Single-photon single ionization of W + ions: experiment and theory
A. Müller, S. Schippers, J. Hellhund, K. Holste, A. L. D. Kilcoyne, R. A. Phaneuf, C. P. Ballance, B. M. McLaughlin
SSingle-photon single ionization of W + ions:experiment and theory A Müller † , S Schippers , , J Hellhund , , K Holste ,A L D Kilcoyne , R A Phaneuf ,C P Ballance , , and B M McLaughlin , ‡ Institut für Atom- und Molekülphysik, Justus-Liebig-Universität Giessen,35392 Giessen, Germany I. Physikalisches Institut, Justus-Liebig-Universität Giessen, 35392 Giessen,Germany Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley,California 94720, USA Department of Physics, University of Nevada, Reno, NV 89557, USA Department of Physics, 206 Allison Laboratory, Auburn University, Auburn,AL 36849, USA Centre for Theoretical Atomic, Molecular and Optical Physics (CTAMOP),School of Mathematics and Physics, The David Bates Building, 7 College Park,Queen’s University Belfast, Belfast BT7 1NN, UK Institute for Theoretical Atomic and Molecular Physics, Harvard SmithsonianCenter for Astrophysics, MS-14, Cambridge, MA 02138, USA
Abstract.
Experimental and theoretical results are reported for photoionizationof Ta-like (W + ) tungsten ions. Absolute cross sections were measured in theenergy range 16 to 245 eV employing the photon-ion merged-beam setup at theAdvanced Light Source in Berkeley. Detailed photon-energy scans at 100 meVbandwidth were performed in the 16 to 108 eV range. In addition, the cross sectionwas scanned at 50 meV resolution in regions where fine resonance structurescould be observed. Theoretical results were obtained from a Dirac-CoulombR-matrix approach. Photoionization cross section calculations were performedfor singly ionized atomic tungsten ions in their s p d ( D )6 s D J , J =1/2,ground level and the associated excited metastable levels with J =3/2, 5/2, 7/2and 9/2. Since the ion beams used in the experiments must be expected to containlong-lived excited states also from excited configurations, additional cross-sectioncalculations were performed for the second-lowest term, d S J , J =5/2, and forthe F term, d s F J , with J = 3/2, 5/2, 7/2 and 9/2. Given the complexityof the electronic structure of W + the calculations reproduce the main features ofthe experimental cross section quite well.PACS numbers: 32.80.Fb, 31.15.Ar, 32.80.Hd, and 32.70.-n Short title: Valence shell photoionization of singly ionized atomic tungsten ions † Corresponding author, E-mail: [email protected] ‡ Corresponding author, E-mail: [email protected] a r X i v : . [ phy s i c s . a t o m - ph ] S e p hotoionization of W + ions
1. Introduction
Tungsten presently receives substantial scientific interest because of its importance innuclear-fusion research. Due to its high thermal conductivity, its high melting point,and its resistance to sputtering and erosion tungsten is the favoured material for thewall regions of highest particle and heat load in a fusion reactor vessel [1]. Inevitably,tungsten atoms and ions are released from the walls and enter the plasma. Withtheir high atomic number, Z = 74 , they do not become fully stripped of electronsand therefore radiate copiously, so that the tolerable fraction of tungsten impurityin the plasma is at most 2 × − [2]. Understanding and controlling tungsten in aplasma requires detailed knowledge about its collisional and spectroscopic properties.Although not directly relevant to fusion, photoionization of tungsten atoms and ionsis interesting because it can provide details about spectroscopic aspects and, as time-reversed photorecombination, helps to better understand one of the most importantatomic collision processes in a fusion plasma, electron-ion recombination.R-matrix theory is a tool to obtain information about electron-ion and photon-ion interactions in general. Electron-impact ionization and recombination of tungstenions have been studied experimentally [3–9] while there are no detailed measurementson electron-impact excitation of tungsten atoms in any charge state. Thus, thepresent study on photoionization of these complex ions and the comparison ofexperimental data with R-matrix calculations provides benchmarks and guidance forfuture theoretical work on electron-impact excitation.Photoabsorption by neutral tungsten atoms in the gas phase has been studiedexperimentally by Costello et al. employing the dual-laser-plasma technique [10]. Afew years later the production of W + and W photo-ions from tungsten vapor wasobserved by Sladeczek et al. [11]. However, no experimental data have been availablein the literature for tungsten ions prior to the present project. Direct photoionizationof W q + ions is included in the calculations by Trzhaskovskaya et al. [12] as time-reversed radiative recombination but is expected to be only a small contribution to thetotal photoionization cross section. Theoretical work using many-body perturbationtheory (MBPT) has been carried out by Boyle et al. [13] for photoionization of neutraltungsten atoms. Theoretical treatment of photoabsorption using a relativistic Hartree-Fock (RHF) approach was reported by Sladeczek et al. [11] in conjunction with theirexperiments. Very recently, Ballance and McLaughlin have carried out large-scaleR-matrix calculations for the neutral tungsten atom [14] using the Dirac-Coulomb R-matrix approximation implemented in the DARC codes. The present study is the firstinvestigation on photoionization of tungsten ions and addresses singly charged W + .Preliminary reports on our ongoing tungsten photoionization project were presentedat conferences [15–17] previously.The ground level of the Ta-like W + ion is p d ( D )6 s D / with an ionizationpotential of (16.37 ± D / ground level, the excited ground-configuration fine-structure levels D J with J =3/2,5/2, 7/2 and 9/2 at excitation energies below 0.8 eV [18], respectively, are alsopopulated in an ion source that produces W + by electron-impact ionization of neutraltungsten. Also the lowest levels of the first excited d and d s configurationshave excitation energies below 2 eV and are likely populated in the ion-source plasma.The energetically lowest configurations p d s , d and d s all have even parityand, hence, all the 118 excited levels within these configurations are long-lived becauseelectric dipole transitions between any of these levels are forbidden. Any strong signal hotoionization of W + ions ± p d ( D )6 s D , p d S and p d s F terms.The direct and resonant photoionization processes occurring in the present energyrange up to 245 eV for the interaction of a single photon with the ground-state andthe lowest metastable configurations of the Ta-like tungsten ion comprise removal orexcitation of either a f , s , p , d or a s electron. For the theoretical description ofW + photoionization suitable target wave functions have to be constructed that allowfor promotions of electrons from these subshells to all contributing excited states. Thisis challenging for a low-charge ion such as W + but becomes simpler for the ions inhigher charge states due to the increased effect of the Coulomb charge of the targetand the slight reduction in the R-matrix box size.This paper is structured as follows. Section 2 details the experimental procedureused. Section 3 presents a brief outline of the theoretical work. Section 4 presents adiscussion of the results obtained from both the experimental and theoretical methods.Finally in section 5 conclusions are drawn from the present investigation. An appendixhas been added to describe corrections of measured photoionization cross sections foreffects of higher-order radiation present in the photon beam particularly at low photonenergies.
