The heavy-particle-stripping reaction in (α, p) reactions

- - ~

NuclearPhysics 3 4 (1962) 5 9 3 - - 6 0 8 ; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprmt or maerofllm without written permxssaon frOm the pubhsher

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 (~, p) REACTIONS (I) The CI2(~, p)NlS(ground state) reaction TSUYOSHI

HONDA

t

Department o/ Physws, Tokyo Inst*tute o/ Technology and HARUO

UI t

The Institute for Sol*d State Phys,cs, Unwers,ty o/ Tokyo R e c e i v e d 3 J u l y 1961 T h e e x p e r i m e n t a l differential cross s e c t i o n for t h e CX~(~, p ) N x5 (ground state) r e a c t i o n a t large a n g l e s is a n a l y s e d m 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 r e a c t i o n f r o m E~ = 16 ~¢IeV t o 40 MeV" t h e f o r m u l a t i o n a d o p t e d h e r e h a s a l r e a d y b e e n p r e s e n t e d in o u r p r e v i o u s p a p e r s in t h e case of (d, p) a n d (d, n) reactions. I t IS s h o w n t h a t n o t o n l y t h e s h a p e of t h e a n g u l a r d i s t r i b u t i o n a t large a n g l e s b u t also t h e m a g n i t u d e of t h e cross section, b o t h of w h i c h v a r y n o n - m o n o t o m c a l l y w i t h t h e e n e r g y of a n i n c i d e n t a l p h a - p a r t m l e , c a n be well e x p l a i n e d b y 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 r e a c t i o n w i t h o u t i n t r o d u c i n g a n y e n e r g y - d e p e n d e n t a d j u s t a b l e p a r a m e t e r s for E~ ~ 17 MeV. I n p a r t m u l a r , g o o d 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 t h e o r y a n d t h e e x p e r i m e n t w i t h res p e c t to t h e s h a r p p e a k a p p e a r i n g m t h e a n g u l a r d i s t r i b u t i o n a t t h e b a c k w a r d d i r e c t i o n a r o u n d E~ = 25 MeV. S o m e dxscusslon is a d d e d c o n c e r m n g t h e i m p o r t a n c e 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 in (~¢,p) r e a c t i o n c o n t r a r y t o t h e case of t h e (d, p) reaction. Finally, it is e m p h a s i z e d t h a t t h e s y s t e m a t i c s t u d y of (0t, p) r e a c t i o n will be a p o w e r f u l tool to m v e s t x g a t e t h e b e h a v i o u r of t h e f o u r - n u c l e o n correlation, t e. q u a s i - a l p h a - p a r t i c l e , n e a r t h e n u c l e a r surface.

Abstract:

1. Introduction Recently, systematic experimental data of (,t, p) reactions have been accumulated for light nuclei, in which the residual nucleus is left to a definite low-lying state. Among them, the C12(~, p)N 15 (ground state) reaction has been most thoroughly investigated: the experiments were performed by Sherr and Ricky 1) at ~-particle energies 30.5 and 40.5 MeV, by Hunting and Wall 2) at 30.5 MeV, by Nonaka, Yamaguchi, Mikumo, Umeda, Tabata and Hitaka 3) at twelve energies between 25 and 39 MeV, by Yamabe, Kondo and Yamazaki 4) at 20, 21 and 22 MeV, and by Priest, Tendam and Bleuler 5) at six energies between 16 and 19 MeV. In the last three experiments, especially in that of t P r e s e n t a d d r e s s : D e p a r t m e n t of P h y s i c s , T o h o k u U m v e r s l t y , Sendai. 593

