A Unified Theory of Nuclear Reactions 2

8/11/2019 A Unified Theory of Nuclear Reactions 2

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ANNALS OF PHYSICS: 19,

287-313 (1962)

A Unified Theory of Nuclear Reactions. II*

The princip al device eml)loyed, :ts in lxlrt I, is 1 IC 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 con-

pound nuclear reactions including the cor~plrtl rquation clewription for direct

rewtions renmins valid in this wider contrxt A Ik~pu1~ ~lcirrls espm sion

m:~y dso be rr:ttlily ot)t:tincd. The conce pt of ch:~nrlel rxtlii is not nretlrd nor is

any dec onipo sition of the LS ive function for thr system into :mgul:tr momen tum

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~ cou pling

schcmc. Since thp

OIIPII

channels can only Iw

~lefirl~tl

in XII asymptotic sense,

the correspond ing 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)hcno mcln:l \vhich c:m occur in nuc.Ic:Lr rewtiow :tnd in

t hc secon d ~)l:tw can c~fle~tiwly rxploit XI intuitive nnclrrst:rnding of the

phenomrnx. li:\-:unplc~ of pr.jwtion opcr:~tors :IIP obtained inclu(lin g one

which lexls to t hr \2ii~npr-15 i~p~Il,llti foun:~lisnr. :mot hpr \vhich is :tppropriate

for the strip[)ing wwtion. :rnd, finally. one \I-hich t alie s the Inu li rxclrision

l)rinc.iplc into accou nt Note, that clxl)licit r.cl)~.(~~(~~~t:~tiolls of the projwtion

oprr:ttors XIP not required for t hr tlrvelopmrnt of general formd results I)rlt

:ir( rrwws:try if, rvrnt u:lll~,, ct~wn tit:ttivc c:llcrll:~ tiolis :~rp tiiatlc .

I. INTI~OI)~~CTION

In this p:tpw the forrnalisn~ dcvclopcd in ?I Inificd Theory of Swlcur

ltesctions (I ) is gcnernlized and irnprovcd. In I wc wcrc principally conwrncd

with developing :L theory of nwlcar reactions from which :l description of direct

and compound nuclear proccssrs, as ~11 a:: tht mnplcs potential model, would

1x1 both cxsily and nat~uxlly alwtractcd. This inwcnpahly led to :I dcri\-ation

* Th is Lvork is supported in part through .4Is:C (ontrwt AT(30 I)-2098, by funds prw

vided t)?; t.ho IV.9. .4tom ic b:nergy Com mission , thr ( )Iiire of T:,v:d I?ese:~rch :tntl thr Ait

Force Oilice of Scien tific I


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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 pro-

jectile and that of the emergent light particlc were the same as they would bt

in an (n, p) or (n, n) process. Stripping and similar prowsses were not treated.

Secondly, although it was possible to show that t,hc ef fects of the Pauli exclusion

principle and exchange scattering would not change the structure of the formal-

ism, 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


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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 Hamiltonian. In another treut-

ment 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 cigen-

functions have complcs eigcnvnlues and are clearly of the Kapur-Irierls 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-Jcierls 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#ential 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 thc 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 differen-

Gates hetwren the narrow compolmd nwlcar resonancw and thr single particle

giant, resonances ; i.e.,

brtween compound and dirrrt nuclrar reactions. In the

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 flexihil itly which is extremely usrful since it allows one to choosr

t,hat projection operator whirh is most convenient, for the problem under in-

vrstigation. In the present paper wr arc for the most part, cwncerned with thr

type of projection 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 are functions of the cnerg,~-, h, of the systenl. i\n e>-p;lnsion of the

twrisit~ion elnployirlg vollll)les

eigcnw llucs which are independent of E is referred to as the

*Siege d -H:m~ldet series. Such :I series is derived in the appendix to this p:~pe~. nor :, yevent

~liscussion see IIumhlet and I~osenfrld 151.


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wave functions are linear superposit,ions of the possible residual targrt, nwlenl

states. The class of these states arc restrict,rd of course by energy c*onservstion

and other 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 thc channel radii are chosen

so that cscept for Coulomb or ot,her long range pot,rntials 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 essential ly 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 appropriate when drtails of the interact,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 ones

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 poten-

Gal 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 thr 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 de-

veloprd 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 oprra-

tors. 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 throry 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


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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 comnmni-

cation.

