Antisymmetrization of a unified nuclear reaction theory

2.A.2

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Nuclear Physics 56 (1964) 6 3 6 - - 6 4 6 ; ( ~ North-Holland Publishing Co., Amsterdam

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

ANTISYMMETRIZATION OF A UNIFIED NUCLEAR REACTION T H E O R Y W. M. M A C D O N A L D

t

Laboratoire Joliot-Curie, Orsay, France Received 26 N o v e m b e r 1963 Abstract: A completely a n t i s y m m e t r i z e d verion o f the unified n u c l e a r reaction theory given in an earlier p a p e r is developed by u s i n g a t h e o r e m d u e to Brenig. T h e theory leads naturally to a shell-model description o f the c o m p o u n d n u c l e a r states.

1. Introduction

In a previous paper 1) (hereafter referred to as I), a unified nuclear reaction theory was formulated which leads directly to an unambiguous procedure for the phenomenological analysis of single-nucleon reactions wherein both direct reaction and compound nuclear amplitudes are present. An attractive feature o f the theory is that the shell model for the compound states o f nuclei above particle emission thresholds receives a natural interpretation permitting its further development and exploration in a manner consistent with reaction theory. The significance of such an achievement can only be appreciated in the light of the realization that while the shell-model calculation of the nuclear states bound against particle emission has been given a plausible interpretation as a procedure for solving a characteristic value problem, no completely satisfactory interpretation has been given for the shell-model states lying in the continuum. Sometimes viewed as a procedure for finding the internal states X u of R-matrix theory, for example, the boundary conditions of the shell-model wave functions can actually be shown to contradict those of R-matrix theory 2). Again, the shell-model states lying in the continuum have been described as approximations over the nuclear volume of resonant scattering states 3), but the justification of the shell-model procedure for finding these states rests on rather approximate arguments. The physical model for the origin of resonances employed in this paper is a further development o f ' t h a t which was so clearly formulated by Newton 4) and which formed the basis for the unsymmetrized theory of (I). In proceeding to a fully antisymmetrized theory for single-nucleon reactions, we shall first provide an independent derivation of an expression for the reaction amplitude given by Brenig 5). Using this result we shall then be able to develop a separation of the reaction amplitude into a direct reaction amplitude and a resonant compound nuclear amplitude, with t O n leave f r o m the University o f M a r y l a n d . 636

UNIFIED NUCLEAR THEORY (II)

637

appropriate definitions of the effective interaction, the compound nuclear states, the resonance energies and the level widths. 2. A New Form for the Reaction Amplitude An expression for the reaction matrix will now be derived in which eigenstates of a symmetric model Hamiltonian are employed. Consider a system of A + 1 nucleons of which A nucleons form a target nucleus upon which the remaining nucleon is incident. The entire system will be described by a Hamiltonian containing a potential V which is the sum of two-body potentials v(i,j). i=A,j=A

V=½

2

v(i,j).

(1)

i=O j = O

The state ~a(¢; 0) will be defined as the product of a plane wave state for the nucleon O and a state qh(~) of the target nucleus which is an eigenstate of the Hamiltonian for A nucleons. As usual, ff~+) will denote the eigenstate of the total Hamiltonian H with outgoing wave boundary conditions. In this paper ~k~+) will be a totally antisymmetric wave function which actually satisfies the Lippman-Schwinger equation ~(+) = dq)~(¢, O) + (E + - H) - '~¢ Vo 0~,

(2)

where ~¢' is the normalized operator which antisymmetrizes the wave function with regard to exchange of the incident and target nucleons. The reaction amplitude is then given by Tb~ = (~/'b(~; 0)1Vol~ + ) ) ,

(3)

where A

Vo -- Z v(0, i). i=l

Now let Pot exchange the 0th and ith nucleon. Then

Tba = and using the antisymmetry of ~k~+), we have Tba = --. Thus the transition amplitude can be written in a more symmetric way as Tba = (,s~/Vo ~b(~; 0 ) l ~ + ) ) •

(4)

In this equation, however, there is still a remnant of the fundamental asymmetry of the asymptotic states. Neither the initial nor the final states of this matrix element

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W. M. MACDONALD

are eigenstates of some Hamiltonian. We shall now attempt to remedy this situation by introducing eigenstates of the symmetric Hamiltonian defined by A

A

H o = - ( h 2 / 2 M ) 2 A,+ Z U(i), i=0

(5)

i=0

which will be called "the model Hamiltonian". The U(i) is defined initially as an arbitrary central potential, to be specified more precisely later. Now observe that the definition of ~b leads to the equation (6)

d VO~b(~, 0) = ( / 4 - - E ) g ~ b ( ¢ , 0).

