PHYSICAL REVIEW C VOLUME 57, NUMBER 3 MARCH 1998 Nuclear breakup of Borromean nuclei G. F. Bertsch and K. Hencken* Institute for Nuclear Theory, University of Washington, Seattle, Washington 98195 H. Esbensen Physics Division, Argonne National Laboratory, Argonne, Illinois 60439 ~Received 1 August 1997! We study the eikonal model for the nuclear-induced breakup of Borromean nuclei, using 11Li and 6He as examples. The full eikonal model is difficult to realize because of six-dimensional integrals, but a number of simplifying approximations are found to be accurate. The integrated diffractive and one-nucleon stripping cross sections are rather insensitive to the neutron-neutron correlation, but the two-nucleon stripping does show some dependence on the correlation. The distribution of excitation energy in the neutron-core final state in one- neutron stripping reactions is quite sensitive to the shell structure of the halo wave function. Experimental data favor models with comparable amounts of s and p waves in the 11Li halo. @S0556-2813~98!03503-1# PACS number~s!: 25.60.Gc, 25.70.Mn, 21.45.1v, 27.20.1n I. INTRODUCTION during the interaction time, provided we take the interaction from nucleon-nucleon scattering. The energy domain around Halo nuclei having a very weakly bound neutron pair ~of- 250 MeV has an additional advantage from a theoretical ten referred to as Borromean nuclei! are interesting objects, point of view: The real part of the NN forward-scattering but they are difficult to study experimentally. Secondary in- amplitude goes through zero in this vicinity, so only the ab- teractions in radioactive beams have been an important tool, sorptive part of the interaction needs to be treated in the with Coulomb excitation providing quantitative data about theory. the excitation properties @1,2#. Nuclear excitation is also im- The nuclear excitation of Borromean nuclei have been portant from an experimental point of view, but the theoret- considered by a number of authors @7–14#. In treating the ical interpretation of nuclear reaction cross sections deserves differential cross sections, it is common to make a number of closer attention. In this work we attempt to make a link, as simplifying assumptions. We list them here. quantitative as possible, between the nuclear excitation ob- (i) Ground state wave function. Neutron-neutron correla- servables and the fundamental properties of a Borromean tions were neglected in Ref. @8#. We shall apply wave func- nucleus. The fact that correlations can play an important role tions that have the full three-particle correlations. It turns out makes this goal more difficult than for a nucleus with a that differential cross sections are quite insensitive to these correlations, except the two-neutron stripping, which does single-nucleon halo. On the experimental side, we have been show an effect. Independent-particle models can only de- inspired by the work on 11Li carried out at Ganil, NSCL, scribe pure configurations, so a mixture of s and p waves RIKEN and most recently at GSI. The extremely large Cou- requires a correlated model. lomb breakup cross section shows the halo character of the (ii) Reaction model. In this work we use an eikonal model nucleus, but the details of its wave function have been con- description of the nuclear reaction, improving on the black troversial. Starting from the shell model, two of us @3# con- disk model of Ref. @8#. structed a wave function that fits many Coulomb excitation (iii) Neutron-core potential. It is important to include the 2 measurements @4#. It had a dominant p1/2 shell configuration, final-state neutron-core potential in calculating the energy or as one expects from Hartree-Fock theory. However, several momentum spectra, as demonstrated in Refs. @11,14#. Refer- measurements ~see, for example, Ref. @5#! and also the spec- ence @9# also included the final-state interaction, using a 11 2 troscopy of the nearby nucleus Be suggest a leading s1/2 zero-range neutron-core potential. Our detailed models de- configuration in 11Li. scribed here use a realistic finite-range potential in both the In principle, a nuclear-induced breakup gives independent initial and final states. information and so it is desirable to calculate the various We shall investigate the validity of these as well as other cross sections and compare with experiment. A recent ex- approximations that are often made. Our main interest is the periment @6# was carried out on a 12C target at sensitivity of experiments to the properties of the halo 280 MeV/nucleon. At that energy it is justified to treat the