Polarization in nuclear excitation by electron impact

Available online at www.sciencedirect.com

ScienceDirect Nuclear Physics A 937 (2015) 65–83 www.elsevier.com/locate/nuclphysa

Polarization in nuclear excitation by electron impact D.H. Jakubassa-Amundsen Mathematics Institute, University of Munich, Theresienstrasse 39, 80333 Munich, Germany Received 22 December 2014; received in revised form 9 February 2015; accepted 12 February 2015 Available online 18 February 2015

Abstract Electron–electron polarization correlations from nuclear excitation by spin-polarized electrons are studied within the distorted-wave Born approximation. Restriction is made to spin-0 and to unpolarized spin- 12 nuclei. When Coulomb scattering is dominant the resulting spin asymmetries may exceed those from elastic scattering. However, at the backmost scattering angles or for large transition current densities where magnetic scattering prevails, there is a strong suppression of the transverse polarization correlations. Pre− dictions are made for the excitation of the lowest 3− and 5− states of 208 Pb and the lowest 52 state of 89 Y by 60–220 MeV electrons. © 2015 Elsevier B.V. All rights reserved. Keywords: Polarized electron scattering; Nuclear excitation; Spin asymmetry

1. Introduction With the design of new powerful electron accelerator and detector systems [1] and the availability of polarized electron beams of high intensity [2–5] the interest of investigating nuclear structure by inelastic electron scattering [6,7] has been revived (see, e.g. [8]). Probing the nucleus by electron impact has the advantage of a weak coupling between the collision partners such that a first-order perturbative treatment is usually sufficient. Early calculations of excitation probabilities and spin asymmetries relied on the plane-wave Born approximation (PWBA) [9–13], but later the use of distorted waves for the scattering electron (DWBA) has become standard for the analysis of experimental cross section data for medium and heavy nuclei ([6], see E-mail address: [email protected]. http://dx.doi.org/10.1016/j.nuclphysa.2015.02.001 0375-9474/© 2015 Elsevier B.V. All rights reserved.

66

D.H. Jakubassa-Amundsen / Nuclear Physics A 937 (2015) 65–83

also [14–18]). Moreover, while photon-impact excitation of heavy nuclei is basically restricted to dipole transitions (for the L = 1 excitation of Pb by polarized photons, see [19]), electron impact allows for the population of nuclear states of arbitrary multipolarity [7]. The use of polarized electron beams or polarized targets and the measurement of spin asymmetries in addition to mere differential cross sections provide a sensitive probe of the nuclear form factors, including their relative phases, which are involved in the excitation of a given nuclear state [11–13]. On the other hand, the respective transition densities (the Fourier–Bessel transforms of the form factors), entering into the DWBA, depend strongly on the underlying nuclear models [20,7]. Hence, a precise experimental determination of the spin asymmetries provides a more stringent test of the nuclear models than the cross section measurements. Up to now, mostly proton targets or light nuclei have been investigated. Measurements of the spin asymmetry for high-energy polarized electrons colliding with heavy targets [21,2,3] and corresponding calculations (see, e.g., [22–24]) have only been performed for elastic scattering. There are predictions for the polarization correlations in inelastic scattering of electrons from light polarized targets, as well as from 181 Ta, probing their sensitivity to different nuclear models [11,12,25]. Only longitudinally polarized electrons were considered, and the collision energies investigated were usually well above 200 MeV. Moreover, it was shown in the context of 180◦ scattering cross sections for 181 Ta that the PWBA used so far [11] even fails qualitatively when compared to DWBA results [26]. It is the aim of the present work to study the polarization correlations pertaining to arbitrary spin-polarized electrons in the initial as well as in the final state, which have not been investigated so far. In order to establish the significance of these polarization correlations and to investigate their dependence on the parameters of the collision system, use will be made of the known transition densities extracted from experiment. Conventionally these transition densities are obtained from the measured cross sections by means of a Fourier–Bessel analysis [27,17,18]. We will concentrate on the excitation of low-lying states of heavy nuclei because there the cross sections are large which is necessary for a precise measurement of the spin asymmetries. Restriction will be made to spinless nuclei, where only a single multipolarity contributes to the excitation, and to spin- 12 nuclei where at most four transition densities come into play. By circumventing a superposition of too many multipolarities, this choice of targets allows for a separate study of the influence of these transition densities on the spin asymmetry. Emphasis will be laid on the backmost scattering angles, apt to the S-DALINAC detection system [28,29], and on collision energies near and below 200 MeV, where one knows from (elastic) potential scattering that the transverse spin asymmetries are large, particularly for heavy (spinless) nuclei [21,30–32]. In contrast, it was shown that for elastic scattering from nuclei carrying spin the presence of magnetic scattering strongly reduces the transverse spin asymmetry as compared to neighboring spin-0 nuclei [36], the more so, the smaller the nuclear charge. This might lead to the conjecture that for a nuclear transition in which only a fraction of the protons participate, the transverse asymmetry is considerably smaller than for elastic scattering. It will be shown below that this is not necessarily the case. The paper is organized as follows. Section 2 provides an outline of the scattering theory, and numerical details are given in Section 3. Results for the cross sections and polarization correlations for electron-impact excitation of 89 Y (5/2− ) at an excitation energy Ex = 1.745 MeV, of 208 Pb (3− ) at 2.614 MeV and of 208 Pb (5− ) at 3.198 MeV are displayed in Section 4. The conclusion is drawn in Section 5. Atomic units (h¯ = m = e = 1) are used unless indicated otherwise. In particular, the electron mass m is retained throughout.

D.H. Jakubassa-Amundsen / Nuclear Physics A 937 (2015) 65–83

67

2. DWBA theory Our theoretical prescription uses the conventions introduced by Heisenberg and Blok [7] and applies the partial-wave representation of the electronic scattering states and of the transition operators according to the formalism provided by Tuan et al. [15]. The electronic spin degrees of freedom are explicitly taken into account, while it is assumed that the nuclear spins are not observed. This implies a considerable simplification of the tensor algebra presented in [11]. 2.1. Differential cross sections Let us consider a nucleus of mass MT , initially at rest, which undergoes a transition from 2 , angular momentum quantum numbers J , M the ground state of total energy Enuc,i = MT c i i

and parity πi to a final state with Enuc,f = Pf2 c2 + MT2 c4 + Ex , Jf , Mf and parity πf . Ex is the excitation energy and P f the momentum of the recoiling nucleus. Simultaneously the electron changes its energy from Ei to Ef and its polarization vector from ζ i to ζ f . Here,  2 c2 + m2 c4 with k the initial electron momentum. Its final momentum k can be Ei,f = ki,f i f obtained from the conservation of the 4-momentum, Q2 ≡ (Ei − Ef )2 − (k i − k f )2 c2 = (Enuc,f − Enuc,i )2 − Pf2 c2 , leading to the result  kf = (bx + bx2 − ax cx )/ax c