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 products 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 encrgetirally allowed are referred to as open channels; the others are

closed cahannels. Let the wave functions dew-ibing t#he energetically possible

residual nllc~lei hc drnot)ed by c$, . Note that, these functions are generally not

mutually orthogonal nor do they nwessarily involve the same number of co-

ordinates. 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

optrator.

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)isymmetri-

cal 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:


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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, satisfy ing 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 por-

tion 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. More-

over a projertion 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 that, 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 the projection opcra-

tor. If 9 is the wave function for t#he system it is clear t,hat. we need only ran-

sider 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 situntions in

which the final reactions products can be arbit8rarily complex. There is no limita-

t,ion on the number of open channrls or on tht 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 C alled generalized optical potcnt,ial in I) . q sat. isfietl

t,Ee equation

6 This option wvas n~entioned in I. w:ts partially described in ref. 4 and hats been emplopcd

in refs. 8 and 9.


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Since @ projeck int#o t,he calosed channel suhspace we can solve (2&b) as

f0110\vs8

(2.8)

SubstitAng Eq. (2.8) in likl. (2.(k) yirlds t,hc Schroedingrr cquation for Pq:

(,B - H)Z% = 0

(2.9)

where N., the effect~ivr Hamiltonian, 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 appl icabil ity as a consequence of the generalizat8ion in t,he

tlcfin ition 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 val id 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. 113 )JE:(TION OIISIL4TOIW

In this scvtion 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 con-

liguratioii 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 thr ~~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: rc not identical with t)he particles in the target nuc~lcl~s. l,c,t8 ht, wnvc


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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 to excite t,he firs t, II excited

states of the target whose wave full&ions are +j , j running from one to n. Then

Inserting this result into I


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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 del l 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 thc proj&ion pro-

c~lurt~ one st#tp further 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,)~ l t r.,) . lomarcl this rnd n-c write

wtrcl , rJ = P.du + QNw

\vhcre I,v is t,hc projec+on operatot

13 = .fX.T

:md

QN = 1 - I.%J

Ihe fun&m 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) = 0

(3.9)

Sotc t,hat 21 will asympt~ot~i~ally clrscrihe thc 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)

(3.10)

111' Q,,,H,,~,Q,\Q~EI

H,,,,(r,, , r,).f(r,)u(r,) - ./(rl)N(ro)u(ru)

\I.11 IT

HCrcl) = (ftrl), Hdro , rdf(rd)

CXllj

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 to dcwrihc a rcxc:tion invoking t,hc> ampti-

t,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


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296

FEHHBACH

N is just 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) beco mes an (qua-

tion for

where

I* = u(ro)40(rl . . .

rJ + QNw(fO, rl)lC0(r2 . . rA)

(3.14)

QN = 1 - 40)(40

The equations for u and QNw are identical in form wit,h Eq. ( 3.10) :

(3.15)

w - (4 , IHP4,)lu = (4lJHlQNW~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 cf fec-

t,ive potSential will be hcrmit,ian; some treatme& drduce nonhtrmitian effec-

tive 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 thc antisymmetrized wave flmction

4drl . . .

r.4) the projectIion operator we require has the following property

P*(ro,

rl . . rl) =

@u(r,)4,dr, ra)

(3.17)

where C% s the antisymmetrization operator and + is antisymmetrized. I f 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)]) = 0

(XlSj

where the integration is over the coordinates which are common to hot,h sides

of the inner product in (3.18). That condition (3.18) ib sufficient can be shown

as follows. From (3.18) we have

I1 In

the

discllssion to follow, 4 is :tn arbitrary function

except t,hnt

it satisfies xl)l)ropri-

nte

boundary

conditions at infin ity.


