Excitation of giant resonances in intermediate energy heavy-ion reactions

Nuclear

Physics

A345 (1980) 263 - 277; @

North-Holland

Publishing

Co., Arnsterabm

Not to be reproduced by photoprint or microfilm without written permission from the publisher

EXCITATION OF GIANT RESONANCES IN INTERMEDIATE ENERGY HEAVY-ION REACTIONS R. A. BROGLIA, C. H. DASSO *, H. ESBENSEN *, A. VITTURI The Niels Bohr Institute,

University

of Copenhagen,

DK-2100

and A. WINTHER

Copenhagen tJI, Denmark

and * NORDITA,

Blegdamsvej

17, DK-2100

Copenhagen Q, Denmark

Received 6 February 1980 Abstract:

Quasielastic and deep inelastic processes in the reaction L60+208Pb have been studied at bombarding energies of lo-80 MeV per nucleon. The excitation probability of the giant modes becomes larger with increasing energies. At a given bombarding energy the multiple excitation of low-lying modes becomes predominant as the angle of observation moves away from the grazing angle. Near grazing, no appreciable background of this character is found for bombarding energies above 30 MeV per nucleon. At these energies the electromagnetic interaction becomes the most effective way to excite the giant modes in the excitation energy interval of lo-20 MeV. The importance of Coulomb excitation for the highest bombarding energies makes the excitation of isovector modes no longer negligible.

1. Introduction

In the study of the nuclear response function, heavy-ion reactions seem to be a promising probe lm4). Both the strength of the Coulomb field and, to some extent, of the nuclear field can be varied over a wide range. Different effective interaction times can be obtained as a function ofthe projectile velocity. These features, combined with the large angular momentum carried by the relative motion of the two ions, appear to open new possibilities. At this time very little is known about the subject and some basic questions have yet to be answered +. A problem of particular interest concerns the energy dependence of both single and multiple (background) excitation of the different collective modes of the reacting nuclei. In the present paper we study the reaction I60 + *“Pb at bombarding energies of 10,20,40 and 80 MeV per nucleon utilizing the coherent surface excitation model of ref. 6). De main steps of the calculation are described in sect. 2. The predictions of the model are presented in sect. 3. An appendix with formulae to calculate Coulomb excitation at high but non-relativistic energies is provided. ’ A study of some of these questions has been carried out in ref. ‘). 263

264

R. A. Broglia et al. / Giant resonances

2. Description of the calculations

The model of heavy-ion reactions used in this paper has been introduced and further ellaborated in a series of short letters. A detailed presentation has also been given 7). In what follows we thus only summarize the main points relevant to the present investigation. The intrinsic degrees of freedom of the nuclei are, in the model, the low-lying collective surface vibrations and the high-lying damped giant resonances. The effective sharp radius ‘) of the nuclei is parametrized as

in terms of the amplitudes Q, of the collective modes. The trajectories of relative motion are constrained to remain in the initial scattering plane. Choosing the z-axis perpendicular to this plane only p-components for which A+ ,Uis even are included in (1). The spectrum of the surface modes utilized in the calculations is displayed in

TABLE1 Parameters defining the modes of I60 and “*Pb included in the calculations zosPb

I60

1

ho, (MeV)

F (MeV)

% EWSR

0+

13.6 4.1 10.8 2.6 17.0 4.3 11.0 24.0 3.3 20.0

2.0 (0.5) 2.0 (0.5) 4.0 (0.5) 2.0 6.0 (0.5) 8.0

100.0

2+ 2+ 334+ 4+ 4+ 5s-

16.0 82.0 15.0 80.0 4.6 23.0 72.0 2.0 39.0

I hoi (MeV) _____._~___~--_-------

F (MeV)

% EWSR

2+ 2+ 3-

6.9 23.0 6.1

(0.5) 10.0 (0.5)

15.0 85.0 8.6

4+

23.0

10.0

25.0

(t = I)

13.16 23.0

2 4

86.0 55.0

For the different multipolarities 1, the energy hw,, the full width at half maximum F and the percentage of the isoscalar energy-weighted sum rule are given. For the low-lying modes one should use an energy dependent damping since they acquire a width only when multiple excited. In the calculations we have treated these modes as undamped, spreading the resulting cross sections over the energy region indicated in parenthesis. For “‘Pb we have included two isovector modes which are only excited through Coulomb excitation. They were considered only in the calculations at 1280 MeV (cf. fig. 6).

