University of Nebraska - Lincon DigitaCommons@University of Nebraska - Lincon Gordon Gaup Pubications Research Papers in Physics and Astronomy 3-26-2007 Resonances and threshod effects in ow-energy eectron coisions with methy haides Gordon A. Gaup University of Nebraska-Lincon, ggaup1@un.edu Iya I. Fabrikant University of Nebraska-Lincon, ifabrikant@un.edu Foow this and additiona works at: http://digitacommons.un.edu/physicsgaup Part of the Physics Commons Gaup, Gordon A. and Fabrikant, Iya I., "Resonances and threshod effects in ow-energy eectron coisions with methy haides" (2007). Gordon Gaup Pubications. 44. http://digitacommons.un.edu/physicsgaup/44 This Artice is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitaCommons@University of Nebraska - Lincon. It has been accepted for incusion in Gordon Gaup Pubications by an authorized administrator of DigitaCommons@University of Nebraska - Lincon.
Resonances and threshod effects in ow-energy eectron coisions with methy haides Gordon A. Gaup and Iya I. Fabrikant Department of Physics and Astronomy, University of Nebraska, Lincon, Nebraska 68588-0111, USA Received 2 January 2007; pubished 26 March 2007 Cross sections for eastic and ineastic eectron coisions with CH 3 X X=C,Br,I moecues are cacuated. For the owest partia wave, the resonance R-matrix theory, and for the higher partia waves, the theory of scattering by dipoar pus poarization potentia, are used. It is shown that the rotationay eastic scattering ampitude for a poar moecue in the fixed-nucei approximation is ogarithmicay divergent for the forward direction, and a cosure formua is derived to speed up the convergence at sma anges. In treating the nucear motion, ony C-X stretch vibrations are taken into account. The dipoe moment as a function of the C-X distance is modeed by a function incorporating the experimenta vaue of the moecuar dipoe moments at the equiibrium distance and the derivatives of the dipoe moments extracted from the experimenta data on infrared intensities. This is suppemented by ab initio cacuations of the dipoe moment function for CH 3 Br using the muticonfigurationa vaence bond method. The resuts for scattering cross sections show pronounced features caused by vibrationa Feshbach resonances and threshod cusps. The features are most noticeabe at the v=6, 7, and 8 threshods in CH 3 C, at the v=3 and 4 threshods in CH 3 Br, and at the v=1 threshod in CH 3 I. DOI: 10.1103/PhysRevA.75.032719 PACS numbers: 34.80.Bm, 34.80.Gs, 34.80.Ht, 34.60.z I. INTRODUCTION The recent deveopment of highy resoved-in-energy eectron sources 1 4 has stimuated further experimenta and theoretica studies of ow-energy eectron coisions with poyatomic moecues 5 14. Other advances in experimenta research have enhanced the abiity to study the rates of coisiona processes for seectivey excited vibrationa eves 15 17. If the ong-range eectron-moecue interaction is strong enough, it can support vibrationa Feshbach resonances VFRs, a specia scattering resonance with the eectron weaky bound to a vibrationay excited state of the neutra moecue 18,19. These resonances have been observed in HF 20,21, CH 3 I 6, CH 2 Br 2 9, N 2 O 22, and moecuar custers 5,10. It is customary to reate VFRs to the eectron interaction with the moecuar dipoe moment, but quite often for exampe, in the case of CH 3 I 6, VFR appears due to the combination of the dipoar and poarization interactions. It is becoming apparent that VFRs are a quite common phenomenon. Recent experiments indicate their importance in bioogicay reevant moecues, particuary uraci and thymine 23,24. They aso pay an important roe in positron-moecue scattering, in particuar, enhancing positron annihiation 25,26. In some systems the eectron-moecue interaction is not strong enough to support a weaky bound state, but is cose to producing it. In this case virtua-state reated cusps are observed at the vibrationa excitation threshods, particuary in CF 3 I 12 and CF 3 Br 13 moecues. Most experimenta information on VFRs and dipoesupported cusps has been extracted from dissociative eectron attachment DEA measurements. Much ess is known how these phenomena show up in eastic eectron scattering and vibrationa excitation VE. Two joint experimenta and theoretica studies iustrate this connection for CH 3 I 6,27. The permanent dipoe moment of the CH 3 I moecue is rather sma, D=0.639 a.u., and the infrared activity is rather weak the transition dipoe moment is about 0.015 a.u.. For other methy haides these quantities are higher, therefore the contribution of direct scattering, both in eastic and VE channes, becomes important. In particuar it is of interest to know how direct and resonant contributions interfere in VE cross section. For detaied investigation of this probem the dipoe moment as a function of