Algebraic treatment of multistep excitation processes in collective nuclei

N&ear Physics A459 (1986) 631-644 North-Holland, Amsterdam

ALGEBRAIC

TREATMENT OF MULTISTEP EXCITATION IN COLLECTIVE NUCLEI

PROCESSES

(III). Medium energy proton scattering G. WENES Los Aiamos

and J.N.

GINOCCHIO

Nafional Laboratory,

Theoretical Division,

A.E.L.

and

Kernjjjsisch

DIEPERINK

Versnelter I~fifuuf,

MS 8283 Los Alamos,

B. VAN

Zernikelaan

DER

25, 9747AA

NM 87545, USA

CAMMEN Groningen,

the ~efher~an~

Received 24 February 1986 (Revised 29 April 1986) Abstract: Channel coupling effects in the excitation of collective nuclear states with medium energy protons are analysed within the framework of the Glauber theory. The collective excitations are treated using the interacting boson model. A comparison is made between experimental and calculated cross sections for excitation of J” = O+, 2+ states in ‘54Sm.

1. Introduction Hadron scattering on nuclei provides an important source of information on nuclear structure such as the matter distribution and the correlations among the target nucleons. In particular for studying dynamical correlations associated with pairing, nuclear deformation, clustering, and so on, medium-energy protons are thought to be superior to low-energy hadronic probes in that the reaction mechanism is believed to be well understood and related in a straightfo~ard way ‘,‘) to the elementary

nucleon-nucleon

scattering

amplitude.

These dynamical correlations manifest themselves in a coupling between elastic and inelastic channels. It has been shown in a number of papers 3, that this coupling crucially affects the elastic and inelastic cross sections both in shape and magnitude. Surprisingly enough, this not only holds for well-deformed nuclei 3,4) - as could have been anticipated - but also to some extent in cases where the inelastic scattering is normally thought of as proceeding by weak coupling ‘) as in “*Pb. Of course in this case the effect occurs at very large angles, much larger than considered in this paper, and additional effects may be important as well “). At these high energies and large momentum transfers, incorporating channel coupling effects by solving the coupled partial wave SchrGdinger equation becomes a complicated task. The conventional partial wave decomposition ‘) of the elastic scattering amplitude, for instance, is usually replaced by an integral over impact parameter b of an impact-parameter S-matrix “) for which there exists an explicit 03759474/86/$03.50 @! Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

632

G. Wenes et al. / Algebraic

treatment

form based on the eikonal approximation ‘). In refs. 9-11), an expression for the T-matrix has been derived in this eikonal approximation appropriate for describing the elastic

and inelastic

states, the latter being

scattering described

of medium

energy

in the framework

protons

to collective

of the interacting

boson

nuclear model

(IBM) 12). This model describes collective nuclear states in medium-heavy nuclei in terms of monopole (s) and quadrupole (d) bosons. Its underlying symmetry group is the unitary group in 6 dimensions, U(6). The U(6) group has three dynamical symmetries, each one corresponding with a chain of subgroups of U(6) and having a geometric analogue. For these three cases of physical interest-the harmonic quadrupole vibrator, the y-soft rotor and the geometric rotor-the authors of refs. 9-11) were able to derive closed analytic expressions for the T-matrix. However, it would clearly be of interest to be able to compute the T-matrix not only for these limiting cases but also for transitional nuclei exhibiting mixed vibrational-rotational phenomena. It is precisely the aim of this paper to present such a computational framework making use of a recently proposed algorithm 13,14)(hereafter referred to as (I) and (II) respectively) for calculating representation matrices for the group U(6). With this formalism established we shall be able to extract the IBM transition operator. The outline of this paper is as follows. Sect. 2 is subdivided into a brief introduction to the Glauber theory and the interacting boson model. We will follow the approach of ref. lo) in showing how these two can be joined together. In sect. 3 we show some first results of applying the formalism to 800 MeV proton scattering to J” = O+, 2+ and give suggestions states in 154Sm. Finally, in sect. 4 we will draw some conclusions for future

work.

2. Medium energy proton scattering and the interacting boson model

2.1. THE GLAUBER

APPROXIMATION

For medium energy hadron scattering the Glauber approximation ‘) gives a good description of the scattering at forward angles. We will briefly review it and reformulate it slightly such as to take into account The interaction of a hadronic projectile hamiltonian H

channel coupling effects. with a nucleus A is described

H = Hp~~j + Hi”, + HA .

by the

(1)

The projectile hamiltonian, Hpro,, contains the kinetic energy of the projectile and an optical-model potential, V(r), where the projectile coordinate r = (r, &C/I) is measured from the centre of mass of the nucleus A. The projectile initial wave vector is k, its final wave vector k’ and the z-axis is parallel with i(k+ k’). The projectile coordinate is decomposed into a vector sum of a two-dimensional impact

G. Wenes et al. [ Algebraic

vector L and a component

treatment

633

along the z-axis, i.e., r=b+zl,.