(2.1)

(2.2)

with ax = A2x − ki2 c2 cos2 θ,

bx = Bx ki c cos θ,

cx = m2 c4 A2x − Bx2 ,

(2.3)

where θ is the scattering angle, Ax = MT c2 + Ei − Ex and Bx = (MT c2 − Ex )Ei + m2 c4 − Ex (MT c2 − Ex /2). The explicit dependence of the energy Enuc,f on the recoil momentum P f = k i − k f enters also into the energy conserving δ-function of the collision process. Its evaluation leads to the recoil factor frec = 1 +

ki Ef cos θ E i − Ex − . 2 MT c k f MT c 2

(2.4)

For the heavy nuclei considered here, such recoil effects are usually small and Ef is close to Ei − Ex . When Ex  Ei , (2.4) reduces to the expressions known from the literature [26,7]. The transition amplitude for the nuclear excitation by electron impact can be written as a coherent sum of two contributions, mag

Af i (Mi , Mf ) = Acoul f i (Mi , Mf ) + Af i (Mi , Mf ),

(2.5)

where the first term arises from the interaction between the charges of electron and nucleus (the Coulomb scattering), and the second one from the current–current interaction of the collision partners (the magnetic scattering). Denoting the charge part of the nuclear transition matrix element by f i (r N ) and the current part by j f i (r N ) where r N is the nuclear coordinate, one has [9,10,15]

68

D.H. Jakubassa-Amundsen / Nuclear Physics A 937 (2015) 65–83

Acoul f i (Mi , Mf ) = − mag

Af i (Mi , Mf ) =

1 c

1 c 



 eik|r e −r N |  (σ )+ dr N dr e ψf f (r e ) ψi(σi ) (r e ) f i (r N ) |r e − r N |  ↔  (σ )+ I eik|r e −r N | (σi ) f dr N dr e ψf (r e ) α ψi (r e ) j f i (r N ), (2.6) |r e − r N |

(σ )

(σ )

where k = Ex /c. ψi i and ψf f denote, respectively, the initial and final scattering states of the electron (with spin projections σi , σf ), r e is the electronic coordinate, α is the vector of Dirac ↔

matrices and I the dyadic unit matrix. The nuclear transition matrix elements are subject to a partial-wave decomposition which relates them to the respective transition densities L and JL,λ [7,33],  ∗ f i (r N ) = (Ji Mi LM|Jf Mf ) L (rN ) YLM (ˆr N ), LM

j f i (r N ) = −i



(Ji Mi LM|Jf Mf ) JLλ (rN ) Y M∗ r N ). Lλ (ˆ

(2.7)

λLM

The YLM and Y M Lλ are, respectively, spherical harmonics and vector spherical harmonics, and the symbol in round brackets is a Clebsch–Gordan coefficient (as defined in [34]). From the angular momentum selection rule (Ji , Jf , L) and parity conservation expressed by means of πf i ≡ (−1)πf −πi = (−1)L for electric and πf i = (−1)L+1 for transverse magnetic transitions, there occur at most four transition densities, L , JL,L±1 and Jλ,λ (with λ = L + 1 or L − 1), for the spin-0 and spin- 21 nuclei. They are calculated from the nuclear models (see, e.g., [20,7]) or, alternatively, are obtained from the measured cross sections at forward angles (L ) or backmost angles (JL,λ ). The differential cross section for the excitation of unpolarized nuclei by spin-polarized electrons is obtained from [35,36] 2 kf 1 4π 3 Ei Ef   dσ 1 Af i (Mi , Mf ) , (ζ i , ζ f ) = 2 d

ki frec 2Ji + 1 c

(2.8)

Mi ,Mf

where it is averaged over the initial and summed over the final polarization of the nucleus. A prefactor 4π is absorbed into the transition densities (see (3.2); note this 4π -ambiguity in the literature). For the evaluation of Af i the z-axis is chosen along k i and the x-axis in the scattering plane, such that k f = (kf sin θ, 0, kf cos θ ). The electronic states are expanded in terms of partial waves [37],   2li + 1  1 (σi ) ψi (r e ) = am i (li 0 mi | ji mi ) i li eiδκi ψκi mi (r e ), 4π 2 1 κ mi =± 2

(σf )+

ψf

(r e ) =



i

∗ bm s

ms =± 12

× (−i)lf e

 κf m f

iδκf

(lf mf − ms

1 ms | jf mf ) 2

Ylf ,mf −ms (kˆ f ) ψκ+f mf (r e ),

(2.9)

where ami , bms are complex coefficients determined by the direction of the spin polarization, δκi and δκf are the phase shifts and

D.H. Jakubassa-Amundsen / Nuclear Physics A 937 (2015) 65–83

ψκm (r e ) =

gκ (re ) Yj lm (ˆr e ) i fκ (re ) Yj l  m (ˆr e )

69

(2.10)

with gκ and fκ the radial Dirac functions, Yj lm the spherical harmonic spinors, and j , l, l  are related to κ = ±1, ±2, . . . by means of j = |κ| − 12 , l = |κ + 12 | − 12 and l  = |κ − 12 | − 12 . Also for the propagators a multipole expansion is made [15]. For the Coulomb amplitude it is given by  eik|r e −r N | (1) ∗ jl (kr< ) hl (kr> ) Ylm (ˆr e ) Ylm (ˆr N ), = 4πik |r e − r N |

(2.11)

lm

(1)

where r< = min{re , rN }, r> = max{re , rN }, and jl and hl denote, respectively, spherical Bessel and Hankel functions of the first kind. The expansion of the propagator for the magnetic amplitude (2.6) is obtained from (2.11) by replacing Ylm with Y m j l , and summing over j , l and m [15]. For later use we define F (Mi , Mf , mi ) by means of  Af i (Mi , Mf ) = ami F (Mi , Mf , mi ), (2.12) mi =± 12

and obtain for the Coulomb amplitude upon insertion of (2.7), (2.9) and (2.11) into (2.6), √ ∞ 4π  ∗  F coul (Mi , Mf , mi ) = bms Ylf ,mf −ms (kˆ f ) c 1 ms =± 2



×

 1 (lf mf − ms ms | jf mf ) SLcoul (L ) 2 1

jf =lf ± 2

with SLcoul (L ) = −ik



lf =0

(2.13)

