Models implemented

This page shows an automatically generated overview over all models defined in jaxrts.models and jaxrts.hnc_potentials. The latter module contains only the potentials relevant for calculating the elastic scattering in the Hypernetted Chain approach.

The following keys are available to add to jaxrts.plasmastate.PlasmaState:

screening length, ionic scattering, BM S_ii, Debye temperature, free-free scattering, chemical potential, ipd, screening, BM V_eiS, free-bound scattering, ee-lfc, form-factors, bound-free scattering, ion-ion Potential, electron-ion Potential, electron-electron Potential

To set a specific model, add it to a jaxrts.plasmastate.PlasmaState, e.g.,
>>> state["free-free scattering"] = jaxrts.models.RPA_DandreaFit()
For more details on an individual model, we refer to the literature, provided in the jaxrts.models.Model.cite_keys attribute. You can easily obtain the bibliographic information by using the jaxrts.models.Model.citation() method, e.g., calling
>>> import jaxrts
>>> print(jaxrts.models.RPA_DandreaFit().citation()
R. G. Dandrea, N. W. Ashcroft, and A. E. Carlsson. Electron liquid at
any degeneracy. Physical Review B, 34:2097–2111, 8 1986.
doi:10.1103/physrevb.34.2097.

screening length

The screening length characterizes the spatial scale over which electric fields are screened by the surrounding electrons.

`jaxrts.models.ArbitraryDegeneracyScreeningLength`()	A screening length valid for arbitrary degeneracy [Baggott, 2017].
`jaxrts.models.ConstantScreeningLength`(value)	A model that returns a constant screening length, given by a user.
`jaxrts.models.DebyeHueckelScreeningLength`()	The standard Debye Hückel screening length.
`jaxrts.models.Gericke2010ScreeningLength`()	The Debye-Hückel screening length.

ionic scattering

The ionic (elastic) scattering component, \(S_{ee}^{\text{el}}(k, \omega)\), represents the coherent scattering of photons from electrons that are tightly bound to ions and follow their motion, and from the screening cloud. This term reflects the static ion–ion correlations in the system and typically gives rise to a sharp elastic peak centered at \(\omega = 0\).

`jaxrts.models.ArkhipovIonFeat`()	Model for the ion feature of the scattering, presented in [Arkhipov and Davletov, 1998] and [Arkhipov et al., 2000].
`jaxrts.models.CHSIonFeat`()	Model for the ion feature of the scattering, presented in [Gregori et al., 2007].
`jaxrts.models.DebyeWallerSolid`(S_plasma, b)	This model approximates the static structure factor (SSF) of a solid at finite temperature as suggested by [Gregori et al., 2006a].
`jaxrts.models.FixedSii`(Sii)	Model for the ion feature with a fixed value for \(S_{ii}\).
`jaxrts.models.Gregori2003IonFeat`()	Model for the ion feature of the scattering, presented in [Gregori et al., 2003].
`jaxrts.models.Gregori2006IonFeat`()	Model for the ion feature of the scattering, presented in [Gregori et al., 2006a].
`jaxrts.models.Neglect`()	Universal model to neglect a contribution and set it to zero.
`jaxrts.models.OnePotentialHNCIonFeat`(rmin, ...)	Calculates \(S_{ab}\) in the Hypernetted Chain approximation.
`jaxrts.models.PeakCollection`(k_pos, ...)	A model for approximating \(S_\text{ii}\) as a sum of peaks.
`jaxrts.models.ThreePotentialHNCIonFeat`(rmin, ...)	Calculates \(S_{ab}\) including electron-ion and electron-electron static structure factors using the Hypernetted Chain approximation.

BM S_ii

When calculating the the Born collision frequencies in jaxrts.models.BornMermin and derived free-free scattering models, one needs a notion of the static ionic structure factor. For a single species, jaxrts.models.Sum_Sii is identical to the model used for the ionic scattering key. However, it is not clear if the sum calculated there also holds true for a multi-species plasma, where we would run an HNC calculation, which cannot be directly used. In this case, jaxrts.models.AverageAtom_Sii might be a reasonable alternative, as it calculates S_ii in an HNC step for an average atom.

`jaxrts.models.AverageAtom_Sii`(rmin, rmax, ...)	This model performs a HNC calculation, assuming one average atom with a given, average charge state.
`jaxrts.models.Sum_Sii`()	This model sums up all \(S_{ab}\) from the HNC and multiplies it with \(\sqrt{x_{a}\cdot x_{b}}\).

Debye temperature

The Debye temperature defines the characteristic temperature below which the vibrational modes of a solid begin to freeze out, marking the transition from classical to quantum behavior in lattice vibrations.

