Low-energy collisions between electrons and BeH^+: cross sections and rate coefficients for all the vibrational states of the ion
S. Niyonzima, S. Ilie, N. Pop, J.Z. Mezei, K. Chakrabarti, V. Morel, B. Peres, D.A. Little, K. Hassouni, ?. Larson, A.E. Orel, D. Benredjem, A. Bultel, J. Tennyson, D. Reiter, I.F. Schneider
LLow-energy collisions between electrons and BeH + :cross sections and rate coefficients for all the vibrational states of the ion S. Niyonzima a,b , S. Ilie a,c , N. Pop a,c , J. Zs. Mezei a,d,e,f , K. Chakrabarti g , V. Morel h , B. Peres h , D. A. Little i , K.Hassouni d , ˚A. Larson j , A. E. Orel k , D. Benredjem e , A. Bultel h , J. Tennyson i , D. Reiter l , I. F. Schneider a,e, ∗ a Laboratoire Ondes et Milieux Complexes CNRS − Universit´e du Havre − Normandie Universit´e, 76058 Le Havre, France b D´epartement de Physique, Facult´e des Sciences, Universit´e du Burundi, B.P. 2700 Bujumbura, Burundi c Fundamental of Physics for Engineers Department, Politehnica University Timisoara, 300223 Timisoara, Romania d Laboratoire des Sciences des Proc´ed´es et des Mat´eriaux, CNRS − Universit´e Paris 13 − USPC, 93430 Villetaneuse, France e Laboratoire Aim´e-Cotton, CNRS − Universit´e Paris-Sud − ENS Cachan − Universit´e Paris-Saclay, 91405 Orsay, France f Institute of Nuclear Research, Hungarian Academy of Sciences, Debrecen, Hungary g Department of Mathematics, Scottish Church College, Calcutta 700 006, India h CORIA CNRS − Universit´e de Rouen − Universit´e Normandie, F-76801 Saint-Etienne du Rouvray, France i Department of Physics and Astronomy, University College London, WC1E 6BT London, UK j Department of Physics, Stockholm University, AlbaNova University Center, S-106 91 Stockholm, Sweden k Department of Chemical Engineering and Materials Science, University of California, Davis, California 95616, USA l Institute of Energy and Climate Research-Plasma Physics, Forschungszentrum J¨ulich GmbH Association EURATOM-FZJ, Partner inTrilateral Cluster, 52425 J¨ulich, Germany
Abstract
We provide cross sections and Maxwell rate coefficients for reactive collisions of slow electrons with BeH + ions on all theeighteen vibrational levels ( X Σ + , v + i = 0 , , , . . . ,
17) using a Multichannel Quantum Defect Theory (MQDT) - typeapproach. These data on dissociative recombination, vibrational excitation and vibrational de-excitation are relevant formagnetic confinement fusion edge plasma modelling and spectroscopy, in devices with beryllium based main chambermaterials, such as the International Thermonuclear Experimental Reactor (ITER) and the Joint European Torus (JET).Our results are presented in graphical form and as fitted analytical functions, the parameters of which are organized intables.
Keywords:
Plasma-wall interaction; electron-impact processes; multi-channel; dissociative recombination; vibrationalexcitation; cross sections and rates. ∗ Corresponding author.
Email address: [email protected] (I. F. Schneider)
Preprint submitted to Elsevier November 8, 2018 a r X i v : . [ phy s i c s . p l a s m - ph ] J a n ontents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42. Theoretical approach of the dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Explanation of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Explanation of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Tables1. List of the parameters used in Eq.(10) for the DR Maxwell rate coefficients of BeH + ( v + i = 0 , , , , ..., + ( v + i = 0 and 1)represented in the upper panels of Graph 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203. List of the parameters used in Eq.(10) for the VE and VdE Maxwell rate coefficients of BeH + ( v + i = 2 and 3)represented in the upper panels of Graph 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214. List of the parameters used in Eq.(10) for the VE and VdE Maxwell rate coefficients of BeH + ( v + i = 4 and 5)represented in the upper panels of Graph 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225. List of the parameters used in Eq.(10) for the VE and VdE Maxwell rate coefficients of BeH + ( v + i = 6 and 7)represented in the upper panels of Graph 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236. List of the parameters used in Eq.(10) for the VE and VdE Maxwell rate coefficients of BeH + ( v + i = 8 and 9)represented in the upper panels of Graph 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247. List of the parameters used in Eq. (10) for the VE and VdE Maxwell rate coefficients of BeH + ( v + i = 10 and 11)represented in the upper panels of Graph. 5.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258. List of the parameters used in Eq.(10) for the VE and VdE Maxwell rate coefficients of BeH + ( v + i = 12 and 13)represented in the upper panels of Graph 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269. List of the parameters used in Eq.(10) for the VE and VdE Maxwell rate coefficients of BeH + ( v + i = 14 and 15)represented in the upper panels of Graph 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710. List of the parameters used in Eq.(10) for the VE and VdE Maxwell rate coefficients of BeH + ( v + i = 16 and 17)represented in the upper panels of Graph 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Graphs1. Dissociative recombination cross sections of ground ( v + i = 0) and excited ( v + i = 1 , ...,
5) BeH + in its electronicground state. Direct mechanism: dashed thick line, total (direct and indirect) mechanism: continuous thin line. 112. Dissociative recombination cross sections of excited ( v + i = 6 , , ...,
11) BeH + in its electronic ground state. Directmechanism: dashed thick line, total (direct and indirect) mechanism: continuous thin line. . . . . . . . . . . . . . . . . . 123. Dissociative recombination cross sections of excited ( v + i = 12 , , ...,
17) BeH + in its electronic ground state.Direct mechanism: dashed thick line, total (direct and indirect) mechanism: continuos thin line. . . . . . . . . . . . 134. Dissociative recombination (DR, thick line), vibrational excitation (VE, thin lines) and vibrational de-excitation(VdE, symbols and thick lines) Maxwell rate coefficients of ground ( v + i = 0) and excited ( v + i = 1 , ...,
5) BeH + in its electronic ground state (total mechanism). For VE, since the rate coefficients decrease monotonically withthe excitation, the lowest final vibrational quantum number of the target is indicated only, and the lower panelsextend the range down to 10 − cm /s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 . Dissociative recombination (DR, thick line), vibrational excitation (VE, thin lines) and vibrational de-excitation(VdE, symbols and thick lines) Maxwell rate coefficients of excited ( v + i = 7 , , ...,
