Test of nuclear currents in transverse electron scattering

Nuclear Physics A402 (1983) 235-246 @ North-Holland Publishing Company

TEST OF NUCLEAR

CURRENTS IN TRANSVERSE SCATTERING*

E. WfiST, Instihd

fCrTheoretische

U. MOSEL

ELECTRON

and i. KUNZ*

Physik, Universitd Giessen, D-6300

Giessen, West Germany

and H.G. Institut

fiir Kerphysik,

ANDRESEN

and M. h&JLLER

Universitiit Mainz, D-6500 Received

23 December

Mainz, West Germany

1982

Abstract: Nuclear

current distributions for rotational states of heavy nuclei obtained in realistic cranked shell-model calculations with BCS correlations are used to calculate the transverse PWBA form factors for inelastic electron scattering. By keeping the same density but changing the velocity fields the sensitivity of the form factors to the dynamics of nuclear motion is explored. It is found that experiments on the O’-+ 2+ transition alone will not allow a satisfactory determination of the nucIear flow but that an analysis also of the O+ + 4+ transition will be necessary. For these transitions optimal q-windows are identified.

1. Introduction The flow pattern of nuclear collective motion has recently been the subject of many theoretical* studies. Although most of these are concerned with nuclear vibrations there exists now a number of microscopic calculations also for rotations. After more exploratory studies by Radomski “) and Gulshani and Rowe 3), the first calculations that were based on realistic shell models with BCS correlations included were reported in refs. 4*5). Th ese calculations that gave the correct moment of inertia, i.e. the correct first moment of the current distribution, showed that the rotational currents exhibited an interesting structure roughly corresponding to a mixture of rigid and irrotational rotation with weak vortices superimposed “). The possibility of analyzing nuclear rotational current distributions by means of inelastic transverse electron scattering has been pointed out several times 6-R).The current contributions to the cross section for rigid and irrotational rotations, respectively, are of the order of IO-* to 10m9mb/sr [refs. ‘,‘)I. These small cross sections can be measured with high-current accelerators, which are presently available or planned for the near future 9). * Present address: Los Alamos National Laboratory, Los Alamos, NM, USA. ’ Work supported by BMFT and GSI Darmstadt. * For a comprehensive collection of theoretical results on flow patterns see ref. ‘). 235

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E. W&t et al. / Nuclear currents

In order to separate the current contribution from the longitudinal contribution, 180” scattering experiments are essential. Even for this experimental condition the radiation tail background of the elastic peak and the still remaining background of the longitudinal component may be seriously limiting factors for the obtainable experimental accuracy. Because of all these reasons experiments to determine the flow pattern will be quite dificuit. Their value will obviously depend strongty on how many details one can actually hope to resolve in the strongest transitions. This problem has been partly touched upon in an exploratory study lo) comparing the PWBA form factor obtained from an approximate angular momentum projection technique 7, with that for a velocity field for rigid rotation lo). The somewhat surprising outcome of this study was the enhancement of almost two orders of magnitude of the transverse form factor obtained from the approximate projection method relative to the rigid rotor results. Stimulated by this latter result it is the purpose of this paper to analyze which experiments are most sensitive to the details of rotational flow. For this purpose the form factors for transverse inelastic scattering have been calculated in PWBA as in refs. 7,10). The calculations are based on current distributions obtained in a realistic microscopic model and include the magnetization contributions. The sensitivity test has been accomplished by the comparison of the form factors for three different physically reasonable velocity fields always using the same density. In this way only the dynamical properties are checked. By calculating also the spin contributions optimal ~-windows for the determination of the convection currents will be discussed. All calculations were performed for ‘@Er, also treated in refs, ‘*IO). 2. Details of the calculations The microscopic current distributions cranking calculation,

were obtained

(N-~~~)ffl(rl.“‘rA;W)=EwZZ-f(r*‘..rA;W),

from a BCS correlated 0)

in which in the singIe-particle hamiltonian a deformed Woods-Saxon potential was used*. The parameters of this potential were taken from ref. ‘l); they yieId an intrinsic quadrupol~ moment of Q$ = 7.1 e * b in quite nice agreement with the experimental value of QYp = 7.6 e - b [ref, ‘“)I. Thus it is assured that the transition density multipole moment for the O+-+2+ transition is correctly described. The moment of inertia, as calculated from the cranking model with pair fields A, = 0.866 MeV and A,, = 1.018 MeV, comes out to be $th r;=27.2 tt* * MeV-I. This is somewhat too small compared with the experimental value of ..9cXl,= 37.5 h2 - MeV-” but this deviation could easily be fixed by a slight change of the ’ In our earlier calculations a Nilsson potential was used. The WS potentiat leads to better density distributions and thus more reliable longitudinal form factors.

