Mon. Not. R. Astron. Soc. 348, 1275–1281 (2004)
doi:10.1111/j.1365-2966.2004.07461.x
Effective collision strengths for transitions in singly ionized nitrogen
C. E. Hudson and K. L. Bell
School of Mathematics and Physics, The Queen’s University of Belfast, Belfast BT7 1NN
Accepted 2003 November 18. Received 2003 November 18; in original form 2003 September 9
ABSTRACT
The R-matrix method has been used to calculate effective collision strengths for the electronimpact excitation of the carbon-like ion N II. The lowest 23 LS target states are included in the
expansion of the total wavefunction, consisting of the nine n = 2 states with configurations
2s2 2p2 and 2s2p3 , along with 14 n = 3 states with configuration 2s2 2p3l (l= s,p,d). The
fine-structure collision strengths have been obtained by transforming to a jj-coupling scheme
using Saraph’s JAJOM program, and have been determined at a sufficiently fine energy mesh
to delineate properly the resonance structure. Effective collision strengths are obtained by
averaging the electron collision strengths over a Maxwellian distribution of velocities. Results
are presented for transitions among the 2s2 2p2 3 P, 1 D, 1 S and 2s2p3 5 S levels for electron
temperatures in the range log T e (K) = 2.0−5.1, appropriate for astrophysical applications.
Key words: atomic processes – line: formation – methods: analytical.
1 INTRODUCTION
Nitrogen, like carbon, is amongst the most abundant of the chemical
elements, and essential to life on Earth. As such, there has been much
interest in nitrogen and its ions over the last few years. Recently,
Henry, Edmunds & Koppen (2000) have analysed the cosmic origins
of carbon and nitrogen, with their nitrogen data being largely based
on N II measurements and analysis from, for example, spiral discs,
‘post-starburst’ galaxies and H II regions at the edge of the Galaxy.
Rubin et al. (1998) have also examined temperature variations in
the N+ region of the Orion Nebula using N II observations from the
Hubble Space Telescope (HST).
Data from specific transitions are employed as diagnostics to determine electron temperature and density. For example, in the optical
spectrum, the 2s2 2p2 1 De2 –2s2 2p2 1 Se0 , 2s2 2p2 3 Pe1 –2s2 2p2 1 De2 and
the 2s2 2p2 3 Pe2 –2s2 2p2 1 De2 transitions, observed at 5756.2, 6549.9
and 6585.2 Å, respectively, give rise to the I(5756.2 Å)/I(6549.9 +
6585.2 Å) line ratio, which is frequently used as an electron temperature diagnostic (Aller 1984).
Lines in the ultraviolet and infrared regions are also used as diagnostics. The 2139.7 and 2143.5 Å lines in the ultraviolet spectrum, arising from the 2s2 2p2 3 Pe1 –2s2p3 5 So2 and 2s2 2p2 3 Pe2 –2s2p3
5 o
S2 transitions, have been used in conjunction with the 6549.9 and
6585.2 Å lines to provide electron temperature and density diagnostics (Czyzak, Keyes & Aller 1986). An electron density diagnostic based on emissions in the infrared region, using the transitions
2s2 2p2 3 Pe1 –2s2 2p2 3 Pe2 and 2s2 2p2 3 Pe0 –2s2 2p2 3 Pe1 with lines at 122
and 205µm, respectively, is also discussed by Keenan et al. (2001).
Hence there is a vast quantity of observable data which require
analysis using theoretical predictions.
E-mail: [email protected]
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2004 RAS
Theoretically, the emission-line ratios are calculated using electron excitation rates which are derived from effective collision
strengths. Previous calculations for the effective collision strengths
among the N II levels include two R-matrix calculations both carried
out in 1994. Lennon & Burke (1994) carried out a 12-state calculation, the targets being of the form 2s2 2p2 , 2s2p3 and 2p4 . Three real
orbitals (1s, 2s, 2p) and four pseudo-orbitals (3s, 3p, 3d, 4s) were
included, and the effective collision strengths were produced over
the electron temperature range of log T e (K) = 3.0−5.0 (i.e. T e in
the range 1000–100 000 K).
