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The heavy-particle-stripping reaction in (α, p) reactions
2.G
[
Nuclear Physics 34 (1962) 609--622, (~) North-Holland Publishing Co., Amsterdam Not to be
reproduced by photoprmt or microfilm
without written
permxsmon from the pubhsher
THE HEAVY-PARTICLE-STRIPPING REACTION IN (a, p) REACTIONS (II) The Li(a, p ) B e and the F(a, p ) N e reactions T S U Y O S H I H O N D A and H A R U O U I
Department of Physws, Tohoku University, Sendal, Japan Received 2 F e b r u a r y 1962 I n t h e same w a y as t h a t p r e s e n t e d in t h e preceding paper, t h e following reactions are analysed in detail in t e r m s of t h e h e a v y - p a r t i c l e - s t r i p p i n g reaction" t h e Li 6 (a, p0)Be' (ground state) reaction, t h e La~ (~, po)BO ° (ground state) reaction and t h e L17(a, p 1 ) B O O (1st excited state) r e a c t i o n a t Ea ~ 30 MeV a n d t h e F19(a, po)Ne *l (ground state) reaction, t h e FI'(~, Pl)NeZ* (1st excited state) reaction a n d t h e FI'(~, p , ) N e 2. (2nd excited state) r e a c t i o n a t a b o u t E~ = 20 MeV. I t is s h o w n t h a t t h e angular distributions of these t h r e e (a, p) reactions for Li nuclei c a n b e satisfactorily explained b y t h e heavy-partxcle-stripping reaction e x c e p t for their b e h a v i o u r a t small angles. ]:t is f u r t h e r s h o w n t h a t t h e differential cross sections a t larger angles for t h e FI'(~, p ) N e n reactions q u o t e d a b o v e are also explained r e a s o n a b l y well b y t h e h e a v y particle-stripping reaction. I n particular, good a g r e e m e n t is o b t a i n e d b e t w e e n t h e o r y and t h e e x p e r i m e n t with r e s p e c t to t h e v e r y s h a r p p e a k a t t h e b a c k w a r d direction appearing in t h e differential cross section for t h e FI'(~, p0)Ne 2~ (ground state) reaction. I n this connection, some r e m a r k s are a d d e d concerning t h e shell m o d e l wave f u n c t m n s of 018 a n d Flg calculated b y Elliott a n d Flowers a n d b y Redlich. The reduced w i d t h {O~O~Op} 2 e x t r a c t e d from t h e e x p e r i m e n t a l d a t a of t h e F19(~, po)N *z reaction is found to be very small in comparison to t h a t of t h e C12(~, pg)N 15 reaction o b t a i n e d m t h e preceding paper.
Abstract:
1. Introduction In the preceding paper 1), which will be referred to as (I), we have analysed in detail the differential cross section for the C1'(~, p0)N 15 (ground state) reaction at large angles from E~ = 16 MeV to 40 MeV in terms of the heavyparticle-stripping reaction by employing the formulation 2, 3), which has been presented previously by us for the (d, p) and (d, n) reactions. It has been shown therein that not only the shape of the angular distribution at large angles but also the magnitude of the cross section, both of which vary nonmonotonically with the energy of the incident 0c-particle can be well reproduced by the theory of the heavy-particle-stripping reaction without introducing any energy-dependent adjustable parameters. In the present paper, we shall extend the previous analysis to some other available data of the (,t, p) reactions, including the LiS(~, p0)Be 9 (ground 609
610
HARUO UI AND TSUYOSHI HONDA
state) reaction, the Li~(x, Po) Be1° (ground state) reaction and the Li~(x, Pl) Be1° (lst excited state) reaction at the incident x-particle energy, E~ = 30 MeV 4) and the F19(x, po)Ne 22 (ground state) reaction, the F19(x, pl)Ne z2 (lst excited state) reaction and the F19(x, p~)Ne z2 (2nd excited state) reaction at about E~ = 20 MeV S-T). We shall treat in the next section the Li(x, p)Be reactions, which m a y be expected to be one of the best examples that exhibits the typical angular distribution for the heavy-particle-stripping reaction as will be discussed in detail there. It will be made clear from our analysis that this is actually the case.
In sect. 3, the F19(x, p)Ne 2z reaction will be treated in detail. It will be shown that not only the angular distributions but also the magnitudes of the cross sections for the F19(x, p0)Ne ~z (ground state) and the F19(x, pl)Ne 2~ (lst excited state) reactions at large angles can be rather reasonably reproduced b y the theory of the heavy-particle-stripping reaction, although both cross sections are quite different from one another. In this connection, some remarks are added concerning the shell model wave functions of F 19 and 0 is calculated b y Elliott and Flowers s) and b y Redlich ~). In analysing the F19(x, P2) Ne22 (2nd excited state) reaction, we shall tentatively adopt two alternative assignments, 2 + and 1% of the spin and parity of the second excited state of Ne 2~. Unfortunately, however, we cannot make a definite assignment of this state from our result of the analysis, although the assignment of 2+ seems to be more plausible than that of 1-. Finally, in sect. 4, some discussion is added concerning the dependence of the cross section on the mass number of the target nucleus and on the incident energy of the x-particles, etc. A remark on the structure of the light nuclei will be presented there in connection with the heavy-particle-stripping reaction *.
2. The Li6(a, Po)Beg, Li'(a, Po)Be l° and Li'(a, pi)Be l° Reactions The experiment of the (x, p) reactions on the Li isotope has been performed b y Klein, Cindro, Swenson and Wall 4) at the incident x-particle energy E~ = 30 MeV leading to several definite states of the residual nuclei. We shall present in this section the result of the analysis of these reactions. The Li(x, p)Be reaction m a y be supposed as one of the best examples which exhibit the typical angular distribution for the heavy-particle-stripping reaction because of the following reasons: firstly, the mass of the target nucleus does not differ so much from that of the incident particle and, secondly, the t T h e s a m e n o t a t i o n is u s e d t h r o u g h o u t t h e p r e s e n t p a p e r a s i n t h e p r e v i o u s p a p e r t).
THE HEAVY-PARTICLE-STRIPPING
REACTION
(II)
611
binding energy of the proton in the ,t-particle (Bp---- 19.81 MeV) is much larger than t h a t of the outer proton in the target nucleus (Bp = 4.655 MeV for Li e and Bp = 10.006 MeV for LIT). Thirdly, the target nucleus has a rather simple structure in the sense t h a t there are no complications in the decomposition of the target nucleus into the core and proton states. Namely, it m a y not be necessary to take inoto account m a n y states of core, which leads to a quite involved calculation of the cross section. For the sake of convenience, we shall treat the LiS(*t, p0)Be 9 (ground state) reaction in subsect. 2.1. and the Li?(*t, p0)Bel° (ground state) and the Li 7 (*t, pl)Be 1° (lst excited state) reactions in subsect. 2.2, respectively. 2 I. THE Li'(~z,p0)Be0 (GROUND STATE) REACTION The cross section for the heavy-particle-stripping reaction presented in sect. 1 of the preceding paper I) has been expressed in terms of t h e / - i coupling prescription. The states of Li e, however, are well-known to be approximately described b y the L-S coupling rather than t h e / - j coupling shell model. It seems, therefore, to be necessary to represent the cross section in terms of the L-S coupling scheme, which can be performed directly by means of the so-called transformation bracket or Wigner's 9-j symbol. Equivalently, we shall make use of the more general expression of the cross section in the 7"-Jcoupling scheme in which several appropriate ~'-values are taken into account on an equal footing: For example, the ground state of LiS(T = 0, J = 1+) can be represented as an SS-state in L-S coupling, which can in turn be described as a mixed configuration between the (lpl) and (Ip~) states in t h e / - j coupling prescription. We shall not take into account the proton emission from the (1 s) state due to its large binding energy. Then, there are two odd parity states of He 5, i.e. the ground f = ~- and the first excited f = 21-- states, both of which m a y be identified as the core in the heavy-particle-stripping reaction. In the following analysis, we shall assume the core to be the ground state of HeS; the first excited state will not be tal~en into account, since this state is a very broad virtual level. Accordingly, the incident *t-particle should be captured by the core with the orbital angular momenta l~ = 0 and 2 to form the residual nucleus, the ground state of B e g ~ = {-). In table 1 are summarized the various values of the quantities which are needed to the analysis. As can be seen from the explicit expression of the cross section, the angular distribution arising from the (1 Pl) state proton emission is exactly the same as that from the (1 p½) state. Further, the energy-dependence of the former cross section is the same as t h a t of the latter cross section, provided the cut-off radii be taken equal to one another. As an example, a typical calculated cross section for the Lie(*t, po)Be 9
612
HARUO
UI
AND
TSUYOSHI
HONDA
(ground state) reaction is shown in fig. 1 together with the experimental data at E= = 30 MeV b y Klein et al. Here, the solid curve represents the calculated
Lie(co,
TABLE 1 P a r a m e t e r s for the Q Bp B= It
p0)Be' reaction
--2.125 MeV 4.655 2.529 1+
/r
t-
Ie
~ - (the g r o u n d s t a t e of He s)
lp
1
l=
0
2
0
2
Gh
1
1
1
1
L,i 8( ,,po) B/A
/
J~,~= 0
,,.