nucear geometry R is necessary. This dependence, for exampe, can significanty affect threshod peaks in vibrationa excitation of hydrogen haides 28. In the present work we study ow-energy eectron coisions with CH 3 X, where X stands for C, Br, and I, with the incusion of ony C-X symmetric stretch vibrations. Previous theoretica studies 6,7,29,30 showed that this approximation is adequate for theoretica description of eectron coisions with these compounds. Therefore we are interested in the dipoe moment as a function of the C-X distance. We cacuate first the dipoe moment function for CH 3 Br and compare the resut with a mode based on experimenta vaues of the equiibrium dipoe moment 31 and the transition dipoe moment extracted from the infrared intensities data 32. We demonstrate then that DEA cross sections are very cose for both dipoe moment functions, and extend our cacuations to CH 3 C and CH 3 I. The cross sections exhibit pronounced threshod structures in VE and eastic cross sections, but these features become weaker for very weaky bound states. II. THEORY For cross section cacuations we empoy the one-poe R-matrix theory 33 with one active C-X stretching mode which has been successfuy appied to description of DEA to CH 3 C 29,30, CH 3 I 6,7, and CH 3 Br 7 moecues. We modify it by incusion of both more accurate dipoe moment functions and direct contributions to eastic scattering and VE. These modifications are described beow. A. Dipoe moment function In our previous DEA cacuations 6,7,29,30 for methy haides CH 3 X we empoyed the dipoe moment functions 1050-2947/2007/753/03271910 032719-1 2007 The American Physica Society
GORDON A. GALLUP AND ILYA I. FABRIKANT TABLE I. Some data for the methy radica. Number of eectrons 9 Mutipicity 2 Dipoe moment 0.2258 a.u. R cacuated for hydrogen haides HX 34, but rescaed them to incorporate the experimenta vaue of the dipoe moment at the equiibrium internucear separation. Specificay, we used the foowing Padé approximant for R: 1+x 3 R = 0 7, 1+ c n x n where x=r R e /R e and R e is the equiibrium internucear separation. This approach can be modified further to take into account the observed intensities in the infrared spectra 32. They generate the transition dipoe moment 01, and this can be used to cacuate, the derivative of the dipoe moment in x at x=0: = 2MRe 01, 2 where M is the reduced mass, and is the vibrationa frequency. generates the coefficient c 1 in expansion 1. To compare the mode dipoe moment function with ab initio resuts, we performed muticonfiguration vaence bond MCVB 35,36 cacuation of CH 3 Br to determine the dependence of the eectric dipoe moment on the C-Br distance. A of the cacuations were done with a 6-31Gd GAMESS 37 definition Gaussian basis set. The MCVB formuation appropriate for this probem invoves a prior cacuation of the spin-restricted open-she Hartree-Fock ROHF structures of CH 3 and Br separatey. Athough it woud be possibe to do the cacuations at each C-Br distance with the methy group in its corresponding equiibrium geometry, in this case we used a rigid geometry corresponding to the anges present in the equiibrium geometry of the whoe CH 3 Br moecue. Additiona cacuations with reaxed geometry show that quaitativey H atoms neary foow the C atom during the vibration. We concude that the approximation of the fixed CH 3 geometry shoud not ater the resuts significanty. Tabe I gives some vaues obtained in our cacuations for the methy radica. The configuration in C 3v is 1a 2 1 2a 2 1 1e 4 3a 1 and the singy occupied orbita 3a 1 is essentiay an sp 3 hybrid pointing away from the H atoms. The Br atom was aso treated aone in the conventiona way with an open she Roothaan treatment. It is aso a doubet system in a 2 A 1 state when considered in C 3v symmetry. The configuration we used is Ar:3d 10 4s 2 4p 5, and it has 35 eectrons. The singy occupied orbita is essentiay of 4p z character. n=1 1 FIG. 1. Dipoe moment as a function of C-X distance for CH 3 X compounds. Lines: mode function, see text. Circes: MCVB cacuations. A together, CH 3 Br has 44 eectrons, and the MCVB cacuations invoved dividing the eectrons into two groups, the inner set of 30 eectrons arranged to generate a static exchange potentia for the remaining 14, which were arranged in various occupations of the outer orbitas. The configurations incuded foow the pattern used successfuy before with singe and doube excitations out of a set of 36 ground state reference state configuration functions. With the symmetry constraints 780 terms were produced. Figure 1 shows the mode dipoe functions for a methy haides and comparison with the MCVB cacuations for CH 3 Br described above. The overa agreement between two functions is quite good. Our scattering cacuations show that the difference between the two dipoe