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\title{Analytic fits for partial photoionization cross sections\thanks{Table
1 is available in electronic form:
see the Editorial in A\&AS 1994, Vol. 103, No. 1}}
\author{
D.A.~Verner\inst{1}
\and
D.G.~Yakovlev\inst{2}
}
\offprints{D.A.~Verner}
\institute{
Kapteyn Astronomical Institute, P.O.~Box 800, 9700 AV Groningen,
The Netherlands
\and
Ioffe Institute of Physics and Technology, St.~Petersburg 194021, Russia
}
\thesaurus{02.01.3, 02.01.4}
\date{Received date; accepted date}
\maketitle

\label{sampout}

\begin{abstract}
We present a compact, uniform and complete set of analytic fits to the
partial Hartree-Dirac-Slater photoionization cross sections for
the ground state shells of all atoms and ions of elements from H to Zn
($Z \leq 30$). Comparison with experiment and theory demonstrates
generally high accuracy of the fits up to energies of 100~keV.
\keywords{atomic data -- atomic processes}
\end{abstract}

\section{Introduction}
Photoionization cross sections of atoms and ions
are required for constructing theoretical models of many objects, from
planetary nebulae to active galactic nuclei
and intergalactic medium. In an earlier paper
(Verner et al. 1993, hereafter
Paper I) we presented the Hartree-Dirac-Slater (HDS)
calculations and analytic
fits for the partial photoionization cross sections
of atoms and ions for all
subshells $nl$ of the ground state species.
The calculations embodied all neutral atoms
from He to Zn and about 70\% of ions
of elements with atomic number
$Z \leq 30$. The calculated cross sections
were fitted by an analytic formula
based on the approximate criterion of
similarity (Kamrukov et al. 1983)
of the photoionization cross
sections for different atoms and ions. 
In Paper I, we performed also the
secondary fitting of the fit parameters
along $Z$ and $N$ (number of electrons) for the atomic species
separated into several groups. This enables us
now to calculate the fit parameters of the photoionization
cross sections of the species omitted
in the initial HDS computations.

\section{Fits to the photoionization cross sections}
In this paper we present a uniform, compact and
complete set of the fit parameters to
the cross sections for all subshells of
atoms and ions of the elements
under study.
The partial photoionization cross
sections $\sigma_{nl}(E)$ will be
described by the fitting formula:
\begin{eqnarray}
    \sigma_{nl}(E)=\sigma_0 \, F(E/E_0) \;\; {\rm Mb}, \nonumber \\
               F(y) = \left[(y-1)^2+y_{\rm w}^2 \right]y^{-Q}
               \left(1+ \sqrt{(y/y_{\rm a})} \right )^{-P},
    \label{eq:1}
\end{eqnarray}
where $n$ is the principal quantum number of the shell,
$l$ =0, 1, 2 (or $s$, $p$, $d$) is the
subshell orbital quantum number,
$E$ is the photon energy in eV, $y=E/E_0$;
$\sigma_0$, $E_0$, $y_{\rm w}$,
$y_{\rm a}$ and $P$ are the fit parameters, and
$Q=5.5+l-0.5P$. As a rule, the parameters $P$,
$y_{\rm w}$ and $y_{\rm a}$
vary smoothly over atomic species for a
fixed subshell $nl$. Accordingly, the cross section shape is
described by a nearly universal function $F(y)$. This reflects
a useful criterion of similarity
of the photoionization cross sections (Kamrukov et al. 1983).
Two other fit parameters, $\sigma_0$ and $E_0$, rescale
the function $F(y)$ for a given cross section. The rescaling
is especially simple in logarithmic variables, $\log \sigma_{nl}$ against
$\log E$: $\sigma_0$ shifts the universal curve $\log F$
along the cross section axis, while $E_0$ shifts
the curve along the energy axis.

