- Email: [email protected]
Momentum distribution of fragments in reactions induced by neutron rich nuclei
27 April 1995 PHYSICS
EISEVIER
LETTERS
B
Physics Letters B 349 (1995) 421-426
Momentum distribution of fragments in reactions induced by neutron rich nuclei l?Banerjee, R. Shyam ’ Saha Institute of Nuclear Physics, Calcutta
- 700064, India
Received 29 December 1994; revised manuscript received 21 February 1995 Editor: C. Mahaux
Abstract
We calculate the longitudinal and transverse momentum distributions of the fragments ‘Li, “Be and ‘*Be emitted in the reactions induced by the neutron rich nuclei ” Li, “Be and 14Be respectively on several targets at different beam energies within a direct fragmentation model. For all the cases, the shapes of the longitudinal momentum distributions are found to depend very weakly on the interactions governing the fragmentation. However, both widths and the absolute magnitudes of the transverse momentum distributions are sensitive to tbe fragment-target interactions. Comparisons with the recent experimental data are made to support our conclusions.
Experiments performed already with the beams of the neutron rich exotic nuclei like “Li, “Be and 14Be have revealed the existence of a “neutron halo structure” in their ground states where the valence neutron(s) extends(extend) too far out in the space with respect to the core [ l-31. Such systems provide a stringent test of the nuclear structure models (which were developed mostly to explain the properties of the stable isotopes), as they involve new structures and surface densities. The halo structure is manifested through the experimental observation of abnormally large total reaction and Coulomb dissociation cross sections [ 4,5], and very sharply forward peaked angular distributions of neutrons measured in coincidence with the core nuclei [ 61. The study of the momentum distributions of fragments emitted in the dissociation of these nuclei is particularly useful in probing the halo structure in their I E-mail: [email protected].
ground state. Distributions in the momentum space are related to those in the coordinate space by Heisenberg’s uncertainty relation. Indeed, in the study of the projectile fragmentation with stable isotopes, the experimental momentum distributions, under certain specific conditions, are shown to be proportional to the square of the ground state wave function of the projectile in the momentum space [ 7,8]. However, the fragment-target interactions lead to deviations from this simple picture. It is, therefore, of interest to investigate carefully the effect of such interactions on the shapes as well as the absolute magnitudes of the momentum distributions of the fragments. In this letter, we calculate both longitudinal and transverse momentum distributions of the fragments 9Li, toBe and ‘*Be emitted in “Li, “Be and 14Be induced breakup reactions respectively on several target nuclei at beam energies below 100 MeV/A. The direct fragmentation model (DFM), formulated in the framework of post form distorted wave Born approxi-
0370-2693/95/$X39.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO370-2693(95)00261-8
422
P. Banerjee, R. Shyam / Physics Letters B 349 (199.5) 421-426
mation (DWBA), is employed to account for the dissociation of the projectile in the Coulomb field of the target nucleus [ 91. The nuclear breakup is calculated within the diffraction dissociation and diffraction stripping mechanisms [lo], which along with the DFM Coulomb dissociation (CD) provide a good account of the experimental data on the target mass and beam energy dependence of the total two-neutron removal cross sections and the angular distributions of the exclusive and inclusive neutrons in “Li, 14Be and “Be induced reactions [ lo]. Our aim here is to see to what extent the Coulomb and nuclear interactions that govern the breakup of the projectile, affect the longitudinal and transverse momentum distributions of the charged heavy fragments. This is expected to highlight the usefulness of these observables in extracting the information about the structure of the projectile from the breakup data. Within the DFM, the double differential cross section for fragment b in the reaction a + A --) b + x + A (e.g. for the fragmentation of “Be, a = “Be, b = “Be, x = n and A is some target nucleus) is given by