E. WCst et al. j Nuclear currents

237

pairing parameters A, and A,. From our earlier studies 4*5)we know that such a change does not significantly alter the current distribution apart from an overall scaling which can easily be taken into account in the form factor (see below). The expectation value of the current for cranking around the x-axis is then given in lowest order perturbation theory by 4,

where j(r) is the usual current operator

and J, is the x-component of the angular momentum. Except for the small and here neglected current contribution from the spin-orbit potential this current (eq. (3)) fulflls the continuity equation. The E, are the quasiparticle energies and the u,, U, the usual BCS amplitudes. The current given in eq. (2) is defined in the body-fixed system. In a study of the observability of the current distributions it is essential to note that there is also a magnetization contribution to the transverse form factors. This “spin-current” is given by C(rt=vx*(rf, (pi = 2.79

&N

for protons, -1.91 @;Nfor neutrons) , (4)

where p(r) is the expectation value of the spin density in the cranking wave functions (including pairing). In order to calculate the transition current for the O’-+f’ state one has two possibilities. One can either describe the rotational states by collective model wave functions (-DLK); in this case the current operator is needed in collective coordinates. Equivalently, one could describe the states microscopically and then directly employ the microscopic current operator 13). We choose here the first approach that was also used by Radomski ‘*i4)b ecause it corresponds directly to the treatment of vibrations in inelastic (e, e’) scattering “). In complete analogy to the definition of the quadrupole operator in collective coordinates we define the current operator as a function of the collective coordinate 0%:

=Tfr)o=

C i=l

(T(r)ei)oi,

(3

238

E. W&r et al. / Nuclear currents

where the liF’s are the cranking wave functions obtained from eq. (1) in first-order perturbation theory, T(r) is a tensor that depends only on r, and the ei are the unit vectors along the Cartesian body-fixed principal axes. The cranking current j’(r) as defined in eq, (2) is just i”(r) = (~($~e~~~. For nuclei that are axially symme~i~ around the z-axis i”(r) = 0, and j’(r) can be obtained by a simple rotation of the coordinate frame, In a next step o is then replaced by an operator: Wj+ ~i/~~3

(6)

where i (= 1,2,3) labels the components of the vectors with respect to the body-fixed principal axes and the Bi are the corresponding principal-axis moments of inertia. This operator can be placed between the symmetrical-top wave functions2*i4); Radomski has shown that it fulfils the continuity equation 2*14). After working out the necessary matrix elements of the angular momentum operator one obtains for the reduced matrix element for an electric transition in a K=Oband

(7) where the jl;c(r> are the multipole components of the calculated current field,

determined by a numerical projection, In deriving eq. (7) the cranking relation &J =

Jr(r + 1)/S,

(9)

with 8 = 8i, = & (axial symmetry), has been used. All calcuIations have been performed with the empirical u-value. The transition operator TEA in eq. (7) contains a phase factor i” Eref. ‘“)I. The restriction to ct = z&l in eq. (7) is a direct consequence of our restriction to terms linear in w (eq. (2)). Parity conservation imposes the additional restriction I = odd whereas the usuaf symmetry of the nucEeus under rotation of the coordinate system by an angle of 71”around the x-axis leads to the relation

E. W&t

In addition, the requirement

et al. / Nuclear

239

currents

of a real current yields jX,;“,(r) = -(->*jL-1

.

(11)

Taking the last two equations (eqs. (lo), (11)) together implies a purely imaginary i;r1. Thus, for example, for the 0” -+ 2* transition in a K = 0 band one is left with ~~Ol~~~AllOO~ = &z( @ J jr(~) Im (i&r (rHr2 dr

In fig. 1 we show first the current distribution j”(r) in the y-r plane as obtained from the cranking calculation. The flow pattern is again extremely similar to the ones previously obtained both in calculations using a Nilsson potential 4, and in a self-consistent calculation 5). The overall pattern, in particular the fact that the deformation of the current field is vertical to that of the density 4, and the presence of weak vortices on the y-axis - which is the same also for other rotational nuclei, is thus fairly “model independent”.

IC I...

I..

2?

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2I.z

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0

L

to

5 yffml

Fig. 1. Current for Ie6Er with the cranked Woods-Saxon potential for (J,. = 2h in the y-r plane in the laboratory frame. The ellipsis represents the half-density contour. The arrow at y = 0.97 fm and z = 4.58 fm corresponds to a current of 0.0375 MeV/A fm’S2.

240

Fig. 2. ~rr~~~~~~~ form factor far 16’Er accwding to eq. (13) for the O’+ 2” transitive for the conviction current (dashed), the spin-current (dash-dotted) and the total current (full line).

