Reconciling ionization energies and band gaps of warm dense matter derived with ab initio simulations and average atom models

Abstract

Average atom (AA) models allow one to efficiently compute electronic and optical properties of materials over a wide range of conditions and are often employed to interpret experimental data. However, at high pressure, predictions from AA models have been shown to disagree with results from ab initio computer simulations. Here we reconcile these deviations by developing an innovative type of AA model, AvIon, that computes the electronic eigenstates with novel boundary conditions within the ion sphere. Bound and free states are derived consistently. We drop the common AA image that the free-particle spectrum starts at the potential threshold, which we found to be incompatible with ab initio calculations. We perform ab initio simulations of crystalline and liquid carbon and aluminum over a wide range of densities and show that the computed band structure is in very good agreement with predictions from AvIon.

Full text

PHYSICAL REVIEW RESEARCH 3, 023026 (2021)
Reconciling ionization energies and band gaps of warm dense matter derived with ab initio
simulations and average atom models
G. Massacrier ,1,*M. Böhme ,2,†J. Vorberger,3,‡F. Soubiran,4and B. Militzer5,§
1Univ Lyon, Univ Lyon1, Ens de Lyon, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, F-69230, Saint-Genis-Laval, France
2Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
3Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstraße 400, D-01328 Dresden, Germany
4École Normale Supérieure de Lyon, Université Lyon 1, Laboratoire de Géologie de Lyon, CNRS UMR 5276, 69364 Lyon Cedex 07,
France and CEA DAM-DIF, 91297 Arpajon, France
5Department of Earth and Planetary Science, Department of Astronomy, University of California, Berkeley, California 94720, USA
(Received 15 June 2020; accepted 19 March 2021; published 8 April 2021)
Average atom (AA) models allow one to efficiently compute electronic and optical properties of materials over
a wide range of conditions and are often employed to interpret experimental data. However, at high pressure,
predictions from AA models have been shown to disagree with results from ab initio computer simulations.
Here we reconcile these deviations by developing an innovative type of AA model, A
VION, that computes the
electronic eigenstates with novel boundary conditions within the ion sphere. Bound and free states are derived
consistently. We drop the common AA image that the free-particle spectrum starts at the potential threshold,
which we found to be incompatible with ab initio calculations. We perform ab initio simulations of crystalline
and liquid carbon and aluminum over a wide range of densities and show that the computed band structure is in
very good agreement with predictions from A
VION.
DOI: 10.1103/PhysRevResearch.3.023026
I. INTRODUCTION
The electronic and optical properties of materials change
drastically with increasing pressure or density, which affects
their equation of state, opacity, and transport properties. These
changes have profound implications for the interior structure
and evolution of stars and planets [1,2]. The modification of
electronic states with temperature and pressure is a very active
research subject [3,4]. The characterization of electronic prop-
erties is especially challenging in the regime of warm dense
matter (WDM) because pressure and temperature are high,
particles are strongly interacting, and the system is partially
degenerate. One relevant phase transition is the dissociation
and metallization of hydrogen [5–8] that changes from an
insulating, molecular substance to a monoatomic, electrically
conducting fluid. This phase change is a manifestation of a
Mott transition [9,10].
Due to the development of high-power lasers and other
facilities, the capabilities to create and diagnose WDM have
increased substantially and a variety of new experimen-
tal results have been obtained [11–17]. Many characterize
*gerard.massacrier@ens-lyon.fr
†m.boehme@hzdr.de
‡j.vorberger@hzdr.de
§militzer@berkeley.edu
Published by the American Physical Society under the terms of the
Creative Commons Attribution 4.0 International license. Further
distribution of this work must maintain attribution to the author(s)
and the published article’s title, journal citation, and DOI.
modifications of the atomic structure due to the plasma en-
vironment. A key quantity is the degree of ionization, which
may be enhanced by the continuum lowering phenomenon,
or ionization potential depression (IPD), that describe the
reduction of the ionization energy compared to the value of
an isolated ion. X-ray Thompson scattering measurements
(XRTS) provide direct access to the IPD [18–20]. The analysis
of the scattering signal is based on the Chihara formalism
[21,22], which relies on the separation of bound and free
electronic states as well as well-defined direct transitions
between the two. There are experimental results both sup-
porting and contradicting the two classic IPD models of
Stewart-Pyatt (SP) [23] and Ecker-Kröll (EK) [24]. SP models
are widely used to predict plasma properties like ionization
balance, equation of state, opacities, electrical and thermal
conductivities.