`jaxrts.models.BohmStaver`()	The Bohm-Staver relation for the Debye temperature, valid for 'simple metals', as it is presented in Eqn (3) of [Gregori et al., 2006a].
`jaxrts.models.ConstantDebyeTemp`(value)	A model of constant Debye Temperature.

free-free scattering

The free–free scattering component, \(S_{ee}^{\text{ff}}(k, \omega)\), describes inelastic scattering from unbound (free) electrons. In the RPA (Random Phase Approximation), electrons are treated as a weakly coupled gas with collective screening effects, while the Born–Mermin approach extends this by including electron–ion collisions and finite damping, providing a more accurate description in partially or strongly coupled plasmas.

`jaxrts.models.BornMermin`(no_of_freq, ...)	Model for the free-free scattering, based on the Born Mermin Approximation ([Mermin, 1970]).
`jaxrts.models.BornMermin_Fit`(no_of_freq, ...)	Model for the free-free scattering, based on the Born Mermin Approximation ([Mermin, 1970]).
`jaxrts.models.BornMermin_Fortmann`(...)	Model for the free-free scattering, based on the Born Mermin Approximation ([Mermin, 1970]).
`jaxrts.models.BornMermin_Full`([RPA_rewrite, KKT])	Model for the free-free scattering, based on the Born Mermin Approximation ([Mermin, 1970]).
`jaxrts.models.Neglect`()	Universal model to neglect a contribution and set it to zero.
`jaxrts.models.QCSalpeterApproximation`([...])	Quantum Corrected Salpeter Approximation for free-free scattering.
`jaxrts.models.RPA`([sample_points])	Model for free-free scattering based on the Random Phase Approximation.
`jaxrts.models.RPA_DandreaFit`([sample_points])	Model for free-free scattering based fitting to the Random Phase Approximation, as presented by [Dandrea et al., 1986].

chemical potential

The chemical potential is the energy required to add one particle to a system at constant temperature and volume.

`jaxrts.models.ConstantChemPotential`(value)	A model that returns a constant chemical potential, specified by a user.
`jaxrts.models.DegenerateElectronChemPotential`()	Chemical Potential of a fully degenerate electron gas, given by the Fermi energy.
`jaxrts.models.IchimaruChemPotential`()	A fitting formula for the chemical potential of an ideal electron gas between the classical and the quantum regime, given by [Gregori et al., 2003].
`jaxrts.models.NonDegenerateElectronChemPotential`()	Chemical Potential of a non-degenerate electron gas.
`jaxrts.models.SommerfeldChemPotential`()	Interpolation function between the low and high temperature limit for the chemical potential of a non-interacting (ideal) fermi gas given in the paper of Cowan [Cowan, 2019].

ipd

Ionization Potential Depression (IPD) refers to the lowering of an atom’s or ion’s ionization energy due to the presence of surrounding plasma, which modifies the effective potential experienced by bound electrons and can lead to phenomena like pressure ionization.

`jaxrts.models.ConstantIPD`(value)	A model that returns a constant value for the IPD, set by the user.
`jaxrts.models.DebyeHueckelIPD`([arb_deg])	Debye-Hückel IPD Model [Debye and Hückel, 1923].
`jaxrts.models.EckerKroellIPD`([C, arb_deg])	Ecker-Kröll IPD Model:[Ecker and Kröll, 1963].
`jaxrts.models.IonSphereIPD`([C, arb_deg])	Ion Sphere IPD Model [Rozsnyai, 1972].
`jaxrts.models.Neglect`()	Universal model to neglect a contribution and set it to zero.
`jaxrts.models.PauliBlockingIPD`()	Pauli Blocking IPD Model [Röpke et al., 2019].
`jaxrts.models.StewartPyattIPD`([arb_deg])	Stewart Pyatt IPD Model [Stewart and Pyatt, 1966].
`jaxrts.models.StewartPyattPrestonIPD`([...])	Stewart Pyatt IPD Model [Stewart and Pyatt, 1966].

screening

Screening refers to the reduction of the effective interaction between charged particles due to the redistribution of surrounding charges.

`jaxrts.models.DebyeHueckelScreening`()	Debye Hückel screening as presented by [Chapman et al., 2015].
`jaxrts.models.FiniteWavelengthScreening`()	Finite wavelength screening as presented by [Chapman et al., 2015], using a using a result from linear to calculate the screening density \(q\):
`jaxrts.models.Gregori2004Screening`()	Calculating the screening from free electrons according to [Gregori et al., 2004].
`jaxrts.models.GregoriCHSScreening`()	Screening model to calculate the screening charge q based on expression for the static structure factors given in [Gregori et al., 2007].
`jaxrts.models.LinearResponseScreening`()	The screening density \(q\) is calculated using a result from linear response:
`jaxrts.models.LinearResponseScreeningGericke2010`()	The screening density \(q\) is calculated using a result from linear response:

BM V_eiS

These models implement potentials which can be when calculating the Born collision frequencies in jaxrts.models.BornMermin and derived free-free scattering models.

