Ionisation of hydrogenic ions by impact of antiprotons have been studied at collision energies varying from 2 to 2000 keV/amu. Total and double-differential cross-sections are calculated employing a final state wave function which incorporates distortion due to Coulomb fields of both the projectile and the target nucleus. The present calculated values are compared with the existing theoretical predictions.Proučava se ionizacija vodiku-sličnih iona pri sudarima s antiprotonima na energijama 2 do 2000 keV/ajm. Ukupni i dvostruko-diferencijalni presjeci se računaju upotrebom valne funkcije konačnog stanja koja sadrži izobličenje Coulombovim poljem projektila i jezgri mete. Dobiveni se ishodi računa uspoređuju s ishodima drugih teorijskih predviđanja

Chandl Sil, Nimai

Chandra Mukherjee, Sunil

Roy, Kanika

Sahoo, Sudhakar

English

Hrčak - Portal of scientific journals of Croatia

ISSN 1330–0008CODEN FIZAE4IONISATION OF HYDROGENIC IONS BY ANTIPROTON IMPACTSUDHAKAR SAHOO, KANIKA ROY, NIMAI CHAND SILand SUNIL CHANDRA MUKHERJEEDepartment of Theoretical Physics, Indian Association for the Cultivation of Science, Jadavpur,Calcutta 700 032, IndiaReceived 18 August 1998; revised manuscript received 25 November 1998Accepted 18 January 1999Ionisation of hydrogenic ions by impact of antiprotons have been studied at collisionenergies varying from 2 to 2000 keV/amu. Total and double-differential cross-sectionsare calculated employing a final state wave function which incorporates distortion due toCoulomb fields of both the projectile and the target nucleus. The present calculated valuesare compared with the existing theoretical predictions.PACS number: 34.50.Fa UDC 535.353, 539.125.46Keywords: hydrogenic ions, ionisation by impact of antiprotons, calculations for collision energies2 to 2000 keV/amu1. IntroductionIonisation of charged particles is a process of great importance in view of its significantrole in understanding large number of processes in atomic physics, plasma physics andastrophysics. For many applications in these areas, it is useful to measure the ionisationcross-sections to a given transition of atomic-ionic system as a result of charged particleimpact. Although collision processes have been studied for the last several decades, it isstill a challenge to understand in detail even the simplest of ionisation process. This isFIZIKA A 7 (1998) 4, 157–163 157SAHOO ET AL.: IONISATION OF HYDROGENIC IONS BY ANTIPROTON IMPACTdue to the complex dynamics of the system which consists of more than two particles thatinteract via Coulomb force.The process of electron emission resulting from ionisation of atoms/ions by fastcharged-particle impact is quite complicated. Ionisation can be probed directly by the useof projectiles of different masses and of opposite charges. In the Bethe-Born theory, ion-isation cross-section depends on the square of projectile charge and is independent of itsmass. Nevertheless, evidence for the charge-dependence dynamical effect, such as Barkas’effect, in stopping-power measurements has been known for some time.Antiproton collision with atomic hydrogen and hydrogenic ions provide a unique andfundamental testing ground for the development of non-perturbative quantal scatteringtechnique. For proton impact, the final state is a superposition of elastic scattering, ex-citation, ionisation and charge transfer, whereas for antiproton impact only the first threeprocesses can occur. This essential difference is obviously a result of change in the chargesign of the projectile which is always available to the theoretical physicist for assessmentto antimatter. From the theoretical point of view, the main difficulty is the representationof many-body Coulomb state. Due to the long-range nature of the Coulomb potential, theplane-wave function describing the continuum state remains correlated even to a macro-scopic distance.The first measurements of antiproton-atom collisions were performed by Andersen etal. [1], in which single- and double-differential ionisation cross-sections of He were mea-sured by antiproton impact. A year later, Fainstein et al. [2] applied a theoretical model tocalculate the total ionisation cross-section of He by antiproton impact. Recently, Knudsenet al. [3] accurately measured the total ionisation cross-section of hydrogen atom takingantiproton as projectile.In the present work, we have calculated the total ionisation cross-sections of hydro-genic ions (He+ to O7+) and double-differential cross-sections of He+ by antiproton im-pact. The results of total ionisation cross-section for He+ are compared with the valuesobtained by Shultz et al. [4].In Section 2, we describe briefly the theoretical method adopted here and in Section 3we present our results with discussion.2. TheoryWe