Isospin symmetry-breaking in compound and precompound nuclear reactions

ANNALS

OF PHYSICS

107, 222-265 (1977)

Compound

lsospin Symmetry-Breaking in and Precompound Nuclear Reactions* R. LEON FEINSTEIN

The Niels Bohr Institute, Uniuersify of Copenhagen, DK-2100 Copenhagen @,Denmark Received November 18, 1976

The preequilibrium formalism of the Unified Theory of Nuclear Reactions serves as the formal framework for studying the consequence of isospin symmetry-breaking. A T-violating generalization of the energy-average fluctuation cross section is obtained by (a) choosing a “chained-partition” representation of the closed-channel Hilbert space, from the “simple” doorway through subspaces of increasing “complexity,” (b) introducing explicitly the symmetry-breaking component of the A-body Hamiltonian, (c) following its effects through the entire formal framework, and (d) applying standard statistical approximations. An isospin-conversion mechanism emerges in an explicit microscopic representation. Its functional relationship to the mean isospin-mixing interaction strength is realized by performing an approximate diagonalization of a realistic subspace and by employing random matrix theory. Our analysis suggests that near “complete” mixing is common in compound as well as some precompound processes and that its energy dependence can be very rapid. Simplifications of the general formula are noted. One by-product is the Tviolating version of the Hauser-Feshbach expression involving a single mixing parameter. The relationship of this parameter to the mixing interaction strength is precisely clarified, and an approximate analytic relation is provided. Employing an extended exciton model with isospin properly incorporated, the statistical theory is applied to a comparative study of the photoalpha-particle reactions **Si(n aJZ4Mg and %i(n QOMg in their giant dipole resonance regions. Although the former is isospin-forbidden, its energy-integrated experimental cross section is about twice that of the latter which is isospin-allowed. This discrepancy is explained.

1. INTRODUCTION Isospin symmetry-breaking in the fluctuation crosssection is a little understood subject which has received surprisingly little attention from the theorist. This is due to the misconception that isospin-mixing is negligible and therefore ineffective. The primary thesis of this paper is that, on the contrary, isospin-mixing can be very effective in compound as well as in someprecompound reactions, and that its proper inclusion into the formalism can significantly affect the theoretical analysis of empirical data. Our motivation for commencing these studies derives from developments in both the

theoretical and experimental fronts. Theoretical progress effectively begins with the * This work was supported in part at the Massachusetts Institute of Technology provided by the Atomic Energy Commission under contract AT(1 l-1)-3069.

through funds

222 Copyright All rights

Q 1977 by Academic Press, Inc. of reproduction in any form reserved.

ISSN

0003-4916

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relatively recent attempts, [l, 21, to modify the Hauser-Feshbach formalism [3] and the Ericson fluctuation theory [4]. Although these papers raised more questions than they answered, they did demonstrate how isospin-mixing could affect the interpretation of experimental results. The unanswered questions were quite evident. In particular, Grimes et al. [1] introduced ad hoc a parameter to account for the ihospinconversion of probability flux in the Hauser-Feshbach expression. A great deal of speculation has since been generated on its meaning: its relationship to the t‘undamental symmetry-breaking interaction strength and its energy dependence. We address ourselves to these issues. On the experimental front, evidence continues to accumulate suggesting substantial breaking of isospin selection rules in the fluctuation cross section. MeyerSchiitzmeister et al. [5] compare the energy-integrated cross sections 28Si(y. I,,) ‘-rMg and “%i(,, x,,) “4Mg where both proceed through the giant dipole resonance region and exit through the decay-channel leaving the residual nucleus in its ground state. Even though the former violates isospin selection rules, its integrated cross section is about twice that of the latter which is isospin-allowed. From their anal>&. hoth reactions proceed almost entirely through the fluctuation cross section. Peschel rt ul. [6] in their comparative analysis of “*Ti(y, a,,) “OCa and j2Ti(y. an) Va find ~111 even greater disparity: The isospin-forbidden cross section exceeds the latter by nearly a factor of 20. The purpose of the present study is twofold: (a) to provide a unified prescription for the isospin-violating energy-average fluctuation cross section that can be employed in an inclusive, microscopic calculation such as the exciton model: and (b) to explore the various consequencesof isospin symmetry-breaking in the analysis and Interpretation of experimental data. In order to motivate our choice of the reaction framework and to illusir:llc its content, we consider the concrete example of the photonuclear reaction initiated by the electric dipole. We shall assumethat the target nucleus is well described 1~) the shell model and that its ground state is a renormalized reference state. In fir>t-order perturbation theory, Ip-lh (1 particle-l hole) collective states are generated by the dipole interaction. As these “entrance-doorway” states are not stationary states of the system, they rapidly decay into other regions of the Hilbert space. In the int=raction picture, the time development is governed by successiveapplications of the residual interaction that manifests in the difference between the model Hamiltonian and the total nuclear Hamiltonian. Assuming two-body interactions. the reaction evolves as illustrated schematically in Fig. 1. By Fermi’s Golden Rule. the decay favorj the denser regions of Hilbert spaceassociatedwith the higher particle-hole conligurations. The correspondence between the duration of the reaction and the complexit\ of the wavefunction follows. It is is also clear that isospin-mixing effectiveness increases with time becausemixing is directly related to the density of the respective region of Hilbert space. We may now complete the picture by introducing the concept of the “exit-doorway” in compliance with time reversal invariance. As the name suggests,it represents that portion of the closed-channel Hilbert spacewhich couples directly to the exit-channel

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RESIDUAL ; INTERACTION: : ksidual

= ~,,,I,,

-

Hmodel

DIPOLE EXCITATION

FIG. 1. Schematic model of the photonuclear reaction in the interaction representation. The squares symbolize the n-particle and n-hole configurations of the Hilbert space, and the arrows reflect the biased flow of probability flux.

by action of the Hamiltonian. If the exit-doorway overlaps considerably with the entrance-doorway, then there is substantial probability that the reaction proceeds rapidly, sampling only, say, the lp-lh subspace. As mixing in this region is weak, the isospin selection rules between initial and final channels are well preserved. Examples of this are found in the direct and intermediate structure of the (y,p) reaction as well as in the proton induced isobaric-analog resonance [I I]. The narrownessof the resonancewidth in the latter casesubstantiates this rather well. In contrast, when the exit-doorway is quite different in structure from the entrance-doorway, the reaction requires time to generate sufficient overlap. An exit-doorway consisting of, say, 4p-4h states would require the probability flux to probe the denser regions where isospin-mixing may be nearly complete; thus, selection rules could be violated without great penalty to the cross section. An example of this is the (y, a,,) reaction described above. Compound and precompound processes,by definition, also satisfy these latter characteristics. We choose as our theoretical framework the Unified Theory of Nuclear Reactions [7] as extended by Feshbach, Kerman, and Koonin (FKK) in Ref. [9]. In the spirit of the doorway formalism developed in [lo], they partition the closed-channel Hilbert space into r-subspaces ~2~, S$ ,..., L!& which satisfy the chaining property that each & can couple only to its nearest neighbors -%?+Iand L??~+~ by the action of the Hamiltonian. In practice, this is accomplished by construction and by reasonable approximations. For the above example we make the identification S1 = (lp-lh], 2, = {2p-2h}, etc. Thus, in analogy with Fig. 1, by following the chain through increasing particle-hole configurations, we approximate the essential flow of probability flux in time. By definition, L& represents the final stage along the chain and includes ail that remains of the closed-channel space that has not been assignedto the previous (r - I)-subspaces. We surmise that due to the increase in complexity, the dependence of ;2, on the initial channel decreaseswith increasing n and is essentially zero for n = r. We are therefore justified in identifying 2,. with Bohr’s Compound Nucleus [16]. The cross section is reconstructed by sampling each 4, subspace to determine its contribution to the exit-doorway.

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Our study of symmetry-breaking begins with the fundamental A-body Hamiltonian. We then follow its effects through the entire reaction formalism. Isospin-mixing within each subspace and across partitions is explicitly taken into account. By employing standard statistical approximations, we arrive at a T-violating generalization of the energy-average fluctuation cross section that provides a unified description of both the compound and precompound processes. The isospin-mixing mechanism is explicit and available for analysis. Its implications are explored. Under specified conditions of applicability, simplifications‘ are noted which can be employed in practice. One by-product is the emergence of a generalized Hauser-Feshbach formula involving a single mixing parameter to account for isospin-mixing within the Compound Nucleus. The relationship of this parameter to the fundamental mixing interaction strength is precisely defined. In the spirit of degenerate perturbation theory, isospin-mixing uaithin each 3, subspacc is realized by an approximate model space diagonalization. Its average properties are then determined through random matrix theory. A functional relationship between the average isospin spreading strength and the mean-square isospinmixing interaction element is obtained. The results are then analyzed to provide answers to the following: the necessary interaction strength for nearly complete isospin-mixing; the dependence of mixing on energy; and the conditions of applicability for the T-violating Hauser-Feshbach expression. And finally, the statistical theory of T-violating fluctuation cross section is applied to a comparative study of the photoalpha-particle reactions YSi(y, a,,) “4Mg and ““Si(y. LX”)?‘jMg (discussed above). This is made possible by adapting the microscopic exciton description of FKK as developed in Ref. [9]. They have modified the traditional model by properly incorporating orbital momentum and by performing the angular integrals correctly. We extend their results by endowing each spinless nucleon with internal isospin degrees of freedom and by performing the appropriate isospin coupling. The empirical results of [5] are explained. We find that isospin violation in YSi does not greatly inhibit the reaction cross section. The essential results of this study are summarized as follows: First, a weak mixinginteraction is very effective in thoroughly mixing a dense region of levels. We argue that the necessary condition is easily satisfied in the compound as well as in some precompound reactions involving medium-weight and heavy nuclei. Second, when isospin-mixing becomes appreciable, its energy dependence can compete in importance with the threshold effects embodied in the decay width, This dependence is particularly important in T-violating processes: under specified conditions, it could be observed directly in the cross section. Third, thorough mixing in the precompound and/or compound cross section can greatly influence the analysis and interpretation of empirical results. And fourth, we show that the Hauser-Feshbach expression generally underestimates the “true” degree of isospin-mixing in the Compound Nucleus. In Section 2, we begin by reviewing the formal reaction framework with isospin invariance preserved. In Section 3, the consequence of isospin symmetry-breaking on the reaction formalism is explored. And finally, in Section 4, the modified exciton model is introduced and applied to the photoalpha-particle reaction.

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2. FORMAL REACTION FRAMEWORK In this section, we develop a suitable reaction formalism to serve as our basic framework. As we shall see in Section 3, the chained-partition formalism of Feshbach, Kerman, and Koonin (FKK), as developed in Ref. [9], is a natural choice for the study of isospin symmetry-breaking in the fluctuation cross section. By way of review, sufficient detail is provided here to detie our terms and to make this paper reasonably self-contained. For the purpose of later application, we focus our discussion on the photonuclear reaction, but this does not confine the general applicability of our results. From the “golden rule” of time-dependent perturbation theory, the photonuclear cross section is

where the transition

T-matrix is given by T,, = (Yj-’

I H, / 0; k,).

