Full two-electron calculations of antiproton collisions with molecular hydrogen
aa r X i v : . [ phy s i c s . a t o m - ph ] O c t Full two-electron calculations of antiproton collisions with molecular hydrogen
Armin L¨uhr and Alejandro Saenz
Institut f¨ur Physik, AG Moderne Optik, Humboldt-Universit¨at zu Berlin, Newtonstr. 15, D-12489 Berlin, Germany (Dated: October 19, 2018)Total cross sections for single ionization and excitation of molecular hydrogen by antiproton impact arepresented over a wide range of impact energy from 1 keV to 6.5 MeV. A nonpertubative time-dependent close-coupling method is applied to fully treat the correlated dynamics of the electrons. Good agreement is obtainedbetween the present calculations and experimental measurements of single-ionization cross sections at highenergies, whereas some discrepancies with the experiment are found around the maximum. The importanceof the molecular geometry and a full two-electron description is demonstrated. The present findings providebenchmark results which might be useful for the development of molecular models.
PACS numbers: 34.50.Gb,25.43.+t
A central point of atomic and molecular physics is the de-scription of charged particles moving in a Coulomb field. Oneof the simplest and most basic systems which provides an in-sight into dynamic processes of charged particles is the colli-sion of antiprotons with atoms. The heavy mass of the antipro-ton allows, first, for a semiclassical theoretical approach and,second, for the investigation of “slow” ionizing collisions. Incontrast to positively charged projectiles, for antiprotons thereis no complication from charge transfer.Further attention is drawn to this topic due to the upcomingFAIR [1] facility with the international collaborations FLAIR[2] and SPARC [3], both intending to investigate antiprotondriven processes and even kinematically complete collisionexperiments. However, the design of FLAIR already requiresa reliable knowledge of low-energy antiproton cross sectionsof residual gases as, e.g., molecular hydrogen. These experi-mental efforts complement the recent intensive studies on an-tihydrogen at CERN aiming to test the CPT invariance and todisclose the nature of antimatter gravity.Over the last decades a remarkable progress in the under-standing of interactions between antiprotons and atoms hasbeen achieved (cf. [4] and Refs. therein). The theoretical de-scription concentrated mainly on hydrogen and helium atoms.It was relatively easy to establish a full treatment of the for-mer target which only consists of one electron and one nu-cleus. For helium atoms, on the other hand, the dynamic electron-electron correlation effects turned out to be decisive;requiring much larger efforts for their correct description. Dueto the above mentioned favorable properties, antiproton colli-sions on helium atoms became a benchmark system for study-ing electron correlation in atoms stimulating a large numberof calculations which employed various theoretical methods.During the last ten years close-coupling calculations using ei-ther a spectral or spatial expansion of the two-electron wave-function [5, 6, 7, 8] yielded the most precise results where thelatter usually take advantage of large-scale computing facili-ties. They have provided cross sections for single and doubleionization which are mostly in agreement with experiment forintermediate to high impact energies while discrepancies stillpersist for low energies.Antiproton collisions with molecules have been studied ex-perimentally in a similar way as atoms concentrating mainly on ionization cross sections [9] and stopping powers [10]where for the latter rather diverse results were obtained atlow energies. In contrast, the theoretical work on collisionsinvolving molecules is still comparably sparse (cf. [11] andRefs. therein). Certainly, the description of a four-particlesystem like a hydrogen molecule , consisting of two electronsand two nuclei, is a further step in complexity compared toa helium atom . Recently, the ionization and excitation crosssections [11, 12] as well as the stopping power [13] for an-tiproton impact on molecular hydrogen were calculated us-ing spherical one-electron models for the hydrogen molecule[14]. They could mostly reproduce the experimental antipro-ton