Partial equilibrium model for nuclear reactions (II)

2.C

I

Nuclear Physics 62 (1965) 673--689; (~) North-HollandPublishing Co., Amsterdam

I

Not to be reproduced by photoprint or microfilm without written permission from the publisher

PARTIAL EQUILIBRIUM

MODEL

FOR

NUCLEAR REACTIONS

(II)

KO IZUMO Bartol Research Foundation of the Franklin Institute, Swarthmore, Pennsylvania t and

Institute for Nuclear Study, University of Tokyo, Tokyo, Japan Received 11 May 1964 Abstract: The gross resonances investigated experimentally at medium excitation energies are analysed by using the concept that only a small fraction of the nucleons are excited in the compound nucleus. Differential cross sections near resonance are determined by this "intermediate resonance" theory. The reaction amplitude contains both intermediate resonance and direct interaction terms. In order to calculate statistical quantities the excited nucleons are treated as a Fermi gas, together with a rough estimate of the effect of nucleon-nucleoninteractions. Level spacings, nuclear temperatures and angular distributions are calculated and are compared with experimental results. It is pointed out that nuclear reactions and excited nuclei at medium excitation can be classified into a few groups characterized by the number of excited nucleons. The Lane-Thomas-Wigner theory is applied to intermediate resonances, and the strength function of this model shows that these resonances should appear not only in excitation curves but also in energy spectra of the reactions.

1. Introduction G r o s s resonances h a v i n g a b o u t 100 keV widths a n d a b o u t 400 keV spacings have been observed b y m a n y a u t h o r s 1-8) in the m e d i u m excitation energy region. They m a y be called " i n t e r m e d i a t e resonances". I n previous papers 9,1 o),t general features o f the i n t e r m e d i a t e resonances were studied b y a s s u m i n g that only a small fraction o f the n u c l e o n s are excited in the c o m p o u n d nucleus, with the r e m a i n i n g n u c l e o n s f o r m i n g a n inert core a n d p r o d u c i n g a n average potential for the excited nucleons. The excited n u c l e o n s m a y be called outer nucleons. M o r e recent extensive experimental data 11-31) have been accumulated, a n d it is n o w possible to discuss the i n t e r m e d i a t e resonances in terms o f such quantities as level spacings, level widths, nuclear temperatures a n d a n g u l a r distributions. I n this paper these quantities are calculated n u m e r i c a l l y in terms o f the f o r m a l i s m given in I1. I f this m o d e l is a good representation of the system at m e d i u m excitation, it will be expected that the outer n u c l e o n n u m b e r should take well defined discrete values, a n d the reactions a n d the excited nuclei at m e d i u m excitation could be classified by * Work supported by the U.S. Atomic Energy Commission. *t Ref. 10) will be referred to as I1 hereafter. The same notation as in I1 is used here. 673

674

K. mtlMO

the outer nucleon numbers. Correlations between the intermediate resonances and compound nuclear resonances, which correspond to core excitations, are given by applying the Lane-Thomas-Wigner theory a2). A new strength function for the intermediate resonances is calculated and it is predicted that a gross structure should appear even in the reaction channels. There are m a n y theoretical view points 3a, 34) for treating the complicated nuclear reactions at medium excitation. The relations and differences between these theories and this model are discussed. 2. Statistical Quantities It was pointed out in I1 that about six outer nucleons were excited in all compound nuclei at medium excitation energy. The excitation energy is shared only a m o n g outer nucleons in this model. The mean excitation energy per nucleon of the outer nucleons will be about ten times higher than that of the usual medium-weight compound nucleus. It is well-known that the nucleons in a highly excited nucleus can be well described by means of the Fermi gas approximation 3s). The outer nucleons will be treated as a Fermi gas, with weak 6-type nucleon-nucleon interactions. 2.1. D E P E N D E N C E

ON MASS NUMBER

When the same number o f outer nucleons are excited in compound nuclei, the same Hamiltonian HM describes all such compound nuclei. The compound nuclei should have very similar intermediate resonances. Their nuclear radii, however, have an effect on excitation of the outer nucleons. The Schr6dinger equation of this model is (1)

HMA~, = EuA~,,

=

-

--

+

...

+

+ v(,,

.....

,K),

(2)

2m where rl . . . . . rK are the coordinates of the outer nucleons * and V is the interaction a m o n g the outer nucleons and between the inert core and the outer nucleons. When the outer nucleons can be treated as a Fermi gas, the dependence of excitation energies g~ on nuclear radius R is given by the following relation: f 2 ~¢~R = constant.