594

TSUYOSHI

HONDA

AND

HARUO

UI

Nonaka et al., the angular distributions were measured in almost all ranges of angles. The main feature of the experimental results quoted above m a y be expressed as follows; At smaller angles, the angular distributions show a pronounced diffraction-like pattern 1-5), while at larger angles there appears in general a sharp peak at 180 ° in the region of lower incident energy E~ of the a-particle say, below 30 MeV 3-5). The peak at the backward directions, however, is no longer observed at higher energy such as above 35 MeV 3). Moreover, the cross section, both as to magnitude and angular distribution, varies nonmonotically with changing incident energy. Such circumstances seem to be more pronounced, as E~ decreases. In fact, the angular distributions 6) vary quite rapidly from E~ = 16 MeV to 17 MeV. In 1957, Butler e) treated the angular distribution observed at 30.5 MeV b y Hunting and Wall 3) as a typical evidence of the knock-on reaction. The sharp peak of the angular distribution at the backward direction in the lower region of E~, however, can hardly be ascribed to the knock-on reaction. Moreover in order to explain the non-monotonic change of the cross section with incident energies, it will at least be necessary to introduce two or three sorts of direct-interaction mechanisms: the phase relations among them should be correctly taken into account. In the previous paper ~, 8), which will hereafter be referred to as (I) and (II), we have developed an antisymmetric treatment of the deuteron stripping reaction in such a w a y as to enable one to treat the stripping and the heavyparticle-stripping reactions on an equal footing and, furthermore, to obtain the energy-dependence of the cross sections directly without introducing any energy-dependent adjustable parameters. in the present paper, the previous formulation of the heavy-particle-stripping reaction will be applied to the analysis of the differential cross section for (~, p) reaction at large angles. Concerning the angular distribution at smaller angles, the knock-on reaction is likely to be more important than the stripping one in (*c, p) reactions, contrary to the case of (d, p) reactions. In the present stage of the nuclear direct-interaction theory, it seems to the authors highly desirable to formulate theories without introducing any energy-dependent adjustable parameters, since systematic experimental data are now being available with a wide range of incident ,c-particle energies. From the view-point mentioned above, previous theories of the knock-on reaction can hardly be said to be satisfactory. Accordingly, the cross section for (*c, p) reaction at smaller angles will be dealt with later in a separate paper. We shall summarize in the following the formula which will be used in the actual analysis in the subsequent section. When the cut-off Born approximation is adopted, the cross section for the

THE HEAVY-PARTICLE-STRIPPING REACTION (I)

~95

heavy-particle-stripping reaction may be represented * as follows tt: da(at, p) (H.P. Stripping) d.O

Ch" Gh" ~ .~6.~p~

(k/K) × U,p(k, Kp, R,)]*[{(KRt)'+(K,,RI)*}-IJz,(R, K,,, R,)]L

(1)

where Ch is a constant factor, given by Ch = 36(Rt/Rf)Ri*.

(2a)

The geometrical factor C~ is expressed as

Ga(It, It, Ie, lp, ½,/'p, l~; 0) ---- (2Ir+ 1) (2/~+ 1) (2/p+ 1) (2/'p+ 1)(--)'t-i-" × ~ (l,l, 001L0)(lj= O01LO)W(Z,i,I,i,;{L)

(2b)

L

× W(jpIcipIc; ItL)W(l,,Icl~,Ic; IrL)Pz(cos 0). Further, O~ is the overlapping integral between the wave function of the internal motion of a-particle 9,(to) and that of the four nucleon correlation (quasi a-particle) ~(oJ) in the residual nucleus near nuclear surface:

o, = f 9,(,o)~* (,o)dco.

(2c)

The amplitudes Ozpjp and Oz= are the dimensionless reduced width amplitude of the emitted proton in the target nucleus and that of the captured at-particle in the residual nucleus, respectively. The quantity Jz(k, K, R) is defined, in terms of the spherical Bessel and Hankel functions/', and k,, by J,(k, K, R) = [(kR)i,_~(kn)+c,(~R)i,(kR)] ,

(2d)

where C~(KR) = --i(KR)[hz_I(iKR)/h,(iKR)] , with Co(KR ) = KR,

Cz+I(KR) = (KR)2/[(2I+I)+C,(KR)].

Further, K" appearing in the last factor of (1) is given by K = K + (M, d M , ) . k, and 0 in PL (cos 0) of Gh denotes the angle between the vectors k and K. t A brief d e r i v a t m n of t h e f o r m u l a will be described in t h e appendix. The detailed f o r m u l a t i o n and discussion h a v e been p r e s e n t e d m (I) for the case of t h e deuteron-induced reaction, w h i c h can be easily modified to the (ct, p) reaction. • * The cross section (1) c o r r e s p o n d s t o t h e s i m p l e s t case in which o n l y one s t a t e of t h e cores is available. A general expression of the cross section has been given in (I) and (It).

