Population modelling of the He II energy levels in tokamak plasmas: I. Collisional excitation model

1 Population modelling of the He II energy levels in tokamak plasmas. I : Collisional excitation model K D Lawson1, K M Aggarwal2, I H Coffey1,2, F P Keenan2, M G O'Mullane3 & JET Contributors* EUROfusion Consortium, JET, Culham Science Centre, Abingdon, OX14 3DB, UK 1 UKAEA/CCFE, Culham Science Centre, Abingdon, OX14 3DB, UK 2 Astrophysics Research Centre, School of Mathematics and Physics, Queen’s University Belfast, Belfast, BT7 1NN, Northern Ireland, UK 3 Department of Physics, University of Strathclyde, Glasgow, G4 0NG, UK * See the author list of X. Litaudon et al., 2017, Nuclear Fusion, 57, 102001 Abstract Helium is widely used as a fuel or minority gas in laboratory fusion experiments, and will be present as ash in DT thermonuclear plasmas. It is therefore essential to have a good understanding of its atomic physics. To this end He II population modelling has been undertaken for the spectroscopic levels arising from shells with principal quantum number n = 1 to 5. This paper focuses on a collisional excitation model; ionization and recombination will be considered in a subsequent article. Heavy particle collisional excitation rate coefficients have been generated to supplement the currently-available atomic data for He II, and are presented for proton, deuteron, triton and α-particle projectiles. The widely-used criterion for levels within an n shell being populated in proportion to their statistical weights is reassessed with the most recent atomic data, and found not to apply to the He II levels at tokamak densities (1018–1021 m-3). Consequences of this and other likely sources of errors are quantified, as is the effect of differing electron and ion temperatures. Line intensity ratios, including the so-called ‘branching ratios’ and the fine-structure β1, β2, β3 and γ ratios, are discussed, the latter with regard to their possible use as diagnostics. 1. Introduction To limit the heat and particle fluxes reaching plasma-facing surfaces, contemporary large tokamak plasma machines employ a divertor box, in which a cold, dense plasma acts as a buffer between the hot thermonuclear plasma and the device’s solid surfaces. The plasma densities will be lower, 1018 to 1021 m-3, than those under which local thermodynamic equilibrium (LTE) applies, with temperatures ranging from ~0.2 to 30 eV. Hence detailed population modelling is required to describe the behaviour of the cold plasma, and allow predictions to be made for tokamaks such as ITER and DEMO. Consequently, it is essential to understand the atomic physics of the fuel and of any seeded gases used to increase the radiation and enhance the cooling of the divertor plasma. To this end, a study has been undertaken of the atomic physics of H plus its isotopes, and of hydrogen-like He (He II). This ion is very similar in behaviour to H, but its atomic physics is more tractable in that the collisions with heavy particles are ion-ion rather ion-neutral atom interactions. It is important to note that He is the proposed fuel for the first phase of ITER, and will be present as ash from the DT fusion reaction in later ITER experiments and also in DEMO. The most complete population model, which will then allow the reliable calculation of line intensities, includes as many populating processes as possible. A difficulty arises in that 2 the atomic data for both H and He II are incomplete. For example, although some authors have used heavy particle collisional excitation rate coefficients in their studies (e.g. Sampson, 1977, Tallents, 1984, 1985) there are no listings of these data available for either H or He II. However, these collisions are important for transitions between the near-degenerate energy levels within a shell defined by the principal quantum number n, i.e. the fine-structure transitions. In cases where the fine-structure components of the observed spectral lines are not resolved, one approach which overcomes this difficulty in low density, non-LTE plasmas is to assume that the fine structure levels within the individual n shells (the levels being defined by the quantum numbers n, l and J) are populated in proportion to their statistical weights, nJln n n J n2 ,, 2 12 + = . Here l and J are the orbital and total angular momenta, respectively. Indeed, this approach has been adopted to the extent that some recent calculations of electron collisional rate coefficients for H do not include fine-structure collisions (Anderson et al., 2000, 2002). Although this assumption is expected to be accurate for higher n shells, for example as considered by Burgess and Summers (1976), it is less certain for the spectroscopic levels within the n = 1 to 5 shells studied here. Given the advances made in the calculation of electron collisional cross sections, it is appropriate to reassess the criterion for which the statistical weighting of the populations within n shells is valid. Sampson (1977), who investigated this topic, used a semi-classical description of both electron and heavy particle cross sections. Tallents (1985) found that significantly higher densities for He II than those of Sampson were needed to ensure a statistical population. In the present study, the semiclassical method of Walling and Weisheit (1988) is used to generate improved heavy particle cross sections, which are then combined with recent R-matrix electron collisional data. A collisional-radiative model has been used which includes electron and heavy particle rate coefficients for both intershell and fine-structure transitions. Such a model is essential when the fine-structure components of the spectral lines are resolved, but is also useful for assessing the importance of different aspects of the population modelling. Here results based on a collisional excitation model are presented for the spectroscopic levels of He II. At low temperatures (Te < ~3 eV) processes such as recombination become important and can dominate the populating channels. These will be considered in a subsequent paper. Section 2 assesses the available electron collisional data for He II, while section 3 presents heavy particle cross sections for He II generated from the implementation of the Walling and Weisheit (1988) method. The collisional-radiative population model, including both electron and heavy particle collisional channels, is described in section 4. This section also discusses the importance of the different channels, and the criterion determined for the validity of the assumption of the statistical population of the n shells. Spectral line intensity ratios can be useful for diagnosing certain plasma parameters, and these are discussed in section 5, while section 6 assesses the effect of differing electron and ion temperatures. Section 7 contains a comparison with previous work, and finally section 8 presents our conclusions. 