INTRODUCTION TO NUCLEAR FISSION

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INTRODUCTION TO NUCLEAR FISSION

K. Dietrich

To cite this version:

K. Dietrich. INTRODUCTION TO NUCLEAR FISSION. Journal de Physique Colloques, 1972, 33 (C5), pp.C5-3-C5-15. �10.1051/jphyscol:1972502�. �jpa-00215104�

JOURNAL DE PHYBIQUE C o l l o q u e C 5 , supplement a u no 8-9, Tome 33, AoGt-Septembre 1972, page C5-3

INTRODUCTION TO NIiCLEAR FISSION K. DIETRICH

P h y s i c s Department, TU - MUnchen

RIsum6 - Nous r6suraons bri&vement les donni'es e x p 8 r i m e n t a l e s f o n d a n t e n t a l e s d e l a f i s s i o n e t l e u r i n t e r p r e t a t i o n a l ' a i d e d ' u n e t h e o r i e s e m i - c l a s s i q u e e t ph8nomenologique. Une

t h e o r i e p l u s compl&te e s t e x p o s e e d a n s l a d e r n i h r e p a r t i e .

A b s t r a c t - ^A s h o r t r e v i e w i s g i v e n of t h e b a s i c o b s e r v a t i o n s i n n u c l e a r f i s s i o n and t h e i r i n t e r p r e t a t i o n i n t e r m s o f a s o m i - c l a s s i c a l phenomenological t h e o r y . A more c o m p l e t e theo- r e t i c a l t r e a t m e n t i s i n d i c a t e d i n t h e l a s t p a r t .

I n t h e f i r s t p a r t we s h a l l r e v i e w t h e main e x p e r i m e n t a l f a c t s on n u c l e a r f i s s i o n i n c l u - d i n g a n ( i n c o m p 1 e t e ) s e l e c t i o n of more r e c e n t r e s u l t s . I n t h e second p a r t , we p r e s e n t a n i n t e r p r e t a t i o n of t h e s e r e s u l t s i n t e r m s of a s e m i - c l a s s i c a l p h e n m e - n o l o g i c a l t h e o r y . I n t h e l a s t p a r t , we make a few r e m a r k s on a more c o m p l e t e t h e o r y which i s y e t t o b e developed.

I n view of t h e v e r y l i m i t e d t i m e , I s h a l l n o t t r y t o b e c o m p l e t e o r t o d o j u s t i c e t o t h e au- t h o r s of a l l t h o b e a u t i f u l work which wils performed i n t h e r e c e n t y e a r s . I n s t e a d I s h a l l s p e n d t i m e o n p o i n t i n g o u t open problems a n d a n a l o g i e s t o o t h e r f i e l d s of p h y s i c s .

I - REVIEW OF THE EXPERIMENTAL RESULTS ON NUCLEAR FISSION

I f a n u c l e u s undergoes ( s p o n t a n e o u s o r

a d i s t r i b u t i o n o f masses, c h a r g e s , and k i n e t i c e n e r - g i e s of t h e f r a g m e n t s i s o b s e r v e d . I n T a b l e 1 t h e

" c l a s s i c a l " r e s u l t s on t h e s e d i s t r i b u t i o n s a r e d i s - p l a y e d . The b a s i c f e a t u r e s a r e well-known :

- p r e p o n d e r a n c e o f asymmetric m a s s - d i v i s i o n f o r low-energy f i s s i o n of n u c l e i h e a v i e r t h a n 2 2 6 ~ a , o f s m e t r i c m a s s - d i v i s i o n f o r n u c l e i below 2 2 6 ~ a ;

r a p i d i n c r e a s e of t h e r a t i o of symmetric mass-divi- *

s i o n a s a f u n c t i o n o f e x c i t a t i o n energjv ; number of e m i t t e d n e u t r o n s and t o t a l y - d o e x c i t a t i o n e n e r g y s a w t o o t h - l i k e f u n c t i o n s of t h e fragment mass.

I n T a b l e 2 a s e l e c t i o n of more r e c e n t f i n - d i n g s i s p r e s e n t e d . The d i s c o v e r y of t h e s o - c a l l e d f i s s i o n isomers by F l e r o v , P o l i k a n o v e t a 1 [I ] and by Michaudon e t a1 [ 2 ] i s c e r t a i n l y t h o most promi- n e n t one and h a s f o s t e r e d enormously t h e r e s e a r c h i n t h e f i e l d of n u c l e a r f i s s i o n . The f i s s i o n i s o m e r s produce a n i n t e r m e d i a t e s t r u c t u r e i n t h e s u b t h r e s h o l d c r o s s s e c t i o n f o r f i s s i o n which i s r e m i n i s c e n t of t h e i n t e r m e d i a t e s t r u c t u r e o b s e r v e d i n c o n n e c t i o n w i t h t h e a n a l o g u e s t a t e s . The "doorway s t a t e s " which produce t h e i n t e r m e d i a t e s t r u c t u r e a r e d i s t i n g u i s h e d from t h e "more c o m p l i c a t e d s t a t e s " i n t h e same e n e r - gy r a n g e by a much s h o r t e r f i s s i o n l i f e t i m e . I t t u r n s o u t t h a t t h e r e a r e 5 l e v e l s o f i n t e r m e d i a t e s t r i l c t u r e : t h e r e a r e g r o u p s o f r e s o n a n c e s w i t h a s p a c i n g of 0.1 t o 1 MeV and a w i d t h of t h e o r d e r of 0.1 MeV . T h e s e l a r g e s c a l e peaks d i s p l a y a 1 s t l e - v e l of f i n e s t r u c t u r e o f an a v e r a g e s p a c i n g of a b o u t 50 eV a n d a w i d t h o f a b o u t 10 e V

