On a formal theory of resonance reactions

ANNALS

OF

PHYSICS:

On

%,

(1964)

401-ko9

a Formal

Theory

of Resonance

LUCIANO Zstituto Zstituto

A3~azionale

di Fisica di Fisica

Reactions

FONDA

Teorica

dell’Universit8, and Nucleare, Sottosezione

Trieste, di Trieste,

A recent theory of resonance reactions is reconsidered riving from it a complex potential model and the decay nuclear states.

Italy Trieste,

Italy

for the purpose of delaw of the compound

I. INTRODUCTION

A new general theory of resonance scattering and reactions, based essentially on an expansion of the scattering amplitude in terms of the eigenstates of the effective Hamiltonian describing the closed channels and not involving the concept of channel radius, has been recently proposed by the present author and by R. G. Newton (1). According to this theory, sharp resonancesare related to complex eigenvalues, of small imaginary parts, of the above mentioned effective Hamiltonian and, in the sense of slightly altered forces, to bound states embedded in the continuum. In other words, all forces which by a small perturbation could lead to a stable (infinite lifetime) bound state of the total Hamiltonian in an energy region where some channels are open, produce a narrow resonance near that energy. Since the bound state may require the coupling between open and closed channels for its very existence, there follows that we may observe a sharp peak even when the coupling to the open channels is strong. A narrow resonance need have no relation to possible bound states of the isolated closed channels, a feature which has been usually adopted in other formal approaches to this subject (see, for example, the Feshbach (2,s) and the AgodiEberle (4) works). In fact, as shown in the Appendix of this paper, a bound state of the total Hamiltonian degenerate with the continuum may exist even when the reduction of the total Hamiltonian to the subspace of the closed channels does not have bound states, i.e., when Eq. (2.16) of ref. 2 (or Eq. (4.1) of ref. 3) does not have normalizable solutions, or the states labelled 1cn} in ref. 4 do not exist.’ Nonetheless also in this case a sharp compound nuclear resonance may be produced by slightly altering the forces. 1 The possibility of expanding the scattering the effective Hamiltonian describing the closed 401

amplitude in terms of the eigenstates of channels, considered by Fonda and Newton

402

FONDA

It is the purpose of this paper to add two considerations to the theory formulated in ref. 1. That is, to obtain formally a complex potential model for the case of many overlapping narrow resonances, and to derive the decay law of the eigenstates of the effective Hamiltonian describing the closed channels. To these states we shall refer in the following as compound nuclear states. II.

FOKRIALISM

Let us first introduce the various notations by summarizing briefly the main results of ref. 1. We split up all channels into two groups o and c and we choose group c having only channels of threshold energy such that in the energy region of interest they are all closed. The total Hamiltonian operator is given by H. After the introduction of the projection operators P, and P, = 1 - P, on channel groups o and c, respectively,’ and after few formal manipulations of the complete Green’s function, the scattering amplitude reads as follows (see ref. 1, Eq. (2.26) ) : Tfi = Tofi + (c$:-““, H&(E,

E)Hcod+“i’)

(1)

d+’ and 4:-’ describe scattering states, satisfying outgoing and ingoing wave boundary conditions respectively, of the reduction of the total Hamiltonian to the subspace of the open channels H, = P&P,. Of course, Ho, = P,HP, and H,, E P,HP, . The index i (or f) stands for the initial (or final) channel and for all quantum numbers that, together with the energy, are necessary to specify the states completely. Tori is the scattering amplitude associated with Ho . se is defined by3 $(z, E) = [z - X,(E)I,-’ with X,(E)

(2)

given by X,(E)

= Hc + i$$Hco E +

1 HOC ie - Ho

which is the effective Hamiltonian for channel group c (see ref. I, Eq. (2.28a) ). $(x, E) is the resolvent for the effective Hamiltonian X,(E), and it appears in the scattering amplitude calculated at x = E + iO+. Since .X,(E) has a in ref. 1, has been considered recently also by Feshbach in ref. 3. All the results of Section IV, C of Feshbach’s paper have been already established by Fonda and Newton in Section II, of ref. 1. The eigenstates and the eigenvalues of the effective Hamiltonian describing the closed channels are called u,(E) and A,(E) in ref. 1 and E,r and Et in ref. 3. 2 A thorough discussion on the introduction of the projection operators P, and P, given by Feshbach in ref. 3, where they are called P and Q, respectively. Channel groups o and e have been labelled by b and a, respectively, in ref. 1. 3 For the inverse of a projected matrix we mean hereafter the inverse of the submatrix under consideration and zeros everywhere else.

