Nuclear structure studies from reaction induced by radioactive nuclear beams

was found to be in the range of -10 to -130 MeV/c for various fragments (a negative sign for

indicates that the fragment speed is less than that of the projectile) . 4. Although the main part of the spectrum is symmetric for the P/i and Pt direction [dP//) = a(Pt) I, a longer tail is observed in the Pt direction. The width of the momentum spread (a(P//)) as well as

A,&

- AJ / (Ap - 1)

(4.8)

9

where uo=90 MeV/c (Goldhaber and Heckman, 1978). a(P//) takes its A F/2. The width (a(Pt)) of the transverse maximum value when AF = momentum distribution of the fragment is found to be equal to o(P//), consistent with an isotropic production of fragments in a frame moving at B/i= -

u1 =

t4 g> .

200 MeV/c.

Goldhaber (1974) explained the 00 based on the nucleon Fermi motion, and associated ug with the Fermi momentum (PF) as,

a$ -
(4.10)

,

4A Gaussian is exp(-P2 /20* ) and it’s three dimentional 302.

mean-square momentum

width is

537

Nuclear Structure Studies

Hiifner and where
-s

@I,

I

d3a

&

’

-dP’ = j-d2bW)jdzjd2P, W(b,z;P,,P,,- < P,, >)

,

(4.11)

where W is the Wigner transform of the one-body density matrix and contains information about the position-dependent momentum distribution of the removed nucleon. The distorting function (D(b)) contains the reaction dynamics and localizes the reaction to the nuclear surface. Under the Glauber model D(b) is essentially the same as the transmission function (T(b)) for the collision between a fragment and the target nuclei. They found that a reasonable fit was obtained only when they used a realistic nucleon momentum distribution. As a different formulation for extending the method used for stripping reactions to many nucleon removal, the momentum width of the projectile fragment is expressed by the separation energy of last nucleons, ti = 2U(&)MA, 1 -A,)

, or

(4.12)

AP

o* = 2u(&$

,

for one nucleon removal.

P Here, u is the atomic mass unit and is an average separation energy of the removed nucleons. If we assume a two-body breakup of a projectile by separation energy (E), Eq. (4.4) gives the momentum of the fragment under the sudden approximation. By approximating the Lorentian by the Gaussian that has the same second derivative at the peak, one can obtain Eq. (4.13). The only difference is replacing by E. It therefore gives the same conclusion as shown in Sec. 4.1, except for a tail part of the distribution. The tail of the momentum distribution is strongly affected by D(b). One can obtain Lorentian tail only when D(b)=1 for all value of the impact parameter b. If D(b)<1 the tail of the Lorentian becomes shorter. The P/l distribution of fragments from halo nuclei l*Li, “Be, and 14Be were recently measured at NSGL and at GSI (Orr et al., 1992; 1993; Kelley et al., 1993; Riisager 1993). All of them show a width o(P//) - 20 MeV/c, much narrower than that observed in the fragmentation of stable nuclei, as shown in Fig. 4.5 as an example.

538

I. Tanihata F’l”“I”‘J

I ,

100

150: (a)

I 1 I I , I I I I -

: (c) o = 20.9 f 0.8 MeVlc -

80 -

o - 21.2 f 0.7 MeVlc _

p

0 -

18.7 * 0.8 MeVlc :

o - 20.9 f 0.6 MeVlc

(MeVk)

Fig. 4.5 Parallel momentumdistributionof 9Li fragmentsfrom 11Li reactions.(a) Be (b) Nb, and (c) Ta targets. (c) High-acceptance mode using a 9Be target. Single Gaussian fits were made for (a), (b), and (c). A tw@Gaussianfit is shown for (d). The wide Gaussianhas u-1 I2fl3

MeVk)

As discussed above, the a(&) measure the same if a high-energy projectile is used. The transverse-momentum distributions of fragments from 1% and llLi reactions with a carbon target were measured at 790A MeV (Kobayashi et al., 1988, Tanihata et al., 1992). Figure 4.6 presents the transverse-momentum distribution of (a) a loBe fragment from *IBe + C reaction and (b) a 9Li fragment from *lLi + C reaction both at 800 MeV/nucleon. Both of the data show a very narrow peak on top of another wider peak. The fitting of the momentum distribution by two Gaussians gives a width onarrow= f 4 MeV/c and In 9Li, onarrow= 21 f 3 MeV/c awide= f 7 MeV/c for the *oBe spectrum. and awide= 80 f 4 MeV/c. Although the separation of the momentum distribution into two Gaussians is somewhat arbitrary, it was shown that the fit neither by a Gaussian nor by a Lorentian for the entire range of momentum is In conclusion, the poor. We will come back to this discussion later again.

539

NuclearSpucture Studies small values of a(P//) and a(Pt) indicate that the removed small momentum fluctuation, and thus has long density tail.

neutron(s)

has a

The narrow width of loBe is 600 l”“l”“l”“~““I’c : 1lBe + C --> loBe + X consistent with a value of 21 MeV/c that was estimated from the separation energy of 400 a neutron (Es (n) = 503 keV) using Eq. (4.12). It is more % complicated for IlLi, because u two neutrons are in the same 200 loosely bound orbitals. However, if two-neutron cluster removal is assumed, 0 the observed narrow width 400 0 200 -200 -400 gives = 0.34 f 0.16 MeV, I”“I”“I”“. which is consistent with the -11Li+C-->9Li+Xat 8OOAMeV separation two-neutron energy: AE[eLi + 2n) - (IlLi)] 1000 = 0.34 f 0.10 MeV. Therefore, both spectra are consistent with the expected narrow 0 - 2 If 3 MeVlc momentum distribution of weakly bound last neutrons, 500 and thus show the formation of a neutron halo.

a

The momentum distribution in of neutrons the fragmentation region may 0 also carry halo information. 400 0 200 -200 -400 Moreover, the correlation of Transversemomentum (MeVk) neutrons in the halo may be studied by the comparison of Fig. 4.6 Transverse momentum distribution of the fragments from halo nucki. the neutron spectrum and the related fragment spectrum (Tanihata et al., 1992). Recently, many measurements of neutrons from 6He, 8He, 9Li, **Li, 1113e,and We have been reported (Kobayashi 1993). However, as already mentioned above, the nucleon distribution does not behave as that of a fragment, because of a complication in the reaction mechanism, such as evaporation p’ocesses. It has already been pointed out by Hiifner and his collaborator that one has to study the fragment momentum distribution, not the nucleon distribution, to

540

I. Tanihata

obtain the internal motion. Neutron Pt distributions from various projectiles at 800A MeV are shown in Fig. 4.7. Let see first the 9Li + C -> n + 8Li + X reaction (neutrons are detected in coincidence with aLi), the Pt distribution is wide and has a width ot = 110 MeV/c. It is consistent with the distribution of neutrons of diffractive scattering on a C target (n+C) with an initial momentum spread on the order of the Fermi motion of a neutron in the projectile. The curve in this figure shows thus calculated distribution.