There are a variety of theoretical approaches to study
WDM and the IPD [25–27]. With ab initio simulations, one
follows the evolution of many ions in a periodic simulation
cell by coupling density functional theory (DFT) [28,29]to
molecular dynamics (MD) [30]. In comparison, average atom
(AA) models are computationally far less demanding because
they place only a single nucleus at the center of a cloud of
electrons and determine the resulting states. Many variations
of the AA approach exist [25,31–33]. However, when these
methods are applied to guide or interpret experiments at ex-
treme densities or highly degenerate plasma conditions, the
results appear to deviate from predictions of ab initio meth-
ods [26,34–36], which provided the initial motivation for this
study.
In this paper, we use DFT calculations to study the evolu-
tion of electronic states of dense C and Al over three orders of
2643-1564/2021/3(2)/023026(9) 023026-1 Published by the American Physical Society

G. MASSACRIER et al. PHYSICAL REVIEW RESEARCH 3, 023026 (2021)
magnitude in density and then reproduce them with a novel,
general-purpose type of AA model. For the latter, we adopt an
alternate view of the continuum. Usually, AA models embed
the ion in some kind of jellium and the continuum of states
starts at the threshold of the potential. We determine that it
is this property that leads to a disagreement with ab initio
results. As a matter of fact, valence and conduction bands
as derived from DFT-MD simulations exhibit no clear link to
such a potential threshold.
II. AB INITIO SIMULATIONS
We performed ab initio calculations for dense carbon
and aluminum with the ABINIT code [37–42]. For both ma-
terials, we performed static DFT calculation using perfect
lattices (T=0 eV) and DFT-MD simulations at a finite tem-
perature (T=12.5 eV). For aluminum, we employed the
local density approximation (LDA) to incorporate exchange-
correlation (XC) effects [43] and the PBE-GGA functional
was used for carbon [44]. For both materials, we generated
new pseudopotentials with very small core radii using the
OPIUM code [45]. The 1s core electrons were frozen in the
norm conserving pseudopotential for aluminium. For carbon,
we even performed all-electron calculations.
As the predicted crystal structures of aluminium and car-
bon change with density, we change the structure at T=0
according to Ref. [46] for carbon and Ref. [47] for aluminium.
For carbon, the order of the phases is diamond (3.52 to 7.66 g
cm−3), BC8 (7.87 g cm−3), simple cubic (sc, 15.9 g cm−3),
simple hexagonal (sh, 23.29 g cm−3), and face-centered cubic
(fcc, 46.36 to 640.83 g cm−3). A 123k-point grid with a
plane wave cutoff energy of 600 Hartrees (Ha) was used for
diamond. The BC8, sc, and sh calculations used a k-point grid
of 323points and a cutoff energy of 700 Ha. The static T=0
calculations for aluminium were carried out for both the fcc
and body-centered cubic (bcc) [48] phases to get a better
understanding on how the phase affects the valence band gap.
For the fcc unit cell a 643k-point grid with 12 bands and a
plane wave cutoff energy of 400 Ha has been used. The bcc
calculations used a 123k-point grid with 20 bands and a plane
wave cutoff at 400Ha. The band gap reported for these solid
phases is the indirect gap over the sampled Brillouin zone.
The MD simulations at T=12.5 eV were run using a
Nosé-Hoover thermostat and with a time step of δt=0.09
fs and with eight atoms in the periodic supercell which was
sampled only at the -point. The number of bands considered
was 450 and the plane wave cutoff was 500 Ha for carbon.
The aluminum DFT-MD simulations considered 630 to 640
bands and a cutoff energy of 400 Ha. Electronic density of
states (DOS) and band-gap results at finite temperatures were
derived by averaging over the DOS of 10 sample configura-
tions obtained via DFT-MD and which where postprocessed
with a static run using 83k-points. These DOS were averaged
after aligning them first at the chemical potential μ(a few eV
below the Fermi energy at these low temperatures).