&#,EL -=
pkbk, .I dS2x (215;‘fi2
d&,d&
k,
I fwL(q,Q)
12t (3)
where
fiv,m(q,Q)
= [WPxq,Q)
+ ~(hAl,Q>]fbl>
i
--Sdg’F(q’,Q)f(Pbcl-q’)S(PIq+q’)
+ 2rka
d2u z&2g=
refer to [ 91. The longitudinal and transverse momentum distributions of the fragment b are obtained by integrating Eq. ( 1) over the appropriate momentum components. The nuclear breakup (NB) cross sections consists of two components; the diffraction dissociation where both fragments b and x are emitted and the target A remains in the ground state during the reaction (elastic breakup) and the diffraction stripping where the fragment x is absorbed by the target nucleus while only fragment b is emitted (inelastic breakup). The double differential cross-section for the “elastic” breakup process (a+A -+b+x+A) isgiven by
J
dfl p(phase) x
(4)
1T’?’ I2’ fl
where Tj5?’ =< ,y-)(kb,Rb)
exp(ik,.R,)
1@‘ah&(+)&&)
>,
1 &(rbx) (2)
and p(phase) is the three body phase space factor [9]. In Eq. (2) ,$-) and ,$+) are the Coulomb wave functions representing the relative motions of the fragments in the respective channels with appropriate boundary conditions. @‘a( rbX) is the ground state wave function of the projectile which is assumed to have a simple Yukawa form, QO(r) = &$% exp( -ar) /r, where LYis related to the separation energy E of the b + x system. We have taken E = 0.30 MeV, 0.50 MeV and 1.325 MeV for ” Li, “Be and 14Be respectively. Although this wave function represents only the external part of the projectile wave function properly, we use it for all the distances since the projectile fragmentation reactions are known to be strongly peripheral with the contribution of the nuclear interior to the matrix element (Eq. (2) > being negligibly small [ 71. For the definition of other quantities in Eq. (2) and details of the evaluation of the T-matrix (Eq. (2) ) we
In Eq. (4) the form factors F( q, Q) and the fragmentnucleus scattering amplitudes f(q) are the same as defined in Ref. [ LO]. The differential cross-section for the diffraction stripping process a + A -+ b + X is given by ,j3 @N,INEL = -
dk;
XI
1
(2T)3 I
s
d2Sx[l-
11 -
%(Sx)
I21
e-ikb’rb( 1- mb(Sb))@a(rbx d3rb12,
(5)
where the profile function Wi (i = b, x has a WoodsSaxon form with radius and diffuseness parameters r-0 and A [ 101. As discussed in details in Ref. [ 101, the possibility of observing fragment b from the diffraction dissociation mechanism has been explicitly excluded while deriving Eq. (5). Therefore, the elastic and inelastic breakup modes are clearly separated from each other and there involves no double counting. As done in earlier studies [IO), we use Q = 1.2 fm and A = 0.5 fm throughout. Since the major contributions to the Coulomb and nuclear cross sections come from mutually non-overlapping regions of the impact
P. Banerjee, R. Shyam / Physics Letters B 349 (I 995) 421-426
bJ
04 I
80 -
423
“Li
’ LI+x .
-
+ A
"Be+
A J
- 100
-LO
-20
20
0
p (MeV/c
II
(c)
I
’
11
Be+A
I
LO
60
I
1
+
0 7, ( MeV/c
-60
I
I
A --"Be
X
100
1
) 1
-+“Be+X
Fig. 1. (a) Longitudinal momentum distributions (LMD) of ‘Li fragment for “Li induced reactions on lEITa, g3Nb and gBe targets at 66 MeV/A incident energy. The dotted (dashed) line represents the Coulomb (nuclear) contribution, whereas the total (Coulomb + nuclear) contribution is represented by the full line curve. The experimental points have been taken from [ 121. (b) Longitudinal momentum distributions of toBe fragment for “Be induced reactions on *‘)sPb and t2C targets at 73 MeViA beam energy. The dotted, dashed and full lines have the same meaning as in Fig. ( 1a). (c) Longitudinal momentum distributions of ‘*Be fragment for 14Be induced reactions on ao8Pb and “C targets. The beam energy is 56.8 MeV/A. The dotted, dashed and full lines have the same meaning as in Fig. ( I a). The experimental points are taken from 1141.
