ST〈TE
CYCLOTRON LABORA工ORY
受入
一秘
高工研図書由
PlOMC FUSlON lN ト11≡1AVY lON C◎LL1S1ONS
AND Tトl1三 CLUSTER1NG CORRELATl◎N
曳
T◎SH1TAl〈A 1〈AJ1N◎，ト”ROSHl T◎K1ond KEN−1Cト11KU8∪
NOVEM8εR1986
MSUCL−579
Novelnber 1986
Pionic Fusion in Heavy
I on
Collisions and the Cl us tering CorFelation
Tosh i taka Ka J ino
National Superconducting Cycl otFon
Labo ra to ry
)(ichigan State Un ivers i ty
East Lansing, NI 48824, V.S .A.
and
ae
De partment of Physics, Tokyo Metropolitan
Un iver s i ty
Fukazawa 2-1-1, Setagaya-ku, Tokyo 158, Japan
an d
Hiroshi Toki and Ken-ichi Kubo
Department of Physics, Tokyo Metropolitan Un iv er s
F ukaz awa 2-1-1, Setagaya-ku, Tokyo 158, Japan
t
Permanent address
i ty
) Be at the
7
4
3
07
subthreshold energy are studied theoretically. Toe cross section
Abstract: Pionic fusion reactions He(
4 3He,1T+) Li and He( He,
is
enhanced largely by the clustering correlation . General trend of targetmass dependence of the A( 3He,1T+)C reaction is also studied.
l
1 . Introduction
One of the recent topics in pion physics is the pion production in
heavy ion collisions at subthreshold energies. Pions more than those
expected In the Independent partrcle modell) are observed in heavy-ion
collisions2 4) although the beam energy per nucleon is significantly lower
than 280 NeV which is the pion production threshold in fFee nucleon-nucleon
collision. To explain the phenomena one has to assume a cooperative or a
coherent mechanism by which almost all kinetic energy of relative motion of
the colliding nuclei is converted into the pion mass . Several recent
5,6)
of inclusive pions also have suggested the existence of the
measurements
collective me9hanism. However, the answer to the question as what kind of
collective mechanism really dominates the subthreshold pion production is
still unclear both experimentally and theoretically.
The first purpose of this aFticle is to propose an interacting cluster
model for the pionic fusion. Our picture is that the high energy and high
momenturn part of the clustering components in the final nucleus cooperates
with the relative motion in the entrance channel to permit coherent pion
production. We are therefore interested in the question whether the pionic
fusion always shows up the clustering corFelation at the short distance. It
is to be noted here that the present model is quite different from the
phase-space model with cluster formation which was proposed by Shyam and
Knoll f.oF the inclusive pion production. OuF model is based on the
micFoscopic cluster model and aims at studying the exclusive pionic fusion
reaction .
.The pionic fusion process at subthreshold energy provides with a good
probe for the study of coherence of pion production since many complicated
processes are excluded by identifying the ground state of the final nucleus.
There is a small energy left for nuclear excitation.
The experimental data
3 He,
+
are restrlcted to the llght nucleaF systems like3 He(
2
6) 8)
Li ,
4He(3 He,+ 7) Li9), etc.
In this paper we study the two reactions
4 3 +7 4 3 07
He( He,1T ) Li and He( He,1T ) Be' in detail.
So far, there have been at least two different theoretical models for
studying the exclusive pion production ; one is a semiempirical model of
10)
Gelmond and Wilkin which describes the ｦ+_pFoduction in terms of the
3He(p,1T+4
) He pFocess, and the other is a A-excitation model of Klingenbeck,
Dillig and Huber . 1 1 ) Although both models refer the significance of the
4He + d OF 3He + t and Li
cluster structure Li
nuclei, the harmonic oscillator wave function was practieally used without
taking account of the strong clustering correlation of these nuclei. In
= = He + t of the final
addition , the distoFtion of the entrance channel and the Pauli exclusion
principle were neglected. These models have not been extended to heavier
systems .
The
econd purpose of this paper is to extensively estimate the
variation of pionic fusion cross sections foF A( 3He,1T+)C with respect to the
tanget mass A, which is varied from He to the heavier nuclei C - Ca.
Although this estlmate could' be worked out micFoscopically in the same
) Li Feaction, we take here several reasonable
4 3 +7
method as foF the He( He,
approximations in order to clearly pick up the target-mass dependence of the
cross section.
The model and the method for practical calculations are explained
briefly in the next section. In sect. 3 we show the calculated results for
4 3 +7
4 3 07
the two reactions He( He,ｦ ) Li and He( He, ) Be, and the Fole of the
clustering correlation is discussed. in detail. The target mass dependence
of the pionic fusion cross section for A(3He,1T+)C is discussed in sect. 4
and finally, the present study is.summarized in sect 5
3
2 . Nodel
4 3 07
3 +7
Let us consider the pronlc fuslons 4He( He,1T ) Li and He( He,1T ) Be
at
subthFeshold energy Elab/nucleon = 88.8 MeV.t This energy corresponds to
'ECM = 152.3 l(eV (-- mlT + 12.3 NeV and 17.3 ,(eV for chaFged and neutral
pions) .
The differential CFOSS Section in the c.m. system Is glven by
do
E7Mf'vkTr(fv 2
d 8 _Ei 2Vrel
1
(1)
k. .k
where T)(fv Is the transltlon matrix, Vrel= E + ET Is the relatlve veloclty
a
m the entrance channel, LD, E E and E7 are the relatlvlstlc energres of
a' 1
4 3
7
7
the pion, He, He and Li(oF Be), Ei = Ea + ET, Ef = E +7 a,, and ki and klT
denote the relative momenta in the entrance and the exit channels,
respectively . The tFansition matrix is defined by
; l
TNfv = <klT JfMf H 1 1/2 v >.
o
(2)
As shown In flg. I , Ho Fepresents the pion pFoduction operator, 1 1 /2v> Is
3 4
the initial scatteFing state of the He ,+ He system with channel spin I 12
and its proJection v, and IJfMf> is the final bound stat.e of Li with total
spin Jf and its proJection Mf' Expllcit forms of these nuclear states will
be defined later .
As for the nuclear states , correlated microscopic cluster model wave
functionsl2, 13) are adopted
4
< a ;T; I JM >
=V 4!3! AaT{[ca( a) c(1/2)(gl) Q iLY(L)(r)]( ) XJL(r)},
tThe corresponding
(3)
beam energy E lab =266.5 MeV was used in the experiment of
Bimbot et al 9)
5
where A T Is the antisyrnrnetrizer of nucleons and ccB(;a) and c (; ) are the
T T
4
inteFnal wave functions of He and He (or trlton) respectively,
X (r)=X
(r)iLY(L)(r) Is the mtercluster relatrve wave functron whlch rs a
.
