Ionization potential depression and dynamical structure factor in dense plasmas

TL;DR

This paper investigates how ionic correlations and fluctuations influence ionization potential depression in dense plasmas, using quantum statistical theory and the ionic dynamical structure factor, with applications to aluminum and CH plasmas.

Contribution

It introduces the ionic dynamical structure factor as a key indicator for ionic micro-fields, highlighting the importance of ionic correlations in ionization potential depression.

Findings

Ionic correlations significantly affect ionization potential depression.

Theoretical results agree with experimental data for aluminum and CH plasmas.

Large mass and charge asymmetry in ion mixtures enhances ionic effects.

Abstract

The properties of a bound electron system immersed in a plasma environment are strongly modified by the surrounding plasma. The modification of a basic quantity, the ionization energy, is described by the electronic self-energy and by dynamical screening within the framework of the quantum statistical theory. Introducing the ionic dynamical structure factor as the indicator for the ionic micro-field, we demonstrate that ionic correlations and fluctuations play a critical role in determining the ionization potential depression. This is in particular true for mixtures of different ions with large mass and charge asymmetry. The ionization potential depression is calculated for dense aluminum plasmas as well as for a CH plasma and compared to the experimental data and more phenomenological approaches used so far.

Tables

Table 1. Table 1: IPD in eV and mean charge for CH mixture at density 6.74 g/cm 3 and T = 86 𝑇 86 T=86 eV Kraus16 . The ionization energies for different charge states are I ​ [ C 3 + ] = 64.5 ​ eV , I ​ [ C 4 + ] = 392.1 ​ eV , I ​ [ C 5 + ] = 490.0 ​ eV formulae-sequence I delimited-[] superscript C limit-from 3 64.5 eV formulae-sequence I delimited-[] superscript C limit-from 4 392.1 eV I delimited-[] superscript C limit-from 5 490.0 eV {\rm I\,[C^{3+}]=64.5\ eV,I\,[C^{4+}]=392.1\ eV,I\,[C^{5+}]=490.0\ eV} . We have taken in our calculation an effective SF, where x 𝑥 x is the carbon ratio. (For Refs. see Fig.  1 .)

	a $C^{3+}$ a	a $C^{4+}$ a	a $C^{5+}$ a	mean charge
DH	261.3	326.7	392.0	4.91
SP	91.7	108.3	123.9	4.18
IS	103.2	119.7	135.2	4.21
mEK	116.0	145.0	174.0	4.24
SF $\left(x=0.75\right)$	237.3	296.6	355.9	4.79
SF $\left(x=1\right)$	99.0	123.7	148.4	4.19
expt.				4.92 $\pm$ 0.15

Equations

$$\displaystyle\Big{[}E(1)+E(2)+\sum_{\bf q}[f(1+{\bf q})+f(2-{\bf q})]V_{12}({\bf q})$$

$$\displaystyle+\Delta V^{\rm eff}(1,2,{\bf q},z)\Big{]}\!\psi(1,2,z)+\!\sum_{\bf q}\!\Big{\{}[1\!-\!f(1)\!-\!f(2)]V_{12}({\bf q})$$

$$\displaystyle+\Delta V^{\rm eff}(1,2,{\bf q},z)\Big{\}}\psi(1+{\bf q},2-{\bf q},z)=\hbar z\,\psi(1,2,z).$$

$$\displaystyle\Big{[}E(1)+E(2)\Big{]}\!\psi(1,2,z)+\!\sum_{\bf q}\!V_{12}({\bf q})\psi(1+{\bf q},2-{\bf q},z)$$

$$=\hslash z\psi (1,2,z)$$

$$\Delta V^{\mathrm{eff}}(1,2,\mathbf{q},z)=-V_{12}(\mathbf{q}){\int }_{-\infty }^{\infty }\frac{d{\omega }^{\prime }}{\pi }\mathrm{Im}{\epsilon }^{-1}(q,{\omega }^{\prime }+i0)$$

$$\times \left[n_{{}_{\mathrm{B}}}({\omega }^{\prime })+1\right](\frac{\hslash }{\hslash z-\hslash {\omega }^{\prime }-E(1)-E(2-\mathbf{q})}$$

$$+\frac{\hslash }{\hslash z-\hslash {\omega }^{\prime }-E(1+\mathbf{q})-E(2)}),$$

$${\Sigma }_c(1,z)$$

$$=\text{fmffile}MW8799\text{fmfgraph}\ast (15,-3)\text{fmfleft}i1\text{fmfright}o1\text{fmf}fermioni1,o1\text{fmf}photon,right,tension=0o1,i1={\Sigma }_c^{\mathrm{HF}}(1,z)+{\Sigma }_c^{\mathrm{corr}}(1,z)$$

$$\mathrm{Re}{\Sigma }_c^{\mathrm{corr}}(p,\omega )=-\mathcal{P}\int \frac{d^3\mathbf{q}}{(2\pi {)}^3}\int \frac{d{\omega }^{\prime }}{\pi }{V}_{cc}(q)$$

$$\times \mathrm{Im}{\epsilon }^{-1}(q,{\omega }^{\prime }+i0)\frac{1+n_{{}_{\mathrm{B}}}({\omega }^{\prime })}{\omega -{\omega }^{\prime }-E_{c,\mathbf{p}+\mathbf{q}}/\hslash }.$$

$$\mathrm{Im}{\epsilon }^{-1}(\mathbf{q},\omega +i0)=\frac{e^2}{{\epsilon }_0{q}^2}\frac{\pi }{\hslash \left(1+n_{{}_{\mathrm{B}}}(\omega )\right)}$$

