J. Plasma Fusion Res. SERIES, Vol.7 (2006) 334-337

Compilation of Wavelengths and Energy Levels ofTungsten, W III through W LXXIVKRAMIDA Alexander E. and SHIRAI Toshizo1∗
National Institute of Standards and Technology, Gaithersburg, MD 20899, USA
1 Naka Fusion Research Establishment, Japan Atomic Energy Research Institute,

Ibaraki 311-0193, Japan（Received: 5 October 2004 / Accepted: 21 December 2005）

Abstract
The methods used to critically evaluate and compile the data on energy levels, wavelengths, and ionization potentials of multiply ionized tungsten and other ions and atoms are described. Some examples of theproblems occurring in the process of data evaluation are given.
Keywords:
wavelength, energy level, ionization potential, highly charged ion, level optimization,
parametric calculation
1．Introduction
Tungsten is important for controlled fusion research
as a prospective construction material for tokamaks. Diagnosing its ions in the tokamak plasma requires detailed knowledge of their spectra. The spectra W I
through W VII were studied the most thoroughly. The
recently completed compilation of energy levels, wavelengths, and transition probabilities of W I and W II [1]contains 771 energy levels and 9887 lines. Critical compilation of spectral data for higher ionization stages oftungsten is currently underway at the National Instituteof Standards and Technology (NIST). For the spectra W
III through W VII, studied using a sliding-spark lightsource, there are now several hundreds of known lines.
Other light sources such as laser-produced plasmas, exploding wires, tokamaks, and electron-beam ion traps(EBITs) produce higher ionization stages. Some experimental data exist now for W13+ , W27+ through W48+ ,and W52+ . Some of the highly ionized species of tungsten were extensively studied by several teams. However, in many cases the information for the energy levels
involved in the observed transitions is absent, fragmentary, or ambiguous.
The available ionization potentials (IPs) for most
species of tungsten were calculated with uncertainties
of about 5 %. This is insuﬃcient for plasma-diagnosticapplications. A semi-empirical method developed atNIST allowed us to obtain the IPs with uncertainties as
small as 0.01 %.
The methods that we use to critically evaluate andcompile the data for tungsten are similar to those used
for other elements. We select the most accurately measured wavelengths and derive the energy levels that areconsistent with them. For complex spectra, the level optimization problem requires a special treatment. A computer code LOPT [2] that implements the least-squares
method to find the optimal energy levels was developedat NIST for this purpose. To resolve line-assignment
ambiguities, we make ab initio and semi-empirical calculations of atomic structure and transition rates. For
the Na-, Ne-, He-, and H-like ions, where there are nomeasurements, we include the published results of accurate calculations, as these spectra are important forplasma diagnostics and benchmark theory-testing experiments.
2．Semi-empirical calculation of IPs
For multiply ionized tungsten, the only available experimental IP values exist for W4+ [3], W5+ [4], W6+[5], and W45+ [6-8]. Accurate values of the IP were obtained for Ni-like W46+ [7] by means of an isoelectronicinterpolation and for He-like W72+ [9] and H-like W73+
[10] by means of precise ab initio calculations. For therest of the tungsten spectra, the best available values of
the IP were calculated by Carlson et al. [11] in a simple
approximation. Their uncertainty is 5 %.
An accurate semi-empirical method of calculating
the IPs [12] was developed at NIST. It is based on thesuite of codes of R. Cowan [13]. These codes are widely
C 2006 by The Japan Society of Plasma

Science and Nuclear Fusion ResearchCorresponding author’s e-mail: alexander.Kramida@nist.gov* Deceased334

Compilation of Wavelengths and Energy Levels of Tungsten, W III through W LXXIV

Ground configuration of A
Et(Aq+14fq+1)

1.025q+1)

I VII
1.020

DEt(exp.)/DEt(th.)

Eav(ADEtq

IP of AXe VIII
Cs IX

1.015

Ba X
La XI

1.010

Sm XV
1.005

1.000
0.000

Ce XIIPr XIII
Nd XIVEu XVII
0.005

0.010

0.015

0.020

2s

1/Z

qEt(A )q

Eav(A )
Ground configuration of A

Fig. 2 Isoelectronic comparison of empirical scaling factors for the 4f binding energies in Ag-like ions. Z sis the number of the spectrum. The error bars represent the experimental uncertainties of the IPs.
The solid line is a linear ﬁt.

qFig. 1 Calculation of IP of an ion Aq . Et are the total energies of the ground conﬁgurations of the successiveions Aq and Aq+1 , Eav are the average energies ofthe ground conﬁgurations.