596

TSUYOSHI HONDA AND HARUO UI

Here, the following notation has been adopted: K and k are the wave numbers of the incident *t-particle and of the emitted proton; Mr, Mt and M~ are the reduced masses of the target and residual nuclei and that of the *t-particle; Kp and K~ are defined b y Bp = (~2%2/2Mp) and B~ = (~2K~2/2M~); Bp and B~ are the binding energy of the emitted proton in the target nucleus and that of the captured *t-particle in the residual nucleus; It, Ir and Ie are the spins of the target and the residual nuclei and that of the core; lp and l~ are the orbital angular momentum of the emitted proton in the target nucleus and that of the captured *t-particle in the residual nucleus; R t and Rf are the cut-off radii of the integrals with respect to the radial coordinates in the initial and final states. It should be noted that the cross section (1) does not contain any adjustable parameters which depend on the incident energy of *t-particle. Before entering the analysis of the experimental data, it seems useful to add some discussion on the qualitative properties of the cross section (1); The angular dependence is determined through ~7 appearing in the last factor of (1) and through PL (COS0) in Gh, whereas J~p (k, Kp, Rt) determines mainly the non-monotonic dependence of the magnitude of the cross section on the incident energy of ,t-particle. It is worth noting that Gh does not depend on the angle only for the case of l~ =- 0 and/or lp = 0: otherwise, its form will become a+b P2 (cos O)+cP t (cos 0 ) + . . . . which leads, in general, to a rather sharp peak at 180 ° in the angular distribution except for high incident energy, say, above 30 MeV *. Consequently, we can obtain some information on the orbital angular momentum l~ of the captured *t-particle directly from the shape of the angular distribution at backward directions. Besides the above points, our formula (1) is more unambiguous than the previous one of Owen and Madansky 9) in a sense that the Butler(cut-off Born) approximation has been adopted in ours, while the Bhatia one was used in Owen and Madansky's. Namely,/'z~(KTR1) appeared in the formula of Owen and Madansky instead of 9rt~(-~, K~, Rt) in our cross section (1). From the mathematical point of view, it is evident that the cut-off Born approximation is better than the Bhatia approximation. In many cases of the deuteron-stripping reaction, the angular distributions calculated b y the Bhatia's formula are known to be not so much different from those of Butler at forward directions. In our case, however, the cross section (1) leads to markedly different angular distribution from that of Owen and Madansky owing to large values of (KTRI). In the next section, the C12(*t, p)N 15 (ground state) reaction at incident a-particle energies of 16 to 40 MeV 1-5) will be analysed in detail b y means of t A s i m i l a r a r g u m e n t h a s b e e n m a d e i n d e p e n d e n t l y b y S. O ka i . The a u t h o r s a re i n d e b t e d t o Dr. S. O k a l for d i s c u s s i o n s on r e l a t e d p r o b l e m s .

THE HEAVY-PARTICLE-STRIPPING REACTION (I)

597

the heavy-particle-stripping reaction. It will be shown that not only the angular distribution at large angles b u t also the energy-dependence of the cross section can be well reproduced b y the formula (1), except for their rapid variation between 16 and 17 MeV 5). In the previous paper (II), it has been both qualitatively and quantitatively shown that the heavy-particle-stripping reaction hardly ever plays an important role in (d, p) and (d, n) reactions in comparison to the stripping reaction except for some special cases, provided the cut-off Born approximation is adopted in the calculation of the cross sections. In view of above situation, therefore, it seems necessary to add some discussion of the different situations between (d, p) and (*t, p) reactions; sect. 4 is devoted to these discussions. Finally in sect. 5, we shall discuss the nuclear structure in connection with (~, p) reaction. In particular, it will be emphasized that the systematic analysis will be one of the most powerful tools for the investigation of the behaviour of the four-nucleon correlation near nuclear surface. Namely, the analysis will make the investigation mentioned above possible through the determination of { 0 ~ } 2 for various nuclear low-lying states.

2. The Cl2(*t, p)N Is (ground state) Reaction We shall analyse the experimental differential cross section for C12 (*t, p)N 15 (ground state) reaction at large angles from E~ = 16 MeV to 40 MeV b y means of the formula (1) for the heavy-particle-stripping reaction. First; for purposes of illustration, the three dimensional view of the experimental data is presented in fig. 1 as a function of E~ and 0p; their main feature has already been mentioned in the sect. 1. The reaction Q-value and the spin and parity of the target and residual nuclei are given b y Q -- - 4 . 9 6 5 MeV,

It = 0 ( + ) , Ir = { ( - - ) .

In the heavy-particle-stripping reaction of the C12(*t, p)N is reaction a proton in the target nucleus C12 is supposed to be emitted from a definite state in its ground state, while the incident ,t-particle will be captured b y the remaining core of C12 to form the residual nucleus, i.e., the ground state of N 16. There are three available states for the proton in the target nucleus in the/'-/" coupling prescription; (ls½), (lp]) and (1P½) states *. Among them, the proton emission from (Is½) state is not dealt with in the present paper, since it will be quite unfavourable to the emission from this state due to its large binding energy. We shall discuss the proton emission from the other two states separately. * I n t h e e x t r e m e case of t h e ideal ]- 7 c o u p l i n g shell model, t h e ( l p t ) S a t e n e e d n o t be t a k e n i n t o account.

