On the theory of nuclear reactions

2.A.2

I

Nuclear Physics 75 (1966) 189--208; (~) North-Holland Publishing Co., Amsterdam

i

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

ON THE THEORY

OF NUCLEAR REACTIONS

(I). Shell-model calculations in the continuum HANS A. WEIDENM~LLER Institut fiir theoretische Physik, Universitiit Heidelberg, Heidelberg, W. Germany Received 26 July 1965

Abstract: By a generalization of a method due to Fang, a theory of nuclear reactions is developed which applies to the elastic and inelastic scattering of nucleons. The method determines the positions and the total and partial widths of the resonances directly in terms of the eigenvahies and properties of a complex symmetric matrix. This matrix is the generalization of the usual shell-model matrix to resonant states. The formalism incorporates the exclusion principle exactly and without complication of the theory. Particular attention is paid to the description of threshold effects and to the possibility of obtaining simultaneously the eigenvalues of bound states and of resonant states. Single-particle resonances are easily incorporated into the formalism without our having to introduce projection operators. The S-matrix is decomposed into a slowly-varying background term plus pole terms. For the latter a centre-of-gravity theorem is proved. A relationship is established which connects the sum over all the poles of the partial widths for a given channel with the total coupling strength of all the states with the continuum (corresponding to that channel).

1. Introduction T h e interest in giant r e s o n a n c e a n d i n t e r m e d i a t e structure p h e n o m e n a has strongly increased recently. This is due b o t h to the e x p e r i m e n t a l discovery o f the i s o b a r i c analogue resonances 1) a n d to the p r o p o s a l b y F e s h b a c h a n d his c o l l a b o r a t o r s 2) t h a t i n t e r m e d i a t e structure m i g h t exist in nuclear excitation functions. Such i n t e r m e d i a t e structure w o u l d be c a u s e d b y the existence o f d o o r w a y states a) which are believed to p l a y a vital role in the f o r m a t i o n a n d decay o f the c o m p o u n d nucleus. I n o r d e r to f o r m u l a t e a t h e o r y o f these p h e n o m e n a w h i c h elucidates the f o r m a l i m p l i c a t i o n s o f the c o n c e p t o f a " m i c r o - g i a n t " 1) o r d o o r w a y resonance a n d which at the same time m a y serve as a basis for actual calculations, we present in this p a p e r a t h e o r y o f nuclear reactions which is b a s e d u p o n the a s s u m p t i o n t h a t the shell-model yields a g o o d d e s c r i p t i o n n o t only o f low-lying b o u n d states, b u t also o f low-energy nucleonnucleus elastic a n d inelastic scattering. O u r starting p o i n t is thus a shell-model d e s c r i p t i o n o f the nuclear m o t i o n , i.e. a s e p a r a t i o n o f the nuclear H a m i l t o n i a n H into a single-particle H a m i l t o n i a n H o a n d a r e s i d u a l i n t e r a c t i o n V. I n c o n t r a s t to the u s u a l shell-model calculations, where the single-particle p o t e n t i a l s are for reasons o f simplicity often c h o s e n to be infinitely deep, we s t a r t o u t f r o m m o r e realistic potentials, like for e x a m p l e a S a x o n - W o o d s well. W e d o this in o r d e r to o b t a i n a set o f eigenfunetions o f H o which have the p r o p e r 189

190

H. A. WEIDENMI3LLER

asymptotic behaviour from the outset. We then proceed in the spirit of the usual shellmodel calculations, i.e. we select according to physical principles a subset of eigenfunctions of Ho, and we diagonalize H exactly in the space spanned by this subset. The only complication which arises in this procedure is the presence of one or several continuous sets of eigenfunctions of Ho, due to the fact that the single-particle potentials have a finite depth, so that scattering functions occur as solutions of Schr6dinger's equation. We treat this problem by means of a method proposed by Fano 3). We generalize this method and show that it is possible to deal with the continuum problem in a way which is completely analogous to the usual shell-model procedure. Whereas that procedure ultimately amounts to the diagonalization of several real and symmetric matrices of finite dimension the eigenvalues and eigenvectors of which approximate the positions and wave functions of the nuclear levels, the method employed here leads to the diagonalization of complex symmetric matrices. The eigenvalues give the poles of the S-matrix and thus the positions and total widths of the nuclear levels. The partial widths can also be obtained directly from the diagonalization procedure. In this way, we directly determine all the relevant matrix elements (partial widths) instead of calculating the wave functions. Naturally, the physical ideas and formal procedures presented here are related to work published by Feshbach 2,4), MacDonald 5), Brenig 6) and others. We believe, however, that our approach, while much less general than that of Feshbach 4) is more suitable to the description of elastic, inelastic and charge-exchange nucleon scattering processes because it is much simpler. This simplification and clarification is mainly due to the fact that in the present formulation the exact incorporation of the Pauli exclusion principle, one of the formidable difficulties encountered in nuclear reaction theories, presents no problem at all because we do not try to derive an equation of motion for the last particle. The basic physical idea of this paper is the same as that used by MacDonald 5), but the formalism employed here is different from that of ref. 5). The essential assumptions used in this paper are twofold. Firstly, we restrict ourselves to elastic, inelastic and charge-exchange nucleon scattering. The difficulties which arise if one tries to give an appropriate description of re-arrangement processes are due essentially to the fact that the set of independent-particle shell-model states is not a good basis for a description of such processes. We also omit states with more than one nucleon in the continuum. Secondly, we assume that the residual interaction V is a sum of one- and two-body operators. This critical assumption is necessary for the concept of doorway states to be useful at all 2). No assumptions about the existence of a well-defined nuclear radius have to be made, however. The formalism developed in this paper appears suitable for a variety of applications. First of all, we may try to calculate the elastic and inelastic nucleon scattering cross sections, particularly for the light nuclei, directly in terms of a shell-model approach. Our procedure seems to greatly simplify previous calculations of this type (see sect. 7). Secondly, the level shifts found experimentally (the most famous example is the difference in excitation energy of the first excited states in ~3C and 13N, res-

NUCLEAR REACTIONS

191

pectively) are calculable directly in terms of nuclear matrix elements. Thirdly, the description of doorway states and the analysis of giant-resonance phenomena is very simple as we shall show in a subsequent paper. It is well-known 7) how one deals with the latter phenomenon in the R-matrix theory of nuclear reactions 8); this approach can be similarly applied to the Kapur-Peierls formulation 9) and to Feshbach's theory 4) of nuclear reactions 10). In the present paper, emphasis is put upon the microscopic properties of the S-matrix. This will allow us later to derive exact formulae from which sum rules can be extracted and to give a criterion for the applicability of the doorway state concept. Finally, the mechanism of the isobaric analogue resonances also seems to be understandable in terms of the formulation presented here.