The calculation of Stafford et al. (1994) incorporated the 13 lowest
LS target states, using five real orbitals (1s, 2s, 2p, 3s, 3p) and three
pseudo-orbitals (3d, 4s, 4p). The target states here were of the form
2s2 2p2 , 2s2p3 , 2s2 2p3s, 2s2 2p3p, and the data were produced for
values of T e in the range 5000–125 892 K [i.e. log T e (K) = 3.7–
5.1].
As noted by Rubin et al. (1998), there are significant differences
in the effective collision strengths from the two calculations. This
was further echoed in the work of Keenan et al. (2001), where some
low-temperature data from the calculation of Stafford were reported
(temperatures down to 1000 K), and the results found to differ by
up to a factor of 2.1. Thus to resolve these discrepancies and to
provide additional low-temperature data, an improved calculation
is presented in this paper.
2 D E TA I L S O F T H E C A L C U L AT I O N
Configuration interaction (CI) wavefunctions were constructed for
the 23 lowest LS target states of N II using the CIV3 code (Hibbert
1975). Each target-state wavefunction is represented by a linear
combination of single-configuration functions i , each having the
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C. E. Hudson and K. L. Bell
same total LS π symmetry as the target-state wavefunction
m
(L S) =
ai i (αi L S),
(1)
i=1
where the single-configuration functions i are constructed from a
set of one-electron orbitals, with the α i representing the coupling of
the angular momenta associated with these one-electron spin orbitals
to form the total L and S. The a i are eigenvector components of the
Hamiltonian matrix having particular LS π symmetry, with elements
Hi j = i |H | j ,
(2)
where H denotes the Hamiltonian operator. The one-electron orbitals used to construct the i each consist of a radial function, a
spherical harmonic and a spin function:
1
m
Pnl (r )Yl l (θ, φ)χm s (σ ).
(3)
r
These orbitals are chosen to be analytic, with the radial part being
expressed as a sum of Slater-type orbitals:
u nlml m s (r, σ ) =
Pnl (r ) =
C jnl r I jnl exp(−ζ jnl r ).
(4)
jnl
In this expression, the powers of r are kept fixed and the coefficients and exponents are treated as variational parameters. 10
orthogonal orbitals are used in this calculation: six real orbitals (1s,
2s, 2p, 3s, 3p, 3d) and four pseudo-orbitals (4s, 4p, 4d, 4f). The
c jnl , I jnl and ζ jnl parameters for the 1s orbital were taken to be the
Hartree–Fock values of Clementi & Roetti (1974) for the N II ground
state 2s2 2p2 3 Pe . The 2s and 2p orbitals were also initially taken to be
the N II ground-state Hartree–Fock values, but were re-optimized on
the energy of the 2s2p3 1 Do state. The 3s orbital was optimized on the
2s2 2p3s1 Po state using a single configuration. The 3p and 3d orbitals
were optimized using two configurations: the 3p was optimized on
the 2s2 2p3p1 Se state using the 2s2 2p2 and 2s2 2p3p configurations;
the 3d was optimized on the 2s2 2p3d1 Do state using the 2s2p3 and
2s2 2p3d configurations.
The pseudo-orbitals were optimized in the following way: a 4s
orbital was optimized on the energy of the 2s2 2p2 3 P state using the
2
2s2 2p2 , 2s2p2 3s, 2s2p2 4s, 2p2 3s2 and 2p2 4s configurations. This 4s
orbital, optimized in this way, allows for the differences in the 2s
orbital between the 2s2 2p2 and the 2s2p3 states. For similar reasons,
a 4p orbital was added and optimized on the energy of the 2s2 2p2 1 S
state using the 2s2 2p2 , 2s2 2p3p and 2s2 2p4p configurations, so here
correcting for the differences in the 2p orbital in the 2s2 2p2 and the
2s2 2p3l states.
A 4d orbital was also used, to allow for coupling between 2s2p3
and 2s2 2pnd states. This was optimized on the 2s2p3 1 Do state using
configurations of the form 2s2p3 , 2s2 2p3d and 2s2 2p4d. Finally, a
4f orbital was also optimized on the same state using the 2s2p3 and
2s2p2 4f configurations. The orbital parameters resulting from these
optimizations are presented in Table 1, along with the Hartree–Fock
parameters for the 1s orbital.