698

TSUYOSHI HONDA AND HARUO U I

2.1. P R O T O N

EMISSION FROM THE

(1 p | ) S T A T E

When the proton is emitted from the (lp~) state, the spin and parity of the remaining core are determined uniquely as Ie = { ( - - ) . We shall assume the core to be the ground state of B u, this assumption being needed only for the calculations of Kp and K=. On the other hand, the orbital angular momentum of the ~-particle in the residual nucleus should be taken to be l= = 2 because of the conservation of the angular momentum and parity. The geometrical factor Gh in (1), then, will

-mb ,2.0

d2

P)

N=

I.$

1.0

0.5

I@

25

¢

Fig. 1. T h e cross section for t h e Cl1(¢~, p ) N x~ (ground s t a t e ) reaction. T h e t h r e e d i m e n s i o n a l view of t h e e x p e r i m e n t a l cross s e c t m n f r o m E= = 16 M e V to 40 MeV. T h e e x p e r i m e n t a l d a t a a r e t a k e n f r o m refs, s-6).

become C~(0, {, {, 1, 1, {, 2; 0) = ½ [ l + P z ( c o s 0)]. The calculated result of the cross section for this case is presented in fig. 2. The cut-off radii R t = R t = 4.9 fm adopted in fig. 2 are chosen so as to obtain an overall fit with the experiments in the region of E= treated in the present paper. If no attempt is made to explain the overall behaviour of the cross sections without introducing any energy-dependent adjustable parameters, we can obtain better agreement with a experiment at fixed energy E= than in

TEE

HEAW-PARTICLE-STRIPPING

REACTION

599

(I)

fig. 2 by making use of the different values of Ri and RI. As a typical example, the calculated cross sections at E, = 25 and 30.5 MeV are presented in fig. 3, together with the experimental data at the respective energies. It is to be noted that, when the cross section for the knock-on reaction calculated by Butler “) is superimposed on that for the heavy-particle-stripping reaction presented in fig. 3, the agreement between the experimental and the theoretical angular distribution seems to be satisfactory at E, = 30.5 MeV

mb sr

Ri

I.C

= 4.90

O.!

0

0

60

90 ep

120

160

CW

Fig. 2. The calculated cross sections for the C1*(a, p)N16(g round state) reaction at Ea = 10.1, 19.0, 22.0, 26.0, 30.6 and 36.0 MeV. In the calculation, the proton is assumed to be emitted from a (1~)) orbit in the ground state of Cl*, i.e. I, = 1 and fr, = Q and 1, = 2. Further, the following values of the parameters are adopted: R, = 4.90 fm, Rf = 4 90 fm and {Ok@,.&,}* = 1.00.

for all ranges of angles. The experiment at 25 MeV can also be well fitted in a similar manner as in fig. 3 by the choice of R, = 7.4 fm; in these cases, however, the agreement at other energies is not good. 2.2. PROTON

EMISSION

FROM

THE

(lpi)

STATE

In the extreme case of the ideal j-j coupling shell model, the proton emission from (lpi) state is completely forbidden, because this state contains no nucleons

TSUYOSHI

600

HONDA

AND

HARUO

UI

in the ground state of C12. The actual situation, however, will not be as simple as mentioned above. In order to explain various properties of Cl2 as well as of its neighbouring nuclei from the standpoint of the shell model, it should at least be necessary to take account of the configuration-mixing between (lp+) and (1~4) states, i.e., the intermediate coupling shell model lo, ll). Namely, the ground state of Cl2 will contain the (lp+) configuration appreciably ll).

C”(oc+) N’” g(H.P.Stripping)

*

p

i

f

i I‘I

L4=2

1.0

0.5

0 0

30

60 ep

90 (C.W

I20

150

II 0

,

Fig. 3. The typical calculated cross sections at E, = 25.0 and 30.6 MeV. The solid curve represents the calculated cross section at Ea = 26.0 MeV, while the dotted broken curve is the calculated one at Ea = 30.5 MeV. The solid dots and crosses (together with the dashed curves) represent the expenmental data at Ee = 25.0 and 30.6 MeV, respectively. The following values of the parameter are adopted in the calculation: lp = 1, jP = 4, le = 2, Ri = 4.80 fm, RI = 3.00 fm and {O&J&,}* = 0.64. Here, we do not intend to explain the peak of 110” at Ea = 25.0 MeV, since this peak may be due to the knock-on reaction.

