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The nuclear breakup of halo nuclei through diffraction and stripping
NUCLEAR PHYSICS A ELSEVIER
Nuclear Physics A654 (1999) 669c-672c
www.elsevier.nl/locate/npe
The Nuclear Breakup of Halo Nuclei through Diffraction and Stripping K. Hencken ~ , G. Bertsch u and H. Esbensen c ~Institut fiir Physik, Universit/~t Basel, Basel, Switzerland blnstitut for Nuclear Theory, U Washington, Seattle, USA CPhysics Department, Argonne National Laboratory, Argonne, USA We calculate the different nuclear breakup cross sections (diffraction, stripping and absorption) within the Serber model for single-nucleon as well as two-neutron halos (socalled Borromean nuclei). In contrast to calculations up to now, we use realistic wave functions as well as a realistic model for the interaction with the target. We show results of our calculations for total as well as differential cross sections. 1. I n t r o d u c t i o n Halo nuclei are a new kind of nuclei that exist close to the proton or neutron drip line. The breakup of these nuclei through Coulomb excitation and nuclear interaction are two of the main experimental methods used to study their properties. Here we concentrate on reactions with light targets, which are dominated by the nuclear interaction. We have used in this work as far as possible a realistic reaction mechanism and realistic wave functions. Total as well as differential cross section can therefore be calculated at the same time. The applicability of some commonly used approximation was tested in this way. The distortion of the ground state properties due to the reaction mechanism was studied. This is essential to understand whether and how the measured quantities relate to the ground state properties of the halo nuclei. The reaction model we use is the Serber model. The interaction of the target with each "particle" - that is, each halo-nucleon and the core - is described by a profile function S(b), depending only on the individual impact parameter b. The different reactions are characterized by the remaining "particles" in the final state: In diffraction all "particles" are present, in the different stripping reactions, either a halo nucleon or the core is absorbed, in absorption the whole halo nucleus is absorbed [1]. The profile function is calculated in the eikonal approximation as
where V is the interaction potential and we integrate along the beam axis. At low energies we use an optical potential for V with both real (elastic) and imaginary (reactive) parts. At high energies only the imaginary part is important and we use a folding approach, giving a potential of the form V(r) = where aNN is the free NN-cross
--i~-c~NNpr(r),
0375-9474/99/$ see front matter © 1999 Elsevier ScienceB.V. All rights reserved.
K. Hencken et al./Nuclear Physics A654 (1999) 669c-672c
670c
Table 1 Total cross section for the different processes are given for 6He and llLi and are compared with experiment [4,5]. p89 and s23 are two different models with 5% and 23% s-wave component in the ground state. 6He 11Li C (th) C(ex) Pb(th) Pb(ex) p89 s23 exp. corrected diffraction 32.3 304-5 127 6504-110 27 33 604-20 94 in-stripping 136 127+14 332 3204-90 121 137 1704-20 175 2n-stripping 16.7 334-23 69 1804-100 6 9 504-10 11 2n-removal 185 1904-18 528 11504-90 154 179 2804-30 280
section and pT(r) the target density. For the core-target interaction we also use a folded potential, here in the form Vc,(r) = f dazV(r + x)pc(x). For the wave function of the proton or neutron in a single-nucleon halo, we solve the SchrSdinger equation in a Wood Saxon potential, where the depth has been adjusted to reproduce the measured binding energy. In the case of the two-neutron halo nuclei 11Li and 6He, we make use of the model of [4]. The wave function is of the form =
E
......