L

c c c Cif (Ji Mi LM | Jf Mf ) (R12 + R21 ) W12 (lf , li , L),

κi

 1 i(δ +δ ) 2li + 1 (li 0 mi | ji mi ) e κi κf . (2.14) 2 is the result of the integration over electron angle,  2li + 1 c M 2L + 1 W12 (lf , li , L) = (−1) (li 0 L0 | lf 0) 4π 2lf + 1  1 1 × (lf μf msi | jf mf ) (li μi msi | ji mi ) (li μi L − M | lf μf ) 2 2 1 Cif = i li −lf

c W12

msi =± 2

(2.15) where M = Mf − Mi , mf = mi − M, μi = mi − msi and μf = mf − msi . From (2.15) with its requirement that (li , L, lf ) with li + L + lf an even number has to hold true, as well as that for any l one has two values of κ (κ = l and κ = −l − 1), one readily derives that to each κf there are 2L + 1 values of κi contributing (except that κi = 0 has to be excluded). These are calculated from

70

D.H. Jakubassa-Amundsen / Nuclear Physics A 937 (2015) 65–83

κi = (−1)n+1 κf + n − (L + 1)

(2.16)

with n an integer running from 1 to 2L + 1. So for L even, one has κi ∈ {κf , −κf ± 1, κf ± 2, ..} while for L odd, κi ∈ {−κf , κf ± 1, . . .}. The radial integrals are given by [15] ∞ c c R12 + R21 =

∞ rN2 drN L (rN )

0



(1) re2 dre jL (kr< ) hL (kr> ) gκf gκi + fκf fκi .

0

mag In a similar way, Af i (Mi , Mf ) and F mag (Mi , Mf , mi ) are, respectively, obtained from   mag and (2.13) by replacing L SLcoul (L ) with the sum λL SλL (JL,λ ), where [15,36] mag

SλL (JL,λ ) = ik

(2.17)



(2.12)

Cif (Ji Mi LM | Jf Mf )

κi

  mag mag mag mag × R12 (gκf fκi ) W12 (lf , li , L, λ) − R12 (fκf gκi ) W12 (lf , li , L, λ) , (2.18) with the radial integrals mag R12 (gκf fκi )

∞ =

∞ rN2 drN

0

(1)

re2 dre jλ (kr< ) hλ (kr> ) gκf fκi .

JL,λ (rN )

(2.19)

0

The angular integral contains the coupling of (σ Y M∗ Lλ ) to the spherical harmonic spinors with ji , li and jf , lf and can be expressed in terms of a 9j -symbol [10,34]. For computational reasons a 4-term sum over Clebsch–Gordan coefficients is, however, more convenient,  2li + 1  3 √ mag  M W12 (lf , li , L, λ) = (−1) 2λ + 1 (l 0 λ0 | lf 0) 4π 2lf + 1 i  1 1 (lf μf msf | jf mf ) (li μi msi | ji mi ) (λml 1 | LM) × 2 2 m ,m sf

si

1 1 × ( msi 1 −  | msf ) (li μi λ − ml | lf μf ). (2.20) 2 2 The definition of M, mf , μi is given below (2.15), while μf = mf − msf , ml = μi − μf and  = M − ml . We note that the different signs as compared to [36] arise from using Y M∗ Lλ in (2.7) ), in concord with [7,33] (which, of course, leads to identical cross sections). (instead of Y M Lλ For the nuclear excitations to be discussed below, the multipole sum in the magnetic scattering reduces to  mag SλL (JL,λ ) = S te(+) (JL,L+1 ) + S te(−) (JL,L−1 ) + S tm (Jλ,λ ), (2.21) λL

where the first two terms refer to the transverse electric (te) contributions, while the last term represents the transverse magnetic (tm) contribution. This last term can be simplified by making tm (l , l  , λ, λ) = −W tm (l  , l , λ, λ) [15]. Recalling that l  + l is odd if use of the relation W12 f i f i 12 f i li + lf is even, the formula (2.16) interrelating κi and κf holds also for JL,L±1 , but has to be replaced by κi = (−1)n κf + n − (L + 1) for JL,L .

D.H. Jakubassa-Amundsen / Nuclear Physics A 937 (2015) 65–83

71

2.2. Polarization correlations Let us assume that the outgoing electron is in a helicity (+) state, i.e. it is longitudinally polarized (in the direction of k f , such that in (2.9), b 1 = cos θ2 , b− 1 = sin θ2 [37,38]). The polarization 2 2 correlations are defined in terms of the cross section asymmetry upon spin-flip [30], P (ζ i ) =

dσ (ζ i , ζ f ) − dσ (−ζ i , ζ f ) dσ (ζ i , ζ f ) + dσ (−ζ i , ζ f )

,

(2.22)

with dσ ≡ dσ/d from (2.8), and the three linear independent ones are the Sherman function S = P (ey ) with ey = kˆ i × kˆ f , the transverse spin asymmetry R = P (−ex ) and the longitudinal asymmetry L = P (ez ). Only S is independent of ζ f and thus most easily accessible by experiment. Also, S = 0 in the PWBA, hence visualizing the higher-order effects [30]. We also note that L = 1 throughout were the electron mass set to zero. Hence, L is usually close to unity for high-energy electron impact (except at the backmost scattering angles). Noting that a 1 = cos α2s e−iϕs /2 and a− 1 = sin α2s eiϕs /2 for ζ i characterized by the spherical coordinates 2

2

(1, αs , ϕs ), we have αs = ϕs = π/2 for S, αs = −π/2, ϕs = 0 for R and αs = ϕs = 0 for L. Using the definition (2.12) one easily derives the cross section for unpolarized electrons from (2.8),



 1 2 1 2 dσ = N0 |F (Mi , Mf , )| + |F (Mi , Mf , − )| , (2.23) d 0 2 2 Mi ,Mf

with N0 being the prefactor in (2.8). Further,  1 1 S = −2 Im {F ∗ (Mi , Mf , ) F (Mi , Mf , − ) } · N0 /(dσ/d )0 , 2 2 Mi ,Mf



1 1 Re {F ∗ (Mi , Mf , ) F (Mi , Mf , − )} · N0 /(dσ/d )0 , 2 2 Mi ,Mf

 1 2 1 2 L = |F (Mi , Mf , )| − |F (Mi , Mf − )| · N0 /(dσ/d )0 . 2 2 R = −2

(2.24)

Mi ,Mf

From this representation it is clear that S and R are sensitive to the relative phases of the transition amplitudes. We also note that if the sums in (2.23) and (2.24) are restricted to a single term, the sum rule  = 1 holds, with  = S 2 + R 2 + L2 ,

(2.25)

as is the case for elastic potential scattering [30,32]. 3. Numerical details 3.1. DWBA code The radial Dirac functions gκ and fκ are calculated with the help of the Fortran 77 package RADIAL of Salvat et al. [39]. Their output, say Gκ and Fκ , where the large component Gκ is asymptotically plane-wave normalized (with unit amplitude), is in our code renormalized according to