2. The Basic Equations As mentioned in sect. 1, we start out from a shell model. The physical picture thus employed for an understanding of nuclear resonance reactions is that of a "doorway state" process 2). The formulation presented in this and the following sec/tions is, in particular, applicable to the light nuclei, where for excitation energies up ~o 10 or 20 MeV probably all states are "doorway" states, and no "more complicated" states (3 particle- 2 hole states in the terminology of Feshbach) exist 2).

(a)

(b)

(c)

Fig. 1. Various eigenstates of the shell-model Hamiltonian H0, depicted schematically. (a) the incoming particle and the nucleus in its ground state, (b) a "doorway" state and (c) one of the "'complicated" modes of excitation. The quantity e is the fermi energy and full (open) circles denote particles (holes).

We first consider those eigenfunctions of Ho in the space of which we want to diagonalize the complete Hamiltonian H = Ho + V. Initially, the target nucleus is in its ground state with respect to Ho, and one nucleon is scattered elastically by the shell-model central (and spin-orbit) potential. The completely antisymmetric eigenfunction of Ho corresponding to this situation, which is shown schematically in fig. la, is denoted by ~a). Here, E serves as a continuous parameter which relates to the energy of the incident particle, so that H o ~ z) = E~b~ex~ and 2 stands for the other quantum numbers needed to specify the state completely. In the following, we confine ourselves for simplicity to the case of purely elastic scattering. We discuss inelastic scattering in sect. 6. Also, we disregard the complications which arise from angular momentum coupling. The functions ~ ) therefore describe one nucleon of fixed angular momentum scattered by the shell-model potential. The exchange of

192

H. A. WEIDENMfOLLER

angular momentum between the incident nucleon and the target nucleus could have easily been incorporated into the formalism presented here, but is omitted because it only complicates the issue. In view of these siniplifications we may omit - except in sect. 6 - the index 2 and simply write @E. We assume that ~z is normalized according to the equation (~kel~ke,) = 6(E-E'). All the radial wave functions occurring in ~k~ can be chosen to be real. In fig. lb we show a state which is bound with respect to H0 and in which the incoming nucleon has exchanged part of its energy with one of the nucleons of the target nucleus. This state differs from the state ~kE in that the occupation numbers of no more than four single-particle states have changed. All states of this type form a finite set, and the completely antisymmetric normalized wave function of any of these states (a doorway state in the terminology of Feshbach 2)) is denoted by ~b~°), so that we have Ho~b~°) = E~t°)~b}°). For the situation shown in fig. lb, E~(°) lies within the continuous spectrum E of Ho defined above. The state vectors ~b}°) are orthogonal upon each other as well as upon all the functions @E. Besides the states ~b~°), there are also more complicated modes of excitation, also bound states with respect to Ho, one of which is shown in fig. lc. These states are those bound eigenstates of Ho which differ from ~kr in the occupation numbers of more than four single-particle states. The completely antisymmetric orthonormalized wave functions of these states, which are orthogonal both upon the functions ~ke and the functions ~b}°), are denoted by s o that Ho(k}1) = E~(~)q~}1). E=O

o oo

E~

F--<>~ × ~ - ' < - ~ - ~ - ~ - E

P

Fig. 2. Schematic plot o f the spectrum o f H o for low excitation energies. The symbols are explained in the text.

A part of the spectrum Ho is shown in fig. 2. We plot the energy E, normalized in such a way that E = 0 at the threshold for elastic scattering. For E < 0, there occur some bound states of Ho, indicated by squares. For E > 0, we have a continuous spectrum, embedded into which there are bound states, namely the bound doorway states q~o), denoted by crosses, and the more complicated states ~b~t), represented by circles. The average density of the states qb}1) is very much larger (smaller) than that of the states q~}O)for very heavy (light) nuclei 2). At some energy/~ above threshold, the first channel for inelastic nucleon scattering opens. There are many more such thresholds which, for clarity, are not indicated in the figure. As we turn on the residual interaction V, all the states q~}~)(~ = 0, 1) are coupled to the continuum of states ~kr through the matrix elements of V. The states q~}') with energies E~(') > 0 thus cease to be bound states and give rise to resonances. If V consists only of one- and two-body interaction terms, the matrix elements (~krl VIq~1)) vanish. The population of the complicated states @}i) is possible only through the states q~}o) as intermediaries, and the doorway picture emerges.

193

NUCLEAR REACTIONS

Thus, the basic set of equations which we want to investigate is the following. The normalization conditions for the functions CE, ~0), ~ 1 ) are <~'~1¢~,> =

a(E-e'), i = 1 . . . . . M~

(~l~bl°)> : 0,
(~ik

-- 0,

i and k = 1 . . . . . M, i = 1,...,M,k

= 1 . . . . . N,

i and k = 1 , . . . , N, <¢,~1¢(~')> = 0,

k = t .....

N.

(2.1)

The dynamical relations are <~kEIHI~kE'> = VEE' <~kElHl~b~°)> = V~E,

i = 1 . . . . . M,

<¢~°)1H1¢(~°)> = e}°)6,k+( 1 - 6 , k ) ~ k ,

i and k - 1 . . . . , M,

<¢~°!1H1¢~1~ > = V,~,

i -- 1, . . . ,

<¢}l)lnl~(k')> = M,k, <~lnl¢~)>

= 0,

M, k = 1.....

N,

i and k = 1 . . . . . N, k = 1.....