results for impact energies E ≥ keV. The findingssuggest, however, that for lower energies molecular as wellas electron-electron correlation effects are important and haveto be considered. An earlier work by Ermolaev [15] turnedout to be unsatisfactory for E ≤ keV reproducing ratheratomic than molecular hydrogen. Furthermore, two calcula-tions, both treating the target as a molecule, were performedby Sakimoto [16] and recently by the present authors [17] formolecular hydrogen ions. It was shown that the calculationof only three orientations of the molecular axis with respectto the projectile trajectory are sufficient to obtain the ioniza-tion cross section [17]. Currently new experimental data forantiproton collisions with molecular hydrogen are producedusing the AD facility at CERN [19].In response to the renewed experimental activity and thelimited theoretical understanding a full two-electron close-coupling method has been developed. Converged cross sec-tions for single ionization and excitation of molecular hydro-gen are provided over a wide energy range from 1 keV to 6.5MeV on a dense energy grid. They demonstrate the impor-tance of a full two-electron description and of the molecu-lar geometry including different orientations of the molecularaxis as well as the differences between atomic and moleculartargets. To the best of the authors’ knowledge no two-electrondescription for antiproton impacts on molecular targets hasbeen introduced before in this energy range.The collision process is considered in a non-relativisticsemi-classical way using the impact parameter method (cf.[20]) which is known to be highly accurate for impact energies E & keV [6]. The quantum-mechanically treated electronsare exposed to the Coulomb potential of the molecular nucleias well as the heavy projectile. The latter is assumed to moveon a straight classical trajectory R ( t ) = b + v t given by theimpact parameter b and its velocity v while t is the time.In the Born-Oppenheimer approximation the total wavefunction of the hydrogen molecule separates into the product(atomic units are used unless stated otherwise) ˜ ψ (Ω) k ( r , r , R nu ) = χ ( k ) νj ( R nu ) R nu Y mj (Θ , Φ) ψ k ( r , r ; R nu ) , (1)where χ ( k ) νj are the eigenfunctions of the molecular vibra-tion, Y mj the spherical harmonics, and Ω ≡ ( ν, j, m ) denotesthe vibrational and rotational quantum numbers. R nu =( R nu , Θ , Φ) and r , are the position vectors of the nu-clei and the electrons, respectively. The wave function ψ k ( r , r ; R nu ) satisfies the electronic part of the time-independent Schr¨odinger equation ( ˆ H e ) for an unperturbedmolecule at a fixed internuclear distance R nu . The ψ are ob-tained together with the eigenenergies ǫ in a full configuration-interaction (CI) calculation [21, 22]. The two-electron con-figurations are constructed from correctly anti-symmetrizedproducts of orbitals which are eigenstates of the molecularhydrogen ion. The orbitals were obtained as in Ref. [17],where the radial part is expanded in B splines and the angu-lar part in spherical harmonics. More details on the extensionof the close-coupling method from one-electron [17, 18] totwo-electron targets are provided in [23].For a fixed R nu the fully correlated wave function Ψ of thetwo-electron target molecule interacting with the antiproton isobtained by the evolution of the time-dependent Schr¨odingerequation in real time, i ∂∂t Ψ( r , r , t ) = (cid:16) ˆ H e + ˆ V int ( r , r , t ) (cid:17) Ψ( r , r , t ) , (2)where the time-dependent interaction between the electronsand the projectile with charge Z p is expressed by ˆ V int ( r , r , t ) = − Z p | r − R ( t ) | − Z p | r − R ( t ) | . (3)The interaction of the projectile with the nuclei, which leadsonly to an overall phase, is not considered here.The time-dependent scattering wave function Ψ( r , r , t ) = X k c k ( t ) ψ k ( r , r ) e − iǫ k t (4)is expanded in the normalized eigenstates ψ k of ˆ H e . Employ-ing this expansion in Eq. (2) and projecting with ψ k leads tothe usual set of coupled equations for every trajectory R ( t ) , i d c k d t = e iǫ k t X j c j h ψ k | ˆ V int | ψ j i e − iǫ j t . (5)The two-electron interaction matrix elements in Eq. (5) can —according to Eq. (3)— be expressed as a sum of one-electronmatrix elements in the orbital basis. The full two-electron in-teraction matrix element between ψ k and ψ j is therefore the sum of products between the CI coefficients of ψ k and ψ j andthe one-electron matrix elements —calculated as in [17]— be-tween the orbitals of the corresponding CI configurations.The coupled differential equations in Eq. (5) are integratedin a finite z -range −
