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JAERI-Conf--97-007
JAERI-Conf97-007
JP9706040
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Japan Atomic Energy Research Institute
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This report is issued irregularly.
Inquiries about availability of the reports should be addressed to Research Information
Division, Department of Intellectual Resources, Japan Atomic Energy Research Institute, Tokai-mura, Naka gun, Ibaraki-ken, 319-11, Japan.
© Japan Atomic Energy Research Institute, 1997
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JAERI-Conf 97-007
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JAERI-Conf 97-007
Proceedings of the Fourth Symposium on Simulation of Hadronic Many-body System January 23 and 24,1977, JAERI, Tokai, Japan
(Eds.) Akira IWAMOTO and Toshiki MARUYAMA
Advanced Science Research Center
Japan Atomic Energy Research Institute
Tokai-mura, Naka-gxm, Ibaraki-ken
(Received April 24, 1997)
The fourth symposium on Simulation of Hadronic Many-Body System, organized by the Research
Group for Hadron Transport Theory, Advanced Science Research Center, was held at Tokai Research Establishment of JAERI on January 23 and 24,1997. The theme of the symposium was the
simulation study of light- and heavy-ion induced nuclear reactions but the reports on heavy-ion
nuclear reaction experiments, large-scale variational study of nuclear structure, the simulation
study of liquid flow and a theory of microcluster collision were also included as important related
topics. Twenty-two papers on current topics presented at the symposium aroused lively discus
sions among forty participants from universities, institutues, industries and JAERI.
Keywords: Proceedings, Simulation, Molecular Dynamics, Hadronic Many-body System
Organizers: A. Iwamoto, T. Kido, S. Chiba, Tbshiki Maruyama, TbmoyuM Maruyama, (Hadron Transport Group,
Advanced Science Research Center, JAERI), K Niita (Research Organization for Information Sci
ence & Technology), Y. Nara (Hokkaido University), A. Ono (Tohoku University) and H. Horiuchi
(Kyoto University)
JAPAN ATOMIC ENERGY RESEARCH INSTITUTE
t 319-11 TOKAI-MURA, NAKA-GUN, IBARAKI-KEN, JAPAN.
JAERI-Conf 97-007
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JAERI-Conf 97-007
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IV
JAERI-Conf 97-007
Contents
1. Quantum Fluctuation Effects on Nuclear Fragment and Atomic Cluster Formation....................1
Akira Ohnishi (Hokkaido Univ.)
2. Extension of Variational Space in the Antisymmetrized Molecular Dynamics.............................. 6
Yuichi Hirata (Hokkaido Univ.)
3. Tree-fold Loop Approximation of AMD..................................................................................................13
Akira Ono (Tohoku Univ.)4. Analytic Continuation of Bound States to Solve Resonance States................................................ 19
Norimichi Tanaka (Niigata Univ.)
5. Application of Stochastic Variational Method to 3-4 Body Systems with Realistic
Nuclear Force........................................................................................................................................... 23Yoshihide Ohbayashi (Niigata Univ.)
6. Quantum Monte Carlo Diagonalization Method as a Variational Calculation............................. 27
Takahiro Mizusaki (Univ. Tokyo)
7. QMD Application of Sub-saturated Nuclear Matter..........................................................................33
Toshiki Maruyama (JAERI)
8. Coulomb Dissociation of Light Unstable Nuclei................................................................................. 40Toshihiko Kido (JAERI)
9. Simulation of Sub-barrier Fusion Process Including Dynamical Deformation............................. 44
Kentaro Hata (JAERI)
10. A Way for Synthesis of Superheavy Elements.................................................................................. 49
Masahisa Ohta (Konan Univ.)11. Search for the Neutron-rich Nuclei in RIKEN-RIPS........................................................................55
Masahiro Notani (Univ. Tokyo)12. Numerical Simulation of 3-dimensional Rayleigh-Benard System by Particle Method............60
Tadashi Watanabe (JAERI)13. Collision of Highly Charged Ion with Clusters :
Simulation Study for Electronic Systems..........................................................................................64
Kazuhiro Yabana (Niigata Univ.)14. Incident Energy Dependence of the Transverse Flow: from SIS/GSI to SPS/CERN................. 68
Yasushi Nara (JAERI)
15. Nuclear Multifragmentation Experiment at the KEK12 GeV PS: The First Results
of the KEK-PS E337 Experiment........................................................................................................74
Kazuhiro Tanaka (KEK)16. Freeze out Temperature on Light Projectile Induced Reaction......................................................77
V
JAERI-Conf 97-007
Jiro Murata (Kyoto Univ.)
17. Parallel Processing of Cascade Simulation for Ultra Relativistic Heavy-ion Collisions........... 82
Kenji Kumagai (Hiroshima Univ.)18. Time-evolution of Dense Hadronic Matter in High Energy Heavy Ion Reactions......................83
Naohiko Otuka (JAERI)
19. Clustering Effect of 12C Fragmentation in p+12C, a +12C and 14N+12C Reactions........................88
Hiroki Takemoto (Kyoto Univ.)
20. The Isovector/isoscalar Ratio for the Imaginary Part of the Medium-energy NucleonOptical Model Potential Studied by the Quantum Molecular Dynamics.....................................93
Satoshi Chiba (JAERI)21. Status of Semi-classical Distorted Wave (SCDW) Model.............................................................. 104
Yukinobu Watanabe (Kyushu Univ.)
22. Photo Nuclear Reactions by QMD.....................................................................................................114
Tomoyuki Maruyama (JAERI)
VI
JAERI-Conf 97-007
1. Quantum Fluctuation Effects on Nuclear Fragment and Atomic Cluster Formation
Akira Ohnishi ° and Jprgen Randrup b
a. Department of Physics, Hokkaido University, Sapporo 060, Japan b. Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA
We investigate the nuclear fragmentation and atomic cluster formation by means of the recently proposed quantal Langevin treatment. It is shown that the effect of the quanta! fluctuation is in the opposite direction in nuclear fragment and atomic cluster size distribution. This tendency is understood through the effective classical temperature for the observables.
1 Introduction
Molecular dynamics presents a powerful tool for elucidating both statistical and dynamical properties of mesoscopic systems. While quantitative insight can be obtained in many cases, the foundation and interpretation of such approaches can be problematic when quantum systems are addressed, since the energy fluctuations are necessarily present in wave packet wave functions whose effects are neglected in molecular dynamics.
We have shown this quantal energy fluctuations significantly affect the statistical properties of nuclei [1, 2], and that effect can be included in dynamical treatments by means of a quantal Langevin force. This quantal Langevin force distributes the ensemble according to the probabilities (exp(~(3H)) and (6(E - H)) in the canonical and microcanonical cases, respectively, within the harmonic approximation [2], while the probabilities are exp(—(3{H)) and 6(E — (H)) with the normal treatment.
In this short report, we apply this quantal Langevin model to the nuclear fragmentation and the atomic cluster formation of noble gases. These two processes have been extensively studied by using molecular dynamics, although the role of quantum effects is different. While atomic nuclei are highly quantal objects, atomic clusters are believed to be described by classical dynamics.
2 Quantal Langevin Model
2.1 Quantal Langevin Equation
We first give a condensed description of the recently introduced quantal Langevin model for the situation when the system can be regarded as being in thermal equilibrium at a given temperature.
The treatment seeks to take account of the energy fluctuations present in a system being described in terms of many-body wave packets. As we have already discussed in detail in Ref.[2], this inherent energy dispersion modifies the statistical weight relative to the naive the classical form,
W0(Z) = (Z\exp(-PH)\Z) / exp(-PH) ■ (1)
Here H = {Z\H\Z) is the expectation value of the Hamiltonian in the given wave-packet state |Z) and thus the last quantity represent the usual classical statistical weight. The complex parameter
1
JAERI-Conf 97-007
set Z = {zi, z2,... za} is related to the phase space coordinates, zn = rn/2Ar + ipn/2Ap, where Ar and Ap are widths of wave packet. By invoking the harmonic approximation, it is possible to obtain a good description of the statistical weight by means of a simple “free energy”,
T0{Z) = -logW^Z) = ^ (1 - exp(-/?Z?)) , (2)
where D = (t%/E* is the effective level spacing. (The energy of the wave packet relative to its ground state is denoted by E* and <r| is the associated variance.)
The relaxation towards this approximate quanta! equilibrium can be described by the following Fokker-Planck equation for the distribution <f>(Z) of wave-packet parameters,
D(f>~Dt <t>, Vi = -VM,
dE«(3)
where g, represents either rn or pn. It is easy to check that the statistical equilibrium distribution, <f>eq = exp(-Ep), is a stationary solution to the above Fokker-Planck equation. Moreover, when the classical statistical weight is employed (i.e. when Ep = 0H), the drift and the diffusion coefficients of the Fokker-Planck equation satisfy the usual Einstein relation, corresponding to a = 1 in (4). On the other hand, when the quanta! statistical weight obtained with the harmonic approximation is used, eq. (2), the relation is modified. For example, if the effective level spacing D does not depend strongly on the wave-packet parameters, the drift coefficient reduced by the factor a,
Q-TJVi = -aP^Mij— , a = 1 - exp (-(3D)
Jb (4)
Since a is smaller than unity, the resulting Fokker-Planck equation gives smaller friction, thus in effect relatively larger fluctuations will arise.
It is convenient to solve the Fokker-Planck transport equation by means of a Langevin method. Within the framework of QMD the Langevin equation becomes
p = f - a/3Mp -(v-u) - j3Mp u + gp ■ tP , (5)r = v + a/3Mr f + gT ■ ? , (6)m 97"i
v ~ ’ , Mp = gp gp , M = g ■ g • (7)
Here r and p are the phase-space centroid parameters for the wave packet, £ is used to denote random numbers drawn from a normal distribution with a variance equal to two, and u is a local collective velocity. In these equations, we have omitted the diffusion-induced drift term and that part of the mobility tensor that connects r and p.
2.2 Thermal Distortion and Observation
In addition to modifying the statistical weight, the energy fluctuation also modifies the meaning of wave packet ensemble, since it causes a thermal distortion of the spectral strength distribution of the energy eigencomponents within each wave packet. The distortion operator exp(—(3H/2) reduces the expectation value of the Hamiltonian in the particular state |Z). The thermal distortion is calculated by replacing the time t by the imaginary time ir in the equation of motion. The resulting “evolution” is then described by a cooling equation,
dpn 2Ap2 drn 2A r2"57------ "57 = ~k~ '
with which the state should be propagated until r = h/3/2. Here, v is again replaced by v — u in order to leave the collective (or cluster) motion unaffected.
- 2 -
JAERI-Conf 97-007
Nuclear Mass Spectra In Box (p=0.012) Cluster Mass Spectra in Box
Grandcan. Q. L. C. L.
T=10 MeV
T=0.9eT=9 MeV
T=6 MeV
1=7 MeV
T=6 MeV
T=0.6e
T=0.5e
80 1000 20 3Fragment Mass Cluster Mass
Figure 1: Nuclear (left) and cluster (right) mass distribution at given temperatures in a box. Solid circles and open triangles show the results of the simulation with the quantal and classical Langevin force. In nuclear case, fragment grandcanonical calculation is also shown (solid line).
3 Application to Fragment Formation Processes
In this section, we show the calculated results of nuclear fragmentation [3] and atomic cluster formation [4] processes by means of Langevin models with and without quantal fluctuations. In a Langevin model without quantal fluctuations, the classical Einstein relation is kept (o = 1), and the thermal distortion does not exist.
In Fig. 1, we show the nuclear fragment and atomic cluster mass distributions at given temperatures. In the nuclear case, we put 40 nucleons in a box with periodic boundary condition, and quanta! or classical (normal) Langevin force is included in the Quantum Molecular Dynamics (QMD) model. In the atomic case, the dynamics of 100 argon atoms in a box interacting via Lennard-Jones potential is simulated.
It is clear that the quantal fluctuation effect on the atomic cluster mass distribution is opposite to that on nuclear fragmentation. Namely, the inclusion of the quantum Langevin force tends to produce more heavy fragments in the nuclear case, and vice versa in atomic cases.
These features are intuitively understood by considering the corresponding effective classical temperature. The effective temperature can be estimated by means of the Einstein relation as the square of the diffusion coefficient divided by the drift coefficient. In the case of atomic clusters, the distances of atoms are much larger than the wave packet width, then the thermal distortion does not modify the cluster configuration. Then the corresponding effective temperature can be obtained from Eq. (5),
reff = VJL = 1 = Df( 1 - e~D'T) >T. (9)av a
This expectation is indeed borne out, as shown in Fig. 2 where we compare the cluster mass distribution obtained with the quantal model at T — 0.5c to the result of the classical treatment carried out at the corresponding effective temperature TeR = 0.62c. The quantitative similarity between the two distributions is remarkable and supports the above discussion.
— 3 —
JAERI-Conf 97-007
Q. L. (T =0.5 t],=0.62e;
p=0.025 o'
Cluster Mass
Figure 2: The cluster mass distribution obtained with either the quanta! Langevin model at T — 0.5c (solid circles) or the classical Langevin model at the corresponding effective temperature Teff = 0.62c (open triangles), at the density p — 0.025 cr~3.
On the other hand, the thermal distortion strongly modify the nucleon configuration in nuclear fragmentation, and the above discussion does not hold. In order to illustrate this feature, we consider here the evolution of the distorted momentum (i.e. the solution to Eq. (8)) which is givenby
Pi(0 = Pi(t,T=^) = e~D/2T Pi(t) . (10)
Thus, in the rest frame of the nuclear fragment, the distorted momenta of the constituent nucleons are governed by a modified Langevin equation,
p7 = e~D/2Tf - auM2 ■ p' + e~Dt2TVvTM • £ . (11)
We can again invoke the Einstein relation and extract an effective temperature for the intrinsic cluster motion,
Tin = e~D/JjT = D/{eDIT - 1) < T . (12)
It has been shown that calculations with classical molecular dynamics at this equivalent temperature Tgff yields results that are very similar to the exact quantal results for the real temperature T, for non-interacting particles in a harmonic potential [2, 5, 6, 7],
In nuclear fragmentation, the normal Langevin model at this effective temperature T'ea gives a similar fragment mass distribution to the quantal Langevin model at T, except the region around the critical temperature. At around the critical temperature, the system is mechanically unstable, and a small fluctuation induces a large difference after the thermal distortion. At higher and lower temperatures than the critical one, the system is mechanically stable, and the above discussion approximately holds.
4 Summary
In the present report, we have applied a recently developed quantal Langevin model to systems of nucleons and argon atoms in thermal equilibrium. The basic features of the quantal Langevin model can be summarized as larger fluctuations(a < 1) and the thermal distortion. The combination of these two points appears as different effects in nuclear fragmentation and atomic cluster formation. In the atomic case, the distortion effects are small and the effective classical temperature becomes higher than the actual temperature. Namely, quantum fluctuations gives a steeper slope in the size
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JAERI-Conf 97-007
distribution. In the nuclear case, however, the distortion strongly modifies nucleon configurations, then the quantum fluctuation enhances fragment yield.
As it was pointed out at this meeting, the quantal fluctuation effect shown in this work seems to be too large in the atomic case, since the distance between atoms is much larger than the width of the wave packet and the classical dynamics is believed to be valid. However, a small width in r-space leads to a large width in p-space. Therefore, the associated energy fluctuation can be non-negligible. In our estimate, the ’’level spacing” is D ss 0.2265 £, where c is the depth of the Lennard-Jones potential. Compared with the critical (classical) temperature at low densities (ss 0.6c), this value is far from negligible. This may be related to the treatment of cluster intrinsic degrees of freedom in Ref. [8]. Ikeshoji et al. treat the cluster-intrinsic degrees of freedom in a different way from the cluster-translational motion. Therefore, we expect that there is still room for taking account for the quantal fluctuation in atomic dynamics.
References
[1] A. Ohnishi and J. Randrup, Phys. Rev. Lett. 75 (1995), 596.
[2] A. Ohnishi and J. Randrup, Ann. Phys. 253(1997), 279; E-print nucl-th/9604040.
[3] A. Ohnishi and J. Randrup, Phys. Lett. B (in press); E-print nucl-th/9611003.
[4] A. Ohnishi and J. Randrup, Phys. Rev. A (in press); E-print cond-mat/9612214.
[5] A. Ohnishi and J. Randrup, Nucl. Phys. A565 (1993), 474.
[6] J. Schnack and H. Feldmeier, Nucl. Phys. A601 (1996), 181.
[7] A. Ono and H. Horiuchi, Phys. Rev. C53 (1996), 2341.
[8] T. Ikeshoji, B. Hafskjold, Y. Hashi, and Y. Kawazoe, Phys. Rev. Lett. 76 (1996), 1792.
— 5 —
JAERI-Conf 97-007
2.
¥E a—, f“, *s mm b=
a. Department of Physics, Faculty of Science, Hokkaido UniversitySapporo 060, Japan
b. Advanced Science Research Center, Japan Atomic Energy Research Institute,Tokai, Ibaraki, 319-11, Japan
c. Department of Social Information, Sapporo Gakuin University,Ebetsu 069, Japan
Abstract
(amd)AMD \Z£-?X. 12C \Z/<-7
Ltz0 $£5f5<D AMD ttiMVtz AMD <7)®Uk~-£whfr\zi-*0
1 Introduction
lx ztzmmmi' ^ i/- 3 >mm [1] zm^xi2ch U iFFWiSS^istt-S. [2]
(Dl/yff A/ • 7A/ • A>f Ax—[3] dynamical fragmentation
fa3[MeV/A]EJrCfc9. Ctl^TAMD^ Ifc D J£^T^s^/jx$ <, mm\stzw^m'b<Dn^<Di&\mms&<%mi> hnxLt 91V*9 AMD [6,14]o £<D J: 9&{&®Jjgx*;l/4^—-fc V^T > 5 tlX V * <5 E~ + 12C —» + ®Be #(0 fragmentation
D AtLTKmL,^ AMD^iAb,
2 Description of Microscopic Models
#4kL AMD £-~%L:?tf'JX]&$i(DM<D&&&Z%ll9AtlXfanVtz AMD ^Vx 9r.xxDm-otzmmy -n-U-ya y@S$rE¥WiSE^fflv>TB±E- '>
2.1 Initialization
awms/<a,i/-v3 >@m (amd, amd) (Dwn&mtzxFox 91:
* E-mail: [email protected] , Fax: +81-11-746-5444
— 6 —
JAERI-Conf 97-007
H-tt, l2C<D?-u's%iM\m±L, ZCfrbWZtiZo MfirBti, E~t12C Omn: Nimegen model D [4] £<6/£U E~ t12C OZfr<Di/3- Uf4 AMD [1]hfltz12C ‘P<Dm?<D density <D overlap o
T©«fc'5K:ro<OAtt*££JirfS0S"(sds) + p(uud) -> A(sud) + A(sud) + 28.3 MeV
£j££ftfcA)6BHi, 28.3MeV0 Q value 1:ZoCO®it\ loOAtf iEooUf1
6 at a $ns= cn<b w '> uu-'>3 >mm<DvmmfttkZo
2.2 AMD
EiSfSo ar,
X8iM<DXl'-?-mmz£’oX&T<DJ:J)\Z-¥X.htiZ0
$
<Ai
VA!det[0,] ,
= E>3/2
exp —I/,(r — Z,/ y/Vi)2 + -z2 Xi
(1)
(2)
CCX\ xM> spin-isospin tSWBTfc 9 *btv^o 9WTOJ: 3 ce#6o
Zi = y/uldi + (3)2hy/Fimm<D&acDMftm&mL. *,as»<DmB£si-Tv>£o Amom#
d BC dCdt dii dz{ = 0SS — 8 J dt C = 0 <
£ s { *|i»Zj|* )/( *1* ) = iH'£ij}dt ' ; 3 dzj
ct&x%z„
,c, = =Y&rr
(4)
(5)
(6)
- 7
JAERI-Conf 97-007
mmmmw [i] vwM*mtzt~fom^)m\z^x<nnn\m^(Dtz#>mstxTom&vw.d&v°imx&s„
(1) # f 7 # f ® ?
