On the excitation of the 2 + 1 state in 12 C in the (e, e ′ γ) reaction
EEPJ manuscript No. (will be inserted by the editor)
On the excitation of the +1 state in C in the ( e, e (cid:48) γ ) reaction D. H. Jakubassa-Amundsen and V. Yu. Ponomarev Mathematics Institute, University of Munich, Theresienstrasse 39, 80333 Munich, Germany Institute of Nuclear Physics, Technical University of Darmstadt, 64289 Darmstadt, GermanyReceived: date / Revised version: date
Abstract.
The excitation of the carbon 2 + state at 4.439 MeV by 70 −
150 MeV electron impact andits subsequent decay to the ground state by photon emission is described within the distorted-wave Bornapproximation. The transition densities are obtained from the nuclear quasiparticle phonon model. Thephoton angular distributions are compared with earlier results and with experiment, including the influenceof bremsstrahlung. Predictions for spin asymmetries in the case of polarized electron impact are also made.
PACS.
With the advent of modern accelerators and efficient spin-polarized electron sources, such as the Darmstadt facil-ity S-DALINAC, coincidence measurements between elec-trons scattered inelastically from nuclei and decay photonsare feasible with high accuracy. This has stimulated thetheoretical reinvestigation of the lowest quadrupole exci-tation of C by electron impact which had been studiedin a pioneer coincidence experiment by Papanicolas et al[1], followed by a theoretical interpretation by Ravenhallet al [2].Nuclear excitation by electron impact is a powerfultool to obtain nuclear structure information [3,4], becauseonly the electromagnetic interaction between the partici-pating particles is involved. The nuclear properties enterexclusively into the electric and magnetic transition den-sities (cid:37) L and J L,L ± . They can be calculated from nuclearmodels.The first theoretical investigation of the coincident nu-clear excitation and decay (ExDec) process dates back toHubbard and Rose [5], who employed the plane-wave Bornapproximation (PWBA). This theory was subsequentlyapplied to the 2 +1 excitation of C [6]. Later, a combina-tion of the distorted-wave Born approximation (DWBA)for the electric transition and the PWBA for the mag-netic transition was used [2]. In these calculations, the nu-clear transition densities were taken in the form of Fourier-Bessel series with coefficients obtained from a fit to earlymeasurements of inclusive electron scattering form factors(e.g. [7,8]). It has been demonstrated that the currenttransition densities J L,L ± are very strong for the C nu-cleus, such that interference phenomena between electricand magnetic excitations are already visible at scatteringangles in the forward hemisphere where the cross sectionsare large. Such interference effects, augmented in coinci- dence experiments, are particularly sensitive to details ofthe nuclear structure.A competitive process to ExDec is the emission ofbremsstrahlung, which contributes coherently to the pho-ton emission from nuclear decay to the ground state [5].Bremsstrahlung calculations at high collision energies areusually performed within the relativistic PWBA. Since C is a spin-zero nucleus, only potential scattering hasto be taken into account [9]. For the radiation of photonswith small frequencies as compared to the collision energy(i.e. for low momentum transfer), it was shown by Betheand Maximon [10] that the PWBA, as limiting case of theSommerfeld-Maue theory, is an appropriate theory, irre-spective of the nuclear charge.Bremsstrahlung may have a considerable influence onthe angular distribution of the emitted photons, and wasalready taken into account in [6], however not in the Cinvestigation by Ravenhall et al [2]. More recently, whenstudying the 2 +1 and 2 +2 excitations in the ( e, e (cid:48) γ ) Zr reac-tion [11], it was shown that the contribution of bremsstrah-lung to the detected photons depends not only on thescattering angle, but also on the resolution of the photondetector, which in general is much poorer than the linewidth of the decay photon.Within a new campaign of the coincidence experimentsin the ( e, e (cid:48) γ ) reaction at S-DALINAC [12], it is planned torevisit the previous measurements in [1] to test the set-up.In the present paper, we extend the theoretical analysisfor this experiment. We employ a full DWBA prescriptionof the ExDec process and add bremsstrahlung coherently.This guarantees a consistent representation of all inter-ference effects, which are absent in PWBA. We will usethe charge and current transition densities from [2] andalso the ones from the random phase approximation ofthe quasiparticle phonon model (QPM [13,14]) to discussthe nuclear structure effects on the ExDec cross sections. a r X i v : . [ nu c l - t h ] F e b D. H. Jakubassa-Amundsen and V. Yu. Ponomarev: On the excitation of the 2 +1 state in C in the ( e, e (cid:48) γ ) reaction Section 2 provides the differential cross section results.We also compare with results where the QPM transitiondensities are replaced by the ones fitted to experiment.Section 3 deals with the Sherman function for polarizedelectrons. A short conclusion is given