Microscopic form factors for inelastic excitation of isovector modes in heavy-ion reactions

Nuclear Physics A378 (1982) 10(1`1 l0 © North-Holland Publishing Company

MICROSCOPIC FORM FACTORS FOR INELASTIC EXCITATION OF ISOVECTOR MODES IN HEAVY-ION REACTIONS C . H . DASSO, T . S . DUMITRESCU' and A . VITTURI r' The" Niels Bohr Institute" and NORDlTA, Blegdam.seej 17, DK-21(X) Capenhagen, Denmark acrd

Oak Ridye National Lahnratort ", Oak Ridye, Tennessee 3783() 3

Received 30 July 1981 Abstract : Microscopic form factors are calculated for the excitation of isovector modes in heavy-ion reactions . They are constructed from RPA model wave functions . For the case of giant dipole resonances, the results are compared with the Goldhaber-Teller and Jensen-Steinwedel macroscopic expressions .

1. Introduction Much effort has been devoted in the last few years to the experimental study of the nuclear response function at excitation energies in the continuum region with both light and heavy ions. These experiments have systematically confirmed the existence of other strongly collective high-lying modes besides the known giant dipole resonance. Clear and systematic evidence has, however, been obtained so far mostly for isoscalar modes, in particular for the quadrupole isoscalar and the breathing mode . Since all giant modes are expected to exhibit large widths and often lie in the same range of excitation energy, it is helpful to have theoretical predictions of the cross section and energy distribution associated with the excitation of the different modes. In the case of inelastic scattering induced by a projectile with N ~ Z, the isovector modes are also excited by the nuclear field. So far, these modes have been excluded from the analysis of the data because the associated cross sections are expected to be small. However, the possibility of exploiting the isovector field has greatly increased over the last few years due both to the number of projectiles and the range of bombarding energies available. In this paper we use the formalism for the microscopic calculation of nuclear inelastic form factors to include the excitation of isovector f Permanent address : Dept . of Fundamental Physics, Institute of Physics and Nuclear Engineering, PO Box MG6, Bucharest, Romania. r' Permanent address : Istituto di Fisica dell'Universita and INFN, Padova, Italy . x Research sponsored in part by the Division of Basic Sciences, US Department of Energy, under Contract W-7405-eng-26 with the Union Carbide Corporation .

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C. H. Dasso et al. / Microscopic form factors

10 1

modes. This may be useful in order to study to what extent they can affect the analysis of the isoscalar modes. Furthermore, in medium and heavy nuclei (N > ~ the nuclear modes can only approximately be associated with an isospin quantum number ' " 2 ), and we can only talk of `mainly isoscalar' and `mainly isovector' modes. As a consequence, all the modes are excited through both the isoscalar and isovector field, and it is interesting to investigate the relative importance of the two mechanisms for the different states . The mixed character ofthe modes makes it convenient to be able to evaluate the form factors without resorting to the introduction of macroscopic models . In sect. 2 we present a microscopic description of the isovector modes based on the random phase approximation (RPA). The formalism which results in the microscopic expression for the inelastic form factor is developed in sect . 3. Examples of calculations for different multipolarities and comparisons with macroscopic models for the case of the dipole mode are shown in sect . 4. 2. Microscopic description of collective isovector modes We construct the form factors for inelastic excitation from a model wave function of the collective isovector modes. The microscopic description of these states results from a RPA calculation. In this section we briefly review the basic ingredients of the formalism and illustrate some features of the results. Details have been given for the quadrupole modes in ref. ' ). The RPA states are generated by simultaneously correlating particle-hole excitations through an isoscalar and an isovector residual interaction of the form hi(.1) _ - zKZ(T

=

0) ~ Qzu(T

= 0)Qxu(T =

- iKZ(T

=

1) ~ Qz,~(T N

=

t<

0)

1)QzN (T = 1) - ~z ~ Qz,~(T = 0)Qz,~(T = w

1)~

(1)

where Qz w(T = 0) and Qz,~(T = 1) are the particle-hole parts of the isoscalar and isovector multipole operators, respectively . They are given by the sum and the difference of the neutron and proton multipole operators rz Yz~,. The coupling term between the isoscalar and the isovector modes in (1) vanishes for nuclei with TZ = 0. In order to take into account the residual interaction (1), one defines the creation operators for the correlated states n as a linear combination of the form l'z,.(n) _ ~ {X~(P, h ; ~)l',ia (P, h)+( - Y`Y~(P, h ; ~)rz_N(P, h)} . p" h