2. Experiment
The measurements on photoionization of W + ions were carried out at the Ion-Photon Beam (IPB) endstation of beamline 10.0.1.2 at the Advanced Light SourceALS in Berkeley, California, USA. The general layout of the experimental setup andthe procedures employed have been described previously by Covington et al [19].Technological developments since these early measurements have been discussedrecently by Müller et al [20]. An overview of the experiment is presented here andaspects specific to the present measurements are discussed in detail.Ta-like W + ions were produced from W(CO) vapour in an electron-cyclotron-resonance (ECR) ion source. The ions were extracted and accelerated to groundpotential by a voltage of 6 kV forming a beam of ions that was composed of acomplex mixture of electrically charged fragments of the initial tungsten hexacarbonylmolecules. The ion beam was steered and focused electrostatically to the entranceaperture of a 60 ◦ -bending-angle dipole magnet which separated the ions with respectto their momentum per charge. A beam of isotopically pure W + ions was selectedby appropriately choosing the field of the analyzer magnet. The W + ion beamemerging from the exit aperture of the magnet was deflected by a hemisphericalelectrostatic 90 ◦ deflector, the ”merger”, onto the axis of the counter-propagatingnarrow-bandwidth photon beam provided at the beamline.W product ions formed as a result of interactions of single photons withsingle W + parent ions were separated from the parent ion beam by a second magnetpositioned downstream of the interaction region (upstream of the photon beam). Thisso called ”demerger” magnet deflected the product ion beam by 45 ◦ so that it couldenter a single-particle detector unit while the parent ion beam, deflected by halfthat angle, was collected by a large Faraday cup. The detector unit consists of ahemispherical electrostatic 90 ◦ deflector, a variable-size aperture based on a four-jawmovable-slit system and a single-particle detector with almost 100% efficiency [21,22].The additional deflection served to decouple the detector from stray particles and hotoionization of W + ions signal count rates the ion beam was tuned for maximumoverlap with the photon beam positioned on the axis of the merging section of theapparatus between the ”merger” deflector and the ”demerger” magnet. The overlap ofthe ion and photon beams was quantified by using three slit scanners at the entrance,exit, and middle position of the photon-ion interaction region which was defined bya metal drift tube with entrance and exit apertures. Each slit scanner could probethe overlapping beams in two directions perpendicular to one another thus providingaccess to the form factor F ( z ) [19, 23] at each position z. The total form factor F = (cid:82) F ( z ) dz characterizing the three-dimensional overlap of the two beams in theinteraction region was then obtained by interpolation of the position-dependent formfactors F ( z ) considering that the trajectories of ions and photons within the interactionregion are straight lines.For maximum reduction of background arising from collisions of W + parentions with residual-gas particles along the whole merging section the pressure inthis section was kept at a minimum. Operating pressures ranged from about 3 to5 × − mbar. For separation of the signal of photoionized tungsten ions frombackground of collisionally-produced W ions the photon beam was chopped atapproximately 6 Hz during the data acquisition so that the background and signal-plus-background count rates could be measured separately. By subtracting the formerfrom the latter the instantaneous signal count rate was recovered. Total signal countingtimes up to hundreds of seconds were used to obtain suitable levels of statistics.Two modes of operation were applied to obtain information about thephotoionization cross section of W + ions. In ”spectroscopy mode” the voltage onthe interaction region was set to 0 V so that signal could be collected from the wholemerging section (approximately 1.4 m length) between the ”merger” and ”demerger”units. At constant but unknown beam overlap the energy dependent W signal countrate R ( E γ ) (obtained by chopping the photon beam) was recorded as a function ofphoton energy E γ . These energy-scan measurements were carried out at 100 meVconstant resolution covering an energy range from 16 to 108 eV in 50 meV steps (atenergies below 27.1 eV the beamline provides beams with lower bandwidths; at a fixedmaximum exit slit width of 1300 µ m the beamline resolution gradually drops from100 meV at 27.1 eV to about 36 meV at 16 eV.) At the same time the primary-ion andphoton fluxes, ˙ N i and ˙ N γ , respectively, were recorded so that a normalized relativecross section R/ ( ˙ N i ˙ N γ ) could be obtained for each energy step. As already mentioned,the ion beam was collected in a large Faraday cup and its electrical current I i = e ˙ N i was measured with a precision electrometer with e being the elementary charge.The photons were collected on the sensitive area of a calibrated photodiode. Thephotocurrent I γ = eQ ( E γ ) ˙ N γ was measured with a similar electrometer where Q ( E γ ) is the known conversion efficiency of the photodiode, i.e., the number of electronsper incident photon. The relative cross section function obtained by energy-scanmeasurements was normalized afterwards to absolute cross sections measured in aseparate phase of the experiment.In ”absolute mode” absolute cross-sections σ ( E γ ) at selected photon energies E γ were determined from σ ( E γ ) = R ( E γ ) qe v i Q ( E γ ) I i I γ η F ( E γ ) (1) hotoionization of W + ions q is the charge state of the primary ions, v i the primary ion velocity and η the detection efficiency of the product-ion detector system. The quantities R , I i and I γ are readily obtained from the experiment, Q ( E γ ) and η are known from separatecalibration measurements. The determination of the conversion efficiency Q ( E γ ) F ( E γ ) theinteraction length has to be experimentally defined. For this purpose the photoionized(W ) ions produced in the interaction region were