In general the energy levels are taken from the NIST (Martin et al. 1995) database. For those levels not available from the NIST database we have calculated theoretical energies using the UCL SSTRUCT program (Eissner et al. 1974). These have been carefully adjusted to provide what we consider to be our "best estimates" of the predicted energy values for individual ions. In those cases where there is no reliable energy, the energy is taken from the target values in the atomic scattering calculation.
The energy levels are generally arranged so that the higher energy levels correspond to a higher index. However, we have found that it is often more useful to use a fixed ordering for the LSJ levels throughout an isoelectronic sequence in order to aid in data evaluation and in data interpolation and extrapolation.
For each ion, one file contains the radiative data including wavelengths, weighted oscillator strengths and A values. The wavelengths are calculated from the energy levels based on observed spectra. If values of the observed energy levels are not available, the wavelength is calculated from the theoretical wavelength but is stored in the file as a negative value to distinguish reliable wavelengths from predicted wavelengths. All wavelengths are vacuum wavelengths. Radiative transition probabilities (A values) are generally taken from the literature. Transitions for which the branching ratio is less than 10-5 have been removed from the files that are generally distributed but are still available.
In a number of cases, the radiative data are not available in the literature and we have used the radiative code SSTRUCT, described in Eissner et al. (1974), to calculate theoretical energy levels and radiative data. A scaled central potential is used to obtain the one-electron radial wavefunctions and the scaling parameters are obtained by minimizing the term energies. Using the code is relatively simple--the main inputs being the charge of the ion and the configurations to be used in the model--however some care has to be taken in the choice of configurations to ensure the accuracy of the energy levels and radiative transition probabilities.
SSTRUCT can calculate electric dipole (E1) and quadrupole (E2), and magnetic dipole (M1) and quadrupole (M2) transition probabilities. For Chianti, the code has been used for three main cases: ions for which no electronic version of the radiative data exists--in such cases the SSTRUCT data was checked against the original data, e.g., Fe X; ions for which the previously calculated radiative data could be improved upon, e.g., Fe IX; individual ion transitions for which radiative data was unavailable, e.g., Fe XIII. Where SSTRUCT has been used, the configurations used in the calculation are documented in the radiative data files.
The electron collisional excitation rate
coefficient (cm3 s-1)
for a Maxwellian electron velocity distribution
with a temperature
(K), is given by:
where 
is the statistical weight of
level i;
Ei,j is the energy difference between levels
i and j;
k is the Boltzmann constant and
is the thermally-averaged
collision strength:
where Ej is the energy of the scattered
electron relative to the
final energy state of the ion. The collision
strength 
is a
symmetric (
),
dimensionless quantity. It is
related to the electron excitation cross-section
, where 
is the area of
the first Bohr orbit and E is the incident
electron energy in
Rydbergs. The electron de-excitation rates are
obtained by the
principle of detailed balance.
Collision strengths are obtained from theoretical calculations. The solution of the electron-ion scattering problem is complex and requires extensive computing resources. The accuracy of a particular calculation depends on two main factors. The first is the representation which is used for the target wavefunctions, the second is the type of scattering approximation chosen.
The target must take account of configuration interaction and allow for intermediate coupling for the higher stages of ionization. The main approximations used for electron-ion scattering are Distorted Wave (DW), Coulomb Bethe (CBe) and the more elaborate Close-Coupling (CC) approximation. The DW approximation neglects the coupling of the channels (target + scattering electron). Since the scattering electron sees a central field potential, the DW approximation is valid for systems which are more than a few times ionized. The University College London (UCL) DW program (Eissner & Seaton 1972) has been extensively used by Nussbaumer, Flower, Mason, Bhatia and co-workers. For high partial wave values of the incoming electron, the CBe approximation (Burgess & Shoerey 1974) is valid, when it is assumed that the scattering electron does not penetrate the target.
The Coulomb Born (CB) approximation has been used extensively in early work. This takes no account of the distortion of the scattering electron's wavefunction due to the central field potential. The Coulomb Born Oppenheimer (CBO) approximation includes exchange (cf. Burgess et al. 1970).
Extreme care must be taken to ensure that the high partial wave contribution is accurately accounted for in both dipole and non-dipole transitions. This has not always been the situation in published results. The Burgess & Tully (1992) scaling method allows one to determine whether or not the electron scattering calculations are tending towards their correct high energy or temperature limits. Burgess et al. (1996) use the Coulomb Born approximation to determine the high energy limits for non-dipole transitions.
In the CC approximation, the scattering electron
sees individual target
electrons, the channels are coupled and a set of
integro-differential
equations are solved. A CC code which has been
extensively used is the
R-matrix package (Burke et al.
