Atomic data for calculation of the intensities of Stark components of excited hydrogen atoms in fusion plasmas
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Atomic data for calculation of the intensities of Starkcomponents of excited hydrogen atoms in fusionplasmas
Oleksandr Marchuk , David R. Schultz and Yuri Ralchenko Forschungszentrum Jülich GmbH, Institut für Energie- und Klimaforschung - Plasmaphysik, Partner of theTrilateral Euregio Cluster (TEC), 52425 Jülich, Germany Department of Astronomy and Planetary Science, Northern Arizona University, Flagstaff, AZ 86011, UnitedStates of America Atomic Spectroscopy Group, National Institute of Standards and Technology, Gaithersburg, MD 20899-8422,United States of America * Corresponding author. Email: [email protected]: date; Accepted: date; Published: date
Abstract:
Motional Stark effect (MSE) spectroscopy represents a unique diagnostic tool capable ofdetermining the magnitude of the magnetic field and its direction in the core of fusion plasmas. Theprimary excitation channel for fast hydrogen atoms in injected neutral beams, with energy in the rangeof 25-1000 keV, is due to collisions with protons and impurity ions (e.g., He + and heavier impurities).As a result of such excitation, at the particle density of 10 -10 cm − , the line intensities of the Starkmultiplets do not follow statistical expectations (i.e., the populations of fine-structure levels withinthe same principal quantum number n are not proportional to their statistical weights). Hence, anyrealistic modeling of MSE spectra has to include the relevant collisional atomic data. In this paperwe provide a general expression for the excitation cross sections in parabolic states within n =3 for anarbitrary orientation between the direction of the motion-induced electric field and the proton-atomcollisional axis. The calculations make use of the density matrix obtained with the atomic orbital closecoupling method and the method can be applied to other collisional systems (e.g., He + , Be + , C + , etc.).The resulting cross sections are given as simple fits that can be directly applied to spectral modeling.For illustration we note that the asymmetry detected in the first classical cathode ray experimentsbetween the red- and blue-shifted spectral components can be quantitatively studied using the proposedapproach. Keywords:
Motional Stark effect, cathode rays, fusion plasmas, plasma spectroscopy, density matrix,ion-atom collisions
1. Introduction
Beam-assisted spectroscopy represents a special class of diagnostics in plasma spectroscopy, as here,in contrast to passive emission spectroscopy, the heavy-particle collisions at the energies of a few atomicunits (1 a.u. ≈
25 keV) play a dominant role [1]. Local measurements of ion temperature, concentrationof impurity ions, plasma rotation, and electric field measurements including polarisation coherence, orcontrol of the plasma current density profile, are the most representative examples of the use of injectionof heating or diagnostic beams in fusion plasmas [2–6]. So, for instance, charge-exchange recombinationspectroscopy, which is based on the capture of bound electrons of the beam atoms by impurity ions, hasbeen exploited for the last few decades on practically all former and present tokamaks and stellarators,including JET, ASDEX, W7-X, KSTAR, etc., [7]. Beam-assisted spectroscopy is also expected to play asignificant role in future fusion devices such as ITER [8–10]. of 13
The Motional (translational) Stark effect (MSE) diagnostic provides an excellent benchmark of atomicdata for the simplest collisional systems, for example, excitation of H atoms by H + [11]. Both Stark effectmeasurements with beam atoms and Zeeman effect measurements for the cold atoms at the plasma edgecan clearly detect the spectral line components using high resolution spectroscopy [12,13]. For instance,for the applied external magnetic field of 1-5 T the energy separation due to the Zeeman effect is a fewtimes larger than the fine-structure splitting of cold atoms in the plasma. In the rest frame of the beamatoms moving in the external magnetic field the bound electron experiences the static electric field ~ F = ~ v / c × ~ B where F and B are the strengths of electric and magnetic field, respectively, ~ v is the velocity of thebeam atom, and c is the speed of light. As can be easily seen for typical