The wavefunctions of the 23 N II target states were constructed
from this orbital set using configurations generated by a two-electron
replacement on the 2s2 2p2 basis configuration, with the 1s shell
being kept closed. Following this scheme, a total of 1520 configurations were generated. Table 2 shows the calculated energy
levels for the LS target states in atomic units (1 au = 1.08993 ×
10−18 J), relative to the 2s2 2p2 3 Pe ground state. These values are
compared with the experimental values from the US National Institute of Standards and Technology (NIST) data base (accessible
at http://physics.nist.gov). The data base values have been taken
from Moore (1993) and references therein (Green & Maxwell 1937;
Moore 1949; Eriksson 1958).
The agreement between the current theory and the observed energies is very satisfactory, with the largest percentage difference
between the two being 6 per cent for state number 2 – the 2s2 2p2
1 e
D level. When compared to the energy levels from the calculation
of Stafford et al. (1994), it is found that the current energies lie closer
to the NIST values, in all cases except for state 12, the 2s2 2p3p3 De
level.
The (N + 1), ion-plus-electron, system is described using the
R-matrix method given by Burke & Robb (1975), using the associated computer codes detailed by Berrington et al. (1987). Configurations describing the (N + 1)-electron symmetries are generated
by the addition of one electron from the orbital set to those configurations previously generated by the two-electron replacement on
the 2s2 2p2 configuration. The R-matrix radius is calculated to be
to be 18.0 au, and 30 continuum orbitals were included for each
orbital angular momentum, ensuring convergence up to an incident
electron energy of 14 Ryd (1 Ryd = 2.17987 × 10−18 J).
Although there is good agreement between the calculated energy
levels of the target states with experiment, the LS-coupled Hamiltonian diagonal matrices are adjusted so that the calculated energy
Table 1. Orbital parameters optimized using the CIV3 code.
Orbital
c jnl
I jnl
ζ jnl
1s
0.91649
0.07678
0.00539
0.01393
−0.00200
1
1
2
2
2
6.42321
10.61430
2.53445
6.15603
1.79167
2s
−0.19796
−0.02487
0.43679
−0.10758
0.66778
1
1
2
2
2
6.51157
9.68443
2.70746
5.77864
1.78563
0.53054
0.18887
0.33066
0.01355
2
2
2
2
10.73127
4.15522
1.49966
1.02543
2p
Orbital
c jnl
I jnl
ζ jnl
Orbital
c jnl
I jnl
ζ jnl
3s
0.11825
−0.49585
1.09576
1
2
3
5.69655
2.07460
0.94717
4s
1.41548
−4.58360
3.80048
−0.70819
1
2
3
4
2.38853
2.31023
2.26604
1.10471
3p
0.28685
−1.03820
2
3
1.81217
0.66607
4p
1.61538
−3.56620
2.14428
2
3
4
1.65662
1.25106
1.10537
3d
0.02354
0.46059
0.53479
3
3
3
1.82592
0.77599
0.65024
4d
1.34081
−0.54915
0.44544
−0.70694
3
3
4
4
2.13964
2.16613
1.34164
0.81866
4f
1.00000
4
2.82164
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2004 RAS, MNRAS 348, 1275–1281
Effective collision strengths for singly ionized nitrogen
Table 2. Target-state energy levels (in au) relative to the 2s2 2p2 3 Pe ground
state of N II. The experiment values of NIST are also shown, along with
those from the calculation of Stafford et al. (1994).