in section 4. Atomicunits ( (cid:126) = m = e = 1) are used unless indicated otherwise. The triply differential cross section for the inelastic scat-tering of an unpolarized electron with (total) initial energy E i and final energy E f into the solid angle dΩ f with thesimultaneous emission of a photon with frequency ω intothe solid angle dΩ k is given by [5,11] d σdωdΩ k dΩ f = 2 π ω E i E f k f k i c f rec (cid:88) σ i ,σ f × (cid:88) λ (cid:12)(cid:12)(cid:12) M (1) fi + M (2) fi + M brems fi (cid:12)(cid:12)(cid:12) , (2.1)where k i and k f are, respectively, the electron momentain initial and final state. Here we have assumed that po-larization is not observed, such that (2.1) includes a sumover the photon polarization (cid:15) λ and over the final elec-tron spin projection σ f , in addition to an average over theinitial-state spin projection σ i . Furthermore, it is assumedthat a spin-zero nucleus decays into its ground state, sothat no further spin degrees of freedom are present. Thefactor f rec is due to the kinematical recoil arising from thefinite mass of the nucleus.The amplitude M (1) fi describes the excitation of thenucleus into a quadrupole state n with energy E x , spin J n = 2 and magnetic quantum number M n , followed byradioactive decay according to the decay width Γ n , M (1) fi = i Z T c π √ ω ω − E x + iΓ n / × J n (cid:88) M n = − J n A exc ni ( M n ) A dec fn ( M n ) , (2.2)where Z T is the nuclear charge number and A exc ni and A dec fn are, respectively, the excitation and decay amplitudes ase.g. given in [11].The second transition amplitude in (2.1), M (2) fi , de-scribes the reversed process where the photon emissionoccurs before the nuclear excitation. This process is, how-ever, suppressed by several orders of magnitude and canbe disregarded.The last term in (2.1), M brems fi , is the contribution frombremsstrahlung photons with the same frequency ω , M brems fi = i c √ ω (cid:90) d x ψ ( σ f )+ f ( x ) ( α(cid:15) ∗ λ ) e − i kx ψ ( σ i ) i ( x ) , (2.3)where ψ i and ψ f are, respectively, the initial and finalscattering states of the electron, while k is the photon momentum and α is a vector of Dirac matrices. In thePWBA, when ψ i and ψ f are expanded in terms of planewaves, the rhs of (2.3) has to be multiplied by the Diracform factor F ( q ), which accounts for the charge distribu-tion of the nucleus [15,16]. The excitation amplitude A exc ni is conventionally calcu-lated with the help of partial-wave expansions [17,18]. Itis composed of the contributions originating from the elec-tric transition density (cid:37) L ( x N ) and the magnetic transitiondensities J L,L ± ( x N ) with L = 2. Their dependence onthe nuclear coordinate x N is displayed in Fig.1, where themagnetization current densities contributing to J and J are shown separately. The transition densities, calcu-lated within the QPM, are presented by solid lines. Theyhave been calculated within the one-phonon approxima-tion by adjusting the strength of the residual interaction toreproduce the experimental value of the B ( E , g.s. → +1 )= 39.7 e fm [19]. Notice that the QPM transition densi-ties deviate considerably from those provided in [2] whichare obtained from a Fourier-Bessel fit to scattering exper-iments. The B ( E , g.s. → +1 ) value obtained from theintegration of the charge transition density [4], B ( E , g.s. → + ) = 5 (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∞ r (cid:37) ( r ) dr (cid:12)(cid:12)(cid:12)(cid:12) , (2.4)when using (cid:37) from [2], equals to 42.41 e fm , which is7% above the experimental value. A stronger peak at thesurface of the charge transition density [2] is compensatedby a slightly stronger tail of the QPM density.For quadrupole excitation there are five magnetic sub-levels M n , which are populated with a probability givenby P ( M n ) = dσ exc /dΩ f ( M n )( dσ exc /dΩ f ) tot , (2.5)where (see, e.g. [20]) dσ exc dΩ f ( M n ) = 2 π E i E f k f k i c ˜ f rec (cid:88) σ i ,σ f × (cid:88) M (cid:48) n = − | A exc ni ( M (cid:48) n ) | δ M n ,M (cid:48) n , (2.6)valid for spin-zero nuclei. The recoil denominator ˜ f rec dif-fers from f rec in (2.1) due to E x in the energy balance.The total excitation cross section in the denominator of(2.5) results from (2.6) with the delta function removed.The calculation of the exact electronic scattering statesby means of the Dirac equation is performed with the helpof the Fortran code RADIAL by Salvat et al [21]. The nu-clear potential of C is generated from the Fourier-Besselexpansion of the ground-state charge distribution [22].The radial integrals in the transition matrix elements areevaluated by means of the complex-plane rotation method . H. Jakubassa-Amundsen and V. Yu. Ponomarev: On the excitation of the 2 +1 state in C in the ( e, e (cid:48) γ ) reaction 3 C, 2 E l e c t r i c t r an s i t i on den s i t y ( f m - ) Radius x N (fm) 0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0 1 2 3 4 5 6 7(b) C, 2 T r an s i t i on den s i t y J ( f m - ) Radius x N (fm) 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0 1 2 3 4 5 6 7(c) C, 2 T r an s i t i on den s i t y J ( f m - ) Radius x N (fm) Fig. 1.