$$\times \left[Z_i^2{n}_i{S}_{\mathrm{ii}}(\mathbf{q},\omega )-2Z_i\sqrt{n_e{n}_i}{S}_{\mathrm{ei}}(\mathbf{q},\omega )+n_e{S}_{\mathrm{ee}}(\mathbf{q},\omega )\right].$$

$$⟨\delta \mathbf{E}\delta \mathbf{E}{⟩}_{\mathbf{q},\omega }=2\pi (Z_i^2{e}^2/q^2)S_{\mathrm{ii}}(\mathbf{q},\omega ).$$

$$S_{\mathrm{ei}}(q,\omega )$$

$$S_{\mathrm{ee}}(q,\omega )$$

$$Z_i{S}_{\mathrm{ii}}(q,\omega )-2\sqrt{Z_i}{S}_{\mathrm{ei}}(q,\omega )+S_{\mathrm{ee}}(q,\omega )$$

$$=$$

$$\mathrm{Re}{\Sigma }_c^{\mathrm{corr},\mathrm{ion}}(p=0,\omega )={\Delta }_c^{\mathrm{ion}}(0,\omega )$$

$$=-\mathcal{P}\int \frac{d^3\mathbf{q}}{(2\pi {)}^3}\int \frac{d{\omega }^{\prime }}{\pi }\frac{V_{cc}(q)}{\omega -{\omega }^{\prime }-E_{c,\mathbf{q}}/\hslash }\frac{\pi Z_i{e}^2{n}_e}{\hslash {ϵ}_0{q}^2}{S}_{\mathrm{ii}}^{\mathrm{ZZ}}(q,{\omega }^{\prime }).$$

$$S_{\mathrm{ii}}^{\mathrm{ZZ}}(q,\omega )\approx S_{\mathrm{ii}}^{\mathrm{ZZ}}(q)\frac{\delta (\omega -{\omega }_{q,i})+\delta (\omega +{\omega }_{q,i})}{1+e^{-\hslash \omega /(k_{B}T)}}.$$

$${\Delta }_{{}_{\mathrm{IPD}}}^{\mathrm{ion}}=-\frac{(Z_i+1)e^2}{2{\pi }^2{ϵ}_0}\cdot \frac{{\kappa }_i^2}{k_{\mathrm{F},i}}{\int }_0^{\infty }\frac{d{q}_0}{q_0^2}{S}_{\mathrm{ii}}^{\mathrm{ZZ}}(q_0),$$

$${\Delta }_{{}_{\mathrm{IPD}}}^{\mathrm{ion}}=-\frac{(Z_i+1)e^2}{2{\pi }^2{ϵ}_0{r}_{{}_{\mathrm{WS}}}}\cdot S({\Gamma }_i),$$

$$S({\Gamma }_i)=F({\Gamma }_i){\int }_0^{\infty }\frac{d{q}_0}{q_0^2}{S}_{\mathrm{ii}}^{\mathrm{ZZ}}(q_0).$$

$$F({\Gamma }_i)=\frac{{\kappa }_i^2{r}_{{}_{\mathrm{WS}}}}{k_{\mathrm{F},i}}={\left(\frac{4}{9\pi }\right)}^{1/3}{r}_{{}_{\mathrm{WS}}}^2{\kappa }_i^2={\Gamma }_i{\left(\frac{12}{\pi }\right)}^{1/3}$$

$$\int_{0}^{\infty}\!dx\cdot\!x^{2}\bigg{\{}\!1\!-\!\exp\left[-\frac{\Gamma_{i}}{x}\!\cdot\!\exp\left(\!-\tilde{\kappa}_{i}(\Gamma_{i})r_{{}_{\rm WS}}\,x\right)\!\right]\!\bigg{\}}\!=\!\frac{1}{3}$$

$$F({\Gamma }_i)=\sqrt[3]{\frac{4}{9\pi }}{r}_{{}_{\mathrm{WS}}}^2{\tilde{\kappa }}_i^2({\Gamma }_i)=\frac{3{\Gamma }_i}{\sqrt{(9\pi /4{)}^{2/3}+3{\Gamma }_i}}$$

$$n_s^{z-1}=\frac{1}{2}{z}^{3/5}(a_0^3{N}_e{)}^{-2/15},$$

$$\epsilon (\mathbf{q},\omega )=1-\frac{1}{{ϵ}_0{q}^2}\Pi (\mathbf{q},\omega )$$

Full text

I Introduction

In the context of new experimental facilities exploring warm dense matter (WDM) and materials in the high-energy density regime, a detailed theoretical investigation of thermodynamic, transport and optical properties of strongly coupled and nearly degenerate Coulomb systems becomes of emerging interest. This is of relevance not only for material science investigating matter under extreme conditions (Mbar pressures, temperatures of 1 eV up to 1 keV), like inertial confinement fusion implosions in laboratory experiments, but also for understanding the structure and evolution of the increasing number of known planets as well of other astrophysical objects.

A fundamental phenomenon is the modification of bound state levels as well as of continuum states owing to the surrounding warm and dense medium. Here, we are interested in the ionization potential depression (IPD) which is relevant for the composition of the plasma, and, in this way, for the thermodynamic and transport properties. We focus on experiments showing the dissolution of spectral lines due to the IPD which determines the ionization degree of WDM. Accurate predictions are necessary for simulation codes such as FLYCHK Chung05 which model plasmas under extreme conditions.