1971; Berrington et al.
1978) developed
at Queens University of Belfast (QUB). The CC
approximation is the
most accurate. It is also the most expensive in
terms of computing
resources and it has sometimes been necessary to
truncate the size of
the target, i.e. the number of interacting
configurations which are
included. Sometimes pseudo states
(non-spectroscopic orbitals) are
used to simulate the missing configurations. In
practice, the accuracy
of any calculation depends on the target as well
as the scattering
approximation, so it should not automatically be
assumed that CC
results are always better than DW results. In
general DW is thought to
be accurate to about 25
and CC to better than
10. The
agreement between CC and DW calculations for
allowed transitions is
usually excellent (Burgess et al. 1989,
1991). Resonance structures
can contribute significantly to the excitation
rates, particularly for
forbidden and intersystem lines and these
generally can be accounted for only by
the CC models. Although resonances can
contribute significantly to the
excitation rates, the process of radiation
damping can reduce their
effect for highly ionized systems. This
importance of this process has
only recently been acknowledged and allowing for
it is still not
straightforward.
It is important that electron scattering calculations for highly ionized systems take account of the spin-orbit effect and other relativistic corrections to the target wavefunctions. For low stages of ionization, the transformation from LS coupling to intermediate coupling (IC) is straightforward - it is just algebraic. However, for higher stages of ionization, the transformation from LS to IC must take account of the relevant relativistic interactions as a perturbation to the non-relativistic Hamiltonian. The scattering calculations are often carried out with target in LS coupling, including some one-body terms, then the target is transformed using term coupling coefficients (TCC) from an atomic structure calculation in IC coupling (cf. (Saraph 1972).
A relativistic DW code has been developed by
Sampson, Zhang and
co-workers (Zhang & Sampson 1994)
based on the multi-configuration
Dirac-Fock program of Grant and co-workers
(Grant et al. 1980). It
does not calculate the resonance structure or
allow for the coupling
between channels. A Dirac-Fock R-matrix (DARC)
package has been
developed by Norrington & Grant
(1987). Comparisons of collision data
between the Dirac-Fock R-matrix, relativistic
distorted wave, Breit-Pauli
R-matrix, and R-matrix with TCC's is generally
good for the iron ions and
elements with Z 
26.
Recent advances in laboratory techniques have enabled more accurate determinations of the electron excitation rate coefficients for ions. This field of research is too extensive to cover in this paper but excellent reviews are given by Henry (1993) and Dunn (1992).
A semi-empirical formula commonly used in
astrophysics is the effective
Gaunt factor or 
approximation (Van
Regemorter 1962). This is
based on the CBe approximation and relates the
collision strength to
the oscillator strength for electric dipole
transitions:
With this formula, it is possible to estimate an electron excitation rate when only the oscillator strength is known.
Younger & Wiese (1979) assessed
the reliability of the effective
Gaunt factor approximation. They found that for
transitions in alkali-like ions (where n is the
principal quantum
number), the Gaunt factor is within 25% of unity
and varies slowly
with energy. They proposed an approximate
expression for the Gaunt
factor as a function of Z and incident energy
for the
case. For the case 
, they found
that the Gaunt factor
varies from 0.05 to 0.7 as a function of energy
and concluded that it
was not possible to produce a reliable
approximation to the effective
Gaunt factor. They also conclude that the
effective Gaunt factor
approximation is unsuitable for forbidden and
intercombination
transitions. These often have resonances in the
collision strength,
which are quite narrow in energy and numerous so
that they can make a
substantial contribution to the excitation rate
in addition to the
nonresonant component. More recently,
Sampson & Zhang (1992) have
re-assessed the use of the Van Regemorter
formula. They found it to be
frequently a very poor approximation, especially
for 
excitation transitions and recommended that with
the present
availability of accurate atomic data, the use of
the Van Regemorter
formula should be abandoned.
Mewe (1972) and Mewe &
Gronenschild (1981)
abandoned the simple Van
Regemorter formula and introduced a polynomial
with four terms
AnE-n and a logarithmic term 
expressing the electron
energy (E) dependence of the electron
excitation Gaunt factor. This
expression can be integrated over a Maxwellian
electron energy
distribution. The coefficients An can be
adjusted to fit both
calculated collision strengths and measured
excitation rates as well as
to estimate collision strengths when only few or
even no calculations
exist. The inclusion of the logE term provides
the capability of
having the correct functional form at high
electron impact energies.