(3)

All compound nuclei have the same intermediate resonances and the same wave functions for the outer nucleons with regard to coordinates xl at the corresponding energies connected by eq. (3), where xi = r JR,

i=

1, . . ., K .

t T h e n o t a t i o n K is used for the o u t e r n u c l e o n n u m b e r instead o f N+n in I I : K = N+n.

(4)

NUCLEAR REACTION MODEL (II)

675

As was shown in I1 the outer nucleons are under the influence of the long-range average potential Ur produced by the inert core, and they are also affected by the mutual interaction Vrr. The potential Ur and the long-range part of Vr, will be approximated by a deep square well potential Vs having a range in proportion to R. The short range part of Vrr will be represented by 6-type interaction. Then eq. (1) is written

2mR2 V~(x,,..

E (L+

.,

x r ) + 2m

~>jh(x,-xj) A, 2mR 2 -

h2

E~A~,,

(5)

where Vo is the strength of the 6-type interaction. The resonance energies E,]+, measured from the bottom of the square well are easily obtained by applying the perturbation method to 6-type interactions: eu = E ] +.(A + n) ~ ~ constant + O((A + n)-~),

(6)

where O is a smallcorrection term, and n is the mass number of the incident particle. Very similar intermediate resonances are expected in all compound nuclei if their mass numbers A+n and their excitation energies E~+, are connected by eq. (6). Further, all quantities related to intermediate resonances will be similar with each other when any two compound nuclei are connected by eq. (6) assuming their outer nucleon number K is the same. It is noted that the quantities related to the intermediate resonances are expected to change gradually from nucleus to nucleus showing no individual nuclear characteristics. 2.2. LEVEL SPACING Eq. (6) is applied to Weisskopf's semi-empirical level density formula 36) D ( e ) = c exp {--2~/a6~},

(7)

where a and c are constants independent of excitation energy d~. This formula (7) is transformed by eq. (6) for the intermediate resonances:

(8) Here the suffix/2 is omitted from E~+,, and ao and c o are constants independent of A +n. The constants a o and Co are determined from the data for level spacings of light nuclei given in I1.

676

K. IZUMO

Eq. (6) is applicable to Newton's level density formula 37). The parameter a for an intermediate resonance is the same for either form of the level density 36, aT):

a=ao

(A;.)' --

•

9

The details of level spacings and comparison with experimental results are studied in sect. 3. 2.3. LEVEL WIDTH

The partial width for channel a is written in terms of eqs. (15) and (16) of II and eq. (6) (~h2)

r.o

n-2mR2vlA~'(xl . . . . . xr))2'

(~b'(xl . . . . .

(10)

where tka is the wave function of channel a. When the long-range interaction V~ or the short range interaction Vo is dominant, the dependence on mass number is

F~,a oc (_4+ n)tF~a(e~,),

(11)

Fua oc (A + n)- +F~o(eu),

(12)

or

respectively. Here F~, and F~, are matrix elements of V~ and f-type interaction and are functions of the excitation energy e~ defined by eq. (6). The total width Fu has the same dependence on mass number as Fu,. The widths will depend more strongly on excitation energy than the usual compound nucleus widths, since the mean excitation energy per nucleon are higher. They are compared with experimental data in sect. 3. 2.4. STRENGTH FUNCTION

The basic relation between the reduced widths of intermediate resonances fist and those of the usual compound nucleus Yas was given by eq. (28) ofI1. This equation can be easily extended to the strength function s~t by means of the Lane-ThomasWigner theory a2): /

\

S#t(E;~' Ee) : \ O ~ O - s / a v , / z , t

=

wl

\2~]

w2

(EA--EIt)2dr'~W 2 (Ee-Et)2--}-~W22

,

(13)

where

W 2 = c(aul(Vp,+Vrc-- Up- Ur)eIAu), W 2 -- C ( • t ] ( V r c -

Vr)21¢t).

(14) (15)

NUCLEAR

REACTION MODEL

(II)

677

Here D~ and D S are the level spacings of the compound nucleus and residual nucleus. The quantity 7~s/DzD, is averaged over within the widths of the intermediate resonance/~ and the gross channel t; W 1 and Wz are the widths of intermediate resonances coming from the deviation of the averaged interactions Up and Ur. Eq. (13) shows that the intermediate resonances should appear not only in excitation curves but also with the same form in energy spectra. The data for the AlZT(a, ~') and AlZV(p, p') reactions 23, 30) show the existence of the gross channels at medium excitation. It is noted that sut changes gradually with mass number satisfying eq. (6). The sut has a quite different dependence on mass number and excitation energy in comparison with the optical model calculations.