1041’

i..‘.1‘..‘1““1”“1’

1”“I”“l”“l”“l‘

8He+C -> n+6He+X

: 6He+C -> n+4He+X

&-

i

:

.

p\

‘\:

. l lLi+C -> n$Li+X _i>,

..= /

7 i

‘. . a

:

Taransverse Mometum P, [MeV] Fig. 4.7 Transverse-momentum distributions of neutrons from fragmentation of neutron-richnuclei at 790A MeV (Kobayashi 1993).

This component is observed in all of the data shown in Fig. 4.7, indicating that it is a general process considered as above. In addition, a narrow peak is seen in each of the spectra on top of this component for others. In these reactions, a projectile has two valence neutrons. One of them become unbound if the other is removed, and is thus emitted simultaneously without any further collisions. This is why one could expect that this narrow spectrum can give information of the internal momentum of the halo neutron.’ Figure 4.8 shows the neutron separation energy (S, or S& dependence of the width. The a0 of the fragments in one- and two-neutron removal shows a monotonic rise for larger Sxn. The

541

Nuclear StructureStudies

solid line in Fig. 4.8-a shows the calculated value by Eq. (4.13), assuming a twoneutron cluster for two-neutron removal (--s&. The line reproduces the data well. On the other hand, the width (o) of the neutron distribution does not follow the same tendency. The width for sHe and 9Li is as small as that for IlLi, which thus show that the other reaction mechanism is important. I I 1 1

I 1 1 1

A sequential decay process is G I a Fragments considered to be important for $ d) 100 the neutron spectrum. In this q process, one neutron is removed by the interaction of a neutron with the target (see Fig. 4.9). The recoil momentum distribution of the “A-l” system is given by the one-neutron separation energy S,. Then a particle unbound “Al” decays by emitting a second neutron having a recoil momentum given by the decay Q value. The mean-square momentum I b Neutron width of the second neutron is 80given by,

a*(n) =

2us, A(A - 1)

,I,,

II

direct breakup

+zuQ(A-2) --. 3

A-l (4.14)

The first term is usually small be0 _ ----__ cause of the factor l/A (A-l). ; two-step breakup Therefore, the width of the neutron spectrum is mainly deterOt,, mined by the decay Q value. Neutron Separation Energy MeV} Using the known excited states shown in Fig. 4.9, Kobayashi ob- Fig. 4.8 Width of the momentum distributions of tained the widths indicated by fragments and neutrons from the reaction of neutron rich nuclei. Solid lines present the prediction by the the dashed line in Fig. 4.8-b. single step direct reaction and dashedlines are for two Although some discrepancies stepprocesses. remain, the over all tendency of -For a more realistic fit one has to include other the o is reproduced well. channels, such as the direct break up and sequential decay from higher excited states. ‘The contributions from these processes give a larger value of o, and thus are consistent with the deviation of the data from the dashed line. However, no realistic calculations including all of those effects have yet been made.

542

I. Tanibata

In contrast to the neutron spectrum, the o for fragments is rather insensitive to this sequential decay process. It is because o is expressed as, (4.15) In this case the first term dominates in most of the situations. In conclusion, it was found that the two-step process (fragmentation and evaporation) is important. In this process, although the fragment momentum spectrum is determined mainly by the fragmentation process, the neutron spectrum is determined mainly by the evaporation Q value.

Scatteredneutron 10 8 6

MeV

:-,-

7Li+2n

Fig. 4.9 Two-step process of fragmentation reactions and the related levels for the emission of neutrons.

The neutron Pt spectra were also measured at lower energies, as shown in Fig. 4.10. They essentially show the same behavior. However, a difference can seen in the spectra with heavy (high-Z) targets. The neutron distribution from the llBe+C ->n+lOBe+X reaction shows no narrow peak, because llBe is one

543

Nuclear StructureStudies

neutron halo and the neutron is scattered by the reaction. However, a narrow This indicates the peak rapidly develops when the target becomes heavier. progressive importance of the Coulomb interactions, because the nuclear It also indicates that the probability of reaction does not change strongly. exciting the projectile by a strong interaction is small. Therefore, the following three different mechanisms are involved in the reaction: 1. scattering or diffraction of halo neutrons, 2. scattering of the core loBe, and 3. Coulomb dissociation. Until now, no firm determination of the relative importance of these mechanisms have been made quantitatively. The Coulomb interaction is discussed in Sec. 5.

4.4 Correlation between neutrons in a halo.

two

41 MeV/u

““0

Be target 0 0

I’ ‘,

4 \

“Be

o

(%a.

.

(“&.‘&+n)

n)

41 MeV/u

,, ‘*

“Be

Au target

Except for 11Be, two neutrons form halos in nuclei. In all cases, the nucleus with one less neutron is not particle stable. It is therefore obvious that the correlation between two halo neutrons is essential to make these nuclei bound. The paring energy, for example, was studied for .I inter-media &mass nuclei. It was Neutron Angle (deg.) found that the paring energy decreases as the neutron number Fig. 4.10 Transverse momentum distribution of neutrons increases. If one extrapolates this from 11Be reactions. result to the neutron dripline, the expected paring energy would be positive, and thus unrealistic. On the other hand, the odd-even regularity of the particle stability observed near the dripline indicates the importance of the correlation. Is it due to the same mechanism observed as a paring correlation near the stability line? Or is it a different correlation, such as a di-neutron cluster?

544

I. Tanihata

An extraction of the strength of the correlation was first attempted by making a comparison of the one-neutron and two-neutron momentum distributions. The sum momentum of two neutrons (PsUm) is expressed in terms of the momentum of the individual neutron momentum, P sum - Pt + P2 .

(4.16)

Then, the broadening of these momenta is obtained by averaging the square of the momentum. Thus,

- +

+2 .