parameters of magnetic fusionplasmas, the energy separations induced by the Stark effect exceed the Zeeman splitting by a significantfactor. Therefore, the representation of atomic structure for these specific conditions should primarilyreflect the electric field effects, and in particular dictate usage of a parabolic basis. Accordingly, for MSEstudies the parabolic quantum numbers offer a good description of the atomic structure and the emissionof spectral lines [14].Calculations of relative MSE line intensities in laboratory plasmas are based on either a statistical(static) or dynamical assumption [15]. In the former case the populations of levels are considered tobe proportional to their statistical weights. The line intensities are then derived using the followingexpression: I a − b ∝ g a A a − b , (1)where a = n a k a m a and b = n b k b m b are the sets of quantum numbers for the atomic eigenstates , k = n − n is the electric quantum number, n ≥ n ≥ m is themagnetic quantum number ( n = n + n + | m | + A a − b is the Einstein coefficient (radiative tranistionprobability) for the transition between the states a and b [16]. The dynamical intensities are calculatedusing the next formula: I a − b ∝ g a A a − b / ∑ c A a − c . (2)It is generally accepted that Eq. (1) is valid for high density plasmas, whereas Eq. (2) is utilised todescribe experiments at low densities where radiative relaxation occurs on timescales much faster thanthe collisional redistribution among the sub-levels. Though extensive atomic structure calculations wereperformed for both types of experimental conditions, the observed line intensities were not alwaysadequately described using tabulated statistical or dynamical intensities. The disagreement amongmeasured and calculated intensities accompanies the Stark effect observation starting from the dawnof quantum mechanics. The historical view of the problem was well characterized in Ref. [17]: "Uponthe whole the problem of the hydrogen intensities showed a very confused picture" . The aim of this paper is topresent a concise tabulated set of atomic data that has been found to be the most successful one, at leastfor fusion plasmas, by applying a density matrix formalism.
2. Theoretical approach
Measurements of hydrogen line intensities in the presence of electric fields are often related toanisotropic excitation. This is the case both for the classical cathode ray experiments and for the MSEmeasurements in fusion plasmas. Figure 1 illustrates the geometrical relationship between the collisionaland electric field directions. In the absence of electric field the quantization axis is normally defined bythe collision axis between incident ions and atoms. However, when an external electric field is applied of 13 to an atom, the spherical symmetry of an isolated atom is replaced by the axial symmetry defined by thedirection of the electric field.For standard problems in atomic scattering, the quantization axis ( z ′ ) is normally selected along theprojectile velocity. This choice results in well-defined spherical eigenstates of the atom φ nlm . However,for MSE conditions, the induced electric field provides a new quantization axis ( z ) which is to beused to define new parabolic eigenstates ψ nkm (see, e.g., Ref. [18] for details). Such modificationrequires calculation of collisional scattering amplitudes and cross sections for transitions betweenparabolic rather than spherical eigenstates. To utilize the standard techniques of atomic collisiontheory, one has to transform the parabolic basis wavefunctions quantized along z to the sphericalbasis wavefunctions quantized along z ′ . This procedure actually requires two steps. The first one isa conventional quantum-mechanical rotation [19] from axis z to axis z ′ by angle α and the second oneis the transformation between the spherical states and parabolic states [20,21] defined along the samequantization axis: ψ nkm = n − ∑ l = | m | C lmnk l ∑ m ′ = − l d m ′ lm ( α ) ϕ nlm ′ (3)where C lmnk is the Clebsch-Gordan coefficient and d m ′ lm ( α ) is the rotation matrix element. By applyingformally the collision operator to the wavefunctions ψ nkm the excitation cross sections can be calculatedusing any applicable theoretical method, for example, atomic orbital close coupling (AOCC) [22],convergent close coupling (CCC) [23], or the Born or Glauber approximations [24,25]. of 13 Figure 1.