Being a long-standing problem in plasma physics, IPD experiments Hoarty13 ; ciricosta12 ; ciricosta16 ; preston13 ; Fletcher14 ; Kraus16 have been performed recently using the new possibility to produce highly excited plasmas at condensed matter densities by intense short-pulse laser irradiation. Comparisons of observed optical spectra with simulations using traditional expressions for the IPD given by Ecker and Kröll (EK) EK63 or Stewart and Pyatt (SP) SP66 have been performed. Neither of them leads to a satisfying description for all of the available experiments. While, on one hand, Hoarty’s results Hoarty13 on the disappearance of spectral lines seem to favor SP, and, on the other hand, the direct measurements on the ionization energy of the K-shell in aluminum and the subsquent Kα lines by Ciricosta et al. ciricosta12 ; ciricosta16 tend to confirm EK, recently reported results by Kraus et al. Kraus16 can not be understood by either of the two approaches. A more systematic and accurate theory is demanded to describe the measurements.

The commonly used expressions for the IPD derived by Ecker and Kröll EK63 or Stewart and Pyatt SP66 interpolate between the Debye (DH) limit for low densities and an ion sphere (IS) expression, see ZM80 , for high densities. They are based on simplified assumptions such as the introduction of an average static potential to perform Thomas-Fermi calculations. A critical discussion of these approaches and their applicability for the experiments given above was presented in Crowley14 . Other approaches use Hartree-Fock-Slater calculations STJZS14 , Monte Carlo simulations Stransky16 , molecular dynamics simulations Calisti15 , density-functional theory calculations VCW14 , microfield concepts and a detailed configuration accounting description IS13 ; IS14 , or the theory of disordered solids where itinerant band electrons become localized below a mobility edge DP92 .

A systematic approach to describe the properties of dense plasmas is given by the quantum statistical many-body theory, in particular the use of the Green function method KKER . It has been applied to optical properties GHR91 by calculating shifts and broadening of spectral lines in a plasma environment. The shift of bound states and the continuum edge in dense plasmas has also been considered in Refs. Seidel ; RKKKZ78 ; ZKKKR78 .

Already some decades ago, the shifts both of the continuum edge and of the bound state levels have been discussed for the electron-hole plasma in excited semiconductors RKKKZ78 ; ZKKKR78 ; KKER ; Zimmermann ; KSK . Depending on the density and temperature of the electron-hole plasma, excitons are modified by medium effects, and merge with the lowered continuum at the Mott density. Thus, an exciton gas is transformed into an electron-hole liquid. A highly sophisticated theory describing dynamical screening and degeneracy effects by the fermionic plasma constituents had been worked out, explaining precise measurements in excited semiconductors. However, because the ions are heavier compared to the effective mass of holes, a simple transfer of the physics of excited semiconductors to WDM is not possible. The ions remain classical within a large density region, forming strong correlations which are described by the dynamical ionic structure factor (SF) $S_{\rm ii}({\bf q},\omega)$.

In the following, we will give a relation between the IPD and the ionic structure factor. Thus, mean-field (average atom) approaches are improved taking into account fluctuations of the ionic microfield. Further systematic improvements would be possible considering higher order Feynman diagrams in the Green function approach.

II The in-medium two-particle problem

We consider a two-particle system, consisting of an electron (charge $-e$, mass $m_{e}$) and an ion (charge $\left(Z_{i}+1\right)e$, mass $m_{i}$) imbedded in a surrounding plasma. In vacuum, the solution of the Schrödinger equation for the Coulomb interaction is well known. Bound states are found at negative energies, whereas a continuum of scattering states is observed at positive energies. The simple case of the hydrogen atom can be generalized to a two-particle system with total charge $Z_{i}e$, consisting of a core ion with charge number $Z_{i}+1$ and an electron, charge number $Z_{e}=-1$. According to these definitions, the notations $Z_{i}$ and $Z_{i}+1$ denote the charge number of the ions before and after the ionization, respectively. (Note, that the charge number $Z_{i}$ is at least by one smaller than the nuclear charge number of the corresponding atom. For neutral atoms before ionization, we have obviously $Z_{i}=0$.)

If the two-particle system is embedded in a plasma, bound state energies and wave functions as well as the scattering states are modified. A systematic quantum statistical approach to describe these medium effects is given by the method of thermodynamic Green functions KKER ; KSK . In particular, the following in-medium Schrödinger equation (or Bethe-Salpeter equation) can be derived RKKKZ78 ; ZKKKR78 ; KKER ; KSK :

[TABLE]

Here, the single particle states $1=\{\hbar{\bf p}_{1},\sigma_{1},c_{1}\}$ are given by momentum, spin and species, respectively, $E(1)=\hbar^{2}{\bf p}_{1}^{2}/(2m_{1})$. In the case considered here, $c_{1}$ and $c_{2}$ denote the electron and the core ion, respectively. For the interaction we assume the Coulomb potential $V_{12}({\bf q})=Z_{c_{1}}Z_{c_{2}}e^{2}/(\varepsilon_{0}q^{2})$ which contains the charge numbers of the interacting particles, in our case $Z_{c_{1}}Z_{c_{2}}=-(Z_{i}+1)$.

The complex variable $z$ describes the analytical continuation of the functions, defined for the Matsubara frequencies, into the entire $z$ plane. Of interest is the behavior of the functions near the real axis, $z=\omega\pm i\epsilon$.

Neglecting in Eq. (II) the medium effects arising from the effective interaction $\Delta V^{\rm eff}(1,2,{\bf q},z)$ as well as the Fermi distribution functions $f(i)=[\exp(\beta(E(i)-\mu(i))+1]^{-1}$ with $\beta=1/(k_{B}T)$ and $\mu(1)$ denoting the chemical potential of species $c_{1}$, the equation

[TABLE]

has eigensolutions $\psi_{n}(1,2)$ at energies $\hbar z=E_{n}$, well known from hydrogen-like ions. For more complex ions consisting of a nucleus and some bound electrons, a pseudopotential can be introduced to describe the effect of the electrons within the core ion.