(2.2)

The ket 10; k,) denotes the initial-channel with the target in its ground state and an incoming k, photon. The bra (Yj--’ represents the total wavefunction of the target nucleus with an outgoing boundary condition in the final channel f (e.g., nucleon or a-particle decay). The interaction of the photon with the nucleus is contained in H, . We have chosen to normalize the wavefunctions on the energy shell. The nuclear A-body Hamiltonian H is assumed Hermetian, rotation invariant, and isospin invariant. (bospin symmetry will be relaxed in Section 3.) The total wavefunction then satisfies (E - H) ( Y;-‘)

= 0

(2.3)

where we set the energy scale so that the ground state energy is zero. Ignoring recoil effects, the excitation energy E corresponds to the incident photon energy. Following Feshbach [7], we split the Hilbert space into the P-space and -%space, and associate with each the projection operators P and Q, respectively. For the moment it suffices to give them the following properties: P projects onto that part of !P;-‘) which includes all the open-channel components near the excitation energy E; and Q projects onto all that remains. Since P + Q = 1, PQ = 0, PP = P, and QQ = Q, the T-matrix (2.2) can be separated into its “direct” and “fluctuation” parts by the usual operator algebra: T,, = T;ydir) + Tj;‘“‘)

(2.4)

with Tjy)

= (#j-’ ( Ha, ( 0; k,),

(2.5)

ISOSPIN-MIXING

where the effective Hamiltonian

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&co is defined by

AfQQ = HQQ t =H

QQ

and the effective photon interaction

i

(2.7a)

W,, HOP

E”)

I _ HPP

HPO

(2.7b)

yt”, by (2.8)

We have adopted the usual subscript abbreviation eigenfunction @‘\ satisfies (E -

HP0 = PHQ, etc. The scattering

Hpp) ’ I/J).-“, = 0.

(2.9)

The .Gi-’ in Eqs. (2.7) and (2.8) denotes E f iv with 71---f O1. As our notation suggests, we require that Tcdir) contribute to the smooth background of the cross section (as a function of energy) and that PUC) be responsible for the fluctuation cross section. Following Feshbach, Kerman, and Lemmer [IO], this is formally achieved by removing from T tfluC) the energy-average. nonresonant contributions and incorporating them into T (dir); this corresponds to an appropriate redefinition of the 9 and 2 spaces. Equation (2.9) then acquires the standard properties of the optical model. Kawai et al. [13] have shown that this separation of the T-matrix, with interference effects eliminated, is always possible in principle. From now on, we shall concern ourselves only with T (*luc). It is important to remember, however, that the direct step and multistep direct processes associated with Tfdir) must be properly included before comparing with observation. As discussed in the Introduction, we can extend the doorway concept by adopting the chained-partition formalism of FKK. We begin by partitioning the &pace into r-orthogonal subspaces Z?*, & ,..., 2,. which satisfy the chaining property that each Z?Vcan couple through the action of H,, to only 5?+r and &+l. By definition, & is the entrance-doorway for the initial channel: It contains only that part of 2 which couples via H, directly to the initial ket. This suggests that the structure of 9,, & ,... is dependent on the choice of the initial channel. We surmise, however, that this dependence will decrease as we move down the chain and will essentially be zero for 2r . Clearly, 2V embodies all that remains of the full 2-space that has not previously been assigned to the other (r - I)-subspaces. In practice, the essential chaining properties are satisfied by construction and reasonable approximations. The illustrative example in the Tntroduction is such a case. Following FKK, we associate with each nth-subspace a projection operator Qn with the properties Completeness: Orthogonality: Tdempotency:

Ql + Q2 + ... L Q,. = Q, for n 7. m, QnQ,ri = 0 0,/P,, - Qn .

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After standard operator algebra, T$ruc), as given by (2.6), is transformed into y4PlUC)

=

fv

i

(2.10)

T:;n)

?L=l

where the partial T-matrix

T:,“’ is defined by

(2.11) with the propagator (2.12) The effective potential Vnn is introduced generated recursively by

to represent the continued fraction that is 1

VT&n =

ez*rt+1

E

_

.&e

n+1.n+1

-

vn+1,n+1

Jf

%+lsn

(2.13a)

with v,, = 0.

(2.13b)

This last equality follows by definition since Z& is the final subspace. Again, we have adopted the notational abbreviation Xn, to represent Q,&‘Q, . In obtaining the above results, we have made use of the “chaining-identity” derived in the Appendix of FKK. Its validity rests on the assumption that the effective Hamiltonian Zoo , defined by (2.7), preserves the nearest-neighbor coupling property along the chain. The source of trouble is found in the W,, term, which permits nonneighbor subspaces to couple via the P-space. The plausibility of this “nearestneighbor assumption” is argued in FKK. It is dependent on there being many open channels near the energy of interest and on the random phase assumption of statistical reaction theory. We interpret the above equations as the natural extension of the doorway formalism introduced by Feshbach, Kerman, and Lemmer [lo]. In fact, their results are recovered by setting r = 2 and retaining only the first partial T-matrix T:j’. The single subspace 9, is then assumed sufficient to serve as the doorway for both the exit and entrance channels. The chained-partition formalism is more general in that it allows for significant differences in structure between the initial and final channels. Beginning on the right in Eq. (2.1 l), the initial channel couples to the %space through the entrance-doorway 9, . The alternating products of interactions and propagators serve to connect Z?l with the remainder of the d-space. On the left, the bra (#j-j I HPn samples each $,-subspace to determine its contribution, Tj:‘, to the full fluctuation T-matrix. This sampling of the entire chain can be interpreted as the effective reconstruction of the proper exit-doorway subspace. The precise rule for terminating the chain is arbitrary. It might reflect the degree of microscopic sophistication one wishes to generate. For example, Shakin and Wang

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[15] choose Y :- 3. While stressing microscopic structure in ~9~and Y, , they introduce energy-average parameters to account for the fine structure in drE3 . For our present investigation into the fluctuation cross section, we extend the configurational complexity at the cost of microscopic sophistication. (Of course, intermediate structure may still be retained if so desired.) As suggestedby FKK, we find that a natural termination point for r is provided by the point along the chain where the property of rapidly increasing level densities begins to deteriorate. We now proceed to the derivation of the energy-average fluctuation cross section (‘l”“. From Eqs. (2.1) and (2.1Oj, we find ufY (3.14) with the partial cross section

Unless >pecified otherwise, the bracket c\ ;, shall always refer to the box-weighted energy-average over the energy interval (E - (LIE/~), E - (dE;2)). Equation (2.14) follows as a result of the property

and the assumption of zero correlations between bilinear products of matrix elements. Condition (2.16) is satisfied by definition as discussed in the paragraph following (2.1). The zero correlation assumption is a standard assumption in statistical reaction theory and is employed throughout this development. As noted in FKK, the energy-average of, T crl)I’?is complicated by the rapid energy dependence of I/,, in the denominator of the propagator G,, (‘seeEqs. (2.12) and (2.13)). Since by construction the level density of 2,, ,m1 is much greater than that of 2,. we expect to find rapid fluctuations of V,, over an interval equivalent to the average level spacing in dn. This difficulty can be dealt with in two ways: First, by recognizing that under certain conditions the problem resolves itself; or second. when these conditions are not met. by replacing the troublesome terms by their energy-average values in a brute force fashion. The problem resolves itself when the natural width I’(n 4. 1). the imaginary component in the denominator on the r.h.s. of Eq. (3.13a), is large enough to impart a slow energy dependenceto I/,,, over the level spacing of 2,. If this is true for all II along the chain, then the “self-averaging” criteria of FKK is satisfied. Clearly. an identical result is obtained artificially by replacing E by E + i(1/2) in the denominator of (‘2.13a). This brute force method, however, ignores correlations between subspacesunder those conditions when they may be of importance. From exciton model calculations performed to date, the self-averaging criteria appears to be satisfied for medium-weight and heavy nuclei, but may be in trouble for light elements.

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The box-weighted energy-average of I r:,“’ I2 can now proceed as described in FKK and [14]. By employing once again the zero correlation assumption and by performing the standard algebra of statistical reaction theory, we transform the partial cross section (2.15) into the transparent form

(n)_-- Gb> FL@ - 1) . . . ___Wl)

Df?J

I-(n)

r(n - 1)

r(l)

(2.17)

uy

where l-(n) = P(n)

+ P(n)

(2.18)

with P(n)

= 1 r:(n). c

(2.19)

The decay width r:(n), associated with the coupling of L& to the open c-channel in B, is defined by (2.20) rk4 = 274CK ( ) I HP% I 44>12>a and the damping width P(n), accounting for the coupling of L& to LS?n+1,by w4

= 277P(~ + l)‘x~(n

+ 1) I sL+l.n I ~(4)12)a,a

(2.21)

where ~(n + 1) is the average level density in Z?n+l in the energy interval of interest. The biorthogonal kets (1 a), / a”)] re p resent the eigenstates of (Z& + V,,) in the denominator of G, in Eq. (2.12), and &‘) is the scattering eigenfunction satisfying Eq. (2.9). The subscripted bracket refers to a state-average involving those states that fall within the interval AE surrounding E. And finally, the y-absorption cross section a, is represented by (2.22)

cv = (dkv2) T,(l) where the transmission either of the forms

coefficient T,(l)

(into the entrance-doorway

(2.23a)

T,(l) = 25741) C(l),

W) ml) = (E - Ea(1))2+ VV)/‘T

=??I)may have

’

(2.23b)

with r;(l)

= 274(4)

I 8 I 0; W12>a

(2.24)

Expression (2.23a) results if there are many doorway states near the excitation energy of interest, while (2.23b) corresponds to a dominant single-doorway state as in the giant dipole resonance. In this latter case, the state-average bracket in (2.24) should be removed. It is important to note that in arriving at the simple representation of r(n) in Eq. (2.18) we have imposed one further condition on the formalism: we have assumed

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that the virtual P-space coupling term in Z, Eq. (2.7) plays a weak role in coupling 9,) to its nearest neighbors L&+l and 9,-, , compared with the direct action of H rc'1.n and H,,,, . This is a much stronger statement than our earlier nearestneighbor assumption which concerns itself with the coupling of 9, to, say, Z& via the Y-space. This “weak-coupling assumption” is a simple criteria for avoiding the phase complications that arise from the non-Hermetian nature of the virtual Y-space coupling term. It is employed in order to make the conservation of probability flux explicit in the energy-average fluctuation cross-section (2.14). Beginning at the right and moving to the left, Eq. (2.17) appears to embody all the essential features of the iIlustrative example in the Introduction. The photon is absorbed into the entrance-doorway with cross section a,, . The system may then either decay into the Y-space, with a probability proportional to P(l), or damp down into the next most complex region L& . Thus, the amount of probability flux reaching .L?~ is depleted by the factor rl(l)/r(l). The product of “depletion factors” to the right of the &h-stage account for the leakage of flux into the open channels. The factor fj(n)/r(/r) on the left represents the probability that 9,, will decay into the particular final channel ,f; or, equivalently, it represents the proportion of exit-doorway contained in &. The total energy-average fluctuation cross section is reconstructed by summing up the contributions from each stage along the chain. As suggested by FKK, it is natural to interpret ~(~1as the compound reaction cross section and zL:t acn) as the precompound cross section. In order to satisfy the properties of Bohr’s Compound Nucleus [16], the compound state 9,. must be essentially independent of the initial channel. Because many applications of the residual two-body interaction are required to generate 9T , we surmise that its complexity is enough to ensure this independence. Further consequences of this interpretation are discussed in Section 3.