.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972502

T a b l e 1 ( " ~ l a s s i c a l ' R e s u l t s )

T y p i c a l f e a t u r e s S c h e m a t i c p l o t s

1. Mass d i s t r i b u t i o n 1 . a ) P r e p o r l d s r a n c e mf a s y m m e t r i c

t

R e l a t i v e number N o f mass d i v i s i o n f o r n u c l e i f i s s i o n e v e n t s a s a f a c - > 2 2 6 ~ a , a n d o r symmetric t o r o f t h e f r a g m e n t mass mass d i v i s i o n f o r n u c l e i a ) f o r low e x c i t a t i o n < 2 2 6 ~ a .

e n e r g i e s f < 10 MeV) F o r maxima o f t h e d i s - t r i b u t i o n f o r symmetric and

a s y m m e t r i c mass s p l i t . Heavy mass peak s t a b l e a t

.

% 140

.

- - . ^{A > 2 7 6}

A = 2 2 6 - - - A < 2 2 6

b) f o r medium and h i g h 1.b) With i n c r e a s i n g e x c i t a t i o n N

e x c i t a t i o n e n e r g i e s e n e r g y t h e d i p f o r symmetric ,)---C. \ mnss d i v i s i o n i s f i l l e d . A t \

s u f f i c i e n t l y h i g h e x c i t a t i o n e n e r g y maximum of d i s t r i b u - t i o n o c c u r s f o r s y m m e t r i c f i s s i o n .

A 1

I -

A2

S p o z t . a . l o w e n . f .

.

-

- -- - - -

-

2. K i n e t i c e n e r g y d i s t r i b u - Ekin a p p r o x i m a t c l y e q u a l t o

t

^{b e d}

t i o n

-

A v e r a g e r e l a t i v e k i n e t i c s c i s s i o n c o n f i g u r a t i o n m ,

e n e r g y gkin as a f u n c t i o n D i p f o r s y m - e t r i c mass d i v i s i o n . 160 of f r a g m e n t mass.

3. C h a r g e d i s t r i b u t i o n A p p r o x i m a t e l y i n t h e same r a t i o a s mass d i v i s i o n :

AI/A2 Z1/Z2

.

INTRODUCTION TO FISSION

4. Distribution of excita- tion energies of frag- ment s

a) Number of emitted a) Avcra[:e number ; of emitted 6 neutrons as a f unc- neutrons 2-2.5

.

tion of fragment tions between 0 and 6 as a

mass. functim of A1 (sawtooth form) 1

b) Average excitation b) Ecx to be obtained from sum- energy of primary ming the energy releases due

fragments as a fune- to neutron emission, y- and

t

tion of fragment mass. &decay. Similar fluctuations

1

as for 4.a)

.

5. Total fission cross- Steplike behaviour, fluctua- section as a function tions due to competition with of excitation energy. other reaction channels.

[-I I

K t h r = T!~reshold of _{l S t} chance f iss.

K nd t :

: = 'I'iireshold of 2 chance f i s s .

6. Angular distribution Anisotropic at energies of fission products. 6 classical fission threshold

Ethr (exp. with alignment, if necessary ).

Isotropic at excitation ener- gies $ 5 MeV above F

thr '

T a b l e 2 (More r e c e n t r e s u l t s )

T y p i c a l f e a t u r e s I . S u b - b a r r i e r r e s o n a n c e s , G r o u p s o f r e s o n a n c e s b e l o w t h e

" f i s s i o n i s o m e r s " c l a s s i c a l t h r e s h o l d f o r f i s s i o n . F i n e s t r u c t u r e w i t h i n t h e s e g r o u p s

~ e f s . [ l , 2 ] ( i n t e r m e d i a t e s t r u c t u r e phenomenon).

F i s s i o n l i f e t i m e s r a n g i n g f r o m 10-lo sec t o lo-' ^{s e c} , odd-even e f f e c t i n f i s s i o n l i f e t i m e s .

4. D e t a i l e d s t u d i e s of a ) A v e r a g e k i n e t i c e n e r g y o f f r a g m e n t s

-- -

2 . S p e c t r o s c o p y o f f i s - R o t a t i o n a l s p e c t r u m a t t r i b u t e d t o s i o n i s o m e r s f i s s i o n i s o m e r s by m e a s u r i n g t h e

c o n v e r s i o n o f y - d e c a y s b e t w e e n

Ref. [3 ] "class 11-states".

3. D e - e x c i t a t i o n o f a ) A n g u l a r momentum d i s t r i b u t i o n f o r 4 E~

d i s t r i b u t i o n s a s a rises v e r y s l o w l y w i t h e x c i t a t i o n e n e r g y , may e v e n d e c r e a s e a t low f a c t o r of e x c i t a t i o n e x c i t a t i o n e n e r g i e s . R e f . [S ] p r i m a r y f r a g m e n t s p r i m a r y f r a g m e n t s . R e s u l t : a v e -

r a g e a n g u l a r momentum o f p r i m a r y f r a p e n t s : 7.E ( t h e r m a l f i s s i o n of "5Lll. R e f . [ 4 ]

b) Even-odd e f f e c t s i n t h e p r o m p t de- e x c i t a t i o n p r o c e s s . R c s u l t : n o n o t i c e a b l e even-odd e f f e c t i n num-

e n e r g y .