energy-tagged by applying avoltage of +500 V to the metal tube defining the interaction region. Parent W + ions were decelerated to an energy of 5.5 keV when entering the interaction region.Product W ions generated inside the drift tube were accelerated to 6.5 keV whenleaving the interaction region thus gaining a net total energy of 6.5 keV compared tothe 6 keV of those W ions which were formed outside of the interaction region. Theinteraction length could thus be defined as the ”inside” length of the potential barrierapplied to the drift tube which was (29.4 ± ions with energies of 6 and 6.5 keV energy.The energy-tagged 6.5 keV W ions unambiguously arise from the defined interactionlength of 29.4 cm and are associated with the measured total form factor F .Typical ion currents in collimated beams used for the absolute measurementswere 15 nA. Photon fluxes strongly depended on the energy E γ and the desired energybandwidth. For consistency, most absolute measurements were performed at 45 meVbandwidth defined by suitable settings of the monochromator slits. In a few casesbandwiths of 50 meV and 70 meV were employed which did not have a significantinfluence on the cross section results. This was expected given the fact that themeasurements were carried out at energies where there were no obvious resonancesin the spectrum. Numbers for the photon flux were about × s − at 16 eV, × s − at 40 eV, . × s − at 120 eV, and × s − at 245 eV. Totalbeam overlaps F varied in the absolute measurements between about 200 cm − and500 cm − . Signal count rates varied from 40 kHz at 40 eV to 0.8 Hz at 245 eV.Background count rates were near 1 Hz.The present measurements were extended to energies down to 16 eV because theground-level ionization energy is expected to be at (16.37 ± + because the photoionization cross sections at twice and three timesthe ground-state ionization threshold energy are relatively large. Fractions of higher-order radiation were determined by separate experiments which are described in theAppendix. From the results of that investigation corrections were derived for themeasured apparent cross sections. The corrections are large for the lowest photonenergies reaching almost a factor of 2 at 16 eV. At 20 eV the correction is down to37% and at 25 eV amounts to only 17% which is less than the typical uncertainty ofabsolute measurements of photoionization cross sections at the ALS IPB endstation.The error budget of cross section measurements has recently been discussed byMüller et al [20] who estimated a systematic uncertainty of 19% for cross sections hotoionization of W + ions et al [20] ( E γ ≥ eV) werefar beyond the region where significant corrections for higher-order effects would berequired. In the present study, however, the uncertainty of the higher-order correctionshas to be considered. Moreover, the photodiode used in the present experimenthad already been exposed to substantial photon doses. Therefore the photon fluxmeasured with the present photodiode was compared with the response of a calibratedpristine photodiode covering the whole energy range investigated here for whichthree different gratings were used in the monochromator. The comparison showedan average reduction of the diode conversion efficiency of 5% for the heavily useddiode. This fatigue effect was corrected for. On the basis of the observed fluctuationsthe additional relative uncertainty of the photon flux measurements with the twophotodiodes was estimated to be 9% and the associated absolute uncertainties wereincluded in quadrature in the total systematic uncertainty. In addition, an uncertaintyof 50% of the difference between the uncorrected and the higher-order-corrected crosssection (see Appendix) was assumed. The associated absolute uncertainty as well asthe statistical uncertainty of the measured count rate R was added in quadrature inorder to obtain an estimate of the total possible error of the cross section. Totalabsolute error bars are shown for absolute cross sections in Sec. 4.In addition, the photon-energy calibration of the present measurements hasuncertainties. The counterpropagation of the photon and ion beams results in smallDoppler effects of . × − of the actual photon energy. Corrections of E γ are thusat most 62 meV at 245 eV. The uncertainties of such corrections are negligible. Theenergy axis was cross-calibrated to known resonance positions in the photoionizationof Ar + ions. Since there are no sharp cross section features in the investigated energyrange the calibration of the energy axis was not a prime issue. We estimate a maximumuncertainty of the energy axis of at most ±
100 meV in the energy range 16 to 80 eVwhere the only structures in the cross section occur.As mentioned in the introduction, the 118 excited levels (plus the ground level) inthe lowest-energy configurations p d s , d and d s of W + are all expected to belong lived. In principle, they can all be populated by energetic electron collisions in theplasma of the ECR ion source [3]. It is not a priori evident, however, which levels arepopulated at what relative weight. Given the small energy splitting of fine-structurelevels within a given term, it is reasonable to assume that all levels within that termare populated and that the population is statistical, i.e., it follows the statisticalweight of each level within the term. In this context it may be worth mentioningthat measurements on the photoionization of W + ions have been carried out over atime range of 5 years during 4 different beamtimes at the ALS. Energy-scan spectrain certain energy ranges were taken repeatedly and with different operation modes ofthe ion source. Nevertheless, the scan measurements and particularly the cross sectionfeatures at certain energies were always reproducible. This is a strong indication forconsistent and reproducible sets of initial-level populations in the primary ion beamsused in all those experiments. The photoionization cross section results suggest thatthere was a strong, maybe even dominant, contribution from ions in the lowest-energy hotoionization of W + ions D term, within the p d s ground-state configuration.