When the proton is emitted from the (lpi) state, the spin of the corresponding core is taken to be I, = 8(-), while the a-particle should be captured by the core with the orbital angular momentum Z, = 0. The geometrical factor G, then, becomes simply unity, viz., G(0,

*, *, I, +, 8, 0; 0) = I.

THE

HEAVY-PARTICLE-STRIPPING

We shall take the core *(-)

REACTION

601

(I)

as the first excited state of B1l: this assumption

will be the most reasonable from Kurath’s wave function 11) of the intermediatecoupling shell model. Our calculated cross sections are presented in fig. 4. The experimental angular distribution at larger angles varies very slowly from E, = 17 MeV to E, = 19 MeV, the peak of 135” at E, = 19 MeV being displaced gradually towards the smaller angle 115’ at E, = 17 MeV 5). The

C’2(q4 s

(H.P.Stripping) ’

*

N”

Qa=o Ri

q

5.20

1.5

/-. {o*z,;;.;:

o.!54

/’

I.0

0.5

0

0

30

60 qa

90

120

150

I80

ma

Fig 4. The calculated cross sections for the Cl* (a, p)N’& reaction at Ea = 19 0,22.0, 25.0, 30.5 and 35.0 MeV. The proton is assumed to be emitted from a (lpi) orbit in the ground state of P, i e. I, = 1. jr, = 4 and la = 0. Further, the following values of the parameters are adopted RI = 6.20 fm, R, = 3.60 fm and {O&,@,}* = 0 54.

above behaviour of the cross section can be satisfactorily reproduced by means of the formula (1). In a similar manner as in fig. 3, we can obtain good agreement with the experiment at a fixed energy by making use of the other choice of Ri and RI. As a typical example, we present in fig. 5 the calculated cross sections at E, = 18 and 19 MeV, together with the experimental data at respective energies. The parameter {0,0&,}2, which is intimately connected with nuclear structure, is adopted to be 1.0 for the (lpq) case and 0.54 for the (1~4) case in figs. 2 and 4, respectively. The above values of the parameter include

602

TSUYOSHI

HONDA

AND

HARUO

UI

implicitly not only the geometrical factor such as cfp but also the mixing ratio between the (lpt) and (lP½) configuratiorl hi tile target nucleus. It should, however, be noted that the magnitude of {O=O~@p}~ thus extracted from the, experiment depends * rather strongly upon Rt. Although the value of Rt adopted in fig. 4 is different from that in fig. 2, we in fig. 6 superimposed curves of figs. 2 and 4, together with the experimental data in fig. 7.

G '2

(o<,p) N"

m__~bd~p( H. P.St ripping ) 8r d=O

A t- 7.?0 Rj: 6.8 0

1.5

~-"~ !

o/~,

{o., *,,.,,..ot ® .-o.,,I~'

\a "~.

E.-IO.O I '

°o

\L'/

o.s

19.0 ',/

o

oo

o

0

0

0

i

I

l

30

60

90

I

120

i

150

180

ep (C.M) F*g 5 Typical calculated cross sections for the C 12 (~, p ) N .5 reaction a t E= = 18.0 and 19.0 MeV. T h e sohd curve represents the calculated cross section a t / ~ = = 18 0 MeV, while the b r o k e n curve is the calculated cross section a t E= = 19.0 MeV. The e x p e r i m e n t a l d a t a at 18 MeV and 19 MeV are s h o w n b y the open a n d solid circles, respectively. The following values of the p a r a m e t e r s are a d o p t e d in t h e calculation. ]p = ½, l= = 0, R i = 7 70 fm, R t = 6.80 f m and {OaOaOp} ~ = 0.98.

As can be seen from figs. 6 and 7, the experimental cross section for the C1. (~¢, p)N 15 (ground state) reaction at large angles can be well explained by the heavy-particle-stripping reaction. The agreement with the experiment seems to be rather noticeable on account of our simple treatment of the cross section, i.e., the cut-off Born approximation. t Careful analysis, therefore, will be needed to c o m p a r e the m a g n i t u d e s of {O=O=@p}z a m o n g v a r i o u s nuclear states. A similar, b u t less p r o n o u n c e d situation a p p e a r s also in the e x t r a c t i o n of t h e nucleon reduced w i d t h s f r o m the e x p e r i m e n t a l cross sections for (d, p) a n d (p, d) reactions.

