Calculation of multiple-excitation cross sections using angle-dependent phase shifts

Nuclear 0

Physics

North-Holland

A366 (1981) 119-124 Publishing

Company

CALCULATION OF MULTIPLE-EXCITATION CROSS SECTIONS USING ANGLE-DEPENDENT PHASE SHIFTS R. LIPPERHEIDE Bereich Kern-und-Strahlenphysik,

+

Hahn-Meitner-Institut fir

Kernforschung

Berlin

and Fachbereich Phystk, Freie Unirersitrit Berlin, Berlin- West, German) H. MASSMANN Facultad de Ciencias. Universidad de Chile. Castlla 653, Santiago, Chile and H. ROSSNER Bereich Kern-and-Strahlenphysik.

Hahn-Meitner-institut fcr Kernforschung Berlin, Berlm- West, German) Received

26 February

198 1

Abstract: Multiple-excitation cross sections for the scattering of 70.4 MeV i2C ions on the even neodymium isotopes are calculated by the method of angle-dependent phase shifts. The results are compared with the data.

In a proper treatment of multiple excitations, as they occur in heavy-ion reactions, one generally has to carry out a full coupled-channels calculation involving all the channels of interest. For a simpler, approximate approach to this problem, the method of angle-dependent phase shifts has been proposed, which is particularly suited for the calculation of the cross sections of inelastic scattering on a rigid rotor ‘92). In the present note we report the results of such calculations for the scattering of 70.4 MeV 12C ions from the even neodymium isotopes 3). The method is based on the sudden approximation, and employs the following picture 2). An incident partial wave impinges concentrically on the rigid deformed nucleus; viewed in different directions from the centre of the target, it reaches the interaction boundary at different radii. It will be reflected with a phase shift which depends on the angle of incidence 0’ relative to the nuclear symmetry axis. Moreover, the incident angular momentum 1 will be spread over a range of outgoing momenta Z’. It is assumed that the phase shift for the transition I + Z’at angle 8’ can be determined approximately by a spherical optical-model calculation for f = +(1+ P), using a potential whose radial distribution depends parametrically on + On leave at the Department

of Theoretical

Physics, 119

Umversity

of Oxford,

Oxford

OX1 3NP, UK.

R. Lipperherde et al. / Multiple excitation

120

The interaction boundary may be rather diffuse, and, for convenience, we also include the deformed Coulomb potential in this formulation [although in this case the approximation becomes rather less well-founded ‘), especially at more forward scattering angles]. Writing these ideas out in terms of formulas, we have found in ref. 2, the following expression for the differential cross section for a transition from a 0’ ground state to a final state I+ : 8’.

I

I

I

142Nd(12C

1

1

I

I

I

12C')142Nd+

E Beom'70.4MeV

20105-

\ .\

A

\ \\ x,

\

10

20

2 I

E

l-

z

2+ 4

1.::

T

3 P

\ \ \ \

O5 02Ol-

‘\ \\\

005-

002-

30

LO

50 ec m

60

70

60

001'

90

[desl Fg. 1

10

' 20

' 30

I

I

I

I

I

I

LO

50

60

70

60

90

ec m

[de91

R. Llpperheide et al. / Multiple excrtation

121

where the or are Coulomb phase shifts, and

1

9’i,, =

dQ’Y,,(Q’)S7(8’).

(2)

44x s

The quantity Sd0’) is given by S-,(0’)= e21&RJ’), where the angle-dependent phase shift ss(&) is calculated for each 8’ using a spherical potential scattering code for the Coulomb-plus-nuclear potential. As shown in ref. 2), this procedure is closely related to the work of Austern and Blair “). We have applied this method to the description of inelastic scattering of 70.4 MeV r2C ions on the even neodymium isotopes, leading to even-parity rotational states. Accurate experimental data are available for this case; these have moreover been analysed with the help of coupled-channels calculations, so that comparisons

E Beam= 70 LMeV

200 100 : 50

50

20

20

10

10

f

I t

T 2 &El a, _,

52l-

:

g

0502Ol-

0 01

L-

10

I

II

I

II II

I

III1 20

I

30

LO %

I

I

I

I

I

50

60

70

80

90

IIT

3

001’ 10

Cdesl Fig.

1 (continued).