50 a . u . ≤ z = vt ≤ a.u. with theinitial conditions c k [ R ( t = − /v )] = δ k , i.e., the target isinitially in the electronic ground state ψ with energy ǫ . Theprobability for a transition into the final state ψ k at t f = 50 /v for a fixed R nu is given by p k ( b, E ; R nu , Θ , Φ) = | c k ( b, v, t f ; R nu , Θ , Φ) | . (6)In accordance with [16], the transition probability p k ( b, E ) = Z (cid:12)(cid:12) χ νj ( R nu ) Y mj (Θ , Φ) (cid:12)(cid:12) (7) × p k ( b, E ; R nu , Θ , Φ) sin Θd R nu dΘdΦ . becomes orientation-independent by integration over R nu .The corresponding cross section σ k ( E ) = 2 π Z p k ( b, E ) b d b , (8)can then be obtained by integration over b as is done foratomic targets which are spherical symmetric. The total crosssections for ionization σ ion and bound-state excitation σ exc ofthe target can be obtained by summing up all partial cross sec-tions σ k into states k [as given in Eq. (8)] with ǫ k > I H and ǫ < ǫ k < I H , respectively, where I H is the first ionizationpotential of the hydrogen molecule.In this work, however, p k is approximated by p k = 13 h p k (0 ,
0) + p k (cid:16) π , (cid:17) + p k (cid:16) π , π (cid:17)i , (9)using only the probabilities p k (Θ , Φ) for three orthogonal ori-entations of the internuclear axis with respect to the trajectoryas is discussed in detail in Ref. [17]. It was shown that thisapproximation is equivalent to Eq. (7), if the integration isperformed with a six-point quadrature formula [24]. Further-more, excellent agreement between the results using Eqs. (7)and (9) were obtained for antiproton collisions with molecularhydrogen ions [16, 17].The dependence of the ionization cross section on the inter-nuclear distance R nu in antiproton collisions with the molec-ular hydrogen ion and the hydrogen molecule was examinedin [16] and [11], respectively. In both cases an approximatelylinear dependence on R nu around the expectation value h R nu i of the ground state was observed leading to Franck-Condonresults which were found to be very close to the exact crosssections obtained by an integration over R nu . In this work, thecalculations are accordingly performed for h R nu i = 1 . a.u. The two-electron basis consists of 3080 singlet stateswith total azimuthal quantum numbers M = 0 , . . . , and amaximum energy of ǫ ≈ a.u.In Fig. 1, the results of the time-dependent close-couplingcalculations for (a) single ionization and (b) bound-state ex-citation of the hydrogen molecule by antiproton impact arepresented. The present results are compared to the data avail-able in literature. The single-ionization cross section in the σ i on ( - m ) (a) Single Ionization σ e x ( - m ) (b) Single Excitation Figure 1: (Color online) Cross sections for (a) single ionization σ ion and (b) excitation σ exc by antiproton impact. Black solid curve withpluses, present results for molecular hydrogen; red solid curve with X , H model [11]; blue dashed–dotted curve, scaled hydrogen atomH scal [11]; green doubly-dotted–dashed curve, hydrogen atom [18];maroon dotted curve, two times hydrogen atom [11]; violet dashedcurve with triangles, H scal by Ermolaev [15]; green squares, CERN94 [9]; red circles, CERN 90 [25]. top panel is in excellent agreement with the experimental mea-surements for energies above 85 keV except for the data pointsat 500 keV. Below 85 keV the experimental data show a smalldiscontinuous step and increase to a higher maximum than thepresent results which is situated in both cases around 40 keV.Note, in the extensive convergence studies performed in thiswork an enlargement of the basis always led to smaller valuesof the maximum. Below the maximum the experimental datafall off steeply in a similar way as the data for helium whichwere measured in the same occasion [9]. For helium, how-ever, the two lowest energy data points were withdrawn aftera recent remeasurement [4]. The currently produced experi-mental hydrogen molecule