DWBA in the shell-model approach to nuclear reactions

I.D.1

]

NuclearPhysics AI05 (1967) 489--521; (~) North-HollandPublishin# Co., Amsterdam

1

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

D W B A IN THE S H E L L - M O D E L A P P R O A C H TO N U C L E A R R E A C T I O N S J. H O F N E R , C. M A H A U X * and H. A. W E I I ) E N M / , ) L L E R

lnstitut fiir Theoretische Physik der Universitiit, Heidelber9, W. Germany Received 7 July 1967 Abstract: The scattering matrix S is decomposed into two parts, the compound nucleus part and

the direct interaction part S(DI). The compound nucleus part S(CN) has uncorrclated partial width amplitudes, so that its energy average has vanishing non-diagonal matrix elements. The direct interaction part is defined as the difference S--S(CN). The decomposition is carried out, in the shell-model approach to nuclear reactions, by a suitable decomposition of the residual interaction, which bears close analogy to the separation proposed recently by Austern and Ratcliff in the frame of a simple model. Under proper statistical assumptions, it is shown that the energy average of S(DI) can often be calculated in first-order perturbation theory aod is then given by the familiar DWBA term, if the channels are weakly coupled. The conditions for convergence of the Born series are investigated. The wave functions entering into the DWBA expressions are detined by optical-model potentials. The potentials are derived, and some of their properties are discussed. It is shown that they are equal to the conventionally defined, phenomenological optical-model potentials inasmuch as they reproduce the energy average of the elastic scattering phase shifts. The modifications of the DWBA due to agiant or microgiant resonance or to strong coupling between some channels are given. The analysis of experimental data is discussed for the case when both compound and direct processes contribute to the cross section. The connection between the present work and the approaches used by other authors is exhibited and discussed.

1. Introduction The distorted wave Born approximation (DWBA) plays an important role in the analysis of nuclear reaction cross sections. The foundation and the conditions of validity of the DWBA are, nevertheless, only poorly known. We feel that a reexamination of the problem connected with the derivation and foundation of the DWBA is justified in view of the recent progress in the lield of nuclear reaction theory. Because of the existence of dynamical approaches to reaction theory ~-8), one is now in a better position to give a definition of a direct reaction; also the concept and properties of the optical-model potential are now better understood 1,2,9,10), though its connection with the optical-model potentials to be used in the DWBA has not yet been worked out. The main purpose of the present paper is to show how the DWBA can be derived and understood in the frame of the shell-model approach to nuclear reaction theory (refs. 4-1o)). t Chercheur agrd6 I.I.S.N.; permanent address: Theoretical Nuclear Physics, University of Liege, Belgium. 489 December 1967

490

s. tlUFNERet

al.

In sect. 2, we discuss in simple terms the definition of the direct and compound nuclear parts of the scattering matrix. This discussion leads to a program for the derivation of the DWBA, which is pursued in the following sections of the paper. The main results are summarized in sect. 8. 2. Outline 2.1. FAST AND SLOW ENERGY DEPENDENCE As pointed out by Austern 1~, a2) and others, one of the fundamental problems in the theory of the DWBA is to give an orderly separation of the direct and compound parts of the scattering matrix S. Such a separation is familiar from the statistical theory of nuclear reaction cross sections. There, one writes 13) S = S ( f a s t ) + S (slow),

(2.1)

where the two terms on the right-hand side refer to the rapidly and slowly energydependent parts of S, respectively. The quantity S (slow) can be defined by an appropriate energy averagc, indicated by a bar: S (slow) = S.

(2.2)

Eq. (2.1) is then the definition of S (fast). It is customarily assumed that S (slow) is associated with the direct reaction process, since the latter involves only a few degrees of freedom and should, therefore, give rise to a slowly energy dependent scattering matrix. Accordingly, S (fast) is identified with compound nuclear reactions. The question arises whether one can identify S with the scattering matrix as given by the DWBA. In the following, we show that this is indeed the case under reasonable statistical assumptions concerning the compound nuclear states. We stress that the usual statistical treatment suggests that one can define the DWBA matrix elements only after having taken the energy average of the S-matrix elements. 2.2. RANDOM PHASES OF THE REDUCED PARTIAL WIDTH AMPLITUDES The statistical theory of nuclear reaction cross sections frcquently takes R-matrix theory as a starting point 14). If the reduced width amplitudes Y;.c are assumed to be random, uncorrelated numbers given by a stationary distribution, one is led to the familiar Hauser-Feshbach expressions for the average cross sections. It was pointed out by Bloch 15) and Lane and Thomas ~4) that in the frame of R-matrix theory, formulae of the DWBA type can be derived only if one introduces correlations among the reduced width amplitudes. Using a perturbation expansion t4-16), one selects long-range correlations in the set {7at} and obtains the DWBA. It is not clear, however, why perturbation theory should apply, or that the "leading terms" which are selected are truly the important ones ~7). Furthermore, the physical reason for the existence of the correlations has not been exhibited. Nevertheless, the underlying reasoning - the existence of correlations - is certainly valid and is related to the

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treatment presented in sect. 3. In the frame of the shell-model approach 4-s), we are thus led to search for a separation of the residual interaction V into two parts, V = Vt + V2;

(2.3)

this separation should have the two main features that (i) for V2 = 0, the quantities analogous to the reduced width amplitudes Y~ occurring in the R-matrix approach are randomly distributed, while a non-vanishing V2 introduces correlations; (ii) it is possible to take the effects of V2 into account by using first-order perturbation theory, at least (see subsect. 2.1) as far as the energy-averaged S-matrix is concerned. 2.3. ROLE OF T H E O P T I C A L - M O D E L P O T E N T I A L

Aside from containing the residual interaction only in first order, the DWBA matrix elements involve wave functions for the entrance and exit channels which are distorted by optical-model potentials. The latter describe the fact that part of the flux entering in a given channel populates the compound states. The optical-model potential in a given channel is usually defined in such a way that it reproduces the energy-averaged elastic scattering phase shift in that channel. The occurrence o f optical-model potentials in the DWBA again points to the fact (see subsect. 2.1) that the DWBA describes energy-averaged S-matrix elements. In terms of the decomposition (2.3), it is clear that the interaction V1 should lead to the formation o f the compound nuclear states and to the optical-model potentials. We finally note that in the DWBA one needs the optical-model wave functions only in the internal region of the target nucleus. 2.4. S U G G E S T I O N F O R A D E R I V A T I O N OF T H E D W B A

From the previous qualitative discussion, the following program emerges. Using the shell-model approach, we look for a separation of the residual interaction V in the form of eq. (2.3), with the properties that VI populates the compound states and leads to a random distribution of the reduced width amplitudes. The energyaveraged elastic scattering amplitude due to V1 alone can be expressed in terms of an optical-model potential. The part V2 of the residual interaction leads to correlations among the reduced-width amplitudes. We must show that it is sufficient for the calculation of energy-averaged S-matrix elements to use first-order perturbation theory in V2. We must also show that the optical-model potential due to V1 detines wave functions which enter into the DWBA matrix elements, i.e. which are correct in the internal region. This program is carried through in the following parts of the present paper. In sect. 3, we deal with the most important problem of finding the decomposition (2.3). We investigate various formulae derived previously and show that one obtains terms resembling the DWBA by extracting from the compound nuclear resonances the

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correlations introduced by V2. In sect. 4, the investigation of the energy-averaged S-matrix elements leads to the optical-model potentials, the discussion of which is preceded by that of the generalized optical-model potentials. The convergence of the perturbation series with respect to V2 is considered in sect. 5, where we also generalize some results obtained in the previous sections. Sect. 6 deals with the analysis of experimental data when both direct and compound processes contribute to the cross section. Sect. 7 is devoted to the comparison of the present treatment with those developed by various authors. The main results of the paper are summarized in sect. 8.

3. Direct and compound processes Formal definitions and the decomposition (2.3) in terms of shell-model matrix elements are given in subsect. 3.1. We are led by the following picture that emerges from the shell-model approach 4-8). The compound nuclear resonances originate in their overwhelming majority from bound states of the shell-model Hamiltonian H o embedded in the continuum. The matrix elements of V can accordingly be decomposed into those between bound states or between bound and continuum states on the one hand (group 1) and those between two continuum states on the other hand (group 2). The treatment of random Hermitean operators developed by Wigner and others ~s) suggests that the matrix elements of group 1 give rise to random properties of the resulting compound nuclear states, while those of group 2 destroy this randomness and introduce the correlations; this then suggests a way to define the decomposition (2.3). In subsect. 3.2, we study the case V2 = 0 and relate it to the statistical treatment familiar from R-matrix theory. The interaction V2 is taken into account in subsect. 3.3, where we show that it does indeed introduce correlations a m o n g the reduced width amplitudes. The extraction of these correlations and the decomposition of the S-matrix into two terms, one of which resembles the familiar DWBA, is carried through in subsect. 3.4. 3.1. DEFINITIONS We introduce the shell-model Hamiltonian H 0 (a sum of A single-particle operators) and a residual interaction V. The bound cigenfunctions of Ho are denoted by q0i (i = 1. . . . , M); they are orthonormalized: ( ~ [ ~ k ) = 3ik" The functions Z~ are defined as antisymmetrized products of two functions, the first of which describes a nucleon of energy E-e.;. > 0 scattered by the single-particle potential contained in Ho, and the second of which, a function of the A - 1 other-particle coordinates, is an eigenfunction of the full Hamiltonian H = H o + V in the subspace spanned by the bound shell-model configurations of A - 1 particles 7), the energy of which we = 3a;.' 6 ( E - E ' ) . denote by e,z. The functions Z~ " ; are normalized according to (7,e,]ZE) ~'" The symbol Zz denotes a standing wave, while 7,ff +) obeys the familiar outgoing wave boundary condition. The index ). characterizes the target state as well as the discrete quantum numbers (spin, parity, etc.) of the nucleon in the continuum. We restrict

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REACTIONS

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ourselves to the space spanned by the functions ~ , ;(~ and thus to states with at most one nucleon in the continuum. We can, therefore, treat only elastic and inelastic nucleon scattering, including charge exchange. The Lippmann-Schwinger equation for the full scattering function ~-(±) in the space of functions q~i, ;(§ reads 7)

= zU + ___1_1t (v, +

(3.0

E ±-Ho

where we define the quantities (in the shell-model representation) ;.'