424
P. Banerjee, R. Shyam / Physics Letters B 349 (1995) 421-426
Table 1 Calculated widths (half width at half maximum) of longitudinal ( l-7) and transverse (8-13) momentum distributions of various fragments. C = Coulomb, N = Nuclear, WC = Calculated width and WE = Experimental width. All the widths are in MeV/c Case
Fragment
Projectile + target
(MeV/A)
C
N
C+N
22 18
22 21 20 20 20 46
18.7f0.8 21.2f0.7 20.9f0.8 _ _ _
WC
4WIl
WE
2 3 4 5 6
9Li 9Li ‘Li ‘OBe ‘OBe ‘*Be
“Li + lR’Ta “Li + ‘“Nb ’ ’ Li + “Be “Be + 2osPb “Be + “C 14Be + 208Pb
66 66 66 73 73 56.8
25
18 21 20 20 20 46
7 8
‘*Be 9Li
14Be + ‘*C “Li + ‘*‘Ta
56.8 66
30
47 15
47 25
44f5 _
9 10 11 12 13
gLi ‘OBe ‘OBe ‘*Be ‘*Be
“Li + 9Be “Be + 2osPb “Be + ‘*C 14Be + *08Pb 14Be + lzC
66 73 73 56.8 56.8
40 20 24 25 42
40 20 24 27 42
_ _ 46f4.4
I
parameters, we have ignored the Coulomb-nuclear interference term (see e.g. [ 111). In Fig. ( la) we show the longitudinal (or parallel) momentum distribution (LMD) of 9Li fragment emitted in the “Li induced reactions on target nuclei and at beam energies as specified. We see that the widths of the LMDs ( WII) are almost independent of the target mass. The reaction mechanism, of course, changes rather drastically as one goes from the lighter target to the heavier ones. The CD contributes negligibly for the 9Be target where the NB makes up the entire cross section. On the other hand, CD dominates the cross sections for 18’Ta with NB contributing only to the extent of 15-20%. However, w11corresponding to Coulomb and nuclear contributions are quite similar (see Table I), leading to nearly target independent shapes of the LMDs of 9Li. Our calculations are in good agreement with the experimental data (taken from [ 121) . Similar observations are made in Fig. ( 1b) where we show the LMD of the ‘OBe fragment in the ’ ‘Be induced reactions on 208Pb and 12C targets at the beam energy of 73 MeV/A. The calculated W’I in this case is in remarkable agreement with the corresponding experimental values reported recently at the beam energy of 63 MeV/A [ 131. In Fig. ( lc), we show the results for the LMD of 12Be in the 14Be induced reactions on the same targets at the beam energy of 56.8 MeVIA as a function of the modulus of the longitudinal momentum. The experimental data are taken from [ 141, In this case too the
20
23 110
w11for 208Pb and 12C targets are the same. It should be noted that due to the larger separation energy ( 1.32 MeV) of valence neutrons in 14Be as compared to that in “Li (0.30 MeV), the NB dominates the spectra of 12Be even for 208Pb target (see Ref. [ lo] for details). This also is the case for “Be. The w11can be approximately related to the separation energy E by E II wi/,u [ 151. From the theoretical widths as shown in Tab Ie 1, we get E = 0.3 1 MeV, 0.50 MeV and 1.321 MeV for “Li, “Be and 14Be respectively (averaged over all the targets), which agree well with the experimental values of E in each case. Although approximate, this shows that E can be obtained from the measured WII independent of the fragmenttarget interaction. Therefore, the LMD of the heavy fragments are almost independent of the interactions governing the breakup process; they reflect only the binding energy of the fragment inside the projectile. Our results are in agreement with the observations made in Ref. [ 161 where only “Li case was considered and calculations were made in an entirely different model. It should, however, be noted that these authors do not include the elastic breakup mode in their NB cross sections; thus underestimating their results by lo-15% (see the discussions given below). In Fig. (2) we show the results for the transverse momentum distributions (TMD) of the 9Li and “Be fragments emitted in the same reactions as in Figs. (la) and (lb). The corresponding widths are shown
I? Banerjee, R. Shyam / Physics Letters B 349 (1995) 421-426
(al
“‘.,
-k 300 200
100
w .= 5
;Y.