functions becomes important ony for highy excited vibrationa states. To iustrate this we present in Fig. 2 DEA cross sections for v=7 and v=11 states, in the energy region where the difference is most noticeabe. Here v stands for the vibrationa quantum number of the symmetric stretch 3 mode. Even for v=11 the difference is not substantia. The step structures are associated with vibrationa excitation threshods. A resuts presented in the rest of the paper have been cacuated with the mode dipoe moment function, Eq. 1. B. Vibrationa excitation cross section We wi assume that the moecue has a fixed orientation during the coision and that the projection of the eectron anguar momentum on the moecuar axis m is a good quantum number. The first approximation works we for energies exceeding the rotationa spacing. The second assumption, stricty speaking, does not appy to nondiatomic moecues, but it can be justified by noticing that the ow-energy resonant scattering by methy haides is dominated by the * resonance, and for the pure dipoe scattering m is conserved. We wi aso assume that the anguar part of the eectron wave function in the outer region is a dipoar anguar harmonic satisfying the equation 38,39 032719-2
RESONANCES AND THRESHOLD EFFECTS IN LOW- FIG. 2. DEA to vibrationay excited states of CH 3 Br. Soid ines: cacuation with the dipoe function from the MCVB cacuations. Dashed ines: cacuation with the mode dipoe function. 2 +2 vv cos + m m +1Z m, =0, = m,m +1,..., Z m, = D m cos e im, 4 2 where is the anguar part of the Lapace operator and vv is the dipoe moment for a given vibrationa state v. Then the ampitude for vibrationa transition from v state to v state is f vv 0, = 2i vv q cos i kk 1/2S vv D 0 0 cos 0 D 0 0 cos, 5 where 0 and are incident and scattering anges reative to the moecuar axis, k and k are initia and fina eectron momenta, q=k k, is the ange between q and the moecuar axis, vv is the matrix eement of the dipoe moment, and S vv is the S-matrix eement for the resonance vibrationa excitation. The first term in Eq. 5 represents the direct excitation in the Born approximation 40 and the second the resonance contribution 41. Since we wi be mosty interested in the transition v v+1, we negect the difference in the diagona dipoe moment in initia and fina state, and assume that the dipoar harmonic D 0 0 is the same for both states. Taking the square of the absoute vaue of the ampitude and averaging over orientations, we obtain for the differentia cross section as a function of the scattering ange = k k A dir + A res + A int, 6 2 where A dir =4 vv /3q 2 40. The method of cacuation of the resonance contribution A res was described esewhere 41. Here we wi concentrate on the interference term A int, 3 A int = 4 vv Re S vv qkk 1/2 cos D 0 0 cos 0 D 0 0 cos, 7 where anguar brackets mean the average over orientations. To cacuate the average, we write cos = k ŝ k ŝ, 8 q where ŝ is a unit vector in the direction of the moecuar axis. Then we expand D 0 0 cos in Legendre poynomias and use the addition theorem for spherica harmonics: 1/2 2 +1 a P cos 2 D 0 0 cos = =2 m 1/2 2 a 2 +1 Y * m kˆ Y m ŝ, 9 where we have performed transformation to the frame with the poar axis aong the vector k. The expansion coefficients a can be easiy obtained by the diagonaization of Eq. 3. Using a simiar expansion for D 0 0 cos 0 and integrating over ŝ, we obtain cos D 0 0 cos 0 D 0 0 cos = k + k p P cos, q 10 1/2 p = a 1 +1 2 +1 2 +3 1/2a +1 + 2 1 1/2a 1. 11 Now the interference term can be written as A int = 2 vv q 2 kk 1/2k + kre S vv p P cos. 12 The interference term integrated over scattering ange can be obtained using the equation 1 1 P x k 2 + k 2 2kkx dx = 1 kk Q k2 + k 2 2kk, 13 where Q x is the Legendre function of the second kind. Finay the integrated cross section is vv = 28 2 vv k 3 n k + k k k + S vv 2 k + k +4 vv kk 1/2 Re S vv p Q k2 + k 2. 2kk 14 Note that the phase factor in vv shoud be consistent with the phase factor in S vv, since the sign of the interference term is determined by the product vv Re S vv. C. Differentia eastic cross section It is we known that resonant features in eastic eectron scattering by poar moecues can be significanty suppressed 032719-3