3. THEORY OF ISOSPIN SYMMETRY-BREAKING

We now turn to the primary concern of this paper: the consequence of isospin symmetry-breaking on the compound and precompound cross sections. The chained partition formalism of Section 2 serves as the basic framework on which this analysis is made. We begin at the source and follow its effects through the entire structure: symmetry-breaking manifests in the fundamental A-body Hamiltonian H; H operates through an effective Hamiltonian %’ which, in turn, emerges in various capacities in the final fluctuation cross section. Two modes of symmetry-breaking are distinguished: intraspace-mixing and interspace-mixing. “Intraspace-mixing” refers to the mixing of states of differing isospin within each 9,-subspace, and “interspace-mixing” denotes isospin-mixing across the partitions of the Hilbert space, between 2 and d, and between nearest-neighbor subspaces in 9. A statistical model is considered to relate intraspace-mixing directly to the fundamental, mixing-interaction strength. The implications on the fluctuation cross section are explored. And finally, under specified conditions of applicability, simplifications of the generalized fluctuation cross section

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are noted which can be employed in practice. One by-product is an isospin-violating version of the Hauser-Feshbach formalism involving only a single, precisely defined isospin-mixing parameter. In Section A, an effective, isospin-mixing interaction is introduced to serve as the fundamental, mixing operator in the context of the reaction framework. The consequences of intraspace-mixing are formally developed in Section B. In Section C, the full T-violating generalization of the energy-average fluctuation cross section is presented with damping and decay widths explicitly modified to allow for interspacemixing. In Section D, a statistical mixing model is considered to provide a functional link between the intraspace-mixing mechanism and the mixing-interaction strength; its implications are explored. And in Section E, we conclude by noting further implications and simplifications of our results: e.g., the generalized Hauser-Feshbach formula is derived. A. Eflective Isospin-Mixing

Interaction

A discussion of symmetry or symmetry-breaking must always begin with the fundamental A-body Hamiltonian H. The consequence of symmetry-breaking on the cross section can then be deduced by following the operator H through the reaction framework. Since its conception, isospon invariance has been understood to be an approximate symmetry: the existence of charge is enough to warn us that H is not invariant to rotations in isospin space. Experience tells us, however, that these noninvariant components are weak compared to the total interaction strength. We isolate these symmetry-breaking components and represent them by the single charge-dependent interaction V, . The isoscalar portion of the Hamiltonian that remains is designated, without confusion, by the isospin-invariant H. Thus, we make the replacement in the chained-partition framework H+H+

V,.

(3.1)

The specific composition of V, does not concern us here. The dominant piece comes from the Coulomb interaction, but the T-violating component of the nuclear interaction as well as the proton-neutron mass difference must also be considered (see [I 11). We are ultimately interested in the isospin-mixing properties of V, . Generally, V, can be transformed into its irreducible components: an isovector and an isotensor of second rank. We recognize, therefore, that V, can couple states of the same isospin quantum number. As these effects are of little interest to us, we relegate them to the domain of the isoscalar H. Formally, this is achieved by introducing isospin projection operators and making the separation explicit. For notational simplicity, however, this distinction is to be implicitly understood; henceforth, V, represents the isospinmixing interaction, and H includes all the nonmixing consequences of symmetrybreaking. The effective Hamiltonian .%?, as given by (2.7), is still applicable, but according to (3.1), with H replaced by H + V, . This new L%?becomes a complicated, symmetry-

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breaking operator with isovector and isotensor components of rank equal to and greater than two. Following the example of (3.1) we separate out the isospin-mixing components of Z and represent them by the single, effective interaction 9: . Thus, as a result of (3.1), x-tce+Y; (3.2) where 2, on the right-hand side is understood to embody all the nonmixing consequences of symmetry-breaking. The separation (3.2) is easily accomplished through standard operator algebra and by introducing isospin projection operators. The significance and interpretation of Vi in the energy-average fluctuation cross section is the main subject of this article and is taken up in detail in the following sections. The reaction formalism contained in Eqs. (2.10)-(2.13) is still valid, but with the substitution of (3.1) and (3.2) properly incorporated. As a result, r’, emerges in two capacities: (a) as the intraspace-mixing interaction (VJ,, which acts to couple states within Y,, of differing isospin; and (b) as the interspace-mixing interaction (K),,,n i-1which couples states in 2, to those in L?,+l that differ in isospin. Tn the denominator of the propagator G, , (2.12), the introduction of (Y i),, results in an isospin-mixed, spectral representation of G, . This leads to a generalized, T-violating propagation factor in the final expression for the fluctuation cross section. As this serves as the dominant mechnism for isospin conversion along the chain. considerable attention is devoted to intraspace-mixing in subsequent sections. The formal development begins in Section B. Interspace-mixing results in a modified version of the damping width rl, (2.21). The arguments employed in its derivation are essentially identical to those of Section 2, but with (2 A YJ,,,+r substituted for %&+i . The fact that both Yi(/nsnil and (m,n i-1 are treated equivalently is responsible for reducing the effectiveness of interspace-mixing as an isospin-conversion mechanism. This is verified in Section E. The explicit form of the modified damping width is given in Section C. We note in passing that the new, effective potential V,, , resulting from (2.13) with the substitution (3.2), does not contribute significantly to the isospin-mixing within 2!, . These effects have been neutralized in the “self-averaging” (or “imposedaveraging”) process as a consequence of the zero correlation assumption and the complexity of the Z?n+l subspace. One advantage of the present formalism is that effects of isospin symmetry-breaking within and in coupling to the P-space are implicitly contained in the effective operators X and V’, . In fact, substantial progress can be made without explicit reference to these effects. The one important exception, however, is the emergence of a T-violating decay width in place of P, (2.19), in the denominator of the propagator. The interaction term that gives rise to this modified width is

w + VJPn (H + VJnPE’f’ _ (; + V,) L PP

.

(3.3)

This comes from the effective, nonmixing Hamiltonian Z on the r.h.s. of (3.2). An explicit representation of the modified decay width is given in Section C.

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B . In traspace-Mixing : Formal Development Intraspace-mixing provides the most important mechanism for isospin conversion in the chained-partition framework. We therefore devote this entire section to its formal development. Isospin-mixing within the &-subspace is made explicit by expressing the propagator G, of (2.12) in its “mixed” spectral representation

where the isospin-mixed coupled set [~,a

biorthogonal - z-z, -

kets 1 CX) and 1 Zi> are eigensolutions v?m - KM

of the

I 4fo> = 0, (3.5)

with cTa(n) = E,(n) - i -$$Q .

(3.6)

This follows directly from (2.12) by the replacement (3.2). As formulated in the previous section, both Z,, and V,, in Eqs. (3.5) are modified from Section 2 to account for the effects of isospin symmetry-breaking, but only (VC),, acts to couple states of differing isospin. Our treatment of intraspace-mixing in the energy-average fluctuation cross section depends on two important properties. First, the isospin-mixing interaction is relatively weak. This suggests that we treat (‘9’& as a local perturbation mm on the isospin-pure spectrum of (X + V),, in Eqs. (3.5). It is “local” in the sense that its effective coupling range is considerably less than the energy interval AE over which the box-weighted energy-average is performed. Second, despite its local nature, is still strong enough to induce considerable mixing in the precompound and mm compound stages of the reaction. This is a natural consequence of the rather large level densities associated with these complex regions of the Hilbert space. Since nondegenerate perturbation theory has limited applicability here, we resort to a rediagonalization in the spirit of degenerate perturbation theory. Both of these properties are substantiated in Section D. The local perturbation effect and rediagonalization approach are the essential ingredients of our “local rediagonalization” approximation. This approximation may be summarized as follows: (a) The spectrum of (9 + V),, is first determined by solving Eqs. (3.5) with (Y’& set equal to zero; (b) a local, finite-dimensional subset is specified by choosing those states which fall within the interval AE surrounding the energy E; (c) the isospin-mixed states are then obtained by diagonalizing the finite-dimensional matrix with off-diagonal elements (l(V’J,, I). The essential approximation in this recipe is that we can ignore the coupling of states of differing

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isospin across the boundaries of the interval LIE; its validity rests on the local coupling range of (“t Jnll as discussed above. One important consequence is that closure can be suitably employed. We insert the mixed representations (3.4) into the partial T-matrix (2.11) and proceed as prescribed in Section 2. This results in a modified version of the energy-average fluctuation cross section. It differs from expression (2.17) by the emergence of a generalized propagation factor M,,,,(n) in place of the isospin-conserving factor l/r(n) where (3.7)

The related functions a=(n) and 6$-(n) are defined by

a,(n) = c Ii”, IIT

where the biorthogonal basis (1 a,), j &)) represent the isospin-pure eigenstates of (Z + YLn and (1 a), I a)} the isospin-mixed eigenstates of Eqs. (3.5). The primed summations in (3.8) denote the restricted space of states which fall within the energy interval dE surrounding E; this meaning is implicit in all the unprimed summations to follow. The subscripted bracket in (3.7) specifies a state-average in the appropriate energy interval. And finally, pT(n) is the level density of isospin-pure states, and the total density PW = c p*(n) 7

(3.9)

is equivalent to the level density of isospin-mixed states. The meaning and significance of the propagation factor MTfT” can best be understood by first examining the properties of c&z). A more transparent form is obtained by employing the “weak coupling” assumption of FKK. The non-Hermetian components of the effective interactions are assumed sufficiently weak to justify our ignoring the biorthogonal phase complications. That is, we assume

ET(n)= a,(n)= c i(&) I &)‘>I”.

(3. IO)

aT

We recognize a&z) as the isospin strength function, a direct measure of isospin-i’ purity in the mixed state. Every mixed state 1a(n)) in the limit V, + 0 must approach some isospin-pure state I aJn)>. It is useful to incorporate this zero isospin-mixing (ZIM) boundary 595/x07/1-2-16

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condition explicitly into our formalism mixed ket 1 G) implies that

Accordingly,

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by attaching

the superscript label T. The

the isospin strength function (3.10) is modified to

where in the ZIM limit (3.13)

By the local rediagonalization approximation, and pure representations require that

the closure properties of both the mixed

(3.14b) The interpretation of & is straightforward. It is the isospin T’ strength in the mixed state ( aT) which in the ZIM limit becomes a pure state of isospin T. At the opposite extreme, it proves equally useful to introduce the concept of complete isospin-mixing (CIM). This is an idealized limit which is defined by the properties of degenerate perturbation theory. In the ideal degenerate case, the partial mixing probabilities on the right-hand side of (3.12) are all equivalent. It follows immediately that c&(n) is proportional to pT’(n) with a constant of proportionality determined from the closure condition (3.14a). Accordingly, the ideal CIM limit is identified by

where p(n) is defined in (3.9). Note that in the CIM limit, the isospin strength function is independent of the ZIM boundary condition (i.e., the superscript T) as might have been expected. We return now to the propagation factor Mr’T” as defined by (3.7). By making the ZIM boundary condition explicit as prescribed in (3.1 I), MTfTp can be approximated by the more transparent expression (3.16)

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with the average mixed width P(n)

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defined by (3.17)

where I-r(n)

~- ~::r,,(n):‘ ,,J .