(2)

The incident projectile energy is much larger than the typical excitation energies of nuclear collective states such that one can neglect the nuclear dynamics described by HA. This corresponds to a sudden approximation in which it is assumed that the nuclear collective degrees of freedom remain fixed (“frozen”) during the collision. Furthermore, the scalar* interaction hamiltonian Hint contains tensor couplings between projectile and nuclear degrees of freedom Hint=C

A

0W9C~AY~,

#J)

* P),

where CF’(@, 4) is the unit spherical tensor of rank h and projection projectile,

cW4 4) =

(5

with Yf’ the spherical harmonic and, 6:’ nucleus,

(3) p for the

t/2

> y3e, 44,

(44

is the moment of rank h of the target

g, WA( I) are the coupling strength times the radial form factors. One is now in the position to write down an impact parameter representation for the scattering amplitude by approximating the Glauber multiple scattering series by an exponentiated first-order optical operator, i.e.

where q = k’- k is the momentum transfer, k is the initial projectile momentum and x(b)=&(l-ip)

+CC dzp(b, z) I --co

(6a)

is the profile function while

are the transition profile functions, u is the total projectile-nucleon cross section, /3 is the ratio of the real and imaginary part of the projectile-nucleon forward * Conventional coupled channel calculations on well deformed nuclei have demonstrated that equivalent predictions of inelastic angular distributions resuft when either spin-orbit terms are fully omitted or fully included ‘*) at least provided equal fits for elastic data were obtained.

634

G. Wenes et al. / Algebraic treatment

scattering

amplitude,

and p(r) is the nuclear

mass A. Furthermore,

matter density normalized

since the major contributions

to the transition

to the atomic

profile functions

(6b) will come from z near 0, one uses the small z/b approximation and evaluthe tensors C’,^’ in eq. (4a) for 19= $r. In this way, eq. (5) has the transition ates 4311S*6) profile functions xi,: with the same angles &= ($T, 4) for each multipole such that it can be rewritten 9P11)as a product of a three-dimensional rotation through Euler A angles b, a simpler transformation in which only the p= 0 term contributes in xi.2 and the inverse three-dimensional rotation, i.e., fi=R(&P’P($),

(7)

where fi = exp

,

-F X:*‘(b) . $*) (

>

fi”‘=exp

.

-a&,‘&’ >

It is straightforward to show that the scattering amplitude fi for the scattering from a target with initial angular momentum Jr = 0: to final angular momentum J and projection M reads (f, Jr= J, Mf= M, k’l&, Ji = 0, Ml = 0, k) OD = ki-M+’ b dbJIM,(qb)[6i,-d~~(3p) I0

e -X(b)(J, M = 01fico’~O+)] ,

where 6,r is the Kronecker delta, JIMi is the Bessel function [MI, q = 2k sin ie and d is the (small) Wigner d-matrix.

of integer

(10)

order

Eq. (10) has now to be extended to include Coulomb scattering following the approach of refs. “-19) in which the nucleus is replaced by a spherically symmetric charge distribution charge distribution.

pch(r) hereby The modified

(f, Jr=J,

neglecting expression

Mf= M, PI&,

Ji=O, Mi=O,

= FR( 8)6i,+ kimM+’ x [aif-

dg&($r)

the effect of the deformation then reads

b db&(qb)

of the

k) e

Ixp,(b)

emXcb)(J, M = 01fico’lO’)] ,

(11)

where 18,19)

(124

xpt(b)= 271ln (kb) , xc(b)=%

~bmd~t’p.n(~){ln1~~~~~b2’f’-~I-b2~~2}.

(12b) (12~)

G. Wenes et al. / Algebraic

Furthermore,

7 is the usual

Sommerfeld 77

v is the velocity

of the proton

635

treatment

parameter,

(124

=Ze'/fiv,

and a,=argr(l+in).

(12e)

Note that pch is normalized to 1. From eqs. (lo), (11) it is clear that the nuclear structure information is largely contained in the matrix element (J, M = 01fi(‘)lO+). In order to calculate this matrix element one has to specify: (i) the nuclear eigenstates IO’), IJ, M = 0) and (ii) the tensors &’ of the target nucleus. To this problem we now turn in the next subsection.