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
N II state
Current
NIST
Stafford
2s2 2p2 3 Pe
0.0000
0.0743
0.1577
0.2057
0.4227
0.5054
0.6675
0.6780
0.6808
0.7194
0.7620
0.7707
0.7803
0.7793
0.7845
0.7993
0.8157
0.8503
0.8528
0.8542
0.8609
0.8639
0.8683
0.0000
0.0698
0.1489
0.2132
0.4203
0.4976
0.6570
0.6790
0.6798
0.7068
0.7500
0.7592
0.7598
0.7695
0.7775
0.7938
0.8128
0.8502
0.8524
0.8542
0.8606
0.8627
0.8663
0.0000
0.0774
0.1590
0.2045
0.4304
0.5183
0.6764
0.7030
0.7063
0.7251
0.7629
0.7691
0.7991
2s2 2p2 1 De
2s2 2p2 1 Se
2s2p3 5 So
2s2p3 3 Do
2s2p3 3 Po
2s2p3 1 Do
2s2 2p3s 3 Po
2s2 2p3s 1 Po
2s2p3 3 So
2s2 2p3p 1 Pe
2s2 2p3p 3 De
2s2p3 1 Po
2s2 2p3p 3 Se
2s2 2p3p 3 Pe
2s2 2p3p 1 De
2s2 2p3p 1 Se
2s2 2p3d 3 Fo
2s2 2p3d 1 Do
2s2 2p3d 3 Do
2s2 2p3d 3 Po
2s2 2p3d 1 Fo
2s2 2p3d 1 Po
ωi ki2
,
πa02
(5)
where ω i is the statistical weight of the initial target level, k2i the
incident electron energy in Rydbergs and a 0 the Bohr radius.
For optically forbidden transitions, the inclusion of all partial
waves L 12 is sufficient to ensure convergence of the collision
strengths. However, for the dipole-allowed transitions, the contributions from the L > 12 partial waves to the collision strength is
approximated by assuming that the partial collision strengths form
a geometric series, where the scaling factor is determined from the
ratio of adjacent terms. Thus, in this way, the dipole-allowed transitions are ‘topped-up’ to account for the higher partial waves. This
‘topping-up’ procedure has been used successfully in similar calculations (see, for example, Bell & Ramsbottom 2000; Ramsbottom,
Bell & Keenan 2001).
For many applications, where it is assumed that the electrons have
a Maxwellian velocity distribution, effective collision strengths are
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2004 RAS, MNRAS 348, 1275–1281
often a more useful quantity than the electron collision strengths.
Effective collision strengths ϒ are formed by averaging the collision
strengths over a Maxwellian distribution. Thus, for excitation from
level i to level f , the (dimensionless) effective collision strength,
ϒ if , at electron temperature T e (in Kelvin) is given by
ϒi f (Te ) =
∞
i f (E f ) exp(−E f /kTe ) d(E f /kTe ),
(6)
0
where E f is the final free electron energy after excitation and k is
Boltzmann’s constant.
Resonant structures dominate the collision strengths and can significantly enhance the effective collision strengths. Thus in order
to obtain a detailed picture of these structures so that they may
be fully resolved for each transition, the collision strength calculations were carried out using a fine energy mesh in the region
between the target thresholds. For this calculation a typical mesh
of 0.0001 Ryd was employed. Above the last target threshold at
1.7326 Ryd, pseudo-resonances arise as a result of the inclusion of
pseudo-orbitals in the target wavefunction representation. In order
to average properly over these pseudo-resonances so that they do
not distort the effective collision strength in the high-energy region,
the non-physical pseudo-resonances lying above the highest target
have been smoothed out using a cubic spline fit (see, for example,
Wilson & Bell 2002a,b).
3 R E S U LT S A N D D I S C U S S I O N
levels match those of the experiment data from NIST. This ensures
the correct positioning of resonances relative to the thresholds included in the calculation. It is noted that, in this work, the order of
two of the target states differs from that of the NIST data base. The
states involved are labelled 13 and 14 in Table 2. For these states,
both energies were adjusted to take the arithmetic mean of the two
NIST values, i.e. 0.7647 au.
In forming the (N + 1)-electron symmetries, all total angular momenta 0 L 12 for both even and odd parities, and for doublet,
quartet and sextet multiplicities are included. The scattering matrices are calculated at each electron impact energy by evaluating
the contributions for each LS π symmetry. The JAJOM program of
Saraph (1978) is then used to transform to LSJ-coupling, so producing collision strengths, , between the J-resolved levels.