Considerable effort has recently been put into
the assessment of
published atomic data, in particular electron
excitation rates. A
workshop on this topic was held in Abingdon in
March 1992 sponsored by
the SOHO CDS and SUMER projects. The
proceedings, edited by J. Lang,
are published as a single volume of the Atomic
Data and Nuclear Data
Tables (Lang 1994). Review papers
are presented on each of the
isoelectronic sequences from H-like to Ne-like,
together with several
Si and S ions and Fe I-Fe XXVI. These reviews
critically assess
published data and provide tables or graphs of
recommended 
values. This work together with the bibliography
of Itikawa (1991) and
the compilation by Pradhan & Gallagher
(1992) provide a comprehensive
survey of the electron excitation rates required
for spectral
diagnostics of astrophysical plasmas. In the
meantime, new
calculations of 's and 's
continue to become available.
The ions of iron pose a difficult challenge for electron scattering calculations, because of the complexity required to accurately represent the n=3 configurations. Mason (1994) has assessed available publications for the coronal ions Fe IX-Fe XIV. She found that existing electron excitation data are severely limited in accuracy. Subsequently, new data for Fe X and Fe XI have been calculated by Bhatia & Doschek (1995a, 1996), which are used here. However, there still remain serious inadequacies with the current atomic data and detailed work using the CC approximation and very accurate targets (including relativistic effects) is underway as part of the Iron Project (Hummer et al. 1993) to remedy this situation. We intend to incorporate these results in future releases of CHIANTI.
Electron excitation data are provided by authors
in many different
formats. Some give as a function of
energy, others give
as a function of temperature. Our task
has been to gather
these electron excitation data together, to
assess it and to present it
in a compressed and easily accessible format.
Burgess & Tully (1992)
provide a very useful method for critically
examining the electron
scattering data and highlighting differences
between calculations. The
method is based on scaling the incident electron
energy and the
collision strength so that they both fall within
a given, finite
range. Each "type" of transition (ie electric
dipole allowed,
forbidden, intercombination) is treated
differently. Taking the case
of dipole (allowed) transitions as an example,
the formulation for the
scaled energy, x, is given by
and the scaled collision strength, y, by
where X is the colliding electron energy in
threshold units and C
is a variable which can be adjusted to suit the
case. The scaled
energy x varies between 0 (threshold) and 1
(infinite energy). The
high energy limit for and high
temperature limit for
is obtained from the Coulomb Bethe
approximation; it is
simply 
, where
is the
weighted dipole oscillator strength and Ei,j
is the energy
difference in Ryd.
Burgess & Tully's interactive graphical programs are written in BBC BASIC. They are not very portable and cannot be run on Unix workstations, although they can be run on PC type machines with an emulator. In this work we have developed IDL routines based on the Burgess & Tully (1992) concept and methods, called BURLY (from BURgess..tulLY). An important aspect of scaling collision strengths by the Burgess and Tully approach is that the scaled values can often be approximated by a straight line so that the extrapolation of the collision strength to threshold and to infinite energy becomes relatively straightforward. A knowledge of the collision strength over the full energy range is needed to determine the integration over a Maxwellian velocity distribution which is carried out with a Gauss-Laguerre method. The collision strengths are finally expressed as a 5 point spline fit to the scaled collision strengths. Burgess et al. (1997) have developed a method for obtaining the infinite energy or temperature values of the collisional data from the Coulomb Born approximation. It is hoped that this will be incorporated into our fitting programs in a future release. We considered it to be very important to ensure that all of the collisional data, each ion and transition, should be visually examined.
The use of the Burgess and Tully scaling has
revealed several problems
with the calculations of 's and
's. The version of
the UCL DW code that has been used until recently
for most of the DW
calculations requires that the energy of the
incident electron be
higher than the energy of the highest level of
the target ion.
Consequently, many DW wave calculations provide
values of at
energies considerably above the threshold so that
extrapolation of the
calculated points back to the threshold can be
uncertain. The value at
the threshold often dominates in determining the
average over a
Maxwellian, especially at low temperature. As
pointed out by Burgess
et al. (1991) and Burgess & Tully
(1992), a potential problem with
collision strength calculations is the tendency
to include an
insufficient number of partial waves, often
because of their cost in
terms of computing resources. The higher partial
wave contributions are
estimated from the Coulomb Bethe or Coulomb Born
approximations or
using a geometrical progression. Inadequate
account of the high
partial wave contribution can lead to either an
overestimate or
underestimate of the collision strength for
allowed transitions,
particularly at high incident energies, so that
the collision strengths
do not approach the correct high energy limit
given by the Bethe
approximation. This can occur with either DW or
CC calculations.
Examples of this have been uncovered and have
been dealt with by
truncating the published values at some energy or
temperature and then
interpolating to the high energy limit.