2.2. THE INTERACTING

BOSON

MODEL

We assume that the collective nuclear states in the target nucleus can be described in the framework of the IBM model in terms of monopole (s) and quadrupole (d) bosons which are thought of as correlated valence nucleon pairs. The IBM hamiltonian contains at most two-body terms in the bosons. A particularly simple IBM hamiltonian which has been shown to provide a good description of a large variety of collective phenomena is

where nld = C d:d,

I$ = &J+

K@2)

is the d-boson

number

. Q(2)

+

Krio)

operator,

. i(l)

)

(13)

oC2) is the quadrupole

operator

@=(s+c?+d+s);‘+~R(d+&;‘, with c?,, = (-l)PdP term is diagonal

and R is the quadrupole with eigenvalues

(14)

structure

parameter.

Since the iC1’ . f(l)

L( L+ 1) it does not affect the wave functions

and

therefore it is omitted from now on. Depending upon the value of ad, K, R the hamiltonian (13) can describe various types of collective spectra: in the absence of a quadrupole-quadrupole interaction (K = 0), H describes the harmonic vibrator (U(5) limit) while a strong quadrupole-quadrupole interaction (ideally E,, = 0) results in either the axially-symmetric deformed rotor (R = -$J35, the SU(3) limit) or in the y-soft rotor (R = 0, the O(6) limit). In between these cases, H describes mixed rotational-vibrational phenomena. For these transitional nuclei the eigenstates Ii) and If) are obtained by diagonalizing H in an appropriate basis (usually the U(5) basis)

in which states are labelled

by the quantum

numbers

U(6) = U(5) = O(5) = O(3) 1 O(2) , [Nl

%

d%l)

L

M

of the group

chain (15)

i.e., (16)

636

G. Wenes et al. 1 Algebraie

freafme~t

In eq. (15), N is the total number of bosons (no distinction is made between proton and neutron bosons), nd is the number of d-bosons, z) the O(5) seniority; L, M are the angular momentum quantum numbers, and nA an additional label needed to uniquely determine the state and taken equal to the number of triplets of d-bosons coupled to spin zero. In the IBM, the electromagnetic transition operators are also specified in terms of the generators of the U(6) group. In particular, the electric quadrupole and hexadecupole operators read &‘=

(s+~+d+s)~‘+J~R,(d+d)~’ ,

(174

Ql”‘=

(d+J);‘,

(17b)

where the electric quadrupole operator (17a) need not be the same as the one given in (14), i.e., R # R, in general. We will, however, use the same notation for both. From the phenomenological point of view, the model has been applied quite successfully for calculating spectra and electromagnetic transition rates in a whole range of the nuclear chart. In particular, it has been shown 20-22) that all even Sm isotopes from the vibrational 14%rn up the rotational *56Sm may be described in terms of the hamiltonian (13) where the parameters &d, K, K’ are smoothly varying functions of mass number. Note that in these calculations R was taken equal to -&!35 while R, was allowed to vary, R, - -1.5. The quad~pole electromagnetic transition density operator FE’{ r) is given by 23*24) &F)(Y) = *,(r)(s+d”+dts)~f+P2(r)(d+d”)~),

(lga)

where the radial functions CQ(r), P*(r) can be considered as phenomenological boson densities (which in a microscopic theory eventually could be related to the correlated nucleon pair density matrix elements). It was furthermore assumed that the variation of az( r), &(r) in a certain mass region is negligible. transition operator (17a) is related to the density operator: &‘=

lrn r”drbF’(r). 0

For the Sm region a fit to electron

scattering

The electromagnetic

(18b)

data to J” = 2: state was carried out ‘“)

in order to determine CQ(Z), &(r). In this paper we will use the Fourier-Bessel decomposition of a*(r), j&(r) as given in ref. 23) and apply it for the Sm nuclei although in view of a study of the neighboring nucleus ls4Gd [ref. “)I some refinement may be needed. An analysis of the radial extension of the hexadecupole transition operator (17b) becomes even more complicated because the role and importance of the hexadecupole (g) boson in the IBM is not precisely known [see, e.g. ref. ‘“)I. It may very well be that the effect of the g-boson in calculating B(E4) in nuclei can be incorporated in the operator (17b) through an effective hexadecupole charge e4 but it remains doubtful whether this will be sufficient to describe inelastic scattering (with electrons and protons) to collective J” = 4’ states in nuclei which probes the

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NOILVMIIXOlIddV

WI8flV-I~

BH.L CINV flvI1 3H.L NLEIM.EItI

B!DVIlIlIVE’i

3H.J. ‘E’Z

G. Wenes ef at. / Algebraic freatment

638

in a straightforward

way

/$2(b)=-a(l-i/3) t, +

dzaZ(b, z)(s+&t&)$” dz&(b, z)(d+d)&*’ .

