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

Journal of Physics B: Atomic, Molecular and Optical Physics PAPER • OPEN ACCESS Synthetic spectra of BeH, BeD and BeT for emission modeling in JET plasmas To cite this article: D Darby-Lewis et al 2018 J. Phys. B: At. Mol. Opt. Phys. 51 185701 View the article online for updates and enhancements. Recent citations Theoretical study on the low-lying electronic excited states and laser cooling feasibility of AuH molecule Mao-Sen Xu et al - Vibrationally resolved electron impact electronic excitation of BeH D Darby-Lewis et al - R-matrix study for electron scattering of beryllium dihydride for fusion plasma Dhanoj Gupta et al - This content was downloaded from IP address 87.218.223.151 on 14/07/2020 at 20:20 Synthetic spectra of BeH, BeD and BeT for emission modeling in JET plasmas D Darby-Lewis 1 , J Tennyson 1,5 , K D Lawson 2 , S N Yurchenko 1 , M F Stamp 2 , A Shaw 2 , S Brezinsek 3 and JET Contributors 4 1 Department of Physics & Astronomy, University College London, Gower St.,London, WC1E 6BT, United Kingdom 2 UKAEA/CCFE, Culham Science Centre, Abingdon, OX14 3DB, United Kingdom 3 Institut für Energie und Klimaforschung—Plasmaphysik Forschungszentrum Jülich GmbH Wilhelm- Johnen-Straße, D-52428 Jülich, Germany E-mail: [email protected],[email protected] and [email protected] Received 28 March 2018, revised 26 June 2018 Accepted for publication 30 July 2018 Published 24 August 2018 Abstract A theoretical model for isotopologues of beryllium monohydride, BeH, BeD and BeT, A P 2to X S+2visible and X S+2to X S+2infrared 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 S+2,A P 2, and C S+2states. The energy levels from these calculations are then used in the program Duo to produce a potential energy curve for the ground state, X S 2, and to fit an improved potential energy curve for the first excited state, A P 2, 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. Supplementary material for this article is available online Keywords: Fusion, emission spectrum, beryllium monohydride (Some figures may appear in colour only in the online journal) 1. 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+2and A P 2states based on explicit potential energy curves (PECs), spin–orbit (LS) Journal of Physics B: Atomic, Molecular and Optical Physics J. Phys. B: At. Mol. Opt. Phys. 51 (2018)185701 (12pp)https://doi.org/10.1088/1361-6455/aad6d0 4 See the author list of ‘Litaudon et al 2017 Nucl. Fusion 57 102001’. 5 Author to whom any correspondence should be addressed. Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s)and the title of the work, journal citation and DOI. 0953-4075/18/185701+12$33.00 © 2018 IOP Publishing Ltd Printed in the UK1 and Λdoubling (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 (Tennyson and Yurchenko 2012)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. 2. 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 2and C S+2electronic 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 1illustrates the links between the steps of this process, each step is presented in detail below. 