11) BeH + in its electronicground state (total mechanism). For VE, since the rate coefficients decrease monotonically with the excitation,the lowest final vibrational quantum number of the target is indicated only, and the lower panels extend therange down to 10 − cm /s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156. Dissociative recombination (DR, thick line), vibrational excitation (VE, thin lines) and vibrational de-excitation(VdE, symbols and thick lines) Maxwell rate coefficients of excited ( v + i = 12 , , ...,
17) BeH + in its electronicground state (total mechanism). For VE, since the rate coefficients decrease monotonically with the excitation,the lowest final vibrational quantum number of the target is indicated only, and the lower panels extend therange down to 10 − cm /s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167. Global (sum over all the possible final states) Maxwell rate coefficients for dissociative recombination (DR),vibrational excitation (VE) and vibrational de-excitation (VdE). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 . Introduction Even though various isotopes of light elements can be coupled to achieve thermonuclear fusion energy release, thenext generation of thermonuclear fusion reactors will use deuterium-tritium (D-T) reactions, by far most efficient andaccessible plasma fuel for fusion reactors and power plants. As discussed in detail elsewhere [1, 2, 3, 4, 5, 6], beryllium(Be) is meant to enter the composition of the wall of the future fusion devices (ITER). Its performance on preventingtritium retention and, meanwhile, still keeping the benefits of a low Z material (low fuel dilution), is currently beingtested in the JET [3, 4]. According to current plans, tungsten (W) will be the plasma facing material in the high heatflux components (the entire divertor). These materials (Be and W) are expected to provide sufficiently low fuel retention,plasma impurity levels, neutron damage, and sufficient heat removal capabilities in the divertor and then, meet ITERrequirements [2]. The key challenge in the use of beryllium as main chamber material for experimental and commercialfusion devices is to understand, predict and control the characteristics of the thermonuclear burning plasma, the plasmaedge regimes that result in acceptable erosion performance and the divertor plasma (heat and particle exhaust, impuritycontrol, lifetime).Due the the low mass ratio between beryllium and the D, T plasma fuel ions, beryllium erodes rather easily underplasma exposure by physical and chemical sputtering, a process which releases Be, Be + , and other impurities into theplasma. Significant fractions of the eroded beryllium will be transported towards the divertor and will form compoundswith the fuel atoms, molecules and/or molecular ions. Therefore, BeH as well as BeD and BeT molecules are expectedto appear in a significant (spectroscopically detectable) amount in the edge and divertor plasmas. Various sourcemechanisms may lead to their formation, either surface or volumetric (particle rearrangement) processes. The particleinterchange reaction [7], Be + + X −→ BeX + X + , (1)where X denotes one of the fuel atoms H, D or T, was suggested as one, possibly dominant, volumetric BeX formationchannel, when X is in its vibrational ground state. However, the exo- or endothermicity will greatly depend upon thevibrational state of X and other channels, including electron transfer channels, may also be open for BeX formationin fusion divertor plasma conditions. The involved particles - atoms, molecules and molecular ions - follow their oftenquite complex transport pathways in the edge plasma and take part in the complex reactions determining the plasmacomposition. Its detailed modeling by taking into account reactions between all present species is necessary, first of all,for interpretation of molecular and atomic line spectroscopy and also to understand and predict the overall plasma edgedynamics and the divertor region behavior in particular.In principle, the rate of beryllium erosion in fusion devices can be measured by spectroscopical techniques of allthe states of the atoms and molecules, so primarily of Be, Be + , Be +2 , BeX, BeX + , BeX , BeX +2 . In order to provide aquantitative interpretation of such spectroscopic measurements, one needs a complete set of rate coefficients for excitation,ionization and the various atomic and molecular ions break-up reactions. The inelastic electron-impact processes ofvibrationally excited beryllium monohydride BeH + play a key role in the reaction kinetics of low-temperature plasmasin general, and potentially also in certain cold regions of fusion reactor relevant (e.g. the divertor) plasmas. In orderto model and diagnose plasmas containing BeH + it is essential to build a complete database of cross-sections andrate coefficients for electron-impact collision processes. Knowing the loss rates of BeH + (and isotopologues) as well as4bsolutely calibrated spectroscopic emission from this molecular ion will allow to draw conclusions also about the BeH + formation rates.The BeH + ion is subject to Dissociative Recombination (DR), competed by Vibrational Transitions (VT) - excitation/de-excitation (VE/VdE) - and Dissociative Excitation (DE) respectively [8]:BeH + ( v + i ) + e − −→ Be + H , BeH + ( v + f ) + e − , Be + + H + e − , (2)( v + i ) / ( v + f ) standing for the initial / final vibrational levels of the cation.Whereas for numerous ions measurements of these reactive collisions have been performed in magnetostatic or elec-trostatic storage rings (multipass experiments using merged electron and ion beams) [8], this is certainly not the casefor BeH + , beryllium being highly toxic.In the current study, we performed large scale computations of cross sections for the reactive collisions DR, VE andVdE displayed in eq. (2), as well as of the corresponding rate coefficients.More specifically, we have used the molecular structure data computed by some of us [9] in order to model the dynamicsof these reactions by our stepwise MQDT-method, neglecting the rotational structure and interactions [10, 11, 12, 13].Whereas our previous works [9, 13] restricted to the ground and three lowest vibrationally-excited levels of BeH + , wehave extended here our analysis to the whole range of its vibrational states, i.e. up to v + i = 17.After briefly reminding the major ideas and steps of our MQDT method, including the main features and parametersof the computational part (section 2), we present the cross sections and rate coefficients below 2 .