.Ni ^-

b ) V a r i a n c e s o f d i s t r i b u t i o n s i n c r e a s e w i t h i n c r e a s i n g e x c i t a t i o n e n e r g y .

b e r o f e m i t t e d n e u t r o n s , a b o u t ^z1

1 MeV even-odd e f f e c t i n t o t a l

e n e r y r e l e a s e d by y - d e c a y . 0 - ^{e v . c v .} f r a g m e n t s

Ref.f6 I. ^X - ^{o d d} - ^{o d d} f r n j i m c n t s

8 = a n g l e b e t w e e n d i r e c t i o n o f l i g h t f r a g m e n t s and a - p a r t .

5. T e r n a r y f i s s i o n A n g u l a r d i s t r i b u t i o n o f a - p a r t i c l e A ~ ( 0 ) ( t w o l a r g e f r a g m e n t s p e a k e d i n a d i r e c t i o n p e r p e n d i c u l a r

t o t h e m o t i o n o f t h e t w o b i g f r a g - a n d o n e l i g h t p a r t i - m e n t s . R e s u l t o f t r a j e c t o r y c a l c u - c l e , p r e f e r a b l y l a t i o n s : Ekin N 20-25 M e V a t a

" p a r t i c l e ) . f r a g m e n t d i s t a n c e o f N 2 0 fm

. ^. A

INTRODUCTION TO FISSION C5-7

*. +

0.1- 0 . 2 MeV F i g . 1

I t depends on t h e e x p e r i m e n t a l t e c h n i q u e which le- v e l of i n t e r m e d i a t e s t r u c t u r e is s e e n . The s p a c i n g a n d w i d t h of t h e f i n e s t s u b s t r u c t u r e i s t h e s a m e a s t h e one s e e n i n ( n t h , y ) - r e a c t i o n s .

I 1

-

I n t h e a b s e n c e of a d i r e c t mechanism o f f i s s i o n w e may w r i t e t h e S-matrix f o r t h e f i s s i o n p r o c e s s a s

"1/2 -112

A s i m p l e s t a t i s t i c a l t h e o r y c o u l d b e a p p l i e d i f t h e p a r t i a l w i d t h s were u n c o r r e l a t e d , i . e . i f

ha1

The f a c t t h a t Eq.(2) was n o t f u l f i l l e d f o r f i s s i o n l e d A. Bohr [7] t o e l a b o r a t e t h e p i c t u r e of " t r a n - s i t i o n s t a t e s " which i s well-known from t h e t h e o r y of c h e m i c a l r e a c t i o n s . The P i c t u r e i s q u a l i t a t i v e l y a s f o l l o w s : i f t h e e x c i t a t i o n e n e r g y E d o e s n o t e x c e e d t o o much t h e c l a s s i c a l t h r e s h o l d e n e r g y

E f '

t h e f i s s i o n i n g s y s t e m h a s t o p a s s a n i n t e r m e d i a t e s t a g e where t h e main p a r t of i t s e x c i t a t i o n e n e r g y i s c o n t a i n e d i n t h e form of a c o l l e c t i v e e n e r g y of d e f o r m a t i o n . To t h i s s t a g e , t h e c o l l e c t i v e model a p p l i e s i n v e r y much t h e same way a s t o low-energy e x c i t a t i o n s of n u c l e i c l o s e t o t h e i r ground s t a t e s h a p e , i . e . we may w r i t e t h e H a m i l t o n i a n H of t h e s y s t e m i n t h e f o l l o w i n g way

( 5,q = s e t of i n t r i n s i c a n d c o l l e c t i v e v a r i a b l e s r e s p e c t i v e l y ).

The p o i n t i s t h a t c l o s e t o t h e s a d d l e p o i n t of n u c l e a r f i s s i o n a n d even s l i g h t l y beyond i t , t h e t o t a l wave f u n c t i o n may b e d e s c r i b e d by a r e l a t i v e l y s m a l l number of e i g e n s t a t e s of Ha

.

t h u s c a 1 c : l l a t e s t h e S - m a t r i x n o t w i t h r e s p e c t t o t h e t r u e f i n a l s t a t e s b u t w i t h r e s p e c t t o t h e e i g e n - s t a t e s of

Ho a s f i n a l s t a t e s ( c ' of E q . ( l ) z t r a n - s i t i o n s t a t e ) . The H a m i l t o n i a n Hcoll(q) i s w r i t t e n a s a sum of a k i n e t i c e n e r g y %(q) of c o l l e c t i v e motion p l u s a p o t e n t i a l e n e r g y U(q)

F o r a l o n g t i m e , U(q) was t a k e n t o b e t h e deforma- t i o n e n e r g y of a s t a t i c l i q u i d d r o p of d e f o r m a t i o n q and t ( q ) was o b t a i n e d from t h e r e q u i r e m e n t of i r r o t a t i o n a l f l o w . I t i s t h r o u g h a m o d i f i c a t i o n of Hcoll t h a t we may u n d e r s t a n d t h e phenomenon o f i n t e r - m e d i a t e s t r u c t u r e and a l s o a c q u i r e a t l e a s t a q u a l i - t a t i v e u n d e r s t a n d i n g o f t h e asymmetric mass d i v i s i o n . T h e r e f o r e we s h a l l s h o r t l y s k e t c h how w e c o u l d d e r i - ve H c o l l from a s e m i - c l a s s i c a l a d i a b a t i c p i c t u r e :

A

Assume t h a t we a r e g i v e n a model H a m i l t o n i a n H wfiich d e p e n d s n o t o n l y on t h e p a r t i c l e d e g r e e s o f freedom b u t a l s o on a l i m i t e d number of p a r a m e t e r s q 1 . .