°
(2)
l,j,n,n I
Here the u(r) are the single-particle wave function of the neutron in the Wood-Saxon potential. The coefficients c). are determined by diagonalizing the Hamilton operator including the neutron-neutron interaction, which is described by a density-dependent delta-function. For details, we refer the reader to [2,31. For SHe the strength of the Wood-Saxon potential was adjusted to the known spectrum of 5He. For llLi we have used different strengths for the Wood-Saxon potential for even and odd values of" L. The strength of the odd L potential was adjusted to give a pl/2 resonance at 0.540 keV. The strength for the even L was varied to give ns different models with different s- and p-wave contents in the ground state. Within our model we were only able to get ground-state wave function with dominantly p-wave content. 2. R e s u l t s
For the single-nucleon halo of 11Be and SB tile calculated total cross section for the ditf~rent reactions agree well with experimental results at different energies and for different targets [1]. For the case of the Borromean nuclei, results are given in Table 1. The cross sections for 6He are in good agreement with the experimental results. The large measured cross section for diffraction on a Pb-target is due to the Coulomb excitation, which dominates in this case. For llLi we compare the results of two of our models with the experiment. The raw data were corrected by us to account for tile limited neutron acceptance of the experiment. The agreement for absorption and neutron-stripping is reasonable, but all our models have diffraction cross section much smaller than tile experiment. The longitudinal m o m e n t u m distribution of tile core after stripping or diffraction of the halo-nucleon has been interpreted as a measure of the m o m e n t u m distribution of the
671c
K. Hencken et al./Nuclear Physics A654 (1999) 669c-672c
1000
1.2
800 0.8
/
"X,
"~
g
(A)
/" 0,6
/"
\, \
(B)
600
400
0.4 200 0.2 0 . . . . . . 3380 3390 3400 3410 3420 3430 3440 3450 k I (MeV/c)
0 -150
-100
-50
0 50 Pl (MeV/c)
100
150
Figure 1. (A) The calculated longitudinal momentum distribution of the core of a B e for the full calculation (solid line) and for the transparent limit (dash-dotted line) is compared with experimental results of [7]. the 5He fragment after stripping of a neutron is compared with the experimental results of [6].
single-nucleon halo wave function. In the transparent limit, where one neglects the coretarget interaction, they are proportional to each other. Of course the distortion due to the stripping mechanism has to be taken into account. Fig. I(A) shows the momentum distribution for a b e . Both full calculation and transparent limit agree well with the experiment. This is not true in the case of 8B. The measured width of 82 + 6MeV/c ([9]) agrees well with the width of the full calculation (88MeV/c), but the transparent limit is to wide by almost a factor of two. This is due to the fact that the proton in SB is in a p-state, in contrast to the s-state of the neutron in a b e . The core-target interaction favors the m = 0-state, which has a smaller momentum distribution, as proton and core have a larger transverse separation in this case, whereas the transparent limit weights all the m-states equally. Fig. I(B) shows the calculated (cm) momentum distribution of the 5He fragment after neutron-stripping on 6He. They agree well with the recent measurement [6]. The transparent limit on the other hand was reported to be to wide. For ULi we have studied the relative energy distribution of the 9Li-n system after the stripping of the other neutron. We have calculated the spectrum for different models (as explained above) and compared them with the experimental results after folding with the experimental resolution. In a second approach we have also used realistic parameterization of the form of the spectrum for l = 0, 1 and fitted them to the data. Results are shown in Fig. 2(A). In the first case we get the best agreement for an s-wave content of 35%, in the second case for 65% and for the pl/2-resonance at about 0.3 MeV. Recently the angular correlation between the (cm) m o m e n t u m of the core-neutron system and the relative momentum after stripping was proposed and studied for 6He [8]. This quantity is sensitive to the correlations between the two neutrons in the ground state. Expanding this correlation in the form ~r(A0) ~ 1 + ai cos(A0) + a2 cos2(A0)
(3)
672c
K. Hencken et al. /Nuclear Physics A654 (1999) 669c-672c
2.5E+05
0.35 0.3
}i}::
•
0.25
.~ ta
"
k
......-
L'=O
"
1.5E+05
0.2
(A)
"
2.0E+05
-
.a
0.15
I.OE+05 5. OE+04
0.1 0.05
O.OE+O0 i
0.5
i
1 Ere I (MeV)
..........
i
1.5
-5.0E+04 0
0.5
I
1.5 2 E (MeV)
2.5
Figure 2. (A) Comparison of the experimental relative energy spectrum of i°Li after stripping. The solid line is the best fit within our model, the dotted line for a parameterization of the spectrum. (B) Contributions of different L to the cross section or(A0) are shown.
we get values of at = 0.04, a2 = 2.33, which compare rather well with the experimental result of at ~ 0, a2 = 1.6. In [8] the smallness of al was explained due to destructive interference near the p3/2 resonance of 5tte at 0.9 MeV. Fig. 2(B) shows the contribution of different PL(cos(AO)) to the cross section. The destructive interference around 0.9 MeV is clearly seen.
