A unified theory of nuclear reactions. II

ANNALS

OF

PHYSICS:

19,

A Unified

287-313

(1962)

Theory

of Nuclear

Reactions.

II*

The principal device eml)loyed, :ts in lxlrt I, is 1IIC project ion olwrator which selehs the open channrl components of the W:IW function. It is found that the forrnd structure of prt. I providing a unified tlrscriptiorr for dirwt :md conpound nuclear reactions including the cor~plrtl rquation clewription for direct rewtions renmins valid in this wider contrxt A Ik~pu1~~l’cirrls espmsion m:~y dso be rr:ttlily ot)t:tincd. The concept of ch:~nrlel rxtlii is not nretlrd nor is any deconiposition of the LS:ive function for thr system into :mgul:tr momentum eigenst,:kt.cs required, so that t hi esprwsiolw for I ransition :tmplitudes and widths are invariant wit,h rrspect to the angul:~r ltloriientrrn~ coupling schcmc. Since thp OIIPII channels can only Iw ~lefirl~tl in XII asymptotic sense, the corresponding projwtion opernt ors :IKP not unique. As :t consequence the projection opcator method has :t flcxihility lvhich in thrs first place is consonant with the \vitlr r:mge of I)hcnomcln:l \vhich c:m occur in nuc.Ic:Lr rewtiow :tnd in t hc second ~)l:tw can c~f’l’e~tiwly rxploit XI intuitive nnclrrst:rnding of the phenomrnx. li:\-:unplc~ of pr”.jwtion opcr:~tors :II’P obtained inclu(ling one which lexls to t hr \2ii~npr-15i~p~Il,llti foun:~lisnr. :mot hpr \vhich is :tppropriate for the strip[)ing wwtion. :rnd, finally. one \I-hich t alies the I’nuli rxclrision l)rinc.iplc into account Note, that clxl)licit r.cl)~.(~~(~~~t:~tiolls of the projwtion oprr:ttors XI’P not required for t hr tlrvelopmrnt of general formd results I)rlt :ir(’ rrwws:try if, rvrnt u:lll~,, ct~wntit:ttivc c:llcrll:~tiolis :~rp tiiatlc. I. INTI~OI)~~CTION

In this p:tpw ltesctions” (I )’ with developing and compound 1x1 both cxsily

the forrnalisn~ dcvclopcd in “?I I’nificd Theory of Swlcur is gcnernlized and irnprovcd. In I wc wcrc principally conwrncd :L theory of nwlcar reactions from which :l description of direct nuclear proccssrs, as ~11 a:: tht mnplcs potential model, would and nat~uxlly alwtractcd. This inwcnpahly led to :I dcri\-ation

* This Lvork is supported in part through .4Is:C (‘ontrwt vided t)?; t.ho IV.9. .4tomic b:nergy Commission, thr ( )Iiire Force Oilice of Scientific I
AT(30 of T:,v:d

I)-2098, by funds prw I?ese:~rch :tntl thr Ait

of resonance reactions which did not involve the concept of the channel radius, nor did it require t,hc decomposition of the wave fmlc+ion of the systrm into orbitma angular moment,um eigenstatcs. although our treatment was successful, t,here wrrc a number of limit,at,ions. We considered only situations in which the mass nunlbcr of the incident projectile and that of the emergent light part’iclc were the same as they would bt in an (n, p) or (n, n’) process. St’ripping and similar prowsses were not treated. Secondly, although it was possible to show that t,hc effects of the Pauli exclusion principle and exchange scatt’ering would not change the structure of the formalism, the discussion was rather indirwt and did not provide a method for the evaluation of these effects.’ Finally the connertion of resonance formalism in I with the boundary condit#ion methods of Wignrr and Eisenbud (2) and I
hy J. S. Bell, the discussion in I was incomplete. are given in Appendix 13 of “Radiation of LOUwritten with I). Uennie and submitted for publi-

NUCLE:AH

ItEACTIONS

289

the direct rtact,ions, for potential scattering and for “single particle” resonances. The rapidl,y varying part, gives rise to t,hc narrow compound nuclear resonances; t#hr widt)hs for these are expressible as matrix element,s involving the “single part,irle” wave funct,ions of the dirwt rewt#ion Hamilt’onian. In another treutment the effcct,ive Hamiltonian is not, broken up into a fast, and slowly varying parts. Instead the transit,ion matrix is expnndcd in t,erms of the eigen solutions of the effective Hamiltonian for the closed channels ohtaincd hy eliminating the open channels in the ahsenw of any incident, wave; the open channel part of t,hew eigcnfunctions sat,isfy an outgoing waI,r 1)oundary condition. The cigenfunctions have complcs eigcnvnlues and are clearly of the Kapur-I’rierls t,ype and t,he rrsulting expansion of the trnnsit,ion matrix hccomes 311 expansion OVPI resonanw::.” There are howcwr several signifirant. differences between this reslllt, and t,hat of the standard Iiapur-J’cierls formalism. These all stem from the fact t,hat in the present work not even the concept, of nuclear radius is used. Our resuli s hold thcrcfore even when t,ht> pot#ent’ial involved does not have a sharp rutoff. As a conse(luerwe t,here is no need to dccompost the wave funcAtion int,o eigenstates of t’hc orbital angular momentum. IGnally, thr widths art expresstd in terms of matrix elrments of the interabon rather than in terms of overlap int,cgrals hetSween inside and outside wave functions at, the nuclear radius. These t,wo expressions for the transit,ion matrix, that of I which we shall call the cffcc%i\-c Hnmiltlonian method and the method of expansion in complex cigenvalues. are of course equivalent,. However it, seems to us that the efiect,ive Hwniltonian method is most conwnicnt for nuclear reactions in that it differenGates hetwren the narrow compolmd nwlcar resonancw and thr “single particle” giant, resonances ; i.e., brtween compound and dirrrt’ nuclrar reactions. In t’he romplrs eigrnvnlur expansion both kinds of rrsonanws arc lulliped togrt.her indiwriminat~ely. The projection operator which srlrcts open channrls is not uniclue sinw it is possihlr to define oprn channels only in terms of the asymptotic behavior of the wave function mhcn the rrwtion products are far apart. This gives thr formalism an additional flexihilitly which is extremely usrful since it allows one to choosr t,hat projection operator whirh is most convenient, for the problem under invrstigation. In the present paper wr arc for the most part, cwncerned with thr type of project’ion operator employed in I in which the open channel war-c flmct~ions are defined by ext,rnding the form of the wave function lvhich is valid asymptotically to all of (*onfiguration spaw. This mrans that ollr open rhannel J These

eigenvalues

twrisit~ion

elnployirlg

*Sieged -H:m~ldet series. ~liscussion see IIumhlet

are functions

of the cnerg,~-, h’, of the systenl. i\n e>-p;lnsion of the which are independent of E is referred to as the Such :I series is derived in the appendix to this p:~pe~. nor :, yevent and I~osenfrld 151.