N,

(2.2)

I f H is time-reversal invariant, all the quantities VEE,, Vie, Wik, V~k and M a occurring in eq. (2.2) can be chosen to be real. Our p r o b l e m is to diagonalize the H a m i l t o n i a n matrix (2.2). This can be accomplished by a m e t h o d due to F a n o a), if we introduce the following assumption. We replace the first of the eqs. (2.2) by the relation (~EIH[~g,> = E r ( E - E ' ) . Since we are here mainly concerned with the description of resonance processes, o u r a s s u m p t i o n a m o u n t s to saying that the residual interactions, if taken in the space o f functions ~g alone, do not give rise to resonances. We could circumvent this a s s u m p t i o n by introducing linear combinations ~e = ~ dE'CE(E')~e' of the functions ~ke in such a way that <~EIHI3E.> = E r ( E - E ' ) holds exactly. F o r the p u r p o s e o f the present paper, we introduce the relation <~&glnl~&E,> =

Er(E--E')

(2.3)

as an additional assumption. We n o w turn to F a n o ' s treatment 3) in a particularly simple case.

3. Reminder of Fano's a) Method In order to give an outline of the m e t h o d to be employed, we simplify the eqs. (2.1)-(2.3) to the case of one d o o r w a y level ~b e m b e d d e d in the continuum, so that

194

H . A . WEIDENblULLER

(~/'EI~'E') = 6(E--E'),

E and E' > 0

= 0, =

(3.1)

1,


lnlV, r > --

(~bElnl4') = VE real, <4'lnlq~> = 8 > 0.

(3.2)

The model scattering phase shift ~z for the elastic scattering due to Ho with angular momentum l is defined in terms of the asymptotic behaviour of @n. Let the radial coordinate r of one of the protons (if we consider elastic proton scattering) or one of the neutrons (if we consider elastic neutron scattering) tend towards infinity. Then, assuming that we deal with a screened Coulomb potential, we have On -+ 1 Q sin (kr-½1~z+~,),

(3.3)

r-+oo r

where Q contains the wave function of the target nucleus, a normalization factor, and the angular wave function of the last particle. The assumption (2.3) implies that the model scattering phase shift 6z introduced by eq. (3.3) is equal to the "potential scattering" phase shift of the exact problem, or that the residual interactions contribute negligibly little to the "potential scattering". In the space of functions ~,e` and 4' the eigenfunction ~/'r of the total Hamiltonian H is given by

~'E = I, / de' av(E')~l,~, + be`4',

(3.4)

it obeys the equation =

(3.5)

Using eqs. (3.1), (3.2) and (3.4), multiplying eq. (3.5) from the left with (~'E,I and with (4'1, we obtain for the coefficients ar.(E') and be the coupled system of equations,

( E - E')ar.(E') = be. Vv,

(3.6)

P

( E - e)be` = j dE' ae`(E') Vn,. The first of these can be solved to give

E --E' + z(e)f(E-- E') ,

(3.7)

where P stands for principal value and z(E) is as yet an unknown function. Insertion of eq. (3.7) into the second of eqs. (3.6) yields a homogeneous linear equation for be,

(e-e)be` = (F(E)+ z(E)V2)b~,,

(3.8)

NUCLEAR REACTIONS

195

where we have introduced

F(E) = P f d E '

V~, .

(3.9)

1 (E-e-F(E)).

(3.10)

d

E-E'

Eq. (3.8) has a non-trivial solution only if z(e) =

Insertion of eqs. (3.7) and (3.10) into eq. (3.4) shows that ~ e has an asymptotic behaviour similar to eq. (3.3), namely that ~ e ~ 1 be exp (iA,)QVe(z(E)+ ir0 sin (kr-½1rc+6t+A,), /-

SO

(3.11)

that the scattering function S is given by (we drop the index l) S = e 2i(~+a).

(3.12)

1- i --2re

(3.13a)

Explicit calculations shows e 21LI =

z(E) + in or

e 2/~ = 1 - i

2~zV2

E-e-F(E)+

(3.13b)

inV2"

Let us discuss these formulae first for the simplest case, where VE and e 2~* can be approximated by constants in the energy region of interest, E ,~ ~. Then, F(E) ,,~ O, and we obtain a resonance at E --- e with width F = 27rV2. We may also express this result by saying that the S-function e 21(~+A) has acquired a pole by the coupling of the state ~b to the continuum ~kE, and that this pole occurs at the complex energy E = e - i r c V 2 with a residue (-e2i~2rfiV2). I f V2 cannot be considered a constant, then eq. (3.13) shows that resonances (poles of S) occur wherever

z(E) = - in.

(3.14)

Inserting this condition into eq. (3.10), we obtain

E = e+F(E)--irrV 2.

(3.15)

The complex solutions of eq. (3.14) or (3.15) define the positions of the poles of the S function. I f both F(E) and Vg are slowly energy-dependent functions, then F(E) gives the level shift, and 2n Vg the level width. The positions and widths of the poles of S can be calculated directly. The generalization of this statement and the derivation of equations corresponding to eqs. (3.14) and (3.15) in more complicated situations is the main aim of this paper. In the next section, we consider explicitly first the case where two bound states are coupled to the continuum, and then the case where only one bound state is coupled to the continuum and where V2 cannot be approxi-

196

H.A.

WEIDENMULLER

mated by a constant, but instead exhibits a single-particle resonance. This situation occurs in practical calculations, when we are sufficiently close to the nucleon threshold or when we consider particles of high,angular m o m e n t u m or protons impinging upon target nuclei of high Z. The results will be used in a subsequent paper on isobaric analogue states. In sect. 5, we consider the case where we are so close to a thresold that the threshold behaviour of V2 is important for the calculation. We show how one can develop a generalized shell-model procedure which treats the truely bound states and the resonances on exactly the same footing: the energies of both are obtained as the solutions of a single equation. In sect. 6, we describe briefly how the method presented here can be generalized to the case of more than one continuum. It is not surprising that eq. (3.14) should determine the poles of the S-function. It can be seen from eqs. (3.7) and (3.4) that for z(E) = - in the function ~ has asymptotically an outgoing wave only. This again suggests the well-known 1~, 12) connection between Gamow-functions and poles of S.