Scheme of the transformation between spherical and parabolic eigenstates. The z ′ axis defines theaxis of symmetry of the collisional process in the absence of an electric field. The z axis is the quantizationaxis defined by the direction of the electric field ~ F . For experiments in fusion plasmas, angle α is π /2assuming the ions are cold (e.g., the ion temperature T i is much lower than the beam energy E b ). of 13 General expressions for the cross sections σ nkm for excitation from the ground state (1 s ) of hydrogento the n =2 [26] and n =3 parabolic states nkm are as follows: σ ± = σ s + cos ( α ) σ p + sin ( α ) σ p ∓ cos ( α ) Re ( ρ p s ) , (4) σ = sin ( α ) σ p + σ p (cid:18) − sin ( α ) (cid:19) , (5) σ ± = σ s + ( + cos ( α )) σ p + ( − cos ( α )) σ p + ( + cos ( α )++ cos ( α )) σ d + ( − cos ( α )) σ d + ( − cos ( α ) + cos ( α )) σ d ∓∓ √ cos ( α ) Re ( ρ p s ) + √ ( + cos ( α )) Re ( ρ d s ) ∓∓ √ ( cos ( α ) + cos ( α )) Re ( ρ d p ) ± ( cos ( α ) − cos ( α )) Re ( ρ d p ) , (6) σ ± = ( − cos ( α )) σ p + ( + cos ( α )) σ p + ( − cos ( α )) σ d ++ ( + cos ( α ) + cos ( α )) σ d + ( − cos ( α ) − cos ( α )) σ d ±± √ ( cos ( α ) − cos ( α )) Re ( ρ d p ) ∓ ( cos ( α ) + cos ( α )) Re ( ρ d p ) , (7) σ = σ s + ( + cos ( α ) + cos ( α )) σ d + ( − cos ( α )) σ d ++ ( − cos ( α ) + cos ( α )) σ d − √ ( + cos ( α )) Re ( ρ d s ) , (8) σ = ( − cos ( α ) + cos ( α )) σ d + ( − cos ( α ) − cos ( α )) σ d ++ ( + cos ( α ) + cos ( α )) σ d . (9)Here σ nlm on the r.h.s (e.g., σ d ) is the excitation cross section from n =1 to the spherical state nlm and ρ nlmn ′ l ′ m ′ is the off-diagonal density matrix element. Note that σ nkm = σ nk − m and σ nlm = σ nl − m for m = n =1 to n =2 and n =3 equals that for the field-free case: σ n = = σ s + σ p + σ p = σ + σ − + σ , (10) σ n = = σ s + σ p + σ d + (cid:0) σ p + σ d + σ d (cid:1) = σ + σ − + σ + ( σ − + σ + σ ) . (11)These expressions are valid for the linear Stark effect only, namely, when the energy splitting dueto the induced electric field is much larger than the fine-structure splitting and much smaller than theenergy separation between the levels belonging to different principal quantum numbers n . This conditionis generally fulfilled only for levels with n . m -resolved cross sections for the spherical states but also the off diagonal elements(real part) of the density matrix [29]. First, the cross sections exhibit a strong dependence on the anglebetween the direction of the electric field and the axis of symmetry of the collisional frame. This wasremarkably demonstrated first in Refs. [30,31] through the study of excitation of highly excited circularRydberg atoms ( k = n −
1) at thermal energies. It is the presence of the coherence terms in Eqs. (4-9)that explains the asymmetry between the red- and blue-shifted lines for Stark effect measurement (note of 13 that the influence of field ionisation is still low [32] under these conditions). Therefore, for instance,depending on the orientation between the cathode rays and the vector direction of the electric field (e.g.,parallel, α =