(2) m? + a -> m? + a,
(3) A -F A —v A -j- A,
2.3 Extended AMD
& UT#-) c ^HtDtVC V*5 [6, 8] H.Feldmeier ?><D Fermionic Molecular Dynamics(FMD) [7, 8] X
AMD aiwimic.ftTomUzi,
*-9tbxwLY)WL^x&Y), s*«.
r amdom^vt^x, mmzitmvi &urns*-?tvtzmoZtlZ&mbtzAMD tbXWA'tZZtiZi-Zo
mmvtz amd amd tmmiz^tz^ommmm^-m^
tfX%ZQ
ss
c
mv&nmzmf'tzrtj *~?t vtzmomwiiim&imTiDx? n&s*
, r*a<f / dt C = 0
Jhd
d dC dt dii
dCdii
o L*HL-®£’ dt dv\ di>i = 0
( »| .7^ - H |* )/( *|* )
— ih dlog( *|* ) dlog( )+ "r
di/j!)-n
(7)
(8)
(9)
}
thY.
3Mog( *|* ) . 0!log( *|* ) .dzidzj -Zj + dziduj Vi
d2 log{ ) . d2log( >dvidzj
-zj +dvidvj
ondzi
dHdl>i
(10)
(11)
xommjm&z&zt, e^me* amd m 19xns £muz^mm^tt mvi<Dmm&<ttfm&Lxm<D$iiti)mmzi\zmwzm-tc£tft>frz>0 Ltz^x
l±E<D#%Efk 13J:oT^ffcfSOT. E<DS6Eti'>5^l/-'>3
- 8 -
JAERI-Conf 97-007
3 Interactions
3.1 Zero Point Energy of the C. M. System
AMD JkXfWM\stz AMD T'tt,„ utz^t . \xr
<D£5 K&Son=<H> -Tcm (12)
e$> 9 N AMD T'iiaTO«£ 9 iC£ELTV>50
Tcm =< 7^ > NF =» < 7gM > A - a„ < 1™ > (A - NF) (13)
CCT, <TfaM > fragmentfit -fZtzftKWAVtzrty NF\$ fragment (D&T*6o
$64 li. < TfcM >ZtXT(OX 9 £5E8Ufco
< >=E,
tCMg(”i)J*L 3 h2tf" = p—BtiBJ?}, B„ =< <6,#j > (14)
IV, 2m,- K + ^
mf&ZWL*)Kfltz EQMD [6] SSCU, &AFmT<DJ: ££i8l/fc„
A^f = ni = ^/ij, g(^) = 1 + goe (a*a * , (15), «. >
fij = exp (-i/t|zi-z, |2) (16)
AMD fragment ^ V ;k a LT##L, fragmentation \Z&WZ-$XZ> t^Z-htlX^Z [1] J&>\mmw&mmmwmx'teto'o
3.2 Effective Interaction
UTtix aTO Volkov No. 1 [5]
vnn = (0.35 + mmaj0P„Pr) x (—83.34exp(—0.3906r2) + 144.86exp(—1.487r2)) , (17)
AMD T#ti> mmojoti, 12C <D$E5KS^ shell ^1:^:6 mmaj0 = 0.576 [1] £ 3 Of cluster mmaj0 = 0.65 [9] —
- 9 -
JAERI-Conf 97-007
parameter 1:mmaj0 = 0.576, ap = 0.772, vt = 0.5, g0 = 0
parameter2:mmajo — 0.65, op — 0.772, vt — 0.2, go = 1.8, M — 12, a = 2.0
ttz, AMD httMbtz AMD (D$]ffl%iftX\t, 0.16,
(#f-A, a-a) [9]?
vaa = -67.12 exp(-0.9353r2) , (18)van = -43.622(0.1 + 0.9&?T)exp(-0.9353r2) (19)
4 Results and Discussion
Figure 1: AMD ilffltfUkXstz AMD A*h&htltz fragment distribution
AMD fragment distribution Extended AMD fragment distribution
(majorana=0.576) (majorana=0.65)
parameterf parameters
parameter! -collision parameterS+collision
number of evaporated Lambda number of evaporated Lambda
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- 10 -
JAERI-Conf 97-007
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— 11 —
JAERI-Conf 97-007
References
[1] A. Ono, H. Horiuchi, T. Maruyama, and A. Ohnishi, Prog. Theor. Phys. 87 (1992), 1185; Phys. Rev. Lett. 68 (1992), 2898; Phys. Rev. C47 (1993), 2652.
[2] S. Aoki et al., Prog. Theor. Phys. 85 (1991), 1287.
[3] S. Aoki et al., Phys. Lett. B 355 (1995), 45.
[4] Y. Yamamoto et al., Prog. Theor. Phys. Suppl. 117 (1994), 281.
[5] A. Volkov, Nucl. Phys. 75 (1965), 33.
[6] T. Maruyama et al., Phys. Rev. C 53 (1996), 297.
[7] H. Feldmeier et al., Nucl. Phys. A515(1990), 147.
[8] H. Feldmeier et al., Nucl. Phys. A586(1995), 493.
[9] Y. Nara, A. Ohnichi, and T. Harada, Phys. Lett. 8346(1995), 217
[10] A. Ono, H. Horiuchi and T. Maruyama, Phys. Rev. C48 (1993), 2946.
[11] T. Yamada and K. Ikeda, Prog. Theor. Phys. Suppl 117(1994), 445.
[12] A. Ohnishi and J. Randrup, Phys. Rev. Lett.75(1995), 596.
[13] A. Ohnishi and J. Randrup, Ann. Phys. 253(1997), 279.
[14] A. Ono and H. Horiuchi, Phys. Rev. C(in press)
- 12 -
JAERI-Conf 97-007
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JAERI-Conf 97-007
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JAERI-Conf 97-007
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References[1] A. Ono, H. Horiuchi, Toshiki Maruyama and A. Ohnishi, Phys. Rev. Lett. 68, 2898
(1992); A. Ono, H. Horiuchi, Toshiki Maruyama and A. Ohnishi, Prog. Theor. Phys. 87, 1185 (1992).
[2] A. Ono and H. Horiuchi, Phys. Rev. C53 2958 (1996).
[3] A. Ono and H. Horiuchi, Phys. Rev. C51 299 (1995).
[4] H. Takemoto, H. Horiuchi and A. Ono, Phys. Rev. C54 266 (1996).
[5] E. I. Tanaka, A. Ono, H. Horiuchi, Tomoyuki Maruyama and A. Engel, Phys. Rev. C52 316 (1995).
[6] J. Aichelin, Phys. Rep. 202, 233 (1991).
[7] G. F. Bertsch and S. Das Gupta, Phys. Rep. 160, 189 (1988).
- 18 -
JAERI-Conf 97-007
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JAERI-Conf 97-007
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JAERI-Conf 97-007
####U <£ &%nW:£ complex scaling method H X 2>o
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[1] V. I. Kukulin, V. M. Krasnopol’skii, and M. Miselkhi, Sov. J. Nucl. Phys. 29, 421 (1979)
[2] T. Vertse, K. F. Pal, and Z. Balogh, Comput. Phys. Comman. 27, 309 (1982)
[3] W. H. Press. S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran Second Edition (Cambridge University Press, Cambridge, 1992), p.194
[4] V. I. Kukulin and V. M. Krasnopolskii. J. Phys. G3, 795 (1977)
[5] Iv. Varga and Y. Suzuki, Phys. Rev. C52, 2885 (1995)
[6] N. Tanaka. Y. Suzuki, K. Varga, and K. Aral. Phys.Rev.C to be published
— 21 —
JAERI-Conf 97-007
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JAERI-Conf 97-007
5. &mm$if}T<r> 3-4
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- 23 -
JAERI-Conf 97-007
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- 24 -
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[1] K. Varga and Y. Suzuki, Phys. Rev. C 52 (1995)
2885.
[2] J. Carlson, Phys. Rev. C 36 (1987) 2026., J. Carl
son, Phys. Rev. C 38 (1988) 1879.
[3] W. Glockle, H. Ivamada, H. Witala, D. Hiiber, J.
Golak, Iv. Miyagawa, and S. Ishikawa, Few-Body
Systems. Suppl. 8 (1995) 9.
[4] C. R. Chen, G. L. Payne, J. L. Friar and B. F.
Gibson, Phys. Rev. C 31 (1985) 2266.
[5] R. B. Wiringa, R. A. Smith and T. L. Ainsworth,
Phys. Rev. C29 (1984) 1207.
[6] R. B. Wiringa and V. R. Pandharipande, Nucl.
Phys. A317 (1979) 1.
[7] K. Varga, Y.Ohbayasi and Y.Suzuki. Phys. Lett. B, to be publishd
Argonne V8 potential
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25 -
JAERI-Conf 97-007
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— 26 —
JAERI-Conf 97-007
6. Quantum Monte Carlo Diagonalization method as a variational calculation
Takahiro Mizusaki and Takaharu Otsuka Department of Physics, University of Tokyo, Kongo, Tokyo 113,
and
Michio HonmaCenter for Mathematical Sciences, University of Aizu
Tsuruga, Ikki-machi Aizu-Wakamatsu, Fukushima 965
ABSTACT
A stochastic method for performing large-scale shell model calculations
is presented, which utilizes the auxiliary field Monte Carlo technique and
diagonalization method. This method overcomes the limitation of the con
ventional shell model diagonalization and can extremely widen the feasibility
of shell model calculations with realistic interactions for spectroscopic study
of nuclear structure.
One of the trends in theoretical nuclear structure physics is a stochas
tic approach for nuclear shell model which describes the dynamics of nu
cleons strongly interacting each other via residual two-body interaction.
The conventional shell model diagonalization clearly meets the difficulty
as the dimension of the Hilbert space increases, while, several stochastic
approaches are proposed in order to overcome this limitation. For instance,
- 27 -
JAERI-Conf 97-007
Horoi et al. [1] proposed a truncation scheme with stochastic criteria for
the conventional shell model diagonalization. Koonin and his collabora
tors [2] developed the Shell Model Monte Carlo (SMMC) method based on
the auxiliary field Quantum Monte Carlo method, which turns out to be
a powerful method to extend the feasibility of shell model calculation for
ground state and finite temperature properties. However, this method is
not free of the sign-problem which is a generic problem in Quantum Monte
Carlo method. Consequently it is not enough for investigating spectroscopic
study of nuclear structure. In a glance, Quantum Monte Carlo method and
exact diagonalization seems to be incompatible each other because they are
based on completely different principles. However, we combine advantages
both of methods and develop a new method called Quantum Monte Carlo
Diagonalization (QMCD) method [3]-[7]. In this article, first we outline this
new method and then present the feasibility of the large-scale shell model
calculations by this method.
The ground state energy Eg can be written as
/3->oo (^| e~PH |»)(1)
where H is a Hamiltonian and |#) is an arbitrary wave function which is
not orthogonal to the ground state. The e~@H is a ground state projector
with a sufficient large (3 and the ground state is expressed as,
= lim e~0H |*>. (2)
In general, Hamiltonian consists of the one and two body interactions. The
latter causes the difficulty in treating Quantum many-body problem. If we
treat it as two-body interaction, exact diagonalization is inevitable and the
ground state projection is often realized by the Lanczos method. In turn, if
we can treat the Hamiltonian as an effective one-body interaction, there exist
- 28 -
JAERI-Conf 97-007
two approaches. One is a well-known mean-field approach. The other is an
auxiliary field approach. The SMMC and QMCD methods utilize this aux
iliary field approach, by which can be shown by the sum of the e~^h^
where h(a) is one-body Hamiltonian parameterized by the auxiliary field a.
In the SMMC method, the ground state energy is evaluated by the Monte
Carlo integration over a. As the action of g-W") to Slater determinant
keeps form of Slater determinant, we can count the number of Slater deter
minant in the SMMC calculation. In the typical SMMC one [2], the ground
state energy can be expressed by several thousand Slater determinants. In
a sense, these Slater determinants can be considered to be potentially good
basis for describing the ground state. Then, we proposed the diagonalization
of the Hamiltonian evaluated by these Slater determinants [3]. For better
efficiency of this method, we utilize a mean-field knowledge and consider
stochastic methods and explicit treatment of symmetries. Details of the
QMCD method are presented in Refs [3]-[7].
To what extend the QMCD method can describe exact wavefuactions
is an important problem. We have examined the validity of the QMCD
method for the shell model calculations of the sd-shell nuclei where the
exact solutions are known. This has been reported in ref [5], and is not
reported here. Instead, we then proceed to full pf shell calculations, which is
a crossover region between the conventional shell model diagonalization and
the QMCD calculation. The largest calculation [8] which has been carried
out by the conventional shell model diagonalization is for 48Cr with the
KB3 interaction [9]. Figure 1 shows the energies of several low-lying states
obtained by the QMCD method, conventional shell model diagonalization
[8], and the SMMC method [2], In the results of conventional shell model
diagonalization, different dimensions mean the different truncation schemes:
The maximum number of particles allowed to jump from the /7/2 orbit to
- 29 -
JAERI-Conf 97-007
Fe (fpd6)Cr (KB3-168v -
Exact QMCDQMCD
SMMCHoroi et al.SMMC
500 1000 1500basis dimensionbasis dimension
Figure 1: Energies of lowest three states of 48Cr and 54Fe plotted as a function of the basis dimension. The results of the QMCD method are plotted by lines, In the left figure, the results of the QMCD method are compared to those of the exact and truncated shell model calculations. In the right figure, the results of the QMCD method are compared to those of the stochastic truncation method which are shown by symbols and lines. The SMMC results are also shown by open square with an error bar.
the remaining ones, denoted t, is given in each truncation differently. In this
figure results for £=0,1,2,3,5 are shown as well as the exact result (i.e. t=8).
The SMMC result corresponds to the finite temperature T=0.5MeV, and
is plotted near the exact results since we cannot define the dimension for
the SMMC calculations. We can see that the QMCD method gives energies
with rather good quality by taking only 600 basis states.
Next we compare the QMCD method to other stochastic methods for
54Fe nuclei. Although there is no exact calculation for this nuclei, two
stochastic methods evaluated the ground state energy of 54Fe nuclei with
the FPD6 interaction [10]. One is a method proposed by Horoi et al.[l]
They truncate the shell model basis based on the unperturbed energies of
the basis states and on the constancy of their spreading widths. Conse
quently the JT dimensions of the order of a few times 106 are reduced to a
— 30 —
JAERI-Conf 97-007
few times 103. The other is the SMMC method. As a realistic interaction
includes both good and bad parts for sign-problem, we need an extrapola
tion method for extracting physical quantities. In this case, the estimated
ground state energy is shown within a certain error bar [2], Note that the
QMCD method is free of sign-problem, it can handle any realistic interaction
without any problem. We also evaluate the same nuclei with the same inter
action by the QMCD method. In the present calculation, the M-projected
QMCD basis is utilized [4]. Fig.l shows the results of above three meth
ods. As the lowest energy of the deformed Hatree Fock method in the same
shell model configuration is -167.622 MeV, above three methods are found
to include certain correlations for the ground state. However, in the view of
variation principle, the QMCD method offers the best value among them.
Furthermore, one sees that the QMCD method can describe excited states
too. Although we do not mention in this short article, we can evaluate the
B(E2), Q moment and so on.
In summary, we present the QMCD method, which is a diagonalization
method by the bases generated by the auxiliary field Monte Carlo method.
It is reported that the present QMCD method is superior to other stochas
tic methods. The present method is useful for the large-scale shell model
calculations.
- 31 -
JAERI-Conf 97-007
References
[1] M.Horoi, B.A.Brown and V.Zelevinsky Phys. Rev. C50, R2274 (1994).
[2] S.E. Koonin, D.J. Dean, K. Langanke (1996): to be published in
Phys. Repts; W. E. Ormand, et al., Phys. Rev. C49 (1994) 1422; C.
W. Johnson et al., Phys. Rev. Lett. 69 (1992) 3147; K. Langanke, D.J.
Dean, P.B. Rad ha, Y. Alhassid, S.E. Koonin (1995): Phys. Rev. C52,
718
[3] M.Honma, T.Mizusaki and T.Otsuka, Phys. Rev. Lett. 75,1284 (1995).
[4] T. Mizusaki, M. Honma, T. Otsuka (1996): Phys. Rev. C53, 2786
[5] Otsuka, M. Honma, T. Mizusaki : Proceedings of the Workshop on
Comtemporary Nuclear Shell model, 29-30 April, 1996, Drexel Uni
versity, USA, to be published in Lecture Notes in Physics (Springer-
Verlag).
[6] T. Mizusaki, T. Otsuka, M. Honma : Proceedings of the XXXI Za
kopane School of Physics, 3-11, Sept., 1996, Zakopane, Poland, to be
published in Acta Polinica.
[7] M. Honma, T. Mizusaki, T. Otsuka : Phys. Rev. Lett.77 (1996) 3315.
[8] E. Caurier, A.P. Zuker. A. Poves, G. Martinez-Pinedo (1994):
Phys. Rev. C50, 225; Private communication with Poves.
[9] A. Poves, A. Zuker (1981): Phys.Rep. 70, 235
[10] W.A. Richter, M.G. vanderMerwe R.E. Julies, B.A. Brown (1991):
Nucl.Phys. A523, 325
- 32 -
JAERI-Conf 97-007
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- 33 -
JAERI-Conf 97-007
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— 34 —
JAERI-Conf 97-007
2.4
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- 35 -
JAERI-Conf 97-007
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- 36 -
JAERI-Conf 97-007
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- 37 -
JAERI-Conf 97-007
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- 38 -
JAERI-Conf 97-007
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[1] D. G. Ravenhall, C. J. Pethick and J. R. Wilson, Phys. Rev. Lett. 27 (1983) 2066.
[2] M. Hashimoto, H. Seki and M. Yamada, Prog. Theor. Phys. 71 (1984) 320.
[3] R. D. Williams and S. E. Koonin, Nucl. Phys. A435 (1985) 844.
[4] C. P. Lorentz, D. G. Ravenhall and C. J. Pethick, Phys. Rev. Lett. 25 (1993) 379.
[5] G. Peilert, J. Randrup, H. Stocker and W. Greiner, Phys. Lett. B 260 (1991) 271.
[6] J. Aichelin and H. Stocker, Phys. Lett. B 176 (1986) 14.
[7] J. Aichelin, Phys. Rep. 202 (1991) 233; and references therein.
[8] G. Peilert, J. Konopka, H. Stoker, W. Greiner, M. Blann and M. G. Mustafa, Phys. Rev. C 46
(1992) 1457.