The in-medium Schrödinger equation (II) describes the influence of the medium by two effects, Pauli blocking and screening. Pauli blocking is caused by the antisymmetrization of the fermionic wave function. States which already are occupied by the medium are blocked and can not be used for the two-particle system under consideration. The blocking is described by the Fermi distribution function. Pauli exclusion principle is acting as Fock shift $\sum_{\bf q}f(1+{\bf q})\,V_{12}({\bf q})$ in addition to the single particle energy $E(1)$ in Eq. (II) (for charge-neutral plasmas, the Hartree term vanishes). Also in the interaction term, Pauli blocking gives the contribution $-\sum_{\bf q}f(1)V_{12}({\bf q})\psi(1+{\bf q},2-{\bf q},z)$. Both in-medium contributions are caused by the degeneracy of the plasma particles. In the plasmas considered here, electrons may be degenerate because of their small mass $m_{e}$. The ions are non-degenerate and can be treated as classical particles.

Considering only the Pauli blocking effects, the effective (non-hermitean) Hamiltonian of Eq. (II) remains real and can be symmetrized. The energy eigenvalue problem can be solved, and the bound state energies as well as the edge of continuum states are shifted. At a certain density, the bound states merge with the continuum of scattering states and disappear. Within this approximation, which is essentially a mean-field approximation, a sharp value for the lowering of the continuum edge and for the IPD can be calculated.

Screening of the interaction by the medium is described by the effective interaction

[TABLE]

where terms $\propto f(1)$, which give corrections in higher orders of the density, are neglected. ${n}_{\rm{}_{B}}(\omega)=[\exp(\beta\hbar\omega)-1]^{-1}$ is the Bose distribution function. The dynamical properties of the surrounding plasma are contained in the dielectric function $\varepsilon(q,z)$ to be taken at the real axis, $z=\omega^{\prime}+i0$. In general, this is a complex, frequency dependent quantity, with a jump of the imaginary part at the real axis. Often the random-phase approximation (RPA) is taken, and in the static limit $\omega\to 0$ the Debye screening is obtained. In this work, we show that these simple approximations have to be improved in a systematic way which is obtained from the quantum statistical approach.

Including the effective potential, the effective Hamiltonian in the in-medium Schrödinger equation (II) becomes complex and frequency dependent. As a consequence, the eigenstates are no longer stationary states with sharp energy levels which are shifted by the polarisation of the medium, but have a finite life time given by the imaginary part of the effective Hamiltonian. This can be interpreted as collisions with the plasma particles and leads to a broadening of the energy levels. The corresponding quantum statistical approach to plasma line shapes based on the treatment of the polarization function has been worked out GHR91 and will not be investigated in the present work.

Subsequently, sharp level shifts and a sharp shift of the continuum edge are only obtained from a mean-field approximation. Any frequency dependence beyond the mean-field approximation gives imaginary parts and, in this way, a broadening of the continuum edge and the energy levels. The latter problem has been considered also earlier ZKKKR78 ; Seidel where both, real part and imaginary part of the energy levels of the in-medium two-particle problem, are calculated. As a consequence, only the spectral function has a unique physical meaning, showing the spectral line profiles and the smooth transition to the continuum. However, within this work we will focus on the shifts that are obtained from the real part of the effective Hamiltonian.

As shown in KKER ; KSK ; RKKKZ78 ; ZKKKR78 ; Seidel , density effects arise from dynamical screening in the effective potential, expressed by the inverse dielectric function $\varepsilon^{-1}(q,z)$ of the medium in Eq. (II). For bound states, Pauli blocking as well as the screening in the self-energy term ($\Delta V^{\rm eff}$ in the first square bracket of Eq. (II)) and the effective interaction partially compensate each other so that the bound state energy levels are only weakly dependent on the density. In contrast, the energy shift of the continuum states is determined only by the self-energy contribution. Therefore, in leading order of the density, the medium modification of the IPD is given by the shift of the edge of continuum states. For a more extended discussion see RKKKZ78 ; ZKKKR78 ; Zimmermann ; Seidel ; KKER ; KSK .

A standard expression for the dielectric function $\varepsilon(q,z)$ is the random phase approximation (RPA). From the real part of the self-energy, the Debye shift of the continuum edge is immediately observed. Here we discuss improvements beyond RPA to evaluate the shift of the continuum edge occurring at ${\bf p}_{1}={\bf p}_{2}=0$. Thus, our approach, which is based on a systematic quantum statistical approach, can be regarded as an improvement of the Debye theory or other approaches using semi-empirical assumptions such as the ion sphere model.