(3.181

An explicit expression for the average pure width r=(n), associated with the pure states of isospin T, is given in the following section appropriately modified to account for the effects of interspace isospin-mixing. We are justified in replacing the averaging bracket in (3.7) by the approximate expression in (3.16), with the numerator and denominator averaged independently, only after the important ZIM correlations have been made explicit. In the context of the full reaction formalism, the conditions for this approximate decoupling are now essentially identical to those discussed by Moldauer [17]. The propagation factor A4,,T- as represented by (3.16) obeys the proper ZI M factor limiting form. Inserting the ZTM limit of &, (3.13), the isospin-conserving of Section 2 is recovered:

In the ideal CIM limit, M7fT* evolves into

where

This follows by substituting the ideal CTM limit of .x:, , (3.15), directly into (3.16) and (3.17). Note that at both the zero-mixing and degenerate extremes, the average of the product strengths in (3.16) is equivalent to the product of the averages. The meaning and significance of MTf7- can be best clarified by comparing the two mixing limits, (3.19) and (3.20). Two essential differences are noted. First. the pure width r,, in (3.19) is replaced by the density-weighted mixed average (3.21): second. the isospin-conserving delta 6,,,- evolves into the density depletion factor pT,/p. In the context of the full reaction formalism, the interpretation of (3.20) is straightforward. Moving down the chain (from right to left), the probability fux enters the 3,-subspace through the isospin channel T” and exists through the T/-channel. While propagating through 2,) the flux spreads its strength among the available isospin channels in proportion to their relative densities. Thus, the initial probability flux is depleted by the amount pT,/p upon departing through the T’-channel. (Tn the general case. the amount of flux depletion depends, of course, on the extent of isospin mixing within L& .) The implication of this result is significant. For example.

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if pT’ is the dominant density in the energy region of interest (i.e., pT’ > pT” for T” # T’), then MTfT- acts to convert most of the T”-channel flux into the final T’-channel. The (isospin-violating) reaction could then proceed with only slight penalty (p=,/p N 1) to the cross section. We conclude this formal section by deriving a simple parametrized version of M T’T” in the two-dimensional space of isospin. By confining the %,-subspace to two isospin degrees of freedom, Tupper (7’>) and Tl,,, (T,), where T, = T< + 1, the four average strength functions (&) are simply related by the four coupled closure equations (3.14). If we introduce the dimensionless isospin-mixing parameter p such that (3.22a j then it follows from Eqs. (3.14) that (4)

= 1 - (P
(4)

= CP>/P) I4

(4)

= 1 - (P>lP) p7

(3.22b)

adopting obvious abbreviations for the isospin labels. Hence, the effects of intraspace-mixing in MT’T” are completely determined by the single mixing parameter ,u if we assume that

(c!&&)

= (L&)(&)

(3.23)

in the expression (3.16). Our particular choice of parameterization (3.22a) is motivated by the ideal isospinmixing limits of CX$: p serves as a direct measure of isospin-mixing with values ranging from zero to one. Between the two extremes, the separability assumption (3.23) tends to underestimate MTrT” for T’ = T” and overestimate it for T’ # T”. The extent and meaning of this error is discussed in greater detail in Section D. This transparent parameterization of MTJT~ is particularly useful when isospinmixing is confined to the compound stage of the reaction. In Section E, a generalized version of the T-violating Hauser-Feshbach formula is constructed. C. Isospin-Violating Fluctuation CrossSection

The assumptions and results of the previous two sections may now be collected into an explicit expression for the T-violating energy-average fluctuation cross section. Aside from the specific isospin symmetry-breaking results given above, the algorithm that we follow is essentially identical to that outlined in Section 2 and will not be repeated here. (See FKK for further details.) Without loss of general applicability, we again confine ourselves to the photonuclear process. As conjectured at the end of Section 2, the energy-averaged fluctuation cross

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section, in the chained-partition framework, may be split into its precompound and compound contributions. Accordingly, in place of Eq. (2.14), flW UfY

PCP Uj./

=

-t

u:,:

where the precompozznd (PCP) component is defined by (3.25)

and the compound (CP) component by cp @fY

:I

13.26)

a,,(r).

We recall from Section 2 that the &subspace is, by construction, the final and most complex stage along the chain; its properties will be further clarified below. The partial cross section now refers to the isospin symmetry-breaking generalization of Eq. (2.17) uf.,(n) =

C TJ,Tw(n) MyI(T,T’,T” ,...)

where the generalized depletion factor DTy DT,&n)

DTfl,&z

-- 1) ... Dyr(l)

u,,~

(3.27)

represents the contraction

== 2 r;,,(n)

M,,#(n).

(3.28)

T

As prescribed in the substitution

(3.2) of Section A, the T-violating

damping width

replaces the isospin-conserving expression (2.21). The T-violating decay width I’:.,(n), according to (3.3) is substituted for the T-conserving result of (2.20):

where the channel label c, the outgoing wave boundary condition, is understood to include the isospin projection quantum numbers of the emitted particle and the residual nucleus. In Eqs. (3.29) and (3.30), the biorthogonal bases {I aT>,. ; d, / represent the isospin-pure eigenstates of (X + V),,* . That is, they are eigensolutions of Eqs. (3.5) with (V& set equal to zero. The subscripted bracket indicates an average over those states of appropriate isospin which fall within the energy interval AE surrounding the excitation energy E. Finally, the y-absorption cross section ‘T?,~ in Eq. (3.27) corresponds to (5Y,T = (dk2) CT(l) (3.31)

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where the transmission coefficient T,,r(l), for absorption 2?13 may have either of the forms T,Tl)

= 277PTU) cm;

= (E - E&N2 +

(r,(1)/2)2

;

into the doorway subspace many resonances,

(3.32a)

single resonance,

(3.32b)

many resonances,

(3.33a)

single resonance.

(3.33b)

with the respective widths ~,,d)

= W(a”dl)

I % IO; kJ12), ;

= 27.r I(a”dl) I % IO; k,Y;

The ket / 0; k,) corresponds to the initial state of a free photon with k, and a target nucleus in its ground state. Here, the photon interaction sY is still defined by Eq. (2.8) but with H replaced by (H + V,). The explicit structure of the total width I’, (defined in (3.18)) is clarified in the next paragraph. Note that the (isospin-pure) single resonance effect embodied in (3.32b) and (3.33b) is associated with a depletion factor &r( 1) in Eq. (3.27) corresponding to zero isospin-mixing within the L&subspace. The generalized propagation factor Mr,r” was introduced in the last section to account for the effects of intraspace isospin-mixing. By imposing the weak-coupling assumption of FKK, we found the approximate form given by (3.16). One further consequence of this assumption is that the isospin-pure width r&z), (3.18) may be approximated by (3.34) r,(n) = r;(n) + rm where (3.35)

and ri(n)

= c r$J(n).

(3.36)

7’

This follows by the same arguments that led to (2.18). The total decay width r;(n) and the total damping width P;(n) are the average widths associated with the pure states of isospin T. The sums on the r.h.s. of (3.35) and (3.36) account for interspace isospin-mixing between L& and B and between L?&and L&+1 , respectively. Inserting (3.34) into Eq. (3.17), the explicit structure of the isospin-mixed width P(n), which appears in the denominator of MT~T-(n), is completely specified. Inserting this back into the depletion factor &J&Z), we see that the conservation of probability flux is explicit in the partial cross section (3.27). This is an important consequence of the weak-coupling assumption. We make a few additional comments about the decay width r:,, , as given by Eq. (3.30). The ket #b-‘) satisfies Eq. (2.9) with Hpp replaced by (H + V,),, as prescribed in (3.1). (See also the denominator of (3.3).) In the coordinate representation, a set of coupled-channel equations results involving nonlocal interactions that mix

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241

states of differing isospin. If we replace these by local, phenomenological interactions which preserve isospin symmetry, as is commonly done in practice, then expression (3.30) becomes

(3.37)

where c(T) represents the c-channel coupled to a total isospin T, and
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A more convenient representation of the compound stage is realized by making its substructure explicit. Formally, this is achieved by extending the chained-partitions into the .??,-subspace %

4

-%P

=

6%

3 -%+I

(3.39)

,...I

where L&. on the r.h.s. represents only that segment of the full compound space LZcp which couples directly to the precompound subspace Z?7--1by the action of (X + VJT,V--l . Following the recipe of Section B, we first diagnolize the full 9,-P space by solving Eqs. (3.5) with Vc = 0 and then perform a local rediagonalization with Vc turned on. The equilibrium assumption is employed in the first step. An ergodic statement of this assumption is that each state (of isospin T, angular momentum J, etc.) in 2,, has an equal probability of being occupied. It then follows that if a$ (CP) represents the strength of isospin T’ in the mixed compound state 1 $(CP)) then

c&(m) = ~fw(m> &(CP);

mar

PTf(cp)

where PT@P> = 2 pT’(m). rn=T Inserting this result into the propagation

(3.41)

factor (3.16), we find (3.42a)

where (3.42b) with (3.42~) and

Note well the absence of the damping width in (3.42b): because 9cp represents the final stage along the chain, V,,, = 0 form 3 r in Eqs. (3.5) by definition (see (2.13b)). There are two important advantages to these results: (a) the decay width r&(CP) is simply described by the density-weighted average (3.42d); (b) we still retain the single strength function &CP) to describe compound isospin-mixing. In other words, it is perfectly suited for the extended exciton model of FKK (to be applied in Section 4) and can be parameterized under appropriate conditions by the compound mixing parameter p as demonstrated in (3.22). The consequences of this treatment are discussed in greater detail in Section E.

ISOSPIN-MIXING

D. Zntraspace-Mixing:

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Model Analysis

For the purpose of application and to complete the microscopic realization of isospin symmetry-breaking, it is necessary that a functional link be found between the propagation factor MTITl and the effective isospin-mixing interaction “y;: . In this section, we consider a statistical mixing model in which intraspace-mixing is approximately realized by diagonalizing an ensemble of random (infinite-dimensional) matrices. We obtain a functional relationship between the average isospin strength function and the mean-square value of the mixing strength. The assumptions and intuitive arguments of preceding sections are analyzed. Further implications of our results are noted such as the dependence of isospin-mixing on energy. Intraspace-mixing manifests in MTrTr through the average isospin strength and product strengths, (c$,) and (c$J&), according to Eq. (3.16). The problem of determining MT,T- is greatly simplified by restricting our model space to two isospin degrees of freedom, Tupper (T>) and Trower (T,), T> = T< + 1. From the closure properties (3.14), it can be easily verified that ,{n,T,a,T-) = (~~,)(a$>

-L (26,~,- - 1) Var(zT,)

(3.43)

with Var(&)

= Van&)

for

T’ f T”

(3.44)

where the variance of x, Var(x), is defined by Var(x) = ((x - ~:.x))~;,.

(3.45)

The usual isospin abbreviation is employed, and the nth-subspace label is implicitly understood. With one additional assumption, one can also show that for T # T’ Var(a&) = (pr,/pT) Var(c$).