0

30

,

I

I I

ii/

\/ /

22.0/

lIll

90

ep (C.M3

60

120

150

2:

i"

/~'

Rt = 4.90 Rf= 4.90

,,/\.~ //2~9/3o,~!'

,,,'

/

E= 19.o,,

•

/

. fo,®j~, ®j,.~'= -. o.s4

Ri = 5 . 2 0 Rf = 5 . 6 0

180

Fig. 6. The calculated cross sections for the heavy-particle-stripping reaction at E~ = 19.0, 22.0 25.0, and 30.5 MeV. The calculated cross sections presented in figs. 2 and 4 are superimposed at 19.0, 22.0, 25.0 and 30. 5 MeV.

o

0.5

1.0

L5

mb $r

C,Z (~,p) N 's

_~p(..p.s,=~ap(~,..0) + d--Kp(l,do . 2)

I

0

I 30

60

/

I

•

/

\.

I

9O

ep (C.MI

z

".,-J\ /z

/

/

I

/

/

/

120

/.

i 22, I

/

Eor--19.0 //

Experiments

150

~'" --y"

180

I

Fig. 7. The experimental data for the CZ*(~¢, p)N~5(ground state) reaction at Ea = 19.0, 22.0, 25.0 and 30.5 MeV.

0

0.5

1.0

1.5

mb sr

C" (o<,p) N"

v

O

0

f~ t~

604

TSUYOSHI HONDA AND HARUO UI

It is, however, found impossible to fit the rapid change of the differential cross section between E~ = 16 MeV and 17 MeV b y means of the formula(l), that m a y be due to the simple treatment of the cross section or m a y be due to other more complicated mechanisms than the direct-interaction. In this connection, it is to be noted that there are several experimental data of the N14(d, p) N is, N14(d, n) 015 and N14(d, d')N 14 reactions in which the excitation energies of the compound nucleus O le are approximately equal to that of the C12 (~, p) NlS-reaction between E~ = 16 MeV and 17 MeV. The differential cross sections for these reactions seem to show some correlations among each other, that m a y suggest the importance of the compound nucleus formation at this energy t. In the present stage of the compound nucleus theory, however, such phenomena can hardly be adequately analysed, since the "statistical hypothesis" will not hold in these energy region. Moreover, the states of the compound nucleus formed b y the ~-induced reaction m a y not necessarily be the same as those excited b y the deuteron-induced reaction owing to the isobaric spin selection rule and the different maximum angular momentum associated with the respective reaction.

3. R e m a r k s in Connection with (d, p) and (d, n) Reactions In the previous paper (II), it has been shown both qualitatively and quantitatively that the heavy-particle-stripping reaction does not almost always play an important role in (d, p) and (d, n) reactions, provided the cut-off Born approximation is adopted in the calculation of the cross sections. The above conclusion will be rather exceptional and solely valid for the deuteron-induced reaction that m a y be explained as follows: the cross section for the stripping reaction is generally quite large clue to the exceptionally small binding energy of deuteron, whereas the cross section for the heavyparticle-stripping reaction will be very small, since the separation energy of a proton from light nuclei is b y a factor 5 to 10 larger than that from the deuteron. In fact, the calculated cross sections for the heavy-particle-stripping reaction have been found to be negligibly small in comparison to that for the stripping reaction except for some special cases. On the other hand, in (~, p) reactions the separation energy of a proton from the ~-particle and that from a light nucleus are of similar magnitude. The cross section for the stripping reaction is, therefore, expected to be very small in comparison to the case of the (d, p) reaction. Actually, the experimental cross section for the (~,p) reaction at small angles is b y a factor 10-1-10 -2 smaller than that for the (d, p) reaction. Further, the knock-on reaction is likely to contribute appreciably to the cross section for the (~, p) reaction at small angles, due to the rigidity of ~-particle. t The authors are much indebted related problems.

to Dr

T. [ s h i m a t s u

and

D r . T. M l k u m o

for discussion

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 (I)

605

Accordingly, the heavy-particle-stripping reaction can take an appreciable part in the (*t, p) reaction even at higher energies of incident *t-particle, contrary to the case of (d, p) reaction. It is to be noted that the calculated results based on the cut-off Born approximation will become rather reliable at such higher energies as treated in the sect, 2, as far as the angular distributions are concerned. 4. R e l a t i o n t o N u c l e a r