GORDON A. GALLUP AND ILYA I. FABRIKANT by the strong direct contribution. Moreover, in the approximation of a fixed moecuar orientation the tota eastic cross section is divergent. For diatomic poar moecues in eectronic states and asymmetric tops this divergence can be removed by incusion of rotations for arge orbita anguar momentum of the incident eectron or sma scattering anges and cacuation of the corresponding scattering ampitudes in the Born dipoe approximation 39. However, for symmetric top moecues, such as methy haides, even incusion of rotations does not average out the permanent dipoe moment, and the tota cross section is sti divergent 42. The reason for this is the presence of degenerate channes couped by the dipoar interaction 43,44. To remove this degeneracy, the inversion spitting shoud be incuded 45, and the integrated cross section becomes proportiona to the ogarithm of the inversion spitting. In a series of moecues such as those in this study, the inversion spitting is very difficut to determine. About a one can say about it is that it must be an exceedingy sma energy difference. The actua motion over the inversion barrier can be viewed as some inear combination of the A 1 norma modes of the harmonic vibrations, but, of course, when inversion spitting is incuded, the nucei are no onger treatabe as cassica partices. Thus, there is a sense in which the whoe system must be considered to have D 3h symmetry. In addition to this compication, a correct treatment woud require the handing of effects of whatever nucear spin states are present. The upshot is that the extremey sma energy difference wi produce a tota eastic cross section that might as we be infinite, as far as any experiment is concerned. Since the range of impact parameters contributing to the tota cross section becomes enormous, we do not think that it is of much practica vaue to cacuate the tota cross section. Since most of the avaiabe experimenta data on eastic scattering by poar symmetric tops were obtained for ange-differentia cross sections, we are cacuating these in the present paper. We base our description of eastic scattering by a poar moecue on Refs. 41,46,47. The differentia cross section for a diatomic moecue can be decomposed in partia cross sections for JM JM transitions, where JM and JM are the rotationa quantum number and its projection in the initia and fina states, respectivey. In the fixed-nucei approximation the cross section summed over JM and averaged over M does not depend on the initia rotationa quantum number J 48. Therefore a nondiatomic moecue in the fixed-nucei approximation can be treated as a diatomic with fictitious quantum numbers JM. Because the independence of JM has been proven, we wi use for simpicity J=M=0. We wi describe eastic scattering by the S matrix in the dipoar harmonics representation given by functions 3. We assume that at,m=0,0 the eastic scattering is described by the matrix eement S 00 cacuated by the R-matrix method which incudes resonance, dipoe and poarization contributions. For,m0,0 we assume scattering by dipoe pus poarization potentia with 49 S m = exp i m b m + ik 2 b m ik 2, 15 b m = 2 m 12 m +12 m +3, 16 where is the isotropic part of the poarizabiity. Then the scattering ampitude in the fixed-nucei approximation is f 0, 0,, = i D m k cos 0 e im S D m cos 0 m D m cos e im 0, 17 where 0, are the incident and scattering poar anges, as in the previous section, and 0, are corresponding azimutha anges. Cacuating the ampitude for the transition 00 JM as a function of scattering anges, in the aboratory frame with poar axis aong the initia momentum k, we obtain 41,47 f JM, = 2i k m m m JM Y M,, 18 where m JM = 1 M+m J 2J +12 +1/41/2 M 0 M J 19 0 m m, m = 1 e i S m a m a m, 20 where a m are coefficients in expansion of dipoar harmonics, Eq. 3, in spherica harmonics, simiar to expansion 9 but for arbitrary and m. The differentia cross section is cacuated then as = f JM, 2. 21 JM Since the ampitude f 11 is divergent as 1/ at 0, the expansion 18 converges very sowy. To speed up the convergence, we rewrite Eq. 18 in the foowing form 47 f JM, = f B JM, + 2i m k mb m JM Y M,, m 22 where f B mb JM is the Born ampitude 47, and is the vaue of m cacuated in the first order in the dipoe moment mb m =2, 1q q +1m,+1 +1, 23 1/2 m + m q m =. 24 2 12 +1 However, for substantia dipoe moments the cosure expression for f 11 is not sufficient. This is because the ampitude f 00 is divergent at 0 as we, athough ony ogarithmicay. We prove this statement in the Appendix by cacuating f 00 in the second Born approximation. Using Eqs. A8 and A9 032719-4
RESONANCES AND THRESHOLD EFFECTS IN LOW- FIG. 3. DEA to vibrationay excited states of CH 3 C. Numbers near the threshod peaks and/or steps indicate the VE threshods. from the Appendix, we can write down the cosure formua for f 00 as f 00 = 2i2 3k + i 2k nsin 2 m m 22 3 III. RESULTS AND DISCUSSION 2 +1 cos. 25 +1P A. CH 3 C VFRs in DEA for methy haides were investigated before 5 7,50. Here we summarize major features for further discussion using CH 3 C as an exampe. Adiabatic potentia energy curve for the CH 3 C anion supports nine weaky bound vibrationa states at the v=0 through the v=8 threshods. Ony three of them, those associated with the v=6, 7, and 8 threshods, are noticeabe in the DEA spectrum. Moreover, the positions of a resonances except the v=8 resonance