(3.46)

This follows from closure (3.14) if we assume that p>H+)

=

(3.47)

p
where the normalized variance K(x) is defined by K(x) = Var(x)/(x)”

(3.48)

Relation (3.47) is justified in the context of random matrix theory [18]; it is exact if we assume, for example, a chi-square distribution for LX;, (T =& I!“) with degrees of freedom proportional to pT’ . As a consequence of closure (3.14) and the results (3.43)-(3.46), intraspace-mixing in M,,, is completely specified by, say, (&? and Var(olz). We now turn to the problem of determining the strength function LX:. According to Eqs. (3.5), we must diagonalize a large non-Hermetian matrix. For our model, we consider the Hermetian Hamiltonian (sP + YJ where replaces (&&

t V,,)

(3.49a)

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and vc = vc

replaces (VJ,,

(3.49b)

in the Eqs. (3.5). As implied in (3.2), $?’ includes terms which break isospin symmetry, but does not act to mix states of differing isospin. We now solve the more tractable equation (cYa- A? - VJ 1 a’) = 0

(3.50)

in the two-dimensional space of isospin. This is achieved formally by expanding / LX>) in terms of the isospin-pure eigenstates of X. We find it useful to partition these bases states into three subspaces a, SY, and V, where a is identified with the single pure state 1a) of isospin T, , B with the entire T,-subspace {I b)), and ‘$7 with the remainder of the T, states {I c): (c 1a) = 01. If A, B, and C represent their respective idempotent projection operators, then it follows by construction that Kc)AA = Kcc)BB = v%c

= v%C

= 0,

(3.51)

After standard operator algebra, we find I a>) = A 1 CL>) + B 1 a') + Cl a')

(3.52a)

with (3.52b)

A I a>> = I ah B I a') =

B E, - xBB (YJB.d A I a?) + (higher order in VJ,

(3.52~)

c 'I

a>' = E, - Xc, - (?'&B(B/(Ea x E fxBB a

- %"))(Fc)BC(c)CB

(+%A A 1 +,

(3.52d)

where E, is the unperturbed energy of j a). Terms of order (8, - E,) have been ignored in anticipation of the statistical symmetry that we later impose. Note that we have chosen 1a) to be the ZIM boundary limit of I a>) as formulated in (3.11): I+& 1a>) = 1a).

(3.53)

A manageable, closed expression for a< emerges by imposing two assumptions: (a) the higher-order terms on the r.h.s. of Eq. (3.52~) can be ignored; and (b) the correlations between matrix elements are negligible. By its definition (3.12), the strength function then becomes (a’ I A I a’> + a= = (cl’ I A 1a’> + (CL’ 1 B I a’) + (a’ 1 c 1a’>

(3.54a)

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with partial strengths approximated

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by

where yba z I@ \ ?; i a:,,“.

(3.55)

The spreading width r,. is parameterized by r, = 2n(y/D,)

(3.56)

where I y

=

\


(3.57)

and D, is the mean level spacing of the T,-levels in 9 (not to be confused with the depletion factor of (3.28)). The denominator of (3.54a) ensures that the normalization (a’ 1 a-,> = 1

(3.58)

is properly obeyed and that the closure property (3.14a) is satisfied. The content of our model diagonalization is now complete. The width r, in the denominator of (3.54d) accounts for the spreading of T,-strength in the subspace % among the T,-levels in 9. Treating I’, at the outset as a function of average parameters is a necessary consequence of our ignoring the higher-order terms in Eq. (3.54~). The particular parameterization (3.56) follows directly from the equal-spacing, constant-y assumption (see the statistical treatment below). The terms that we ignore in (3.54~) account for second- and higher-order coupling between 1a) and the ‘I’,-levels of g via the T,-levels of V. Because p< > p> in all cases of interest, this approximation is quite reasonable. We interpret these results as follows. Each IF’,-state (including / a)) spreads its strength as though it were an isolated state resting in a sea of T,-levels. This occurs through ) and I’,, in (3.54c) and (3.54d), respectively. The mixing of T,-levels among themselves is permitted only with respect to the coupling of / LI to the %-space in the partial strength (~9 1c 1 c?). This occurs through the convolution of the two appropriate strength functions on the r.h.s. of Eq. (3.54d). We shall now assume that the auerage properties of the mixed states can be replaced by the statistical properties of the single state ( CL>‘>.To determjne the mean and variance of CL:, we consider an ensemble of infinite-dimensional random matrices, each diagonalized according to (3.54). The isospin strength function is then interpreted as a random functional with the y-strengths and level-spacings treated as random functions, the distributions of which are obtainable from the theory

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matrices [18]. Both (012) and Var($) are then approximated by (a) expanding az and (as)” in a Taylor series about the mean values of their arguments; (b) performing the appropriate weighted averages; (c) retaining in our final expressions terms up to the second power of the inverse density ratio l/R where (3.59)

R = P = D,ID<.

The results of this program are given in Appendix A. For our present computations, we adopt the standard distributions of random matrix theory. The Porter-Thomas distribution for the y-strengths with (3.60)

K(Y) = 2 and the Wigner distribution

for the level-spacings with K(D3

= K(D>)

(3.61)

= 0.27

where the normalized variance is defined in (3.48). Inserting these results into (A3) and (A4), both ((II:) and Var($) are completely specified by the mean y-strength and the mean level-spacing. In the analysis below, we use the full expressions (A3) and (A4). However, these are too cumbersome to be employed in practice. If we adopt the separability assumption (3.23), then the simple analytic expression

cOW(~2/R>(r/D<2>) ‘oI=’ = coth((r2/R)(y/D<2))

+ R

(3.62)

suffices to relate MTtT” to Yc. This is just the equal-spacing, constant-y result of (A5) which follows directly from (3.54) by replacing all the y-strengths and levelspacings by their mean values and by assuming an infinite-dimensional space of T< and T, states. By neglecting the corrections due to the variance of y and DT , (3.62) slightly overestimates (a<) near the zero-mixing limit and underestimate it near the complete-mixing limit. The error, however, is never greater than 20 %. The reliability of the separability assumption is examined below. We return now to the generalized propagation factor MT’=” as approximated by (3.16). According to Eqs. (3.43)-(3.46) and the closure properties (3.14), MT,=” is now completely determined. For our quantitative analysis, we focus our attention as a function of y112 and D, , where M,“y is the ideal CIM on the ratio M,,/M,c:M limit identified in (3.20). This choice of normalization is particularly convenient because it effectively cancels out the density related complications embodied in the widths, r, and r, , in the denominator of M,, . In fact, this cancellation is made exact by imposing the arbitrary constraint r< = r,.

(3.63)

We do this in the present context to emphasize the features of intraspace-mixing. In practice, of course, the properties of r, and I’, must be properly included, and the constraint (3.63) relaxed.

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0.8

0.6

M<> Mc:Y

1.0 0.8 0.6 0.L 0.2

0.0I

0.05 0.1

&ID<

0.5

I

’

FIG. 2. Propagation factor M< :. in relation to its ideal complete-mixing limit is plotted on an inverted semilog scale as a function of ~l/~jD< for two values of the density ratio R. The solid curves correspond to the proper inclusion of the variance of the isospin strength, while the dashed curves follow from the separability assumption.

In Fig. 2, MJM,, crM is plotted on an inverted semilog scaleas a function of yliz/D, for two different values of the density ratio R. The solid curve representsthe realistic case with Var (cY’,,)properly included as prescribed in Eq. (3.43). The dashed curve corresponds to the neglect of Var(c&) as required by the separability assumption (3.23). Before interpreting these results, we make a few comments on their reliability. First, by neglecting higher-order terms on the r.h.s. of Eq. (3.54). the rate of ascent of M,.. ;iM,CY towards its asymptotic limit is slightly underestimated. However, this error is partially rectified by our neglect of higher-order moments in the Taylor expansion of Appendix A. Both of these approximations are dependent on the condition R > I. Second, the Taylor expansion truncation results in a slight underestimation of Var(olZJ. Consequently, the gap between the solid and dashed curves is conservatively estimated. Third, an order of magnitude variation in the constraint (3.63) produces at most a 30 y{, variation in the value of M,,/M;y in the critical

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region between the two mixing extremes; the qualitative features are essentially unaffected. And finally, the intent of the “local rediagonalization assumption” of Section B is still retained inspite of our introducing an infinite-dimensional model. The spreading widths associated with isospin-mixing are still much smaller than the energy interval LIE over which the cross section is energy-averaged. Overall, we believe that the qualitative features of our results as manifest in Fig. 2 are not limited to the approximations and constraints of our model. but would remain in a more exact treatment. We devote the remainder of this section to a discussion of the important properties of intraspace-mixing implied from our model. Most of the material is extracted directly from Fig. 2. Further implications in the context of the reaction cross section are explored in Section E. (i) Efectiveness of a weak mixing interaction. We first note the effectiveness of a weak mixing interaction K to thoroughly mix a dense region of levels. Depending on R, isospin mixing is about 90 y0 complete for y1f2 between 0.6 and 0.9 of the T< mean level-spacing. For example, with p< = 10 KeV-l and R = 100, mixing is more than 90 o/o complete when y1J2 3 80 eV.

(3.64a)

This corresponds to a spreading width, (3.56), r > 0.4 KeV.

(3.64b)

Densities of the order 10 KeV-l are not uncommon in compound reactions involving medium-weight nuclei at excitation energies above 10 MeV. For heavy nuclei, substantially greater densities are expected even at the precompound stages of the reaction. (See exciton model in Section 4, and in [9].) The essential point is that near complete isospin-mixing is likely to be a common feature of the fluctuation cross section. From P-decay studies, for example, Bloom [8] has found that I(/ %< I)1 generally falls between 1 and 40 KeV. His calculations, however, are based on the coupling of “known” states of “simple” structure, using first-order perturbation theory. As we, on the other hand, are interested in the mixing of “complex” states lying reasonably close in energy, Bloom’s estimates can only serve as upper-limits. For our present purposes, we surmise that the root-mean-square value of I(1 YC I)1 generally falls within the range 0.1 KeV < y112 < 10KeV

(3.65)

depending, of course, on the size of the nucleus and the complexity of the subspace. This is partially substantiated in the application example of Section 4. The range of yl/‘J will be more precisely localized after the present theory is more extensively applied.

