v 1 vt N p ¼ N p N o p Dt ¼ X X Nq W q/p N p W p/q N p W p/p 0 q p 0 X p 00 N p W p/p 00 þ X p 0 N p 0W p 0 /p þ X p 00 N p 00W p 00 /p ð1þ where W are temperature- and density-dependent rates for all relevant processes (neglecting double ionization). In matrix form, this can be written: v vt N ¼ L$N ð2þ where L is a large block-diagonal matrix, the order of which, L tot, is the total number of levels in the model including all possible charge states. This can be extremely large, so to make the problem more tractable, we propose to split the L matrix into several smaller matrices: (a) Diagonal blocks square matrices M Q of order M Q (the number of levels pertaining to charge state Q). M Q can be typically a few thousands, including numerous doubly excited states necessary for accurate evaluation of dielectronic recombination. The elements of these matrices are: M pq ¼ W q;p M pp ¼ 1 Dt XM Q q XM Q0 W p;q p 0 X M Q 00 W p/p 0 W p/p 00 ð3þ p 00 (b) Rectangular matrices R Q,Q 0 connecting neighboring charge states by recombination R p,p 0¼W p 0 /p etc. and S QQ 00¼ W p 00 /p for ionization. (c) Later we will also consider a smaller matrix Z, of order Z (number of charge states necessary), the elements of which are the global effective rates, and from which the total population of the charge states: N Q ¼ XM Q p¼1 N p will be directly computed. We now introduce the reduced populations: n Q p ¼ N p=n Q. Note that the global rate W Q/Q 0 is the contribution of the total population of one ion to the total population of another. According to the well-known rule, one takes the average over the initial states and the sum over the final states. W Q/Q 0 ¼ X! 1 X N p W p/p 0 ¼ X N p 0 Q p 0 p X n Q p W p/p 0 Thus, the global rates depend only on the reduced populations. Hence, through the solution of the Z matrix, the total populations N Q can be obtained also from the reduced populations. This is one reason the latter were introduced. Conversely, once the total populations are given, the solution of the M matrices equations with a right hand side describing the population gain from neighboring ions gives the reduced populations. Thus, the two sets of variables are coupled, but there are two ways to approach the problem. In the present work, we choose to start from the reduced populations Basic equation Let us concentrate on one charge state Q, in relation to its neighbors Q 0 and Q 00. Eq. (1) can be written explicitly: p

3 M. Klapisch, M. Busquet / High Energy Density Physics 3 (2007) 143e ¼ N Q 0R Q n Q0 þ N Q 00S Q n Q00 þ N Q M Q n Q þ C ð4þ with vector C p ¼ N o p ð1=dtþ R and S are the rectangular matrices mentioned above. Left multiplying by M 1, we obtain the reduced population n p of any excited level p as a linear combination of the adjacent charge states populations: n Q N Q ¼ N Q 0A Q $n Q 0 N Q 00B Q $n Q 00 M 1 C where A Q ¼ M 1 Q R Q; or B Q ¼ M 1 Q S Q n p ¼ a p N Q 0=N Q þ b p N Q 00=N Q þ c p =N Q where the vectors a¼{a p }, b¼{b p } and c¼{c p } are obtained by multiplying A, B, by the respective corresponding reduced populations, and C by M 1. This equation is linear in n p unless the elements of matrix M depend of the populations of the same charge state, for example due to a Net Radiative Bracket formalism [9] but it remains valid even if non-linear. But a, b, depend on the reduced populations of the other charge states, so iterations will be necessary Linearization of reduced populations We have now two coupled sets of unknowns, the ion populations N Q on the one hand, and the reduced populations n Q on the other hand. We look for a linearization, where the increments for each set will be obtained while the other set is fixed, and the coupling will be achieved by combined iterations. Let us now add and subtract from Eq. (4) the quantity: ~G Q ¼ G ~N Q ; ~n Q ¼ ~N Q 0R Q ~n Q0 þ ~N Q 00S Q ~n Q00 þ ~N Q M Q ~n Q þ C which is