Mon. Not. R. Astron. Soc. 322, 779±784 (2001)
Electron collisional excitation of Si viii: (1s22s22p3 4 So3=2 ; 2 Do3=2;5=2 ; 2 Po1=2;3=2 †
fine-structure transitions
K. L. Bell,w A. Matthews and C. A. Ramsbottom
Department of Applied Mathematics and Theoretical Physics, The Queen's University of Belfast, Belfast BT7 1NN
Accepted 2000 October 25. Received 2000 October 16; in original form 2000 August 30
A B S T R AC T
Effective collision strengths for electron-impact excitation of the N-like ion Si viii are
calculated in the close-coupling approximation using the ab initio R-matrix method. The 22
fine-structure levels arising from the 11 lowest LS target states: 2s22p3, 2s2p4, 2p5 and
2s22p23s are retained in the present calculation. The effective collision strengths for all
forbidden transitions within the 1s22s22p3 4 So3=2 ; 2 Do3=2;5=2 ; 2 Po1=2;3=2 ground configuration
levels are obtained by averaging the electron collision strengths, for a wide range of incident
electron energies, over a Maxwellian distribution of velocities. Results are presented for
electron temperatures in the range log T…K† ˆ 3:3 to log T…K† ˆ 6:5; applicable to many
laboratory plasmas and in particular to astrophysical plasmas. We believe that the results are
the most accurate to date and, indeed, the only data available which take account of
resonance effects.
Key words: atomic processes ± line: formation ± plasmas.
1
INTRODUCTION
Emission lines from nitrogen-like ions with 2s22p3 ground
configurations have long been known to be useful as density
diagnostics of solar and nebular plasmas. The intensity ratio of
forbidden lines that arise from transitions within the ground
configuration of O ii was first suggested by Aller, Ufford & Van
Vleck (1949) as a diagnostic for gaseous nebulae. Feldman et al.
(1978) pointed out that the O ii ratio is not a useful diagnostic for
the solar atmosphere, but that forbidden lines of the ions Mg vi,
Si viii, S x, and Ar xii that belong to the same isoelectronic
sequence as O ii could be used to determine electron densities in
the inner corona. More recently, Doschek et al. (1997) have
derived electron densities in the solar polar coronal holes from a
forbidden spectral line ratio of Si viii. They employed lines
corresponding to transitions 4 So3=2 ±2 Do5=2 and 4 So3=2 ±2 Do3=2 : It is
noted however that this ratio, whilst primarily depending on the
electron density, may also have some sensitivity to the electron
temperature. Recent work by Keenan et al. (1999), in employing
the same ratio in a study of the emission lines of [O ii] in the
optical and ultraviolet spectra of planetary nebulae, clearly shows
that the ratio has sensitivity to the electron temperature. In this
paper we extend our earlier work for Mg vi (Ramsbottom & Bell
1997) and S x (Bell & Ramsbottom 1999) to consider electronimpact excitation for all forbidden transitions within the ground
configuration levels of Si viii.
Surprisingly little attention has been paid to this ion. Davis,
Kepple & Blaha (1976) and (1977) used the distorted wave
w
E-mail: [email protected]
q 2001 RAS
approximation, neglecting exchange, for a few dipole allowed
transitions from the 2s22p3 4So, 2Do levels and parametrized their
effective collision strengths. The only calculation, to the best
knowledge of the present authors, which has been performed for
the fine-structure forbidden transitions considered in this paper is
that of Bhatia & Mason (1980). They also employed the distorted
wave method, together with a transformation of the K-matrices
obtained in LS-coupling, into pair-coupling using algebraic recoupling coefficients, which allow for spin-orbit interaction and
other relativistic effects. 13 fine-structure levels were considered,
namely those arising from the seven LS-coupling states associated
with the 2s22p3 and 2s2p4 configurations, and limited configuration-interaction was included in the target state representation.