Transition densities (a) (cid:37) , (b) J and (c) J for the2 +1 excitation of C at 4.439 MeV as a function of x N . ———–, QPM calculations (consisting in (b) and (c) of magnetizationand convection currents). · · · · · · , contribution of the magneti-zation current to J and J . Also shown are the transitiondensities of Ravenhall et al ( − · − · − ): (cid:37) and J are takenfrom [2], J is obtained from the continuity equation [4]; note,however, the reversed sign of J as compared to the definitionin [4]. (CRM) introduced in [23] and applied to electron scatter-ing in [24].The M n -sublevel populations for a collision energy of70 MeV as a function of scattering angle ϑ f are displayedin Fig.2. It is seen that at scattering angles below 40 ◦ alllevels have similar occupation probabilities, in particularthe pairs (+ M n , − M n ). However, in the backward hemi-sphere, it is just M n = 0 and M n = 1 which remain im-portant, M n = 1 taking over for ϑ f → ◦ . This is due to -4 -3 -2 -1
0 20 40 60 80 100 120 140 160 1800 1-22 -1 M n s ub s he ll popu l a t i on Scattering angle (deg)
Fig. 2.
Subshell population probabilities P ( M n ) for the 2 +1 excitation of C by 70 MeV electrons as a function of scat-tering angle ϑ f . − · − · − , M n = 0; ———-, M n = 1; · · · · · · , M n = 2; − · · · − · · · − , M n = − − − − − − , M n = −
2. Theprobabilities sum up to unity. the strong influence of the magnetic transitions at smallelectron–nucleus distances (corresponding to a large mo-mentum transfer, respectively to large scattering angles).
In all subsequent results, a coplanar geometry is chosen,where the photon is emitted in the scattering plane, spannedby k i (which is taken as z -axis) and k f . Thus the az-imuthal angle ϕ between k i and k f is 0 ◦ or 180 ◦ .For the carbon 2 +1 state at E x = 4 .
439 MeV, theground-state decay width is Γ n = (1 . ± . × − eV [25]. The ground-state decay amplitude A dec fn is medi-ated solely by the current transition densities J L,L ± fromFig.1b,c. According to the different occupation probabili-ties of the M n -substates, the intensity of the emitted decayphotons depends on M n as well. Fig.3 shows the triply dif-ferential cross section for the excitation of the M n -subshelland its subsequent decay, defined according to (2.1) by d σdωdΩ k dΩ f ( M n ) = 2 π ω E i E f k f k i c f rec (cid:88) σ i ,σ f × (cid:88) λ (cid:12)(cid:12)(cid:12) M (1) fi ( M n ) (cid:12)(cid:12)(cid:12) , (2.7)where M (1) fi ( M n ) is obtained from (2.2) if the sum over M n is dropped, corresponding to the excitation of justone substate M n . In that case, the photon angular dis-tribution is symmetric with respect to θ k = 180 ◦ and isa superposition of dipole and quadrupole patterns [11].This symmetry is lost in the total cross section where all M n subshells are added coherently. In particular, thereare angles θ k where the total cross section is well belowany M n -subshell cross section (for a scattering angle of ϑ f = 80 ◦ near e.g. θ k = 40 ◦ , see Fig.3a). At backward an-gles (Fig.3b), the total cross section is mainly composed D. H. Jakubassa-Amundsen and V. Yu. Ponomarev: On the excitation of the 2 +1 state in C in the ( e, e (cid:48) γ ) reaction -4 -3 -2 -1
0 50 100 150 200 250 300 350tot -11 2-2 0(a) 80 o C r o ss s e c t i on ( b / M e V s r ) Photon angle (deg)10 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2
0 50 100 150 200 250 300 350(b)-2-1 201 tot 175 o C r o ss s e c t i on ( b / M e V s r ) Photon angle (deg)
