Abstract
Superdense plasmas widely exist in planetary interiors and astrophysical objects such as browndwarf cores and white dwarfs. How atoms behave under such extremedensity conditions is not yet well understood, even in singlespecies plasmas. Here, we apply thermal density functional theory to investigate the radiation spectra of superdense iron–zinc plasma mixtures at mass densities of ρ = 250 to 2000 g cm^{−3} and temperatures of kT = 50 to 100 eV, accessible by doubleshell–target implosions. Our ab initio calculations reveal two extreme atomicphysics phenomena—firstly, an interspecies radiative transition; and, secondly, the breaking down of the dipoleselection rule for radiative transitions in isolated atoms. Our firstprinciples calculations predict that for superdense plasma mixtures, both interatomic radiative transitions and dipoleforbidden transitions can become comparable to the normal intraatomic Kαemission signal. These physics phenomena were not previously considered in detail for extreme highdensity plasma mixtures at superhigh energy densities.
Introduction
Extreme material conditions, such as superhigh density and warm or hot temperatures, can be widely found in the universe. For example, browndwarf cores and white dwarfs^{1,2,3} can have a mass density of ρ = 10^{3}–10^{7} g cm^{−3} and temperatures up to ~10^{6} K. Thanks to technological advancements, such extreme states of matter can now be created in the laboratory using powerful lasers^{4,5,6,7} and/or pulsedpower machines^{8}. For instance, deuterium and tritium contained in a millimetersize inertial confinement fusion (ICF) target can be squeezed to ρ = 10^{2}–10^{3} g cm^{−3} by powerful lasers through laserdriven compression and spherical convergence^{9,10,11,12,13}. Using doubleshell implosions^{14,15}, mid/highZ materials can be squeezed to superhigh densities ranging from ρ = 10^{3}–10^{4} g cm^{−3} with a temperature ranging from tens to hundreds of electron volts (1 eV ~ 11,604 K). Understanding how matter behaves at such extreme conditions is the purview of highenergydensity physics, inertial confinement fusion, planetary science, and astrophysics.
Under superdense conditions, atoms and molecules—the fundamental building blocks of matter—can have drastically different properties from those found under ambient conditions. For instance, because of pressure ionization, the binding energy of core electrons of atoms might significantly shift in dense plasmas^{16,17,18} when compared with the case of isolated atoms. By probing the energy level changes in these systems, one can infer the denseplasma conditions if one knows precisely beforehand how atoms behave in highdensity environments. Moreover, such an extreme environment experienced by embedded ions can also alter the characteristics of atomic wavefunctions because of closely encountered neighboring ions. This can have profound implications for understanding radiation transport in such dense plasmas. For example, the dipoleselection rule for isolated atoms can break down in extremely dense plasmas. Most interestingly, if a plasma mixture is compressed to very high densities above 10^{3} g cm^{−3}, wavefunction overlapping of deeply bound electrons between different atomic species may occur. A schematic diagram of such a scenario is depicted in Fig. 1, in which the iron (Fe) and zinc (Zn) ions in the mixture closely interact with each other in superdense plasma. As a result of the short distance (d) between the two species, their outer electrons on n = 3 and n = 4 levels can be pressure ionized and their 2s and 2p states might also be significantly distorted by each other. The significant overlapping of n = 2 states could enable a physics phenomenon—interspecies radiative transitions (IRT)—to occur.
As Fig. 1 illustrates, if 1s holes of both Fe and Zn ions are created by either radiation pumping^{19} or energetic electron collisions^{20,21}, the 2s and 2p electrons of one species (e.g., Fe) could radiatively transition to the 1s hole of the other species (e.g., Zn), giving interspecies Kα emission. On the other hand, if the 2p state is no longer fully occupied and the 1score state is filled, the interspecies Kα absorption could occur in such extremely denseplasma mixtures. To the best of our knowledge, this phenomenon of IRT between bound states has not been considered in emissivity/opacity calculations of plasma mixtures^{22,23,24,25,26}, even though interCoulombic Auger decay was discovered in large molecules and clusters^{27,28,29}, and collisioninduced absorption and emission between atomic gases was discussed^{30,31,32}. Furthermore, the significant distortion of the 2s state resulting from closest neighboring ions will make both intraatomic and interatomic 2s−1s transitions possible, which are dipoleforbidden for an isolated atom and relatively lowdensity systems when deeply bound 2s and 1s states preserve their ideal s symmetry. To classify various transitions, we use the word of intraatomic for transitions of electron having both initial state and final state belong to the same atom, while interatomic transitions involve two atoms that can be either the same type or different species.