2. Published atomic data for He II In addition to the previously mentioned collisional excitation rate coefficients, a collisional-radiative model requires transition probabilities for spontaneous emission (A- values) for each transition. These radiative decay rates for the intershell transitions are found in the NIST database (Kramida et al., 2015), which incorporates the extensive listing of 3 Wiese and Fuhr (2009). Additional A-values for the fine-structure transitions within an n shell are given by Aggarwal et al. (2017) and Jitrik and Bunge (2004). Aggarwal et al. adopt the fully relativistic GRASP (General-purpose Relativistic Atomic Structure Package) code to generate the atomic structure, which gives wave functions, energy levels and A-values, while Jitrik and Bunge use a point-nucleus Dirac Hamiltonian model. For He II, the available data are expected to be reliable, being more accurate than the collisional results and therefore not limiting the accuracy of the population model. For electron collisions, R-matrix calculations are expected to be more accurate than older methods and hence we focus on these. The earliest R-matrix calculation for He II is that of Aggarwal et al. (1991), while Callaway (1994) reviews the various calculations for H and hydrogen-like ions available up to the time of publication, the latter forming the basis of the 1997 dataset within the ADAS database (Summers, 2004). Both Aggarwal et al. and Callaway present data for the so-called 'spectroscopic levels', n = 1-5. These data are lresolved, therefore including 15 levels, and are for the intershell transitions only. Aggarwal et al. cover a temperature interval of 0.43 to 8.6 eV, while Callaway's formula for the effective collision strengths range from 0.5 eV up to a high temperature limit of 10 keV. For the IRON project, Kisielius et al. (1996) performed a J-resolved He II calculation for the 16 levels of the n = 1-4 shells. Data are presented for both intershell and fine-structure transitions, although only for a very limited temperature range from 0.137 to 1.72 eV. The more recent work by Ballance et al. (2003) is an l-resolved, 15 level calculation providing data for intershell transitions for n up to 5, covering a temperature range of 1-86.2 eV. Unusually, Hamada et al. (2010) present data only for the fine-structure transitions. The calculations are J-resolved, including the 25 levels of the n = 1-5 shells for a wide range of atomic number Z. For He, the temperature range covered is 0.86-8.62 eV. Of most use for the present study are the calculations of Aggarwal et al. (2017), with the 25 J-resolved levels of shells n = 1-5 included, and both intershell and fine-structure transitions treated over the temperature range 0.34-21.7 eV. The scattering calculation is performed using the DARC (Dirac Atomic R-matrix Code) program to calculate collision strengths, Ω, as a function of incident projectile energy. All authors present their collisional data in terms of the effective collision strength, Υ, since this provides a more concise listing than is possible for collision strengths or cross sections, σ, given the very fine energy resolution necessary to capture as many resonances as possible, particularly near the threshold energy. The collision strength of a transition from a lower level i to an upper level j is given by H i i ij ij I E a ω π σ 2 0 =Ω where Ei is the initial free electron energy, ωi the statistical weight of level i (ωi = 2J + 1), ao is the Bohr radius and IH the Rydberg unit of energy. It is a dimensionless parameter and symmetrical between the initial and final states. To obtain the effective collision strength, Ωij is convolved with a Maxwellian electron energy distribution, this implicitly supposing a Local Kinetic Equilibrium (LKE), Υij =Ωij exp(−Ej kTe )d(Ej kTe ) 0 ∞ ∫ . Here Ej is the final free electron energy after excitation, Te the electron temperature and k the Boltzmann constant. From Υij the collisional excitation rate coefficient qij can be derived using the expression 4 qij =2 π 1/2 α ca0 2IH kTe ⎛ ⎝ ⎜⎞ ⎠ ⎟ 1/2 Υij ω i exp(−ΔE kTe ) with the corresponding deexcitation rate coefficient qji given by qji =qij ω i ω j exp(ΔE kTe ) . These are the parameters used in the collisional-radiative model, where α is the fine-structure constant, c the velocity of light and ΔE the transition energy. As previously noted, there are no listings of heavy particle collisional excitation data available for He II. 3. Heavy particle collisional data To provide heavy particle collisional excitation rate coefficients for this study, the method of Walling and Weisheit (1988) for ion impact bound-state excitation of ions has been implemented fully for electric dipole and electric quadrupole transitions. This semi-classical treatment extends the work of Seaton (1964), the trajectory of the ions being determined from classical equations, but with the probability of excitation from quantum mechanical formulae. The semi-classical cross section, a function of the relative velocity v, is obtained by integrating the excitation probability, Pij, over all impact parameters, b, i.e. ∫∞ =0),(2)( dbbbvPv ijij πσ . It is found that at small impact parameters, the first-order expression for the excitation probability becomes invalid, with Pij > 1, and hence it is necessary to limit Pij to obtain meaningful results. Seaton (1964) imposed a limit of Pij ≤ 0.5 on the first order probability into any one state. A further consideration is the projectile penetrating the target ion and, in this situation, the probability will again be overestimated. Another limit was therefore imposed by Seaton that Pij ≤ the smaller of 0.5 and PT, where PT = Pij(bT), the distance of closest approach being greater than the size of the target ion for all impact parameters less than bT. The cross section can now be defined as ∫∞+= c bccijij PbdbbbvPv 2 ),(2)( ππσ , where Pc is a limiting excitation probability and bc the corresponding limiting impact parameter. For numerical calculations it is convenient to change the integration variable from the impact parameter, b, to the Coulomb trajectory's eccentricity, ε, and introducing the multipole strength factor χij and the adiabaticity parameter ξ, where v E r ! Δ =0 ξ , r0 being half the distance of closest approach in a collision at zero impact parameter. The cross section can be written as 5 σ ij (v)= π r0 2 χ ij ( λ )Ψ( λ )( ε c, ξ )+( ε c 2−1)P c ⎡ ⎣⎤ ⎦ . In this equation λ indicates the multipole of the transition, 1 corresponding to an electric dipole transition and 2 an electric quadrupole transition, while εc is the eccentricity corresponding to the limiting impact parameter bc and ∫∞ =Ψ c dR c ε λλ εεξεξε ),(2),( )()( where R(λ) is a relative excitation probability. To facilitate the implementation of their approach, Walling and Weisheit (1988) provide tables of log10R(λ) and log10Ψ(λ) as functions of ε and ξ. Initially, the tabulated values were used, but the necessary two-dimensional interpolation of these led to cross sections being generated that did not vary smoothly. Consequently, the method was implemented fully allowing log10R(λ) and log10Ψ(λ) to vary smoothly as continuous functions of ε and ξ. The tables of log10R(λ) were reproduced exactly, but small differences were found with the tabulated values of the integral log10Ψ(λ). These were generally ≤3%, although could be larger (up to ~40%) when the logarithm changed sign, and in the case of electric dipole transitions for large ε and small ξ. The present integrations are preferred. As for the electron data, the cross sections are converted into collisional excitation rate coefficients for use in the collisional-radiative model. In this case, there is an additional factor involving the reduced mass µ, where tp mm 111 += µ , qij =2 α c ( π IHkTp)1/2 me µ ⎛ ⎝ ⎜⎞ ⎠ ⎟ 1/2 exp(−ΔE kTp ) σ ij Eiexp(−Ej kTp )d(Ej kTp ) 0 ∞ ∫ , and me, mp, mt are the electron, projectile and target masses, respectively, and Tp the projectile temperature. The multipole strength factor, χij, depends only on the atomic properties and requires knowledge of the line strength, which in turn can be calculated from the A-value. These are available for 58 of the 70 fine-structure transitions in the n = 2-5 shells. Of the 12 transitions for which there are no A-values, 10 transitions are electric dipole between near-degenerate levels. A reasonable estimate of the A-value of one of these, the 2p 2P1/2 - 2s 2S1/2 transition, was made via a comparison of proton impact cross sections for He II with the results of Zygelman and Dalgarno (1987) as described below. Estimates were also made for the 2P1/2 – 2S1/2 transitions in the higher n shells, as detailed in section 4, although these are expected to have a greater uncertainty. In total, cross sections were generated for 20 electric dipole and 25 electric quadrupole transitions. The other transitions are either between near-degenerate levels for which there are no A-values, higher electric multipole transitions, which are expected to be less important, or magnetic multipole transitions. Figure 1 shows d- and α-particle collisional excitation data which have been generated for the fine-structure transitions within the n = 2 shell of He II. There are two electric dipole transitions, 2p 2P1/2 - 2s 2S1/2 and 2s 2S1/2 - 2p 2P3/2, and an electric quadrupole transition, 2p (2P1/2 - 2P3/2) in this shell. Table 1 details the energy levels 6 used in the present calculations with energies taken from NIST (Kramida et al., 2015). Those in units of cm-1, which are given to a higher precision than those listed in eV, are of use in determining the energy separation of the near-degenerate levels. The ordering of the levels used is that given by NIST apart from 4d 2D5/2 and 4f 2F5/2, where that of Aggarwal et al. (2017) is followed. Tables 2 to 5 list the collisional excitation rate coefficients for the fine-structure electric dipole transitions of He II for collisions with the projectiles p, d, t and α-particles. These optically allowed transition rate coefficients are both larger, and expected to be more reliable than those for the forbidden transitions. We note that heavy particle collisional data for the intershell transitions were also checked, but their rate coefficients are small compared with the electron rate coefficients, being at least two orders of magnitude lower at the highest temperature considered in the present study (21.7 eV). This is because the rate coefficients tend to peak when the momentum of the projectile, either electron or heavy particle, matches the change of momentum of the electron being excited. In electron collisions, with the projectile and target electrons having the same mass, this condition is met when the projectile energy, which must be sufficient to excite the transition, is ~ kTe, this being of the order of the intershell transition energies. For heavy particles with their greater mass, the most favourable velocities are a factor me/mp smaller, this corresponding to energies that are lower by this fraction. Heavy particles are therefore most efficient at exciting transitions with smaller transition energies. Significant differences are found between the proton impact results of Zygelman and Dalgarno (1987) and the present calculations for the 2p (2P1/2 - 2P3/2) transition of He II. Hence it is important to consider the expected accuracy of the semi-empirical approach and the effect on the He II level populations of any discrepancies. Some indication is provided via comparisons with full quantal calculations, and several have been undertaken by Walling and Weisheit (1988). Their first is with Heil et al. (1982) for the proton impact excitation of the 2s22p (2P1/2-2P3/2) fine-structure ground state transition of O IV, for which there is excellent agreement, to within a few per cent (see figure 4 of Walling and Weisheit). A second comparison is with the quantum mechanical calculations of Heil et al. (1983) for excitation of the