vollll)les eigcnwllucs

wave functions are linear superposit,ions of the possible residual targrt, nwlenl states. The class of these states arc rest’rict,rd of course by energy c*onservstion and ot’her conservation nlles which may be pertinent. We also briefly discuss another projection operat#or which introduces the channel radius into the theory. The projection operator here is unitmy for regions in configuration spaw whew the partirlcs are separated by distanws great,er than t,he channel radius, and is zero elsewhere. if t’hc channel radii are chosen so that cscept for Coulomb or ot,her long range pot,rnt’ials the interaction vanishes where the projection oprrator is unity, we are directly led to t#hc Wigner-ICisrnbud (2) boundary conditions. These two types of projection operators have essentially complementary domains of usefulness. The Wignrr-Kisenbud projection is most, convrnient when details of the internal region are not under scrutiny. Thrre is no difficulty with the Pauli prinriplr and tschangc scattering. The resonance energies have a definit,ion indcpcndrnt of the incident, energy. The project,ion operator used in I is appropriat’e when drtails of the int’eract,ion arc of inter&. It is the natural ext,ension to thr ront~inuum of methods employed for bound statr problems and at, t,he same time it can be easily joined on t,o t#he high rnergy multiple scuttrring limit of Krrman rt al. (6). Therr is t,hus no nwd to cahunge thr formalism as one’s at#tent,ion changes from negative to pos;iGvc and finally to high rnergirs. The closr relat,ion of t,he shrll model pot,rntial and t.he real part of t#hr complex potenGal hecomrs manifrst and one rralizes t,hat# the residual potential is rrsponsible for t,hr variegat,ed phenomrna which occur in nuclear reactions. On the other hand with t#his projection oprrator type t’hr trcatmc~nt of the Pauli principle (discllssion Section III) is not’ simple although it’ dors srrm mnnageablr. Srvrral writrrs havr cont~ributetl to thr eclui\-alent Hamiltonian method since the publication of I:’ Wr me&on Hwnig (7’), Srn-ton and Iconda (8), &Ygodi and Eberlr (9), and most rrccnt,ly Lipprrheide (IO). Each of thrsc havr developrd material whirh overlaps somr of our work as n-e shall duly note Mow. Lnnc and Thomas (11) and G. Rreit (12) give ewrllent, reviews of thr Wignrr R matrix theory and reacation theorirs in grneral, whilr the most recent, treatment of t,hc rspansion in resonanws is givrn by Rosrnfeld and Humblrt (5). Wr conclude this introdw+on with a hricf description of the rontrnts of the paprr. In Sec%ion II t,hr general throry is drveloprd in terms of projr&on oprrators. Specific forms of thew operat,ors arc deli\-cd in SecLtion III for the stripping reaction, for incorporating t,he Pnuli principlr. and for thr Wigner formalism. In Section IV, rrsonancc t’hrory is devrloped and an expansion in terms of rwonances of a Kapur-Peiwls type is dcri\.rd. Section IV dors not drpend upon t,he d&ails of S&ion III so t,hat it can br read dirwtly after II. In an A1pprndis, t,he Scigrrt,-Humblet, description is dwivrd. Thrw will hr some re\-iew of the 4 See I for

the c:trlisr

refrrsnces

earlier paper on t,his subject, 1, bot#h for tht purpose of completeness as well as to pxmit some further remarks which round out the earlier discussion. The interaction of gamma rays with nuclei will br discussed in another comnmnication. II.

THE

WFIXTIWS

HAMILTONIAN

When a projectile a st#rikes a target nwleus X, a \-xi&y of reactions can occur. Ela4ic scat,tering in which the final product,s are agzlin a and X will always tak.e place. In addition t#hcrr might be various transmutations in whic&h the final product’s h and I’ (b can itself be composite) differ in some int,rinsic* respect’ from a and X respectively. Those reactions (including elastic swt,tering) which are encrget’irally allowed are referred to as open channels; the others are closed cahannels. Let t’he wave functions dew-ibing t#he energetically possible’ residual nllc~lei hc drnot)ed by c$, . Note that, these functions are generally not mutually ort’hogonal nor do they nwessarily involve the same number of coordinates. Asympt80tically, when the rrac.t8ion prodw+ are well separat8ed, t,hr wave fun&)n for the spstem will take on the antisymmetrized form:

whcre,f, arc functions of t,hc wordinnt,cs not contained in +i and of course satisfy appropriat#e asymptotic bolmdary cwndit8ions. @ is the :~ntisymmrtrization optrat’or. It is now possible to define t,hc projection operutor which sclrct,s t,hc open channels. It. is any projection opcratx f’ which, operat,ing on any ant)isymmetrical fun&m x, satisfying t,hc same asymptotic boundary wndit,ions as q, b111, otherwise arbitrary, yields a fuwt~ion which asympt~otically is of the form ( 2.1 ) : 12.“) The quanGtirs 11,are fuwt~ions of the (wrdinates not wntained in 4! satisfying the same boundary conditions as ,/“, , as well as c~onditions imposed by invariant principles. A$ simple example of such an opc&or is t.hc one PwE which, as we shall show in Swtion III, leads to t,hr migncr-F:iscllt)lltl formalism:

292

FESHBACH

Another example is furnished by the projection operator P1 employed in 1 which we shall now suitably generalize. Consider the set of all antisymmekzrd wave functions Q, satisfying the same asymptot,ic boundary conditions as q which have t,he form: a = @c

ZL$#JL

(2.41

The uL approach t,he Ui of Iiki. ( 2.2) axymptBotCirally. The set (a] subtends a portion of Hilbcrt’ space (let’ us call it the “open-channel subspace”), associated wit.h the Hamilt.onian and the other pert,inent’ operators of the system. Moreover a projert’ion operator I’ I must exist which project#s on t,o t,his open channel subspace. In other words if x is an arbit(rary stat,c vector, P1x helongs to the set @‘; that is, P1x can he expressed as a sum over the open channel residual &ate C/J~. We shall obtain explicit expression for PI in Section III. Sate that, Pw, and P, are only two examples of posnihle projection operat,ors which satisfy condition (2.2). IJIany others are possible; wh.ich one is to be used will be determined by t,he physics of thr phenomenon under investigation. It, is important to realize t’hat, t,he remainder of the discussion in t,his section, as well as t,hat in Section IV, requires only th.c exist,ence of t’he projection opcrator. If 9 is t’he wave function for t#he system it is clear t,hat. we need only ransider FP in order to obtain t#he asymptot,ic behavior of + and thus the scattering and t#ransit,ion amplitudes. It, was t.his fact’ which was employed in I for the case considered there and it’ will be useful principally for completeness to rephrase some of t,he results of I in terms of an explicit’ project,ion operator formalism.” We notIe t#hat’ we have already oht#ained the generalization of I to situnt’ions in which the final reactions products can be arbit8rarily complex. There is no limitat,ion on the number of open channrls or on t’ht nat.ure of t,he emergent part#icles. All we need to do is construct t,he appropriak projeck)n operator. Let us rephrase the calculation in I up t)o t#he point, where we obt8ain the cqui\ralent, Hamiltonian Ccalled generalized optical pot’cnt,ial in I) . q sat.isfietl t,Ee equation

6 This option wvas n~entioned in refs. 8 and 9.

in I. w:ts partially

described

in ref.