Fig. 3. M n -subshell cross sections d σdωdΩ k dΩ f ( M n ) for the 2 +1 excitation of C by 70 MeV electrons and subsequent de-cay for scattering angles (a) ϑ f = 80 ◦ and (b) ϑ f = 175 ◦ ,with azimuthal angle ϕ = 0 between electron and photon,as a function of photon angle θ k . − · − · − , M n = 0; ————-, M n = 1; · · · · · · , M n = 2; − · · · − · · · − , M n = − − − − − − , M n = − . Also shown is their coherent sum,the total cross section (thick solid line). of the M n = 0 and M n = 1 contributions according to theoccupation probabilities from Fig.2.In order to display the importance of electric and mag-netic excitation, Fig.4 shows the contributions from po-tential scattering (arising from (cid:37) ) and from magneticscattering (due to J , J ) entering into the excitationamplitude A exc ni . Of course, the decay amplitude A dec fn iskept unchanged in both cases. In the forward hemisphere,even up to scattering angles ϑ f ∼ ◦ , the excitation bythe electric force is largely dominant at all photon angles(Fig.4a). Only a little shift of the minima to smaller θ k isobserved when the excitation by the current interaction isincluded. Coulomb distortion effects, measured by meansof the difference between the DWBA and the PWBA re-sults, can basically be neglected for light nuclei such as C for not too large scattering angles. We also note thatat 140 ◦ the photon angular distribution has still the sameregular quadrupole pattern as for the smaller angle 80 ◦ from Fig.3a. At the backmost angles (Fig.4b for 175 ◦ ),the magnetic scattering gives an essential contribution tothe cross section, which modulates the quadrupole pat-tern considerably. This leads to a shift of the minima by -5 -4 -3 -2 -1
0 50 100 150 200 250 300 350(a) magelectot 140 o C r o ss s e c t i on ( b / M e V s r ) Photon angle (deg)10 -5 -4 -3
0 50 100 150 200 250 300 350175 o totelecmag(b) C r o ss s e c t i on ( b / M e V s r ) Photon angle (deg)
Fig. 4.
Triply differential cross section for the nuclear ExDecprocess of the C, 2 +1 state by 70 MeV electrons (a) at ϑ f = 140 ◦ and (b) at ϑ f = 175 ◦ , with ϕ = 0, as a functionof photon angle θ k . Total cross section: —————, DWBA; · · · · · · , PWBA. Also shown are the DWBA electric contribu-tion ( − − − − − ) and magnetic contribution ( − · − · − ) to thetotal cross section. about 20 ◦ as compared to potential scattering. Also theCoulomb distortion effects are considerably larger, up to10 percent.In Fig.5a the DWBA results from the QPM densitiesare compared with those based on the Ravenhall et al [2]densities included in Fig.1. At the parameters of the mea-surements [1], a collision energy of 66.9 MeV and a scatter-ing angle of 80 ◦ , the Ravenhall cross section is enhancedby a factor of 3.5. This results from the higher transitiondensity (cid:37) , since electric excitation is dominating at thisangle. Included in the figure are results for potential scat-tering within the PWBA, where one of the minima in thephoton angular distribution coincides with the angle θ q which the momentum transfer q = k i − k f forms withthe z-axis ( θ q = 312 . ◦ ). This results in an angular distri-bution which is azimuthally symmetric with respect to θ q [2]. The shift between the minima of the electric PWBAand the full DWBA is about 2 ◦ , which is verified by theexperimental data [1]. These data are measured on a rel-ative scale and are in Fig.5a normalized to the respectivetheories. It follows from Fig.4a that the shift in angle isbasically due to magnetic scattering and not to distor-tion effects. In Fig.5b the scattering angle is increased to170 ◦ . At this angle, the maxima of the Ravenhall results . H. Jakubassa-Amundsen and V. Yu. Ponomarev: On the excitation of the 2 +1 state in C in the ( e, e (cid:48) γ ) reaction 5 -4 -3 -2 -1
0 50 100 150 200 250 300 350(a) 66.9 MeV e + C 80 o C r o ss s e c t i on ( b / M e V s r ) Photon angle (deg)0.0000.0010.0020.0030.0040.0050.006 0 50 100 150 200 250 300 350(b) 66.9 MeV e + C 170 o C r o ss s e c t i on ( b / M e V s r ) Photon angle (deg)
Fig. 5.