where r.iplY+h) _ ~, ~.% pmp.%hmhl~~~CP r lrlp .llló

102

C. H. Dasso et al. / Microscopic form factors

represents the operator associated with p-h excitations coupled to angularmomentum ~l and projection ~ . The RPA equations result from approximate boson commutation relations between the operators Tx~(n), together with the equation of motion

characteristic of harmonic modes. Explicit expressions are then obtained for the forward and backward amplitudes X(p, h ; ~,) and Y(p, h ; ~,) [cf., e.g., ref. 1 )] . In the present calculations the isoscalar coupling constant tcx(T = 0) was chosen equal to the self-consistent value for shape oscillations of a nucleus described by a harmonic oscillator mean field z) : xx(t

_

0)

_ 4~c Mwó 2,1+1 A(rzz-z )

The strength of the isovector coupling is estimated from the isovector component in the static nuclear potential. Following ref. 1) the value xx(i = 1 )/xz(i = 0) _ -0.5 (2~,+3) was used for the ratio of the two coupling constants. For the isoscalarisovector coupling constant, we used the value xx = - xa(i = 1)C(2TZ/A) with C = 0.5. A modified harmonic oscillator potential a) was utilized to construct the singleparticle states . The energies of the levels are given by eN,1 = hcuo{(N+Z)-x[j(j+1)-I(l+1)-á] -N~[I(1+1)-~}N(N+3)]} ficoo = 41A -} C1 ±3

N-Z 1 A)

~+

neutron proton .

The following values of x and ~ [ref. 3)] were used : x = 0 .077, ~ = 0.011 for 4°Ca, and rc = 0 .0604, ~ = 0.0379 (protons), x = 0.0636, ~' = 0.0233 (neutrons) for zoaPb. The RPA formalism with velocity-independent interactions preserves the multipolarity strength as measured by the energy-weighted sum rule (EWSR). This holds separately for both the isoscalar and isovector strength accumulated by the RPA roots. In fig. 1 we display the distribution of isovector strength for dipole and quadrupole states in 4°Ca and z°BPb. The strength distribution is pushed toward higher energies compared with the unperturbed response which is also shown in the figure . The identification of the giant isovector modes is straightforward in all cases, although a small fractionation of the i = 1 strength occurs already at the RPA level. Selfconsistent calculations using "realistic" interactions and taking into account the particle continuum °) found a larger fragmentation of strength in the resonance region . The overall picture, however, is consistent with the present calculations .

103

C . H. Dasso et al . / Microscopic form factors

vC b

v U L

U á

0

û0 cm > :° co

' o rv

O

V CD ~ C N '~.

0 "+ N U 4. O

W

4~

L~ ~.. O .

o

0

K

:+ m ~ d

O'

O

O

ó

O

N L ~ L ~ ~ (dó

O

~ ~

La

Y Vi

O

~ aL+

> y L ~ s ... c ócv

W

W

a .~ . o_ .. ~

°~

a

d

.y ~ b T

0

o

Ó

O

O

A

aN K,i

Ó

_ Op m w~

Vl

104

C. H. Dasso et al. / Microscopic form factors TABLL I

Energy and fraction of isovector EWSR for giant dipole and quadrupole isovcctor modes in zoePb i.* aoCa zoePb

Energy (MeV)

°°Ca

and

Fraction of EWSR t = l

I2+

22 .7 40.1

98 '% 98 °,

12'

13 .6 23 .2

83 ',%~ 64 ',

In table 1 we give the relevant parameters characterizing the isovector states that were further used in the form-factor calculations. For a°Ca the resulting wave functions are consistent with the fact that in light nuclei the isoscalar/isovector character ofthe giant resonance is quite pure. It follows from the coupling defining the isospin nature of the residual interaction (I) that nuclei with neutron excess will have modes with a mixed character . A method of displaying the isospin impurity of the modes is by means of the transition density for protons and neutrons aPx" ) ( r)

_

SP~")(r) =

~

~Jp~~ YzIL1niLX"(P, h ; ~)- Y"(p, h ; ti)~RP(r)Rn(r),

~

~.1p~~ YxIIJniLX"(p, h ; ~)- Y"(p, h ; ti)~RP(r)Rt,(r~

P.btP~otons)

p,h (rcuuone)

from which the isoscalar and isovector transition densities are constructed by

The functions R(r) are the radial wave functions for particle and hole states . In figs. 2a, b we show the transition densities for the giant dipole states in a°Ca and zoapb . The choice of examples has been made to illustrate the difference between a light nucleus with N = Z, and a heavy one with large neutron excess . Also shown in figs. 2c, d are the corresponding macroscopic transition densities in the GoldhaberTeller s) (GT) and Jensen-Steinwedel e) (JS) models. A discussion on the dipole transition densities in these approximations has been given by Satchler in ref. '). The curves in fig . 2 were calculated using ,. 8,d~ (r)