3. Theory
For comparison with the measurements made at the ALS, state-of-the-art theoreticalmethods using highly correlated wavefunctions were applied that include relativisticeffects. An efficient parallel version [26] of the DARC [27–29] suite of codes was appliedwhich has been developed [30–32] to address electron and photon interactions withatomic systems providing for hundreds of levels and thousands of scattering channels.These codes are presently running on a variety of parallel high performance computingarchitectures world wide [33, 34]. Recently, DARC calculations on photoionization oftrans-Fe elements were carried out for Se + , Kr + , Xe + , and Xe ions [20, 31, 32, 35]showing suitable agreement with high resolution ALS measurements.Photoionization cross section calculations on Ta-like W + ions were performed for10 selected levels in the ground and the lowest excited configurations s p d s , s p d and s p d s . The atomic structure calculations for the W production were carried out using the GRASP code [36–38]. We included 573 levelsin our close-coupling calculations resulting from the 6 configurations s p d , s p d s , s p d p , s p d d , s p d s and s p d to representthe atomic structure of the W residual ion. The 573-level approximation is thesimplest approximation which allows for the opening of both the d and p subshell.Alternatively, a 449-level model based on the 6 configurations s p d , s p d s , s p d p , s p d d , s p d s and s p d s p was used to see possibledifferences in the representation of the atomic structure of W levels.In Table 1 we show a comparison of the 449-level and 573-level targetapproximations obtained from the GRASP code with the tabulated values from NIST.Note the difficulty of accurately describing the energy levels in the two approximationsfor near neutral states of tungsten. Larger target expansions (allowing for two-electronpromotions to higher lying residual orbitals, and the opening of further inner shells)would naturally bring the theoretical results in better agreement with experimentbut would be prohibitive for scattering and photoionization calculations. In otherwords, extending the basis set for describing photoionization of W + with its open d and s subshells by including more configurations of the W product ion wouldquickly increase the number of levels and thus require a computational effort thatwould go well beyond the limitations set by presently available computing resources.Therefore, the 573-level approximation has to be considered a reasonable compromisebetween adequate representation of the tungsten ion structure and feasibility of thephotoionization computations at the present technical limit of ab initio close-couplingtreatment.The cross section calculations for this 573-level model were carried out in theDirac-Coulomb approximation using the DARC codes [31, 32] for photon energiesfrom the ionization thresholds up to 150 eV. The R-matrix boundary radius of 10.88Bohr radii was sufficient to envelop the radial extent of all the n=6 atomic orbitalsof the residual W ion. A basis of 16 continuum orbitals was sufficient to spanthe photon energy range chosen for the calculations. Since dipole selection rulesapply, total ground-state photoionization cross sections require only the bound-freedipole matrices, J π = 1 e → J π = 1 ◦ , ◦ , ◦ . Whereas for the excited metastablestates then, J π = 3 e → J π = 1 ◦ , ◦ , ◦ and J π = 5 e → J π = 3 ◦ , ◦ , ◦ , J π = 7 e → J π = 5 ◦ , ◦ , ◦ , J π = 9 e → J π = 7 ◦ , ◦ , ◦ are necessary. hotoionization of W + ions Table 1.
Comparison of the NIST [18] tabulated data with the present theoreticalenergies obtained by using the GRASP code. Relative energies with respect tothe ground state are given in eV. A sample of the 19 lowest NIST levels of theresidual W ion are compared with two different GRASP calculations, 449- and573-level approximations. . Level CONFIG Term NIST GRASP GRASP ∆ ∆ Energy † Energy a Energy b Energy c Energy d (eV) (eV) (eV) (eV) (eV)1 d D d D d D d D d D d P2 d P2 d P2 d ( F )6 s F d ( F )6 s F d ( F )6 s F d ( F )6 s F d ( F )6 s F d F2 d F2 d F2 d H d H d H † Energies from the NIST Atomic Spectra Database [18]. a GRASP theoretical energies from the 449-level approximation. b GRASP theoretical energies from the 573-level approximation. c ∆ energy difference (eV) of the 449-level approximation with NIST [18] values. d ∆ energy difference (eV) of the 573-level approximation with NIST [18] values.For the ground and metastable initial states of the tungsten ions studied here, theouter region electron-ion collision problem was solved (in the resonance region belowand between all thresholds) using a fine energy mesh of 10 − Rydbergs ( ≈ d s D / ground level and 10 − Rydbergs ( ≈ jj -coupled Hamiltonian diagonal matrices were adjusted sothat the theoretical term energies matched the recommended NIST values [18]. Wenote that this energy adjustment ensures better positioning of resonances relative toall thresholds included in the calculation [31, 32].In the present work the DARC PI cross-section calculations for Ta-like tungsten hotoionization of W + ions Cross section ( Mb )
P h o t o n e n e r g y ( e V ) g + W fi W + e Figure 1. (Colour online) Photoionization of W + ions measured at energyresolution 100 meV. Energy-scan measurements (small open circles with statisticalerror bars) were normalized to absolute cross-section data represented by large(cyan) shaded circles with total error bars and smaller full (red) circles withstatistical uncertainties. The (blue) vertical bar at 16.37 eV indicates the ground-state ionization potential. The data have been corrected for the effects of higher-order radiation (see text and appendix). The uncorrected energy scan is shownas a dotted (grey) line. ions were convoluted with Gaussian profiles of 50 or 100 meV FWHM simulating theexperimental photon energy bandwidths.
4. Results and Discussion
Fig. 1 presents the measured cross section for single photoionization of W + ions ona double-logarithmic scale. The measurements have been corrected for the effectsof higher-order radiation as described in the appendix. For comparison, also theuncorrected energy scan is displayed. The difference between the corrected anduncorrected scan data is within the total error bars of the experiment. In the energyrange investigated, 16 to 245 eV, the cross section spans more than three orders ofmagnitude in size. It is dominated by very broad features with relatively small andnarrow resonances occurring in certain energy ranges. The energy-scan data are shownby small open circles with error bars which are mostly so small that they can only beseen between 16 and at most 18 eV and again at energies above 90 eV where the signalcount rates were small. The absolute cross sections are shown twice, once as smallsolid (red) points with statistical error bars and once as large (cyan-)shaded circleswith their total absolute uncertainties. At 245 eV the statistical uncertainty is the hotoionization of W + ions Cross section ( Mb )