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(u(r&O(rl f . rlj , [*irOrI . . rl) - @n(rOMrl . . r.dl) = 0

(3.19)

and sinw the qknnt,i t,y in the squarr hrwkct~s is antisymmetSrical, it follows

tmhat,

(Ctu(rO)ddrl . . r.4), [\k(rOrl . . r..,) - Uu(rOMO(rl . . r..,]) = 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 this we nerd first . to sol\-c 1:(1. (3.18). Iw-

forming t,hc indicated opclrat,ions Eve oht,ain

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

(32lj

I(ru) = (&(rl . .

h), *)

( 3.22)

integral rq~c~t,ion of t,he lrcdholm type

Trj, +drm . r.4))

(3.~2)t

Ilr,,) = u(rJ - (K(ro ) rl)w(rl))

( 3.21)

lhc krrnel K has several imporbant, properties. It, is hermitian. Since 4 is the

wave function for a hound st,ate, K decreases cxpownt~ially as either

rl

or

ru

hwon~c~ infinite. K is hounded:

lhc 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 ohtain when c#,) s a Slntcr dct,erminant composed of

mut~unlly ort,hogonsl rlcments wi Then

.A

K = c w,*(t)w,(rO)

1

(3.X)

Bcfor(~ solving t,he inhomogencous tquat,ion (3.21) it, is wnvenient t,o consider

the eigrnvaluc problem of the wrmsponding homogeneous form :

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

(:r.ar,,

From the proprrtirs of K it follows t)hat# X, are real and

/ X, 1 1 1 and c 1 = A

a

(3.26)


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298

FF:SHH..(H

The eigenfunc%ion u, will form a11 orthonormal stlt, which is not necessarily

complete. However, 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 the 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 . h)) = 0

From this equation we have

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


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As indicated, the terms in u,l--tShc amplitSude of the 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 h)

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

(Ixv, ( 1 - r)*) = 0

Ihtw two propert,ies haw t,hc consequrnw 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

+,J(+,, and thw antisymmctSrizc.

This resldt, is no surprise from the swond

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

cul llot. 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


14/27

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 to exchange

interactions in the cfiect,ivc Hamiltonian. Kot,r that the cf fwt ivc 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 multiplying 1%~. (2.9) from t,he lef t

by

+O*(rl . . .

r,,l) and integrating over coordinat#es rl . . . r.i . The prop&k of

the projection operat,or permit some important simplifications in t#he resultant

equation. Xote that Eq. i3.18) can be ren-rit,ten

7(r0) = (40(r1 ... rA), *) = (40(r1 ... r.,), T*) (3.18)

This equation is valid for Amy antisy~~~wtrical function q. Rwarw of this it, is

most convenirnt~ t,o consider t,hc equatJion for C which, in lirtuc of (3.18) and

the symmetry of H hccomes:

This equation 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 uni ty 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 to show.

To keep thc 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

for all (Y (3.38)

= 0 if 1 RT - ra 1


15/27

Here RT is the ccntSer of mass of t#hc residual nucleus, and ru thc caoordinatc of

the emerging light particle. I f 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 thr Iartli principle is trivially sat.isficd, a \-cry im-

portant 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 dif ficwlty 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 singrdarities 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

19

and @P. I& IZ. thr lwunclarp c*ondit,ion oprrator, he defined

as follows :

Here p and v are arbit,rary constants while d./&~, is the normal derivative to t,he

surfaw pa =

n. Thrn Eq. (2.6) is replawd hy

[B - /XI - B]/W = -ncpP

(3.41)

[IT - Q%XQ - K]CpP = --RI*

( 3.42)

The f:wt i,hnt. siugularitics on hot,11 sides of rwh equatJion mlwt. matc*h lrads im-

mediat,r ly to the corAnuity of slope and valws in uossing thr dividing surfwe

Pa =

a. 1%~ rffec%ivr Hamiltoninn for I* replac*ing l


16/27

302 FESHBACH

By integrating over an infinit,esimal region encslosing the surface pa = a one (aan

immcdiat~ely show t,hat# Sk satisfied the e(luation and boundary condit,ions

(f?h - (3xc))Xx = 0

ax,

(3.44)

v ~ = /Lxx

an,

at pa = a

In thc sune way, Hamilt~oninn ( 3.43) (*orresponds t,o the following continllit,y

equat,ion

where q(a) just means t,hat, all pa equal a. The expansion for (E - QHQ) PI in

terms of Xi is not snf fkient ly 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 the 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 particular advantagc in obtaining the results of the

Wigner-Eisenbud theory in this 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.

Perhaps, most important, of all, t,hr separatSion of a

surface region as suggested

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 on the observat,ion that when t#he compound &ate has a

very long lifetime the probabilit,y of a particle re-entering an open channel is

very small. Therefore, to a first approximation, t,he wave function describing

t,he compound state is a hound st#at,e solut,ion of t#hc homogeneous form of Ey.