[9] T. Maruyama, A. Ono, A. Ohnishi and H. Horiuchi, Prog. Theor. Phys. 87 (1992) 1367.
[10] K. Niita, S. Chiba, T. Maruyama, T. Maruyama, H. Takada, T. Fukahori, Y. Nakahara and
A. Iwamoto, Phys. Rev. C 52 (1995) 2620.
[11] K. Niita, JAERI-Conf 96-009 (1996) 22.
[12] S. Hama, B. C. Clark, E. D. Cooper, H. S. Scherif and R. L. Mercer, Phys. Rev. C 41 (1990) 2737.
- 39 -
JAERI-Conf 97-007
msfftmiW
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tE-mail:kido@hadron01. tokai.jaeri.go.jp
- 40 -
JAERI-Conf 97-007
n - > V -v v{r) fi
v(r) = K,,(r) + ^/(r)
= 1 + e(r-R)/a
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— 41 —
JAERI-Conf 97-007
3.
8B O^-o
8B +208 Pb —>7 Be + p -f208 Pb E=46.5MeVr/nucleon
Q E2 #@(7)^4-1$ V^ cf3f4^vN □ —^nBe(^N n — ^ -'b ^f~ 503keV) (D ? ~ n y]$b?|SlC jo
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— 42 —
dP/d
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[MeV
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JAERI-Conf 97-007
0.015 -
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0.005
0.0000.015
0.010
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0.000
Energy Dist. of Break-up Prob.-i---1---1---1---[—i---1---1--- f---1---1---1---1---r
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P0u(QuadnjpoJe) = 0.029
Relative Momentum dist. (b=12fm)
0.15 -
- e 0.10
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0.00
E [MeV] k [fm1]
[1] T. Kido, K. Yabana, and Y. Suzuki, Phys. Rev. C50 (1994) R1276.[2] T. Kido, K. Yabana, and Y. Suzuki, Phys. Rev. C53 (1996) 2296.[3] K. Ieki et al., Phys. Rev. Lett. 70 (1993) 730.[4] T. Nakamura et al, Phys. Lett. B331 (1994) 296.[5] T. Motobayashi et al., Phys. Rev. Lett. 73 (1994) 2680.[6] T. Kikuchi et al., Phys. Lett. B391 (1997) 261.
- 43 -
JAERI-Conf 97-007
9. >b mm±m
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44 —
JAERI-Conf 97-007
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zkT= @ 3 (C d tl 6 Vm}n(z) £ E o XthS £ tlfcfolded cross section (C HIT 5 HI!&zmmmtmzTFto 6(5)iU: =kx^ ^6avoidedcross section, SB12 '> ^ zl 1/ - 7 3 > tC <£ -3 X# 6 tlXcIfSfEX &5„ wKBm^EoxtfSu^^^Exsu^o a^icTK^nxv^^u, fimmm ^#(DH#(D7l;f->v^;i/|$E(Df&@^^LXV^6. £fc®4,l;:7H$nX^£Eshfl^ #^EE(DSMdlf#B<7)^£^lET5£&t;:&$nfcx*;i/^-77 MZ)EX&5 = m M&X&Z40Cad58N{> j3J:U(lt^^S(7)/h$V^4Ni(DESM^'7V^Xti, 1 MeV^ r<DX^;i/^-'>7 hx>^al/-ya 7tc=t;5ff#m2SiM£¥ifS6:< SSLXi^6.
— 45 —
JAERI-Conf 97-007
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1) J. Schneider and H.H. Wolter: Z.PhysA 339 (1991) 177-188.2) P. Moller and A Iwamoto: Nucl.Phys. A575 (1994) 381-411.3) P. Moller, J.R.Nix, W.D.Myers and WJ.Swiatecki: Atomic Data and Nuclear Data
Tables 59 (1995) 185-381.4) E.Thomasi, et al.: Nucl.Phys. A373 (1982) 341-348.5) M.Beckerman, et al.: Phys.Rev. C25 (1982) 1581-1589.6) M.Beckerman, et al.: Phys.Rev. C25 (1982) 837-849.7) M.Beckerman, et al.: Phys.Rev. C28 (1983) 1963-1969.
Mf#
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JAERI-Conf 97-007
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JAERI-Conf 97-007
10. A WAY FOR SYNTHESIS OF SUPERHEAVY ELEMENTS
M. Ohta, Y. Aritomo, T. Wada, and Y. Abe*
Department of Physics, Konan University, Kobe 658, Japan * YITP, Kyoto University, Kyoto 606-01, Japan
Abstract
Fusion-fission process in heavy systems are analyzed by Smoluchowski equation taking into account the temperature dependent shell correction energy. The evaporation residue cross sections of superheavy elements have been shown to have an optimum value at a certain temperature, due to the balance between the diffusibility for fusion at high temperature and the restoration of the shell correction energy against fission at low temperature. The essential element which realize an significant yield for the (HI, 4-5n) reaction in superheavy mass region is found to be the characteristic time for cooling by neutron evaporation.
IntroductionRecent several years, new heavy elements have been synthesized by the so called cold
fusion reaction [1] in which the target of Pb or Bi isotopes near doubly magic nucleus are bombarded by Ni or Fe isotopes, and the findings of the heaviest elements from 110 to 112 are reported with the cross section of several picobarns [2], The cold fusion reaction is aimed to obtain a high survival probability against fission, but it suffers a great loss of fusion probability into compound nuclei. On the other hand, a symmetric target-projectile combination can produce a rather cold heavy compound nucleus due to the interplay of the potential barrier and the Q-value [3], But it turned out that there is a fusion hindrance, i.e., a necessity of extra- push energy [4]. Therefore, also in those systems, there are the conflicting requirements of high fusion probability and high survival probability. The purpose of the present work, thus, is to find out an optimum condition compromising the two requirements for synthesis of superheavy elements with massive target-projectile combinations [5, 6, 7].
Since there is no pocket around the spherical shape in the potential of the liquid drop model, and thereby no barrier, there is no formula for fusion probability, neither for fission decay probability of superheavy elements (remind that Bohr- Wheeler [8] as well as Kramers [9] formulae are not valid for cases without barrier). Therefore, we have to employ a new dynamical description at least for the early stage from the di-nucleus complex to the spherical compound nucleus formation [5, 6] and for its decay before the temperature becomes low enough for the barrier to appear. In this paper, we describe the whole process by dissipative dynamics from the contact of two incident nuclei to the formation of the compound nucleus as well as to the reseparation, namely, fission back into the symmetric fragments. From the analysis of pre-fission neutrons and fragment kinetic energies, a strong dissipation comparable to one-bodv model is recommended [10], which permits us to use Smoluchowski equation for fusion-fission dynamics as an approximation of Kramers or Langevin equation.
With this diffusion model, we can immediately expect that an optimum condition exists for residue cross sections. Qualitatively, in formation process, higher temperature is favorable due to large diffusibility into the compact configuration from the di-nucleus one at contact,
— 49 —
JAERI-Conf 97-007
while in decaying process, lower temperature is favored for larger residue probability because of the higher fission barrier caused by the restored shell correction energy as well as smaller diffusion coefficient. Therefore, a balance between the above two requirements give rise to an optimum temperature or excitation energy of compound system for the synthesis of super heavy elements.
The position of the optimum energy crucially depends upon how fast the cooling due to the neutron evaporation is. If the cooling is fast, so the restoration of the shell correction energy is, and thus the fission barrier arises rapidly to prevent fission decay of the superheavy compound nucleus. On the contrary, if it is slow, so the growth of the fission barrier due to the restoration of the shell correction energy is, and thus the compound nucleus decays through fission according to the barrier height at the initial temperature. In the former case, the evaporation residue cross section is mainly controlled by the formation process, namely higher temperature is preferable. In the latter case, it is mainly controlled by the decaying process, namely lower temperature is preferable. The neutron evaporation width is determined by the neutron separation energy. Thus, it is very interesting to investigate the isotope dependence of the evaporation residue cross section in the present framework.
FormulationThe brief description for the formula used in the present calculation is given following
[5, 6, 7]. The evolution of the probability distribution P(x,l\t) in the collective coordinate space is assumed to follow Smoluchowski equation;
d_dt P(x,l;t)
\__d_ HP dx
dV(x,l\t)dx P(x,l\t) > +
T d2 HP dx2
P(x,l]t). (1)
The coordinate x is defined as x = Rem — |roAly/3 so that x — 0 corresponds to the spherical shape, where Rcm denotes the separation distance between the center-of-mass of the nascent fission fragments in the case of symmetric fission, A the mass number of the nucleus, and z’o =1.16 fm. The angular momentum of the system is expressed by l. Both the inertia mass parameter /z and the reduced friction ft are assumed to be independent of the shape of nucleus in the present calculations. The parameter /t is taken to be the reduced mass for the symmetric separation and p is 5xl021sec-1 corresponding to the weakest value of one-body dissipation in a series of shapes.
The time dependent potential energy curve appeared in Eq. (1) is defined as follows;
!'(*,!;<) = VDM(x) + + Kw,(!)*(<), (2)
where I(x) is the moment of inertia of the rigid body at deformation x. VDM and Vsheii are the potential energy of the finite range droplet model and the shell plus pairing correction energy at T — 0, respectively. Both are calculated with the code developed by Moller [11]. The potential energy curve along the minimum valley is calculated with the e-parametrization [12]. The temperature dependent factor $(t) in Eq. (2) is parametrized as;
4(()=expf-^lj, (3)
following the work by Ignatyuk et aZ.[14], where a denotes the level density parameter of Tolte and Swiatecki [15]. The shell-damping energy Ed is chosen as 20 MeV according to the
— 50 -
JAERI-Conf 97-007
above results. The cooling curve T(t) is calculated by the statistical model code SIMDEC[13]-
Concerning the initial condition, we assume that the kinetic energy of the relative motion in the entrance channel dissipates completely just inside the contact distance. The initial probability distribution P(x,l;t = 0) has a sharp Gaussian shape and is imposed at x0 =Xcom — 0.5 fm, where xcont is the contact distance evaluated as xcont = 2r0 (f) ^ — |r0AV3.
The evaporation residue cross section is defined as the probability which is left inside the fission barrier in the final stage of the cooling process and is proportional to the quantity d(T0, /; t) at t = oo;
/x»adP(x,l;t)dx. (4)
-OO
Here, To is the initial temperature and stands for the first saddle point. The evaporation residue cross section gev = cr(HI, yn) is calculated as;
= 5—F- B2' + Wo, ';< = »), (5)ZllQE/cm I
where fiQ denotes the reduced mass in the entrance channel and Ecm the incident center-of- mass energy.
Excitation function of the evaporation residue cross sectionAs an example of reactions forming the doubly closed superheavy nucleus, we consider
the reaction g49La + ^49La —► 298114. The time dependent feature of the probability d(To, l — 10; t) are plotted in Fig.l for five different incident energies which correspond to To = 0.68, 0.79, 0.96, 1.11, and 1.24 MeV. Up to the time of around 30xl0-21sec, the probability density in the region of the compact configuration is supplied by diffusion from the contact region and its yield increases rapidly. But during that time, the main part of the probability initially at xq has descended down the slope of the potential and thus, the supply ceases. After f~30x 10-21sec, the probability density accumulated in the compact configuration area diffuses back over the fission barrier arising from the restoration of the shell correction energy. At low temperature such as To = 0.68 MeV, 60% of the correction energy is restored and the fission barrier is about 6 MeV. Therefore, the fission width is very small and d(T0,l]t) becomes flat quickly. On the contrary, in the case of To = 1.24 MeV, the restoration takes time for being enough to prevent the system from fissioning, during which the yield accumulated in the compact configuration area diffuses out rapidly as shown in Fig.l.
The height of the peak around 30xl0-21sec is essentially determined by the diffusibility into the compact configuration area, while the decrease from the peak value to the final yield at = 2000xl0-21sec is determined by how fast the fission barrier glows enough by the restoration of the shell correction energy. Thus, the final yield surviving in the compact configuration area is determined by the two factors; the diffusibility depending on the temperature and the restoration of the shell correction energy.
In terms of the obtained values of d(To,/;too), we can calculate the evaporation residue cross section oev with Eq. (5) for several friction parameters; 3 — 2.5 x 1021 sec-1, 5.0 x 1021 sec-1, and 7.5 x 1021 sec-1 [5], The excitation function of gev for the 149La + 149La —► 298114 reaction is shown in Fig.2. The corresponding excitation energy to the Bass potential [16] is indicated by the arrow and it should be emphasized that the optimum cross section can be
- 51 -
JAERI-Conf 97-007
realized fairly above the Bass barrier in this reaction system and thereby can be observed experimentally.
i 1 1 1 I i i i 1 I 1 i i i
I I I I | I ! I I | I I I I J 1 I 1 I
t( 10*1 sec)
FIG. 1 The time evolution of the probability density in the compact configuration region d(To, l — 10; t). The curves for five initial temperatures are plotted; Tq = 0.68 (short- dashed), 0.79 (long-dashed), 0.96 (solid), 1.11 (dot-dashed), and 1.24 MeV (dot- dot-dashed).
FIG. 2 The excitation function of the evaporation residue cross section for ^La + 579La —► 298114 reaction calculated from d(7b,Z;too). Results for three values of reduced friction parameter /? are plotted:/? = 2.5xl021 sec-1 (circles), 5.0xl021 sec-1 (squares), and 7.5xlO21 sec-1 (triangles). The corresponding Bass potential barrier is indicated by the arrow.
Isotope and Z dependenceIt is commonly accepted that the use of neutron rich beam will enhance the evaporation
residue cross section of superheavy elements because of the large Fn/F/ ratio. The smaller neutron separation energy accelerates the cooling by the neutron emission and enhances the survival probability against fission due to the restoration of the shell correction energy.
We calculated the evaporation residue cross section for a series of Z — 114 isotopes from N = 176 to 184. We used different cooling curves for each isotope while we neglect the isotope dependence of the energy surface. Figure 3 shows the calculated evaporation residue cross section for ^5,147,148.149^ + 145,147,148,149^ ^290,294,296,298^24 reactions as functions of initial excitation energy. The isotope dependence of evaporation residue cross section is found to be very strong.
The theoretical neutron separation energies averaged over 4 neutron emissions {En) for the corresponding composite systems, 290,294,296,298224, are 7.0, 6.0, 5.5, and 5.0 MeV, respectively. The cooling curves for these isotopes are shown in Fig.4. It can be seen that the bell shape structure of the excitation function disappears when the neutron binding energy becomes greater than 7.0 MeV. The average neutron separation energy in shown in Fig.5. We can see that the cold fusion and the hot fusion may be separated in this plane by the contours line of 6 or 7 MeV.
A systematic calculation of the evaporation residue cross sections to form the compound nucleus with the atomic number from Z = 102 to Z = 114 through symmetric entrance channel for 8 = 5.0xl021 sec-1 are shown in Fig.6, where the neutron number of the compound nucleus is selected so that the average neutron separation energy becomes about 5 MeV. This means that the cooling curves in these systems are similar each other. We can see
- 52 -
JAERI-Conf 97-007
that the evaporation residue cross sections decrease as the atomic number of the compound nucleus increases. It is prominent, however, that the enhancement around Ex ~ 25 MeV in Z ~ 114 becomes to be distinguished, where the yield of the cross section is reduced up to the pico barns order. This visible enhancement is coming from the strong shell correction energy around the nucleus with Z = 114.
r x
\ 182
• 180•176
FIG. 3 The isotope dependence of the excitation function of the evaporation residue cross section for Z = 114. Figures denote neutron numbers.
* GSI• Our cal.
neutron number
I I I I—|—HTT T-r-r-r
FIG. 4 The cooling curve calculated by the code SIMDEC [13].
A- 106m- 108
! -•©— 112
E'(MeV)
FIG. 5 The average neutron separation energy. The cold fusion is only possible for the experiment of GSI because of the large neutron reseparation energy.
FIG. 6 The excitation function of the evaporation residue cross section that forms the compound nucleus from Z = 102 to Z = 114 through symmetric entrance channel.
- 53 -
JAERI-Conf 97-007
SummaryA diffusion model which takes into account dynamical evolution of a distribution includ
ing statistical fluctuations in the deformation parameter space is shown to be a necessary and appropriate way to describe fusion-fission process for systems without, as well as with, pocket. With the model, it is shown for the synthesis of superheavy elements that there exists the optimum temperature or the excitation energy of compound system due to the balance between the diffusibilitv for fusion and the restoration of the shell correction energy against fission. Roughly speaking, the optimum temperature is around the restoration temperature of the shell correction energy. The absolute value of the cross section, of course, depends on the friction coefficient 7 = p/3 as well as the initial condition, etc. It is confirmed quantitatively that the separation energy of neutron affects the characteristic time of the cooling, therefore the formation of neutron rich composite system is called for decisively in the synthesis of superheavy elements. To get an appreciable yield for the evaporation residue cross section the in hot fusion, the neutron separation energy of the composite system should be less than ~ 6 MeV, i.e. a neutron rich system is called for.
References[1] P. Armbruster, Ann. Rev. Nucl. Sci., 35 (1985) 135;
G. Miinzenberg, Rep. Prog. Phys., 51 (1988) 57.[2] S. Hofmann et al, Z. Phys., A350 (1995) 277; S. Hofmann et al, ibid. 281;
S. Hofmann et al, Z. Phys., A354 (1996) 299.[3] Y.T. Oganessian, Lecture Notes in Physics, 33 (Springer-Verlag, Berlin, 1974) p.221.[4] W.J. Swiatecki, Phys. Scripta 24 (1981) 113;
W.J. Swiatecki, Nucl. Phys., A376 (1982) 275.[5] Y. Aritomo, T. Wada, M. Ohta and Y. Abe, Phys. Rev., C55. in print.[6] Y. Aritomo, T. Wada, M. Ohta and Y. Abe, Proceedings of the RNB4, Omiya, Japan,
Jun.4-7, 1996.[7] Y. Aritomo, T. Wada, M. Ohta and Y. Abe, in preparation.[8] N. Bohr and J.A. Wheeler, Phys. Rev., 56 (1939) 426.[9] H.A. Kramers, Physica (Utrecht), 7 (1940) 284.[10] T. Wada, Y. Abe and N. Carjan, Phys. Rev. Lett., 70 (1993) 3538.[11] P. Mdller et al., Atomic Data and Nuclear Data Tables, 59 (1995) 185.[12] R.W. Hasse and W.D. Myers, Geometrical Relationships of Macroscopic Nuclear Physics
(Springer-Verlag, Berlin, 1988).[13] M. Ohta, Y. Aritomo, T. Tokuda and Y. Abe, Proc. of Tours Symp. on Nuclear Physics
II (World Scientific, Singapore, 1995) p.480.[14] A.V. Ignatyuk, G.N. Smirenkin and A.S. Tishin, Sov. J. Nucl. Phys., 21 (1975) 255.[15] J. Tdke and W.J. Swiatecki, Nucl. Phys., A372 (1981) 141.[16] R. Bass, Nucl. Phys., A231 (1974) 45.
- 54 -
JAERI-Conf 97-007
11.