`jaxrts.models.DebyeHueckel_BM_V`()	A Debye Hückel potential, using the `jaxrts.plasmastate.PlasmaState.screening_length()` method to get the screening length.
`jaxrts.models.FiniteWavelength_BM_V`()	Uses finite wavelength screening to screen the bare Coulomb potential, i.e., \(V_{s}=\frac{V_\mathrm{Coulomb}}{\varepsilon_\text{RPA}(k, E=0)}\)

free-bound scattering

The free–bound scattering component, \(S_{ee}^{\text{fb}}(k, \omega)\), represents inelastic scattering events in which a free electron recombines into a bound state, emitting a photon. In thermodynamic equilibrium and for \(k \ll k_0\), its spectral shape is directly related to the bound free component, \(S_{ee}^{\text{bf}}(k, \omega)\), through the principle of detailed balance [Böhme et al., 2023].

`jaxrts.models.DetailedBalance`([sample_points])	Calculate the free-bound scattering by mirroring the free-bound scattering around the probing energy and applying a detailed balance factor to the intensity.
`jaxrts.models.Neglect`()	Universal model to neglect a contribution and set it to zero.

ee-lfc

The electron–electron local field correction (LFC) modifies the electron response to account for short-range correlations and exchange effects beyond the mean-field approximation.

Local field corrections \(G\) describe exchange correlation effects. The full linear susceptibility can be written as

\[\chi_\text{ee} = \frac{\chi_\text{ee}^\text{0}}{1 - V_{\text{ee}} (1 - G) \chi_\text{ee}^\text{0}}\quad\]

where \(V_{\text{ee}}\) is the Coulomb potential between two electrons and \(\chi_\text{ee}^\text{0}\) is Lindhard’s function. While this is formally correct, \(G\) is practically often unknown and has to be approximated.

Currently, only static and effective static approximations of the local field corrections are treated in jaxrts.

`jaxrts.models.ESLFCDornheim2021`()	Effective static approximation (ESA) of the local field correction model by Dornheim et al. [Dornheim et al., 2021].
`jaxrts.models.LFCConstant`(value)	A constant local field correction which can be defined by the user.
`jaxrts.models.SLFCFarid1993`()	Static local field correction model by Farid at al.
`jaxrts.models.SLFCGeldart1966`()	Static local field correction model by Geldart and Vosko [Geldart and Vosko, 1966].
`jaxrts.models.SLFCInterpFortmann2010`()	Static local field correction model that interpolates between the zero temperature result by Farid [Farid et al., 1993] and the Geldart result [Geldart and Vosko, 1966] for high temperatures.
`jaxrts.models.SLFCInterpGregori2007`()	Static local field correction model that interpolates between the zero temperature result by Utsumi and Ichimaru [Utsumi and Ichimaru, 1982] and the Geldart result [Geldart and Vosko, 1966] for high temperatures.
`jaxrts.models.SLFCUtsumi1982`()	Static local field correction model by Utsumi and Ichimaru [Utsumi and Ichimaru, 1982].

form-factors

Form factors describe how the spatial distribution of charge within an atom or ion affects scattering, modulating the scattering intensity as a function of momentum transfer \(k\).

`jaxrts.models.FormFactorLowering`([...])	Form factor lowering model as introduced by [Döppner et al., 2023].
`jaxrts.models.PaulingFormFactors`()	Analytical functions for each electrons in quantum states defined by the quantum numbers n and l, assuming a hydrogen-like atom.

bound-free scattering

The bound–free scattering component, \(S_{ee}^{\text{bf}}(k, \omega)\) , arises from inelastic scattering events in which a photon ejects an electron from a bound state into the continuum.

`jaxrts.models.Neglect`()	Universal model to neglect a contribution and set it to zero.
`jaxrts.models.SchumacherImpulse`([r_k])	Bound-free scattering based on the Schumacher Impulse Approximation [Schumacher et al., 1975].
`jaxrts.models.SchumacherImpulseColdEdges`([r_k])	Bound-free scattering based on the Schumacher Impulse Approximation [Schumacher et al., 1975].
`jaxrts.models.SchumacherImpulseFitRk`()	Bound-free scattering based on the Schumacher Impulse Approximation [Schumacher et al., 1975].

ion-ion Potential

The potential modelling the interaction between two ions.