fine-structure ground state 3s23p (2P1/2-2P3/2) transition of Fe XIV by protons, the heavy particle collision first studied by Seaton (1964). Agreement is again within a few per cent at collision energies less than the ionization potential of Fe XIV (392 eV) the difference increasing to only ~10% at energies of twice this value (see figure 5 of Walling and Weisheit). A limitation of the semi-classical approach is evident in the treatment of the excitation of the fine-structure transitions within the 3s23p2 3P multiplet of Fe XIII. Since to first order the 3s23p2 (3P0-3P1) transition does not occur, the two-state semi-classical cross section is zero. A close-coupling calculation for this three-state problem allows the 3P0-3P1 transition to be treated via the intermediate J = 2 state, and Faucher (1975) provides quantal close-coupled results for proton impact excitation of these transitions. At low collision energies, up to the ionization potential of 361 eV, agreement for the 3s23p2 (3P1-3P2) transition is within ~10%. However, for the 3s23p2 (3P0-3P2) transition there are larger differences at all energies, the loss of flux due to the coupling with the forbidden 3s23p2 (3P0-3P1) transition having a greater effect (see figure 6 of Walling and Weisheit). A more recent paper by Burgess and Tully (2005) note an error in Faucher’s derivation of the Born formula used to give the high energy behaviour of his cross sections. Burgess and Tully consider high energy proton excitation of electric quadrupole transitions, combining a semi-classical approach with the Born approximation (the initial and final states of the projectile being defined by plane waves) to give the correct high energy behaviour. The same limitation of the semi-classical approach found for Fe XIII is seen for 7 transitions within the 2s2p 3P multiplet of C III. Ryans et al. (1998) calculate the heavy particle rates for the fine-structure transitions within this multiplet using a close-coupled impact parameter method. Again, the 2s2p (3P0-3P1) transition cannot be treated with the Walling and Weisheit (1988) semi-classical approach and, as for Fe XIII, a larger discrepancy is found for 3P0-3P2 than for 3P1-3P2. In this case the comparisons are valuable in that they can be made for four projectiles, namely protons, deuterons, tritons and α-particles, and similar results are obtained in all cases. Typically the cross sections agree within ~25% and 40% up to collision energies of ~150 eV for the 2s2p (3P1-3P2) and 2s2p (3P0-3P2) transitions, respectively, and the rate coefficients within ~20% and 35% up to temperatures of 90 eV, respectively. Figure 2 shows a comparison of the deuteron and α-particle excitation cross sections and rate coefficients for some representative transitions. Of particular relevance to the present study is a comparison with the quantum mechanical calculations of Zygelman and Dalgarno (1987) for the proton impact excitation of the n = 2 shell fine-structure transitions of several H-like ions, including He II and Ar XVIII. As seen in figure 1, all three transitions occur in first-order and comparisons for Ar XVIII made by Walling and Weisheit (1988; see their figure 7) reveal good agreement (within 10%) for the 2p 2P1/2 - 2s 2S1/2 transition, whose cross section is by far the largest. There is poorer agreement for the weaker excitations, particularly the 2p (2P1/2-2P3/2) quadrupole transition, the discrepancies decreasing with energy as the close-coupling effects diminish. For He II the discrepancies for the 2s 2S1/2 - 2p 2P3/2 transition are similar to those found for Ar XVIII. However, there is no A-value available for the strong He II 2p 2P1/2 - 2s 2S1/2 transition and an estimate is made by assuming good agreement with the results of Zygelman and Dalgarno, this being found for the transition in Ar XVIII. As can be seen in figure 3, which compares results for all three transitions of He II, more severe discrepancies are found for the weak electric quadrupole 2p (2P1/2-2P3/2) transition than for Ar XVIII. Consequently, checks must be made in the population modelling for sensitivity to the quadrupole transitions. 4. The collisional-radiative population model The energy levels within an ion are populated chiefly by electron collisional excitation and deexcitation and by radiative decay, as illustrated in the schematic energy level diagram shown in figure 4. Within an n shell, heavy particle collisions may also play a role. The population of the ith level, ni, is therefore given by the rate equation dni dt =nenjqji −neniqij j≠i ∑ j≠i ∑+nhp njqji hp j≠i ∑−nhp niqij hp j≠i ∑+njAji j>i ∑−niAij j<i ∑+nenz+1 α Ri −nenzsi . Here ne and nhp are the electron and heavy particle densities, respectively, and Aij the A-values of a transition from level i to j. The terms nz and nz+1 are the densities of the hydrogenic species and fully stripped ions, respectively, with αRi and si representing recombination and ionization rate coefficients to and from level i, respectively. These processes are not included in the present analysis, and will be discussed in a subsequent paper. Recombination in particular is expected to have a significant effect on the level populations at temperatures below ~ 3 eV. The adjustment of the populations within an ion due to collisions and radiative decay is usually rapid compared with processes such as the transport of ions within the plasma. Hence, steady-state will be assumed, i.e. .0= dt dni This leads to a set of simultaneous linear equations, which can be solved by matrix inversion 8 for ni / ng, where ng is the ground-state population. Figure 5 shows the populations as a fraction of the ground-state population calculated for levels within the n = 2-4 shells of He II for a range of densities (ne = 1016-1022 m-3) at a temperature of 3.4 eV, and in the case of the n = 4 shell these are normalized to ne. We note that α-particle collisions are included in the calculation, with the assumption that the He ion density is 47% of the electron density. This allows both for the small number of He particles that are not fully stripped and for the low impurity concentrations normally observed in tokamak discharges fuelled only by He. It can be seen that the 2s 2S1/2 level is metastable, the radiative decay to the 1s 2S1/2 ground-state being