ISOSPIN-MIXING

IN

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REACTIONS

249

(ii) Rapid energy dependence of isospirt-mixing. We observe directly from Fig. 2 a rapid rise in intraspace-mixing over a critical interval of Y~/~/D, . If y is assumed reasonably constant, then at the appropriate density an order of magnitude increase in pi results in a rapid transition in M,, from 0.1 to 0.9 of its CIM limit. It is through p< that isospin-mixing acquires its sensitive dependence on excitation energy. It is well established, for example, that the Compound Nucleus level density increases exponentially with energy and that in the precompound stages of the reaction, the relevant level densities have a high-power energy dependence [23]. (This latter dependence is argued in the context of the equidistant spacing model where the density associated with the excitation of p-particles and h-holes is proportional to the ( p -+ h - I)-power of energy.) In the 4p - 4h subspace, for example, the above critical transition can occur over an interval of 4 MeV in excitation energy. Tn the compound space, this can correspond to an interval of 2 MeV. This effect has important implications on the analysis and interpretation of the fluctuation cross section. We recall from the paragraph following (3.21) that for R > I, MCtM <> permits isospin-channel conversion without significant penalty to the cross section; M:y, on the other hand, is zero. Correspondingly, M$.y is just the isospin-conserving propagator of Section 2 while MzlM inhibits the flux by about l/R. Consequently, an increase in isospin-mixing efficiency can produce a significant rerouting of probability flux along the chain in Eq. (3.27). (The 7’,-channel is favored for R > 1.) The associated energy dependence can therefore be expected to compete in importance with the energy dependence embodied in the damping and decay widths in the denominator of MTzT,, . This is particularly important for T-violating reactions. It is conceivable that under certain circumstances the energy dependence of isospinmixing could be observed directly in the T-violating compound cross section. The necessary conditions are (a) the excitation energy must be high enough for the compound level density to achieve the critical density, but low enough to ensure negligible isospin-mixing in the precompound stages; (b) the rapid increase in the total compound decay width P, (3.42c), due to threshold effects, does not significantly detract from the rapid rise in T-violating flux due to the increase in isospin-conversion. (iii) Underestimation of mixing eficienq~ by separabilitJ3 assumption. The gap between the dashed and solid curves in Fig. 2 reflects the overestimation of $I<,, due to the separability assumption (3.23). Actually, this follows directly from Eq. (3.43): by dropping Var(&), M,, (and M,,) is overestimated, and M,, (and /M,,) is underestimated. Comparing Figs. 2a and 2b. we see that the error, decreases as R increases. We recall that the separability assumption was employed at the end of Section B to reduce MTpT” (in two-dimensional isospin space) to a function of single parameter. The isospin-mixing parameter p was intentially defined in (3.22a) to serve as a convenient measure of mixing efficiency: p = 0 in the ZIM limit, and p == I in the CIM limit. However, if p o~served is the value obtained from fits to experimental data (see Hauser---Feshbach expression (3.74) below). then, due to the inflated influence of h4 , Pobserred

‘: Plrur

(3.66)

250

R. LEON

FEINSTEIN

Here, fitrue reflects the “true” degree of mixing resulting from the proper inclusion of Var($). The equality is satisfied near the two mixing extremes as can be verified from simple statistical analysis. In the critical intermediate region, p&served = 0.4 could correspond to an actual mixing of ptrue = 0.7. These implications are particularly important in Section E where the separability assumption is used to derive a T-violating version of the Hauser-Feshbach formalism. E. Further Consequences In the formal development and model analysis of the preceding sections, a common theme has emerged: isospin symmetry-breaking can appreciatively influence the analysis and interpretation of the fluctuation cross section. In this concluding section, we explore further characteristics and implications of our results. Substantial simplifications are noted which can be employed in practice under specified conditions. In Eqs. (3.24)-(3.27), the T-violating generalization of the energy-average fluctuation cross section is given with its precompound and compound contributions formally separated. To facilitate the present discussion, we now distinguish between two modes of isospin-mixing: precompound-mixing, and compound-mixing where, as the names suggest, the former applies to mixing within and across the chained-partitions of the precompound stages, and the latter refers to mixing within the compound subspace only. (Note that interspace-mixing between L.& and L&. is considered precompound.) While precompound-mixing can significantly affect the compound crosssection, compound-mixing confines its influence to the compound cross section only. In practice, the distinction relies critically on the extent to which the precompound and compound cross sections can be identified in the theoretical analysis of empirical data. (This improves with the size of the target nucleus.) The relative importance of precompound-mixing and compound-mixing is directly correlated with the characteristic nuclear relaxation time for equilibrium, r, . If by 7Gwe denote the characteristic interaction time for V, to mix states of differing isospin, is permitted (the Compound Nucleus then for TV> r7, only compound-mixing begins to compete can last much longer than TV); for TV - 7, , precompound-mixing with compound-mixing in influence; and for rc < r, , precompound-mixing emerges as the dominant mechanism for isospin-conversion in the reactions evolution (see below). Because T, increases with the size of the system, precompound-mixing is most favored in reactions involving medium-weight and heavy nuclei. E. 1 Precompound-Mixing In the language of the time-independent, chained-partition formalism, precompound-mixing results when the precompound level densities are large enough for the mixing interaction Yc to have a significant effect. We surmise that this is indeed the case for most reactions involving medium-weight and heavy nuclei. (See the model analysis of Section D, part (i), and the application example of Section 4.) In the fluctuation cross section, it manifests through both M,y(n) and &y(n) for n < r. The implications of MT,,- have been thoroughly analyzed in Sections B and D.

ISOSPIN-MIXING

IN

NUCLEAR

REACTIONS

251

We now explore in greater detail the properties and significance of the depletion factor D,,,- . Its properties are best illustrated by considering its form in the two intraspace-mixing limits: for zero-mixing

&“(iZ) D%(n) = XT,,,r&,(n) f cc qT”(n)

(3.67)

while for complete-mixing

Both follow directly from Eq. (3.28) by inserting (3.19) and (3.20), respectively. We make the following observations. (i) Interspace-mixing plays minor role. This is a direct consequence of our treating V’Jn,n+l and =%.n+1 on an equal basis, while treating (VJ,, as a “local” perturbation on the L%?&spectrum. For example, if we make the conservative estimate that (3.69) and R = 100 (see Section D, part (i)), then from (3.29) and (3.67)

In words, most of the 7’,-channel probability flux entering d, leaves through the T,-channel when interspace-mixing is the only conversion mechanism available. It is conceivable that in some heavy nuclei, R might be large enough for interspacemixing to be noticeable, but in most applications the correction does not justify its inclusion. Although this is a consequence of our particular treatment, it does not affect the viability of our results in principle. First, we are free to partition the Hilbert space in such a way as to reduce the importance of interspace-mixing. And second, but most important, this is a statistical theory which relies only on average properties. The consequences of symmetry-breaking in the proximity of &, are adequately taken into account through the mechanism of intraspace-mixing. (ii) Isospin conversion is biased. Most of its preferences are dictated by the property 1. First, conversion occurs mostly from the T,-channel to the Z”,-channel. Because DCIM 1 -22 (3.71) DCIM = F << R>

the probability j9j/IO7/1-Z-17

flux entering LJ& in the T,-channel

can be expected to remain there.

252

R.

LEON

FEINSTEIN

Thus, on moving down the chain in Eq. (3.27), isospin-conversion from the T< to T, channel is highly unfavorable. Second, conversion is most likely to occur at the earliest well-mixed stage of the reaction’s evolution. This can be seen directly from the relation D CIM 1 >> (3.72) DC’M CT+F * <>

In words, only l/R2 of the flux passing through the well-mixed 9,-subspace remains in the T,-channel, leaving little for Z?n+l to convert. This implies that when the precompound-mixing occurs, it emerges as the dominant mechanism for isospin-conversion. Altogether, these properties result in a substantial reduction in the sum over isospin channels in Eq. (3.27). (iii) Energy dependence of precompound-mixing pIays a minor role. We observed in Section D, part (iii), that M,, may have a rapid energy dependence due to isospinmixing. This dependence should be most effective in M,,(n) and Do(n) for n < r as a result of the relation P(n) > P(n). That is, the threshold effects in the decay width P are dwarfed by the comparatively slow energy dependence embodied in P. However, according to the conversion bias discussed above, a rapid growth of, say, D,,(n) results simply in a shift of the conversion mechanism from the L&+r to the 9, subspace. The precompound and compound cross sections would be only slightly affected. E.2 Compound-Mixing:

Generalized Hauser-Feshbach

In light and medium-weight nuclei, isospin-mixing should be confined to the compound stage of the reaction. For this condition, a simplified version of the compound cross section is obtained. From the parameterization of (3.22) and the separability assumption (3.23), we find

where TcT 3 2%-r;,T(cP)

p&P)

(3.75a)

and T

T _ Y

riT
-

l> 1)

The compound propagation factor (3.42) is implied in (3.74). Explicit expressions for rkT ) c,T ) and r, are realized in (3.29), (3.30), and (3.34), respectively. The doorway transmission coefficient T,(l) is given in (3.32). And finally, we have introduced the

ISOSPIN-MIXING

IN NUCLEAR REACIlONS

253

symbols p: and PL to replace the average strength functions. According to (3.22). these are both related to the isospin-mixing parameter p: (3.773) (3.77b)

pi :x = (p,jp)p.

We note in passing that by imposing isospin symmetry within the P-space and by adopting the ergodic description of equilibrium, (3.75a) can be further specified by

as follows directly from (3.37) (3.38), and (3.42d). We recognize Eq. (3.74) as the T-violating generalization of the Hauser-Feshbach formalism [3].l its validity rests on three conditions: (a) that s’r is indeed the equilibrium state and, as such, is effectively independent of the initial mode of formation; (b) that there be negligible precompound-mixing; and (c) that the separability assumption is a reasonable approximation. Arguments in support of condition (a) have been offered at the end of Section 2 and in the paragraph preceding (3.39). Condition (b) is necessary in order to remove the complications of precompound isospin-conversion (see the previous section). And condition (c), the separability assumption, has been extensively analyzed in Section D, part (iii). The isospin strength PJ corresponds to the parameter first introduced by Grimes e’t al. [i] to account for the fraction of probability flux that is removed from the T,-channel and added to the &-channel. In fact, their Hauser-Feshbach expression is recovered ifr form by setting p’ = 0 in Eq. (3.74). This is not unreasonable considering the relation pp -= (l/R)pL in (3.77). Our derivation, however, provides two important advantages. First, the conditions of applicability are more thoroughly understood (see the above paragraph); and second, the microscopic meaning of pLTis clearly defined. This latter advantage is manifest in Section D where a statistical model is employed to relate the average strength function to the mean-square value of the mixing interaction strength. From the equal-spacing, constant-y model, the isospin-mixing parameter p can be approximated by ' = coth((r2/R)(r/o<"))

+ R'

This follows from (3.62). Its validity rests on the condition

R > 1.

4. APPLICATION The statistical theory of the T-violating fluctuation cross section developed in Section 3 is applied in this section to a comparative study of the photoalpha-particle reactions ?Si(y. 01~)24Mg and 3oSi(y, ao) 2sMg; both proceed through the giant dipole 1 An expression similar in form to Eq. (3.74), but slightly different in content, has been independent-

ly derived by Terasawaet al. [20].