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## 1$#\ Beaumel Didier\ Sm iE#\ ## #t
6> Sergei Loukianovc x Yuri Penionzhkevichc
aDepartment of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113, Japan
b Institute of Physical and Chemical Research (RIKEN), 2-1 Hirosawa, Wako, Saitama
351-01, Japan
c JINR, 141980 Dubna, Moscow region, Russian Federation
Abstract
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34Ne 0.6 - 3 0.8 - 4.0
36Na 0.4 - 1 0.6 - 1.6
$ 2. Estimated production cross sections and yields
under present experimental conditions with the 181 Ta
target for three nuclei, 33Ne, 34Ne, and 36Na.
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- 57 -
JAERI-Conf 97-007
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- 58 -
JAERI-Conf 97-007
References
1) T. Kubo et al.: Nucl. Instrum. Methods Phys. Res., B70, 309 (1992)
2) H. Sakurai et al.: Phys. Rev. C., in press.
3) M. Robinson et al.: Phys. Rev. C53, 1465 (1996)
4) D. Guillemaud-Mueller et al.: Phys. Rev., C41, 937 (1990).
5) M. Notani et al.: RIKEN Accel. Prog. Rep. 29, 50 (1996)
6) M. Notani et al.: RIKEN Accel. Prog. Rep. 30, 48 (1997)
7) H. Sakurai et al.: RIKEN Accel. Prog. Rep. 30, 49 (1997)
8) J. A. Winger,B. M. Sherrill and D. J. Morrissey: Nucl. Instrum. Methods,B70,380 (1992)
9) Y. Doki: JAERI-M 94-028, p. 70; Master thesis, Tokyo University (1994)
10) Peter. E. Haustein: Atomic Data and Nuclear Data Tables, 39, 185(1988)
11) H. Iwasaki et al.: RIKEN Accel. Prog. Rep. 30, 50 (1997)
59 -
JAERI-Conf 97-007
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[1] Bird G. A.: “Molecular Gas Dynamics and the Direct Simulation of Gas Flows”, (Clarendon, Oxford, 1994).
[2] Watanabe T., Kaburaki H. and Yokokawa M., Phys. Rev. E, 49, 4060(1994).
[3] Chandrasekhar S: “Hydrodynamic and hydromagnetic stability,” (Clarendon, Oxford, 1961).
[4] Watanabe T., Kaburaki H., Machida M. and Yokokawa M., Phys. Rev. E, 52, 1601(1995).
[5] Watanabe T. and Kaburaki. H, Proc. 8th Int. Conf. Phys. Computing, 180(1996).
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- NEAR THE SIDE WALL- CENTER- HORIZONTAL AVERAGE
HYDRODYNAMIC CRITICAL RAYLEIGH NUMBER (Re-1706)
RAYLEIGH NUMBER (-)£
Fig.l. Mid-elevation temperature in the steady state. Fig.2. The average characteristic length.
Fig.3. Steady state temperature distribution at the midelevation: left R=3414, center R=4527, right R=8103.
0.30
V
0.25 ■
0.20
15
0.10
0.05
0.00-1.0
Oe Increase A e decrease
-------tin fitting X
4/a'''v
-----3Z—
-0.5 0.0e
0.5 1.0
Fig.4. Distribution of vertical velocity. Fig.5. Maximum vertical velocity.
— 63 —
JAERI-Conf 97-007
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— 64 —
JAERI-Conf 97-007
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(1)
p(r,t) = Y^\ip(f,t)\2 (2)i
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CwO^gias.sAT&o,
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— 66 —
JAERI-Conf 97-007
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Reference
1. H. Flocard, S. Koonin, M. Weiss, Phys. Rev. C17(1978) 1682.2. B. Walch et.al, Phys. Rev. Lett. 72(1994)1439.3. C. Guet et.al, Phys. Rev. Lett. 74(1995)3784.3. S. Takami, K. Yabana and K. Ikeda, Prog. Theor. Phys. 94(1995) 1011.
10. K. Yabana and G.F. Bertsch, Phys. Rev. B54(1996)4484.5. J.R. Chelikowsky, N. Troullier, K. Wu and Y. Saad, Phys. Rev. B50 (1994) 11355.
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14. Incident Energy Dependence of the Transverse Flow: from SIS/GSI to SPS/CERN
Y. Nara“, N.Ohtsuka", Tomoyuki Maruyamaa, K.Niitaai> Toshiki Maruyama"and A. Ohnishic
“Advanced Science Research Center, Japan Atomic Energy Research Institute b Research Organization for Information Science and Technology
c Division of Physics, Graduate School of Science, Hokkaido University
Abstract
BUU 7 AW TO
1 liDtotC (7n-A'64'(C»'i7bA,5A>)
—fiY Jr -'OCO g 6^Ji±> quark gluon plasma(QGP) 0 ^
yu-iDX^ £ii> & L QGP ft [aUzWl§ (softening of the EOS)^ GAGeV'/c & b t Hf £ [1, 8]c
QGPEOSCollaboration(E895,E866) [2] HZ
I), 400MeV/c^^ 10GeV/c
~ ~Jl ^ yx KO-7 7 (Hadronic cascade, BUU, QMD) H Z h Z. tl Z ~C(iJilTWZ 9 U4bZ>o 7 7 7 7 7 JV h 7' RQMD (relativistic quantum moleculardynamics) [3] Au+Au#^WZ9^Km^^#V^-eU7n-^#%f6^f
B.A.Li,C.M.Ko(7)ART(A relativistictransport) model [4] H Z h 7 D — L T T 7 '> Y O' § < x Jj X7-- (^U9^7a-^2^#V^V^?#LTk^o
/\Ra^v7y7%ir-K#+#:^, QMD:t#-e(i, QGP&iR^L-CV'&V'Wt\ [5, 6]= L
A^ZT^BUU/QMD:t#:-Mi. T;U^wTO#^#fWTO#^RC^L/::t#:
Au+Au^^Z9l:±#^^T(i,(A matter) [7]. ^ Ra^ y
HZ b. BUU W?#^^WZ 9 6 =
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2 /\ KP — "J y7$BU)$t$5y. — BUU - collision term and potentials
buu y-f"j MW L TEW So tT\ AGS ^^)Pf-(-10AGeV)(i, /<Vt7###M^f(7)m^^(7)##l:*S(7)T,
Wo 7hV7^^^(0^^T7)*^2S(!:##fSc 2#
9 ^^7 7^$r#x.So
• Resonace excitation:
# + #-»#(') + &('), A(') + A('), AT + AT
• string excitation:N + N —> string + string, N + string
• s-channel scattering for M-B:
M + B-^R, R — N*i A(*)
mViBli, 2GeV/c2£ T(DAm,N*%t'<X%fiiK\,'tl' f 4?r ya^7vs 7^T<7)^#^7/fv h L/:o Lund ^A/-yt:ZSFritiofmodel[9] ^ h V>^(7)7 7^7 77—^3 >(i, AMOR(Artru-Mennessier Off-shell Resonance model) [10] (2 L-fzfr'^X fetf) S c AMORTli, 7, h V 7 7
22T, <mt7 hT, dT(iH$M-C^)6o 2(7)Z9^T-f;i/T.T#So
Skyrme potential
Mrt = a(^)+/j(^) (1)
U — &nucl H~ H” ^resonance i (2)
C C. T\ Uq = aps + bp's, B = nucly A, resonance... -6
r = «S {opbary + PPbary} (3)
2 2T, P6ary = /)nucf + PA + PTe„ 2(7)#^T(i, Eq.(3)^^^#^o A(7)^f7vT7l/(7)im^^#%MT, -30MeV 1:^6i^l:mA,^[ll]o #T7v-Y;i/?)0%UA f(7)v.]L;i/(7)n-7 77Bf'yT#J^/:
t (0^#9 =
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Rapidity distribution in Si-t-Al collsion at 14.6AGeV/c
Ylab
7 Exp.
Ylab
El 1: Proton and tt* rapidity distributions for central 14.6AGeV/c Si+Al collision with experimental data of E802.
Au+Au l.OAGeV BUU soft b=4fm Au+Au lO.OAGeV BUU soft b=4fmHucieonT
delta(!232) baryon resonances
0.2 -
time (fm/c)
nucleons delta(1232)
baryon resonances
time (fm/c)
[U 2: The ratio of particle to baryon number for Au+Au at impactparameter b=4fm. Left window corresponds to the case of incident energy lAGeV and righ for lOAGeV.
3
^ ^7 $ < Si(14.6AGeV/c)+Al t
/\pn-<
L /: =
El 2i;t±, Au+Au KlET:a v T y^RS^S, (& i> t >) coit-E-KfYl, lAGeV^10AGeV^#^^BUUr:t#L^t^^
E)3T$)6c A#^^;hf-^iAGeV(o#^-u,10AGeV(O#^(i,
b s-Sy A — 9 o L/p £ l\ 6 = Giessen V + — 7°
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BUU soft b=4fmlGeV2GeV3GeV4GeV5GeV9GeV
lOGeV
Time (fm/c)
HI 3: Time evolution of the relative abundance of baryon resonances for Au+Au at b=4fm.
BUU Hard Au+Au b = 4fm BUU soft Au+Au b = 4fm
O.lSGeV ---- h0.40GeV —•X
lGeV ....*•4GeV —&
lOGeV
O.lSGeV ---- h-0.40GeV —-X-
lGeV ....* -3GeV —B-
lOGeV
HI 4: Left windows shows the transverse momentum distribution for Au+Au b=4fm collsion with hard EOS, right with soft EOS.
t:Z 0, lAGeV-COx A*°t b [11], *) %lOAGeVm,
Transverse flow HI 4'C, (Hard EOS) Z 6 (soft EOS) £X. £ T 7 V + IV-eitS L £ 7E L c Hard <7)^(ix X + t & < px >
Soft(7)^iZ, 3AGeV#ia-rm%LTV^Z9l:Mx.6.-2> Z , efe h -I- + "C 7 O — li— teIC t£ o TV-4 & Z •) -& C0"C, Hard HSH&Z
+ lAGeVWT1Z, Z0 ##£ir+fcibiztt, MStistflftr 7 v + + £fflv' fz%\%£ L%iftU£% b £v'0
HI 512 Au+Au directed transverse flow tf — A zr. + ;l/4r'— 7>[tll|& Z L T, 7°D v hLf:o SoftEoSKi, -flZ^(7)^T7v+7l/^V^/::t#:-C, SoftEoS2U, T&f
^^2Z^T%K6o SoftEoS2##^/J'5W\b Zl-2> o
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O>oIsoc
<uIZ)
<U>
g
1IT3
CASCADE -----1—hard EoS —-X—soft EoS 1 X -soft EoS2 0
Au+Au b = 4frn
Beam energy per nucleon (AGeV)
HI 5: The directed transverse flow as a function of incident energy
4 t t&b
modelUZb. A,
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JAERI-Conf 97-007
[1] D. H. Rischke, Nucl. Phys. A610, 88c (1996); D.H.Rischke, et al., Columbia preprint CU-Tp-695,nucl-th9505014.
[2] EOS Collaboration nucl-ex9607008.
[3] H. Sorge, H. Stocker and W. Greiner, Nucl. Phys. A498, 567c (1989); Ann. Phys. 192, 266 (1989). R. Mattiello, H. Sorge, H. Stocker and W. Greiner.
Phys. Rev. Lett.63(1989) 1459,
[4] Bao-An Li, Che Ming Ko, Phys.Rev.C52 2037 (1995); Phys.Rev.C53:22-24,1996.
[5] Bao-An Li, Che Ming Ko, Nucl.Phys.A601:457-472,1996.
[6] L.A. Winckelmann, et al., Nud.Phys.A610,116c (1996).
[7] H. Sorge, Phys. Rev. C49 (1994) 1253; M. Hofmann, R. Mattiello, H. Sorge, H. Stocker, W. Greiner, Phys. Rev. C51 2095,(1995).
[8] L. Bravina, et. al, Phys. Rev. C50, 2161 (1994).
[9] B. Andersson, G. Gustafson and B. Nilsson-Almquist, Nucl. Phys. B281, 289 (1987); B. Nilsson-Almquist and E. Stenlund, Computer Phys. Comm. 43 (1987), 387.
[10] K. Werner, Z. Phys. C42, (1989) 85; K. Werner, Phys. Rep. 232, (1993) 87; X. Artru and G. Mennessier, Nucl. Phys. B70, 93 (1974).
[11] W. Ehehalt, W. Gassing, A. Engel, U. Mosel and Gy. Wolf, Phys. Lett. B298 (12993) 31.
[12] G. F. Bertsch and S. Das Gupta, Phys. Rep. 160 (1988) 189; W. Gassing, V. Metag, U. Mosel and K. Niita, Phys. Rep. 188 (1990) 363.
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15. Nuclear Multifragmentation Experiment at the KEK 12 GeV PS:(The first results of the KEK-PS E337 Experiment)
K.H. Tanaka"1", Y. Yamanoi, M. Haseno1, Y. Tanaka1, Y. Okuma2, Y. Sugaya3, F. Kosuge4,Y. Shibata4, K. Nakai4, Y. Inotani5, H. Ochiishi5, S. Morinobu5 R. Muramatsu6,
J. Murata6, K. Yasuda6 and T. Murakami6 (E337 Collaboration)
KEK, National Laboratory for high Energy Physics, Tsukuba, 305, Japan Nagasaki Institute for Applied Science, Nagasaki, 851-01, Japan institute of Physics, Tsukuba University, Tsukuba, 305, Japan
3Faculty of Technology, Tokyo University of Agriculture and Technology, Koganei, 184,Japan
4Faculty of Science and Engineering, Science University of Tokyo, Noda, 278, Japan 5Faculty of Science, Kyushu University, Fukuoka, 812-81, Japan
6Faculty of Science, Kyoto University, Kyoto, 606-01, Japan +Corresponding author. E-mail [email protected].
A KEK-PS Experiment E337 “Angular correlation of intermediate mass fragments emitted from the target multifragmentation reactions with 12 GeV protons” is an extension of the E288 which we performed a few years ago. The E288 revealed that the proton-induced target multifragmentation (TMF) reactions at 12 GeV showed quite interesting phenomena1 such as 70° peaking angular distributions for intermediate mass fragments (IMF; Z>3). In order to understand the phenomena we planed to measure detailed angular correlations among several IMFs and multiplicity-gated angular distributions of IMFs in the E337. For this purpose we have developed a new large acceptance Bragg Curve Counter (BCC2) array, which consists of 25 BCCs enclosed in one gas volume and can cover nearly 13% of entire solid angle. Additional 12 BCCs is prepared in the common horizontal plane which included the target in order to measure the accurate angular distribution (see Fig. 1). In total the BCC system can cover about 20% of 4tt.
In December, 1995, the test experiment with 12 GeV protons was started at KEK using this newly constructed counter array of 37 BCCs. The beam line used was EP1-B line, which was also newly constructed exclusively for the primary beam experiments. The intensity of primary proton beam focused on the target was about 4x109/spill with the spot size of approximately 5 mm in diameter. The typical thickness of targets used was about 500 jxgjcn?. The data were taken under the minimum bias condition, i.e. a sum of 37 self triggers from individual BCC. A newly developed data taking system consisted of the 68EC030 Auxiliary Crate Controller and the VME on-board Sparc 5CE computer enabled us to treat such a large number of triggers as about more than 2000 per sec. The main production experiment was performed in April and May in 1996 after debugging the new counter system and DAQ system, as well as EP1-B beam line. Data with Au, Tm, Sm and
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Ag targets were successfully accumulated. Now the date are in the analysis stage. Several interesting features of high energy nuclear reactions have already been seen. Typical Bragg peak (charge of particles) vs. kinetic energy scatter plots of IMFs obtained by one of the 37 BCCs are shown in Fig. 2. It can be clearly seen that heavier IMFs can be produced in heavier target reactions. Preliminary results of angular correlations of IMFs as well as energy spectra, a charge distribution, angular distributions of IMFs with and without multiplicity gate were presented in the 4-th Symposium on Simulation of Hadronic Many-Body System. Final results will be published soon.
References1 K.H. Tanaka et ah, Proceedings of the 7-th Varenna Conference on Nuclear Reaction
Mechanisms, Varenna, Italy (1994) p.643.K.H. Tanaka et al., Nuclear Physics A 583 (1995) p.581.T. Murakami et ah, Perspective in Heavy Ion Physics, World Scientific (publisher), (1996) p.152.
2 H. Ochiishi et ah, Nuclear Instruments and Methods A 369 (1996) 269.
(Cone type 0CC)12QoV proton Beam
Fig. 1; E337 setup. Vertical (left) and horizontal (right) views.
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Brag
g Pe
ak (A
DC
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nts)
Br
agg
Peak
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ount
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Bragg Peak vs Energy Plots for Various Targets
?3000
100 150_ „ Energy (MeV)Tm Target
3000
100 150Energy (MeV)
Ag Target
Fig. 2; Two dimensional charge versus kinetic energy spectrum of IMFs produced in the Au,Tm, Sm and Ag(p,X) reaction at Ep=12 GeV obtained by one of 37 BCCs.
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16. Freeze Out Temperature on Light Projectile Induced Reaction
J.Murata1, M.Haga2, M.Haseno3, Y.Inotani4, H.Ito1 *, K.Kimura3, F.Kosuge5, S.Mihara1, S.Morinobu4,T.Murakami1, R.Muramatsu1, K.Nakai5, H.Nakamura4, H.Ochiishi4, Y.Ohkuma2, Y.Okuno3, S.Sawada1 t,
T.Shibata6, Y.Shibata5, Y.Sugaya7 K.H.Tanaka8, Y.Tanaka3, K.Ushie3, Y.Yamanoi8, K.Yasuda1(MULTI Collaboration)
1 Kyoto University, Kyoto 606, Japan 2Tsukuba University, Tsukuba 305, Japan
3 Nagasaki Institute of Applied Science, Nagasaki 851, Japan * Kyushu University, Fukuoka 812, Japan
5 Science University of Tokyo, Chiba 278, Japan6 Institute for Nuclear Study, University of Tokyo, Tokyo 188, Japan7 Tokyo University of Agriculture and Technology, Tokyo 184, Japan
8 KEK, National Laboratory for High Energy Physics, Tsukuba 305, Japan
Nuclear temperature was deduced for 12GeV proton induced target multi-fragmentation reactions on Au,Tm,Sm,Ag targets. Using isotope yield ratios, clear target mass dependence was obtained for high-multiplicity events. Deduced temperatures for light targets have higher value than those for heavy targets.
Over the past few decades a considerable number of studies have been made on extracting nuclear temperatures from experimental data of intermediate or relativistic energy heavy ion collisions. The main aim of the temperature measurement is a search for a signal of nuclear liquid gas phase transition [1,20-23], Although temperature extraction using slope of energy spectra has been an object of study for a long time, usually they showed much higher temperatures than expected. Is is mainly because dynamical effects exist [8-17]. To avoid the difficulty, relative populations of excited states were used. However this method was experimentally hard to determine the yield of the excited states [13-19], Isotope yield ratio has been recently brought to light by the determination of the “caloric curve” [1], One advantage of this method is the simplicity. Isotope temperature, as a probe for a chemical freeze out temperature, can be obtained only by calculating isotope yield ratios [2-5]. This paper is intended as an investigation of deducing geometrical temperature distribution in light particle induced reactions by means of isotope temperature method.