Here, we present interspecies radiative transition results from firstprinciples calculations by thermal densityfunctional theory (DFT) using the ABINIT software package^{33,34} in the planewavebased projector augmentedwave (PAW) approach. All electrons are considered as evolving—no frozen core approximation—and spin–orbit coupling effects are explicitly included. As an example for midZ elements presented in browndwarf cores, a denseplasma mixture of Fe and Zn was considered with an equal atomic fraction for each species (50:50). We varied the Fe–Zn plasma density from ρ = 250 to 2000 g cm^{−3} and temperatures of kT = 50 to 100 eV. For a chosen plasma condition, we first ran orbitalfree DFTbased molecular dynamics^{35,36} to obtain the equilibrium ionic configurations. We then took several snapshots of uncorrelated ionic configuration for the electronic structure calculations using ABINIT. Once the electronic structure of a dense plasma is determined from the ABINIT calculations, we created a 1shole state by removing the occupation of the 1s state for both Fe and Zn ions. Finally, we calculated the dipole matrices to determine the emission spectra of superdense Fe–Zn plasmas with the Kubo–Greenwood formalism. More numerical details and convergence tests can be found in the “Methods” and Supplementary Information.
Results
Interspecies radiative transition in warm and superdense plasmas
For a superdense and warm Fe–Zn plasma of ρ = 1000 g cm^{−3} and kT = 50 eV with 1s vacancies of both Fe and Zn ions, the calculated emission coefficient as a function of photon energy is shown by the solidred line in Fig. 2. To identify the IRT features, we also plotted the spectra of singlespecies Fe (dashed–dotted green line) and Zn (dashed blue line) plasmas in Fig. 2, respectively. Again, these pure plasmas have the same density and temperature conditions as that of the Fe–Zn mixture.
From Fig. 2, one can clearly see that four additional spectral peaks appear in the superdense Fe–Zn plasma mixtures (highlighted by the dashed ellipse): the two emission lines located at hν ≈ 8666 eV and hν ≈ 8816 eV correspond to transitions from the 2s and 2p states of the Fe ion to the 1s hole of the Zn ion, while the other two peaks at hν ≈ 5838 eV and hν ≈ 6012 eV belong to radiative transitions of 2s/2p electrons of the Zn ion to the 1s vacancy of Fe. Besides these interatomic Kα emissions, the dominant intraatomic Kα lines for each species, are of course, present in the emission spectra in Fig. 2. The vertical dotted black lines mark the normal intraatomic Kα locations of ambient Fe and Zn, respectively. The red shift of the intraatomic Kα line is caused by the increased electron screening resulting from the dense plasma environment^{19}. In addition, the intraatomic 2s → 1s transitions for each species, although being about three orders of magnitude weaker than the normal intraatomic Kα lines, also appear as a consequence of the breaking down of the dipoleselection rule due to the densityinduced distortion of 2s states. Finally, the continuum emissions from free electrons filling 1s holes of Fe and Zn ions are also present in the emission spectra, as expected (shown by Fig. 2).