consider the ionisation of hydrogenic ions by collision with antiproton of chargeZP. We use the perturbative approach [5] which has already been successfully applied tothe proton-hydrogen system. A brief discussion of the approach may be useful to study thepresent case. We adopt the impact parameter formalism, where the motion of the electronis subjected to the quantum mechanical laws while the internuclear motion is classicallytreated as~R= ~p+~vt, where~R is the internuclear distance,t the time,~p the impact parame-ter and~v the relative velocity of the nuclei. Let the positions of the target and the projectilebe at T+ and P , respectively, as shown in Fig. 1. The midpoint of the line joining thepoints T+ and P  is taken to be at rest and is chosen as the origin. Hence, P  and T+ movewith equal and opposite velocities~v=2 and ~v=2, respectively. The perpendicular (with158 FIZIKA A 7 (1998) 4, 157–163SAHOO ET AL.: IONISATION OF HYDROGENIC IONS BY ANTIPROTON IMPACTrespect to~v) distance between T+ and P  is the impact parameter,~p. Time is measuredfrom the instance when the two nuclei are closest to each other.P-e-T+pr rTrPR+v/2-v/2Fig. 1. The collision diagram for antiproton - hydrogen system.Let~rP; ~rT and~r denote the position vectors of the electron relative to the projectile,target and origin, respectively. The HamiltonianHe may be written (atomic units are con-sidered throughout) asHe= 12∇2r  ZPrP ZTrT; (1)whereZP= 1 andZT varies from 2 to 7.The nucleus-nucleus interaction term is omitted here as this interaction can be removedby a canonical transformation and as such it should not affect any transition probability fora particular impact parameter.The development in time of the transition amplitude for ionisation with the ejection ofelectron having momentum~k can be written asdakdt=Z Ψ kcHe  iddtΨid~r; (2)with initial condition that att =  ∞; ak = 0. The ionisation probability corresponds tojak(t =+∞)j2.The continuum-state wave function in the present work is represented by the productof two Coulomb wave functions,Ψ kc= N1N2ei~k~r1F1(iαP;1; i(kPrP+~kP ~rP))1F1(iαT;1; i(kTrT +~kT ~rT))e ik2=2; (3)where αP = 1=kPe and αT = ZT=kTe, and kPe and kTe are the velocities of electron inthe projectile and the target frame of reference andN1 = exp παP2Γ(1+ iαP) and N2 = exp παT2Γ(1+ iαT) :FIZIKA A 7 (1998) 4, 157–163 159SAHOO ET AL.: IONISATION OF HYDROGENIC IONS BY ANTIPROTON IMPACTThe above wave functionΨ kc asymptotically satisfies the Schr¨odinger equation 12∇2r  ZPrP ZTrT  iddtΨ kc = 0: (4)Considering Eqs. (3) and (4), the ground-state wave function of the hydrogen atom aroundthe target nucleus and the contour representation [6] for the confluent hypergeometric func-tion, we may write Eq. (2) asak(p) =CkIdt1dt2 t 1 iαP1 (t1 1)iαP t 1 iαT2 (t2 1)iαTZeiEtdt (5)Ze ~K~r 1rTe λrT ei(t1kPrP+t2kTrT)ei(t1~kP~rP+t2~kT~rT)d~r;where~K =~k+~v=2; E = k2=2  v2=2  ε0 andCk = (λ3=2N1N2)=(4π5=2). The detailedsteps of evaluation of Eq. (5) has been described in the previous paper [5].The double-differential cross-section and the total cross-section are calculated by per-forming the following integralsd2σdEedΩe= kZd2~p jak(p)j2andσtotal = 2π∞Z0d2σdEedΩesinθedθedEe3. Results and discussion3.1. Total ionisation cross-section (TICS)In Fig. 2, we compare the present calculated results for total ionisation cross-sectionas function of collision energy with the values of Shultz et al. [4] for antiproton impacton He+. The present results for total ionisation cross-section show good agreement withthe values calculated by applying continuum distorted-wave eikonal initial-state (CDW-EIS) calculation [4], throughout the energy region considered. Below 150 keV, the presentresults, however, fail to agree with the computed results of Ref. 4 obtained from the time-dependent Schr¨odinger equation (TDSE), classical trajectory Monte Carlo (CTMC) andhidden crossing (HC) calculations. The present approximation is essentially a high-energyapproximation and does not predict accurate cross-sections in the low-energy region.Figure 3 displays the values of total ionisation cross-sections as a function ofcollision energy for antiproton impact on Li2+, Be3+, B4+, C5+, N6+ and O7+.The magnitude of the cross-section is multiplied by a factor 10 and compared with160 FIZIKA A 7 (1998) 4, 157–163SAHOO ET AL.: IONISATION OF HYDROGENIC IONS BY ANTIPROTON IMPACTthe results for He+ [4]. It is evident from the figure that the values of the cross-Total Cross Section (10  cm )-16      2Energy (keV)10-110 010 110 210 310-1710-1810-1910-20Fig. 2. Total ionisation cross-sections for ionisation ofHe+ by antiproton impact: presentresults: (——-) , CDW-EIS: (– – – –), HC: (- - - - -), CTMC: (— —  —) and TDSE:(   ).Total Cross Section (10  cm )-16      2Energy (keV)(a)(b)(c)(d)(e)(f)10-110 