3. Groups of Nuclear Reactions It was shown in sect. 2 that when the same number of outer nucleons are excited, the same intermediate resonances will appear in all compound nuclei at the excitation energies related by eq. (6). On the other hand, the intermediate resonances among

'°~Ii r

' .... ~- i~FeS6(p,p;)O=90 ~ '

,

.

.

.

.

-

. . . . . I0

[ II

;

; i

i

:;

, 13

~

i NiSS(p, eo)

14

E59 MeV Fig. 1. Excitation curves assigned to the group with excitation o f 5 outer nucleons. Excitation energies have been transformed to A + n = 59 using eq. (6). Vertical lines indicate the positions o f intermediate resonances.

those compound nuclei with different numbers of outer nucleons being excited should be uncorrelated. I f the outer nucleon number takes discrete values analogous to a quantum number, nuclear reactions and the corresponding compound nuclei at medium excitation energy will be clearly different according to their outer nucleon numbers. They can then be classified into certain groups characterized by the outer nucleon number. When almost all intermediate resonances of any two reactions are found to correspond after application of eq. (6), they will be assigned to the same group. The

6

-

"

)°a

8

9

'

!

I0

II

No23+n

:: 12

i

" ;

AI

i e,~o~

13

, 14

~V~'//~,\

27 (p,,,,)

Si29(n,ao )

1,5

-/L\ ~

M926(,.

16

~,

17

E28

18 MeV

~L v L , \ ~ / ] ~k/i ~ .

!

i!;

i!: ~

!

~i ' ~ : : ,

i

l~p~'(p, yo ) 8 = 9 0 ='

P i (P~,To) 19=90 ° :

~

19

20

, 21

: , 2 ;)

23

: E :; 24

:.~r.~p.%~:o-,v ~ _ ; ~ 1 2 Z : ~ . . _ ~ o:y:~ :.' i ~ : : : : : :tp', , ) ~ : 9 0 ' :

rX)

25

:

8: 149"

27

,

8= 149"

26

27

12 )s , ,;C (0 ,o 4 ) e = 1 4 9 °

/

?C"(O',°e~,+3I

i.'i/~] ''

,:

) 8=149 °

Fig. 2. Excitation curves assigned to the group with excitation o f 6 o u t e r nucleons. Excitation energies have been t r a n s f o r m e d to A+n = 28 using eq. (6). Vertical lines indicate t h e positions o f intermediate resonances.

0.1 1 7

I0

2 I0

,d

REL.

~05

I0

NUCLEAR REACTION MODEL {II)

679

resonance energies will be distributed somewhat randomly among the reactions assigned to different groups. Three reaction groups have been tentatively identified by this procedure. In figs. 1-3, the excitation curves assigned to the three groups are shown and the reactions are listed in tables 1-3, respectively. Several inter-

24

IX / I L - . , - / \

8

9

'lVJlVv"!

I0

II E25MeV

2

13

Fig. 3. Excitation curves assigned to excitation o f 7 o u t e r nucleons. Excitation t r a n s f o r m e d to A + n = 25 using eq. (6). ,

,

,

,

Ne÷n'o'

14 energies have been

,

500 . D/.*. keV

I00I

ol I,IrllllI0 ,Ill 12 I, IIIJII I 8

E59MeV

Fig. 4. M e a n level spacing o f intermediate resonances for t h e group with excitation o f 5 o u t e r n u c l e o n s as a f u n c t i o n o f excitation energy E59 t r a n s f o r m e d to A + n = 59 by eq. (6). Level s c h e m e is s h o w n by vertical lines. Theoretical curves have been calculated u s i n g eq. (8). T h e best choice for t h e o u t e r n u c l e o n n u m b e r is 5 for this reaction group. T h e level spacings o f intermediate resonances h a v e been averaged u s i n g a sliding average o f f o u r adjacent spacings.

mediate resonances are composed of a number of small peaks, which will be the usual c o m p o u n d nuclear resonances or their fluctuations. Presumably, the intermediate resonances should be clearly seen when the excitation curves are averaged over an energy spread about 100 keV, that is the widths of intermediate resonances, as is done in the NiSS(p, p)Ni s8 reaction 12).