To further understand the emission spectra of Fig. 2, we have computed the density of states (DOS) for the three denseplasma cases. The results are plotted in Fig. 3, in which Fig. 3a, 3b are for Feonly and Znonly plasmas, respectively. One finds that the outer bound states of 3s, 3p, 3d (or 4s) states of Fe and Zn atoms have merged into the continuum because of pressure and thermal ionization. Note that the continuum states below and above the Fermi level (E_{F}) (i.e., chemical potential) are partially occupied. Clearly, the discrete states of 1s, 2s, and 2p of Fe and Zn ions are evidenced in Fig. 3a, 3b. By looking into the occupations on states below the Fermi energy, the estimated average ionizations are <Z> ≈ 17.3 and <Z> ≈ 19.1, respectively, for Feonly and Znonly cases. This indicates that the 2p state of Fe begins to be partially occupied. When Fe and Zn plasmas are mixed together, their discrete states of 1s, 2s, and 2p are slightly red/blue shifted by ~15 to 30 eV in Fig. 3c when compared with the corresponding pureplasma cases. This shift can be attributed to the interactions between the two species. Now, if their 1s states become empty, i.e. a hole/vacancy is created, the radiative transitions from 2s/2p electrons of Fe and Zn ions to fill 1s holes give rise to the corresponding emission lines in Fig. 2. The breaking down of dipoleselection rule for the 2s → 1s transitions is caused by nonspherical character in the 2s state due to densityinduced distortions. Finally, the emission from transitions of continuum to the 1s hole can also be explained.
Density dependence of interspecies radiative transition (IRT)
To explore how density change affects the interspecies radiative transition in superdense plasmas, we have performed similar firstprinciples calculations by varying the Fe–Zn density from ρ = 250 g cm^{−3} to ρ = 2000 g cm^{−3} but keeping kT = 50 eV. The DFTpredicted emission spectra are plotted in Fig. 4 for three different Fe–Zn plasma densities of ρ = 500, 1000, and 1500 g cm^{−3}. Again, the IRT peaks are highlighted by the dashed ellipses in each panel of the figure.
At a relatively lower density of ρ = 500 g cm^{−3}, Fig. 4a shows that the interatomic Kα emission is significantly weaker than the normal intraatomic Kα emission by ~3 to 4 orders of magnitude. They are even lower than that of dipoleforbidden intraatomic 2s → 1s transitions. It is noted that the spin–orbit coupling–induced splitting of Kα1 and Kα2 is clearly seen for Zn, but for Fe they are merged into one peak because of density/temperature broadenings. As the Fe–Zn plasma density increases to ρ = 1000 g cm^{−3} and ρ = 1500 g cm^{−3}, the interatomic Kα emission peaks drastically rise in amplitude and their widths increase as a result of strong density broadening (Fig. 4b, c). At an extremely high Fe–Zn density of ρ = 1500 g cm^{−3}, Fig. 4c indicates that the peak amplitude of interatomic Kα from 2p(Fe) → 1s(Zn) transition can approach ~10% of the intraatomic Kα from 2p(Zn) → 1s(Zn) transition, which should be readily detectable in experiments. It is noted that these interatomic transitions become even stronger than the dipoleforbidden intraatomic 2s(Zn) → 1s(Zn) transition.
One interesting feature seen in Fig. 4 is that the interatomic Kα emission from 2p(Fe) → 1s(Zn) transition is always stronger than that of 2p(Zn) → 1s(Fe). To further explore this asymmetry and the overall trend of IRT versus plasma density, we have plotted the DFTpredicted ratio of the interatomic Kα signal \({\mathrm{K}}_\alpha ^{{\mathrm{Zn}}\, \to \,{\mathrm{Fe}}}\) or \({\mathrm{K}}_\alpha ^{{\mathrm{Fe}}\, \to \,{\mathrm{Zn}}}\) to the corresponding intraatomic emission \({\mathrm{K}}_\alpha ^{{\mathrm{Fe}}\,}\) (or \({\mathrm{K}}_\alpha ^{{\mathrm{Zn}}}\)) (by the red diamond symbols in Fig. 5) as a function of interatomic distance between Fe and Zn ions. For a chosen Fe–Zn plasma density varying from ρ = 250 g cm^{−3} to ρ = 2000 g cm^{−3} at the same temperature of kT = 50 eV, we derived the Fe–Zn distance (d) from the orbitalfree DFTMD runs, in which d corresponds to the peak location of the pair distribution function—g(r). The fullwidth halfmaximum of g(r) peak gives the plausible range of Fe–Zn distance (“error bar” of d in Fig. 5).