small (5.3×102 s-1). This corresponds to the two photon decay (Tallents 1984) instead of the magnetic dipole transition for which the A- value is 2.6×10-3 s-1. Consequently, with decreasing density and fewer collisions, the population of this level flattens and becomes independent of density, rather than matching the 2p 2P1/2 level population as occurs at higher densities. Despite its enhanced population, 2s 2S1/2 has no significant effect on the radiation given the very low transition probabilities for its radiative decays. Its fractional population reached at low densities in this steady-state solution is dependent on the coupling of this level to 2p 2P1/2, which was found as described in section 3. Similarly, the 3s 2S1/2, 4s 2S1/2 and 5s 2S1/2 level populations are sensitive to the corresponding 2P1/2-2S1/2 transition in the higher shells. Although no A-values are published for these transitions, estimates have been made by assuming a similar 2P1/2-2S1/2 / 2S1/2-2P3/2 ratio in the cross sections as has been found for the n = 2 shell, and α-particle excitation rate coefficients calculated accordingly. The full and dashed lines in figure 5 indicate the differences found when these rates are included in the population model. 4.1 The importance of fine-structure transitions The most reliable populations will be those derived using as many populating channels as possible. Consequently, it is of interest to determine the errors when the available atomic data are incomplete and without the imposition of any assumptions. Two cases will be considered. The first when only intershell electron collisional data are available, the second when both intershell and fine-structure electron collisional rate coefficients are included in the population model, but heavy particle rate coefficients are not used. Comparisons will be made with the full population model, which includes electron intershell transitions, plus electron and α-particle impact rate coefficients for the fine-structure transitions. The αparticle impact rate coefficients for the 2P1/2-2S1/2 transitions, which particularly affect the populations of the 2S1/2 levels, are determined as described above. Figure 6 shows ratios of populations in the n = 2 and 3 levels for the two cases considered along with the results from the full population model. It is clear from the figure that there are very significant differences (up to two orders of magnitude) when the atomic data are incomplete. Much reliance is therefore being placed on the assumption of the populations within an n shell being in proportion to the statistical weights when incomplete sets of atomic data are used. 4.2 Sensitivity to transitions between near-degenerate levels Another source of error investigated to ensure it will not seriously affect the accuracy of the results of the full population model is the sensitivity of the populations to the collisional rate coefficients between near-degenerate levels. An inspection of figure 5 reveals that including the rate coefficients of the transitions between the near-degenerate 2P1/2 and 2S1/2 levels in the population model affects the derived populations. The 2D3/2 and 2P3/2, 2F5/2 and 2D5/2 and 2G7/2 and 2F7/2 levels also form pairs with near-degeneracy. These levels are better connected to neighbouring ones and, hence, transitions between them are expected to be less 9 important than in the case of the 2P1/2 and 2S1/2. However, the question arises as to whether the high rate coefficients associated with these small energy transitions could affect the results. A check was undertaken by assuming that the same rate coefficients applied to these transitions as for 2P1/2 - 2S1/2 in the particular shell. Figure 7 shows the ratios of populations at a temperature of 3.4 eV for levels in the n = 3 and 4 shells, both including and excluding rate coefficients for all transitions between the near-degenerate J = 3/2, 5/2 and 7/2 levels. It can be seen that at densities above ~1018 m-3 the differences are < 5%, and this is the case for most temperatures. The largest differences occur at either low (~0.34 eV) or high (~22 eV) temperatures, typically of up to ~10%, although greater discrepancies are found for the 4f 2F5/2 and 4d 2D3/2 levels (~17%) and for 3d 2D3/2 (~20%) at the highest temperatures considered. These discrepancies rapidly decrease with increasing density. 4.3 Sensitivity to electric quadrupole transition rate coefficients A significant difference was found between the results of Zygelman and Dalgarno (1987) and the present heavy particle calculations for proton impact of the 2p (2P1/2-2P3/2) electric quadrupole transition of He II, illustrated in figure 3c. This can result in up to two orders of magnitude difference in the derived excitation rate coefficients. To check the sensitivity of the population modelling to the electric quadrupole transition rate coefficients, comparisons were made in which all electric quadrupole rate coefficients were increased by a factor of 100. Comparisons of populations for levels in the n = 3 and 4 shells at a temperature of 3.4 eV are shown in figure 8. The differences are smaller than 10%, and above densities of ~1018 m-3 the discrepancies are typically ~5% or less. 4.4 Criterion for the statistical population of the n shells Given the significant differences found when the atomic data are incomplete, as illustrated in section 4.1, it is important to reassess the validity of the assumption that the level populations within an n shell are in proportion to the statistical weights for the densities encountered in large plasma machines. Figure 9 shows the minimum densities required to ensure a statistical population, plotted as a function of temperature, with the full lines the results of the present population modelling. In all shells, it is the 2S1/2 level that sets the limit needed to achieve a statistical population. However, note that transitions from this level are weak and, therefore, contribute little to the observed line radiation. Consequently, the densities required to ensure that the 2P1/2 level was populated statistically compared with all levels apart from 2S1/2 was checked, but were found to be very similar. Since there are various uncertainties in the calculations, it is reasonable to allow a 5% error margin in the