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Comparative Study of the Electron-And Positron-Atom

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Comparative Study of the Electron-And Positron-Atom Comparative study of the electron- and positron-atom bremsstrahlung V. A. Yerokhin,1, 2, 3 A. Surzhykov,1, 2 R. M¨artin,4, 2 S. Tashenov,1 and G. Weber4, 2 1Institute of Physics, Heidelberg University, Im Neuenheimer Feld 226, D-69120 Heidelberg, Germany 2GSI Helmholtzzentrum f¨ur Schwerionenforschung, Planckstraße 1, D-64291 Darmstadt, Germany 3Center for Advanced Studies, St. Petersburg State Polytechnical University, Polytekhnicheskaya 29, St. Petersburg 195251, Russia 4Helmholtz-Institut Jena, Fr¨obelstieg 3, D-07743 Jena, Germany Fully relativistic treatment of the electron-atom and positron-atom bremsstrahlung is reported. The calculation is based on the partial-wave expansion of the Dirac scattering states in an external atomic field. A comparison of the electron and positron bremsstrahlung is presented for the single and double differential cross sections and the Stokes parameters of the emitted photon. It is demon- strated that the electron-positron symmetry of the bremsstrahlung spectra, which is nearly exact in the nonrelativistic regime, is to a large extent removed by the relativistic effects. PACS numbers: 34.80.-i, 34.50.-s, 41.60.-m, 78.70.En When a charged particle traverses an atomic field, a bremsstrahlung [5, 13]. part of its energy may be converted into radiation. This is the atomic bremsstrahlung, one of the fundamental col- lision processes. It is one of the important mechanisms I. THEORY of the energy loss in hot plasmas and in particle beams traversing thick targets. The process of the electron-atom Relativistic theory of the electron-atom bremsstrahlung has been extensively studied in the liter- bremsstrahlung is nowadays well established and ature. The single and double differential cross sections can be found, e.g., in the review article [14]. For our of this process were tabulated [1, 2] and the polarization bremsstrahlung calculations we use the density-matrix properties of the emitted radiation were calculated [3–6]. formalism described in detail in our previous paper [5]. Much less is known on the positron-atom In this section, we will extend the formulas obtained bremsstrahlung. Several reported investigations of previously to the case of the positron bremsstrahlung. this process [7–9] were dealing primarily with the Such extension can be done by exploiting the fact bremsstrahlung energy loss. It was shown that, for high that in the QED theory, an incoming positron with a energies of the incident and final projectile, the cross four-momentum pi and a helicity mi can be described as sections of the electron and positron bremsstrahlung are an outgoing electron with a four-momentum −pi and a very much similar. However, when the energy of the final helicity −mi (see, e.g., books [15, 16]). projectile decreases, the cross section of the positron We now consider the transition from the electron-atom bremsstrahlung becomes increasingly suppressed as to the positron-atom bremsstrahlung in more detail. The compared to the electron one, because of the strong amplitude of the electron-atom bremsstrahlung is Coulomb repulsion between the positron and nucleus at short distances. The ratio of the positron-to-electron el (+)† Mif (λ)= dr Ψ (εi, pi,mi) bremsstrahlung stopping power was the main subject of Z ik·r (−) those early works. × α · uˆλ e Ψ (εf , pf ,mf ) , (1) To the best of our knowledge, the angular dependence (+) of the cross section and the polarization properties of where Ψ (εi, pi,mi) is the wave function of the inital the relativistic positron-atom bremsstrahlung have never electron state with the energy εi, the momentum pi, the been studied. They are becoming subjects of exper- helicity mi, and the asymptotics of an outgoing spherical arXiv:1209.5532v1 [physics.atom-ph] 25 Sep 2012 (−) imental interest today, with the advent of techniques wave, and Ψ (εf , pf ,mf ) is the wave function of the for the production of highly polarized positrons beams final electron state with the energy εf , the momentum [10] and the detection of polarization correlations in the pf , the helicity mf , and the asymptotics of an incoming bremsstrahlung radiation [11, 12]. spherical wave, and k and λ are the momentum and po- In the present investigation, we make a comparative larization of the emitted photon, respectively. The Dirac study of the positron- and electron-atom bremsstrahlung, scattering states Ψ(±)(ε, p,m) are given by their partial- by analyzing the double differential cross section and the wave expansion [17, 