The experiment (KEK-PS E337) was performed at a newly constructed EP1-B primary beam line of the KEK-PS with 12GeV proton beam. Four targets (Au,Tm,Sm,Ag) were used. The aim of E337 was to detect IMFs and collect information about the excitation mechanism of high-temperature nuclear matter with small compression system. IMFs emitted from the targets were detected by Bragg Curve Counters (BCCs) for 3 < Z < 30, and determined their charge numbers, kinetic energies and ranges in the counter gas [24,25],
There were 37 BCC-channels surrounding the foil targets. Total acceptance of the counters was about 20% of 4ir. Aiming to measure angular distributions and in-plane correlations, 12 BCC-channels were located within a horizontal plane from 30° to 150° in laboratory angles at step of 20°. In order to collect information about out-of-plane correlation and IMF multiplicity, a large cone-type BCC which had 25 channels were located above the target.
IMFs make their loci in Bragg peak-energy-range plot. The loci are separated from each other with the distance about 9<r in Bragg peak. The excellent Bragg peak resolution split the loci of light fragments (Z < 4) according to their mass difference. In our experiment, 6Li,7Li,8Li,9Li,7Be,9Be,10Be fragments can be separated from each other in range-energy plot. In order to estimate the total yield of each fragments, function forms of moving source model are used for fitting the energy spectra. Some of fitting parameters were fixed to adequate values. It is because the dynamic range of kinetic energy is not wide enough to make complete fitting.
It is necessary to explain the procedure of isotope temperature. According to simple classical statistical dynamics, all single ratio R can be expressed as below [2].
Ri = P''pFPnFc‘iexP-fL (1)
where Ri is the t'-th single ratio, and pPF,PnF are free proton and free neutron densities in nucleons and fragments mixed ideal gas of equilibrium temperature T. The index means r)i = Z1 — Z2 and & = (Ai — Z{) — (A2 — Z2).
‘Present address: Nippon Telegraph and Telephone Corporation (NTT)1 Present address: Institute for Nuclear Study,University of Tokyo 1 Present address: Research Center for Nuclear Physics (RCNP),Osaka University
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JAERI-Conf 97-007
Here Aj,Zj indicate mass number and charge number of j-th particle. AS, is binding energy differences between and the factor a means a, = (A7’jv/2)Ai""A2(A1/A2)3/'2u)(A1, Z1)/oj(A2, Z2). Here XTn is the thermal nucleon wave-length. Internal partition function u> = £V(2sj + 1 )exp(Ej/T) of the fragment (A, Z) will contain all known 7-unstable states in the following calculations, where s; and Ej are spins and energies of j-th states.
Three unknown quantity T,ppp and pnF can be extracted from three single ratios. By selecting single ratios whichhave AZ =f Z\ - Z2 = 0, we can get only two quantity, T and pnp using two single ratios. It is because % = 0 for all Ri with AZ = 0. The condition AZ — 0 is indispensable to systematic error reduction. There are 377 combinations which can deduce T and 36 of them satisfy AZ = 0. In this restriction, T can be written as below.
T= ^ aiABi/ln J] (%/a,r (2a)1=1,2 1=1,2
where ai = £2,a2 = —£1. or simply
T —B
ln(R/a(T)) (2b)
Because of the exponential function form of Eq.2b, \dR/dT\ shows small value for large B. So,the ratio which has large B should produce relatively small temperature fluctuations. 6 ratios has been selected due to their large B from the ratios of AZ = 0. Finally we selected one ratio to be used as a thermometer in this paper as below.
yWy(?ae)~ Y(7Li)3Y(l0Be) W
This ratio has the smallest statistical error in the 6 selected ratios.R should be corrected because of sequential decay effects, and the corrected ratio can be simply written as Reoi = tcR
[3]. We compare corrected temperature for FNAL inclusive experiment (80 GeV/c p + Xe, Tcoi = 3.9 ± 0.4 MeV [3] for 76° counter [7]) and the mean temperature of all targets (T = 3.08 ± 0.20 MeV for multiplicity = 1,70* counter). It is because the calculated temperatures have small target mass dependence for low multiplicity events detected by side-ward BCC channels. Then we got k = 0.235 ± 0.168 using following relation, 1 /T = 1 /Tcoj + lnn/B. Using Reoi instead of R, Eq.2b are calculated numerically. If w account only their ground states, Eq.2b gives T directly. Considering all known 7-unstable states, Eq.2b must be solved numerically. Corrected temperatures obtained for inplane BCC channels are shown in Fig.3 as a function of laboratory angles with IMF multiplicity selection for all targets. Independent to multiplicity, side-ward channels show nearly constant temperatures as a function of angles. Although clear target mass dependences can be seen for high multiplicity events, side-ward channels for low multiplicity events show small target mass dependences. High multiplicity events show higher temperatures than low multiplicity events for almost all channels and targets.
High multiplicity events should have large centrality on the collision. So they are natural to show high temperatures than low multiplicity events. The jump between multiplicity = 2 and 3 have been observed already in Fig.2 for single ratios. In Fig.3, we can find strong target mass dependence for only high multiplicity events. Low multiplicity events should also have strong target mass dependences as seen in single ratios. The information about the target mass dependence for low multiplicity events have been lost in the process of calculating multi-ratios. It is because the single ratio Y(8Li)/Y(7Li) has opposite target mass dependence for low multiplicity events.
The anisotropic distributions imply the chemical freeze out had been established before the total remnant reached to thermal equilibrium. What must be noticed is that even forward channels show high temperatures as well as backward channels. One explanation may be that, this is the trace of fire ball which penetrate the target nuclei with projectile [26]. On this assumption, energy should be deposited in proportion to the path length of the fire ball region, that is, diameter of the target nuclei. If the deposited energy diffused in the target nuclei, the energy density can be roughly written as below.
Ex2 vqA1/3
47rrgA(/3 - zd22r0Alt^3 (4)
here At is the target mass number and d is the radius of penetrated columnar region, which must be almost independent to the target mass. The denominator of Eq.4 means the volume of the remnant. r0 is normal nuclear radius. We should not forget that Eq.4 has only first-order reliability, because they regard the temperature distribution in the remnant as isotropic. Now we should consider the relation between energy and temperature. It is well known that
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JAERI-Conf 97-007
T oc E1 11!2 in normal nuclear matter. But as be seen in “caloric curve [1]”, T is nearly constant during the mixed phase and T oc E in gas phase on liquid-gas phase transition. So here we put the “phase index” parameter a and use the relation T oc E° with Eq.4 to fit the results of target mass dependence for high multiplicity events. Obtained value of d and cr are plotted in Fig.4 and they are almost constant around d ~ 3ro and a ~ 1. The mean value of d is (3.17 ± 0.15)ro_ It seems a bit large comparing to the target radius especially for Ag target. It must be noted that we regard the “tunnel” as column in the volume estimation. So adequate radius should be somewhat smaller. The phase index parameter <7 have their mean value 1.06 ± 0.015. This is a clear evidence for the fact that they are in nuclear gas phase. We should notice that the isotope temperature method suppose the free nucleons and fragments mixed ideal gas as the system. According to this assumption, we may not use isotope temperature as a probe for complete free nucleon gas. So if we use isotope temperature, “gas phase” should be defined as phase index equal to 1. To determine the real temperature of such nucleon gas which existed before the chemical freeze out, further efforts are needed.
There is a further point which needs to be mentioned. The starting assumption of the tunnel formation is the anisotropy in the angular distribution of temperature. On the other hand, we cannot find such anisotropy for the temperature distribution obtained by another combination of ratios. They have large error bars compared to those of the selected combination, and we can say nothing about the angular distributions using such another ratios. Even we can regard the anisotropy as denominations, good agreement with Eq.4 will support the assumption.
In summary, we have studied isotope yield ratios resulting from 12 GeV proton induced target multi-fragmentation reactions. Observed isotope yield ratios show a strong linear correlation with target mass numbers. Isotope temperatures were derived from the yield ratios. Sequential decay effects and the influence of 7-unstable states to the partition functions has been considered. Strong target mass dependence has been observed for both temperatures and single ratios. It can be explained that the deposited energy diffused in the remnant when chemical freeze out established.
[1] J.Pochodzalla et al.,Phys.Rev.Lett. 75, 1040 (1995).[2] S.AIbergo et al.,Nuovo Cimento A 89, 1 (1985).[3] M.B.Tsang et al.,MSUCL-1035 JULY 1996[4] M.B.Tsang et al.,Phys.Rev.C 53, R1057 (1996).[5] J.A.Hanger et al., Phys.Rev.Lett. 77, 235 (1996).[6] A.M.Poskanzer et al.,Phys.Rev.C 3, 883 (1971).[7] A.S.Hirsh et al.,Phys.Rev.C 29, 508 (1984).[8] B.V.Jacak et al.,Phys.Rev.Lett. 51, 1846 (1983).[9] W.A.Friedman and W.G.Lynch Phys.Rev.C 28, 950 (1983).
[10] H.Stocker et al.,Z.Phys.A 303, 259 (1981).[11] S.Nagamiyaet al.,Phys.Rev.C 24, 971 (1981).[12] J .Cosset et al., Phys.Rev.C 18, 844 (1978).[13] F.Zhu et al., Phys.Rev.C 52, 784 (1995).[14] C.Schwarz et al., Phys.Rev.C 48, 676 (1993).[15] T.K.Nayak et al.,Phys.Rev.C 45, 132 (1992).[16] H.M.Xu et al.,Phys.Rev.C 40 186 (1989).[17] J.Pochodzalla et al., Phys.Rev.C 35, 1695 (1987).[18] G.J.Kunde et al.,Phys.Lett.B 272, 202 (1991).[19] C.B.Chitwood et al.,Phys.Lett.B 172, 27 (1986).[20] S.Ban-Hao and D.H E.Gross Nucl.Phys.A 437, 643 (1985).[21] S.D.Gupta and A.Z.Mekjian Phys.Rep. 72, No.3 131 (1981).[22] L.P.Csernai and J.I.Kapusta Phys.Rep. 131, No.4 223 (1986).[23] J.Kapusta Phys.Rev.C 29, 1735 (1984).[24] H.Ochiishi et al.,NIM A 369, 269 (1996).[25] K.H.Tanaka et al.,Nucl.Phys. A 583, 581 (1995).[26] K.Nakai et al.,Phys.Lett.121, B, 373 (1983).
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JAERI-Conf 97-007
5 10 15 20 25 30 35
Energy (MeV)FIG. 1. Isotope separation using BCC. The loci of Li and Be are split according to their mass differences.
10 Be/Be 9 Be/Be 7Li/Li 7Li/Li 7U\/Ur~ 4 r
0.5 ~-0.5
M-1M-2M*3
FIG. 2. Target mass dependence of single isotope yield ratios. Mean value of 50° to 130° counters are plotted.
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JAERI-Conf 97-007
> 90 Target= Ag Multiplicity= 3 ▼Sm 2 a
<D 40
LAB’ TargetFIG. 3. Target mass dependence and angular distribution of temperature. Plot of temperature vs. A for each angle are
placed in a row. The line indicate the results of fitting.
iI
43.5
32.5
21.5
10.5
0 20 40 60 80 100 120 140 160 180® lab@QQ-)
..........................t.......A.......................V............... 1 A
R(mean)m3.105±0.0995
j-
1 1 '_1 1 1 1 1 _i_i ip (mean)-1.060±0.0149
_i i i I i_i i i_t_j_ilii_i_L„ i. i.
FIG. 4. Results of fitting. Each channels show almost same values.
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JAER
I-Con
f 97-00
7
m
e
*K
A
V
•tiniff
ti
SICNJ00
JAERI-Conf 97-007
±muBab * > *Eisa
b a kd vnaw^t^/u-y'%*####%*#%**
(April 17, 1997)
Time evolution of hadronic resonance matter in ultrarelativistic nucleus-nucleus collisions are
studied in the framework of cascade models. We investigate the role of higher baryonic resonances
during the time evolution of hot and dense hadronic matter at AGS energies. Although final hadronic
spectrum can reproduced well with and without higher baryonic resonances, the inclusion of higher
resonances is shown to prevent the temperature from going beyond 200 MeV.
18.
*y®?3<7)±E&3v-7V<7)—T'lV'y t^xr)AGS <DJZ * (10-15AGeV) JCfcV>TJi#fiflS?&
E<0^ Kn y%M^I%'C'&&fztiTi'Z0 >1(0 AGS
Tit 2-? <D&<DM4Stc^v^TEV'cojitf^EihL^vx fc
Z itlf.
* >E$li. m#(0#e%(0 6»V'L10 f§ClLTv^0
RQMD [1,2], ARC [3], ART [4], QGSM [5],
BSD [6]> UQMD [7] <D «t ? <ak>3'/ SrffiV'
ra«J®«*K<OltKS:a»LJ: 9 i:+ *&**** $*IT
§fc0 ffbf fL(0## J:^A6/\ KnygSflEliH* 4o fOltf RQMD^UQMD ^@52GeV t
>S6Eh®*1.8GeV h V
>y^^T#-tr(01:#LT. ARTtiPS^tL^etiEx BP*.
A(1232), A*(1440), AT*(1535), A, E, ir, p(770),w(782), r?, A
co*4-/x'V t>66Ek LT#/VTV£0 HvvGj *>(0 & -1 n± o x/<') * y b £ t *> ^mt
ARTffv z t tr J;oT-E-(07FS.5-1SoTv^0
*ibtz< $A(066E5-«A
&&. -#±t a 2 -e-nii** *> & oo.
fL6l:Wta(76#f(oe$) «ffa»:«®LTV'40 bp*,,
l < f xo#LTIi^tveti
o A: i> <D ic^iLT^i t #£ b tli <DT$>&0
&±TV' < ofrofivvo; * :/#i*#&Wa9ST*a c
JIBE fcfcl h tlT V>*o mx. fI* > V y A( 1232)
ct [3]0 t /c 7T > '/ > (OSZibS^ (0 7*,. #mm#(0#V.#mi:MLT(iA(1232) (0JW#TA
i [2]0 -E-tlTIl i <0 Sv'£q|v<') * y ti AO t 7 frESti£Mtz IT^'Z<nfz?> 7 fr?
li AGS (11.6AGeV/c) ^#^<0
* x-y-
ten. BPt%#2GeV f -C(7)*q|y< V * y (Model-A) t
e-mail: [email protected]
JAERI-Conf 97-007
A(1232),Ar*(1440), AT*(1535) CO.ACOAV *
O## (Model-B) Z <Oit$<£TI CT AGS
ajjbfrlltz Z tfr'Z(Dfflft(DimT$)Z0
II. ££!
A. ilT.'T- h'8t*C7)2-0<7)t£3y
Z<D^-T)V\t. 511 2GeV f Tco*q|/<'J * > 6 6$
5GeV) Tti RQMD h V > V £ c
. Model-A .. Model-Bn— Cal hszopm
f BBSS{ Eeee ra(.
6 E877 rat
4—i—1
I ■ I ■ I ■ 1 I . I ■ I ■ I . L02460246
Rapidity y,*, Rapidity y*
FIG. 1. &(11.6AGeV/c) 4-£*g§f'/><07
t 4 -Mcotmilt£mto ^'Model-A, Model-B
IZXZttn&MT&Zo -^mA#^ovV:6li%##(BNL-AGS esgg.es??) xhha ±: ht-, 4=: tt"> v*>\ T : ,+ > y>.
Model-A t > - /<V t >
B{ + Bj <-> B{ + Bj, N{ -)- Nj *-<■ TV* + R
Ni + Nj *-> Ri + Rj, Ni + V) «—> TV,- + Vj,
TV:#f. Y:A,E,
R : A(1232)~£>(1950),TV*(1440)~Ar*(2190) Z
(O9 t - #f%&(0
9 1:
i&Aitc 2<o#l:A'/>-/<>V> A/>
Ixd£>
M + S] <—► B2
7T + 7T <-> p
t-aSflTv^o C CT
M : n,p, I<, rj,u,T]', K,E51 : iV,A(1232)~D(1950),A*(1440)~7V*(2190),
A*( 1440)~A* (2350), £*(1385)~£* (2250),E
52 : A(1232)~A(1950),iV*(1440)~A4*(2190),
A*(1520)~A*(2350), E*(1660)~£*(2250)
4'56’t" o—# Model-B t ti
AT TV e- NR, NN <-* AA, ATV*(1440),
NN —* NNp, TVTVw, AAtt, TV TV —► AAp,
NN -* AT Aw
G21:TV:#f. fl : A(1232),A'*(1440),A4*(1535)
M + Bi «-*■ Bj
??+ TV ~ Ar*(1535)
7T -(- 7T <-> p
UJti.Tv>i0 ;:r M : Jr, p
B : TV, A( 1232),TV*(1440),TV*( 1535)
Sr^to Model-B 1:#^ C K6
ART [4] & CD t fWJ
M#@U:Z-3-CBt#$fl/:^(11.6AGeV/c +A) (07
#^#^3)7 & f fl f dl FI G. 1, FI G. 2 IZiji
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JAERl-Conf 97-007
to BNL-AGS 60 E866 [8], E877 [9] 1: Z & 6 <0
■T-L'Cv E°t*d tJytMXM mTti
1 E + pz
HIT = \jvi + I'l + m2
FIG. 2. £(11.6AGeV/c) + A
±>' Model-A, T7)? Model-B \Z l h%\
(BNL-AGS E866)
B.
? t-f 2 & l:i
10%~20%?)#&@T#ma-<V';5. #%*L X li Model-A h i ( "'StL'CV'1 Z 60
1:#LT Model-B Tli1:T#w*<7)
CcO^
Boltzman ijtff'Cifc Z> 11illi\
ffkil/cHf.ti; (7 V -X77 h) T<0 Model-B <OSE*5\
Model-A t%SU n
rExp:E866(BNL-AGS)Nucl.Phys.A.553(1993)810c
Model-A
i | i -i1 1-1-4 I I I IrExp:E866(BNL-AGS)Nucl.Phys.A.553( 1993)81 Oc ]
Cal b<2.0[fmf
Model-By= 3.0-3.1
J__ I__ L
Transverse Mass mrm[GeV
> v -v x, p)(
-TlogH(T,V»)
^gas — ^ ^ »i
n,(t, y, p) = ±ty ^ iog(i ? exp
2 C K i itz E = \/m2 + p2(c =
/k = 1) #l:p = 0 ti
^<0Z 7 :
f(T) = Ts — pd figas ^ FIgasdf~) ~ ~v~
=(»*+H bt,+o<t,>CCTng, nF li -E- tl t't-1 - X>2^. n 7 j- )V X tit-<Og
tiSSvtto -(Di&frb'%L=t-<D6EfeE^±it£t^-z-bti tz x * /u df - ^E17 W L x SE o3 ii i) cog SE^-a 3 y i &^ t x^-^E &Ew
------ A/,A,l,A,-,it,K,r|
$ 0.2
® 0.1
Energy Density[GeV/fm3]
— 85
JAERI-Conf 97-007
FIG. 3. 5$<OMS:i: LX(DX.^)1
(ft^'-Fr > yf&p = Oo '/>«§6$<0A^£>&iE> tfxx : /<v t>A£JMt d V>AM
mw##: /<vtyAi®, 7 7>Al|«lfi®i-J;y;A Hide'S*
hi.
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17x afSG vuuwi 7 i:IW:#%<o#2Ssa^ttSLT.
#T#ja2 117ZoT@^^^^<c<f ad ta.
fCTe/rti.