":'•
30MeV /.: l o°
4 o~o oo
3
°m
.,o o° eL
Z
•
•
•
•
°°° •
o@
I •
~~R0.---~ 60frn 0
=o
4O
e0
00
100 tZ0
140 180 too
ep ( c. M.) Fig. 1. The cross section for the Ll°(~, p0)Be 9 (ground state) reaction a t E= = 30 MeV. The sohd curve represents the calculated cross section for the heavy-particle-stripping reaction, while t h e p o i n t s are the e x p e r i m e n t a l d a t a b y Klein et al. 4). As the m a g n i t u d e of the cross section w a s n o t m e a s u r e d b y Klein et a l , the scale of the ordinate is w r i t t e n in a r b i t r a r y units. The scale of the abscissa denotes t h e angle of the emitted p r o t o n in the centre-of-mass s y s t e m m e a s u r e d in degrees. I n the calculation, t h e following values of the p a r a m e t e r s are a d o p t e d R 1 = 6 0 fro, lp = l, jp = ~ and t a n d l~ = 0. The core is a s s u m e d to be t h e g r o u n d s t a t e of He 5.
THE HEAVY-PARTICLE-STRIPPING REACTION (If)
613
cross section for the heavy-particle-stripping reaction, in which we adopt the cut-off radius Rt = 6.0 fm in addition to the values of the various parameters listed in table 1. Further, in the calculated cross section presented in fig. 1, we have taken into account only the ~-particle capture with l~ = 0, the ,tparticle capture with l~ = 2 being disregarded. The value of Rt adopted here is not necessarily unique. One could determine it rather unambiguously if the experimental data were available in such a wider range of E~ as in the case of the CZ2(~, po)N is (ground state) reaction treated in (I). Moreover, since even the magnitude of the cross section has not yet been measured, we cannot extract from the experiment the parameter {0~ O~ Op}~ which would provide important information on the nuclear structure. 2.2. T H E
LiT(0c, p ) B e t° R E A C T I O N S
Next, we shall analyse the LiT(,c, po)Be 1° (ground state) reaction and the LiT(~, pz)Bel°(lst excited state) reaction at E~ = 30 MeV. We shall simply assume the core to be the ground state of H e e ( J = 0+): the first excited state of He6(J = 2+) will not be taken into account, since the state is unstable against particle emission. Then, the proton is emitted from the (lp~) orbit in the ground state of Li T, while the proton in the (lP½) orbit can not take part in the heavy-particle-stripping reaction because of the conservation of the angular momentum. For the LiT(x, Po)Be1° (ground state) reaction, the incident ~-particle should be captured b y the core with an orbital angular momentum l~ ---- 0 to form the ground state of Be 1°, while the ~-particle should be captured with l~ = 2 for the LiT(~, pz)Be 1° (lst excited state) reaction. In table 2 are summarized the various values of the quantities which are needed in the analysis. TABLE 2 P a r a m e t e r s for t h e Li~(ct, p ) B e lo r e a c t i o n LiT (~, Po) Bez° Q Bp
-2.566
Lx~ (m, Pt) BeZ°
MeV
--5.934 MeV 10.006
B~
7.439
~-
It Ir Io lp
0+ 2+ 0 + (the ground state of H e e) 1
lp t~ Gh
o 1
t 2 5
A typical calculated cross section for the LiT(~, P0) Be1° (ground state) reaction is shown in fig. 2 together with the experimental data at E~ = 30 MeV
O
20
0
(d.Po)
BelO
'
2o
'
4o
eo
so
'
'
"
'
'
6Afro
..
~00 Izo =40 ~eo tso
ep (C.M.)
'
•
I
£== 30MeV ~(H.P.Stripping).]
L 7i
Fig 2. The cross section for the Li T(~, Po) BeZ° (ground state) reaction at E~ = 30MeV. The sohd curve represents the calculated cross section for the heavy-particle-stripping reaction, while the p o i n t s are the experimental d a t a b y Klein et el. 6) I n the calculation, the following values of the p a r a m e t e r s are adopted: Rt = 6.4 fm, lp = 1, Ip = }:and l~ = 0 The core is a s s u m e d to be the g r o u n d s t a t e of He ~
•ID
"-
L
so
~
60
I00
•~.
J~
120
m
140
180
leo
o
40
~6.0fm
eo eo ~oo ,20 ~,~ 160 e , (c.M.)
.""".
do" ~,(H.P.Stripping)
eo
°
£'~,= 30 MeV
LO'(,,,p,)
Fig. 3 The cross s e c t m n for the LiT(ot, p l ) B e lo ( I s t excited state) reaction at Ea = 30 MeV. T h e solid curve represents the calculated cross section for the h e a v y - p a r t i c l e - s t r i p p i n g reaction, in which the following values of the p a r a m e t e r s are adopted. R l = 6.0 fm, lp = 1, ?p = ~ and la = 2. J u s t the same as m fig. 2, the core is a s s u m e d to be the g r o u n d s t a t e of He e. The points are the e x p e r i m e n t a l d a t a of K l e m et al. 4).
Ib
i._
!,,,
120C
0
,4
o
THE HEAVY PARTICLE-STRIPPING REACTION (II)
615
b y Klein et al., in which the solid curve represents the cross section calcUlated b y the use of the cut-off radius Rl = 6.4 fm in addition to the values of the other parameters listed in table 2. Here, the calculated cross section is normalized so as to obtain a good fit to the experimental data at larger angle. B y making use of the different normalization but the same Rt as that ~in fig. '2, we can obtain satisfactory agreement with the experiment with respect .to the peak 90 ° appearing in the angular distribution. Next, we present in fig. 3 the calculated cross section for the LiT(~, p~)Be 1° (lst excited state) reaction together with the experimental data at E~ -~ 30 MeV b y Klein et al. Here, we have adopted the cut-off radius R1 ---- 6.0 fm. Because of the same reason as mentioned in subsect. 2.1, we cannot determine the value of Rt very uniquely. Nevertheless, it will be possible to conclude from figs. 1-3 that the experimental angular distributions for the Li (~, p) Be reactions observed b y Klein et al. can be satisfactorily explained b y the heavy-particlestripping reaction *) except for their behaviour at small angles which will, of course, be explained b y the stripping and the knock-on reactions.
3. T h e Fig(a, p)Ne ~ Reactions The experiment was performed b y Priest, Tendam and Bleuler 5) at the incident ,t-particle energy, E~ = 18.9 MeV, of the differential cross sections for the F19(a, po)Ne .2 (ground state), reaction, F19(~, pl)Ne 2. (lst excited state) reaction and the F19(~, p2)Ne 22 (2nd excited state) reaction. The former two cross sections have also been obtained b y Kondo, Yamazaki and Yamabe s) at E~ = 22.2 MeV and the last one b y Martin, Sampson and Miller 7) at E~ = 21.9 MeV. The main feature of the experimental differential cross sections m a y be expressed as follows: the angular distribution for the "P0 group" shows a very sharp peak in the extreme backward direction, while such a sharp peak is not observed in the "Pl and p, groups". Moreover, the magnitude of the cross section for the P0 group is b y a factor of about 10 smaller than that for the Pl * As h a s b e e n m e n t i o n e d in (I) we c a n n o t in g e n e r a l o b t a i n t h e s h a r p p e a k a t t h e b a c k w a r d d i r e c t i o n w h e n t h e h-particle is c a p t u r e d b y a core w i t h o r b i t a l a n g u l a r m o m e n t u m l~ = 0. I n t h e (~, p) r e a c t i o n s t r e a t e d in t h i s section, h o w e v e r , we h a v e a c t u a l l y o b t a i n e d t h e s h a r p p e a k in t h e b a c k w a r d d i r e c t i o n b y a d o p t i n g l~ = 0 t h e s e a r e r a t h e r e x c e p t i o n a l d u e m e r e l y to v e r y l i g h t w e i g h t of t h e t a r g e t nuclei as will be d i s c u s s e d in t h e final section. F u r t h e r , it will be n o t e d t h a t t h e d e c o m p o s i t i o n of t h e t a r g e t n u c l e u s i n t o core a n d p r o t o n s t a t e s e m p l o y e d in s u b s e c t s 2.1. a n d 2.2. m i g h t n o t n e c e s s a r i l y be sufficient. H o w e v e r , if few e x c i t e d s t a t e s of t h e core b e t a k e n i n t o a c c o u n t m a d d i t i o n to i t s g r o u n d s t a t e , t h e q u a h t a t i v e f e a t u r e of t h e a n g u l a r d i s t r i b u t i o n will n o t be c h a n g e d so drastically, p r o v i d e d t h e cnt-off radix R t be t a k e n p r o p e r l y for e a c h case; t h i s is d u e also to t h e v e r y l i g h t w e i g h t of t h e t a r g e t nuclei. I n fact, as will b e clear in t h e l a t t e r sections, t h e a n g u l a r d i s t r i b u t i o n s a r e in g e n e r a l q u i t e s e n s i t i v e t o t h e a n g u lar m o m e n t u m of t h e core F o r h e a v i e r p-shell nuclei, t h e d e c o m p o s i t i o n m a y be m o s t c o n v e n i e n t l y p e r f o r m e d b y t h e m e t h o d of K u r a t h 10).