populations from a strict statistical population. These results are also shown in figure 9 together with the results of Sampson (1977) and Tallents (1985); the latter two will be discussed in section 7.2. It can be seen from the figure that for the densities of interest, ~1018-1021 m-3, the populations, particularly in the lower shells, are not in statistical equilibrium even allowing for a 5% error margin in the populations. In applications of the theoretical results to observed spectra, calculations of line intensities are important rather than those for the level populations. The spectrally-unresolved He II lines consist of a number of components, and it is expected that the transitions forming these may to some extent compensate for deviations in the individual level populations. It is therefore appropriate to consider the errors that would result in the intensities from making the assumption of statistical populations, even at densities where this does not hold. Figure 10 shows the ratio of line intensities when this assumption is made to those when the full population model is used for the first four lines of the Lyman series at the lowest density of 16 determine reliable line intensities. Heavy particle collisional excitation data has been generated using the method of Walling and Weisheit (1988), and are listed for the optically allowed fine-structure transitions with projectiles p, d, t and α-particles. Population modelling demonstrates the importance of fine-structure transition rate coefficients, and allows the assumption of the statistical population of levels within an n shell to be reassessed with the most recent atomic data. This assumption is often made when the atomic data for the fine-structure transitions are not available. In agreement with the study of Tallents (1985), it is found that this assumption is not valid for the spectroscopic levels (within shells n = 1–5) at tokamak densities (1018-1021 m-3). However, fortuitously, different components within the spectroscopically-unresolved He II lines do tend to compensate, allowing this technique to be used at densities at which the assumption no longer applies. The effect this has on line intensities is quantified, as are a number of other possible sources of error in the population modelling. Mechanisms such as ionization and recombination will be considered in a subsequent paper. Line intensity ratios can provide a more accurate comparison with experiment than absolutely-calibrated line intensities, and the intershell, the so-called ‘branching ratios’, plus the β1, β2, β3 and γ fine-structure ratios are discussed. Of the latter, only β1 is likely to be of diagnostic use at tokamak densities, while the intershell ratios are found to be temperature dependent. It is shown that differing ion and electron temperatures are not expected to lead to significant errors in the calculated ratios beyond those expected from other sources. Comparisons of previously published R-matrix calculations of electron collisional rate coefficients highlight significant variations, which in turn lead to differences in the resultant line intensities for all but the n = 2 shell. These differences, typically ~20% for n = 3 levels, ~25-35% for n = 4 and a factor of ~2 for transitions involving n = 5, are larger than other expected uncertainties. Of the available calculations, the results of Aggarwal et al. (2017) are considered to be the most reliable, these authors suggesting an accuracy of better than 20% for the majority of the transitions. It is noted, however, that all the calculations agree to within 10% for the intensity of the Lyman-α line and there is good compensation between the two components of this line when a statistical population is supposed. This is an important result given the contribution of this line (~90%) to the total He II line power, which is widely used to predict the power loss term in transport modelling. This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 under grant agreement No. 633053 and from the RCUK [grant number EP/P012450/1]. To obtain further information on the data and models underlying this paper please contact [email protected]. The views and opinions expressed herein do not necessarily reflect those of the European Commission. 17 References Aggarwal K M, Berrington K A, Kingston A E and Pathak A, 1991, J. Phys. B, 24, 1757 Aggarwal K M, Igarashi A, Keenan F P and Nakazaki S, 2017, Atoms, 5, 19 Anderson H, Ballance C P, Badnell N R and Summers H P, 2000, J. Phys. B, 33, 1255 Anderson H, Ballance C P, Badnell N R and Summers H P, 2002, J. Phys. B, 35, 1613 Ballance C P, Badnell N R and Smyth E S, 2003, J. Phys. B, 36, 3707 Burgess A and Summers H P, 1976, Mon. Not. R. Astron. Soc., 174, 345 Burgess A and Tully J A, 2005, J. Phys. B, 38, 2629 Callaway J, 1994, ADNDT, 57, 9 Clarke R E H, Sampson D H and Goett S J, 1982, Astrophys. J. Suppl., 49, 545 Faucher P, 1975, J. Phys. B, 8, 1886 Golden L B, Clark R E H, Goett S J and Sampson D H, 1981, Astrophys. J. Suppl., 45, 603 Hamada K, Aggarwal K M, Akita K, Igarashi A, Keenan F P and Nakazaki S, 2010, ADNDT, 96, 481 Heil T G, Green S and Dalgarno A, 1982, Phys Rev A, 26, 3293 Heil T G, Kirby K and Dalgarno A, 1983, Phys Rev A, 27, 2826 Jitrik O and Bunge C F, 2004, J. Phys. Chem. Ref. Data, 33, 1059 Kisielius R, Berrington K A and Norrington P H, 1996, Astron. Astrophys. Suppl. Ser., 118, 157 Kramida A, Ralchenko Yu, Reader J and NIST Team, 2015, NIST Atomic Spectra Database (ver. 5.3), http://physics.nist.gov/asd, National Institute of Standards and Technology, Gaithersburg, MD Lawson K D, Aggarwal K M, Coffey I H, Keenan F P, O'Mullane M G, Ryć L and Zacks J, 2011, Plasma Phys. Control. Fusion, 53, 015002 Ryans R S I, Foster-Woods V J, Copeland F, Keenan F P, Matthews A and Reid R H G, 1998, ADNDT, 70, 179 Sampson D H, 1977, J. Phys. B, 10, 749 Seaton M J, 1962, Proc. Phys. Soc. 79, 1105 Seaton M J, 1964, Mon. Not. R. Astron. Soc., 127, 191 Shevelko V P, Skobelev I Y and Vinogradov A V, 1977, Phys. Scr. 16, 123 Summers H P, 2004, ‘The ADAS User Manual, version 2.6, http://adas.phys.strath.ac.uk’ Tallents G J, 1984, J.Phys. B, 17, 3677 Tallents G J, 1985, J. Phys. B, 18, 3299 Walling R S, Weisheit J C, 1988, Physics Reports, 162, 1 Wiese W L and Fuhr J R, 2009, J. Phys. Chem. Ref. Data, 