(4.17)

Here, the correlation appears in the last term, . Since two halo neutrons are identical, and cP& are equal. The value of
12C+6He-> 4He+X -

2.0

(p&2 shell model

-

I

di-neutron cluster model experiment r /

1.5 1.0

\

I’ \ p *\’ I *+

-

0.5 0 -200

-100

0

loo

200

Fig. 4.11 Transverse momentum distribution of a 4He fragment from a 6He reaction. Neither of the simple models reproduce the data if the observed radius of 6He is fitted.

54.5

Nuclear Structure Studies

making a hybrid model, which is a linear combination of the cluster orbital shell model and the di-neutron model, they could fit the experimental data quite well. Although this fitting was rather artificial, it shows the importance of selecting the correlation of neutrons. In fact the density distribution of halo neutrons is related not only to the singleThe particle wave function, but also to the position correlation . momentum distribution is then related to , as already mentioned. Because these two quantities are inter-related by wave mechanics, one can expect to extract the correlation of halo neutrons in a rather model-independent way from a comparison of the density distribution and the fragment However, no serious analysis has yet been made. momentum distribution. For 6He, the microscopic three-body calculation is expected to provide accurate information, because the core 0.08 4He is extremely tightly bound. Such a calculation has been reported by many authors; they all reproduce the momentum distribution of the 4He fragment well. Here, as an example, the results of three-body calculations by Zhukov et al.. [Zhu931 are presented. Figure 4.12 shows a twoparticle correlation density plot of The two-neutron density 6He. shows two separate peaks: one corresponding to a cigar geometry and the other corresponding to a di-neutron configuration. Although no direct information of such a correlation has been observed yet, it would be interesting to see experimentally whether such configurations are mixed in a nucleus. In the same figure the calculated Pt distribution of 4He from a fragmentation of 6He is shown (solid -250 curve). It shows a quite good fit to the data. Recently measurements of all three final state particles (9Li and two neutrons) after the fragmentation of **Li were made at MSU and at

6He Spatial correlations

10

--_ 6He

-150 4He

-> 4He

breakup at 400A MeV

-50

50

150

_I 250

transverse momentum

Fig. 4.12 Two-neutron distribution of the 6He wave function

in a three-body

momentum model.

distribution

calculation of 4He

and the

in the same

546

I. Tanihata

RIKEN and two-neutron relative-momentum spectrum was studied. However, the spectrum shows a statistical behavior presenting a importance of the final-state interactions. The difficulty here is due to the fact that the interesting relative momentum of two neutrons is much smaller than the inverse of the range of the neutron-neutron interactions. Therefore, an experiment that has weaker final-state interactions, such as (p, pn) quasi-free nucleon-nucleon scatterings, would be suitable for studying the correlation of the neutrons in a halo state. This is because the kicked out neutron has a momentum that is very far from the momenta of both the fragment 9Li and the partner neutron. However, no such measurement has yet been reported. In spite of several trials, no definite conclusions between halo neutrons have been obtained so far.

concerning

the correlation

In addition to the data shown above, many measurements have been reported for **Li reactions. These are summarized in Table 411 for the two-neutron removal cross sections and in Table 4111 for the momentum distributions (they were compiled by Riisager 1993).

Table 4-11 Two-neutron removal cross sections for 1lLi in barn.

1. Ieki, K. et al., (1993), 2. Riisager, K. et al., (1992). 3. Blank, B. et al., (I 993). 4. Kobayashi, T. et al.. (1989).

547

Nuclear Struuure Studies

Table 4-U Deduced width (FWHM) of the momentum distribution for ltLi reactions.

1. Ieki, K. et al., (1993). 2. Riisager, K. et al., (1992). 3. Humbertet al., to be published (1994) 4. Kobayashi et al., (1993). 5. Orr, N. et al., (1992). 6. Oeissel, H. et al., (1993). 7. Kobayasbi, T. et al., (1988). 8. Tanihata, T. et al.. (1992).

.

Other related cross sectrons (p

.

.

cross sectrons.I

Does the proton distribution change when a neutron halo is developed? Since all of the measurements for unstable nuclei so far use strong interactions, no direct information on the charge distribution or proton distribution is available. The neutron halo is considered to be formed only by neutrons from several indirect measurements. The separation energy of a proton in a neutron-rich nucleus is large so that the Also, as wave function is expected not to have a long asymptotic tail. discussed in the previous section, the narrow distribution of fragments was one of the decisive evidence of the halo. This narrow width is observed only in fragments in which only one or two neutrons are removed, depending on the single-neutron halo or two-neutron halo, respectively. Other fragments, in which at least one proton is removed, do not show such a narrow distribution. Therefore, a proton does not show a halo-like distribution in neutron halo nuclei.

548

I. Tanihata

Recently, Blank et al. (1992) measured the charge-changing cross I section of Li isotopes. If we believe the geometrical model, the charge changing cross section gives the lOOOoverlapping size of a proton. In 800practice, however, a proton may also be emitted from an excited fragment that is produced by neutron removal. This process gives 400an additional cross section to the charge changing. Therefore, the 1 I I charge-changing cross section is 200 8Li 9Li 1lLi considered to give the upper limit of the proton radius. The measured Fig. 4.13 Charge changing cross section and the charge-changing cross sections do reaction cross sections of Li isotopes at 80A MeV on C target. (Blank et al., 1992) not change from 7Li to IlLi, as shown in Fig. 4.13. It therefore indicates that the proton distribution does not change very much from 8Li to 11 Li.

bi

Table 4-11 Electromagnetic moments of Li isotopes and 8B. nucleus

Spin

Magnetic Quadrupole moment (n.m.) moment (mb) 7Li 3/2 3.2564 -40.0 f 0.3 8Li 2 1.6536 31.1 f 0.5 9Li 3/2 3.4391 f 0.0006 -27.4 f 1.0 1lLi 3/2 3.6678 f 0.0025 -31.2 f 4.5 8B 2 1.0355 68.3 f 2.1 (Arnold et al., 1992, Minamisono et al., 1992)

The magnetic moment and the quadruple moment of a nucleus provide a good In particular, the to study the static-electromagnetic properties of nuclei. quadrupole moment is proportional to
Nuclear Strum

Studies

549

All of the measurements so far are consistent with the persistence of the proton distribution before and after the formation of a neutron halo. Therefore, the known strong attractive p-n interaction does not affect the proton distribution to any visible degree. However, it is stlll an open question as to whether the proton distribution is completely decoupled from the halo or not.

