Synthetic spectra of BeH, BeD and BeT for emission modeling in JET plasmas

A theoretical model for isotopologues of beryllium monohydride, BeH, BeD and BeT, A 2 Π to X 2 Σ + visible and X 2 Σ + to X 2 Σ + infrared rovibronic spectra is presented. The MARVEL procedure is used to compute empirical rovibronic energy levels for BeH, BeD and BeT, using experimental transition data for the X 2 Σ + , A 2 Π , and C 2 Σ + states. The energy levels from these calculations are then used in the program Duo to produce a potential energy curve for the ground state, X 2 Σ , and to fit an improved potential energy curve for the first excited state, A 2 Π , including a spin–orbit coupling term, a Λ-doubling state to state (A–X states) coupling term, and Born–Oppenheimer breakdown terms for both curves. These, along with a previously computed ab initio dipole curve for the X and A states are used to generate vibrational-rotational wavefunctions, transition energies and A-values. From the transition energies and Einstein coefficients, accurate assigned synthetic spectra for BeH and its isotopologues are obtained at given rotational and vibrational temperatures. The BeH spectrum is compared with a high resolution hollow-cathode lamp spectrum and the BeD spectrum with high resolution spectra from JET giving effective vibrational and rotational temperatures. Full A–X and X–X line lists are given for BeH, BeD and BeT and provided as supplementary data on the ExoMol website.


Introduction
In order to predict the erosion, migration and re-deposition of the Be first wall in fusion devices such as JET, and in the future ITER, and in view of impurity production and lifetime of components, an understanding of the release and transport of Be is an essential requirement. BeT is also important for modeling tritium retention in beryllium containing fusion devices. BeD X release was shown to contribute more than 50% to the total erosion in certain cases in JET D plasmas in a limiter configuration deduced from BeD emission spectra (Brezinsek et al 2014). Detailed studies of molecular spectra (Duxbury et al 1998), such as those for BeH, BeD, and BeT A-X bands can provide valuable input to codes used for erosion modeling such as ERO (Borodin et al 2011, Lasa et al 2018. To this end, we develop a full spectroscopic model for BeH, BeD and BeT X S + and Λ doubling (L + ) couplings, Born-Oppenheimer breakdown (BOB) terms, and ab initio dipole curves. The PECs, couplings, and BOB terms are derived using accurate experimentally recorded transition frequencies. A previous PEC fitting for the X state by Le Roy et al (2006) has been improved upon by Dattani (2015), and further refined in this work by fitting procedures involving the X-A state transitions. The A state PEC was previously studied by Le Roy et al (2006) and is also improved upon here. Accurate ab initio calculations for the X state were performed by Koput (2011). The fitting in Duo (Yurchenko et al 2018) results in a full set of accurate transitions for BeH, BeD, and BeT from a single set of PECs, couplings, and BOB terms (Le Roy 2017). These data are used to determine temperatures in the experimental spectra. This method of using variational nuclear motion calculations with BOB terms to link isotopologue data differs from the methods based on perturbation theory generally employed up until now, see Duxbury et al (1998) and Hornkohl et al (1991). These older methods use spectroscopic constants to calculate energies but cannot join isotopologue data and thus do not provide predictions for yet-to-be observed isotopologues. Our approach is similar in spirit to that adopted by Parigger et al (2015) to model laser ablation and by McGuire et al (2016) to model plasma ablation.
Our line lists for BeH, BeD and BeT improve on those of Yadin et al (2012) produced as part of the ExoMol project  both in having greater range and improved accuracy in the transition frequencies. The present paper aims to give a more accurate fit to the spectrum of the BeD A-X transition than was achieved by Duxbury et al (1998) and Björkas et al (2013) by using more accurate transition data and allowing separated treatment of vibrational and rotational temperatures. The use of different rotational and vibrational temperatures, which implies there is no local thermodynamic equilibrium (non-LTE), is expected to lead to a more accurate description of the experimental spectrum.
We compare theoretically produced synthesized spectra to new experimental BeD spectra recorded on JET and BeH spectra recorded in Be hollow-cathode discharges in Forschungszentrum Jülich. Both of these spectra were recorded employing spectrometers with high spectral resolution in the visible range, sufficient to resolve the rotational lines.