They presented collision strengths only at three values of the
electron energy: 10, 15 and 20 Ryd. In their review article,
Pradhan & Gallagher (1992) gave a percentage uncertainty to
these collision strengths of less than 30 per cent. However, a more
significant error can arise in calculating the effective collision
strength. The distorted wave method neglects resonances and it
has been shown that even the inclusion of additional resonances
can affect fine-structure forbidden transitions by as much as a
factor of two (Ramsbottom, Bell & Keenan 1997).
Clearly there is a need for a large and sophisticated closecoupling calculation, which includes extensive configurationinteration wavefunctions to represent the target states and which
takes full account of resonance effects.
2
METHOD
The configuration-interaction code civ3 (Hibbert 1975) was used
780
K. L. Bell, A. Matthews and C. A. Ramsbottom
to calculate the wavefunctions for the nitrogen-like energy levels
of Si viii in LS coupling. Each term was represented by
wavefunctions of the form
C…LS† ˆ
m
X
ai Fi …ai LS†:
…1†
iˆ1
The electron configuration and angular momentum coupling
scheme of the ith configuration are denoted by a i. The optimal
coefficients {ai} are the components of appropriate eigenvectors
of the Hamiltonian matrix, with elements
H ij ˆ kFi jHjFj l;
1s22s22p3p2, 1s22s22p4s2, 1s22s22p4d2, 1s22p33s3p, 1s22p33p4s,
1s22p33p3d and 1s22p33p4d. A total of 330 configurations formed
in this way were retained in the wavefunction expansion for both
odd and even parity states. Table 3 presents the calculated energy
levels in Ryd relative to the 2s22p3 4So ground state. Comparisons
are made with the observed data of Martin & Zalubas (1983), and
the SSTRUCT intermediate coupling calculations of Bhatia &
Mason (1980). The agreement between the present results and
experiment is excellent with the largest difference of 3 per cent
occurring for the odd parity states.
The electron scattering calculations were performed using the
established R-matrix approach. The theory has been described by
while the corresponding eigenvalues are upper bounds to exact
energies of the ionic states. The configuration state functions {Fi }
are built from a set of one-electron functions, each consisting of a
product of a radial function, a spherical harmonic and a spin
function
1
l
unlml ms …r; s† ˆ Pnl …r†Y m
l …u; f†xms …s†:
r
Table 2. The orbital parameters (c,I,z ) of
the radial wavefunctions.
Pnl …r† ˆ
1
…2zjnl †I jnl ‡ 2
jˆ1
‰…2I jnl †!Š 2
cjnl
1
r I jnl exp…2zjnl r†:
Table 1. Optimization of orbitals.
3s
State
2
3s D
4
2
2
2
2
2
1s
0.95053
0.02448
0.00281
0.03316
20.00097
1
1
2
2
2
13.69090
23.27600
6.34003
11.82660
5.09759
2s
20.28083
20.00391
0.16176
20.15017
0.97338
1
1
2
2
2
13.69090
23.27600
6.34003
11.82660
5.09759
3s
0.20121
20.92219
1.41013
1
2
3
10.69512
4.42218
3.04899
4s
3.80438
231.46523
56.75035
229.15708
1
2
3
4
2.31952
2.31952
2.83052
3.33569
2p
0.44352
0.10041
0.47571
0.00345
2
2
2
2
5.70890
8.96174
4.67900
18.30000
3p
4.53333
25.17268
2
3
2.95695
3.80368
3d
1.00000
3
2.84975
4d
1.15439
20.74364
3
4
5.89100
3.35311
4f
1.00000
4
7.82820
Table 3. Target state energies (in Ryd) relative to the 2s22p3 4So
ground state of Si viii.