R. A. Broglia et al. / Giant resonances

265

table 1 and is essentially the same as used in previous cases ‘). The surface-surface interaction between the ions was described in the proximity approximation of ref. ‘). The uncertainty in the nuclear radius due to the quanta1 fluctuations in the ground state of the collective surface modes is measured by the spread

(2) where D,, is the mass parameter and oni the frequency of the mode (n, 2). It is an open question which modes should be included in the summation (2). We have only considered the contributions of the low-lying modes, which lead to a fluctuation in a surface-surface distance of 0.42 fm. The effect of these fluctuations was taken into account in the calculations by a set of random initial conditions as discussed in ref. lo). The differential inelastic cross sections were constructed in the following way. For each interval [P-+-dp] of impact parameters a finite set of classical trajectories were calculated. The initial conditions for the amplitudes and momenta of the surface vibrational modes were chosen as random numbers compatible with the gaussian semi-classical phase-space distributions. For a given interval [0, 0 + de] of center of mass scattering angles the average number of phonons Ni(p, 0) excited in the various collective modes i = (n, ,I) were calculated. The occupation number probability Plni associated with the mode i was determined using the Poisson distribution

P,i(i, P, 0) = The probability

@Yi

(Ni(P,

e-

Nitp,

0)

mi !

of exciting an energy E for given p and 8 is then given by

where the sum extends over all the occupation numbers of the various surface modes. In the calculations, the distribution (4) was, for each set {mi} of occupation numbers, folded with a gaussian distribution of width r = (& mJf)f, where Ti is the damping width associated with the mode i. For the low-lying modes which were considered undamped, a fictitious width of Ti = 0.5 MeV was ascribed in calculating the sum over i (cf. table 1). The double d~erential inelastic cross section is now obtained by adding the contributions from all impact parameters

d2d40) = dBdE

pdp

sin

@,

fww P, 0).

(5)

266

R. A. Broglia et al. / Giant resonances

Here, t(p, 0) represents the fraction of trajectories at an impact parameter p that emerges with a scattering angle within the interval [0,8 + do]. In the factor t we also include the effect of the depopulation of the I60 channel due to particle transfer. Most of the cross section is associated with deep inelastic events with a broad energy distribution which is determined by the energy loss partly due to the excitation of surface modes and partly due to the transfer of particles. The energy spectrum of the I60 ions will include contributions from events in which particle transfer has taken place. In this investigation we assume that such events will lead to a structureless background spectrum. For the rate of transfer reactions we use the expression 11)

where f(r) is the form factor for single-particle transfer and r N Ju/r, is the collision time. The quantity a is the diffusivity of the ion-ion potential and Y,,is the acceleration of the relative motion at the distance of closest approach. For the form factor we use

“f(r)=

I+

,:r-R, ’

where R = 1.25(@ + Ai) fm and K = 1 fm- r.The coefficient F depends on the singleparticle configurations in the two nuclei. An average over typical form factors 12) gives F of the order of 5 MeV. The factor N indicates the effective number of transfer channels with favourable Q-value which contribute. For each interval of impact parameters and scattering angles we have calculated the average integrated rate of transfer reactions (n,(p, 0)) using eq. (6). The corresponding probability exp (- (n,(p, 0))) to remain in the 160 channel was included in the factor t(p, 0) in eq. (5).

3. Main features of the results In fig. 1 we show the standard deviation for the deflection angle and for the final energies of relative motion as a function of the impact parameter. The full drawn curves represent the results of the calculations utilizing the average trajectories. For these trajectories the initial conditions for all the modes are a+, = nnAp= 0. It is seen that the range of impact parameters between the two rainbow angles is much smaller at the lower energies than at the higher. This is because with increasing bombarding energies we move further away from the orbiting condition and thus increase the cross section for quasielastic processes. The figure also shows that the fluctuations in scattering angle are relatively larger at smaller energies. These two results are to some extent related. The zero-point amplitudes lead to significant fluctuations in the surface-surface distance at the turning point. This can be viewed

R. A. Broglia et al. 1 Giant resonances

_- \“‘\

-.-

4‘

.-

.-

f

-.--

i,

J

I



54

2

(Aawl 3

-

‘.