).

r

(zrlHolZ~) = v

(~lHol~)

EOav6(E-E ), -

r

t

= E~6u,

,;.

(~IHolZE) = 0;

(3.2a)

M

1"1 = Y~ I,/',>(~,lVl,/,~>(,l, jl+ l, j = 1

dE'Iz~;>(Z~;IVIq'~> i, ;.',, ¢C

x (~1 +

zf

i, 2'

V2 =

p

H

).'

3.'.

2"

dE'l~e)(~el

glz~-;)(z~;I;

(3.2b)

t:A'

.a."

r f= dE f ° dE Izr)(Xe.IPIzv)(Z~,,I.

(3.2c)

We have decomposed the interaction V occurring in the Lippmann-Schwinger equation into two terms in keeping with the program outlined in sect. 2 and at the beginning of sect. 3. The hat on l? in the expression of V2 indicates 7) that the matrix element, when reduced to that between four single-nucleon orbitals, always involves the two states with the nucleons in the continuum. 3.2. THE

CASE

Vz =

0

If V2 = 0, the Lippmann-Schwinger integral equation reduces to an algebraic equation, because V1 is a sum of separable terms 5-7). We write 8) 2M

~<+) = z~<+)+SE= I I~>, where the state vectors Is) and

I~) = 1,~),

(sl

, E-E,

(3.3)

(sl are given by

ILion,

V I~j)(¢'~.I

+ ~,,

dg'<4',l

V

3." ,~" IZE,>
(s = 1 . . . . , M ) ;

(3.4a)

A' t / e ) . ,

(sl = ( e , - ~ I ,

[g) =

dE'lY'ea • ~ e A,

-E'

(z~:lgl~s-M)

(s

= M + I . . . . . 2M).

12,

(3.4b)

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494

The solution of eq. (3.3) amounts to the construction of the resolvent Ft of the operator I~, defined by the equations 2M

(3.5a)

-q = Z I >
(1 +F~)(1 - I a ) = (1 --Ia)(l + F , ) = 1,

(3.5b)

since

~{+) =

(3.6)

(1 + F O z ~ (+).

Algebraic manipulation leads to 2M

F~ =

[g>(M-~),,
Z

(3.7)

$,1=1

where

M,o = 6uo-

M = (M.v);

(u, v = 1. . . . . 2M).

(3.8)

The S-matrix is easily obtained from eqs. (3.6)-(3.8) by consideration of the asymptotic behaviour of T~ (+). We introduce the notations ~,u -- ,

= <¢~lVlz~>,

v?(e')vf(e')

F~(E) =

dE' - ...... t.~.

d,/e) =

,

(3.9a) (3.9b)

E + -- E'

(3.9c)

E 2.

The expressions for the S-matrix elements read 5,6,s) S¢1) 2#

=

exp (2i6z)6;.,-- 2in exp (i6;.+ i6,) ~ Via(E)(d - l)i j Vf(E), • .

(3.10)

GJ

where the quanties 6z are the potential scattering phase shifts due to H o. Although the statistical assumptions will be fully introduced and discussed only at a later stage (subsect. 4.3), we want to show that, under suitable statistical assumptions (1) the energy-average or- ~~;., vanishes for 2 # p. We can write _,.,S(. 1) as the ratio of two determinants 6) S zU/ t ) = -2inexp(igz+ig~,)D-;_: "_ D

(2 # p),

(3.11a)

ViZ(E) 0 /1"

(3. l 1b)

with D = det (dig),

D;., = det \[V fdij (E)

In order to introduce the statistical assumptions and calculate the statistical averages, we use the formal analogy 19) between eqs. (3.11) and the results familiar from R-matrix theory. There, one has 14,19) expressions analogous to (3.11) with 6¢, D and D¢~, replaced by cp¢, D R and D~,, R respectively, (we follow for a short moment

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495

the notation of refs. ~*' 19) and call c, c' the channel indices, and ;t the index referring to the R-matrix levels). Here, ~oc is the hard-sphere scattering phase shift, and D R = det (d~.),

Dc~, = det (

dR,

(2P ~'~.c .

(3.12)

The quantities Pc are the penetration factors, and dip, a level matrix defined in ref. 19). Thus, the quantities (2P¢)-~7~.c and V•(E) formally play a similar r61e. We shall later make full use of this analogy which suggests that we assume that the quantities V?,(E) are random, uncorrelated numbers, with the property* that

< v?(E)v,"(e')> = 3~.,(I/,* (E)V,~-( e )' ) ,

(3.13)

where ( ) means the ensemble average. We define the transmission coefficients f a by o~j- _ 4rt2(V~a(E)2~, d

(3.14)

where d is the average spacing between the energies E i. If

~ . << 1,

(3.15)

we obtain, in the usual fashion, \"-'j-,a / c ~ ' ) \/

~

--J.p. .~(0

O.

(3.16)

We recall that the bar indicates the energy average. The average cross section is given by the familiar Hauscr-Feshbach formula 1,) /07--./02--J =

3.3. I N C L U S I O N

-

~ .

(3.17)

O F F~

If V2 # 0, the Lippmann-Schwinger equation (3.1) reads with the help of tile notation (3.4)

~<+) --- zg +>+ Z I~)(sl~(+))+

-----

~=1

E+-H~

V2

(3.18)

We define the integral operator 1

Iz

-

e ;-Ha

I/2.

(3.19)

The problem amounts to the construction of the rcsolvent F defined by (1 +F)(1 - I x - / 2 ) = (1 - I ~ - 1 2 ) ( 1 + F ) = 1,

(3.20a)

Actually, the quantities which must be assumed to be random and uncorrelated are the v~a(E) detined in subsect. 4.3 where a more detailed discussion is given.

j. HOFNER et al.

496

since (3.20b)

+ ) = ( t + v)x . +

Let F 2 be the resolvent of I2 (1 +F2)(1 - / 2 ) = (1 -I2)(1 +F2) = 1.

(3.21)

Using the fact that I~ is a sum of separable terms, one finds 7, s) 2M

F = F z + ( I + F 2 ) Z Ig)(M-')st(tl( 1 +Fz),

(3.22)

$~1=1

where .hTl = (/~,,v);

Muv = 6 , w - ( u ] l +F2[O)

(u, v = 1. . . . . 2M).

(3.23)

Eq. (3.22) served as the starting point s) of a numerical calculation of the reaction 15N(n, n')lSN *. It leads to a decomposition of the S-matrix into two parts, given in eq. (2.8) of ref. s). The first part results from the term F 2 in eq. (3.22); it corresponds to the S-matrix stemming from the interaction V z alone. Unless narrow singleparticle resonances are present in some channels, this contribution is a smooth function of energy. The second part arises from the second term in eq. (3.22), which is similar in form to the expression (3.7); hence, it describes compound nucleus formation. One might therefore be tempted to believe that the decomposition of F given in eq. (3.22) leads to a decomposition of the S-matrix in the sense of eq. (2.1), and that the first term should be identified with the DWBA, while the second term would be the compound nucleus contribution. Such reasoning is incorrect, however, because (i) the imaginary part of the optical-model potential as used in the DWBA accounts for compound nucleus formation (see subsect. 2.3). Since V2 has nothing whatever to do with the compound nucleus, the part of the S-matrix resulting from the term F2 alone in eq. (3.22) cannot possibly be identified with the DWBA. This is exemplified by eq. (3.26) below; the first term on the right-hand side of this equation gives the explicit form of the F 2 term in the special case where F2 is approximated by the first term in the Born series expansion of ( 1 - 1 2 ) (ii) The energy average of the part of the S-matrix which corresponds to the second term in eq. (3.22), although it describes compound nuclear resonance contributions does not vanish if we use the assumptions (3.13). Indeed, we show below that the appearance of the operator F 2 in this expression introduces correlations among the partial width amplitudes of the compound nuclear levels. These width amplitudes are linear combinations of the quantities V/(E). They do not, in general, fulfill equations analogous to eqs. (3.13). It is therefore necessary to extract these correlations from the second term in eq. (3.22). Moreover, this must be done in such a way that the r61e of the optical-model potential in the DWBA becomes transparent.

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In order to demonstrate our claims (i) and (ii), it suffices to assume that V2 is sufficiently small to be treated in first-order perturbation theory. More specifically, we assume that F 2 is given by the first term in the Born series: F2 ~ 12.

(3.24)

The conditions of validity of eq. (3.24) have been discussed elsewhere 7). We can then derive the expression for the scattering matrix a) from eqs. (3.20b) and (3.22). We define the quantities V~'~" z'~'' = (XE' ;" lJ2 " X~"), " ;"

(3.25a)

~.,) ,, ~.,,(e t; ) G¢ja"(E) = fiX) dE' fOG dE" Vi~"(E ¢)V~,E,,Vj .

(E +

+ -v:')

(3.25b) '

adE) = dalE)- Y. Gi~ (E), a~'(E) = V,~(E)+

zf dE' V~(E')V~,;'e u -~z

/~ = det (dli),

bx. -- -2iz~ det

E +-E'

(3.25c)

(3.25d) '

(\a~(E) ~ii at(E)] 0 ] "

(3.25e)

The element Sa. of the S-matrix reads 8) for 2 ~ p • • 2 g S~.u = - 2ire exp(t6z+ z6u)(ZE[ V2IX~) + exp

(i6~ + i6u) ~lz .