616
HARUO
UI
AND
TSUYOSHI
HONDA
group. The cross sections for the PI and P2 groups seem to be similar to one another with respect to both magnitude and angular distribution. Since the cross sections for the P0 and Pl groups at E~ = 22.2 MeV differ very little from the corresponding ones at E~ = 18.9 MeV, we shall present the result of the analysis at E~ = 18.9 MeV. From the standpoint of the shell model, the ground state configuration of F 19 is reasonably described as an appropriate combination of (ld~)~=t, (ldj)~(2s½)1= t and (2s½)~_t apart from the rigid core of (Is) ~ (lp) 12. We shall not take account of the proton emission from the rigid core, because it has been found that the cross section for the heavy-particle-stripping reaction becomes very small for a proton with such a large binding energy. Further, for the sake of convenience, we shall discuss separately the two cases of the proton emission from the (ld4) and the (2s½) states. 3.1. PROTON EMISSION FROM THE (ld|) STATE
From the conservation of the angular momentum, we have
lp=2,
5 Ie---- 2 + or 3+. -~,
The core 3+ will not be taken into account, because there is no 3+ state in the bound state of 0 is. Further, the core 2+ will be assumed to be the first excited state of 0 is , the assumption being needed only for the calculation of the binding energy of the emitted proton in the target nucleus and that of the captured a-particle in the residual nucleus. The incident x-particle is captured by the core with a definite orbital angular momentum l~ = 2 for the F19(:c, po)Ne 22 (ground state) reaction. For the Fla(0t, pl)Ne z2 (1st excited state) reactions, the available values of l~ are 0, 2 and 4. As the spin and parity of the second excited state of Ne 22 has not yet TABLE 3 Values ot l~ and Gu for proton emission from the ld t state Ir
l~
Gu
po-group
0+
2
pl-group
2+
0
Po 12
2 4
p,-group
2+
1-
12
P o + 12 P~ + 4--~ P~
0 2 4
(completely the same as above)
1
3
4
3
3
32
Pz) 2
~ ( P o +'ff~ Pz + ~ P , )
017
T H E HEAVY-PARTICLE-STRIPPING REACTION (II)
been definitely determined, we shall adopt the most plausible two alternatives, 2+ and 1% which have been inferred from the intensity of the gamma-transitions. In table 3 are summarized the available values of l= and the corresponding geometrical factors Gb for the Po, Pl and P2 groups.
F'9(4, Po )N/2
5.00
%,
/°
L¢'=~= 1 8 . 9 M e V /
_
4.0C
~.RS~pbng) 1~= 2
t,.
/ / /
Ri =5.1 frn =I,
/
RJ. = 3 . 6 f m
t~
¢0
{0.,%-,%,
/
=o.o33
1 /
.4.-
I
"7.
I I
/ .~,~=
.Q=(= 0 I /
Ri. • §.1 f m o
/ /
"o
l.OC ooo o
o
t ') ,.~_ %/°°
* o
O o o
* o o
o
ii °
oI I
0
0
Oo
50
I00 ep ( c . M.)
00
150
Fig. 4. The cross section for t h e F I ' (x, po)Ne~Z(ground state) reaction a t E= = 18.9 MeV. The solid c u r v e r e p r e s e n t s the calculated cross section for t h e h e a v y - p a r t i c l e - s t r i p p i n g reaction, while the p o i n t s are the e x p e r i m e n t a l d a t a b y Priest et al. 6) The scale of t h e o r d i n a t e is t a k e n to be the s a m e as t h a t of Priest et a l , t h a t is, 1 u m t in t h e figure c o r r e s p o n d s a p p r o x i m a t e l y to 8.5 #b/sr. I n the calculation, the following values of t h e p a r a m e t e r s are a d o p t e d . Rt = 5.1 fm, lp = 2, jp = ~ a n d l= = 2 The core is a s s u m e d t o be the first excited s t a t e of 018. F u r t h e r , the reduced wxdth {O=O=Op} z is d e t e r m i n e d as 0.083, if the cut-off r a d i u s Rt is t a k e n t e n t a t i v e l y to be 3.6 frn. The b r o k e n curve r e p r e s e n t s the cross section for t h e h e a v y - p a r t i c l e - s t r i p p i n g reactzon for the case discussed in t h e n e x t subsection, in which t h e core is a s s u m e d t o be t h e g r o u n d s t a t e of O xS. The values of the p a r a m e t e r s a d o p t e d are as follows" R t -~ 5.1 fm, l v = 0, jp = { a n d l~ = 0. The b r o k e n curve is n o r m a h z e d to the solid curve a t 180 °.
We present in fig. 4 a typical calculated cross section for the F19(~, p0)Ne 22 (ground state) reaction together with the experimental data at E= = 18.9 MeV b y Priest et al. Here, the solid curve represents the calculated cross section for the heavy-particle-stripping reaction, in which we adopt the cut-off radius RI = 5.1 fro.
6]8
HARUO U I AND T S U Y O S t t I H O N D A
Further, when Rt = 3.6 fm is adopted, we can determine the magnitude of {O~O=Op}2 as 0.033, which would be intimately connected with the nuclear structure. The determination of {O=O=Op}2, however, is not necessarily unambiguous, since it is impossible here to determine Rt uniquely. In order to determine the value of Rt and, consequently, the {0=O~Op}2 rather unambiguously, experiments are highly required in a wider range of E~ as has been discussed in detail in (I).
22
Ne =
s.9 M e V
d~r -~-~- ( H. P.Str! pping)
2.00
•
S.,
~== o
O
0
O0
00000
CD
?2 r.
0
1.00
0
0o
a.
]"
T
0 0
00
0 0 0 0 o O0
50
0
00
0o 0
I00 ep (C.M.)
15(3
Fig. 5. The cross section for t h e Fl0(~, pl)Nem2(lst excited state) reaction a t E= = 18.9 MeV. The sohd curve represents the calculated cross section for t h e heavy-particle-stripping, while t h e p o i n t s are the e x p e r i m e n t a l d a t a b y Priest 8t al. 6). E a c h u n i t in the o r d i n a t e corresponds a p p r o x i m a t e l y to 85.0/Jh/sr Notice this scale is larger b y a factor 10 t h a n t h a t used in fig. 4. The following v a l u e s of t h e p a r a m e t e r s are a d o p t e d in the calculation: R I z 5.1 fro, ~ ~ 2, ]p z ~ a n d l= = 0. The core is a s s u m e d to be the h r s t excited s t a t e of O is. The c o n t r i b u t i o n s f r o m l~ = 2 a n d 4, w i n c h are also allowed b y the conservation of the a n g u l a r m o m e n t u m , are disregarded here.
The accurate analysis of the F19(0t, pl)Ne 22 (lst excited state) reaction and the F19(,¢, p~)Ne 22 (2nd excited state) reaction will be quite involved in comparison to that ot the F19(~, po)Ne a2 (ground state) reaction, because several values of l= are available in these reactions from the conservation of the angular
THE HEAVY-PARTICLE-STRIPPING REACTION {II)
619
momentum. Associated with these l¢ will generally be different values of {0~0~}. If these {0~0~} are all taken to be adjustable parameters, we could of course fit the experimental data quite satisfactorily by mal6ng use of the "least squares method". Such a procedure as mentioned above m a y be of little meaning until the more detailed knowledge on the cluster structure of an individual nuclear level be provided from various sources in nuclear physics. In view of above situation, therefore, we present in fig. 5 the calculated cross section for the F19(~, pl)Ne .2 (lst excited state) reactioA together with the experimental data at E~ = I8.9 MeV by Priest et al., in which only l, = 0 is taken into account, the contributions from l~ = 2 and 4 being disregarded. Further, we adopt the cut-off radius Rt ----- 5.1 fm in the calculation. Although the agreement between theory and the experiment seems to be not satisfactory in fig. 5, it will at least be concluded that the contributions from l~ ~ 2 and 4 should be much smaller than that from l~ = 0, because the large contributions from l, = 2 and 4 will necessarily lead to a sharp peak at 180 ° in the angular distribution, similarly to the case of the Po group, which is evidently contradictory to the experimental data. Further, it can be inferred from our analysis that the different magnitudes of the cross sections for the P0 and Pl groups can be explained naturally if the reduced width {0,0~} 2 for the F19(x, pl)Ne ~2 (lst excited state) reaction has approximately the same value as that for the F19(e, po)NeZZ(ground state) reaction. By similar reasoning to t h a t above, we cannot determine definitely from our analysis the spin and the parity of the second excited state of Ne z~. It seems, however, to be more plausible to adopt the assignment 2 + rather than 1from our calculations, although these are not presented here in the graphical form. A more definite assignment might be possible if the experimental data were available in the wider range of E~ as has been repeatedly mentioned. 3 2. P R O T O N E M I S S I O N F R O M T H E (2st) S T A T E
In an analogous way we have ip=
1 ~p----T ~, 0, I e = 0
+ or 1+.