are coser than 0.1 mev to the corresponding threshods. This means that the account of moecuar rotations wi turn them into the virtua-state cusps 51. Figure 3 iustrates these features by showing the behavior of the DEA cross section for scattering from the v=5, 6, and 7 states in the vicinities of the v=6, 7, and 8 threshods, respectivey. We aso observe a strong increase of cross sections with v which resuts in strong temperature effect discussed before 7,29,30. In Fig. 4 we present VE of CH 3 C for a series of the v v+1 transitions. The first noticeabe threshod peak appears in the v=5 6 transition. With the further increase of v the peak deveops further and reaches maximum vaue exceeding 5010 16 cm 2 at v=7. This is expected, since the v=8 threshod generates the most pronounced feature in the DEA cross section. An interesting shape is observed for the v = 6 7 cross section. At the v = 7 threshod we see the virtua-state cusp, and then a broader VFR associated with FIG.4.VEofCH 3 C from vibrationay excited states. the v=8 threshod. Cusps and step structures seen at higher threshods are simiar to those observed before in VE of hydrogen haides and CH 3 I 21,27,52,53. From the above discussion it is cear that VFRs and threshod cusps can be seen in eastic cross sections ony for scattering from excited states starting from v = 5. Before presenting the corresponding cross section, we show in Fig. 5 differentia eastic cross section as a function of ange for scattering from the ground vibrationa state and compare it with experiment 54. It appears that cacuated cross sections are becoming too arge at arge scattering anges, greater than 100. Ab initio compex Kohn variationa cacu- FIG. 5. Eastic scattering from the vibrationay ground state of CH 3 C as a function of the scattering ange at E=0.5 ev. Soid ine: present cacuation with fu incusion of poarization. Dashed ine: cacuation with no poarization incuded in the partia wave =1. Dotted ine: dipoe-born approximation. Squares: experimenta data 54. 032719-5
GORDON A. GALLUP AND ILYA I. FABRIKANT ations 55 do not exhibit growth at higher scattering anges. The partia wave anaysis of our cross section indicates that our method perhaps overestimates the poarization contribution to the scattering matrix for =1 which approximatey corresponds to =1. Indeed, Eq. 15 for the scattering matrix is vaid if the poarization potentia can be considered as a perturbation compared to the centrifuga potentia, simiar to the equation of O Maey et a. 56. For the poarizabiity of CH 3 C, =31 a.u., and E=0.5 ev, =1 this is becoming a very crude approximation. To check this we aso performed another cacuation whereby the poarization contribution for =1 was negected. Agreement with experiment 54 and with the compex Kohn cacuations 55 becomes much better, which is perhaps somewhat fortuitous, but this indicates the important roe of the p scattering by the poarization potentia even at such a ow energy as 0.5 ev. Apparenty this is an upper bound for our resonant approach which does not empoy accurate methods for cacuation of contribution of higher partia waves. Therefore in our further discussion of eastic scattering we wi consider ony energies beow 0.5 ev. We shoud add, however, that our method gives reiabe resuts at higher energies for resonance processes such as DEA and VE. Another important observation foowing from Fig. 5 is the high sensitivity of the cross section to the poarization contribution even at reativey sma anges about 30 where the conventiona wisdom says that the dipoar interaction shoud dominate. Fu incusion of poarization according to Eq. 15 gives the resut which is higher than the dipoe Born approximation, whereas eimination of poarization in Eq. 15 at =1 gives ower cross section which is coser to the experimenta vaue. We concude that inaccurate incusion of poarization can ead to an overestimation of the cross section at ow anges. This might be reevant to disagreement at ow anges between the compex Kohn cacuations and experiment discussed in Ref. 55. In Fig. 6 we present ow-energy eastic differentia cross sections for CH 3 C at =100 and compare them with the dipoe-born resuts. The present cacuations give cross sections which are much smaer than the Born resuts above the first VE threshod, but they increase significanty towards smaer energies. Pronounced step structures and cusps are observed at the v=7 and v=8 threshods. B. CH 3 Br In CH 3 Br there are five dipoe-supported states associated with the v=0 through v=4 threshods 7. Ony the highest v=4 feature appears as a pronounced VFR. Its existence was recenty confirmed by experimenta observations 50, where the integrated VE cross sections for CH 3 Br were aso cacuated. Threshod peaks appear even in the v =0 1 transition and become more pronounced