III Shift of single particle states

III.1 Self-energy of single particle states

In the single-particle picture, the influence of the plasma environment on the properties of the investigated particle is merged into the self-energy $\Sigma_{c}(1,z)$. It can be represented by Feynman diagrams, in lowest approximation by the diagram (also known as $V^{s}G$ or $GW$ approximation) with the dressed propagator $G$ and the screened potential $V^{s}$

[TABLE]

The Hartree-Fock (HF) contribution to the self-energy has been investigated elsewhere, see KKER , and will not be discussed here. The correlation part of the self-energy $\Sigma^{\rm corr}_{c}(1,z)$ contains the contribution of the interaction with electrons, as well as the interaction with ions. We are interested in the real part of the self-energy since it describes the continuum shift. It follows from Eq. (II) by renaming, e.g., $\hbar z-E(2)=\hbar\omega$ in the last term of (II). Then we have

[TABLE]

(${\cal P}$ denotes the principal value.) In general, the dielectric function is connected to the dynamical SF via the fluctuation-dissipation theorem. For a two-component plasma (free electrons with charge $-e$, ions with effective charge $Z_{i}e$ and charge neutrality $Z_{i}n_{i}=n_{e}$), the imaginary part of the inverse dielectric function can be expressed via the dynamical SFs, see also Chihara ,

[TABLE]

The dynamical SFs $S_{cd}({\bf q},\omega)$ characterize the plasma in response to any perturbation. For instance, they have been investigated to describe X-ray Thomson scattering, see Ref. GRHGR07 . Other plasma properties such as the electrical conductivity are also governed by the dynamical SF. The dynamical SFs are related to the density-density correlation functions $\langle\delta n_{c}({\bf r},t)\delta n_{d}(0,0)\rangle$ via Fourier transformation. Note, that it is also connected to the symmetrized correlation function of the longitudinal microfield fluctuations $\langle\delta{\bf E}\delta{\bf E}\rangle_{{\bf q},\omega}$ KSK

[TABLE]

For further discussion of the general expressions (III.1) and (6), we perform exploratory calculations using model approaches for the dynamical SFs. Following the relations for the dynamical SFs reported in Refs. Chihara ; GRHGR07

[TABLE]

the decomposition of the dynamical SF as introduced in Eq. (6) can be divided into $S_{\rm ee}^{0}(q,\omega)$ of the fast moving free electrons and the ionic part $S_{\rm ii}^{ZZ}(q,\omega)$ which includes also the screening cloud of the slowly moving electrons following the ionic motion, denoted by ${q}_{\rm sc}(k)$,

[TABLE]

with $S_{\rm ii}^{ZZ}(q,\omega)=\left(1-{q}_{\rm sc}(k)/Z_{i}\right)^{2}S_{\rm ii}(q,\omega)$. The electronic contribution to the continuum lowering is described by the electronic SF $S_{\rm ee}^{0}$ and has been widely discussed, see Refs. KKER ; KSK . Results in the Montroll-Ward approximation are well known. Compared to the ionic contribution $(\sim Z_{i}^{2}e^{2})$, the electronic contribution $(\sim e^{2})$ is usually quite small for highly charged states.

Following Eqs. (III.1) and (6), we now discuss the ionic contribution to the correlation shift of the continuum edge ${\rm Re}\,{\Sigma}^{{\rm corr}}_{e}(0,\omega)+{\rm Re}\,{\Sigma}^{{\rm corr}}_{i}(0,\omega)$. It is expressed as

[TABLE]

Thus, the ionic contribution to the continuum shift is related to the dynamical SF of the ions. The quasiparticle shift has to be defined self-consistently at $\omega={\Delta}^{{\rm ion}}_{c}(0,\omega)$, but this shift is compensated in the denominator of the integrand by the energy $E_{c,{\bf q}}$, which is shifted too. Then the ionic contribution ${\Delta}^{{\rm ion}}_{c}(0,\omega)$ is given by ${\Delta}^{{\rm ion}}_{c}(0,0)$, later denoted as ${\Delta}^{{\rm ion}}_{c}$.

III.2 Plasmon pole approximation

Under WDM conditions considered here, the ions are strongly coupled, so that the SF $S_{\rm ii}^{ZZ}$ should not be taken in the Debye limit. However, the plasma ions can be treated classically. Therefore, for ${\Delta}^{{\rm ion}}_{c}(0,0)$, see Eq. 10, we consider the limit $\hbar\rightarrow 0$ in the propagator $1/[-\omega^{\prime}-\hbar q^{2}/(2m_{c})]$. In addition, the ions move very slowly in comparison to the electrons, which indicates that it is reasonable to replace the dynamical SF of ions by the static SF within some approximations. We use the plasmon pole approximation ${\rm Im}\,\varepsilon^{-1}_{\rm ion}(q,\omega)=-\pi\omega_{i}^{2}\{\delta(\omega-\omega_{q,i})-\delta(\omega+\omega_{q,i})\}/(2\omega_{q,i})$, where $\omega_{q,i}^{2}=\left(q^{2}\,\omega_{i}^{2}\right)/\left(\kappa_{i}^{2}\,{S}^{{\rm ZZ}}_{\rm\rm ii}(q)\right)$ is fulfilling the f-sum rule GRHGR07 with the ionic plasmon frequency $\omega_{i}^{2}=Z_{i}^{2}n_{i}e^{2}/(\epsilon_{0}m_{i})$ and the inverse Debye screening parameter $\kappa_{i}^{2}=\omega_{i}^{2}m_{i}/k_{B}T$. Then we find the following expression

[TABLE]

The physical meaning of the replacement of the dynamical SF by the static SF in Eq. (11) is that the ions are considered to have a fixed distribution in the plasma neglecting temporal fluctuations.