P h o t o n e n e r g y ( e V ) g + W fi W + e Figure 2. (Colour online) Detail from Fig. 1 with data shown on linear cross-section and energy scales. The symbols are the same as in Fig. 1. The first sevenlowest-energy data points of the energy scan are enlarged and their statisticalerror bars are emphasized by bold (brown) lines with endcaps. The statisticaluncertainties of the absolute data points are negligibly small in the energy rangeof the figure. This is also true for the energy-scan data except for the range 16 to17 eV. Total uncertainties of absolute data points are indicated by the solid errorbars with large endcaps. The total relative error of the point at 16 eV is about60% due to the uncertainty of the correction for higher-order effects (see text). dominating source of possible error.The most interesting features in the photoionization cross section of W + are foundin the energy range from 16 to about 70 eV. For better display of the cross sectiondetails Fig. 2 highlights the interesting energy range and shows the experimental dataon linear cross-section and photon-energy scales.The apparent onset of the cross section is near 16 eV which is close to the ground-level ionization potential. It should be mentioned, however, that below 16 eV nomeasurements were possible. Although the cross section almost goes to zero at 16 eVwithin the large uncertainties of these low-energy measurements there might still be asizable cross section contribution below 16 eV which would arise from some of the manymetastable levels that are within reach of the ion source. Towards higher energies astrong increase of the cross section above 36 eV indicates the onset of new ionizationchannels beyond the removal of a s or d electron. We assign the first big peak inthe cross section to excitation and ionization of a f electron. The next strong peakwith its onset at around 45 eV is attributed to the opening of the p subshell.Similarly strong peaks are predicted by theory for photoionization of W + ionsfrom all the initial levels investigated. The theoretical results for the ten energeticallylowest initial levels are shown in the next three graphs. Fig. 3 illustrates thephotoionization results for each individual fine-structure component within the term d s D with total angular momentum quantum numbers J = 1/2, 3/2, 5/2, 7/2and 9/2. Fig. 4 displays the photoionization cross section for the only level within the d S term with J = 5/2. Finally, Fig. 5 shows the theoretical results for all levels hotoionization of W + ions
02 04 06 0 D D
02 04 06 0 D
02 04 06 0
Cross section ( Mb ) D
02 04 06 0 D P h o t o n e n e r g y ( e V )
Figure 3. (Colour online) Theoretical photoionization cross sections from lowest-term W + (5 d s D J ) ions with total angular momentum quantum numbers J = 1/2, 3/2, 5/2, 7/2 and 9/2 individually specified in each panel. The theoreticaldata were obtained from 573-level DARC calculations and then convoluted witha 100 meV FWHM Gaussian profile. hotoionization of W + ions S Cross section ( Mb )
P h o t o n e n e r g y ( e V )
Figure 4. (Colour online) Theoretical photoionization cross sections of secondlowest-term W + (5 d S / ) ions. The theoretical data were obtained from 573-level DARC calculations and then convoluted with a 100 meV FWHM Gaussianprofile. within the term d s F with J = 3/2, 5/2, 7/2 and 9/2. All theoretical crosssections were convoluted with a Gaussian of 100 meV full width at half maximum inorder to simulate the experimental conditions of the data displayed in Fig. 1.The excitation energies of the excited d s D J levels investigated in Fig. 3 are0.188 eV for J = 3/2, 0.393 eV for J = 5/2, 0.585 eV for J = 7/2 and 0.762 eVfor J = 9/2 [18]. At energies right above the associated ionization thresholds andbelow 18 eV theory predicts sharp resonance features. With increasing photon energya region of smooth energy dependence is predicted up to about 35 eV where narrowresonances start again to appear. The smoothly decreasing cross section arises fromdirect photoionization mainly of one of the d electrons. The narrow resonancesoccurring predominantly in the photoionization of the D / level are most likely dueto f excitations which add up to a strong increase of the cross section above 35 eVwhile producing broad peak features. It is worth noting that smooth broad crosssection peaks are also seen in the theoretical results before convolution. Apparently,the extreme density of excited states accessible by excitation of a f electron in W + leads to overlapping and interacting resonances similar to the behaviour found recentlyin photorecombination of tungsten ions in charge states around q = 20 [3, 7, 9]. Athigher photon energies the onset of p excitation and then also direct ionization ofthe p subshell is expected. However, the signature for opening the p subshell is notclearly evident from the theoretical results displayed in Fig. 3. Beyond 55 to 60 eV thecalculated cross sections rapidly drop from a level of 60 Mb at 55 eV to about 2 Mbat 100 eV. Qualitatively, the theoretical cross sections shown in Figs. 3 and 4 displaya similar overall behaviour.Fig. 4 illustrates the calculated photoionization cross section of the lowestterm, S , of the first excited configuration ( d ) comprising only one level withthe spectroscopic notation S / . Its excitation energy from the ground level is0.920 eV [18]. Slightly different from the d s D J levels in the ground configurationmore structure is predicted in the cross section between 45 and 55 eV. Also the decreasetowards higher photon energies is not equally steep.The predictions of the photoionization cross sections from the next higher levelsin W + , associated with the d s F term, are displayed in Fig. 5. The excitationenergies above the ground level of W + are 1.080 eV for J = 3/2, 1.401 eV for J = 5/2and 1.663 eV for J = 7/2 [18]. The d s F / level is not listed with this notation inthe NIST database. This is attributed to the increasing importance of mixing effects hotoionization of W + ions
02 04 06 0 F
02 04 06 0
Cross section ( Mb ) F
02 04 06 0 F F P h o t o n e n e r g y ( e V )
Figure 5. (Colour online) Theoretical photoionization cross sections of F - termW + (5 d s F J ) ions with total angular momentum quantum numbers J = 3/2,5/2, 7/2 and 9/2 individually specified in each panel. The theoretical data wereobtained from 573-level DARC calculations and then convoluted with a 100 meVFWHM Gaussian profile. to be considered and the associated difficulty to unambiguously assign a level notation.The present calculations give 1.318 eV for J = 9/2. The calculated spectra for the F J levels are very similar to the photoionization cross section of W + ( S / ) shown inFig. 4.All ten calculated cross sections show the same most prominent features. Narrowresonances at energies up to about 18 eV are followed by a smoothly decreasingfunction of energy which can be associated with s and d outer-shell ionization.In all cases a steep increase of the cross section is caused by the opening of the f subshell and the onset of vacancy production in the p subshell produces a furtherstrong peak feature above 45 eV. At energies beyond about 55 eV all cross sectionsdecrease with increasing photon energy. They all show indications of additional smallstructures between 65 and 70 eV and a smooth dependence up to 125 eV (where therange between 100 and 125 eV is not shown in the figures).The interpretation of the experimental results shown in Fig. 1 is complicated by hotoionization of W + ions
02 04 06 0 e x p e r i m e n t
02 04 0