The explicit form of $F(y)$ given by Eq. 1 was proposed
in Paper I. The form appears to
be very accurate for most cases under study.
All the parameters of $F(y)$ have a simple
physical meaning. The $F(y)$-curve possesses a resonant-like
dip of width  $y_{\rm w} $ at $y \approx 1$ (for small $y_{\rm w}$).
For higher $y$ ($1 \ll y \ll y_{\rm a}$), $F(y)$ decreases
with $y$ as $F(y) \propto y^{-Q}$. Thus $P$ defines
the power-law index, and $y_{\rm a}$ determines the upper boundary
of $y$ for this power-law regime
(that is realized provided $y_{\rm a} \gg 1$). Finally, when
$y \gg y_{\rm a}$ our
formula reproduces the well-known
(e.g., Bethe \& Salpeter 1957) non-relativistic
asymptote $\sigma_{nl}(E) \propto E^{-3.5-l}$
for high energies. In addition to each set of
the fit parameters, one should specify the
threshold energy $E_{\rm th}$. In many cases, the
latter is higher than the energy of
the resonant dip, so that the dip does not
appear in the actual cross section. However
in some cases (mostly, for the outer shells of neutrals),
$E_{\rm th}$ is lower or comparable to the
dip position. This allows us to describe the sophisticated
behaviour of the cross section curve just above the
threshold in these cases. The advantage of the proposed fit is
that it is not related directly to the
cross section parameters at the threshold (see Paper I for details).

Equation 1 allows us to describe
correctly the background photoionization cross
sections near the threshold, at
intermediate energies, and at high non-relativistic
energies. The fit
parameters have been taken from Paper I
for the species with the calculated HDS
cross sections (75\% of data) or calculated
from the the secondary fits of
Paper I for non-hydrogenic species omitted
from the HDS calculations (23\%
of data). As compared to Paper I, we have improved
the $1s$-shell fits for He~{\sc i} and 
Li~{\sc i} by including new high
energy cross sections in the fitting.
We have found that the $1s$-shell photoionization
cross sections for Li~{\sc ii}
in Paper I were calculated with the photoelectron wavefunction
corresponding to erroneous electron energy.
This resulted in the overestimation of the cross section
near the photoionization threshold by a factor of 1.5 although the
overestimation factor decreased with photon energy.
In the present paper, we give the
fit for the correct Li~{\sc ii} cross section.
Using Eq.~(1), we have also fitted
the hydrogen-like cross sections (2\%
of data) calculated from the analytic expression
(see, e.g., Bethe \& Salpeter 1957).

In addition to the fit parameters, we present
also the ionization threshold energies.
For the outer subshells, we list the
experimental ionization energies taken
from Kelly (1987), Sugar \& Corliss
(1985) and Moore (1970).
For the inner subshells of neutral atoms, we present the
experimental energies from Sevier (1979).
This does not
violate the fit self-consistency, since
the fits are not based on the threshold values.
For the inner subshells of ions, we list
the theoretical ionization energies from
Paper I. All the fit parameters and ionization energies are given in Table 1.
The columns are:

\begin{tabular}{lcl}
$Z$ & -- & atomic number\\
$N$ & -- & number of electrons\\
$S$ & -- & subshell\\
$E_{\rm th}$, eV & -- & subshell ionization threshold energy\\
$E_0$, $\sigma_0$, $y_{\rm a}$, $P$, $y_{\rm w}$ & -- & fit parameters\\
\end{tabular}

The rms relative error of the hydrogen-like
fit is $\delta_{\rm av}=0.0019$ for 21 photon energy
points placed logarithmically equidistant from the
threshold energy $E=E_{\rm th}$ to $100E_{\rm th}$,
with the maximum error $\delta_{\rm max}=0.0044$
at the threshold. The fitting
accuracy for the non-hydrogenic
species evaluated from comparison with the
calculated HDS cross sections is
also rather good. The rms relative errors
averaged over all the atomic species involved
are within 0.7\% for the $1s$, $2s$,
$2p$, $3s$ and $3p$ subshells, and
within 2.3\% for the $3d$ and $4s$ subshells
(see Paper I for details). Below we compare
our analytic fits with other
theoretical calculations and experimental data.

\section{Comparison of the fits with experiment and theory}
We have made extensive comparisons of
the fits with
experimental and theoretical data available.
The cross sections have been measured
mostly for the neutral atoms (see Samson et al. 1991
for a recent review of
absolute atomic photoionization cross sections). Compilations of
experimental and theoretical data for
all atoms with $Z=1-94$ are given by
Henke et al. (1982) at $E=30-9887$~eV
and by Veigele (1973) at $E=1-1000$~keV.
The theoretical Hartree-Slater total
photoionization cross
sections for the atoms and ions with
$Z \leq 30$ at $E=5-5000$~eV are given in
tabular form by Reilman \& Manson (1979; hereafter RM).
The recent Opacity
Project (OP) calculations (Seaton et al. 1992)
give, in particular, the
cross sections for
the ground state of atoms and ions of
elements with $Z \leq 14$ and $Z=16$,
18, 20, 26. The OP calculations are based
on the high level methods of atomic physics such as the R-matrix
method (Burke et al. 1971) that reproduces many
autoionization resonances near the ionization thresholds.