4. Treatment of Single-Particle Resonances In this section we describe how the simple treatment of the last section has to be modified if we may not assume V~ = constant, but if instead the single-particle scattering solutions ~bE and the potential scattering function e 2~ exhibit a single-particle resonance. In preparation of this treatment we sketch first the treatment of two bound doorway states, q51 and q62, coupled to the continuum ~kE. We use eqs. (2.1)-(2.3) and the method described in sect. 3. Writing the total wave function ~E in the form 2

~r. =

dE' aE(E')OE'+ E b,r.~,,

(4.1)

i=1

we obtain in analogy to eq. (3.8) the equation 2

(E-- e,)b,(E) = E {(Fik(E) + z(E)ViE VkE)bkE+ W~kbke'},

i = I, 2.

(4.2)

k=l

The matrix Wik in eqs. (4.2) is defined to have only one non-vanishing element, W12 -- W21, and the matrix Fig is defined in analogy to eq. (3.9) by

F,k(E)

f. EE, V,E, = PIdE' d E--E'

(4.3)

The scattering function is again given by eqs. (3.12) and (3.13a). The function z(E) is determined by the requirement that eqs. (4.2) have a nontrivial solution. We define the determinant

D(z, E) = E-et-Fxl(E)-z(E)V2r. -- W,2-F,2-z(E)V1EVz~

--W12--F12--z(E)V1EV2E E--e2--F22(E)--z(E)V2~

.

(4.4a)

NUCLEAR REACTIONS

197

Then, the equation

D(z, E) = 0

(4.4b)

defines z(E) in such a way that eqs. (4.2) have a solution for real energies. In analogy to eq. (3.14), the poles of the S-function are determined by the equation

D(E) = E-et-FI'(E)+ircV2E -W12-F12q-inVIEV2E

- W12--F12"~iTtV1EV2E E-e2-Fz2(E)q-iTtV~E

= 0,

(4.4)

which for constant matrix elements V1E, V2~ is a quadratic equation for the positions of the two poles. The residues can be calculated as follows. It is clear from eq. (4.4a) that D(z, E) is a linear function of z(E),

D(z, E) = A(E) + z(E)B(E).

(4.5)

Since from eq. (4.4) D ( - i ~ , E) = D(E), we have 1

_

1

z(E) + iTz

~D(z,E) = _ ~

D(E)

dz

B(E).

(4.6)

D(E)

If Vie and V2E are constant, then B(E) is a linear function of energy. Eq. (4.4) has two complex roots #1 and #2 so that D(E) = ( E - # I ) ( E - # 2 ) , and we get finally with - B ( E ) = ctE-fl from eq. (4.6) 1

z(E)q- iTz

~i

- -

+

E -I11

52 . - -

(4.7)

E--/d 2

where gl and ct2 are defined by the two equations gl + ctz = ct, cq# 2 + ct2#1 = ft. Eqs. (3.12), (3.134) and (4.7) give the complete decomposition of the S-function. We have two poles, and a unitary S-function. Clearly, this procedure including eq. (4.7) can easily be generalized to the case where M states qS~°) are coupled to the continuum, provided that the ViE, i = 1. . . . . M can be considered constant. Let us now come back to the case where we have just one bound state q~ coupled to the continuum, but where V2 cannot be approximated by a constant. We assume instead that the single-nucleon phase shift ~ defined by eq. (3.3) displays a singleparticle resonance, and that it is possible to approximate the model scattering by a resonance plus a slowly varying background term, which is approximately constant within the energy region of interest. We write e2i~ = e2,~ E - (E~)*,

(4.8)

E_E • where E ~ = Re(E~')-½iF gives the position and width of the resonance, and ~ is a constant. (We assume that we are sufficiently far away from the threshold so that we may neglect the threshold behaviour, see sect. 5.) In order to discuss the analytical behaviour of the real function l/E, we recall that the scattering functions fin

198

H. A, WEIDENMfJLLER

defined above are related to the functions ~k~+ ) or ~k~e -) usually employed in scattering theory through the relations (valid for s-waves) ~/2-~i~kr = -- e-'O~k~+) = e + iaff(g-).

(4.9)

Consequently, we may write VE = i/~/2-~ e-~n(q~]VI~k~+)). The function ~k~e+) can be continued analytically. It has a pole at E = E', but no singularity at E = (E0*. The real function Vr can therefore also be continued analytically, if the product Vl~b) falls off more strongly at infinity than the scattering function ~k(~+) diverges z a). (The function ~O~+) increases exponentially if continued analytically to complex wave numbers k, ~b~+) ,~ exp[Im k[r for r ~ oo, where k 2 ,,- E). In practical cases, this is no serious difficulty z a), at least for narrow single-particle resonances, in view of the binding energies occurring in q~. We may therefore write (R and T are two complex constants) Vr = e - " \(TE-E'--~iFR + R) e'¢.

(4.10a)

Using the reality of V~ = V* and eq. (4.8), we obtain from eq, (4.10a) the conditions that T and R must be real. Therefore, V~ has the form V2 = ( T - Re E ' R + ER) 2

(4.10b)

iE-e'l ~

Eq. (4.10b) defines V~ as an energy-dependent quantity, the description of which requires four constants, T, R, Re(E ") and F. In practical cases, it is not difficult to compute these constants. We may now calculate F(E) from eq. (4.10b) and obtain for [ImE'[ << ReE ~ r(E)-i~zV2 =

T-iu

vr 2Fg

R] 2 1 ] E-E ~

iuR z.

(4.11)

We insert this result into eq. (3.15) and obtain (4.12) Eq. (4.12) may be written in the form E-E •

--

-~ T +

r27r

= 0.

(4.13)

E - - e + ircR2

A comparison with eq. (4.4) shows that a single-particle resonance leads to a determinantal eq. (4.13) which has the same form as eq. (4.4) which one obtains for two