72

D.H. Jakubassa-Amundsen / Nuclear Physics A 937 (2015) 65–83

gκ (re ) = λGκ (re ),

fκ (re ) = −λFκ (re ),

λi,f =

Ei,f + c2 1 , πEi,f ki,f re

(3.1)

for the initial and final states, respectively. These functions, calculated on a grid up to re = R0 where R0 exceeds the range of the short-range part of the nuclear potential, are subsequently spline-interpolated to continuous values of re . Outside R0 the radial Dirac functions are constructed in terms of a superposition of regular and irregular Coulomb–Dirac functions, solutions to the Coulombic part of the nuclear potential [40,39]. The nuclear potential is obtained from the nuclear ground-state charge density, which is available in terms of a Fourier–Bessel expansion [41,18]. (The coefficients Aμ published in [18] have to be multiplied by a factor of 3.1 to give the correct normalization of the ground-state charge density.) The transition densities, when extracted from experiment, can also be represented in terms of a Fourier–Bessel expansion [7,17]. Correcting printing errors, √ 4π  rN L (rN ) = Aμ qμ(L−1) jL (qμ(L−1) ), R μ R  √ rN JL,L+1 (rN ) = − 4π kRj Bμ jL+1 (qμ(L) ), (3.2) Rj μ where R and Rj are cutoff radii outside of which the transition densities are zero. The number (λ) qμ is the μ-th zero of the spherical Bessel function jλ , and Aμ and Bμ are the tabulated expansion coefficients. (In [18], the sign of Aμ should be reversed if (3.2) is used.) With the help of the continuity equation which interrelates the electric transition densities [7], 

L+1 L+2 d kL (rN ) = − + JL,L+1 (rN ) 2L + 1 drN rN 

L L−1 d + − (3.3) JL,L−1 (rN ), 2L + 1 drN rN and using the recursion relations of the spherical Bessel functions, one obtains   4π  √ rN JL,L−1 (rN ) = k − 2L + 1 Aμ jL−1 (qμ(L−1) ) L R μ  √ rN + L + 1 Rj Bμ jL−1 (qμ(L) ) . Rj

(3.4)

If JL,L+1 is not available in terms of a Fourier–Bessel expansion, JL,L−1 can be calculated from the continuity equation (3.3) in the following way. Using the fact that (3.3) is a linear inhomogeneous differential equation for JL,L−1 , its solution is given by R1 JL,L−1 (rN ) =

−rNL−1  +

dx x L−1

rN

L+1 L





2L + 1 k L (x) L



d L+2 + JL,L+1 (x) , dx x

(3.5)

where R1 = max{R , Rj } and the boundary condition JL,L−1 (R1 ) = 0 was used. An integration by parts leads to the simpler form,

D.H. Jakubassa-Amundsen / Nuclear Physics A 937 (2015) 65–83

73



JL,L−1 (rN ) =

L+1 JL,L+1 (rN ) L  R1   2L + 1 L−1 dx  − (x) + (L + 1)(2L + 1) J (x) . rN kx  L L,L+1 L xL rN

(3.6) For the evaluation of the radial integrals (2.17) and (2.19) we make use of the fact that the transition densities are zero outside R1 to split the infinite electronic integral according to ⎛ ⎞ ∞ Rm ∞ ⎜ ⎟ re2 dre h(1) + ⎠ re2 dre h(1) (3.7) λ (kre ) D(re ) = ⎝ λ (kre ) D(re ), rN

rN

Rm

where D(re ) denotes the product of radial Dirac functions and Rm ∼ 4R1 is well outside the range of the transition densities. The second integral is independent of rN such that the corresponding double integral in (2.17) or (2.19) reduces to a product of single integrals. Our DWBA code for the cross section (dσ/d )0 was tested against the plane-wave Born approximation, where the cross section can be expressed in closed form,

PWBA

  1 θ  te 1 dσ = σMott + tan2 |FL (q)|2 + |Fλtm (q)|2 , |FLc (q)|2 + d

frec 2 2 0 σMott =

1 cos2 θ2 , c2 4ki2 sin4 θ2

(3.8)

with q = |k i − k f | and kf from (2.2). The functions FLc , FLte and Fλtm are, respectively, the Coulomb and transverse electric and magnetic form factors which are the Fourier–Bessel transforms of the respective transition densities [7]. In particular, FLte reflects a coherent superposition of JL,L+1 and JL,L−1 . In the Mott cross section σMott inelasticity and electron mass are neglected. This is of no serious consequence for the magnetic scattering. On the other hand, the contribution from Coulomb scattering in (3.8) vanishes at θ = 180◦ . Therefore σMott × |FLc (q)|2 /frec should be replaced by coul

kf 1 (Ei + c2 )(Ef + c2 ) c dσ (Z = 0) = |FL (q)|2 [1 + 2x cos θ + x 2 ] (3.9) d 0 ki frec c4 (q 2 − k 2 )2 with x = c2 ki kf /[(Ei + c2 )(Ef + c2 )]. This expression is obtained from (2.6) by substituting (σi )

relativistic plane waves for the Dirac functions ψi  dr e eiqr e

eik|r e −r N | 4π . = eiqr N 2 |r e − r N | q − k2

(σf )

and ψf

and by using the formula (3.10)

The importance of Coulomb scattering at 180◦ has been pointed out in several papers [42,26,43], showing within the DWBA that its contribution to the cross section may in some cases even be comparable to the magnetic scattering. The agreement between (3.8) with the replacement (3.9) and the DWBA result for a nuclear charge Z = 0.01 is about 1% or below for all angles (while, without the replacement (3.9), there may be deviations of 10% or more already at θ = 178◦ ). We note that the rapid decrease of the Coulomb scattering cross section for θ → 180◦ is due to a