_

~

9

>

-4V '° (2Mco 1

f

8p~(r) - 2

A

} ZN

ZN)

Az

d

dr P°(r),

~z 1 127c ZN ~r (r), (2M fiu~, (rz~z ,q s ) Po

C. H. Dasso et al. / Microscopicform factors

105

106

C . H. basso et al. / Microscopic form factors

where po(r) is the density calculated from the harmonic oscillator wave functions defining the ground state and Pico t is the energy of the resonance. One can see that the microscopic transition densities resulting from the RPA calculations are closer to the GT than to the JS model prediction. This could be expected from the fact that we have used a coupling rxYat, as residual interaction. In the self-consistent calculations 4), where the strength of the giant dipole mode is split into several peaks, the transition density appears to be closer to the JS prediction in the lower part of the energy spectrtun, whereas it looks more like a GT transition density for the higher states . 3. Formalism for the inelastic form factor During a heavy-ion reaction, a target nucleon in an initial state ß may undergo a transition to a final state a, exploiting the matrix elements, Faß(r) _ ~ ~ ~drt~a(rt,Qt)V(~r - riI)~b(rt,Qi),

(10)

of the projectile isoscalar and isovector fields V(rt°) = Yo(rt°)±

N° °Z° Vi(rt°)~ A

(ll)

where the upper and lower sign applies for neutron and proton states, respectively . The notation and definition of the variables are illustrated in fig. 3.

Fig. 3 . Coordinates used in the evaluation of the form factor .

In the expression (10) the states of a nucleon bound to the target A are defined as ~a(ri, Qt) = R°at~1e(r) ~ <1~sivLi°m°i Ii .r~(~1)X~.(Qi ), NsY

(12)

where we use the notation aQ, lQ, ja, ma, to to specify the state a. Besides the angular momentum quantum numbers la, ja, ma, we set explicitly the projection of isospin t° to denote proton orneutron states . The label a° represents all other quantum numbers which are necessary to define the state.

C . H. Dasso et al. / Microscopic form factors

10 7

Making use of the symmetries of the problem, the sum over the spin variable and one of the angular integrations in (10) can be carried out analytically . This leads to the explicit multipole expansion of the function Faß(r), J(ZJa+ Fae(r) ° ~n ~ ( -)m°+~ 1)~(2j~±l~ ~jßzja - il~oXlßmplamal~h> .~/(2~. + 1) zk x'S(Ia + lß + .i., even) x

i ~ridr, duRa*°i~°(rl)V( rl+rZ-2rrlu)Raoin~e(r l)P x(u Yxk(~). CJ o J i )J

(13)

In general, we are not interested in single-particle transitions like the one induced by the matrix element (13), but rather in the coherent superpositions of p-h excitations that define the states of the nucleus once the residual interactions have been taken into account . Using a complete set of single-particle states, we define the onebody operator associated with inelastic excitations as

k(r) _ ~ Fae(rká c a . aß

(14)

As the matrix elements in (14) are zero unless to = t~, the operator (14) separates in a proton and a neutron component . Each of these terms in turn contains the isoscalar and isovector parts which result from the substitution of (11) into (13) . The form . factor for inelastic excitation from the ground state Ig.s.) of the target to an excited state IL11~ of angular momentum L and projection M is given by F(r) - ~LMI~(r)Ig.s.~.

(15)

The microscopicexpression for the form factor results from the structure information contained in the wave functions describing the ground state and the excited state. For example, using the expression (2) for the creation operator of a vibrational state in a closed-shell nucleus, one obtains s' 9) F(r) _ ~ ~ ~( - ~° +~° +~ J(21a+ 1~(21Q + 1) J(2L+ 1) a°~°J° ~° Qaiaia

x S(L + la + Iß, even)[X(ßß1a jota, ßß1 a jata ; L) - Y(aala jat°, aß1ß j ßta ; L)]

i x { ~r;drl duR,*°,~°(rl)Rap,p~e(rl) [Vo( ri +rz -2rrlu) (o J J i +2ta

N'A

Z" Vi( ri+r 2 -2rr lu)