Triply differential cross section for the nuclear ExDecprocess of the C, 2 +1 state by 66.9 MeV electron impact at(a) ϑ f = 80 ◦ and (b) ϑ f = 170 ◦ , with ϕ = 0, as a functionof photon angle θ k . —————–, DWBA results; − − − − − ,PWBA results (in (b)); · · · · · · , PWBA results for electric exci-tation (all with QPM densities); − · − · − , DWBA results withRavenhall densities. In (b), the Ravenhall results are scaleddown by a factor of 0.27 to display the differences in shape.The experimental data ( ◦ ) in (a) are taken from Papanicolaset al [1]. The same data ( (cid:4) ) are scaled down by a factor of0.285 to fit the QPM results. do not coincide anymore with those from the QPM densi-ties. Although the importance of potential scattering hasdecreased, the Ravenhall cross sections are still a factor of3.7 above the QPM ones. This may be caused by an en-hanced current J near the surface of the nucleus. Notethat the nuclear radius of C is R N = 1 . A / = 2 .
75 fm,which has to be compared to the distance of closest ap-proach during the electron-nucleus encounter, determinedby the inverse momentum transfer ( q − = 1 .
53 fm for 66.9MeV and 170 ◦ ). In order to give predictions for the contribution of brems-strahlung to the nuclear ExDec process it is importantto account for the finite resolution ∆ω of the photon de-tector. As far as the nuclear ExDec process with its reso-nant behaviour is concerned, the averaging over the detec-tor resolution leads basically to a reduction of intensity, but not to a change in the photon angular distribution.Bremsstrahlung, on the other hand, due to its weak de-pendence on ω , is hardly affected by the averaging pro-cedure. When both contributions are considered, the av-eraged photon intensity at the peak frequency ω = E x iscalculated from (cid:28) d σdωdΩ k dΩ f (cid:29) ∆ω ≈ π E i k i c (cid:88) σ i ,σ f (cid:88) λ × ∆ω (cid:90) E x + ∆ω E x − ∆ω dω (cid:48) ω (cid:48) E f k f f rec (cid:12)(cid:12)(cid:12) M (1) fi + M brems fi (cid:12)(cid:12)(cid:12) , (2.8)such that the different ω -behaviour of M (1) fi and M brems fi leads to a change in the θ k -distribution which stronglydepends on ∆ω . This feature is displayed in Fig.6a wherea resolution of ∆ω/ω = 3% is taken, corresponding to aLaBr photon detector to be used in experiments, while inFig.6b, ∆ω/ω = 0 .
5% is assumed. Again, the experimen-tal parameters, E e = E i − c = 66 . ϑ f = 80 ◦ and ω = 4 .
439 MeV, have been chosen.The bremsstrahlung angular distribution is characterizedby the narrow double-peak structure near θ k = 0 and near θ k = ϑ f for ω (cid:28) E e . These structures dominate the pho-ton distribution from the ExDec process. In addition, thebremsstrahlung photons fill the minima of the quadrupolepattern, the more so, the poorer the detector resolution.We note that the experimental data points are slightlybetter reproduced with a resolution near or below 1%.In order to study the influence of bremsstrahlung atother geometries we display in Fig.7 photon angular dis-tributions at two scattering angles in the backward hemi-sphere, ϑ f = 140 ◦ and 170 ◦ , and two collision energies, 70MeV and 150 MeV. In all subfigures, an average is takenwith ∆ω/ω = 3%. Comparing Figs.6a, 7a and 7b, it isseen that, away from the bremsstrahlung peaks, the in-fluence of bremsstrahlung decreases with scattering angle,favouring the backmost angles. Also, profiting from a weakdependence of the nuclear ExDec process on collision en-ergy [11], while bremsstrahlung is strongly decreasing with E e , bremsstrahlung is the more suppressed, the higher E e ,see Figs.7c and 7d. Previous investigations of the ExDec process were restric-ted to unpolarized beam electrons. However, a more strin-gent test of the nuclear models is achieved if additionallythe spin degrees of freedom are taken into account. Anappropriate measure of the spin asymmetry is the Sher-man function S [26,27] which requires a beam polarizationperpendicular to the scattering plane. It measures the rel-ative difference in intensity when the direction of the beampolarization is switched.In the discussion of the spin asymmetry we will disre-gard bremsstrahlung, since actual measurements will al-ways be performed at photon angles where the influence of D. H. Jakubassa-Amundsen and V. Yu. Ponomarev: On the excitation of the 2 +1 state in C in the ( e, e (cid:48) γ ) reaction -12 -11 -10 -9 -8 -7 -6 -5 -4
0 50 100 150 200 250 300 350(a)3 % 80 o C r o ss s e c t i on ( b / M e V s r ) Photon angle (deg)10 -11 -10 -9 -8 -7 -6 -5 -4
0 50 100 150 200 250 300 350(b)0.5 % 80 o C r o ss s e c t i on ( b / M e V s r ) Photon angle (deg)
Fig. 6.