In fig. 2 the square of the form factor fur transverse electron scattering is shown as a function of momentum transfer 4. It is defined as

The form factor F$ reaches a maximum of about 5 x 10W9at 4 =0.4 fm-“, then drops down to a sharp minimum at CJ= 1.35 fm-* and thereafter rises again. Also shown in fig. 2 are form factors of the pureIy transverse spin current (eq. (4)) and the convection current &me (note that the total form factor is the cu~ere~t sum of spin- and convection-current cantributions). The spin contribution starts to rise at 4 = 0.5 fm-’ and dominates the total form factor for 4 3 1 .Ofm-“. measurements that aim to determine the convectional flow pattern must thus be performed below 9 =0,6fm-‘. In order to study the sensitivity of the form factors on the details of the current di§trib~~o~s we have also performed cafculations for the convection current fields:

E. Wiist et al. / Nuclear j”

= ptf,ig

currents

= C@p(SZx X r> )

241 (14bI

where r is a deformation-dependent constant. In these calculations exactly the same density and the same w as above were used thus ensuring agreement of the form factors for 4 + 0 [ref. “)].

Fig.

3.

Form factor F: for the rigid-rotor current (dashed), the irrotational-flow and the convection current {full line).

current (dash-dotted)

Fig. 3 shows the comparison of the squared form factors (calculated without spin current) for k-rotational, rigid and realistic flaw. The main result of this comparison is that irrotational and realistic flow lead to quite similar form factors in the interesting region CJs 0.6 fm-’ (see above) whereas that for rigid rotation starts to deviate drastically already at 4 ~0.3 fm-‘, reaching a first minimum at 9 = 0.6 fm-’ [see also ref. ‘“)I. The reason for the similarity between the realistic and the irrotationaf flow Iies in the fact that the current distribution indeed contains a large irrotational component 16). That urig leads to such a different behavior, on the other hand, is due to the quite different r-dependence of the two transition currents. Qual~tatively~

242 this

E. Wiist el Eal.i .KwAw cm

efmenh

be seen as foffsws. The rigid rotor Geld {for rotation around the x-axis) is

given by ap,ig=w(O?-X* y)-r(Y:t;l and that for incompressible,

- Y:$_I) *

irrotational flow by

J’(r) = (PO(r)+ P2~~)‘Ir"zaW)~,

(17)

TS+ER fp&r] * dp,&% is the tra~s~t~o~ density, Noaf, for &E O”+ 2, ~~~~~ti~~ on@ the quadrupols component of j contributes. For the rigid rotor velocity, that is a pure J = 1 field (eq. (lS)), a transition is, therefore, only possible because of the second term in eq. ( 17) giving a tra~~s~t~o~current,

that is strongly surface peaked (because of pz(r) - dp&dr), On the other hand, z>iTr is a J = 2 field (eq. fl6)) so that now the ~a~s~t~on current, . li,,-r~a(r)‘cpzCr)Y20(~~))11Y~~1

+cLl~,

cw

afsb contains a volume part (-p&)).

This explains the large di&rence between the ~~otat~o~~ zrnd reahstic flow on one side and the rigid Bow nn the other, Xn order to get an intuitive picture of what details of the current can actually be “seen” in transverse (e, e’) scattering to the 2* state we show in fig, 4 the current distribution generated only from the ~~rnpone~~ts jzzr and jZIf, i.e.

This is the current distrib~tjon &at oTIr3 Eim &t best hqx to determine from. m accurate measurement of the 0” --, 2” transverse scattering cross section. As can be seen from fig. 4 this flow pattern resembles to a large extent that of irrotational Aow due to the fact that &(r)
243

E. W&t et at, / Ntrclenr currents -r-

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,

,

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0

-5

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Fig. 4 Convection cwrenf

seen

in fee, e’) scattering

far

the t)* -, 2” transition. The scale is the samt: as

in fig. 1.

large q. Tfie 0“ I, 4+ form factor is smaller by more than one order of magnitude compared to the O”* 2*; this strong decrease is probably connected with the unusually small &, deformation in 166Er. The spin ~~~t~~butio~ to the O’+ 4’ transition starts to rise at 4 = 0.6 fm-” and becomes dominant at large 4 GS1.2 fm-‘. However, already at g = 0.7 fm-’ Fg (4) starts to deviate significantly from the form factor without spin contribution. Fig. 6 shows again the comparison of the realistic form factor with those calculated under the assumption of irrotational and rigid rotation (eqs. (14a, b)). Again, as for the O*+ 2* transition, the rigid rotation leads to SLq-dependence that is quite different from that of the other two cases. What is remarkable, however, is that contrary to the 0” + 2” transition, now also the irrotational flow gives a form factor that dif’lers si~Rj~~ant~y from that for the realistic case in the “spin window” 4 i 0.6 fm-‘. In fig. 7, finally, the form factor for the O* -N6’ transition is shown. Although this form factor is nearIy as strong as that for the 0’-+4* transition [see also ref, “)I it is dominated by the spin contribution over the entire q-range. Thus observation of the transverse O++ 6* transition would be of no help in determining the convection current’s flow pattern.

