Excitation of highly charged hydrogen-like ions by the impact of equivelocity electrons and protons: a comparative study
aa r X i v : . [ phy s i c s . a t o m - ph ] F e b Excitation of highly charged hydrogen-like ions by the impact of equivelocity electronsand protons: a comparative study
B. Najjari and A.B.Voitkiv Institut Pluridisciplinaire Hubert Curien, Universit´e de Strasbourg,23 rue du Loess, BP 28, 67037 Strasbourg Cedex 2, France ∗ Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany † Abstract
We consider excitation of highly charged hydrogen-like ions by the impact of equivelocity electrons and protons. The kineticenergy of the protons is more than three orders of magnitude larger than that of the equivelocity electrons. It is shown, however,that despite this fact, the electrons can be much more effective in inducing excitation at collision velocities (slightly) above thetheshold for electron impact excitation. The basic reason for this is the strong distortion of the motion of the electron by theattractive field of the nucleus of the highly charged ion.
PACS numbers: PACS:34.10.+x, 34.50.Fa
I. INTRODUCTION
Excitation of highly charged ions by the impact ofcharged particles (projectiles) is an interesting physi-cal process which may also have many applications. Inparticular, in case of electron projectiles the studies ofthis process are of importance for the physics of high-temperature plasmas produced in laboratories and exist-ing in astrophysical sources.Collisions of energetic highly charged ions with atomsrepresent an important field of research at modern accel-erators of heavy ions. In collisions of a highly chargedion with neutral atoms the ion can also be excited. Ifthe momentum transferred to the atom in the collision ismuch larger than the typical momenta of the electrons inthe atom the excitation process can be regarded as oc-curring due to the incoherent interactions of the electronof the ion with the nucleus and the electrons of the atomwhich behave with respect to each other as (quasi-) freeparticles [1]. In the rest frame of the ion the excitationthen can be viewed as induced by the incoherent impactsof the “independent” beams of the atomic nucleus andatomic electrons.If the collision velocity is much larger than the Bohrvelocity in the K -shell of the atom, the contribution σ N to the excitation cross section caused by the interactionwith the nucleus of the atom is very simply related to thecross section σ p for excitation by proton impact: σ N = Z A σ p , where Z A is the charge of the atomic nucleus.We thus see that the excitation of a highly charged ionin collisions with atoms can, under certain conditions,be reduced to two basic processes: excitations in colli-sions with an equivelocity electron and proton. In thisrespect a question arises about the relative effectivenessof these two types of projectiles in producing the excita-tion. Note that althought excitation of ions by electronic ∗ Electronic address: [email protected] † Electronic address: [email protected] and protonic (nuclei) projectiles has been studied (seee.g. [2]-[11] and references therein), to our knowledgethese studies were always done separately for electronsand protons.From the perspective of atomic physics the differencesbetween the electrons and protons mainly include i) thehuge difference in masses and also ii) the opposite signof their charges. For instance, due to the first point,the kinetic energies of equivelocity electrons and protonsdiffer by about a factor of 2000.A trivial consequence of this fact is that in the rangeof impact velocities v below the threshold velocity v th for incident electrons, where these electrons do not haveenough energy to excite the ion, the protons do have andare capable of producing excitation. Besides, it is alsoquite natural to expect that at sufficiently high impactvelocities, where the kinetic energy of the incident elec-tron is much larger than the excitation energy of the ion,the cross sections for excitation by equivelocity electronsand protons will converge.What, however, can can say about the relative effec-tiveness of these two projectiles in case when the impactvelocity is already above the threshold velocity v th butthe kinetic energy of the incident electron is not yet muchlarger than the excitation energy? Below in this article,where excitation of highly charged hydrogen-like ions bythe impacts of equivelocity electrons and protons is con-sidered for a broad range of the atomic numbers of theions, we shall address this question.The article is organized as follows. In the next sec-tion we briefly outline the basic physics of proton-ionand electron-ion collisions and discuss how one can cal-culate the corresponding excitation cross sections. Insection III we present results for excitation cross sec-tions of hydrogen-like ions of nickel, xenon, erbium, bis-muth and uranium (the corresponding atomic numbersare: Z I = 28, 54, 68, 83 and 92). The main results aresummarized in section IV.Atomic units (¯ h = m e = | e | = 1) are used throughoutthe paper except where otherwise stated.1 I. THEORY
To an excellent approximation the nucleus of the ion,which has a charge Z i ( Z i ≫ S fi = − ic Z d x Z d y j µ ( x ) D µν ( x − y ) J ν ( y ) . (1)Here, j µ ( x ) and J ν ( y ) ( µ, ν = 0 , , ,