The collision strength if between an initial target level, i, and a
final level, f , is related to the total cross-section σ if by the expression
i f = σi f
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Effective collision strengths have been calculated over the electron
temperature range log T e (K) = 2.0 − 5.1, in steps of 0.1 dex, for
the 13 transitions arising from among the first six LS levels, i.e.
the 2s2 2p2 3 P, 1 D, 1 S and 2s2p3 5 S levels. The previous work of
Lennon & Burke (1994) provides data for electron temperatures in
the range log T e (K) = 3.0–5.0 for these 13 transitions. Stafford et al.
(1994) provide effective collision strengths for these transitions for
temperatures in the range log T e (K) = 3.7–5.1. Thus this calculation
gives effective collision strengths at lower temperatures than the
previous works. These data were produced at lower temperatures
since they may be applicable to the interstellar medium (see Gry,
Lequeux & Boulanger 1992).
Figs 1 to 9 present the effective collision strengths, and a comparison with the previous works of Lennon & Burke (1994) and Stafford
et al. (1994) is made. A selection of the data from these graphs is
also given, in tabular form, in Table 3. For conciseness in this table,
transitions from the three ground-state levels, 3 P 0 , 3 P 1 , 3 P 2 , to the
final states of 1 D, 1 S and 5 S have been grouped together as 3 P–1 D,
1
S, 5 S. The effective collision strengths for the transitions from the
individual ground-state fine-structure levels may be obtained from
the following formula, which holds closely for all the transitions
noted:
3
2J + 1 3 S
ϒ P J −S L J =
(7)
ϒ P − L J ,
9
where 3 P J represents an individual 3 P fine-structure level, i.e. one
of 3 P 0 , 3 P 1 or 3 P 2 , and S L J represents a final state from 1 D 2 , 1 S 0 or
5
S 2 . The effective collision strength data, over the complete range
of the calculation [log T e (K) = 2.0–5.1], are available, along with
the collision strengths, by contacting the authors.
3.1 3 P–3 P ground-state fine-structure transitions
Effective collision strengths for the ground-state transitions of
2s2 2p2 3 Pe0 –2s2 2p2 3 Pe1 , 2s2 2p2 3 Pe0 –2s2 2p2 3 Pe2 and 2s2 2p2 3 Pe1 –
2s2 2p2 3 Pe2 are shown in Figs 1 to 3 and are compared with the
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C. E. Hudson and K. L. Bell
Table 3. Comparison of the current N II effective collision strengths among the 2s2 2p2 3 P, 1 D, 1 S and 2s2p3 5 S levels
with the earlier calculations of Lennon & Burke (1994) (LB) and Stafford et al. (1994) (S), at a selection of electron
temperatures.
Transition
3.0
3.4
3.8
4.2
4.6
5.0
0.409
0.384
0.413
0.451
0.429
0.439
0.509
0.491
0.482
0.555
0.550
0.528
0.228
0.226
–
0.251
0.250
0.249
0.301
0.301
0.289
0.354
0.352
0.338
0.360
0.350
0.350
0.953
0.898
–
0.971
0.930
–
1.077
1.041
1.076
1.240
1.123
1.199
1.433
1.405
1.363
1.502
1.473
1.447
Current
LB
S
2.541
2.484
–
2.582
2.516
–
2.665
2.588
3.031
2.801
2.700
2.994
2.986
2.844
2.993
3.055
2.967
3.019
2s2 2p2 3 Pe –2s2 2p2 1 Se0
Current
LB
S
0.299
0.281
–
0.302
0.283
–
0.310
0.288
0.376
0.327
3.000
0.365
0.360
0.324
0.364
0.380
0.352
0.372
2s2 2p2 3 Pe –2s2p3 5 So2
Current
LB
S
1.114
1.197
–
1.113
1.194
–
1.117
1.188
1.148
1.144
1.205
1.153
1.168
1.230
1.157
1.075
1.155
1.098
2s2 2p2 1 De2 –2s2 2p2 1 Se0
Current
LB
S
1.124
1.164
–
1.057
1.071
–
0.900
0.918
0.500
0.723
0.761
0.508
0.635
0.683
0.539
0.621
0.657
0.590
2s2 2p2 3 Pe0 –2s2 2p2 3 Pe1
Current
LB
S
0.369
0.334
–
0.367
0.339
–
2s2 2p2 3 Pe0 – 2s2 2p2 3 Pe2
Current
LB
S
0.219
0.214
–
2s2 2p2 3 Pe1 –2s2 2p2 3 Pe2
Current
LB
S
2s2 2p2 3 Pe –2s2 2p2 1 De2
0.40
Effective Collision Strength γ
Effective Collision Strength γ
0.6
0.5
0.4
0.3
0.2
2.0
log T e (K)
2.5
3.0
3.5
log T (K)
4.0
4.5
5.0
0.35
0.30
0.25
0.20
2.0
2.5
3.0
3.5
log T (K)
4.0
4.5
5.0
Figure 1. Effective collision strengths as a function of temperature for
the 2s2 2p2 3 Pe0 –2s2 2p2 3 Pe1 transition [solid line: current results; dot-dashed
line: Lennon & Burke (1994); dashed line: Stafford et al. (1994)].