This result can be used in eq. (4) to introduce the model Hamiltonian into the transition amplitude:

Tba =

( ( H o - - ~ ) ~ ¢ ~ b [ ~(a + ) ) + ( ~ ' d ~ b [

~Pa(+)) •

(7)

The quantity q/is defined by

(8)

ql -- ( H - E ) - ( H o - e ) ,

where E is the physical energy and e is an eigenvalue of the model Hamiltonian and will be found from an eigenvalue problem to be specified. From eqs. (2) and (6) follows ~//(a+ ) = " ~ a

At-( E + -- n ) - l ( n - g ) d ~ a

,

(9)

and with the identity ( E + - H ) -1 = (e+--HM)-I + ( e + - - H M ) - ~ : I I ( E + - - H ) - ' ,

(10)

one can write the result

Tba = ( ( e - - - H o ) - X ( H o - - Q ~ b I [ I + q l ( E + - - H ) - ~ ] ( H - - E ) d ~ , ) .

(11)

We have assumed ( e - - H o ) -1 to be the hermitian adjoint of the model Green function (e + - H o ) - ~ in this matrix element. At this point we introduce the state Xg±) as the solution to the equation )~(b+ ) --= d ~ b + (e- -- Ho)- l(Ho - e)~C~b •

(12)

Clearly X~+) is an eigenstate of the model Hamiltonian corresponding to the eigenvalue (Ho-e)Z~±' = O. (13) The boundary condition we choose for the solution is that in the entrance channel X(b~) have the asymptotic form ~ikro

Zff(¢, 0) ,~ .Eb(¢)(e'k''°+ f ( k ' " k)) v

, ro

UNIFIED NUCLEAR THEORY (11)

639

whcre

E -- hZ/2M, and --~bis a model state of the target nucleus. This relation corresponds to the definition of the zero of energy as that for which the target nucleus is in its physical ground state and the incident nucleon is at rest at infinity. The model state Eb(~) will be a discrete eigenstate of the model Hamiltonian for A nucleons corresponding to the eigenvalue eb: Ho~ b

=

Ebb, b .

The total energy e of the model state X~±) is therefore e = eb+h2k2/2M, and in general e will not be equal to the total energy E of the state ~b~+) of the physical system. If the state X~b-) defined by eq. (12) is now introduced into the expression given by eq. (12), the following result can be obtained after a certain amount of algebraic manipulation: Tb, = (X[-)IHo-eI~CC'R)+(Z~-)I~IW~ +) )+Rab, where

((H--E)d#bld#a)--(d#bl(H--E),~#a)

Rab =

+ ( a-~'~C#bl(E + - H ) - 1( H - E ) d # a ) - ( d ~ b l ~ ( E + -- H ) - t ( H - E ) d ~ a ) . The quantity Rab can now be shown to vanish for single-nucleon reactions. From eq. (8) q / i s seen to be i=A,j=A

= ½ ~ /=0, j=0

A

v(i,j)-- ~ U ( i ) - ( E - e ) i=O

and is therefore hermitian. Thus the last two terms of Rab cancel immediately. Note that the first two terms are equal by eq. (6) to

<(H-E)~C#b(~; 0)ld#a(~; 0)>-(dq~b(~; 0)(H-E)~C#a(~; 0)> ---- (dVotPb(~; 0)[.~¢tPb(~; 0 ) ) - - ( d t P b ( ~ ; 0)ldVoCa(~; 0)). But by change of integration variables one can show that

= ( ~ b ( ~ ,

0)[ V0 ~a(~; 0))

= (dgo¢.(~O;)1~.(~;

0)).

The first two terms of Rab also cancel. Therefore, Rab vanishes, and the equation for the transition matix element becomes Tba

=

+.

(14)

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w.M. MACDONALD

This result leads immediately to a generalization to the case of more than one open channel of an expression for the S-matrix for single-nucleon reactions first given by Brenig 5): Sba = (X(-)l~(a+)) --2~i(z(b-)l Ok'[~a+)f(Ea-Eb).

(15)

In this equation, E, and E b are the total initial and final state energies, these being related to the model energies ea and eb in the manner described in the discussion following eq. (13). The first term of this equation for Sau is just the S matrix for the model Hamiltonian Ho. This term will be non-zero only for elastic scattering, and it is the analogue of the shape elastic scattering term often mentioned in discussions of nuclear reactions. Now, however, that here the exchange terms for shape elastic scattering will also be included. The second term of eq. (15) will contain the resonances arising from the discrete states of the model Hamiltonian, and these will be discussed next.