JL JL
solution of the RGM equation of motionl4)
J l,)RGM(+ 'I WD(?)6(; ; )} XJL(rt) drt
,
= E N RGM
(r +,
r')
X JL(r')
'
dFf ,
where H (r F,'
.. ) and NRGM (; ;,) are well known Hamiltonian and
normalization kernels. The Hamiltonian kernel consists of two parts , the
kinetic energy of nucleons with the c.m. energy subtracted and the
interaction energy. As for the nucleon-nucleon effective inteFaction, the
modified-Hasegawa-Nagata ( central part) and the Nagata (spin-orbit paFt)
forcesl5) aFe used . Coulomb force is taken into account to examine the
4 3 +7
effect of charge sy!nrnetry breaking in the two reactions He( He,1T ) Li and
4He(3He,1T
07) Be. In addltlon to the Feal Interactlon we take account of a
local imaginary potential iWD(r) between the two clusters. WD(r) has the
same radial dependence as the real paFt of the nuclear direct folding
potential and the depth is set equal to WD(O) = -25 MeV for the scattering
7
states and WD(O) O for the bound (Jf 3/2 and 1/2 ) states of Li and
7Be. WD(O) = -25 NeV is a reasonable value obtained in the DWBA analysis
16)although there are observed data only for lo
for .the (3He,t) reactions
energy region (E
CM
40 MeV). The internal wave functions of the
constituent clusters c and c aFe assumed to have the highest spatial
a T
sym:netFles (Os)4[4] and (Os)3[3] with Gaussian radial dependence. Different
6
4 3
◎sci1ユa七◎r parame七ers B and B are ad◎pted for He and He （◎r 七ri七◎n），
α τ
・ep…七・1y．B・thp…m・七・…ユ・・…七i的肋・…i・七i…1・七・bi1i七y’
c◎nd it i◎ns and 七he wave fuηctions repr◎duce 七he ◎bserved charge radi i and
12，13）
bエnding energies very we1ユ．
In ◎rder ヒ◎ cユarify 七he r◎1e ◎f 七he cユus七ering c◎rre1a七ion ◎n the
7
pi◎n ic fus ion， we c◎ns ider an◎七her wave func七ion f◎r Li assuming 七he she1ユ
・。・・1・・・・・・・・・・…（・・）4（・・）3［・・1・・・・・・・・・・・・・…ll・1・・…㎝・・…。・・
一2 7
se七 equaユ 七◎ O．32 fm s◎ 七ha七 七he ◎bserved charge rad ius of Li i s
Pepr◎duced by this e◎nfigura七i◎n． N◎te 七hat the cluster wave functi◎n （3）
is equivaユent 七◎ 七he she1ユ modeユ wave func七i◎n ωhen 七he c◎rreユa七ion be七ween
the 七w◎ c1usters is c◎mple七eユy s凹itched ◎ff in 七he l imi七 ◎f B ◆ B and B “
α 7 τ
37・Thesheユ1㎜◎d・1・…∼・・七i㎝p・mi七・㎝1y七heF・mi㎜◎七ionina
harm◎n ic ◎sc iユ1a七◎p weユ1 w i七h◎u七 any ◎七her c一◎rre！a七i◎ns．
The in i七iaユ sca七七er ing wave wi七h channeユ spin S ： 1／2 in eq． （2） is
c◎nstruc七ed by superp◎sing 七he clus七er wave func七i◎ns （3） as
“
＜ζ …； r11／2 v〉
α 11
・Σ州／2L・1（LO1／2・lJ・）〈ξζ引洲〉．
（5）
α τ
L，J
I・七hi・・・…七h・i・七…1・就・…1・乞i・・囎・・f…七i㎝XJL（・）i・…1・七i・・
◎f eq． （4） sa七isfying 七he scat七ering b◎undary c◎ndi七i◎n．
F◎「七h・pi㎝p「・d・・七i㎝・p…七g・HO…ユy肋…nt「ibuti◎nofsingle
nucユe◎n pr◎cess is c◎nsidered． Aユ七h◎ugh 七here are severa1 f◎pms ◎f 七he ¶NN
17）
vertex ， we ad◎pted pseudovec七◎r c◎upl ing 七erm which g ives a Gaユiユ6an
invaria皿七 f◎r㎜
フ
=m J
H V41Tf 7 )6(;1T
n'{(1i=
7 )
;n
o
c
nc } n.$(?1T)d;1T'
(6)
n= 1
Here .o and Tn are spin and isospin operators of nu leons, $(; ) Is the plon
n
* pion and nucleon . The coupling
field, and r and rn are the coordinates of
f2=
. Slnce we work In the c.m. system of the
constant is taken to be
O 08
pion and 7Li (or 7Be), the following equation was used in eq. (6);
ｦn
= (1+=r)V
-u nC'
co*C )(N
7MN
where
,
and
(7)
are derivatives with respect to the Felatrve
n ITC nC
coordinates between pion and nucleon , pion and Li (or Be), and nucleon and
7
7Li (or Be)
, Fespectively. Recoil term was neglected in the ealculation.
It is to be noted that the p-wave ooupling term of two nucleon process as
displayed in fig. 2 is included automatically in the present calculation
because both of the initial and final state correlations between nucleons
are fully taken into account by the adopted clusteF wave functions . The
remaining two-body terms, i.e. the s-wave coupling and A-isobar intermediate
coupling terms , aFe neglected here .
The created pions at subthreshold energies have a long mean free path
XIT 5 fm and, in addi_tion, the nuclear systems to be consideFed here are
12He3He
40ca) . 'It is therefore
relatively light (3 He4 +3He,
+ '+
C,...
expected that the pions created in the nuclear interio? are not stFongly
affected by the distortions like reabsorption effect. Plane wave solution
is assumed for the pion wave function.
Inserting eqs. (3), (5) and (6) into eq. (2), we can express the
transition matrix TMfv in the following form;
8
=
TM fV
( )
A
A+ ( Mf
v )
- ( Mf V )
A
/4
x
Y(X)
2Li+1
/
L i'Ji (2ｦ)3/2 J
x
(k )
(Li O 1/2 v l Ji v) (Ji v X Mf v I J M )
ff
f
Li 1/2 J.
1 Xl}
JiJfA {
< 1/21 I
o
(1)
l 1 1/2><Lf l lO
(o) I ILi>,
(8)
Lf 112 Jf
0=A:!:i
(a)
in the ?educed matFix element consists of impulse
where the opeFatoF O
and recoil terms defined by
O( o )
(o) (o)
= O Imp
. +0Fec
O( o )
V41Tf ! /2 l
i mp
m 1 IJ
,
( I
u)
7
=7MN ) (
iklT)
Fec
(o) (pn) '
n= 1
IT
O(o) _ V lTf {V21}
4ｦioJo(k Pn) Y
o( oo lo I ko)
(u¥
M /
o (oololAo)
N
7
4
(o)'
iAJX(k Pn) [Y(X)( n) Q (1)pn]
(9)
n=1
In eqs . (8) and (9), a puFe LS-coupling scheme was assurned in the cluster
model wave functions and the following equations were used for the pion
f ield ;
<k 1$(;1T)i0> c exp{ I IT'(;1T
C)}
<klTIHol0>=
mlT ･{(1i
n 1 )(-i
n
) -
7MN NN
9
Vpn} {V21 }
exp {-iklT'Pn}
* , ,
( Io)
where the coefficient /2(1) is for positive (neutral) pions, and p =rnn
C
is the coordinate of the nucleon from the c.m. of 7Li. Since two clusteFs
He and He (or He and triton) have different oscillator paFameteFs BG! and
B from each other, the Pauli operator A makes it Very difficult to
al
calculate the reduced matFlx element <Lfl 10 1 1 Li>. Several practlcal
12,18,19)
techniques
have been exploited in OFdeF to Femove spurious c.m.
motion from the :natrix element, and we used those techniques in the present
calculat ion .