254

R. LEON

FEINSTEIN

resonance (GDR) region and decay through the cu,-channel leaving the residual nucleus in its ground state. The reaction involving 2*Si is isospin-forbidden (the initial and final channels have different isospin), while the reaction involving 3OSi is isospin-allowed. The ratio of their energy-integrated cross sections, however, suggests that the former reaction is less inhibited by a factor of about two. It is this discrepancy that we wish to explain. The relevant widths are generated with the microscopic exciton description of Griffin [24] as extended by Feshbach, Kerman, and Koonin (FKK) in Ref. [9]. Considering an interacting system of identical spinless nucleons, FKK modify the traditional exciton model by properly incorporting orbital momentum and by performing the angular integrals correctly. We take the next step towards reality and endow each nucleon with two internal degrees of freedom: neutron and proton. Their respective scattering wavefunctions are then distinguished by the presence of the Coulomb barrier. In order to determine the residual interaction matrix elements, isospin coupling is correctly performed, and the important isospin correlations of the realistic shell model are properly taken into account. The a-particle decay is described by “fusing” together the statistical properties of the exciton model with the collective features associated with barrier penetration. This crude treatment is justified for the comparative study at hand. As most of the derivational methods and results of FKK remain essentially intact, we confine our present discussion to the particular complications associated with isospin and with the a-particle. We refer the reader to [9, Section VI] for further details. A. Extended Exciton Model For our quantitative study, the photoalpha-particle reaction is described by the following specifications: The target nucleus is in its ground state E = 0, with ordinary spin I = 0, isospin T = To, and isospin projection MT = To . By absorbing a k, gamma ray, it is excited into its GDR state at energy E = ED with spin and parity Jr (= I-) and isospin T. As the reaction evolves, the excited system decays through the c-channel by emitting a spinless nucleon (or an a-particle in its spinless ground state). The emitted particle is identified with its appropriate isospin t, and isospin projection m, . The residual nucleus is left with intrinsic energy U, spin 1, , isospin TR . and isospin projection (TO - m,). The reduced center of mass carries away a kinetic energy Edand orbital momentum 1. Conservation of energy requires that E, = (AC)k, ‘v U + B, + E,

(4.1)

where B, is the binding energy of the emitted particle in the target nucleus. The c-channel decay is also identified by it wavevector k, kC2 = 2p,4fi2 . where pe is the reduced mass. For our particular application,

the T-violating

(4.2) fluctuation

cross section of (3.24)

ISOSPIN-MIXING

IN

NUCLEAR

is simplified to allow only compound-mixing. section then assumes the form

REACTIONS

The GDR (.I” = l-) fluctuation

255 cross

(4.3a)

where (4.3d)

(4.3e)

(4.3f) with

The factor C(i)represents theclebsch-Gordon coefficient.lnside the averaging bracket. p(U) is the (I = I,, T = T,) level density of the residual nucleus at energy U. and r:,,(U) corresponds to the isospin-conserving c-channel decay width of J!n leaving the residual energy U. The damping width rjT is the isospin-conserving width of (3.29a). And finally, the compound propagation factor IV&~(T) is specified by (3.42). and its relationship to the mixing interaction $2 is delineated in Section 3D. The remaining symbols were introduced in the preceding paragraph. We partition the Hilbert space as in the illustrative example of the Introduction (See Fig. 1). The nth-subspace now denotes the space of n-excitons where n =. 17 --- /r (‘p-particles and h-holes) and p = h. Moving “down” the chain (with increasing n) the subspace level-density increases rapidly. Adopting the rule of FKK, the compound space begins at the last subspace to satisfy

For our computation, we assume that the subspace level density (‘for states of definite parity) satisfies the expression

256

R

LEON

FEINSTEIN

From Ericson [23], the equidistant spacing model gives g(E’)“-l

PDdE)= 2p! h! (n - l)!

(4.5)

with E’--E-Ash, A,, = f( p2 + A3 + i(p - 4 - (h/2), g = gn + g, 5 where A,, is the Pauli Principle correction factor of Williams [25], and g,( g,) corresponds to the single particle neutron (proton) level density. The factor 2 in the denominator of (4.5) implies an even splitting of total parity. The angular momentum and isospin distribution multipliers are represented by 25 + 1

R,(J) = 2(27r)1/2 (n(m2>)312 e-((J+(1/2))*/zn
(4.6)

where (m2> is the mean-square single particle magnetic quantum number, and 2T+

1

R,(T) = (n + 2T + 2) 2”-I i T +‘;n,2) 1 with .x ! x 0Y = y!(x - y)! -

Note that both (4.6) and (4.7) are the “observable” distributions associated with particular projection quantum numbers. The derivation of R,(J) is based on the pioneering work of Bethe [27]. R,(T) follows in a similar manner by assuming a binomial distribution for the probability that rz-excitons have a resultant isospin projection. And finally, the shift parameter 6(T) in (4.4) corrects for the neglect of symmetry correlations in the exciton model. It is well known that in a realistic shell model, the residual interaction matrix is strongly correlated with isospin due to its spatial symmetry. It is the nature of this relationship that results in a substantial shifting of T, levels to higher energies. We choose 6(T) equal to the energy gap between the ground state and the lowest lying level of isospin T in the “realistic” model. The representation (4.7) is valid for large nuclei with proton and neutron numbers nearly equivalent. It is instructive to note that

is exactly equivalent

to the probability

that (n/2)-protons and (n/2)-neutrons

couple

ISOSPIN-MIXING

IN

NUCLEAR

REACTIONS

251

to a total isospin T. This can be verified by summing over the appropriate product of squared Clebsch-Gordon coefficients. The damping and nucleon decay widths are calculated in a manner prescribed by FKK. It is assumed that both %&,.r and HPn are represented by the single residual interaction V(r, , r2) = Vo(QrR3) P(r,

- r2)

(4.9)

with nuclear radius R = 1.4 A1/3 fm. and where the overall strength I’, is a free parameter. As energy, angular momentum, and isospin are treated semi-independently in our statistical model, the damping widths and nucleon decay widths factorize into the forms (4.10) r:,(n) = W:T(E) Y,‘(Ej -2%~ and

The derivation of Y and Z is outlined in [9, Section VI]. The symbols (:)- distinguish the three types of decay interaction considered by FKK: (+) refers to particle-hole creation; (0) to particle-particle and particle-hole interactions; and (-) to particlehole annihilation. The function Y,i(Ynr) represents the average final density of states in --fL2 (in the residual nucleus) to which a typical state in S, can couple (while emitting a nucleon of energy E - B, - U). The 2 function accounts for the effects of angular momentum and isospin coupling. For the particular interaction (4.9), the isospin dependence is averaged away, and the results of FKK are recovered. The isospin-weighting factor W emerges naturally to account for the proper shell model (SM) correlations mentioned above. It corrects in an average way for the missing correlations in the exciton model (EM). Ideally,

but by the prescription (4.4), we assume that wiT(E)

Similarly,

=

P~+I.~+LE PP+Lh+lw

WN

(4.12a)

.

the decay weighting factors become

wt+;(r/> = P~.h+l(~ - 6(Td) w$~R(u) @.t.#>

= =

PD,h+du

'

ppr--lAU - WR)) Pv--l.hW)

1

p9-2.h--1(U PD--?J--1(U)

-

6(TR))

(4.12b) (4.12c) .

(4.12d)

258

R. LEON

FEINSTEIN

In order to determine the average-square matrix elements of the relevant widths, radial overlap integrals must be performed. Following FKK, we make the crude estimate that the bound and continuum radial wavefunctions be constant inside the nuclear volume. However, to account for the charge of the emitted particle, we modify the continuum radial wavefunction to allow for Coulomb barrier effects. Normalizing on the energy shell, we surmise that &k&>

= 5W4

pcKTW

k,r

(4.13)

where TC is the penetration coefficient for a square-well of radius R derived in Blatt and Weisskopf [28]. For neutrons, we use the Riccati-Bessel and Riccati-Neumann functions, respectively, for the regular and irregular solutions in the exterior region. For protons and a-particles, we require the regular and irregular Coulomb functions. The interior wavenumber is defined by Kc2 = &4~,

(4.14)

+ u,)lfi’

where U, is the square-well potential depth. And finally, the “continuum correction factor” 5 amends for the artificial coherency imposed on the radial overlap integrals by the constant wavefunctions and for the redundancy associated with the partial overlap of the 9 and B spaces. This is a free parameter in our calculation which is clearly less than one. The remaining symbols are defined at the beginning of this section. The a-particle requires further clarification. Due to the dominance of nucleon decay, we need only consider the a-particle decay width associated with the final ol,-channel (tm = mar = 0, U = 0, IR = 0, and TR = To). For our present needs, the kinematic and spectroscopic features are adequately described by the factorized representation ~,o~.r=~,r=~o(4 0~ T,ntexl PXE) W,“.

(4.15)

The factor T” is the square-well penetration factor discussed above. P;Z(E) is the probability that with E spread among n-excitons, four of the particles (above the Fermi sea) have the proper amount of energy for a,-decay. We propose that P; (E) = where BB--d.h is the minimum

pas@

-

L-4.J

P,,@)

energy required to realize a (p - 4, h) configuration:

&h = (l/d[$tP2

+ W + &(P -

41.

The factor W; is the probability that the four chosen particles have their isospin properly correlated to form a t, = mar = 0 particle. For a large nucleus consisting of an equal number of neutrons and protons, we find P-2

(p - 4)! 3!

w?T = “T2 (v - 2)! (p - v - 2)! 2p

(4.17)

ISOSPIN-MIXING

259

IN NUCLEAR REACTIONS

where p = n/2. The constant of proportionality in (4.15) embodies all the remaining complications that we wish to avoid. We make the crude assumption that it varies little between 2sSi and 3oSi and between one subspace and another. This latter assumption is not too serious because, as implied in (4.16), the oi,-decay requires a configurational complexity of at least four particles-four holes. B. Calculation and Analysis We employ the formula and assumptions of Section A to compute the ratio of the energy-integrated cross sections ?‘Si(y, ao) 24Mg and 3oSi(y, ao) 26Mg. The relevant energies, angular momentum, and isospin are illustrated in the level schemes of Figs. 3 and 4. The displacement between the T L= 1 and T = 2 GDR states of 3oSi are calculated from the schematic model of Akyiiz and Fallieros [29], while the gaps 6(T) of (4.4) are determined from Janecke’s formula [30]. The GDR energies are extracted from the experimental data of Singh et al. [31] and Katz et al. [32] for e8Si and 3oSi, respectively. We assume that the integrated giant dipole strengths of 2sSi and 3oSi are equal (about 60 % of the total dipole strength). However, in 3oSi this strength is split between the two isospin components, T = 1 and T = 2. We estimate the fractional distribution by appealing to the model of Fallieros and Goulard [33]. The remaining parameters are chosen as follows: Both g and
GDR

E19.5

-19.0

l;T=l

17.18 ?z;T=

5/z+, T = 3/z '/i

-15.0

OTT=2

1158 + ?z, T=l/z

9.99 O:T=O

27Si(+n)

- 9.0 OfT-1

27Al(+ p)

2LMg(c CL)

0.0 (t&VI f=O+,T=O

“Si FIG. 3. Schematic energy-level diagram

for YG and relevant neighboring

nuclei.

260

R.

LEON

FEINSTEIN

GDR

13.51 5/z:T=3/2 -411.0

10.65

10.61

-bfT='/2

OfT=2

O:T=l

%U+p)

2gsi(+“)

26M~(+ al

0.0 (McV) J<=O:T=l

3oSi FIG. 4. Schematic energy-level diagram for %i and relevant neighboring

nuclei.

5 are determined by fitting two empirical quantities: (a) the YSi GDR width r, = 4 MeV; and (b) the total integrated photoneutron cross section up to 30 MeV excitation in YSi equals 90 MeV-mb. And finally, we adjust our last free parameter r1j2, the root-mean-square mixing strength, by normalizing the ratio of the integrated cross sections to fit the empirical results. The relevant experimental cross sections are found in the paper by Meyer-Schtitzmeister et al. [5] and reproduced in Table I. The final results of our calculation are contained in the parameter values of Table II. Both 5 and Y,, are substantially greater than the values found by FKK. We attribute this to the difference in size of our respective systems: FKK consider nuclei substantially heavier than 2sSi. In light nuclei, we expect the imposed coherency in TABLE Experimental

I

Energy-Integrated Cross Sections Reproduced from Meyer-Schiitzmeister et al. [5]

Target nucleus

Range of integration NV)

esSi

16.65-22.65

Y3i

14.1 -20.1

sutibgYdE (mb-MeV) 4.34 2.23

ISOSPIN-MIXING

IN NUCLEAR REACTIONS TABLE

II

Parameters Employed in Fitting Empirical Parameter

261

Results

Value

Relevant formula

55 MeV 70 MeV 3.8 1.5 l/l5 0.8 MeV 180eV

(4.14) (4.14) (4.5)

-~-.