is

ON

A

FORMAL

THEORY

OF

RESONANCE

40~3

REACTIONS

negative definite skew-hermitian part, the resolvent $(z, E’), considered as a function of the complex variable x for fixed E, can have poles in the third aud fourth quadrants of the first sheet of its Riemann surface, besides having a normal threshold cut starting from the threshold of the lowest channel in group c. sC( E’, E:> has, on the contrary, as a function of the complex variable E, a normal threshold cut starting from the threshold of the lowest open channel and, of course, no poles on the first sheet of the Riemann surface. The poles x = A,(~FJ’I of $(x, E) correspond to normalizable eigenstates of the effective Hamiltonian X,(E) : XC(EbCIL(E) The imaginary

part of A,(B) Im A,(E)

= AL(E)ucn(E).

is readily

= -7r(uen(lC), =

-7r,E

obtained H,6(E

J (ucn(-u,

C-1)

(see ref. 1, Eq. (2.12b)) -

HA”’

:

H,)H,,u,,(E’)) (P, El)

j2 5 0

Cc)) -1

where uC,‘(E) has been taken normalized. p indicates all the continuous and discrete quantum numbers which, together with the energy, are necessary to specify 4Ai’ completely. I#$~’satisfies either an outgoing or an incoming wave boundary condition. Introducing the spectral resolution for the HamiltonianX,(B), we immediately obtain an expansion of the scattering amplitude in terms of the eigenstates u,,(E) {see ref. 1, Eq. (2.14) ) , exploiting in this manner all narrow compound nuclear resonances. From considerations based on this expansion of the scattering amplitude and on the properties of the eigenstates u,%(E’) of X, (E), the interpretation of the resonance mechanism has been given in ref. 1 according to the line of thought sketched in the Introduction. III.

COMPLEX

POTENTIAL

MODEL

Let us now consider the possibility that several discrete eigenvalues of the nonhermitian Hamiltonian X, lie very close to each other so as to give rise to a region of overlapping narrow resonances. What has been proved to be particularly useful in this caseis the concept of complex potential (5) and we are going to give a close expression for it using our formalism. We have therefore a group of narrow spaced and sharp resonances in the proximity of the energy E:. We are interested in the energy average of the scattering amplitude over an energy region ti sufficiently broad to cover this group of resonancesbut narrow enough not to alter appreciably the energy dependence of the other quantities in the scattering amplitude. The introduction of the complex potential follows then after the determination of the wave function whose asymptotic behavior leads to the averaged scattering amplitude.

404

FONDA

Introducing the spectral resolution its resolvent the expression :

for the Hamiltonian

X,(E’),

we write for

c&(x,E’) = 2 z Qm(E’) n _ A,(E,) + se”‘(z,-m where

Q,,(E’)= 14~‘))(~c,(~‘)1 (f&n u-v,ucn(E’)) u,, (IX’) being the eigenstate of X,f (23’) belonging to the eigenvalue A,*(E’). (v,,( E’) , ~~~(23’) ) = 0 for nz # n. The summation runs over the eigenvalues of X, which describe the considered group of narrow spaced and sharp resonances. SF contains all the other contributions to $jC. Since there is an implicit energy dependence in the Hamiltonian X,(E’), when we vary the energy E’ over the resonance region the eigenvalues A, (E’) and the eigenstates u,, (23’) of x,(E’) vary consequently. However, this region is narrow so that it is expected that this energy dependence does not matter much for the average process and we can therefore take the Hamiltonian %.$(I?) at a fixed energy in the considered resonance region. Apart from the wildly energy varying denominators in $(x, E’), representing the group of closely spaced and narrow resonances, everything is then slowly energy dependent over the resonance region. The resonances are supposed to be sharp so that the poles of $jCwhich describe them are very cIose to the real axis: Im A, M 0. In performing the average we can therefore make the approximation: 1

1

E’ - An(F)

cz

E’ - Re ~4,

- i&(E

- Re A,)

where 8 stands for “principal value.” The first term, when summed over n and if the number of resonances is sufFiciently large, will give a small contribution after the average and will be neglected. Writing Re A, = E after the average, and defining the operator

&c(E)= &c

12&m(E),

we get for the averaged scattering amplitude 1 a

E+(AE/Z)

sB-(AED)

!&(E’)

dE’ = To/i(E)

+ (#Yf'(E), The complex potential Xii)

H&r

- i~Qcl&,~:""~'(E)).

model wave function is then easily seen to be: = +:+)tn + G,H,,[$’ - ivrQ,]Hcoqd+t)‘i’,

(7)

(8)

ON

A

FORMAL

THEORY

OF

RESONANCE

REACTIONS

where G, = lim [E’ + ic - H,]--I. C-HI+ In order to obtain the Schrbdinger similar to that given by Feshbach favor of $R:+“~‘:

equation for xii’ we use a formal technique, in ref. 3, consisting in solving Eq. (8 ) in

= ( 1 + G&,&Y

PCS’

-

i?rQ,]H,<,]-l~;i).