.\

‘\\

0

268

R. A. Brvglia et al. / Giant resonances

as sampling the deflection function with a range of impact parameter Ap. The steeper the deflection function the broader the range of scattering angles. Furthermore, from the relation between impact parameter p and distance of closest approach b valid for Coulomb orbits

where Z,Z,e’ a’ = YE,,,,



(9

it is seen that this effect is amplified at the lower energies. Also in fig. 1 is displayed the probability P,, = e-Q of not being removed from the 160 channel by mass transfer [cf. eq. (6)]. We have extended the calculations only to the range of impact parameters where P,, # 0.The contribution to the inelastic cross section from impact parameters which essentially lead to Coulomb trajectories was evaluated utilizing the analytic expressions in the appendix. In what follows we shall refer to this component of the inelastic cross section as Coulomb excitation. In fig. 2 is shown the double differential cross section d20/dEd0 for the cases of 10 and 40 MeV per nucleon. While there is a rather clear separation between quasielastic and deep inelastic events in the 10 MeV per particle reaction an almost continuous transition is observed at 40 MeV per nucleon. This difference reflects the steeper energy loss function associated with the lower energy (cf. fig. 1). In this, as well as in the following figures, the events in which l 6O was excited above 10 MeV were removed since these events will lead to particle evaporation from 160. One of the most conspicuous features of the present calculations is the change in the role of Coulomb excitation and multi-phonon excitation as a function of the bombarding energy. This is exemplified in fig. 3. For all three energies an observation angle near the grazing angle was chosen. While at 10 MeV per nucleon the contribution of Coulomb excitation in the energy domain of the giant resonances (E > 10 MeV) is negligible it becomes the dominant mechanism at 40 MeV per nucleon. In the lower part of the figure we observe an appreciable reduction of the multiple excitation cross section as the energy increases. At 640 MeV the energy spectrum is formed essentially by single-phonon excitations. The growing importance of the electromagnetic interaction for the higher bombarding energies is indeed expected from simple considerations. As the velocity of the projectile increases, the effective interaction time resulting from the Coulomb coupling is reduced. It thus becomes possible to move the adiabatic cut-off up in energy and make the giant resonance region at about 12 MeV accessible to Coulomb excitation. Quantitative results for Coulomb excitation supporting this argument are shown in fig. 4. The probability of one phonon excitation for dipole, quadrupole and octupole

R. A. Broglia et al. / Giantresonances

l

f

THETA 175

188

I

r

269

160 -

I

+

208PB

C.M

I160 I

I

MEVI ’

I

I



I

+ +I+

+ 1

*:

1 *** l

l

+ f*

+ +

l

*

THETA

C.M

Fig. 2. Final kinetic energy of relative motion versus scattering angle for an ensemble of trajectories at 10 and 40 MeV/nucleon. Each cross is associated with a trajectory with impact parameter in the range 7.5 fm < p < 10.5 fm for E = 160 MeV and in the range 9 fm d p Q 12 fm for E = 640 MeV.

0

1

2

0

ENERGY-IMeVl

I t&V)

EXCkATlON

20

ENERGY

IO EXCJTATION

--

30

EXCITATION

10 EXCUATION

ENERGY

ENERGY

20

IMeV)

( MeVl

3.0

0 0

0

10 EXCITATION

IO EXCITATION

20 (Me’4

20 ENERGY IMet’)

ENERGY

30

30

Fig. 3. Distribution of cross section as a function of excitation energy for the bombarding energies of 140, 315 and 640 MeV. In each case an observation angle slightly forward of the corresponding grazing angle was chosen. These cross sections were calculated taking into account the removal of flux from the I60 channel due to particle transfer. In (A)-(c) the total inelastic cross section is shown in comparison with the contribution due to the Coulomb excitation as defined in the text. In (D)-(F), the contribution to the cross section due to events where multiplephonon excitation has taken place is shown.

S$i

2

: r” -

rt

=

271

R. A. Broglia et al. / Giant resonances I

I Pl

Dipole

I

I

!

Quadrupole

I.

I

I

Octupole

Fig. 4. Probabilities of one-phonon Coulomb excitation as a function of the energy of the mode for I = 1, 2 and 3. In all cases a transition matrix element of one single particle unit was used for a trajectory leading to a distance of closest approach of 15 fm. The different curves for each multipolarity are labelled by the bombarding energy in MeV/nucleon.