4.6

Proton Halo?

At the limit of the bound-proton rich side, many nuclei have separation energies of less than 1 MeV. Do they have proton halos? Recently, the large quadrupole moment of sB was claimed as an evidence for a proton halo (Minamisono et al., 1992). They compared the newly measured quadrupole moment of 88 and that of 8Li. Under the assumption that the effective charges of the proton and neutron are equal to those of neighboring nuclei, they concluded that the rms radius of the proton distribution should be large,
550

I. Tauibata

not extend as much as that of neutrons. In fact the tail extends only up to the one that correspond to about Es=1 MeV of the neutron distribution. The Therefore, the density Coulomb barrier in this case is about 1 MeV. distribution is same as that with an effective separation energy, E,eff = lMeV, that is the depth counted from the barrier height.

r(fd

r (fm)

Fig. 4.14 Single-particle density distributions of a nucleon in a WoodsSaxon potential.

NuclearSmtcture Studies

551

if we compare the neutron density tail for different 1, the tail is shorter for higher 1 orbitals. It is also true that the behavior of the tail is similar for the same effective separation energy. The addition of a Coulomb barrier to a high I orbital is again shown in the figure.

Now,

2.2 Now let us relate the above mentioned tendency to the 2.0 observed neutron halo nuclei. First of all, there is 1.8 only one example of a single neutron halo, **Be. It shows 1.6 a pronounced halo, as already discussed in sec. 4. It 1.4 is expected because the last neutron has lq 12 as the main component of the wave 1.2 function On the other hand, I?, a candidate of a neutron 1.0 112 712 halo, shows no enhancement -312 -l/2 312 512 Ne F 0 N C B of ol in a measurement at *z intermediate energies (Fig. 4.15). As already shown in Fig. 4.15 The radius parameter rg determined from various sec. 4, the enhancement of q measurements. (Ozawa et al., 1994) No big enhancement is appears sensitively at a suggested in spite of its small neutron separation energy. The low energy data of t7B suggests a large enhancement of lower energy. It therefore the cross section. indicates that *7C may not have a large neutron halo in spite of its small separation energy (0.729 MeV). The single-particle orbital of the last neutron is expected to be 065/z in *7C, and is thus consistent with the large centrifugal barrier. All other known cases (6He, l*Li, IdBe), and the possible case of 17B have two neutrons coupled to jn = 0. In a two neutron halo, although single-particle orbitals include I= 0, 1, and 2, the centrifugal barrier sensitively depends on the configuration of these two neutrons. For example, if they are coupled to a OS di-neutron, no centrifugal barrier takes place. On the other hand, if they are in a pure shell-model orbital without any correlation, the centrifugal potential is the same as that felt by a single particle in the same orbital. It, therefore, requires a detailed microscopic calculation to determine the real strength of the barrier. However, the existence of a halo suggests that the centrifugal barriers for these nuclei are smaller than expected from the single-particle orbital alone. The situation is different for proton halo candidates. The candidates, sB, 12N, and 17F, have a single proton in the last loosely bound orbitals. Therefore, both the Coulomb barrier and the centrifugal barrier come into play, and thus the

552

I.Taniha

tail of the density distribution does not extend very much. It is therefore consistent with a small or no enhancement of the interaction cross section (q). The 17N nucleus is an exception in which two protons exist in the last orbital. In fact, an enhancement of aI has recently been observed in 17Ne + C collision (Ozawa et al., 1994). Thus may indicate that the centrifugal barrier is very low These situations are in this nucleus, and a proton halo may be developed. summarized in Table 4111. Table 4-111Halo candidates and their characteristics

candidates

S, or S, (MeV)

Sn or S2P

01

Orbital

enhancement

(MeV) 6He 1lLi V3e 14Be 17B 19B 17C W 8B 9C 12N 17F *7Ne

0.97

1.051 0.503

0.247 1.28 (2.45) (0.87)

0.739 (0.22) 0.14 1.299 0.601 0.600 0.96

@p3/2)2 (Opl/2orh/2)2 h/2 (Op1/20rh/212 G&/2)2 @k/2)2 f-W2 O&i/z

Yes Yes Yes Yes prob. yes

oP3/2

no

no

@p3/2)2

OP1/2 Od5/z 1so

(O&nor

ls1/2)2

prob. no no yes

All known halo candidates behave as expected based on the barrier effects. Those candidates with no barrier l*Be clearly shows the halo. In addition, all neutron halo candidates with two neutrons in the last orbitals show an enhancement of aI. Among them is the well-established halo nucleus IlLi. All candidates that have a single particle in the last orbital with 1>1 show no enhancement of a~. However, the barrier height is sensitively determined self consistently by the wave function. Therefore, the barrier height depends on the density distribution at the surface region of the nucleus, and becomes lower when the wave function has a longer tail.

Nuclear Struchne Studies

553

5. Soft Mode of Collective Excitation Because of the excess of neutrons on a surface, either the skin or the halo, new types of excitation modes can be expected. In particular, when a core and surface neutron are decoupled as

(5-l) some of the collective strength should be shared with the relative movement of the core and the surface neutrons. Although several predictions have been made for higher multipole excitations, such as E2, the study of such collective excitation is in its very primitive stage, and experimental searches are made only on the El mode of excitation.

10

A 100

Fig. 5.1 Target mass dependence of the interaction cross section and the two-neutron removal cross sections of neutron-rich nuclei. Large enhancements are seen in heavy-target reactions.