NUCLEAR REACTIONS

199

bound states coupled to the continuum. If we formally equate R with V2~., x/F/2~ with Vt~ (this is reasonable in view of eq. (3.13)) and remember E , = ReE~- i~(F/2rc), then eqs. (4.13) and (4.4) agree in form. Since R is a constant, the level shift matrix Fig(E) is expected to be very small, and we see that x/2~/FT ~ W12. This analogy is not surprising because in the complete solution of the scattering problem, there is no way of telling whether the resonances originated from the single-particle resonances or from bound states embedded in the continuum. (This distinction hinges upon our choice of He, and the exact solutions are independent of this choice.) The result contained in eq. (4.13) can be expressed in another way. In the classical shell-model diagonalization procedure, one considers all states as bound. In our example, both the state corresponding to the resonance at E ~ (a single-particle excitation in the shell model) and the state q9 would have been used for the diagonalization. The result would have been an equation of the type (4.13), with IIIIE = --- 0, and R = 0. In this context, we note that limr-.o x/(2zc/F)T is finite. The limit can, for example, be taken by adding to the single-particle Hamiltonian a repulsive barrier, the height of which tends towards infinity. Then the single-particle resonance becomes a bound singleparticle state, and limr~o x/(2zc/F)Tequals the matrix element W12 between this state and the bound doorway state ~b. This follows from eq. (IV, 27b) of ref. 12), from our eqs. (4.9) and (4.10a) and from the reality of T. This statement is of great practical importance in that it shows how shell-model calculations can easily be modified to yield the positions and the widths of the nuclear levels. The quantity x/(2n/F)T can be taken directly from the usual calculations; the single-particle width F can be determined from the single-particle Hamiltonian, and the constant R can be obtained by calculating the matrix element VE for a value of E which obeys the condition [E-ReE~I >> F. It is now easy to show that we can decompose the scattering function in the form (4.7). Also, one can immediately see that S = exp(2i6 + 2iA) no longer has a pole at E ~ or zero at (E~) *. We show that the method which leads to eq. (4.13) is not restricted to only one bound state embedded in the continuum. The method applied in sect. 3 leads for the problem formulated in eqs. (2.1)-(2.3) to the following determinantal equation, analogous to eq. (4.4):

(E--e~°))ftk--F,k+ irCVIEVkE--Wtk(1--rlk) -- Vim

-

-

Vkn

= O.

(4.14)

Ec~mn -- M m n

In this equation, the matrix indices l and k go from 1 to M, and the matrix indices m and n from M + 1 to M+N, in accord with eqs. (2.1) and (2.2). The matrix Fik is defined in analogy to eq. (4.3). If a single-particle resonance dominates the scattering, we are again led to eq. (4.10b) for the matrix elements V~Ewhich we now write in the form Vie Vke = (T~ -- Re E~R,+ ER,)(Tk -- Re E~Rk+ ERk) (4.15)

200

n. A.

WEIDENM/.)LLER

with real constants R~, T~. This leads correspondingly to the expression analogous to eq. (4.11),

- - E I E ~ .IVan--

in V ~ R I ) ( V - ~

Tk--izc

V ~-R k )

-izcR, Rk.

(4.16)

We insert this expression into eq. (4.14). The resulting determinant can be seen to be proportional to the following one, which differs from the determinant of eq. (4.14) in that one row and one column have been added (this row and column are written down explicitly):

iv iRk

--

0

- f Tk

(E--e~°))6,k - Wtk(1--6,k)+ izcRtRk

-- Vk.

-- Vl~

E6m.- Mm,

__0

(4.17)

Indeed, by multiplying the first row of this determinant by (E-E*) -1 and by subsequenfly adding the first row to the M following rows in such a way that the first elements in all these rows vanish, one obtains eq. (4.14). Again, eq. (4.17) is the generalization of a shell-model determinant. 5. Treatment of Thresholds In this section we wish to demonstrate that the treatment presented heretofore is not restricted to resonances with sufficiently narrow widths, but can be extended to yield a simultaneous treatment of bound states (below the first nucleon threshold) and resonances. We again confine ourselves to one continuum, and assume that we have two bound doorway states at e~°) and e~o), with e~°) < 0 and e~°) > 0, where E = 0 is the continuum threshold. We first calculate with the wave function (4. I) for E > 0 the determining equation for the resonance. This turns out to be eq. (4.4). We now discuss the solutions of eq. (4.4) by analytical continuation of Fjk(E) and V iEVkE (j, k = 1 or 2) to complex energies. This is necessary because e~°) < 0 means that we have to continue both functions analytically to negative values of E. By the same token, we may no longer assume F~k(E) and V~VkE to be equal to zero or constant, but rather have to take into account the threshold behaviour of Vm. Again, we assume that Vm can be continued analytically sufficiently far to render our discussion meaningful. We define the functions

G}I)(E) = lim /o odE' VJI~'VkE" ~-*+o E+ia--E'

--

Fjk--i~VjEVkE.

(5.1)

NUCLEAR REACTIONS

201

These functions are - under suitable conditions on Vjr and Vkr` -- defined for all complex values of E, and have a cut along the positive real E-axis. The discontinuity o f G).~)(E) along the cut is given by lim

{ G~lk~(E+ itr)- G~)(E- itr)} = - 2i~VjE VkE

(5.2)

cry+0

for E > 0. Eq. (5.2) means that the function G~)(E+itr), continued analytically across the positive teal E-axis into the next sheet where I m E < 0, differs from G~)(E-itr) by the additive function, -21ziVjeVkn. The function obtained by continuing G~)(E) analytically by crossing the real positive E-axis in a clockwise motion is called G~Z)(E). Then, from eq. (5.2) jkk

(5.3)

/~-

Now, we continue G~2) in the same manner, by moving clockwise around the origin. After having crossed the positive real axis again, we enter a third sheet, with a corresponding function G~3)(E). Again, we have

G~3k)(E) = G~2)(E)- 2inVje Vk~.

(5.4)

E ~ O, VfE is proportional to

Now we use the fact that for

V2~ ~ (E) '+~

for all j.

(5.5)

This follows if we assume that ~bE is the scattering function of a particle with angular momentum l in the combined nuclear and Coulomb potential, provided that we use screening for the Coulomb field. Using eq. (5.5) in eqs. (5.4) and (5.3) we obtain G(3)twa j k 1,."t''/

~

~(1) , '~'Jjk

(5.6)

which shows that "jk~(1)and "-'jkr-(2)are the two branches of a two-valued function Gjk(E). The two-valuedness of this function corresponds to the branch-point at E = 0 introduced through eqs. (5.2) and (5.5). These arguments show how it is possible to continue the functions Fjk(E)-ircVjr.l/kr. occurring in eq. (4.4) analytically, so that the determination of the roots of eq. (4.4) and the subsequent calculation of the S-function becomes possible. This calculation may proceed along the lines described in sect. 4 if we introduce the channel wave number k instead of the energy E. We chose h = 1 and put the reduced mass of the scattered particle equal ½ so that k 2 = E. We describe briefly the application of the method discussed above to the case where the energy-dependence of the matrix elements ViE is given by k2l + 1