Configurations
2
z jnl
…3†
The powers of r, {Ijnl}, are held constant, but the coefficients
{cjnl} and the exponents {z jnl} are treated as variational
parameters when determining the radial functions. Nine orthogonal orbitals were used in the calculation, five `spectroscopic'
(1s, 2s, 2p, 3s, 3d) and four pseudo-orbitals (3p, 4s, 4d, 4f). The
parameters cjnl, Ijnl, z jnl for the 1s, 2s and 2p functions were taken
to be the Hartree±Fock values obtained by Clementi & Roetti
(1974) for the 2s22p3 4So Si viii ground state. For the remaining
orbitals, Table 1 gives the configurations used, together with the
state employed to obtain the orbital parameters by the minimization of the associated energy level. The pseudo-orbitals are
included to represent the effects of electron correlation omitted by,
for example, the Hartree±Fock method. The coefficients for all of
the orbitals are given in Table 2.
The configuration-interaction wavefunctions of the Si viii target
states were then constructed from this orbital set using configurations obtained from one electron replacement in the basis
reference set 1s22s22p3, 1s22s2p4, 1s22p5 and 1s22s22p23s, the 1sshell being kept closed, together with the configurations
Orbital
Ijnl
…2†
In our calculations, we choose these orbitals to be analytic,
expressed as sums of Slater-type orbitals
k
X
cjnl
function
1s 2s 2p 3s, 1s22s2p4
2
4
2
4
3p
3s P
1s 2s 2p 3s, 1s 2p 3s, 1s 2s2p ,
1s22s22p3s3p, 1s22p23s3p2
3d
3d 4P
1s22s22p23d, 1s22s2p4, 1s22s22p23s,
1s22p43s
4d
2s22p3 4So
1s22s22p3, 1s22s2p33d, 1s22s2p34d
4s
2s2p4 2S
4f
2s22p3 2Do
1s22s2p4, 1s22s22p23s, 1s22s22p24s,
1s22s22p23d, 1s22s22p24d, 1s22s2p33p
1s22s22p3, 1s22s22p24f
Si viii
State
2s22p3 4So
2s22p3 2Do
2s22p3 2Po
2s2p4 4Pe
2s2p4 2De
2s2p4 2Se
2s2p4 2Pe
2p5 2Po
2s22p23s 4Pe
2s22p23s 2Pe
2s22p23s 2De
Martin & Zalubas
(1983)
Present
LS Energy
Bhatia & Mason
(1980)
0.0000
0.6317
0.9637
2.8684
3.9218
4.5985
4.8407
7.4915
13.068
13.236
13.553
0.0000
0.6522
0.9895
2.8844
3.9619
4.7025
4.8979
7.5593
13.066
13.242
13.578
0.0000
0.6581
0.9495
2.8540
4.0112
4.6685
5.0071
±
±
±
±
q 2001 RAS, MNRAS 322, 779±784
Electron collisional excitation of Si viii
Burke & Robb (1975), and Seaton (1987), and the associated codes
given by Berrington et al. (1987). 35 Schmidt-orthogonalized
continuum orbitals were included, so that a converged R-matrix
was obtained up to the highest electron energy considered,
namely, 160 Ryd. The R-matrix radius was taken to be 6 atomic
units. The LS-coupled Hamiltonian diagonal matrices were adjusted
so that the theoretical term energies matched the recommended
values of Martin & Zalubas (1983). We note that this energy
adjustment ensures the correct positioning of resonances relative
to all thresholds included in the calculation. All total angular
momenta up to and including L ˆ 12 were included together with
all possible total spin, so that for the forbidden transitions
considered in this paper, convergence of the collision strengths
was fully achieved. Having obtained the scattering K-matrices
within the framework of the LS-coupling scheme, we utilize the
program of Saraph (1978) to transform to LSJ intermediatecoupling including fine-structure mixing of the target terms and
thus produce collision strengths, V, between the J-resolved levels
[McLaughlin & Bell (2000) provide fuller details of the
transformation].
We note that the cross-section between an initial target level i
and a final level f is related to the collision strength (Vif) by the
expression
sif ˆ Vif
pa2o
cm2 ;
vi k2i
…4†
where v i is the statistical weight of the initial target level, k2i the
incident electron energy in Ryd and ao the Bohr radius. While the
collision strength is often useful in its own right, more frequently
in application it is assumed that the scattering electrons have a
Maxwellian velocity distribution. The effective collision strength
is then the appropriate quantity. For excitation from level i to level
f, the (dimensionless) effective collision strength, Yif, at electron
temperature Te (in K) is given by
…1
Yif …T e † ˆ Vif …Ef † exp…2Ef =kT e † d…Ef =kT e †;
…5†
0
where Ef is the final free electron energy after excitation and k is
Boltzmann's constant.