3) are the elec-tromagnetic transition 4-current densities generated bythe electron of the ion at a space-time point x andby the projectile at a space-time point y , respectively,and D µν ( x − y ) is the propagator of the electromagneticfield which transmits the interaction between these par-ticles. The contravariant a µ and covariant a µ a µ = ( a , a ) and a µ = ( a , − a ). The met-ric tensor g µν of the four-dimensional space is defined by g = − g = − g = − g = 1 and g µν = 0 for µ = ν .In (1) the summation over the repeated greek indices isimplied.Provided the transition currents in (1) (and in the cor-responding exchange contribution to the amplitude) areevaluated using the relativistic description of the boundand free particles, the treatment of the excitation pro-cess is fully relativistic. In particular, in this treatmentthere is no upper limit on the collision energy and alsoexcitation of most heavy ions may be considered. A. The effect of the field of the nucleus of the ionon the motion of the incident and scattered particle
Since we suppose that the nucleus of the ion has ahigh charge, its field can in general strongly influencenot only the motion of the bound electron but also thatof the incident (and scattered) particle. A simple esti-mate for the magnitude of the effect of this field on themotion of the incident electron and/or proton in the pro-cess of excitation can be obtained in the following way[1]. Assume that there is a particle with a charge q andmass m which is incident with a velocity v on the nucleus Z i . One can estimate the effect of the field by using theratio ς = δp/p i , where p i = mγv ( γ = 1 / p − v /c )is the initial momentum of the incident particle and δp is the change in the momentum of this particle causedby the field of Z i . This change is roughly given by δp ∼ Z i q/ ( bv ), where b is the impact parameter. Forthe problem of excitation the typical impact parametersare of the order of 1 /Z i or larger. Therefore, ς ≃ | q | mγ Z i v . (2) In order to make the process of excitation in collisionswith electrons energetically possible, one roughly needs Z i /m e v < ∼
1. Therefore, it follows from (2) that for theimpact velocities of interest for the present article theparameter ς is very small in the case of proton projectiles( ς < ∼ /m p γ < − ). As a result, the field of the ionicnucleus does not affect the motion of the proton whichcan be regarded in the initial and final states as a freeparticle. In contrast, for electron projectiles ς may beclose to 1 ( ς < ∼ /m e γ < ∼
1) which means that the field ofthe nucleus can very strongly distort the motion of theelectron. Indeed, it will be seen below that this distortioncan have a crucial impact on the process of excitation.
B. Excitation in collisions with protons
The treatment of excitation of a highly chargedhydrogen-like ion by protons is based on the followingmain points (see e.g. [1]).First, the charge of the proton is much smaller thanthat of the highly charged nucleus of the ion. As a result,the interaction between the proton and the electron of theion in the process of excitation is much weaker than theinteraction between the electron and the ionic nucleus.Therefore, it can be regarded as a weak perturbation andmay be taken into account within one-photon exchange(first-order perturbation theory).Second, as was already mentioned in the previous sub-section, due to the relatively heavy mass of the proton thedistortion of its motion caused by the field of the nucleusof the ion can be ignored. Then, regarding the protonas a Dirac particle, one can approximate the initial andfinal states of the proton by (Dirac) plane-waves.
C. Excitation in collisions with electrons
Let us now briefly discuss the treatment of excitationof a highly charged hydrogen-like ion in collisions withelectrons (see e.g. [2]-[5], [1]).Like in the case of collisions with protons, the inter-action between the incident electron and the electron ofthe ion is comparatively very weak. Therefore, the de-scription of this interaction in the excitation process canbe reduced to just single-photon exchange between theelectrons.However, in contrast to the excitation by protons, theinteraction between the incident electron and the ion ingeneral cannot be treated within the first-order pertur-bation theory. The reason is that the motion of the in-cident (and scattered) electron can be very substantiallyaffected by its interaction with the nucleus of the ion.This point can be addressed by describing not only thebound but also the continuum electron as moving in theCoulomb field of the nucleus of the ion.2urther, the electrons are indistinguishable and, there-fore, the exchange effect has to be taken into account byincluding an additional diagram (the so called exchangediagram) into the treatment of electron-impact excita-tion.Below we shall refer to the treatment, which (i) de-scribes the continuum electron as moving in the Coulombfield of the nucleus of the ion, (ii) takes into account theinteraction between the continuum and bound electronswithin first-order perturbation theory and (iii) includesthe exchange effect, as Approach I.In addition, in the next section we shall present resultsfor electron impact excitation obtained by using another– simplified – treatment. This simple treatment – termedApproach II – also describes the interaction between thefree and bound electrons within first-order perturbationtheory but neglects the distortion of the continuum elec-tron states by the field of the ion (approximating themby Dirac plane waves) and does not take into account theexchange effect between the free and bound electrons.Note that both these approaches do not take into ac-count the channel of resonance excitation. This chan-nel may become effective when the energy of the initialconfiguration of the electrons (the incident electron plusthe electron bound in the ground state of a hydrogen-like ion) closely matches an energy of a doubly excitedbound state of the correspoding helium-like ion. Undersuch conditions the excitation of a hydrogen-like ion mayproceed via formation of a doubly excited bound state ofthe corresponding helium-like ion which then decays dueto autoionization into an excited state of the hydrogen-like ion (see e.g. [7]).