3. S u m m a r y and Conclusion The Serber model is able to calculate total as well as differential cross section for the different nuclear breakup reactions in good agreement with experiments in most cases. Therefore this model seems to be well suited to study breakup reactions of halo nuclei. The use of the transparent limit (that is, neglecting the core-target interaction), was found to fail in some cases, especially those, where angular m o m e n t a I ¢ 0 are important. Calculations of m o m e n t u m and relative energy distrihutions were compared with experimental results and were used to estimate the s- and p-wave contents of 11Li. Recently it has become possible to make more detailed measurement on halo nuclei and therefore there is a need for more refined theoretical calculations in this field.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.
K. Hencken, G. Bertsch, and H. Eshensen, Phys. l{ev. C54, 3043 (1996). H. Esbensem G. Bertsch, and K. Hencken, Phys. Rev. C56, 305:1 (1997) G. Bertsch, K. Hencken, H. Esbensen, Phys. Rev. C57, 1366 (1998) G . F . Bertsch and H. Esbensen, Ann. Phys. 209, 327 (1991). M. Zinser el; al.: Nucl. Phys. A619, 151 (1997) Aleksandrov et al., Nuch Phys. A6aa, ~.34 (1998). J . H . Kelley et al., Phys. Rev. Lett. 74, 30 (1995). L . V . Chulkov and G. Schrieder, Z. Phys. A359, ?.31 (1997) W. Schwab et al., Z. Phys. A350, 285 (1995).
Nuclear Physics A654 (1999) 669c-672c
www.elsevier.nl/locate/npe
The Nuclear Breakup of Halo Nuclei through Diffraction and Stripping K. Hencken ~ , G. Bertsch u and H. Esbensen c ~Institut fiir Physik, Universit/~t Basel, Basel, Switzerland blnstitut for Nuclear Theory, U Washington, Seattle, USA CPhysics Department, Argonne National Laboratory, Argonne, USA We calculate the different nuclear breakup cross sections (diffraction, stripping and absorption) within the Serber model for single-nucleon as well as two-neutron halos (socalled Borromean nuclei). In contrast to calculations up to now, we use realistic wave functions as well as a realistic model for the interaction with the target. We show results of our calculations for total as well as differential cross sections. 1. I n t r o d u c t i o n Halo nuclei are a new kind of nuclei that exist close to the proton or neutron drip line. The breakup of these nuclei through Coulomb excitation and nuclear interaction are two of the main experimental methods used to study their properties. Here we concentrate on reactions with light targets, which are dominated by the nuclear interaction. We have used in this work as far as possible a realistic reaction mechanism and realistic wave functions. Total as well as differential cross section can therefore be calculated at the same time. The applicability of some commonly used approximation was tested in this way. The distortion of the ground state properties due to the reaction mechanism was studied. This is essential to understand whether and how the measured quantities relate to the ground state properties of the halo nuclei. The reaction model we use is the Serber model. The interaction of the target with each "particle" - that is, each halo-nucleon and the core - is described by a profile function S(b), depending only on the individual impact parameter b. The different reactions are characterized by the remaining "particles" in the final state: In diffraction all "particles" are present, in the different stripping reactions, either a halo nucleon or the core is absorbed, in absorption the whole halo nucleus is absorbed [1]. The profile function is calculated in the eikonal approximation as
where V is the interaction potential and we integrate along the beam axis. At low energies we use an optical potential for V with both real (elastic) and imaginary (reactive) parts. At high energies only the imaginary part is important and we use a folding approach, giving a potential of the form V(r) = where aNN is the free NN-cross
--i~-c~NNpr(r),
0375-9474/99/$ see front matter © 1999 Elsevier ScienceB.V. All rights reserved.