4 and hats been emplopcd

Since @ f0110\vs8

projeck

int#o t,he calosed channel

suhspace

we can solve

(2&b)

as

(2.8) SubstitAng

Eq. (2.8) in likl. (2.(k)

yirlds

t,hc Schroedingrr

(,B - H)Z% = 0 where N., the effect~ivr Hamiltonian,

cquat’ion for Pq:

(2.9)

is

Eyuat~ion (2.9) is identical in form t)o Eq. (17) in ref. 4 and Eq. (2.15) of 1 but has a wider applicability as a consequence of the generalizat8ion in t,he tlcfinition of 1’ given above. ?clortover, bccsause the form of 1’:q. (2.9) is identical to t,hat of Ect. (2.15) of I, tht deductSions in I, including t,he resonance formulas, the complex poknt,ial model, and the direct, rea&ons based on Eq. (2.15) of I, remain valid in the present contest. We shall discuss tbcse at grcukr length in Section IV. For the prrscnt let us render Eq. (2.10) less formal by determining I’ for some csamples of rnlclrnr reactions. However let. us emphasize that the formal rcsl&s of Sect,ion I\’ do not8 rrquire explicit, expressions for 1’. The need for such expressions o~urs when cvaluatSions of t#hr various matrix clcments for various ;~ir~clear models or cvent~llnlly from nn~lcon~nll~leon forces arc being made. I I I. 1’13 )JE:(‘TION

OI’ISIL4TOIW

In this scvt’ion the projec%ion operators of t#hr type I’, corresponding t,o various possible final st.:lt,es arc esplicitl? T given. Wr have chosen to csprcss t#hese in conliguratioii spac*cl. How-ever it is also possible, and it is often more convenient, to use t,hc language of second cluniit~ization. ’ Thr translat.ion, configuratlion space to SWOIK~ cluantization is cluitc straightforward for t#hc cases considcrcd hrrc, so that, w will lravr it to the Icader to rcwritc our resuks in the latter form. We c*onc*ll& this svction wit,h a tliscsllssion of I’ WE and t’hr ~~ignc~r-I~~is~nhud for malism. ( a) WC considrr first the vast whcrc the inc*ident, and cmtrgcnt, part,iclc art sam(‘ 1~11:arc not identical with t)he particles in the target nuc~lcl~s. l,c,t8tht, wnvc

294

FESHBACH

function for the ground state of the target nucleus he do and suppose that the incident particle is sufficiently energetic, as to he able t’o excite t,he first, II excited states of the t’arget whose wave full&ions are +j , j running from one to n. Then (3.1) Inserting

this

result

into

I
and

(3.10)

and

let,ting

udrd 3 (+i(rl . . . r,)QCrOrl . . . r.4)) we obtain

where

H,j(ro)

= (+t( rl .

r,)H(rorl

. r.A)+j(rl ,.

. r,))

(3.4)

I
r.,)

= .f’(rl)#dr2

rA)

1’ is simpAy

(3.6)

1’ = $“)(lc/O ISquation

(2.8)

(X.-,) ‘,

komes

[B - Hodro , rl)]wcru , rl) = 0 70 = (&I ) *‘)

(3.7)

Ho, = ($dWo) IO The particle to he “picked up ” is not antisynlnletrized with respect to the particles the residual (-1-l) nuclear wave funct,ion & This :rns:ltz implicitly assumes that particles act essentially as :t spectator particle, not particip:Ltiny in the reaction.

in :l-2

This is the form from \vhkh most, direct intcrx%ion theories of t,he picak up process &art. Tbrre is one import,ant diffcrenc*c. lfquation (3.7) cor&ins t,he resonance phenomena as dell for t#he incident neutrons or if me are disc:lSng t,he inverse process, &ripping, the resonance phenomena for the incident dcuteron. We shall discuss the rcsonanc( terms in Section IV. Our discussion is not yet compl&. We should continue t’hc proj&ion proc~lurt~ one st#tp furt’her so as to isolate t,hat, part, of t,hc have funct,ion w(r,, , r,) I\-hich describes t,he process in \\-hich tbc residual nuc:lc~~s wave’ funckion is 4,)~rl t r.,) . l’omarcl this rnd n-c write wtrcl , rJ \vhcre I’,v is t,hc projec+on

= P.du + QNw

operatot 1’3 = .fX.T

:md QN

‘I’he fun&m

=

1 -

I’.%J

IU t,hrn assumes :he dcsirecl form zdr,, , rI) = u(r,,).f(r,)

+ t&w

(3.8)

whrrc ‘I = (,f(r1)2u(ru , rd Sllld (.f,

QNW)

=

(3.9)

0

Sotc t,hat 21 will asympt~ot~i~ally clrscrihe t’hc iwidcnt ( if any) and emergent, nac~lcon waves, while Q,“w, thtx remainder of w, will provide a similar description of the dcwtrron. Tht fmwtions u ant1 QNw sat,isfy t#he following 1,G of couplet1 f~clliations [E - H’(rJ]u(r,,)

= (.f(rJ,

Hdr,,

rl)QNw)

111'- Q,,,H,,~,Q,\Q~EI = H,,,,(r,, , r,).f(r,)u(r,)

(3.10’) - ./(rl)N’(ro)u(ru)

\I.11 IT H’Crcl) = (ftrl),

Hdro

CXllj

, rdf(rd)

11:quat~ion (3.10) is wry similar in stxuc%tuc~ tSo13~1.(3.3) and suggests that as a gencml princ4plc it, will lx> possihlc t’o dcwrihc a rcxc:tion invoking t,hc> amptit,llcles for vnriolw opcw ~hnnnrls iu trrms of c~~uplecl S~hrocdinger type equations involving cfic~ctiw HaInilt,oiiialls. We consider nest the> more realistic* situation in \I-hkh 4,) is not proport8ion;t1 to +bo; i.t,.. 1<(1. (:<..i) is not valid. Then P == &‘(rl

. r,j)(4,,‘(r1

r.,) + $dr?

r,))($drt’

.

r41

(3.11’)

whew 4”’ = N[qh - $,I(+” , &)I;

(Ql,,‘, $h) = 0

(3.13)

296 N is just

FEHHBACH

a normalization

chosen so that (40’ , 40’) = 1

Kate that projection operator (3.12) reduces to (3.6) when +0 is proportional to $0 . With (X13), and with suitable rearrangements FCq. (2.9) becomes an (quation for

I*

= u(ro)40(rl . . . rJ

+ QNw(fO, rl)lC’0(r2 . . rA)

(3.14)

where (3.15)

QN = 1 - 40)(40 The equations

for u and QNw are identical w -

(4” , I’HP4,)lu

in form wit,h Eq. ( 3.10) : = (4lJ’Hl’QNW~O)

where H is given in I’,({. (2.10). (c) The inclusion of the Pauli principle into the theory of rract,ions has been the subject of several papers (10, 11, 14-16). In t,hr prwent, treat,ment \vc avoid the subtleties and complexities of the t,imc dependent, treatment, in which t,he behavior at infinite tIimcs must, be examined with considerable care. Our cffect,ive potSential will be hcrmit,ian; some t’reatme& drduce nonhtrmitian effective potentials. For simplicity we again consider the cast n-here the incident, particle is a nucleon. Moreover we assume t#hat’ it is not energetic enough t#o excite t,hr target, nucleus so t#hat the only process cncrget’ically allowed is elast,ica scattering. If the ground state of the target nucleus is t’hc antisymmetrized wave flmction operator we require has the following property 4drl . . . r.4) t’he projectIion

P*(ro,

rl

. . rl)

= @u(r,)4,dr,

(3.17)

ra)

where C%is the antisymmetrization operator and + is antisymmetrized. If 9 is the wave function of the system, t.hen t)he function u( r,,) asymptotically yields both the direct and exchange scattering amplitude. The operator 1’ selec+ ollt of JP that part which describes the target’ nucleus in t#he ground st,atc. The equationI determining u is

(40(r1 . . r.,), [Wrorl . . ra) - @u(r0)4dfl . . . rl)]) where t’he integration is over the coordinates of the inner product in (3.18). That condition as follows. From (3.18) we have I1 In the discllssion to follow, 4 is :tn arbitrary nte boundary conditions at infinity.