(3.26)

The form of the first term in eq. (3.26) substantiates the remark (i). In order to show that the energy average of the second term in eq. (3.26) vanishes, one would need, in analogy to eq. (3.13), the relation

(at(E)a~(E)) = 6a,(at(E)a~'(E)). We now show, with the help of eq. (3.25d) that eqs. (3.13) and (3.27) are patible. For this purpose, we write

V/(E) = f~(E)r/t;

(3.27)

incom(3.28)

this separation has been justified in ref. ~9) and is analogous to the separation of a penetration factor in R-matrix theory. The quantities ~ff"are [cf. eq. (3.13)] real random numbers, independent of energy, so that = 6;.,((r/i" ) ). F r o m eq. (3.25d), we find for 2 -~ /~

(3.29)

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498

(a~'(E)a~(E)) = ,5,~uf~(E)Z((rl})=> + ((tl~)Z)fl'(E) x £ ; d E ,¢m'E'''7"~' . , t _ , , ~'_~: + ((rl~)2)f,.(E )Q= dE' f " ("E ' ) V['.'~ .. E+-E ' , z E+-E ' /'~

+ z.., ( ( q l ) ) |

/*oa

dE [. dE f

l z2"a

(E)f

(E)~¥

--v,

;.'~

VE"e + Ett

(3.30)

Hence, the Kronecker symbol 6~.u occurs only in the first term on the right-hand side of eq. (3.30). Clearly, the continuum-continuum coupling V2 introduces correlations a m o n g the quantities a~(E). We make this result more conspicuous by considering the following simple model. We assume that all states ,/)~ have non-vanishing matrix elements (of VI) with only one continuum, ~ say, and that ~ oC

.~t

t

Otlt

dE' [~i (E )V~,E ~ -,zrl/i" "~'(E)V~.E.'" ~= E+-E '

(3.31)

Eq. (3.25d) reduces to =

v,

(3.32)

This result shows that in this model the partial widths in all channels are proportional to one another! It corresponds to the physical picture that the channels 2 -~ c~ feel the existence of the resonant states only through the continuum-continuum coupling; a decay of a resonant state primarily feeds the channel ~, which in turn may populate channels 2 ¢ ~ through the interaction V 2. While we have shown that eqs. (3.13) and (3.27) are incompatible with one another * for V2 -¢ 0, it is not clear, of course, which of the two relations provides the best starting point for a statistical treatment. We face here a familiar dilemma. We know that part of the reaction described by DWBA is not statistical. There is, however, always a certain ambiguity in dividing the set of states of the Hilbert space into two subsets, one of which is treated statistically, while the other one is not. Furthermore, such a division is not invariant under a change of representation. We know, on the other hand, that the statistical assumption (3.27) does not lead to the D W B A and hence does not correspond to experimental evidence. We hope that our previous remarks have made the use of the statistical assumption (3.13) plausible and have shown that the interaction I/2 is at least a plausible cause for the occurrence of correlations as indicated by DWBA. We shall proceed on the basis of this hypothesis. We stress, however, that further correlations could result from deviations from the validity of eq. (3.13). Such deviations are known to occur, for example, on a scale of several tens keV in the case of isobaric analogue resonances. We return to this case in sects. 5 and 6. From the foregoing discussion, it follows that eqs. (3.13) and (3.27) are compatible if V2 has only diagonal matrix elements; this will be used in sect. 5.

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3.4. PERTURBATION TREATMENT OF V2 As suggested by the discussion in subseet. 3.3, we want to extract the correlations from the second term in eq. (3.22). This should be done in such a way that the remaining part of the scattering matrix is just given by ~(1) (considered in subsect. 3.2), ~2u the energy average of which vanishes under the assumptions (3.13). A comparison between eqs. (3.7) and (3.22) shows that this can be accomplished by expanding the second term in eq. (3.22) into a power series with respect to F 2. In this power series, we only keep terms up to first order in F 2 and, furthermore, use eq. (3.24). The assumptions amount to the use of first-order perturbation theory in V2; the validity of this procedure for the calculation of the energy-averaged S-matrix will be discussed in sect. 5. We obtain the result S~.,

~ 9 ) + ~(2)

(3.33)

where S~1,) is given by eq. (3.11), while )./z S(2)

---~

(Z~(-~[ V2[Z~.(+)).

(3.34a)

Here the distorted wave due to the compound nuclear effects Contained in F x is [Z~(+)) = (1 + FI)]Z~(+)),

(3.34b)

Z~ (-) is obtained from Z~i(+) by time reversal. Eq. (3.33) holds only in first-order perturbation theory with respect to V2. The full expression for the S-matrix is given by the identity

S~, ~ = o~,~()'±/7~"-)V' 2+V2(E+-H;-V,7I-x.~E

-- V2)-JV2IZ~.(+)).

(3.35)

The identity (3.35) was considered by Rosenfeld zo) and Newton 21) who suggested that it might serve as a starting point for the derivation of the DWBA. It holds, of course, for any separation of V into two terms Vl and Va. In the light of the discussion given in sect. 2, the derivation of the DWBA from eq. (3.35) implies, aside from the use o f the statistical assumptions (3.13), the proof of the following statements: (i) In eq. (3.35), first-order perturbation theory in Vz is justified for the energy-

averaged quantities. (ii) The energy-averaged functions Z~ (+) are solutions of a one-channel Schr6dinger equation with a complex optical-model potential. The substantiation of these statements is the main content of the following two sections.

4. Generalized optical-model potential and optical-model potential The eigenfunctions Z~.(+) of H I = H o + V1 or at least the continuum parts of these functions are needed for the calculation of the DWBA matrix elements S~.2) defined in eq. (3.34a). We shall study the functions Z~ (+) in subsect. 4.1 by means of the generalized optical-model potential (GOMP). The G O M P will be defined as the

500

J.

HIJPNER e t al.

potential which appears in the SchrSdinger equation for the continuum part of Z~ ~+). The interest in the G O M P is twofold. First, it makes easier the comparison with other treatments like Lipperheide's 9, 1o). Second, it provides the basis for our definition of the optical-model potential (OMP). The OMP will be defined (subsect. 4.2) as the potential which reproduces the energy average S(~ ) of S,(..~) and gives the matrix element Sa(2), if the eigenfunctions of the O M P are substituted for the functions Z~ (+) in eq. (3.34a). Some properties of the OMP will be studied in the appendix. Subsect. 4.3 will be devoted to the OMP in case the statistical assumptions (3.13) hold. Only then does the familiar DWBA expression emerge. 4.1. THE GENERALIZED OPTICAL-MODEL POTENTIAL We use the separation H = (H~+ V,)+ V2 = H~+V2

(4.1)

defined in eqs. (3.2). In some cases, it is actually more convenient to use a different separation of V; this happens when a few channels are strongly coupled or when the OMP is required to reproduce the full elastic energy-averaged scattering amplitudes. Both situations are dealt with in subsect. 5,2. The scattering functions Z~ (+) of H t can be written

Z~(*)

=

Z

dE az,a,(E)zz,,

(4.2)

-

i=1

).,

where the sum over 2' includes all channels. Substituting eq. (4.2) into the LippmannSchwinger equation

Z~(+ ) = Z~(+)+

V,Z~(+),

(4.3)

dE'a~u,(E )vJ '(E') = 0,

(4.4)

+

1

E -Ho

t

we get the relations

M b~(E)(e, i j - E 3 , i ) + i = 1

~." t l ~ ) . ,

a~,x,(E) = 5a;., 3 ( E - E 1) +

1

M

V~ (E )b,(E). E + - E 'I=1 ~ ;'" ' ~ "

(4.5)

Using the matrix A.~k, defined by

(A - X(E))~k = Er~R-- e,~,

(4.6)

we express through eq. (4.4) the coefficients b~(E) in terms of the a~.,a,(E) and insert them into eq. (4.5). We get

ak .(E) =

J

z'

-,

- f~

- E+_E, Z V, (E )A,i(E ) . i , i , 2"

~e?.,,

iI

dE

,~

)."

.rt

Vj (E ).

(4.7)

NUCLEAR

REACTIONS

501

Eq. (4.7) has the form of a Lippmann-Schwinger equation for the continuum part of

Z~ ~+), We call the potential which appears in eq. (4.7) the generalized optical-model potential (GOMP). In the shell-model representation, it has the following matrix elements V, (E)Au(E)V (4.8) wG:.(e) = E " j ( E ") . . o ipj

The G O M P as defined by eq. (4.8) is a Hermitean operator. It does not depend on the entrance channel 2, which is defined by the boundary condition for Z~ ~+). The dependence on E of the G O M P is entirely contained in the matrix A. The matrix has singularities at the (real) eigenvalues of the matrix eu. It may be appropriate to compare the present definition of the G O M P with that recently given by Lipperheide 9). While we define the G O M P as the potential which reproduces the continuum part of Z~ ~+) in all channels, Lipperheide requires that it reproduces only the entrance channel part of Z~ C+), i.e. only the coefficient aE,a, (E) with 2' = 2. It will turn out that it is the G O M P of eq. (4.8) which is of interest for the DWBA rather than that defined by Lipperheide. The expression (3.10) for S~)u) is easily obtained from eqs. (4.4) and (4.5). Indeed, we can insert the coefficients a~,a,(E) ofeq. (4.5) into eq. (4.7) and get a system of linear equations for the coefficients b~(E):

Z b~'(E)[(A-'(E))ik-- ~, FI~(E)] = Vd'(E). i

(4.9)

X"

The matrix d a defined in eq. (3.9c) can be written in the form

d,~(E) = (A-t(E)) u - Z Ft.i(E).