3. Applications to ‘“Sm The 800 MeV (p, p’) reaction on 154Smhas been studied experimentally

29) quite extensively and the data have been analysed in a conventional coupled channels calculation 29) and in the framework of the analytic stationary-phase method ujr2’). Both calculations - as well as others - have clearly demonstrated the importance of multi-step excitation processes. In fig. 1 we show the results of our calculation for the JT = O:, 2: states and compare them with the experimental data 29). The parameters: the matter and charge distribution is a 2 parameter Fermi distribution with radius R, = 1.07A”’ and thickness a, = 0.68 fm; the IBM hamiltonian is the one given in ref. *‘) while the transition densities fxz, p2 are given in ref. 23). Furthermore, we took u = 41.0 mb and p = -0.17 (eq. (6a)). The agreement with experimental data is reasonable although we seem to underestimate the inelastic cross section for the J,” = 2: state by a factor 1.5 to 3 in the region 6” d 0,,, =S7.5”. This feature is quite persistent (that is, is not easily remedied by changing the input parameters) and occurs also in the calculation of ref. 29). Related with this, we also note that for inelastic scattering, we calculate minima which are too deep compared with experiment. This discrepancy has been discussed in several papers [see, e.g. refs. 3”,3’)]. It was found 30) that these diffractive minima in the spin-independent cross sections are not significantly affected by the elementary spin-orbit amplitudes although it was also pointed out that at some level one has to take into account effects of nuclear ground state deformation, transitions to magnetic substates that are not easily reached through the collective mechanism, contributions from other intermediate nuclear states and so on. Our calculation only partially addresses these suggestions: Although we take into account the nuclear ground states defo~ation and coupling to i~te~ediate nuclear states we are not yet sure that (due to the small z/b approximation we have made in deriving the T matrix) we have not been neglecting some effects which may influence the population of the different magnetic substates. In order to demonstrate the effect of including the coupling to excited states we also show in fig. 1 the results of calculations without them (g2= 0) for the elastic scattering (optical model) and of including them only to first order (Born term) for the scattering to the JT =2: state. Coupled channel effects are clearly important at large 9 especially in the elastic channel.

G. Wenes et al. / Algebraic f~eafmenf

5

IO

I5

639

20

-@,,(degrees)~ Fig. 1. Calculation Full curve includes

of cross sections for the J,” = O:, 2: state for 800 MeV (p, p’) scattering on “*Sm. channel coupling effects; dashed cxrve is either the optical model (in the case of J,” = 0:) or the Born ‘limit (J,” = 2:).

In fig. 2 we show the predicted cross sections for excitation of the JT = 2& states (the “p vibrational” 2+ state 2& at 1.178 MeV and the “-y-vibrational” 2’ state, 2;, at 1.440 MeV). Also shown is the contribution to da/d0 from the Born term. However, at this point, we would like to warn against taking the predicted cross sections for scattering from the 1” =2;, 2: states at face value. Indeed, IBM-l calculations 20) do not correctly reproduce the LJ+ 0 limit or B(E2,0:+ f” = 2&) - which are weak in any case - such that it may be anticipated that the curves shown in fig. 4 must be, at best, multiplied with the same factor by which the calculated B(E2)‘s are off from the experimental values. The discrepancies for the B(E2,0+ + 2,, 2y) may be due to deficiencies in the IBA-1 wave functions or in the quadrupole operator (17a). Indeed, since the Ot+ 2$,, strengths originate from an appreciable cancellation between the dts and d’d terms in (17a), it is conceivable

G. Wenes et al. / Algebraictreatment

I

1

I

I

I

I

I

I

I

I

I

I

I

10-s. 5

IO

15 -

&,,

20

(degrees) -

Fig. 2. Calculated cross sections for the I: = 2& states in ‘s“Sm for 800 MeV (p, p’) scattering. Dashed cmve is the Born limit, full curve includes channel coupling effects.

that a small change in either the IBM hamiltonian or in the quadrupole operator could greatly improve transition strengths. The main motivation for iiberhaupt showing these curves is to illustrate that nuclear structure effects may result in appreciable qualitative and quantitative differences in cross sections for inelastic scattering from states belonging to different bands and that consequently it is an impo~ant tool for studying nuclear collective motion. In any case, for *54Gd for which an experimental study is now under way (including an investigation of scattering from side bands) [ref. “)I we intend to perform a more careful analysis. Also we investigate the sensitivity of the excitation of the beta-vibrational O+state to a monopole term (A = 0) in the transition profile function given in (6b). The simplest way to do this is to replace p(r) by a density operator p^(r), P^(r)=p,,fr)+NAp(r)+cu,(r)n^,,