(2.6h)

(8, - 3&Q)@, = 0

(4.1)


17/27

9, is of cwwse not, the exact, c~~mpo~md nuclear state since it, has an infinite life-

time, 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 Hamil-

tonian, 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

111 he 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 the 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& &


18/27

304

FESHBACH

The t,ransit,ion matrix, 7( ,6 j 0)) giving the amplit,ude for a transit,ion t,o a final

date p is

Herr 9~ - is the solut,ion of (4.6) with incoming wave boundary condit,ion in

which the incident, wave is in channel @. IlP(/3 1 0)

is t,he t,ransition amplitude

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. I f t#hc rcsonanee can

he characterized, beside the energy, by a specific set. of charact#crist,ic nmnhers

such as total angular momentSum, J, helic ity, et#c., then 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

not necessary to int,roduee a sharp channel radius,

hut, it was not, necessary to

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 this most, directly if we cvaluat,c t.he tnacket in the

dcnominat,or, part,icularly its imaginary part, more cspl icit ly. In this connect,ion

note t#hat, the solmions of (4.6) for diffrrent~ channels having the same energy

are orthogonnl (18) :


19/27

of it tlif lcrential partial width which mcwww the rrl:lt,ivc prohabili tSy of emission

into n giwn solid angle for n given channel fi :

(4.15 )

By rxpanding (as .jtiyP $L) in trrms of the spin, angulw momentum rigcn-

function 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 that 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 this 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 iso-

lzlt,ed rcsonnnw, it may it,sclf he luidcrgoing :L single partklc Ipsoii:~nce with :L

witlt8h wry much 1xg;cr thnn I. In thta 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, ~1 ew modes, t#hc single partick :md fine struckrc rwon~nw

mwy not, overlap. Hcnw WC ~w~11tl find that, 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 enrrgy ovrr the width of t.he rcsonnnw into Yn ,

t#hc sep:trat,ion of l int80 tShrw two parts is no longer wnwnicnt. In this wsr it,

will not, sufliw to mu& tsplicit. thr c4kt~ of t#he st,ate a,? whose energy E, is close

t#o E. One or more neighboring &ten must, br iwludrd; t,he number involwd

king th: it rccluirrd t,o rrmow t.hr rapid wrrgy drprndrnw of TP . E:clll~t~ion

(4.5) hcc~ottlcs

Ifi Th is has been verified tw direct calcu lation t)y 1,. %:tmick. (lompar~ with remarks in

Ref. (i).

I7 The se: rcnxtrks are p:trtidarly relrv:rnt whn~ we are dea ling with :L few coup led

equations such as ocwr in the description of dirwt nuclear reactions. or their counterparts

in txxryon reaction s of elementary p:Lrt,icle phys ics.


20/27

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 terms 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 notat,ion. The result#s art that

T zz lTNp + TNn

(4.15)

where

TN,

is t#hr transition matrix which follows from t#he Hamiltonian

H.

T,

has t,he form

(4.16)

Here $0 and $,+ are appropriaCe scattering eigenfwkons of H; wi are linear

combinations of

Wl = c xy.Dr (4.17)

where the _Yji are solutions of t,he secular equat.ion which follows from 1Sq.

(4.14) :

c [( Ej - E,)6,, - wpy]Xti) = 0 (4.18)

where

(4.18)

The E, in (4.16) and (4.18) are the complex eigenvalurs of the secular dekwni-

nant following from Eq. (4.18) .

Note that, t,he expression for t,hat, part, of

T, T,

which varies rapidly with

energy is expressed as a sum over resonances.

The numerat~ors are factorable

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 the 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 to 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


21/27

terms of rcsonm~~es, wh.ivh. is idcnt,ical tmohat diwussed lwlmv in Section IV, C.