data for low impact energies mayhelp to clarify the trend below the maximum.The literature results obtained using a model potential anda hydrogen atom with scaled nuclear charge Z = 1 . [11, 12, 14] are able to approximate the present calculationsfor energies above 50 and 100 keV, respectively. Though, theyare throughout lower than the latter for these energies. Forlower energies the models yield evidently too large cross sec-tions and in both cases show rather an atomic than a molecularslope by what they reveal their atomic nature. Below the max-imum also the lack of electron-electron correlation effects canbe expected to become severe as in the case of the heliumatom [4, 23]. The calculations by Ermolaev [15], using also a σ i on ( - m ) x 4 Figure 2: (Color online) Comparison of single-ionization cross sec-tions for antiproton impact on different targets. Black solid line withpluses, present results for molecular hydrogen; red dashed curve withcircles, helium atom [23]; blue dot–dashed curve with stars, molecu-lar hydrogen ion [17] (times 4); green doubly-dot–dashed curve withsquares, hydrogen atom [18]. scaled hydrogen atom, are not satisfactory, since they followfor intermediate energies rather the data for a hydrogen atommultiplied by a factor 2.The lower panel of Fig. 1 compares the present excitationcross sections for molecular hydrogen with the existing liter-ature, i.e., the two already mentioned models [11, 18]. Obvi-ously, the scaled hydrogen atom is not capable of reproducingthe excitation cross section for molecular hydrogen despite itsreasonable results for ionization for
E > keV. The modelpotential, on the other hand, is again an excellent approxi-mation for energies above 50 keV. This might have been ex-pected since the bound state energies and oscillator strengthsof the model potential were found to be in good agreementwith those of the hydrogen molecule in contrast to the onespredicted by the scaled hydrogen atom [14]. Note, for ener-gies above the maximum the cross section for excitation of thehydrogen molecule is quite similar to that of atomic hydrogenwhile for ionization it is rather comparable to twice the crosssection of the hydrogen atom.In Fig. 2, the single-ionization cross section of molecularhydrogen by antiproton collision is compared to results for thehelium [23] and hydrogen atom [18] and the molecular hydro-gen ion [17], where the latter is scaled by a factor 4. The com-parison shows that below the maximum the curve for molec-ular hydrogen decreases much faster with decreasing energythan is the case for the hydrogen and helium atom . The molec-ular hydrogen curve is on the other hand qualitatively similarto that of the molecular ion. That is, especially for these ener-gies the qualitative shape of the ionization cross section seemsto be different for atomic and molecular targets. At low ener-gies ionization occurs mainly in a small region close to thenuclei where the electronic density and the expectation valueof the electron velocity are high. In a close encounter of theantiproton on a molecular target the electron cloud might bemore efficiently moved away from the projectile towards theother nuclei since there is in contrast to atoms always one pos-itive particle which is not neutralized by the antiproton. σ ( - m ) (ii)(i) b b bz’ (iii) y’x’ (a) (b) Figure 3: (Color online) Cross sections for (a) single ionization and(b) excitation of molecular hydrogen by antiproton collisions for dif-ferent molecular orientations. Black pluses, orientationally averaged;red circles, ( i ); blue squares, ( ii ); green triangles, ( iii ). The insetshows a sketch of the three orientations in the molecule-fixed frame. At high energies the single-ionization cross section for he-lium and 4 times the molecular hydrogen ion are both similarto the curve of the hydrogen atom while the results for molec-ular hydrogen are in good agreement with twice the curve forthe hydrogen atom [cf. Fig. 1(a)]. For these energies distantencounters are dominating the ionization process. Accord-ingly, details of the targets like the exact distribution of thepositive charges become less important and the cross sectionsare mostly determined by the ionization potential.In Fig. 3, the dependence on the orientation of the molecu-lar axis with respect to the antiproton trajectory is presented.The cross sections σ (Θ , Φ) for (a) ionization and (b) excita-tion are given for the three orthogonal orientations (Θ , Φ) = ( i ) (0 , , ( ii ) ( π/ , , and ( iii ) ( π/ , π/ (cf. the sketch inFig. 