Cross section ( Mb ) D t e r m a v e r a g e
02 04 0 F t e r m a v e r a g e S t e r m ( a n d l e v e l )
P h o t o n e n e r g y ( e V )
Figure 6. (Colour online) Comparison of experimental and term-averagedtheoretical photoionization cross sections of W + ions at 100 meV energyresolution. The three lower panels show the theoretical results obtained from573-level DARC calculations for photoionization from the energetically lowestterms d s D , d S and d s F , respectively. the possible presence of long-lived excited levels in the primary ion beam used for thecross section measurements. Since photon energies below 16 eV were not accessible bythe experimental setup it was not possible to draw final conclusions on beam fractionsof those ions from the observation of photoionization signal below the ground-levelionization potential. With the measured cross section being as little as (0.5 ± ± + . According to the NIST level energies [18] mentioned above, the D term averaged ionization potential is (15.86 ± + are 16.0 eV for the D term and 15.0 eV for both the S and F terms. So theexperiment is in agreement with the calculation for the D term. The fact, that theexperimental cross section is so small at 16.0 eV even when considering the very large hotoionization of W + ions W + ions for the 486-cm path from the ECR ionsource to the center of the photon-ion interaction region is readily determined to beabout 62 µ s. An estimate of decay probabilities (equivalent to partial lifetimes) ofthe dipole-forbidden transitions between all levels within the d s , d s and d configurations, all with even parity, has been obtained by employing the GRASPcode. Ten configurations ( d s , d p , d d , d s , d p , d d , d , d s p , d s d , and d p d ) were used as a basis for representing the structure of W + (seesection 3). The results show that even the fastest E2 transitions have partial lifetimesexceeding the ions’ time of flight by more than a factor of three. The less probable M1transitions have lifetimes of even more than 0.1 s. As a result of these comparisonsone would expect that levels within the d s , d s and d configurations almostall survive the flight time of the excited ions after their extraction from the ion source.However, beside their flight time there is also a drift time of the ions before theyleave the source volume. This has been observed previously [39] for an unambiguouscase, the metastable s s S level in heliumlike Li + with a known lifetime of about0.5 ms [40]. In electron-impact ionization experiments with Li + ions Borovik et al [39]found no evidence for the presence of Li + (1 s s S ) in the parent ion beam althoughthe ion flight time was only about 8 µ s, i.e., over 60 times less than the lifetime of the s s S level. The conclusion was that the ions spend a much longer time driftingwithin the ion source between production and extraction than traveling from the ionsource to the interaction region, giving them enough time to even let the s s S level decay in spite of its 0.5 ms lifetime. This interpretation was also supported byother ionization experiments with a focus on metastable s s heliumlike ions in whichthe same type of ion source was employed [41, 42]. On the basis of such previousobservations, one may speculate that in the present photoionization experiment aconsiderable fraction of the long-lived levels in the lowest configurations of W + mayhave decayed before the ions reached the photon-ion interaction region - in accordwith the findings reported in the previous paragraph.Clearly, one cannot expect that only the lowest level within a given termis populated in the ion-source plasma. The electron energy distribution in anelectron-cyclotron-resonance heated plasma in which also multiply charged ions canbe produced must be expected to contain components with at least several tens of eV.Compared to this high energy, the fine-structure energy splitting within a J -multipletis negligibly small. The only reasonable assumption about the population of the fine-structure levels of W + ions within a given term is that of a statistical distributionwith the statistical weights given by J + 1 . Calculated lifetimes of the levels withinthe ground term are between several days and a few years. Therefore, comparison ofexperimental cross section data with term-averaged theoretical cross sections is themost meaningful.Fig. 6 thus shows the experimental data from Fig. 1 in comparison with the term-averaged theoretical results for the d s D , d S and d s F terms. If W + ions in a given level are present in the parent ion one has to assume that all otherlevels belonging to that same multiplet are also present and that they are statisticallypopulated. Hence, the remaining question is which multiplets or terms are populatedin the ion source and survive the time of flight of the ions from the ion source to hotoionization of W + ions e x p e r i m e n t a t 5 0 m e V r e s o l u t i o n D t e r m a v e r a g e
Cross section ( Mb ) S t e r m ( l e v e l ) a v e r a g e F t e r m a v e r a g e P h o t o n e n e r g y ( e V )
Figure 7. (Colour online) Comparison of experimental and term-averagedtheoretical photoionization cross sections of W + ions at 50 meV energy resolutionin an energy range where narrow resonances occur in the cross section. Thethree lower panels show the theoretical results obtained from 573-level DARCcalculations for the energetically lowest terms d s D , d S and d s F ,respectively. The theoretical spectra are shifted in energy by -3.2 eV, -2.5 eV and-2.1 eV, respectively. For more details see text. the photon-ion interaction region. As discussed above, there are at least 118 excitedlevels in the first three lowest-energy configurations that are expected to be metastablesince they all have the same (even) parity. In principle, all these levels might havecontributed to the experimental result (together with the d s D / ground level).To perform cross-section calculations for 109(=119-10) more levels of the W + ionwould require an enormous computational effort and dedication of resources. Thus,the experiment can only be compared to a limited set of theoretical data. In spite ofthis limitation one can state a number of observations of theory describing prominentfeatures in the experiment.Theory predicts narrow resonances at energies up to about 18 eV. Although thestep width is too coarse in the experiment there is clear indication of rapid oscillationsin the cross section at low photon energies. A relatively smooth energy dependence of hotoionization of W + ions D term. This is furtherevidence of a substantial fraction of ions in their lowest-energy term present in theexperiment.All calculations show the rapid increase of the cross section at photon energiesabove 35 eV that also characterizes the experimental result. The cross sectionmaximum reached in each of the calculations is in close proximity of the experimentalmaximum. In the energy range 40 to 55 eV the details of the experimental cross sectionstructure are not closely reproduced by the calculations or a reasonable combinationof contributions from the investigated terms. These differences in the details areascribed to the still very limited basis set of the calculations which was chosen to keepthe computational effort managable.Beyond 55 