2. Theoretical approach of the dynamics
In this paper, we use an MQDT-type method to study the electron-impact collision processes:BeH + ( v + i ) + e − −→ BeH ∗ , BeH ∗∗ −→ Be + HBeH + ( v + f ) + e − . (3)resulting from the quantum interference between the direct mechanism - involving the doubly excited resonant statesBeH ∗∗ - and the indirect one - occurring via Rydberg singly-excited predissociating states BeH ∗ .A detailed description of our theoretical approach was given in [13]. Its main steps are the following: i) Building of the interaction matrix V : It is based on the computed [9, 13] Rydberg-valence couplings within aquasi-diabatic representation of molecular states of the neutral system. The matrix elements of this matrix correspondto - and are accordingly indexed with - all the possible pairs of channels. More specifically, for a given electronic totalangular momentum quantum number Λ and a given symmetry (total electronic spin singlet/triplet) of the neutral, ourformalism rely on ionization channels - labelled by the vibrational quantum number of the cation v + and the orbitalquantum number l of the incident/Rydberg electron - and on dissociation channels labelled d j .5 i) Computation of the reaction matrix K : it is performed by adopting for the Lippman-Schwinger integral equationthe second-order perturbative solution [14, 15, 16], written in operatorial form as: K = V + V E − H V , (4) H being the Hamiltonian of the molecular system under study in which the Rydberg-valence interaction is neglected. iii) Computation of the eigenchannel wavefunctions: It relies on the eigenvectors and eigenvalues of the reactionmatrix K , i.e. the columns of the matrix U and the elements of the diagonal matrix tan( η ) respectively : K U = − π tan( η ) U (5)where the non-vanishing elements of the diagonal matrix η are the phaseshifts introduced into the wavefunctions by theshort-range interactions. iv) Frame transformation from the Born-Oppenheimer representation to the close-coupling one: It is performed viathe matrices C and S , built on the basis of the matrices U and η and on the quantum defect characterizing theincident/Rydberg electron, µ Λ l ( R ). v) Construction of the generalized scattering matrix X , eventually split in blocks associated to open and/or closed(o and/or c respectively) channels: X = C + i SC − i S X = X oo X oc X co X cc . (6) vi) Construction of the physical scattering matrix S , whose elements link mutually the open channels exclusively,given by [17]: S = X oo − X oc X cc − exp( − i π ν ) X co . (7)Here the matrix exp( − i π ν ) is diagonal and relies on the effective quantum numbers ν v + associated to the vibrationalthresholds of the closed channels. vii) Computation of the cross-sections: Given the target cation on its level v + i , its impact with an electron of energy ε results either in dissociative recombination or in vibrational excitation/de-excitation according to the formulae: σ diss ← v + i = (cid:88) Λ ,sym σ sym , Λdiss ← v + i , σ sym , Λdiss ← v + i = π ε ρ sym , Λ (cid:88) l,j | S d j ,lv + i | , (8) σ v + f ← v + i = (cid:88) Λ ,sym σ sym , Λ v + f ← v + i , σ sym , Λ v + f ← v + i = π ε ρ sym , Λ (cid:88) l,l (cid:48) | S l (cid:48) v + f ,lv + i − δ l (cid:48) l δ v + i v + f | , (9)where, ρ sym , Λ stands for the ratio between the multiplicity of the involved electronic states of BeH and that of the target,BeH + .