680

K. zztn~o

The coincidence of intermediate resonances in figs. 1-3 is satisfactory considering the rough approximation used. There are, however, many resonances which deviate from others. The systematic deviation of the Ne 2° reactions in fig. 3 could come from TABLE 1 R e a c t i o n s a n d c o m p o u n d nuclei assigned to the g r o u p with excitation o f 5 o u t e r nucleons Compound nucleus

Reaction Ti4a(p, p'z)Ti 48. Cr52(p, p ' z ) C r 5~* FeSS(p, p ' l ) F e 56. NPa(p, p , 0 N p a .

0 = 90 ° 0 = 90 ° 0 = 90 °

NPa(p, ao)CoS5 NP°(p, p)NP ° Nia0(p, p,x)Np0*

0 = 90 °

SraS(p,p,z)SrSa,

i

.

.

.

.

i

.

.

.

.

i

.

.

Ref.

V 4a Mn~a CoS~ CuSa

z) z) z) 2)

Cu59

11)

Cu61 Cu61

is) xl)

ys9

xs)

.

.

I

,

1000

D~ keY

\ \\.,....

7

, I, i . ] .] 1]. I JI, I[, I/,liLI tl t I II,llill ]1.11]1f[LII I0

I5

E28 MeV

20

25

Fig. 5. M e a n level spacing o f intermediate r e s o n a n c e s for t h e g r o u p with excitation o f 6 o u t e r n u cleons as a f u n c t i o n o f excitation energy Es8 t r a n s f o r m e d to A q- n = 28 by eq. (6). Level s c h e m e is s h o w n by vertical lines. Theoretical curves have been calculated u s i n g eq. (8). T h e best choice for t h e o u t e r n u c l e o n n u m b e r is 6 for this reaction group. T h e level spacings o f intermediate r e s o n a n c e s h a v e been averaged u s i n g a sliding average o f f o u r adjacent spacings.

the relatively small nuclear radius. The greater part of these deviations will be due to secondary effects caused by inexactness of the transformation eq. (6) and by the detailed individual nuclear characteristics. An exception may be the C12+O 16 reaction data shown in fig. 2, for which the usual compound nucleus formation may be dominant rather than the partial equilibrium.

NUCLEAR REACTION MODEL (II)

681

C a l c u l a t e d l e v e l s p a c i n g s o f i n t e r m e d i a t e r e s o n a n c e s u s i n g eq. (8) a r e c o m p a r e d w i t h t h e e x p e r i m e n t a l m e a n l e v e l s p a c i n g s o f t h e t h r e e r e a c t i o n g r o u p s i n figs. 4 - 6 . T h e b e s t c h o i c e o f o u t e r n u c l e o n n u m b e r s i n figs. 4 - 6 a r e 5, 6 a n d 7, r e s p e c t i v e l y . TABLE 2 Reactions and compound nuclei assigned to the group with excitation of 6 outer nucleons Reaction F19(p, n)Ne19 F19(p, %)O t° Ne2°(y, po)F 1~ NC°(e, poe')F t~ OlS(~, no)Ne~t Naaa(n, total) NaSS(n, po)Ne2S Na2a(n, Oco)F20 NaSa(p, ~o)Ne 2° Naa3(p, ~'I)Ne z°* C12(O l°, ~0)Mg 64 Cxa(O le, ~' 1, • •~'4)Mg 24. SiSS(y, p, 1)A127. AlaT(p, p'l)A167* Ala7 (p, ?,o+x)Si2S* A127(p, oco)MgS4 A127(p, ~,l)MgS4, SiSS(n' O~o)Mga6 Si2a(n ' ~,l)Mg26, pat(p, ?o)Sa2 K41(p, p,x)K4t* K41(p, C~,l)Aa6*

Compound nucleus 0 = 70 ° 0 = 76 ° 0 = 76 °

0 0 0 0

= = = =

90 ° 90 ° 149 ° 149 °

0 = 90 °, 165 °

0 = 90 °

Cuea(p, Cto)Ni6O Cuea(p, ~,l)NiOO*

Ref.

NeSO Ne 2° Ne a° Ne s° NeZZ NaS4 Na24 Na24 Mg ~4 Mg ~4 Si 26 Si as Si~S Si ss SiSS Si2S Si2a SiaO

14) x~) lo) io) 17) 16) ls) 19) so) 20) 21) El) 3, aa) 63) 4) ~, 64) 5, 64) as)

SiaO

66)

Sas Ca42 Ca42 Zn64 Zn64

so)

67) ZT) so) ae)

TABLE 3 Reactions and compound nuclei assigned to the group with excitation of 7 outer nucleons Reaction

The agreement

Compound nucleus

Ref.