38, 565 Zygelman B and Dalgarno A, 1987, Phys Rev A, 35, 4085 18 Table 1. He II energy levels with energies from the NIST database. Index Configuration Level Energy (cm-1) Energy (eV) 1 1s 2S1/2 0.0 0.0 2 2p 2P1/2 329179.29359 40.8130305 3 2s 2S1/2 329179.76197 40.8130886 4 2p 2P3/2 329185.15075 40.8130305 5 3p 2P1/2 390140.82463 48.3712970 6 3s 2S1/2 390140.96418 48.3713143 7 3d 2D3/2 390142.55723 48.3715118 8 3p 2P3/2 390142.56009 48.3715122 9 3d 2D5/2 390143.13567 48.3715835 10 4p 2P1/2 411477.12241 51.0166608 11 4s 2S1/2 411477.18141 51.0166681 12 4d 2D3/2 411477.85334 51.0166608 13 4p 2P3/2 411477.85456 51.0167514 14 4d 2D5/2 411478.09737 51.0167816 15 4f 2F5/2 411478.09693 51.0167816 16 4f 2F7/2 411478.21894 51.0167967 17 5p 2P1/2 421352.67859 52.2410737 18 5s 2S1/2 421352.70883 52.2410774 19 5d 2D3/2 421353.05282 52.2411201 20 5p 2P3/2 421353.05345 52.2411202 21 5f 2F5/2 421353.17753 52.2411355 22 5d 2D5/2 421353.17776 52.2411356 23 5g 2G7/2 421353.23988 52.2411433 24 5f 2F7/2 421353.24000 52.2411433 25 5g 2G9/2 421353.27737 52.2411479 19 Table 2a. Proton collisional excitation rate coefficients for He II (m3s-1). Temperature (eV) 0.273 0.43 0.69 1.09 1.72 2.73 4.33 6.86 Lower index Upper index 2 3 3.52e-11 3.86e-11 4.00e-11 3.95e-11 3.78e-11 3.52e-11 3.21e-11 2.89e-11 3 4 4.33e-12 7.28e-12 1.09e-11 1.48e-11 1.86e-11 2.19e-11 2.44e-11 2.58e-11 5 6 2.75e-10 2.82e-10 2.78e-10 2.65e-10 2.47e-10 2.26e-10 2.04e-10 1.83e-10 5 7 3.46e-11 4.91e-11 6.44e-11 7.90e-11 9.12e-11 9.97e-11 1.04e-10 1.04e-10 6 8 5.30e-11 7.64e-11 1.03e-10 1.29e-10 1.54e-10 1.72e-10 1.83e-10 1.87e-10 8 9 1.10e-10 1.33e-10 1.50e-10 1.62e-10 1.66e-10 1.63e-10 1.56e-10 1.46e-10 10 11 1.06e-09 1.06e-09 1.02e-09 9.56e-10 8.79e-10 7.98e-10 7.18e-10 6.42e-10 10 12 1.93e-10 2.59e-10 3.28e-10 3.90e-10 4.39e-10 4.70e-10 4.81e-10 4.77e-10 11 13 2.64e-10 3.59e-10 4.60e-10 5.56e-10 6.35e-10 6.89e-10 7.17e-10 7.20e-10 12 15 4.01e-10 4.56e-10 4.90e-10 5.03e-10 4.97e-10 4.76e-10 4.47e-10 4.12e-10 13 14 5.63e-10 6.58e-10 7.26e-10 7.61e-10 7.66e-10 7.46e-10 7.09e-10 6.63e-10 14 16 6.42e-10 6.76e-10 6.81e-10 6.63e-10 6.27e-10 5.81e-10 5.31e-10 4.81e-10 17 18 2.87e-09 2.81e-09 2.67e-09 2.48e-09 2.27e-09 2.05e-09 1.85e-09 1.64e-09 17 19 6.62e-10 8.64e-10 1.06e-09 1.23e-09 1.36e-09 1.43e-09 1.45e-09 1.43e-09 18 20 8.68e-10 1.14e-09 1.42e-09 1.66e-09 1.86e-09 1.98e-09 2.03e-09 2.03e-09 19 21 1.53e-09 1.71e-09 1.82e-09 1.86e-09 1.83e-09 1.75e-09 1.65e-09 1.53e-09 20 22 1.82e-09 2.07e-09 2.23e-09 2.30e-09 2.29e-09 2.22e-09 2.10e-09 1.96e-09 21 23 1.58e-09 1.61e-09 1.58e-09 1.51e-09 1.40e-09 1.29e-09 1.17e-09 1.06e-09 22 24 2.40e-09 2.51e-09 2.51e-09 2.43e-09 2.30e-09 2.14e-09 1.96e-09 1.78e-09 24 25 2.02e-09 1.98e-09 1.88e-09 1.75e-09 1.60e-09 1.45e-09 1.30e-09 1.16e-09 20 Table 2b. Proton collisional excitation rate coefficients for He II (m3s-1). Temperature (eV) 10.87 17.23 27.31 43.29 68.61 108.74 136.90 Lower index Upper index 2 3 2.57e-11 2.28e-11 2.00e-11 1.75e-11 1.52e-11 1.31e-11 1.21e-11 3 4 2.61e-11 2.55e-11 2.42e-11 2.26e-11 2.08e-11 1.90e-11 1.80e-11 5 6 1.62e-10 1.43e-10 1.26e-10 1.09e-10 9.41e-11 8.02e-11 7.37e-11 5 7 1.00e-10 9.48e-11 8.82e-11 8.11e-11 7.37e-11 6.64e-11 6.27e-11 6 8 1.85e-10 1.78e-10 1.68e-10 1.56e-10 1.44e-10 1.30e-10 1.24e-10 8 9 1.34e-10 1.22e-10 1.10e-10 9.81e-11 8.70e-11 7.63e-11 7.12e-11 10 11 5.69e-10 5.00e-10 4.36e-10 3.76e-10 3.20e-10 2.69e-10 2.46e-10 10 12 4.61e-10 4.38e-10 4.09e-10 3.76e-10 3.42e-10 3.06e-10 2.88e-10 11 13 7.04e-10 6.74e-10 6.35e-10 5.88e-10 5.37e-10 4.82e-10 4.54e-10 12 15 3.76e-10 3.40e-10 3.05e-10 2.70e-10 2.37e-10 2.06e-10 1.91e-10 13 14 6.11e-10 5.57e-10 5.03e-10 4.48e-10 3.95e-10 3.44e-10 3.19e-10 14 16 4.32e-10 3.85e-10 3.39e-10 2.97e-10 2.57e-10 2.19e-10 2.01e-10 17 18 1.45e-09 1.27e-09 1.10e-09 9.39e-10 7.91e-10 6.61e-10 6.02e-10 17 19 1.38e-09 1.31e-09 1.22e-09 1.12e-09 1.01e-09 8.95e-10 8.37e-10 18 20 1.97e-09 1.88e-09 1.76e-09 1.62e-09 1.46e-09 1.30e-09 1.21e-09 19 21 1.40e-09 1.27e-09 1.14e-09 1.01e-09 8.77e-10 7.53e-10 6.94e-10 20 22 1.81e-09 1.65e-09 1.48e-09 1.31e-09 1.14e-09 9.82e-10 9.05e-10 21 23 9.45e-10 8.37e-10 7.33e-10 6.35e-10 5.43e-10 4.58e-10 4.19e-10 22 24 1.60e-09 1.43e-09 1.26e-09 1.09e-09 9.36e-10 7.91e-10 7.23e-10 24 25 1.03e-09 9.00e-10 7.81e-10 6.69e-10 5.65e-10 4.73e-10 4.31e-10 21 Table 3a. Deuteron collisional excitation rate coefficients for He II (m3s-1). Temperature (eV) 0.273 0.43 0.69 1.09 1.72 2.73 4.33 6.86 Lower index Upper index 2 3 3.25e-11 3.73e-11 4.01e-11 4.10e-11 4.03e-11 3.84e-11 3.56e-11 3.25e-11 3 4 2.77e-12 4.95e-12 7.83e-12 1.13e-11 1.51e-11 1.89e-11 2.22e-11 2.47e-11 5 6 2.63e-10 2.82e-10 2.88e-10 2.83e-10 2.69e-10 2.51e-10 2.29e-10 2.08e-10 5 7 2.40e-11 3.60e-11 5.01e-11 6.52e-11 7.97e-11 9.19e-11 1.00e-10 1.05e-10 6 8 3.58e-11 5.44e-11 7.71e-11 1.03e-10 1.29e-10 1.54e-10 1.72e-10 1.84e-10 8 9 8.75e-11 1.12e-10 1.35e-10 1.53e-10 1.64e-10 1.68e-10 1.65e-10 1.58e-10 10 11 1.04e-09 1.08e-09 1.07e-09 1.03e-09 9.70e-10 8.96e-10 8.18e-10 7.39e-10 10 12 1.36e-10 1.94e-10 2.59e-10 3.27e-10 3.89e-10 4.39e-10 4.72e-10 4.88e-10 11 13 1.83e-10 2.63e-10 3.56e-10 4.56e-10 5.51e-10 6.33e-10 6.92e-10 7.27e-10 12 15 3.38e-10 4.07e-10 4.61e-10 4.96e-10 5.09e-10 5.03e-10 4.84e-10 4.57e-10 13 14 4.59e-10 5.68e-10 6.62e-10 7.31e-10 7.68e-10 7.75e-10 7.60e-10 7.27e-10 14 16 5.90e-10 6.53e-10 6.86e-10 6.91e-10 6.72e-10 6.37e-10 5.93e-10 5.45e-10 17 18 2.89e-09 2.92e-09 2.85e-09 2.71e-09 2.53e-09 2.33e-09 2.12e-09 1.91e-09 17 19 4.78e-10 6.57e-10 8.54e-10 1.05e-09 1.22e-09 1.36e-09 1.45e-09 1.49e-09 18 20 6.18e-10 8.57e-10 1.13e-09 1.40e-09 1.65e-09 1.86e-09 2.01e-09 2.09e-09 19 21 1.29e-09 1.53e-09 1.72e-09 1.83e-09 1.88e-09 1.86e-09 1.80e-09 1.71e-09 20 22 1.51e-09 1.82e-09 2.07e-09 2.24e-09 2.33e-09 2.33e-09 2.28e-09 2.18e-09 21 23 1.51e-09 1.60e-09 1.63e-09 1.60e-09 1.53e-09 1.43e-09 1.33e-09 1.21e-09 22 24 2.22e-09 2.43e-09 2.53e-09 2.54e-09 2.47e-09 2.35e-09 2.20e-09 2.04e-09 24 25 2.02e-09 2.05e-09 2.01e-09 1.91e-09 1.79e-09 1.64e-09 1.50e-09 1.35e-09 22 Table 3b. Deuteron collisional excitation rate coefficients for He II (m3s-1). Temperature (eV) 10.87 17.23 27.31 43.29 68.61 108.74 136.90 Lower index Upper index 2 3 2.93e-11 2.62e-11 2.32e-11 2.05e-11 1.79e-11 1.56e-11 1.45e-11 3 4 2.61e-11 2.64e-11 2.59e-11 2.47e-11 2.32e-11 2.15e-11 2.06e-11 5 6 1.87e-10 1.67e-10 