2.1. MARVEL MARVEL (Furtenbacher et al 2007, Furtenbacher and Császár 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 2and C S+2, with assigned quantum numbers taken from the literature (Shayesteh et al 2003,LeRoyet 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 1shows 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=+1uptov″=6 and some transitions for C–X with v″=0–2 and v′=6–10. [H2]Shayesteh et al (2003)infrared, rovibrational transitions, were duplicated for Σ=±0.5 giving 314 valid transitions. [D1]Le Roy et al (2006)contains A–X transitions Δv=0 up to v″=6, Δv=+1uptov″=5 and some transitions for C–X with v″=0 and v′=8–12. [D2]Shayesteh et al (2003)infrared, rovibrational transitions, were duplicated for Σ=±0.5 giving 328 valid transitions. [T1]Le Roy et al (2006)contains only A–X transitions Δv=0. A small portion of the MARVEL input file for BeH is shown in table 2where 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 2 J. Phys. B: At. Mol. Opt. Phys. 51 (2018)185701 D Darby-Lewis et al 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 (XS+2,A P 2or 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 and Herzberg 1979, Bernath 2005). Figure 2shows 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. 2.2. 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 Figure 1. Flow diagram showing the production of high accuracy synthetic spectra, starting in the top left corner with collated experimental transitions and resulting in, bottom center, synthetic spectra with the best fit rotational and vibrational temperatures. Table 1. Input transitions for MARVEL online. * See text for discussion on the comments. Isotope Tag Range (cm −1 )States Transitions Largest network Comments * BeH Le Roy et al (2006)15 132–20 822 A–X, C–X 1887 1264 [H1] Shayesteh et al (2003)1802–2239 X–X 160 [H2] BeD Le Roy et al (2006)15 164–20 619 A–X, C–X 2276 1495 [D1] Shayesteh et al (2003)1240–1680 X–X 167 [D2] BeT Le Roy et al (2006)19 824–20 424 A–X, Δv=0 only. 524 215 [T1] 3 J. Phys. B: At. Mol. Opt. Phys. 51 (2018)185701 D Darby-Lewis et al 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),isrefitted using energy levels output from MARVEL. A new PEC, of the extended Morse oscillator type, is fitted to the A P 2excited state using energy levels from output from MARVEL. The A P 2PEC 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 2on D, to 2 3on 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 3and 4show 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 ¹v0transitions 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 Table 2. Table of MARVEL input transitions for BeH with column labels and explanations. 1 2 3 4 5 6 7 8 9 10 111213 ˜ vD ˜ vState′v′¢+ () J1 2P′Σ′State″v′+ ( ) J1 2P″Σ″ID 20.3282 0.001 1 0 1 −−0.5 1 0 1 +0.5 MAGIC.001 15 132.42 0.1 3 0 13 +0.5 1 9 14 −0.5 LeRoy.00001 15 204.53 0.1 3 0 12 −0.5 1 9 13 +0.5 LeRoy.00002 15 224.66 0.1 3 0 22 −0.5 1 8 23 +0.5 LeRoy.00003 15 273.77 0.2 3 0 11 +0.5 1 9 12 −0.5 LeRoy.00004 15 335.3 0.1 3 0 21 +0.5 1 8 22 −0.5 LeRoy.00005 15 339.62 0.1 3 0 10 −0.5 1 9 11 +0.5 LeRoy.00006 15 401.44 0.1 3 0 9 +0.5 1 9 10 −0.5 LeRoy.00007 15 433.71 0.1 3 0 13 +0.5 1 9 12 −0.5 LeRoy.00008 15 458.91 0.1 3 0 8 −0.5 1 9 9 +0.5 LeRoy.00009 15 485.93 0.1 3 0 12 −0.5 1 9 11 +0.5 LeRoy.00010 Column Notation 1 ˜ vTransition frequency (cm −1 ) 2D ˜ vEstimated uncertainty in transition frequency (cm −1 ) 3 State′Initial electronic state, 1=XS + 2,2=AP 2,3=CS + 2 4v′Initial vibrational quantum number 5¢+ () J1 2Initial total angular momentum quantum number plus 0.5 6P′Initial parity quantum number 7Σ′Initial electron angular momentum quantum number 8 State″Final electronic state, 1=XS + 2,2=AP 2,3=CS + 2 9v″Final vibrational quantum number 10 + ( ) J1 2Final total angular momentum quantum number plus 0.5 11 P″Final parity quantum number 12 Σ″Final electron angular momentum quantum number 13 ID