arXiv:1209.5532v1 [physics.atom-ph] 25 Sep 2012 xed u rvoscluain fteelectron-atom the work of This calculations momentum. previous asymptotic our the with of extends states value continuum fixed Dirac a performed the of is partial- the representation calculation on wave based The approach relativistic fully the photon. within emitted projectile the incident and the the and between section correlations cross differential polarization double the bremsstrahlung, analyzing electron-atom by and positron- the of study the in 12]. correlations beams [11, polarization positrons radiation of techniques bremsstrahlung polarized detection exper- of the highly of and advent of [10] subjects the production becoming the with for are today, of interest They properties imental polarization studied. never the have been bremsstrahlung and positron-atom section relativistic the cross the of of positron-to-electron subject the works. main early of the strong those was ratio power the The stopping at of bremsstrahlung nucleus and because distances. positron as short one, the between electron positron suppressed repulsion Coulomb the the increasingly of to section compared becomes cross final the bremsstrahlung the of cross decreases, energy the the are projectile when However, bremsstrahlung the projectile, positron similar. much final and very with electron and the high incident primarily of for sections that, the dealing shown of of was It were energies investigations loss. [7–9] energy reported bremsstrahlung process Several this bremsstrahlung. rpriso h mte aito eecluae [3–6]. calculated polarization were the radiation and emitted sections 2] the cross [1, of tabulated differential properties were double process and this of single liter- The the in beams studied ature. particle extensively in been the and has of bremsstrahlung plasmas mechanisms process The hot important targets. in the thick traversing loss of energy one the is of It This col- fundamental processes. radiation. the of lision into one converted bremsstrahlung, be atomic the may is energy its of part ntepeetivsiain emk comparative a make we investigation, present the In dependence angular the knowledge, our of best the To uhls skono the on known is less Much a field, atomic an traverses particle charged a When 1 oprtv td fteeeto-adpsto-tmbrems positron-atom and electron- the of study Comparative nttt fPyis edlegUiest,I Neuenheim Im University, Heidelberg Physics, of Institute 2 ASnmes 48.i 45.s 16.m 78.70.En 41.60.-m, t 34.50.-s, by 34.80.-i, removed numbers: extent PACS large a to is bremsst the regime, of nonrelativistic symmetry the parame brems electron-positron Stokes positron the the and that and strated electron sections cross the differential th of double of comparison and expansion A partial-wave the field. on atomic based is calculation The S emotznrmfu cwroefrcug Plancks f¨ur Schwerionenforschung, Helmholtzzentrum GSI ul eaiitcteteto h lcrnao n posi and electron-atom the of treatment relativistic Fully .A Yerokhin, A. V. 3 etrfrAvne tde,S.Ptrbr tt Polytech State Petersburg St. Studies, Advanced for Center 4 emot-ntttJn,F¨blte ,D073Jn,Ge Jena, D-07743 Fr¨obelstieg 3, Jena, Helmholtz-Institut ,2 3 2, 1, oyehihsaa2,S.Ptrbr 921 Russia 195251, Petersburg St. 29, Polytekhnicheskaya .Surzhykov, A. positron-atom electron-atom ,2 1, .M¨artin,R. Ψ 18] [17, expansion wave aiaino h mte htn epciey h Dirac The Ψ respectively. states photon, scattering emitted the of larization p peia ae and wave, spherical lcrnsaewt h energy the with state electron nleeto tt ihteenergy the with state electron final hr Ψ where uhetnincnb oeb xliigtefact a with the positron exploiting incoming an by theory, done QED four-momentum bremsstrahlung. the be obtained positron in formulas can the that the of extension extend case Such will [5]. the paper we to our previous For previously section, our this in [14]. and density-matrix detail In article in the review described established use the formalism we in well calculations e.g., bremsstrahlung found, nowadays be can is bremsstrahlung helicity 13]. [5, bremsstrahlung ae n Ψ and wave, helicity mltd fteeeto-tmbestaln is bremsstrahlung electron-atom The the detail. of more amplitude in bremsstrahlung positron-atom the to noton lcrnwt four-momentum a with electron outgoing an where f ( eaiitcter fteelectron-atom the of theory Relativistic enwcnie h rniinfo h electron-atom the from transition the consider now We ± h helicity the , ) ( ,2 4, rFl 2,D610Hiebr,Germany Heidelberg, D-69120 226, Feld er ε, aln pcr,wihi eryeatin exact nearly is which spectra, rahlung | eso h mte htn ti demon- is It photon. emitted the of ters εκµ rß ,D621Drsat Germany Darmstadt, D-64291 1, traße erltvsi effects. relativistic he M p ia cteigsae na external an in states scattering Dirac e m rnao rmsrhugi reported. is bremsstrahlung tron-atom − (+) .Tashenov, S. m , if m i el taln speetdfrtesingle the for presented is strahlung n h smttc fa ugigspherical outgoing an of asymptotics the and ,