1.47e-10 1.29e-10 1.12e-10 9.66e-11 8.91e-11 5 7 1.05e-10 1.02e-10 9.74e-11 9.13e-11 8.44e-11 7.70e-11 7.32e-11 6 8 1.90e-10 1.89e-10 1.83e-10 1.75e-10 1.64e-10 1.51e-10 1.44e-10 8 9 1.49e-10 1.37e-10 1.26e-10 1.14e-10 1.02e-10 9.04e-11 8.47e-11 10 11 6.62e-10 5.88e-10 5.18e-10 4.50e-10 3.86e-10 3.27e-10 3.00e-10 10 12 4.88e-10 4.77e-10 4.56e-10 4.29e-10 3.96e-10 3.59e-10 3.40e-10 11 13 7.39e-10 7.30e-10 7.05e-10 6.67e-10 6.19e-10 5.64e-10 5.35e-10 12 15 4.24e-10 3.90e-10 3.54e-10 3.17e-10 2.81e-10 2.46e-10 2.29e-10 13 14 6.85e-10 6.36e-10 5.82e-10 5.26e-10 4.69e-10 4.12e-10 3.84e-10 14 16 4.96e-10 4.47e-10 3.99e-10 3.52e-10 3.07e-10 2.64e-10 2.44e-10 17 18 1.71e-09 1.51e-09 1.32e-09 1.13e-09 9.61e-10 8.05e-10 7.35e-10 17 19 1.48e-09 1.45e-09 1.38e-09 1.29e-09 1.18e-09 1.06e-09 9.97e-10 18 20 2.11e-09 2.07e-09 1.98e-09 1.86e-09 1.71e-09 1.54e-09 1.45e-09 19 21 1.60e-09 1.47e-09 1.33e-09 1.19e-09 1.05e-09 9.09e-10 8.41e-10 20 22 2.05e-09 1.90e-09 1.73e-09 1.55e-09 1.37e-09 1.19e-09 1.10e-09 21 23 1.10e-09 9.82e-10 8.69e-10 7.60e-10 6.56e-10 5.57e-10 5.11e-10 22 24 1.86e-09 1.67e-09 1.49e-09 1.31e-09 1.13e-09 9.61e-10 8.82e-10 24 25 1.21e-09 1.07e-09 9.32e-10 8.05e-10 6.86e-10 5.77e-10 5.27e-10 23 Table 4a. Triton collisional excitation rate coefficients for He II (m3s-1). Temperature (eV) 0.273 0.43 0.69 1.09 1.72 2.73 4.33 6.86 Lower index Upper index 2 3 3.06e-11 3.60e-11 3.97e-11 4.14e-11 4.13e-11 3.99e-11 3.74e-11 3.44e-11 3 4 2.17e-12 4.01e-12 6.51e-12 9.65e-12 1.33e-11 1.71e-11 2.07e-11 2.36e-11 5 6 2.52e-10 2.77e-10 2.89e-10 2.89e-10 2.79e-10 2.63e-10 2.43e-10 2.21e-10 5 7 1.96e-11 3.02e-11 4.32e-11 5.78e-11 7.28e-11 8.63e-11 9.67e-11 1.03e-10 6 8 2.89e-11 4.50e-11 6.56e-11 8.98e-11 1.16e-10 1.42e-10 1.64e-10 1.80e-10 8 9 7.69e-11 1.01e-10 1.25e-10 1.46e-10 1.60e-10 1.67e-10 1.68e-10 1.63e-10 10 11 1.02e-09 1.08e-09 1.09e-09 1.06e-09 1.01e-09 9.42e-10 8.66e-10 7.89e-10 10 12 1.13e-10 1.64e-10 2.25e-10 2.92e-10 3.58e-10 4.16e-10 4.59e-10 4.86e-10 11 13 1.50e-10 2.21e-10 3.07e-10 4.03e-10 5.02e-10 5.93e-10 6.66e-10 7.17e-10 12 15 3.04e-10 3.76e-10 4.39e-10 4.84e-10 5.08e-10 5.13e-10 5.01e-10 4.78e-10 13 14 4.05e-10 5.16e-10 6.19e-10 7.02e-10 7.56e-10 7.81e-10 7.79e-10 7.56e-10 14 16 5.54e-10 6.30e-10 6.79e-10 6.98e-10 6.90e-10 6.63e-10 6.23e-10 5.78e-10 17 18 2.85e-09 2.95e-09 2.92e-09 2.81e-09 2.65e-09 2.46e-09 2.26e-09 2.05e-09 17 19 3.98e-10 5.61e-10 7.49e-10 9.48e-10 1.14e-09 1.30e-09 1.42e-09 1.49e-09 18 20 5.13e-10 7.27e-10 9.80e-10 1.25e-09 1.52e-09 1.76e-09 1.95e-09 2.08e-09 19 21 1.16e-09 1.42e-09 1.63e-09 1.79e-09 1.87e-09 1.89e-09 1.86e-09 1.79e-09 20 22 1.35e-09 1.67e-09 1.96e-09 2.17e-09 2.31e-09 2.36e-09 2.35e-09 2.28e-09 21 23 1.44e-09 1.57e-09 1.64e-09 1.63e-09 1.59e-09 1.50e-09 1.40e-09 1.29e-09 22 24 2.09e-09 2.35e-09 2.51e-09 2.57e-09 2.54e-09 2.45e-09 2.32e-09 2.16e-09 24 25 1.99e-09 2.06e-09 2.05e-09 1.98e-09 1.87e-09 1.74e-09 1.59e-09 1.45e-09 24 Table 4b. Triton collisional excitation rate coefficients for He II (m3s-1). Temperature (eV) 10.87 17.23 27.31 43.29 68.61 108.74 136.90 Lower index Upper index 2 3 3.12e-11 2.80e-11 2.49e-11 2.21e-11 1.94e-11 1.69e-11 1.57e-11 3 4 2.56e-11 2.65e-11 2.64e-11 2.57e-11 2.44e-11 2.28e-11 2.20e-11 5 6 2.00e-10 1.79e-10 1.59e-10 1.40e-10 1.22e-10 1.06e-10 9.76e-11 5 7 1.06e-10 1.05e-10 1.01e-10 9.61e-11 8.98e-11 8.27e-11 7.89e-11 6 8 1.89e-10 1.92e-10 1.90e-10 1.83e-10 1.74e-10 1.61e-10 1.55e-10 8 9 1.55e-10 1.45e-10 1.34e-10 1.22e-10 1.10e-10 9.81e-11 9.22e-11 10 11 7.11e-10 6.35e-10 5.61e-10 4.91e-10 4.23e-10 3.60e-10 3.30e-10 10 12 4.97e-10 4.93e-10 4.78e-10 4.54e-10 4.23e-10 3.87e-10 3.68e-10 11 13 7.45e-10 7.50e-10 7.36e-10 7.06e-10 6.62e-10 6.08e-10 5.79e-10 12 15 4.48e-10 4.15e-10 3.79e-10 3.42e-10 3.05e-10 2.68e-10 2.50e-10 13 14 7.20e-10 6.75e-10 6.23e-10 5.67e-10 5.08e-10 4.49e-10 4.19e-10 14 16 5.29e-10 4.80e-10 4.31e-10 3.82e-10 3.35e-10 2.89e-10 2.67e-10 17 18 1.84e-09 1.64e-09 1.43e-09 1.24e-09 1.06e-09 8.86e-10 8.10e-10 17 19 1.52e-09 1.51e-09 1.45e-09 1.37e-09 1.27e-09 1.15e-09 1.08e-09 18 20 2.14e-09 2.14e-09 2.09e-09 1.98e-09 1.83e-09 1.66e-09 1.57e-09 19 21 1.69e-09 1.57e-09 1.43e-09 1.29e-09 1.14e-09 9.97e-10 9.24e-10 20 22 2.17e-09 2.02e-09 1.86e-09 1.68e-09 1.49e-09 1.30e-09 1.20e-09 21 23 1.18e-09 1.06e-09 9.43e-10 8.29e-10 7.18e-10 6.12e-10 5.62e-10 22 24 1.99e-09 1.80e-09 1.61e-09 1.42e-09 1.24e-09 1.06e-09 9.70e-10 24 25 1.30e-09 1.16e-09 1.02e-09 8.82e-10 7.54e-10 6.36e-10 5.81e-10 25 Table 5a. α-particle collisional excitation rate coefficients for He II (m3s-1). Temperature (eV) 0.273 0.43 0.69 1.09 1.72 2.73 4.33 6.86 Lower index Upper index 2 3 5.87e-11 7.78e-11 9.49e-11 1.08e-10 1.16e-10 1.19e-10 1.18e-10 1.12e-10 3 4 1.93e-12 4.41e-12 8.31e-12 1.39e-11 2.12e-11 3.02e-11 4.04e-11 5.11e-11 5 6 5.42e-10 6.56e-10 7.47e-10 8.06e-10 8.29e-10 8.21e-10 7.88e-10 7.41e-10 5 7 2.34e-11 4.08e-11 6.49e-11 9.58e-11 1.33e-10 1.74e-10 2.16e-10 2.52e-10 6 8 3.39e-11 5.90e-11 9.45e-11 1.42e-10 2.01e-10 2.71e-10 3.44e-10 4.15e-10 8 9 1.18e-10 1.74e-10 2.39e-10 3.08e-10 3.73e-10 4.26e-10 4.62e-10 4.80e-10 10 11 2.34e-09 2.71e-09 2.97e-09 3.10e-09 3.11e-09 3.02e-09 2.87e-09 2.69e-09 10 12 1.48e-10 2.36e-10 3.54e-10 5.03e-10 6.78e-10 8.66e-10 1.05e-09 1.21e-09 11 13 1.94e-10 3.10e-10 4.70e-10 6.75e-10 9.22e-10 1.20e-09 1.48e-09 1.73e-09 12 15 5.19e-10 7.11e-10 9.18e-10 1.12e-09 1.28e-09 1.40e-09 1.45e-09 1.46e-09 13 14 6.57e-10 9.23e-10 1.22e-09 1.53e-09 1.81e-09 2.03e-09 2.18e-09 2.24e-09 14 16 1.11e-09 1.39e-09 1.64e-09 1.84e-09 1.95e-09 1.99e-09 1.96e-09 1.88e-09 17 18 6.90e-09 7.75e-09 8.26e-09 8.44e-09 8.34e-09 8.04e-09 7.60e-09 7.08e-09 17 19 5.48e-10 8.38e-10 1.22e-09 1.69e-09 2.22e-09 2.79e-09 3.33e-09 3.80e-09 18 20 6.98e-10 1.07e-09 1.57e-09 2.19e-09 2.91e-09 3.69e-09 4.48e-09 5.19e-09 19 21 2.02e-09 2.72e-09 3.46e-09 4.16e-09 4.75e-09 5.17e-09 5.41e-09 5.49e-09 20 22 2.29e-09 3.12e-09 4.03e-09 4.92e-09 5.70e-09 6.30e-09 6.69e-09 6.87e-09 21 23 3.10e-09 3.72e-09 4.22e-09 4.55e-09 4.69e-09 4.67e-09 4.53e-09 4.31e-09 22 24 4.23e-09 5.24e-09 6.13e-09 6.80e-09 7.21e-09 7.37e-09 7.31e-09 7.08e-09 24 25 4.76e-09 5.38e-09 5.77e-09 5.92e-09 5.86e-09 5.65e-09 5.34e-09 4.97e-09 32 Figure 2. 33 Figure 3. 34 Figure 4. 35 Figure 5. 36 Figure 6. Figure 7. 37 Figure 8. Figure 9. 38 Figure 10. Figure 11. 39 Figure 12. Figure 13. 40 Figure 14. 41