.

U ( q ) 3 (W 1W ) = e n e r g y of t h e Hartree-Fock

0 0 0

ground s t a t e Iwo ) ;

+,av = c r e a t i o n a n d a n n i h i l a t i o n o p e r a t o r s ;

E = s i n g l e p a r t i c l e e n e r g i e s i n a deformed s i n g l e p a r t i c l e p o t e n t i a l .

The d e f o r m a t i o n i s d e s c r i b e d by t h e p a r a m e t e r s q (vl v2 IV ^Iv 3 ^v 4 ⁾ r e s i d u a l i n t e r a c t i o n

: : = normal product

.

We introduce the stationary eigenstates ("adiabatic states") of H for Eixed values of q

and expand the solution '4 of the time-dependent SchrJdinger equation

in terms of these states ,t

n

Substitution of ( 8 ) in ( 7 ) leads to a coupled set of differential equations for the amplitudes c (t) If we treat the terms containing t V as a smal!

perturbation and if we require that the unperturbed motion is deszribed by the lowest adiabatic state we obtain the following result for the expectation value (I(t)lH"l~(t))

Eq.(lO) represents the "cranking approximation1' of the collective inertia. The energy vo(q) of the lowest adiabatic state is seen to play the role of a potential energy of collective motion. A s indica- ted in Eq. (9), we may interpret (Y(t) 1; ^Iv(ft) ) as a classical Hamiltonian function of collective mo- tion. We may quantize 8 by replacing the classi- cal kinetic energy by 4 2 /2 times the Laplacian

n 2

in curvilinear coordinates (Z(q) -+ 'C(q) z - Y l / L r l ( q ) ) (the quantization procedure is known to be not uni- que, Ref.[8]. The result of the quantization may be shown to be unique up to a scalar potential which does not contribute in t.he WKB limit and which may be thought to be incorporated in g(q). ) t and thus obtain a Hamiltonian operator of collective motion

We emphasize that our procedure hinges upon the assumption that the coupling between the Intrinsic and collective dekrees of freedom is small and that the collective dynamics can be treated semi-classi- cally. Let us assume teat the potenLia1 energy

i5' (q) can be written as a functional of the nuclear o

density p : ro(q) 1 ro[pl

A : number of nucleons.

We now assume the existence of a smooth average density F ^{such that} may be expan- ded in terms of A - ~ ' ~ , i.e. that it may be repre- sented by the energy ELD(q) of the static liquid drop of deformation q :

This can be shown to be possible if - p is almost constant in the nuclear interior and drops to zero in a surface layer of thickness d << nuclear radius.

In view of E q . ( 1 3 ) , a Taylor expansion of eoJpl in terms of powers of the density fluctuation Lp a p - p snggests itself :

If we calculate the term linear in 6p within the Hartree-Fock approximation and if wc generate the average density - p f r m the Hartree-Fock density by a special smoothing procedure, we obtain the well- known shell-correction term oE Strutinski (9,101

Here, A is about the Fermi energy and g(&) is a smooth level distribution generated by smearing the single-particle levels E,, over an energy inter- val comparable to the spacing of major shells. The shell correction ESC depends on the density of single-particle levels close to the Fermi energy and is thus a function of the shape df the nucleus and of the proton and neutron numbers. It is usual- ly calculated on the basis of phencrmenological shell model potentials. The validity of the Strutinski

INTRODUCTION M FISSION

t h e o r y is s e e n t o h i n g e e s s e n t i a l l y on t h e n e g l e c t of t e r m s 9 6 p 2 ) , on t h e v a l i d i t y o f t h c LD-model, a n d , a s f a r a s t h e p r a c t i c a l a p p l i c a t i o n s are c o n - c e r n e d , o n t h e q u e s t i o n of how w e l l we c a n r e p r e - s e n t t h e t r u e s i n g l e - p a r t i c l e s p e c t r u m by phenome- n o l o g i c a l s i n g l e p a r t i c l e p o t e n t i a l s .

T h e d e p e n d e n c e o f t h e s i n g l e - p a r t i c l e s p e c t r u m o n t h e s h a p e o f t h e o b j e c t , i.e. on t h e b o u n d a r y c o n d i t i o n s , i s a v e r y g e n e r a l p r o b l e m w h i c h was b e a u t i f u l l y t r o . l t e d i n t h e l a s t p a p e r s o f C. B l o c h a n d R. B a l i a n [ll]. Dr. B a l i a n w i l l g i - v e a t a l k o n t h i s s u b j e c t a n d w i l l show you t h a t t h e

" s h e l l - c o r r e c t i o n t e r m " o c c u r s i n a s i m i l a r way i n e l e c t r o d y n a m i c s a n d a c o u s t i c s whenever t h e wave- l e n g t h s o f t h e i n t e r e s t i n g e i g e n m o d e s a r e compara- b $ e t o t h e d i m e n s i o n o f t h e o b j e c t . R c t u r n i n g t o t h e f i s s i o n p r o b l e m , we d e f i n e t h e c o l l e c t i v e p o t e n - t i a l e n e r g y U(q) t o b e t h e sum o f t h e l i q u i d d r o p e n e r g y ELD and t h e s h e l l - c o r r e c t i o n

E s ~

nnd t h e c o l l e c t i v e l t a m i l t o n i a n t o b e

= a d d l e 2 n d saddle /

#

T h e s y m m e t r i c s h a p e s t u r n o u t t o b e u n s t a b l e w i t h r e s p e c t t o a. l i n e a r c o m b i n a t i o n of '30 a n d YS0 d c f e r m a t i o n s .