= 0

which are common (3.18) ib sufficient function except

t,hnt

(XlSj

to hot,h sides can be shown

it satisfies xl)l)ropri-

(u(r&O(rl and sinw tmhat,

f . rlj,

the qknnt,it,y

[*irOrI

. . rl)

-

in the squarr

@n(rOMrl

hrwkct~s

. . r.dl)

(3.19)

= 0

is antisymmetSrical,

(Ctu(rO)ddrl . . r.4), [\k(rOrl . . r..,) - Uu(rOMO(rl . . r..,])

it follows

= 0

( 3.20)

I:rom this last ccllutSion \ve SW t#hat# ~rwe (i2.18) is solved, WC can hrcak up q into t.wo mutually ort8hogonal parts of which one is t,he desired form (i3.17). It’ is &ill ncwssury to show that the rrsult,ant, form R( u#J,,) can hc writ,trn as a projrboll opcrut#or 011 \lr. But for t’his we nerd first. to sol\-c 1’:(1. (3.18). I’wforming t,hc indicated opclrat,ions Eve oht,ain

(32lj

I:(r0) = u(rd - -4($drlr:! . . r4), +O(r0r2 . . r~4jn(r1)) I’(ru)

= (&(rl

integral Trj,

I’lr,,)

= u(rJ

( 3.22)

h), *)

..

rq~c~t,ion

of t,he l’rcdholm

+drm

. r.4))

- (K(ro ) rl)w(rl))

‘l’hc krrnel K has several imporbant, properties. wave function for a hound st,ate, K decreases hwon~c~ infinite. K is hounded:

type (3.‘~‘2) at

( 3.21’)

It, is hermitian. Since 4” is the cxpownt~ially as either rl or ru

‘l’hc traw of K is finite and equal to 11. It will he useful to keep in mind as an r~.u~wple t,he lwrnrl K WC oht’ain when c#,) is a Slntcr dct,erminant composed of mut~unlly ort,hogonsl rlcments ‘wi Then

.A K = c w,*(t)w,(rO) 1 Bcfor(~ solving t,he inhomogencous tquat,ion the eigrnvaluc problem of the wrmsponding

(3.X)

(3.21) it, is wnvenient homogeneous form :

u,(roJ = A, (K(ro j rl) u,(n) ) From

the proprrtirs

of K it follows / X, 1 1

t,o consider

(:r.ar,,

t)hat# X, are real and 1 and

c

1

a

= A

(3.26)

298

FF:SHH.‘.(‘H

The eigenfunc%ion complete. However,

u, will form a11 orthonormal stlt, which is not necessarily K can he expanded in terms of normalized 21, as follows (3.27)

For example, (3.X), { Us} = (,w,} and A, = I for t,ht cnt,ire set. The case where t’he X, eyual u&y requires some special considerations. Denoting these eigenfunctions by 71al , wc note that they sat,isfy IL,I = (K(r0 I rdud) 01

($O(rl ... r.4), ~~~,l(roi40(rl From this equation

. h))

= 0

we have

(~t~d(rO)+O(rl

r.<), Ckl(rdddrl

.

r,,))

= 0

or @zh(rO)+drl

(3.28)

. . rA) = 0

For example (3.24) ; t#his is just t,he determinant,al t8heorem which states that the determinant vanishes when two columns are identical. Marc generally it, may be shown t.hat# if condit.ion (3.28) is satisfied, $0 itself is given by hl(rl

. . . rA) = CkI(r1)1c/(r2

. . r.d

where J/ is antisymmetrical. Hruce by examining 4,) OIlC can pick out those u, , u,, , whose eigenvnlue X, = 1. Auother consequence of immediak interest, is that, ( 11,l ) I’)

= 0

(3.29)

To prove this, simply note t,hnt t#he uhore bracket (uaI(rOb(rl and since \Ir is antisymmctrkal

. . r.$), *)

it, is proport,ional (@u,,(hb#dr,

is ident8ical wit,h

to

. r.4), *)

which in virt,ue of (3.28) is zero, proving (:‘,.39). We now return t,o Eq. (8.21’). Employing t,he expansion the orthonormality of t#he set, lu,} we find ( lLa, 11) =

(3.27)

for K, and

A, # 1

(3.30)

l“illtlllJ',

(3.81)

As indicated, the terms in u,l--tShc amplitSude of t’he eigenfunctions whose eigcnvalw X, are unity-arc not2 dct,ermined hy IQ. (3.31). However, hecar~sc of ( 3.28) t,hc function of interest, a( 2~4”)) does not c*ont,ain these terms. In fact : U7c(rfl)$drl

.r.,)

= Qu’(rll)+O(rl

14nallp, making 1~ of tht> antisymmc%ry

h)

of q TVPha~c

I’ is th.c dwirttd projection operator which projc&s out of a fun&m t#hat, part. which cm tw writkw c~u+~ (cf. 11:(1.(3.17) ). To prow that it is a projcc%oll opc>rator \vo nwd only note that I’ is hcrmitiau , and from I~Zq. ( 3.20) that. (I’xv, ( 1 - r’)*)

= 0

‘I‘htw two propert,ies haw t,hc c’onsequrnw that, II” = I’, proving that I’ is a projwtion operat80r. It is amusing t,o note that when 9” is a Slater dekrminant @(u+~) is just, ~t+~)(+, awording to (3.35). This result’ tells us that in that. cxamplc, t,o project ollt the ~dcpendenw on +, we need only employ t,hc naive praj&ion opcrat,or This resldt, is no surprise from the swond +,J(+,, and thw antisymmctSrizc. cluantization point8 of \-iew sinw t#hc appropriat)cl projt4on opcmtors in this cxsc can tw writt,en simply as a product, of crration operators for the st8at8rs c*orrrsponding to w, ac*t,ing 011 thcl “vacuum. ” Thr mow general operator (3.36) cwrresponds to the errntion operator in which t,hr A psrticlcs arc creat#ed in the stat,c &, AUthough tbc simplicity of this rwnlt for the simple dcterminant~al 4” culllot. I I? maintairwl for mow complicat~rd 4,) there appears to he very litt,lc difficwltp in solving Eel. (32.25) for uu and X a as long as $3” may he writt,en as a liner cwmhinat~ion of Slatcr dtt~erminants. WP have thrwforr provided a dirwt and prwt~iwl met,hod for tbc det crminat.ion of the projwt,ion operator which

300

FESHBACH

selects out that’ part’ of * which represents the t,arget nucleus in it,s ground statt,. Equat~ion (2.9) now becomes an equat~ion for B( u+,) . The existence of sucsh an equat#ion is all t,hat we need for the derivat,ion of t,he resonance formulas, tht optical model, ct#c. The antisymmctrizatio11~11 operators give rise t’o exchange interactions in the cfiect,ivc Hamiltonian. Kot,r that the cffwtivc Hamiltonian operat,or is hermitian and that from the asymptot,ic behavior of u(rO) me can obtain the ela& scatCtcring amplit,ude which inc~ludcs both t,he dirwt and exchange scattrring amplitudes. It may he convenient, to derive an equat,ion involving only ON wordinutr, say r. . This quat~ion can be obt#ained by mult’iplying 1%~. (2.9) from t,he left over coordinat#es rl . . . r.i . The prop&k of by +O*(rl . . . r,,l) and integrating the projection operat,or permit some import’ant simplifications in t#he resultant equation. Xote that Eq. i3.18) can be ren-rit,ten 7’(r0)

=

(40(r1 ... rA), *)