Averaged triply differential cross section for the co-incident ( e, e (cid:48) γ ) process by 66.9 MeV electrons scattered at ϑ f = 80 ◦ , with ϕ = 0, as a function of photon angle θ k .The detector resolution is (a) ∆ω/ω = 3% (correspondingto ∆ω = 133 keV), and (b) ∆ω/ω = 0 .
5% (correspondingto ∆ω = 22 . − − − − − , photons from the nuclearExDec process; · · · · · · , bremsstrahlung; —————, coherentsum. The experimental data ( (cid:4) ) are from Papanicolas et al [1]and are normalized to the full lines. bremsstrahlung is small. In that case, the Sherman func-tion can alternatively be obtained from the transition am-plitude M (1) fi . Denoting the coefficients of the initial-statepolarization vector ζ i in the standard basis (cid:0) (cid:1) and (cid:0) (cid:1) by a m i , M (1) fi is formally written as [18] M (1) fi = (cid:88) m i = ± a m i F ( m i ) , (3.1)and the Sherman function results from [20] S = − (cid:80) λ Im { F ∗ ( ) · F ( − ) } (cid:80) λ (cid:2) | F ( ) | + | F ( − ) | (cid:3) . (3.2)The denominator is proportional to the total cross sectionfor unpolarized particles, obtained by summing (in addi-tion to m i ) over the two photon polarizations (cid:15) λ and overthe projections of the two spin polarization vectors ζ f ofthe final electron (note, however, that F ( m i ) is indepen-dent of σ f if ζ f is taken parallel, respectively, antiparallelto k f ). -12 -11 -10 -9 -8 -7 -6
0 50 100 150 200 250 300 35070 MeV 140 o (a) C r o ss s e c t i on ( b / M e V s r ) Photon angle (deg)10 -11 -10 -9 -8 -7
0 50 100 150 200 250 300 350(b) 170 o
70 MeV C r o ss s e c t i on ( b / M e V s r ) Photon angle (deg)10 -12 -11 -10 -9 -8
0 50 100 150 200 250 300 350(c) 140 o
150 MeV C r o ss s e c t i on ( b / M e V s r ) Photon angle (deg)10 -13 -12 -11 -10 -9
0 50 100 150 200 250 300 350150 MeV 170 o (d) C r o ss s e c t i on ( b / M e V s r ) Photon angle (deg)
Fig. 7.
Averaged triply differential cross section for the coinci-dent ( e, e (cid:48) γ ) process by (a), (b) 70 MeV and (c), (d) 150 MeVelectrons as a function of photon angle θ k . The scattering angleis (a), (c) ϑ f = 140 ◦ and (b), (d) ϑ f = 170 ◦ , at ϕ = 0. The de-tector resolution is ∆ω/ω = 3%. − − − − − , photons from thenuclear ExDec process; · · · · · · , bremsstrahlung; —————-,coherent sum.. H. Jakubassa-Amundsen and V. Yu. Ponomarev: On the excitation of the 2 +1 state in C in the ( e, e (cid:48) γ ) reaction 7 -0.0005 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0 50 100 150 200 250 300 350(a) 140 o S p i n a sy mm e t r y S Photon angle (deg)-0.002-0.001 0 0.001 0.002 0.003 0.004 0.005 0 50 100 150 200 250 300 350(b) 170 o S p i n a sy mm e t r y S Photon angle (deg)
Fig. 8.