<C^M%T:E^ $ %&#### &#%-&&
%l±JaiT<7) 1 7 &&OT&&0
A1+A2
A 1 + A2
= E1=1
' exp
r+co
1 pm(2ttZ,,)3/2 m,
(* - x,(<))2 + {p,(<) • (x - Xi(t))/mt)'22 L,
/(*.p)= 53 /=(x-p)
,.2/i(x,p) = AT exp ( ~~ ) 6(p-pi)6(p2 - m2)20(p°)
r""M = ^ ^pp"p"/(x,p)
1 pr(^p^oA1 + A 2
= E 7:exp
(2ttL,)3/2 m,
(a; - »t-(<))2 + {p,-(<) • (x - Xi{t))/mi}2
2 L3
z > v ;f itmvMjpmit&Ttf
*). 0m%4?##TU:&v'ia
:0di IF^'—t'ESE^A Infeto^NiEE^ i^hZ- a a 71 7. 17
4 Team
u^(x)AI"(r)
$TSEti^B'r^ihM 17 It aSE (47c* U> K0; -SK^) tKtlfrh
P(x) = p(z)T(r) L=F Ai°(z)T(z)
*,T = -(* - Xi(<))2 - {Pi(<) • (x - Xi{t))/mi}2
ttmmMtmitiz t> ^-e [io]x ci;h,tin-i/>yxtf
7-T&Z0 ±T4x.^ 1/>7X*
^<cV'6(7)a LT#a2 ^^T&ao
$TE<7)53'3!flMSc5-fflV^T4 7C* U> h
ihMf- > y IF (iL«*-r > vifco 4^7C^P^^coK;?g)T(DZ 7
N^ix) = j dApp»f(x,p)
Kn >*'X 17 L^ ^ (0"C* a0
#^(D^&l7&ct< &(11.6AGeV/c) +^(DEm#^T
(OIL4.IF^'-SS • ia*(OB#H|fBSS- Model-A, Model-B
[70V'-tFIG.4t7/Ttto Model-A x EE** Model-B
17 40 ^.B$r5l7d3lt^7L^;V4r-&JS
m?#Lv^l7MLTia$17ciV'-cij:gv'*i#xu
# Model-A &oTua
- 86 -
JAERI-Conf 97-007
250 -
o 150
t= 100
-e— Model-A ♦ — Model-B
X 50
Energy Density e [GeV/fm3]
FIG. 4. &(11.6AGeV/c + &«£ I:-fc It-Z> ^ ^ h
Ofm/c ~25 fm/c)0 Model-A,
It Model-B ~C<h -50
IV.
T, A (10.6AGeV) + &C7)®|*] * X tr - KfHfcfrffofco
i> p+pmLTcD7t°r'fT'ffrft, #3# i^KMLT^®HB£W£ai-£ i ') Hlfi§*I£4x£o
iSSli,
i^tL/Vcti^tSCJxS < &o£0
[1] H. Sorge, H. Stocker and W. Greiner, Nucl. Phys. A498,
567c (1989); Ann. Phys. (N. Y.) 192, 266 (1989).
[2] H.Sorge, Phys. Rev. C49, R1253 (1994)
[3] Y. Pang, T.J. Schlagel, S.H. Kahana, Nucl. Phys. A544,
435c (1992); Phys. Rev. Lett. 68, 2743 (1992);
[4] B.A. Li, C.M. Ko, Phys. Rev. C52, 2037 (1995);
[5] L. Bravina, L.P. Csernai, P. Levai, and D. Dtrottman,
Phys. Rev. C 51, 2161 (1994).
[6] W. Ehehalt and W. Gassing, Nucl. Phys. A602, 449
(1996).
[7] L A. Winckelmann, et al., Nucl. Phys. A610, 116c
(1996).
[8] F.Videbaek et al. (E866 Collaboration), Nucl. Phys.
A590, 249c (1995).
[9] R.Lacasse et al. (E877 Collaboration), nucl-ex/09001
(1996).
[10] C. Fuchs, H.H. Wolter Nucl. Phys. A589, 732 (1995)
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JAERI-Conf 97-007
19. Clustering effect of 12C fragmentation in p + 12 C, a + 12C and14N + 12C reactions
Hiroki Takemoto, Hisashi Horiuchi Department of Physics, Kyoto University, Kyoto 606-01, Japan
Akira OnoDepartment of Physics, Tohoku University, Sendai 980-77, Japan
In general, self-conjugate 4n-nuclei have anomalous excited states with the excitation energy 10-15 MeV, which are recognized to be generated by the change of the structure from the shell-model-like one into the cluster one due to the activation of the clustering degrees of freedom [1]. In 12C case these anomalous levels, O2 at 7.65 MeV and 2^ at 10.3 MeV, have been recognized to have 3a structure. Since the excitation energies of these levels are usually near or above the threshold of the breakup into constituent clusters, it is natural that the clustering effect is expected to play an important role in heavy ion reactions.
We analyze p + 12C at 55MeV, a + 12C at 22.5 MeV/u and 14N + 12C at 35 MeV/u reactions using antisymmetrized molecular dynamics (AMD) to investigate the activation of alpha-cluster degrees of freedom in 12C fragmentation by various projectile reactions.
The formalism of AMD was described in detail in Ref. [2], and only the outline of AMD is explained below. In AMD, the wave function of A-nucleon system |$) is described by a Slater determinant
m =1
det (j)],
where
Vi = <t>z,Xai (a, = P T,P 1,n T, n |),
(1)
(2)
<t>Zi = exp (3)
Xai and <f>z, represent the spin-isospin wave function and the spatial wave function of the i-th single particle described by the Gaussian form, respectively, u is the width parameter which is independent of time and in the following calculations u = 0.16 fm~2. Z — {Z,} represent the positions of the centers of Gaussians. Thus A-body wave function |<3>) is parameterized by Z. The time development of Z is determined by the time dependent variational principle which leads to the following equation of motion for Z,
dCicjT^ZjT
JT
dH%
and c.c., (4)
c,„„ = gzfg log (*(Z)|*(Z)), (5)
where cr, t = x,y, z and 7i is the expectation value of Hamiltonian H.When we apply AMD to heavy ion reactions, nucleon-nucleon collision processes should
be included. In AMD, as is made in QMD, they scatter stochastically when the spatial
- 88 -
JAERI-Conf 97-007
distance between two nucleons is small. But due to the effect of antisymmetrization the centers of Gaussians Z don’t always have the meaning as the positions and momenta of nucleons. So we have to transform coordinates Z to the physical coordinates W = {WJ which can be interpreted as the positions and momenta of nucleons
W, (6)
where
Oij=a(zr^)log(®(z)|4(z))' (7)
In AMD we treat fermionic nature of nucleons exactly, because the wave function of A-body system is antisymmetrized by a Slater determinant. Hence Pauli principle has been fully incorporated in AMD.
We calculate the time development of the system with AMD till a certain time t — taw when the produced fragments are thermally equilibrated. At this time many excited fragments exist, and these thermally equilibrated fragments evaporate particles or 7 ray with a long time scale. We calculate the evaporation process after t3W by multi-step statistical decay code [3] which is similar to the code of Puhlhofer [4]. In this paper we call the process before t3W “the dynamical process” and the one after taw “the statistical decay process”.
It should be noted that the AMD method is very well suited for this kind of study, because AMD describes the time development of the system wave function and, hence, it can describe quantum mechanical features such as shell effects and cluster degrees of freedom.
Recently we investigated the difference between 12C and 14N fragmentation in 14N + 12C reaction at 35 MeV/u [5]. Since 12C and 14N have almost the same mass number, if the fragmentation from each nucleus is different, it indicates the existence of the fragmentation mechanism which is related to the nuclear structure. The results is following.
1. 4He fragments from 12C are more numerous than those from 14N and this abundance from 12C mainly originates from the dynamical process.
2. Energy spectrum of 4He fragments from 12C produced during the dynamical process has the peak near the incident energy.
3. 4He fragments are produced most frequently at semi-peripheral collisions during the dynamical process.
4. 12C breaks up into 3a particles most probably in all production events of 4He fragments during the dynamical process.
We concluded that the above features of 12C fragmentation are originated from the activation of alpha-cluster degrees of freedom by indicating the excitation energy spectra of 12C before its breakup at the dynamical stage. Excitation energy spectrum before 12C breakup into any fragments consists of two components. One distributes in the range 7—15 MeV and the other in the range above 15 MeV. Most of the former component results from the events that 12C breaks up into 3a particles during the dynamical process and these excitation energies, 7—15 MeV, corresponds to excited levels of 12C which are considered to have the cluster structure. Accordingly the features' mentioned above is related to those excited states of 12C that is considered to have the cluster structure.
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JAERI-Conf 97-007
AMO —AMD —EXP -----EXP —
,4n + '*cm 35MeVAi
AMOAMO —a ♦ '*C * 22.5 MPVAI EXP —
r io'
,4N ♦ l2C at 35MeVAi
Mass NunDef A Mee Number A
FIG. 1. Mass distribution in p + l2C at 55 MeV and isotope distributions in a + 12C at 22.5 MeV/u and 14N + 12C at 35 MeV/u. Solid lines indicate the distributions of the AMD calculations and dashed lines indicate the experimental data.
In this report we investigated the projectile-mass dependence of the clustering effect of 12C by comparing 12C fragmentation in p + 12C at 55 MeV, a + 12C at 22.5 MeV/u and 14N + 12C at 35 MeV/u. Figure 1 shows the comparison of AMD results with the experimental data for p + 12C at 55 MeV [6], a + 12C at 22.5 MeV/u [7] and 14N + 12C at 35 MeV/u [8]. Solid and dashed lines indicate the AMD results and the experimental data, respectively. As is seen, the AMD calculations reproduce the experimental data well in the whole mass range but underestimate the production cross section of fragments with A = 7 in all reactions.
We show mass distributions in p + 12C at 55 MeV (left panel), a + 12C at 22.5 MeV/u (central panel) and 14N + 12C at 35 MeV/u (right panel) in Fig. 2. As is shown clearly by histograms, intermediate-mass fragments are merely produced before statistical decay in the proton induced reaction, while in a and 14N induced reactions many fragments are produced before statistical decay. Especially 4He fragments are much produced during the dynamical process and the high yield of fragments with A=8 with respect to the yields of neighboring elements is seen in a and 14N induced reactions. In addition, the capture process by the 12C nucleus appears in the a induced reaction at 22.5 MeV/u, which reflects the low relative velocity between projectile and target compared with the other two reactions.
10*
103
„ 102 |% 10'
10°
10''
FIG. 2. Mass distributions in p + 12C at 55 MeV (left panel), a + 12C at 22.5 MeV/u (central panel) and 14N + 12C at 35 MeV/u (right panel). Solid lines indicate mass distribution after statistical decay and histograms indicate those before statistical decay.
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before decay
Mass Number A
<x + C at 22.5 MeV/u -+•— N + 12C at 35 MeV/u
after decay
Mass Number A
FIG. 3. Mass distributions normalized by the breakup cross section of 12C. Solid, dashed and dotted-dashed line indicate those in proton, a and 14N induced reactions. Left and right panel display those before and after statistical decay, respectively.
We compare mass distributions in proton-, a- and 14N- projectile reactions directly in Fig. 3. Since it is meaningless to compare mass distributions one another directly due to the geometrical effect from different radius of each projectile, the production cross sections are normalized by the breakup cross section of the 12C nucleus. Left and right panels show mass distributions before and after statistical decay, respectively. It is clearly seen that the mass distribution of the proton induced reaction is quite different from other two reactions. The normalized production cross sections of the fragments with A = 2,3,4 in the proton induced reaction is smaller than those in a and 14N induced reactions. This difference appears more remarkably before statistical decay. Especially the normalized production cross sections of 4He fragments before statistical decay are 3.05 x 10-3, 2.06 and 1.27 in proton, a and 14N induced reactions, respectively. We can say that alpha-cluster degrees of freedom in the 12C nucleus are more hardly excited during the dynamical process in the proton induced reaction than in a and 14N induced ones. In addition, there is the difference between a and 14N induced reactions before statistical decay. The normalized production cross section of the fragments with A = 8 in the a induced reaction is larger than that in the 14N induced reaction. This indicates that the a-cluster degrees of freedom are excited more easily in the a induced reaction than in the 14N induced one because the fragments with A = 8 are almost3Be fragments. But, since incident energies in these three reactions is different, this difference of the excitation of alpha-cluster degrees of freedom is not due to the projectile-mass dependence but due to the incident energy dependence.
Finally we display in Fig. 4 the excitation energy spectra of 12C before its breakup into any fragments during the dynamical process in proton, a and 14N induced reactions. Long- dashed, solid and short-dashed lines indicate those in proton, a and 14N induced reactions, respectively. As is seen clearly, the proton induced reaction is quite different from other two reactions. In a and 14N reactions excitation energy spectra have two component. One distributes in the region of 7—15 MeV and the other distributes in the region above 15 MeV. As was shown in Ref. [5], most of the former distribution results from the events of 12C breakup into 3a particles and these excitation energies correspond to excited levels of 12C
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FIG. 4. Excitation energy spectra of 12C before its breakup into any fragments during the dynamical process. Long-dashed, solid and short-dashed lines indicate those in proton, a and 14N induced reactions,respectively.
” 0 10 20 30 40 50 60 70 80 90Excitation Energy [ MeV ]
which have cluster structure. Therefore it is expected that alpha-cluster degrees of freedom is excited in o- and 14N-projectile reactions during the dynamical process. On the other hand, the excitation energy spectrum of 12C in proton induced reaction has only one component which distributes above 15 MeV. This energy roughly corresponds to the threshold of the one nucleon emission from the 12C nucleus. Hence in proton induced reactions it is difficult for alpha-cluster states to be excited during the dynamical stage and it is expected that shell-model-like excited states are mainly excited. This conjecture is also supported by the normalized production cross sections of a nucleon shown in Fig. 3. This cross section is the largest in the proton-projectile reaction, so a nucleon emission from the 12C nucleus most probably in the proton induced reaction. Moreover the cross section in the 14N-projectile reaction is larger than that in the a-projectile reaction. This may result from the difference of the effect of nucleon-nucleon collisions, because the 14N nucleus constitute of more nucleons than the 4He nucleus and the contribution of nucleon-nucleon collisions becomes large in the 14N-projectile reaction. The different size of projectile also changes the mean-field effect, which is expected to excite alpha-cluster degrees of freedom. But, as was mentioned above, the incident energies of all three reactions analyzed here is different, so we can not conclude immediately that these differences results from the projectile-mass dependence. It is one of future problems to investigate the incident energy dependence in each projectile reaction.
AcknowledgmentsMost of calculations for this research project were performed with the Fujitsu VPP500
of RIKEN, Japan. We thank Dr. I. Tanihata and Dr. S. Ohta for their arrangements for using this computer system. 1 2 3 4 5 6 7 8
N + C at 35 MeV/A
[1] K. Ikeda, H. Horiuchi, S. Saito, Y. Fujiwara, et al, Prog. Theor. Phys. Suppl. 68 (1980).[2] A. Ono, H. Horiuchi, Toshiki Maruyama and A. Ohnishi, Prog. Theor. Phys. 87, 1185 (1992).[3] Toshiki Maruyama, A. Ono, A. Ohnishi, and H. Horiuchi, Prog. Theor. Phys. 87, 1367 (1992).[4] F. Puhlhofer, Nucl. Phys. A280, 267 (1977).[5] H. Takemoto, H. Horiuchi, A. Engel and A. Ono, Phys. Rev. C 54, 266 (1996).[6] C. T. Roche et al, Phys. Rev. C 14, 410 (1976).[7] M. Jung et al, Phys. Rev. C 1 (1970)[8] A. Kiss et al, Nucl. Phys. A499, 131 (1989).
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20. The isovector/isoscalar ratio for the imaginary part of the medium-energy nucleon optical model potential studied by the Quantum Molecular Dynamics
Satoshi Chiba. Koji Niita, Tokio Fukahori, Tomoyuki Maruyama, Toshiki Maruyama and Akira Iwamoto Japan Atomic Energy Research Institute,
Tokai-m.ura, Naka-gun, Ibamki-ken 319-11, Japan
Abstract
Energy dependence of the ratio of the isovector and isoscalar strengths in the imaginary part of the nucleon optical model potential at the medium energy range was extracted from an analysis of proton and neutron induced total reaction cross sections on 11 Li with a theoretical framework called quantum molecular dynamics (QMD). The iso vector / isoscalar ratio was found to be about 0.8 at 100 MeV, and decreased almost linearly in log(E) to 0 at several hundred MeV. This result was consistent with an estimate at lower energy, and was also in good accord with the values used by Kozack and Madland for the analysis of nucleon + 208Pb reactions.
I. INTRODUCTION
In recent years, the intermediate-energy nuclear reactions have come to be more and more important for various reasons related with not only the basic but also the applied research fields [1]. It is obvious that the optical model potential (OMP) remains to be an important quantity in the researches of nucleon-induced nuclear reactions at medium energy range as was the case at the lower energy region.
The nucleon optical potential has been studied intensively in the past by many authors from many points of view [2-19]. It is known that the proposed potentials give excellent results in many cases. However, it is also recognized that they are still far from perfect in many aspects, and there are many ambiguities which prevent the OMP to be defined uniquely. If we look into the status of the imaginary isovector part of the OMP, the situation seems to be particularly poor: In the low energy region where the surface absorption is dominant, the strength of the surface imaginary isovector potential is distributed in the range from 9 MeV [10] to 16 MeV [9]. This shows that the imaginary isovector strength has an ambiguity as large as a factor of 2 in spite of the huge efforts to define the potential at low energy region where both the neutron and proton data are available. At the intermediate energy region where the volume absorption becomes dominant, the proposed global potentials do not give the isovector volume imaginary potential explicitely [3 12] (except one by Kozack and Madland [13]). This is against the idea of the Lane model [20] (and its relativistic extent ion [21]) on which the OMPs have been based. It is true that the difference of the nucleon- nucleon (N-N) interaction between the identical (p-p, or n-n) and non-identical (p-n) pairs
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of nucleons, that is the origin of the isovector term in OMP, becomes smaller and smaller as energy increases, and finally reaches to zero at several hundred MeV. This fact, together with small asymmetry ((;Y — Z)/A) range spanned by most of the stable nuclei, may make the net effect of the isovector part less and less significant in the intermediate-energy region. However, it cannot be a justification of ignoring the imaginary isovector term from the basic point of view.
For exotic nuclei such as 11 Li, the effects of imaginary isovector term will be significant because they have large asymmetry parameters. In the case of 11 Li, there are 8 neutrons and only 3 protons. Due to the big difference between the proton and neutron numbers, the isovector effects will be magnified in these nuclei. Furthermore, these exotic nuclei arc known to have an outer region consisting only of neutrons, i.e., the neutron halo or neutron skin [22]. Due to this structure, the incident particles will interact firstly with only neutrons, so it is expected for these nuclei to respond quite differently depending on the (z-component of the) projectile isospin. Therefore the effects of the isospin-dependent N-N interaction will still be noticeable for such exotic nuclei at intermediate energy while the Coulomb correction is kept to be negligible.