( 3 ) T h e r e i s a c r i t i c a l e l o n g a t i o n beyond t h e 2nd s a d d l e , where t h e s h e l l c o r r e c t i o n s f a v c u r a Hcoll(q) 1 - y -nd A(q) + U(q) . ^{( 1 7 )} n e c k i n g - i n o f t h e n u c l e u s r a t h e r t h a n a f u r t h e r

e l o n g a t i o n ( s c i s s i o n s t a g e ) . T h i s may b e c o n n e c - t e d w i t h t h e o c c u r r e n c e o f a s c i s s i o n b a r r i e r . O u r t h e o r y d o e s n o t g i v e a p r e s c r i p t i o n o f how t o

f o r m u l a t e t h e i n t r i n s i c a n d t h e c o u p l i n g H a m i l t o n i a n ( S e e F i g . 2 ) .

i n Eq. ( 3 ) . However, t h e q u a l i t a t i v e u n d e r s t a n d i n g ( 4 ) T h e s c i s s i o n s t a t e , i . e . t h e s t a g e o f r e l a t i v e - of t h e i n t e r m e d i a t e s t r u c t u r e i s i n d e p e n d e n t o f t h e l y weak c o n t a c t o f t h e t w o n a s c e n t f r a g m e n t s , d e t a i l e d f o r m of t h e s e o p e r a t o r s . seems t o b e o f g r e a t i m p o r t a n c e f o r t h e d i s t r i -

b u t i o n of m a s s e s , k i n e t i c e n e r g i e s a n d i n t r i n s i c T h e s h e l l c o r r e c t i o n ESC o n d u l a t e s a -

r o u n d t h e smooth l i q u i d d r o p p o t e n t i a l ( s e e F i g . 2 ) . E x t e n s i v e i n v e s t i g a t i o n s of t h e p o t e n t i a l l a n d s c a p e

U(q) h a v e b e e n p e r f o r m e d by v a r i o u s g r o u p s [12]

a n d h a v e l e d t o v e r y i n t e r e s t i n g r e s u l t s .

F u r t h e r m o r e , o n e c a n u n d e r s t a n d t h a t t h e i n t e r m e d i a t e s t r u c t u r e o b s e r v e d i n s u b t h r e s h o l d f i s s i o n i s a c o n s e q u e n c e o f t h e 2 n d v a l l e y i n U(q).

L e t u s q u o t e t h e main r e s u l t s .

( 1 ) T h e s e c o n d v a l l e y s a r e m o s t p r o n o u n c e d i n t h e r e g i o n o f t h e h e a v y a c t i n i d e s a n d t e n d t o d i s - a p p e a r f o r l i g h t e r e l e m e n t s b e c a u s e o f t h e s h a r p r i s e o f t h e l i q u i d d r o p e n e r g y ELD(q)

.

( 2 ) T h e s h a p e o f t h e n u c l e u s a t t h e 2 n d s a d d l e i s a s y m m e t r i c ( p e a r - s h a p e d ) f o r t h e n u c l e i known t o u n d e r g o p r e f e r e n t i a l l y a s y m m e t r i c f i s s i o n .

-

e n e r g i e s o f t h e f i s s i o n f r a g m e n t s . H y p o t h e s e s o f a c o m p l e t e [13], o r o f a p a r t i a l [14] t h e r m a l e q u i l i b r i u m a t t h e s c i s s i o n p o i n t h a v e b e e n made.

I n o u r model l a n g u a g e t h i s c o u l d b e d e s c r i b e d by t h e a s s u m p t i o n t h a t t h r o u g h t h e i n t e r a c t i o n Hic t h e e n e r g i e s 8 ( q ) o r a s u b g r o u p o f them are s t a t i s t i c a l l y p o p u l a t e d i n t h e v i c i n i t y o f t h e s c i s s i o n c o n f i g u r a t i o n qsc

.

C5-10 K. DIETRICH

some r e c e n t m o d e l s on h e a v y i o n r e a c t i o n s . O b v i o u s l y , i f t h e s c i s s i o n b a r r i e r a n d a c o r r e s p o n d i n g 3 r d v a l l e y e x i s t t h e y would b e t h e c o u n t e r p a r t o f t h e q u a s i - m o l e c u l a r s t a t e s w h i c h w e r e s u g g e s t e d t o b e r e s p o n s i b l e f o r a c e r t a i n k i n d o f i n t e r m e d i a t e s t r u c t u r e w h i c h i s o b s e r v e d i n r e a c t i o n s b e t w e e n l i g h t n u c l e i . (5) I m e n t i o n j u s t b r i e f l y t h a t i t i s q u i t e common-

l y a g r e e d upon t h a t t h e C o r i o l i s i n t e r a c t i o n c o n t a i n e d i n H. i s n o t s t r o n g enough t o d e s -