Figure 2. Effective collision strengths as a function of temperature for
the 2s2 2p2 3 Pe0 –2s2 2p2 3 Pe2 transition [solid line: current results; dot-dashed
line: Lennon & Burke (1994); dashed line: Stafford et al. (1994)].
effective collision strengths from the calculations of Lennon &
Burke (1994) and also of Stafford et al. (1994).
For the 2s2 2p2 3 Pe0 –2s2 2p2 3 Pe1 transition, see Fig. 1, the current
calculation is found to agree with the Lennon & Burke (1994) data
to within 10 per cent at the low-temperature end of the scale [log
T e (K) = 3.0]. These two sets of data then converge so that at the
high-temperature region of the calculation [log T e (K) = 5.0] they
agree to within 1 per cent. The Stafford et al. (1994) data also appear
to give good agreement over the temperature range for which the
calculation is valid, with the two agreeing to within 5 per cent over
the entire range log T e (K) = 3.7–5.1.
In Fig. 2, for the 2s2 2p2 3 Pe0 –2s2 2p2 3 Pe2 transition, the current
results also agree well with both of the previous calculations. The
current data agree with the Lennon & Burke data to within 3 per
cent over the entire temperature range, and for the Stafford et al.
data the agreement is to within 5 per cent everywhere.
The 2s2 2p2 3 Pe1 –2s2 2p2 3 Pe2 transition is shown in Fig. 3. Here the
effective collision strength of Lennon & Burke is found to differ by
about 6 per cent at lower temperatures and by about 2 per cent at
higher temperatures. The Stafford et al. data are found to agree to
within about 1 per cent at the low-temperature extremity, but this
rises to about 4 per cent at higher temperatures. In all three of these
graphs, the shape of the effective collision strengths is found to follow that of the Lennon & Burke work, whereas the Stafford et al. data
seem to cross over the current results in the region of log T e (K) =
3.8, so that in one region the Stafford et al. data overestimate the
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Effective collision strengths for singly ionized nitrogen
1.2
Effective Collision Strength γ
Effective Collision Strength γ
1.6
1.4
1.2
1.0
0.8
0.6
2.0
2.5
3.0
3.5
log T (K)
4.0
4.5
1.0
0.9
0.8
0.7
2.5
3.0
3.5
log T (K)
4.0
4.5
5.0
Figure 5. Effective collision strengths as a function of temperature for the
2s2 2p2 3 Pe1 –2s2 2p2 1 De2 transition [solid line: current results; dot-dashed
line: Lennon & Burke (1994); dashed line: Stafford et al. (1994)].
0.40
2.2
Effective Collision Strength γ
Effective Collision Strength γ
1.1
0.6
2.0
5.0
Figure 3. Effective collision strengths as a function of temperature for
the 2s2 2p2 3 Pe1 –2s2 2p2 3 Pe2 transition [solid line: current results; dot-dashed
line: Lennon & Burke (1994); dashed line: Stafford et al. (1994)].