18] polarization correlations between the incident projectile (±) 1 l ±i∆κ jµ ∗ and the emitted photon. The calculation is performed Ψ (ε, p,m)= i e C 1 Ylm (pˆ) |εκµ , p |ε| lml, 2 m l within the fully relativistic approach based on the partial- Xκµ wave representation of the Dirac continuum states with p (2) a fixed value of the asymptotic momentum. This work extends our previous calculations of the electron-atom where |εκµ are the Dirac continuum states with a given 2 angular-momentum quantum number κ and angular- density matrix of the final (photon) state, momentum projection µ, j = |κ|− 1/2, l = |κ + 1/2|− 1/2, (0) and ∆κ = σκ − σκ is the difference between the asymp- ′ ∗ ′ hkλ|ρf |kλ i = dΩf M (λ) Mi′f (λ ) totic large-distance phase of the Dirac-Coulomb solution Z if miXm ′ mf and the free Dirac solution (see book [17] for details). i In order to obtain the amplitude of the positron-atom × hpimi|ρi|pimi′ i , (4) bremsstrahlung from Eq. (1), we need to make the fol- lowing substitutions in the wave functions: ε → −ε, where Ωf is the solid angle of the scattered positron, p → −p, m → −m, Ψ(±) → Ψ(∓), and to interchange and hpimi|ρi|pimi′ i is the density matrix of the initial the initial and the final state. The resulting amplitude positron state. Note that the inital density matrix is the of the positron-atom bremsstrahlung is given by same for positrons and electrons (since it depends only on the spin of the particle and not on its charge). The pos (+)† photon direction kˆ = k/k will be characterized by the M (λ)= dr Ψ (−εf , −pf , −mf ) if Z Euler angles (θ, φ), with the z axis directed along the mo- ik·r (−) mentum of the initial-state projectile p . The final-state × α · uˆλ e Ψ (−εi, −pi, −mi) , (3) i density matrix (4) contains all the information needed where pi(pf ) and mi(mf ) are the momentum and helicity for calculating the differential cross section and the po- of the initial (final) state positron, respectively. larization correlations of the bremsstrahlung radiation. Assuming that the final-state positron is not observed Extending the derivation given in Ref. [5] to the case in the experiment, we introduce the two-by-two reduced of the positron-atom bremsstrahlung, we obtain ∗ ′ ′ ′ − − i∆κ ,−ε −i∆κ′ ,−ε ′ ′ ′ 1/2 4 g ˆ κ (i) li li L+L i i i i kλ|ρf |kλ = 8(2π) Dγ1γ2 (k) (−1) ρκ,γ1 i e [L,L , ji, ji,li,li,g,κ] ′ ′ 1 2 κiXκiκf LLXκgt γXγ 1/2 1/2 κ ′ 2 ji−jf +li+g+κ gγ t0 t0 L jf ji ′ ′ ′ ′ × (−1) CL −λ ,Lλ Cli0,l 0 Cg−γ1,κγ1 ′ ′ ji ji g i ji g L ′ li li t ′ ∗ p ′ p′ (p) ′ (p ) × (iλ) (−iλ ) −εiκi α · aL − εf κf −εiκi α · aL′ − εf κf , (5) ′ D E D E Xpp L (i) where DMλ is Wigner’s D function [19], ρκ is the spheri- case, it is convenient to transform the negative-energy cal tensor of the initial-state density matrix [see Eq. (10) states to the positive-energy ones by using the symmetry (p) of the radial Dirac equation. One can show that the up- of Ref. [5]], aL are the magnetic (p = 0) and electric (p = 1) operators defined by Eqs. (15)-(17) of Ref. [5], per and lower radial components of the negative-energy Dirac solution with the potential V , gV and f V , can [x1, x2,...] ≡ (2x1 + 1)(2x2 + 1) .... The states |− εiκii −ε,κ −ε,κ be expressed in terms of the components of the positive- and | − εf κf i are the spherical-wave continuum-state energy solutions with the potential −V as Dirac wave functions with a negative energy, −εi < −m and −εf < −m, where m is the electron rest mass. V −V g−ε,κ(r)= fε,−κ(r) , (6) In our calculations of the final-state density matrix, f V (r)= g−V (r) . (7) we used the method described in Ref. [5], with radial in- −ε,κ ε,−κ tegrals evaluted numerically and the integration contour This approach to the evaluation of the negative-energy rotated to the imaginary axis. We performed calculations Dirac states was previously used in Ref. [20]. for two types of the target, (i) the bare nucleus and (ii) the neutral atom. In the latter case, the electronic struc- ture of the atom was represented by a static screening II. RESULTS AND DISCUSSION potential obtained by the Dirac-Fock method. The negative-energy continuum-state Dirac wave func- We begin with calculating the single-differential tions |− εκi for the point Coulomb potential can be cal- cross section of the electron and positron-atom culated by using their analytic representation in terms of bremsstrahlung. The results can be conveniently rep- the Whittaker M function [18], similarly to that for the resented in terms of the scaled cross section σ ≡ positive energies. For the neutral atoms, however, the (k/Z2) dσ/dk, where Z is the nuclear charge and k is wave functions have to be computed numerically.
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