3. Resonance Expansion of the Transition Amplitude The resonance expansion can now be developed along the lines established in (I). The essential link is provided by equations connecting Z(a±) with ~(~±) which are derivable from eqs. (2) and (12): Va(±

=

+__/_/)-1

(16)

=

--/-/0) -1

(17)

Use of the first equation enables us to replace the second term of eq. (14) by the "reduced-transition operator" defined by 9-- - q/ + q l ( E + - H ) - l q l ,

(18)

giving Tba = (Z(b-)ln0 --el~a) + (gb(-) l~lX~a+~ ).

(19)

From eq. (17) follows the implicit equation for the reduced transition operator ~7-= q/+~//(e+

Ho)-l~---,

(20)

which exhibits the resonance behaviour generated by the discrete states of Ho. This last equation again permits us to define an "effective interaction" J - by removing the dicsrete terms in the bilinear expansion of the model Green function. Let rc be the operator which projects onto the discrete states of the model Hamiltonian. Then define the effective interaction by the equation = °/-/+°/-/(e+--Ho)-l(1--x)J .

(21)

UNIFIED NUCLEAR THEORY (11)

641

As in (I) we follow the procedure of solving eq. (21) for ~ and substituting into eq. (20) to obtain

= O(e+--Ho)+O,.~ ',

0 ----,.qP:p-+-Ho- (l-n)3"] -1

This expression is solved for ~J" and the solution written in the form

= O(l--O)-l(~+--Ho), by using the commutativity of 0 with (I-0) -I. In this way the reduced transition operator is expressed in terms of the effective interaction: J - = ~- + ~-rr(e + - H o - n~'~r)- 1n ~ ' .

(22)

The inverse operator will be diagonalized on the set o f discrete eigenstates of the model Hamiltonian. Let

X~, = Z A~ln),

(23)

n

where In) is a discrete state o f Ho, and determine the coefficients of A~ for which

(e#- H o - zr,~-zOX. = O.

(24)

The eigenvalues of this equation are complex:

5. - E . - ½ i t . ,

(25)

and the imaginary part is the width of the resonance produced by the compound state X,. The width F~ can be shown to be given in fact by a formula closely related to the time rate of decay of the state X~, which would be produced by ~ - treated as a transition operator:

r

=

2zfdcl(X.lJ'lx+)12h(,-c).

(26)

As in (I) this equation follows from an optical theorem for J - , easily proved as in the appendix of that paper, Im J~ = - 7r~-* (1 - lt)b(e- Ho)(1 - n)ff'--. The remark should be made that the dual state vectors to the X~ are the state vectors )~u satisfying the equation (e* -- Ho -- lr~-~trr))~u = 0.

(27)

Note also that although the real part J - a of J - and the imaginary part J-~ of J " are hermitian, the J - = J - R + i J ' t is not hermitian. Tiae imaginary part of J - is a

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w.M.

MACDONALD

reflection of the fact that the states Xu of positive total energy are not stable states of the total Hamiltonian but that they will decay, in fact, through their interaction with the continuum states. The final equation for computing the transition matrix element for a singlenucleon reaction is now found by combining eqs. (19), (22), (23) and (24): TD. =

(Z~-)lHo--el~/~)+
(Z~-)I~'IX")(~"'IY'IX(a+)>

(~-~+i~)

(28)

We shall now proceed to discuss the significance of the various terms in this expression and the properties of the effective interaction. 4. The Effective Interaction

In eq. (28) the first term is just the transition matrix element for the model system described by the symmetric Hamiltonian Ho containing the central potential U. The only reaction describable by such a Hamiltonian is elastic scattering and therefore this term will vanish for all other reactions. This amplitude is in fact a generalization of the shape elastic scattering amplitude, which now contains both direct and exchange contributions. The second term of eq. (28) is the matrix element o f the effective interaction ,Y-, which may have properties considerably different from those of the "residual interaction" V - U due to the fact that ~¢- includes the effect of interaction between the discrete states and the continuum states of the model. Eq. (20) reveals that only in lowest approximation is the effective interaction equal to the residual interaction, and that in general one might expect ~ " to be a non-local, many-body and energy dependent interaction. By construction, however, J - is expected to be free of sharp resonances, and the usefulness of the present formalism is determined to a large extent by the degree to which this expectation is fulfilled. In this case the second term o f Tba becomes a direct-interaction matrix element. It should be observed, however, that this is the matrix element of a complex interaction between distorted waves o f a real potential. In the usual direct-interaction theories the matrix element is that of a real operator between distorted waves in a complex well. The difference arises because the present theory describes all single-nucleon reactions simultaneously rather than schematizing all reactions other than the one of interest by an absorption of the wave function in the incident channel. The last term in eq. (28) now involves a prescription for the calculation of a set of compound nuclear states, the resonance energies, the partial widths and the total width. These quantities are also all to be calculated by employing the effective interaction. Inspection of eqs. (23) and (24) reveal that the compound nuclear states are found by diagonalization of the effective interaction on a discrete set of bound states. This is then a shell-model calculation.