For the ionization process $i_{{}_{Z_{i}}}\rightarrow e+i_{{}_{Z_{i}+1}}$, the IPD can be given by the difference between the self-energy before and after the ionization of the investigated system, i.e., ${\Delta}^{{\rm ion}}_{\rm{}_{IPD}}={\Delta}^{{\rm ion}}_{i}-({\Delta}^{{\rm ion}}_{e}+{\Delta}^{{\rm ion}}_{i+1})$. We assume that the ionic structure of the plasma environment does not change during the ionization. Therefore, we insert expression (11) into ${\Delta}^{{\rm ion}}_{c}$, see Eq. (10). Performing the approximations as discussed in context with Eq. (10), we obtain for the IPD

[TABLE]

where $q_{0}=q/k_{\mathrm{F},i}$ is the reduced wavenumber with $k_{\mathrm{F},i}=\left(3\pi^{2}n_{i}\right)^{1/3}$. Considering the ion-ion SF ${S}^{{\rm DH}}_{\rm ii}(q)=q^{2}/(q^{2}+\kappa_{i}^{2})$ of a one-component plasma (OCP), valid in the low density and the high temperature limits, the DH result ${\Delta}^{{\rm ion}}_{\rm{}_{DH}}=-(Z_{i}+1)e^{2}\kappa_{i}/(4\pi\varepsilon_{0})$ is recovered for the ionic contribution to the IPD. The expression (12) shows a strong dependence on the temperature indicated by the inverse Debye length $\kappa_{i}$ appearing in the frequency $\omega_{q,i}\sim\kappa_{i}^{-1}$ in Eq. (11), and also by the static ionic SF. The screening parameter $\kappa_{i}^{2}\propto 1/(k_{B}T)$ follows from the linearized Debye theory for classical systems. Nevertheless, with increasing coupling parameter, the plasma starts to crystallize and forms a periodic structure. In this case, the frequency $\omega_{q,i}$ is determined by the Wigner-Seitz radius $r_{{}_{\rm WS}}=(4\pi n_{i}/3)^{-1/3}$, as discussed, e.g., in Ref. Kalman . Consequently, the parameter $\kappa_{i}^{2}$ occuring in $\omega_{q,i}$ should be replaced by a more general expression $\tilde{\kappa}_{i}^{2}(\Gamma_{i})$ depending on the ionic coupling parameter $\Gamma_{i}=Z_{i}^{2}e^{2}/(4\pi\epsilon_{0}k_{B}Tr_{{}_{\rm WS}})$. We can express Eq. (12) in the form

[TABLE]

introducing the parameter function

[TABLE]

From the Debye-Hückel theory follows

[TABLE]

valid for weakly coupled system $\Gamma_{i}\ll 1$. For strong coupling, a similar type of expression, $F(\Gamma_{i})=\sqrt[3]{4/(9\pi)}\cdot r^{2}_{{}_{\rm WS}}\,\tilde{\kappa}_{i}^{2}(\Gamma_{i})$ can be defined and will be discussed in detail in the next section. One should keep in mind that, for a fixed charge state $Z_{i}$, the parameter function $S(\Gamma_{i})$ should gradually tend to a constant due to crystallization of the plasma with increasing coupling parameter Kalman . At a fixed temperature and density, the parameter function $S(\Gamma_{i})$ slightly depends on the charge number since the dependence on charge number $Z_{i}$ in the static ionic SF ${S}^{{\rm ZZ}}_{\rm\rm ii}(q_{0})$ compensates with that of the function $F(\Gamma_{i})$.

The approach, presented in this work, shows a close connection of the IPD to the detailed structure of the plasma system. The general expression (13) with (14) should work within the valid range of the fluctuation-dissipation theorem for both equilibrium and non-equilibrium systems described by the static SF of the quantum many-body system. Once the SF is known from other methods, for instance, simulations or Thomson scattering measurements, the IPD can be directly evaluated. In this work, the local thermodynamic equilibrium is assumed for the calculation. Further investigations are needed to describe non-equilibrium situations, for instance, after irradiations by strong short-pulse laser beams.

IV Results and discussion

IV.1 Model calculation: comparison to other approaches

To determine the function $F(\Gamma_{i})$ in the parameter function $S(\Gamma_{i})$, Eq. (14), the implicit normalization relation KKER

[TABLE]

according to the non-linear Debye theory is used, which avoids negative densities of the screening cloud. The Debye-Hückel theory can be recovered by expanding the exponential function outside of the square brackets up to the first order in $\Gamma_{i}$, see Eq. (15). For intermediate and strong coupling, Eq. (16) has to be solved numerically. In this work, we introduce the following expression

[TABLE]

as an approximation which reproduces the Debye-Hückel limit (15) as well as the numerical solutions of Eq. (16) in the strong coupling regime of interest.

In general, the pair correlation function exhibits a peak near $r_{{}_{\rm WS}}$ when approaching the liquid state, which would be reasonably well described by a Percus-Yevick SF. In the intermediate density region, an interpolation formula for the ionic SF can be applied, see Ref. GRHGR07 . In the following, we use expression (13) together with Eq. (17) and the static ionic SF as given in Ref. GRHGR07 to evaluate the ionic contribution to the IPD in the plasma.

As an exploratory calculation in order to compare to other theoretical models, we consider the IPD of the ion Al11+ $(Z_{i}=11)$ at a temperature of 600 eV. Fig. 1 shows the IPD calculated using different theoretical models. It can been seen, that the IPD from SP SP66 , original EK (oEK) EK63 and our result are in good agreement with the DH shift in the low density region. Above the critical density ${n}^{{\rm crit}}_{\rm{}_{EK}}=3/(4\pi)\cdot(4\pi\varepsilon_{0}k_{B}T/(Z^{2}e^{2}))^{3}$ with the nuclear charge $Ze$, the underestimation of the IPD by the SP model and the overestimation by the modified EK (mEK) model ciricosta12 can be seen in comparison to the original IS (oIS) model. Note that, with increasing density, corresponding to increasing coupling parameter $\Gamma_{i}$ ($\Gamma_{i}=0.16$ for the density $0.001$ g/cm3 and $\Gamma_{i}=7.28$ for the density $100$ g/cm3), our result shows, on one hand, a transition from SP at low densities (weakly and moderately coupled) to mEK at large densities (strongly coupled), and, on the other hand, a good agreement with the oIS model in the intermediate density region.