0; transverse, α = π /2; or anti-parallel, α = π ) blue-shifted, red-shifted, or symmetricpatterns were detected for Balmer- α components of the emission [17]. In addition, it has to be mentionedthat detailed beam-foil experiments firmly established the value of employing the density matrix methodin interpreting these measurements [33,34].The first calculations of excitation cross sections in parabolic states relevant to fusion plasmaswere performed using the Born approximation [24] and later the Glauber approximation [18]. It wasexplicitly shown how sensitive the ion-atom parabolic cross sections are to the orientation between thefield direction and the ion-atom collision axis. However, at the collisional velocity of 1-2 a.u. noneof perturbative two-state approximations could adequately describe the measured cross sections. Thissituation was improved by introducing the AOCC calculation for n =2 and n =3 states [22]. Furthermore,the collisional radiative model in parabolic states was extended from n =5 to n =10 demonstrating theeffect of field ionisation [26]. Recently new extensive CCC calculations of m -resolved cross sections anddensity matrix elements became available [23]. Despite the somewhat stronger oscillations of the crosssection as a function of collision energy, the magnetic-quantum-number resolved AOCC cross sections ofRef. [22] and the new CCC results agree quite well .Here we report accurate fits to the m -resolved cross sections (diagonal elements of the density matrix)and the coherence terms (off diagonal elements of the density matrix) of Ref. [22] in the energy range of20-2000 keV using the following formulas [36] : σ ( E ) = A e − A / E ln ( + BE ) E + A e − A E E A + A e − A / E + A E A ! , (12) σ ( E ) = A e − A / E E + A e − A E E A + A e − A / E + A E A ! . (13)The energy E is in keV and cross sections are given in units of π a ≈ · − cm where a ≈ · − cm is the Bohr radius. The derived coefficients are given in Table 1, and Figs. 2 and 3 show the originaldata and the fits for excitation to n =2 and the coherence terms for excitation to n =3, respectively.Using the data of Table 1 and Eqs. (4-9) one may calculate the excitation rate coefficients to parabolicstates even at conditions when the ion temperature is comparable to the beam energy, and also analysethe Stark effect spectrum at an arbitrary angle of electric field direction.
3. Lines ratio of Stark components in the low density limit
We analyse in detail here the influence of the coherence elements on the line ratios of Starkcomponents. For this purpose the collisional-radiative (CR) model NOMAD [37] was adopted. Theanalysis has been performed in the low density limit, i.e., neglecting both the collisional redistributionbetween the excited states and the collisional ionisation. We also neglect the effect of radiative cascadesabove n =3. This approximation was used to better highlight the effect of direct-excitation coherent termson line intensities. It is expected to be generally valid for densities below about (1-5) × cm − . Fordenser plasma simulations, a CR model has to include the above mentioned processes. The AOCC excitation cross sections to 3d and 3d levels shown in Fig. 2 of Ref. [23] are surprisingly lower than in the originalpublication [35]. of 13 A A A A A A A A A B2s p p d d d Table 1.
Table of fit coefficients for the excitation cross sections and the real part of coherence terms forexcitation to n =2 and n =3. Expression (12) is applied to fit the AOCC data for all elements except the 2p ,3p , and 3d cross sections in which case the expression (13) must be used. We note that the real part ofall coherence terms is negative except for s d excitation. C r o ss s e c t i on , p a Energy, keV/a.m.u. -Re( r p s )2s Figure 2.
Excitation cross sections for n =2 states and real part of the coherence term [22] (thick lines withsquares). The results of the fit are shown as thin lines using the same colours. The new CCC data at 50keV are shown as triangles using the same colours (Table 1 of Ref. [23]). of 13 C ohe r en t t e r m s , p a Energy, keV/a.m.u. -Re( r s p )Re( r s d )-Re( r p d )-Re( r p d ) Figure 3.