Figure 1shows the DOS, weighted by the Fermi-Dirac
factor 1/(1 +e(ε−μ)/kT ), at T=12.5 eV and for densities a
few times the solid density. For illustration purposes the DOS
were smoothed using a Gaussian with a half width at half
maximum (HWHM) of 0.8 eV. For carbon, the peaks below
0
0.5
1.0
1.5
f
FD
x DOS (1/eV)
0
0.1
0.2
3.51 g/cc DFT-MD
21.2 g/cc DFT-MD
3.51 g/cc AvIon
21.2 g/cc AvIon
0
0.5
1.0
1.5
f
FD
x DOS (1/eV)
0
0.1
0.2
8.03 g/cc DFT-MD
22.0 g/cc DFT-MD
8.03 g/cc AvIon
22.0 g/cc AvIon
-250 -200 -150 -100 -50 050
-125 -100 -75 -50 -25 025
HP (eV)
(a) C
(b) Al
1s
2s
2p
V(R)
V(R)
FIG. 1. Density of states (multiplied by the Fermi-Dirac factor)
at various densities and T=12.5 eV for liquid (a) carbon and (b)
aluminum. Energies are counted from the chemical potential. Results
are from DFT-MD or A
VION simulations as shown in the legends.
The position of the threshold potential V(R) for each A
VION simula-
tion is also shown as vertical bars at the top of each panel. For better
readability, we introduced different yscales for the valence bands
(left yaxis) and the continuum states (right yaxis).
−250 eV represent the 1s states while the 2s and 2p states
have already merged with the continuum at these conditions.
With increasing density, we find that the continuum edge
shifts towards lower energies when compared with the Fermi
energy, consistent with earlier findings [49]. For the following
discussion, we define the valence band gap (VBG) of carbon
to be the gap between the 1s states and the bottom of the
continuum band.
For aluminum at the lowest density of 8.03 g cm−3, the 2s
and 2p states form two well-separated peaks. At 22.0 g cm−3,
these peaks have broadened. At yet higher densities, they will
merge (not shown). The VBG of aluminum is defined to be
the gap between the uppermost 2s/2p state and the continuum
band.
Our simulations show consistently that the continuum edge
shifts towards lower energies, well below the Fermi energy,
as density increases. Eventually gaps between valence and
continuum bands close. In traditional AA models, the bottom
of the continuum begins at the maximum of the potential and
a state is pressure-ionized as soon as it crosses this thresh-
old. This leads to inconsistent results between the DFT and
AA approaches because in DFT one derives the free-particle
spectrum without introducing such a threshold. To resolve
this discrepancy and to reproduce the fully self-consistent
023026-2

RECONCILING IONIZATION ENERGIES AND BAND GAPS … PHYSICAL REVIEW RESEARCH 3, 023026 (2021)
many-body DFT calculations, we developed a novel type of
AA approach, A
VION, that is much more efficient than DFT-
MD and can be readily applied to arbitrary materials and
thermodynamic conditions. As we will show below, A
VION
reproduces the DFT-MD predictions very well. It also explains
apparent discrepancies between the usual AA models and
DFT-MD simulations.
III. AVION MODEL
In ab initio many-body simulations, the band structure
is obtained from eigensolutions of the Schrödinger equation
in a periodic box for an effective potential resulting from
a collection of electrons and nuclei, while taking advantage
of the Kohn-Sham scheme [50]. In our A
VION approach, we
intend to solve the same equations to obtain the band struc-
ture and wave functions around a single nucleus in a limited
volume. The only connection with the plasma environment
arises from the boundary conditions of the wave functions at
the surface of that volume. In an actual many particle system
these boundary conditions are highly complex and vary from
ion to ion depending on the environment. Still these variations
can be represented well enough with approximate methods
as long as the predictions are validated through comparisons
with many-body simulations.
A
VION is a spherical model with a neutrality radius R.To
obtain agreement with many-body simulations, we introduce
novel boundary conditions at Rthat allows us to derive bound
and free states within a common scheme. As we will show,
this has deep implications for the band structure of the contin-
uum states and for the magnitude of band gaps. Conversely,
traditional AA models set the boundary conditions at infinity.
This leads to a clear distinction between a bound (discrete)
and a free (continuous) spectrum, which is not compatible
with predictions of many-body simulations at high density.