3. Results
Using the available molecular data - quasi-diabatic potential energy curves and electronic couplings for Π, Σ + and ∆ states displayed in Figure 1 of [13] (for more details see as well [9, 18]) - we have extended our previous calculations6 initially restricted to the ground and weakly excited vibrational states - to all vibrational levels (up to v + i = 17) ofthe ground electronic state. The energy of the electron is inferior to 2 . coherently added to indirect) cross section is characterized by resonant captures into Rydberg states, but thatthey do not contribute too much in average.Similar features characterize the cross sections of competitive processes, VE and VdE, but we do not display themhere, restricting ourselves to show only their Maxwell rate coefficients.Indeed, graphs 4, 5, and 6 give the whole ensemble of rate coefficients available for the state-to-state kinetics ofBeH + . They illustrate the dominance of the DR in its v + i = 0 − v + i > global - i.e. coming from the sum over allthe possible final levels - vibrational transitions rate coefficients. One may notice that the excitation process becomes anotable competitor to DR and VdE above 1000 K only.In order to allow the versatile implementation of the rate coefficients shown in Graph 4-6 in kinetics modelling codes,we have fitted them with generalized Arrhenius-type formulas: k P ( BeH + ,L ) ( T e ) = A L T α L e exp − (cid:88) j =1 B L ( j ) j T je , (10)over the electron temperature range 100 K ≤ T e ≤ v + i or v + i → v + f ) respectively. The coefficients for Dissociative Recombination (when P corresponds to DR) A v + i , α v + i and B v + i ( j ) are displayed in Table 1, and those for Vibrational Transitions (when P corresponds to VT, i.e. VE andVdE) A v + i → v + f , α v + i → v + f and B v + i → v + f ( j ) are given in Tables 2-10. The values interpolated using the equation (10) agreewith the MQDT-computed ones within a few percent, and are represented in Graphs 4-6.
4. Conclusions
The present paper provides a complete state-to-state information of the BeH + reactive collisions with electrons,illustrating quantitatively the competition between the vibrational transitions and dissociative recombination. We displaythe cross sections or/and the Maxwell rate coefficients for the molecular ion in all of its initial vibrational states and forthe entire range of energies of the incident electron below the ion dissociation threshold.Arrhenius-type formulas are used in order to fit the rate coefficients as function of T e , the electronic temperature.These rate coefficients strongly depend on the initial vibrational level of the molecular ion.7hese data are addressed to the fusion community - being relevant to the modeling of the edge of the fusion plasmaas well as for divertor conditions - and, more generally, to the modelers of any beryllium-containing-plasmas, pro-duced in laboratory experiments, industrial processing and natural environments. No experimental work concerning theelectron / BeH + collisions can be found in the literature.Further studies, devoted to higher energy and consequently taking into account the dissociative excitation [19], aswell as others, extending to isotopomers of BeH + (BeD + , BeT + ) are the object of ongoing work. All these data, as wellas the presently displayed ones will be of huge importance for modeling of the plasma/wall interaction [3]. Acknowledgments
We acknowledge the French LabEx EMC , via the project PicoLIBS (No. ANR-12-BS05-0011-01), the BIOENGINEproject (sponsored by the European fund FEDER and the French CPER), the Fdration de Recherche Fusion par Confine-ment Magntique - ITER and the European COST Program CM1401 (Our Astrochemical History). AEO acknowledgessupport from the National Science Foundation, Grant No PHY-11-60611. In addition some of this material is based onwork done while AEO was serving at the NSF. ˚AL acknowledges support from the Swedish Research council, Grant No.2014-4164. NP and SI acknowledge the Sectoral Operational Programme Human Resources Development (SOP HRD),ID134378 financed from the European Social Fund and by the Romanian Government. ReferencesReferences [1] G. Federici, C. H. Skinner, J. N. Brooks, J. P. Coad, C. Grisolia, A. A. Haasz, A. Hassanein, V. Philipps, C. S.Pitcher, J. Roth, W. R. Wampler and D. G. Whyte, Nucl. Fusion 41 (2001) 1967.[2] J. Pam´ela, G. F. Matthews, V. Philipps, R. Kamendje and JET-EFDA Contributors, J. Nucl. Mater. 363-365(2007) 1.[3] R. Celiberto, R. K. Janev, and D. Reiter, Plasma Phys. Control. Fusion 54 (2012) 035012.[4] R. A. Anderl, R. A. Causey, J.W. Davis, R. P. Doerner, G. Federici, A.A. Haasz, G. R. Longhurst, W. R. Wamplerand K. L. Wilson, J. Nucl. Mater. 273 (1999) 1.[5] M. Shimada, D. Campbell, V. Mukhovatov, M. Fujiwara, N. Kirneva, K. Lackner, M. Nagami, V. Pustovitov, N.Uckan, J. Wesley, A. Asakura, A. Costley, A. Donn, E. Doyle, A. Fasoli, C. Gormezano, Y. Gribov, O. Gruber, T.Hender, W. Houlberg, S. Ide, Y. Kamada, A. Leonard, B. Lipschultz, A. Loarte, K. Miyamoto, V. Mukhovatov,T. Osborne, A. Polevoi, and A. Sips, Nucl. Fusion 47 (2007) S1.