3 and Ref. [17]) revealing the following results. First,the curves for the three different orientations generally dif-fer considerably especially around the maximum. Second, thecalculation of only the parallel orientation ( i ) reproduces forenergies above the maximum the orientationally-averaged re- sults with less than 3 % relative deviation. Third, for lowerenergies at which close collisions become dominant the con-sideration of the molecular geometry is inevitable. In contrastto the findings for the molecular hydrogen ion [17], the curvesof orientation ( i ) are close to those of ( ii ) for ionization be-low the maximum and of ( iii ) for excitation below 5 keV. Ingeneral, the differences among the cross sections for the threeorientations are less pronounced than for the molecular hydro-gen ion. This might be due to the smaller internuclear distanceand the two electrons of the hydrogen molecule making it amore spherical target.In conclusion, theoretical data are presented for single ion-ization and excitation of molecular hydrogen by antiprotonimpact for a wide energy range obtained with a two-electrontime-dependent close-coupling method. The experimentaldata are in good agreement with the present calculations athigh energies but are larger around the maximum. For en-ergies below the maximum the ionization cross section de-creases with decreasing energy much faster than in the casesfor the hydrogen and helium atom but in a similar way asfor the molecular hydrogen ion revealing the differences be-tween atoms and molecules. Furthermore, the importance ofthe molecular geometry and a full two-electron description isdemonstrated. The present work should motivate new experi-mental efforts for molecular targets at low impact energies toconfirm and further extend the gained insight. Additionally,it provides benchmark data for molecular collisions in generaland for single ionization and excitation of molecular hydrogenby antiproton impact, in particular, which might be useful forthe development of molecular modes.The method should be further exploited to extract alsoquantities like, e.g. electron-energy spectra or differential ex-citation cross sections. Such results may allow, especially atlow energies, for the elimination of the diversity of results forthe stopping power obtained in different experiments [10, 13].The authors gratefully acknowledge stimulating correspon-dence with H. Knudsen. This work was supported by theBMBF (FLAIR Horizon), the Stifterverband f¨ur die deutscheWissenschaft , and the
Fonds der Chemischen Industrie . [1] Facility for Antiproton and Ion Research.[2] Facility for Low-energy Antiproton and Ion Research.[3] Stored Particle Atomic Research Collaboration.[4] H. Knudsen et al. , Phys. Rev. Lett. , 043201 (2008).[5] T. G. Lee , H. C. Tseng, and C. D. Lin, Phys. Rev. A , 062713(2000).[6] A. Igarashi et al. , Nucl. Instrum. Methods Phys. Res. B ,135 (2004).[7] D. R. Schultz and P. S. Krsti´c, Phys. Rev. A , 022712 (2003).[8] M. Foster, J. Colgan, and M. S. Pindzola, Phys. Rev. Lett. ,033201 (2008).[9] P. Hvelplund et al. , J. Phys. B , 925 (1994).[10] E. Lodi Rizzini et al. , Phys. Rev. Lett. , 183201 (2002).[11] A. L¨uhr and A. Saenz, Phys. Rev. A , 032708 (2008).[12] A. L¨uhr and A. Saenz, Hyperfine Interact. (2009). [13] A. L¨uhr and A. Saenz, Phys. Rev. A , 042901 (2009).[14] A. L¨uhr , Y. V. Vanne, and A. Saenz, Phys. Rev. A , 042510(2008).[15] A. M. Ermolaev, Hyperfine Interact. , 335 (1993).[16] K. Sakimoto, Phys. Rev. A , 062704 (2005).[17] A. L¨uhr and A. Saenz, Phys. Rev. A , 022705 (2009).[18] A. L¨uhr and A. Saenz, Phys. Rev. A , 052713 (2008).[19] H. Knudsen, private communication (2009).[20] B. H. Bransden and M. R. C. McDowell, Charge Exchange andthe Theory of Ion-Atom Collisions (Clarendon, Oxford, 1992).[21] A. Apalategui, A. Saenz, and P. Lambropoulos, Phys. Rev. Lett. , 5454 (2001).[22] A. Apalategui and A. Saenz, J. Phys. B , 1909 (2002).[23] A. L¨uhr and A. Saenz (to be published).[24] L. F. Errea et al. , J. Phys. B , 3855 (1997). [25] L. H. Andersen et al. , J. Phys. B23