- Email: [email protected]

North-Holland Publishing Co., Amsterdam

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

T H E O R Y OF N U C L E A R R E A C T I O N S III. Physical Quantities and Channel Radii j. H U 3 I B L E T

University o[ Ligge, Belgium Received 2 October 196I

A b s t r a c t : I t is shown t h a t the physical quantities introduced in the first p a p e r of this series, such as the collision m a t r i x and the energies, total and p a r t i a l widths of the r e s o n a n t states, although they are expressed in termb of a r b i t r a r y channel radii, are in fact independent of these p a r a m e t e r s and define intrinsic properties ~)f the nuclear system.

1. Introduction In the first paper 1) of this series, the resonant states associated with a nuclear reaction have been i n t r o d u c e d as the eigenstates of the c o m p o u n d system for which there is formally no incoming wave in a n y of the channels. In terms of such a set of eigenstates, we were able to analyze in a v e r y simple way the resonant part of the collision matrix. For each channel c, we have i n t r o d u c e d a channel ra.diu: ae and a cut off of the Coulomb field at a distance be ( ) \ ae); b y definition, ae is such t h a t if the distance re of the pair of nuclear f r a g m e n t s in channel c is larger t h a n ae, the nuclear interaction between these fragments is negligible: t h e y interact only through the Coulomb field, if t h e y are b o t h charged, in the interval ae ~

re ~

be.

On the one hand, it is clear from a physical point of view t h a t the channel radii are not sharply defined. On the other hand. the choice of a well defined set of channel radii is necessary from the m a t h e m a t i c a l point of view. Accordingly, if the nuclear interaction between the f r a g m e n t s of channel c is negligible for re > ae, a n y physical q u a n t i t y , such as the position of the resonances and their total or partial width, must be insensitive to the choice of a channel radius a'c larger t h a n ae. This must be true in a n y channel c and we shall say more briefly t h a t the above physical quantities m u s t be defined in such a way as to be a-independent. I t is the main purpose of this paper to discuss from this point of view all the physical quantities which have been i n t r o d u c e d in the first paper of this series; there, proofs of a-independence were given in the elastic scattering case only. f44

THEORY

OF

NUCLEAR REACTIONS (III)

545

2. T h e R e s o n a n t S t a t e s a n d t h e I n t e r i o r R a d i a l F a c t o r s

Because of its very definition (I.6.2) ¢, the radial factor ~e - - ~e(ae) is ind e p e n d e n t of ae,, ae,,, . . . . but it does of course depend on ae, i.e. #c(ae) =fi #e (a'e), where #e (a'c) = f 9c * (re }/",nt),.=.,o dSe.

(2.1)

For re > ac, the exterior radial factor Ue has the following form: ue(re, ke) = xcOc(re, ke) + ycle(re, ke).

(2.2)

It satisfies, like Oe and Ie, an equation of the form X"+

(ke'

l(l+1) ,e 2

2~e) ~ 1 x = o.

(2.s)

Because of the continuity of the wave function on the surface re = a'e, we have #c(a'e) = xeOc(a'e, kc)+yeXe(a'c, kc).

(2.4a)

#'e(a%) = xeO'e(a'e, k e ) + y e I ' e ( a ' e , ke).

(2.4b)

Similarly we find

Adopting the Wronskian notation (I.l.16), eqs. (I.6.7), (I.3.18) and (2.4) immediately give W(Oe, Oc; ac, ke) = W ( O e , Oe; a'e, kc) = --2ikcyc, W(~¢, Ie; ae, kc) = W(~e, Ie; a'o,

k0):

-2ikexc.

(2.5a) (2.5b)

It is clear, therefore, that the two equations oo: ao, ko) : o,

W(¢o,O,, 0o: a'o, kc) = 0

(2.6)

have the same set of solutions, which amounts to say that the total energies

e.

=

E.--lir~

(2.7)

Ke.--iye.