010 110 210 310-1710-1810-1910-20Fig. 3. Total ionisation cross-sections for ionisation of (a)Li2+, (b) Be3+, (c)B4+, (d) C5+, (e) N6+, (f) O7+, Present results (——–) for (a), (b) , (c), (d), (e)and (f) and total ionisation cross-sections of He+ (Ref. 4): CDW-EIS: (– – – –),HC: (- - - - -), CTMC: (– –  –) and TDSE: (   ).sections for different hydrogenic ions show the same trend as that for He+, though theFIZIKA A 7 (1998) 4, 157–163 161SAHOO ET AL.: IONISATION OF HYDROGENIC IONS BY ANTIPROTON IMPACTmagnitude in each case differs from one another.3.2. Double-differential cross-section (DDCS)In Fig. 4, we display the present calculated results of double-differential cross-sections for 300-keV antiproton impact on He+ for fixed emission angles of 0Æ and10Æ. The double-differential cross-sections display prominent features in the ejectedspectra (a sharp exponential dip in the region of the peak due to electron cap-ture to continuum (ECC) in the forward direction (Fig. 4a) and remains an im-portant feature even at 10Æ (Fig. 4b)). The exponential dip arises from ejectedODCS (m /eV Sr)2Electron Energy (eV)10 010 010 110 110 210 210 310 310-2210-2310-2410-2510-2610-2710-28Fig. 4. Double-differential cross-sections ofHe+ as a function of ejection energy byimpact of antiproton of incident energy 300keV for a fixed emission angle of (a)0Æ and(b) 10Æ. Present results (———).electrons travelling with nearly the same velocity as that of the projectile. The dip in ejectedspectra can be considered as a two-center electron-emission effect (TCEE), i.e., as therelated effects on the electron-emission spectra due to the motion of ejected electron underthe influence of a two-centre potential (Coulomb fields of the projectile and the residualtarget) in the final continuum state. In the present case, the normalization factorN1 of thefinal continuum state wave function (Eq. (3)) appears in the cross-section asjN1j2=2παPexp(2παP) 1:It is responsible for producing different structure in DDCS, as for a negatively chargedprojectile (ZP < 0),limkPe!0jN1j2! 2παPexp( 2παP)! 0162 FIZIKA A 7 (1998) 4, 157–163SAHOO ET AL.: IONISATION OF HYDROGENIC IONS BY ANTIPROTON IMPACTgives rise to a singular structure which leads to a pronounced dip in DDCS. Fainstein et al.[7] extensively studied the emission spectra for both positive and negative ion impact on Heatom. They reported that at forward angles, instead of maximum in DDCS due to increasedattraction by heavy ion, one obtains minimum due to the repulsion between the electronand the negatively-charged projectile, and as the angle increases the dip disappears.4. ConclusionIn this work, the single-electron ionisation of hydrogenic ions by impact of antiprotonshave been calculated. The results for the total ionisation cross-section for He+ are found tobe in a reasonably good agreement with CDW-EIS values and also interpret well the valuesof other theoretical calculations, especially in the high-energy region. We also present thetotal cross-section values for other hydrogenic ions (Li2+ to O7+) and compare them withthe results for He+ [4]. The double-differential cross-section for He+ is calculated and theejected spectra are discussed to some extent.The present theoretical approach is comparatively simple and easier to tackle numer-ically. Fairly good results can be predicted for the cross-sections in the intermediate andhigh-energy region.AcknowledgementsThe authors would like to express their thanks to the International Atomic EnergyAgency (IAEA), Vienna, for encouragement under the Research Agreement No. 8003/CF.References1) L. H. Andersen, P. Hvelplund, H. Knudsen, S. P. Mollar, K. G. Rensfelt and E. Uggerhoj, Phys.Rev. Lett57 (1986) 2147;2) P. D. Fainstein, V. H. Ponce and R. D. Rivarola, Phys. Rev. A36 (1987) 3639;3) H. Knudsen, U. Mikkelson, K. Paludan, K. Kirsebom, S. P. Mollar, E. Uggerhoj, J. Selvin,M. Charlton and E. Morenzoni, Phys. Rev. Lett.74 (1995) 4627;4) D. R. Shultz, P. S. Kristic, C. O. Rienhold and J. C. Wells, Phys. Rev. Lett.76 (1996) 2882;5) S. Sahoo, K. Roy, N. C. Sil and S. C. Mukherjee, Physica Scripta58 (1998) 126;6) A. Nordsiek, Phys. Rev.93 (1954) 785;7) P. D. Fainstein, V. H. Ponce and R. D. Rivarola, J. Phys. B21 (1988) 2989.IONIZACIJA VODIKU-SLI ČNIH IONA UDAROM ANTIPROTONAProučava se ionizacija vodiku-sliˇcnih iona pri sudarima s antiprotonima na energijama 2do 2000 keV/ajm. Ukupni i dvostruko-diferencijalni presjeci se raˇcunaju upotrebom valnefunkcije konačnog stanja koja sadrˇzi izobličenje Coulombovim poljem projektila i jezgrimete. Dobiveni se ishodi raˇcuna uspored–uju s ishodima drugih teorijskih predvid–anja.FIZIKA A 7 (1998) 4, 157–163 163

Ionizacija vodiku-sličnih iona udarom antiprotona

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