The core with 1+ will not be dealt with, since there are no 1+ states in the lower excited states in 01°. Then, corresponding to table 3 for the preceding case, we have table 4 for the present one. As can be inferred immediately from table 4, the differential cross section for the F19(,¢, po)Ne 22 (ground state) reaction shows no sharp peak in the extreme backward direction, since the angular momentum of the captured ~-particle is necessarily zero in this case. For purposes of illustration, we have presented also in fig. 4 the calculated cross section for the heavy-particle-stripping reaction for this case by the broken curve. The cut-off radius Rt has been taken
6~0
HARUO UI AND T S ~ O S H I
HONDA
as 5.1 fm and, further, the angular distribution has been normalized to the calculated cross section for the proton emission from the (ld|) state, which has been represented by the solid curve in fig. 4, for the sake of illustration. TABLE 4
Values of l~ and Gh for proton emission from the 2s½ state
3.3.
I,
h
Gh
po-group pt-group
0+ 2+
0 2
1 5
P2"gr°up
2+ 1-
2 1
5 3
COMPARISON WITH EXPERIMENT
As can be seen from fig. 4, we cannot obtain good agreement with experiment if the main contribution arises from emission from the (2s½) state. In order to explain the experimental results, therefore, it seems to be quite necessary t h a t the probability of the proton emission from the (ld~) state should be much larger than that from the (2s½) state. Namely, the amplitude of the (2s½) configuration in the ground state of F 1" should be much smaller than that of the (ld~) configuration. Detailed calculations based on the shell model have been performed by EUiott and Flowers s) and by Redlich 9) of the nuclei of mass number A = 18 and 19 including F 1" and 0 Is. In their calculations the most dominant configuration in the ground state of F 1" is ascribed to (2s½)~_p other configurations being smaller. Namely, when the ground state of F 1" is decomposed into a proton plus remaining core, the greatest parentage overlap should be attributed to the case discussed in the preceding subsection. From their result the contribution from proton emission from the ld t state must be smaller than that from the 2s½ state, in contradiction to the result of our analysis. It should be noted, however, that the parentage overlap obtainable from the work of Elliott et al. is the so-called spectroscopic factor in the reduced width of a proton. The dynamical factor depending on the details of radial part of the nuclear wave function has not yet been investigated; the reduced width In the bound state will be expected to depend rather strongly on the binding energy, while in almost all shell-model calculations radial wave functions in the harmonic oscillator potential have been used. In this connection, it will be noted that a similar discrepancy m a y also be seen in the Fl"(p, d)F is and Fl"(n, d)O is reactions, as has been discussed by Macfarlane and French 11). 4.
Discussion
In this section we shall discuss some qualitative properties of the cross section for the heavy-particle-stripping reaction such as its dependence on the mass
THE
HEAVY-PARTICLE-STRIpPING
REACTION
(II)
621
number of the target nucleus and on the binding energy of the emitted proton in the target nucleus. Further, some discussion is added on the relation between the backward peak in the differential cross section for the heavy-particlestripping reaction and the structure of low-lying states of light nuclei. First of all, we shall discuss briefly what role the mass number of the target nucleus plays in the cross section. Intuitively speaking, the heavy-particle-stripping mechanism m a y be considered as a sort of recoil effect 2). Namely, if the target nucleus is sufficiently heavy, the angular distribution for the heavy-particle-stripping reaction can be shown to be symmetric about 90 ° direction 2) and, moreover, the magnitude of the cross section can be proved to vanish in the limit of infinitely heavy target nucleus: this suggests that the heavy-particle-stripping reaction will become unimportant as the mass number of the target nucleus increases appreciably. On the other hand, the heavy-particle-stripping reaction will play a very important role when the mass of the target nucleus is not so much different from that of the incident particle. If the cut-off Born approximation is applicable to the calculation of the cross section for the heavy-particle-stripping reaction, such circumstances as mentioned above can be clearly seen from the explicit expression of the cross section given in ref. 1) through the factor (K~Rt), the momentum transfer in the heavy-particle-stripping reaction multiplied b y the cut-off radius R 1. In the Li(~, p)Be reactions treated in the sect. 2, the relation between the binding energies of the proton in the incident and in the target nuclei is quite favourable to the heavy-particle-stripping rather than to the stripping reaction 1-3). We can therefore expect a large contribution from the heavy particlestripping reaction in the Li(a, p)Be reaction. As has been shown in figs. 1-3, it is actually the case. It is further noted that we can obtain quite satisfactory agreement with the experimental data of the LiS(~, po)Be 9 (ground state) reaction and the Li~(~, po)Be 1° (ground state) reaction, in both of which the angular distribution shows a rather sharp peak in the backward direction, b y means of the heavy-particle-stripping reaction with l= = 0. As has been discussed in detail in (I), we cannot obtain in general the sharp peak in the backward direction b y adopting l~ = 0, since the geometrical factor Gh is constant in this case. The Li(a, p) Be reactions are rather exceptional cases due to very light weight of the target nucleus and to the binding energy relation mentioned previously. In fact, in the Cl~(x, p0)N 15 reaction and the F19(~, P0)Ne z2 reaction, it has been found impossible to reproduce the sharp peak in the backward direction b y means of the heavy particle-stripping reaction with l= = 0. For the cases of l, v~ 0, the geometrical factor Gh plays an essential role in determining the shape of the angular distribution at large angles. Even if l~ va 0, there are several other special cases in which the geometrical
622
HARUO UI AND TSUYOSHI HONDA
factor Gh does not contain any angle-dependence. Among them, the most interesting is the case in which the proton is emitted from the s-orbit in the target nucleus, i.e. lp = 0. A typical example is the F19(~, p0)Ne ~ reaction in which the proton is assumed to be emitted from the (2s½) state, as can be seen from the table 4 in sect. 3. Therefore, the discussions presented in sect. 3 in connection with the shell model wave function b y Elliott and Flowers and b y Redlich is persistent, independent o/the choice of other parameters, as far as the very sharp peak in the extremely backward direction in the F19(c¢, P0) Ne z2 reaction can be interpreted in terms of the heavy particle-stripping reaction. It m a y be interesting to note that the sharp peaks in the backward direction have all been reproduced by means of the cross section formula (eq. (1) of ref. 1)) with approximately the same value of (~TRt) in our calculations in the previous. and the present papers. It seems then to be probable that the cut-off Born approximation will be applicable to the calculation of the cross section for the heavy-particle-stripping reaction in the energy region treated here as far as the angular distribution is concerned. In this connection, it is to be noted that Owen, Madansky and Edwards 13) have performed detailed analysis of the angular distribution for the Cis (He s, ¢¢) Clz reaction at 2.0 and 4.5 MeV 13) b y means of the stripping and the heavyparticle stripping reactions, getting fairly good agreement with the experiment. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)
T. H o n d a and H. U1, Nuclear Physics 34 (1962) 593 T. H o n d a and H. UI, Prog. Theor Phys. 25 (1961) 613 T. H o n d a and H. Ui, Prog. Theor. Phys 25 (1961) 635 P R Klein, N. Cindro, L. W Swenson and N. S. Wall, Nuclear Physics 16 (1960) 374 J. R. Prmst, D. J. Tendam and E. Bleuler, Phys. Rev. 119 (1960) 1301 M. Kondo, T. Yamazaki and S. Yamabe, J. Phys. Soc. J a p a n 16 (1961) 1091 H. J Martin, M. B. Sampson and D W. Miller, Phys. l~ev. 121 (1961) 877 J. P. Elliott and B. H Flowers, l=Yoc. Roy Soc. 229 (1955) 536 M. G. Redlich, Phys. Rev. 110 (1958) 468 D. Kurath, ANL-6312 (1961), unpubhshed; D K u r a t h and L. P16man, Nuclear Physics 10 (1959) 313 11) M. H. Macfarlane and J. B. French, Rev Mod. Phys. 32 (1960) 567 12) G. E. Owen, L Madansky and S. Edwards Jr., Phys. Rev. 113 (1959) 1575 13) H. D. Holmgren, Phys. Rev. 106 (1957) 10O; Holmgren, Greer, Johnston and WolJcki, Phys. Rev. 106 (1957) 102
[
Nuclear Physics 34 (1962) 609--622, (~) North-Holland Publishing Co., Amsterdam Not to be
reproduced by photoprmt or microfilm
without written
permxsmon from the pubhsher
THE HEAVY-PARTICLE-STRIPPING REACTION IN (a, p) REACTIONS (II) The Li(a, p ) B e and the F(a, p ) N e reactions T S U Y O S H I H O N D A and H A R U O U I
Department of Physws, Tohoku University, Sendal, Japan Received 2 F e b r u a r y 1962 I n t h e same w a y as t h a t p r e s e n t e d in t h e preceding paper, t h e following reactions are analysed in detail in t e r m s of t h e h e a v y - p a r t i c l e - s t r i p p i n g reaction" t h e Li 6 (a, p0)Be' (ground state) reaction, t h e La~ (~, po)BO ° (ground state) reaction and t h e L17(a, p 1 ) B O O (1st excited state) r e a c t i o n a t Ea ~ 30 MeV a n d t h e F19(a, po)Ne *l (ground state) reaction, t h e FI'(~, Pl)NeZ* (1st excited state) reaction a n d t h e FI'(~, p , ) N e 2. (2nd excited state) r e a c t i o n a t a b o u t E~ = 20 MeV. I t is s h o w n t h a t t h e angular distributions of these t h r e e (a, p) reactions for Li nuclei c a n b e satisfactorily explained b y t h e heavy-partxcle-stripping reaction e x c e p t for their b e h a v i o u r a t small angles. ]:t is f u r t h e r s h o w n t h a t t h e differential cross sections a t larger angles for t h e FI'(~, p ) N e n reactions q u o t e d a b o v e are also explained r e a s o n a b l y well b y t h e h e a v y particle-stripping reaction. I n particular, good a g r e e m e n t is o b t a i n e d b e t w e e n t h e o r y and t h e e x p e r i m e n t with r e s p e c t to t h e v e r y s h a r p p e a k a t t h e b a c k w a r d direction appearing in t h e differential cross section for t h e FI'(~, p0)Ne 2~ (ground state) reaction. I n this connection, some r e m a r k s are a d d e d concerning t h e shell m o d e l wave f u n c t m n s of 018 a n d Flg calculated b y Elliott a n d Flowers a n d b y Redlich. The reduced w i d t h {O~O~Op} 2 e x t r a c t e d from t h e e x p e r i m e n t a l d a t a of t h e F19(~, po)N *z reaction is found to be very small in comparison to t h a t of t h e C12(~, pg)N 15 reaction o b t a i n e d m t h e preceding paper.