Spin asymmetry for the nuclear ExDec process fromthe carbon 2 +1 excitation by 70 MeV (—————) and 150MeV ( − − − − − ) perpendicularly polarized electrons as afunction of photon angle θ k for scattering angles (a) ϑ f = 140 ◦ and (b) ϑ f = 170 ◦ at ϕ = 0 . The maxima of S in (a) amount to0.031 for 70 MeV and to 0.066 for 150 MeV (using a step sizeof ∆θ k = 1 ◦ in the plot). Also shown is the spin asymmetryfrom the excitation process alone ( − · − · − ,
70 MeV; · · · · · · ,
150 MeV).
Fig.8 provides examples for the spin asymmetry in caseof some geometries from Fig.7. The total cross section be-ing in the denominator of (3.2), S has extrema at photonangles where the minima of the cross section are located(see also Fig.9). In the forward hemisphere, and even atscattering angles up to 140 ◦ , the cross section has verydeep minima and consequently, the maxima of S are verysharp. In a true experimental situation the excursions of S at such angles will be reduced since bremsstrahlung tendsto fill the cross section minima. At the backmost scatteringangles, diffraction effects come into play and modulate thesign of S . Such diffraction effects occur when the electronis sufficiently energetic to penetrate the nuclear surfaceand to scatter off the individual protons.In Fig.8 we have included the spin asymmetry resultingfrom the mere excitation process as horizontal lines. It iscalculated by replacing M (1) fi with the amplitude A exc ni , forwhich an equation of type (3.1) also holds. In (3.2), thesum over λ has to be changed into a sum over M n [20].For excitation it is well known (and verified in Fig.8) that | S | decreases globally with E i (at fixed ϑ f ) and increases -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0 50 100 150 200 250 300 35070 MeV 140 o (a) C r o ss s e c t i on ( b / M e V s r ) and S Photon angle (deg)-0.02-0.015-0.01-0.005 0 0.005 0 50 100 150 200 250 300 350(b) 170 o
150 MeV C r o ss s e c t i on ( b / M e V s r ) and S Photon angle (deg)
Fig. 9.
Spin asymmetry for nuclear excitation and decay ofthe C, 2 +1 state by perpendicularly polarized electrons of (a)70 MeV at ϑ f = 140 ◦ and (b) 150 MeV at ϑ f = 170 ◦ , with ϕ = 0, as a function of photon angle θ k . Shown is S from thenuclear ExDec process using the QPM densities (—————) and the Ravenhall densities ( − · − · − ). Also shown is thecorresponding (unaveraged) triply differential cross section (in bMeV sr ) from the QPM densities ( − − − − − ) and from theRavenhall densities ( · · · · · · ). In (a), the Ravenhall cross sectionis scaled down by a factor of 0.26, in (b) the QPM cross sectionis scaled up by a factor of 3 to display the differences in shape. with scattering angle (at fixed E i ). It is only the latterfact which remains true for the nuclear ExDec process.In order to demonstrate the greater sensitivity of S to the choice of nuclear models as compared to the per-ceptivity of the cross section for unpolarized particles, wedisplay in Fig.9 the results obtained from the QPM tran-sition densities on the one hand, and from the Raven-hall transition densities on the other hand. At a beamenergy of 70 MeV and ϑ f = 140 ◦ (Fig.9a) the Raven-hall cross section is by a factor of 3.85 higher, but theshape of the angular distribution is nearly identical. TheSherman function, however, differs visibly. In particular,the maxima in S from the QPM prescription have turnedinto weak minima in the Ravenhall picture. The sensi-tivity to details in the transition densities increases withenergy. In Fig.9b a collision energy of 150 MeV is chosen,together with a backward scattering angle of 170 ◦ whichincreases the spin asymmetry in the regions between thesharp peaks considerably. In this geometry, the Ravenhallcross section is enhanced by a factor of 3, and the angular D. H. Jakubassa-Amundsen and V. Yu. Ponomarev: On the excitation of the 2 +1 state in C in the ( e, e (cid:48) γ ) reaction distribution is slightly modulated and shifted. The Sher-man function, on the other hand, shows considerable devi-ations in the two prescriptions. The maxima of the QPMmodel have now turned into deep minima. These extremaare nevertheless wide enough to make a detection feasi-ble. Moreover, as becomes clear from a comparison withFig.7d, bremsstrahlung plays no role except in a small re-gion around θ k = 170 ◦ , so that the extrema in S are notinfluenced. We have calculated the triply differential cross section forthe simultaneous observation of the scattered