1 C

t r o y t h e c o n s e r v a t i o n o f t h e K-quantum number on t h e p a s s a g e f r o m s a d d l e t o s c i s s i o n . T h i s f a m o u s a n d o l d ~ u g g e s t i o n by A . Bohr l e a d s t o d e f i n i t e p r e d i c t i o n s on t h e a n g u l a r d i s t r i b u - t i o n o f f i s s i o n f r a g m e n t s , w h i c h s h o u l d c o i n c i - d e w i t h t h e d i s t r i b u t i o n of t h e n u c l e a r symme- t r y a x i s a t t h e 2 n d s a d d l e p o i n t . T h e d i s t r i - b u t i o n s h o u l d t e n d t o become i s o t r o p i c w i t h i n c r e a s i n g e x c i t a t i o n e n e r g y , b e c a u s e more a n d more t r a n s i t i o n s t a t e s are o c c u p i e d a b o v e t h e d a d d l e p o i n t a n d t h e a n g u l a r d i s t r i b u t i o n i s a n i n c o h e r e n t sum of t h e t e r m s

WiK

A n i s o t r o p i c d i s t r i b u t i o n s may b e o b s e r v e d i f a n l y a f e w t r a n s i t i o n s t a t e s a r e p o p u l a t e d a n d i f t h e m a g n e t i c s u b s t a t e s M are n o t a l l e q u a l - l y p o p u l a t e d . I f t h e 2nd s a d d l e i s l o v e r t h a n t h e l s t , t h e K quantum number i s l i k e l y t o b e n o l o n g e r c o n s e r v e d , e v e n i f t h e e n e r g y i s c l o - se t o t h e t h r e s h o l d . T h e r e f o r e , e x p e r i m e n t s on t h e a n g u l a r d i s t r i b u t i o n o f f r a g m e n t s a s a f u n c t i o n o f t h e e x c i t a t i o n e n e r g y are v e r y i n - t e r e s t i n g .

( 6 ) L i f e t i m e s w i t h r e s p e c t t o s p o n t a n e o u s f i s s i o n w e r e c a l c u l a t e d o n t h e b a s i s o f

H c o l l a l o n e , u s i n g t h e WKB a p p r o x i m a t i o n . I t i s shown i n

(7) L e t u s f i n a l l y t u r n t o t h e d i s c u s s i o n of t h e i n t e r m e d i a t e s t r u c t u r e o b s e r v e d i n ( n , f ) ,

( d , p f ) e t c . r e a c t i o n s : i n t e r m e d i a t e s t r u c - t u r e i s known t o a r i s e , i f a s i m p l e c o n f i g u r a - t i o n i s w e a k l y c o u p l e d t o a h i e r a r c h y of more c o m p l i c a t e d s t a t e s . S o t h e r e m u s t b e a mecha- n i s m w h i c h p r o m o t e s a s i m p l e c o n f i g u r a t i o n i n t o a n e n e r g y r a n g e w h e r e a l a r g e number o f c o m p l i - c a t e d c o n f i g u r a t i o n s e x i s t s , a n d t h e r e must b e a p h y s i c a l r e a s o n w h i c h p r e v e n t s t h e s i m p l e c o n - f i g u r a t i o n f r o m b e i n g c o m p l e t e l y d i s s o l v e d i n t o t h e n e i g h b o u r i n g c o m p l i c a t e d s t a t e s . I f t h e s i m - p l e c o n f i g u r a t i o n s a r e " c o u p l e d t o t h e c o a t i - nuumtt, i . e . t o t h e e n t r a n c e o r e x i t c h a n n e l of a r e a c t i o n , t h e y a c t a s "doorway statcs" a n d e x h i b i t t h e m s e l v e s by a n i n t e r m e d i a t e s t r u c t u r e o f t h e c r o s s - s e c t i o n . T h e b e s t e x a m p l e s o f c o n - f i g u r a t i o n s p r o d u c i n g i n t e r m e d i a t e s t r u c t u r e a r e t h e a n a l o g u e s t a t e s a n d t h e f i s s i o n i s o m e r s .

One c a n u n d e r s t a n d t h e i n t e r m e d i a t e s t r u c - t u r e o b s e r v e d i n s u b t h r e s h o l d f i s s i o n o n t h e b a s i s o f t h e H a m i l t o n i a n ( 3 ) a n d a p o t e n t i a l e n e r g y U(q) o f t h e t y p e shown i n F i g . 2

.

0 1

T h e p o t e n t i a l s U a n d U a r e d e f i n e d i n F i g . 1 .

T h e r e a r e e i g e n s t a t e s

4

-

s t a t e s ) w h i c h a r e l o c a l i z e d i n t h e 1 s t v a l l e y o f U(q) a n d e i g e n s t a t e s z l ( q ) w h i c h a r e l o c a l i z e d i n t h e 2nd v a l l e y . F u r t h e r m o r e , we i n t r o d u c e t h e

I I I

e i g e n s t a t e s qI'(c) , q ( f ) which d e s c r i b e d i s c r e t e I'

i n t r i n s i c e x c i t a t i o n s i n t h e 1 s t a n d 2 n d v a l l e y r e s - t h e s e c a l c u l a t i o n s t h a t s h e l l e f f e c t s a r e n o t

p e c t i v e l y , a n d c o n t i n u o u s e i g e n f u n c t i o n s q e ( < ) o n l y i m p o r t a n t i n U(q) b u t a l s o i n t h e i n e r -

w h i c h d e s c r i b e t h e e n t r a n c e c h a n n e l , f o r i n s t a n c e t i a l t e n s o r gpv

.

a t h e r m a l n e u t r o n nth s c a t t e r e d b y t h e t a r g e t nu- f i s s i o n l i f e t i m e i s t o o l a r g e by s e v e r a l o r d e r s

c l e u s . o f m a g n i t u d e , b u t t h e r e l a t i v e c h a n g e s a s a

f u n c t i o n o f n e u t r o n a n d p r o t o n n u m b e r s , e s p e - T h u s o u r b a s i s c o n s i s t s o f t h e f o l l o w i n g c i a l l y t h e odd-even e f f e c t s are e n c o u r a g i n g l y product :

r e p r e s e n t e d by t h e s e c a l c u l a t i o n s .