J

PL(u)~ Yiar(~)~

(16)

l 08

C. H. Dasso et al. / Microscopic form factors

Similarly, one can derive the expressions for a nucleus with two particles outside the closed shell or extend the formalism to include superfluid nuclei . Microscopic calculations of form factors for single-particle transitions have been also reported in ref. 1 °) . The isoscalar modes are characterized by the fact that protons and neutrons move in phase with each other. Therefore, the form factor for inelastic excitation will mostly result from the V° term in (16), as the contributions from the isovector term V1 will tend to cancel . Conversely, the form factor for an isovector mode will be basically built from the isovector part of the interaction. In what follows, we will use the terms isoscalar and isovector form factor to refer to the result of expression (16) calculated setting V1 or V° equal to zero, respectively . 4. Applications We have calculated the isovector form factors for the states listed in table 1, in the case ofreactions induced by 18 0 as a projectile . In the calculations we have used for the nucleon-projectile field (11) the value V1 = -0.5 V° with a standard geometry (see caption to fig. 4). The form factors for the giant dipole states of a °Ca and Z°8 Pb are shown in fig. 4. They are compared with the macroscopic form factors that follow from the GT and JS models . These were not obtained by folding the macroscopic transition densities

Fig . 4 . Comparison between microscopic and macroscopic form factors for the excitation of the giant dipole resonance in 4°Ca and ~ °8 Pb by an `s0 projectile . In the construction of the microscopic form factor a Woods Saxon potential was used for the nucleon-projectile interaction (11) with V° _ -50 MeV, V, _ -25 MeV, a = 0.63 fm, r° = 1 .25 fm . For details on the macroscopic form factors see text .

C. H. Dasso et al. / Microscopic form factors

10 9

shown in fig. 2, but rather by expressing the results in terms of the ion-ion potential [cf. ref. 7)] . In this step, one assumes that the changes of the density distribution follow the variation of the associated optical potential. We have usedthe parametrization given in ref. 9) for the ion-ion potential. We have, however, used an isovector component which is only a factor of two larger than the one consistent with (11) to acxount for the neutron excess in the surface. In the case of a °Ca, the RPA calculation concentrated the dipole isovector strength in essentially one state. The dipole strength for z°BPb was, instead, split into two roots within an interval of 0.7 MeV. The second root accumulates, however, only 5 of the isovector energy-weighted sum rule . We have calculated the microscopic transition density associated with this state and found it to be essentially equal to the one displayed in fig. 2d, except for the scaling factor which follows naturally from the normalization of Sp~ = t , namely ß/e3. In a situation like this, the fractionation of strength at the RPA level could be taken into account by renormalizing the form factor calculated with the state in table 1 by the factor s + s . To use this prescription, the energy interval between the roots should be small compared with the width of the states . A decrease of the residual interaction by 6 ~ (which does not modify the energy of the states by more than 2 ~) would change the fraction of the EWSR exhausted by the two roots to 75 ~ and 11 ~, respectively . However, due to the previous argument, the form factor would be practically unchanged . The agreement between the macroscopic and the microscopic approaches is quite good in the tail, which is the region of the form factor which is probed in heavy-ion reactions. zoaPb ( le~~ le~ ) zoePb.

{

2 ( 2a2 MeV1

10~

,___

-Z= 1 -- -Z= 0

10~

_1

lo

10

~

b)

3

lo` I 0

,

,~, 10

`c,I 20

Fig. 5 . Form factors for the excitation of the isovector giant quadrupole resonance in ° °Ca and Z°sPb by's0 . For the Z°sPb case the isoscalar and isovector contribution to the form factor are explicitly shown .

l 10

C. H. Dasso et al. / Microscopicform factors

We show in fig. 5 the isovector form factors for the isovector giant quadrupole modes of the same nuclei . For the case of a° Ca, it follows from the pure isovector character ofthe state that the corresponding isoscalar form factor is zero . This is not the case for a°BPb, and we compare in fig. Sb the magnitude of the isoscalar and isovector components of the form factor. Even if the isoscalar impurity of the state is only of the order of a few percent, the two form factors turn out to be of the same order ofmagnitude. These examples have been calculated utilizing t 80 as a projectile, which has a relative neutron excess (Na -Z,)lA Q of about 0.1 . Although there are projectiles which could better exploit their isovector field, none is expected to change the qualitative nature of these results. One of us (T.S.D .) acknowledges the kind hospitality of the Niels Bohr Institute and financial support from the Danish Research Council. References 1) 2) 3) 4)

5) 6) 7) 8) 9) 10)

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