Structure

As has been shown in the previous section, we can determine directly from experiments the parameter {O~O~Op}2 which is intimately connected with nuclear structure. Its magnitude obtained in sect. 2, however, seems to be extremely large, that is due to the use of the cut-off Born approximation in our calculations. In fact, as has been discussed in detail in (I) and (II), we can expect to obtain a rather reasonable magnitude for it, if we employ the distorted approximation instead of the cut-off Born approximation 7, s). The calculations b y means of the distorted wave approximation, however, are quite cumbersome. Accordingly, it seems to be rather preferable for the first step of the analysis of the many experimental data to adopt the cut-off Born approximation in the calculation of the cross section, since the relative ratios of the parameters corresponding not only to excited states of the same nucleus but also to various nuclear states of light nuclei will provide us with much information on the nuclear structure. In extracting the parameters from experiments, we should use the experimental differential cross sections at higher incident energies of the ~-particles, since at lower E~ more complicated mechanisms other than the direct-interaction m a y take place in (*t, p) reaction. The relative ratios of the single particle reduced widths, i,e., the "empirical" reduced widths Op are now available among various nuclear states from (d, p) and (p, d) reactions; this has been most elaborately and completely reviewed b y Macfarlane and French 12). Then, the behaviour of the four-nucleon correlation, I.e., quasi ,t-particles, near the nuclear surface can be inferred from the magnitudes of {0Jg~} 2 extracted directly from experiments. The systematic studies shall be presented later in the near future. On the other hand, it should be noted that the theory of the nuclear manybody problem so far developed especially b y Brueckner is) has been concerned mainly with infinite nuclear matter. Recently, Brueckner and his co-workers have applied his treatment of the many-body problem to a finite nucleus b y adopting a Thomas-Fermi like model 14). Although they have obtained a rather reasonable value of the nuclear binding energy, their nuclear wave function will not necessarily correspond to the actual wave function of the nucleus. In fact, it m a y be inferred from the work of Weisskopf and his co-

606

TSUYOSHI HONDA AND HARUO UI

workers 1~) that the clustering of nucleons m a y be quite important in the nuclear surface region, because the "healing distance" will become very large at nuclear surface in comparison to that in its interior due to the weakness of the effect of the Pauli principle. Recently, several experiments of high energy proton induced reaction have appeared, in which the probability for ,t-particle emission is found to be surprisingly large is, 17). Although these experimental facts m a y indicate the importance of the clustering of nucleons in nuclei, we cannot obtain from them any information on the structure of an individual nuclear level, since only the total emission probability for the *t-particle is determined from these experimental data. In view of above situation, therefore, the systematic studies of (*t, p) reactions leading to nuclear low-lying states will be one of the most powerful tools to obtain the behaviour of the four-nucleon correlation near the nuclear surface. The last part of the present work was performed at the Institute for Nuclear Study, University of Tokyo. The authors would like to express their sincere thanks to the members of the theoretical division for their valuable discussions.

Appendix THE

DERIVATION

OF THE

CROSS SECTION

The matrix element of the heavy-particle-stripping reaction in' the cut-off Born approximation can be represented in terms of the final state interaction V(p, c) as follows: M(H.P. S t r i p p i n g ) : fext drpdo f drcdra × e x p ( - - i k • r'p)Z~,,(p ) ~r* (*t, c) × V(p, c)~rJt(p, c)9~(ro)exp (iK-~),

(A.1)

where r' o = r p - - ( M ~ Q + m c r . ) / M v

The wave function of the target nucleus ~(/t(P, c) m a y be decomposed into the sum of the product of a core and the corresponding proton state in the external region 18): ~rJt(P, c) = • (/p {~pVpl~'p/~)(jplcl~pMc[ItMt) × ~zoM.(C)(2Mp/~2Rf)½7,p,#Ztv~(P)

(A.2)

× [h,p(iKprp)[h6(iKpRf)JY,,,np(Qr,).

In the same way, the wave function of the residual nucleus ~r(¢¢, c) m a y be represented as a sum of the product of a core and the qausi-*t particle VJ~(eo)

607

THE HEAVY-PARTICLE-STRIPPING REACTION (I)

in the nuclear bound state 7-',(~,

c)

18):

= ~E

(&Icm,,MdI,M,)~,.M.(c)

X ~v~(to) (zM,,/l~Rt)½ 7,~

(A. 3)

X [h,~,(iK~,p)/h,~,(iu:~,R,)]Y,~,,,,o,(QQ). By inserting (A.2) and (A.3) into (A.1) and, further, by eliminating the interaction V(p; c) by means of the SchrSdinger equation of the target nucleus, we obtain M(H.P. Stripping)

= -- (Ep+Bp)X(lp{mpvpIipl~)({plct~Mc[ItMt)(l~,Icm~,Mcllfl~Ir) x (2Mp/l~2R~)½7,,,, X foxt drp[h6(iKprp)/ht,(iKpRt)]Y'p'%(Qr,)exp(--ik"

rP)

(A.4)

X (2M,/hZRt)½~,t, / d(ag,(eo)~o~*(eo)

x fo.t de[h'~(iK~p)/h',(iK'R')]V*~'~(Qe)exp(iff"e), where ~p = r p - - C m o / M ~ )

• ro,

K =

K--i- ( M ~ . / M ~ ) k .