modes is displayed as a function of the frequency of the modes and for several bombarding energies. In all cases a transition matrix element corresponding to B(EL) = 1 single-particle-unit was used and the probability was evaluated for a Coulomb trajectory leading to a distance of closest approach of 15 fm. In the energy interval 10 MeV < Rw < 15 MeV, enhanced excitation probabilities result for the higher bombarding energies, The cross section due to Coulomb excitation for some of the modes included in the calculations is shown in fig. 5 as a function of bombarding energy. It represents the summed contribution of all the trajectories with a distance of closest approach larger than 15 fm. In fig. 5 the cross section associated with the isovector giant dipole resonance at 13.16 MeV is also displayed. Given the strong increase in cross section at the higher energies we have included this mode in a calculation for 80 MeV/nucleon. The corresponding Coulomb excitation spectrum shown in fig. 6 constitutes the overwhelming fraction of the total inelastic cross section. The contribution of the dipole mode is explicitly displayed. In fig. 4 one observes that the probability for the excitation of low-lying modes decreases for the higher bombarding energies. It happens because for these modes, which are basically unaffected by the adiabatic cut-off, the action integral is approximately proportional to the collision time. This result explains why the multiplephonon background disappears for the higher reaction energies (cf. fig. 3). As one

212

R. A. Broglia et al. 1 Giant resonances

I

I

I

I

1‘ (*‘*Pb,

IO

13.16 MeV I-

si E

I

b

0.1

0.01

50

100

100

Elob /A (MeV) Fig. 5. Coulomb excitation cross sections for some of the modes considered in the calculations. Only contribution from trajectories leading to distances of closest approach larger than 15 fm were included. The cross sections, associated with the giant dipole (I 3. I6 MeV) and the giant isovector quadrupole mode (23 MeV) of “sPb are also shown. They exhaust 86 y0 and 55 oAof their corresponding energy-weighted sum rules, respectively.

x t-l x

2

1500 I

1280 MeV

t

8 = 2.7

1000

@$#

I- f*‘*Pb,

13.2MeV)

I

1

2

E 500

bC D-0

0 0

5

IO

EXCITATION

15

ENERGY

20

25

30

IMeV)

Fig. 6. Distribution of Coulomb excitation cross section as a function of excitation energy for 80 hieV/ nucleon. The contribution of the giant dipole resonance of 20sPb is explicitly shown.

5

z”,

0

0

10

EXCITATION

IO

EXCITATION

ENERGY

ENERGY

20

20

( MeV 1

(MeV)

30

30

0

0

0

10

EXCITATION

IO

EXCITATION

ENERGY

ENERGY

20

( MeV)

I MeV )

20

30

30

Fig. 7. Distribution of purely inelastic cross section as a function of excitation energy at 20 MeV/nucleon for different observation angles. The contribution to this quantity due to multiple-phonon excitation is displayed. (A) differs from the equivalent figure in ref. 4, partIy because the contribution to Coulomb excitation from large partial waves (p 3 11 fin) was then missing and partly because of the different spreading widths quoted in table 1.

0

10

x 2

3

R. A. Broglia et al. / Giant resonances

274

moves away from the grazing angle by decreasing the impact parameter, the overlap and therefore the interaction between the nuclei becomes larger, leading also to larger collision times. The probability of exciting the low-lying modes increases accordingly, giving rise to a larger multiple excitation background, as shown in fig. 7 (cf. also fig. 3). The angular distribution associated with the structures of the spectrum at excitation energies 8 MeV < E* < 16 MeV and of 16 MeV < E* < 24 MeV, and for different bombarding energies are displayed in fig. 8. For 315 MeV, the number of channels N appearing in (6) was adjusted to fit the elastic scattering cross section [cf. ref. ‘)I. The resulting value N = 6 corresponds to the label MTR = 1. It is expected that for increasing bombarding energies the number of transfer channels with favourable Q-value increase. A linear extrapolation was used to adjust MTR for the bombarding energies in which no elastic scattering is available. For the sake of comparison all differential cross sections were also calculated utilizing MTR = 1. We have calculated the distribution of cross section versus energy for the events that remain in the purely inelastic reaction channels. The experimental spectrum will contain the contribution of events where mass transfer has taken place but still ended up in the 160 channel. To the extent that this particle-transfer background can be assumed to be structureless it is possible to compare the predicted spectra to the results of the experiments after the background has been subtracted. One cannot however exclude that some of the peaks observed may be remnants of the combined structure of inelastic and transfer processes [cf. for example ref. “)I.