for higher v. The argest cross section, v =2 3 reaches very high vaue at the peak, 9610 16 cm 2, not very far from its unitary imit, 15810 16 cm 2. This is perhaps the argest VE cross section ever predicted. In Fig. 7 we show VE cross sections at two anges and compare them with those from the direct dipoe contribution. FIG. 6. Eastic scattering from vibrationay excited states of CH 3 C at =100. Soid ines: present cacuations. Dashed ines: dipoe-born approximation. The atter becomes more important at ower anges, athough in the case of the v=2 3 transition the resonant process strongy dominates even at =30. Another interesting feature is a pecuiar shape of the 2 3 cross section. It is dominated by the virtua-state effect just above the v=3 threshod and VFR beow the v=4 threshod, and is simiar to the v=6 7 excitation in CH 3 C presented in Fig. 4. In Fig. 8 we present eastic differentia cross sections for scattering from the v=2 and v=3 states and compare them with the dipoe-born resut. At =30 both curves are quite cose to each other, whereas a noticeabe difference is observed for =100. C. CH 3 I In CH 3 I there are two weaky bound states associated with v=0 and v=1 threshods. The atter appears as pronounced VFR in DEA 6,7. Since the dipoe moment of CH 3 I is reativey weak, the incusion of poarization is becoming crucia for obtaining the correct shape of VFR 57. The tota VE cross section and the resonance contribution to the eastic cross section was compared to experimenta data at =135 in Ref. 27. Here we show a more direct comparison by cacuating differentia cross sections at 135. In Fig. 9 we present v=0 1 and v=1 2 cross sections and compare them with arbitrary normaized experimenta data of Aan 27. Quaitativey a of the main features in the experimenta cross section are reproduced by theory, athough the theoretica features threshod peak and structures at higher threshods are sharper, apparenty due to imited experimenta energy resoution. Due to substantia popuation of the first excited vibrationa state in experiment about 8%, the shape of the experimenta cross section is apparenty affected by the v=1 2 transition. Overa, the present comparison does not differ noticeaby from that presented in 032719-6
RESONANCES AND THRESHOLD EFFECTS IN LOW- FIG. 7. Differentia cross sections for VE of CH 3 Br. Soid ines: present cacuations. Dotted ines: dipoe-born approximation. Ref. 27 since the energy dependence of the integrated cross section and differentia cross section at arge anges is very simiar. Note that the direct contribution is very sma in this case and its peak vaue is 0.003 2410 16 cm 2 /sr at E=0.081 ev. In Fig. 10 we present eastic differentia cross section at =135. In contrast to the CH 3 Br case, the present cross section differs substantiay from the dipoe-born contribution, apparenty because of the smaer dipoe moment and arger poarizabiity in this case =54 a.u.. The accurate account of poarization becomes important even at reativey ow energy. In order to check the poarization effect, we, as in the case of CH 3 C, performed additiona cacuation without incusion of poarization in the p wave. The cross section has become even higher. The difference with the first cacuation represents the uncertainty of our resuts in the present case. With regard to comparison with experiment, the experimenta cross section grows not as fast as the cacuated when approaching the zero energy. The disagreement becomes particuary striking beow the excitation threshod, party because of the rapidy deteriorating quaity of the incident eectron beam 27. The cacuated dip at the excitation threshod is more pronounced than experimenta. Party this can be expained by the negect of couping with other vibrationa FIG. 8. Eastic scattering from vibrationay excited states of CH 3 Br at =30 and 100. Soid ines: present cacuations. Dotted ines: dipoe-born approximation. 032719-7
GORDON A. GALLUP AND ILYA I. FABRIKANT FIG.9.VEofCH 3 Iat=135. Soid and dashed ines: present cacuations. Dotted ine: arbitrariy normaized experimenta data 27. modes. On the other hand, the theory reproduces very we the shape of VFR in DEA channe 6, therefore obviousy there coud be severa reasons for disagreement. We concude that the mode empoyed in the present paper is much better suited for studies of resonance processes, whereas cacuation of eastic scattering requires more accurate description of the direct process and higher partia waves. Finay, in Fig. 11 we present comparison of the integrated resonance contribution to the eastic scattering with the tota integrated resonant cross section incuding DEA and VE cross sections. We observe that the feature at the v=1 threshod is strongy suppressed in the tota cross section. This might be reevant to observation of VFR in uraci 23, where pronounced