If the quantization axis of angular momenta is taken to be the direction of K, the overlap integrals with respect to rp and Q appearing in (A.6) can be easily carried out. For example, we have

fextdrp [h,p (iKp rv)/h,p (iupRf) ]Y,p mp(t2rp)exp (-- i k " rp) = 4zd-'pYtpmp(g2k)[Rt/(k2+Kp~)]J,,(k, Kp, Rt),

CA.5)

where ~2k denotes the solid angle of k measured from K. Inserting (A.4) into da(~, p) MpU¢, k 1 d.(2 (H.P. Stripping) -- (2~2)~K 2 I t + l

~

[M(H.P Stripping)p,

M t M r Pp

and performing explicitly the summation over the various z-components of angular momenta, we get the expression of the cross section (1), provided only one term in the summation with respect to the cores in (A.2) and (A.3) is taken into account. Strictly speaking, when several core states are needed in the calculation, each term in the summations in (A.2) and (A.3) should be multiplied by the corresponding "c.f.p" and the mixing ratio, as has been discussed in detail in our previous papers.

608

TSUYOSHI HONDA AND

HARUO UI

Note added in proo]: In eq. (A.4), the recoil effect has been taken into account approximately. If all recoil effects are accounted for exactly in the calculation of the cross section, the wave number k appearing in J~p(k, ,%, Rt) of the cross section (1) should be correctly replaced by following angle-dependent quantity: ]c = [k+(Mp/Mt). K I, and the angle 0 appearing in PL(COS0) denotes the angle between the vectors and K instead of the angle between k and K,. After employing above correction, the numerical result presented in figs. 2--6 changes very little, though slightly better agreement with experiment than that in the figures can be obtained. References 1) R. Sherr and M P~ckey, Bull Amer Phys. Soc Ser. II, 2 (1957) 29; Ann. Progr. Rep. Washington Umversity (1957) 2) C. E. Hunting and N. S Wall, Bull Amer. Phys. Soc. Ser. II, 2 (1957) 181 3) I. Nonaka, H. Yamaguchi, T Mikumo, I. Umeda, T. Tabata and S. Hltaka, J. Phys. Soc. Japan 14 1959) 1260 4) S. Yamabe, M. Kondo and T. Yamazaki, J. Phys. Soc Japan, to be pubhshed 5) J. R. Priest, D. J. Tendam and E. Bleuler, Phys Rev. 119 (1960) 1301 6) S. T Butler, Phys. Rev. 106 (1957) 272 7) T. Honda and H U1, Prog. Theor. Phys 25 (1961) 613 8) T Honda and H. U,, Prog. Theor. Phys. 25 (1961) 635 9) G. E. Owen and L. Madansky, Phys Rev. 105 (1957) 1766, K. Hasegawa and Y. H. Ichlkawa, Prog. Theor. Phys. 21 (1959) 569 10) J . M . Elliott and A NL Lane, in Handbuch der Physik, Band X X X I X (Sprlnger-Verlag, Berlin, 1968) 11) D. Kurath, Phys. Rev. 101 (1956) 206; private communication to A. Arima. 12) M. H Macfarlane and J. B. French, Rev. Mod. Phys. 32 (1960) 567 13) K A. Brueckner and J. L. Gammel, Phys. Rev. 109 (1958) 1023; and refs quoted thereto 14) K A Brueckner, J. L Gammel and H. Weitzner, Phys Rev. 110 (1958) 431 15) L.C. Gomes, J. D Walecka and V. F. Weisskopf, Ann. of Phys. 3 (1958) 241, A de Shallt and V. F. W'eisskopf, Ann of Phys 5 (1958) 282 16) P. E Hodgson, Nuclear Physics 8 (1958) 1 17) V. I Ostroumov and R. A. Fllov, Sovmt Physms J E T P 10 (1960) 459 18) H U1, lJrog Theor. Phys. 18 (1957) 163