Abstract:
1. Introduction In the preceding paper 1), which will be referred to as (I), we have analysed in detail the differential cross section for the C1'(~, p0)N 15 (ground state) reaction at large angles from E~ = 16 MeV to 40 MeV in terms of the heavyparticle-stripping reaction by employing the formulation 2, 3), which has been presented previously by us for the (d, p) and (d, n) reactions. It has been shown therein that not only the shape of the angular distribution at large angles but also the magnitude of the cross section, both of which vary nonmonotonically with the energy of the incident 0c-particle can be well reproduced by the theory of the heavy-particle-stripping reaction without introducing any energy-dependent adjustable parameters. In the present paper, we shall extend the previous analysis to some other available data of the (,t, p) reactions, including the LiS(~, p0)Be 9 (ground 609
610
HARUO UI AND TSUYOSHI HONDA
state) reaction, the Li~(x, Po) Be1° (ground state) reaction and the Li~(x, Pl) Be1° (lst excited state) reaction at the incident x-particle energy, E~ = 30 MeV 4) and the F19(x, po)Ne 22 (ground state) reaction, the F19(x, pl)Ne z2 (lst excited state) reaction and the F19(x, p~)Ne z2 (2nd excited state) reaction at about E~ = 20 MeV S-T). We shall treat in the next section the Li(x, p)Be reactions, which m a y be expected to be one of the best examples that exhibits the typical angular distribution for the heavy-particle-stripping reaction as will be discussed in detail there. It will be made clear from our analysis that this is actually the case.
In sect. 3, the F19(x, p)Ne 2z reaction will be treated in detail. It will be shown that not only the angular distributions but also the magnitudes of the cross sections for the F19(x, p0)Ne ~z (ground state) and the F19(x, pl)Ne 2~ (lst excited state) reactions at large angles can be rather reasonably reproduced b y the theory of the heavy-particle-stripping reaction, although both cross sections are quite different from one another. In this connection, some remarks are added concerning the shell model wave functions of F 19 and 0 is calculated b y Elliott and Flowers s) and b y Redlich ~). In analysing the F19(x, P2) Ne22 (2nd excited state) reaction, we shall tentatively adopt two alternative assignments, 2 + and 1% of the spin and parity of the second excited state of Ne 2~. Unfortunately, however, we cannot make a definite assignment of this state from our result of the analysis, although the assignment of 2+ seems to be more plausible than that of 1-. Finally, in sect. 4, some discussion is added concerning the dependence of the cross section on the mass number of the target nucleus and on the incident energy of the x-particles, etc. A remark on the structure of the light nuclei will be presented there in connection with the heavy-particle-stripping reaction *.
2. The Li6(a, Po)Beg, Li'(a, Po)Be l° and Li'(a, pi)Be l° Reactions The experiment of the (x, p) reactions on the Li isotope has been performed b y Klein, Cindro, Swenson and Wall 4) at the incident x-particle energy E~ = 30 MeV leading to several definite states of the residual nuclei. We shall present in this section the result of the analysis of these reactions. The Li(x, p)Be reaction m a y be supposed as one of the best examples which exhibit the typical angular distribution for the heavy-particle-stripping reaction because of the following reasons: firstly, the mass of the target nucleus does not differ so much from that of the incident particle and, secondly, the t T h e s a m e n o t a t i o n is u s e d t h r o u g h o u t t h e p r e s e n t p a p e r a s i n t h e p r e v i o u s p a p e r t).
THE HEAVY-PARTICLE-STRIPPING
REACTION
(II)
611
binding energy of the proton in the ,t-particle (Bp---- 19.81 MeV) is much larger than t h a t of the outer proton in the target nucleus (Bp = 4.655 MeV for Li e and Bp = 10.006 MeV for LIT). Thirdly, the target nucleus has a rather simple structure in the sense t h a t there are no complications in the decomposition of the target nucleus into the core and proton states. Namely, it m a y not be necessary to take inoto account m a n y states of core, which leads to a quite involved calculation of the cross section. For the sake of convenience, we shall treat the LiS(*t, p0)Be 9 (ground state) reaction in subsect. 2.1. and the Li?(*t, p0)Bel° (ground state) and the Li 7 (*t, pl)Be 1° (lst excited state) reactions in subsect. 2.2, respectively. 2 I. THE Li'(~z,p0)Be0 (GROUND STATE) REACTION The cross section for the heavy-particle-stripping reaction presented in sect. 1 of the preceding paper I) has been expressed in terms of t h e / - i coupling prescription. The states of Li e, however, are well-known to be approximately described b y the L-S coupling rather than t h e / - j coupling shell model. It seems, therefore, to be necessary to represent the cross section in terms of the L-S coupling scheme, which can be performed directly by means of the so-called transformation bracket or Wigner's 9-j symbol. Equivalently, we shall make use of the more general expression of the cross section in the 7"-Jcoupling scheme in which several appropriate ~'-values are taken into account on an equal footing: For example, the ground state of LiS(T = 0, J = 1+) can be represented as an SS-state in L-S coupling, which can in turn be described as a mixed configuration between the (lpl) and (Ip~) states in t h e / - j coupling prescription. We shall not take into account the proton emission from the (1 s) state due to its large binding energy. Then, there are two odd parity states of He 5, i.e. the ground f = ~- and the first excited f = 21-- states, both of which m a y be identified as the core in the heavy-particle-stripping reaction. In the following analysis, we shall assume the core to be the ground state of HeS; the first excited state will not be tal~en into account, since this state is a very broad virtual level. Accordingly, the incident *t-particle should be captured by the core with the orbital angular momenta l~ = 0 and 2 to form the residual nucleus, the ground state of B e g ~ = {-). In table 1 are summarized the various values of the quantities which are needed to the analysis. As can be seen from the explicit expression of the cross section, the angular distribution arising from the (1 Pl) state proton emission is exactly the same as that from the (1 p½) state. Further, the energy-dependence of the former cross section is the same as t h a t of the latter cross section, provided the cut-off radii be taken equal to one another. As an example, a typical calculated cross section for the Lie(*t, po)Be 9
612
HARUO
UI
AND
TSUYOSHI
HONDA
(ground state) reaction is shown in fig. 1 together with the experimental data at E= = 30 MeV b y Klein et al. Here, the solid curve represents the calculated
Lie(co,
TABLE 1 P a r a m e t e r s for the Q Bp B= It
p0)Be' reaction
--2.125 MeV 4.655 2.529 1+
/r
t-
Ie
~ - (the g r o u n d s t a t e of He s)
lp
1
l=
0
2
0
2
Gh
1
1
1
1
L,i 8( ,,po) B/A
/
J~,~= 0
,,.