INTRODUCTION TO FISSION C5-11

x', $,

4'

( P ~ The "complicated" c l a s s 11 s t a t e s a r e i n t r i n s i c ,y,

.p:'

A f i n e s t r u c t u r e of a somewhat l a r g e r s c a l e c o u l d z e g t r a n c e channel s b )

.

v a l l e y d i f f e r e n t from t h e f i s s i o n mode ( r o t a t i o n a l We may n e g l e c t t h e d i r e c t channel-channel c o u p l i n g s t a t e s b u i l t on t h e v i b r a t i o n a l c l a s s I 1 s t a t e s ) .

( (3

IH

c o u p l i n g between c l a s s 11 s t a t e s and t h e e n t r a n c e c ) T h e r e is a 2nd l e v e l of f i n e s t r u c t u r e ,(channel ((3 IH ^{12)- 0 ) . If} we c a l c u l a t e t h e S-matrix (width 0.1 e V , s p a c i n g 1 eV) of t h e i n t e r m e d i a t e

i n t h i s model space, we f i n d t h e f o l l o w i n g b a s i c peaks j u s t mentioned which i s produced by t h e weak f e a t u r e s : a ) t h e r e i s a g r o s s s t r u c t u r e (width 0.1 c o u p l i n g between c l a s s 11 s t a t e s 12 ) and complicated MeV , s p a c i n g I MeV) produced by t h e c o u p l i n g of c l a s s I s t a t e s 11 ) of a neighbouring energy ( i . e . c l a s s I 1 s t a t e s t o t h e f i s s i o n continuum through through t h e m a t r i x elements (1 12 ) ). I t depends t h e d i f f e r e n c e of t h e t r u e and t h e a u x i l i a r y bar- on t h e experiment which h i e r a r c h y of f i n e s t r u c t u r e r i e r p o t e n t i a l s w i l l be seen. The f i n e s t h i e r a r c h y i s of c o u r s e w i -

ped o u t , i f t h e c o u p l i n g between 11 ) ^{and 12} ) - s t a t e s i s s t r o n g , The d i f f e r e n t c a s e s of c o u p l i n g have been

1 e x t e n s i v e l y s t u d i e d by Lynn.

IU(ql- U(q13 f o r 9 < qsa

.

The analogy t o t h e analogue resonances i s d i s p l a y e d i n T a b l e 3. I t would be s t i l l more per- W e c a l l t h i s g r o s s s t r u c t u r e t h e ( ' t r a n s m i s s i o n re-

f e c t , i f we c o u l d d i s t i n g u i s h c l a s s I and c l a s s I 1 sonances". They correspond t o t h e p o t e n t i a l r e s o -

s t a t e s by t h e v a l u e of a quantum number. T h i s i s nances i n t h e s c a t t e r i n g from an o p t i c a l model

only p o s s i b l e by r e c u r r i n g t o u n r e a l i s t i c models p o t e n t i a l .

b ) There i s a 1 s t level of f i n e structu- l i k e t h e SU model. There, t h e symmetry c h a r a c t e r 3

r e (width 20-50 eV , s p a c i n g 100 eV - ^{1 kev). his} ( h , p ) would p r o v i d e such a c l a s s i f i c a t i o n . We n o t e i s due t o t h e c o u p l i n g between simple and compli- t h a t t h e i s o s c a l a r p a r t of t h e E2 t r a n s i t i o n vani- c a t e d c l a s s I 1 s t a t e s , i . e . due t o (2 b ^{12' )}

.

T a b l e 3

Subthreshold f i s s i o n Analogue s t a t e s

s i m p l e s t a t e s c l a s s 11 s t a t e s analogue s t a t e s T+

IN,z)

complicated s t a t e s c l a s s I s t a t e s

-

T s t a t e s c l o s e t o t h e energy of t h e analogue

< s t a t e promotion of simple

s t a t e s i n t o energy s u r f a c e energy of d i f f e r e n c e of Coulomb e n e r g i e s of n u c l e i i n r a n g e of complicated t h e l i q u i d drop t h e same i s o s p i n m u l t i p l e t

c o n f i g u r a t i o n r e d u c t i o n of t h e

c o u p l i n g between d i f f e r e n t deformation of d i f f e r e n t i s o s p i n , i . e . c o u p l i n g only through simple and compli- c l a s s I and c l a s s I 1 s t a t e s t h e i s o s p i n v i o l a t i n g p a r t of t h e i n t e r a c t i o n s c a t e d s t a t e s

approximately con- o n l y i n s i m p l i f i e d models.

s e r v e d quantum Example : SU symmetry cha-

3 i s o s p i n T

number r a c t e r (1, p) of represen-

t a t i o n

I11

-

( i f I t i s based on a Hamiltonian HI (Eq. ( 5 ) ) with s u p e r f l u o u s d e g r e e s of freedom.