-200
-100
0
pl (MeV/c
100
200
1
Fig. 2. Transverse momentum distributions (ThID) of (a) ‘Li fragment for “Li induced reactions on tstTa target at 66 MeV/A beam energy, (b) same as in (a) but for ‘Be target, (c) ‘OBe in “Be induced reaction on 208F%target at 73 MeVlA beam energy, (d) same as in (c) but for **C target. The doffed, dashed and full lines have the same meaning as in Fig. (la). in Table 1. We see that, in contrast to the case of lon-
gitudinal momentum distributions, the widths of the transverse momentum distributions (WI) depend substantially on the target mass. In this case the Coulomb and nuclear contributions have different widths. The WI for lighter target are relatively broader than those on the heavy targets, indicating that the diffractive fragment-target nuclear interactions lead to the broadening of the TMDs. For heavier targets the CD dominates the momentum distibution. As this contribu-
425
tion is strongly confined to the narrow angles in the forward direction, the corresponding widths are narrower. Therefore, the TMDs, being quite sensitive to the fragment-target interactions and the reaction mechanism, can not be related to the structure of the projectile in a simple way. We would like to point out that the calculated WI for all the targets are considerably narrower than those seen in the fragmentation of stable isotopes. This is true even for the case of 14Be where the value of E is 4-5 times larger than that of “Li. This confirms that this nucleus too has a neutron halo structure despite a smaller rms halo radius as compared to that of 12Be core. On the other hand, we fail to corroborate the larger WI reported for 9Li fragment in ‘lLi induced reaction on lead target in Ref. [ 171. However, it should be stressed that TMDs could also contain contributions from reaction mechanisms other than those considered by us [ 181 which may broaden these distributions to some extent. The nuclear breakup cross sections (in Figs. ( 1) and (2)) are dominated by the inelastic breakup mode; the elastic breakup contributes only to the extent of lo-12%. In many other cases of the projectile fragmentation reactions, similar observation has been made (see e.g. [ 10,16,19] ), which has even been verified experimentally for stable isotopes [7]. The large elastic breakup component - almost the same in magnitude as the inelastic breakup - reported by Bertulani et al. [ 201 for the inclusive ( “Li, 9Li) reaction on heavy targets at 800 MeV/A beam energy could result from their choice of prior form DWBA which is known to overestimate this mode [ 71. Furthermore, the “non-elastic” breakup as discussed by Yabana et al. [21] includes only those processes where the target is excited during the reaction; this is only a part of our inelastic breakup mode. In summary, we calculated the longitudinal and transverse momentum distributions of fragments 9Li, “Be and t2Be emitted in the “Li, “Be and 14Be induced reactions on several target nuclei and at different beam energies. We found that for all projectiles the longitudinal momentum distributions of the heavy charged fragments are rather independent of the interactions that govern the breakup process, and hence they are better suited for probing the halo structure in these projectiles from studies of their fragmentation. On the other hand, the transverse momentum distributions are sensitive to the reaction mechanism.
426
P. Banerjee, R. Shyam /Physics Letters B 349 (1995) 421-426
Therefore, TMD of the fragments can not be easily related to their momentum distributions inside the projectile. A possible improvement of our calculations could be to use a better projectile ground state wave function [ 221. However, the overall conclusions of our paper regarding the target mass dependence of the momentum distributions are unlikely to be altered. One of us (RS) would like to thank Prof. R. C. Johnson and Dr. I. J. Thompson for their kind hospitality at the University of Surrey, where a part of this work was done. References [ 1] See e.g., H. Toki, I. Tanihata and H. Kamitsubo, Proc. Int. Conf. on Nucleus-Nucleus Collisions IV, Nucl. Phys. A 533 (1992).
[ 2 ] I. Tanihata, Nucl. Phys. A 522 ( 1991) 275~; T. Kobayashi, Nucl. Phys. A 553 (1993) 465c, and references therein. [3] PG. Hansen and B. Jonson, Europhys. L&t. 4 (1987) 409. [4] I. Tanihata et al., Phys. Lea. B 206 ( 1988) 592; B. Blank et al., 2. Phys. A 340 (1991) 41; M. Fukuda et al., Phys. I&t. B 268 (1991) 339.