":'•
30MeV /.: l o°
4 o~o oo
3
°m
.,o o° eL
Z
•
•
•
•
°°° •
o@
I •
~~R0.---~ 60frn 0
=o
4O
e0
00
100 tZ0
140 180 too
ep ( c. M.) Fig. 1. The cross section for the Ll°(~, p0)Be 9 (ground state) reaction a t E= = 30 MeV. The sohd curve represents the calculated cross section for the heavy-particle-stripping reaction, while t h e p o i n t s are the e x p e r i m e n t a l d a t a b y Klein et al. 4). As the m a g n i t u d e of the cross section w a s n o t m e a s u r e d b y Klein et a l , the scale of the ordinate is w r i t t e n in a r b i t r a r y units. The scale of the abscissa denotes t h e angle of the emitted p r o t o n in the centre-of-mass s y s t e m m e a s u r e d in degrees. I n the calculation, t h e following values of the p a r a m e t e r s are a d o p t e d R 1 = 6 0 fro, lp = l, jp = ~ and t a n d l~ = 0. The core is a s s u m e d to be t h e g r o u n d s t a t e of He 5.
THE HEAVY-PARTICLE-STRIPPING REACTION (If)
613
cross section for the heavy-particle-stripping reaction, in which we adopt the cut-off radius Rt = 6.0 fm in addition to the values of the various parameters listed in table 1. Further, in the calculated cross section presented in fig. 1, we have taken into account only the ~-particle capture with l~ = 0, the ,tparticle capture with l~ = 2 being disregarded. The value of Rt adopted here is not necessarily unique. One could determine it rather unambiguously if the experimental data were available in such a wider range of E~ as in the case of the CZ2(~, po)N is (ground state) reaction treated in (I). Moreover, since even the magnitude of the cross section has not yet been measured, we cannot extract from the experiment the parameter {0~ O~ Op}~ which would provide important information on the nuclear structure. 2.2. T H E
LiT(0c, p ) B e t° R E A C T I O N S
Next, we shall analyse the LiT(,c, po)Be 1° (ground state) reaction and the LiT(~, pz)Bel°(lst excited state) reaction at E~ = 30 MeV. We shall simply assume the core to be the ground state of H e e ( J = 0+): the first excited state of He6(J = 2+) will not be taken into account, since the state is unstable against particle emission. Then, the proton is emitted from the (lp~) orbit in the ground state of Li T, while the proton in the (lP½) orbit can not take part in the heavy-particle-stripping reaction because of the conservation of the angular momentum. For the LiT(x, Po)Be1° (ground state) reaction, the incident ~-particle should be captured b y the core with an orbital angular momentum l~ ---- 0 to form the ground state of Be 1°, while the ~-particle should be captured with l~ = 2 for the LiT(~, pz)Be 1° (lst excited state) reaction. In table 2 are summarized the various values of the quantities which are needed in the analysis. TABLE 2 P a r a m e t e r s for t h e Li~(ct, p ) B e lo r e a c t i o n LiT (~, Po) Bez° Q Bp
-2.566
Lx~ (m, Pt) BeZ°
MeV
--5.934 MeV 10.006
B~
7.439
~-
It Ir Io lp
0+ 2+ 0 + (the ground state of H e e) 1
lp t~ Gh
o 1
t 2 5
A typical calculated cross section for the LiT(~, P0) Be1° (ground state) reaction is shown in fig. 2 together with the experimental data at E~ = 30 MeV
O
20
0
(d.Po)
BelO
'
2o
'
4o
eo
so
'
'
"
'
'
6Afro
..
~00 Izo =40 ~eo tso
ep (C.M.)
'
•
I
£== 30MeV ~(H.P.Stripping).]
L 7i
Fig 2. The cross section for the Li T(~, Po) BeZ° (ground state) reaction at E~ = 30MeV. The sohd curve represents the calculated cross section for the heavy-particle-stripping reaction, while the p o i n t s are the experimental d a t a b y Klein et el. 6) I n the calculation, the following values of the p a r a m e t e r s are adopted: Rt = 6.4 fm, lp = 1, Ip = }:and l~ = 0 The core is a s s u m e d to be the g r o u n d s t a t e of He ~
•ID
"-
L
so
~
60
I00
•~.
J~
120
m
140
180
leo
o
40
~6.0fm
eo eo ~oo ,20 ~,~ 160 e , (c.M.)
.""".
do" ~,(H.P.Stripping)
eo
°
£'~,= 30 MeV
LO'(,,,p,)
Fig. 3 The cross s e c t m n for the LiT(ot, p l ) B e lo ( I s t excited state) reaction at Ea = 30 MeV. T h e solid curve represents the calculated cross section for the h e a v y - p a r t i c l e - s t r i p p i n g reaction, in which the following values of the p a r a m e t e r s are adopted. R l = 6.0 fm, lp = 1, ?p = ~ and la = 2. J u s t the same as m fig. 2, the core is a s s u m e d to be the g r o u n d s t a t e of He e. The points are the e x p e r i m e n t a l d a t a of K l e m et al. 4).
Ib
i._
!,,,
120C
0
,4
o
THE HEAVY PARTICLE-STRIPPING REACTION (II)
615
b y Klein et al., in which the solid curve represents the cross section calcUlated b y the use of the cut-off radius Rl = 6.4 fm in addition to the values of the other parameters listed in table 2. Here, the calculated cross section is normalized so as to obtain a good fit to the experimental data at larger angle. B y making use of the different normalization but the same Rt as that ~in fig. '2, we can obtain satisfactory agreement with the experiment with respect .to the peak 90 ° appearing in the angular distribution. Next, we present in fig. 3 the calculated cross section for the LiT(~, p~)Be 1° (lst excited state) reaction together with the experimental data at E~ -~ 30 MeV b y Klein et al. Here, we have adopted the cut-off radius R1 ---- 6.0 fm. Because of the same reason as mentioned in subsect. 2.1, we cannot determine the value of Rt very uniquely. Nevertheless, it will be possible to conclude from figs. 1-3 that the experimental angular distributions for the Li (~, p) Be reactions observed b y Klein et al. can be satisfactorily explained b y the heavy-particlestripping reaction *) except for their behaviour at small angles which will, of course, be explained b y the stripping and the knock-on reactions.
3. T h e Fig(a, p)Ne ~ Reactions The experiment was performed b y Priest, Tendam and Bleuler 5) at the incident ,t-particle energy, E~ = 18.9 MeV, of the differential cross sections for the F19(a, po)Ne .2 (ground state), reaction, F19(~, pl)Ne 2. (lst excited state) reaction and the F19(~, p2)Ne 22 (2nd excited state) reaction. The former two cross sections have also been obtained b y Kondo, Yamazaki and Yamabe s) at E~ = 22.2 MeV and the last one b y Martin, Sampson and Miller 7) at E~ = 21.9 MeV. The main feature of the experimental differential cross sections m a y be expressed as follows: the angular distribution for the "P0 group" shows a very sharp peak in the extreme backward direction, while such a sharp peak is not observed in the "Pl and p, groups". Moreover, the magnitude of the cross section for the P0 group is b y a factor of about 10 smaller than that for the Pl * As h a s b e e n m e n t i o n e d in (I) we c a n n o t in g e n e r a l o b t a i n t h e s h a r p p e a k a t t h e b a c k w a r d d i r e c t i o n w h e n t h e h-particle is c a p t u r e d b y a core w i t h o r b i t a l a n g u l a r m o m e n t u m l~ = 0. I n t h e (~, p) r e a c t i o n s t r e a t e d in t h i s section, h o w e v e r , we h a v e a c t u a l l y o b t a i n e d t h e s h a r p p e a k in t h e b a c k w a r d d i r e c t i o n b y a d o p t i n g l~ = 0 t h e s e a r e r a t h e r e x c e p t i o n a l d u e m e r e l y to v e r y l i g h t w e i g h t of t h e t a r g e t nuclei as will be d i s c u s s e d in t h e final section. F u r t h e r , it will be n o t e d t h a t t h e d e c o m p o s i t i o n of t h e t a r g e t n u c l e u s i n t o core a n d p r o t o n s t a t e s e m p l o y e d in s u b s e c t s 2.1. a n d 2.2. m i g h t n o t n e c e s s a r i l y be sufficient. H o w e v e r , if few e x c i t e d s t a t e s of t h e core b e t a k e n i n t o a c c o u n t m a d d i t i o n to i t s g r o u n d s t a t e , t h e q u a h t a t i v e f e a t u r e of t h e a n g u l a r d i s t r i b u t i o n will n o t be c h a n g e d so drastically, p r o v i d e d t h e cnt-off radix R t be t a k e n p r o p e r l y for e a c h case; t h i s is d u e also to t h e v e r y l i g h t w e i g h t of t h e t a r g e t nuclei. I n fact, as will b e clear in t h e l a t t e r sections, t h e a n g u l a r d i s t r i b u t i o n s a r e in g e n e r a l q u i t e s e n s i t i v e t o t h e a n g u lar m o m e n t u m of t h e core F o r h e a v i e r p-shell nuclei, t h e d e c o m p o s i t i o n m a y be m o s t c o n v e n i e n t l y p e r f o r m e d b y t h e m e t h o d of K u r a t h 10).