Finally, we give a conHnent on the nucleon-nucleon and intercluster
correlation. Since an adopted effective nuclear force has a soft-core as
shown in fig. 3, the short range correlation between nucleons is taken into
account in the constructed cluster wave function (3). In paFticular the
inteFcluster corFelation between 4 He and 3He (or 4He and triton) is
maximally taken into conside ation in the present model. In fact, the
7
observed electromagnetic form factors of Li are explained very well even at
high tFansfer momentum up to 3 fm 1 by the present wave function . 12) On the
other hand, the short range correlation between nucleons in the intra-
4 3
clusters He, He and triton is not considered here explicitly although the
so called clustering correlation is taken into account. However , the
assumed Gaussian functions reproduce the observed electromagnetic form
factors Qf these constituent nuclei at q
fm 1 owing to the use of
different oscillator parameters. Even if one uses the correlated wave
function instead of the Gaussian function, the form factor is not modified
strongly at q ( 3 fm though enhanced veFy much at the momenturn Fegion
13)
igher than 3 ftn 1 . As the transfer momentum per nucleon m the
subthreshold (Elab/nucleon 88 8 MeV) pronlc fuslon Is about I ftn 1 , we
lO
presume that the present Gaussian functions
good approximation .
ll
f o r
4He
an d
3He (or
tr iton ) ar e
4 3 07
3 The 4He(3He IT+)7Li and He( He,1T ) Be Reactlons
We fiFst study the 4He(3He
+)7Ll reactron at Elab/nucleon 88 8 r4eV
for which the obseFved data exist. Since the ground state ( 3/2 ) is not
separated expeFimentally from the 0.48 MeV first excited level ( 1/2 ) 9) we
show the surn of the two contributions in fig. 4. Solid and dotted eurves
are the calculated Fesults given by using respectively the cluster model and
7
the shell model wave functions for Li. The same scatteFing wave functions
in the entrance channel were used for the two cases. Partial waves up to
Li = 7 are necessary to obtain convergence for the angular distFibu.tion,
while the total cross section oan be calculated with paFtial waves up to
Li = 3' Note that the higher partial waves (Li 8) aFe not necessary
because the conservation law of total spin and parity makes these
contFibutions very small i･n the pionic fusion leading to the low lying
states. This point is in a remarkable contrast with the usual hadronic
process which needs many partial waves . The pionic fusion cross section
calculated with the cluster wave function (solid curve in the case of WD = 25 rieV in fig. 4) explains the observed data fairly well . Although the
oalculated result slightly underestimates the absolute cross section , this
lack is _attributed to the two-body mechanism which has not fully been
considered in the present calculation, namely the s-wave and A-intermediate
20 )
It is worthwhile pointing o,It the followl!ng two facts
state couplings.
from fig. 4-(a) . First, the clusteF model wave function provides much
larger cross section by an oFder of magnitude than the shell model wave
function . Second, the relative magnitude is not changed much by the
distortion effect of the imaginary potential in the entreince channel .
Bertsch has first pointed out that a simple colliding Fermi-sphere model for
heavy ion collisiops gives only 1/10 - 11100 of the observed inclusive cross
sectron at sub-threshold energie I ) OuF result wlth the shell model wave
12
function is consistent with the conclusion. The question is, then, what
kind of mechanism enhances the pion,ic fusion cross section when we use the
cluster model wave function.
To answer this question we first deoompose the tFansition matFix TMfv
into four different terms as displayed in fig. 5. Eacri term from fig.
5-(a) to (d) consists of fuFther three different graphs when one explicitly
writes up all seven nucleon-lines with the antisymmetrization effect being
considered. All four contFibutions to the pionic fusion cross section are
shown in fig . 4-(b) for the case with the cluster model wave function. Most
of the calculated cross section is exhausted by the (a)-type among them,
namely pion is created from "3He-t sidet' . Based on thls fact we
approximate the tFansition matrix (2) as
Th(fv = <k ; A {c c Xf(?)}lHolAaT{cac3HeXi(?)}>, (11-a)
IT aT a T
.4
He><Xf(;) IVN exp{ l
<c
/N 1 ;)>
T Ih
OT lc3 -7- ｦ ･;
} Xi(
' (11-b)
>j/N
I
X;(;) exp{i(
T OT 3He i 7
1T)'F}d;,
where the coupling scheme of angular momentum and associated summation over
the quantum numbers are omitted for simplicity. hoT Is the pion production
opeFator acting on the inteFnal variables of 3He cluster whlch Is deflned m
Appendix A, and .r is the intercluster relative coordinate. The Feduced
width amplitude is defined by
V X(;) =N 1/2
(r r -) -'
X(r)-'
dr.-' (12)
J
RGM '
13
The adopted approximation is threefold : First , only th impulse term of Ho
is taken into account. Second , the normalization kernel is substituted for
antis metFizer A in eq ( 1 1-b) as usuall used in the ortho onalit
condition model21) ThlFd the mltlal scatterlng wave /N x (r) at Ech( =
152.3 MeV is approximated by the plane wave in eq. ( 11-c). As the result,
the tFansition matrix TMfv is factored out into twp Parts; transition matrix
element for 3He* + + t and the Fourier transfoFm of the reduced width
amplitude of 7Li with transfer momentum
=
i
4/7
' Such a
factorization is Justified for 7Li because the 7Li nucleus has a remarkable
4
probability of finding the tFiton and He clusters in the gFound state.
This is a well known fact observed in the (7Li,a) and 3H(a,Y)7Li reactions.