= (40(r1 ...

r.,),

This equation is valid for Amy antisy~~~wtrical function most convenirnt~ t,o consider t,hc equatJion for C’ which, the symmetry of H hccomes:

T’*)

(3.18’)

q. Rwarw of this it, is in lirtuc of (3.18’) and

This equat’ion has t,hr important virt,ue that, t,he modifications arising dirwtly from the inclusion of t,he Pauli principle do not, intSrodrwr any additional explicit energy dependence. (d) In a cert,ain sense t#hc simplest projection operator, which projects outS from the exact solution q a part, which has t,he same asymptotic behavior as \Ir, is one which is unity outside the region of interaction and zero inside said region. In other words, I* is just the asymptotic dependence of Q outside of the region in configuration space in which interactions occur. I3 As we have asserted earlier this is the proj&ion operator employed by the Wigner-Eiscnhud (2) formalism for nuclear rcact)ions as we shall now proceed t’o show. To keep t’hc discussion simple wc shall only vonsidcr t#he sit#uations in which the only possible reaction is elast8iv scattrring. The procedure is easy enough to generalize and we shall indicak how this can he done but shall not carry out, the details. In the clustic scattering cast I’( r,, , rl

. ra)

= 1

if

j RT -

ra / >a

= 0

if

1 RT -

ra 1
‘3 Interact ions here refer to specifiwllv range forces we, :ts is customary, included in the asymptotic region.

nurlwr interact in the descript.ion

for all

ions.
(Y

(3.38)

and other long of the particles

Here RT is the ccntSer of mass of t#hc residual nucleus, and ru t’hc caoordinat’c of the emerging light particle. If we are dealing with the case of many final channels, then 1’ zz zp c ( 3 .:w i where I’, is a projcc%ion operator as defined in I a t,hv wave fun&ms for the renc*t8ion prodlwts do not overlap. Ah n (‘OllSP(~llCn~C t’hr I’artli principle is trivially sat.isficd, a \-cry important advantage of the holmdary wndition formulat~ion. The applkation of t,ht kinct,ica energy operator will generally give rise to singularit8ics. We shall avoid this difficwlty by defining Yz,,PP whcrc pa is the relnt,iw wordinate RT - ra as the limit ‘?I* as pa - u approa~hcs zero from the positi.re side, while CzJJ+ is the limit8 of ?Q4 as pa - a approaches zero from the negative side. However we now need to insure the wntinuity of +, i.e., 1% and QP must have the same value and normal slope at pa = n. This would have hwn :t cv~~~sequmw of the cclrutions if we had allowed the singrdarit’ies in ?f’\k to remain; hut it is more wnvrnient to have thcsc joining ronditions explicitly acw)lmted for. This ran he ac~hicvcd by introduckg t,he appropriate cwupling l~~ctwcn 1’9 and @P. I& IZ. thr lwunclarp c*ondit,ion oprrat’or, he defined as follows :

Here p and v are arbit,rary constants while surfaw pa = n. Thrn Eq. (2.6) is replawd [B -

/‘XI’

-

[IT -

Q%XQ -

d./&~, is the normal hy”

B]/W

derivative

= -ncpP

K]CpP =

to t,he (3.41)

--RI’*

( 3.42)

The f:wt i,hnt. siugularitics on hot,11 sides of rwh equatJion mlwt. matc*h lrads immediat,rly to the corAnuity of slope and valws in uossing thr dividing surfwe a. 1%~ rffec%ivr Hamiltoninn for I’* replac*ing l<(l. (2.10) is Pa =

It can hr rradily verified that this rfl”wti\-r lcru this pwpwe, c;orrec? joining ecluation. homogeneolw form of Eq. ( 3.42) . [tr:x Ii Singular

I~3rmdar\-

v:rluc

opr:ttors

Q;KlQ are

:ilso

Hamilt~oniun wr nrrd thr R]Si wctl

leads dirwt~ly eigenftmvtions

= 0 t)y

13lwtr

to the of the

P,
t.

302

FESHBAC’H

By integrating over an infinit,esimal immcdiat~ely show t,hat# Sk satisfied

region encslosing the surface pa = a one (aan the e(luation and boundary condit,ions

(f?‘h -

(3xc))Xx

ax,

v ~ = /Lxx an, In t’hc sune equat,ion

way,

Hamilt~oninn

( 3.43)

at

= 0

pa = a

(*orresponds

t,o the following

(3.44)

continllit,y

where q(a) just means t,hat, all pa equal a. The expansion for (E - QHQ) PI in terms of Xi is not snffkiently convergent t,o be used to obtain &Plan, Actually, since B involves just boundary operators there is no need to do so. Indeed 13~1. (3.45) is just, the fundamental equation of t’he Wigner-Eiscnbud theory and leads directly t.o t)he R matrix IlpcJn scparat,ion of \k int,o angular momcntmn eigenstates. There is of course no part’icular advant’agc in obtaining the results of the Wigner-Eisenbud theory in t’his fashion. We have shown t#hat it forms a rat,hcr natural procedure within our formalism. Within the framework of t,hc boundary condition model another insight, int,o t#hc direct’ reaction process becomes possible. region as suggested Perhaps, most important, of all, t,hr separatSion of a “surface” by Thomas (17) is easily managed by reducing a, with the consequence that interactions will occur in the region beyond c(. This does int8roduce compli~at~ions becanst of the possibilitSy of chamx~l (JVerhp. We shall not cntcr int#o these here, but, refer the reader t,o the discussion in the Thomas paper.

We can now rephrase the derivations of t,hc t,ransit,ion amplit,ude for resonance reactions given in I in terms of the projertion operat,or formalism. We shall first consider the isolatrd resonance case in which the width of the lcvcl F is rnllch smaller than t,hc energy difference, 11, between resonances. A.

ISOLATED

~~so~s~ck:

Our analysis is based very long lifet’ime the very small. Therefore, t,he compound state is (2.6h)

on the observat,ion that when t#he compound &ate has a probabilit,y of a particle re-entering an open channel is to a first approximation, t,he wave function describing a hound st#at,e solut,ion of t#hc homogeneous form of Ey. (8, -

3&Q)@,

= 0

(4.1)

9, is of cwwse not, the exact, c~~mpo~md nuclear state since it, has an infinite lifetime, a ~onsec~uence of dropping t,he ~~J~\lr krms of (2.Gh) which permitted t#hc decay of Q\lr into Z’\zI. However, lx~~ause the compound state has a very long liktinw WV may expect, t#hut tht compound stut.c wave func%ion will he closrly npprosimatcd by @,?and its resonance rwrgy by &,, . Thcsc considerations lcad to t,hcl following discussion of the efkct8irc Hamiltonian, H, FL{. (2.8) of t#hc open c*hanrwl mavc funct,ion /II, satisfying E:y. (3.9). WC expand the oprrat,or (fC - 3~uu)P1 111the complet8c set. of wave fmwtions dcfiacd 1)~ 11;q. (4.1) so that’”

whrre G is the wnt8inuum energ.y c~igcnvalue of (4.1) and where cr denot8es the eigcnvaluc which, in addition to t,hc cncrgy, is rcquircd to describe t’he st.ate a( c,, a). I~kumine now t#hr tx)havior of H in t#he ucighborhood of G,? It is quite clear t,hat, we can break up H int80 two part’, *, onr of which varies slowly in this neighborhood t,he other rapidly, viz: H = H’ + [&&,)(a?,&,,]

(I:’ - 8,)