0.35
0.30
0.25
0.20
2.0
1279
2.5
3.0
3.5
log T (K)
4.0
4.5
5.0
2.0
1.8
1.6
1.4
1.2
1.0
2.0
2.5
3.0
3.5
log T (K)
4.0
4.5
5.0
Figure 4. Effective collision strengths as a function of temperature for the
2s2 2p2 3 Pe0 –2s2 2p2 1 De2 transition [solid line: current results; dot-dashed
line: Lennon & Burke (1994); dashed line: Stafford et al. (1994)].
Figure 6. Effective collision strengths as a function of temperature for the
2s2 2p2 3 Pe2 –2s2 2p2 1 De2 transition [solid line: current results; dot-dashed
line: Lennon & Burke (1994); dashed line: Stafford et al. (1994)].
current work, and then in another region underestimate the current
work. However, one can now assume that, for these transitions, the
effective collision strengths have an accuracy of a few per cent.
For the Stafford et al. calculation, the effective collision strength
is found to agree well in the high-temperature region of the graph.
However, at low temperatures the Stafford et al. data clearly begin
to deviate, with a difference of 15 per cent being observed at log T e
(K) = 3.7.
3.2 The 3 P–1 D 2 fine-structure transitions
Figs 4 to 6 display the effective collision strengths for the 2s2 2p2
2s2 2p2 3 Pe1 –2s2 2p2 1 De2 and 2s2 2p2 3 Pe2 –2s2 2p2 1 De2
transitions. Transitions from fine-structure levels arising from the
same LS state to a particular final level all have the same shape, as
illustrated by Figs 4 to 6, with their relative proportions being given
by equation (7). Thus the differences observed between the current
calculation and those of Lennon & Burke (1994) and of Stafford
et al. (1994) for one of the transitions given in Figs 4 to 6 will be
the same as for the other two transitions. Therefore the following
analysis relates to all three of the transitions 2s2 2p2 3 Pe0,1,2 –2s2 2p2
1 e
D2 .
The effective collision strength from the current calculation is
found to be in excellent agreement with that of Lennon & Burke. The
two curves closely follow each other in shape, and in magnitude they
differ by 2–3 per cent over the range of the Lennon & Burke data.
3 e
P0 –2s2 2p2 1 De2 ,
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2004 RAS, MNRAS 348, 1275–1281
3.3 The 3 P–1 S 0 fine-structure transitions
Fig. 7 shows the effective collision strength for the 2s2 2p2 3 Pe0 –
2s2 2p2 1 Se0 transition. This collision strength is representative of all
the transitions from the ground-state levels to the 2s2 2p2 1 Se0 state,
with the same shape of collision strength being observed for all three
transitions, and so, to avoid excess repetition, graphical representations for the 2s2 2p2 3 Pe1,2 –2s2 2p2 1 Se0 transitions are not explicitly
included. The magnitude of the collision strengths for the 2s2 2p2
3 e
P1,2 –2s2 2p2 1 Se0 transitions can be extracted by applying equation
(7). Hence the observations noted here for Fig. 7 are common to all
three of these transitions.
Close agreement between the current results and those from the
calculation of Lennon & Burke (1994) is again observed, with the
two sets of values agreeing in the region of 6–7 per cent over
C. E. Hudson and K. L. Bell
0.06
1.4
0.05
1.2
Effective Collision Strength γ
Effective Collision Strength γ
1280
0.04
0.03
0.02
0.01
0
2.0
2.5
3.0
3.5
log T (K)
4.0
4.5
5.0
1.0
0.8
0.6
0.4
0.2
2.0
2.5
3.0
3.5
log T (K)
4.0
4.5
5.0
Figure 7. Effective collision strengths as a function of temperature for
the 2s2 2p2 3 Pe0 –2s2 2p2 1 Se0 transition [solid line: current results; dot-dashed
line: Lennon & Burke (1994); dashed line: Stafford et al. (1994)].
Figure 9. Effective collision strengths as a function of temperature for the
2s2 2p2 1 De2 –2s2 2p2 1 Se0 transition [solid line: current results; dot-dashed
line: Lennon & Burke (1994); dashed line: Stafford et al. (1994)].
the temperature range log T e (K) = 3.0–5.0. Also apparent from
this set of transitions is the way in which the Stafford et al. (1994)
data deviate at low temperatures. Whilst the agreement at the hightemperature extremity is within a few per cent of the current work,
at the lower temperature limit of the Stafford et al. calculation
[log T e (K) = 3.7], a difference of 22 per cent is observed.
to lower temperatures there would be a definite and rather sharp
divergence in the effective collision strength.