NeaO(p, p,1)Nea0. Ne2°(n, total) Ne20(n, ~0)O1~

Naat Ne21 NeSt

e) ss) 26)

Ne2O(n, Qc,i)O17,

Ne2t

ss)

MgS4(p, p ' l ) M g 64*

A1ss

SiSa(p,p,1)SiSS,

pea

between calculations and experimental

~)

s,sa)

r e s u l t s is g o o d . T h e t h r e e

r e a c t i o n g r o u p s will b e c h a r a c t e r i z e d b y e x c i t a t i o n o f 5, 6 a n d 7 o u t e r n u c l e o n s . T h e a b s o l u t e v a l u e s o f K, h o w e v e r , h a v e _ 2 a m b i g u i t y b e c a u s e t h e s t a t i s t i c a l t r e a t m e n t h a s b e e n a p p l i e d t o c a l c u l a t e s u c h s m a l l K v a l u e s i n s u b s e c t . 2.2.

682

J¢. IZUMO

W h e n the excitation energy is t r a n s f o r m e d b y eq. (6) the i n t e r m e d i a t e resonances o f all reactions assigned to the same g r o u p are expected to be specified b y a u n i f i e d level scheme. T h e unified level schemes o f the three reaction groups are s h o w n in fig. 7. The g r o u p c o r r e s p o n d i n g to excitation o f 6 outer n u c l e o n s differs f r o m the other groups. The greater p a r t o f i n t e r m e d i a t e resonances assigned to this g r o u p

oof rr

o 7

8

I0

12

E25

14

MeV

Fig. 6. Mean level spacing of intermediate resonances for the group with excitation of 7 outer nucleons as a function of excitation energy E25 transformed to A - k n = 25 by eq. (6). Level scheme is shown by vertical lines. Theoretical curves are calculated using eq. (8). The best choice for the outer nucleon number is 7 for this reaction group.

l

I

I

I

I

1

t

I

' li' 1

7

8

i

I

9

I0

i

12

t

14

"

,

16

,

i

18

i

I

20

'

'

7'2

'

'

24

i

i

,

26

E28 MeV

Fig. 7. Unified level schemes of intermediate resonances for the groups with excitation of 5, 6 and 7 outer nucleons downward, respectively. Each excitation energy has been transformed to A ÷ n = 28 using eq. (6). c a n n o t be made to coincide with those assigned to the other groups even b y shifting their excitation energies. However, c o m p a r i s o n between the groups with excitation o f 5 and 7 outer n u c l e o n s is impossible since they have n o o v e r l a p p i n g energy region i n fig. 7.

683

NUCLEAR REACTION MODEL (II)

The relation o f the outer nucleon numbers to nuclear shell structure is shown in table 4. The closed shell at 20 nucleons m a y divide the groups with excitation o f 7 and 5 outer nucleons, but there is generally no relation between the shell structure and the excitation o f outer nucleons. TABLE 4

Outer nucleon number K related to inert cores and compound nuclei Number of neutrons or protons in core

2 -+ 7

8 -+ 19

20 --+ 27

28 ~ 49

50

Even-mass compound nuclei

6

6

6

?

?

Odd-mass compound nuclei

7

7

5

5

?

The reaction groups and compound nuclei characterized by excitation of 7 and 5 outer nucleons are separated by the closed shell at 20 nucleons. The widths o f intermediate resonances in figs. 1-3 seem to increase with excitation energy and decrease slightly with mass number. I f they do, the short-range interaction should be d o m i n a n t as is shown in subsect. 2.3.

4. Angular Distributions It was shown in I1 that the angular distributions o f various reactions at m e d i u m excitation energy were determined by the intermediate resonance terms and the direct interaction term interfering with each other. The differential cross section for the reaction f r o m channel a to channel b was given by eq. (19) o f I i : da(a, b) _ dO

rc ( 2 s + 1 ) _ 1 Z Ia(a, b ) + B ( a , b)l 2,

(16)

vv"

where A(a, b) = i

~.

(21+ l ) ~ ( s l v O I J M ) ( s ' l ' v ' m ' l J M )

~JMll'm'

×

E - E~ - ½iF u

Y,,m,(O,

B ( a , b ) = ( k k ' ] ~ ( rn ] T,2) \ ~ ] \2rch2 ] "ha • Here Fu,, Fub and Fu are the widths o f the incoming channel, o f the outgoing channel and the total width o f the intermediate resonance, respectively *. The primed quantities relate to the outgoing channel. The transition matrix ~'r(e)barepresents all the other contributions except those t h r o u g h intermediate resonances; this matrix will give mainly the contribution o f direct interaction. t These resonance quantities are clearly defined when the representation of J. Humblet and L. Rosenfeld ag) is chosen.