To further see how the interatomic Kα emission qualitatively changes with d, we have used two simple models to estimate the intertointra Kα ratio through the following dipolematrix elements calculations:
where the unit vector e defines the direction of dipole moment. Depending on how we choose the initial and final states in the nominator of the above equation to estimate the matrix element, we have two simple models: (a) the independentatom model (IAM) which simply takes the 2p and 1s hydrogenic wavefunctions of independent Fe and Zn atoms as the initial and final states (namely \(\left {\left. {\psi _{{\mathrm{initial}}}^{{\mathrm{IAM}}}({\mathbf{r}})} \right\rangle } \right. = \left {\left. {\psi _{2p}^{{\mathrm{Zn(Fe)}}}({\mathbf{r}})} \right\rangle \quad {\mathrm{and}}\quad \left {\left. {\psi _{{\mathrm{final}}}^{{\mathrm{IAM}}}({\mathbf{r}})} \right\rangle } \right. = \left {\left. {\psi _{1s}^{{\mathrm{Fe(Zn)}}}({\mathbf{r}})} \right\rangle } \right.} \right.\)); and (b) the 2pvalencebonding model in which the initial and final states have a form of symmetrized products of hydrogenic 1s and 2p wavefunctions of both Fe and Zn atoms, that is, \(\left {\left. {\psi _{{\mathrm{initial}}}^{{\mathrm{2p  valence  bonding}}}({\mathbf{r}}_1,{\mathbf{r}}_2)} \right\rangle } \right. = \left[ \left {\left. {\psi _{2p}^{{\mathrm{Fe}}}({\mathbf{r}}_1)} \right\rangle } \right.\left {\left. {\psi _{2p}^{{\mathrm{Zn}}}({\mathbf{r}}_2)} \right\rangle } \right. + \left {\left. {\psi _{2p}^{{\mathrm{Fe}}}({\mathbf{r}}_2)} \right\rangle } \right. \left {\left. {\psi _{2p}^{{\mathrm{Zn}}}({\mathbf{r}}_1)} \right\rangle } \right. \right]/\sqrt 2\) and \(\left {\left. {\psi _{final}^{{\mathrm{2p  valence  bonding}}}({\mathbf{r}}_1,{\mathbf{r}}_2)} \right\rangle } \right. = \left[ {\left {\left. {\psi _{1s}^{{\mathrm{Fe(Zn)}}}({\mathbf{r}}_1)} \right\rangle } \right. \left {\left. {\psi _{2p}^{{\mathrm{Zn(Fe)}}}({\mathbf{r}}_2)} \right\rangle } \right. + \left {\left. {\psi _{1s}^{{\mathrm{Fe(Zn)}}}({\mathbf{r}}_2)} \right\rangle } \right.\left {\left. {\psi _{2p}^{{\mathrm{Zn(Fe)}}}({\mathbf{r}}_1)} \right\rangle } \right.} \right]/\sqrt 2\), which is analog to the covalent bonding in molecules. For the latter model, sixdimensional integration over r_{1} and r_{2} is needed for evaluating the dipole matrix. Using these two models, we can qualitatively estimate the above ratios as a function of the interatomic distance d. The results are plotted as the dashedblue and solidred lines in Fig. 5a and b, respectively, for the independentatom model and the 2pvalencebonding model. One can see that these simple models qualitatively give the overall trend of increasing interatomic Kα emission as interatomic distance decreasing; Quantitatively, both models show orders of magnitude differences from manybody DFT calculations. Nevertheless, the 2pvalencebonding model is better than the independentatom model, which manifests the molecularbonding nature among atoms in such superdense systems. Molecular bonding involving more than two atoms might account for the discrepancies between the two simple models and DFT calculations. It is noted that the multicenter wavefunction nature^{37} was properly accounted for by DFT. At the highest density explored (ρ = 2000 g cm^{−3}), Fig. 5b shows that the interatomic \({\mathrm{K}}_\alpha ^{{\mathrm{Fe}}\, \to \,{\mathrm{Zn}}}\) emission can reach over 11% of the regular intraatomic \({\mathrm{K}}_\alpha ^{{\mathrm{Zn}}}\) signal. The asymmetry that the interatomic Kα emission from 2p(Zn) → 1s(Fe) transition is always weaker than that of 2p(Fe) → 1s(Zn) is caused by the fact that the 2p state of Fe ion \(\left[ {\left {\psi _{2p}^{{\mathrm{Fe}}}({\mathbf{r}}) > } \right.} \right]\) spreads much more than \(\left {\psi _{2p}^{{\mathrm{Zn}}}({\mathbf{r}})} \right. > \) (because the former is less bounded) so that it can have significant overlap with the 1s hole of Zn (see Fig. 1). This consideration is further confirmed by looking into the orbital (wavefunction) overlap in such superdense situations.