781
1
(a)
0.8
0.6
0.4
0.2
0
0
10
20
30
Incident Electron Energy (Ryds)
1.5
(b)
1
0.5
0
0
10
20
Incident Electron Energy (Ryds)
30
Figure 1. Collision strength as a function of incident electron energy in
Ryd for the fine-structure transitions: (a) 2s22p3 4 So3=2 ±2s2 2p3 2 Do3=2 and
(b) 2s22p3 4 So3=2 ±2s2 2p3 2 Do5=2 : Present results are lines while the dots
represent Bhatia & Mason (1980).
0.5
(a)
0.4
0.3
0.2
0.1
3
R E S U LT S A N D D I S C U S S I O N
In order to evaluate accurately the effective collision strengths, it
is imperative to know the energy dependence of the collision
strength over a wide energy region and for a large number of
energies. Therefore, to properly delineate the complex autoionizing resonances in the cross-sections, we have utilized a
sufficiently fine mesh of incident impact energies (typically
0.005 Ryd) across the range 0±160 Ryd. Pseudo-resonances
arising from our inclusion of pseudo-orbitals in the wavefunction
representation (Burke, Sukumar & Berrington 1981) have been
smoothed out for energies above the highest lying threshold
included (2s22p23s 2De).
Figs 1±5 graphically present the collision strength for all 10
fine-structure forbidden transitions between the 2s22p3 levels in
Si viii and comparison is made with the earlier results of Bhatia &
Mason (1980). For most transitions the agreement between the
present results and the distorted wave data is at least as good as
the 30 per cent accuracy indicated by Pradhan & Gallagher (1992)
as the accuracy for the Bhatia & Mason calculation; indeed,
for the two transitions 4 So3=2 ±2 Do5=2 and 2 Do3=2 ±2 Po3=2 the accord is
q 2001 RAS, MNRAS 322, 779±784
0
0
10
20
30
Incident Electron Energy (Ryds)
0.2
(b)
0.15
0.1
0.05
0
0
10
20
Incident Electron Energy (Ryds)
30
Figure 2. Collision strength as a function of incident electron energy in
Ryd for the fine-structure transitions: (a) 2s22p3 4 So3=2 ±2s2 2p3 2 Po1=2 and (b)
2s22p3 4 So3=2 ±2s2 2p3 2 Po3=2 : Present results are lines while the dots
represent Bhatia & Mason (1980).
782
K. L. Bell, A. Matthews and C. A. Ramsbottom
1
1
(a)
(a)
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
10
20
30
0
Incident Electron Energy (Ryds)
10
20
30
Incident Electron Energy (Ryds)
0.5
0.5
(b)
(b)
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0
10
20
Incident Electron Energy (Ryds)
30
Figure 3. Collision strength as a function of incident electron energy in
Ryd for the fine-structure transitions: (a) 2s22p3 2 Do3=2 ±2s2 2p3 2 Do5=2 and
(b) 2s22p3 2 Do3=2 ±2s2 2p3 2 Po1=2 : Present results are lines while the dots
represent Bhatia & Mason (1980).
0.5
(a)
0.4
0.3
0.2
0.1
0
0
10
20
30
Incident Electron Energy (Ryds)
0.3
(b)
0.2
0.1
0
0
10
20
Incident Electron Energy (Ryds)
30
Figure 4. Collision strength as a function of incident electron energy in
Ryd for the fine-structure transitions: (a) 2s22p3 2 Do3=2 ±2s2 2p3 2 Po3=2 and
(b) 2s22p3 2 Do5=2 ±2s2 2p3 2 Po1=2 : Present results are lines while the dots
represent Bhatia & Mason (1980).