Unique ID for transition with source label and counting number 4 J. Phys. B: At. Mol. Opt. Phys. 51 (2018)185701 D Darby-Lewis et al outputs two files for the line list of each isotopologue, one containing the list of the states involved in the transitions and the other being a list of the transitions between states and their A-values; this corresponds to the format of the ExoMol database (Tennyson et al 2013,2016). The start of the states file and the start of the trans file for BeH are shown respectively in tables 5and 6followed 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). 3. Experiment 3.1. 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 4shows the experimental spectra of the BeH A– X transition as well as the best-fitting simulated spectrum. 3.2. 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. 4. Spectral analysis The final step in generating an assigned synthetic spectra is performed using a program called ExoCross (Yurchenko et al 2018). ExoCross produces cross-sections for the absorption or emission of photons by molecules. It uses rotational, vibrational and electronic temperatures to produce a statistical (Boltzmann)population model. The equation for Figure 2. Output rovibronic energy levels from MARVEL for the largest spectroscopic networks of BeH, BeD, and BeT, against nuclear rotational quantum number, N, times nuclear rotational quantum number plus one. The rovibrational states of X S + 2are the lower set and the AP 2rovibrational 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. 5 J. Phys. B: At. Mol. Opt. Phys. 51 (2018)185701 D Darby-Lewis et al Table 3. Comparison of excitation energies as a function of vibrational quanta, v, computed with Duo and obtained by MARVEL for: BeH and BeD X S + 2at J=0.5, Ω=0.5, parity=+; BeT X S + 2at J=2.5, Ω=0.5, parity=−. vXS 2 BeH BeD BeT Duo MARVEL Duo MARVEL Duo MARVEL 0 0 0 0 0 49.2756 50.3 1 1986.3054 1986.4169 1488.4401 1488.8472 1323.9046 2 3896.8004 3896.8707 2935.6045 2936.1953 2568.5846 3 5729.2711 5729.2613 4340.7423 4341.3802 3782.8768 4 7480.4545 7480.4219 5702.8860 4966.2478 5 9145.4221 7020.7372 6118.0189 6 10 716.6777 8292.5346 7237.3131 7 12 182.7098 9515.8791 8322.9914 8 13 525.5493 10 687.4899 9373.5704 9 14 716.2767 11 802.8549 10 387.1133 10 15 705.6510 12 855.7188 11 361.0802 11 16 402.4285 13 837.299 12 292.1239 12 16 664.9980 14 735.0110 13 175.8010 13 15 530.2285 14 006.1479 14 16 194.0499 14 775.0274 15 16 679.3277 15 471.0599 16 16 918.4202 16 077.7733 17 16 956.8145 16 570.2919 18 16 910.3608 19 17 056.2519 20 17 076.5405 Table 4. Comparison of excitation energies as a function of vibrational quanta, v, computed with Duo and obtained by MARVEL for: BeH and BeD A P + 2at J=1.5, Ω=0.5, parity=+; BeT A Π + at J=2.5, Ω=0.5, parity=−. vA 2 Π BeH BeD BeT Duo MARVEL Duo MARVEL Duo MARVEL 0 20 050.8587 20 092.2658 20 048.3803 20 071.3872 20 089.3230 20 090.7 1 22 056.7376 22 097.0590 21 552.7700 21 575.8314 21 378.1542 2 23 978.5785 24 017.3355 23 011.6534 23 034.2030 22 634.0613 3 25 813.4908 25 850.6970 24 423.8044 24 445.6549 23 856.2579 4 27 558.2734 27 594.5151 25 788.0151 25 043.9940 5 29 208.9935 29 243.2969 27 102.9758 26 196.4966 6 30 760.6586 30 793.4022 28 367.1779 27 312.9196 7 32 206.9278 29 578.8347 28 392.3027 8 33 539.8057 30 735.8129 29 433.5357 9 34 749.2511 31 835.5682 30 435.3273 10 35 822.6125 32 875.0780 31 396.1765 11 36 743.7529 33 850.7620 32 314.3426 12 37 491.6119 34 758.3833 33 187.8099 13 38 037.6829 35 592.9147 34 014.2541 14 38 