For an element with nuclear charge Zand at given temper-
ature T,theA
VION approach self-consistently determines the
electronic density profile ne(r) and an effective potential V(r)
within the Kohn-Sham scheme [50]. The spherical potential
V(r)=−
Ze2
r+vH(r)+vxc[ne(r)] (1)
is the sum of the Coulombic nuclear attraction, the Hartree
potential vHresulting from the electronic density through
Poisson equation, and an exchange-correlation part in the
LDA approximation. (We use Refs. [51–53].) The potential
vHinside Rcan be written, up to a constant, as a function of
the sole charge inside this same volume:
vH(r)=vH(0) −e2r
0
Q(r)
r2dr,(2)
where Q(r)=r
04πr2ne(r)dris the number of electrons
inside radius r. We do not need to assume any definite value
for V, and leave vH(0) unspecified.
The electronic density (including all electrons) is, in
turn, determined from the spherical eigenstates φεm=
1
rPε (r)Ym
(ˆ
r) in the potential V. Their radial parts obey
d2Pε
dr2=(+1)
r2+2m
¯h2[V(r)−ε]Pε (3)
(with mthe electron mass, ¯hthe Planck constant), are normal-
ized to unity inside the ion sphere, and populated according to
a Fermi-Dirac distribution. Accordingly,
4πr2ne(r)=

2(2+1) dεg(ε)
e(ε−μ)/kT +1|Pεl(r)|2,(4)
where the g(ε) are partial DOS for the angular momentum
. The chemical potential μis adjusted so that Zelectrons
exactly pile up inside of radius R, i.e., Q(R)=Z.From
electro-neutrality one has dV
dr (R)=0 (neglecting vxc). The
value V(R) defines the potential threshold on the energy scale.
The distinctive feature of A
VION is in the way the DOS is
constructed. The continuum in traditional AA models starts
above that threshold [setting V(rR)=0], and includes
all ε>V(R) solutions, matching them to a combination of
Bessel functions (free solutions) at the Rboundary. We show
in this paper that the band structure inherent to condensed
matter, which persists for crystals under WDM conditions as
shown recently by Bekx et al. [54] (see, in particular, their
Fig. 7), or in DFT simulations, challenges this simple view,
yet can be recovered with a modified AA model.
For bound states, the concept of bands instead of isolated
states [which give a δ-function contribution to some g(ε)]
has already been used in a number of AA models [55–58].
Regarding states above threshold, some preliminary works
have explored the modification of the DOS due to neighbors
using multiple scattering theory [59–61]. While this is in the
spirit of our work, our aim is to avoid the huge complexity it
adds.
In A
VION for every angular momentum, , the partial DOS
g(ε)=kgk(ε) is a sum over successive energy bands k.
For given , all wave functions within the kth band have
k−1 nodes in the interval ]0,R[. In addition, the lower ε−
k
and upper ε+
kenergy limits of this kth band result from the
eigensolutions of the radial Schrödinger equation [Eq. (3)]
with the respective boundary conditions d(Pε/r)
dr (R)=0 and
Pε (R)=0. With these two conditions, we intend to mimic the
bonding and antibonding character of typical molecular wave
functions. Note that the Pε (R)=0 condition for the upper
band limits does not imply that there is a hard wall at radius
R, but that the total (over the whole plasma) wave function, a
part of which we solve for in the ion sphere, has a node at R.
We emphasize that we no longer make a distinction between
bound (below threshold) and free (above threshold) states, as
is done in traditional AA models. Still we need to make an
assumption for the DOS inside each band.
Between the respective limits ε±
k, we choose in this work
the Hubbard functional form gk(ε)=2
πδ2[(ε+
k−ε)(ε−
ε−
k)]1/2, where δ=1
2(ε+
k−ε−
k)[62]. This is undoubtedly
oversimplified. The DOS of the bands is actually controlled
by a complex interplay between the ion and its neighbors, and
its computation requires involved techniques. For instance the
recent work of Starrett and Shaffer [61] used a multiple scat-
tering approach, but requires around one hour of CPU time.
Our goal is precisely to bypass this computationally highly
demanding step: A
VION necessitates a few seconds per point
[63]. With this DOS at hand and wave functions computed for
a set of energies inside the bands, the electronic density can
be obtained from Eq. (4).