74

D.H. Jakubassa-Amundsen / Nuclear Physics A 937 (2015) 65–83

strong mutual cancellation of the component of the radial integral involving gκf gκi on one hand and fκf fκi on the other. This mutual cancellation is independent of L, but increases with θ and decreases with impact energy. 3.2. Convergence In particular for low multipoles, severe convergence problems may occur in the DWBA code. These concern not only the convergence of the radial integral (3.7) with the choice of an upper cutoff (re ≤ Rmax ), but also the convergence of the transition amplitudes with the cutoff lf,max of the partial-wave series in (2.13). Thereby the Yennie et al. [40] convergence acceleration for the sum over the spherical harmonics, as provided in [15], is used. Such an acceleration (up to 4-fold) is mandatory for the extraction of the spin asymmetries. Using lf,max ∼ 15–25 satisfactory convergence (with a 5% accuracy or better for the polarization correlations) is achieved for L > 2 up to impact energies Ei,kin ∼ 130 MeV (L = 3) and 220 MeV (L = 5), respectively, with a computation time of a few minutes on a conventional work station. Concerning the radial integrals there is no absolute convergence for nonzero excitation energies. Recalling that for re → ∞ the product of Dirac functions behaves like |D(re )| ∼ 1/re2 (1) while hl (kre ) ∼ 1/re [44], the integrand in (3.7) decays only ∼ 1/re . There have been various attempts in the literature to cope with this problem, such as introducing an exponential convergence generating factor into the radial integrals [45] or averaging over a wide range of Rmax [46,15], the latter method being improved by including an estimate for the tail contribution [26]. In our code we avoid such manipulations and, following [46], take a fixed integration limit Rmax which is considerably larger than the one used in early work. That this method actually works (for l ≥ 2) can be understood in the following way. For elastic scattering (k = 0) there is rapid l /r l+1 in (2.11) instead of the product of Hankel convergence for all l ≥ 1, since there occurs r< > and Bessel functions (see, e.g. [47]), such that the integrand behaves asymptotically ∼ 1/rel+1 . Therefore, provided Ex is small and Rmax not too large, the radial integral may show stability with Rmax in a certain regime (which we have found to be 0.006 au  Rmax  0.015 au for all collision energies considered, where 1 au = 5.29177 × 104 fm), and this local convergence improves with L. There remain, however, considerable convergence problems occurring for L ≤ 2 (and in particular for L = 1) due to the presence of the S te(−) contribution relating to JL,L−1 . We note that for those L where convergence is marginal, too high values of lf,max may lead to spuriously high cross sections, invalidating also a multiple convergence acceleration. 4. Results In this section we provide results for the differential cross sections for electron impact excitation of low-lying nuclear states with multipolarity 2 ≤ L ≤ 5, and we give predictions for the respective polarization correlations S and R. As reference data, results from elastic scattering are included. Let us start with considering the lowest 3− and 5− states of the doubly magic nucleus 208 Pb with ground state 0+ . The 3− state is strongly collective [7,33]. Since Ji = 0 and πf i = −1, only L and JL,L±1 (with L = 3, respectively, L = 5) contribute to the excitation. Apart from a series of cross section measurements there have been repeated theoretical investigations for their interpretation, based on the calculation of the transition densities from various nuclear models and using a DWBA analysis for the scattering process (for a compilation of the 3− results, see [27,16], for the 5− results see, e.g. [48,17]).

D.H. Jakubassa-Amundsen / Nuclear Physics A 937 (2015) 65–83

75

dσ ) in fm2 /sr (= 10−2 b/sr), (b) Sherman function S and (c) transverse polarFig. 1. (a) Differential cross section ( d

0 ization correlation R for the excitation of the 5− state of 208 Pb (———), as well as for elastic scattering from the 0+ ground state of 208 Pb (− · − · −) at the angle θ = 178◦ as a function of kinetic electron energy Ei,kin = Ei − c2 . Included are the results for pure Coulomb excitation (Sc and Rc ) by setting J5,4 = J5,6 = 0 (− − − − −), as well as the transverse electric contribution to the excitation cross section (· · · · · ·).

4.1. The 5− state of 208 Pb at 3.198 MeV In our calculations we have used the experimentally extracted transition densities 5 and J5,6 as given in [17], from which J5,4 is obtained with the help of (3.6). Fig. 1(a) shows the energy dependence of the excitation cross section at a scattering angle θ = 178◦ , together with the separate contributions for Coulomb and magnetic scattering. For such a backmost angle the transverse electric contribution is already largely dominant over the Coulomb contribution. Comparison is made with the cross section for elastic electron scattering from 208 Pb, and it is seen that beyond 160 MeV the excitation has even a higher probability than elastic scattering (except in the diffraction minima). In fact, the diffraction structures are considerably more pronounced for inelastic scattering. In Fig. 1(b) the corresponding results for the Sherman function are given. Clearly the strong magnetic scattering leads to a considerable reduction of this transverse spin asymmetry, except in the diffraction minima of the cross section where Coulomb scattering has retained some importance. This quenching of S is also present for elastic scattering from nuclei with spin as soon as the magnetic scattering prevails [36]. As compared to elastic (potential) scattering from 208 Pb [22,32], the reduction of S for the 5− excitation may, however, amount to

76

D.H. Jakubassa-Amundsen / Nuclear Physics A 937 (2015) 65–83

dσ ) for the excitation of the 5− state of 208 Pb (———) and for elastic scattering Fig. 2. (a) Differential cross section ( d

0 + from the 0 ground state (− · − · −) by 167 MeV electrons as a function of scattering angle θ . Also shown are the Coulomb (− − − − −) and the transverse electric (· · · · · ·) contributions to the excitation cross section. The experimental data are from [48] (•, including the data between 105◦ –120◦ ) and from [17] (). The datum point at 160◦ is for 163.7 MeV. (b) Sherman function S for the excitation of the 5− state of 208 Pb as a function of scattering angle θ for the collision energy of 205 MeV (———), respectively 167 MeV (− · − · −). Included is the Sherman function Sc from pure Coulomb scattering (− − − − −, 205 MeV; · · · · · ·, 167 MeV).

several orders of magnitude at such backmost angles, whereas the spin asymmetry from inelastic Coulomb scattering alone (Sc ) is nearly comparable to the one from elastic scattering. Fig. 1(c) displays the results for the spin asymmetry R from transversely in-plane polarized electrons. This asymmetry is considerably larger than S and may reach 10% or more, even with the quenching by the magnetic scattering taken into account. However, its experimental determination requires the measurement of the (longitudinal) spin polarization of the scattered electrons. In Fig. 2(a) the angular dependence of the inelastically scattered electrons is displayed. A collision energy of 167 MeV has been chosen because of the strong suppression of the Coulomb scattering near this energy (by about two orders of magnitude at 178◦ , see Fig. 1(a)) and because of the availability of experimental data [48]. It should be mentioned that the transition densities used in our work were extracted from a different data set [17]. Hence the agreement between DWBA and experiment for θ  100◦ reflects the compatibility of the two data sets, which, however, is no longer the case at the higher angles. Included is one datum point at a backward angle (160◦ ), which is measured for a slightly lower energy (163.7 MeV) where theory predicts a cross section about 30% lower than at 167 MeV. It is clearly seen that the magnetic scattering gains importance at angles θ  160◦ while the Coulomb contribution drops sharply towards 180◦ . Fig. 2(b) shows the angular dependence of the Sherman function at 167 MeV as well as at 205 MeV where S has a local maximum (see Fig. 1(b)). While the spin asymmetry Sc pertaining to Coulomb scattering alone is strongly peaked very close to 180◦ (at θ = 179.5◦ , Sc = −0.212 for 167 MeV and Sc = 0.115 for 205 MeV), the extremum of S is much reduced and shifted to smaller angles, the more so, the more important the magnetic scattering. The change of the minimum to a maximum when increasing the energy from 167 MeV to 205 MeV is due to the diffraction structures and provides a sensitive way to test nuclear models (see, e.g. [32]). 4.2. The 3− state at 2.614 MeV For the excitation of this state only the transition charge density 3 is available in the literature because the current transition densities are too weak for an experimental extraction. However, one

D.H. Jakubassa-Amundsen / Nuclear Physics A 937 (2015) 65–83

77

Fig. 3. Radial dependence of the transition charge density 3 (− − − − −, scaled down by a factor of 0.1) and the transition current densities J3,2 (———) and J3,4 (− · − · −) for the excitation of the 3− state of 208 Pb. Note the √ multiplication by 4π as compared to [33].

can use the concept of vorticity, introduced in [33], and the relation between the vorticity density ωLL (rN ) and the transition current density JL,L+1 (rN ), 

L 1 JL,L+1 (rN ) = − 2L + 1 c rNL+2

rN x L+2 dx ωLL (x).