Electromagnetic dissociation(EMD) provides important information on collective excitation of nuclei. For stable nuclear projectiles, observations of the EMD process at energies from 1 (Heckman and Lindstrom, 1976; Olson et al.,

554

I. Tanihata

1981; Mercier et al., 1986; Hill et al., 1988) to 200 GeV/nucleon (Hill et al., 1988a; Price et al., 1988) have been reported, and enhancements of fragmentation due to the EMD process have been clearly seen, particularly in the single-nucleonremoval channels on high-i! targets. This phenomenon was interpreted as resulting from the excitation and decay of an El giant dipole resonance (GDR) in a projectile by virtual photons associated with a rapidly changing Coulomb field. Kobayashi et al. (1989, 1989a) observed the EMD process from the target dependence of the interaction cross sections (or) for the first time for unstable nuclei. Figure 5.1 shows the target mass dependence of the interaction cross sections, as well as the neutron (single- or two-) removal cross sections, for selected neutron-rich nuclei. The estimated nuclear parts of the cross sections are also shown in the figure by solid curves for a1 and dashed lines for
I

- . . -*“‘I

’ ’ ”

3 2 _ ,Iti 3 8He zro i f ‘g “Bel I”Be f 12Be v, ! 101 ’ PLi 8 . ,a_B,MD/z; * ,oO l e:m/zzp ‘% !! f I..: I * . . . . . . . I i IO 1 Separationenergy (MeV)

In order to compare the observed EMD Fig. 5.2 Separation energy dependence of the EMD cross sections. The EMD cross cross section of neutron-rich nuclei with a section increases rapidly with a decrease typical cross section for stable nuclei, the in the separationenergy. total EMD cross section, o$MD( *2c+Pb), was estimated to be m35mb (Heckman and Lindstrom, 1976). If the projectile charge dependence of uEMDisassumed to be Zpr$, the enhancement of aEMD(**Li) compared with aEMDQ2C) is extremely large, by a factor of 80.

Nuclear Struchlre Studies

555

An EMD process is considered to occur due to the excitation of a projectile nucleus by the virtual photon field of a target nucleus. The EMD cross section is then calculated by,

(5.2) where IVYis the virtual photon spectrum and or is the photo-nuclear cross It is known that the electric giant dipole resonance (GDR) dominates section. or in high-energy collisions. It has been shown (Heckman and Lindstrom, 1976; Olson et al., 1981; Mercier et al., 1986) that the observed EMD cross section for stable nuclei can be well reproduced by this equation from the known photonuclear cross section, o+y+B+X), and theoretically calculated NY.

.

Soft El mode of excitation

Concerning the GDR of extremely neutron-rich nuclei, K. Ikeda (1992) suggested the possibility that the GDR could split into two components for nuclei with a neutron halo. One component would correspond to the oscillation of core protons against core neutrons with a normal frequency; the other correspond to the oscillation of the core against the skin and/or halo neutrons. Since the restoring force of the dipole oscillation is roughly proportional to the derivative of the density distribution, the frequency of the latter mode (denoted as the soft GDR) is expected to be very low. Within the framework of the GDR model, the energy of the soft GDR can be estimated. The GDR at normal excitation energies tEnorma GDR = 22 MeV estimated from the systematics) contributes to aIEMDonly by about 80 mb that agrees well with EMD cross section of 12C. Therefore, the main part of qmD is due to the excitation of the soft GDR. The integrated strength of the soft GDR was estimated to be 10% of the total El strength. (Since the El strength is roughly proportional to the square of the number of neutrons involved in each mode; the strength of the soft GDR : normal GDR = 22 : 62 for the llLi case.) In order to reproduce the observed 01EMDPLi+Pb), the energy of the soft GDR had to be E,ft GDR = 0.9+Os54.3 MeV if r,ft GDR was assumed to be 0.7 MeV. The energy (E,h GDR)was rather insensitive to the choice of r,,ft ~-J-JR. Thus, the excitation energy of the soft GDR turned out to be remarkably low. Qualitatively, the most essential conclusion drawn here is that the observed large EMD cross section, together with the rapidly decreasing virtual-photon spectrum, requires an appreciable strength of the El strength (resonance and/or continuum) to be located at a very low excitation energy, irrespective of the detailed structure.

5.56

I. Tanihata

Bertsch and Foxwell (1990) extended the Hartree-Fock model to calculate the El strength for IlLi. (Remember that his model explains the neutron halo.) The Random Phase Approximation (RPA) was used to calculate the dipole strength. The calculation shows a softening of the dipole mode, as shown in Fig. 5.3. However, their model predicts a smaller strength than needed to explain the observed enhancement in the EMD cross section. The calculated cross section is only about the 30% of the observed value. Even including the uncertainty in the experimental determination of the EMD cross section, more enhancement is needed. It is, therefore, suggested that any theory has to deal with the correlation between halo neutrons. Suzuki and To&a (1990) extended the cluster-orbital shell-model calculation for soft dipole excitation in the 9Li + n + n system. They found that quite a large strength (10% of the energy-weighted sum rule, or 30% of the non-energyweighted sum rule) is located at very low energy. They also obtained the EMD cross section to be 600 mb that agrees reasonably well with observed cross section. I “U

’

Dipole

1

’

1

I

‘2

Response

Free RPA

.... o.of

0

’

’

’

’

’ 10

’

m ’

Energy

’

’ 20

’

’

‘.......__) B -l 30

’

(MeV)

Fig. 5.3 Electric dipole strength distribution of 1*Li calculated by a RPA calculation.

Experimentally, an excited state of IlLi, as a possible candidate from the soft GDR, was observed at the excitation energy E, =1.2 fO.l MeV, from an experiment on double-charge-exchange (DCX) reaction induced by pions, llB( It-, rr+)“Li, at T, =164 MeV by Kobayashi et al. (1992). The spin and the parity of the state were assigned as being l/2+, 3/2+ or 5/2+ from the measured angular distribution, so that the state can be excited by the El transition.

557 NuclearStructureStudies However, it was not clear whether this state is the main component of the soft GDR or additional components are necessary, especially since the DCX reaction does not favor to excite a real collective state. Recently, the excitation distribution of l*Li EMD was measured by two groups: one at MSU (Ieki et al., 1993) and the other at RIKEN. (Shimoura et al., 1994) Also, the excitation distribution of llBe was measured in RIKEN (Nakamura et al., 1994). The El strength distribution in llBe has a special meaning because it has a single neutron halo, and thus no confusion due to neutron correlations arises. Also, it is known that a strong El amplitude exists between the first excited state f&=0.3198 MeV, In =1/2-j. It therefore shows that the expected strength of the low energy El resonance is in the bound state, and therefore the El strength observed in the EMD process is most likely due only to the nonresonant continuum. Figure 5.4 shows the El strength distribution. The direct breakup model calculation which give the El distributionas,

= gexp(21cr,) l+mo

Fig. 5.4

xez n*~(

2

* @(E,

0A

- E$‘* E,’

,

El strength

distribution

deduced

from the EMD cross section of 11Be.