(4.6) (4.13) (4.9) (A3) and (A4)

the radial overlap integral has greater validity, and the redundancy associated with the continuum and bound wavefunctions is less serious. The single particle density g is slightly larger than the value suggestedby Gilbert and Cameron [26]. This difference is due to their reliance on low-energy empirical data whereas we allow for the influence of the higher single particle orbitals which are expected to be important at the present energies of excitation. However, there still remains a significant enough uncertainty in g to affect the reliability of our final yllJ. (We recall from Section 3D that yliz/D< emergesas the essential parameter in determining the isospin-conversion efficiency of the cross section.) Until a more reliable estimate of g is provided we are reasonably safe in interpreting y1i2 in Table II as a lower bound. The primary advantage of this crude model calculation is that all the essential physical characteristics of the reaction process are explicit and available for analysis. The discrepancy in the energy-integrated cross sections of Table 1 can be explained as follows: (i) Minor penalty imposed for T-violation in 2sSi. Over the energy-integrated interval, the 28Sicross section is inhibited on the average by a factor of about 0.5 due to isospin-conversion in the compound space. This corresponds to an average value of 0.7 for the mixing parameter rutrue (see Section 3D part (iii)). That is, compoundmixing is about 0.7 of its ideal complete-mixing limit. (ii) Excess neutrons inhibit a-decay in 3OSi. Due to the 7 MeV drop in neutron binding energy in 3oSi(seeFigs. 3 and 4), the total neutron decay width in the denominator of the compound propagator suffers a substantially increase in relation to the analogous width in 28Si.This results in a decreaseof the a,,-particle decay probability in 3oSiby a factor of about 0.3, on the average. (iii) Coulomb barrier inhibits a-decay in 3oSi. The ,a-particle penetrability in 3oSi is lower on the average by a factor of about 0.7 compared with that in 2%. This is due to the larger a-particle binding of about 1.5 MeV and to the 2.5 MeV drop of the “OSiGDR region in excitation energy.

262

R. LEON FEINSTEIN

(iv) GDR strength is displacedin 3oSi. Approximately 0.4 of the 3oSi GDR strength is displaced up to the T = 2 state at E = 21.5 MeV. This reduces the flux by a factor of about 0.9 in the lower energy-integrated region. To emphasize the credibility of our results, we compare with the recent analysis of Shikazono and Terasawa [20]. They compute the compound cross section of 28Si(y, 01~)24Mg using a T-violating generalization of the Hauser-Feshbach formula identical in form to Eq. (3.74). By normalizing their results with the empirical data, they obtain

which is held constant over the 28Si GDR region. If the product of depletion factors is assumed to be 0.8 (a realistic estimate from the present results), then PJ = 0.31. As discussed in Section 3D part (iii), this corresponds to ptrue between 0.5 and 0.6. Within the error associated with these analyses [19], their result agrees reasonably well with our estimate above. Despite its obvious crudeness, the extended exciton model apears to embody most of the essential physical characteristics of the “exact” shell-model. At the cost of greater complexity, a lot of the simplifications imposed here can be removed in a straightforward manner. The prospects for the exciton model in more sophisticated treatments are very encouraging. 5. CONCLUSION We have provided substantial evidence to support our thesis that isospin-mixing in the fluctuation cross section cannot be ignored and that its inclusion can greatly influence the theoretical analysis and interpretation of experimental results. The T-violating generalization of the energy-average fluctuation cross section was given in Section 3C. The implications of symmetry-breaking on the cross section were explored in Sections 3D and 3E. In Section D, with our statistical mixing model, we explored the effectiveness of the mixing interaction strength and the energy dependence of the mixing mechanism. In Section E, further simplifications were noted; for example, the T-violating version of the Hauser-Feshbach formalism in Eq. (3.74). And finally, in Section 4, we provided a simple microscopic realization of our theory in the exciton representation and applied it to explain the experimental discrepancy between the cross sections 28Si(y, ao) 24Mg and 3oSi(y, 01~)24Mg. We demonstrated how isospin selection rules may be broken without great penalty to the T-violating probability flux. APPENDIX

A

In this appendix, we derive the approximate expressions for (cyz) and Var($) which are employed in the analysis of Section 3D. The final results are given in Eqs. (A3) and (A4), respectively.

ISOSPIN-MIXING

IN NUCLEAR REACTIONS

263

We begin with the expression (3.54) and the assumption that az is a random functional with y-strengths and level-spacings treated as random functions, the distributions of which are obtainable from random matrix theory [18]. If F(x, , xz ....) is a “well-behaved” random functional, then it can be expanded in a Taylor series about the mean values of its arguments. The mean and variance of F can then be expressed in terms of the central moments of (x,):

]+

.o=

(higher-order terms), (Al)

Var-F) _ c Var(xj) [g lZ=,Z,]z + (higher-order terms). 3 j The variance of x, Var(x), is defined in (3.45). > In our final expressions,however, we only keep It is this recipe that we apply to CX>. terms up to order R-2, where R = D,/D, . It is the condition R > I that ensures the rapid convergence of both (Al) and (A2). Generally, these expansions do not converge rapidly (or at all). After a straightforward, but lengthy exercise, we obtain ,’ >\ ‘k~>,,’

Var($)

_

-> - ix> + K(y) Cl + K(D,) C” - K(D,) C,,

(A3)

= K(y) C, + ND<) C, ,

644)

where 6; := a;((~:‘, (D
(A51 R

is just the equal-spacing, constant-y result corresponding to the zeroth-order term in (A 1). For convenience, we have introduced the function M = (dR)(ylD,*)

W)

and the normalized variance K(X) = Var(x)/(x,?“. The C-coefficients are complicated functions of y/DC2 and R: Cl = ,v3 1[25X(4)] &,, c, = Iv3 1t-3

+ [P’.X(4) &.A1 )I R-” (+I’!

. 257(1, 4)] (+I

2

6474

-t t2%(2, 3) - 3 * 2s.x(2) q,(l, 4>1(+J

t- [21ob4(1) $2, 3) - 3 * 2gh(2) &.,(l) $1, 4)] R-2 (+-j’(,

(A%)

264

R. LEON FEINSTEIN

c, = N3 /[29’(2)

c&l,

+ [212A3(2) t’dl,

3) - 2’.x2(2) tM(l, 2)] R-2 (+)” 3) - 21°h3(2) h(1,

2)l R-2 (+)4/,

(A7c)

C, = N4 25X(4) (+)2,

(A74

C- = N4 2gq(2, 3) (+)2.

(-474

The following functions have been extensively used: 1 N = TM coth(rrM)

+ TMR

and m

h(J9 = b;. [2b :

&4(P) = e=l g

[c2 :

1p ’

M2]P ’

These results are extremely good in the critical regions that concerns us. Their reliability decreases, however, with increasing y/Dd2 and decreasing R. With R = 100, we find an error of about 10 % in (cy~) and Var($) at y1f2/D, = 2. ACKNOWLEDGMENTS The author extends his sincere gratitude to Professor H. Feshbach for his suggesting the original problem and for his stimulating comments throughout the course of the investigation. The author also acknowledges the hospitality of the Niels Bohr Institute where this work was completed. REFERENCES 1. S. M. GRIMES, J. D. ANDERSON, A. K. KERMAN, AND C. WONG, Phys. Rev. C5 (1972), 2. D. ROBSON, A. RICHTER, AND H. L. HARNEY, P&s. Rev. C 8 (1973),153.

85.

ISOSPIN-MIXING

IN

NUCLEAR

REACTIONS

265

3. W. HAWSER AND H. FESHBACH, Whys. Rev. 87 (1952), 366. 4. T. ERICSON, Ann. Physics 23 (1963), 390. 5. L. MEYER-SCH~~ZMEISTER, Z. VAGER, R. E. SEGEL, AND P. P. SINGH, Nucl. Phys. A 108 (l968), 180. 6. R. E. PESCHEL, J. M. LONG, H. D. SHAY, AND D. A. BROMLEY, Nuc/. Phys. A 232 (1974), 269. 7. H. FESHBACH, Ann. Physics 5 (1958), 357; 19 (1962), 287. 8. S. D. BLOOM, Isobaric spin conservation and @decay, in “Isobaric Spin in Nuclear Physics” (J. D. Fox and D. Robson, Ed.), p. 123, Academic Press, New York, 1966. 9. H. FESHBACH, A. K. KERMAN, AND S. KOONIN, to appear. (Referred to as FKK in text.) 10. H. FESHBACH, A. K. KERMAN, AND R. H. LEMMER, Ann. Physics 41 (1967), 230. 11. N. AIJERBACH, J. HUFNER, A. K. KERMAN, AND C. M. SHAKIN, Rev. Modern Phys. 44 (1972), 48. 12. F. L. FRIEDMAN AND V. F. WEISSKOPF, in “Niels Bohr and the Development of Physics” (W. Pauli, Ed.), p. 134, Pergamon Press, New York, 1955. 13. M. KAWAI, A. K. KERMAN. AND K. W. McVou, Ann. Physics 75 (1973), 156. 14. R. L. FEINSTEIN, Doctoral Thesis, M.I.T., 1975. 15. C. M. SHAKIN AND W. L. WANG, Phys. Rev. C 5 (1972), 1898. 16. N. BOHR, Nature (London] 137 (1936), 344. 17. P. MOLDAUER, Rev. Modern Phys. 36 (1964), 1079. 18. M. L. MEHTA, “Random Matrices,” Academic Press, New York, 1967. 19. T. TERASAWA, private communications. 20. N. SHIKAZONO AND T. TERASAWA, Nucl. Phys. A 250 (1975), 260. 21. A. BOHR AND B. MOTTEISON, “Nuclear Structure: Volume 1,” Benjamin, New York, 1969. 22. N. T. POFULE, J. C. PACER, J. WILEY, AND C. R. Lux, Phys. Rev. C9 (1974), 2171. 23. T. ERICSON, Advan. Phys. 36 (1960), 425. 24. J. J. GRIFFIN, Phys. Rev. Lett. 17 (1966), 478. 25. F. C. WLLLIAMS, Nucl. Phys. A 166 (1971), 231. 26. A. GILBERT AND A. G. W. CAMERON, Cunad. J. Phys. 43 (1965), 1446. 27. H. A. BETHE, Phys. Rev. 50 (1936), 332. 28. J. M. BLATT AND V. F. WEISSKOPF, “Theoretical Nuclear Physics,” Wiley, New York, 1952. 29. R. 0. AKY~~Z AND S. FALLIEROS, Phys. Rev. Lett. 27 (1971), 1016. 30. J. JANECKE, Nucl. Phys. 73 (1965), 97. 31. P. P. SINGH, R. E. SEGEL, L. MEYER-SCHOTZMEISTER, S. S. HANNA, AND R. G. ALLAS, Nucl. Phys. 65 (1965), 577. 32. L. KATZ, R. N. H. HASLUM, J. GOLDEMBERG, AND J. G. V. TAYLOR, Canad. J. phys. 32 (1954), 580. 33. S. FALLIEROS AND B. GOULARD, Nucl. Phys. A 147 (1970), 593. 34. H. G. WAHSWEILER AND W. GREINER, Phys. Rev. Let?. 19 (1967), 131.