244

E. Whistet al. / Nuclear currents UT -

.s _

N

*__-..--‘--

0

0” 0” ;: N

0

E. W&it et al. / Nuclear currenfs 10-8

r

I

I

I

245 /

I

I

FT2 O+-, 6

+

WS

IO+/

IO-" r

IO

-12

I

0

Fig. 7. Form factor

02

0.4

06

08

IO

F$ for the O++ 6+ transition.

1.2

IL

16

q I tm+l

For details see fig. 5.

In summary then, three main points have emerged from this study: first, the spin-current contributions tend to mask the convection current for q > 0.6 fm-’ for all transitions considered. Since for smalf q, on the other hand, the transverse form factors are determined by the density multipole moments, information about the convection current can be obtained only in a fairly narrow window. For the Ot + 2+ transition the optimal momentum transfer is q = 0.4 fm-‘; for the 0’-+4’ transition it is q -0.6 fm-‘. These are the values where the form factors are largest and the spin contributions are stiI1 negligible. Second, whereas it should be possible to distinguish between the rigid and real flow from the observation of the transverse O++ 2’ transition alone [although the from irrotational flow difference here is not as large as in ref. *‘)I, the distinction will be much harder. However, for the transverse 0’+4+ transition the differences from irrotational flow are considerably larger so that observation of this transition should be crucial for a determination of the flow pattern. Our results show that for both transitions the sensitivity is largest for the momentum transfer q = 0.4 fm-‘. The form factor for O+ -+ 4’ is smaller by about an order of magnitude than that for O’+ 2+. It may, however, actually be more easily observable because of its

246

E. Wi& et al. / Nuciear currents

higher transition energy which makes it easier to separate this transition from the elastis scattering. Third, a reconstruction of the actual flow pattern is impossible since the “magnetic” contributions of the current, i.e. the functions j,,+(r), are not determined in experiments on (even, even) nuclei: only simultaneous measurements of the transition currents in (even, even) and neighbouring (even, odd) nuclei could lead to a determination of the detailed structure of the flow pattern provided the polarization by the odd particle can be neglected. In many of the cases discussed the strong diffraction minima might seem to be ideal test regions because here the differences between the different flow patterns are largest. One has to keep in mind, however, that DWBA calculations will undoubtedly smear these minima out. Such calculations also have to be performed to check whether the q-windows identified above are still optimal when the distortion effects and the interference terms between longitudinal and transverse components contained in a DWBA treatment are taken into account. In addition, the effects of the radiation-tail background have to be considered. A quantitative investigation of those effects is presently under way. We would finally Iike to point out that also transverse scattering to low-lying vibrational states would be of considerable interest as recent theories have predicted quite complicated flow patterns for these states r6,17). References 1) Proc. Topical Meeting on nuclear fluid dynamics, Trieste, Italy, Oct. 1982, ed. M. di Toro ef al., 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17)

IAEA (Vienna), in press M. Radomski, Phys. Rev. Cl4 (1976) 1704 P. Gulshani and D.J. Rowe, Can. J. Phys. 36 (19783 468,480 J. Kunz and U. Mosel, Nucl. Phys. A323 (1979) 271 J. Fteckner, J. Kunz. U. Mosel and E. Wiist, Nucl. Phys. A339 (1980) 227 J. Heisenberg, Adv. Nucl. Phys. 12 (1981) 61 E. Moya de Guerra and S. Kowalski, Phys. Rev. C22 (1980) 1308 H.G. Andersen et al., Proc. Symp. on perspectives in electro- and photo-nuclear physics, Saclay (1980), Nucl. Phys. A358 (1981) 36% H. Herminghaus, Proc. Linear Accelerator Conf., Santa Fe, USA, 1981, LA-9234-C, ed. R.A. Jameson and L.S. Taylor, p. 22 E. Moya de Guerra, Nucl. Phys. A366 (1981) 259 J. Dudek and T. Werner, J. of Phys. 64 (1978) 1543 A.S. Goldhaber and G. Scharff-Goldhaber, Phys. Rev. Cl7 (1978) 1171 E. Moya de Guerra, Ann. of Phys. 128 (1080) 286 M. Radomski, PhD thesis, Stanford University (1973), unpublished T. de Forest, Jr. and J.D. Walecka, Adv. Phys. 15 (1966) 1 J. Kunz, A. Schuh, U. Mosel and E. Wiist, in ref. ‘) A. Schuh, J. Kunz and U. Mosel, in: Proc. 5th Kyoto Summer Institute, 1982 on microscopic theories of nuclear collective motion; Suppl. Progr. Theor. Phys., in press