684

K. IZUMO

An example of the simplest numerical calculations is shown here. It can be seen from fig. 1 that the gross resonances measured in the reactions NiSS(p, p~)Ni ss*,

1 (P, eo)CO

Ni

-10.4

/

O.e

!o.4 ~ 1 ~ o.o

7 .

0 90 180 OC.M.(deg)

9.7 "

~o.s

9.6

,.2 ,,o,

b =~ 0'4 •.1~. o o

~.,~"

.0

9t

~q..O

O 90 180 eC.M.(deg)

Fig. 8. A n g u l a r distributions o f t h e NPS(p, c¢0)Co ~5 reaction. Vertical arrows indicate positions o f intermediate resonances.

1

do"__

8 --30 ° /

•

--

60 °

,'~

•

--

1

~

dr~

mb/srlO

0

o - - 90 °

2s°

t'

8

I0

9

II

Ep in MeV Fig. 9. Excitation curves o f t h e NPS(p, p ' x ) N P 8. reaction. A r r o w s indicate positions o f i n t e r m e d i a t e resonances.

685

NUCLEAR REACTION MODEL (II)

NiSS(p, so)Co 5s at Ep = 9.41 MeV and the Ni6°(p, p l ) N i 6°* reaction at Ep = 7.69 MeV can be identified with the similar intermediate resonance at Es9 = 12.83 MeV assigned to the group with excitation of 5 outer nucleons. This means that the intermediate resonances in the three reactions have the same resonance parameters. The excitation curves of the three reactions are shown in figs. I, 8 and 9. These figures also show that the resonances are isolated and that the main contribution will come from only a single resonance. The direct interaction will be neglected in the Ni 5s (p, so)Co ss reaction since the measured angular distributions in fig. 8 show no forward peak, while the other two reactions in figs. 9-11 show a direct interaction contribution. The matrix ,7"(2)bais assumed to be described by the surface direct interaction only using the Born approximation for the reactions Ni 58 (p, p~)Ni6O, and Ni 6 o (p, p~)Ni 6o.. .

,

,

Ni

,

60

r

I0

d6~

t

(p,p~) Ni

60*

d~

T~

dn

in

in

mb/sr

mb/sr 5

0

r E p = 14.3 M e V

~ 0°

60 °

i20"

Ep= 14.3 MeV 180°

0 Fig. 10. Angular distributions of the NP*(p, P'I) NP 8. reaction. The main form of the angular distribution does not change with energy.

30

90 8*

120

Fig. 11. Angular distributions of the Ni6°(p, P'z) Nie°* reaction. The main form of the angular distribution does not change with energy.

Eq. (16) is rewritten by means of the general S matrix formalism as) for the reactions Ni58(p, p l ) N i 58. and Ni6°(p, p])Ni6°*: da(p, Pl) d£2

7~ -

2~-2 ,~'ZIA(p, p l ) + B ( p ,

P l )']

2,

where A(p, pl) =

~, (21+l)~(½1v0lJM)(s'l'v'm'lJM) Mll'm's"

x (r"pr~p")~ Y,'m'(O, E -- E~ -- ½iF~

B(p,

pl) = Cei÷(kp kpq)~j2(QR),

Q2 = k 2 + k2,1_ 2kpkp,~ cos O,

(17)

686

K. IZUMO

and for the NiSS(p, % ) C o s5 reaction

da(p, %) dO

(18)

-- ~ , ~ BL(p,%)PL(COS0 ) ,

where BL(p, %) = ¼ ~, ( - 1)~-s'Z(lJIJ,

½L)Z(l'Jl'J,s'L)

l l ' s'

F~p F~= o

(E-E.)2 +¼£~ "

Here c and ~b are real phase parameters, PL Legendre polynomials, and Z the Z coefficients defined by Lane and Thomas as). o

1.0

58 55 Ni (p, eo)Co

.

Ep=9374 MeV [ 0.5

do" m

d~

o

Ep=9468 MeV

o o o

6

O.t 20

/

m b/s'r"

A

58 58* Ni (p, p,)Ni

I ~

E

Ep*9444 MeV

=9374 p MeV

I0

60 60 * Ni (p, p')Ni 15

I0

~0

Ep=769 MeV

Ep=76[ MeV

o Ep=787 MeV

. . . .