It is the basic idea of this work to use the feature of the exotic nuclei as an amplifier of the isospin-dependence in the N-N interaction to investigate the imaginary isovector OMP. For this aim 11 Li was selected as the target nucleus, and the total reaction cross sections for neutron and proton projectiles were calculated by a theoretical framework called quantum molecular dynamics (QMD) [23-25]. We use a QMD framework developed at JAERI [26], which has been used intensively for investigations of light-ion induced reaction mechanisms at intermediate energy region [27-30]. This framework, however, was not satisfactory in several senses. We then modified it for the present purpose as 1) to be Lorentz covariant [31], 2) to include the momentum dependence in the effective N-N interaction, 3) to include the Pauli potential to simulate the Fermion nature of the nucleon system better, and 4) to include a revised N-N collision term. As a check of the new framework, we have carried out an analysis of total reaction cross sections for carbon target with various kinds of projectiles, including “Li, for which experimental data are available.
II. BRIEF EXPLANATION OF THE QMD
The details of the formulation we adopted will be given elsewhere [32], so only a simple explanation is given in this paper. We start from representing each nucleon (denoted by a subscript i) by a Gaussian wave packet in both the coordinate and momentum spaces. The total wave function is assumed to be a direct product of these wave functions. Thus the one-bodv distribution function is obtained by the Wigner transform of the wave function,
/(r,p) = ]T/,(r,p) = 52 8 • exp(r-R,)2 2L(p-Pi)5
2 L(1)
where L is a parameter which represents the spacial spread of a wave packet, R, and P, corresponding to the centers of a wave packet in the coordinate and momentum spaces, respectively. The equation of motion of R, and P is given, on the basis of the time- dependent variational principle, by the Newtonian equation:
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R, — dHdPi'
dHdRi'
(2)
and the stochastic N-N collision term [26,36]. The Hamiltonian H was taken to be a sum of the zero-th component of the 4-momentum vector of each particle [31]:
H = E P? = E + m? + ZmiUj (3)
The scalar potential U, consists of the Skyrme-type effective N-N interaction [35], Coulomb and symmetry energy terms, the Pauli potential and the momentum-dependent potentials:
r, 1 A IBt'- 2fr<Pi> + lT^pZ
a
< pi >T + ^ E c- ci erf (a^/v^Z)* j(&) q,j
+ 2]^ E (! ~ 2|c, - Cj|) + Ej(/«)
+yd)
Z2po 1 + [Apij/fii]
j(^i) VC2)
2 P’J -----~2P° j(^,) 1 + [Apu/p2] 2 P'i (4)
where ''erf denotes the error function, and c, is 1 for proton, and 0 for neutron. Other symbols in this equation are defined as
<Pi> =12 P'i = E / drpiWpjir)
= E (4ttL)™3/2 exp [- A$"/4L]
A = /(2rfj$/',r'P)
l3
f<1
exp2 ql 2pi
^T, Tj
Agf, = -A# +,2 , (Ag.jP.j)2V
Ap2 = -Ap2 +
Ptj(ApjjPtj)
P>j(5)
and
At?,-, — q, — qjAPij = p, - pj
Pij = P, + Pj (6)
The <p and p, are the coordinate and momentum of particle i in the 4-vector representation, respectively. It is easy to note that the variables and Ap2 defined above are Lorentz scalars. In addition, this form of Hamiltonian gives the equation-of-motion equivalent with the Relativistic QMD [33,34] with a special choice of the time-fixation [31].
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The parameters in the above Hamiltonian were determined to reproduce the saturation dentisv p — p0 = 0.168 /m~3, minimum energy E/A — -16 MeV, the energy dependence of the real optical model potential, and the effective mass m"/m = 0.8 at the Fermi surface. Furthermore, the parameters of the Pauli potential were chosen for the kinetic energy of the QMD to be equal to the total energy of the free Fermion systems [24]. From these conditions, the following values were determined: A = -127.68 MeV, B — 204.28 MeV, r = 4/3, C, = 25 MeV, lA11 = -258.54 MeV, V™ = -375.60 MeV, p, = 2.35 MeV, p-2 = 0.4 MeV, L — 1.75 fm2, Vp = 140.0 MeV, p0 = 120.0 MeV, and qQ = 1.644 fm.
i i rt njEnergy dependence of real OMP
-P to -
8 - He "Be
■ Exp.—— QMDVJ*
QMO TotalHamaiHama2
I i.i i.nlI i i__i__i I i i i__i__LJ_l_L—L101Mass Number A
Fig. 1 Energy dependence of the real part of OMP Fig. 2 Binding energy per nucleon
Fig. 1 shows the energy dependence of the real part of the OMP. The solid curve shows the potential depth calculated from Eq. (4) without the Coulomb and the Pauli potentials. It is understood that the present parameterization reproduces the energy dependence of real OMP obtained experimentally [12] fairly well. The binding energies per nucleon of several stable nuclei calculated with QMD are compared with experimental data in Fig. 2. This figure shows that the QMD calculation gives a very good description of such basic nuclear structure information.
Li density distribution' r1— QMD Particle
QMD Gaussian vtin. l::i Exp.
Li + C—>®Li + X at790 MeV
■ QMD ------- Exp.------0 = 21 MeV/c -------o = 80 MeV/c
-200 0 200 Transverse Momentum [MeV/c]
Fig. 3 Nucleon density distribution of 11 Li Fig. 4 Transverse momentum distribution of 9Lifrom the 11 Li + C reaction at 790 McV/A
The nucleon density distribution of 11 Li is shown in Fig. 3 with the experimental data [22]. The smooth curve denotes the nucleon distribution calculated by QMD while the broken curves denote the upper and lower bounds of the experimental uncertainty [22]. In the same
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figure, we show the ” particle” distribution by the histogram, which means the distribution of the center of the Gaussian wave packets. The ’’particle” distribution shows the existence of valence neutrons outside a core (9Li). The neutron halo structure is reproduced well by the present calculation. The binding energy for 11 Li was calculated to be 44.89 MeV, which is consistent with the experimental value of 45.54MeV. Fig. 4 shows the transverse momentum of 9Li for the reaction 11 Li + C at 790 MeV per nucleon. The 2 components in the experimental data by Tanihata et al. [22] are reproduced excellently by the QMD calculation.
III. CALCULATION OF REACTION CROSS SECTIONS FOR ,2C TARGET AS A VERIFICATION OF THE COMPUTATIONAL METHOD
The reaction cross section was calculated based on the following formula, which is equivalent with the optical limit of the Glauber approximation:
oR = 2irjb( l -T(b))db (7)
where the T{b) denotes the transparency, i.e., the probability that the projectile having the impact parameter b causes no interaction with the target nucleons. In the Glauber approximation, such quantity is evaluated along a straight line trajectory, while in QMD it is calculated on a more realistic trajectory determined by the mean-field described by the effective two-body potential Eq.(4) including the Coulomb interaction.
TT ! rTTJT- I I I M |1200 -
----- QMD — QMD$ Exp. • Exp.
600 -
I i I l 111-Li i ini
o + 12c
----- QMDS' 800 S' 1000• Exp.
'T 600------- QMD
• Exp.
.1 I I l.U-l_L L-L-L
E/A (MeV)E/A (MeV)
Fig. 5 Total reaction cross sections of ,2C for incident proton (left-top), deuteron (left-bottom), o (right-top) and 12C (right-bottom).
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The reaction cross sections for 12C were calculated for projectiles of proton, deuteron, a and 12C, and are shown in Fig. 5 with experimental data [37-40]. These figures confirm that the QMD gives satisfactory descriptions of the reaction cross sections of various projectiles on carbon even though no parameter was adjusted for this purpose. The only exception is the case of d -f 12C for which the QMD overestimates the reaction cross section noticeably. The reason of this was found to be related with the stability of deuterium in the QMD simulation: Deuterium is a nuclei in which a proton and neutron bind each other with the biding energy of 1 MeV per nucleon. Such system is not stable enough in QMD calculation, so it breaks into a neutron and proton when it reaches to the carbon target and feels the mean-field (real OMP) of the target without causing any N-N collision (which is the origin of the imaginary OMP). In other cases, it could be concluded that the QMD calculation to be reliable.
The total reaction cross sections for 12C target induced by several Li isotopes are shown in Fig. 6. Again, the QMD calculation reproduces the basic feature of the experimental data [41,42]. The agreement is particularly good for 11 Li on 12C case.
Li (800 MeV per nucleon) + 12C =e— QMD$ Tanihata et al. ___ft
10 11
Fig. 6 Total reaction cross sections of I2C for incident ALi, where A = 6, 7, 8, 9 and 11.
IV. EXTRACTION OF THE ISOVECTOR/ISOSCALAR RATIO OF IMAGINARY NUCLEON OMP AT INTERMEDIATE ENERGY
Based on the success of the previous section, we proceed to extraction of the isovector/isoscalar ratio of the imaginary OMP. Firstly we define a quantity a to be
<Jr(p) - gfl(n) zoxcrfl(p) + oR{n)
where on{i) denotes the total reaction cross section for incident particle i. The quantity o was calculated at 100, 200, 400 and 800 MeV for 11 Li target by QMD, and is shown in Fig. 7. This quantity is found to be about 0.27 at 100 MeV, so the isospin dependence in the N-N collision really affects the total proton and neutron cross sections significantly. Such difference, however, becomes smaller and smaller as energy increases, and the effect is negligible at several hundred MeV. This is an intuitively understandable behavior.
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Then, we take the 1st order expansion of aR with respect to the imaginary OMP {W) around Wo;
aR{W — Wo ±eWi) = aR(W0) ± fW\-^ar{Wq) (9)
where + applies to incident protons and - to neutrons, W0 denotes the imaginary isoscalar strength, the W\ the isovector strength, and ( = (N — Z)/A. By using this formula, the quantity a can be written also to be
(Wi^aniWo)vr{Wq) (10)
The ratio W\/Wq is calculated by putting the 2 o’s in Eqs. (8) and (10) equal;
Ifi _ <?r(p) ~ <rx(n) (Tr(W0) 1
W0 aR(p) + aR{n) -^aR{WTo)eW0
The first factor of the right hand side was already obtained by the QMD calculation (Fig. 7). We have then calculated the 2nd factor by employing the following classical expression for oR obtained with the Glauber approximation for uniform sphere of radius R [43,44],
aR(W) = 7ri?2 ftl - (1 + 2RkW/E)e~2RkWIE (:IRkW/E?
(12)
where k denotes the wave number and E the projectile energy. The imaginary isoscalar strength W0 was taken from Finlay’s parameterization [45],
Wo15.353(E - 80)2
+ (E - 80)2 + 137.82 (13)
The ratio Wi/Wo calculated based on Eq. (11) is shown in the left part of Fig. 8. This figure shows that the ratio Wi/Wo has a value of about 0.8 at 100 MeV, then decreases almost linearly in log (E), and reaches to 0 at several hundred MeV. This energy dependence is consistent with that of the difference of the cross sections between the identical and non- identical pairs of nucleons. The error bar was obtained from the statistical uncertainty in the factor o, and by assuming (rather extremely) the error of W0 to be 50 %. The main source of error comes from the uncertainty in Wq at 100 MeV, while the statistical error is dominant at 800 MeV. Anyway, the results are rather insensitive to the choice of the H o parameter.
In the right part of Fig. 8, the low-energy limit of this ratio was calculated with the Walter-Guss potential [8] at 10 MeV, and plotted with the presently obtained results. The energy dependence obtained in this work extrapolates smoothly to the lower energy value. The result calculated from the parameters of Kozack and Madland [13] are shown by the dash-dotted curve in the same figure. These values were obtained by adding the (Lorentz) scalar and vector imaginary potential strengths for each of the isoscalar and isovector components. Their value is slightly higher than the present estimate at 100 MeV. However, these 2 curves become closer as energy increase, and finally they are consistent at 300 to 400 MeV region.
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E (MeV)
Fig. 7 The quantity a (defined in Eq. (8)) calculated by QMD for 11 Li
T—l"l I IIRatio of W, to W0
—#— Present • Walter-Guss
------ Kozack and Madland
0.5 -
j_i i 1111
E (MeV)
Ratio of W, to W0 for "Li reaction
E (MeV)Fig. 8 The isovector / isoscalar ratio for imaginary nucleon OMP derived as Eq. (11). The left
figure shows present result, while the right one includes the lower-energy estimate [8] and valuesused by Kozack and Madland [13].
V. CONCLUDING REMARKS
A QMD (quantum molecular dynamics) framework was used to extract information on the imaginary isovector term in the intermediate-energy nucleon OMP. The 11 Li was selected as an amplifier of the isospin-dependence in the nucleon-nucleon interaction which is the origin of the isovector potential. The difference in the reaction cross sections induced by neutron and proton on 11 Li indicated that the imaginary isovector potential plays a noticeable effect on the observables for such exotic nuclei at intermediate energy. The present result were found to be consistent with a lower energy estimate and with the values used by Kozack and Madland for the analysis of N + 208Pb observables.
It must be noted that there is a slight inconsistency in the generation of ground states for stable nuclei and 11 Li. For the stable nuclei, the ground states were generated by the Metropolis method with the temperature of 2.5 MeV by using the parameters for the effective
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nucleon-nucleon interatcion as given in the text. On the contrary, the temperature was set to be 0 MeV and another parameter set was used for 11 Li. Although this inconsistency does not alter significantly the conclusion of this work, a consistent description is definitely preferred. This will be one of the future subjects on this work.
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Phys. A 534 (1991) 720[35] T.H.R. Skyrme, Nucl. Phys. 9, 615(1959).[36] J. Cougnon, private communication.[37] W. Bauhoff, Atomic Data and Nuclear Data Tables, 35, 429(1986).[38] A. Auce, R.F. Carlson, A.J. Cox, A. Ingemarsson, R. Johansson, P.U. Renberg, O.
Sundberg and G. Tibell, Phys. Rev. C 53, 2919(1996).[39] M. Nolle, H. Machner and J. Bojowald, Phys. Rev. C 36, 1312(1987).[40] A. Ohnishi, Microscopic Simulation of Nuclear Reaction as a Tool to Evaluate Nuclear
Data, NRDF Annual Report 93, Hokkaido University (1994) (in Japanese).[41] I. Tanihata, H. Hamagaki, O. Hashimoto, Y. Shida, N. Yoshikawa, K. Sugimoto, O.
Yamakawa, T. Kobayashi and N. Takahashi, Phys. Rev. Lett. 55, 2676(1985).[42] I. Tanihata, T. Kobayashi, O. Yamakawa, S. Shimoura, K. Ekuni, K. Sugimoto, N.
Takahashi, T. Shimoda and H. Sato, Phys. Lett. B206, 592(1988).[43] H. Bet he, Phys. Rev. 57, 1125(1940).[44] M.S. Hussein, R.A. Rego and C.A. Bertulani, Phys. Rep. 201, 279(1991).[45] R. W. Finlay, Precision Total Cross Sections and the Optical Model at Intermediate
Energy, Proc. Int. Symp. on Fast Neutron Physics, 9-13 Sept. 1991, Beijing, China, World Scientific, p.299(1992).
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21. Status of Semi-Classical Distorted Wave(SCDW) Model
Y. Watanabe, H. Higashi, R. Kuwata, M. Kawai°, M. Kohnofc Department of Energy Conversion Engineering, Kyushu University
a Department of Physics, Kyushu University
bPhysics Division, Kyushu Dental College
1 Introduction
Nucleon-induced preequilibrium reactions have been studied extensively to enhance the understanding of non-equilibrium phenomena in excited nuclei. In the preequilibrium reactions at energies of more than tens of MeV, multistep direct (MSD) processes into continuum become dominant and the energy spectra with smoothly forward-peaked angular distributions are observed at the intermediate outgoing energy region. Various MSD models have been proposed and applied to analyses of experimental data[l], i.e., phenomenological models including several versions of the exciton model and statistical quantum-mechanical (SQM) models such as the Feshbach-Kerman-Koonin (FKK) model[2], the Tamura-Udagawa-Lenske (TUL) model[3] and the Nishioka-Weidenmuller-Yoshida (NWY) model[4]. More recently, microscopic simulation methods based on the Quantum Molecular Dynamics (QMD)[5] and the Antisymmetrized Molecular Dynamics (AMD)[6] have been applied as a new approach to study the nucleon-induced preequilibrium reactions
As an alternative SQM model, we have proposed the semi-classical distorted wave (SCDW) model[7, 8, 9]. The SCDW model is based on DWBA series expansion of the T-matrix and its energy average in a given energy bin of the exit channel as the other SQM models. It is greatly simplified by the local density Fermi-gas model to describe nuclear states, a local semi-classical approximation to the distorted waves, and the Eikonal approximation to intermediate state Green functions. Under these assumptions and approximations, the double differential emission cross sections of each MSD step can be expressed in a simple closed form in terms of the distorted waves, the nucleon-nucleon scattering cross sections, and the nucleon density distribution, as will be described in the next section. If these quantities are given either empirically or theoretically, no free adjustable parameter is involved in the SCDW model. Furthermore, the expressions for the cross sections allow us a simple intuitive interpretation which gives a justification for a basic assumption of the intra-nuclear cascade (INC) mo del [10] that the reaction proceeds via sequential nucleon-nucleon collisions in a nucleus and each reaction path has no interference.
In our previous report[ll]1, we presented some results of the SCDW model calculations for 1- and 2-steps of (p,p'x) and (p, nx) reactions at energies up to 200 MeV, and pointed
'There were numerical errors in the calculated 2-step cross section by an order of magnitude in Figs. 1 through 3.
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out that the calculations involving higher-step processes (at least 3-step) were necessary for comparisons with the other model calculations. In this paper, we report on the present status of the SCDW model, showing some new results of the SCDW calculations for the leading three MSD steps of (p,p'x) reactions.
2 Outline of the SCDW model
The detail of the SCDW formulation has been described elsewhere}?, 8, 9]. Only the final SCDW formulae are given below. The 1-step cross section is expressed by
{dEfdnfd2a
dEf(r)dttf(r) Xr), (1)
where A is the target mass number, kc and A:c(r) (c=i and /) are the wave number at infinity and the local wave number in the initial (z) and final(/) channels, xt(Xf) is the distorted wave in the initial (final) channel, p(r) is the nucleon density, and
4mfc/(r)KdEf(r)dttf(r)) h2ki(r)(4Tr/3)kF(rY
/ (2)
k<kF(r
is the local average differential cross section of N-N scattering where k — (k,(r) — k)/2 (k1 = (ky(r) — k')/2) is the relative momentum in the two-nucleon c.m. system where k(k') is the momentum of the struck target nucleon in the initial (final) state. The local average cross section, however, is put to zero if the Pauli principle, kj(r0) and k' > fcp(r0), is violated. The local kinetic energy, Ef(r) = %2 k j(t0)2/p, and direction, Oy(r) = ky(r0), of emission correspond to Ej and fly at infinity, respectively. In Eq. (2),
' dadi1*/ NN
(W2)2(2irh2y
J dxu(x)e ,q x (3)
is the N-N scattering cross section with the momentum transfer q = k' — k — ky(to) — ki(r0), where m is the nucleon mass and v(x) is the two-body potential acting between two nucleons.
In the calculation of 2- and 3-step cross sections, the Eikonal approximation to the intermediate Green function is made as an additional approximation[8]:
< r21
~Em — A — Um + ii7 n >% 2P exp(ikm |r2 - n|) h2 |r2 — i*i | (4)
where iq(r2) is the first (second) collision point, and km = [(2p/%2)2(Em — Um)Y^2 = Km+z7m is the complex local wave number in the intermediate state.