K. Hencken et al./Nuclear Physics A654 (1999) 669c-672c
670c
Table 1 Total cross section for the different processes are given for 6He and llLi and are compared with experiment [4,5]. p89 and s23 are two different models with 5% and 23% s-wave component in the ground state. 6He 11Li C (th) C(ex) Pb(th) Pb(ex) p89 s23 exp. corrected diffraction 32.3 304-5 127 6504-110 27 33 604-20 94 in-stripping 136 127+14 332 3204-90 121 137 1704-20 175 2n-stripping 16.7 334-23 69 1804-100 6 9 504-10 11 2n-removal 185 1904-18 528 11504-90 154 179 2804-30 280
section and pT(r) the target density. For the core-target interaction we also use a folded potential, here in the form Vc,(r) = f dazV(r + x)pc(x). For the wave function of the proton or neutron in a single-nucleon halo, we solve the SchrSdinger equation in a Wood Saxon potential, where the depth has been adjusted to reproduce the measured binding energy. In the case of the two-neutron halo nuclei 11Li and 6He, we make use of the model of [4]. The wave function is of the form =
E
......
°
(2)
l,j,n,n I
Here the u(r) are the single-particle wave function of the neutron in the Wood-Saxon potential. The coefficients c). are determined by diagonalizing the Hamilton operator including the neutron-neutron interaction, which is described by a density-dependent delta-function. For details, we refer the reader to [2,31. For SHe the strength of the Wood-Saxon potential was adjusted to the known spectrum of 5He. For llLi we have used different strengths for the Wood-Saxon potential for even and odd values of" L. The strength of the odd L potential was adjusted to give a pl/2 resonance at 0.540 keV. The strength for the even L was varied to give ns different models with different s- and p-wave contents in the ground state. Within our model we were only able to get ground-state wave function with dominantly p-wave content. 2. R e s u l t s
For the single-nucleon halo of 11Be and SB tile calculated total cross section for the ditf~rent reactions agree well with experimental results at different energies and for different targets [1]. For the case of the Borromean nuclei, results are given in Table 1. The cross sections for 6He are in good agreement with the experimental results. The large measured cross section for diffraction on a Pb-target is due to the Coulomb excitation, which dominates in this case. For llLi we compare the results of two of our models with the experiment. The raw data were corrected by us to account for tile limited neutron acceptance of the experiment. The agreement for absorption and neutron-stripping is reasonable, but all our models have diffraction cross section much smaller than tile experiment. The longitudinal m o m e n t u m distribution of tile core after stripping or diffraction of the halo-nucleon has been interpreted as a measure of the m o m e n t u m distribution of the
671c
K. Hencken et al./Nuclear Physics A654 (1999) 669c-672c
1000
1.2
800 0.8
/
"X,
"~
g
(A)
/" 0,6
/"
\, \
(B)
600
400
0.4 200 0.2 0 . . . . . . 3380 3390 3400 3410 3420 3430 3440 3450 k I (MeV/c)
0 -150
-100
-50
0 50 Pl (MeV/c)
100
150
Figure 1. (A) The calculated longitudinal momentum distribution of the core of a B e for the full calculation (solid line) and for the transparent limit (dash-dotted line) is compared with experimental results of [7]. the 5He fragment after stripping of a neutron is compared with the experimental results of [6].