Finally, we shall discuss the radiativetononradiative decay branching ratio for 1scorehole states created in such superdenseplasma mixtures. We shall point out that for superdense plasmas considered here, the nonradiative Auger decay channel is hard to measure because Auger electrons will quickly thermalize inside the superdense plasma; On the other hand, the radiative decay can be easily probed by measuring the escaped K_{α} photons through spectrometers. To calculate the ratio of K_{α} emission to Auger decay, we have used the atomic kinetic modeling code PrimSPECT^{38}, which is extensively used in the plasma physics community. For the concerned plasma densities varying from 250 g cm^{−3} to 2000 g cm^{−3} and kT = 50 eV, the averaged ionizations of Fe and Zn ions are about <Z> = 15.8~16.2 and <Z> = 19.5~20.1, respectively. Namely, such superdense plasmas mainly consist of neonlike ions of Fe^{16+} and Zn^{20+}, which both have the dominant electronic configuration of 1s^{2}2s^{2}2p^{6} that is close to what is shown by Fig. 3. Now, if external radiative/collisional pump creates 1scorehole state (1s^{1}2s^{2}2p^{6}) of both ions, we can use PrimSPECT to compute the decay rate coefficients. The calculations give a decay rate of \({\mathrm{\Gamma }}_{{\mathrm{rad}}} = 5.4 \times 10^{14}\,{\mathrm{s}}^{  1}\) for the radiative channel of 1s^{1}2s^{2}2p^{6} → 1s^{2}2s^{2}2p^{5} (K_{α} emission) for Fe^{17+} ions, while its Auger decay rate is about \({\mathrm{\Gamma }}_{{\mathrm{Auger}}} = 9.8 \times 10^{14}\,{\mathrm{s}}^{  1}\) for the dominant transition of 1s^{1}2s^{2}2p^{6} → 1s^{2}2s^{2}2p^{4}. Thus, the radiativetoAuger branching ratio for Fe^{17+} ions is about \({\mathrm{\Gamma }}_{{\mathrm{rad}}}/{\mathrm{\Gamma }}_{{\mathrm{Auger}}} \approx 0.55\). For Zn^{21+} ions, PrimSPECT calculations give the two decay rate coefficients of \({\mathrm{\Gamma }}_{{\mathrm{rad}}} = 1.01 \times 10^{15}\,{\mathrm{s}}^{  1}\) and \({\mathrm{\Gamma }}_{{\mathrm{Auger}}} = 1.07 \times 10^{15}\,{\mathrm{s}}^{  1}\), respectively, which results in a branching ratio of \({\mathrm{\Gamma }}_{{\mathrm{rad}}}/{\mathrm{\Gamma }}_{{\mathrm{Auger}}} \approx 0.94\). These calculations indicate that the radiative decay channel has the same order of probability as the nonradiative Auger decay for intraatomic transitions. In other words, one third of Fe^{17+} corehole ions will decay radiatively, while one half of Zn^{21+} corehole ions will emit K_{α} photons through intraatomic transitions. Given the same physics nature of radiative versus nonradiative decay for both intraatomic and interatomic transitions, we expect the similar branching ratio should hold between the interatomic radiative transition and the interatomic Coulombic decay^{39,40,41}. Once again, the interatomic Coulombic decays^{39,40,41} certainly occur within such superdense plasmas, although they may not be measured as easily as the interatomic radiative transitions.