FIFst let us conslder the FouFrer tFansform. Fig. 6-(a) illustrates
the He + t reduced width amplitude of Li in momentum space . At
ECM 152 3 NeV the bombarding 3He nucleus carries a momentum ki =3'5 ] 1
whereas the created pion takes off a small amount of momentum k = O .3 fm 1
VeFy large momentum q = ki
4/7k
= 3 '3 fm
1
is therefoFe transferred into
7Li. Let us presume here that the intercluster Felative motion between
4
triton and He receives the whole transfer momentum. Tae dominance of fig. _
5-(a) agrees with this idea. The cluster model wave function has a high _
momentum component in the relative motion much more than the shell model
wave function, as cleaFly shown in fig. 6-(a). This is the first reason why
the pion production oross section is enhanced veFy much with the cluster '
4
model wave function. We expanded the reduced width amplitude of the He + t
cluster model wave function in the complete set of harmonic oscillator
functions with B = 0.32 fm 2
It is found in fig '6-(b) that the high
14
㎜◎㎜en七um co㎜p◎nen七 is ◎rigina七ing fr◎㎜ 七he higher n◎daユ aηd high energy
comp◎nent ◎f 七he in七ercユus七er reユaセive m◎セi◎n which is in七r◎duced by ヒhe
4 ．
・h・・tdi・七・・・・・…e1・七i・hb・t鵬・・七砒◎P釦dH・・Thi・・・・…y
in七eresting finding in addi七ion 七◎ 七he fae七 七ha七 七he higher noda1 c◎mp◎nen七s
c◎heren七1y enhance 七he 1◎w m◎㎜entu㎜ c◎mp◎nen七 ◎f 乞he wave func七i◎n as is−
a皿n◎unced very ◎f七en in the s七udy ◎f セhe c1us七er ing phen◎㎜ena． I七 is a1s◎
in七erest ing 七◎ see t；he s i㎜ilar resu1七 in 七he quark cユuster m◎de1 calcu1ation
22）
七ha七 七he shor七 range c◎rre1a七i◎n be七ween nucユeons m◎difies 七he high
m◎鵬n七㎜co㎜p㎝en七s◎f the eユec七r◎magnetic f◎m fac七◎rs◎f de此er㎝．
The second s◎urce ◎f 七he enhancemen七 ◎f 七he pi◎nic fusi◎n cross sec七i◎n
is 七he first facセ◎r iη eq． （11■c）， i．e． むhe ma七r ix eユemen七 ◎f is◎vec七◎r M1
3 3 ＋
m◎de f◎r 七he He “ H ＋ π 七rans i七i◎n． 工七 is 七〇 be n◎ted here 七ha七 七he
nuc1e◎ns in 七he 七r i七〇n c1us七er have very 1arge ㎜o㎜en七u㎜ in c◎mparis◎n wi七h
7
七h◎se in 七he she11 mode1 wave func七ion ◎f Li because 七r i七〇n has sma1ユer
． 7
nucユear s1ze than Li． Theref◎re， 七he firs七 七er㎜ in eq． （11＿c） aユso
increases 七he pi◎n ic fus i◎n cr◎ss sec七i◎n ωhen 七he cユus七er m◎deユ wave
func七ion is used as 七he fina1 st；a七e． Fig． 7 sh◎ws 七he enhance㎜en七． Since
we neg1ect 七he Pauユi princ ipユe in eq． （11・c）， the in七ernaユ degrees of
3 4
freed◎m of 七he セw◎ c1u−s七ers H and He are c◎㎜p1ete1y separated fr◎m the
i・te㏄1uste・・eユa乞i・e㎜◎七i◎・・工ηt肚・apP・◎・i叫i㎝，肋efi・・t七e㎜i・eq・
一1
（11−c）represen七s七he Ml ma七rix eユe㎜e帥砒h七ransfer m◎ment㎜k・O．3伽 ．
¶
・1
H◎wever， in fact， the wh◎ユe 七ransfer ㎜◎mentum q：3．3 f血 is shared by aユ1
parセicipating nuc1e◎ns ◎n acc◎un七 ◎f an interp1ay between 亨he intra・c1uster
aj＝ld in七ercユus七er degrees ◎f fr・eed◎m by ㎜eans ◎f 七he Pauユi pr incip1e： The
−
shared m◎㎜entum per nucユe◎n bec◎㎜es near1y 1．1 fm ． A七 七his m◎㎜en七um 七he
Ml f◎r㎜ fac七◎r ◎f 七he 乞r it◎n cユuster i s much 1arger 七han．tha七 expec七ed in
七he she11 m◎de1 wave funcちion．
十 〇
In fig． 8 we c◎mpare 七he ・π produc七i◎n cr◎ss sec七i◎n Ni七h 七ha七 ◎f ¶ a七
七he same kenima七ical condi七i◎n E ： 152．3 MeV ◎f 七he inciden七 channeユ． The
CM
15
ratio is very close to I /2 . This is a consequence of the charge symmetry
4 3 +7
4 He,ｦ
-3 O) Be.
7
between the two reactions He( He,ｦ ) Li and He(
Quantitatively , however, there is a small deviation fFom I /2 in
th e
calculated ratlon because of the small mass difference between IT + and
16
IT
o
4. Tapget.Mass Dependence of Pionic Fusion Cross Section for the He,
(3 +¥/
Reaotion
Let us assume that the pionic fusion process A(3 He, +)C is dominated by
the slmllar meehanrsm to the 4He(3He,It+)7Li reaction whatever target nuclei
4
we may use instead of He. We then adopt the same approxlmatlon as eq
( 1 1-c) in order to see the general trend of taFget-mass dependence of the
cross section;
do EA+3 ' klT 1
d
81T2Ei Vrel 2
x ex
i
-> / icA
: Xf(?) - ).? d;12 (13)
l<cTlhoTlc3
P{ ( i A+3 ｦ } '
f , v He
where we assurne the triton + A structuFe of final nucleus C as will be
defined soon later. This approximation gives substantially a good result
4 3 +7
within a factor of three for the He( He,1T ) Li(3/2 +1/2 ) reaction; 11.4
nb/sr at OcmO' at Elab/nucleon 88 8 NeV Is predrcted by uslng the cluster
model wave function for V
Xf( ) in eq. (13). This number 1 1 .4 nb/sr is to
be compared with 31.5 nb/sr which was obtained in the RGM calculation in the
case of WD= 25 rieV (see fig. 4-(a)). As for target, we consider only 4N-
nuclei, namely 4He,
1 C,
2 1O,6 Ne.
20 . . and 40ca, because It Is one of the
essential ansatzs for the appFoximation ( 13) that the nucleus A has no spinisospin structure.
First, we estimate the cross section, adopting the harmonic oscillatot
shell model wave function for the composite nucleus C: The ground state
wave function is approximated by the lowest shell model configuration with
17
the highest
patial permutation sy!rrnetry [4N3] . This wave function is
equlvalent to the cluster-coupling wave functlon
tp(A+t) V A(
;), AAt{cA(;A) ct(;t) Xf(;)}
(14)
where the mternal wave function of the target nucleus cA rs descFibed by
the SU(3) shell model wave function with oscillator parameter B which is the
same as the trlton wave function ct (see table I ) . The (Os)3[3]
configuration is assumed for triton. The intercluster relative wave
function xf( ) also is described by the shell model wave function with the
3A B For the 7Li 15N 19F and 43Sc nuclei the
oscillator parameter A+3 ' ' ' '
constructed wave functions ( 14) correspond to the unique SU(3)
Fepresentation ( LL) as listed in table I . This fact makes it easy to
calculate the eigen values of normalization kernel by means of several
technlques24) as Fegards the SU(3) propertres of the normallzatlon kernel
As the Fesult, the reduced width amplitude VN
Xf (?) in eq. ( 13), which was
defined by eq. ( 12), ' turns out to be simply the harmonic oscillator wave
N
functioh multiplied by the square Foot of the eigen value n(AIA)
Iabelled by
(Au) and the total quanta N of valence nucleons;
V
N I 12
Xf(?)={ }
B)'
Xu)(F･3A
unl 'A+3
( 15)
where unl Is the harmonlc osclllator wave functlon wlth Fadlal node n and
orbltal angular nomentum I For the 23Na nucleus, the cluster coupllng wave
18
6
func tion ( 14) contains sever l SU(3) representations of (80) x (60) =
(14-2k k) , where 20Ne is described by the conf iguration
(Os)4(Op) 12( Is ,od)4[45
J (80). Extending the definition (15) to this
k=3
sys tem ,
we define the reduced width amplitude as
v
xf(;) ; { <(80)o,(60)21 l(xv)2>2
N= 14 1 1 12
n(Xu) I
Au
- 23B),
60
x u (r'
(16)
2d '
where <(80)O (60)21 1 (Au)2> is the Feduced Wigner coefficient of the SU(3)
N= 1 4
gFoup The elgen values n(Xu) of normallzatlon kernel are calculated m
terms of the techniques formulated by HoFiuchi24) and FuJlwara and
25 ) . For the heavier nuclei 27Al 31P 35Cl and 39K, one can define
Horiuchi
the reduced width amplitude in the same way as eq. ( 16). However, it is not
easy to calculate the eigen values of normalization kernel for these systems
because there are many possible SU(3) representations resulting from the
coupllng (X'u' ) x (60) where (X'u') is the SU(3) representation of the
28 32S and36Ar. We, therefore, assume
constituent 4N-nuclei A' =24Mg, Si,
the f¥mction
V
Xf(?)