(51 D. Sackett et al., Phys. Rev. C 48 (1993) 118; T. Nakamura et al., Phys. Lett. B 331 (1994) 296. [6] R. Anne et al., Phys. Len. B 250 (1990) 19; Nucl. Phys. A 575 (1994) 125. [7] G. Baur, E Riisel, D. Trautmann and R. Shyam, Phys. Rep. 111 (1984) 333. [8] J. Htifner and M.C. Nemes, Phys. Rev. C 23 (1981) 2538. [9] R. Shyam, P. Banerjee and G. Baur, Nucl. Phys. A 540 (1992) 341. [lo] P Banejee and R. Shyam, Nucl. Phys. A 561 (1993) 112; Phys. Lea. B 318 (1993) 268. [ 1 I ] CA. Bettulani, L.F. Canto and M.S. Hussein, Phys. Rep. 226 (1993) 281. [ 121 N.A. Orr et al., Phys. Rev. I&t. 69 (1992) 2050. [13] J.H. Kelley et al., Phys. Rev. Lett. 74 (1995) 30. [ 141 M. Zahar et al., Phys. Rev. C 48 (1993) R1484. [ 151 W.A. Friedman, Phys. Rev. C 27 (1983) 569. [ 161 CA. Bertulani and K.W. McVoy, Phys. Rev. C 46 (1992) 2638. [ 17 ] T. Kobayashi et al., Phys. Rev. Len. 60 ( 1988) 2599. [ 181 T. Fujita and J. Hilfner, Nucl. Phys. A 343 (1980) 493. [19] Y. Ogawa, K. Yabana and Y. Suzuki, Nucl. Phys. A 543 (1992) 722; Y. Suzuki et al., Nucl. Phys. A 567 (1994) 957. [20] C.A. Bertulani, G. Baur and M.S. Hussein, Nucl. Phys. A 526 (1991) 751. [Zl] K. Yabana, Y. Ogawa and Y. Suzuki, Nucl. Phys. A 539 (1992) 295. [22] M.V. Zhukov et al., Phys. Rep. 231 (1993) 151.
EISEVIER
LETTERS
B
Physics Letters B 349 (1995) 421-426
Momentum distribution of fragments in reactions induced by neutron rich nuclei l?Banerjee, R. Shyam ’ Saha Institute of Nuclear Physics, Calcutta
- 700064, India
Received 29 December 1994; revised manuscript received 21 February 1995 Editor: C. Mahaux
Abstract
We calculate the longitudinal and transverse momentum distributions of the fragments ‘Li, “Be and ‘*Be emitted in the reactions induced by the neutron rich nuclei ” Li, “Be and 14Be respectively on several targets at different beam energies within a direct fragmentation model. For all the cases, the shapes of the longitudinal momentum distributions are found to depend very weakly on the interactions governing the fragmentation. However, both widths and the absolute magnitudes of the transverse momentum distributions are sensitive to tbe fragment-target interactions. Comparisons with the recent experimental data are made to support our conclusions.
Experiments performed already with the beams of the neutron rich exotic nuclei like “Li, “Be and 14Be have revealed the existence of a “neutron halo structure” in their ground states where the valence neutron(s) extends(extend) too far out in the space with respect to the core [ l-31. Such systems provide a stringent test of the nuclear structure models (which were developed mostly to explain the properties of the stable isotopes), as they involve new structures and surface densities. The halo structure is manifested through the experimental observation of abnormally large total reaction and Coulomb dissociation cross sections [ 4,5], and very sharply forward peaked angular distributions of neutrons measured in coincidence with the core nuclei [ 61. The study of the momentum distributions of fragments emitted in the dissociation of these nuclei is particularly useful in probing the halo structure in their I E-mail: [email protected].
ground state. Distributions in the momentum space are related to those in the coordinate space by Heisenberg’s uncertainty relation. Indeed, in the study of the projectile fragmentation with stable isotopes, the experimental momentum distributions, under certain specific conditions, are shown to be proportional to the square of the ground state wave function of the projectile in the momentum space [ 7,8]. However, the fragment-target interactions lead to deviations from this simple picture. It is, therefore, of interest to investigate carefully the effect of such interactions on the shapes as well as the absolute magnitudes of the momentum distributions of the fragments. In this letter, we calculate both longitudinal and transverse momentum distributions of the fragments 9Li, toBe and ‘*Be emitted in “Li, “Be and 14Be induced breakup reactions respectively on several target nuclei at beam energies below 100 MeV/A. The direct fragmentation model (DFM), formulated in the framework of post form distorted wave Born approxi-