(4.10)

2

We obtain

bt(E) = ~, (d-'(E)) u I/):'(E),

(4.11)

J

aZ,;c(E) = 6~.z,6 ( E - E') + . . . . .1 E Vi~'(E')( d- I(E))u Vf'(E). E+ - E ' i.j

(4.12)

The last equation leads to eq. (3.10). 4.2.

THE

OPTICAL-MODEL

POTENTIAL

We want to find an OMP, say ~-, which has the following two properties: (i) The corresponding scattering matrix S ~ is equal t to ~a.,

~,, = -s~'; - ,J./t °

(4.13)

(ii) The matrix element ~2) obtained by replacing in _~S! 2) the eigenfunctions Z~5 +) o f / t I by the eigenfunctions 2~ t ÷) of H~ + ~-~ is equal to S ~2) ~(2)

S(2) - /-z~(-) v 17.(+)\

(4.14)

t We write a hat on any quantity pertaining to the OMP; we recall that a bar indicates the energy average.

s. HIJFNER et at.

502

In what follows, we first define an OMP and then show that it fulfills requirements (i) and (ii). Using the GOMP of eq. (4.8), we define the OMP by the relation

= ~ E,E,,(E+ iI),

(4.15)

or A ).,),,

,~,

t

~¢'~'E"(E) = Z VI'(E )A,j(E + iI) V}"(E").

(4.16)

G,I

It should be noted that "/~ is not the energy average of the GOMP since, as mentioned above, the GOMP has poles on the real axis and therefore has no well-defined energy average. The definition (4.16) generalizes the expression proposed by Lipperheide 10) for the case of purely elastic scattering. One can solve the Lippmann-Schwinger equation for 2~ (+) by algebraic means and obtains the coefficients ~,~,(E) of the continuum part of Z~(+) 1 a~,z,(E) = 6zz,6(E-E')+ E+_~E-, E V,;"( E' )(d- - 1 (E)),sV}(E),

(4.17)

i,j

where the matrix

~ljk(E) = (E + iI)6sk -- e,Sk -- Z Fs~k(E) a

(4.18)

is obtained from the matrix djk(E) [eq. (4.10)] by replacing E by E + i l except in the quantities F~(E). The corresponding scattering matrix reads "~, = exp (i(6;+6,)){6~.,-2~i

.~. V~a(E)(d - ~(E))~j V](E)}.

(4.19)

l,J

Comparing eq. (4.17) with eqs. (3.9) and (3.10), we see that the relation

~;.u = S~.~)(E) - S~.~)(E+ iI)

(4.20)

holds, provided the phase shifts 6z(E ) are smooth functions of energy and provided that the conditions

V}k(E + il) = rask(E),

V](E + iI) = V~(E)

(4.21)

are satisfied for all channels 2 and all bound states ~i, ~k. The energy dependence of the quantities Ff~(E) and V~(E) has been discussed in refs. 7-io,19). It was shown that these quantities are smooth functions of the energy except in the neighbourhood of a narrow single-particle resonance in one of the channels or near threshold. In the present section, we exclude these possibilities. They are discussed in sect. 5. Then, the potential "/P~defined by eq. (4.15) fulfills requirement (4.13). We now investigate whether ~ also satisfies eq. (4.14). A sufficient condition for this would be 2§(+)

aoc ~7;.(+) E+il.

(4.22)

The operator Pc projects onto the continuum states of H~. Clearly, relation (4.22) ;.(+) cannot be fulfilled by any potential ~P" of finite range, because the function 7~E+~t

NUCLEAR REAC'TIONS

503

increases exponentially in channel 2 for large separation distance, while 2~ (+) remains bounded. However, the integration in the matrix element ~(2) is actually restricted to a small region of configuration space, corresponding to the volume of the target. This is obvious from eq. (3.2c). Thus, the equality (4.22) is needed only in a small region of space, henceforth called the internal region by analogy with the terminology used in R-matrix theory. The continuum parts of the functions ~E 74(++)u and 2~(+)~ are D 7 2( + ) = *¢,--e+u

,,2( + ) .31_ E b b ( E + iI /~E+U i~z"

,z,

~oc

dE' . V, ( E ) z e, E+ iI - E ' )."

t

,

(4.23)

).,

2~ (+) = Z ~ + ) + ~ b~(E) dE'.Vi (E )ZE" i,x' ~)., E+--E '

(4.24)

The coefficients b~'(E+ iI) and b~(E) are equal if eq. (4.21) holds. We notice that both requirements, eqs. (4.21) and (4.22) (in the internal region) amount to the statement that the wave function X~ and the one-particle Green function do not change appreciably in the internal region, when one replaces E by E+il. Obviously, the rdependence of these functions will not be modified by this replacement since the kinetic energy of the nucleons inside the potential well is of the order of 50 MeV, while I ~ 100 keV. The main energy dependence is introduced by the normalization of the single-particle scattering functions. It is known 19), however, that this normalization does not change appreciably with energy as long as one is far from threshold and far from narrow single-particle resonances. A discussion of some properties of the OMP as defined by eq. (4.15) is given in the appendix. 4.3. STATISTICAL ASSUMPTIONS AND THE OMP The OMP ~/~ defined in eq. (4.16) differs from the OMP frequently used in that it gives rise to elastic and inelastic scattering. We now show that, under suitable statistical assumptions, the OMP becomes diagonal in the channel indices. Several authors 14,22) discuss how to calculate the average T-matrix under the assumption that the reduced width amplitudes are random numbers. Though these authors work in the frame of R-matrix theory, we can use their results because of the formal analogy between eqs. (3.10) and (3.12). Let us call d~ the real, orthogonal matrix which diagonalizes e,ik [eq. (3.%)] and coy the corresponding eigenvalues. We define the quantities

(Pi = ~ Ci/bj; J It was shown in ref.

19) that

v~'(E) =

<,,o, IVxlz~>.

(4.25)

one can write

#(E) =gt 'E'o , ,

(4.26)

504

~. tli.)FNERet al.

where the quantities O{ are independent of energy. We assume that

).

=

(4.27)

We obtain for the ensemble average of the T-matrix (TQ)5, = -- inbua ((O~)2)-- [g*(E)]2---u* d

(4.28)

where d is the average level spacing. This result involves a number of approximations [refs. ~4,22)] and is true to first order in the ratio fi-F-a--) = 2n ((O]')2)[g;'(E)12 d d

(4.29)

One can easily calculate the expression of the OMP ~ ' r which reproduces the Tmatrix \* ,/T,,Lu t~)'~ I ~ eq. (4.28). The result is

_i=((ot)=> g'(e)d'(e') ,~;.z, TEE" ~ (~2)."

d 0o I -- i~ (({3}~')2> i "; Et 2 [g'( )] d E ' d~ E+-E '

(4.30)

As expected, this OMP is diagonal in the channels. It is not equal to the ensemble average of the OMP ~" given by , ^ ).'2"

<

EE> =

~ Vj (E )uj (E ) ~

Z.

74, " /

-

=

-

irc¢~2.,;.,,((0~")2) ga(E')g;"(E")

(4.31)

d

However, the difference between the expressions (4.30) and (4.31) is only of second order in ((O[')2)/d. Such terms have already been neglected in the calculation of *u). / and correspondingly in ~"r. We therefore feel justified to calculate (_~;. T(t)\/ (7(1)\ from ( ' / ? ) . This procedure will be adopted in sect. 5. 5. Convergence of the Born series for the continuum-continuum interaction

From sects. 3 and 4, we conclude that the DWBA is justified if first-order Born approximation is valid for the treatment of the continuum-continuum interaction V2. We show in sttbsect. 5.1 that the Born series probably diverges for the S-matrix itself. However, the Born series is likely to converge so rapidly for the energy-averaged Smatrix that ill'st-order Born approximation is often valid. In subsects. 5.2 and 5.3, two cases are analysed where tirst-order Born approximation fails for the energyaveraged, S-matrix. (i) The target states which are excited by inelastic scattering are strongly collective. In this case, we may have to resort to a coupled-channels calculation (subsect. 5.2).

NUCLEAR REACTIONS

505

(ii) A narrow single-particle resonance occurs in one of the continua Z~. The Born series then does not converge very well in the neighbourhood of the energy of this resonance. Furthermore, the approximation (4.21) is not valid, so that our derivation of the DWBA does not apply (subsect. 5.3). 5.1. C O N V E R G E N C E O F T H E B O R N SERIES

The conditions for the convergence of the Born series were studied in refs. 7,23). For a given energy E, the Born series converges if and only if all eigenvalues q,.(E) defined by ( E - H ' o - V~)-' kzCm = r/,.(E)~,, (5.1) have modulus less than unity, Ir/,,(E)l < 1

for all m.