ViE VkE -

1 + ( a k ) 2t+~"

Rj R k = ft(k)Rj Rk,

(5.7)

where the Rfs and a are real constants. The numerator in eq. (5.7) gives the correct threshold behaviour, and the denominator guarantees the convergence of the integrals

202

H . A . WEIDENMf2LLER

defining the functions Fjk(E). The eq. (5.7) may be looked upon as a special case, a n d it can be generalized to include single-particle resonances as discussed in sect. 4 etc. The functionft(k) contained in eq. (5.7) can be written in the form

fl(k)

21+4 1 --Y

~m

,,-z=l akin k - k i n

,

(5.8)

where the km's are the 2 l + 4 roots of the equation (akin) 21+4 = - 1 , a n d a m = [(21+4)a2~+3km] -1. It is then simple to show by contour integration that Fjk(E) is given by _ ~_~2 f jk(E ) = RjR~ _rcitt am aklm=l k - k m

2l+4 ~mm} ~ k_-~k . m=l+3

(5.9)

Here, the zeros km have been ordered according to increasing argument, so that all km with m < I + 2 lie in the upper, all k,n with m > l + 3 in the lower half k plane. Obviously, Fig(E) is even in k. The inclusion of single-particle resonances in this treatment produces the complication that single-particle resonances (or bound states) do not occur symmetrically at k = k ~ and k = - k ~. Indeed, bound states correspond to poles of V~e at positive imaginary values of k, virtual states to poles at negative imaginary values of k, and resonances occur symmetrically both at k = k s and at k = - ( k S ) * for negative values of the imaginary part of k. The asymmetry produced between the upper and lower half k-plane by the behaviour of ViE is thus a consequence of the threshold behaviour, eq. (5.5), and of the possible existence of single-particle poles. This leads to some complications in the description of resonances near threshold as discussed in ref. 14). The attractive feature of the generalization described in this section is that we are now able to treat true bound states and resonances on exactly the same footing. The positions of the bound states and resonances and the widths of the latter are obtained from a single equation, which in turn is a generalized form of the shell-model determinantal equation as was discussed before. The resonances correspond to those solutions of this equation for which I m k < 0, Re k > 0, and the bound states to those solutions for which Re k = 0, I m k > 0. Obviously, the extension of the treatment described here to more than two bound states is straightforward.

6. T w o Continua and Several R e s o n a n c e s

In order to show that the foregoing discussion is not restricted to one continuum, we now give the formulae for two continua. These might for instance correspond to elastic and inelastic scattering, so that one continuum of functions ~k(e1) describes a nucleon scattered by the nucleus in its ground state with regard to Ho, whereas the other continuum of functions ~,(e2) describes a nucleon scattered by the nucleus in an excited state with regard to Ho; or else, ~k(~~) and ~(2) might both refer to elastic seat-

203

NUCLEAR REACTIONS

tering with the target in the ground state, but with the scattered particle in different states of angular momentum etc. In the case of two continua, the basic equations of sect. 2 apply correspondingly. The matrix elements V~ ) now have an upper index 2 = 1, 2 to distinguish between the two continua, and it is possible that some of the functions ~0) have vanishing matrix elements V[~ ~ with the first continuum, but not with the other, and vice versa. At any rate, we define the "complicated" states qb~1) by the condition that (~k~)IHI qtkl1)) = 0 for 2 = 1, 2. In addition to eq. (2.3) which is now supposed to be valid for both continua, we have to make the additional requirement that (~b(nl)lHl~b~2)) = 0

(6.1.)

for the following discussion. A term of the type (6.1) might be included into our treatment by means of a method described by Fang 3). It is to be expected that resonance phenomena of the type discussed here should not be strongly affected by a "direct" term of the form (6.1), but this expectation may fail to hold true for sufficiently strong interactions V contained in H. The two continua are, of course, assumed to be orthogonal, (@(EX)l@(ff))= 0. The treatment of eqs. (2.1)-(2.3) with the method of sect. 3 suffers one complication. Since we deal with two continua, we obtain from eq. (3.6) two equations of the form (3.7), for the two coefficients a~X)(E'), with two functions z(a)(E). Correspondingly, there exist two linearly independent solutions ~E~ to each energy. One of them, ~E1, is defined by the requirement that there is only an outgoing wave in channel 2, so that z(E)(E) = --iTr. The function z(1)(E) is then determined from the requirement that the eqs. corresponding to eq. (3.8) should have a solution. Conversely, ~E2 is the function which has only an outgoing wave in channel 1, so that z(1)(E) = - i n . This then determines z(2)(E) uniquely. The determinantal equation obtained in analogy to eq. (4.4a) can be written in the form 0 = D(z O), z (2), E) ka-,]rlE

=

a

rice-

x -

(6.2)

~m

E6r, n

-- Mmn

From this equation, the condition determining zCX)(E) can be obtained by putting z(2)(E) = - i n , and vice versa. We now confine ourselves to the case where we may put v(a), kg ="~ constant, and thus F[~) = 0. For the treatment of more complicated cases, we refer to sects. 4 and 5. We first discuss the elastic scattering, and confine ourselves to channel 1, so that z(2)(E) = - i n . By interchanging the indices 1 and 2, the following arguments can be made to apply also to the elastic scattering in channel 2. A derivation similar to that in sect. 3 shows that the S-matrix element St ~ is still of the form (3.12), $11 = e21("+a'),

(6.3)

204

H.A.

WEIDENMULLER

where e 2izll =

1- i

2n z~l)(E)+ in"

(6.4)

In contrast to eq. (3.13a), z(1)(E) is no longer real, reflecting the fact that inelastic scattering is possible. The poles of $I a are given by those values E = / ~ for which eq. (6.2) is fulfilled with z (1) = - i n , D(-in,

- i n , E) = 0.