n=
-1 12
= n unl
* . 3A
(r
' A+3
B),
( 17)
0.3,
The adopted value of n might be Feasonable
N=6
eigen values of no rmal izat ion kernel s are n( 60 ) =0.3462 and
foF the A' + triton systems
be caus e th e
19
N=9 _ O' 1636 for the triton
N=7 _ 0.5049 for the tFiton + 16O system'and n(90)
n(70)
40
+ Ca system. Harmonic oscillator parameter B is deteFmined so that the
observed charge radius of the composite nucleus C is reproduced by the
adopted configuration for each nucleus . They are listed in table I .
Table 2 shows the pFedicted variation of the oross section for
3 +
A( He,1T )C at forward angle (ecm=0') at Elab/nucleon 88 8 MeV whlch was
obtained by using the shell model wave functions . We can find at least
three interesting features of the pionic fusion cross section in this table.
Flrst the cross section for the Feaction leading to 15N decreases veFy
7
much in comparison with that for Li (as 0.92 nb/sr * 0.016 nb/sr) although
the beam energy per nucleon is common and the momentum distribution of
7 15
nucleons also is expected to be almost the same: Both Li and N are
p-shell nuclei and the observed charge radii indicate almost the same
oscillator parameter B=0 .32 fln 2 This decrease becomes even more
Femarkable when we compaFe the predicted cross section O .016 nb/sr for
12 3 + 15N with 1 1 4 nb/sF for 4He(3He IT+)7Li which was obtained by
using the cluster model wave function instead of the shell model wave
7
function for Li. It is advantageous to work in the c.m, system in order to
understand the laFge difference of the cross section between the two
reactions. Available beam energies in the c.m. system are i52.3 MeV for
4He(3He ｦ+)7Ll and 213 2 MeV for 12c(3He IT+)15N Consequently the tFansfer
momentums are quite different fFom each other, i .e. q=3.3 ftn 1 and 4 .3 fm 1
for the Feactions leading to 7Li and15N, respectively. In the present
model, the intercluster relative motion between triton and partner nucleus
Feceives the whole tFansfer momentu 1. Therefore, the overlapping of the
intercluster relative wave functions of the initial and final states becomes
smaller for the 12c(3He,ｦ+)15N reaction than that foF the 4He(3He,1T+)7Li
reaction 'due to the gFeater momentum mismatch for the former reaction. This
20
is the' reason why the cross section for the 12.C(3He,1T+)15N reaction
decreases very much.
We have an experimental data' verifying the above discussion
27 ) for the 12c(3He, +)15N
quantitatively . The measured cross section
reaction at subthpeshold energy' Elab/nucleon=78.3 eV is 102(+50,-36) pb/sr
at el =20'.
The present model predicts 70.3 pb/sr at e = O' at the same
ab
eneFgy, and the angulaF dependence of the differential CFOSS section is very
4 3 +7
weak as for the He( He,1T ) Li reaction.
Second, Iet us examine the variation of the cross section for' the
Feactions leading to the sd-shell nuclei. Since Elab/nucleon = 88.8 MeV
corresponds to Eem=220-240 MeV for these Feactions , the transfer momentum
does not differ very much from each other; q = 4.5-4 .9 fm 1 on such
conditions, relatively larger cross sections are pFedicted for the
Ne( He,ｦ ) Na and 24Mg( He,
) Al reactions. Nucleons in Na and Al
move faster than those in the otheF sd-shell nuclei because these nuclei
have the oscillator parameters B=0.31 frn 2 and 0.30 frn 2 whlle the otheFS
have B=0.27-0.29 fm . This gives rise to an appearance of the faster
Felative motion between triton and Ne (or triton and Mg) . Namely, the
probability of finding tFiton and 20Ne (OF triton and24Mg) at the high
momentum q = 4 .5L4.9 fm 1 is larger than that of finding triton and the
partneF nucleus in the other sd-shell nuclei. Hence, the cross section for
the above two reactions become laFger than the other reactions .
The third remaFkable fact suggested by table 2 is the periodic
variation of the cross section reflecting the shell effect. Although the
cross section decreases gFadually as the mass of the final nucleus becomes
large up to the end of each she 1, it incFeases again at the beginning of
the next shell The Increase of the cross sectlon for 40ca(3He IT+)43Sc Is a
t ypical example. This is due to the fact that the wave function of 43Sc has
21
high momentum component much more than the light.er nuclear systems because
40
the intercluster relative wave function between triton and Ca has a larger
radial node as shown in table I . According to the shell model calculation
28), the intensity of the a_ssumed SU(3) shell nodel state
(Os)4(Op)12(Is Od)24(Of)3[4103] (Xu)=(90) is only 18 in the Fealistic wave
function of Sc . If only the triton + Ca configuration could be
responsible for the pionic fusion reaotion at subthreshold energy as
appFoximated in the pFesent study, the predicted cross section 0.56 nb/sr
should be multiplied by O. 18. The resulting CFOSS Section 0.101 nb/sr is
3 1 35 39
still larger than those for the reactions leading to P , Cl and K .
In the present estimate, we neglected the distortion effect of pion
field and the initial channel was described by plane wave. Although these
4 3 12 3
approxlmations may work well foF Iight nuclear systems like He+ He, C+ He
and F He, they might not do for heavieF systems. It was reported by
Stachel et al. that the 65 -80
of created pions in N+ Ni collisions
might encounter the reabsorption effect in the nuclear interior. The
present authors, tneFefore, estimated the reabsorption effect for the
nuclear systems discussed here with the use of the mean free path of pions
derived fFom the pion-nucleus optical potential29)
based on the analysis of
30 )
. NeaFly 35 -55 pions are
pion distorted waves by Kaps and Bertsch
reabsorbed in the nuclear interior. In addition, we assurned the SU(3) shell
model wave functions for the composite nuclei. The adopted configuFations
for the 15N, sd-shell and 43
Sc nuclei (in table i) exhaust the intensities
and･ 18 28)
more than31)8085 50
, 28,32)
- , Fespectively,
in the realistic
wave functions of these nuclei. Our prediction for heavier systems,
t erefore, could be considered as upper limit of the pionic fusion CFOSS
sect ion .