VFR is observed in the DEA channe FIG. 10. Eastic scattering from CH 3 Iat=135. Soid ine: present cacuation with fu incusion of poarization. Dashed ine: cacuation with no poarization incuded in the partia wave =1. Dash-dotted ine: dipoe-born approximation. Dotted ine: experimenta data 27. FIG. 11. Anguar-integrated cross sections for scattering from CH 3 I. Soid ine: resonance contribution to eastic scattering. Dashed ine: tota resonance contribution incuding VE and DEA. whereas measurements of the tota cross section by the use of the eectron transmission spectroscopy technique do not give any visibe feature at the v=1 threshod. IV. CONCLUSION We have presented anaysis of the resonance and threshod effects in eastic and ineastic scattering of eectrons by methy haides. The resuts of this paper are of genera significance due to the importance of VFRs in many systems incuding bioogica moecues and positron-containing systems. In contrast to coisions with hydrogen haides, studied in Refs. 21,28,52,53, where the DEA process is endothermic with the exception of HI, coisions with methy haides invove the DEA process which is exothermic. In spite of the possibiity of predissociation after capture in a dipoe supported state, VFRs in methy haides can have narrow width. Moreover, for the dipoe-supported states with very sma binding energies beow 0.1 mev, both contributions to the resonance width, due to autodetachment and predissociation, become so sma that from the practica point of view the resonance disappears competey. This is pertinent to resonances in CH 3 C with v6 and in CH 3 Br with v3. Therefore, in spite of the supercritica vaue of the dipoe moment of CH 3 C and CH 3 Br, no noticeabe threshod features are observed in scattering by CH 3 Cv for v6 and by CH 3 Brv for v3. We shoud emphasize that this concusion has nothing to do with rotationa motion of the moecue. Of course, if we are interested in energies very cose to the threshod so that the distance to the threshod is comparabe to the rotationa spacing, the rotationa motion shoud be taken into account. In this case VFRs with binding energies beow the rotationa spacing wi be converted into virtua states 51, but a major features in the cross sections wi remain unaffected even with the best energy resoution about 1 mev avaiabe in current experiments. 032719-8
RESONANCES AND THRESHOLD EFFECTS IN LOW- In the CH 3 I moecue the ony VFR is associated with the v = 1 threshod. This makes it easier for observations. Indeed, the pronounced features were observed in DEA 6 and VE 27 cross sections, and, somewhat ess pronounced in the eastic scattering 27. Our predictions for VE and eastic scattering from excited states of CH 3 C and CH 3 Br wi await for experimenta confirmation. It shoud be stressed that recent observation 50 of VFRs in DEA to CH 3 Br gives us confidence in reiabiity of our resuts for this moecue. ACKNOWLEDGMENTS The authors are gratefu to H. Hotop for many stimuating discussions. This work has been supported by the Nationa Science Foundation through Grant No. PHY-0354688. APPENDIX: ROTATIONALLY ELASTIC SCATTERING AMPLITUDE We are interested here in rotationay eastic ampitude for a diatomic moecue and its convergence at 0. For simpicity we wi consider the eastic scattering ampitude f 00 for pure dipoar scattering and wi obtain its expicit form in the second Born approximation. This wi prove its divergence at 0 and wi aow us to derive the cosure expansion 25 for the rotationay eastic scattering case. For J=0, Eq. 18 becomes f 00 = i m 2k P cos, A1 m m = 1 expi m a m 2, A2 where we use S m =exp i m for a pure dipoe potentia. Stricty speaking, this equation requires a modification for a supercritica dipoe moment and =0, but for the present case this is not reevant since we are concerned with the high- behavior. We wi expand now m in powers of and wi keep ony the first nonvanishing term, that is the term proportiona to 2. First we obtain the coefficients a m and the eigenvaues m in the first nonvanishing order a m q m q +1m = +, 1 +,+1 +1, m = + 22 2 +1 q 2 m q 2 +1m +1, where q m is given by Eq. 24. From Eq. A2 m =1 expi m + 1 expi m 1 a m 1 2 A3 A4 + 1 expi m +1 a m +1 2. A5 Using Eqs. A3 and A4, we obtain from here m = 2i2 2 +1 q 2 m q 2 +1m +2 +1 2 q m 2 For summation over m we use the equation m= q 2 m = This eads to the resut 2 + q 2 +1m +1 2. A6 1 m + m = 2 12 +1m= 3. A7 m= m = 22 3 1 + 1 +1. A8 Note that imaginary part of m turns to 0 in the second order in. Finay, summation over gives the foowing expression for the scattering ampitude: f 00 = 2i2 3k nsin 2. A9 We concude that the imaginary part of the rotationay eastic scattering ampitude diverges ogarithmicay in the forward direction. This divergence is consistent with the optica theorem since the tota integrated cross section is divergent due to the J J±1 transitions. On the other hand, since the divergence is ony ogarithmica, the integrated cross section for the rotationay eastic cross section is finite. 