(5.2)

As described in ref. 7), ~/7~ vanishes at a pole 8~ of Szu. -(1) It follows that the inequality (5.2) is violated - for an isolated resonance - in a region around ~ of the complex E-plane which is topologically equivalent to a circle. In the present case, we deal with a large number of poles of S(~) _,u some of which are very close to the real axis. The region where condition (5.2) is violated includes the poles of S]I ,) so that the Born series is expected to often diverge for real (physical) values of the energy. This is not necessarily true for the energy average of the S-matrix given by

S~.u : S:~u ( E + il),

(5.3)

i.e. given by the value of the S-matrix elements at a distance I above the real axis. Since all poles U^~I J;./~ c~,) lie below the real axis, the point E + iI will, for sufficiently large /, lie outside the "dangerous region" where condition (5.2) is violated, and the Born series will therefore converge. The argument can be put into a more quantitative form. For I > 0, the operator (E+iI-H~V i ) - 1 V2 is, under reasonable assumptions concerning Vi and Vz, a Hilbert-Schmidt operator. Hence, a sufficient (but not necessary) condition for the convergence of the Born series is that the Hilbert-Schmidt norm N fulfills the condition N = I I ( E + i I - i 4 ' o - V 1 ) - l V 2 [ ] < 1. (5.4) A first and crude upper limit for N is given by

N < IlVzll

(5.5)

1 which again shows the importance of the averaging interval I for the convergence of the Born series. An explicit expression for N is obtained by the bilinear expansion of the operator ( E + i I - H [ ~ - V i ) - l V 2 into eigenfunctions of the operator H i. We recall that the continuum eigenfunctions of H 1 are denoted by Z~ (+). We introduce

506

s. |IlJFNERe t

al.

the bound eigenfunctions 0i of H~ and have N = v

2 (O~lgil@j> + Z • IE+it-E 9

(5.6)

dE' IE+i~Z~E'i y

~,.~,

•

The sum over the bound states ¢'i is expected to give only a small contribution to N, for two reasons. First, only the (small) continuum admixtures contained in Oi contribute to the matrix element (¢~1VzZlOi), since V2 is the continuum-continuum interaction. Second, the denominators are large because Ej < 0. Therefore, we feel justified to neglect the tirst sum in eq. (5.6). The second term in this equation contains a sum over all channels 2'. Because of the appearance of the threshold energies ~::, in the lower limits of integration, the open channels, i.e. those 2' with E - e ~ , > 0 will give the most significant contribution to N, while the closed channels with faraway lying thresholds can safely be neglected. Hence, we have

N ~- ~°°dE' :.',~:.,
(5.7)

where the sum over 2' is restricted as discussed above. The energy-averaged matrix elements defined by the last relation of eqs. (5.7) can be calculated by contour integration. The matrix elements (Z~!+)[ V2IZ~}!+)) have poles in the upper and in the lower half-plane. They originate from the compound resonances of S~.~). Their contribution to the contour integration can be shown to be small, of the order of
7~., /7),(+)i172 72,(+)\

-

I ,."

NL"E+il

I• 2

L'E+II

/"

(5.8)

In the spirit of sect. 4, we may now replace ~e+~rwa'~+)by the corresponding opticalmodel wave function £~'t+). The matrix elements obtained by this replacement are not typical DWBA matrix elements because the DWBA matrix elements are of the ~z.(+) rather than 2~ '(-) form (2~'(-)1V212~'(+)>, while here we have the functions ~E on the left-hand side of the matrix elements. The effect of the absorptive potential which helps to decrease the DWBA matrix elements is therefore lacking in the matrix elements of eq. (5.8). We tried to estimate the expression (5.7) by replacing the function Z~ (+) by the function Z~.(+). The justification of this replacement is the identity

f dE ~ ,'~Z~'+))(Z~ ' +)Pc' : ~ f

dEIz:'+))
(5.9)

Since we integrate in eq. (5.7) over an energy interval I >> F, d, we hope that this interval already exhausts the relation (5.9). Having replaced the functions Z~ (+) by the functions ~.e ,,a(+) , we can omit the integral over the energy altogether, since the func-

NUCLEAR REACTIONS

507

tions Z~(+) are smooth over the interval I. Using particle-hole wave functions in a real Woods-Saxon potential and a standard residual interaction s), we estimated the expression jaw' = f

dE"(Z~:I V21Z~::)(Z~:'.[VzlT.~:)

(5.10)

and found it to be of the order of or less than a few keV. Although the sum over 2' and 2" in 7t jz,z.' U _-__- Z (5.11)

12",)."

will contain only a few important channels, the number of such channels enters quadratically. Therefore, the sum over 2' in eq. (5.8) is likely to be of the order of several 100 keV. This result suggests that a DWBA approach (for which we should have N << 1, since we keep only the first term of the Born series) is possible only for values of I of the order of several MeV, in contrast to experience which shows that D W B A calculations can be performed successfully with usual optical-model wave functions defined in terms of an averaging interval I of a few hundred keV. The reasons for this apparent disagreement are quite clear. First, the condition N<< 1 is sufficient but not necessary for DWBA to hold. Second, the inequality N << ! assures us that we can use DWBA for the calculation of any inelastic scattering process including those which start from excited states of the target. In applications, we are only interested in a restricted set of transitions which start from the ground state of the target. Then the condition N << 1 is probably too restrictive. Third, it is known that for strongly coupled channels the DWBA may fail and must be replaced by a coupledchannels calculation. After the strongly coupled channels (and the part of 112 which couples them) have been treated explicitly, the chance is improved that for the remainder of the transitions induced on the target in its ground state, D W B A is applicable. In this case one must, however, modify the DWBA expressions (subsect. 5.2). In the light of these remarks, we feel that the estimate for N given above is probably realistic. 5.2. S T R O N G C O U P L I N G O F S O M E C H A N N E L S

One of the cases where DWBA probably fails, though the Born series may still converge, is that of strongly coupled channels. This occurs for instance in the excitation of collective states through inelastic scattering 24). The experimental results are then analysed in terms of an explicit coupled-channels calculation. The equations contain an optical-model potcntial in each channel and an interaction between the channels. We now show that this procedure follows from sect. 4 with a Hermitean channel-channel coupling. We also indicate how the wave functions resulting from a coupled-channels calculation should be used in D W B A matrix elements. The latter describe transitions to channels which are not strongly coupled to the entrance channel.

508

et al.

J. HiJFNER

The procedure consists in adding to Vt those parts of 112which connect the strongly coupled channels. Instead of V = V1+ V2, we now write V = V~+ V~, where ' = vi+2

Z .'

A"EA(2')

;."

~.')

f ?f;d .....

=

dE' "

// I)~E')(~E" V ZE">(ZE"I, ' '"

(5.12a)

A'

,~,

dE [ZE')(Z~' V Zz'")(Z~" •

5.12b

,v ~ ; . , ,

The set A(2') contains the channels which are strongly coupled to 2'. The G O M P corresponding to the separation (5.12) can be calculated along the lines of subsect. 4.1. One finds a"//'~."ff;(E) = ~_~ V/"(E')A,j(E)Vj'"(E")+ V ~ ; ~ ; t . , a ( x . ) . i,j

(5.13)

The OMP is again defined by " X'2" ).' ; . " • a~"~,e,,(E) = a'//'E,E,,( E + d)

(5.14a)

= X Vka'(E')AkJ(E + iI) V]'"(E") + V~fff,',bz,,,A , Z ' )

"

(5.14b)

k,j

When we apply the statistical assumptions (4.27) to (5.14b), the sum over k , j becomes diagonal in 2 and equal to the imaginary part of the optical-model potential in the variouschannels, see eq. (4.31). The second term in (5.14b), the Hermitean continuumcontinuum interaction, gives rise to transitions between strongly coupled channels. The wave functions a2ff +) defined in terms of the ensemble average of (5.14b) have non-vanishing components in all channels strongly coupled to 2. The asymptotic e(1) which is diagonal for behaviour of these functions defines a scattering matrix a~,;.u .u ¢ A(2) and has non-diagonal elements f o r / l ~ A0.). It remains to be shown that the wave functions calculated from the ensemble average of (5.14b) can be inserted into DWBA matrix elements for the calculation of transition rates fi'om channel 2 to channel/z, if/l ¢ A(2). The proof follows the lines of subsect. 4.2. There we used the fact that V~ is separable; the DWBA could then be obtained under the condition (4.21). To extend the proof, we note that the nonseparable part V~~ of V; in eq. (5.12a)is a Hilbert-Schmidt operator. Such an operator can be approximated arbitrarily well by a separable interaction, e.g. by 23) CO

V~o,s =

.

~

Ce

V, =ll ")
(5.15)



Replacing V~~ in V~ by V~¢'~, we can repeat the arguments of subsect. 4.2 and reach the same conclusions. A case where the conditions (4.21) are violated is discussed in subsect. 5.3. Even if all channels are only weakly coupled, it is desirable to include part of V2 into V~ in the sense of eq. (5.12). In the weak-coupling case, DWBA applies to all energy-averaged transition matrix elements. Naturally, the DWBA also has diagonal

"NUCLEAR REACTIONS

509

terms, so that our definition of O M P does not coincide with the usual, phenomenological one. The latter defines the optical model in terms of the energy-averaged elastic scattering phase shift, while our O M P gives this average scattering phase shift minus the DWBA contribution. The situation can easily be remedied by the requirement that the sum in eq. (5.12a) should contain all terms with ;t" = ;t', i.e. the diagonal part of Vz in channel space. This redefinition of V~ and Vz does not affect the statistical independence of the reduced widths in different channels as can be seen from eq. (3.25d). Hence, the Hauser-Feshbach treatment still applies to the scattering matrix a S el). We use the representation 2 = {l,j, J} where I is the orbital angular momentum, j the total angular m o m e n t u m of the incident nucleon and J the total angular m o m e n t u m of the system. In this representation, both a3¢~ and the phenomenological optical-model potential are diagonal. 5.3. N A R R O W SINGLE-PARTICLE RESONANCES

Two difficulties arise with the occurrence of a narrow single-particle resonance in some channel )-0- First, the quantities F~)°(E) and V~°(E) display a resonance 5), and the conditions (4.21) are violated. Second, the resonance also disturbs the convergence of the Born series 7). It can be seen as follows. The operator ( E - H o ) - t displays a resonance at the energy Eg with I m ER < 0, so that lira

{(E-ER)(E-H'o) -1}

= [R)(R*I,

(5.16)

E--~ER

where JR) is the G a m o w function. The interaction V t in the operator ( E + iI+ It~ - V 1) couples the bound states cbi to the continuum. One expects the single-particle resonance to be distributed over a large number of compound states and thus to give rise to a giant resonance in S ~ ). The giant resonance is centered around E~. The giant resonance phenomenon persists if we take V2 into account, but the centre of the resonance is shifted. The shift cannot 7) be obtained by using first-order Born approximation for Vz. Can we again define the O M P and use DWBA? We first discuss the convergence of the Born series. It can be improved by again including part of V2 into V 1, In a fashion similar to the one described in ref. 7), we subtract from Vz a separable interaction V~ so that (E+iI-H~)-I(Vz-V~) no longer displays the resonance; the residues cancel if we choose