3.4 The 3 P–5 S 2 fine-structure transitions
Fig. 8 gives the effective collision strength for the 2s2 2p2 3 Pe0 –2s2p3
transition. The transitions from the other two ground-state finestructure levels (2s2 2p2 3 Pe1,2 ) to the 2s2p3 5 So2 state replicate this
shape and so equation (7) is again applicable. For the 2s2 2p2 3 Pe0,1,2 –
2s2p3 5 So2 transitions, the current results again mimic the shape of
the Lennon & Burke data, and the two agree to within 6–7 per cent
over the range of the Lennon & Burke data. For the Stafford et al.
data, the agreement with the current calculation appears to be better
than this, with the two differing by only 2–3 per cent over the temperatures of the Stafford et al. data [log T e (K) = 3.7–5.0]. At the lowtemperature extremity of the Stafford et al. data, however, there
appears to be a turning point, so that if the data were extrapolated
5 o
S2
Effective Collision Strength γ
0.15
0.14
0.13
0.12
0.11
0.10
2.0
2.5
3.0
3.5
log T (K)
4.0
4.5
5.0
Figure 8. Effective collision strengths as a function of temperature for the
2s2 2p2 3 Pe0 –2s2p3 5 So2 transition [solid line: current results; dot-dashed line:
Lennon & Burke (1994); dashed line: Stafford et al. (1994)].
3.5 The 1 D 2 –1 S 0 transition
Fig. 9 shows the effective collision strength for the 2s2 2p2 1 De2 –
2s2 2p2 1 Se0 transition. The data from the Lennon & Burke (1994)
calculation are in excellent agreement. At the low-temperature extremity of log T e (K) = 3.0, the agreement is 4 per cent, and at the
upper end of temperature range, at log T e (K) = 5.0, the agreement is
about 6 per cent, with much of the intermediate region having agreement better than this. For the Stafford et al. (1994) work, the effective
collision strengths almost coincide at the the high-temperature limit
of log T e (K) = 5.1, but from this point down to lower temperatures
the Stafford et al. data diverge quite drastically. At a temperature
of log T e (K) = 3.7, the Stafford et al. results are found to differ by
44 per cent. At lower temperatures, where the current results flatten
out and assume a value of about 1.1, it is easy to see that a factor of
2 difference will occur, as cited by Keenan et al. (2001).
4 CONCLUSIONS
A 23-state R-matrix calculation for the electron-impact excitation
of N II is reported, with effective collision strengths being presented
for the 13 transitions among the 2s2 2p2 3 P, 1 D, 1 S and 2s2p3 5 S finestructure levels, over a wide temperature range log T e (K) = 2.0–
5.1. The discrepancies between two previous R-matrix calculations
(Lennon & Burke 1994; Stafford et al. 1994) have been resolved,
with the current data supporting the work of Lennon & Burke.
For transitions within the ground state, there is little difference between the three calculations, and all the effective collision strengths
agree to within a few per cent, over the common temperature range
of log T e (K) = 3.7–5.0. Lennon & Burke provide data down to
log T e (K) = 3.0, and here differences of not more than 10 per cent
are found for the ground-state transitions.
For the remaining transitions, the work of Lennon & Burke shows
excellent agreement, to within 7 per cent, or better, for all transitions
over the entire Lennon & Burke range of log T e (K) = 3.0–5.0. The
effective collision strength of Stafford et al., whilst displaying good
agreement at high temperatures, suffers significant differences as
low temperatures are approached.
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2004 RAS, MNRAS 348, 1275–1281
Effective collision strengths for singly ionized nitrogen
AC K N OW L E D G M E N T S
This work has been supported by PPARC, under the auspices of a
Rolling Grant. The calculations were carried out on the ENIGMA
supercomputer at the HiPerSPACE Computing Centre, UCL, which
is funded by the UK Particle Physics and Astronomy Research
Council.
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