(4.1)

0

With the help of the continuity equation (3.3), JL,L−1 can therefore be calculated from R1 JL,L−1 (rN ) =

−rNL−1 rN

dx x L−1



2L + 1 k L (x) − L



 L + 1 ωLL (x) , 2L + 1 c

(4.2)

where we have used the experimental 3 and ω33 from an RPA calculation [49] as provided in [33]. The upper integration limit R1 = max{Rc , Rω } where Rω is the radial extension of ω33 . The so obtained transition densities are shown in Fig. 3. Note that the charge transition density 3 is strongly peaked at the nuclear surface, which is related to the collectivity of the 3− state [7], and it is up to a factor of 30 larger than J3,2 . The oscillations reflect the nodes of the nuclear orbits participating in the collective excitation, in particular the 3s1/2 proton orbit [7]. J3,4 is the smallest contribution (although in the numerical calculation of the spin asymmetries the most stable one). It should be remarked that the convergence properties, in particular for the contribution from J3,2 , are considerably worse than for the L = 5 case, such that the polarization correlations can only be extracted for collision energies up to 130 MeV. Fig. 4 shows the transverse spin asymmetries S and R at a scattering angle of 175◦ . Due to the smallness of J3,2 and J3,4 the cross section is basically given by the Coulomb contribution (which only loses importance at angles much closer to 180◦ ), while the transverse electric contribution is nearly an order of magnitude (for Ei,kin  80 MeV) or more than two orders of magnitude (above 90 MeV) below. Therefore, S is very close to Sc at all energies, and there are also only slight deviations between R and the pure Coulomb asymmetry Rc which occur at the lower energies. As compared to the respective polarization correlations from elastic electron scattering, the inelastic ones are of comparable magnitude. With a Sherman function of the order of |S| ∼ 0.1 in the energy region 85–90 MeV (and a cross section of ∼ 10−6 –10−7 mb/sr, two orders of magnitude

78

D.H. Jakubassa-Amundsen / Nuclear Physics A 937 (2015) 65–83

Fig. 4. Transverse spin asymmetries (upper curves for S, lower curves for R) for the excitation of the 3− state of 208 Pb (———) and for elastic scattering from the 0+ ground state (− · − · − for S, - · - · - for R) as a function of kinetic electron energy Ei,kin at θ = 175◦ . Also shown is the spin asymmetry Rc when only 3 is retained (− − − − −). Included is the result for S when, instead of the experimental 3 , the corresponding RPA transition charge density from [33] is used (· · · · · ·).

below the elastic one) this offers the chance of experimental observation. At the extremum (179◦ ) for 85 MeV, we predict S = −0.16 and (dσ/d )0 = 4.6 × 10−8 mb/sr. In order to display the sensitivity to nuclear models, we have calculated S from (4.1) and (4.2), but substituted the experimental 3 by the theoretical charge transition density from an RPA calculation [33] using the Ring and Speth wavefunctions [49]. The result, shown in Fig. 4, implies both a shift of the diffraction structures and a reduction of the minimum near 90 MeV. For example near 80 MeV where the cross section changes only by 30%, |S| changes by more than a factor of two in addition to a sign change in S. In Fig. 5 the angular dependence of the transverse spin asymmetry R is displayed at 85 MeV. For elastic scattering, R reaches nearly −1 at angles close to 180◦ . In contrast, for the 3− excitation, the pure Coulomb asymmetry Rc is reduced to |Rc | ≤ 0.55 and, due to the increasing importance of the transverse electric contribution at increasing angles, the total asymmetry is reduced to |R| ≤ 0.36. In concord with the behavior of S, the minimum is also shifted to increasingly smaller angles when compared to the transverse spin asymmetry from elastic scattering. The angular dependence of S is much alike the one shown for R in Fig. 5 (with the extremum values |S| = 0.5 for elastic scattering, and, at θ ≈ 179◦ , |Sc | = 0.25 and |S| = 0.16 for the 3− excitation). 4.3. The 5/2− state of 89 Y at 1.745 MeV This excitation from the 1/2− ground state of 89 Y, which is a near-closed shell nucleus with a 2p1/2 proton outside the 88 Sr core, can to a large extent be described in terms of a single-particle excitation (2p1/2 → 1f5/2 ), but pair correlation and core polarization must also be taken into account [18]. The parity conservation πf i = +1 implies that the multipoles L = 2 and 3 contribute to the excitation in terms of 2 , J2,1 , J2,3 and J3,3 . The 5/2− state is also a nice candidate for investigating the effect of the magnetic scattering on the polarization correlations because it is expected to be very large (being already large in elastic scattering [36]). Moreover, the three transition densities 2 , J2,3 and J3,3 are all available in terms of Fourier–Bessel expansions, ob-

D.H. Jakubassa-Amundsen / Nuclear Physics A 937 (2015) 65–83

79

Fig. 5. Transverse polarization correlation R for the excitation of the 3− state of 208 Pb (———) and for elastic scattering from the 0+ ground state (− · − · −) at an impact energy of 85 MeV as a function of scattering angle θ . Also shown is the result Rc for pure Coulomb excitation (− − − − −).