For all neutral atoms, we have compared
the photoionization cross sections
obtained from the present analytic fits with
those tabulated by Henke et al. (1982) and
Veigele (1973),
and with the published experimental data of
Angel \& Samson (O{\sc i}, 1988), Samson \&
Angel (N{\sc i}, 1990), Samson et al. (Cl{\sc i}, 1986; Ne{\sc i}, 
Ar{\sc i}, 1991),
Bizau et al. (Ca{\sc i}, 1987), Hudson \& Carter (Li{\sc i}, Na{\sc i}, 1967),
Mehlman et al. (Li{\sc i}, 1982), Krause et al. (Mn{\sc i}, 1984),
Jimenez-Mier et al. (Mn{\sc i}, 1989),
Parthasaradhi et al. (Ti{\sc i}, Ni{\sc i}, 1988), 
West \& Marr (He{\sc i}, Ne{\sc i}, Ar{\sc i}, 1976),
Marr \& West (He{\sc i}, Ne{\sc i}, Ar{\sc i}, 1976), Cant\'{u} et
al. (C{\sc i}, 1981), Kohl \& Parkinson (Al{\sc i}, 1973), 
and Roig (Al{\sc i}, 1975).
We have also made
a comparison with all the OP ground state
cross sections available through the
TOPbase 0.6 database (Cunto et al. 1993),
and with all the data tabulated by RM.
The general conclusions are the following:
\begin{enumerate}
\item The present fits summed over subshells are generally in a very
good agreement with the total
cross sections from RM.
This is not a surprise, since our analytic formula
fits accurately the HDS
cross sections, and RM use the same
(Hartree-Slater) method for calculating the
photoionization cross sections
at non-relativistic photon energies.
We have found some omissions in the RM data. For S~{\sc viii},
RM do not add the contribution from the outer $2p$-shell
to the total cross section. Thus, for S~{\sc viii} RM give the partial
$2s$-shell cross section instead of the total one at all energies from
the first threshold to the K-shell edge at 2700~eV. RM apparently assume
that V~{\sc ii}, Co~{\sc ii} and Ni~{\sc ii} possess $4s$ electrons.
However the ground state electronic
configurations of these ions are non-standard. Consequently the RM cross
sections are inaccurate at $E \la 80$~eV
for V~{\sc ii}, at $E \la 130$~eV for Co~{\sc ii}, and at $E \la 160$~eV
for Ni~{\sc ii}. Moreover the RM data are inaccurate at $E < 330$~eV for
Ar~{\sc v},
at $80 < E < 540$~eV for P~{\sc iii}, and at $E > 3400$~eV for Li~{\sc i}
and Li~{\sc ii}. Excluding these cases from the comparison, we get
an rms deviation of our fits from the RM data (averaged over all species)
less than 0.03.
\item The present fits do not reproduce the resonance
structures. However they are in a
fairly good agreement with the OP and
the experimental background cross sections.
The agreement begins to deteriorate near the
thresholds of the outer shells of neutral and
low-ionized atoms. 
Note that the main goal of the OP is to give reliable data
required for opacity calculations. The major efforts of the
OP collaboration are to get highly accurate photoionization
cross sections at low energies. In the meantime, the OP database TOPbase 0.6 
gives photoionization cross sections with simple power-law tails at high
energies. Comparison with experiment (Fig. 1a-b) shows that these tails
are much less accurate than the present
analytic fits. The high-energy cross sections provided by
the TOPbase 0.6 database appear to be inaccurate
for the neutral atoms as well as for ions (Fig. 1c-d).
Moreover, the OP calculations
do not include inner shell ionization.
\item The present fits agree very well with the
cross sections from Henke et al. (1982) and
Veigele (1973) which lie far from the
thresholds of the outer shells. The agreement
remains good as long as $E \la 100$~keV, i.e.
as long as the ejected photoelectron is
non-relativistic. This is provided by the
correct non-relativistic asymptote of
Eq.~(1).
\end{enumerate}
These conclusions are in accord with the expected
accuracy of the HDS method, which
should be accurate for multicharged ions,
for the deep shells of neutral and
low-ionized atoms, and also far from the
thresholds of the outer shells of the neutral
and low-ionized atoms. The method may give
significant errors near the
thresholds of the outer shells of the neutral and
low-ionized atoms where the correlation
effects and the effects of atomic shell rearrangement
during the photoionization
process are important (Amusia 1990).