I~or most8 purposes, howe\-cr, it, is more mnvcnicnt tmo eep the nnmhcr of rwo-

nancc tStrms to a minimum and to cxprcss the remainder of 1, l, , as the CWWC-

q~~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 infi iii t,c l 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 Iciwls (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-lcitrls asc they do. We shall

develop the Kapln-lricrls 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-lcicrls type of cqmnsion c*an he readily o\,t,ainrd in the present

forrnnlisni 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,

wllc~rf~ I&

.1s t,he solr&on of the homogeneous form of (2.63) vhosrn so as t#o

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) :

WC wri iiow obtain Q* and s1thst,itutr for Q* in (4.21 ) and so obtain I*. Tli tx

rcsult,ing expression for the tmnsitmion mnt,ris is

where l,, is t,hc scat,tcring asswiat~cd with the 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)


22/27

308

FESHBSC'HI

Zt and E, form a biorthogonal set. These functions are solut8ions of

Et - x

1

QQ %Qp E+ _ xpp &'Q

1

E,0

(4.35 )

In tJernls 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 to (4.86) , thus using a different

orthogonal set as i7 changes. For a further discussion of this 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 the energy dependence is carried by the Hamiltonian H. But of

course t,he k-o methods arc complct,cly cquivalent8. The principal advantage of

development in Section IV, B is the explic it, scparatSion of fine and gross struct,lu-e

resonances.

The function EL is given by:

m .4 3*

z-=t

= at

(4.27)

To obtain t#hc imaginary partf of Et ,

multip ly (4.a.sj from the left by ( Sr) *

and int,egrate

Im E, =

-&XQp6 (E - &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 ?tIon~r,'~

As described in 1, t,he pokntial must8 account, for the average trxlsition ampli-

tude (I):

(2) = Tp - 2, C

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

J i

(4.29)

=

TP - $jj C

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

P

I8 The result, Ey. (4.33)j ol~t:rir~etl in this sec tion was first derived by M. Paranger (pri-

vst,e communication).


23/27

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:

[

1

1Ix;+)1 i*p+H,,PQXQP-"

kinally, operat,ing with E -

H on

hot#h sides we (JkLiil

( f$ _ HN) ---.--.~L.

1

lJ = 0

____-

1 - %a j;+ _ H,,


24/27

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 the 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 the 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 scatterittg 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 Hamil-

tonian 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

fie ld theorrt,ic Hamiltonian in which case ottr prowdrur is c~los;c~lyelat~etl to the

Iamn-Dan&f nirt,hcxf

APPESI~IS

The liapur-leierls t,ype of expansion sttflcrs from t#hc diflicrtl t,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 com-

plex eigcnvalucs arc of course independent. of thc energy E of the systcm.

However it, may he shown (32 ) t,hat, t,he complcs eigenvalucs of t#hc Iiapttr-

Icierls 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 interaction nxty very \\-cl1 includ e mtny-body pote ntials; hut,

t hrse art= usu:~Ily neglec ted.


25/27

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 total HamiltIonian x of

the system, the wave fuwtions being srthjcd t#r) he rsdiat,ion boundary wndit,ion:

where I:, has t.hc as1~~1 relation t:) the channel energy and pn is the radial w-

ordinat~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 / 0.

The cwmples cigenvnlws of the syskm cwrrespond t,o k, n-it#h ImI


26/27

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, satisf its the cquation

CR - ;K1) qs = xH3up Zl+h ( A-4)

The transitSion amplit,llde T(P 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 de-

pendence of the Greens 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 thc possible 2, would lead to convergence problems since in such

a series we expand a bounded function in t,erms of funct,ions which have cxpo-

nent,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. Th is paper as well as I wag initi-

ated by a number of questions he raised. The import,ance of a proper definition of open

channe ls was emphasized by IX. P. Wigner. The discuss ion 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 critica l reading of the man uscript, and for

many cogent suggestions. The author x-ould also like to thank his students B. Block, I,.

Estrada, and I,,. Zamick for their h elp in many o f the formulation s presented above.

RECEIVED: March 2, 1962


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NUCLEAR REACTIONS 313

REFERENCE8

f. H. FESIIBAC'H, ;lnn. Phys. (:\lY ) 6, 357 (1958).

2. E. P. WIGNER AND L. EISENBUD, Phys. Rev. 72, 29 (1947); T. TEICHMANN AXD E. P.

WIGNER, Phys. Rev. 87, 123 (1952).

5. P. I. KAPI~RAN D R. PEIER I,S,PTOC . Roy. Sot. A166,27 7 f.1937).

4. H. FESHBAC H, dn/l. Kev. Xuclertr Sc i. 8, 49 (1958); ilnn. Kev. Zm. (Stanford).

5. J. HI-GIBLET AKD I,. ROSENFELD, Nuclear Phys. 26, 529 (1961).

6. -4. I

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A Unified Theory of Nuclear Reactions 2