As given in Ref.[8], the final form of the 2-step SCDW cross section is expressed by
' avz)
dEfdSlj(aTi) /*'/*
/ d2a \dEfdrt2 />(r 2)
h/kf(r 2) ki/ki(r 0
exp(-27rn |r2 - n|)
X)
a2<r2 - ri f(n) (5)
ri
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where Em = h2k^n/2n and is the direction of | r2 — r% |. The local average cross sections are given by Eq. (2) with approximate substitutions of coordinates and momenta.
The extension to the 3-step is straightforward and the cross section is deduced to the following expression:
x
X
dEmX J dEm2 J drx j dv2 J dr: h (r3) kj ' k (rx)kt
( 92(7 \ ( \ exp t 2Tm2 |r3 - raj] / d2a|r3-rf
exp [—27mi |r2 - rx\] ( d2a \|r2 — rj|2 \dEmXdSlmX)r/[ U
p(r 2)
(6)
In the (p,p'x) reaction, there are two types of the 2-step processes, {p,p"){p",p') and (p, n)(n,p'), in accordance with the kind of intermediate fast particle, either proton or neutron. Similarly, the 3-step process consists of four different paths; (p, p")(p",p"'){p'", p'), (p,p")(p",n)(n,p'), (p, n)(n,p")(p",p') and (p, n)(n, n')(n', p'). Hence the final expressions of the 2- and 3-step cross sections are given by the incoherent sum of each contribution.
The primary physical quantities necessary for the SCDW calculation are (a) the distorting potentials, (b) the two-nucleon scattering cross sections and (c) the nuclear density distribution, as shown in Eqs. (1), (5) and (6). Basically, the same input data as in [9] are used for those quantities. The global optical potentials of Walter and Guss[12] are adopted as (a) for energies less than 80 MeV and those of Schwandt et al. [13] for energies above 80 MeV. For neutrons of the intermediate fast particles, however, the real part of the optical potential parameters of Ref.[13] is modified according to Madland’s method[14]. The nonlocality of the distorting potentials is considered by means of the well-known Percy factor [15]. Note that the Percy factor is unity for bound state wave functions in the Fermi gas model because of normalization. The nonlocality range (3 in the Percy factor is taken to be 0.85 fm[15]. For (b), two options are took into account: two-nucleon scattering in free space or in nuclear medium. The free N-N cross sections are given by the same empirical formula of the differential N-N cross section as in Ref.[9], he., the parameterized cross sections taken from Ref.[16] and angular distributions given in Ref.[17]. In-medium N-N cross sections calculated in nonrelativistic Brueckner approach using the Paris potential by Kohno et al.[24] are used to take into account two-nucleon scattering in nuclear medium with the framework of the SCDW model. For (c), the nuclear density distribution of the Woods-Saxson shape with Negele’s geometrical parameters[18] is employed.
3 Results and discussion
3.1 Comparison of calculated angular distributions with experimental data
Using a Monte Carlo integration method with quasi-random numbers[19, 20], we have implemented SCDW calculations for 58Ni(p,p'x) reactions at incident energies of 65, 120, and 200 MeV, and 90Zr(p,p'x) at 160 MeV.
Figures 1 and 2 show comparisons of the SCDW angular distributions of 58Ni(p, p'x) reactions for 120 and 200 MeV calculated using the free N-N cross sections with the experi
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mental ones[2l]. In these figures, each MSD step is decomposed to see how the contribution varies with proton emission energy; the dotted line, the dash-dotted, and the dashed lines represent each component of 1-step, 2-step and 3-step processes. Their sum is represented by the solid lines. The SCDW calculation with no free parameters is in overall agreement with the experimental data except at very small and large angles, although underprediction is seen over the whole angular region at the lowest emission energy for both incident energies. For the highest emission energy, the SCDW calculation has a peak around 20° and overestimates the experimental data by a factor of about two. Similar peaks are also seen in the angular distributions at other emission energies, and they become broader with decreasing proton emission energy. Similar results were obtained for the other reactions.
Comparing the stepwise components in Figs. 1 and 2, we can see that proton emission via the 1-step process is dominant in the intermediate angular region and contributions from the 2- and 3-step processes become appreciable with increase in emission energy and angle. It is found, however, that the MSD components cannot compensate enough the discrepancies between the 1-step cross sections and the experimental data at backward angles. That is possibly because higher momentum components of target nucleons above the Fermi momentum cannot be taken into account properly by the degenerate Fermi-gas model assumed in the SCDW model. The 1-step angular distributions fall off steeply toward 0° at all outgoing energies as shown in our previous calculation]!)]. On the other hand, the 2-step and 3-step cross sections have smoothly forward-peaked angular distributions and their values are not zero at 0°.
3.2 Comparison with other model calculationsWe compare the SCDW calculations with the results of the other models (AMD[6], QMD[5], and FKK[5, 22, 23]) to see similarities and differences among them in Figs. 3 and 4.
First, a comparison with the AMD[6] is given for 58Ni(p,p,x) at 120 MeV in Fig. 3. Agreement is generally good, although the SCDW 1-step and 2-step cross sections are smaller than the AMD ones. It is interesting to note that the AMD 1-step cross sections also show peaks near the angle corresponding to the quasi-elastic scattering(QES), though slightly shifted forward. Such the peaks do not appear in the calculations based on the other models (QMD and FKK) as will be shown later.
Second, we compare the SCDW angular distributions with the QMD ones[5] for 58Ni(p, p'x) at 120 MeV in Fig. 3 and 90Zr(p,p'x) at 160 MeV in Fig.4. From comparisons between two model predictions, we notice that the shape of 1-step angular distributions is different, especially at very small angles and backward angles. The QMD calculations show the forward peaked 1-step angular distributions without the steep fall-off near 0°. According to Ref.[5], the behavior of the 1-step cross sections near 0° is strongly influenced by the refraction effect due to the mean field potential. The SCDW model also takes into account the refraction effect by the distorting potentials of the entrance and exit channels. It is, however, is expected to be weak because the depth of the real potentials including the Coulomb potentials becomes shallow with increasing proton energy. The difference seen at backward angles is due mainly to the momentum distribution of target nucleons as discussed in Ref.[5]. The QMD calculation includes the target nucleons with higher momenta than the Fermi momentum, whereas the SCDW model assumes the degenerate Fermi-gas model with the zero-temperature. As for the 2 and 3-step cross sections, both calculations are almost similar, though the SCDW
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yields rather smaller the cross sections at the highest emission energy than the QMD.Third, a comparison of the SCDW with the FKK[5] is made in Fig.4. It should be noted
that the FKK calculation includes an adjustable parameter Vq to fit the magnitude of experimental data. Both the models are based on the DWBA approach. However, a considerable difference is shown between both predictions of 1-step cross sections, although the multistep components do not differ much. The FKK gives steeper 1-step angular distributions than the SCDW. As a result, the relative contributions from the 2- and 3-step processes for the FKK are large at backward angles compared with the SCDW. In addition, one can compare with the FKK analysis of 58Ni(p,p'x) by Richter et al. [22]. They showed that the QES component based on the DWIA calculation was necessary to reproduce the experimental angular distributions especially at high emission energies. According to Ref.[22], the cross sections in the angular region of 40° to 70° for the emission energy of 100 MeV in Fig. 1 cannot be reproduced well by the FKK calculation alone, but the discrepancy can be resolved by the addition of the QES contribution. Our SCDW model prediction is in excellent agreement with the experimental data at the corresponding angular region.
From the comparisons among those three model predictions, we conclude that the shape of the 1-step angular distributions depends strongly upon the models, but the multistep components are not different much in shape among those models. A general trend shows that the relative contribution to each step is almost similar .
3.3 In-medium effect on SCDW calculationWe have recently calculated in-medium N-N cross sections from G-matrix of the nonrela- tivistic Brueckner approach[24]. The result is shown in Fig.5. The solid curves are obtained from the parametrization of the in-medium N-N cross sections as a function of the incident energy and the nuclear density. As the incident energy increases, the in-medium N-N cross sections become close to the free N-N cross sections presented by the closed circles. This trend is different from that of in-medium cross sections calculated in the relativistic framework by Li and Machleight[25] which are reduced from free ones even at intermediate energies. We understand that the difference does not appear as one of pure relativistic effects, but is rather due to the flux renormalization represented by an effective mass m*. The appearance of m* is related to the nonlocality of the single particle potential. In SCDW, the nonlocality is taken into account by the Perey factor as mentioned in the section 2. Thus, one should use the in-medium cross sections without the flux renormalization, namely the effective mass m* being replaced by the bare mass m, to avoid double-counting the nonlocality effects in the SCDW calculations.
The SCDW calculation in which the free cross sections are replaced by the in-medium ones is shown in Fig.7. From the comparison with Fig.l, there is found to be no appreciable difference between both calculations. Similar comparison was also made for the reaction 58Ni(p, p'x) at a low incident energy of 65 MeV, where the in-medium cross sections are largely reduced from the free ones at normal density p0. However, the in-medium effect on the SCDW calculation does not appear obviously, because the predominant 1-step process occurs mainly in the peripheral region of a nucleus where the density is enough low and the in-medium cross sections are close to the free ones.
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4 Summary
The SCDW model to describe the preequilibrium MSD reaction was extended so as to include the 3-step process. The MSD calculations of 58Ni(p,p'x) at energies of 65, 120 and 200 MeV and 90Zr(p, p'x) at 160 MeV were carried out using the extended SCDW model and compared with the experimental data. The calculations with no free parameter showed overall good agreement with the experiment, although underprediction is seen at very small and backward angles. We found that the 2- and 3-step contributions were not so large enough to compensate the difference between the 1-step cross sections and the experimental ones at backward angles. The discrepancies seen at very small and large angles is possibly responsible for the local Fermi-gas model which does not work well in the nuclear surface region. The comparisons of the SCDW calculations with the AMD, QMD and FKK calculations led to an interesting result that the differences in the shape of 1-step angular distributions are remarkable among the models, but the multistep components are rather similar in the shape of angular distributions and the step-wise contribution is not so much different. The in-medium N-N cross sections were calculated in the nonrelativistic Brueckner framework with the Paris potential, and were parametrized as a function of the incident energy and the nuclear density. The SCDW calculation with the in-medium N-N cross sections was not so different from that with the free ones.
Acknowledgments: We wish to thank Drs. S. Chiba, T. Maruyama, K. Niita, and A. Iwamoto in JAERI, and Drs. Horiuchi and E.I. Tanaka in Kyoto University for their correspondences about QMD and AMD calculations, respectively, and many stimulating discussions. The financial aid of RCNP, Osaka University, for the computation is gratefully acknowledged. The work was supported in part by Grant-in-Aid for Scientific Research of the Ministry of Education, Science, and Culture (No.07640416).
References
[1] See, for example, E. Gadioli and P. E. Hodgson, Pre-Equilibrium Nuclear Reactions, (Oxford University Press 1992).
[2] H. Feshbach, A.K. Kerman and S. Koonin, Ann. of Phys. 125, 429 (1980).
[3] T. Tamura, T. Udagawa and H. Lenske, Phys. Rev. C 26, 379 (1982).
[4] H. Nishioka, H.A. Weidenmuller and S. Yoshida, Ann. of Phys. 183, 166 (1988).
[5] S. Chiba, M B. Chadwick, K. Niita, T. Maruyama, T. Maruyama and A. Iwamoto, Phys. Rev. C 53, 1824 (1996).
[6] E.I. Tanaka, A. Ono, A. Horiuchi, T. Maruyama and E. Engel, Phys. Rev. C 52, 316 (1995).
[7] Y.L. Luo and M. Kawai, Phys. Lett. B235, 211 (1990); Phys. Rev. C 43 2367 (1991).
[8] M. Kawai and H.A. Weidenmuller, Phys. Rev. C 45, 1856 (1992).
[9] Y. Watanabe and M. Kawai, Nucl. Phys. 560, 43 (1993).
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[10] M.L. Goldberger, Phys. Rev. 74, 1269 (1948).
[11] Y. Watanabe, Proc. of the Second Symp. on Simulation of Hadronic Many-Body System, Nov. 30 to Dec. 2, 1994, Tokai, Japan., JAERI-Conf 95-012 (1995), pp. 119.
[12] R.L. Walter and P.P. Cuss, Int. Nat. Conf. Nuclear Data Basis and Applied Science, Santa Fe, New Mexico, 1985, ed. P C. Young, (Gordon and Breach, New Yor, 1986) p.1075.
[13] P. Schwandt, H.O Meyer, W.W Jacobs, A.D. Bacher, S.E. Vigdor, and M.D. Kaitchuck, Phys. Rev. C, 26, 55 (1982).
[14] D.G Madland, in Proceedings of a Specialists’ Meeting on Preequilibrium Reactions, Semmering, Austria, 10-12 February 1988, edited by B. Strohmaiier (OECD, Paris, 1988), p.103.
[15] F.G. Perey and B. Buck, Nucl. Phys. 32, 353 (1962).
[16] K. Kikuchi and M. Kawai, Nuclear matter and nuclear reactions (Oxford Science Publication, Oxford, 1992).
[17] H.W. Bertini, Oak Ridge National Laboratory Report No. ORNL-3383 (1963).
[18] J. W. Negele, Phys. Rev. C 1, 1260 (1970).
[19] C.B. Haselgrove, Math. Comp. 15, 323 (1961).
[20] Y. Akaishi, private communication (1993).
[21] S.V. Fortsch, A.A. Cowley, J.J. Lawrie, D M. Whittal, J.V. Pilcher, and F.D. Smit, Phys. Rev. C 43, 691 (1991).
[22] W.A. Richter, A.A. Cowley, R. Lindsay, J.J. Lawrie, S.V. Fortsch, J.V. Pilcher, R. Bonetti, and P.E. Hodgson, Phys. Rev. C 46, 1030 (1992).
[23] W.A. Richter, A.A. Cowley, G.C. Hillhouse, J.A. Stander, J.W. Koen, S.W. Steyn, R. Lindsay, R E. Julies, J.J. Lawrie, J.V. Pilcher, and P.E. Hodgson, Phys. Rev. C 49, 1001 (1994).
[24] M. Kohno, M. Higashi, Y. Watanabe, and M. Kawai, in preparation (1997).
[25] G.Q. Li and R. Machleight, Phys. Rev. C 48, 1702 (1993).
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Ni(p.p'x) Ep=120MeV Ep'=100MeV
FreeNN
— 1 +2+3step— 1step
■ - 2step- 3step
Exp.
Ni(p,p'x) Ep=120MeV Ep'=60MeV
FreeNN. exd.0 101+2+3stepi
-1 step- - 2step
E 10 — - 3step
5 10
Ni(p.p'x) Ep=120MeV; Ep'=40MeV
FreeNN
E 10
D 10
- 3step
100©(deg)
Ni(p,p'x) Ep=200MeV Ep'=170MeV
FreeNN
-1 step- - 2step
----3step1+2+3step
■o 10
BNi(p,p'x) Ep=200MeV Ep’=120MeV\FreeNN. =xp.
1 +2+3step---- 1step- - 2step
--------3step
■ i ■
0 SO 100 150
Ni(p,p'x) Ep=200MeV: Ep'=60MeV
FreeNN ]
1+2+3step
----3step
0 (deg.)
Figure 1: Comparison between theoretical and Figure 2: The same as in Fig.l but 58Ni(p,p'x) measured angular distributions for the reaction at an incident energy of 200 MeV. The experi- 58Ni(p,p'x) at an incident energy of 120 MeV mental data are taken from Ref. [21] for various emission energies. The experimental data are taken from Ref. [21]
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JAERI-Conf 97-007
lNi(p,p'x) Ep=120MeV Ep'=60MeV
SCDW Foertsch et al.
— SCDW total
-- - SCDW tstep
- - SCDW 2step
• - SCDW 3step
■ I .
0 30 60 90 120 150 180
'Ni(p,p'x) Ep=120MeV Ep'=60MeV
• QMD • Foertsch et al.------------QMD total
**-. ..................QMD 1 step
QMD2stepv, x
\ V— — - QMDSstep
0 30 60 90 120 150 180
’Zr(p,p'x) Ep=160MeV Ep'=80MeV
SCDW• Richter et al.-------- SCDW total......... SCDW tstep------- SCDW 2step------- SCDW 3step
5 10
0 50 100 150
'Zr(p,p'x) Ep=160MeV Ep'=80MeV
QMDRichter et al.
— QMD total
... QMD tstep
- - QMD 2step
• - QMD Sstep
j__i_L
0 50 100 150t i i i r 1 I 1 1
'Ni(p,p'x) Ep=120MeV Ep'=60MeV
AMD e Foertsch et al.'’vX -----------AMD total
~ .............. AMD tstepyi ----- AMD 2step
"x. — - AMDSstep
■ ■ I0 30 60 90 120 150 180
@(deg)
’Zr(p,p'x) Ep=160MeV Ep'=80MeV
FKKRichter et al.
— FKK total --- FKK tstep- - FKK 2step
■ - FKK Sstep
% 10'
6 (deg.)Figure 3: Comparison between theoretical and measured angular distributions for the reaction 58Ni(p,//x) at an incident energy of 120 MeV for an emission energy of 60 MeV: (a) SCDW, (b) QMD [5], and (c) AMD [6]. The experimental data are taken from Ref. [21]
Figure 4: Comparison between theoretical and measured angular distributions for the reaction 90Zr(p,p'z) at an incident energy of 160 MeV for an emission energy of 80 MeV: (a) SCDW, (b) QMD [6], and (c) FKK [6]. The experimental data are taken from Ref. [24]
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JAERI-Conf 97-007
(b) p-n scattering
P-Pq/2
co 200
100 150lab. energy (MeV)
(a) p-p scattering
E 150p = Po 12
to 100
100 150lab. energy (MeV)
Figure 5: In-medium cross sections for (a) pp scattering and (b) np cross section. Results of Brueckner calculations in nuclear matter at the normal density p0 = 0.18 fm-3 and are shown by the filed squares and open circles, respectively. The filled circles stands for the cross section in free space. The curves are obtained by the parametrization.
1 1 rT i—pi r“”j i r
lNi(p,p'x) Ep=120MeV Ep'=60MeV KohnoNN
------ 1 +2+3step........1step— - - 2step— - 3step • Exp.
j_I_i_i—L
30 60 90 120 150 180®(deg)
Figure 6: SCDW calculation with in-medium cross sections for the reaction 58Ni(p,p'x) at an incident energy of 120 MeV for an emission energy of 60 MeV.
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22. QMD
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References
[1] J. Aichelin, Phys. Rep. 202 (1991) 233, and reference therin.
[2] K. Niita et al., Phys. Rev. C52 (1995) 2620.
[3] M B. Cadwick et al., Phys. Rev. C52 (1995) 2800.
[4] S. Chiba et al., Phys. Rev. C53 (1995) 1824.
[5] S. Chiba et al., Proc. of Int. Conf. on Nuclear Data NEA/NSC Specialist Meeting on
Intermediate Energy Nuclear Data, May 30 - June 1, 1994, France, (1995) p.137,
[6] S. Chiba et al., Phys. Rev. C July (1996)in press.
[7] T. Maruyama et al., Prog. Theo. Phys., Vol. 96 in press.
[8] M. Kanazawa et al., Phys. Rev. C35 (1987) 1828.
[9] T. Emura et al., Phys. Rev. Lett.73 (1994) 404.
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