341.2611 36 348.3556 34 790.9884 15 37 017.4704 35 514.9038 16 37 591.4032 36 182.3870 17 38 059.0891 36 789.2084 18 38 406.3136 37 330.3688 19 38 614.1568 37 799.8866 20 38 190.4670 21 38 493.0018 22 38 695.7499 23 38 779.4876 6 J. Phys. B: At. Mol. Opt. Phys. 51 (2018)185701 D Darby-Lewis et al the population of a state: P e,v,J where e=electronic state, v=vibrational quanta, and J=total angular momentum quantum number is =--- -==-== == () () () Pe e e 1 evJ,, EevJ EevJ kT EevJ Eev J kT Eev J kT ,, ,, 0 rot ,, 0 , 0, 0 vib ,0,0 ele in which the temperature dependence is split into three exponentials with exponents relating to the rotational, vibrational and electronic state temperatures respectively. The first term is a function of the rotational temperature, T rot , and pure rotational energy, -= ( )EE evJ evJ,, ,, 0, the second the vibrational temperature, T vib , and pure vibrational energy, - = ( EevJ,, 0 == )E ev J,0,0 , and the third the electronic state temperature, T ele , and the pure electronic state energy, == E ev J,0,0 . Einstein A-values, here provided by Duo, along with the populations are used to generate transition intensities, ¢¢¢ I ev ev,, ,, . ExoCross then uses line positions, also from Duo, and generates Gaussians with a width of the spectrometer’s instrument function, w≈1.5 Å(≈0.6 cm −1 ), and an area of the calculated transition intensity, ¢¢¢ I ev ev,, ,, . These individual Gaussians are summed, giving the emitted intensity at a given energy. 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, Figure 3. Fitted PECs for the X S + 2state and the A P 2state with vibrational energies at J=0.5 for BeH, BeD, and BeT. 7 J. Phys. B: At. Mol. Opt. Phys. 51 (2018)185701 D Darby-Lewis et al 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. 4.1. BeH analysis Figures 4and 6compare 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 Table 5. Section of the states file produced by Duo for BeH with column format explanation. 1234567 891011 nEmJP +/− P e/f State vΛΣ Ω 1 0 16 0.5 +eX2Sigma+0 0 0.5 0.5 2 1986.305 446 16 0.5 +eX2Sigma+1 0 0.5 0.5 3 3896.800 417 16 0.5 +eX2Sigma+2 0 0.5 0.5 4 5729.271 094 16 0.5 +eX2Sigma+3 0 0.5 0.5 5 7480.454 45 16 0.5 +eX2Sigma+4 0 0.5 0.5 6 9145.422 113 16 0.5 +eX2Sigma+5 0 0.5 0.5 7 10 716.677 675 16 0.5 +eX2Sigma+6 0 0.5 0.5 8 12 182.709 789 16 0.5 +eX2Sigma+7 0 0.5 0.5 9 13 525.549 349 16 0.5 +eX2Sigma+8 0 0.5 0.5 10 14 716.276 652 16 0.5 +eX2Sigma+9 0 0.5 0.5 Column Notation 1nRovibronic counting number 2EEnergy of rovibronic state relative to ground state (cm −1 ) 3mmultiplicity, including nuclear spin degeneracy 4JTotal angular momentum quantum number 5P +/− Parity in +/−notation 6P e/f Parity in e/fnotation 7 State Electronic state 8vVibrational quantum number 9ΛProjection of electronic orbital angular momentum quantum number 10 ΣProjection of electron spin angular momentum quantum number 11 ΩProjection of total electronic angular momentum quantum number Table 6. Section of the trans file produced by Duo for BeH with column format explanation. 12 3 4 n″n′A ˜ v 71 1 4.58E-10 16 933.174 781 89 1 5.63E-02 38 037.683 830 65 1 8.17E-11 16 699.528 089 58 1 2.58E-07 12 198.141 744 95 1 6.48E-03 38 492.659 618 83 1 1.95E-01 32 206.945 906 51 1 1.12E-09 20.326 390 Column Notation 1n′Initial rovibronic state number, see table 5 2n″Final rovibronic state number, see table 5 3AEisenstein A-coefficient (s −1 ) 4 ˜ vTransition wavenumber (cm −1 ) 8 J. Phys. B: At. Mol. Opt. Phys. 51 (2018)185701 D Darby-Lewis et al