4. Conclusions Inelastic scattering at intermediate bombarding energies (20 MeV/nucleon < E seems to be a useful probe of the nuclear response function. The associated spectra at grazing angles correspond to the excitation of single phonons built on the background due mainly to particle transfer. The proper description of this background is the main open problem in this study. < 80 MeV/nucleon)

Appendix COULOMB

EXCITATIONS

IN HEAVY-ION

COLLISIONS

A treatment of the relativistic Coulomb excitation has been given in ref. 13). In the present context it is sufficient to use their results in the non-relativistic limit, and include some modifications to be discussed. Thus, at an impact parameter p, the square of the excitation amplitude for exciting a collective target state (n, 2) is (A.1)

R. A. BrogIia et al. / Giant resonances

-0

UP

(Js/qw) op

(JS/qW)

=

DP

276

R. A. Broglia et al. / Giam resonances

where Z, is the charge of the u is the relative velocity. Eq. Using instead a hyperbolic for large impact parameters,

projectile, B(EI, n) is the B-value of the excitation and (A.l) is the exact result for a straight line trajectory. orbit in the Coulomb field, the main correction, valid is a factor exp (- rr<) where Z ZAe2

---co

5=

64.2)

nA’

mov3

Moreover, if we replace p by as, where E = l/sin @ and Z Z,e2

a=+

(A.3) MOO

is half the distance of closest approach in a head-on collision, we obtain

’ &B(El,

(K,W2) n) f 2Ae-“~ i WC -A @+p)!(jl--CL)! 0

(A-4)

This expression is equivalent to eq. (H.14) of ref. i4), valid for 9*,/m zz= 1. A further improvement of eq. (A.4) is obtained using the symmetrized quantities [cf. VI.8 in ref. ‘“)I: Z ZAe2

(J=L-,

%.4Vf

5=L

Z ZAe2 A (llvf

(g

=

-

(A3

l/S)7

(g/(vivf,.

vi and v, being the initial and final relative velocities of the two nuclei. The probability for exciting one phonon of type (nn) is 0) = lal(rO12exp (- 1 b,W12).

P&l,

64.6)

n’l’

The sum over n’ and 1’ extends over all vibrational states in the projectile and target nucleus. The corresponding symmetrized differential cross section is

da&Y =

dQ

+A,

6) f

~

1

(A.7)

4 (sin+@4’

I

Neglecting the exponential factor in eq. (A.6) one obtains

B(Ei,n)

“(W

5 2Ae-n5A K,, ,(x)K,- 1(+wp(x))2

0a

c

#=-A

Q+p)!(A-p)!



(A

@

*

R. A. Broglia et al. / Giant resonances

271

for the one-phonon cross section summed over all angles smaller than a given angle &,,,where x = {/sin @,,,. Note that in this expression one should use the symmetrized quantities defined in eq. (AS).

References 1) R. R. Bet&, S. B. Di Cenzo, M. H. Mortensen and R. L. White, Phys. Rev. Lett. 39 (1977) 1183; H. J. Gils, H. Rebel, J. Buschman and H. Klewe-Nebenius, Phys. Lett. 68B (1977) 427 2) M. Buenerd, D. Lebrun, J. Chauvin, Y. Gaillard, J. M. Loiseux, P. Martin, G. Perrin and P. de Santignon, Phys. Rev. Lett. 40 (1978) 1482 3) R. Kramermans, J. van Driel, H. P. Morsch, J. Wilczynski and A. van der Woude, Phys. Lett. 82B (1979) 221 4) P. 13011,13. L. Hendrie, J. Mahoney, A. Menchaca-Rocha, D. K.. Scott, T. J. M. Symons, K. van Bibber, Y. P. Viyogi and H. Wieman, Phys. Rev. Lett. 42 (1979) 366 5) A. M. SandorB, Brookhaven National Laboratory report 26598 6) R. A. Broglia, C. H. Dasso and A. Winther, Phys. Lett. 53B (1974) 301; 61B (1976) 113 7) R. A. Broglia, C. H. Dasso and A. Winther, Int. School of Physics Enrico Fermi, on nuclear structure and heavytion collisions, Varenna, July 1979, eds. R. A. Broglia, C. H. Dasso and R. R&‘(NorthHolland, Amsterdam), to be published 8) J. Blocki, J. Randrup, W. J. Swiatecki and C. F. Tsang, Ann. of Phys. lOS(1977) 427 9) R. A. Broglia, G. Pollarolo, A. Vitturi, A. Winther, C. H. Dasso and H. Esbensen, Phys. Lett. lt!JB (1979) 22 10) H. Eabensen, A. Winther, R. A. Broglia and C. H. Dasso, Phys. Rev. Lett. 41(1978) 296 11) R. A. Broglia and A. Winther, to be published 12) R. A. Broglia, R. Liotta, B. S. Nilsson and A. Winther, Phys. Reports 29C (1977) 291 13) A. Winther and K. Alder, Nucl. Phys. A319 (1979) 518 14) K. Alder and A. Winther. Electromagnetic excitation (North-Holland, Amsterdam, 1975)