Real parts of the excitation coherence terms to n =3 (thick lines with squares). The results of thefit are shown as thin lines with the same colours. The new CCC data at 50 keV are shown as trianglesusing the same colours (Table 1 of Ref. [23]). The calculations were carried out for the neutral beam energy of 50 keV/u and electron temperatureof 3 keV. The density of ions and electrons were equal in the calculations. Here we report the lineintensities of the Lyman- α , β and Balmer- α lines for different values of α (Figs. 4-6). The angle betweenthe line-of-sight and the direction of the electric field is θ = π /2. In this case both σ ( ∆ m= ±
1) and π ( ∆ m = 0) transitions are observable. In the statistical limit, ∑ σ I σ = ∑ π I π = α line at four different values of α . The most striking effect is astrong asymmetry for the blue- and red-shifted components at α = π /2. Indeed, depending on the anglebetween the field and the projectile velocity (parallel or anti-parallel) either the blue- or the red- shifted π component dominate the spectrum. The blue-shifted component is a factor of two higher compared tothe red-shifted one in Fig. 4.a. In Figure 4.b, at the angle α = π /4, the ratio is reduced only marginally. InFig. 4.c, with α = π /2 that corresponds to the MSE conditions in fusion plasmas, the intensities of bothcomponents are equal. Finally, in Fig. 4.d the red-shifted π component is stronger than the red-shiftedone. As mentioned above, this asymmetry between the components π ± is due to the presence of thecoherence term Re ( ρ p s ) in the expression for the cross sections. of 13 -0.4-0.2 0 0.2 0.4 -2 0 2a) (p)(s) I n t en s i t y , a . u . -2 0 2b) -2 0 2Displacement of L a componentsc) -2 0 2Displacement of L a componentsd) Figure 4.
The Stark effect for the Lyman- α line excited by proton impact at collision energy of 50 keV/uand by electrons with temperature of 3 keV. The results of the calculation are shown for angles α = α = π /4 (b), α = π /2 (c) and α = π (d) between the axis of collision and the direction of electric fieldtaking into account the cascades from n =3. For the statistical intensities the ratio between components π ± : σ = 1:2, and for the dynamical intensities all three components are equal. The σ component is shownwith negative sign for better visibility. The displacement is given in units of hF π m e ec , where ¯ h is a Plankconstant, F is the strength of electric field, m e is the mass of electron, e is the electron charge, and c is thespeed of light. Experimental evidence of such asymmetry was obtained from cathode ray experiments under muchmore complex experimental conditions. In this case, the line emission is affected by electron-impactexcitation, dissociation of molecules, and charge exchange between protons and atoms. Nevertheless,the physical picture of the asymmetry is attributed to screening of the trajectories of bound electronsfrom incident protons or electrons by nucleons at rest as was already pointed out by Bohr [38]: the chargedistribution of the majority of parabolic states is asymmetrical relative to the plane with z = 0 (see, forinstance, Figure 8 of Ref. [15]). In this case the states with k = n - n >0 cannot be as efficiently excited asthe states with k < z axis from - ∞ to + ∞ . Because the direction of theelectric field points against the direction of cathode rays (e.g., in the opposite direction to the protons) itleads to the dominance of the red-shifted emission (Figure 4.d). In contrast to the strong variation of the π ± component the variation of the σ component is negligibly small. Its value only varies from 0.38 to0.42. For all excitation angles ∑ σ I / ∑ π I <
1. We note that observed asymmetry in Lyman- α spectrumcannot be reproduced using a simple two-state approximation. Calculations using the Born or Glauberapproximations give Re (cid:16) ρ p s (cid:17) =
0, so that the symmetrical picture is expected in a MSE observation[18,39]. In the latter case the ratio between the the π ± and σ lines is closer to the dynamical limit.Similar behaviour is found for the Lyman- β line (Fig. 5). Here, as in the case of Lyman- α emission,the variation of intensity between the blue- and the red-shifted π and σ components is found. The ratiobetween the σ and π components for MSE conditions is quite close to the dynamical limit. The ratio isfound to be 21:29 against 23:27. As for excitation by proton-impact of the Lyman- α line, ∑ σ I / ∑ π I < α line (Fig. 6), however, the π ± line remains rather insensitive to variation of theangle between the incident protons and the direction of the electric field. This component originates fromthe state (300) that contains only the even term Re ( ρ d s ) so that this component remains symmetrical forany angle. Also for this line ∑ σ I σ / ∑ π I π <
1. In the low density limit for MSE conditions the ratios are -0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 -6 -3 0 3 6a) (s)(p) I n t en s i t y , a . u . -6 -3 0 3 6b) -6 -3 0 3 6Displacement of L b componentsc) -6 -3 0 3 6Displacement of L b components d) Figure 5.