tained from experiment [18], while J2,1 is calculated from the charge–current conservation (3.4). Therefore, as a further test of our DWBA code we have recalculated the energy dependence of the differential cross section at θ ≈ 178◦ in the region 71.1 MeV ≤ Ei,kin ≤ 263.6 MeV and have found good agreement with earlier DWBA results for the separate transverse electric and transverse magnetic contributions as published in [18]. The transverse electric contribution, which arises predominantly from J2,1 , has a deep minimum near 180 MeV. Since the convergence properties, in particular with lf,max , are poorest for this contribution we have chosen impact energies where it plays a minor role in order to investigate the polarization correlations. Fig. 6(a) displays the separate Coulomb, transverse electric and transverse magnetic contributions to the total cross section at an energy of 213.3 MeV, and it is seen that at all angles, the total cross section is either dominated by the Coulomb contribution (at the smaller angles) or by the transverse magnetic contribution (for θ  110◦ ). Included is also the result for elastic scattering from 89 Y, which is very close to the excitation cross section at angles beyond 140◦ . The respective calculations are described in [36]; however, here we have used the unscaled current density J11 (as provided by [50]), since this gives a better fit to the high-energy experimental data. In Fig. 6(b) the corresponding transverse spin asymmetry R is shown, in comparison with either the pure Coulomb contribution Rc or with the respective spin asymmetries from elastic scattering. It is seen that for the 5/2− excitation the influence of magnetic scattering starts at angles as low as 80◦ , and R gets strongly suppressed above 120◦ , whereas Rc proceeds to its deep minimum at the backmost angles (in a similar way as shown in Fig. 5). In the elastic case the magnetic scattering is much less effective, and the reduction of the respective polarization correlation as compared to its value from potential scattering occurs only beyond 110◦ . Thus, near 170◦ , the spin asymmetry from elastic scattering has decreased to about 0.5%, while the spin asymmetry from excitation is nearly completely quenched. In order to show the sensitivity of R to the ground-state charge distribution we have performed the calculations for elastic scattering not only for a Fourier–Bessel charge density (which is also used for the wavefunctions entering into the excitation formalism and is taken from [18]), but also for a Fermi-type charge distribution ([41], which is used in [36]). When employing the superior Fourier–Bessel charge density, the diffraction structures are shifted to smaller scattering angles, combined with a considerable increase of |R| at the larger angles.

80

D.H. Jakubassa-Amundsen / Nuclear Physics A 937 (2015) 65–83

dσ ) for the excitation of the 5/2− state of 89 Y (———) and for elastic scattering Fig. 6. (a) Differential cross section ( d

0 from the 1/2− ground state of 89 Y (− · − · −) by 213.3 MeV electrons as a function of scattering angle θ . Also shown are the Coulomb (− − − − −), the transverse electric (· · · · · ·) and the transverse magnetic (- · - · -) contributions to the excitation cross section. The experimental datum point for inelastic scattering (•) is from [18], that for elastic scattering (◦) is from [50]. (b) Transverse polarization correlation R for the excitation of the 5/2− state of 89 Y (———) and its value Rc for pure Coulomb excitation (− − − − −) by 213.3 MeV electrons as a function of θ . Also shown is R for elastic scattering from the 1/2− ground state (upper curves) and from pure potential scattering (lower curves) if the ground-state charge distribution is of Fourier–Bessel type (− · − · −) or if it is a Fermi distribution (· · · · · ·). (c) Sherman function S for the excitation of the 5/2− state of 89 Y (———) and its value Sc for pure Coulomb excitation (− − − − −) by 172.9 MeV electrons as a function of θ . Also shown is S for elastic scattering from the 1/2− ground state (− · − · −) and for pure potential scattering (· · · · · ·). In (a) and (c), a Fourier–Bessel ground-state charge distribution is used throughout.

Due to convergence problems we did not find it possible to extract the Sherman function at this energy. Therefore we have chosen in Fig. 6(c) a lower energy (172.9 MeV) to give some estimates of S (with an accuracy as low as 10% in the angular interval 110◦ –160◦ , but considerably worse when |S| is close to zero), where the transverse electric contribution to the cross section decreases smoothly with θ and is strongly suppressed at the backmost angles. At this lower energy (as compared to Fig. 6(b)) the onset of magnetic scattering affects the spin asymmetry only at much higher angles (beyond 120◦ ). However, since the Sherman function Sc for pure Coulomb scattering remains small up to the backmost angles this late onset of magnetic scattering does not help to increase the spin asymmetry. At the backmost angles where transverse magnetic scattering is largely dominating, the Sherman function has nearly vanished completely (|S|  10−5 near 170◦ ).

D.H. Jakubassa-Amundsen / Nuclear Physics A 937 (2015) 65–83

81

Table 1 Sum rule  for the polarization correlations from excitation of the 3− and 5− states of 208 Pb at collision energies of 85 MeV and 205 MeV as a function of scattering angle θ . θ

85 (3− )

85 (5− )

205 (5− )

160 165 170 175 178 179 179.3 179.5

1 1 1 0.998 0.970 0.849 0.739 0.637

1 0.999 0.997 0.988 0.974 0.970 0.970 0.969

1 1 1 0.997 0.988 0.979 0.977 0.976

4.4. Sum rule We have found that for the nuclear excitations studied here, the sum rule (2.25) for the polarization correlations,  = 1, is to a high degree satisfied except at the backmost scattering angles. For example, for the 5− excitation of 208 Pb,  > 0.97 at θ = 178◦ for all collision energies considered, with a global increase to unity with increasing Ei,kin , but with some local oscillations due to the diffraction effects. In Table 1, where the angular dependence of  is shown for some examples, it is evident that  deviates from unity as θ approaches 180◦ . This is related to the fact that the longitudinal spin asymmetry L (a measure of the electron’s helicity conservation) no longer drops down towards −1 in the limit θ → 180◦ if magnetic scattering is present. This same feature is observed for elastic scattering from nuclei carrying spin [36]. On the other hand, strong magnetic scattering (induced by large transition currents or high collision energies) enforces the conservation of helicity up to very high angles, and the (modest) drop-down will only occur at θ extremely close to 180◦ . Therefore, at fixed (but not ultimately high) angles,  tends to increase when the magnetic effects become larger (i.e. when proceeding to the next column to the right in Table 1). 5. Conclusion We have applied the DWBA scattering formalism to give estimates for the transverse spin asymmetries from nuclear excitation by electron impact. The impact energy was restricted to values below 250 MeV in order to produce sizeable spin asymmetries S and to justify the onephoton exchange approximation for the description of the scattering process. For the heavy spin-0 nuclei, represented by 208 Pb, the following results were found. Like for elastic (potential) scattering, the transverse spin asymmetries are tiny in the forward hemisphere but increase at scattering angles beyond 150◦ . For strongly collective excited states (with many protons participating in the transition) or for those collision energies where the Coulomb scattering is largely dominating, the spin asymmetries may be quite large, comparable or even larger than for elastic scattering. However, as soon as magnetic scattering becomes relevant, e.g. for angles close to 180◦ , the spin asymmetries are strongly reduced. Thus the interplay between nuclear charge and current induced transitions leads to a reduction and shift of the backward extremum of S and R to smaller angles. Its precise position and sign is also influenced by the diffraction structures, i.e. varies with the collision energy. For nuclei carrying spin the quenching of the spin asymmetries is enhanced by the simultaneous presence of both transverse electric and transverse magnetic transitions. For our example,