nucleus. In a previous work @15# we developed models of the target-projectile interaction in the sudden approximation, us- 11Li ground-state wave function with differing amounts of s ing the NN forward scattering amplitude for the interaction. wave. One of our objectives is to see how well the amount of Thus we may neglect the evolution of the wave function s wave can be determined by the observables in a breakup reaction. The observables we consider are integrated cross sections for diffraction and one- and two-nucleon removal *Present address: Department fu¨r Physik und Astronomy, Univer- and the differential cross section for the excitation energy in sita¨t Basel, CH-4056 Basel, Switzerland. the 9Li1n final state when one neutron has been removed. 0556-2813/98/57~3!/1366~12!/$15.0057 1366 © 1998 The American Physical Society 57 NUCLEAR BREAKUP OF BORROMEAN NUCLEI 1367 II. REACTION MODEL over r1 and r2 become independent in a shell-model repre- sentation of C such as Eq. ~11!. Another simplifying ap- The sudden approximation leads to the eikonal model for 0 proximation is the transparent limit, defined here by setting nucleus-nucleus interactions. In previous studies, we have the factor S2(R ) equal to one inside the expectation value of applied the model to the nuclear-induced breakup of single- n 2 nucleon halo nuclei @16#. Here we apply it to the breakup of Eq. ~2!, thus neglecting the absorption of the second neutron. a two-neutron halo nucleus. The effect of the interaction with These two assumptions yield the cross section the target is to multiply the halo wave function by the profile functions S(Ri) for each particle, where Ri denotes the im- 2 2 2 s1n-st,trans52 d RSc~R!^12Sn~R1r1'!&. ~5! pact parameter of particle i with respect to a target nucleus. E The halo nucleus 11Li has two neutrons and a 9Li core, Note that this cross section is identical to the sum of the requiring two profile functions Sn and Sc associated with neutrons and the core, respectively. There are three inte- one-neutron-stripping cross section and two times the two- grated cross sections that leave the core intact, namely, the neutron-stripping cross section, diffractive, the one-nucleon stripping, and the two-nucleon stripping cross sections. These can be written s1n-st,trans5s1n-st12s2n-st . ~6! 2 2 We will see later that the two-neutron-stripping cross section sdif5E d Rcm$^@12Sc~Rc!Sn~R1!Sn~R2!# & is rather small, so the transparent limit is a good approxima- tion for this cross section. 2 2^@12Sc~Rc!Sn~R1!Sn~R2!#& % , ~1a! Of course, much more information about the halo is con- tained in differential cross sections. The diffractive cross sec- tion has three particles in the final state, but that distribution 5 d2R @^S2~R !S2~R !S2~R !& E cm c c n 1 n 2 is beyond what we can calculate, requiring three-particle 2 continuum wave functions for many partial waves. The one- 2^Sc~Rc!Sn~R1!Sn~R2!& #, ~1b! neutron stripping leaves two particles in the final state and the differential cross section for that state is amenable to s 52 d2R ^S2~R !S2~R !@12S2~R !#&, ~2! computation. The expression for the momentum distribution 1n-st E cm c c n 2 n 1 associated with the relative motion of the two surviving par- ticles is 2 2 2 2 s2n-st5 d Rcm^Sc~Rc!@12Sn~R1!#@12Sn~R2!#&, E dsst 52 d2R @12S2~R !# d3r uM~R ,r ,k!u2, ~3! d3k E 1 n 1 E 2c 1 2c ~7! where Rcm is the impact parameter of the halo nucleus with respect to the target nucleus and ^& denotes a ground-state where r is the center-of-mass coordinate of the remaining expectation value. Our ground-state wave function C is ex- 2c 0 neutron-core system with respect to the stripped neutron; the pressed in terms of the relative neutron-core distances r and 1 associated impact parameter with respect to the target r . An example of the needed expectation values is the one- 2 nucleus is denoted by R , R 5R 1r . The amplitude nucleon stripping integral 2c 2c 1 2c' M is given by ^S2S2~12S2!&5 d3r d3r uC ~r ,r !u2S2~R ! c n n E 1 2 0 1 2 c c 3 * M5 E d r2ck ~r2!Sc~Rc!Sn~R2!C0~r1 ,r2!. ~8! 2 2 3Sn~Rc1r2'!@12Sn~Rc1r1'!#. ~4! Here ck(r2) is the continuum wave function of the surviving The integrations are here performed for fixed Rcm so Rc neutron-core system, normalized to a plane wave at infinity. depends on the integration variables Rc5Rcm2(r1' The coordinates Rc , R2 , and r1 are expressed in terms of the 1r2')/(Ac12), where Ac is the mass number of the core integration variables as Rc5R2c2r2' /(Ac11), R25R2c nucleus. 1r2'Ac /(Ac11), and r152r2c1r2 /(Ac11). The six-dimensional integration in Eq. ~4! is very time The numerical calculation of Eq.