023026-3

G. MASSACRIER et al. PHYSICAL REVIEW RESEARCH 3, 023026 (2021)
FIG. 2. Evolution of band energies computed with AVION as a function of density for carbon (first column) and aluminum (second column),
for the temperatures T=12.5 eV (upper row) and T=0.1 eV (lower row). In panels (a) and (b), energies are counted from the potential
threshold V(R); in (c) from the top of the 1s band for carbon; in (d) from the top of the 2p band for aluminum. Lowest s,p,anddbands are
in red, blue, and green colors, respectively. The black line is the potential threshold V(R); the black dots mark the chemical potential μat
computed points. The valence band gap (VBG) and the pseudo-ionization potential (PIP) of the C 1s and Al 2p bands are shown as vertical
arrows. In panel (c), an arrow indicates the carbon K-edge, which is in good agreement with Hu’s predictions [26]forT=1.3 eV (blue dots).
As a summary, the three parameters (Z,T,R) entirely
determine an A
VION calculation. The iterative procedure is
initiated with a trial electronic density as input. The poten-
tial is determined through Eqs. (1) and (2). Bands and wave
functions are obtained from the solutions of Eq. (3). The
resulting electronic density is constructed from Eq. (4) with
μadjusted for electroneutrality. The result is compared with
the input. If different, a new loop is initiated with a mix of
the two densities, and the procedure is continued until full
convergence is achieved.
To compare with results at given ion density ni(or mass
density ρ=Amuni, where Ais the atomic mass of the ele-
ment), we set in this paper Requal to the Wigner-Seitz radius,
RWS =[3/(4πni)]1/3. Note that this is only one plausible
choice among other possibilities that we do not explore in this
work [64].
As an illustration of our approach, Fig. 2shows the evo-
lution of band energies over a large density range for carbon
and aluminum at temperatures T=12.5 eV and T=0.1eV.
The reference energy in Figs. 2(a) and 2(b) is the potential
threshold V(R). To better visualize the VBG, energies are
counted from the top of the 1s band for carbon in Fig. 2(c),
and the top of the 2p band for aluminum in Fig. 2(d).Atlow
density, bands below the threshold appear as sharp, localized
bound states. With increasing density, they widen as they
approach the potential threshold, which they eventually cross
in a continuous way. Above a density of a few g cm−3the
bottom of the continuum is associated for carbon first with the
2s and later with the 2p band, while for Al, it is first the 3s
and later the 3d band. The void region between the top of the
valence band (1s for C, 2p for Al) and the continuum defines
the VBG, as shown with arrows in the panels.
Our A
VION model does not distinguish bound and free
states. States rather evolve from being localized to delocal-
ized. To compare with traditional AA and IPD models, we
introduce a pseudo-ionization potential (PIP) in A
VION, which
023026-4

RECONCILING IONIZATION ENERGIES AND BAND GAPS … PHYSICAL REVIEW RESEARCH 3, 023026 (2021)
we define to be the difference between a state’s energy and the
threshold potential V(R), even though the PIP is not linked
to any physical quantity in our approach. We show the PIP
for the 1s state of C in Figs. 2(a) and 2(c), and for the 2p
stateofAlinFigs.2(b) and 2(d). Both decrease rapidly with
density. The VBG predicted by our A
VION approach remains
much larger because the region void of states extends well
above the potential threshold. This explains the discrepancies
reported in Ref. [34] between very small gaps in AA models
and large VBGs in DFT-MD results, as is demonstrated by
further analysis in the following section.
In A
VION, we can predict the K-edge by subtracting the 1s
state’s energy from the chemical potential (or Fermi energy),
as is illustrated for carbon in Fig. 2(c). In the same panel,
we included as blue dots the values computed from DFT
simulations in Ref. [26]forT=1.3 eV. The agreement with
the K-edge inferred from our A
VION approach is excellent.
The increase of the K-edge energy emerges as a “competi-
tion between continuum lowering and Fermi surface rising”
[26,35,36] when including the potential threshold as an inter-
mediate step in the analysis. Still in A
VION and in DFT, the
K-edge energy can be determined directly.