0
10
20
Incident Electron Energy (Ryds)
30
Figure 5. Collision strength as a function of incident electron energy in
Ryd for the fine-structure transitions: (a) 2s22p3 2 Do5=2 ±2s2 2p3 2 Po3=2 and
(b) 2s22p3 2 Po1=2 ±2s2 2p3 2 Po3=2 : Present results are lines while the dots
represent Bhatia & Mason (1980).
excellent. However, more significant differences are found for the
two transitions 4 So3=2 ±2 Do3=2 and 4 So3=2 ±2 Po1=2 and therefore
significant differences will be found for any effective collision
strength calculated by the two sets of data, even ignoring the
resonance structure in the present calculation. The present work
reveals, for all transitions, a wealth of resonance structure
dominating the collision strength in the low-energy region. The
effect of these resonances converging to the target state thresholds
is to enhance the corresponding effective collision strengths,
particularly in the low-temperature region. Such resonant
enhancement has been consistently found (cf. Bell & Ramsbottom
1999 and references therein) and is a common characteristic for
forbidden transitions of this kind.
To illustrate the effect of resonances on the effective collision
strength and to help in assessing the accuracy of the results we
consider two examples in Figs 6 and 7. In both figures we plot the
effective collision strength using the full (including resonances)
calculated collision strength data as well as the effective collision
strength obtained by extracting a non-resonant background
(equivalent to the distorted wave approximation) from the collision strength. In both cases the effect of including resonances is
clear and, in Fig. 7 for the 2 Do3=2 ±2 Do5=2 transition, the shape of the
effective collision strength plotted against log(T) changes with a
factor of about 4 increase at the maximum owing to the inclusion
of resonances. For log(T) less than about 5.7, the increase comes
entirely from the resonance structure lying below the 2s2p4 2P
state, while at higher temperatures a further increase of less than
10 per cent comes from the resonance structure lying at higher
incident electron energies.
Table 4 presents the effective collision strengths for all 10
q 2001 RAS, MNRAS 322, 779±784
Electron collisional excitation of Si viii
783
2.5
0.16
0.15
0.14
2
Effective Collision Strength
Effective Collision Strength
0.13
0.12
0.11
0.1
0.09
0.08
1.5
1
0.07
0.5
0.06
0.05
0.04
3.3
3.8
4.3
4.8
5.3
Log T (Kelvin)
5.8
0
3.3
6.3
Figure 6. Effective collision strength as a function of log temperature in K,
for the 2s22p3 4 So3=2 ±2s2 2p3 2 Po3=2 fine-structure transition. Solid line:
present results including all resonant structure; dashed line: present results
using non-resonant background collision strength.
3.8
4.3
4.8
5.3
Log T (Kelvin)
4 o
S3=2 ±2 Do3=2
4 o
S3=2 ±2 Do5=2
4 o
S3=2 ±2 Po1=2
4 o
S3=2 ±2 Po3=2
2.01621
2.04721
2.20721
2.28921
2.33421
2.36321
2.38221
2.40321
2.40921
2.40521
2.37121
2.30421
2.20621
2.08121
1.94721
1.83221
1.72121
1.56721
1.35721
1.11621
3.02321
3.07121
3.31021
3.43421
3.50121
3.54421
3.57321
3.60421
3.61321
3.60821
3.55721
3.45621
3.30921
3.12221
2.92121
2.75221
2.59321
2.37221
2.06321
1.70121
6.48722
6.48422
6.64822
6.79022
6.84222
6.84222
6.82122
6.77122
6.73522
6.71322
6.72722
6.76222
6.70322
6.48922
6.19322
5.92622
5.61622
5.09822
4.35222
3.49722
1.29721
1.29721
1.33021
1.35821
1.36821
1.36821
1.36421
1.35421
1.34721
1.34321
1.34521
1.35221
1.34121
1.29821
1.23921
1.18721
1.13021