( 4 3)

l~:quati:)n ( 2.9) now may tw writkli ( I< - H’) /‘*

= [x& &<)(@Jc ,p] ’ ( I< - E,)

and a solut.ion IGng the nwth.ods of I may hr easily ohtninrd cigrnflmctions of H’. If & “’ is the outgoing wa\~ solution of (I:’ - H’)&,” thc11

= 0

(4..;j in terms of t,hc ( 4.6)

304

FESHBACH

The t,ransit,ion date p is

matrix,

7’( ,6 j 0)) giving

the amplit,ude

for a transit,ion

t,o a final

Herr 9~(-’ is the solut,ion of (4.6) with incoming wave boundary condit,ion in amplitude which the incident, wave is in channel @. IlP(/3 1 0) is t,he t,ransition which would follow from Eq. (4.6). It is t,hc so-called direct, reaction amplitude while t,he stcond term is the resonant’ amplit,ude in the Brrit,-Wigncr form as shown in 1. Tp varies smoot,hly with energy in the region of t,he isolat(ed resonance, the rapid variation lxing described by the resonant, term. If t#hc rcsonanee can he characterized, beside the energy, by a specific set. of charact#crist,ic nmnhers such as tot’al angular momentSum, J, helicity, et#c., t’hen the second term just, reduces t,o the customary Breit-Wigner form. This is shown in I and need not he rcpcat.ed here. It, should he emphasized that’ in deriving t,he expression (4.8) it was not only hut, it was not, necessary to not necessary to int,roduee a sharp “channel radius,” carry out, a dccomposit~ion imo par%icular angular momentum st#ates or to ttsc any specific coupling scheme. Formula (4.8) is valid in any coupling scheme. In addit#ion (4.8) suggests what seems t,o us to he a ttseful extension of the widt,h concept. WC are led txo t’his most, directly if we cvaluat,c t.he tnacket in the dcnominat,or, part,icularly its imaginary part, more csplicitly. In this connect,ion note t#hat, the solmions of (4.6) for diffrrent~ channels having the same energy are ort’hogonnl (18) :


follows

that, A’“’ and I’(‘) defined

are given

E+

_

by \

1 @?T XQP

&hQ

(4.9)

@s,/

\, = A’“’ _ p/y) -

,-

(4.10)

by I A'"'

=

1 @s XQP 6 E _ H,

&Q

\ @A,,,/

(4.1 I )

where the last, eXpiTSSkJrlS wc have chosen t#o make tQe integration over tShc possible final angles, R, of emission explicit’. These results suggest the definit,ion

of it tliflcrential partial width which mcwww the rrl:lt,ivc into n giwn solid angle for n given channel fi :

prohabilitSy of emission

(4.15 ) By rxpanding (as .jtiyP $L”) in trrms of the spin, angulw momentum rigcnfunction a,ppropriut#e to a given coupling whcmr, t,he rrlnt,ion of (1.13) to the more cust~omary cxprrssion for t,hr widt,h m:ly lx> readily established. 1;inally let us point8 out, th:lt t,hr drri\-ation of (1.8) does not inyolvc any approsimat,ioils, contrary t,o what n1igh.t h3w hcrn indiwt,ed by I .I6 Homevcr the st,:ltement t’hat’ Tp is slowly vnryin g with energy is corrwf only when the rcwnxmw is isolat8ed. When t,hc rcson;Inws Art to overlap YP is no longer slowly va.rying, and Eq. (4.8)) :dt,hollgh cwrrcct, 1, ‘9 110 longw ~YJllv~~iliellt~. The rcsuks which :~pply ilr t’his situation will 1~ discussed shortly. Krturning to T, WC note t,hat, even when Tp dors vtlry slowly OVCP t,hr widt,h r of :I narrow isolzlt,ed rcsonnnw, it’ may it,sclf he luidcrgoing :L “single partklc” Ipsoii:~nce with :L witlt8h wry much 1xg;cr thnn I‘. In t’hta raw c:f nwlwr rcac+ons, this wrrrsponds to n large amplitlldc of fi,, inside t,he nl~c*lr~w lexling immtdint,cly to :L qr~alit:~t~ivc understanding of the so-c&d gi:mt’ rcsonanw. However, whtn the dcnsitSy of tine st,ruc*t~urc reson:mws is small, :1x it wn well 1~ in systrms which arc simplrl t,han the wmpoiind nuclear wse in t,hr sense t.hat, Q+ (~11 hc reuson:~hly ~~11 rc~prtscnt8c~d 1)~ just, ~1few modes, t#hc single partick :md fine struckrc rwon~nw mwy not, overlap.” Hcnw WC ~w~11tl find t’hat, there ;Lrc rwonnnws which :wr closely :wori:lt,cd with thr bound stfat8tl \V:IVP flmrt~ions of thr c*losed chamlcls ad “single part,klr” resoncmws which arc :L co~~se~~wnw of t.hc :Iction of :t pc)tcnt,i4 which nt, most, wrirs slowly n-it,h cnrrgg.

When Tp vuries rapidly wit,h t#hc sep:trat,ion of l’ int80 tShrw will not, sufliw to mu& tsplicit. t#o E. One or more neighboring king th:it rccluirrd t,o rrmow (4.5)

enrrgy ovrr the widt’h of t.he rcsonnnw into Yn , two parts is no longer wnwnicnt. In this wsr it, thr c4kt~ of t#he st,ate a,? whose energy E, is close &ten must, br iwludrd; t,he number involwd t.hr rapid wrrgy drprndrnw of TP . E’:clll~t~ion

hcc~ottlcs

Ifi This has been verified tw direct calculation t)y 1,. %:tmick. (lompar~ with remarks in Ref. (i). I7 These: rcnxtrks are p:trtidarly relrv:rnt whn~ we are dealing with :L few coupled equations such as ocwr in the description of dirwt nuclear reactions. or their counterparts in txxryon reactions of “elementary” p:Lrt,icle physics.

306

FE,SHRACH

where the sum CJVCr p is jnite and H” is given by (4.4)) except that in the sum over n, all t.hc t’erms on the rightbhand side of (4.14) are omit,tcd. Equation (4.14) has been discussed in I and there is no need, in view CJf the example afforded hy the preceding section, t#o rephrase t,his calculation in otlr present, projcc+on operat#or not’at,ion. The result#s art t’hat T zz lTNp + TNn

TN, is t#hr t’ransit’ion T”, has t,he form

where

matrix

which

follows

(4.15) from

t#he Hamiltonian

H”.