Theory
There are four main steps in generating assigned synthetic spectra fitted with rotational and vibrational temperatures. Experimental transitions are inverted to give vibronic energy levels using the online implementation of MARVEL (measured active rotation vibration energy levels). These are then used in Duo to fit PECs which can accurately reproduce all the empirical energy levels. Empirical energy levels belonging to the X S + 2 , A P 2 and C S + 2 electronic states are provided by our MARVEL treatment, although PECs were fitted only for the X and A states. This is because of a lack of transition data for the C state, which in any case does not give rise to a significant feature in the JET emission spectrum. To date, the molecular spectrum is dominated by the A-X band. Duo uses an ab initio X state to A state transition dipole (Pitarch-Ruiz et al 2008) to produce Einstein A-coefficients for the observed X to A transitions. The output from Duo contains all transitions between states within a given wavenumber range (parameter in the Duo input, see supplementary data available online at stacks.iop.org/JPB/51/185701/mmedia) and all the Einstein A-coefficients associated with those energy levels. These are used by ExoCross to generate synthetic spectra with varying rotational and vibrational temperatures. These spectra are compared to experimental spectra to obtain a metric for the fit. The flowchart in figure 1 illustrates the links between the steps of this process, each step is presented in detail below.

MARVEL
MARVEL (Furtenbacher et al 2007, Furtenbacher andCsászár 2012) is a program with an online user interface which takes highly accurate experimental transition energies and calculates spectroscopic networks of energy levels. For BeH, BeD and BeT we used transitions involving X S + 2 , A P 2 and C S + 2 , with assigned quantum numbers taken from the literature (Shayesteh et al 2003, Le Roy et al 2006. In particular Le Roy et al (2006) give transition data compiled from many sources including Shayesteh et al (2003), Colin et al (1983), Focsa et al (1998) and De Greef and Colin (1974). Table 1 shows the transition data sources used for input to MARVEL. Comments on individual sources are as follows: [H1] Le Roy et al (2006) contains A-X transitions Δv=0 up to v″=6, Δv=+1 up to v″=6 and some transitions for C-X with v″=0-2 and v′=6-10.
A small portion of the MARVEL input file for BeH is shown in table 2 where the column format is explained. The number of quantum numbers used for assignment here is 5. The thresholds for changing uncertainties and for deletion were both set to 3. The full files are given in the supplementary data.
These transitions are run through the program and various unlinked spectroscopic networks of energy levels are generated. Separate networks can be joined with 'linking' transitions, e.g. joining the degenerate spin up and spin down states of the ground state with a 'transition' of zero energy from one degenerate state to another. The result of this process is to give a large set of accurate energy levels with quantum number assignments. For BeH and BeD the infrared data of Shayesteh et al (2003) brings together separate vibrational networks. This results in two large networks separated along the quantum number Σ, where Σ=−0.5 for one network and Σ=0.5 for the other. These two networks are joined with a linking transition with a 'magic' wavenumber, between states with opposing spin, as shown for BeH by the transition labeled MAGIC.001 in table 2. This magic transition is an artificial transition calculated from empirical, effective Hamiltonian energy levels belonging to separate networks. The frequency of the magic transition is calculated to produce degeneracy between the states with differing spin at low J.
The 'good' quantum numbers in this application are the total rotational quantum number (J) and total parity (+ or −). Quantum numbers of operators not strictly conserved in this application can still be used in state assignment. They are electronic state (X S + 2 , A P 2 or C S + 2 ), vibrational quantum number (v), nuclear rotational angular momentum (N), projection of total spin angular momentum (Σ), projection of total orbital angular momentum (Λ), and the projection of total electronic angular momentum Ω=Λ+Σ. This places our representation in Hund's case b (Huber andHerzberg 1979, Bernath 2005). Figure 2 shows the rovibrational energy levels of the largest component network for BeH, BeD and BeT. The almost straight lines with isotopologuedependent gradients are vibrational bands and the gradients depend on the rotational constants (∝ 1/reduced mass). Since the electronic states being represented here are doublets each point for an energy level shown in this figure actually corresponds to two spin degenerate states.