IV.2 Numerical results for experimental conditions

We now discuss the application of our model calculation to conditions observed in experiments. In the experiments of Hoarty et al. Hoarty13 ; Hoarty13a , the spectral lines emitted from Al11+ were observed. The investigated density range is 1.2 to 9 g/cm3 at electron temperatures in the range of 550 to 700 eV. The disappearance of the ${\rm He}_{\beta}$ line would be due to the dissolution of n=3 levels. The assumption of local thermodynamic equilibrium is believed to be valid for the high densities Hoarty13a , which implies the ionic coupling parameter is estimated to be in the range of 2-4. In such a moderate coupling regime, the SP and IS models should result in the best agreement with the experiment, as can be seen by looking at the relevant density range in Fig. 1.

The latter is shown in Fig. 2 for a more detailed discussion. The horizontal line denotes the unperturbed ionization potential (220 eV) of the upper level of the Al11+ ${\rm He}_{\beta}$ line nist . The density range, in which the disappearance of ${\rm Ly}_{\beta}$ and ${\rm He}_{\beta}$ lines in aluminum plasma Hoarty13 was measured, is marked as solid line. It occurs at a density somewhere between 5.5 and 9 g/cm3, which is in reasonable agreement with the predictions by FLYCHK Chung05 using the SP model. According to calculations based on a generalized ion-cell model by Crowley Crowley14 , for this range of densities, the modified IS (mIS) model is most suitable. This is consistent with predictions for spectra using the CASSANDRA opacity code with an IS model for the IPD Hoarty13a , where the dissolution of lines from $n=3$ levels is indicated to take place between the density of $6\sim 8$ g/cm3. As shown in Fig. 2, the EK model results in much larger IPD values in comparison to the SP model, and hence leads to a disappearance of spectral lines at a lower density of about 2 g/cm3. A similar estimate was given in the calculation by Crowley Crowley14 . Our approach, predicting a critical density between $7\sim 8$ g/cm3 for the disappearance of $n=3$ levels, gives also an excellent agreement with the experimental data and with the predictions by the CASSANDRA opacity code Hoarty13a .

Using FEL ciricosta12 ; ciricosta16 , further experiments have been performed recently. An aluminum sample at solid density was isochorically heated up to electron temperatures of 200 eV, indicating a strongly coupled plasma, and the IPD was directly measured for different charge states. The LCLS pulse duration in this experiment was estimated to be less than 80 fs. The ionic plasma frequency ${\omega}^{{\rm ion}}_{\rm pl}$ in this laser-produced plasma is found to be in the order of $10^{14}$/s. In contrast, the response of the electrons to the laser field is much faster and can be described by the electron frequency ${\omega}^{{\rm el}}_{\rm pl}\sim 10^{16}$/s. In comparison to the laser pulse, the electrons have enough time to exchange energy between each other and with the laser field and are isochorically heated to a high temperature. Because of the large mass of the ions, the response of the ionic subsystem to the external fluctuations is so slow that the ions in the plasma are weakly excited by the photons and by fast moving electrons, which implies that the ions are colder than the electrons. Of essential importance in the measurement is that the IPD for distinct charge states, inferred from the triggering energy of the photoionization, is measured at different time stages. This fact indicates that the ions are heated during the time evolution and local thermal equilibrium may be achieved.

Fig. 3 shows the experimental results in comparison to several calculations using different theoretical models. The direct measurement of the IPD in aluminum plasma can be explained reasonably well by the mEK model as discussed in Ref. ciricosta12 . Vinko et al. VCW14 performed detailed calculations on electronic structures of Al ions in a plasma via the finite-temperature DFT method. They found that the IPD for a given charge state could be well understood in terms of the electronic structure of valence electron states near core-excited ions within a pseudo-neutral atom approximation. The results from the two-step Hartree-Fock calculations by Son et al. STJZS14 and from the calculations by Crowley Crowley14 are less satisfying. However, as shown in Fig. 3, the experimental data can also be reproduced by our approach, where the effect of the surrounding plasma on the ions is directly accounted for by the screened ionic SF. In our calculation, the LTE condition was assumed. This might not be suitable for the experimental measurements where the ions remain relatively cold because of the femtosecond nature of the X-ray pulse VCW14 . For this non-equilibrium case, the ionic SF under non-LTE conditions should be taken into account. However, detailed calculations of the ionic SF in the non-LTE case are rather intricate and are still in progress.

The application of simple IPD models (e.g. SP) to a mixture of different ions is problematic as displayed by recent measurements on a CH mixture at NIF Kraus16 . The obtained mean charge state can not be explained by either the SP or the mEK models, as shown in Tab. 1. Although the DH shift is inappropriate under the experimental conditions (strong coupling of the carbon ions $({\Gamma}_{\rm C}\sim 4)$), it results in larger IPDs and therefore gives a more reasonable agreement with the experiment than all other models. This fact can be attributed to the deficiency to account for strong correlation and fluctuation effects in these models.