616
HARUO
UI
AND
TSUYOSHI
HONDA
group. The cross sections for the PI and P2 groups seem to be similar to one another with respect to both magnitude and angular distribution. Since the cross sections for the P0 and Pl groups at E~ = 22.2 MeV differ very little from the corresponding ones at E~ = 18.9 MeV, we shall present the result of the analysis at E~ = 18.9 MeV. From the standpoint of the shell model, the ground state configuration of F 19 is reasonably described as an appropriate combination of (ld~)~=t, (ldj)~(2s½)1= t and (2s½)~_t apart from the rigid core of (Is) ~ (lp) 12. We shall not take account of the proton emission from the rigid core, because it has been found that the cross section for the heavy-particle-stripping reaction becomes very small for a proton with such a large binding energy. Further, for the sake of convenience, we shall discuss separately the two cases of the proton emission from the (ld4) and the (2s½) states. 3.1. PROTON EMISSION FROM THE (ld|) STATE
From the conservation of the angular momentum, we have
lp=2,
5 Ie---- 2 + or 3+. -~,
The core 3+ will not be taken into account, because there is no 3+ state in the bound state of 0 is. Further, the core 2+ will be assumed to be the first excited state of 0 is , the assumption being needed only for the calculation of the binding energy of the emitted proton in the target nucleus and that of the captured a-particle in the residual nucleus. The incident x-particle is captured by the core with a definite orbital angular momentum l~ = 2 for the F19(:c, po)Ne 22 (ground state) reaction. For the Fla(0t, pl)Ne z2 (1st excited state) reactions, the available values of l~ are 0, 2 and 4. As the spin and parity of the second excited state of Ne 22 has not yet TABLE 3 Values ot l~ and Gu for proton emission from the ld t state Ir
l~
Gu
po-group
0+
2
pl-group
2+
0
Po 12
2 4
p,-group
2+
1-
12
P o + 12 P~ + 4--~ P~
0 2 4
(completely the same as above)
1
3
4
3
3
32
Pz) 2
~ ( P o +'ff~ Pz + ~ P , )
017
T H E HEAVY-PARTICLE-STRIPPING REACTION (II)
been definitely determined, we shall adopt the most plausible two alternatives, 2+ and 1% which have been inferred from the intensity of the gamma-transitions. In table 3 are summarized the available values of l= and the corresponding geometrical factors Gb for the Po, Pl and P2 groups.
F'9(4, Po )N/2
5.00
%,
/°
L¢'=~= 1 8 . 9 M e V /
_
4.0C
~.RS~pbng) 1~= 2
t,.
/ / /
Ri =5.1 frn =I,
/
RJ. = 3 . 6 f m
t~
¢0
{0.,%-,%,
/
=o.o33
1 /
.4.-
I
"7.
I I
/ .~,~=
.Q=(= 0 I /
Ri. • §.1 f m o
/ /
"o
l.OC ooo o
o
t ') ,.~_ %/°°
* o
O o o
* o o
o
ii °
oI I
0
0
Oo
50
I00 ep ( c . M.)
00
150
Fig. 4. The cross section for t h e F I ' (x, po)Ne~Z(ground state) reaction a t E= = 18.9 MeV. The solid c u r v e r e p r e s e n t s the calculated cross section for t h e h e a v y - p a r t i c l e - s t r i p p i n g reaction, while the p o i n t s are the e x p e r i m e n t a l d a t a b y Priest et al. 6) The scale of t h e o r d i n a t e is t a k e n to be the s a m e as t h a t of Priest et a l , t h a t is, 1 u m t in t h e figure c o r r e s p o n d s a p p r o x i m a t e l y to 8.5 #b/sr. I n the calculation, the following values of t h e p a r a m e t e r s are a d o p t e d . Rt = 5.1 fm, lp = 2, jp = ~ a n d l= = 2 The core is a s s u m e d t o be the first excited s t a t e of 018. F u r t h e r , the reduced wxdth {O=O=Op} z is d e t e r m i n e d as 0.083, if the cut-off r a d i u s Rt is t a k e n t e n t a t i v e l y to be 3.6 frn. The b r o k e n curve r e p r e s e n t s the cross section for t h e h e a v y - p a r t i c l e - s t r i p p i n g reactzon for the case discussed in t h e n e x t subsection, in which t h e core is a s s u m e d t o be t h e g r o u n d s t a t e of O xS. The values of the p a r a m e t e r s a d o p t e d are as follows" R t -~ 5.1 fm, l v = 0, jp = { a n d l~ = 0. The b r o k e n curve is n o r m a h z e d to the solid curve a t 180 °.
We present in fig. 4 a typical calculated cross section for the F19(~, p0)Ne 22 (ground state) reaction together with the experimental data at E= = 18.9 MeV b y Priest et al. Here, the solid curve represents the calculated cross section for the heavy-particle-stripping reaction, in which we adopt the cut-off radius RI = 5.1 fro.
6]8
HARUO U I AND T S U Y O S t t I H O N D A
Further, when Rt = 3.6 fm is adopted, we can determine the magnitude of {O~O=Op}2 as 0.033, which would be intimately connected with the nuclear structure. The determination of {O=O=Op}2, however, is not necessarily unambiguous, since it is impossible here to determine Rt uniquely. In order to determine the value of Rt and, consequently, the {0=O~Op}2 rather unambiguously, experiments are highly required in a wider range of E~ as has been discussed in detail in (I).
22
Ne =
s.9 M e V
d~r -~-~- ( H. P.Str! pping)
2.00
•
S.,
~== o
O
0
O0
00000
CD
?2 r.
0
1.00
0
0o
a.
]"
T
0 0
00
0 0 0 0 o O0
50
0
00
0o 0
I00 ep (C.M.)
15(3
Fig. 5. The cross section for t h e Fl0(~, pl)Nem2(lst excited state) reaction a t E= = 18.9 MeV. The sohd curve represents the calculated cross section for t h e heavy-particle-stripping, while t h e p o i n t s are the e x p e r i m e n t a l d a t a b y Priest 8t al. 6). E a c h u n i t in the o r d i n a t e corresponds a p p r o x i m a t e l y to 85.0/Jh/sr Notice this scale is larger b y a factor 10 t h a n t h a t used in fig. 4. The following v a l u e s of t h e p a r a m e t e r s are a d o p t e d in the calculation: R I z 5.1 fro, ~ ~ 2, ]p z ~ a n d l= = 0. The core is a s s u m e d to be the h r s t excited s t a t e of O is. The c o n t r i b u t i o n s f r o m l~ = 2 a n d 4, w i n c h are also allowed b y the conservation of the a n g u l a r m o m e n t u m , are disregarded here.
The accurate analysis of the F19(0t, pl)Ne 22 (lst excited state) reaction and the F19(,¢, p~)Ne 22 (2nd excited state) reaction will be quite involved in comparison to that ot the F19(~, po)Ne a2 (ground state) reaction, because several values of l= are available in these reactions from the conservation of the angular
THE HEAVY-PARTICLE-STRIPPING REACTION {II)
619
momentum. Associated with these l¢ will generally be different values of {0~0~}. If these {0~0~} are all taken to be adjustable parameters, we could of course fit the experimental data quite satisfactorily by mal6ng use of the "least squares method". Such a procedure as mentioned above m a y be of little meaning until the more detailed knowledge on the cluster structure of an individual nuclear level be provided from various sources in nuclear physics. In view of above situation, therefore, we present in fig. 5 the calculated cross section for the F19(~, pl)Ne .2 (lst excited state) reactioA together with the experimental data at E~ = I8.9 MeV by Priest et al., in which only l, = 0 is taken into account, the contributions from l~ = 2 and 4 being disregarded. Further, we adopt the cut-off radius Rt ----- 5.1 fm in the calculation. Although the agreement between theory and the experiment seems to be not satisfactory in fig. 5, it will at least be concluded that the contributions from l~ ~ 2 and 4 should be much smaller than that from l~ = 0, because the large contributions from l, = 2 and 4 will necessarily lead to a sharp peak at 180 ° in the angular distribution, similarly to the case of the Po group, which is evidently contradictory to the experimental data. Further, it can be inferred from our analysis that the different magnitudes of the cross sections for the P0 and Pl groups can be explained naturally if the reduced width {0,0~} 2 for the F19(x, pl)Ne ~2 (lst excited state) reaction has approximately the same value as that for the F19(e, po)NeZZ(ground state) reaction. By similar reasoning to t h a t above, we cannot determine definitely from our analysis the spin and the parity of the second excited state of Ne z~. It seems, however, to be more plausible to adopt the assignment 2 + rather than 1from our calculations, although these are not presented here in the graphical form. A more definite assignment might be possible if the experimental data were available in the wider range of E~ as has been repeatedly mentioned. 3 2. P R O T O N E M I S S I O N F R O M T H E (2st) S T A T E
In an analogous way we have ip=
1 ~p----T ~, 0, I e = 0
+ or 1+.