(4.16) Here $0 and combinations

$,+ are appropriaCe of

scattering Wl = c

where (4.14)

the _Yji’ are solutions : c

xy.Dr

of t,he secular

[( Ej -

E,)6,,

of H”; wi are linear

eigenfwkons

-

(4.17) equat.ion

wpy]Xti)

which = 0

follows

from

1Sq. (4.18)

where (4.18) The E’, in (4.16) and (4.18) are the complex eigenvalurs of the secular dekwninant following from Eq. (4.18) . with Note that, t,he expression for t,hat, part, of T, T,” which varies rapidly The numerat~ors are factorable energy is expressed as a sum over resonances. produck in which one facbr depends only on t#hc init,ial st,at,e (is independent of the final st,att) , t#hc second depends only on t#he final st,ate. However, vavh of the resonance trrms is not, in t’he Rrrit-Wignrr form. I:or csamplc, f:x pure elastic scnt#tcring ( no rcact,ion processes possihle) t,hr imaginary part, of Rj is not. simply proport.ional t’o the corresponding numerator. Inskad wrt,ain sum rules are obeyed. These arc given in ref. 19. One of these will be needed here for lat,eI discussion: (4.20) There are a minimal number of tjerms which must be included in the finite sum over ,A in I&l. (4.14). However, t#here is no limit, on the number of term:: greater than t#his minimum, so t,hat it is possible to ohtain an expansion of T in

terms of “rcsonm~~es,” wh.ivh. is idcnt,ical tmothat diwussed lwlmv in Section IV, C’. I~‘or most8 purposes, howe\-cr, it, is more mnvcnicnt tmokeep the nnmhcr of rwonancc tStrms to a minimum and to cxprcss the remainder of 1’, l’, , as the CWWCq~~cnw of the :wt,im of a potmtmtial as is implied l).v it s slm- ~arintSion with. energy. (‘.

li.4I~~h:-~‘~I~l~I~8

l’,xrassros

I’,sp:msions of 7’ int.0 an infiiiit,cl rwonaiiw series ham ~Y.YW obtained by a nwthod suggest.cd by Sicger? (ZO) and inxmtigntctl by Hun~hlct~ (21)) as well as 1)~ Iiapur and I’ciwls (5) with improvements hy Hrown ct al. ( ~2). For a review of these results SW IAIIC and Thomas ( 11 ) In t,hc Sic~;crt-I-lllnlhltt, type of cxpnnsion t hc rcsonmw mcrgits do not. drpmd tarpon E, the mcrgy, whereas in the Iiapw-l’citrls asc they do. We shall develop the Kapln-l’ricrls t,ypr here since it, is intirnnt~cly cwnnwtrd wit,h the discussion in Swt~ion IV, 13 :wd shall relrgat~c a drrivatim of the Siegcrt,-HuInblct, c~spnnsioll to an appmdis. The li,:lpur-l’cicrls type of cqmnsion c*an he readily o\,t,ainrd in the present f’orrnnlisni hy nppr0priat.c analysis of the fundarncnt~al ccluations ( 2.::) and (2.4). Tow:wd t,his cmd WC shall obt,nin :tn esuc+ cxprrssion for t,hc transiti~,n niat,ris. I:irst solve (2.&L) for PP:

1% = $4” + F+ ! x Xry Q* J P,’ .1s t,he solr&on of the homogeneous form of (2.63) vhosrn so as t#o I&“’ snt,isfy appropriat,e boundary wmditSiom. Insertming t#his cciuation into (2.6b) leads tl) an tffeckirc Harnilt~onian ecluation for Qq of a form similar to (2.9) :

wllc~rf~

WC wri iiow obtain Q* and s1thst,itutr for Q* in (4.21 ) and so obtain I*. rcsult,ing expression for the tmnsitmion mnt,ris is

Tlitx

where l’,, is t,hc scat,tcring asswiat~cd with t’he honmgrncor~s form of Eq. (2.tia) . It, is natural t,o expand 7’ in terms of t,hr eigenfunctions Z of thr effcc~t,ivr Harniltonian for Qq. The corresponding rigen~alucs E,Y are complex wit,h n negati\x, imaginary part, as WC shnll shortly show. T then \wmmws

(4.24)

308

FESHBSC'H L I

Zt” and E, form a biorthogonal

set. These functions

Et - x QQIn

tJernls

%Qp

1 E+ _ xpp

&'Q

are solut8ions of

1E,=0

(4.35 )

of Eq. (2.3) and ( 2.4) E1 is t,he solution for (sq while

Note t,here is no incident’ wave. p?lr thus satisfies the Iispur-Peierls t#ype of boundary condit,ion (,3) ; we hare been able t#o specify this without t#he use of a “radius.” In this formalism, however, one has t#o pay the price of shifting the boundary condition as E changes according t’o (4.86) , thus using a different’ orthogonal set as i7 changes. For a further discussion of t’his point, see Brown (22) The main difference between t#his development and t#hat, of Sect#ion IV, B is that in t,he latt,er t’he energy dependence is carried by the Hamiltonian H’. But of course t,he k-o met’hods arc complct,cly cquivalent8. The principal advantage of development in Section IV, B is the explicit, scparatSion of fine and gross struct,lu-e resonances. The function EL” is given by: m .4 3* (4.27) z-=t = at To obtain t#hc imaginary and int,egrate

Im E’, =

partf of Et , multiply

-&XQp6(E

-

(4.a.sj from t’he left by ( Sr) *

&p)&Q~:t

Kot,e that ImE, is less than zero. We have thus obtained the major results of the Iiapur-Yeierls formalism using a very simple almost, algebraic procedure. 50 radius or separat,ion int#o partial waves was required. D.

THE

COMPLEX

I'OTENTI.ZL

As described in 1, t,he pokntial tude (I’):

?tIon~r,'~

must8 account, for the average trxlsition

(2’) = Tp - 2, C J i = TP I8 The result, Ey. vst,e communication).

(4.33)j

ol~t:rir~etl

$jj C P in this

ampli-

(~~-)XpgWi)(WiX~p~~+))

(4.29) {~~-'XPQ~,,)(~~XQP~~~+))

section

was first

derived

by M.

Paranger

(pri-

where we have made use of I’:q. ( 1. I(i) and P:q. (4.20) . AR is t,he energy interval over which T is averaged. I& the complex pokntial wave func:tSion which is t#o give rise t,o W29) he cp. Then cp must sat,isfy t,hr rcluakm:

-It-" x;+) = 1- i*p+ _ H,, XPQ AXQP 1 1 1

[

k’inally,

operat,ing

with ( f$

_

E -

H” on hot#h sides we

HN)

---.--.~L.

1

-

%a

____-_ 1

j;+

(JkLiil

lJ = 0 H,,


01 lf

-

[It’ - H” - Hci,+ HP,,+ = -iiacxpQ

= 0

( 4.32J

1

( 4.x5 )

H” gives rise to the direct processes and the additional term gives the averaged influcncc of t#he compound nuclear processes on t#hr direct,. H” is of course very closely related to the shell model potential. Xot’e t’hat, tbis suggests that, c(JInpkx pottnt’ial models can be employed not only for t’hc scat’brring of nwleons where it has hccn so extensively applied, hut also t’r) ot,her reactions sllch as, for example, t,hc entire syndrome of rract8ions which occur when dclltcrons are the incident or emerging projectile including elast,ic, incla&v, and stripping and pickup processes. An approximate evaluation for neutron ind1wcd reactions will he discussed in a paper in prepnratSion. Our final remarks are concerned wi-ith t’he Hamilt80nian x of Ii:q. (2.5). ikcaustic of the singular hard core interactions h&wccn t#wo n~wlrons x is ~~cjfthe