50

90

150

30

90

150

30

90

150

8" Fig. 12. Angular distributions of the reactions NpS(p, p'x)Ni ~s*, NihS(p, %)Co 55 and NiS°(p, p'l)Ni 8°* in the neighbourhood of the intermediate resonance at E59 = 12.83 MeV shown in fig. 2. Theoretical curves are calculated by using the same resonance parameters for these three reactions.

NUCLEAR REACTION MODEL (II)

687

Comparison of the calculations with experiment is shown in fig. 12. The best fit parameters are E, = F,=

9.41 MeV for NiSS+p, 7.69 MeV for N i 6 ° + p , 66keY,

F,~o = 0.13,

J=~,

•=3,

q ~ = 0 °,

c = 11.6.

/~/zpl

The contribution of the intermediate resonance term is 37 9/ooof the total differential cross section for the reactions Ni~a(p, pl)Ni 58. and Ni6°(p, p~)Ni 6°*. The agreement is quite good for the NiSS(p, p~)Ni 5s* reaction. The theoretical calculations predict a rapid change o f angular distributions in the NiSa(p, co)Co s5 reaction within an energy interval of about 250 keV.

6 5 (1 !

•

~A

• • 4&

•

4

MeV-I 3 2 I .

.

.

.

s"o . . . .

,;o . . ~A+n

.

.

.

. . . 150

.

.

.

.

. . 200

250

Fig. 13. Level density parameter a measured by the (e, p) reactions with 30 MeV e-particles. Previous data analysed using Weisskopf's formula 31) are shown by circles. Recent data analysed using N e w t o n ' s formula are shown by triangles. The theoretical curve has been calculated using eqs.

(8) and (9).

It is noted that the same resonance parameters are used for the three different reactions shown in fig. 12. This indicates that such intermediate resonances are very similar and supports the idea o f reaction groups characterized by outer nucleon number. The reactions NiSS+ p and Ni 6 o + p will be assigned to the same reaction group with excitation of 5 outer nucleons in accordance with the results of sect. 3. It is also noted that the interference between intermediate resonance and direct interaction is important for the reactions NiSS(p, p~)Ni ss* and Ni6°(p, p~)Ni 6°*. Rapid change of angular distributions at backward and forward angles near the intermediate resonance will be difficult to explain without taking interference into account.

688

K . IZUMO

5. Nuclear Temperature When the excitation energy of the compound nucleus is as high as ,,~ 30 MeV, the level spacings of the intermediate resonances will become smaller than their widths. At such high excitation the intermediate resonances will be observed indirectly through their effect on the nuclear temperature. As is shown in subsects. 2.2 and 2.4, high nuclear temperatures and a gradual change of the level density parameters with mass number are generally expected, in contrast to Newton's statistical treatment 37). The level density parameter a calculated by eqs. (8) and (9) is compared with the (c~, p) data 31) in fig. 13. High nuclear temperatures and the dependence on mass number are accounted for quite well by this model. This is indirect support for our prediction of subsect. 2.4 that the intermediate resonances should appear in energy spectra. The outer nucleon number used in this calculation, however, is about eight times higher than those obtained from direct observation of intermediate resonances. This descrepancy may come from neglecting the change of widths with excitation energy.

6. Discussion It is pointed out that intermediate resonances should be a kind of gross resonance having a definite spin, parity, width and other resonance quantities and they should appear not only in excitation curves but also in energy spectra. Almost all the measurements show this resonance property indirectly; however, more data are needed in order to prove directly the resonance character of this gross structure. The reaction groups in sect. 3 have been analysed theoretically. For further proof o f their resonance character it will be necessary to check the transformation eq. (6) experimentally and to show that intermediate resonances connected by eq. (6) have the same resonance parameters in many reactions. If these reaction groups are certified, all excited nuclei should have unified level schemes showing no individual nuclear characteristics and characterized by outer nucleon numbers, and all reactions could be described by means of the simple reaction mechanism of outer nucleon excitation in the medium excitation energy region. Then the outer nucleon number should behave as a quantum number. The essential difference between the fluctuation theories 33) and this nuclear model is that the gross structure o f excitation cross sections is identified with "intermediate resonances" having definite resonance quantities. If the gross structure has a clear resonance property it will be difficult to explain it by means of random fluctuation of the compound nuclear resonances because the compound nuclear resonances should then be correlated. On the other hand, the "doorway state" theory 34) seems to have similar results as this model. Small fixed numbers of particles or quasi-particles are responsible for the intermediate resonances in both theories. The main difference between this theory and the present model is the number of excited particles or