(6.5)

It is remarkable that this condition gives the position of the poles not only of S~1, but also of the other elements $22, S~2 and $21 of the S-matrix. This is obvious for $22 and will be shown later for S~2 and $21. Eq. (6.5) is the equation for the eigenvalues of a complex symmetric matrix. Such a matrix can be diagonalized is) by a complex orthogonal matrix. It follows from the invariance of the trace under such transformations that the complex eigenvalues/~i of eq. (6.5) are connected with the eigenvalues e~o) and e~l) of the shell-model problem through the relation M+N

M

N

2

E #, = Ee} °)+ Ee~ 1 ) - i n E i=1

i=1

i=1

2=1

M

E(V/~)) 2"

(6.6)

/=1

Here, we have used M u = e~a). Eq. (6.6) is a "centre-of-gravity-theorem". The eq. D ( z °), - i n , E) = 0 which determines z¢l)(E) is linear in z(1)(E). Hence, we may write 0 = D(z 0), - in, E) = D( - in, - in, E) + (z 0 ) + ir~)B(X)(E) N+M

= 1-I ( E - P J ) + ( z t l ) + i n ) B ~ ' ( E ) • (6.7) j=l

The quantity B~I)(E) is clearly a polynomial of degree N + M - 1 in E, and has the form M B(1)(E) = --EN+M-1 E (V/(E1)) 2 + . . . . (6.8) /=1

where the dots indicate terms of order N + M - 2 or less in E. It follows from eqs. (6.7) and (6.8) that eq. (6.4) can be written in the form (we assume that the eigenvalues are not degenerate) N+M

~(1)

N+M

e 2'al = 1 _ i t ~ ~ ;__~l__ with /'/t

M

~ a~1) = 2n E (Vt~)) 2. t= 1

(6.9)

l= 1

In other words, the total strength with which the resonances are coupled to the continuum is equal to the strength of the coupling of the shell-model states to the continuum. Eq. (6.9)is the generalization of eq. (4.7), and corresponding decompositions hold for more than two eontinua. It now follows from eqs. (6.6) and (6.9) that M+N

2

-- E I m 2 / h = ~ l=1

2=1

M+N

E a}~), I=1

(6.10)

NUCLEAR REACTIONS

205

i.e. a connection between total and partial widths. It is to be stressed that the relations (6.6), (6.10) and the second of eqs. (6.9) are strictly true only for constant matrix elements V~(~);near thresholds, they cease to be exactly fulfilled. Similar relations have been obtained previously by Brenig 6) and Feshbach 4). We now turn to the inelastic scattering and to the unitarity relations. The matrix elements SIE - Sx-.2 and $2~ = Sz-.~ of the S-matrix are obtained by comparing the amplitude of the incoming wave, normalized to unit flux, with the amplitude of the outgoing wave in the other channel, also normalized to unit flux. Using the asymptotic behaviour as in sect. 3, properly modified for the two-channel problem, and the normalization of the function ~h(E ~) as defined in eq. (2.1), one obtains M

-- 2ui exp

(i51+ i52) E

S12 =

M

uIEh(1)v(I/~ 2)"

l=l

'

I=1 M

--2hi exp (i5 x q- i52) E UlE'-(z)t"(1)" Ie l=l $ 2 1 ~-

M

(z(~)(E) + in)l~=,b~ ) V~2)

(6.11)

The coefficients b~E are defined in analogy to eq. (4.1). The upper index 2 in ~,,r1"(a)refers to the two solutions, 2 = 1 for z(2)(E) = - i n , ). = 2 for z(~)(E) = - i n . In order to evaluate the expressions (6.11), we have to perform a number of transformations. The matrix

= (d°% + \ v~,.

M,..!

(6.12)

occurring in eq. (6.2) is real and symmetric and can therefore be diagonalized by an orthogonal transformation Ost. Let the resulting eigenvalues be 2 i with i -- 1 , . . . , M + N . Then, we can calculate D(z ('), z (z), E) from eq. (6.2). We define the quantities M

V(E~)

Z i=i

M+N

-.(z)

tifi) tT(k)

~=l E - 2 s

(6.13)

and obtain

D(z0), z(2), E) M+N

= [ YI ( E - 2s)]{(A, 1 A22 - Ax2 A12)z°)(E)z(2)(E)- z(1)(e)A,t - z(Z)(e)A22 + 1}. s=l

(6.14) In terms of the quantities defined in eq. (6.13), we may also calculate the expressions ~ - - 1 b~ V[rk) which appear in eq. (6.11) by solving a linear system of equations anal-

206

H.A.

WEIDENMOLLER

ogous to eq. (3.6). We find with det A = A l l A 2 z - A t 2 A t 2

AI2 exp (/h a + ih2)

-2in $12

-

the expressions

in det A + A l l

(z ~1)(E) + in)

(6.15)

--2in A21 exp (ihl +ih2) + in) in det A+A22

s21 =

F r o m eqs. (6.2) and (6.14) we m a y calculate z ~1~(E) and z(2)(E), respectively. We get

Slz = Sza =

-2inA12 exp (ifil +ih2) 1 - n 2 det A + in(Ala +A22) "

(6.16)

The denominator vanishes whenever eq. (6.5) is fulfilled. This shows that the poles of $12 are also determined by eq. (6.5). The matrix S is according to eq. (6.16) symmetrical which is the result required from time-reversal invariance arguments. In the same fashion, we may write down exp ( -- 2i61 ) S

z O ) - in

1 + n 2 det A + i n ( A z 2 - A 11)

z(x)+ in

1 - n z det A + i n ( A a t +A22) '

1a =

exp ( - 2it52 ) $ 2 2

(6.17)

1 +re 2 det A + in(All - A z ~ ) =

1 - n 2 det A + in(A 11 + A25)"

The unitarity of the S-matrix can now easily be checked by direct calculation; the fact that Islxl < 1 follows from I m z ~1) > 0 which is an immediate consequence of eq. (6.17). Writing SH and $22 in the form of eq. (6.9), we obtain finally from eqs. (6.16) and (6.17) the following expressions: N+M ~v ~Sjk

exp (-- i6 i -- i l k )

=

fijk --

i ~ Ftj Ftk, ~=1 E - p c

(6.18a)

where we have used the relations Flp = a[ p) = 2n

(in det A + Amo)E= m O/dE(1 - n z det A + in(All +A22))E=u, ' N+M

M

l"tp = 2n ~ T,,(p)r;(p) tiE r IE , I=1

19 = 1,2

/=1

(6.18b)

in det A + A ~ Flzl

'

i~z det A+A22]E=m

The last two expressions agree because of the validity of eq. (6.5) at E = Pl.