ing energies. The scaling coeﬃcients are determinedfrom other spectra where the IPs are known experimentally. An example of such determination is presented inFig. 2. For the Ag I sequence, the isoelectronic behavior of the scaling coeﬃcients is very smooth and proportional to 1/Z s2 , where Z s is the spectroscopic chargeof the ion (for Ag-like W XXVIII, Z s = 28). From
this dependence, the binding energy of the 4f electron
can be determined with relatively small uncertainty forany member of the sequence. The IP of W27+ determined from the scaled 4f, 4d, and 5s binding energies is(7110±13)·103 cm−1 . Using similar methods of scaling
the binding energies, the IPs of all stages of ionizationof tungsten were calculated with greatly improved accuracy.

used to calculate the energy levels and transition ratesof atoms and ions. It is well known that the accuracyof ab initio calculations done with these codes is notvery high (for energy levels and IPs, it is usually a few
percent). However, these codes proved to be extremely
successful in analyses of complex spectra because of
their capability to use empirically adjusted parameters(the Slater integrals). With empirically adjusted parameters, Cowan’s codes can reproduce the experimentally determined energy levels with uncertainties of afew hundredths of a percent, which is often impossible to achieve with any existing ab initio code. Calculation of the IP of an arbitrary ion Aq by means ofCowan’s codes is illustrated in Fig. 1. The quantity ∆E t ,called the binding energy of the ground configurationof Aq , is the diﬀerence of the total energies of theground configurations of the two successive ions Aqand Aq+1 . Cowan’s code RCN calculates the total energies in the single-configuration Hartree-Fock approximation with approximate relativistic corrections. The
fine-structure splitting of the ground configurations iscalculated by Cowan’s code RCG. The shift of the energy of the ground state relative to the center of gravityof the ground configuration is called the average energyof the ground configuration. Then the IP is calculated
as IP(Aq ) = ∆Et + Eav (Aq ) − Eav (Aq+1 ). The binding energy often represents the largest contribution to the IP.
The uncertainty of its calculation with the RCN codeis usually between 1 % and 5 %. To reduce these uncertainties, we use semi-empirical scaling of the bind-3．Classification of lines
Many of the highly-ionized species of tungsten wereexperimentally studied in recent years. Consequently,
there are many unique observations that were not repeated and verified by other teams. In such cases weverify the line identifications by calculating the energy
levels, transition wavelengths, and radiative rates for thestudied spectrum.
An example of this situation is the Pm-like spectrum
W XIV. Hutton et al. [14] recently reported an observation and identification of the 4f14 5s 2 S − 4f 14 5p 2 Po3/2, 1/2
doublet at 365.3 ± 0.2 Å and 258.2 ± 0.2 Å in an EBIT.
Their identification was based on the prediction ofthe collapse of the 4f shell along the Pm I isoelectronic sequence made by Curtis and Ellis [15] and on
Dirac-Fock calculations by Theodosiou and Raftopou335

Compilation of Wavelengths and Energy Levels of Tungsten, W III through W LXXIV

23Table 1 Energy levels of the 4s 4p conﬁguration of W XLIILevel(103 cm−1 )JPercentage composition.

0.0
719.6
762.1
[808]
[1537]

3/2
3/2
5/2
1/2
3/2

47 % 2 Po + 27 % 2 Do + 26 % 4 So55 % 4 So + 45 % 2 Do
100 % 2 Do
100 % 2 Po
52 % 2 Po + 28 % 2 Do

los [16]. Curtis and Ellis [15] stated that the groundstate of Pm-like ions, starting with W13+ , is 4f14 5s
2S1/2 . This would make the spectrum very simple
and similar to other single-electron spectra. However,Theodosiou and Raftopoulos [16] predicted that theground state of W13+ and Ir14+ would be the odd-parity
4f13 5s2 2 Fo5/2 . We calculated the W13+ spectrum bymeans of Cowan’s codes [13], and the results were inperfect agreement with those of [16]. Hence, this spectrum should be much more complex, with the 4f14 5s2S − 4f 14 5p 2 Po3/2,1/2 lines being weaker than the numerous lines of the 4f 13 5s2 − 4f 13 5s5p and 4f 12 5s2 5p −4f 12 5s5p2 transitions dominating the spectrum in thisregion. Therefore, the identifications of Hutton et al.
[14] need confirmation, and we do not include them in
our tables.
The situation with the As-like spectrum W XLII illustrates cases where it is necessary to carefully consider how the levels are to be designated. Asmussen etal. [17] observed the lines at 46.9 Å, 60.6 Å, 64.9 Å,and 131.8 Å in a tokamak. They identified the firsttwo lines as blends of W41+ and other stages of ionization. The line at 64.9 Å was classified as the4s2 4p3 4 So3/2 − 4s4p4 4 P5/2 transition, the lower level being the ground state. The line at 131.8 Å was classified
as the magnetic-dipole (M1) transition 4s2 4p3 4 So3/2 −4s2 4p3 2 Do5/2 . Table 1 shows the level composition ofthe 4s2 4p3 configuration resulting from our parametric fitting. It is diﬃcult to assign distinct names to theJ = 3/2 levels. In this case we have chosen to assign
the names given in the last column of Table 1. Although
not perfect, this seems to be the most reasonable way ofdesignating these levels. Nevertheless, it is importantfor users to be aware of these problems and to consult
the percentage composition of the levels.
In addition to the level-naming problem, the J = 5/2
assignment of the upper level of the M1 line at 131.8 Å
does not agree with our calculations of the transitionrates. Our calculations indicate that the 4s 2 4p3 2 Do3/2 −4s2 4p3 4 So3/2 transition (i.e., between the ground state
and the first excited level) has a much greater radiative rate. The identifications of the allowed transitions