NUCLEAR REACTION MODEL (II)

689

quasi-particles, namely the difference of the number of degrees of freedom. This model predicts that intermediate resonances should appear commonly in all types of nuclear reactions in the medium excitation energy region and they should show little individual nuclear characteristics. The outer nucleon number K, however, may be 3 because of its ambiguity. Then both theories may give identical results. The author would like to thank Dr. D. M. Van Patter for helpful discussions and is much indebted to Professor S. Yoshida and Professor T. Pinkston for their kind comments and discussions. References 1) 2) 3) 4) 5)

6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39)

F. D. Seward, Phys. Rev. 114 (1959) 514 S. Kobayashi et al., J. Phys. Soc. Japan 15 (1960) 1151 K. Shoda et aL, J. Phys. Soc. Japan 16 (1961) 1031; 1807 M. Kimura et al., J. Phys. Soc. Japan 15 (1960) 1128; Nuclear Physics 23 (1961) 338 G. E. Fischer, V. K. Fischer, E. A. Remler and M. D. Tatcher, Phys. Rev. 110 (1958) 286; I. Kumabe et aL, J. Phys. Soc. Japan 14 (1959) 713; H. Ogata et aL, J. Phys. Soc. Japan 15 (1960) 1719 S. Kobayashi, Institute for Nuclear Study Report 33 Japan (1960) N. Yamamuro et al., Institute for Nuclear Study Report 30 Japan (1960) Y. Oda et al., J. Phys. Soc. Japan 15 (1960) 760 S. Igarasi, K. Izumo and T. Udagawa, Prog. Theor. Phys. 18 (1957) 103 K. lzumo, Prog. Theor. Phys. 26 (1961) 807 I. Kumabe et al., Nuclear Physics 46 (1963) 437 L. L. Lee and J. P. Schiffer, Phys. Lett. 4 (1963) 104 K. Matsuda, private communication J. G. Jenkin, L. G. Earwaker and E. W. Titterton, Nuclear Physics 44 (1963) 453 K. L. Warsh, G. M. Temmer and H. R. Blieden, Phys. Rev. 131 (1963) 1690 W. R. Dodge and W. C. Barber, Phys. Rev. 127 (1962) 1764 J. K. Blair and H. B. Willard, Phys. Rev. 128 (1962) 299 J. H. Towle and W. B. Gilboy, Nuclear Physics 32 (1962) 610 C. F. Williamson, Phys. Rev. 122 (1961) 1877 K. L. Warsh, G. M. Temmer and H. R. Blieden, Nuclear Physics 44 (1963) 528 M. L. Halbert, F. E. Durham, C. D. Moak and A. Zucker, Nuclear Physics 47 (1963) 353 K. Shoda et al., J. Phys. Soc. Japan 17 (1962) 735 J. Kokame, private communication K. L. Warsh, G. M. Temmer and H. R. Blieden, Nuclear Physics 44 (1963) 329 R. Potenza, R. Ricamo and A. Rubbino, Nuclear Physics 41 (1963) 298 G. Dearnaley, private communication R. D. Sharp et al., Phys. Rev. 124 (1961) 1557 R. E. Shamu, Nuclear Physics 31 (1962) 166 M. W. Brenner, A. M. Hoogenboom and E. Kashy, Phys. Rev. 127 (1962) 947 H. W. Fulbright, N. O. Lassen and N. O. Roy Poulsen, Mat. Fys. Medd. Dan. Vid. Selsk. 31, No. 10 (1959) W. Swenson and N. Cindro, Phys. Rev. 123 (1961) 910; W. Swenson, private communication A. M. Lane, R. G. Thomas and E. P. Wigner, Phys. Rev. 98 (1955) 693 T. Ericson, Advan. in Phys. 9 (1960) 425; Phys. Rev. Lett. 5 (1960) 430 H. Feshbach, Ann. of Phys. 19 (1962) 287; B. Block and H. Feshbach, Ann. of Phys. 23 (1963) 47 S. Hayakawa, M. Kawai and K. Kikuchi, Prog. Theor. Phys. 13 (1955) 415 J. M. Blatt and V. F. Weisskopf, Theoretical nuclear physics (Wiley, New York, 1952) T. W. Newton, Can. J. Phys. 34 (1956) 804 A. M. Lane and R. G. Thomas, Revs. Mud. Phys. 30 (1958) 257 J. Humblett and L. Rosenfeld, Nuclear Physics 26 (1961) 529