1 D. Kar, M.-W. Ruf, and H. Hotop, Chem. Phys. Lett. 189, 448 1992. 2 D. Kar, M.-W. Ruf, and H. Hotop, Meas. Sci. Techno. 5, 1248 1994. 3 S. V. Hoffmann, S. L. Lunt, N. C. Jones, D. Fied, and J.-P. Ziese, Rev. Sci. Instrum. 73, 4157 2002. 4 J. P. Ziese, N. C. Jones, D. Fied, and L. B. Madsen, Phys. Rev. Lett. 90, 083201 2003. 5 H. Hotop, M.-W. Ruf, M. Aan, and I. I. Fabrikant, Adv. At., Mo., Opt. Phys. 49, 852003. 6 A. Schramm, I. I. Fabrikant, J. M. Weber, E. Leber, M.-W. Ruf, H. Hotop, J. Phys. B 32, 2153 1999. 7 R. S. Wide, G. A. Gaup, and I. I. Fabrikant, J. Phys. B 33, 5479 2000. 8 D. Kar, M.-W. Ruf, I. I. Fabrikant, and H. Hotop, J. Phys. B 34, 3855 2001. 9 A. Schramm, M.-W. Ruf, M. Stano, S. Matejcik, I. I. Fabrikant, and H. Hotop, J. Phys. B 35, 4179 2002. 10 I. I. Fabrikant and H. Hotop, Phys. Rev. Lett. 94, 063201 2005. 11 I. I. Fabrikant, H. Hotop, and M. Aan, Phys. Rev. A 71, 022712 2005. 12 S. Marienfed, I. I. Fabrikant, M. Braun, M.-W. Ruf and H. Hotop, J. Phys. B 39, 105 2006. 032719-9
GORDON A. GALLUP AND ILYA I. FABRIKANT 13 S. Marienfed, T. Sunagawa, I. I. Fabrikant, M. Braun, M.-W. Ruf, and H. Hotop, J. Chem. Phys. 124, 154316 2006. 14 R. Čurik, J. P. Ziese, N. C. Jones, T. A. Fied, and D. Fied, Phys. Rev. Lett. 97, 123202 2006. 15 M. Küz, M. Kei, A. Kortyna, B. Schehaass, J. Hauck, K. Bergmann, W. Meyer, and D. Weyh, Phys. Rev. A 53, 3324 1996. 16 A. Rosa, F. Brüning, S. V. K. Kumar, and E. Ienberger, Chem. Phys. Lett. 391, 361 2004. 17 M. Braun, F. Gruber, M.-W. Ruf, S. V. K. Kumar, E. Ienberger and H. Hotop, Chem. Phys. 329, 148 2006. 18 W. Domcke and L. S. Cederbaum, J. Phys. B 14, 149 1981. 19 J.-P. Gauyacq and A. Herzenberg, Phys. Rev. A 25, 2959 1982. 20 G. Knoth, M. Gote, M. Räde, K. Jung, and H. Ehrhardt, Phys. Rev. Lett. 62, 1735 1989. 21 M. Čižek, J. Horáček, M. Aan, I. I. Fabrikant, and W. Domcke, J. Phys. B 36, 2837 2003. 22 M. Aan and T. Skaický, J. Phys. B 36, 3397 2003. 23 A. M. Scheer, K. Afatooni, G. A. Gaup, and P. D. Burrow, Phys. Rev. Lett. 92, 068102 2004. 24 P. D. Burrow, G. A. Gaup, A. M. Scheer, S. Denif, S. Ptasinska, T. Märk, and P. Scheier, J. Chem. Phys. 124, 124310 2006. 25 L. D. Barnes, S. J. Gibert, and C. M. Surko, Phys. Rev. A 67, 032706 2003. 26 G. F. Gribakin and C. M. R. Lee, Phys. Rev. Lett. 97, 193201 2006. 27 M. Aan and I. I. Fabrikant, J. Phys. B 35, 1025 2002. 28 J. Horáček, M. Čižek and W. Domcke, Theor. Chem. Acc. 100, 311998. 29 I. I. Fabrikant, J. Phys. B 27, 4325 1994. 30 D. M. Pear, P. D. Burrow, I. I. Fabrikant, and G. A. Gaup, J. Chem. Phys. 102, 2737 1995. 31 CRC Handbook of Chemistry and Physics, edited by D. R. Lide CRC Press, Boca Raton, 2004. 32 D. M. Bishop and L. M. Cheung, J. Phys. Chem. Ref. Data 11, 119 1982. 33 I. I. Fabrikant, Phys. Rev. A 43, 3478 1991. 34 J. F. Ogivie, W. R. Rodwe, and R. H. Tipping, J. Chem. Phys. 73, 5221 1980. 35 G. A. Gaup, R. L. Vance, J. R. Coins, and J. M. Norbeck, Adv. Quantum Chem. 16, 229 1982. 36 G. A. Gaup, Vaence Bond Methods Cambridge University Press, Cambridge, 2002. 37 M. W. Schmidt et a., J. Comput. Chem. 14, 1347 1993. 38 M. H. Mitteman and R. E. von Hodt, Phys. Rev. 140, A726 1965. 39 I. I. Fabrikant, Sov. Phys. JETP 44, 771976. 40 K. Takayanagi, Supp. Prog. Theor. Phys. 40, 216 1967. 41 I. I. Fabrikant, Phys. Lett. 77A, 421 1980. 42 O. H. Crawford, J. Chem. Phys. 47, 1100 1967. 43 P. C. Engeking, Phys. Rev. A 26, 740 1982. 44 P. C. Engeking and D. R. Herrick, Phys. Rev. A 29, 2425 1984. 45 G. Herzberg, Infrared and Raman Spectra of Poyatomic Moecues Van Nostrand, New York, 1945. 46 M. H. Mitteman, J. L. Peacher, and B. F. Rozsnyai, Phys. Rev. 176, 180 1968. 47 I. I. Fabrikant, J. Phys. B 16, 1253 1983. 48 I. Shimamura, Phys. Rev. A 23, 3350 1981. 49 I. I. Fabrikant, J. Phys. B 12, 3599 1979. 50 M. Braun, I. I. Fabrikant, M.-W. Ruf, and H. Hotop, J. Phys. B 40, 659 2007. 51 I. I. Fabrikant and R. S. Wide, J. Phys. B 32, 235 1999. 52 A.-Ch. Sergenton, L. Jungo, and M. Aan, Phys. Rev. A 61, 062702 2000. 53 M. Čižek, J. Horáček, A.-Ch. Sergenton, D. B. Popović, M. Aan, W. Domcke, T. Leininger, and F. X. Gadea, Phys. Rev. A 63, 062710 2001. 54 X. Shi, V. K. Chan, G. A. Gaup, and P. D. Burrow, J. Chem. Phys. 104, 1855 1996. 55 T. N. Rescigno, A. E. Ore, and C. W. McCurdy, Phys. Rev. A 56, 2855 1997. 56 T. F. O Maey, L. Spruch, and L. Rosenberg, J. Math. Phys. 2, 491 1961. 57 E. Leber, I. I. Fabrikant, J. M. Weber, M.-W. Ruf, and H. Hotop, in Dissociative Recombination: Theory, Experiment and Appications IV, edited by M. Larsson, J. B. A. Mitche, and I. F. Schneider Word Scientific, Singapore, 2000, pp. 69 76. 032719-10