the residue of that same equation computed with the ion populations estimated at the previous iteration. We obtain: 0 ¼ R Q N Q 0n Q0 ~N Q 0 ~n Q0 þ S Q N Q 00n Q00 ~N Q 00~n Q00 þ G ~ Q þ M N Q n Q ~N Q ~n Q Using the relation: Nn ~N~n ¼ N n ~n þ N ~N ~n ¼ NDn þ DN~n we have: 0 ¼ DN Q 0R Q ~n Q0 þ DN Q 00S Q ~n Q00 þ M$ N Q Dn Q þ DN Q ~n Q þ N Q 0R Q Dn Q0 þ N Q 00S Q D~n Q00 þ ~ G and, left multiplying by M 1 and reordering, ð5þ ð6þ Dn Q ¼ 1 DN Q 0A Q ~n Q0 þ DN Q 00B Q ~n Q00 þ DN Q ~n Q þ M 1 G ~ N Q þ N Q 0A Q Dn Q0 þ N Q 00B Q Dn Q00 ð7þ which is our basic equation. It is a coupling between the increments of the reduced populations of ion Q and the increments of the total populations of charge states Q, Q 0, and Q 00 at that iteration, coupled with the reduced populations of the previous iteration. The reduced populations increments of neighboring charge states appearing on the second line of Eq. (7) may be unknown at the time of solving them. Either sub-iterations over charge states can be done, or an estimate of these increments can be obtained by extrapolation from previous iterations. We will discuss this issue below. The new total ion populations N, on the other hand, are easily obtained at this stage, as explained in the following section Obtaining the total ion populations The Z matrix As mentioned above, let us consider the matrix Z of the global rates: Z QQ 0 ¼ X X n p 0W pp 0 ¼ RQ n Q0 p p 0 Z QQ 00 ¼ X X n p 00W pp 00 ¼ S Q n Q00 p p 00 Z QQ ¼ X n p W p0 p þ X! n p W p 00 p ¼ SQ 0n Q RQ 00n Q pp 0 pp Time-dependent case The time evolution of N Q can now be written in matrix form: v vt Nz N N 0 1 Dt ¼ Z$N ð8þ ð9þ where N 0 is the result at the preceding time step. Let us now introduce the matrix: Z ¼ Z I$1=Dt Eq. (9) can be written as: Z$N ¼ N0 Dt ð10þ Since the matrix is tri-diagonal, the solution of Eq. (10) is trivial.

4 146 M. Klapisch, M. Busquet / High Energy Density Physics 3 (2007) 143e Time independent case We now have: Z$N ¼ 0 ð11þ This matrix is singular by construction, since there is no creation of population. A well-known way to deal with the singularity, which enforces normalization, is to replace one line, e.g. Q 0 with an arbitrary constant a, for instance one of the diagonal elements of Z. The right hand side is now a vector, the elements of which are N Q ¼ adðq; Q 0 Þ. Now, this modified matrix is not tri-diagonal, but it is still small and sparse, since there are seldom more than a dozen ionization stages that really matter for each temperature and density. Consequently, any standard method will do. The new populations and the increments DN Q ¼ N Q ~N Q can now be inserted into Eq. (7) Algorithm The iterative process can be initialized with an estimate of the level and total populations e e.g. with the RADIOM model [10]. The former can be used in Eqs. (10) or (11) to obtain new total populations, thus enabling one to solve Eq. (7). Let us note that in order to obtain good results, we neglected the terms in the second line of that equation. Indeed, it can easily be checked that sub-iterations generate numerical instabilities for all ions such that N Q 0=N Q [1 and N Q 00=N Q [1, i.e. for ions on each side of the maximum of the charge distribution. These two cases are bound to happen often, because the charge distribution is peaked. Because these factors can be very large, extrapolation from previous iterations can lead to reduced populations greater than 1, due to amplified numerical errors. The fact that we achieve significant results without these contributions is a sign of the robustness of the algorithm. 