F~ = 2 f~°dE' f ~dE'' [Z~';)(Z~;II~IR)(R*IPIZ~"')(Z~':;[ x', ~.,'a~,

~ ~:.,,

(5.17)

(R*I~IR)

We then define v; =

+

(5.18)

=

The G O M P corresponding to V; can be constructed as in subsect. 5.2. F r o m the analysis of the resulting equations, one can exhibit the analogy of the giant resonance

510

J. HUFNERet

al.

to a doorway-state phenomenon 25). The scattering matrix ~,~.u¢(a)due to V~' also displays a giant resonance 25,26); moreover, St;) is not diagonal, because the Gamow function JR) cannot, in general, be treated on the same statistical basis as the states ~ . (The width F R of the single-particle resonance is much larger than that of the compound states.) Similarly,. the ensemble-average of the function -~+u7~(+)displays a giant resonance and has non-vanishing components in several channels. It seems difficult for arbitrary values of I to find an OMP with solutions 2~ ~+) so that the conditions (4.22) hold approximately in the internal region. However, the OMP can be defined as described in subsect. 4.2 if FR ~> L It is of particular interest as it applies to the isobaric analogue resonances (sect. 6). Difficulties similar to the ones just described arise in the treatment of inelastic scattering processes near threshold. Here, the functions ~g+it 7;.(-') and 2ff +) are quite different; moreover, our method to calculate S by replacing E by E + / / i s probably not justified. Of course, these difficulties disappear if I << B, where B is the height of the (Coulomb plus angular momentum) barrier in the opening channel.

6. Analysis of experimental data We consider a reaction proceeding from channel 2 to channel p. Under omission of kinematical factors, the cross section is given by ax, -- [S~,,]2

(,i, ~ p).

(6.1)

The quantity of physical interest is the energy-averaged cross section

cry., = Isa, I- = IS~,i 2 + {l&.f-Is~,iz}.

(6.2)

In the previous sections of this paper, we have only been concerned with S~., and have shown that, under suitable conditions, this quantity is approximated by the familiar DWBA. We must now discuss the second term in eq. (6.2), which is the fluctuating part of the cross section. It contains an interference between "direct" and "compound" processes. In most applications of the DWBA, the fluctuating part is, fortunately, of little interest, because the branching ratio for decay of the compound nucleus into channel p is extremely small, since many channels are open. Conversely, reactions in which the fluctuating part is important often have a very small DWBA matrix element and are thus describable by the Hauser-Feshbach theory; this is due to poor overlap between initial and final channels. While the interference between direct and compound processes is of little practical interest for a large number of reactions, it does play a rrle in certain reactions 27). We consider the following three cases * which we believe to be of some interest: (i) the statistical assumptions (4.27) hold, and therefore ~;(i3 --2,u = 0

for

), :~ p;

t The case where a few channels are strongly coupled to channel 2 or (and) # has been considered in subsect. 5.2.

NUCLEAR REACTIONS

511

(i;) a few isolated resonances are superimposed upon a smooth background, i.e. a low-energy reaction; (iii) a doorway-state mechanism operates, so that the statistical assumptions (4.27) are violated; this case applies, for example, to the isobaric analogue resonances. (i) In the case where the statistical assumptions of subsect. 4.3 hold, we know that (2 # ~,) S (li,~# = 0 ;

-)./a'~(2)= Sau(DWBA).

(6.3)

The average cross section can be written in the form

~,. = IS~)l ~ + {IS;..'(') I ~ + IS,~(z)I ~- IS~)l' + 2 Re

,~,(1).~t 2)* ~

(6.4)

From the discussion in sects. 3 and 4, we know that S (~) i is given by the HauserFeshbach theory. Thus, the difficulty in the calculation of the fluctuating part of the cross section comes from the remaining term

(2)

.~(2) 2

Ii~(~5 2-4-9 Re ~(1)~t2),

(6.5)

It arises because S~Z,) is itself a fluctuating quantity and corresponds to the fact that the DWBA only gives the energy average of Sa~2u). It will be shown in the next section that this difficulty is present in every approach to the DWBA suggested so far. The quantity af,uc (2) t is generally not symmetric around 90 ° and never vanishes exactly. In cases where the compound nucleus corrections to a DWBA treatment are small but not negligible, both [S~., (L)]2 and vfiuc ,.(2) t are relevant, particularly at the deep minima of the differential cross section. In these minima, the cross section may be orders of magnitude smaller than at the highest peaks. Such a fluctuating contribution might be important for the interpretation of j-dependences (Lee-Schiffer effect) in direct reactions. We emphasize the existence of this problem because it seems to have received little attention thus far. Naturally, it is also present in a coupled-channels calculation. We feel that a truly unified description of direct and compound processes is not possible without the explicit treatment of the fluctuating part of --,,# .~!2)• It is interesting to note that the direct part of a photonuclear reaction amplitude as defined by Shakin 28) has the same poles as the compound nucleus part. (ii) In the case where a few isolated resonances are superimposed upon a smooth background, the latter being due to some direct process, we see no real justification for the use of the separation (3.33) of S~, into S! 1) and ,~(2) Indeed, in this case, no compound nuclear resonances exist which could give rise to an optical-model potential aside from the few isolated resonances just mentioned. Therefore, our definition in eq. (3.34a) would imply that S~.~) varies with energy and displays the same resonances as S(~)~.u"As stressed before, the separation (3.33) is useful only if we are interested in averages over energy. This is not the case in the present situation, where we want to

512

J. HOFNER et aL

analyse the isolated resonances individually. One is therefore interested in defining a background term which is smooth over the resonances. Rather than resorting to the procedure discussed by Thomas 29) and Yoshida 3o) and applied by Buck and Satchlet al), one should, in our opinion, use eq. (3.22). In the latter expression, the background is caused by F2, which in turn can be approximated by the first-order term in V2 as shown in refs. 7,8). The background is then given by the first term on the right-hand side of eq. (3.26). It has the form of a direct interaction matrix element; the distortion, however, is due to the real shell-model potential contained in H o and not to a complex optical-model potential. The resonance part can in the usual fashion be written as a sum of Breit-Wigner terms (with complex partial widths 6)). (iii) The third case is the one where we encounter a doorway state at some energy E 0. Let us first turn to the situation where the doorway state is coupled only to one channel which we denote by c~, while the coupling of the compound nuclear resonances to all channels but ~ can be treated statistically. The equations given in subsect. 5.3 show that this problem is analogous to the case of a single-particle resonance in channel ~. In the appendix, we give the expression of the O M P for the case that, in channel ~, the complicated nuclear states are coupled to the continuum only through one doorway state. Then, the O M P in channel c~ displays a resonance of width F ~ [eq. (A.19)]. Consequently, the optical-model wave function 21 (+) is also resonating; the same holds true for the DWBA term ¢(2) if either 2 or p equals "~. We disagree on this point with the method of analysis used in refs. 32, 33) where a smooth DWBA expression was used. At the end of the present section, we point out, however, that in some cases the resonance contribution of 2~ C+) to the D W B A may nevertheless be small. A similar situation arises in the case of the isobaric analogue resonances. There, however, the spectroscopic factors of the analogue state can be appreciably different from zero in several open proton channels. By using the statistical assumptions described in refs. 34), one tinds that the functions 7~'~+) have components in all these ~E+il proton channels; the relative amplitude of these components is determined by the spectroscopic factors. The corresponding expressions for the DWBA amplitudes are so complicated that an analysis of (p, p') reactions through the isobaric analogue resonances becomes an almost forbiddingly difficult task if the DWBA term gives a significant contribution to the background of the (p, p') cross section. Both the DW BA and the resonance contribution depend upon the resonance parameters; the dependence of the former involves the spectroscopic factors of all proton channels. The analysis of a doorway state situation thus might be difficult. A simplification arises if, as in ref. 3~), the partial waves contributing to the resonance do not contribute significantly to the D W B A term. This can for instance be checked by omitting the corresponding partial waves in the D W B A calculation. If the effect of these partial waves on the D W B A is small, an analysis of the (p, p') reaction in terms of a non-resonant D W B A expression plus Breit-Wigner terms is probably quite meaningful.

NUCLEAR REACTIONS

513

7. Comparison with other treatments

The work of Austern and Ratcliff 12) bears some close analogy with the present paper. They study the interaction of a particle with a system which has a finite number of states. In addition, they introduce bound states embedded in the continuum of the unperturbed problem in order to simulate the resonances. Neglecting antisymmetrization and some orthogonality problems, they derive a system of coupled equations. The interaction is divided into V1 and Vz just as in the present paper. The compound part of the scattering matrix is defined by putting V2 = 0, while the direct part is obtained in the limit V1 = 0. The coupled system of equations can be solved explicitly if it is furthermore assumed that the interaction between particle and system is restricted to the surface, in which case V2 is separable. It is clear that this model shares many of the physical concepts of the present approach which is, however, more general and, at the same time, leads to the DWBA in the proper sense. The convergence of the Born series is not discussed in ref. 12). The shell-model approach to reaction theory developed by MacDonald 3) is closely connected with that used here and in refs. 5-1 o). MacDonald also separates the Hamiltonian H into a shell-model Hamiltonian H o and a residual interaction V. The essence of his further treatment can be described as follows. He first constructs formally the resolvent operator F 2, defined in eq. (3.21). Then, the interaction between the bound states embedded in the continuum and the shell-model continua is given in terms of an effective non-Hermitean interaction which involves both V 1 and V 2. MacDonald's separation of the scattering amplitude into a slowly-varying part and a compound nucleus part corresponds to the separation of the full resolvent operator as expressed by the right-hand side of eq. (3.22). It follows that in his definition the compound nucleus resonances have the correlations caused by F 2. Feshbach and his collaborators 1,2,35) introduce the operator Q which projects onto the closed channels, and P = 1 - Q. Eq. (2.3') of ref. as) reads, with H~p = P H P etc., { E - H e e - H e e ( E - Hat?)- ' Hee}P~P~ = 0. (7.1) There is a close analogy between Hee and our operator H~ + V2 on the one hand, and between ( H a e + H p e + H e p ) and our H o + V1 on the other hand. In eq. (2.13) of ref. 35), the S-matrix is written as a sum of two terms Sa,

~pot.~R

(7.2a)

with (7.2b) .