Newlabel
2Do
S
2 o
D
2 o
P
2 oP
4 o

were confirmed by Radtke et al. [18] except for thelevel designations. Using an EBIT, they re-measured
the wavelengths with lower uncertainties and succeeded
to distinguish the emission of W41+ from the neighboring stages of ionization that were blending the lines observed in [17]. Radtke et al. used jj-coupling notation
for all levels. Their designations, although unambiguous, are not easy to identify with those of [17].
Utter et al. [19] re-measured the W41+ lines observedby Radtke et al. with much lower uncertainties using
an EBIT. They observed three more lines that they associated with the 4s2 4p3 − 4s2 4p2 4d array. However,their designation of the levels is ambiguous. For example, the upper level of the line at 51.091 Å was denoted as 4p1/2 4p3/2 4d3/2 (J = 3/2). The two p electrons can combine diﬀerently, so their aggregate angular momentum can be either 1 or 2. As this intermediate state is not specified, the reported designation isincomplete. In order to determine what level is actuallyassociated with this and other transitions, we calculated
the W41+ spectrum with Cowan’s codes [13] and founda very good agreement of predicted and observed wavelengths and relative intensities of the lines. The upper
level of the above-mentioned transition turned out to be
4s2 4p2 (3 P1 )4d(1, 3/2)3/2 .
Similar problems are encountered in Br-, Kr-, Rb-,Se-, Ge-, and Ga-like spectra. In all cases, Cowan’s
codes proved to be a very powerful tool in the data evaluation.
4．Level optimization
Complex spectra often have a large number of lines
connecting a much lower number of energy levels witheach other. In such cases, deriving the energy levels
from the wave numbers of the observed transitions is
a non-trivial task. Even when the energy levels werealready determined by the authors who made the spectrum analysis, we often proceed to re-determine the levels in order to verify the energy values and to detect
possible misidentifications. For this purpose, there are
two diﬀerent least-squares fitting codes, ELCALC [20]and LOPT [2]. The well-known and widely used EL336

Compilation of Wavelengths and Energy Levels of Tungsten, W III through W LXXIVCALC code makes iterations, on each cycle adjusting
the energy levels so as to minimize the sum of squaresof weighted deviations of Ritz wave numbers from theobserved ones. The LOPT code solves the least-squares
problem by another method. The problem is represented by a system of linear diﬀerential equations expressed in the form of a matrix equation. This equation is solved by inverting the matrix. The advantage ofthis method is that, simultaneously with the level optimization, it calculates the uncertainties of the Ritz wave
numbers. Thus, in a single run of the code, not only
the energy levels are obtained, but also the uncertainties of the predicted wavelengths of all observed lines.
If the newly optimized energy levels reasonably agreewith the original values determined by the authors ofthe analyses, we accept the original energy levels. Otherwise, we revise the levels and use our optimized energies and Ritz wave numbers.

[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]

5．Conclusion
By using the set of powerful analytical and dataprocessing tools available at NIST, we critically evaluated and compiled an extensive set of energy levels,
wavelengths, and IPs of all stages of ionization of tungsten. The data tables, details of the analysis, and bibliography on these spectra will be published elsewherein full reports and given in the NIST Atomic Spectra
Database (http://physics.nist.gov/PhysRefData/ASD).
The authors express their gratitude to Dr. H. Kubo
who assisted in transferring the data preliminarily assembled by Dr. T. Shirai to NIST.
This work was supported in part by the Oﬃce of Fusion Energy Sciences of the U. S. Department of Energy.

[1]

[12]
[13]

[14]
[15]
[16]
[17]
[18]
[19]

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