Possible experiments on interatomic Kα emissions
Experimental verification of these firstprinciples predictions of interatomic Kα emissions can possibly be conducted at the Omega Laser Facility utilizing the platform of doubleshell implosions^{14,15,42,43}. In a doubleshell target, the inner metal shell can be made of midZ Fe–Cu or Fe–Zn alloys with a core of D2gas fill. When a lowZ outer shell (beryllium or polystyrene) is driven symmetrically by the 60beam OMEGA laser to spherically impact on the inner shell, it can cause the inner Fe–Zn (or Fe–Cu) shell to implode. A small convergence ratio of C_{R} = R_{initial}/R_{final} ∼8–10 of the metal shell could give rise to a mass density of ρ = 500 to 2000 g cm^{−3} for the inner Fe–Zn (or Fe–Cu) shell at its stagnation^{42,43}. To create the 1s holes of Fe and Zn/Cu ions, one option is to use the highintensity OMEGA EP beam to generate MeV electrons that can remove some of the 1s electrons of Fe and Zn/Cu ions through collisions. The other option is to fill the doubleshell target with midZ gases, such as Ar and Kr. As a result, the hotspot selfemission with a certain amount of hard xrays could ionize the 1s electrons of Fe and Zn/Cu ions by radiation pumping. The latter method has been successfully demonstrated in singleshell implosions on OMEGA. In both ways, the created hollow Fe/Zn/Cu ions in such extremely dense plasmas will give rise to interatomic Kα emissions, as we have predicted here. These interatomic Kα emissions can be measured by spectrometers with a dynamic range of 100 to 1000.
Discussion
The two phenomena predicted from our firstprinciples DFT calculations, which are the interspecies radiative transition and the breaking down of dipoleselection rule in extremely denseplasma mixtures, can have significant implications to highenergydensity (HED) science, ICF, and astrophysics. For plasma opacity/emissivity calculations, the crosstalk between different species and dipoleforbidden transitions have generally been ignored so far by the HED science community. Our firstprinciples results show that these interatomic radiative transitions can become significant and even comparable with normal intraatomic transitions. The overall trend of IRT can be qualitatively understood by the independent atom model and the 2pvalencebonding model; while transient multiatom molecular bonding could account for the enhancement of IRT in superdense plasmas. One would expect that these emission/absorption channels, opened up in the warm and extremely dense regime, could affect the radiation transport in ICF (e.g., doubleshell targets) and astrophysical objects such as browndwarf cores. It is noted that the interatomic radiative transitions shall occur in superdense singlespecies plasmas, although they might be indistinguishable to the normal intraatomic transitions.
Methods
Our DFT calculations were performed with the ABINIT software package^{33,34}, in which electrons are treated quantummechanically with a planewave finitetemperature KohnSham DFT description. The electrons and ions are in thermodynamic equilibrium with an equal temperature (Te = Ti). The electron–nucleus interaction is described in the PAW approach by a pseudopotential generated with a very small matching radius (rc = 0.2 bohr). All electronic wavefunctions are explicitly computed in the thermal DFT formalism. For the electronic exchange and correlation interactions, we use the generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE) functional^{44}. It is noted that the PBE functional has been widely used in DFT calculations for warm/hotdense plasmas^{12,45,46,47} that showed good agreements with HED experiments; Our results presented here are insensitive to the choice of exchangecorrelation functional, for which the localdensity approximation (LDA) gives essentially identical results, except for small energy shifts (see Supplementary Information). To sample the denseplasma configurations, we have conducted moleculardynamics simulations based on orbitalfree DFT. Namely, under the Born–Oppenheimer approximation, the selfconsistent electron density is first determined for an ion configuration. Then, the classical ions are moved by the combined electronic and ionic forces, using Newton’s equation. This moleculardynamics procedure is repeated for thousands of time steps, from which optical property (Xray emission/absorption) can be directly evaluated. Note that we have applied the periodic boundary condition to our firstprinciples calculations, with a box size determined by the Fe–Zn density and the number of atoms used. Convergent results for Kα emissions were reached by using 32 atoms in a super cell, the Baldereschi mean value point for the Brillouin zone sampling^{48}, and the highest planewave energy cutoff of Ecut ≈68 keV. This highenergy cutoff is necessary to accurately sample the deeply bound 1score electrons. Detailed convergence tests can be found in the Supplementary Information.