III. NUMERICAL RESULTS AND DISCUSSION
Here we shall consider excitation of Ni (1s),Xe (1s), Er (1s), Bi (1s) and U (1s) ionscaused by the impacts of equivelocity electrons and pro-tons. We restrict ourselves to the excitation into the L -shell only (for which the cross sections are much largerthan for the higher shells). The corresponding results areshown in figures 1 - 5 where the calculated cross sectionsfor excitation by electron impact are displayed by solidand dash curves and those for excitation by protons aredepicted by dot curves.The main conclusion, which can be drawn from the fig-ures, is that, despite the huge difference in kinetic energy,the electrons are overally not less effective than protonsin inducing the excitation. One more important pointfollowing from the figures is that the relative effective-ness of the electron projectiles compared to that of theprotons substantially increases when the atomic numberof the ion grows.The very large difference in kinetic energies betweenequivelocity electrons and protons makes the phase spacefor the final states of the outgoing electron (the cross sec-tion is proportional to the volume of this space) much electron impact energy (keV) (c) c r o ss s e c t i on ( b ) (b) (a) FIG. 1:
Cross sections for excitation of hydrogen-like nickel( Z I = 28) by equivelocity electrons and protons given as a func-tion of the electron kinetic energy. Sections (a), (b) and (c) showthe cross sections for the 1 s / → s / , 1 s / → p / and 1 s / → p / transitions respectively. Solid and dash curves show theresults for excitation by electron impact obtained by using Ap-proach I and Approach II, respectively. Dot curve displays theresults for excitation by protons. Experimental data for electronimpact excitation are from [8].
25 50 75 10004812036902468 c) electron impact energy (keV) b) c r o ss s e c t i on ( b ) a) FIG. 2:
Same as in figure 1 but for excitation of hydrogen-like xenon ( Z I = 54). Experimental result for electron impactexcitation is from [6]. smaller compared to that of the scattered proton. Sincethe volume of this space is proportional to ∼ k f dk f ∼ k f ε f dε f /c , where k f and ε f are the momentum and to-tal energy of the outgoing electron, it becomes especiallysmall when the impact energy of the incident electronapproaches the excitation threshold.This is why the cross section calculated using the sim-plified Approach II, in which the incident and scatteredelectron is described by plane waves, increases from zeroat the excitation threshold. Contrary to Approach II,however, the more sophisticated Approach I leads to thecross sections which have their maxima at the electronimpact energy equal to the excitation energy [12].3 a)b) e xc i t a t i on c r o ss s e c t i on ( b ) c) electron impact energy (MeV) FIG. 3:
Same as in figure 1 but for excitation of hydrogen-likeerbium ( Z I = 68). a)b) e xc i t a t i on c r o ss s e c t i on ( b ) c) electron impact energy (MeV) FIG. 4:
Same as in figure 1 but for excitation of hydrogen-likebismuth ( Z I = 83). Such a behaviour is the consequence of the well knownsingularity which is present for the continuum states ofan electron moving in an attractive Coulomb field withan asymptotic momentum k →
0. This singularity com-pensates for the smallness of the phase space of the out-going electron. Thus, it is the distortion of the motionof the unbound electron by the attractive Coulomb fieldof the ion which makes the electronic projectiles so ef-fective in exciting the ion. This distortion is especiallystrong for the low-velocity electrons which results in thefact that at the excitation threshold and slighly above itthe electrons can be even much more effective than theequivelocity protons.
IV. CONCLUSION
We have considered excitation of highly chargedhydrogen-like ions in collisions with equivelocity electrons e xc i t a t i on c r o ss s e c t i on ( b ) a)c) electron impact energy (MeV) FIG. 5:
Same as in figure 1 but for excitation of hydrogen-likeuranium ( Z I = 92). and protons. We have shown that the electronic projec-tiles are not less effective in inducing the excitation thanthe protons. Moreover, according to our results the rela-tive effectiveness of electronic projectiles increases whenthe atomic number of the ion increases.The differences between these two types of projectiles,which influence the process of excitation, lie in the verylarge differences in their masses and also in the sign ofcharge.The large mass of protonic projectiles in generalfavours the process of excitation. Indeed, it furnishesa large phase space for the scattered proton and alsostrongly diminishes the effect of the repulsion betweenthe proton and the nucleus of the ion enabling the pro-ton to come closer to the electron of the ion (compared,say, to an equivelocity positron) increasing their interac-tion.The small mass of electronic projectile has a two-foldinfluence on the excitation process. One the one hand,compared to a proton an equivelocity electron has muchless kinetic energy which per se would make the elec-trons substantially less effective in inducing excitationclose to the threshold compared to equivelocity protons.However, due to the smallness of the electron mass themotion of the incident and scattered electrons may bevery strongly affected by the field of the nucleus of theion. For electrons this field is attractive and pulls in theincident electron closer to the electron of the ion whichincreases the chances for excitation.Based on the results of this study one can also makea (rather obvious) conclusion that at the threshold apositron projectile would be very inefficient in inducingthe excitation because of its strong repulsion by the nu-cleus. In particular, in collisions with positronium theeffect of excitation at velocities slighly above v th wouldfully come from the electron while the positron wouldbe merely a spectator. In this respect it is interestingto note that such a situation seems to take place even4n collisions of a positronium with a neutral atom [13]where the repulsion effect is much weaker than in case ofcollisions with a highly charged ion. Acknowledgement
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