0370-2693/95/$X39.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO370-2693(95)00261-8
422
P. Banerjee, R. Shyam / Physics Letters B 349 (199.5) 421-426
mation (DWBA), is employed to account for the dissociation of the projectile in the Coulomb field of the target nucleus [ 91. The nuclear breakup is calculated within the diffraction dissociation and diffraction stripping mechanisms [lo], which along with the DFM Coulomb dissociation (CD) provide a good account of the experimental data on the target mass and beam energy dependence of the total two-neutron removal cross sections and the angular distributions of the exclusive and inclusive neutrons in “Li, 14Be and “Be induced reactions [ lo]. Our aim here is to see to what extent the Coulomb and nuclear interactions that govern the breakup of the projectile, affect the longitudinal and transverse momentum distributions of the charged heavy fragments. This is expected to highlight the usefulness of these observables in extracting the information about the structure of the projectile from the breakup data. Within the DFM, the double differential cross section for fragment b in the reaction a + A --) b + x + A (e.g. for the fragmentation of “Be, a = “Be, b = “Be, x = n and A is some target nucleus) is given by
&#,EL -=
pkbk, .I dS2x (215;‘fi2
d&,d&
k,
I fwL(q,Q)
12t (3)
where
fiv,m(q,Q)
= [WPxq,Q)
+ ~(hAl,Q>]fbl>
i
--Sdg’F(q’,Q)f(Pbcl-q’)S(PIq+q’)
+ 2rka
d2u z&2g=
refer to [ 91. The longitudinal and transverse momentum distributions of the fragment b are obtained by integrating Eq. ( 1) over the appropriate momentum components. The nuclear breakup (NB) cross sections consists of two components; the diffraction dissociation where both fragments b and x are emitted and the target A remains in the ground state during the reaction (elastic breakup) and the diffraction stripping where the fragment x is absorbed by the target nucleus while only fragment b is emitted (inelastic breakup). The double differential cross-section for the “elastic” breakup process (a+A -+b+x+A) isgiven by
J
dfl p(phase) x
(4)
1T’?’ I2’ fl
where Tj5?’ =< ,y-)(kb,Rb)
exp(ik,.R,)
1@‘ah&(+)&&)
>,
1 &(rbx) (2)
and p(phase) is the three body phase space factor [9]. In Eq. (2) ,$-) and ,$+) are the Coulomb wave functions representing the relative motions of the fragments in the respective channels with appropriate boundary conditions. @‘a( rbX) is the ground state wave function of the projectile which is assumed to have a simple Yukawa form, QO(r) = &$% exp( -ar) /r, where LYis related to the separation energy E of the b + x system. We have taken E = 0.30 MeV, 0.50 MeV and 1.325 MeV for ” Li, “Be and 14Be respectively. Although this wave function represents only the external part of the projectile wave function properly, we use it for all the distances since the projectile fragmentation reactions are known to be strongly peripheral with the contribution of the nuclear interior to the matrix element (Eq. (2) > being negligibly small [ 71. For the definition of other quantities in Eq. (2) and details of the evaluation of the T-matrix (Eq. (2) ) we
In Eq. (4) the form factors F( q, Q) and the fragmentnucleus scattering amplitudes f(q) are the same as defined in Ref. [ LO]. The differential cross-section for the diffraction stripping process a + A -+ b + X is given by ,j3 @N,INEL = -
dk;
XI
1
(2T)3 I
s
d2Sx[l-
11 -
%(Sx)
I21
e-ikb’rb( 1- mb(Sb))@a(rbx d3rb12,
(5)
where the profile function Wi (i = b, x has a WoodsSaxon form with radius and diffuseness parameters r-0 and A [ 101. As discussed in details in Ref. [ 101, the possibility of observing fragment b from the diffraction dissociation mechanism has been explicitly excluded while deriving Eq. (5). Therefore, the elastic and inelastic breakup modes are clearly separated from each other and there involves no double counting. As done in earlier studies [IO), we use Q = 1.2 fm and A = 0.5 fm throughout. Since the major contributions to the Coulomb and nuclear cross sections come from mutually non-overlapping regions of the impact
P. Banerjee, R. Shyam / Physics Letters B 349 (I 995) 421-426
bJ
04 I
80 -
423
“Li
’ LI+x .