We have also compared our fits with
the fits to the K-shell (i.e., $1s$)
photoionization cross sections
given by Daltabuit \& Cox (1972).
The latter fits are frequently used in astrophysical computer codes.
These fits give errors of
about 30\% at the thresholds, $E=E_{\rm th}$.
The errors decrease between
$E=E_{\rm th}$ and $1.5E_{\rm th}$
but grow noticeably with
increasing $E$, since the fitting formula of Daltabuit \& Cox (1972)
does not reproduce the proper high-energy asymptote. At high
energies, the fits of Daltabuit \& Cox (1972) give
cross sections which are different from
experiment by large factors, whereas our fits are in error
by a few percent only. The more accurate
Seaton's formula used by Daltabuit \& Cox (1972) for the H-like
and He-like ions gives errors within
$2-3$\% in the energy range from once the threshold energy
to $1.5E_{\rm th}$ and more than 10\% at
twice the threshold energy, and it gives cross sections which are
different from experiment by large factors at high energies.

As an illustration, Figs. 2a-c and 3a-c compare
the present fits with experiment and theory
for two astrophysically important elements,
N ($Z=7$) and Fe ($Z=26$). Both
elements are taken in three ionization
stages: neutral, intermediate, and high ionization.

\section{Conclusions}
Comparisons with experiment and theory
show the high accuracy of the
present fits for the partial photoionization
cross sections of the ground
states of atoms and ions up to energies
of 100~keV. 
The cross sections used for fitting are less accurate
at very low energies just above ionization thresholds
of the outer shells
of neutral and low-ionized atoms.
The fits can be used, for instance, in
computer codes which calculate
the plasma ionization state.
Note that the inner shell ionization may lead
to subsequent ejection of Auger
electrons. The probabilities of Auger processes
after the photoionization from
inner shells of all elements with $Z \leq 30$
are listed by Kaastra \& Mewe (1993).

\acknowledgements
DAV is grateful to the Netherlands Organization
for Pure Research (NWO) for
financial support through the visitor programs, and to
the Leids Kerkhoven-Bosscha Fonds for travel support.
We acknowledge the use of the
TOPbase 0.6 database through the Centre
de Donn\'{e}es Astronomiques de Strasbourg.
We wish to thank J. Kaastra for providing us with the
tables of Reilman \& Manson in electronic form, and
for helpful discussions. We are grateful to M. Trzhaskovskaya for
recalculating the photoionization cross sections of He~{\sc i},
Li~{\sc i} and Li~{\sc ii}.

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\newpage
\begin{center}
\bf{Figure captions}
\end{center}

\vspace{0.5cm}
{\bf Fig. 1a-d.} Total photoionization cross
section (solid line) from the present fit
as compared
with theoretical calculations of the Opacity Project
(dotted line) and Reilman \& Manson 1979 (crosses), and with
experimental data of Angel \& Samson 1988 (circles)
and West \& Marr 1976
(triangles): {\bf a} O~{\sc i}, {\bf b} Ne~{\sc i}, {\bf c} Si~{\sc ii},
{\bf d} Ca~{\sc vi}.

\vspace{0.5cm}
{\bf Fig. 2a-c.} Total photoionization cross
section (solid line) from the present fit
as compared with the K-shell fits of
Daltabuit \& Cox 1972 (dashed
line), with theoretical calculations of Reilman \& Manson
1979 (crosses) and the
Opacity Project (dotted line),
with experimental data of Samson \& Angel 1990
(triangles) and with the compilation of Veigele 1973
(circles): {\bf a} N~{\sc i}, {\bf b} N~{\sc iii},
{\bf c} N~{\sc vi}. 


\vspace{0.5cm}
{\bf Fig. 3a-c.}
Total photoionization cross
section (solid line) and the $3d$-shell
cross section (dash--dotted
line) from the present fits as compared
with theoretical calculations of
Reilman \& Manson 1979 (crosses) and
the Opacity Project (dotted line),
and with experimental data from
the compilations of Henke et al. 1982
(triangles) and Veigele 1973 (circles):
{\bf a} Fe~{\sc i}, {\bf b} Fe~{\sc iv},
{\bf c} Fe~{\sc xxiv}.
\end{document}