3. Results We performed rate computations with HULLAC [1] for the NLTE4 workshop on Carbon at several temperatures and densities, and used these for testing the present algorithm. However, we will not present those here, because they were obtained with a fitting to the cross sections that led many excitation rates to be negative. This problem was compounded with an incidental error in the code for this case. This in turn yielded badly conditioned matrices, and some results were incorrect. For the case presented here, we used the new fitting procedure of Busquet [11]. With this new fitting procedure, no negative or zero excitation rates were found Presentation of the cases The cases presented here relate to Xe at temperatures of 40e50 ev, and electron densities from to cm 3. We used two atomic models: a small model in which the ions Xe 7þ to Xe 16þ were included, and the states were non-relativistic average configurations, for a total of 2368 configurations; and a large model including the ions Xe 8þ to Xe 16þ in which the states were relativistic sub-configuration averages for a total of 39,683 configurations. Both models were calculated by HULLAC. In both cases the models included for each ion all possible single excitations within the n ¼ 4 shell, and from the latter to all n ¼ 5 and 6, as well as double excitations to the same shells. The number of configurations and rates for the large model is detailed in Table Results Fig. 1 shows the evolution of the ion distribution of Xe at T e ¼ 50 ev and N e ¼ cm 3 as the iterations proceed. The final number of iterations was 783. Here the convergence criterion was rather strict, i.e. the RMS of the relative variation of the reduced populations of all configurations was required to be less than This is an example of the fact that the algorithm gives meaningful results at every stage of the iteration process, and that one can visualize the progress of the computation. Fig. 2 shows the final spectra obtained at two different densities. It is interesting to see that the spectra are not dramatically different, the smoother appearance of the higher density one coming only from different intensities of satellites. This is in part due to the fact that the average Z* does not significantly change with density, as shown on Fig Analysis We performed computations for the Xe large model for a number of densities between till cm 3. Fig. 3 shows the number of iterations required to attain the same convergence criterion as above, as a function of electronic density. The dramatic decrease of this number shows that using the Boltzman distribution at an effective temperature as an initial approximation was not good for low densities. On the other hand, the average charge does not change much with density. Now following the global rates as a function of iteration number, see Fig. 4, we are able to show that the reason for slow convergence is mainly the poor estimate of the ionization into Xe 13þ. That means that the excited states of the dominant Xe 12þ are not well described by this initial effective Table 1 Detail of the number of levels and rates for the Xe large model Ion Levels Radiat. Transfer Coll. Excit. Ioniz. Radiat. Rec. Autoioniz. Xe IX , ,159 15,008 31,351 Xe X ,727 36, ,340 Xe XI ,026 49,505 11, Xe XII , ,854 18,341 28,915 Xe XIII , ,412 21,936 29,057 Xe XIV , ,180 22,176 25,313 Xe XV , ,312 21,101 22,199 Xe XVI , ,522 16,665 15,833 Xe XVII ,799 86,096 e e Total 39, , , , ,388

5 M. Klapisch, M. Busquet / High Energy Density Physics 3 (2007) 143e Fig. 1. Evolution of the distribution of charge states with iteration index for Xe at T e ¼ 50 evand N e ¼ cm 3 for the large model with 39,683 configurations. temperature T z. We verified this by running the same case with an initial estimate of T z evaluated from the final charge distribution. Again, the algorithm requires many (albeit less) iterations. As the density increases, the effective temperature approximation gets better, since the plasma is approaching LTE conditions. On the other hand, this proves the point stated in Section 1 that the contribution of excited states to ionization has to be considered. For recombination, however, they do not seem to be important. 