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Determination of internal thermodynamic properties of the species in a plasma is relevant for both theoretical and experimental aspect, also in non-equilibrium conditions. A critical aspect is to calculate the contribution to internal levels to the system enthalpy and to the equilibrium constants of relevant reactions 1. The most accurate approach is to perform the Boltzmann summation on the entire level spectrum, truncated to avoid the divergence of the partition function [2, 3]. This calculation is very time consuming, needing the evaluation of a number of exponentials of the order ranging from many hundreds to hundreds of thousands.
The paper presents the general theory to calculate thermodynamic properties of atomic and molecular species lumping all the levels in a small number of groups, each one characterised by a statistical weight and energy, and applying the Boltzmann statistics [4,5].
For atomic species the method is based on the Taylor series of the exponential function [4] and it has shown that the convergence properties strongly depend on the variance of the level distribution. This approach gives very good accuracy for many atoms such as nitrogen, oxygen, hydrogen, carbon [4-6]. However, difficulties arise when the method is applied to atoms with electrons in the angular momentum quantum number $\ell\ge 2$, such as metals, and the accuracy of the lumped level approach is lost. To overcome this difficulty, the theory has been generalised, showing that a multi-group strategy is capable to improve the accuracy. To account for the cutoff of the partition function, the statistical weight and the mean energy of the last group have been fitted as a function of the ionization lowering to account for the dependence on the number of levels included in the calculation of the thermodynamic properties [4]. The multi-group approach has been applied to Cu, Cu$^{+}$ and Cu$^{+2}$, validating the method by comparison between the multi-group and the exact calculation.
Figure 1. Internal contribution to partition function (top), internal energy (middle) and specific heat (bottom) for Cu (left), Cu$^{+}$ (center) and Cu$^{+2}$ (right) for different pressures, calculated with the self-consistent cutoff. Comparison between the values obtained with the real levels (symbols) and multi-group approach (line). The curve label represents the value of the pressure in bar.
The results with the full level set and with the multi-group approach have been compared calculating the equilibrium composition accounting for the self-consistent cutoff. The single species internal partition functions, energy, and specific heat has been reported in Fig. 1 for different values of the pressure. The thermodynamic properties obtained by the multi-group approach (continuous lines) well represents the results obtained with the Boltzmann sum of the full level system (symbols) in the whole range of pressure and temperature, and only small differences are observed for Cu$^{+2}$ for $P$=1~bar and $T\ge$40000~K. It should be noted that all the quantities calculated with the multi-group approach are continuous function of the temperature, crossing the piecewise shape of the curves obtained with the full level model.
The idea has been adapted to the diatomic molecules [7] considering that a given number of ro-vibrational levels can be approximated with the truncated harmonic oscillator and rigid rotor. All the other levels can be lumped in a multi-group structure as in the case of atomic species. It is difficult to determine where is the upper limit of the levels approximated by the harmonic oscillator and rigid rotor. For this reason, a best fitting procedure has been implemented to find the coefficients of the approximate expression.
The method adopted for diatomic molecules has been extended to polyatomic molecules [7], taking into account that there are different vibrational modes and 2-3 rotational axes.
The proposed approach has been demonstrated to be accurate in a wide range of temperature, limiting the calculation of the thermodynamic properties to a small number of exponentials. Moreover, the lumping level approach allows to include the ionisation lowering in the atomic partition function, very relevant when electron density grows. The method can be easily extended also for conditions where the energy of levels depends on the electron density [8].
References
1] J. D. Anderson, Hypersonic and High Temperature Gas Dynamics, American Institute of Aeronautics and Astronautics, 2006; 10.1007/978-3-319-05200-7
[2] D. Landau and E. Lifshitz, Statistical Physics, Pergamon Press, Oxford, 1986; 10.1515/9783110648485-015
[3] M. Capitelli, G. Colonna, and A. D’Angola, Fundamental Aspects of Plasma Chemical Physics: Thermodynamics, in Springer Series on Atomic, Optical, and Plasma Physics, volume 66. Springer Science \& Business Media, 2011; 10.1007/978-1-4419-8182-0
[4] G. Colonna and M. Capitelli, A few level approach for the electronic partition function of atomic systems, Spectrochimica Acta Part B: Atomic Spectroscopy, 64 (2009) 863-873; 10.1016/j.sab.2009.07.002
[5] G D'Ammando, G Colonna, LD Pietanza, M Capitelli, Computation of thermodynamic plasma properties: a simplified approach, Spectrochimica Acta Part B: Atomic Spectroscopy 65 (2010) 603-615; 10.1016/j.sab.2010.05.002
[6] G. D’Ammando, G. Colonna, and M. Capitelli, A simplified approach to calculate atomic partition functions in plasmas, Physics of Plasmas, 20 (2013) 032108; 10.1063/1.4794286
[7] G. Colonna, A. D’Angola, and A. Laricchiuta, Thermodynamic and transport properties of complex plasmas, in Plasma Modeling (Second Edition) Methods and applications, ch- 21 IOP Publishing Bristol, UK, 2022; 10.1088/978-0-7503-3559-1ch21
[8] M. Capitelli, D. Giordano, and G. Colonna. The role of {Debye-H\"uckel} electronic energy levels on the thermodynamic properties of hydrogen plasmas including isentropic coefficients. Physics of Plasmas,
15 (2008) 082115; 10.1063/1.2967490
Summary
A lumped level approach has been applied to atoms and molecules for the calculation of single species thermodynamics.