After we ran the calculations for thousands of OFMD steps, we obtained a sufficiently long trajectory of ionic configurations. We then chose several uncorrelated snapshots from these ionic configurations to calculate the Xray emission spectra of dense Fe–Zn plasmas by using the Kubo–Greenwood formalism^{49,50}. Because of the underestimated bandgap by the PBE functional due to electron selfinteraction, the resulting spectra were shifted by a constant of δω ≈110 eV (~1.5% of the 1s–2p bandgap) to match the Kα locations of ambient Fe and Zn. This is justified by comparing the Hartree–Fock calculated energy 1s–2p gap with the PBEDFT results. The similar matching technique has shown to work well for the measured Kα emission in warm dense Cu experiments on OMEGA EP.
In the K_{α}emission calculations, the dipole approximation has been invoked. For the concerned photon energy range of hν = 6.0–8.8 keV, the corresponding electromagnetic waves have wavelengths of λ ≈1.4–2.1 Å (2.6–3.97 Bohr). Taking an isolated Fe atom as an example, Hartree–Fock calculations give a size of 2s and 2p states (<2s r 2s> or <2p r 2p>) about ~0.12–0.14 Å, which is one order of magnitude smaller than the wavelength of Kα emissions so that the dipole approximation holds well for intraatomic transitions. For interatomic K_{α} emissions in superdense Fe–Zn plasmas (ρ ≥ 1000 g cm^{−3}) concerned here, the interatomic Fe–Zn distance is around d = 0.8–1.0 Bohr. Taking this emitting entity of Fe–Zn as a whole, its size is still about ~3–5 times smaller than the Kα wavelength. Nevertheless, this prompts us to consider highorder contributions such as the electric quadrupole emission, which is examined by computing the contribution of electric quadrupole term with the independentatom model for different densities (i.e., different interatomic Fe–Zn distances). The results indicated that the relative contribution ratio of quadrupole to dipole is overall less than ~3.2% (see Supplementary Information).
Data availability
The data that support the findings of this study are available from the corresponding author upon request. They can be immediately shared through email or any other filesharing systems.
Code availability
The codes for K_{α}emission calculations and IRT models are available from the corresponding author upon request.
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Acknowledgements
This material is based upon work supported by the US Department of Energy National Nuclear Security Administration under Award Number DENA0003856, the University of Rochester, and the New York State Energy Research and Development Authority. This work is partially supported by US National Science Foundation PHY Grant No. 1802964 for S.X.H. and V.V.K. This report was prepared as an account of work (for S.X.H., V.V.K., and P.M.N.) sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any agency thereof.
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S.X.H. conceived the project, performed the DFT calculations, and wrote the initial paper. V.V.K. created the allelectron PAW pseudopotentials and modified the ABINIT absorption code to calculate emission spectra. V.R., N.B., and M.T. have implemented the spinorbital calculation for optical properties in ABINIT, and provided the code and help on running it. P.M.N. and S.X.H. discussed the possible experimental designs. All authors discussed the results and revised the paper.
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Hu, S.X., Karasiev, V.V., Recoules, V. et al. Interspecies radiative transition in warm and superdense plasma mixtures. Nat Commun 11, 1989 (2020). https://doi.org/10.1038/s41467020159163
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