4. Conclusion Fig. 3. Number of iterations (left axis) required for convergence as a function of the electronic density for Xe (large model) at T e ¼ 50 ev. Right axis: estimate of Z* (dotted line) and final Z* (broken line) as a function of electron density. generate databases of emissivities and opacities without the statistical average approximation [12]. This scheme is more physically meaningful than those purely numerical algorithms, like LINBCG [4], which do not take into account the underlying physical processes. It is preferable over other schemes where the ground states populations are tracked, instead of the total charge state, because including excitedto-excited transitions gives the correct Saha-Boltzman charge distribution in the LTE limit. We obtain directly the charge state distribution, which in many cases is the only quantity desired. In addition, it is a simple way to compute accurate We have presented here a CRM equation solver that can be used on a large atomic model (in this case more than 39,000 configurations). It is our aim to use it, whenever possible, to Fig. 2. Computed spectra of Xe for two electronic densities (10 16 and cm 3 ) at 50 ev. The smoother appearance of the higher density spectrum comes from the larger number of significant transitions. Fig. 4. Evolution of the global rates of dominant ions with iteration number. The Xe small model is used for T e ¼ 50 ev, N e ¼ cm 3. The rates are ionization into the ion stated. The lower dotted lines are the three body and radiative recombination rates into the same ions, in the same order from top to bottom.

6 148 M. Klapisch, M. Busquet / High Energy Density Physics 3 (2007) 143e148 temperature- and density-dependent global rates that can be tabulated and used in other programs. Finally, the sum of charge states populations is equal to the total ion density by construction. This fact enables checking and/or tracking numerical errors that are more difficult to determine when ground states are used. Thanks to the possibility of following many physical quantities as the iterations progress, we noticed that the initial estimate of the reduced populations may be critical to obtain quick convergence. We are presently working on an improved model of excitation temperature. Also it may be noticed that the convergence is not quadratic, as might be expected from a Newtonlike algorithm. We are investigating another formulation of the same idea that will possibly possess superlinear convergence. Acknowledgement We gratefully acknowledge the support of the Dr. S. Obenschain, Laser Plasma Branch Head, Naval Research Laboratory, through a grant of the USDOE. References [1] A. Bar-Shalom, M. Klapisch, J. Oreg, J. Quant. Spectrosc. Radiat. Transfer 71 (2001) 169e188. [2] J. Abdallah, R.E.H. Clark, J.M. Peek, et al., J. Quant. Spectrosc. Radiat. Transfer 51 (1994) 1e8. [3] R.W. Lee, J.K. Nash, Y. Ralchenko, J. Quant. Spectrosc. Radiat. Transfer 58 (1997) 737e742; H.K. Chung, K.B. Fournier, R.W. Lee, High Energy Density Phys. 2(7e15) (2006). [4] William H. Press, Brian P. Flannery, Saul A. Teukolsky, et al., Numerical recipes in fortran 77, The Art of Scientific Computing, second ed. Cambridge University Press, Cambridge, UK, [5] S. Kaushik, P.L. Hagelstein, J. Comput. Phys. 101 (1990) 360e367. [6] S.D. Loch, C.J. Fontes, J. Colgan, et al., Phys. Rev. E 69 (2004) [7] M. Busquet, J.P. Raucourt, J.C. Gauthier, J. Quant. Spectrosc. Radiat. Transfer 54 (1995) 81e87. [8] A.L. Osterheld, R.S. Walling, B.K.F. Young, et al., J. Quant. Spectrosc. Radiat. Transfer 51 (1994) 263. [9] D. Mihalas, Stellar Atmospheres, second ed. W.H. Freeman and Co., San Fransisco, 1978, p [10] M. Busquet, Phys. Fluids B: Plasma Phys. 5 (1993) 4191e4206. [11] M. Busquet, High Energy Density Phys. 3 (2007) 48e51. [12] A. Bar-Shalom, J. Oreg, M. Klapisch, Phys. Rev. E 56 (1997) R70eR73.