Let us discuss the effect of clustering correlation. It was shown in
the previous section that the clustering corFelation enhances the pionic
22
4 3 +7
ftlsion cross section fo the He(_ He,ｦ ) Li reactlon We aFe now mterested
in the question whether it shows up in any pionic fusion process . The
16O(3 He, +) 19F Feactlon seems to provide a very good example to answer thls
19
question because there are several low exoited states of F which aFe very
strongly populated by the triton-transfer reactions. In fact, acconding to
33)
19
, the excited 3/2 (6.09
the theoretical study of F by Sakuda and Nemoto
MeV) state which is the band-head of the rotational levels is a remarkable
16
triton + O clustering state and the higher nodal states of the clustering
components ( XO) with X 8 aFe admixed by nearly 30 even in the gFound
1/2+state. The SU(3) shell model conflguratlons wlth the lowest allowed
quanta aFe (60) and (70) for the gFound 1/2+ and excited 3/2 states,
respectively. We here compare the predicted pionic fusion cross sections
which aFe obtained by using the cluster model and shell model wave functions
of 1 9F fFom each otheF . The clusteF model wave functions calculated by
Sakuda and Nemoto were used in the present calculation . As for the shell
model wave function of the excited 3/2 state, the oscillator parameteF 8
was determined so as to give the same charge radius as that given by the
cluster model wave functiont since the calculated mean intercluster
distance 5 .04 frn of the excited state is larger than 3.95 frn of the ground
state, the value of B (fm 2) for the excited state is smaller than that for
the 'ground state, as shown in table I . FOF the gFound state of 19F, both of
the cluster model and shell model wave functions give the same charge radius
as the observed one.
t
In order to avoid the spurious c.m. component, the charge radius of the
excited 3/2 state was calculated in the following equation, corresponding
to the clusteF-coupling wave function ( 14) ;
23
V< r2c >
= /19
2t 9 82
1 uo
32 8 2
<r > + {<r t >=3249<rhel>
2
where V<F > and /<rl60>
are the charge radii of triton and
1 60 ,
and V<r2 >
rel
16
See ref. 33 for
is the r ,m.s. intercluster distance between triton and O
details. Note that the charge radii of the ground states of al l nuclei in
table I were calculated exactly without approximation.
24
In figs. 9-(a) and (b) we compare the trlton + 160 cluster model and
harmonic oscillator shell model wave functions for the ground 1/2+ and
exci･ted 3/2 states, separately, as m flg 6 (a) As Is well known the
clustering component predominates the low q reglon for both of I /2+ and 3/2
states whlle at 1 5 fm 1 ( ･q ( 3.5 ftn I for the gFound 1/2+ state and 2 O
frn I ( of ( 4.0 fm 1 for the 3/2 excited state the shell model component
domlnates. However, at q=4.51 fm I and 4 53 fm I whlch are the transfer
3 +) reactions leading to the ground 1/2+ and
momentums for the 16
O( He,
excited 3/2 states, respectively, the clustering component prevails again.
Therefore, the pionic fusion cross section may be enhanced as shown in table
2 by the use of the cluster model wave function.
+ 15
Quite recently, the pion production reaction12C( 3He,
) N has been
observed34) at the Indiana University and It was found that the unFesolved
excited states of the 15N nucleus in the 6 .5-10.5 MeV excitation energy
range are strongly populated by this reaction. From the theoretical
viewpo int
31)
12
, there are many triton + C clustering states in the energy
range 5-15 rteV . Experirnentally, these excited states are known to have
large reduced tFiton-width as obseFved by the tFiton-transfer reactions.
High resolution measurement which can separate each excited level is highly
requ ired . '
The clustering component prevails over harmonic oscillator shell model
at very high momentum region . This should be the case because the short
distance correlation between two clusters is fully taken into account in the
cluster model wave function. Thus the pionic fusion cross section should be
enhanced in most cases, reflecting the clusteFing correlation in the final
states .
25
5 . Summary and Conclusion
4 3 +7
We have calculated-the pionic fusion cposs.section for He( He,1T ) Li
4 3 07
and He( He,ｦ ) Be at subthreshold eneFgy Elab
/nucleon = 88.8 MeV in the
cluster model. In this model we exactly take account of the strong
clustering correlation of the A=7 nuclear systems, the distortion of the
entrance channel and the Pauli exclusion principle between nucleons. We
have obtained the result that the pionic fusion cross section is enhanced
very much by the clustering correlation. We have also found that it
originates from the high momentum component of the intercluster relative
4
wave function between triton and He.
Assuming that the pion pFoduction is dominated by the high momentum
component of the intercluster relative motion, we extended the analysis of
pionic fusion to heavier systems and estimated the variation of the pion
production cross section . Several kinds of shell effects were clearly
16 +¥1 *
predicted. It was found that the cross section for the O(3He,ｦ / 9*
reaction also might be enhanced by the clustering correlation.
A very simple pion production operator was assumed in the pFesent
calculation. There are, however , pFobably important s-wave coupling and A20 )
excitation teFms
, which we have neglected here. A quantitatively
satisfactory description of the pionic fusion will be fulfilled by
considering these terms with the strong clustering correlation being taken
into account as in the present study. Although a qualitative aspect- is
stressed in this paper , we have learned that the short distance correlation
between heavy ions might be observed systematically in the pionic fusion
reactions by vaFying the combination of projectile and target.
The nuclear cluster model has ever enjoyed a success for relatively low
momentum phenomena. For example, the microscopic clusteF model has
succ eded in predicting the radiative f¥Ision CFOSS Sections for
4 3 7 4 3 7
He( He,Y) Be and He( H,Y) Li at low energies which affect seveFal
26
astrophysical problens very much.
35-37 )
In the present paper, it wa
suggested that the cluster model provides a powerful tool for the study of
pionic fusion reactions, too, although these are the dynamical phenomena
accompanying with high transfer momentum.
The authors wish to thank Profs. H. Ikegami , I. Tanihata, t(. Ishihara,
K. Yazaki, K. Shimizu and A. Arima for many valuable discussions . One of
the authors (T.K.) also is grateful to Profs. G. Bertsch, A. Brown, W.
Benenson and S. Austin for useful oonversations and comuents . Numerical
calculations were performed by using VAX-11 at the National SupeFconducting
Cyclotron Laboratory, )(ichigan State University and M680H at the Computer
Center, UniveFsity of Tokyo and financially supported by Institute for
Nuclear Study, University of Tokyo and ReseaFch Center for NucleaF Physics ,
Osaka UniveFsity . This woFk is also financially supported by the
Grant-in-Aid for Scientific Research of Ministry of Education, Science and
Culture, Japan .
27
Appendix A
The following
eq ua t i on
for the
p ron produc tion
operator Ho was used in eq.
(11);
8n'{(1+ u) )(_i
7MN
mlT n = 1
) M
{hol+S11' r}exp{-i_ k4
pn}exp{-i
' n} Tn
. }+{h + . }exp{i-3
oa la r
';
( A- I )
hOT 3exp{-''{A(-i
}
n IT)+B
1 - Y]
nT}Tnl
nT } (A-2)
{* }=
hlT n=1 1-3B8n '
hoa 7
{- }=
4exp{-i
hla
'{A(-i IT)+B
.
}
}TY{ n } (A-3)
-7!S{ on
where A=1+=r
B=co
LD , .nT(.na) is the coordinate of nucleon fFom the c.m.
7)(
N ' NN
3 4
of He ( He) and 'V,
nl
( na) is the derivative with Fespect to
*nT( *;na) ･ Note
that one can always divide one-body operator into two parts w i thou t
approximation acting respectively on the internal variables of the
constituent clusters
(like (h h ) and (h h )
oT' IT oa' Ia
relative coordinate *F.