UNIFIED NUCLEAR THEORY (If)

643

In the shell-model calculations of Elliott and Flowers 4), Brown 7), Gillet a) and Vinh-Mau 9), the interaction used for the diagonalization procedure is assumed to be two-body, local and energy independent with exchange properties determined to fit the observed resonance energies in 0 16, for example. That this interaction must be regarded as the effective interaction J " can be seen from the formalism of Fano. The Fano theory provides a foundation and a justification for the shell-model calculations employing the residual interaction, where this is interpreted as the difference between the sum of the free nucleon-nucleon interactions and an average Hartree field. Yet an essential aspect of the theory is that the resonances observed in a reaction do not occur at the eigenvalues obtained from the diagonalizatiort of the residual interaction upon a set of discrete states. The resonance energies are displaced from the eigenvalues by the interaction between the discrete states and the continuum. As pointed out by Ferrell 3) the energy shifts Call be expected to be of the same order of magnitude of the widths of the resonances. In this light, therefore, the astonishingly good agreement between the shell-model eigenvalues and the resonance energies would be cause not for exultation but for considerable suspicion. However, the interpretation of the shell-model interaction as the effective interaction of the present theory resolves at the same time two problems, (1) the striking deviation in exchange properties from the free nucleon-nucleon interactions and (2) the excellent agreement between the shell-model energies and the resonance energies. The only shift one should anticipate between these two sets of energies is that generated by the imaginary part of the effective interaction. The optical theorem for the effective interaction indicates that these shifts are of fourth order in the real part of the effective interaction, rather than the second-order shifts in the residual interaction which follow from the interpretation of the shell model based on Fano's theory. The analysis of level shifts within the framework of the present theory will be presented in another paper.

5. Model Potential and the Properties of ,~" The expression in the reaction amplitude given in eq. (28) is exact and independent of the choice made for the model potential U. Two considerations enter into the choice for the model potential. The first arises from the fact that the initial and final states employed in the evaluation of the transition matrix element are continuum eigenstates of the model Hamiltoniart which satisfy the boundary condition of containing a part which is asymptotically equal to the product of a bound model state for A target nucleons and of a plane wave for an incident nucleon corresponding to the physical energy E. Since the bound model state for the target is of course an eigenstate of the model Hamiltonian, the potential U must be chosen to yield at least as many bound states for the target as the physical Hamiltonian. The model Hamiltonian may actually be chosen to have m o r e bound states than

644

W.M.

MACDONALD

physically exist since the effective interaction can displace the extra bound states into the continuum where these states will be observed only as resonances in the reaction cross section. Thus a certain arbitrariness is permitted in the choice of the model Hamiltonian which will enable us to describe in a simple way resonance effects which have been heretofore viewed as arising from scattering states for the unperturbed problem which exhibit resonance scattering occurring near threshold. For example in 016 a calculation of the location of the "unperturbed single-particle states" 7) from separation energies yields the result that none of the excited configurations for a proton are bound and only the states of the ( P 0 - 1 d_~, (p~)-12s½ and the (Pl)-1 2% neutron configuration are bound. It is not correct, however, to interpret these energies as the single-particle energies for the unperturbed shell model of the present theory. These energies refer to asymptotic states which are not eigenfunctions of the symmetrical Hamiltonian Ho which generates the discrete shellmodel states. Only after one has included the effect of the effective interaction in mixing and displacing the eigenstates of the Ho can the resulting spectrum of states be compared with the "unperturbed single-particle energies". Some of the bound states of the model Hamiltonian will now be found to have been displaced into the continuum where they give rise, by means of the resonant term of eq. (28), to the sharp resonances observed in elastic scattering. The second important consideration in the selection of the model potential is that the effective interaction given by eq. (21) should have as much as possible the desirable property of being the sum of two-body, local, and energy independent interactions. The equation for J - can be expressed in terms of the true Hamiltonian for the system in order that the choice of a model Hamiltonian may be more clearly related to these properties of J ' . By solving eq. (21) for J - one obtains the explicit equation J - = q l + q./(1 - r r ) ( E + - H ) - I ( 1 -z0Y/.