( i i ) The t h e o r y does n o t p r o v i d e a well-defined form of t h e c o u p l i n g between c o l l e c t i v e and i n t r i n - sic e x c i t a t i o n s .

( i i i ) The t h e o r y i s s e m i - c l a s s i c a l by c o n s t r u c - t i o n and t h u s does n o t permit t o i n v e s t i g a t e whether t h e s e m i - c l a s s i c a l p i c t u r e i s a t a l l a p p l i c a b l e ,

A t h e o r y which, i n p r i n c i p l e , i s a b l e t o ovetcome t h e s e d i f f i c u l t i e s i s t h e g e n e r a t o r coor- d i n a t e method ( G C M ) which was developed l o n g ago by s e v e r a l a u t h o r s 1171 w i t h t h e more g e n e r a l purpose of f o r m u l a t i n g a microscopic t h e o r y of c o l l e c t i v e motion. Recently, N6renberg a p p l i e d t h i s t h e o r y t o t h e d e s c r i p t i o n Of n u c l e a r f i s s i o n [l8]. I n a number of very r e c e n t p u b l i c a t i o n s t h e GCM i s used f o r t h e d e s c r i p t i o n of heavy i o n r e a c t i o n s . I d o t h i n k t h a t t h e GCNI may p r o v i d e a frame f o r a u n i f i e d d e s c r i p - t i o n of n u c l e a r f i s s i o n and of heavy i o n r e a c t i o n s . The t h e o r y f a c e s s t i l l a c o n s i d e r a b l e amount of c h a l l e n g i n g problems f o r a l l d i r e c t i o n s of t a s t e . W e would l i k e t o p r e s e n t i n a few words t h e b a s i c c o n c e p t s of t h i s approach i n o r d e r t o show t h a t i t p r o v i d e s a v e r s a t i l e and powerful language going be- yond t h e s e m i - c l q s s i c a l d e s c r i p t i o n .

We may a v o i d t h e u s e of s u p e r f l u o u s de- g r e e s of freedom by u s i n g t h e s t a t e s lGn(4)> a s a system of (non-orthogonal) b a s i s f u n c t i o n s , and i n t e g r a t i n g over t h e c o l l e c t i v e parameters q :

n +

I t i s important t o o n l y u s e a r e s t r i c t e d number of

" i n t r i n s i c " e x c i t a t i o n s n f o r any given s e t of c o l l e c t i v e v a r i a b l e s q, because t h e system of b a s i s f u n c t i o n s would o t h e r w i s e be overcomplete, Ix) ^is

one of t h e s t a t e s ]c ) , ^{l c r} ) , l h ) t o b e d e f i n e d be- low.

I n o r d e r t o d e s c r i b e a r e a c t t o n l i k e ( n , f ) i n a microscopic model we need 3 t y p e s of b a s i s func- t i o n s :

f i ) s t a t e s Ic ) which d e s c r i b e t h e e n t r a n c e channel. I n our c a s e t h e s e wsuld be s c a t t e r i n g s t a - t e s of 1 neutron from t h e t a r g e t nucleus.

( i i ) S t a t e s / A ) which r e p r e s e n t compound s t a - t e s of c l a s s I ( (&,I ) f and c l a s s I1 ( IX,II))

.

( i i i ) S t a t e s Ice

>

IC ⁾ and ] c * ) correspond t o t h e p o t e n t i a l s c a t t e r i n g s t a t e s of a nucleon, t h e s t a t e s

Ix)

Each of t h e s e t h r e e c l a s s e s of f u n c t i o n s i s o b t a i n e d by " d i a g o n a l i z i n g r ' t h e fundamental Hamiitonian

b

H = ^{~ ( i )} +

$

i z l i , j

i n a s u i t a b l y chosen subspace of b a s i s f u n c t i o n s lGn(q) )

.

with 1 nucleon i n t h e shell-model c o n t i n u u s and w i t h deformation parameters q c l o s e t o t h e ground- s t a t e deformation of t h e t a r g e t nucleus. For o b t a i - n i n g t h e compound s t a t e s 11) we d i a g o n a l i z e H i n a subspace of bound s t a t e s Sn(q) , with d e f o r - mation parameters encompassing a l l n u c l e a r shapes from t h e ground s t a t e shape t o t h e shape of t h e n u c l e u s c l o s e t o t h e 2nd s a d d l e p o i n t . From t h i s d i a g o n a l i z a t i o n we may e x p e c t t o o b t a i n two c l a s s e s of s t a t e s : c l a s s I - s t a t e s , IX,I ) c h a r a c t e r i z e d by a '*weight f u n c t i o n " gn(q) l o c a l i z e d around t h e deformation of t h e ground s t a t e and c l a s s 1 1 - s t a t e s

/ X , I I ) with a weight f u n c t i o n l o c a l i z e d i n t h e second v a l l e y . F i n a l l y t h e " f i s s i o n s t a t e s ' ' ] c r )

would have t o be c o n s t r u c t e d from b a s i s f u n c t i o n s

which a r e p r o d u c t s of Hartree-Fock s t a t e s of t h e emerging fragments 1 , 2 (fragment deformation ql,

q~ , i n t r i n s i c e x c i t a t i o n s n1,n2 ). We n o t e t h a t t h e d i s t a n c e between t h e fragment c e n t r e s would e n t e r a s a g e n e r a t o r c o o r d i n a t e i n t h i s p i c t u r e .

Having o b t a i n e d t h e b a s i s s t a t e s f c ) , Ix) , ^{[ c *}

>

IY)