[6] J. N. Brooks, D. N. Ruzic, and D. B. Hayden, Fusion Eng. Des. 37 (1997) 455.[7] D. Nishijima, R. P. Doerner, M. J. Baldwin, G. De Temmerman and E. M. Hollmann, Plasma Phys. Control.Fusion 50 ( 2008) 125007.[8] I. F. Schneider, O. Dulieu, and J. Robert (editors), EPJ Web of Conferences 84 (2015) (Proceedings of the 9 th International Conference ’Dissociative Recombination: Theory, Experiment and Applications’, Paris, July 7-122013). 89] J. B. Roos , M. Larsson, ˚A. Larson, and A. E. Orel, Phys. Rev. A 80 (2009) 012501.[10] A. Giusti, J. Phys. B 13 (1980) 3867.[11] I. F. Schneider, I. Rabad´an, L. Carata, L. H. Andersen, A. Suzor-Weiner, and J. Tennyson, J. Phys. B 33 (2000)4849.[12] B. Vˆalcu, I. F. Schneider, M. Raoult, C. Str¨omholm, M. Larsson, and A. Suzor-Weiner, Eur. Phys. J. D 1 (1998)71.[13] S. Niyonzima, F. Lique, K. Chakrabarti, ˚A. Larson , A. E. Orel, and I. F. Schneider, Phys. Rev. A 87 (2013)022713.[14] V. Ngassam, A. Florescu, L. Pichl, I. F. Schneider, O. Motapon, and A. Suzor-Weiner, Eur. Phys. J. D 26 (2003)165.[15] A. I. Florescu, V. Ngassam, I. F. Schneider, and A. Suzor-Weiner, J. Phys. B 36 (2003) 1205.[16] O. Motapon, M. Fifiring, A. Florescu, F. O. Waffeu Tamo, O. Crumeyrolle, G. Varin-Br´eant, A. Bultel, P. Vervisch,J. Tennyson, and I. F. Schneider, Plasma Sources Sci. Technol. 15 (2006) 23.[17] M. J. Seaton, Rep. Prog. Phys. 46 (1983) 167.[18] C. Str¨omholm, I. F. Schneider, G. Sundstr¨om, L. Carata, H. Danared, S. Datz, O. Dulieu, A. K¨allberg, M. afUgglas, X. Urbain, V. Zengin, A. Suzor-Weiner, and M. Larson, Phys. Rev. A 52 (1995) R4320.[19] K. Chakrabarti, D. R. Backodissa-Kiminou, N. Pop, J. Zs. Mezei, O. Motapon, F. Lique, O. Dulieu, A. Wolf andI. F. Schneider, Phys. Rev. A (2013) 022702. 9 xplanation of GraphsGraph 1-3. Dissociative recombination cross sections for all the vibrational levels of BeH + in its groundelectronic state. Ordinate Cross section in cm Abscissa Collision (electron) energy in eVBlack solid line Total (direct and indirect) processRed dashed line Direct processGreen vertical line Precision limit, for details see the text
Graph 4-6. Dissociative recombination, vibrational excitation and de-excitation Maxwell rate coefficientsfor all the vibrational levels of BeH + in its ground electronic state. Ordinate Maxwell rate coefficient in cm · s − Abscissa Electron temperature in KThick black line DR Maxwell rate coefficientThin coloured lines Vibrational excitation rate coefficientsColoured lines with symbols Vibrational de-excitation rate coefficientsGreen vertical line Precision limit, for details see the text
Graph 7. Global (summed-up for all possible final states) dissociative recombination, vibrational excita-tion and de-excitation Maxwell rate coefficients for BeH + in its ground electronic state. Ordinate Maxwell rate coefficient in cm · s − Abscissa Electron temperature in K10 .01 0.1 1v i+ =010 -16 -15 -14 -13 -12 C r o ss s ec ti on ( c m ) Total processDirect process i+ =1 0.1 1v i+ =2 0.1 1v i+ =3 Energy (eV) i+ =4 0.1 1v i+ =5 Graph 1 . Dissociative recombination cross sections of ground ( v + i = 0) and excited ( v + i = 1 , ...,
5) BeH + in its electronic ground state.Direct mechanism: dashed thick line, total (direct and indirect) mechanism: continuous thin line. .01 0.1 1v i+ =610 -16 -15 -14 -13 -12 C r o ss s ec ti on ( c m ) Total processDirect process i+ =7 0.1 1v i+ =8 0.1 1v i+ =9 Energy (eV) i+ =10 0.1 1v i+ =11 Graph 2 . Dissociative recombination cross sections of excited ( v + i = 6 , , ...,
11) BeH + in its electronic ground state. Direct mechanism:dashed thick line, total (direct and indirect) mechanism: continuous thin line. .01 0.1 1v i+ =1210 -16 -15 -14 -13 -12 C r o ss s ec ti on ( c m ) Total processDirect process i+ =13 0.1 1v i+ =14 0.1 1v i+ =15 Energy (eV) i+ =16 0.1 1v i+ =17 Graph 3 . Dissociative recombination cross sections of excited ( v + i = 12 , , ...,
17) BeH + in its electronic ground state. Direct mechanism:dashed thick line, total (direct and indirect) mechanism: continuos thin line. -10 -9 -8 -7 -6 Electronic temperature (K)
100 1000100 100010 -14 -13 -12 -11 -10 -9 -8 -7 -6 R a t e c o e ff i c i e n t s ( c m s - ) v i+ =0 v i+ =1 v i+ =2 v i+ =3 v i+ =4 v i+ =5 VdE DRDRDRDRDRDRDR DR DR DR DR DR v i+ - v f+ Graph 4 . Dissociative recombination (DR, thick line), vibrational excitation (VE, thin lines) and vibrational de-excitation (VdE, symbolsand thick lines) Maxwell rate coefficients of ground ( v + i = 0) and excited ( v + i = 1 , ...,
5) BeH + in its electronic ground state (total mechanism).For VE, since the rate coefficients decrease monotonically with the excitation, the lowest final vibrational quantum number of the target isindicated only, and the lower panels extend the range down to 10 − cm /s. -10 -9 -8 -7 -6 Electronic temperature (K)