2

1

&

3

[MeVl

(5.1)

558

I. Tanibata

is shown by the solid curve. Here, S is the spectroscopic factor for the 291/z state, K =@i?y/n , p is the reduced mass, Es is the separation energy, and QJ is the radius of the square-well potential \ relevant to the halo P neutron. The curve shows a good over-all fit to the data, thus indicating that x the transition is Elb 0. dominated and nona resonant. The angular distribution of the *oBe fragment in the recoil *lBe frame is shown in Fig. 5.5. The angular distribution is Fig. 5.5 Angular distribution of neutron emission by the EMD well reproduced by a shape process of the 1IBe + Pb -> l%e +n +X reaction. The angle confirming the dominance (+ is measured from the reaction plane. The angular distribution is consistent with a calculation based on the El transition. of El transition.

In MSU and also in RIKEN, two neutrons in coincidence with the 9Li fragment from *lLi + I% -> 9Li + n + n + X reaction wre measured. The upper part of Fig. 5.6 shows the obtained excitation-energy distribution by Ieki et al. (1993). The solid line in the figure shows a model calculation based on the BrightWigner distribution of a resonance at E&l.7 MeV with a width lYo=O.&O MeV. The dotted line in the figure is a prediction based on the direct break-up process by a Coulomb interaction. It shows that a better fit is obtained by the resonance picture as far as the excitation distribution is concerned. However, at the same time, they observed a systematic difference in the velocity between the neutrons and the 9Li fragment. Figure 5.6-c shows a velocity difference spectrum. The lithium-9 fragment has larger velocity than They claimed that this is due to a post do neutrons on the average. acceleration of the 9Li after the *lLi break up. From the observed difference in the velocity, llLi should breakup into 9Li and neutrons at a distance very close to the target. The resonance has its own lifetime, and thus decays much later after its production. They, therefore, concluded that this velocity difference is an evidence of a direct breakup.

559

Nuclear StructureStudies

However, recent theoretical studies by Bertsch and Bertulani (1993) show that the breakup is instant, and occurs at close distance independent of the mechanism, resonance or direct breakup. So far, several experiments seem likely to give controversial results for our understanding of the origin of the soft El strength in IlLi. Understanding the dipole strength is also important from the view point of correlations of halo neutrons. Further accurate measurements are thus awaited.

0

1.00

0.50

1.50

Ed (decay energy) [MeV] 400 200 0 3 0 !I 3100 50 0

-0.04

-0.02

0

0.02

0.04

V/C Fig. 5.6 Spectra of the 1lLi EMD process at MSU. (a) Decay energy spectrum of the 9Li+n+n system. (b) Spectrum of the longitudinal component of the center of mass velocity of the 9Li+n+n system, (c) Spectrum of the longitudinal component of the relative velocity VPV2n. The histogram shows the result of a Monte Carlo simulation without Coulomb acceleration.

6.

ELASTIC

AND

INELASTIC

SCATTERING

OF RADIOACTIVE

NUCLEI 6.1

Elastic scatterinv of 11h

A recent development in the intensity of radioactive beams has enabled us to study the elastic and inelastic scatterings of neutron-rich nuclei on proton target. Because a beam is the nucleus of interest, the reaction has to be studied by inverse kinematics. In an inverse kinematic measurement, the energy

560

I. Tan&a

resolution is not determined by the energy broadening of secondary beams, but by the energy resolution of the detected proton. As long as we use a thin target we can obtain a good energy resolution that is independent of the secondary beam resolution. Elastic scatterings were measured for 9Li + p, *lLi+p, *lLi+C, and llLi+Si at intermediate energies. Although a neutron halo has been observed in 11 Li from studies of high-energy collisions, an elastic scattering is expected to provide more detailed information about the structure of the halo. The result of the elastic-scattering angular distribution is shown in Fig. 6.1, together with the proton elastic scattering of other Li isotopes (6Li, 7Li, and 9Li) (Moon et al., 1992). The cross sections are similar for 6Li, 7Li, and 9Li; the absolute values are almost the same and the diffraction minimum systematically shifts to smaller angles for heavier isotopes, reflecting the increase in the matter radii. For l*Li, however, the cross flct..,.‘.,‘,‘....‘. .,.I “” section is smaller by 50% compared “‘“20 30 40 50 60 70 with that of other isotopes. An Center of mass angle (degree) optical potential fitting of the data was reported by two groups. In the Fig. 6.1 Proton elastic scattering angular distributions of Li isotopes. Although the cross first one, a global parameter set was sections show systematic behavior for 7Li, *Li, used as the starting point. Although and 9Li, sudden decrease is seen in 1 1Li this parameter set was determined spectrum. from reactions with mass numbers larger than 40 Warner et al., 1991), the parameter set without any modification fit the 6Li, 7Li, and 9Li data amazingly well as can be seen in Fig. 6.2-a, where an angular distribution is plotted as a ratio to the Rutherford scattering cross section However, it fails to reproduce the llLi cross section [the dashed curve in Fig.6.2-b]. Two different approaches were applied to reproduce the data. One is to change the parameters of the imaginary part of the potential. As shown by the dotted line (set A) in Fig. 6.2-b, a reasonable fit was obtained. The other is, contrarily, to change the real part of the potential. It also gives an equally good fit to the data, as indicated by the solid line (set B) in the figure. The largest difference between this potential is the depth of the real potential and the diffuseness of the imaginary potential. Set A has long tail in the imaginary part of the potential, while set B has a very shallow real potential. The shallow potential is considered to reflect the breakup process. Because of

Nuclear StructureStudies

561

the extremely small separation energy of the valence neutrons in **Li, the polarization potential is reflected as the repulsive force in the real potential. Because these two extreme selections of parameter sets can fit the data equally well, one can imagine that many other different sets of parameters can reproduce the data. Therefore, no unique optical potential can be obtained by this method. In a second paper, a more intuitive method was applied (Hirenzaki et al., 1993). Here, the optical potential was separated into two parts. One is the potential from the core (SLi), the other is that from the neutron halo. The optical potential of 9Li was first determined by fitting the 9Li + p data using the initial values of the parameter from the global potential. Then, an imaginary potential having the 100 same shape as the density distribution of the neutron halo determined from interaction cross section measurements, was added for *lLi. Then, the depth of the potential was treated as the fitting parameter. A reasonable fit to the data was obtained. With slight 5 adjustments of the other part of the potential, the fitting becomes 20 60 80 excellent. Therefore, a model that San4’ @es) explicitly includes the halo effect Fig. 6.2 Ratio of the elastic scatterings to the Rutherford also reproduces the data well. scattering cross sections. See text for the fitting by various models. Dot dashed curve is the results of Aleixo et al., (1991)