-
+ A
"Be+
A J
- 100
-LO
-20
20
0
p (MeV/c
II
(c)
I
’
11
Be+A
I
LO
60
I
1
+
0 7, ( MeV/c
-60
I
I
A --"Be
X
100
1
) 1
-+“Be+X
Fig. 1. (a) Longitudinal momentum distributions (LMD) of ‘Li fragment for “Li induced reactions on lEITa, g3Nb and gBe targets at 66 MeV/A incident energy. The dotted (dashed) line represents the Coulomb (nuclear) contribution, whereas the total (Coulomb + nuclear) contribution is represented by the full line curve. The experimental points have been taken from [ 121. (b) Longitudinal momentum distributions of toBe fragment for “Be induced reactions on *‘)sPb and t2C targets at 73 MeViA beam energy. The dotted, dashed and full lines have the same meaning as in Fig. ( 1a). (c) Longitudinal momentum distributions of ‘*Be fragment for 14Be induced reactions on ao8Pb and “C targets. The beam energy is 56.8 MeV/A. The dotted, dashed and full lines have the same meaning as in Fig. ( I a). The experimental points are taken from 1141.
424
P. Banerjee, R. Shyam / Physics Letters B 349 (1995) 421-426
Table 1 Calculated widths (half width at half maximum) of longitudinal ( l-7) and transverse (8-13) momentum distributions of various fragments. C = Coulomb, N = Nuclear, WC = Calculated width and WE = Experimental width. All the widths are in MeV/c Case
Fragment
Projectile + target
(MeV/A)
C
N
C+N
22 18
22 21 20 20 20 46
18.7f0.8 21.2f0.7 20.9f0.8 _ _ _
WC
4WIl
WE
2 3 4 5 6
9Li 9Li ‘Li ‘OBe ‘OBe ‘*Be
“Li + lR’Ta “Li + ‘“Nb ’ ’ Li + “Be “Be + 2osPb “Be + “C 14Be + 208Pb
66 66 66 73 73 56.8
25
18 21 20 20 20 46
7 8
‘*Be 9Li
14Be + ‘*C “Li + ‘*‘Ta
56.8 66
30
47 15
47 25
44f5 _
9 10 11 12 13
gLi ‘OBe ‘OBe ‘*Be ‘*Be
“Li + 9Be “Be + 2osPb “Be + ‘*C 14Be + *08Pb 14Be + lzC
66 73 73 56.8 56.8
40 20 24 25 42
40 20 24 27 42
_ _ 46f4.4
I
parameters, we have ignored the Coulomb-nuclear interference term (see e.g. [ 111). In Fig. ( la) we show the longitudinal (or parallel) momentum distribution (LMD) of 9Li fragment emitted in the “Li induced reactions on target nuclei and at beam energies as specified. We see that the widths of the LMDs ( WII) are almost independent of the target mass. The reaction mechanism, of course, changes rather drastically as one goes from the lighter target to the heavier ones. The CD contributes negligibly for the 9Be target where the NB makes up the entire cross section. On the other hand, CD dominates the cross sections for 18’Ta with NB contributing only to the extent of 15-20%. However, w11corresponding to Coulomb and nuclear contributions are quite similar (see Table I), leading to nearly target independent shapes of the LMDs of 9Li. Our calculations are in good agreement with the experimental data (taken from [ 121) . Similar observations are made in Fig. ( 1b) where we show the LMD of the ‘OBe fragment in the ’ ‘Be induced reactions on 208Pb and 12C targets at the beam energy of 73 MeV/A. The calculated W’I in this case is in remarkable agreement with the corresponding experimental values reported recently at the beam energy of 63 MeV/A [ 131. In Fig. ( lc), we show the results for the LMD of 12Be in the 14Be induced reactions on the same targets at the beam energy of 56.8 MeVIA as a function of the modulus of the longitudinal momentum. The experimental data are taken from [ 141, In this case too the
20
23 110
w11for 208Pb and 12C targets are the same. It should be noted that due to the larger separation energy ( 1.32 MeV) of valence neutrons in 14Be as compared to that in “Li (0.30 MeV), the NB dominates the spectra of 12Be even for 208Pb target (see Ref. [ lo] for details). This also is the case for “Be. The w11can be approximately related to the separation energy E by E II wi/,u [ 151. From the theoretical widths as shown in Tab Ie 1, we get E = 0.3 1 MeV, 0.50 MeV and 1.321 MeV for “Li, “Be and 14Be respectively (averaged over all the targets), which agree well with the experimental values of E in each case. Although approximate, this shows that E can be obtained from the measured WII independent of the fragmenttarget interaction. Therefore, the LMD of the heavy fragments are almost independent of the interactions governing the breakup process; they reflect only the binding energy of the fragment inside the projectile. Our results are in agreement with the observations made in Ref. [ 161 where only “Li case was considered and calculations were made in an entirely different model. It should, however, be noted that these authors do not include the elastic breakup mode in their NB cross sections; thus underestimating their results by lo-15% (see the discussions given below). In Fig. (2) we show the results for the transverse momentum distributions (TMD) of the 9Li and “Be fragments emitted in the same reactions as in Figs. (la) and (lb). The corresponding widths are shown
I? Banerjee, R. Shyam / Physics Letters B 349 (1995) 421-426
(al
“‘.,
-k 300 200
100
w .= 5
;Y.