E - dJ.

Here, kug(+) is the scattering function due to H~,e alone; its asymptotic behaviour defines S~, t. The decomposition (7.2) is analogous to eq. (3.22). Therefore, the partial width amplitudes in the resonance term sR, are correlated. If one wishes to obtain the

514

J. ]-IiJFNERet

al.

DWBA amplitude from eq. (7.2), one has to extract these correlations from S~u, following the procedure of sect. 3. The effective interaction

= Hee + HI, Q(E - HQO)- a He e

(7.3)

corresponds to the GOMP given in eq. (5.13), if A includes all channels. We know from subsect. (5.2) that the OMP derived from this GOMP leads to a coupled channels problem. Rosenfeld 2o) has studied the problem of fnding the expression for the opticalmodel potential and of deriving the scattering matrix for a direct reaction. He defines a generalized optical-model potential which reproduces the diagonal elements of the exact scattering matrix (without any energy average); this potential is identical to Feshbach's ,,2). The definition (3.7) of ref. 20) for the transition matrix for a direct reaction, with "//" replaced by the complete operator "t/'+$P(E+-G)$/', is just the identity (3.35). Eq. (3.7) of ref. 2o) is therefore valid irrespective of the definition of the optical-model potential. The identity (3.35) was also used by Sano, Yoshida and Terasawa 36) who write the full Hamiltonian H in the form

H= Ho+U+(V-U),

(7.4)

where U is an auxiliary complex potential to be specified below. Eq. (3.35) then gives

+

V)(E+- H ) - ' ( V -

(7.5)

.or

Sz~ = (1)S;.~+ (2)S~.,+ (3~S;.~.

(7.6)

Here, (l)Sx, = Sv.;., = (Z~(-)[U[z~ ~+)> is the scattering matrix element corresponding to Hv = H o + Uand ~0~t+) is the eigenfunction of H U with appropriate boundary conditions. The potential U is chosen in such a way that (2)8;./, + (3)$2~ u = 0;

(7.7)

this implies, under the assumption that (')S;.u is smoothly energy dependent, that S~, = t~)S~., = (')S~,.

(7.8)

Thus, (1)S~.u is different from zero for 2 ¢ g, and the potential U is nccessarily nondiagonal in the channcl indices. The description of ref. 36) therefore implies that the direct transition amplitude has to be calculated from a coupled-channels procedure and not from DWBA. Moreover, thcre does not exist a simple connection between the potential U and the familiar optical-model potential derived from the average total cross section. The term (3)S~, contains, of course, the nacrow compound nuclear

515

N U C L E A R REACTIONS

resonances and can be written in the form (7.9)

(3)Sz, = - i ~, 7z,,7,, - - . , E-E,+½ir,

In eq. (7.9), the partial width emphasized by the tilde sign, (7.5). Under the assumption and with the help of eq. (7.7),

amplitudes ~ are different from the familiar ones as because they depend upon U as is obvious from eq. that (l)S;., and (2)Sa, depend smoothly upon energy the averaged cross section is given by

a;.; ---- [(')Sx.j 2 + {li3)S~ul2 -li3)S~,12}.

(7.10)

In ref. 36), the fluctuating part F = l(3)Sa,I 2 - l(3)S;.u] 2 is evaluated by using a statistical assumption about the quantities ~;.,. This leads to a moditied Hauser-Feshbach formula for F. In the light of the discussion in sects. 3 and 6, we do not believe that the quantities "~a, can be treated as uncorrelated quantities, because the resulting formula for the cross section does not display the interference between the compound resonance part and the direct part of the scattering matrix. This remark is substantiated by eq. (7.7) which requires that the average of (3)S~.u be different from zero and equal to - ( 2 ) S z u . It is incompatible with uncorrelated partial width amplitudes ~7~,. In a recent paper, Buck and Rook 37) show that a DWBA-type expression can arise from a single assumption. The latter essentially amounts to the use of tirst-order Born approximation for the direct part --/./2 S! z) defined in eq. (3.34a). Since Buck and Rook do not introduce energy averages, their optical-model potential has the same strong energy dependence as our GOMP. Furthermore, it is doubtful (see subsect. 5.1) whether the Born approximation is applicable before averaging. No discussion of the interference between direct and compound processes is given in ref. 37). [t is somewhat difficult to give a simple connection between the present approach and the clegant treatment presentcd by Bloch 15, ,6) and, in a slightly modified form, by Brown 17) and Lane and Thomas 14). In these references the Kapur-Peierls and Wigner-Eisenbud formalisms are uscd. We believe, however, that the two approaches share the same essential features. Bloch ~5,16) starts from the Born series for the Green function 1

1

. . . . . . . . e--H e-Ho

1

+ - ~-H

1

(V~ + I/2) - - o e-Ho

+ ....

(7.11)

The sum converges for sufficiently large values of lm(e). It is, however, possible that Ira(e) has to be of the order of ten MeV or more in order to ensure convergence of the series (7.l I), because the f u l l rcsidual interaction is treated as a perturbation. In the present paper, we rather apply perturbation theory to V2. Bloch selects "leading terms" from the expansion (7.1 l). Although Bloch does not explicitly distinguish between V1 and V2, it appears from eq. (87) of ref. ~5) that he treats V1 to all orders in the frame of statistical assumptions analogous to those used in subsect. 4.3 of the

516

~. HOFNERet aL

present paper. In each term of the series (7.11), only first-order corrections in V2 are kept. Thus, the long-range correlations of the R-matrix reduced width amplitudes are seen to be related to our V2. As stressed in ref. 14), it is difficult in that kind of treatment, to discuss the compound nucleus contribution to the cross sections. 8. S u m m a r y

Starting from the shell-model approach to nuclear reaction theory developed in refs. s-a), we have investigated the DWBA and the optical-model potential related to it. We have shown that the DWBA can only give the average S-matrix. It is not related in any simple way to the conventional resonance expansion of the scattering matrix. The latter has the form 38)

S;.. = Qa~,-t-SxRu,

S).~ = - i ~

Y;..Yu. , E - E. + ½iFn

(8.1)

where Q;.. is a smooth function of energy except near thresholds. An expansion of the form (8.1) is derived in most approaches to nuclear reaction theory t-3,8). We have shown that, in general, the energy-average of S~u does not vanish, and that the DWBA matrix elements are equal to S~.u(DWBA) = Q~.~,+ S}; : $ 2

(2 ¢ #),

(8.2)

where the bar denotes the energy-average. If we define the fluctuating part S~. of the scattering matrix Sa. by 13)

(8.3)

S~.r. = S ~ . - S~..,

we are led to the conclusion that sF, cannot be written as a sum of resonance terms without background. Indeed, we have S;,~ = - i ~

7).,Y~,

E - E, +-~iF,

-a-

S).~.

(8.4)

The quantity S~u differs from zero, whenever a direct coupling connects the channels 2 and/~. Then, the partial widths are correlated in the two channels, which means that the statistical assumptions used in fluctuation theory 13) are not valid whenever the direct processes become significant. The difference sR~ between S;,~ (DWBA) and Q~., corresponds precisely to the compound nucleus effects which, in the DWBA, are taken into account by the optical-model potentials in the entrance and exit channels. The resulls are obtained by decomposing the residual interaction V of the shellmodel into two parts, V1 and V2. Here, V2 is the continuum-continuum coupling and V 1 the remainder of V. The decomposition implies a similar formula for the S-matrix

S~. •

=

~(1)~_c(2)

~2.u

'

~"JA/a"

(8.5)

NUCLEAR REACTIONS

517

If we assume that the matrix elements of Vt are random numbers, it follows that S(~i 2g = 0

for

2 #/t.