( ( i = ) ε λ r h ia otnu ttswt given a with states continuum Dirac the are ( se .. ok 1,16]). [15, books e.g., (see, i − = ) , ) p m ( p i p ε m , × f f i Z p ia University, nical ( 1 , n h smttc fa incoming an of asymptotics the and , n helicity a and ± α p k | i ε d .THEORY I. ) stewv ucino h inital the of function wave the is ) f ( 1 | r · and ε, m , rmany u ˆ X Ψ n .Weber G. and κµ p λ (+) f m , e λ stewv ucino the of function wave the is ) i i l k † r h oetmadpo- and momentum the are r ie yterpartial- their by given are ) e · ( r ε ± ε i Ψ i i , ∆ h momentum the , strahlung p ( m − κ i m , ) C i ( ε lm ε a edsrbdas described be can jµ f f ,2 4, i ) , l h momentum the , , p 1 2 f m m , Y lm ∗ f − ) l p ( , i p ˆ p ) n a and i | the , εκµ (1) (2)

, 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. In this the photon energy. This part of our calculations can 3 be compared with the previous work by Feng, Pratt, eters is rather weak, the results shown were obtained for and Tseng [8]. Very good agreement with their cal- the bare nucleus targets. We again observe that for the culation was found. So, for the inital kinetic energy light targets (carbon), the polarization of the positron E = 50 keV, the fractional energy carried by the pho- bremsstrahlung is almost identical to the electron one. ton k/E = 0.6, and the bare-nucleus target, we obtain For the gold target, however, the relativistic effects break the ratio of the positron-to-electron bremsstrahlung cross the electron-positron symmetry of the bremsstrahlung ra- sections σ+/σ− = 0.61498 for Z = 8 and 0.003839 for diation. The difference between the electron and positron Z = 92, whereas Ref. [8] reports 0.615 and 0.00384, re- bremsstrahlung spectra becomes increasingly more pro- spectively. nounced when the initial projectile energy is enlarged. Our numerical results for the cross sections of the The second polarization correlation that is of the ex- positron and electron bremsstrahlung are presented in perimental interest today is the Stokes parameter of the Fig. 1 for two targets, carbon and gold, and the ini- emitted photon P2. It manifests itself as a rotation of tial projectile energy of 100 keV. Carbon is an essen- the polarization ellipse of the emitted radiation in the tially nonrelativistic system. In this case, the positron plane perpendicular to the photon momentum. The ra- bremsstrahlung is only slightly suppressed as compared tio of P2 and P1 yields the tilt angle χ of the polarization to the electron one and the difference between the bare ellipse (P2/P1 = tan2χ), which can nowadays be mea- nucleus and neutral atom is very small (i.e., the screen- sured very precisely [11]. Our numerical results for the ing effect of the atomic electrons is nearly negligible). Stokes parameter P2 for the initially longitudinally polar- For gold, on the contrary, the relativistic binding effects ized projectile are presented in Fig. 5. Since P2 is purely are large. Within the classical-physics picture, one can a relativistic effect, its typical numerical values are very expect that the strong Coulomb potential of the nucleus small for low-Z target but grow rapidly when the initial significantly changes the projectile velocity at the point projectile energy and the nuclear charge are increased. of the closest approach (where the photon emission is An important observation is that the numerical values most probable) and thus breaks the symmetry between of P2 for the initially longitudinally polarized electrons the electron and the positron spectra. Indeed, our calcu- and positrons are of the opposite sign. This could have lations for gold show a large suppression of the positron been anticipated from the fact that P2 scales linearly with bremsstrahlung in the region k/E > 0.5 and also a signif- Z [4] and, therefore, changes its sign after the substitu- icant screening effect of the target electrons. We observe tion Z → −Z. The consequence of the opposite sign of − + that the screening effect reduces the cross section for P2(e ) and P2(e ) is that the rotation of the polariza- the case of the electron bremsstrahlung but enhances it tion ellipse of the emitted radiation for the longitudinally for the positron bremsstrahlung. Remarkably, in the re- polarized electron and positron beams will occur in the gion k/E ∼ 1, the screening effect enhances the positron opposite directions. − + bremsstrahlung by an order of magnitude as compared Beside this effect, we observe that P2(e ) and −P2(e ) to the pure Coulomb field (as noted already in Ref. [8]). nearly coincide for light targets but become increasingly But, as the resulting cross section is still very small, the different as the nuclear charge and the initial projectile effect is difficult to observe experimentally. energy are enlarged. We also note that the difference be- The numerical results for the double-differential cross tween the