(21)

where N is the total number of bosons. Here, we follow a phenomenological approach in determining pCore, Ap, and oO: (i) in a first step, one relates the matrix element of p^(r) for elastic scattering to the matter density for a nucleus with mass number A, i.e.

where

G. Wenes et al. / Algebraic

treatment

which should hold for all Sm isotopes. For the vibrational such that

641

14%m, one has n’46= 0

~l~(r)=~or,(r)+7Ap(r),

which leads to an extraction of Ap(r),

(23)

i.e.,

In a second step one applies eq. (22a) to ‘54Sm resulting in (25) which leads to the final expression for p^(r) used in our calculations p^(r) = i%(r) + %(r)&

(26)

where &f(r) = -4Po&)+~P ” 146(r) .

(27)

For p,,,(r) we take the matter distribution of ‘32Sn which is parameterized - as well as I’ve and pls4(r) - in terms of a two-parameter Fermi distribution. In fig. 3 we show the results of our calculations of the cross sections for scattering to the 17 = 0: and 0: state in ‘54Sm with a diffusity a (core) = 0.58 fm, a (14%rn) = 0.62 fm, a ( ‘54Sm) = 0.68 fm and R = r,,A1’3where r. = 1.07 fm. The full curve includes monopole and quadrupole excitations while the dashed one includes only monopole excitations. The dotted-dashed curve is the one that is obtained using eq. (11) including a quadrupole excitation term only. For the elastic scattering there is no perceptible difference between full and dotted-dashed curve which is why the latter is not shown in fig. 3. This is of course directly related with our constru~ion of g(r) where we have required that in first order we obtain identical results [eq. (23)f. However, as could have been expected, the effect of including monopole excitations for the J” = 0; state is more dramatic. In particular, we would like to draw attention to the fact that effects which show up only modestly in the elastic channel are more pronounced and often in a reversed way for the J” = 0; state. For instance, switching the quadrupole coupling on results in shifting the maxima to smaller angles (by approximately 0.7”) in contrast with the elastic scattering where cross sections are shifted to slightly (by 0.3”) larger angles when quad~pole excitations are included. This general feature of having more pronounced and often opposite effects in the 0; channel as compared with the elastic channel is also demonstrated in fig. 4 where we have investigated the relative importance of the matter distributions. Shown in fig. 4 are the results for a ( ls4Sm) = 0.68 fm and r. = 1.07 fm (full curve), a ( ls4Sm) = 0.66 fm and r. = 1.07 fm (dashed curve) and for a ( ‘54Sm) = 0.66 fm and r, = 1.11 fm (dotted-dashed curve). We have not inciuded in these calculations a possible quadrupole term.

642

G. Wenes et al. / Algebraic treatment

800

-

MeV

( p,d)

Be.,, (degrees)-

Fig. 3. Calculated cross sections for the .J: = OIZ states in ‘%3m for 800 MeV (p, p’) scattering including monopole (dashed curve), quadnrpole (dashed-dotted curve) only and both monopole and quadrupole (full curve) type of excitations. See text for further discussion.

4. Conclusions First

proton

quantitative scattering

results

were obtained

to J” = Of, 2+ states

for the description

of medium

energy

in ls4Sm in the framework of the IBM. Agreement with experiment was reasonable although some discrepancies remained. We made predictions for the inelastic cross sections for scattering to the p- and y-vibrational 2+ states and found that nuclear structure effects are quite important. The effect of multi-step excitations which can be included in an exact way in our formalism was investigated and found to be important for momentum transfers may be q B 1.5 fm-l. It was pointed out that valuable nuclear structure information extracted from scattering to the 0; state. Furthermore, the techniques discussed in this paper are applicable to a whole class of transitional nuclei. Finally, some suggestions and improvements for future work could be made: A study of the hexadecupole degree of freedom, extension to IBM-2 and a precise determination of the relation between the transition densities deduced from electron scattering and the hadronic ones.

G. Wenes et al. f Algebraic

-Bcm

treatment

643

(degrees)-

Fig. 4. Calculated cross sections for the J: = O;, states in r5%rn for 800 MeV (p, p’) scattering for different parametrizations of the Fermi distributions: a = 0.68 fm, r, = 1.07 fm (full curve), a = 0.66 fm, r, = 1.07 fm (dashed curve), a = 0.66 fm, r, = 1.ll fm (dotted dashed curve). See text for further discussion.

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