NUCLEAR

REACTIONS

207

We have thus shown that it is possible to generalize the formalism of the preceding sections. The case of more than two channels is more cumbersome, but eventually also leads to expressions of the type (6.18). The inclusion of thresholds and singleparticle resonances can be carried through as in sects. 4 and 5. We conclude by emphasizing that in all cases considered the poles and the residues of the S-matrix are fully determined by the properties of the determinant (6.2) and by the condition (6.5). Apparently, the determinant (6.2) plays in our problem the role of the determinant of the Jost matrix which is pertinent to the theory of many-channel scattering 12,13, ~6). 7. Summary and Conclusions The formalism presented in this paper allows the calculation of nuclear reaction amplitudes for a wide variety of cases, by means of mathematical operations which involve only the diagonalization o f a complex symmetric matrix and techniques of linear algebra. It seems therefore suitable both for a discussion of certain basic features of nuclear reaction amplitudes and for numerical calculations. It is restricted by our fundamental assumption that the single-particle shell model provides a good definition of the set of basic states in the frame of which the Hamiltonian is diagonalized. This assumption implies, in particular, that the centre-of-mass of all the nucleons cannot move freely and thus leads to the well-known spurious states. The present formalism is essentially a continuation of Fano's method 3) to complex energies and an exploitation of the simplifications which arise. The present approach is closely related to the Humblet-Rosenfeld 11) expansion in that it gives a dynamical prescription for the calculation of the positions of the poles of the S-matrix and of the residues within a restricted energy interval. In particular, no channel radii etc. have to be introduced. For the case that can be treated with this method, namely elastic, inelastic and charge-exchange nucleon scattering, our formulation seems to be simpler than the formulation of Feshbach 4) mostly because it is not difficult to satisfy the Pauli exclusion principle. Another feature which seems attractive is the close relationship that exists between our theory and usual shell-model treatments. Such a direct relationship does not exist in Feshbach's formulation because of his introduction of projection operators, which distinguish between open and closed channels. The present formalism has the disadvantage that it relies rather heavily upon the shell model and thus may be more restricted in its applicability to actual nuclear problems. It is furthermore restricted by the fact that our choice of the basic set of states makes it virtually impossible to treat transfer, three-body breakup reactions or processes induced by projectiles other than neutron and proton. Regarding possible applications, we have already mentioned in the introduction the generalization to the continuum of shell-model calculations for the light nuclei. Calculations of that kind have been carried through by Shakin and Lemmer a7) and by Lovas as); we hope that in the present frame such calculations can be made more transparent. If the resulting complex symmetric matrices should turn out to be too

208

H . A . WEIDENMULLER

large, the usual approximation procedures (the random-phase approximation, for example) can also be applied to our treatment. Another approach to this problem, also based upon Fano's method, has recently been proposed by Bloch and Gillet 19). The reader will have noticed that it is possible to generalize the present approach to nuclear models other than the shell model. Thus, within the framework of the alphaparticle model it is, for example, possible to treat elastic and inelastic alpha-particle scattering. Another possible application seems to be the statistical theory of nuclear reactions 20). We have the impression that the analytical determination of statistical properties of the S-matrix may be more feasible in the present framework than in the frame of R-matrix theory 21). Finally, the investigation of the doorway-state phenomenon 2) becomes rather transparent in the present framework, and a paper on this subject is in preparation. The author is indebted to Dr. C. Mahaux for several stimulating discussions. He is grateful to him and to Dr. K. Dietrich for a careful reading of the manuscript. References 1) J. D. Anderson and C. Wong, Phys. Rev. Lett. 7 (1961) 250; J. D. Anderson, C. Wong and J. W. McClure, Phys. Rev. 126 (1962) 2170; J. D. Fox, C. F. Moore and D. Robson, Phys. Rev. Lett. 12 (1964) 198; C. Bloch and J. P. Schiffer, Phys. Lett. 12 (1964) 22 2) B. Block and H. Feshbach, Ann. of Phys. 23 (1963) 47; A. Kerman, L. Rodberg and J. Young, Phys. Rev. Lett. 11 (1963) 422; A. Lande and B. Block, Phys. Rev. Lett. 12 (1964) 334; H. Feshbach, A. Kerman and R. H. Lemmer, Contrib. C 141 Paris Conf., 1964 Paris (1964) p. 693; H. Feshbach, Revs. Mod. Phys. 36 (1964) 1076 3) U. Fano, Phys. Rev. 124 (1961) 1866 4) H. Feshbach, Ann. of Phys. 5 (1958) 357; 19 (1962) 287 5) W. M. MacDonald, Nuclear Physics 54 (1964) 393, 56 (1964) 636 6) W. Brenig, Nuclear Physics 13 (1959) 333 7) A. M. Lane and R. G. Thomas, Revs. Mod. Phys. 30 (1958) 257 8) A. M. Lane, R. G. Thomas and E. P. Wigner, Phys. Rev. 98 (1955) 693 9) G. Brown, Revs. Mod. Phys. 31 (1959) 893 10) H. Feshbach, A. Kerman and R. H. Lemmer, private communication (January 1965) 11 ) J. Humblett and L. Rosenfeld, Nuclear Physics 26 ( 1961 ) 529 12) H. A. Weidenmtiller, Ann. of Phys. 28 (1964) 60 13) W. GlOckle, Ph.D. Thesis, Heidelberg (1965); E. T. Whittacker and G. N. Watson, A course of modern analysis (Cambridge University Press, 1950) chapt. V 14) H. A. Weidenmtiller, Nuclear Physics 69 (1965) 113 15) J. Heading, Matrix theory for physicists (Longman, London, 1958) p. 61 16) R. G. Newton, J. Math. Phys. 2 (1961) 188 17) C. M. Shakin and R. H. Lemmer, Ann. of Phys. 27 (1964) 13 18) I. Lovas, Nuclear physics (in the press) 19) C. Bloch and V. Gillet, Phys. Lett. 16 (1965) 62 20) N. Rosenzweig, Lectures given at the Brandeis Summer School, (Benjamin, New York, 1962) 21) P. A. Moldauer, Phys. Rev. 135 (1964) B642, 136 (1964) B947