28
) and the inteFcl uster
Referenceき
1）
G．F． Ber七sch， Phys． Rev．」≡…ユ亘 （1977） 713．
2）
W・Benens㎝e七aユ．，Phys．Rev．L嚇．坦（1979）683・
3）
S・Naga㎜iyae七aユ．，Phys．Rev．Le沈．坦（1982）1780．
4）
T． J◎hanss◎n e七 a1．， Phys． Rev． Let七． 48 （1982） 732．
5）
M． Prakash， P． Braun−Munz1nger and J． S七ache1， Phys． Rev． C33 （1986）
937．
6）
J． Sセache1 e七 aユ．， Phys． Rev． C33 （1986） 1420．
7）
R． Shyam and J． Kn◎1ユ， Phys． Le七t． 136B （1984） 221； Nucユ． Phys． A426
（1984） 606．
8）
Y・LeBomece七a1．，Phys．Rev．Le七七．虹（1981）1870．
9）
L． Bi㎜b◎t e七 a1．， Phys． Le七七． 114B （1982） 311．
10）
J． Germ◎nd and C． W1ユk1n， Phys． Le七七． 106B （1981） 449．
11）
K．K1i㎎enbeck，M．Dil1igandM．Huber，Phys．Rev．Le眈．坦（1981）
1654； M． Huber， K． Kユ1ngenbeck aハd R． Hupke， Nuc1． Phys． A396 （1983）
191c．
12）
T． KaJ1n◎， T． Matsuse and A． Ar：Lma， Nuc1． Phys． A413 （1984） 323．
13）
T．KaJ1n◎，T．Ma七suse and A．Ar1ma，Nuc1．Phys． A414（1984） 185．
14）
J・A・冊eele・，Phys．Rev．里（1937）1083，l107・
15）
SupP1・P・og・The◎・・Phys・幽（1980）P．215・
16）
G． Bruge e七aユ．，Nucユ．Phys． A129（1969） 417．
17）
P．Hecki㎎，R．Br◎c㎞am and W．Weise，肋ys．Le眈．72B（1978）432．
18）
A・T◎hsaki−Su・uki，P・・9．肺e◎・．Phys．型（1978）1261・
19）
M． Ka㎜i1−lura， SupP1． Pr◎9． Theor． Phys． N◎． 62 （1977） 236．
20）
E・ Ose七， H． T◎k i and W． Weise， Phys． Rep．旦；…」（1983） 273・
21）
S・S・i七・・P・・g・Th・…Phy・・卑（1968）893；ibid・旦（1969）705・
22）
M． Oka and K． Yazaki， Nuc1． Phys． A402 （1983） 477．
23）
M． Harvey， Advances 1n Nuc1ear Phys1cs Vo1． 1 （ed． by H． Baranger and
E． V◎9七， 1968） 67．
24）
H． H◎r1uch1， SupP1． Pr◎9． Theor． Phys． N◎． 62 （1977） 90．
25）
Y・F・ji胞・♀・・dH・H・汕・hi・P・…Th・…Ph…坦（19gO）895・
29
26 )
C. Jager, H. Vries and C. Vries, Atomic Data and Nucl. Data Tables 14 _
(i974) 479.
27 )
L. Bimbot et al., Phys. Rev. C30 (1984) 739.
28 )
29 )
J.B. NcGrory, Phys. Lett. 47B (1973) 481.
J.A. Carr, H. McManus and K. Stricker-Bauer, Phys. Rev. C25 ( 1982) 952.
30 )
R. Kaps , preprint; G. Bertsch, private corrmunication.
31 )
Y. Goto and H. Horiuchi, PFog. Theor. Phys. 62 (1979) 662.
32 )
33 )
B.A. Brown, private communication.
T. SakLrda and F. Nemoto, Prog. Theor. Phys. 62 (1979) 1274 and 1606.
34 )
W. Schott et al., Phys. Rev. C34 (1986) 1406.
35 )
36 )
T. KaJino and A. Arima, Phys. Rev. Lett. 52 (1984) 739.
T. Kajino, Nucl . Phys. A460 ( 1986) 559.
37 )
T. KaJino, H. Toki and S. Austin, preprint of Michigan State
University ( 1986) , submitted to Astrophys. J.
30
丁自b1e 1． A‘＝lopted ha㎜■＝1nic◎sei1ユa七〇rε㎏1ユmode1旧ve fしInction．
d1理ge ◎sc工］ユa七◎r intぼc1uster Su（3）鞭entation◎f S口（3）nep㈱nta七i◎n◎f
紬。。† 因㎜瞼幽・。。戯i㎝雌。。戚・胸ω由讐雌如・㎏ω脳
鬼。（航）・（〆）冷 （〃） （畑）
J¶
7 −
Li3122．蜘
0．契
1p
0，32
lp
0，32
lp
0，29
3s
0，26
3P
㌔3／2＋2．｝
0，31
射
27 ＋
虹5／23．06
0，30
困
31 ＋
P1／23．20
0，29
3s
35 ＋ ＋
C13／23．30
0，28
困
39 ＋
K3／23．蜘
0，27
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0．26、
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3／2・（6．09剛ま
43 一 十
Sc7／23．50
4距（oo）
4He（00）
12
C（04）
16
0（00）
16
0（00）’
牝（80）
24
㎏（釧）
28．
S1（120）
弩（㎎）
36吐く08）
蜘Ca．（00）
（30）
（30）
（01）
（60）
（70）
㎜並記
．○
”
，1
”
〈90）
十
竿削射蛾・紬ii（爬f・26）・
’十
鯛帽ユueS．
±池紬お。帥・鉦七伽。16・d晦㎏、㈱。・1キ．（・。。㈱舳。汕．）
讐 3
AdoPted ぼiton wave fしrcti◎n （Os） 工3］ is a Su（3） s◎≡1ユar （00）．
A(3He,1T+)C reaction pred icted
Table 2 . Pionic fusion cross section for the
m eq. ( 13) at ecm=0' at E lab /nucleon='88 . 8 MeV
3
A( He, IT +)C
Shel l
JIT( cgr . )
9i
d
t
(nblsr) JIT(C )
do (O' )
d
( nb/sr )
model wave function foF C
c
4He
12c
160 -
20Ne
24
Mg
28si
32s
36 Ar
4 O ca
Clus te F
7Li 3/2 +1/2
1 5N I 12
1 9F I /2+
23Na 3 /2+
0.92
0.016
o . 064
3/2 (6.09 NeV)
0.11
3/2
0.41
o . 62
27Al " 5/2+
3 1 P I /2
0.28
o . 030
35cl 3/2+
39K 3/2+ '
43Sc 7/2
o . 043
0.016
0.56
model wave function for c
A
c
4He
7Li
160
19F
3/2
1 /2+
+ I 12
11 .4
2.3
(6.09 MeV)
Figure captions
Fig. 1
7L i ( Jf )
4 3 He *
Schematic Fepresentation of the mechanism of the He +
+ reaction H represents the pion production operator and C
+ IT
' O
denotes the clustering coFrelation containing the Pauli exclusion
pr incipl e .
P-wave coupling teFm among two nucleon processes of the pion
production operator, which is included automatically in the present model
caloulation in addition to the single nucleon process (see text) .
Fig . 2
Fig . 3
Effective nucleon - nucleon interaction
Nagata force) adopted in the pFesent calculation .