(2.S)

and the channel wave numbers ke. =

corresponding to the resonant state are a-independent. As in ref. 1), let us again add an index n to a q u a n t i t y which is to be considered at the resonance, i.e. for k e = kc,. Since Ye~ : 0, we have ~e.(ae) = xe.Oo.(ae), t A n o t a t i o n s u c h a s (I.6.2) r e f e r s t o f o r m u l a

~e.(a'e) = Xe.0c.(a'e) (6.2) of ref. 1).

(2.9)

546

j. HUMBLET

a n d h e n c e w e see that, a t the r e s o n a n c e , t h e ratio #e/Oe is an a-independent quantity:

en(a'e)

--

een (ac)

(2.10)

Oen(a'o) Oen(ae)" The same argument also applies to the ratio q~'e/O'e.

3. T h e C o l l i s i o n M a t r i x , I t s R e s i d u e s

at t h e P o l e s a n d t h e Q u a n t i t y v n

The a-independence of the collision matrix element Uc,c defined by eq. (I.7.2) immediately follows from the relations (2.5) applied respectively to the channels c and c', when there is no other entrance channel than c. Consequently, the residue rc'cn = lim [(8--o~n)Uc,c] = --i

t=t,

~2

__

1

,'

1

t

~bc,n ~bcn

x/M¢, Me vn kc'nkcn Oe'n Oen

(3.1)

is also a-independent. Comparing this expression with the property (2.10), it appears that the quantity vn defined b y eq. (I.10.11) must be a-independent. This may also be verified by a direct computation, which has the advantage of showing that such a property of vn is consistent with our approximate treatment of the configuration space (we disregard the possible splitting of the compound system into more than two nuclei). Since the expression fo

~2

dLen

(3.2)

is completely symmetrical with regard to the channel index, it is sufficient to verify its a-independence by increasing the radius of a single channel. Let this be channel c and let o~e be that part of the configuration space which is limited by the surfaces re ---=ae and re = a'e. According to eq. (3.2), we have vn(a'c) = vn(ac) + f ?/~ +2Me--

ta e

T-*n~ndw dLcn (a'e)

1~2

dd°n

2Me

~cn 2 (ae)

dLcn (ae)

(3.3)

dg°n

The second term on the right-hand side is computed by means of eqs, (I.5.7), (2.2), where ke = ken, (I.9.7) and (2.9); using the notation -

~ d F, 2Me de*

(s.4)

raEORV oF ~UCL~Aa~EACr~ONS(ml

~47

we obtain

v,~(a'e) = vn(ae) + \Oen(ae)/

. [Oe.(re)]~dre

+ EOen(a'e)]'Len(a'e)-[Oe,,(ae)]2Le.(ae)}

(3.5)

and, introducing the definition (I.3.22) of Le, ~,(a e) -----v,(ac) + \Ocn(ae)]

c [Ocn(rc)]2dre

+w(o~, 0~; a'~, ko)-w(0o, 0o; ao, k~)}.

(3.6)

From the eq. (2.3) satisfied by Oe, we derive successively the relations

O,c, -~- (ke2

l(l+1) Yes

2r/~¢/ Oo+Oe = O,

(3.7)

re /

d

h,e (Oe0'o--O'e0~) = --0~.

(3.S)

The last equation integrated between ae and a'e shows that eq. (3.6) indeed reduces to v.(a'e) = vn(ae ), (3.9) which proves the a-independence of vn. Consequently, the quantities Go., Oen, ~0., ~c. defined in subsect. 10.2 of ref. a) are also a-independent. 4. T h e Partial W i d t h s and the Q u a n t i t y / ~ n The total width/'~ is a-independent because it is proportional to the imaginary part of the eigenvalue (2.7) which has itself this property. The real and imaginary parts of the wave-numbers ken are also a-independent. Consequently, because of eq. (2.10), establishing the a-independence of the partial width 1 ?~2Kc~ #e~ ~ Fo. -