1.03521
8.92822
7.24922
5.41421
6.30821
1.186
1.605
1.865
2.017
2.099
2.142
2.109
2.026
1.804
1.595
1.426
1.267
1.104
9.49621
8.14921
6.87021
5.57921
4.33821
Do5=2 ±2 Po3=2
2 o
P1=2 ±2 Po3=2
2.88421
2.89021
3.04021
3.19721
3.30521
3.38621
3.46121
3.63621
3.85821
4.19621
5.07021
5.89521
6.37521
6.40221
6.04921
5.54621
5.04321
4.53421
4.02221
3.55421
1.36521
1.37621
1.56621
1.77821
1.93421
2.06421
2.19321
2.47621
2.77221
3.12921
3.78121
4.14721
4.18321
3.95521
3.58321
3.23421
2.94321
2.61121
2.19321
1.73721
3.3010
3.3979
3.6990
3.8751
4.0000
4.0969
4.1761
4.3010
4.3979
4.5000
4.7000
4.9000
5.1000
5.3000
5.5000
5.7000
5.9000
6.1000
6.3000
6.5000
log T
3.3010
3.3979
3.6990
3.8751
4.0000
4.0969
4.1761
4.3010
4.3979
4.5000
4.7000
4.9000
5.1000
5.3000
5.5000
5.7000
5.9000
6.1000
6.3000
6.5000
q 2001 RAS, MNRAS 322, 779±784
2
Do3=2 ±2 Po1=2
1.06221
1.06421
1.12421
1.19021
1.23721
1.27121
1.30321
1.37321
1.45921
1.58821
1.92221
2.24321
2.44021
2.46421
2.33421
2.14421
1.95521
1.76721
1.58121
1.41421
2
Do3=2 ±2 Po3=2
1.67021
1.67421
1.74921
1.82021
1.86621
1.90121
1.93521
2.02121
2.13921
2.32321
2.80021
3.23521
3.46221
3.44521
3.23921
2.96321
2.68621
2.40021
2.10421
1.82521
6.3
Figure 7. Effective collision strength as a function of log temperature in K,
for the 2s22p3 2 Do3=2 ±2s2 2p3 2 Do5=2 fine-structure transition. Solid line:
present results including all resonant structure; dashed line: present results
using non-resonant background collision strength.
Table 4. Effective collision strengths for Si viii as a function of the decimal logarithm
of the electron temperature in K.
log T
5.8
2
Do5=2 ±2 Po1=2
1.21621
1.21921
1.27121
1.31921
1.35021
1.37321
1.39621
1.45721
1.54021
1.67221
2.01421
2.32321
2.47921
2.46121
2.31121
2.11221
1.91121
1.70121
1.48421
1.27821
2
2
Do3=2 ±2 Do5=2
784
K. L. Bell, A. Matthews and C. A. Ramsbottom
forbidden fine-structure transitions for electron temperatures in
the range log T…K† ˆ 3:3 to log T…K† ˆ 6:5; applicable to many
laboratory plasmas and in particular to astrophysical plasmas. The
accuracy of the present data for the effective collision strengths is
difficult to predict and can only properly be assessed by
comparison with more sophisticated calculations and/or through
application to the interpretation of astrophysical plasma data.
However, from the discussion above and from our experience
from similar and other calculations (Bell & Ramsbottom 1999 and
references therein) would suggest that an accuracy in the region of
<10 per cent can be anticipated in the low-temperature region
where all the open channels are included in the close-coupling
calculation. However, a lower accuracy will certainly be the case
for the high-temperature results. The most serious error in the
present data arises for the omission of higher-lying target states in
the wavefunction representation of the Si viii ion. The present
calculation is the most elaborate to date and to include higherlying target states with extensive configuration-interaction wavefunctions poses considerable problems. We are confident that the
accuracy of the present data will meet the requirements of most
plasma diagnostics.
AC K N O W L E D G M E N T S
The work reported in this paper has been supported by PPARC,
under the auspices of a Rolling Grant PPA/G/O/1997/00693. Our
thanks are also due to the referee, J. Dubau, for comments which
contributed to the improvement of this paper.
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