( the
3E ,
modified - Hasegawa -
1E, 3O
and
10
represent the triplet even, singlet even, triplet odd and singlet odd
channels, respectively.
4 3
Fig.4 (a): Angular distribution of the He + He * 7Li(3/2 +
1/2 ) + IT+ reactron cross sectron at the Incident energy Elab/nucleon 88 8
MeV. Solid and dashed curves are the calculated results obtained by using
7
respectively the cluster model and the shell model wave functions of Li .
WD denotes the adopted 'strength of the imaginary part of the optical
potential in the entrance channel (see text). Black dots with erFor bars
are the observed data fFom ref.9. (b) : Decomposition of the calculated
differential cross section in the case of WD = 25 MeV with the cluster wave
function (solid curve in Fig.4-(a)) into four different contFibutions
displayed in Fig.5-(a) to 5-(d).
FouF different terms of the transltron matrix for
Fig . 5
7 +
Li
reaction. An oval box denotes the antisy!runetrizer of nucleons.
IT
4 3
Fig.6
7
the 4He + 3He *
(a): Reduced width amplitude of the He + H component of
Li(3/2 ) in momentum space. Solid and dashed curves are cases foF the
cluster model and the shell model wave functions, respectively. q = 3.3
fin 1 is the tFansfer momentum of the 4He + 3He * 7Li + + reaction at
Elab/nucleon = 88.8 h(eV. (b): Decomposition of the reduced width amplitude
of the cluster model wave function (solid curve in Fig.6-(a)) into harmonic
oscillator functions u* ( ) with 8 - O 2 fm 2 Solid and dotted curves
*,* q 7 - '3 '
denote Fespectively the positive and negative amplitudes . Note that the Op
- amplitude does not vanish bec use the different size parameters Ba = 0.574
= 0.460 fm 2 aFe used for He and 3H nuclel respectrvely
Ratio of the squared transition matFix appearing in eq . (11-c) of
Fig . 7
3H +
3
the He +
IT
+ process for the cluster model wave function to that for
the shell mo del wave function as a function of transfer momenturn.
Fig . 8
Compar ison of the calculated angular distribution of
- 7Li(3/2
7Be(3/2
+ 112 ) +
+ 1/2 ) +
'
4
t reaction cross section with that of He +
at Elab
/nucleon
optical potential has the stFength WD
4 3
the He + He
He *
88.8 )teV . The adopted WD of the
- 25 MeV.
Fig.9 (a) : Reduced width amplitude of the 16
O + H3 component of the
ground state ( 1/2
) of19F in momentum space. Solld and dashed curves are
the same as in Fig.6-(a) ; cluster model and shell model wave functions,
respectively . q = 4.5 fm 1 is the transfer momentum of the
16O +3H * 19F +
+ reactron at E /nncleon 88 8 MeV b ' The same as in Fig.9-(a) for
lab = ' ' ( ) '
the 3/2 (6.09 )ieV) excited state of19F.
MSu-86-383
FIGURE l
7Li ( Jf )
+
7E
/
Ho
/
/
/
/
/
/
/
/
/
/ ((A), -)
//
_K7r)
/
c
4He
5 He
FIGURE 2
//.
/
/
ll
/
Ho
/
/
/
/
MSU-86 - 384
FIGURE 3
1000
3E
b
.
IE
,
>o
' .
500
t
t
, ¥
t
Lzz
t
'
o
'
,
,b
I ,__ 1' l' 1"
lp
2
,. d.
s
rNN (fm)
-250
,
.>
o
10
t. ,
50
'
30
100
f¥
'b
'I
'
¥
'
'I'
1l
1lb
'
,l
･Ib,b, .bl･1-_,1lbl･_.1,-1-1･-1-ll-1-･.-1･-.
Lz
J
1
-50
2
s
rNN (fm)
' LL lur¥L
d07d
(nblsr)
a
ELab/N = 88.8 MeV
1 02
4He (SHe, 7c+ ¥/ 71Li312 +v2
101
1-1--1"I,'-d'Q1-..11･,I
11･a･*-_.
1
lbll･l
O_ d,1_'_dP
.,I
, 1 ,,tb ¥
¥
¥¥
¥¥
¥¥
¥¥
1lplPd d 11--'--""I-
¥ ' bl'dPl"I
/ d
¥
/.
'b'_dl'd'
1
11- 11-
Ib
' I tb, 'l'
.
1 0 1
1 2 O'
CM (deg)
1 8 Oo
MSU＿86＿386
FIGuRE 4 b
d0；7dΩ‘（nblSr）
2
10
’WD＝一25
1d1
（O）
11
（b）
10
Oo ．．
0 120◎
◎。。（d・g）
0
180
MSU - 86-387
FIGURE 5
(a)
(b)
7C
/1
/
/
/
/
+
3H
/
ll
/
i
1
3 He
3 He
(c)
4 He
7c+
(d)
+ 4He
7c
¥
¥
¥
¥
¥
¥
4 He
7C
4
He
/
/
+
- J
q ' f : XJL(q)
, MSU-86-388
JL
He + 3H =7Li 312
q = 5.5 fm 1
1
/
/
/
,,
/
/
/
/
/
¥
¥
¥
1
/
¥
i
101
¥
l :
¥
l
¥
¥
¥
¥
-2
10
¥
¥
¥
¥
103
q (fm 1)
4
MSU-86- 389
FIGURE 6 b
4He + 5H = 7L13/2
q' N XJL(q)
lp
1
/,-¥'
l
1 0 1
l
l
l
,,
f
I
V
l
l
l
-2
10
l
l
l
,
/ ¥
I
,
l
¥
l
¥ ¥¥
ll
¥ ¥
ll ¥
l
l t
10 o
1
2
q (fm 1 )
7 p
¥
/
4
FIGURE7
一・φ。1り。τ1φ・。べ、s、、、肌
1・φ。1h㎡1細、・1；。、岨肌
一
◎
⊥
一
◎
’
．o
ヨ∼
⊥
）
ぴ
ζ
ω
⊂
I
oo
①
I
｝
⑩
O
MSU-86-391
FIGURE 8
d /d
(nb/sr)
E Lab
/N = 88.8 MeV
1 02
4 He ( 3He x+ )7LI 312 + v2
101
1
4 He (
101
Cf
120'
CM
(deg)
1800
MSU-86-375
FIGURE 9(a)
I q . Jj
160 + 3H = 19Fv2
XJL(q) I
q = 4.5 fm I
I
-'b
I t
It
/
I
l
l
f
f
I
l
l0 I
fb
/
I
ll
Il
/
¥
// ¥ ¥
¥ /
l
ll
ll
¥
¥
¥
tl
ll
ll
¥
ll
¥
Il
l
ll
I
¥
¥
¥
¥
¥
¥
I0 2
'
¥
L
L
l0 3
o
2
q (fm 1)
MSU-86-374
FIGURE 9 b
l q . Jj
160 + 3H = 19F3/a
XJL(q) l
q = 4.5 fm l
l
l
l
/
1
l
X I
/
I
I
l
/*¥
/
'¥
l
1
ll
tl
ll
/
/
ll
l
i
l
l
I
l0 l
l
/ b¥
¥
¥
¥
¥
¥
¥
¥
¥
¥
¥
l0 2
¥
l
¥
l0 3
o
2
q (fm 1)
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