Stokes parameters is generally smaller at the 2 section, dσ ≡ (k/Z ) dσ/(dk dΩk), are presented in forward angles and larger at the backward angles. This Fig. 2. The calculation was performed for neutral atomic effect can be understood from simple classical-physics ar- targets (carbon and gold) and several energies of the guments. In order to scatter backwards off the positively- incoming projectile. We observe that the dominant charge nucleus, the electron has to go all the way around difference between the electron dσ− and positron dσ+ the nucleus, whereas the positron just enters the nuclear cross sections comes from the suppression of the positron field and bounces back. So, for the back scattering, the bremsstrahlung. As can be seen from the picture, the paths within the region of the strong field (and, therefore, suppression grows with increasing the nuclear charge and the accumulated relativistic effects) are very different for decreasing the energy of the initial projectile. This is the electron and the positron. For the forward scatter- in agreement with Ref. [9], which concludes that, for a ing, however, mainly the straight trajectories contribute large range of the kinetic energy of the incoming projec- and so the paths of the electron and the positron in the tile Ekin, the overall suppression factor is a nearly linear strong Coulomb field are almost the same. 2 function of Z /Ekin. In summary, we perfomed a calculation of the electron- Next, we compare the polarization properties of the atom and positron-atom bremsstrahlung within the rig- electron and positron bremsstrahlung. The most impor- orous relativistic approach based on the partial-wave tant polarization property is the Stokes parameter of the expansion of the Dirac wave functions in the exter- emitted photon P1 for the initially unpolarized projec- nal atomic field. Comparison between the electron and tile. (In this case, P1 yields also the degree of the linear positron bremsstrahlung spectra is made for the single polarization of the emitted radiation.) The numerical re- and double differential cross sections and the Stokes pa- sults for P1 are presented in Figs. 3 and 4. Since the rameters of the emitted radiation. It is demonstrated screening effect of atomic electrons on the Stokes param- that for the low-Z targets, the polarization of the elec- 4 tron and positron bremsstrahlung radiation is very much The work reported in this paper was supported by similar (except for the polarization correlations vanish- the Helmholtz Gemeinschaft (Nachwuchsgruppe VH- ing in the nonrelativistic limit). However, for heavy rel- NG-421). S.T. acknowledges the support by the German ativistic targets and high impact energies, the positron Research Foundation (DFG) within the Emmy Noether bremsstrahlung becomes significantly suppressed and dis- program under contract No. TA 740 1-1. torted as compared to the electron bremsstrahlung.

[1] R. H. Pratt, H. K. Tseng, C. M. Lee, and L. Kissel, At. Yerokhin, and Th. St¨ohlker, Nucl. Instr. Meth. Res. B Data Nucl. Data Tables 20, 175 (1977). 279, 155 (2012). [2] L. Kissel, C. A. Quarles, and R. H. Pratt, At. Data Nucl. [14] R. H. Pratt and I. J. Feng, The electron bremsstrahlung Data Tables 28, 381 (1983). spectrum from neutral atoms and ions, In: C. F. Barnett [3] H. K. Tseng and R. H. Pratt, Phys. Rev. A 3, 100 (1971). and M. F. Harrison (eds.), in Applied Collision Physics. [4] H. K. Tseng and R. H. Pratt, Phys. Rev. A 7, 1502 Academic Press, NY, 1984. (1973). [15] J. D. Bjorken and S. D. Drell, Relativistic Quantum [5] V. A. Yerokhin and A. Surzhykov, Phys. Rev. A 82, Fields, McGraw-Hill, NY, 1965. 062702 (2010). [16] C. Itzykson and J. Bernard Zuber, Quantum Field The- [6] D. H. Jakubassa-Amundsen, Phys. Rev. A 82, 042714 ory, McGraw-Hill, NY, 1980. (2010). [17] M. E. Rose, Relativistic Electron Theory, John Wiley, [7] R. J. Jabbur and R. H. Pratt, Phys. Rev. 129, 184 NY, 1961. (1963). [18] J. Eichler and W. Meyerhof, Relativistic Atomic Colli- [8] I. J. Feng, R. H. Pratt, and H. K. Tseng, Phys. Rev. A sions, Academic Press, San Diego, 1995. Note the mis- 24, 1358 (1981). printed overall sign in Eq. (4.115). [9] L. Kim, R. H. Pratt, S. M. Seltzer, and M. J. Berger, [19] D. A. Varshalovich, A. N. Moskalev, and V. K. Kher- Phys. Rev. A 33, 3002 (1986). sonski˘i, Quantum Theory of Angular Momentum, World [10] G. Alexander et al., Phys. Rev. Lett. 100, 210801 (2008). Scientific, Singapure, 1988. [11] S. Tashenov, T. B¨ack, R. Barday, B. Cederwall, J. En- [20] A. N. Artemyev, T. Beier, J. Eichler, A. E. Klasnikov, ders, A. Khaplanov, Y. Poltoratska, K.-U. Sch¨assburger, C. Kozhuharov, V. M. Shabaev, T. St¨ohlker, and V. A. and A. Surzhykov, Phys. Rev. Lett. 107, 173201 (2011). Yerokhin, Phys. Rev. A 67, 052711 (2003). [12] R. M¨artin et al., Phys. Rev. Lett. 108, 264801 (2012). [13] G. Weber, R. M¨artin, A. Surzhykov, M. Yasuda, V. A. 5

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