Duo
Duo (Yurchenko et al 2016) is a fully-coupled rovibronic nuclear motion code which generates rovibronic energy levels and wavefunctions for diatomic molecules from PECs, couplings between PECs, and Born-Oppenheimer correction terms This program contains an iterative fitting procedure  where PECs and other terms can be fitted to experimental data. In this way the potential form of the ground state PEC, a Morse long range type potential (Le Roy et al 2011) as given by Pitarch-Ruiz et al (2008), is refitted using energy levels output from MARVEL. A new PEC, of the extended Morse oscillator type, is fitted to the A P 2 excited state using energy levels from output from MARVEL. The A P 2 PEC is modified by the addition of an LS coupling curve and an X to A state L + (Λ-doubling) coupling curve. Both the ground and excited state PECs are modified by the fitting of adiabatic and non-adiabatic BOB curves (Le Roy 2017) using the BeD and BeT isotopologue energy level data from MARVEL. Using BOB correction terms makes these data applicable to all isotopologues of BeH. This allows data from all three isotopologues to improve the fit from the same set of PECs and coupling terms. The fitting of BOB terms is achieved using an iterative process in Duo. The non-adiabatic BOB term is an additive correction to the PEC which has an increasing effect with increasing isotope mass (from zero effect on H, to 1 2 on D, to 2 3 on T). However due to the limited data for BeT it is initially fitted to the BeD data and adjusted iteratively for BeT. The adiabatic BOB term is a multiplicative factor applied to the centrifugal potential and has greatest effect on the lighter isotope; it is therefore initially fitted to BeH and adjusted to model BeD and BeT. When a suitable fit is produced for both states using the available energy levels from MARVEL, Duo allows fittings to be performed using the transitions. This allows a larger set of data to be used in the fitting routine, as some of the energy levels output by MARVEL cannot be connected to the main spectroscopic network. Tables 3 and 4 show some of the energy levels generated by Duo for BeH, BeD and BeT for both the X and A states, they are each followed by the energies from MARVEL used in the fitting for those levels. As shown earlier, in table 1, only the v=0 component of BeT joins the main network of transitions, as there are no measured D ¹ v 0 transitions for BeT. Overall the root mean square of the fit for each isotopologue is: BeH, 0.542 cm −1 ; BeD, 0.614 cm −1 ; BeT, 0.384 cm −1 ; for low-lying states much higher accuracies were achieved. PECs modified and fit by Duo for the X and A states of BeH, BeD and BeT are shown in figure 3. The zero point energy of these curves are: 1022.0292 cm −1 for BeH; 742.7911 cm −1 for BeD; 626.2844 cm −1 for BeT. The Duo input containing the PECs, couplings and BOB terms is given for BeH, BeD and BeT in the supplementary data. Predissociation is not included in this model.
Fully fitted PECs, couplings and BOB terms are used in conjunction with an ab initio transition dipole curve (Pitarch-Ruiz et al 2008) for the A to X state to generate Einstein A-coefficients for the rovibronic transitions. Duo  (Tennyson et al 2013. The start of the states file and the start of the trans file for BeH are shown respectively in tables 5 and 6 followed by an explanation of their column formats. The entirety of the line list states and trans files for each isotopologue can be found in the supplementary data and on the ExoMol website (www.exomol.com).

BeH hollow-cathode discharge spectrum
Experimental BeH spectra were recorded using a high resolution visible spectrometer, in cross-dispersion arrangement (grating and prism) covering the spectral range between 373 and 680 nm simultaneously in more than 50 orders with an almost constant resolving power of l/DL ≈200 00, as show by Brezinsek et al (2008). The spectral source was a beryllium hollow-cathode discharge lamp with a neon/hydrogen mixture as a working gas. Inside the lamp, the metallic Be target plate is biased, resulting in it being bombarded by the impurity ions with Be being sputtered, either as Be or BeH X . The current and voltage can be varied changing the plasma characteristics as well as the impact energy of the impinging ions. The released Be or BeH is then excited by electron impact leading to the emission of Be I, Be II as well as BeH light. Figure 4 shows the experimental spectra of the BeH A-X transition as well as the best-fitting simulated spectrum.

BeD JET edge emission spectra
An experimental BeD A-X spectrum was measured in four consecutive JET 2.4 T, 2.0 MA discharges with comparable conditions during the limiter phase of the pulse at 4.7-5.1 s. During this time, the plasma was limited by the inner poloidal limiter and BeD radiation was emitted from this region. It was recorded with a high resolution visible spectrometer (KS3), which had a Czerny-Turner arrangement and directly observed the low density edge plasma close to the inner poloidal limiter. The spectral wavelength range of the spectrometer does not cover the whole spectrum and so four consecutive discharges were chosen to form a joined image, see figure 5, with comparable plasma conditions as well as assumed comparable rovibrational populations.