IV. COMPARISON BETWEEN METHODS
The DOS as obtained from AVION are compared in Fig. 1to
DFT-MD results at T=12.5 eV and for the lowest densities.
To ease comparison, the DOS have been convolved using a
Gaussian distribution for the neutrality radii Rwith a HWHM
equal to RWS/40, which slightly broadens the valence peaks.
As A
VION is based on a quite simple prescription for the DOS
shape inside a band, a perfect match is not expected. Never-
theless the model reproduces remarkably the DFT-MD results
and their trends, especially since it requires a few seconds
against hundreds of hours for DFT. The continuum edges and
the valence band positions agree quite well for carbon, and
for aluminum at the lowest density of 8.03 g cm−3.ForAl
at 22.0 g cm−3the agreement on the positions of both the 2s
and 2p bands and the continuum edge can be made better by
upshifting with ∼10 eV the A
VION DOS (which amounts to a
shift of the chemical potential).
The position of the potential thresholds V(R)arealso
drawn as vertical bars in the top of the panels. For carbon,
the continuum starts just below the threshold at 3.51 g cm−3,
and just above at 21.2 g cm−3. For aluminum, it is just below
the threshold at 8.03 g cm−3. These situations would be
hard to distinguish experimentally from the usual equality
in traditional AA models. In contrast, for aluminum at 22.0
gcm
−3, the continuum edge starts some 40 eV above the
threshold. These results are easily visualized through inspec-
tion of the upper panels of Fig. 2.
In Fig. 3, we compare results of DFT, A
VION, and SP
methods over three orders of magnitude in density. For both
T=0 conditions and fluid states at T=12.5eV,weshow
results from DFT(-MD) simulations and A
VION calculations.
The latter use T=0.1 eV instead of T=0 to avoid numerical
issues, but as a matter of fact the A
VION results are fairly
similar in these low temperature range except for a slight
difference for Al at low density.
100101102103
0
50
100
150
200
250
300
350
400
Energy (eV)
(a)
DFT VBG T=0eV
DFT-MD VBG T=12.5eV
AvIon VBG T=0.1eV
AvIon VBG T=12.5eV
AvIon PIP T=12.5eV
SP C4+ (shifted)
SP C4+ T=12.5eV
SP C5+ T=12.5eV
100101102103
Density (gcm−3)
0
20
40
60
80
100
120
Energy (eV)
(b)
DFT VBG T=0eV BCC
DFT VBG T=0eV FCC
DFT-MD VBG T=12.5eV
AvIon VBG T=0.1eV
AvIon VBG T=12.5eV
AvIon PIP T=12.5eV
SP Al3+ (shifted)
SP Al3+ T=12.5eV
SP Al4+ T=12.5eV
(a)
(b)
C
Al
FIG. 3. Valence band gaps (VBG) of (a) carbon and (b) alu-
minum. We compare static DFT calculations of solids (T=0),
DFT-MD simulations of liquids at T=12.5eV,andA
VION results.
The pseudo-IP (PIP) from A
VION is compared with SP predictions
for the IP of various ionization states, as well as a shifted curve for
the less ionized ion (dash-dotted). The SP C4+curve in (a) has been
shifted by −67 eV, and the SP Al3+curve in (b) by −21 eV.
For carbon, AVION reproduces the DFT band gaps very
well but a small deviation remains for the highest densities.
A
VION predicts a gap closure at a density of ≈500 g cm−3
while the gap in static DFT calculations of fcc carbon has not
yet closed. DFT-MD simulations of liquid carbon predict gap
closure at a density of ≈620 g cm−3. (We define the point of
gap closure in DFT-MD when the standard deviation of the
distribution of gap values overlaps with zero.)
For aluminum, the agreement is very good up to a density
of around 20 g cm−3. Then the AVION band gap increases to
higher values than is predicted with DFT methods. However,
the following decrease and eventual band gap closure at ≈190
gcm
−3occurs in a range compatible with DFT predictions.
The difference in the latter may be traced back to the different
crystal or fluid state in the DFT simulations, while the envi-
ronment in A
VION is always with spherical symmetry. Note
that A
VION shows a marked change of the band-gap slope
around 32 g cm−3, which is linked to the switch from the 3s to
the 3d band used to define the continuum edge [see Fig. 2(d)].