In the analysis presented above, however, it is not clear whether it is really necessary to take into account the halo structure in order to reproduce the data. In particular, it is not clear whether the effect of the breakup process is essential to the halo density shape or not. To answer this question, Suzuki et al., (1993) calculated the cross section based on the four-body model, (gLi+n+n)p, in order to take into account the halo structure. The Eikonal and adiabatic approximations were used to derive an optical potential that includes the breakup effect of halo neutrons to continuum states. They could reproduce the reduction of the cross section satisfactory. Figure 6.3 shows the optical

562

I. Tanhta

potential of 9Li and llLi thus obtained. The p-“Li optical potential has a much longer tail than that of gLi+p. They concluded that both the breakup effect of the halo neutron and the n-p exchange force are important in obtaining a good agreement with the da la. The elastic scatterings of IlLi were also measured with 12c (Kolata et al., 1992) and 28Si (Lewitowicz et al., 1992). As shown in Fig. 6.4, three-body calculations show fair agreement with the data if inelastic scatterings are taken into account (Thompson et al., 1993a). However, because of the uncertainty in the inelastic scattering, as well as the large error bars in the Fig. 6.3 Optical potential of I * Li and 9Li data, discrimination of the model could determined by the four-body model. not be made. This situation is similar to that for the %i case. Further experimental and theoretical studies are necessary to isolate the effect of the halo.

6.2

Elastic and Inelastic scatterinp of a&

We now consider the data of 8He at 73A MeV. Figure 6.5 shows an inclusive proton distribution from the 8He+p reaction as a function of the excitation energy of 8He at CICrnfrom 25 to 60 degrees (Korsheninnikov et al., 1993). One can see a strong peak for the ground state, reflecting the elastic scatterings. At around 3.7 MeV is a peak that corresponds to an inelastic scattering to a 8He When protons are detected in coincidence with forward *He, excited level. only the peak corresponding to the elastic scattering is seen. When protons are detected in coincidence with 6He, the elastic peak disappears and the peak at an excitation energy of 3.6 MeV dominates the spectrum. This state is also seen in several other coincidence conditions p+n, p+bHe+n, and p+7He, where 7He is identified by the invariant mass of the detected However, this level is not detected when a coincidence 6He+n system. includes 4He, as shown in Fig. 6.5-F. because both the 6He+n+n and 4He+4n channels are open, it indicates a small branching to 4He+4n, namely r4He+4n/rtotal<4%.

Nuclear StructureStudies

563

-

100

elastic only .. FaddeevM3 YabanaY1 (Op1/2)2 FaddecvQS FaddeevL6A Faddeev22 Core!potential only ..“.“....“_

8 a

_._I

---

”

-___

t

10-l 101

t

I+

elastic + inelastic - - - FaddeevQS FaddeewL6A - - - - Faddeev22 Core potential only

Fig. 6.4 Quasi-elastic scattering measurementof 1lLi +C and theoretical fits.

J.

564

I. Tanihata

3 80 60 40 20 0 -2

2

6

10 14 18

EIGHTMeV

-2 2

6

10 14 18

EIGHTMeV

Fig. 6.5 Inelastic scattering spectrum of 8He+p.

7.

STUDIES OF PARTICLE UNSTABLE NUCLEI OFF THE DRIP LINES

In addition to bound drip-line nuclei, studies of unbound states and unbound nuclei near the drip line are of great importance for understanding the Also, studies of nuclei beyond the drip line structure of drip-line nuclei. provide a unique opportunity to study nuclei under extreme conditions, such as an extreme Z/N ratio. By using the radioactive beams near the drip lines, one can now study unbound nuclei near to or even beyond the drip lines. Among the method are transfer reactions of the usual kind, such as (d,p) and (d?He) for neutron rich nuclei and the (p,d) and (d,n) reactions for the proton rich side, that provides direct accesses to unbound nuclei off the drip line. Also, invariant mass measurements can be efficiently used for producing unbound states and nuclei. Here, we see some recent studies by invariant mass measurements.

7.1

Invariant-mass

565

NuclearStructureStudies suectroscoDV

The development of high-intensity heavy-ion accelerators has enabled us to produce drip-line nuclei in large quantity and to use them as secondary beams. Interesting examples among them are unbound He isotopes (5He, ‘He, 9He, loHe) and excited states of bound He isotopes. In particular, loHe is interesting because it is a double magic-number nucleus beyond the neutron drip-line and has the largest IV/Z ratio ever observed. The invariant-mass measurement is a widely used method to search for particle resonances. In this method, several particles in the final state after the reaction are detected in coincidence, and the invariant-mass distribution is measured. The invariant mass is calculated from the observed four momenta (Pi) as,

where pi is the three dimensional momentum vector of the i-th particle. This method has two advantages: first, the mass (or excitation energy) is fully determined by the energy-momentum of particles in the final state so that the mass resolution is independent of the energy spread of the incident beam, and is thus suitable for secondary-beam application. Second, a thick target can be used because one detects only highenergy particles and thus has less effect from the energy loss in the target. For the same reason, it is more sensitive to a small production cross section.

140 120 100 $

go

8 s

60 40 2. 0 -2-l

0

12

3

4

5

6

J%EIe-n-n 9 MeV

Such experiments have recently been Fig. 7.1 Invariantm-s spectrumof *He +n+n system after the fragmentation of l ILi. The carried out at RIKEN using a magnetic peak at 1.2 MeV indicates the first spectrometer and neutron detectors observation of loHe. Lines show phase space (Korcheninnikov et al., 1994). A calculations. charged fragment and neutrons are detected in coincidence for the observation of unbound He isotopes. An invariant-mass spectrum of the *He +2n final state from the **Li fragment is shown as a function of the excitation energy from sHe +n+n (E*~H~~+J in Fig. 7.1, in which detector acceptance is corrected.