100 1000100 100010 -14 -13 -12 -11 -10 -9 -8 -7 -6 R a t e c o e ff i c i e n t s ( c m s - ) v i+ =6 v i+ =7 v i+ =8 v i+ =9 v i+ =10 v i+ =11 VdE DRDRDRDRDRDRDR DR DR DR DR DR v i+ - v f+ Graph 5 . Dissociative recombination (DR, thick line), vibrational excitation (VE, thin lines) and vibrational de-excitation (VdE, symbolsand thick lines) Maxwell rate coefficients of excited ( v + i = 7 , , ...,
11) BeH + in its electronic ground state (total mechanism). For VE, sincethe rate coefficients decrease monotonically with the excitation, the lowest final vibrational quantum number of the target is indicated only,and the lower panels extend the range down to 10 − cm /s. -10 -9 -8 -7 -6 Electronic temperature (K)
100 1000100 100010 -14 -13 -12 -11 -10 -9 -8 -7 -6 R a t e c o e ff i c i e n t s ( c m s - ) v i+ =12 v i+ =13 v i+ =14 v i+ =15 v i+ =16 v i+ =17 VdE DRDRDRDRDRDRDR DR DR DR DR DR
13 14 15 16 1713 14 15 16 17 v i+ - v f+ Graph 6 . Dissociative recombination (DR, thick line), vibrational excitation (VE, thin lines) and vibrational de-excitation (VdE, symbolsand thick lines) Maxwell rate coefficients of excited ( v + i = 12 , , ...,
17) BeH + in its electronic ground state (total mechanism). For VE, sincethe rate coefficients decrease monotonically with the excitation, the lowest final vibrational quantum number of the target is indicated only,and the lower panels extend the range down to 10 − cm /s. -9 -8 -7 -6 -5 R a t e c o e ff i c i e n t ( c m s - ) v i+ =0v i+ =1v i+ =2v i+ =3v i+ =4v i+ =5v i+ =6v i+ =7v i+ =8v i+ =9v i+ =10v i+ =11v i+ =12v i+ =13v i+ =14v i+ =15v i+ =16v i+ =17
10 100 1000
Electronic temperature (K)
10 100 1000
DR VE VdE
Graph 7 . Global (sum over all the possible final states) Maxwell rate coefficients for dissociative recombination (DR), vibrational excitation(VE) and vibrational de-excitation (VdE). xplanation of Tables Table 1 List of fitting parameters according to eq. (10) with ’P’=’DR’ and ’L’= v + i ’ calculated for dissociative recombi-nation for all vibrational levels of the ground electronic state of BeH + . v + i Initial vibrational level of BeH + A v + i , α v + i , B v + i ( j = 1 , . . . ,
7) Fitting parameters
Tables 2-10 List of fitting parameters according to eq. (10) with ’P’=’VE’ or ’VdE’ and ’L’= v + i → v + f ’ calculated forvibrational transitions (excitation and de-excitation) between the vibrational levels of the ground electronic state of BeH + . v + i → v + f Vibrational transition of BeH + v + i → v + f Stands for VdE A v + i → v + f , α v + i → v + f , B v + i → v + f ( j = 1 , . . . ,
7) Fitting parameters18 a b l e L i s t o f t h e p a r a m e t e r s u s e d i n E q . ( ) f o rt h e D R M a x w e ll r a t ec o e ffi c i e n t s o f B e H + ( v + i = , , , ,..., ) d i s p l a y e d i n G r a ph s , , a nd . v + i A v + i α v + i B v + i ( ) B v + i ( ) B v + i ( ) B v + i ( ) B v + i ( ) B v + i ( ) B v + i ( ) . × − . - . × . × - . × . × - . × . × - . × . × − - . . × - . × . × - . × . × - . × . × . × − - . - . × . × - . × . × - . × . × - . × . × − - . . × - . × . × - . × . × - . × . × . × − - . × . × - . × . × - . × . × - . × . × . × − - . × - . × . × - . × . × - . × . × - . × . × − - . × . × - . × . × . × - . × . × - . × . × − - . × . × - . × . × - . × . × - . × . × . × − - . × . × - . × . × - . × . × - . × . × . × − - . × . × - . × . × - . × . × - . × . × . × − - . × . × - . × . × - . × . × - . × . × . × − - . × - . × - . × . × - . × . × - . × . × . × − - . × - . × . × . × - . × . × - . × . × . × − - . × - . × . × . × - . × . × - . × . × . × − - . × - . × - . × . × - . × . × - . × . × . × − - . × - . × - . × . × - . × . × - . × . × . × − - . × . × - . × . × - . × . × - . × . × . × − - . × . × - . × . × - . × . × - . × . × a b l e L i s t o f t h e p a r a m e t e r s u s e d i n E q . ( ) f o rt h e V E a nd V d E M a x w e ll r a t ec o e ffi c i e n t s o f B e H + ( v + i = nd )r e p r e s e n t e d i n t h e upp e r p a n e l s o f G r a ph . v + i → v + f A v + i → v + f α v + i → v + f B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) → . × − - . × + . × + - . × + . × + - . × + . × + - . × + . × + → . × − - . × + . × + - . × + . × + - . × + . × + - . × + . × + → . × − - . × + . × + - . × + . × + - . × + . × + - . × + . × + → . × − - . × + . × + . × + - . × + . × + - . × + . × + - . × + → . × − - . × + . × + . × + - . × + . × + - . × + . × + - . × + → . × − - . × + . × + - . × + - . × + . × + - . × + . × + - . × + → . × − - . × + . × + . × + - . × + . × + - . × + . × + - . × + → . × − - . × + . × + . × + - . × + . × + - . × + . × + - . × + → . × − - . × + . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + . × + . × + - . × + . × + - . × + . × + - . × + → . × − - . × + . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + . × + . × + - . × + . × + - . × + . × + - . × + → . × − - . × + . × + . × + - . × + . × + - . × + . × + - . × + → . × − - . × + . × + - . × + . × + - . × + . × + - . × + . × + → . × − . × + . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + . × + - . × + . × + - . × + . × + - . × + . × + → . × − . × + . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − - . × + . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + a b l e L i s t o f t h e p a r a m e t e r s u s e d i n E q . ( ) f o rt h e V E a nd V d E M a x w e ll r a t ec o e ffi c i e n t s o f B e H + ( v + i = nd )r e p r e s e n t e d i n t h e upp e r p a n e l s o f G r a ph . v + i → v + f A v + i → v + f α v + i → v + f B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − - . × + . × + - . × + . × + - . × + . × + - . × + . × + → . × − . × + . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − - . × + . × + - . × + . × + - . × + . × + - . × + . × + → . × − - . × + . × + - . × + . × + - . × + . × + - . × + . × + → . × − - . × + . × + . × + - . × + . × + - . × + . × + - . × + → . × − - . × + . × + . × + - . × + . × + - . × + . × + - . × + → . × − - . × − . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + → . × − . × + - . × + . × + - . × + . × + - . × + . × + - . × + a b l e L i s t o f t h e p a r a m e t e r s u s e d i n E q . ( ) f o rt h e V E a nd V d E M a x w e ll r a t ec o e ffi c i e n t s o f B e H + ( v + i = nd )r e p r e s e n t e d i n t h e upp e r p a n e l s o f G r a ph . v + i → v + f A v + i → v + f α v + i → v + f B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) → . × − - . × - . × . × - . × . × - . × . × - . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × − - . × . × - . × . × - . × . × - . × → . × − . × . × . × - . × . × - . × . × - . × → . × − - . × . × . × - . × . × - . × . × - . × → . × − - . × − . × . × - . × . × - . × . × - . × → . × − . × . × . × - . × . × - . × . × - . × → . × − . × − . × . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × . × - . × . × - . × → . × − . × − . × . × - . × . × - . × . × - . × → . × − - . × . × . × - . × . × - . × . × - . × → . × − - . × . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × − - . × - . × . × - . × . × - . × . × → . × − - . × - . × . × - . × . × - . × . × - . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − . × . × . × - . × . × - . × . × - . × → . × − . × . × . × - . × . × - . × - . × . × → . × − - . × − . × - . × . × - . × . × - . × . × → . × − . × . × - . × . × - . × . × - . × . × → . × − . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × . × - . × . × - . × . × - . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − . × - . × . × - . × . × - . × . × - . × a b l e L i s t o f t h e p a r a m e t e r s u s e d i n E q . ( ) f o rt h e V E a nd V d E M a x w e ll r a t ec o e ffi c i e n t s o f B e H + ( v + i = nd )r e p r e s e n t e d i n t h e upp e r p a n e l s o f G r a ph . v + i → v + f A v + i → v + f α v + i → v + f B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × - . × . × - . × . × - . × . × - . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − - . × − - . × . × - . × . × - . × . × - . × → . × − - . × − . × . × - . × . × - . × . × - . × → . × − . × . × . × - . × . × - . × - . × . × → . × − . × − . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × - . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × − - . × . × - . × . × - . × . × - . × → . × − - . × - . × . × - . × . × - . × . × - . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − - . × - . × . × - . × . × - . × . × - . × → . × − - . × . × . × - . × . × - . × . × - . × → . × − . × − . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × − . × - . × . × - . × . × - . × . × a b l e L i s t o f t h e p a r a m e t e r s u s e d i n E q . ( ) f o rt h e V E a nd V d E M a x w e ll r a t ec o e ffi c i e n t s o f B e H + ( v + i = nd )r e p r e s e n t e d i n t h e upp e r p a n e l s o f G r a ph . v + i → v + f A v + i → v + f α v + i → v + f B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) → . × − - . × - . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − . × - . × . × - . × . × . × - . × . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − . × − - . × . × - . × . × - . × . × - . × → . × − - . × - . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − - . × − . × . × - . × - . × . × - . × . × → . × − . × . × - . × . × - . × . × - . × . × → . × − . × . × . × . × - . × . × - . × . × → . × − - . × − . × . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × − . × - . × . × - . × . × - . × . × → . × − . × . × - . × . × - . × . × - . × . × → . × − - . × - . × - . × . × - . × . × - . × . × → . × − - . × . × - . × - . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − . × − - . × . × - . × - . × . × - . × . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − - . × . × . × - . × . × - . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − - . × − . × . × - . × . × - . × . × - . × → . × − - . × − . × - . × . × - . × . × - . × . × → . × − . × . × - . × . × - . × . × - . × . × → . × − . × . × - . × . × - . × . × - . × . × → . × − . × . × . × - . × . × - . × - . × . × → . × − . × − . × . × - . × . × - . × . × . × a b l e L i s t o f t h e p a r a m e t e r s u s e d i n E q . ( ) f o rt h e V E a nd V d E M a x w e ll r a t ec o e ffi c i e n t s o f B e H + ( v + i = nd )r e p r e s e n t e d i n t h e upp e r p a n e l s o f G r a ph . .. v + i → v + f A v + i → v + f α v + i → v + f B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) → . × − - . × - . × . × . × - . × . × - . × . × → . × − - . × . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − . × - . × . × - . × . × . × - . × . × → . × − . × - . × . × . × - . × . × - . × . × → . × − - . × − - . × . × - . × . × - . × . × - . × → . × − - . × − - . × - . × . × - . × . × - . × . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × . × - . × . × - . × . × - . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × − . × - . × . × - . × . × - . × . × → . × − - . × - . × - . × . × - . × . × - . × . × → . × − - . × . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − . × - . × . × . × - . × . × - . × . × → . × − . × − - . × - . × . × - . × . × - . × . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − . × − - . × . × - . × . × - . × . × - . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × . × . × - . × . × - . × . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − - . × . × . × - . × . × - . × . × - . × → . × − - . × . × . × - . × . × - . × . × - . × → . × − - . × . × . × - . × . × - . × . × - . × a b l e L i s t o f t h e p a r a m e t e r s u s e d i n E q . ( ) f o rt h e V E a nd V d E M a x w e ll r a t ec o e ffi c i e n t s o f B e H + ( v + i = nd )r e p r e s e n t e d i n t h e upp e r p a n e l s o f G r a ph . v + i → v + f A v + i → v + f α v + i → v + f B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × - . × . × - . × . × - . × . × - . × → . × − - . × . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − . × − - . × - . × . × - . × . × - . × . × → . × − . × . × - . × . × - . × . × - . × . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − . × - . × - . × . × - . × . × - . × . × → . × − . × − - . × . × - . × . × - . × . × - . × → . × − . × − - . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × . × - . × . × - . × . × - . × → . × − - . × . × . × - . × . × - . × . × - . × → . × − - . × . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × - . × . × - . × . × - . × . × - . × → . × − - . × . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − . × - . × - . × . × - . × . × - . × . × → . × − . × − - . × - . × . × - . × . × - . × . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − . × − - . × . × . × - . × . × - . × . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − - . × - . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × . × - . × . × - . × . × - . × a b l e L i s t o f t h e p a r a m e t e r s u s e d i n E q . ( ) f o rt h e V E a nd V d E M a x w e ll r a t ec o e ffi c i e n t s o f B e H + ( v + i = nd )r e p r e s e n t e d i n t h e upp e r p a n e l s o f G r a ph . v + i → v + f A v + i → v + f α v + i → v + f B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × - . × . × - . × . × - . × . × - . × → . × − - . × . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − . × . × - . × . × - . × . × - . × . × → . × − . × . × - . × . × - . × . × - . × . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − . × - . × - . × . × - . × . × - . × . × → . × − - . × - . × . × - . × . × - . × . × - . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × - . × . × - . × . × - . × . × - . × → . × − - . × . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − . × . × - . × . × - . × . × - . × . × → . × − . × − - . × . × - . × . × - . × - . × . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − - . × − - . × . × - . × . × - . × . × - . × → . × − - . × − - . × . × - . × . × - . × . × - . × → . × − - . × - . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × a b l e L i s t o f t h e p a r a m e t e r s u s e d i n E q . ( ) f o rt h e V E a nd V d E M a x w e ll r a t ec o e ffi c i e n t s o f B e H + ( v + i = nd )r e p r e s e n t e d i n t h e upp e r p a n e l s o f G r a ph . v + i → v + f A v + i → v + f α v + i → v + f B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) B v + i → v + f ( ) → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × - . × . × - . × . × - . × . × - . × → . × − - . × - . × . × - . × . × - . × . × - . × → . × − - . × . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − . × . × - . × . × - . × . × - . × . × → . × − - . × - . × - . × . × - . × . × - . × . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − - . × - . × - . × . × - . × . × - . × . × → . × − - . × . × . × - . × . × - . × . × - . × → . × − - . × - . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × - . × . × - . × . × - . × . × - . × → . × − - . × . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − . × . × - . × . × - . × . × - . × . × → . × − . × - . × . × - . × - . × . × - . × . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − . × - . × . × - . × . × - . × . × - . × → . × − - . × - . × . × - . × . × - . × . × - . × → . × − - . × - . × . × - . × . × - . × . × - . × → . × − - . × - . × . × - . × . × - . × . × - . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . × → . × − - . × . × - . × . × - . × . × - . × . ×19