(9)

Operating then with E: - H, on both sides of Eq. (9) and using a simple operator identity, we get (E - H, -

U,)&’

= 0

(10)

with U” = - i~ffocQcHco( 1 + GoH&,nr

-

kQc]H,,)

-’

+ HocS:‘H,o{ 1 + GJL,,[s:’

-

~‘rr&,]H,,)-~

(11)

The first term at the right hand side of Eq. (11) contributes to the compound nuclear potential UC”. Contributions in the “second order” to this potential come also from the second term, which contains also a part ITD giving rise to direct nuclear processes and, eventually, to “single particle” resonances. The separation of these two contributions is straightforward, we get :

with Al-N

Cl

= --i?r(l

-

UDGo)H,,Q,H,,{

1 + GZ,,[$’

-

inQ,]Heo] -I

U" = H,,$'H,,( 1 + GcHoc~~~H,o)-'.

(Ma ) (13b)

Contributions to the potential UD come both from the continuum states of X, and from those eigenstates u,, which have been excluded from the sum in Eq. (6) and have been considered as giving rise to a slowly varying energy dependence in the average of the scattering amplitude. IV.

J)ECAY

LAW

OF

THE

COMPOUND

NTiCLEAR

STATES

The decay law of a compound nuclear state u cll camrot be obtained in complete generality, as is usual in all theories of unstable particles. As is customary, we will make the hypothesis that the coupling, in this case H,, ,4 is weak. Mathematically, this amounts to perform a double limiting process, i.e., t -+ + oc and H,, -+ 0, with the requirement that the product of quadratic terms in H,, bimes 4Hoc is

related

to

H,,

through

a kind

of time

reversal

invariance.

406

FONDA

t remains finite. The approximation involved is such that it makes natural the choice of the energy E, appearing explicitly in uen(E), as E zz Re A,,(E) . Besides, u,,(E) clearly approaches a bound state of the Hamiltonian H, , a state which we must now assume to exist. If at t = 0 the state is ucn(E), the probability amplitude that at the time t we have still the same state is given by : m?m = (u,,(E),

P,e-ixtPcuc,(E).

(14)

The evaluation of PCe--zH1P, is best made by means of the complete resolvent 6(x) = [z - HI-‘. The various projections of 6(x) are given as follows (see ref. I, Eq. (2.24)) :

where C&(x)

CL(~)

= Sc(% 2)

CL(z)

= sc(z, x)HcoGo(~)

@L(z)

= Go(z)Ho&c(z,

&(z)

= G,(z)

= P,Cd((z)P,,

= ki

(15)

2)

+ Go(z)H&c(z,

x)H,oGo(x),

etc. For mnnn we therefore

/

r

dz(u,,(E),

where r is an anticlockwise contour threshold cut of a(x). By using the operator identity

which

s&,

have

~)ucn(E>)e-iZt

encloses the poles and the normal

sc(z,z) = sc(z, E) + sc(x,z)[Wc(z) -

Wc(Wlsc(z, E)

(16)

where W,(E) = H,,G,(E)H,,, , we get an expression for 311,, consisting of two terms, the second of which being linear in W, and therefore quadratic in H,, . According to our hypothesis, this term shall vanish in the considered limiting process, so that for t large we have:

mn,, - 2ai A- L &(u,,(E),

Sc(z, E)um(E)

)e-i”t

(17)

The contour r has been chosen in such a way as to include all the poles of $(x, E), otherwise other terms would appear in Eq. (17). It is easily seen that the right hand side of Eq. (17) can be expressed in terms of eUiXc’E)t; for t large we therefore have (ucn(~), e--i&tE)tuen(E)) = e--iReA,(E)te-lI”A”‘E’I~ (18) mm Equation