eV the experimental cross section drops off rapidly. A similarly rapiddecrease is only seen in the calculations for the D term. The bump at about60 eV in the experimental cross section is also seen in the theoretical data for the S and F terms. At energies beyond 70 eV theory overestimates the experimentalsingle ionization cross section. Parts of the calculated ionization contributions mayin fact end up in multiple-ionization channels after relaxation of the photoionizedintermediate state formed by the removal of a single electron from W + .In energy ranges where narrow resonances could be observed, additional energyscans of the cross section were measured at 50 meV resolution. The most prominentoccurrence of narrow features is in the energy range between about 30 and 36 eV. Thetop panel of Fig. 7 shows energy-scan results normalized to the absolute cross sectionsshown in Fig. 1. Detailed resonance structure can be seen with the strongest peakfeature occurring at about 35.5 eV just before the steep rise in the cross section dueto the opening of the f subshell. The associated energy ranges of features in thetheoretical cross sections are shown in the three lower panels. The energy axes of thecalculated data were shifted in order to match certain features in the experimentalcross section. It was felt that the experimental peak at 35.5 eV might correspond tothe relatively broad resonance structures in the D and S calculations occurring justbelow the steep rise in the cross section. Therefore, the theoretical energy scales wereadjusted by -3.3 and -2.5 eV, respectively. In the F calculations no correspondingpeak could be found. The energy axis of the F spectrum was shifted by -2.1 eV tomatch the steep onset of the f contribution to the cross section. Again it is thecalculation for the D initial term of W + ions that matches best with the experimentalthough the fine details of the measurements are not reproduced by theory.Although, the overall shapes of all the calculated cross section curves are verysimilar to the experimental results it is obvious that a conclusive comparison betweentheory and experiment cannot easily be made. Over much of the displayed photonenergy range, the calculated theoretical cross sections are consistently larger than theexperimental cross section by factors 2–3 or more (in particular, at higher energies).A previous isolated case, the photoionisation of singly-charged selenium [32] alsorevealed large discrepancies between experiment and theory, however this must beviewed in the context of Ar + , Kr + and Xe + results, which have similar initialconfigurations and exhibit very good agreement between theory and experiment.Converging the theoretical wavefunction through additional configuration interaction hotoionization of W + ions
5. Summary and Conclusions
Experimental and theoretical photoionization cross sections for W + ions are presented.The experimental cross sections were measured on an absolute scale employing thephoton-ion merged-beam facility at the Advanced Light Source. The theoretical datawere obtained from large-scale close-coupling calculations within the Dirac-CoulombR-matrix approximation (DARC). The comparison of the measured and calculatedresults is complicated by the possible presence of long-lived excited states in the parention beams used for the experiments. More detailed modeling of the experimentaldata by theory would require calculations for at least all the 119 levels in the lowestconfigurations of the W + ion which is presently beyond the availability of computerresources. There are indications, though, in the measured cross section that mostof the parent ions were in the ground-state D term. Given the existing limitationsand considering the complexity of Ta-like tungsten with its open d subshell, onecan conclude that the main features of the experimental results are reasonably wellreproduced by the theoretical calculations. This result for a complex singly chargedion where strong electron-electron correlation effects are to be expected is encouragingfor applying a similar theoretical approach to other more highly charged tungsten ionswhere the relative importance of electron-electron interactions is reduced. Acknowledgments
We acknowledge support by Deutsche Forschungsgemeinschaft under project numberMu-1068/20 in addition to grants from the US Department of Energy (DOE)under contracts DE-AC03-76SF-00098 and DE-FG02-03ER15424. C P Ballance wassupported by NASA and NSF grants through Auburn University. B M McLaughlinacknowledges support by the US National Science Foundation through a grant toITAMP at the Harvard-Smithsonian Center for Astrophysics, Queen’s UniversityBelfast for the award of a visiting research fellowship (VRF) and the hospitality of AM,SS and the University of Giessen during a recent visit. The computational work wascarried out at the National Energy Research Scientific Computing Center in Oakland,CA, USA and at the High Performance Computing Center Stuttgart (HLRS) of theUniversity of Stuttgart, Stuttgart, Germany. This research also used resources of theOak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory,which is supported by the Office of Science of the U.S. Department of Energy underContract No. DE-AC05-00OR22725. The Advanced Light Source is supported by theDirector, Office of Science, Office of Basic Energy Sciences, of the US Department ofEnergy under Contract No. DE-AC02-05CH11231.
Appendix
It is well known that synchrotron light is usually accompanied by higher-orderradiation. Transverse oscillations of electrons in an undulator insertion device hotoionization of W + ions λ n = λ /n where λ is the wavelength of the fundamental [44].In particular, spherical grating monochromators (SGM) disperse these higher- n harmonics in n th order and due to the efficiency of the grating for the 1 st and higherorder radiation, there is a non-zero contribution to the photon beam on the opticalaxis of the monochromator. Higher harmonic radiation has characteristic angularpatterns. The even harmonics radiate with angular distributions that have zero densityat zero angle and peak at a finite (but very small) angle. The odd harmonics peakon axis and have relatively high brightness. The grating and all optical elements ofthe monochromator transporting the radiation to the experimental station have aninfluence on the mixture of harmonics that is present in an experiment, with designspecifications tailored to minimize the higher order components delivered to the target.Higher-order components of the photon beam are often used for energy-calibrationpurposes over wide energy ranges. At the expense of intensity they can be used to getinformation about cross section features whose energy is beyond the available rangeof the first harmonic. While there are ways of making positive use of the presenceof higher-order radiation, it has adverse effects on the measurement of absolutephotoionization cross sections. Such effects have to be considered and correctionsmade to the measurements. Appendix A.1. Effects of higher-order radiation components in the photon beam