Spectral analysis
The final step in generating an assigned synthetic spectra is performed using a program called ExoCross ( are the lower set and the A P 2 rovibrational states start at 20 000 cm −1 . Each of the energy levels represented here actually corresponds to two spin degenerate energy levels as both the X and the A states are doublets.  In taking different rotational, vibrational and electronic temperatures ExoCross allows us to more accurately fit non-LTE spectra. This procedure is useful in the case where LTE has not been reached by the molecule producing the spectra. In such an instance the different spacing between electronic, vibrational, and rotational energy levels means that they adapt to changing temperatures and plasma conditions at different rates. In the cases discussed here the necessary time to reach equilibrium is too long and the plasma density too low for LTE conditions. The ExoCross calculation is repeated for rotational and vibrational temperatures varying independently from 500 to 10000 K. For each combination of vibrational and rotational temperatures a program 'diffspec' is used to integrate the area under both curves, the overlap, and the area between the curves, the difference. The metric minimized to find the best fit temperature is the difference divided by the overlap. The theoretical and experimental spectra are both normalized, in the first instance, to have Q-branch peak values of 1.0. A multiplicative factor is then required to allow the peak values to differ. The experimental spectrum also contains a background which must be removed before matching to the theoretical spectrum, which necessitates a threshold. Also a higher weighting should be placed on fitting to the more intense parts of the experimental spectra which have the smallest experimental uncertainties in measurement. To this end, there are three inputs which adjust the intensity to control the nature of the fitting: factors, background thresholds and weight. The factors are applied to the synthetic spectrum by multiplying the intensity by a series of factors, here 0.8-1.2 in steps of 0.05. This allows the normalized spectra of arbitrary intensities to have different maximum values with respect to one another. For each factor thresholds are applied to the experimental spectra, moving each one up or down by an amount which varies from 0.0 to 0.1 in steps of 0.01. This allows the background in the experimental spectrum to be accounted for even if it differs between spectra recorded in different discharges. The last control input is a weight, allowing a higher weighting to be given to the higher intensity portions of the spectrum for both the difference and the overlap. This is achieved by weighting each intensity by a power, here a weighting of 2.0 is used, the area effectively being squared. This ensures a sensible relation between the uncertainty in an intensity measurement and the magnitude of that intensity. There are also input parameters which are set to ignore certain, polluted, regions of the spectra from the area summation.

BeH analysis
Figures 4 and 6 compare experimental BeH spectra with the theoretical spectra generated by the computational methods described in section 2. An invasive H-atom line, H β , is marked in figure 4; this region was excluded from the temperature fitting procedure in diffspec as described above. The best fit temperature for this spectra has T rot = (540±70) K and T vib =(3300±300) K meaning that the emission is from a  J Total angular momentum quantum number 5 P +/− Parity in +/− notation 6 P e/f Parity in e/f notation 7 State Electronic state 8 v Vibrational quantum number 9 Λ Projection of electronic orbital angular momentum quantum number 10 Σ Projection of electron spin angular momentum quantum number 11 Ω Projection of total electronic angular momentum quantum number non-LTE plasma. The fitting metric for this combination of rotational and vibrational temperatures=0.208. Uncertainties were obtained from the greatest variation in temperature within 10% increase of the metric. The extreme difference between the vibrational and rotational temperatures is a product of the method by which the spectrum was produced. The need for a much higher vibrational temperature is illustrated by the presence of the 1-1 vibrational band head around 20050 cm −1 in figure 4 which is absent for T vib = T rot =540 K. This would be the spectrum of BeH in LTE at 540 K and it clearly shows all of the primary peaks, that is all the transitions from fundamental vibrational quanta v′=0 to v″=0. What is missing are all the lower intensity peaks which are produced by transitions from higher vibrational states. When the vibrational temperature is brought up to the best fit temperature of 3300 K the lower intensity, higher vibrational, components of the spectrum are brought sharply into alignment with the experimentally observed spectrum. The degree of matching to the experimental spectra is highlighted by the close-up view of the R-branch shown in figure 6. The transition assignment labels here show that the vibrational quanta v′=0 to v″=0 transitions are more intense and those of higher vibrational quanta are lower in intensity. The assignments show in order left to right: v′=upper state vibrational quantum number, N′=upper state nuclear rotational quantum number, v″=lower state vibrational quantum number, N″=lower state nuclear rotational quantum number. The heights of the transition lines in this figure are proportional to the A-values of the transitions not to the transition intensities. Hence, these lines do not necessarily correspond one to one to the height of the peaks in the synthetic spectrum which are dependent on temperature based populations, see equation (1), as well as A-values.