I

20

1

30

I

LO

1

50

I

60

ecm [deal

I

70

I

80

1

90

122

R. Lipperheide

et al. / Multiple

excitation

with the present method can be made 3). The same optical-model parameters as in sect. V of ref. 3, were used for the various isotopes of Nd, with the parametric angular dependence of the half-strength nuclear radius parameter given by

4%u = 4x1+ pyY,,(@, O)+ BYY,,(@, ON,

(4)

and similarly for the charge distribution of the target. The values of the deformation parameters E,” coincide, both for the Coulomb (C) and for the nuclear (N) deformation, with those given in ref. 3). The Coulomb interaction was taken into ac,

I

(

I

1

I

E Beom=70

I

I

I

70

60

90

LMeV

200II1oc 5a I-

20/10I-

5

-5

2

2

iz -_ : a

1 05

02 01 005

002 OO'._ I"

20

30

LO

50 %m

60 [de91

Fig. 1 (continued). Fig. 1. Elastic and inelastic differential cross sections for the scattering of 70.4 MeV “C ions on the even isotopes of Nd. The data points are from ref. ‘). The curves were calculated by the method of anglec - 1.25fm (solid curves) and r: = 1.I fm (broken dependent phase shifts for rotational excitation with rD curves). The deformation parameters are (a) 142Nd: /I; = 0.091,fi = 0.096;(b) 144Nd:/?‘: = E = 0.10, s; = /I?$= 0.05; (c) ‘46Nd: /3’: = 0.12, fi = 0.13, a; = /I$ = 0.05; (d) r4’Nd: /3; = 0.16, E = 0.17, /?: = /?$ = 0.02; (e) IsoNd: fi: = 0.25, /?$ = 0.26, /$ = fi = 0.03.

R. Lipperheide

et al. / Multiple e.ratation

123

count in the form of the electrostatic potential due to a deformed diffuse charge distribution 5). The calculated differential cross sections are compared with the data 3, in fig. 1. Using the same parameters as in the coupled-channels calculations of ref. 3), we obtain qualitative agreement with the data, but generatly, the theory yields too much inelastic excitation (solid curves). We attribute this discrepancy to inadequacies in the treatment of the rather long-range Coulomb inelastic interaction. Indeed, in a test calculation involving only nuclear excitations (pi = 0), we found that the method of angle-de~ndent phase shifts yields cross sections which are in close agreement with the corresponding coupled-channels results, as shown in fig. 2.

b

1 0

50

20 10 52l?

0.5-

2, &

oz-

a;

=

2 6-

Ol-

oosooz001 I

10

20

30

1

LO

t

I

,

,

1

50

60

70

80

90

8~ ITI

[deQ]

Fig. 2. Comparison of the differential cross sections for purely nuclear excitation of f50Nd(~~,4 = 0) calculated by the method of angle-dependent phase shifts (broken curves) and by a coupled-channels procedure (solid curves).

As mentioned in ref. ‘) mentum fin the calculation to an overestimate of the angles. We correct for this

(subsect. 4.3), the use of a singfe, average angular moof the angle-dependent phase shift si(@‘)generally leads Coulomb excitation probabilities, except at backward in a heuristic way by reducing the value of the Coulomb

124

R. Lipperheide et al. / Multiple excitation

radius parameter from the value Y,, ’ - 1.25 fm 3, to a value giving the best agreement with experiment. It was found that using in all cases the value F$ = 1.1 fm leads to a Iit of the data which is nearly of the same overall quality (less so for the lower Nd isotopes) as that attained by the coupled-channels results reported in ref. ‘). The calculation shows that the method of angle-dependent phase shifts may provide a useful tool for the description of multiple excitation of rotational states. We believe that nuclear excitations are adequately accounted for in the present scheme, while a more satisfactory treatment of the Coulomb excitation would surely involve the replacement of si(@) by a “non-diagonal” phase shift S,,(f3’) and perhaps also a correction for the sudden approximation. The authors thank E. E. Gross for kindly providing them with the original data of ref. 3). References 1) J. 0. Rasmussen and K. Sugawara-Tanabe, Nucl. Phys. A171 (1971) 497 2) H. Massmann and R. Lipperhelde, Ann. of Phys. 123 (1979) 120 3) D. L. Hillis, E. E. Gross, D. C. Hensley, C. R. Bingham, F. T. Baker and A. Scott, Phys. Rev. Cl6 (1977) 1467 4) N. Austern and J. C. Blair, Ann. of Phys. 38 (1965) 15 5) H. J. Krappe, Ann. of Phys. 99 (1976) 142