single-nucleon halo wave function. In the transparent limit, where one neglects the coretarget interaction, they are proportional to each other. Of course the distortion due to the stripping mechanism has to be taken into account. Fig. I(A) shows the momentum distribution for a b e . Both full calculation and transparent limit agree well with the experiment. This is not true in the case of 8B. The measured width of 82 + 6MeV/c ([9]) agrees well with the width of the full calculation (88MeV/c), but the transparent limit is to wide by almost a factor of two. This is due to the fact that the proton in SB is in a p-state, in contrast to the s-state of the neutron in a b e . The core-target interaction favors the m = 0-state, which has a smaller momentum distribution, as proton and core have a larger transverse separation in this case, whereas the transparent limit weights all the m-states equally. Fig. I(B) shows the calculated (cm) momentum distribution of the 5He fragment after neutron-stripping on 6He. They agree well with the recent measurement [6]. The transparent limit on the other hand was reported to be to wide. For ULi we have studied the relative energy distribution of the 9Li-n system after the stripping of the other neutron. We have calculated the spectrum for different models (as explained above) and compared them with the experimental results after folding with the experimental resolution. In a second approach we have also used realistic parameterization of the form of the spectrum for l = 0, 1 and fitted them to the data. Results are shown in Fig. 2(A). In the first case we get the best agreement for an s-wave content of 35%, in the second case for 65% and for the pl/2-resonance at about 0.3 MeV. Recently the angular correlation between the (cm) m o m e n t u m of the core-neutron system and the relative momentum after stripping was proposed and studied for 6He [8]. This quantity is sensitive to the correlations between the two neutrons in the ground state. Expanding this correlation in the form ~r(A0) ~ 1 + ai cos(A0) + a2 cos2(A0)
(3)
672c
K. Hencken et al. /Nuclear Physics A654 (1999) 669c-672c
2.5E+05
0.35 0.3
}i}::
•
0.25
.~ ta
"
k
......-
L'=O
"
1.5E+05
0.2
(A)
"
2.0E+05
-
.a
0.15
I.OE+05 5. OE+04
0.1 0.05
O.OE+O0 i
0.5
i
1 Ere I (MeV)
..........
i
1.5
-5.0E+04 0
0.5
I
1.5 2 E (MeV)
2.5
Figure 2. (A) Comparison of the experimental relative energy spectrum of i°Li after stripping. The solid line is the best fit within our model, the dotted line for a parameterization of the spectrum. (B) Contributions of different L to the cross section or(A0) are shown.
we get values of at = 0.04, a2 = 2.33, which compare rather well with the experimental result of at ~ 0, a2 = 1.6. In [8] the smallness of al was explained due to destructive interference near the p3/2 resonance of 5tte at 0.9 MeV. Fig. 2(B) shows the contribution of different PL(cos(AO)) to the cross section. The destructive interference around 0.9 MeV is clearly seen.
3. S u m m a r y and Conclusion The Serber model is able to calculate total as well as differential cross section for the different nuclear breakup reactions in good agreement with experiments in most cases. Therefore this model seems to be well suited to study breakup reactions of halo nuclei. The use of the transparent limit (that is, neglecting the core-target interaction), was found to fail in some cases, especially those, where angular m o m e n t a I ¢ 0 are important. Calculations of m o m e n t u m and relative energy distrihutions were compared with experimental results and were used to estimate the s- and p-wave contents of 11Li. Recently it has become possible to make more detailed measurement on halo nuclei and therefore there is a need for more refined theoretical calculations in this field.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.
K. Hencken, G. Bertsch, and H. Eshensen, Phys. l{ev. C54, 3043 (1996). H. Esbensem G. Bertsch, and K. Hencken, Phys. Rev. C56, 305:1 (1997) G. Bertsch, K. Hencken, H. Esbensen, Phys. Rev. C57, 1366 (1998) G . F . Bertsch and H. Esbensen, Ann. Phys. 209, 327 (1991). M. Zinser el; al.: Nucl. Phys. A619, 151 (1997) Aleksandrov et al., Nuch Phys. A6aa, ~.34 (1998). J . H . Kelley et al., Phys. Rev. Lett. 74, 30 (1995). L . V . Chulkov and G. Schrieder, Z. Phys. A359, ?.31 (1997) W. Schwab et al., Z. Phys. A350, 285 (1995).