/zn Me

Oen

(4.1)

amounts to proving this property for/t~, which is defined by eq. (I.10.18). Let us proceed as we did for vn, we easily get

a ' :/zn(ao ) + fa de I~en(re)l~dr~ /zn(o) C

+N~.(a'o) ~o.(a'~)a, ~ N~.(a~) ~on(a~) 3. (4.2) }0o. (~) 0o. (a~)

548

j. HUMBLET

Considering first the case in which none of the fragments in channel c is a neutron (~c :/: 0), the eq. (2.10) and the definition (1.3.28) of N ~ give

m.( a'~) = m.(a~) +

~o.(ao) Oo.(a~) u{f"°° t0~"(r~)[~dr~

+ l b . [Oc,(rc)l~ dr~e/~t0

f.'° 0

(4.3a)

/

IO~.(re)lUdre =/~.(a~).

To complete the proof of the a-independence of/z., let us now assume that one of the two fragments in channel c is a neutron. In this case, according to eq. (I.3.28), we have

(4.4)

N,.(o~) = W,.(ao) - fo°le'~°.'°t~d.o,

where M/~z,(ae)is defined b y eq. (1.1.43). According to eqs. (1.1.42) and (I.1.41), we also have 1

=

1

l)[)"/ ~/P(1)* ~/f(1) •

ken)

(4.5)

and therefore 1

1

N,.(a.) = -2to. - + k~.-k*~ w(~ei')*, ~i"; ao, ko.).

(4.6i

Hence, we find

#.(a'e) = #,,(ae) + i-O~c,

o [Jg~11(kc.re)[~dre

(4.7) +

kCn •_

[W/:~/f(1)* ~fi(1).a

t

ken)__Wl:~/f(1)*

#f(1).

ken)]}

* 2n ~k C

From the relation

,

[],$2 b2 ~" ~//~{1) ;,/~{I)* •~~v / FI ( 1t) * _ ~~./F(1) ol ~;~/~(1)"* L( -- 1 ~.*~cn) --*~cn] ~ I. o'¢~ [ ,

(4.8)

one sees immediately that the eq. (4.7) reduces to /~,,(a'e) = #n(ae).

(4.3b)

5. T h e O b s e r v a b l e P a r t i a l W i d t h s a n d t h e E x p a n s i o n of t h e C o l l i s i o n M a t r i x

The definition (4.1) of the partial width has been adopted with the double purpose of introducing a quantity which is a-independent and satisfies exactly

THEORY

OF NUCLEAR

REACTIONS

(III)

549

the relation

(5.1)

v. = X Vc.. c

This last requirement is responsible for the appearance of the factor q, -----i~./[v,[ in the expansions (I.10.22) of the collision matrix. Using a terminology which was also adopted by Lane and Thomas 2) in similar circumstances *, we are justified in defining an observable partial width by

rio _

1

~2

~c. 2

(5.a)

this is also an a-independent quantity and it satisfies the relation

r . = q. ~ r 0 .

(5.2)

C

Then, the quantity Fcn defined by eq. (4.1) m a y be called the ]ormal partial width. The expansions (1.10.22) m a y be written in terms of the observable partial widths: the factors qn then no longer appear explicitly (although they are still contained in the total widths occurring in the resonance denominators). Since in the expansions of the type (I. 10.22) (with either formal or observable partial widths) each resonance term is composed of factors which are a-independent, the background functions Qc'c and Qc'c also have that property. The author is indebted to Professor L. Rosenfeld for helpful discussions. Grants from the University of Liege and the Institut Interuniversitaire des Sciences Nucldaires are gratefully acknowledged. t See ref. s), p. 327.

References 1) J. Humbler and L. Rosenfeld, Nuclear Physics 26 (1061) 529 2) A. M. Lane and R. G. Thomas, Rev. Mod. Phys. 30 (1958) 257

Copyright © 2023 COEK.INFO. All rights reserved.