(8.6)

We have called (somewhat arbitrarily) S~, ) the compound nucleus amplitude; it is given by a resonance expansion in terms of uncorrelated partial widths ~z, without background, i.e. by the equation

= -i Z

~a,P,,

(8.7)

E - ~ , +.} ilO "

The hat on the resonance parameters indicates that they do not contain the influence of//2 and are therefore different from those appearing in eq. (8.1). All effects from V z are contained in ~(z) 'JZ,u • While the perturbation series in V2 may diverge for real energies, we have shown that it always converges for sufficiently large values o f I m E > 0. From our estimate of the norm N of the operator ( E + / / - H ~ - - V I ) -a V2, it is apparent that first-order perturbation theory is expected to give a good approximation only in the case of weakly coupled channels. Assuming this and excluding the occurrence of isolated single-particle resonances in either channel, we have shown that the first-order perturbation expression for x (.z) ~(2) ( E + / / ) equals the familiar DWBA. The wave functions Z~(+) appearing in the matrix elements o f S (z) are defined in terms of a generalized optical-model potential (GOMP) which involves only V1. The values of these functions at E + / / a r e in the internal re#on given by optical-model wave functions defined for real energies in terms of an optical-model potential (OMP). The OMP is the ensemble-average of the GOMP at E + i I . By a suitable redefinition of Va and V2 (subsect. 5.2) we showed that the OMP is equal to the familiar optical-model potential defined in terms of the energyaverage of the elastic scattering phase shifts. It follows that Sa,(DWBA) = ~'zu~(z)',

(8.8)

a comparison with eqs. (8.4) and (8.7) shows that S 2tL (2) is itself a fluctuating quantity. The occurrence of these fluctuations has led us to discuss the intcrference between "compound" t"~(1)1 a x , . and "direct" [S~.Z~)] processes in the energy-averaged cross secticn (sect. 6). We believe that this problem deserves further study. In the case of a few strongly coupled channels or of a doorway state phenomenon (the isobaric analogue resonances), the DWBA formulae must be replaced by the proper expressions as discussed in sects. 5 and 6. A similar remark applies to the case of single-particle resonances. Care has also to be taken in the application of the DWBA to the description of the background term in resonance phenomena (sect. 6). Actually, the question treated in the present paper involves two different problems. The first one relates to the correlations of the partial width amplitudes in different channels caused by V z. The second one concerns the convergence of the Born series in Vz. The average S-matrix (including the correlations) can always be found from a

518

J. HUFNER

et al.

coupled channels problem, provided one uses statistical assumptions about the matrix elements of V,. The present treatment can formally be extended to reactions induced by composite particles by using, for instance, projection operator techniques. However, the explicit construction of the projection operators and the Green functions involves the full complication of the three-body problem. The authors would like to thank Dr. M. B6hning and Professor R. H. Lemmer for a stimulating discussion, and Professor N. Austern for useful comments.

Appendix PROPERTIES OF THE OPTICAL-MODEL POTENTIAL One can derive a simple expression for the quantity

U~ = 1 - Z Ig~.ui2,

(A.I)

which measures the deviation fi'om unitarity of the g-matrix. In terms of the ?'-matrix, defined by , ~ , = exp (i6x+ i6u){6;.j,-2ni~'~,},

(A.2)

U;. = -47r{Im Taa+7~ Z 17~.~]2}•

(g.3)

eq. (A.I) reads

In order to find an expression for Ua, we start from the equation =

E)+(E

-E)

~;.,;.(E),

(A.4)

where ~'ff;.(E) is the ~-matrix element for the scattering E2 - , E ' 2 ' ("off-shell" matrix element). We multiply eq. (A.4) from the left with (2?;~,(E))*, sum over L' and integrate over E'. We take the imaginary part of the result and obtain

U z = --4rt lm

dE' a~w(E)(T~.~I,(E))*.

(A.5)

z"

Because of the relations =

d E az,;.,(E)ze,, •'

~)~:~I,(E)=

dE"~;'~u,~,,~.,,,(E),

(A.6)

2

eq. (A.5) can be written in the form U~. = - 4 n Im

<2§(+)i~12~(+)>.

(A.7)

The quantity U z can be given a more explicit form by using eq. (4.16) for ~/". We obtain

NUCLEAR REACTIONS ,J:

519~ .!

~"

tt

vs (E)v i ( E )

Im ~ , ' , ' , ( E ) = - I ~ [ E _ %7))2+ i~.

(A.8)

Introducing eq. (A.8) into eq. (A.7) and taking account of the fact that 2ff +~ has only continuum parts, we have

Ua = 21 ~

zs

(A.9)

• (E-~j)2+/2'

where Cj = 2=1(~0~.1V,12~(+>>12.

(A.10)

If the distribution of the {sa. is stationary with respect to the index j, we get u , = 2=

(a.ll}

d where (¢~') is the ensemble average of the quantities {j:. Since Ua < I, we obtain the inequality (~") < d/2n. (A.12) We emphasize that the quantity (¢z) is not the average value of the familiar partial width in channel 2, since the function 2~ (+) is the optical-model wave function. Using eq. (4.24) for 2~ ~+) and the statistical assumption of subsect. (4.3). we obtain for (l-~.)/d<
= 0,

Im ~2"~§(E) = - ( F z ) , 2d

( F a ) = 2n((v~)2).

(A.13)

It is only in the case of isolated resonances that the quantity (F~.) is equal to the average partial width in channel 2. Let us call ( F ) the average value of the total widths. The sum rule ( F ) = Z (F;.) (A.14) always holds, one has therefore the sum rule

z

lm ~/'E~(E) aa - . . . . . . . (. F ) • 2d

(A.15)

We turn to case (ii). Let us call • o the wave function of the doorway state, and O: (i > 1) those of the more complicated compound states. We diagonalize the matrix

520

J. Hi3rNERet al.

elk (i, k > 1), call the eigenfunctions ~ i the c o r r e s p o n d i n g eigenvalues &~. T h e average c o u p l i n g strength between the d o o r w a y state ~ o a n d the c o m p l i c a t e d states ~ is Vz = ( [ ( ~ 0 ] V l l ~ i ) 1 2 ) .

(A.16)

W e recall t h a t 0 is the o r t h o g o n a l m a t r i x which diagonalizes the .full m a t r i x e~k (i, k > 0). I f only one open channel e.g. 20 exists, we have zs) Co~ , = L _ r' 2zt ( @ - E o ) 2 + ¼(1"*) 2 '

(A.17)

where Eo =

<~olHllCPo>,

(A.~8)

F ~ = 2r~ V2/d. I n s e r t i n g eq. (A.17) into eq. (4.16) a n d assuming F ~ >> 21, one gets ~'aoaorr~

EE 112,]

1 F~ 2~ E - E o + ½ i F *

'

I m ~/P~Z°(E) = - 1 . l'tF * 4n ( E - E o ) 2 + ¼ ( F * ) 2 '

(A.19a) (A.19b)

where F r = 2rcl(~ol VIIZe~.o)1 2. Hence, the o p t i c a l - m o d e l potential displays a microgiant resonance due to the e n h a n c e m e n t o f the feeding o f the c o m p o u n d nucleus in the vicinity o f the energy o f the d o o r w a y state. T h e width o f this m i c r o g i a n t resonance is the s p r e a d i n g w i d t h F * . T h e c o n s i d e r a t i o n s can be extended to the case o f a d o o r w a y state s u p e r i m p o s e d u p o n a b a c k g r o u n d by using the m e t h o d s d e v e l o p e d in

refs. 26, 39). References 1) H. Feshbach, Ann. of Phys. 5 (1958) 357 2) H. Feshbach, Ann. of Phys. 19 (1962) 287 3) W. M. MacDonald, Nuclear Physics 54 (1964) 393, 56 (1964) 636 4) C. Bloch and V. Gillet, Plays. Lett. 16 (1965) 62 5) H. A. Weidcnmiiller, Nuclear Physics 75 (1966) 189 6) H. A. Weidenmfiller and K. Dietrich, Nuclear Physics 83 (1966) 332 7) W. GlOckle, J. Hiifner and H. A. Weidenmiiller, Nuclear Physics A90 (1967) 481 8) W. Ebenh6h, W. GlOckle, J. Hi.ifner and H. A. Weidenmfiller, Z. Phys. 202 (1967) 302 9) R. Lipperheide, Nuclear Physics 89 (1966) 97 10) R. Lipperheide, Z. Phys. 202 (1967) 58 11) N. Austern, in Selected topics in nuclear theory., ed. by F. Janouch (IAEA, Vienna, 1963) 12) N. Austern and K. F. Ratcliff, Ann. of Phys. 42 (1967) 185 13) T. Ericson and T. Mayer-Kuckuk, Ann. Rev. Nucl. Sci. 16 (1966) 183 14) A. M. Lane and R. G. Thomas, Revs. Mod. Phys. 30 (1958) 257 15) C. Bloch, Nuclear Physics 4 (1957) 503 16) C. Bloch, Nuclear Physics 3 (1957) 137 17) G. E. Brown, Revs. Mod. Phys. 31 (1959) 893 18) C. F. Porter, Statistical theories of spectra: fluctuations (Academic Press, New York, 1965)

NUCLEAR REACTIONS

19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39)

521

C. Mahaux and H. A. Weidenmiiller, Nuclear Physics A97 (1967) 378 L. Rosenfeld, Nuclear Physics 26 (1961) 594 R. G. Newton, Scattering theory of waves and particles (McGraw-Hill, New York, 1966) p. 186 P. A. Moldauer, Phys. Rcv. 135 (1964) B642 S. Weinberg, Phys. Rev. 131 (1963) 440 D. M. Chase, L. Wiiets and A. R. Edmonds, Phys. Rev. 110 (1958) 1080 C. Mahaux and H. A. Wcidenmiiller, Nuclear Physics A91 (1967) 241 C. Mahaux and H. A. Weidenmiiller, Nuclear Physics A94 (1967) 1 B. Lawergren and 1. V. Mitchell, Nuclear Physics A98 (1967) 481; P. E. Hodgson, Proc. Conf. on nuclear reactions, Rosscndorf (1966) p. 71 C. Shakin, Ann. of Phys. 22 (1963) 54 R. G. Thomas, Phys. Rev. I00 (1955) 25 S. Yoshida, Proc. Kingston Conf. (University of Toronto Press, 1960) p. 336 B. Buck and G. R. Satchler, ibid, p. 355; B. Buck, doctoral thesis, University of Oxford T. Tamura and T. Terasawa, Phys. Lett. 8 (1964) 41 J. Lowe and D. L. Watson, Phys. Lett. 23 (1966) 261 H. A. WeidenmiJller, Nuclear Physics A99 (1967) 269, 289 H. Feshbach, A. Kerman and R. H. Lcmmer, Ann. of Phys. 41 (1967) 230 M. Sano, S. Yoshida and T. Terasawa, Nuclear Physics 6 (1958) 20 B. Buck and J. R. Rook, Nuclear Physics A92 (1967) 513 J. Humblet and L. Rosenfeld, Nuclear Physics 26 (1961) 529 F. Iachello, MIT preprint (1967)