For the CH mixture, the influence of a different chemical species, the protons from the fully ionized hydrogen, on the properties of the carbon ions is, within SP and EK models, described by an additional electron density. In our approach, this effect can be more consistently taken into account by the ionic SF, which includes the response of all charged particles in the plasma. We applied the linear mixing rule DG09 for the SF of a multi-component plasma, ${S}^{{\rm ZZ}}_{\rm{}_{\rm ii}}(q_{0})=x\,{S}^{{\rm ZZ}}_{\rm{}_{\rm CC}}(q_{0})+(1-x)\,{S}^{{\rm DH}}_{\rm{}_{\rm HH}}(q_{0})$. For the ratio $x=0.75$ of carbon, an estimated mean charge of 4.79 and therefore a close match with the experimental value of 4.92 $\pm$ 0.15 Kraus16 is found. Under the experimental conditions Kraus16 , the carbon ions are strongly coupled while the protons are weakly correlated. The SF of the protons modifies the structure of the integrand in Eq. (14) leading to higher IPD values for the carbon ions, and therefore push the carbon ions to a higher charge state.

Calculations for a pure C plasma at the same conditions (same ionic density of carbon and same temperature), lead to the mean ionization degree of 4.2. For the CH plasma, the asymmetry of the charges and masses of protons and carbon ions lead to strong fluctuations and hence significantly enhance the ionization. Future discussions on experiments with pure C targets may test this effect.

More recently, a new experimental study on the ionization states of warm dense aluminum ($T_{e}\sim 20-25$ eV and $\rho\sim 2.7$ g/cm3) was performed Mo17 . It was found that the observed time-dependent absorption spectra are better described using the mEK model for the IPD than using SP and IS models. This result agrees with our findings. For the given experimental conditions, the ion charge states Al4+ and Al5+ are clearly seen, which indicates an ion coupling parameter of $\Gamma_{i}\sim 7$. As discussed for Fig. 1, in such strongly coupled systems, the mEK model should lead to a better description for the IPD.

V Conclusions and further improvements

We treated the in-medium two-particle problem (II) within a quasiparticle approach and obtained the contribution of the shift of the continuum edge to the IPD. In addition to the continuum edge, also the bound state energy levels are shifted. Although their shifts are small as compared to the continuum lowering, see Seidel ; RKKKZ78 ; ZKKKR78 ; KKER ; KSK , these bound state level shifts should also be considered in a detailed calculation for the IPD. Note that the shift of bound state levels has been observed in the shift of spectral lines, and quantum statistical calculations GHR91 agree well with experimental data.

A more serious problem is the use of the quasiparticle approximation. Within a sophisticated Green function approach, the quasiparticle propagators are replaced by spectral functions, see, e.g., Fortmann09 , which describe also the finite life time of the quasiparticle excitations. This leads to the fact that the energy gaps between the optical lines describing bound state transitions are washed out (Inglis-Teller effect ingtel39 ).

In his monograph, Griem Griem64 described the broadening of spectral lines by the Stark effect leading to a shift of the observed series limit. The latter is described by

[TABLE]

with $n_{s}$ - main quantum number, $N_{e}$ - electron number density. Eq. (26) was determined by a fit to a Holtsmark profile holts19 and corresponds to Eq. (4) in Ref. ingtel39 . Griem mentions that the shift of the series limit where lines fully overlap does not have a direct relation to the lowering of the ionization potential (last paragraph of section 5.7 in Griem64 ). As discussed in Sec. II, definite values for the plasma parameters, where the ionization potential vanishes, can only be given within a quasiparticle (mean field) approximation which gives sharp energy levels. As soon as the imaginary part of the effective Hamiltonian (II) is taken into account, the sharp energy levels become broadened as a consequence of their finite life time owing to collisions with the plasma particles. Consequently, the rigorous discrimination between bound states (having a finite life time) and continuum states (including resonances) is no longer possible, and, strictly speaking, the concept of IPD based on sharp quasiparticle energy levels becomes obsolete.

We performed exploratory calculations using a simple model for the dynamical SF (11). As a main result, we found, that correlations which are described by the ionic SF are indeed relevant for the IPD. As proposed, it would be of interest to perform experiments with pure substances like C. Compared to the large IPD seen in CH experiments Kraus16 , a lower IPD is expected for a pure C plasma. More details of the ionic subsystem may be incorporated, in particular the relaxation of the ionic subsystem and collective excitations (plasmons, phonons) can be treated. For a discussion see also Ref. Crowley14 .

Our approach is based on a Born approximation for the interaction of the two-particle system with the plasma ions. The internal structure and dynamics of the plasma is described by the dielectric function which contains the polarization function $\Pi({\bf q},\omega)$,

[TABLE]

Improving the RPA expression for the polarization function, two-particle correlations are included, see also RD . In particular, the ionic dynamical structure factor is taken into account if the cluster decomposition of the polarization function is considered, here the two-ion distribution. Similar approaches have been used for the optical spectra GHR91 where also a cluster decomposition of the polarization function has been considered.

This discussion gives a conception of how to improve our approach. The Born approximation has to be completed accounting for multiple interaction (so-called T matrix). A more general diagram for the self-energy looks like

where the double line denotes the two-ion propagator, and the screened interaction with the investigated particle is considered in ladder approximation. The approximation (4) for the self-energy results from the first contribution of the ladder sum which contains only two electron-ion interaction lines.

Starting from the general expression (4), we obtain a rather simple formula (13) for the IPD containing the ionic static structure factor. We emphasize that this result could now be improved by systematically removing again some of the approximations for the dynamical SF (11). In particular, the plasmon pole approximation in handling the dynamical SF is a model assumption which can be improved, e.g., by numerical simulations. Finally, an advantage of our quantum statistical approach is that any degeneracy effect can be taken into account in a systematic way, which becomes of interest at increasing densities.

Acknowledgement: One of the authors (CL) would like to thank S. Vinko for making experimental data available and D. Kraus, D. Hoarty, and Y. Hou for helpful discussions. This work is supported by the German Research Foundation DFG within SFB 652.