The core with 1+ will not be dealt with, since there are no 1+ states in the lower excited states in 01°. Then, corresponding to table 3 for the preceding case, we have table 4 for the present one. As can be inferred immediately from table 4, the differential cross section for the F19(,¢, po)Ne 22 (ground state) reaction shows no sharp peak in the extreme backward direction, since the angular momentum of the captured ~-particle is necessarily zero in this case. For purposes of illustration, we have presented also in fig. 4 the calculated cross section for the heavy-particle-stripping reaction for this case by the broken curve. The cut-off radius Rt has been taken
6~0
HARUO UI AND T S ~ O S H I
HONDA
as 5.1 fm and, further, the angular distribution has been normalized to the calculated cross section for the proton emission from the (ld|) state, which has been represented by the solid curve in fig. 4, for the sake of illustration. TABLE 4
Values of l~ and Gh for proton emission from the 2s½ state
3.3.
I,
h
Gh
po-group pt-group
0+ 2+
0 2
1 5
P2"gr°up
2+ 1-
2 1
5 3
COMPARISON WITH EXPERIMENT
As can be seen from fig. 4, we cannot obtain good agreement with experiment if the main contribution arises from emission from the (2s½) state. In order to explain the experimental results, therefore, it seems to be quite necessary t h a t the probability of the proton emission from the (ld~) state should be much larger than that from the (2s½) state. Namely, the amplitude of the (2s½) configuration in the ground state of F 1" should be much smaller than that of the (ld~) configuration. Detailed calculations based on the shell model have been performed by EUiott and Flowers s) and by Redlich 9) of the nuclei of mass number A = 18 and 19 including F 1" and 0 Is. In their calculations the most dominant configuration in the ground state of F 1" is ascribed to (2s½)~_p other configurations being smaller. Namely, when the ground state of F 1" is decomposed into a proton plus remaining core, the greatest parentage overlap should be attributed to the case discussed in the preceding subsection. From their result the contribution from proton emission from the ld t state must be smaller than that from the 2s½ state, in contradiction to the result of our analysis. It should be noted, however, that the parentage overlap obtainable from the work of Elliott et al. is the so-called spectroscopic factor in the reduced width of a proton. The dynamical factor depending on the details of radial part of the nuclear wave function has not yet been investigated; the reduced width In the bound state will be expected to depend rather strongly on the binding energy, while in almost all shell-model calculations radial wave functions in the harmonic oscillator potential have been used. In this connection, it will be noted that a similar discrepancy m a y also be seen in the Fl"(p, d)F is and Fl"(n, d)O is reactions, as has been discussed by Macfarlane and French 11). 4.
Discussion
In this section we shall discuss some qualitative properties of the cross section for the heavy-particle-stripping reaction such as its dependence on the mass
THE
HEAVY-PARTICLE-STRIpPING
REACTION
(II)
621
number of the target nucleus and on the binding energy of the emitted proton in the target nucleus. Further, some discussion is added on the relation between the backward peak in the differential cross section for the heavy-particlestripping reaction and the structure of low-lying states of light nuclei. First of all, we shall discuss briefly what role the mass number of the target nucleus plays in the cross section. Intuitively speaking, the heavy-particle-stripping mechanism m a y be considered as a sort of recoil effect 2). Namely, if the target nucleus is sufficiently heavy, the angular distribution for the heavy-particle-stripping reaction can be shown to be symmetric about 90 ° direction 2) and, moreover, the magnitude of the cross section can be proved to vanish in the limit of infinitely heavy target nucleus: this suggests that the heavy-particle-stripping reaction will become unimportant as the mass number of the target nucleus increases appreciably. On the other hand, the heavy-particle-stripping reaction will play a very important role when the mass of the target nucleus is not so much different from that of the incident particle. If the cut-off Born approximation is applicable to the calculation of the cross section for the heavy-particle-stripping reaction, such circumstances as mentioned above can be clearly seen from the explicit expression of the cross section given in ref. 1) through the factor (K~Rt), the momentum transfer in the heavy-particle-stripping reaction multiplied b y the cut-off radius R 1. In the Li(~, p)Be reactions treated in the sect. 2, the relation between the binding energies of the proton in the incident and in the target nuclei is quite favourable to the heavy-particle-stripping rather than to the stripping reaction 1-3). We can therefore expect a large contribution from the heavy particlestripping reaction in the Li(a, p)Be reaction. As has been shown in figs. 1-3, it is actually the case. It is further noted that we can obtain quite satisfactory agreement with the experimental data of the LiS(~, po)Be 9 (ground state) reaction and the Li~(~, po)Be 1° (ground state) reaction, in both of which the angular distribution shows a rather sharp peak in the backward direction, b y means of the heavy-particle-stripping reaction with l= = 0. As has been discussed in detail in (I), we cannot obtain in general the sharp peak in the backward direction b y adopting l~ = 0, since the geometrical factor Gh is constant in this case. The Li(a, p) Be reactions are rather exceptional cases due to very light weight of the target nucleus and to the binding energy relation mentioned previously. In fact, in the Cl~(x, p0)N 15 reaction and the F19(~, P0)Ne z2 reaction, it has been found impossible to reproduce the sharp peak in the backward direction b y means of the heavy particle-stripping reaction with l= = 0. For the cases of l, v~ 0, the geometrical factor Gh plays an essential role in determining the shape of the angular distribution at large angles. Even if l~ va 0, there are several other special cases in which the geometrical
622
HARUO UI AND TSUYOSHI HONDA
factor Gh does not contain any angle-dependence. Among them, the most interesting is the case in which the proton is emitted from the s-orbit in the target nucleus, i.e. lp = 0. A typical example is the F19(~, p0)Ne ~ reaction in which the proton is assumed to be emitted from the (2s½) state, as can be seen from the table 4 in sect. 3. Therefore, the discussions presented in sect. 3 in connection with the shell model wave function b y Elliott and Flowers and b y Redlich is persistent, independent o/the choice of other parameters, as far as the very sharp peak in the extremely backward direction in the F19(c¢, P0) Ne z2 reaction can be interpreted in terms of the heavy particle-stripping reaction. It m a y be interesting to note that the sharp peaks in the backward direction have all been reproduced by means of the cross section formula (eq. (1) of ref. 1)) with approximately the same value of (~TRt) in our calculations in the previous. and the present papers. It seems then to be probable that the cut-off Born approximation will be applicable to the calculation of the cross section for the heavy-particle-stripping reaction in the energy region treated here as far as the angular distribution is concerned. In this connection, it is to be noted that Owen, Madansky and Edwards 13) have performed detailed analysis of the angular distribution for the Cis (He s, ¢¢) Clz reaction at 2.0 and 4.5 MeV 13) b y means of the stripping and the heavyparticle stripping reactions, getting fairly good agreement with the experiment. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)
T. H o n d a and H. U1, Nuclear Physics 34 (1962) 593 T. H o n d a and H. UI, Prog. Theor Phys. 25 (1961) 613 T. H o n d a and H. Ui, Prog. Theor. Phys 25 (1961) 635 P R Klein, N. Cindro, L. W Swenson and N. S. Wall, Nuclear Physics 16 (1960) 374 J. R. Prmst, D. J. Tendam and E. Bleuler, Phys. Rev. 119 (1960) 1301 M. Kondo, T. Yamazaki and S. Yamabe, J. Phys. Soc. J a p a n 16 (1961) 1091 H. J Martin, M. B. Sampson and D W. Miller, Phys. l~ev. 121 (1961) 877 J. P. Elliott and B. H Flowers, l=Yoc. Roy Soc. 229 (1955) 536 M. G. Redlich, Phys. Rev. 110 (1958) 468 D. Kurath, ANL-6312 (1961), unpubhshed; D K u r a t h and L. P16man, Nuclear Physics 10 (1959) 313 11) M. H. Macfarlane and J. B. French, Rev Mod. Phys. 32 (1960) 567 12) G. E. Owen, L Madansky and S. Edwards Jr., Phys. Rev. 113 (1959) 1575 13) H. D. Holmgren, Phys. Rev. 106 (1957) 10O; Holmgren, Greer, Johnston and WolJcki, Phys. Rev. 106 (1957) 102