I. Tanibata

566

The spectrum is shown for a CDztarget. It shows a pronounced peak at 1.2 MeV with a small background at higher excitation energies. The phase-space calculation of the three-body final state including final state interactions are

loBe (l%, l%)lOHe, Elab=334.4 MeV 160 t...l.“‘““““““.-.‘-~

lOHe

z

8

2.7” < 81ab C 3.9” 8

80

20 a

40

60

80

100

120

Channel

Fig. 7.2 10~~ spectrum by a two-body reaction. In addition to tbe ground state, two excited states are suggested

shown by the curves in the figure. None of them including other phase space curves not shown here reproduce a peak near to the observed one, but

567

Nuclear Structure Studies

reproduce the shape at higher excitation. Therefore, the peak at 1.2 f 0.3 MeV is attributed to the *OHestate. Since the width of the spectrum is determined totally by the experimental resolution, they could determine the upper limit of the width as being I’c1.2 MeV. However, this width is already sufficiently small to reject the 2s1/2orbital for the valence neutrons. Helium-10 was also recently observed by nucleon-exchange two-body reactions (Ostrowski et al., (1994). Three levels are observed (see Fig. 7.2) and the lowest one (1.07 f 0.07 MeV above the 2n threshold) is consistent with the data by Korcheninnikov. Two other levels are located at excitation energies of 3.20 and 6.82 MeV from the lowest state.

Production of nuclei beyond the drip lines 1.

Invnriant ma58 spectroscopy ‘9-k: “Li -> %e+n+n 13Li : 14Be -> “ti +n+n ‘6Be . I78 -> ‘$e+ n+ n 18Be :. 19B

.. ... 2.

-> ‘$e+n+n+n+n

Transfer rpactiolrr (d, 3He) reaction

”

ti(d, 3He)‘%e, (d,p) reaction “Ii(d

‘%e(d,%e,‘h, .

... .

3. lhder reactions (p,d) lcactiom gc(PPk (d,n) wxtions ‘tJdsl)‘%,

4

. .. . ..

Invariant mass speetros~py 6Be -> a+p+p ‘ON -> 9c+tl

Fig. 7.3 List of the reactions for producing nuclei beyond the dripline. mass method is promising

Among them, the invariant

for many cases both beyond the proton and the neutron drip lines.

The invariant mass was also measured for the 6He +2n and 6He+n final states, and gives consistent energies of the excited levels with previous measurements based on other methods (Korsheninnikov et al., 1993). Invariant-mass spectroscopy is a powerful tool for studying unbound excited states and unbound nuclei. It is actually the best tool, at present, to study nuclei beyond the drip line, as demonstrated by the observation of loHe. It can also be used in the proton-rich side. Such a possibility is summarized in Fig. 7.3 for both limits

568

I. Tanihata

of the nuclear chart. Studies of 12J3Li, 15J6J7r18Be beyond the neutron drip line and Qe, IoN beyond the proton drip line would be possibile in the near future. Transfer reactions could also be used to study those nuclei beyond the drip line. Transfer reactions may provide more information than the invariantmass spectrum, if they are detected, because angular distributions can be studied in detail. One can also utilize the selectivity of the reaction by applying an appropriate reaction. In particular, (p,n) and (d,n) reactions have advantages in the proton-rich side because very low-energy neutrons can be detected without any energy loss problem in the target that determines the energy resolution for the detection of a charged-particle final state. Acknowledgment The author would like to express his thanks to Prof. T. Yamazaki for recommending that he write this article. He also gives thanks for all helps from Dr. D. Hirata and Ms. Odai for preparing figures and other materials.

Nuclear StructureStudies

569

References Alba, R. et al. (1993). Proc. of the Third Int. Conf. on Radioactive Nuclear Beams, Michigan State University, edited by D. J. Morrisey, Editions Frontiers, p31. Ann, R et al. (1993). Physics Letters m 55. Artukh, A. G. et al., (1993). Proc. of the Third Int. Conf. on Radioactive Nuclear Beams, Michigan State University, edited by D. J. Morrisey, Editions Frontiers, p45. Arnold, E, J. Bonn, A. Klein, R. Neugart, M. Neuroth, E. W. Otten, P. Lievens, H. Reich. W. Widdra. and ISOLDE Collaboration. (1992)Phys. I&. m 16. Asahi, K. et al., (1992). Hyperfine Interactions E 101. Bang, J. M, and I. J. Thompson. (1992) .Phys. Left. ~ 201-206. Becchetti, F. D. et al., (1991). NucI. In&r. and Meth. B56/57 554. Belmont, J. et al., (1992). Int. Workshop on “Physics and Techniques of Secondary Nuclear Beams” March 1992, Dourdan (France) Edited by J. F. Bruandet, B. Fermamdez, M. Bex, Editions Frontiers, 1992, p407. Bertsch. G. F, A. Brown, and H. Sagawa. (1989). Phys. Rev,C 39 1154. Bert&. G, and J. Foxwell. (1990). Phys. Rev w 1300. Bertsch, G. F, and H. Esbensen. (1991). Ann. Phys. 2pe 327. Bertsch, G. F, and C. A. Bertulani. (1993) . Nucl. Phys A556 136. Bibber, K. Van. et al. (1979) Phys. Rev. I&. g 840. Blank, B. et al. (1992)Z Phy. A343 375. Blank, B. et al. (1993). Nucf. Phys. m 408. Boyd, R. (1993). Physics Report 227 57. Brohm, T. et al. (1992). Nucl. Phys. m540. Buchmann et al., (1992). Proc. of the Second Int. Conf. on Radioactive Nuclear Beams, Louvain-la-Neuve 1991, edited by Delbar, Adam Hilger 1992, ~109. Conf. Proc. of the First Int. Conf. on Radioactive Nuclear Beams, Berkeley 1989, edited by W. D. Myers, J. M. Nit&e, and E. B. Norman, World Scientific 1990. Proc. of the Workshop on the Science of Intense Radioactive Ion Beams, Los Alamos, 1990, LA-11964-C UC-413. Proc. of Int. Symp. on Structure and Reactions of Unstable Nuclei, Niigata, Japan 1991, edited by K. Ikeda and Y. Suzuki, World Scientific. Proc. of the Int. Workshop on Unstable Nuclei in Astrophysics, Tokyo, Japan 1991, edited by S. Kubono and T. Kajino, WorId Scientific. Proc. of the Second Int. Conf. on Radioactive Nuclear Beams, Louvain-laNeuve 1991, edited by Delbar, Adam Hilger 1992. Proc. of the international Conference on Nucleus-Nucleus Collision IV, Edited by H. Toki, I. Tanihata, and H. Kamitsubo, Special issue of Nuclear Physics, North-Holland, Nucl. Phys. A538 (1992).

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