many-body Hatnilt,oniatt hut a sttibthly modified nonsingular \-et&n, t hc prwiw det~ermittatiott of which involws a self-consi,+xtt proccdrw of the Br~vxknet t,ypc. I’f>Y t’he present p~wposcs it is sufficirttt~ to note that itt thr limits of lo\\ and high energy good estimates of x are nvailat~lc. &it low energies, ,K cotiaists of t,hc shell model Hamilt80ttian p111s the residual potctttktl. The lat,ter will generally include two-body potcttt,ials as well as c~ollec%i\-clmodel itttcrac.tiotts.“’ At, suitably high encrgits it is pcrmissihlc to nrgltct, eschangc scat,terittg and c*onsequcttt~ly to treat t’he incidcntS nucltott (or ~~~wlcotts) as tlistdttguishnl-,It~ from t#hosc of t,h.e t,argct ttuc~lcus. Then x can he written as a sun of the Hamiltonian for t,he t,argct, tturleus and an effective poktt&l describing t#hr interaction of the incident nucleon and t#he target, nuclr~w where t ht. latt,er is detwmined via the multiple scat’terittg approximutSion as dcvelopcd for esamplc in tvf. 6. Ottcc the cffcct,ive Hamiltonian is gi\.en, a quantSitative t,rcatmcttt, of rcwt~iotts twcomcs c:onceivahle and it is for this reason that, t,hr explicit csprcssiott for the projection opcmtors was ohtainrd in Section III. It, is prohuhly not1 necessary to add that, the formalism dcvcloped in this paper has a wide range of applical)ilit9y extending heyotttl jttst nuclear reactSions i~potl which ottr :tt,t,etttiott has heett focused. Thtl natm-c of the system involved has been kept, open except, for the assumption that a Hamilt~otiian exists, t#he Hamiltonian itself remaining mnspccified. The formalism can f~fJ1lSfY~l~fYl~~y bc applied t(o rcsctions involving many-body systems. The Humiltottiutt may he given itt c~ottfiguratiott space and describe a system of ittkrac%ing nwleons or it, may 1~ a field theorrt,ic Hamiltonian in which case ottr prowdrur is c~los;c~lyrelat~etl to the ‘I’amn-Dan&f nirt,hcxf APPESI~IS

The liapur-l’eierls t,ype of expansion sttflcrs from t#hc diflicrtlt,y that. the complex eigenvalucs h’,, are fun&ons of the incident, energy. Rigorously the resonances of a physical syst,em correspond to poles in t,hc S-matrix which ocwr at, energies (we shall call these co~t~pkx rcso~~~~zcc cnr~~~ics) wit,h a ttegat,i\-(1 imaginary part, corresponding t80 the half widt,h of the resottanw. These complex eigcnvalucs arc of course independent. of t’hc energy E of the syst’cm. However it, may he shown (32 ) t,hat, t,he complcs eigenvalucs of t#hc IiapttrI’cierls cspunsion arc c~losc to the complex resonance energies as long as we art concerned mit,h t#he complex resonancr energies whose real part, is near the energy i? and whose imaginary part is small. This is in fact the merit, of the trcatment~ in Section IV, B since the rrsonance dcscriptSiott is reserved for only t#hwr krms. It may he occasionally cottvrnirttt. to express t,he resonance terms directly in 20 Actually the resitlud t hrse art= usu:~Ily neglected.

interaction

nxty

very

\\-cl1 include

mtny-body

potentials;

hut,

terms of the complrs resonanw energies and the relakd wave fwwt,ions. This we shall now proceed t,o do. The complc~s resonanw energies arc eigrnvalucs of t,he tot’al HamiltIonian x of the system, the wave fuwtions being srthjcd t#r)the rsdiat,ion boundary wndit,ion:

where I:, has t.hc as1~~1 relation t:) the channel energy and pn is the radial wordinat~t for c*hanncl CY.The cigrnvalws of X suhjwtS t,o t,hc boundary cwnditSion (A.1 ) inc~l~de t,hr hormd staks in which case /<, is pure imaginary, Imli, > 0. The cwmples cigenvnlws of the syskm cwrrespond t,o k, n-it#h ImI<, < 0 while Kck, 5 0, t,hc t,wo signs cvrresponding to ingoing and outbgoing wa\~s. In spit,e of t#hc f:wt that t,hr cigrnflmc+ions correspondin g to these complex cigenralws grow espo wntially as pa approarhcs infinit#y, t#hey form an orthornormal set,.” If the eigenvallles are E, and the cigenflmc+ons $2, , the ort8hogon:rlit.y and normalizat ion c9iidit,ions are :

The optmtor ,K operaks only on 12, There is no difhcwltSy in evaluatSing t,he inverse operator (It’,xj -’ for large ~alucs of p a since here the int,craction vanishes. ‘The volrmw element8 c1r signifies t#he J-olumc clement, in configurstion space while c/S, is the siwfacc element. Kot,e that, the surface integral which has the fmwt~ion of canceling the divergent part, of the 11sual normalization integral is c~al~lat8ed at infinite separation, that. is, in t#he region where the asgmptSotic forms of !2, and 62, are valid. The n:)rmalixation and orthogonnlity given hy 151. (AZ) permit, us tBoproject orIt’ of any func*tiorl that part which is proportional to a part8icular II, and ort,hogonal to t,he other 0, as we11 as the holmd states. Wit(h this de\-elopnwntS in hand it hecomes possible to ohtNain a complex cigcnxxhw rcprcsrnt.ation of the t’esonanw tIerms in t)hc t’ransition matrix. WC writ,c t,he cvmplrtSc wave fmwtion ft)r t#ht system, 9, in terms of an “in&wt,” wave $,, and a wattrred n-a\.r \lr,<:

where J/” is detincd to he the solut8ion of the eqllation

21 We shdl following.

not

discuss

their

completeness

:I moot

l)roprrty

as we shall

not

need

it, in the

subject t#o t,h.c wndition t#hat, +,I asymptotically approach an incident’ plane wave in one of t,hc open ch.annels. Since t#here is no coupling t#o the closed channels Q#,, = 0. +,, will of course wnt,ain outgoing waves which describe scatt#ering and reactSions. We shall call the corresponding transit,ion amplit,ude 7’“( /3 1 0) whwe /3 dcnot,cs :L possible open exit, channel. q’, satisfits the cquat’ion CR The

transitSion

amplit,llde

T(P

;K1) qs = xH3up Z’l+h

( A-4)

1 0) is the sum of ?‘,,Cp 1 0) and T,( @ j 0) :

T, ( p 1 0) = (*b~‘FK urJJ$o) or

Employing the orthonormal conditions (AZ) it is possible t#o pick out the dependence of t’he Green’s function (E’ - no)-’ on a particular Q,, . It is however useful t#o consider only those st,ates Q,, whose imaginary part is relatively Pmall. Including all t’hc possible 2, would lead t’o convergence problems since in such a series we expand a bounded function in t,erms of funct,ions which have cxponent,ial gr0wt.h. We therefore writme

So surface t,ernw have been included, an omission which is valid if xPcd decreases with increasing pa more rapidly than rxponent,ially. Thus, for a finit$e range xKlp4, or one which is given by a gnussian, no surface t#erms will be present. Kate that, the sum over p is over a finite number of terms and bhat, 7’:“’ represents t-he rrnlaindcr. Again we note that, t,he dcvclopment leading to t,his result did not, rccllGrc the not.ion of cahanncl radii or a decomposition in orbitma angular momcnt~um eigenstCatScs. An expansion of 7’ in 3 complex eigcnvaluc expansion i,s also obt8aintd by Iiosenfcld and Humhlet (5) but, only :aft,er separat#ing int#o partial waves.

The author is very much indebted to V. F. Weisskopf. This paper as well as I wag initiated by a number of questions he raised. The import,ance of a proper definition of open channels was emphasized by IX. P. Wigner. The discussion of the Kapur-Peierls formalism was developed in the course of a number of conversations with Richard Lemmer. The aut,hor is grateful to Norman Francis for his critical reading of the manuscript, and for many cogent suggestions. The author x-ould also like to thank his students B. Block, I,. Estrada, and I,,. Zamick for their help in many of the formulations presented above. RECEIVED:

March

2, 1962

NUCLEAR

REACTIONS

313

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