82

D.H. Jakubassa-Amundsen / Nuclear Physics A 937 (2015) 65–83

the excitation of the lowest 5/2− state of 89 Y, the magnetic scattering is particularly large, thus dominating the cross section already beyond 130◦ (at Ei  170 MeV). The strong reduction of S and R, most effective at large collision energies or backmost angles, is hence similar to the quenching of S and R for elastic electron scattering from nuclei carrying spin. However, it seems to occur already at considerably smaller angles than in the case of elastic scattering and leads to a complete disappearance of the backward extremum. By using different types of ground-state or transition charge densities, available in the literature, we have shown that the polarization correlations are very sensitive to such changes, considerably more than the differential cross sections. This opens up a new possibility to study details of nuclear structure effects. For a proof-of-principle experiment we suggest to investigate a collective excited state of a heavy spin-zero nucleus which has a large excitation probability, but small transition current densities, in order to obtain a large spin asymmetry at the backmost scattering angles. The lowest 3− state of 208 Pb at 2.614 MeV seems to be a promising candidate, its Sherman function amounting to 10% or more for collision energies around 85 MeV at the backmost scattering angles (176◦ –179◦ ). Acknowledgements I would like to thank J. Enders for stimulating this project. I am particularly grateful to V. Ponomarev for many helpful discussions and for providing DWBA cross section test results. I would also like to thank the Referee for directing my interest to his early papers on the 180◦ scattering. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

T. Suda, Proc. DAE Symp. Nucl. Phys. 58 (2013) 19. S. Abrahamyan, et al., Phys. Rev. Lett. 108 (2012) 112502. S. Abrahamyan, et al., Phys. Rev. Lett. 109 (2012) 192501. K. Aulenbacher, I. Alexander, V. Tioukine, Nuovo Cimento 35 (2012) 186. J. Enders, et al., J. Phys. Conf. Ser. 295 (2011) 012152. H. Überall, Electron Scattering from Complex Nuclei, Academic Press, New York, 1971. J. Heisenberg, H.P. Blok, Annu. Rev. Nucl. Part. Sci. 33 (1983) 569. K.S. Jassim, A.A. Al-Sammarrae, F.I. Sharrad, H.A. Kassim, Phys. Rev. C 89 (2014) 014304. K. Alder, A. Bohr, T. Huus, B. Mottelson, A. Winther, Rev. Mod. Phys. 28 (1956) 432. R.S. Willey, Nucl. Phys. 40 (1963) 529. T.W. Donnelly, A.S. Raskin, Ann. Phys. 169 (1986) 247. E. Moya de Guerra, Phys. Rep. 138 (1986) 293. A.S. Raskin, T.W. Donnelly, Ann. Phys. 191 (1989) 78. T.A. Griffy, D.S. Onley, J.T. Reynolds, L.C. Biedenharn, Phys. Rev. 128 (1962) 833. S.T. Tuan, L.E. Wright, D.S. Onley, Nucl. Instrum. Methods 60 (1968) 70. D. Goutte, J.B. Bellicard, J.M. Cavedon, B. Frois, M. Huet, P. Leconte, Phan Xuan Ho, S. Platchkov, Phys. Rev. Lett. 45 (1980) 1618. J. Heisenberg, J. Lichtenstadt, C.N. Papanicolas, J.S. McCarthy, Phys. Rev. C 25 (1982) 2292. J.E. Wise, F.W. Hersman, J.H. Heisenberg, T.E. Milliman, J.P. Connelly, J.R. Calarco, C.N. Papanicolas, Phys. Rev. C 42 (1990) 1077. N. Pietralla, et al., Phys. Lett. B 681 (2009) 134. H.C. Lee, Atomic energy of Canada, Rep. AECL-4839, Ontario, 1975. J. Sromicki, et al., Phys. Rev. Lett. 82 (1999) 57. E.D. Cooper, C.J. Horowitz, Phys. Rev. C 72 (2005) 034602. M. Gorchtein, C.J. Horowitz, Phys. Rev. C 77 (2008) 044606. O. Moreno, E. Moya de Guerra, P. Sarriguren, J.M. Udías, J. Phys. G 37 (2010) 064019.

D.H. Jakubassa-Amundsen / Nuclear Physics A 937 (2015) 65–83

[25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50]

83

J.E. Amaro, J.A. Caballero, T.W. Donnelly, E. Moya de Guerra, Nucl. Phys. A 611 (1996) 163. M. Nishimura, E. Moya de Guerra, D.W.L. Sprung, Nucl. Phys. A 435 (1985) 523. H. Rothhaas, J. Friedrich, K. Merle, B. Dreher, Phys. Lett. 51B (1974) 23. C. Lüttge, et al., Nucl. Instrum. Methods Phys. Res. A 366 (1995) 325. N. Ryezayeva, et al., Phys. Rev. Lett. 100 (2008) 172501. J.W. Motz, H. Olsen, H.W. Koch, Rev. Mod. Phys. 36 (1964) 881. P. Uginˇcius, H. Überall, G.H. Rawitscher, Nucl. Phys. A 158 (1970) 418. D.H. Jakubassa-Amundsen, R. Barday, J. Phys. G 39 (2012) 025102. D.G. Ravenhall, J. Wambach, Nucl. Phys. A 475 (1987) 468. A.R. Edmonds, Angular Momentum in Quantum Mechanics, 2nd edition, Princeton University Press, Princeton, 1960, §5.9. F. Gross, Relativistic Quantum Mechanics and Field Theory, Wiley, New York, 1993, p. 286. D.H. Jakubassa-Amundsen, J. Phys. G 41 (2014) 075103. E.M. Rose, Relativistic Electron Theory, Wiley, New York, 1961, §15, 19. D.H. Jakubassa-Amundsen, Nucl. Phys. A 896 (2012) 59. F. Salvat, J.M. Fernández-Varea, W. Williamson Jr., Comput. Phys. Commun. 90 (1995) 151. D.R. Yennie, D.G. Ravenhall, R.N. Wilson, Phys. Rev. 95 (1954) 500. H. De Vries, C.W. De Jager, C. De Vries, At. Data Nucl. Data Tables 36 (1987) 495. E. Moya de Guerra, Phys. Rev. C 27 (1983) 2987. P. Sarriguren, E. Moya de Guerra, Phys. Rev. C 30 (1984) 2105. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover Publications, New York, 1964. R. Huby, J.R. Mines, Rev. Mod. Phys. 37 (1965) 406. H.T. Fortune, T.J. Gray, W. Trost, N.R. Fletcher, Phys. Rev. 179 (1969) 1033. J.F. Prewitt, L.E. Wright, Phys. Rev. C 9 (1974) 2033. J. Friedrich, Nucl. Phys. A 191 (1972) 118. P. Ring, J. Speth, Nucl. Phys. A 235 (1974) 315. J.E. Wise, et al., Phys. Rev. C 47 (1993) 2539.