As discussed in Sec. 2 merged-beam photoionization cross sections for ions aredetermined from measurements by making use of Eq. 1 which is based on theassumption of a monoenergetic photon beam. When higher-order light is present in thephoton beam the measured count rate of photo-ions consists of several contributions.In principle, fractions of n th order radiation with n = 2 , , , ... are possible in a beamof predominantly 1 st order light. This is especially the case at the IPB endstation ofbeamline 10 at the ALS when the first (the low-energy) spherical grating is usedbecause the first-order efficiency drops significantly at low photon energies. Thepresence of different photon-beam components produces a count rate of photo-ions R ( E γ ) = I i η F ( E γ ) qe v i (cid:88) n σ ( nE γ ) I ( n ) γ Q ( nE γ ) . (A.1)The photodiode conversion efficiencies Q ( nE γ ) denote the numbers of electronsprovided by the photodiode per incident n th order photon of energy nE γ . Themeasured total photon-induced diode current is I γ = (cid:88) n I ( n ) γ (A.2)which is a sum of individual currents induced by the n th order fractions of photons inthe beam I ( n ) γ = ˙ N ( n ) γ Q ( nE ) e (A.3)with ˙ N ( n ) γ the number of n th -order photons per second. Defining the fractions f n ( E γ ) of n th -order photons in the incident photon beam one can rewrite the individual photonflux components as ˙ N ( n ) γ = f n ˙ N γ (A.4) hotoionization of W + ions ˙ N γ = (cid:88) n ˙ N ( n ) γ . (A.5)When a photoionization cross section is measured, one first does not consider thehigher-order fractions of photons present and therefore an apparent cross section σ app is determined which needs correction later on. The measured count rate ofphotoionized ions is associated with σ app via R ( E γ ) = I i η F ( E γ ) I γ qe v i Q ( E γ ) σ app . (A.6)The photo-ion rate R ( E γ ) of Eq. A.6 is identical with the rate expressed by Eq. A.1provided the n th order components of the photon beam have identical beam profiles.Assuming that this is the case, σ app can be determined to be σ app ( E γ ) = (cid:80) n σ ( nE γ ) f n ( E γ ) (cid:80) n Q ( nE γ ) f n ( E γ ) Q ( E γ ) . (A.7)There are two effects on the measured cross sections due to the presence of n th -orderradiation in the photon beam:(i) the apparent cross section σ app ( E γ ) includes contributions from different energies nE γ ;(ii) the apparent cross section σ app has a problem with normalization to the photonflux. With the presence of higher order radiation the assumption of the conversionefficiency being determined by Q ( E ) rather than a weighted sum of Q ( nE γ ) makesthe resulting cross section σ app ( E γ ) deviate from σ ( E γ ) .To obtain σ ( E γ ) from the measured σ app ( E γ ) the fractions f n ( E γ ) of n th orderradiation have to be known. These fractions most often have to be determined byseparate experiments. Appendix A.2. Assessment of relative fractions of higher-order radiation
Possible procedures to determine fractions of higher-order radiation in a photon beamare the measurement of photoelectron energies and the identification of characteristicphotoionization cross-section features found in first order at photon energy E (1) γ andthen again in n th order at energies E (1) γ /n with n = 2 , , , ... . Both procedures havebeen employed previously to correct cross sections measured at the IPB endstationof beamline 10 at the ALS (see for example Refs. [24, 25]). For the correctionof the present W + photoionization cross section measurement the latter procedurewas employed using a number of different ions and taking advantage of the existingcapability of the IPB endstation for absolute cross section measurement. The followingcross section measurements were carried out:(i) photoionization of He + with observation of the ionization threshold at 54.4 eV infirst, second and third order;(ii) photoionization of Xe with observation of the dominant broad d s f P o resonance at 122.1 eV [20] in first, second and third order;(iii) photoionization of Xe with observation of strong d → f excitations centeredat 87.0 eV [45] in first, second and third order;(iv) photoionization of Xe + with observation of the strong d → p excitations atenergies between 55 and 57.5 eV [46] in first, second and third order;(v) photoionization of Si with observation of the strong p → nl excitations at hotoionization of W + ions + included) are carried out using almostthe full photon beam - just with halos cut off.Cross section ratios r n = σ app ( E γ /n ) /σ app ( E γ ) for n th order cross sectioncontributions at a given photon energy E γ /n relative to st order measurements atthe associated photon energy E γ are readily obtained from the above experiments.The measurements with Si ions show ratios r : r : r : r : r : r = 100 : 1 .
06 :2 .
64 : 1 .
39 : 0 .
70 : 0 . . With the exception of an increase between r and r thesequence appears to indicate that higher order fractions substantially decrease with n . By assuming that the fractions with n ≥ can be neglected and that the fractions f and f (and hence also f ) are smooth functions of E γ one finds that f is a bellshaped function of E γ with a maximum of approximately 3% at about 45 eV and only1% at 27 and 66 eV. Consistent with previous observation [25] the dominant fraction f is about 6% at 20 eV but drops off more slowly with increasing photon energythan previously assumed. Instead of being negligible at energies beyond 30 eV f isconsistently found both from the measurements (ii) and (v) to be still about 3% near40 eV. Appendix A.3. Correction of cross section measurements for higher-order radiationeffects
From Eq. A.7 one can formally derive the true cross section σ ( E γ ) from the measuredapparent cross section to be σ ( E γ ) = 1 f ( E γ ) × (A.8) × (cid:34) σ app ( E γ ) (cid:80) n max n =1 Q ( nE γ ) f n ( E γ ) Q ( E γ ) − n max (cid:88) n =2 σ ( nE γ ) f n ( E γ ) (cid:35) where n max is the index of the highest order of radiation to be considered. Thefirst term on the right side of Eq. A.8 corrects for the wrong normalization appliedto obtain σ app ( E γ ) . The second term represents the admixtures to the measuredapparent cross section due to higher order radiation. Obviously there are unknowncross sections σ ( nE γ ) on the right side. Since their contributions are weighted byfew-percent fractions f n of higher order radiation an iterative approach can be usedto solve Eq. A.8. It turns out that one iteration in which σ ( nE γ ) is replaced by σ app ( nE γ ) is a sufficiently good approximation given the uncertainties of the energydependent fraction f n ( E γ ) of higher-order radiation.For the present correction of W + cross sections only first-, second-, and third-order radiation are considered. Corrections are only necessary for the energy range ofthe first grating, i. e., in the energy range 16 to 80 eV. The estimated total uncertainty hotoionization of W + ions | σ ( E γ ) − σ app ( E γ ) | (see Sec. 2). References [1] Neu R, Arnoux G, Beurskens M, Bobkov V, Brezinsek S, Bucalossi J, Calabro G, Challis C,Coenen J W, de la Luna E, de Vries P C, Dux R, Frassinetti L, Giroud C, Groth M, Hobirk J,Joffrin E, Lang P, Lehnen M, Lerche E, Loarer T, Lomas P, Maddison G, Maggi C, MatthewsG, Marsen S, Mayoral M L, Meigs A, Mertens P, Nunes I, Philipps V, Pütterich T, RiminiF, Sertoli M, Sieglin B, Sips A C C, van Eester D, van Rooij G and JET-EFDA Contributors2013
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