The Stark effect for the Lyman- β line excited by proton impact at a collision energy of 50 keV/uand by electrons with temperature of 3 keV. For the statistical intensities the ratio between components σ ± : π ± = 25:25, and for the dynamical intensities the ratio is 23:27. Other notations are the same as inFigure 4. I σ / I σ = 0.53, I π / I π = 1, and I π / I π = 0.45. It should be noted that the principal mechanism of excitationis via proton collisions while electrons contribute at the level of only 20-30%. Nevertheless, not all thestates within n =3 can be excited by electrons at temperatures of 100 eV or more. For instance, the ratio ( I σ / I σ ) >0.35 is due to preferential excitation of the states (3 ±
11) by electrons. The states (302) and (300)cannot be excited by electrons at all. For this reason the electrons should not be neglected completely inthe analysis of the data as they lead in general to a different distribution than from unidirectional protonimpact.
4. Conclusion
In this work we provided a set of proton-hydrogen excitation cross sections for calculation of the lineintensities for radiative transitions between n ≤ × cm − dependingon the beam energy. For denser plasmas, however, one has to develop a full collisional-radiative modelwith account of collisional and radiative transitions between excited states. Unfortunately, it is not easy torepresent the cross sections for transitions with ∆ n=0 in closed form as they also depend on the magneticfield.The exact expressions for the cross sections in parabolic states for n =3 excitation are given for thefirst time for an arbitrary angle between projectile velocity and direction of electric field. The expressionsinclude, in particular, coherence terms of the density matrix. Using this approach one can efficientlydescribe the excitation for typical MSE conditions. In addition, this method can be used to model theasymmetry of spectral-line emission observed, for instance, in cathode ray experiments where ionisationby the electric field can still be neglected. We also note that the asymmetry observed in the line shape ofMSE, e.g. the obvious difference between π ± components is connected with the geometry of observations,for instance the beam width on the order of tens of centimeters results in a stronger attenuation ofedge beamlets. At the same time the different values of magnetic field at both positions (assumingthe observation port at the outer wall of fusion device) results in a shape asymmetry of all the spectral -0.2-0.1 0 0.1 0.2 0.3 -4 -2 0 2 4a) (s)(p) I n t en s i t y , a . u . -4 -2 0 2 4b) -4 -2 0 2 4Displacement of H a componentsc) Figure 6.
The Stark effect for the Balmer- α line excited by proton impact at a collision energy of 50 keV/uand by electrons with temperature of 3 keV. For the statistical intensities the ratio between components π ± : π ± : π ± : σ ± : σ = 17.8:24.4:7.7:20.5:58.2, and for the dynamical intensities the ratio equals to15.6:18.2:16.2:12.5:74.6. The weak transitions σ ± , σ ± and π ± are omitted from the graph. Other notationsare the same as in Figure 4. components. We plan to analyze the recently measured KSTAR MSE spectra [5] in the nearest future aswell to provide the data for excitations by He + , Be + and C + .In general, the modeling of Stark effect stimulated by electron excitation in gas discharges alsorequires density matrix calculations for dissociative recombination, electron-impact excitation, andcharge exchange. Moreover, to the best of our knowledge, such calculations are not available or werenot required until now. We hope that the present work will motivate such calculations in the future. Funding:
This research was funded by the Program-oriented Funding (PoF) of the Helmholtz-GemeinschaftDeutscher Forschungszentren (HGF).
Acknowledgments:
This work was carried out within the framework of the EUROfusion Consortium and hasreceived funding from the Euratom research and training program 2014–2018 and 2019–2020 under Grant agreementNo. 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission.
Conflicts of Interest:
The authors declare no conflict of interest
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