BeD analysis
A match was made between the experimental BeD spectrum from JET and a theoretical spectrum by varying the vibrational and rotational temperatures. Figure 5 shows an assigned synthetic spectrum generated at T rot =(3800±700) K and T vib =(4700±800) K using our BeD line list. The fitting metric for this combination of rotational and vibrational Measured BeH spectrum, shown in red, was recorded with a high resolution visible spectrometer from a hydrogen doped lamp with a beryllium target. Assigned synthetic spectrum of BeH, in black, is generated with T rot =540 K and T vib =3300 K using our BeH line list. Figure 5. Measured BeD spectra, in red, green, blue and orange lines with pulse numbers and times in the legend, were recorded with a high resolution spectrometer, T e ≈30 eV and n e ≈10 −18 m −3 . Assigned synthetic spectrum, in black, is generated with T rot =3800 K and T vib =4700 K using our BeD line list. temperatures=0.254. Uncertainties were obtained from the greatest variation in temperature within 10% increase of the metric. This combination of experimental BeD spectra has also been fitted assuming LTE (T rot =T vib ). This gave a result of T rot =T vib =(4300±600) K with the fitting metric=0.274. This temperature fitting was also repeated while excluding the under-fitted vibrational 0-0 band head for both LTE and non-LTE assumptions. This gave a result of T rot = T vib = (4400 ± 600) K for LTE with the fitting metric = 0.252 and T rot = (4100±700) K and T vib =(4700±800) K for non-LTE with the fitting metric=0.236.
Measured BeD spectra were recorded during the limiter phase of discharges 92 493-92 496. The vibrational and rotational temperatures fitted to the joined spectrum for BeD are much closer than in the BeH spectrum above, meaning the plasma conditions were much closer to LTE. There are four features, around ≈20 000 cm −1 in the JET spectra which are not reproduced in the synthetic spectrum. These are invasive features from other species, two being impurity lines of remaining nitrogen in the plasma. Figure 7 shows a close-up section of the Q-branch with black drop lines at every transition energy in the region. This figure demonstrates the high degree of accuracy present across the range of these calculations.
The paper by Duxbury et al (1998) shows fittings for several molecular features in JET spectra including a BeD spectrum of the A to X transition. They fit a synthetic spectrum to an observed spectrum, which is generated using molecular constants. These constants are only valid for each isotopologue individually. By visual comparison, our work shows an improvement in line positions and in intensities. Figure 8 shows a predicted synthetic spectrum of the A-X transition of BeT. The rotational and vibrational temperatures used to generate this spectrum are those found for BeD in the  JET discharges discussed before. This is the BeT rovibronic spectrum expected to be observed in JET during a pure tritium campaign in discharges similar to those in which the BeD spectra were observed. The degree of accuracy in the results for BeT, and any isotopologue of BeH, is expected to be as seen in figure 6 for BeD. This will be compared with future JET and ITER spectra with their D/T fuel mix.

Conclusions
We construct line lists for the species BeH, BeD and BeT by using a mixture of fits to empirical energy levels to obtain PECs, coupling terms and beyond Born-Oppenheimer corrections. We conclude from a comparison of a synthetic spectrum fitted to experimental BeH and BeD spectra that our line lists reproduce the spectra of BeH, BeD and BeT to good accuracy. This accuracy is shown in figures 6 and 7 and is assured for BeT by the isotopologue consistency of our model. The comparisons show discrepancies in the intensities, for example, in the case of the BeD spectrum mostly seen in the Q-branch v=0-0 band head. In the synthetic spectrum, the rotational temperature is too high to match this band head, which increases in relative intensity with decreasing rotational temperature. There are two possible explanations for this issue: firstly the theoretical model uses a statistical population model, which assumes a thermal equilibrium; the second possibility is that the line of sight for the experimental spectrum passes through different temperature regions, all radiating and giving a cumulative result. The first of these issues will be addressed in a future publication (Darby-Lewis and Tennyson 2018) by introducing a full collisional-radiative population model utilizing vibrationally averaged R-matrix results from the calculations of Darby-Lewis et al (2017) extended over bond lengths. The issue with the line of sight can be solved with a line of sight integration calculation applied to the results of the full population model. A possible third explanation is self absorption but given the very thin (less than 1 cm) layer of plasma containing BeD in the limiter region observed this is not thought to be able to significantly contribute to the effect seen on the 0-0 vibrational band head, this assumption will be tested in future work when we include a full collisional-radiative population model.
Finally, in a comparison with the previous work on BeD spectra modeling at JET by Duxbury et al (1998), we conclude that a more accurate and more comprehensive model has been achieved for two reasons. Firstly, the accuracy of the transition frequency fitting is much increased in our work. Secondly, our model is built using a single dataset derived from all the experimental transition data to model the three isotopologues. Not only does this improve the accuracy of the model, but it also enables accurate predictions to be made for the sparsely observed BeT isotopologue.