`jaxrts.hnc_potentials.CoulombPotential`([...])	A full Coulomb Potential.
`jaxrts.hnc_potentials.DebyeHueckelPotential`([...])	The Debye-Hückel screening potential as defined, e.g., in [Wünsch et al., 2008].
`jaxrts.hnc_potentials.DeutschPotential`([...])	Quantum diffraction potential suggested by C.
`jaxrts.hnc_potentials.HNCPotential`([...])	Potentials, intended to be used in the HNC scheme.
`jaxrts.hnc_potentials.KelbgPotential`([...])	Quantum diffraction potential suggested by Kelbg.
`jaxrts.hnc_potentials.PauliClassicalMap`()	Calculates the Pauli potential from the exact non-interacting UEG pair distribution function by inverting the Hyper-Netted-Chain calculations:cite:DharmaWardana.2012.
`jaxrts.hnc_potentials.PotentialSum`(...)	A sum of several `HNCPotential` s.
`jaxrts.hnc_potentials.ScaledPotential`(potential)	A `HNCPotential`, scaled with a constant factor.
`jaxrts.hnc_potentials.SpinAveragedEEExchange`([...])	See [Wünsch et al., 2008], Eqn (18).
`jaxrts.hnc_potentials.SpinSeparatedEEExchange`()	See [Wünsch et al., 2008], Eqn (17), Citing [Huang, 1987].
`jaxrts.hnc_potentials.YukawaShortRangeRepulsion`(sigma)	Yukawa Potential with a short-range repulsion as used in [Fletcher et al., 2015].

electron-ion Potential

The potential modelling the interaction between electrons and ions.

`jaxrts.hnc_potentials.CoulombPotential`([...])	A full Coulomb Potential.
`jaxrts.hnc_potentials.DebyeHueckelPotential`([...])	The Debye-Hückel screening potential as defined, e.g., in [Wünsch et al., 2008].
`jaxrts.hnc_potentials.DeutschPotential`([...])	Quantum diffraction potential suggested by C.
`jaxrts.hnc_potentials.EmptyCorePotential`()	The Empty core potential, which is essentially a `CoulombPotential` for all radii bigger than \(r_\text{cut}\).
`jaxrts.hnc_potentials.HNCPotential`([...])	Potentials, intended to be used in the HNC scheme.
`jaxrts.hnc_potentials.KelbgPotential`([...])	Quantum diffraction potential suggested by Kelbg.
`jaxrts.hnc_potentials.KlimontovichKraeftPotential`([...])	Quantum diffraction potential suggested by Klimontovich and Kraeft.
`jaxrts.hnc_potentials.PauliClassicalMap`()	Calculates the Pauli potential from the exact non-interacting UEG pair distribution function by inverting the Hyper-Netted-Chain calculations:cite:DharmaWardana.2012.
`jaxrts.hnc_potentials.PotentialSum`(...)	A sum of several `HNCPotential` s.
`jaxrts.hnc_potentials.ScaledPotential`(potential)	A `HNCPotential`, scaled with a constant factor.
`jaxrts.hnc_potentials.SoftCorePotential`([beta])	Comparable to `EmptyCorePotential`, but to circumvent the hard cutoff, rather introduce a soft, exponential cutoff.
`jaxrts.hnc_potentials.SpinAveragedEEExchange`([...])	See [Wünsch et al., 2008], Eqn (18).
`jaxrts.hnc_potentials.SpinSeparatedEEExchange`()	See [Wünsch et al., 2008], Eqn (17), Citing [Huang, 1987].
`jaxrts.hnc_potentials.YukawaShortRangeRepulsion`(sigma)	Yukawa Potential with a short-range repulsion as used in [Fletcher et al., 2015].

electron-electron Potential

The potential modelling the interaction between two electrons.

`jaxrts.hnc_potentials.CoulombPotential`([...])	A full Coulomb Potential.
`jaxrts.hnc_potentials.DebyeHueckelPotential`([...])	The Debye-Hückel screening potential as defined, e.g., in [Wünsch et al., 2008].
`jaxrts.hnc_potentials.DeutschPotential`([...])	Quantum diffraction potential suggested by C.
`jaxrts.hnc_potentials.HNCPotential`([...])	Potentials, intended to be used in the HNC scheme.
`jaxrts.hnc_potentials.KelbgPotential`([...])	Quantum diffraction potential suggested by Kelbg.
`jaxrts.hnc_potentials.PauliClassicalMap`()	Calculates the Pauli potential from the exact non-interacting UEG pair distribution function by inverting the Hyper-Netted-Chain calculations:cite:DharmaWardana.2012.
`jaxrts.hnc_potentials.PotentialSum`(...)	A sum of several `HNCPotential` s.
`jaxrts.hnc_potentials.ScaledPotential`(potential)	A `HNCPotential`, scaled with a constant factor.
`jaxrts.hnc_potentials.SpinAveragedEEExchange`([...])	See [Wünsch et al., 2008], Eqn (18).
`jaxrts.hnc_potentials.SpinSeparatedEEExchange`()	See [Wünsch et al., 2008], Eqn (17), Citing [Huang, 1987].
`jaxrts.hnc_potentials.YukawaShortRangeRepulsion`(sigma)	Yukawa Potential with a short-range repulsion as used in [Fletcher et al., 2015].