This change in slope is also seen in our DFT simulations. A
similar slope change, though less dramatic, occurs for carbon
023026-5

G. MASSACRIER et al. PHYSICAL REVIEW RESEARCH 3, 023026 (2021)
at around 35 g cm−3due to the switch of the lowest continuum
band from 2s to 2p.
In Fig. 3, we also plot the PIP computed with A
VION for
T=12.5 eV (it is almost the same for T=0.1eVonthis
density range.) Its value goes to zero at already ≈170 g cm−3
for C and ≈30 g cm−3for Al. From the point of view of tra-
ditional AA models, this would imply that the involved bands
have already merged with the continuum at these conditions.
However, both our A
VION and DFT results predict there is still
a sizable VBG present under these conditions.
It is useful to compare when A
VION and SP models predict
a given state to become pressure ionized. From around 1 g
cm−3up to the density where the valence band crosses the
threshold, carbon has two localized electrons and aluminum
has ten. We hence plot in Fig. 3the modified IP, E0−SP,
for the 1s state of C4+and C5+, and for the 2p state of Al3+
and Al4+, where E0is the isolated ion spectroscopic value
and SP the IPD predicted by the SP model. It shows little
resemblance to the VBG curves computed with A
VION and
DFT. SP models predict pressure ionization to occur at lower
densities, at values in between the point where A
VION’s PIP
goes to zero and A
VION and DFT predict the VBG to close.
It should be noted that AA models and DFT simulations
involve approximations which do not allow reproduction of
the measured energies of isolated ions, which are input pa-
rameters in SP models. By shifting SP curves for C4+and
Al3+, i.e., adjusting E0, it is possible to match the evolution of
A
VION’s PIP over most of the density range (see lines labeled
“shifted” in Fig. 3.) In these degenerate conditions, A
VION and
SP models make similar predictions for screening effects and
SP reduces to the ion sphere expression, 3ze2/(2R), where
z=4 for C and z=3 for Al. The residual discrepancy for
the densities where both curves go to zero may be explained
by noticing that the PIP is counted from the top of the valence
band; the SP model rather refers to an eigenstate that would be
computed with the condition Pε (∞)=0. Hence it is situated
in the band below its top and is ionized at a higher density.
V. CONCLUSION
In this paper, we developed a novel type of AA approach,
A
VION, to obtain consistency with much more computation-
ally demanding many-body DFT simulations. We studied
solid and liquid carbon and aluminum over three orders of
density. With our A
VION method, we find excellent agreement
with DFT predictions for the continuum edge shift below the
Fermi energy, the carbon K-edge increase and the valence
band gaps. Traditional AA models on the other hand predict
valence bands to enter the continuum at far too low densities.
Still, our A
VION results agree with a continuum lowering
predicted by SP model up to a few times solid density. But
at higher densities, drastic errors in the SP models become
apparent as the continuum band shifts away from the potential
threshold that is incorrectly equated with the beginning of the
continuum band in the SP model.
Such predictions may be tested in experiment at high
energy laser facilities like the National Ignition Facility or
Laser Mégajoule (LMJ) [65] that are capable of producing
compressed matter of Gbar pressures [66] and multiple times
solid density [67] and may be combined with x-ray diffraction
[68] and x-ray Thomson scattering [69].
Since our A
VION approach is as efficient as traditional
AA models and can deliver DFT compatible predictions in
only a few CPU seconds, it allows to fit and interpret exper-
imental data and infer the temperature and density without
resorting to SP models. The capability of A
VION will make
joint experimental-theoretical explorations of WDM more
efficient.
ACKNOWLEDGMENTS
G.M. acknowledges support from the Programme Na-
tional de Physique Stellaire (PNPS) of CNRS/INSU, M.B.
from the Center of Advanced Systems Understanding (CA-
SUS), F.S. from the European Union through a Marie
Skłodowska-Curie action (Grant No. 750901) and B.M.
from the National Science Foundation-Department of Energy
(DOE) partnership for plasma science and engineering (Grant
No. DE-SC0016248) and by the DOE-National Nuclear Se-
curity Administration (Grant No. DE-NA0003842). The DFT
and DFT-MD simulations were performed on a Bull Cluster
at the Center for Information Services and High Performace
Computing (ZIH) at Technische Universität Dresden.
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