-200
-100
0
pl (MeV/c
100
200
1
Fig. 2. Transverse momentum distributions (ThID) of (a) ‘Li fragment for “Li induced reactions on tstTa target at 66 MeV/A beam energy, (b) same as in (a) but for ‘Be target, (c) ‘OBe in “Be induced reaction on 208F%target at 73 MeVlA beam energy, (d) same as in (c) but for **C target. The doffed, dashed and full lines have the same meaning as in Fig. (la). in Table 1. We see that, in contrast to the case of lon-
gitudinal momentum distributions, the widths of the transverse momentum distributions (WI) depend substantially on the target mass. In this case the Coulomb and nuclear contributions have different widths. The WI for lighter target are relatively broader than those on the heavy targets, indicating that the diffractive fragment-target nuclear interactions lead to the broadening of the TMDs. For heavier targets the CD dominates the momentum distibution. As this contribu-
425
tion is strongly confined to the narrow angles in the forward direction, the corresponding widths are narrower. Therefore, the TMDs, being quite sensitive to the fragment-target interactions and the reaction mechanism, can not be related to the structure of the projectile in a simple way. We would like to point out that the calculated WI for all the targets are considerably narrower than those seen in the fragmentation of stable isotopes. This is true even for the case of 14Be where the value of E is 4-5 times larger than that of “Li. This confirms that this nucleus too has a neutron halo structure despite a smaller rms halo radius as compared to that of 12Be core. On the other hand, we fail to corroborate the larger WI reported for 9Li fragment in ‘lLi induced reaction on lead target in Ref. [ 171. However, it should be stressed that TMDs could also contain contributions from reaction mechanisms other than those considered by us [ 181 which may broaden these distributions to some extent. The nuclear breakup cross sections (in Figs. ( 1) and (2)) are dominated by the inelastic breakup mode; the elastic breakup contributes only to the extent of lo-12%. In many other cases of the projectile fragmentation reactions, similar observation has been made (see e.g. [ 10,16,19] ), which has even been verified experimentally for stable isotopes [7]. The large elastic breakup component - almost the same in magnitude as the inelastic breakup - reported by Bertulani et al. [ 201 for the inclusive ( “Li, 9Li) reaction on heavy targets at 800 MeV/A beam energy could result from their choice of prior form DWBA which is known to overestimate this mode [ 71. Furthermore, the “non-elastic” breakup as discussed by Yabana et al. [21] includes only those processes where the target is excited during the reaction; this is only a part of our inelastic breakup mode. In summary, we calculated the longitudinal and transverse momentum distributions of fragments 9Li, “Be and t2Be emitted in the “Li, “Be and 14Be induced reactions on several target nuclei and at different beam energies. We found that for all projectiles the longitudinal momentum distributions of the heavy charged fragments are rather independent of the interactions that govern the breakup process, and hence they are better suited for probing the halo structure in these projectiles from studies of their fragmentation. On the other hand, the transverse momentum distributions are sensitive to the reaction mechanism.
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P. Banerjee, R. Shyam /Physics Letters B 349 (1995) 421-426
Therefore, TMD of the fragments can not be easily related to their momentum distributions inside the projectile. A possible improvement of our calculations could be to use a better projectile ground state wave function [ 221. However, the overall conclusions of our paper regarding the target mass dependence of the momentum distributions are unlikely to be altered. One of us (RS) would like to thank Prof. R. C. Johnson and Dr. I. J. Thompson for their kind hospitality at the University of Surrey, where a part of this work was done. References [ 1] See e.g., H. Toki, I. Tanihata and H. Kamitsubo, Proc. Int. Conf. on Nucleus-Nucleus Collisions IV, Nucl. Phys. A 533 (1992).
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