(29)

Using the eigenstates of the total Hamiltonian to represent the Green function, one has J " = °h'+ E ~ ' ( 1 - ~ z ) ~ " ) ( t / " " ( 1 - n ) ~ ,,

E-

E.

+fda

°k'(l-n)hu~+))(~a+)(1-n)~ E + - E.

Note that (1 - n ) projects on the continuum states of the m o d e l Hamiltonian. We can anticipate qualitatively the properties which the effective interaction will display in (1) shell-model calculations and (2) direct-interaction matrix elements. If the discrete model states, or linear combinations thereof, provide good approximations to the true bound states, the contribution of both the discrete and continuum states of H to the energy dependence, non-locality and many-body aspects of the effective interaction will be small. However, the existence of discrete model states which do not correspond to true baund states, but rather to resonances in the

UNIFIED NUCLEAR THEORY (n)

645

reaction cross section, will introduce these properties of the effective interaction, particularly through the last term. In fact, it has already been shown that J - has an imaginary part which arises in this way from the interaction of the model states with the continuum. Therefore the price of exhibiting the resonance behaviour of the reaction cross section by introducing discrete shell-model nuclear states is that the effective shell-model interaction becomes increasingly a non-local and many-body interaction with each additional resonance which is described in this way.

6. Summary A unified nuclear reaction theory has been presented which employs properly antisymmetrized shell-model states to describe compound nuclear resonances. The complete reaction amplitude is calculable in terms of an effective interaction which determines the resonance energies, the partial and total nucleon widths and the direct interaction matrix elements. The theory differs from that of Feshbach lO) in that the use of shell-model states for initial and final nuclear states and for compound nuclear states is not an approximation which is made to permit the construction of open and closed channel projection operators as in the work of Lemmer and Shakin 11). Instead these states are used in an exact calculation of the reaction amplitude wherein the effects of correlations and energy shifts produced by the nucleon-nucleon interaction are expressed in terms of an effective interaction. The properties of this interaction can be determined theoretically from the solution of a certain integral equation, or hopefully phenomenologically from the analysis of experimental cross sections. Although the theory provides the basis for a shell-model calculation of compound nuclear states which are unbound against nucleon emission, important corrections of the usual shell-model calculations are indicated. The imaginary part of the effective interaction will enter such calculations and will provide level shifts which have been neglected in the usual shell-model calculations. These shifts result from the interaction between the discrete states and the continuum states, but they are unlikely to be as large as those suggested on the basis of an incomplete formulation of the problem. Another result indicated by the theory is that "refinements" of shell-model calculations obtained by introducing more configurations into the shell-model calculations are not likely to be consistent with the assumption of a local, two-body, and energy independent effective interaction in the shell model. Such calculations also appear to be inconsistent with the use of a direct interaction matrix element employing a model potential of the proper depth to yield shape-elastic scattering. Further investigation on these questions clearly needs to be carried out within the framework of a consistent and unified theory. The author began this work while on a NATO fellowship at the Laboratoire Joliot-Curie. Many of the essential ideas were developed during a p~riod spent at

646

w.M. MACDONALD

the U n i t e d K i n g d o m A t o m i c E n e r g y Research L a b o r a t o r y at Harwell. T o Dr. A. M. Lane the a u t h o r especially owes m a n y of the s t i m u l a t i n g discussions a n d critical c o m m e n t s which led to the present development.

References 1) W. M. MacDonald, Nuclear Physics 54 (1964) 393 2) A. M. Lane, Revs. Mod. Phys. 32 (1960) 519 3) R. A. Ferrell, Eastern Theoretical Physics Conf., p. 53 (Gordon and Breach, New York, 1963) 4) R. G. Newton, Ann. of Phys. 4 (1958) 29 5) W,. Brenig, Nuclear Physics 13 (1959) 333 6) J; P. Elliott and B. H. Flowers, Proc. Roy. Soc. A242 (1957) 57 7;)! G. E. Brown, L. Castillejo and J. A. Evans, Nuclear Physics 22 (1961) 1 8) V. Gillet, Thesis, University of Paris (1962) 9) N. Vinh-Mau, Thesis, University of Paris (1962) 10) H. Feshbach, Ann. of Phys. 5 (1958) 357, 19 (1962) 287 11) R. Lemmer and C. Shakin, Ann. of Phys" 27 (1964) 13