(18) is the requested

decay law for the compound

nuclear

state

OiX

A FORMAL

THEORY

OF

RESONANCE

407

RE.4CTIONS

in the considered approximation. The lifetime of this state is therefore given by half the inverse of the distance of the pole x = A,(E) of the resolvent $(z, h’) from the real axis. The “energy” of the state is Re A,(E) z E. Let us now evaluate the transition amplit,ude from the compound nuclear state Us% to a generic continuum state +,(E’). In this case we must use C4J,,(x). Also, using again the operator identity (16) and neglecting thereafter the term quadratic in H, ,5 we have ‘3?L,(E’)

= (&(E’),

M &s,,

PoePP,u,,(E))

dz(c#~o(E’), Go(z) H,,$(x,

E)u,,(E))e-“’

= (&(E’), Hocucn (E)) a IT (2 - *g: Disregarding, in the limit as t --+ + 00, the exponential the pole x = A,(E), we have: lim ei”‘%Xon(E’)

which exhibits, as a function Re A,,(E) rz E.

ME’),

Z

t-+rJ

E’ - Re A,(E)

of E’, the familiar

*4,(E))

decreasing

residue at

Hocucn(E)) + 2’1Im A,(E) j

(19)

Weisskopf-Wigner

shape around

APPENDIX

Let us consider the case of two channels, 1 and 2, with separable interactions and choose nonrelativistic kinematics. The projection operators PI = I’,, and I’2 = P2, can be written in matrix notation as

(t“0) and c:)1

respectively. Introducing the matrices and

P2, =

we write:

H=K+V Kij

=

6ijPii

Vij

=

-qiqjPij

5 The approximation

/

d3p

SJ

(a

+

$J

1 p>(p

1

(A.1)

d3P d3P’.fifi(P).fj(P’) I P>(P’ I

used here is different from the one made in the evaluation

of ‘3En,,.

408

FONDA

) p) is an eigenstate of relative momentum p. Ei and pi are the threshold energy and the reduced mass for channel i. fi(p) is a real function. From the hermiticity of H there follows that either g1 and g2 are both real or they are both purely imaginary. Note that gi2 positive or negative corresponds to attractive or repulsive Vii, respectively. We have therefore either Vn and V2, both repulsive or both attractive. The proper values of H are the solutions of the equation h(E)

= 1 - 44-Q

(A.21

with

h(E) = -!$ gi21 d3pE

fi2CP)

+ is - Ei - (p2/2pi)

(A.3)

We are now interested in seeking solutions of (A.2) in the energy interval El < E < E2 when the corresponding problem with g1 = 0 does not exhibit normalizable eigenstates for E < E, ; that is, when there are no bound states of channel 2 uncoupled from channel 1. We choose therefore the case of VI1 and Vt, repulsive (gl’ and gt2, negative). It is immediately seenthat, due to the great degree of freedom left to the parameters appearing in (A.2), we can certainly make a bound state embedded in the continuum to appear at the energy E = E, falling between the two thresholds. The cutoff function fi(p) must be chosen such that it vanishes at E, and such that 41(E) changes its sign in the interval El < E < Ez . We have then only to play on the values of g1 and g2 to obtain the desired normalizable state. Note the trivial fact that the result is independent of the sign of glg2and that there are no normalizable states ----

FIG. 1. The solid line represents the function +, (IS). The dashed line represents the funcby the vanishing of the cutoff tion 1 - c$~(E). The intersection at E = E, , accompanied function f, (p) gives rise to a hound state embedded in the continuum.

ON

A

FORMAL

THEORY

Olf

RESONANCE

409

REACTIONS

with energy E < E1 . The situation is illustrated in Fig. l.‘j By varying properly the parameters we can even obtain more than one bound state degenerate wit’h the continuum in the indicated energy interval. This example clearly shows that one can reproduce a situation in which, even though channel 2 uncoupled from channel 1 does not exhibit any bound states for E < E2, a bound state for the total Hamiltonian can appear embedded in the continuum in the energy interval El < E < E2 .7 REFERENCES 1. L. 2. H. 3. H. 4. A. 5. H.

FONDA AND FESHBACH, FESHBACH, AGODI AND FESHBACH,~.

IL G. NEWTON, Ann. Phys. (N. Y.) 10, 4!10 (1960). Ann. Phys. (il;. Y.) 6, 357 (1958). Ann. Phys. (A-. Y.) 19, 287 (1962). E. EBERLE, Nuovo Cimento 18, 718 (1960). E. PORTER, ANDY. F. WEISSKOPF, Phys.Rev.96,448

6 The behavior of I&(E) has been Rimini, Phys. Rev. 133, 196 (1964). 7 The proof given here generalizes

given that

in general given

by L. Fonda,

in Section

V of ref.

(1954). G. C. Ghirardi, 1.

and il.