electron andthe emitted photon in the ( e, e (cid:48) γ ) C reaction. The nuclearquasiparticle phonon model was used for the excitationof the 2 +1 state, while electron scattering was describedwithin the distorted-wave Born approximation. Compar-ing with earlier results using experimental nuclear tran-sition densities, large changes in the photon intensity arefound, but only slight shifts of the angular distribution,even at backward scattering angles. The measured rela-tive photon distribution is well reproduced in both pre-scriptions.Confirming earlier results on quadrupole excitation of Zr, the M n -sublevels of the C, 2 +1 excited state areapproximately equally populated for scattering angles inthe forward hemisphere, while the M n = 0 and M n = 1substates largely dominate at the backmost angles. Con-sequently, at the smaller angles the photon angular distri-bution has a regular quadrupole structure, while there aresubstantial dipole-type modifications (from the M n = 1contribution) at scattering angles close to 180 ◦ .Including bremsstrahlung within the PWBA, a theorywell justified for low-energy photons and a light nucleuslike C even for large scattering angles, it was found thatfor photon angles in the forward direction or close to thescattering angle, bremsstrahlung spoils the visibility of thenuclear decay photons, the more so, the smaller the scat-tering angle, the lower the collision energy and the poorerthe resolution of the photon detector.Finally we have investigated the Sherman function whichis a measure of the spin asymmetry occurring for po-larized electron impact. In contrast to its behaviour forelastic scattering or excitation where the spin asymme-try exhibits a global decrease with collision energy (whichmay be modulated by diffraction structures), the ExDecprocess will lead to considerably higher spin asymmetrieswhen E e is increased. Furthermore, by comparing the re-sults from the two considered types of nuclear transitiondensities, we have demonstrated that the Sherman func-tion is much more sensitive to such changes than the triplydifferential cross section. The large deviations of S in thetwo models at high collision energies, combined with itshigh absolute values at the backmost scattering angleswhere the influence of bremsstrahlung is negligible, makesuch a geometry a promising candidate for nuclear struc-ture investigations. V.Yu. P. acknowledges support by the Deutsche For-schungsgemeinschaft (DFG, German Research Foundation)- Projektnummer 279384907 - SFB 1245. References
1. C.N.Papanicolas et al, Phys. Rev. Lett. , 26 (1985).2. D.G.Ravenhall, R.L.Schult, J.Wambach, C.N.Papanicolas,and S.E.Williamson, Ann. Phys. , 187 (1987).3. H. ¨Uberall, Electron Scattering from Complex Nuclei (Aca-demic Press, New York, 1971).4. J.Heisenberg and H.P. Blok, Ann. Rev. Nucl. Part. Sci. ,569 (1983).5. D.F.Hubbard and M.E.Rose, Nucl. Phys. , 337 (1966).6. H.L.Acker and M.E.Rose, Ann. Phys. , 336 (1967).7. H.L.Crannell, Phys. Rev. , 1107 (1966).8. J.B.Flanz, R.S.Hicks, R.A.Lindgren, G.A.Peterson,A.Hotta, B.Parker and R.C.York, Phys. Rev. Lett. ,1643 (1978).9. H.Bethe and W.Heitler, Proc. Roy. Soc. (London) A ,83 (1934).10. H.A.Bethe and L.C.Maximon, Phys. Rev. , 768 (1954).11. D.H.Jakubassa-Amundsen and V.Yu.Ponomarev, Phys.Rev. C , 024310 (2017).12. Atomkerne: Von fundamentalen Wechselwirkungen zuStruktur und Sternen , Deutsche Forschungsgemeinschaft(DFG, German Research Foundation) - Projektnummer279384907 - SFB 1245.13. V.G. Soloviev,
Theory of Atomic Nuclei: Quasiparticlesand Phonons (Institute of Physics, Bristol, 1992).14. N.Lo Iudice, V.Yu.Ponomarev, Ch.Stoyanov, A.V.Sushkovand V.V.Voronov, J. Phys. G , 043101 (2012).15. E.S.Ginsberg and R.H.Pratt, Phys. Rev. 134, B773 (1964).16. D.H.Jakubassa-Amundsen , Phys. Lett. A , 1885(2013).17. S.T.Tuan, L.E.Wright and D.S.Onley, Nucl. Instr. Meth. , 70 (1968).18. E.M.Rose, Relativistic Electron Theory (Wiley, New York,1961), § , 1 (2001).20. D.H.Jakubassa-Amundsen, Nucl. Phys. A , 65 (2015).21. F.Salvat, J.M.Fern´andez-Varea, and W.Williamson Jr.,Comput. Phys. Commun. , 151 (1995).22. H.De Vries, C.W.De Jager, and C.De Vries, At. Data Nucl.Data Tables , 495 (1987).23. C.M.Vincent and H.T.Fortune, Phys. Rev. C , 782 (1970).24. D.H.Jakubassa-Amundsen and V.Yu.Ponomarev, Eur.Phys. J. A : 48 (2016).25. F.Ajzenberg-Selove and C.L.Busch, Nucl. Phys. A , 1(1980).26. N.Sherman, Phys. Rev. , 1601 (1956).27. J.W.Motz, H.Olsen and H.W.Koch, Rev. Mod. Phys.36