Extending the Kawai-Kerman-McVoy Statistical Theory of Nuclear Reactions to Intermediate Structure via Doorways

. Kawai, Kerman, and McVoy have shown that a statistical treatment of many open channels that are coupled by direct reactions leads to modiﬁcations of the Hauser-Feshbach expression for energy-averaged cross section [Ann. of Phys. 75 , 156 (1973)]. The energy averaging interval for this cross section is on the order of the width of sin-gle particle resonances, ≈ 1 MeV, revealing only a gross structure in the cross section. When the energy-averaging interval is decreased down to a width of a doorway state, ≈ 0 . 1 MeV, a so-called intermediate structure may be observed in cross sections. We extend the Kawai-Kerman-McVoy theory into the intermediate structure by leveraging a theory of doorway states developed by Feshbach, Kerman, and Lemmer [Ann. of Phys. 41 , 230 (1967)]. As a by-product of the extension, an alternative derivation of the central result of the Kawai-Kerman-McVoy theory is suggested. We quantify the e ﬀ ect of the approximations used in derivation by performing numerical computations for a large set of compound nuclear states.


Introduction
One way of analyzing low-energy nuclear cross sections is by varying the experimental energy resolution, or alternatively, by numerical energy-averaging of high-resolution data.It is known that different energy resolutions may reveal different features in the cross section.Such features may provide insight into the dominant processes contributing to the cross section.
Extremely high energy resolution, on the order of fraction of an eV, reveals compound nuclear resonances, often referred to as fine structure.In the other extreme, when the energy resolution is on the order of MeV's, single-particle resonances may remain the only visible feature of what is commonly referred to as a gross structure [1].For an intermediate energy resolution, on the order of 100 keV, a so-called intermediate structure emerges, for which a theory of doorway states was developed in [2].Doorway state concept was used to construct a non-local optical potential in [6].
In this work we will consider the intermediate and the gross structures of low-energy cross sections.The latter is the realm of the optical potentials and statistical theories of nuclear reactions, of which we a e-mail: arbanasg@ornl.govNotice: This manuscript has been authored by UT-Battelle, LLC, under contract DE-FC02-09ER41583 (UN-EDF SciDAC Collaboration) with the U.S. Department of Energy.The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a nonexclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.
will focus on the Kawai-Kerman-McVoy (KKM) theory [1], which will be outlined in Sect. 2. We will leverage some of the formal expressions derived in [2] to extend the KKM theory into the realm of the intermediate structure in Sect.3. We will test the validity of some approximations used in derivation of KKM results using a simple numerical model in Sect. 4.

Gross Structure
In this section we outline some key ingredients of the KKM theory [1].Alternative expositions of the KKM theory can be found in [7] and [5].The theory uses the Feshbach's projection operator formalism to divide the Hilbert space into continuum (P) and the compound (Q) subspaces: where P and Q are unitary Hermitian projection operators.Projecting the Schrodinger equation HΨ = EΨ into the two subspaces, and using a two-potential formula to find the T -matrix, yields a backgroundplus-resonant expression 1 : where T P is the T -matrix of the PHP ≡ H PP , and 2 where a complex symmetric operator describes coupling of compound states to the continuum which gives compound states finite widths.Expanding the T Q in terms of energy-dependent eigenvectors and (complex) eigenvalues where partial width amplitudes ĝqc are given by where |c (−) is an eigenfunction of H PP in channel c with incoming boundary condition.A disadvantage of the background-plus-resonant separation in Eq. ( 2) is that computation of its energy-averaged cross section will contain a non-vanishing energy-average of the product T P T Q .The KKM theory solves this problem by deriving an alternative separation of the T -matrix into its optical (i.e., energy-averaged) and fluctuating parts: where T opt is the T -matrix of the optical Hamiltonian H opt and 1 To simplify the notation, energy dependence of the T -matrix will not be displayed in this paper. 2In a conventional notation subscripts refer to subspace projection operators: H PQ ≡ PHQ, W QQ ≡ QWQ, etc.

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where I is the energy-averaging interval that is on the order of 1 MeV for optical potentials.The KKM expression for T fluct is: where W opt QQ is obtained by substituting H PP → H opt PP and H PQ → V PQ in Eq. ( 4), and where Expanding the T fluct in terms of energy-dependent eigenvectors and (complex) eigenvalues, yields where partial width amplitudes g qc are given by and where |c (−) opt is an optical wave-function in channel c with incoming boundary condition.The advantage of this expression is that the energy-average of the fluctuating term is approximately zero because by construction the optical T opt is the energy-average of the T -matrix: Computation of energy-averaged cross section, being proportional to |T | 2 , is therefore simplified because the energy-average of the cross term is negligible.Using this feature, the KKM derived an expression for the energy-averaged cross section: where the Satchler's penetrability matrix P is computed for H opt .The same feature was also used in derivation of the Kerman-McVoy energy-averaged cross section for two-step (direct-plus-compound) reactions [3], and also in derivation of the Feshbach-Kerman-McVoy statistical theory of multistep pre-equilibrium reactions [4].

Intermediate Structure
We set out to express the T -matrix of intermediate structure as a sum of its average and fluctuating parts, analogous to Eq. ( 8) of the gross structure.We leverage the separation of the T -matrix into its background-plus-resonant parts, which was derived using Feshbach projection operators formalism in [2].The Hilbert space is projected into continuum (P), doorway (D), and compound (F) subspaces: where D+F = Q of Sect. 2. Projecting the Schrodinger equation HΨ = EΨ into these three subspaces, and allowing compound subspace to couple to doorways only (PHD ≡ H PD 0, H DF 0, and H PF = 0), it was shown in [2] that a T -matrix can be formally written as CNR*11 which is the intermediate structure analogue of the background-plus-resonant T -matrix in Eq. ( 2).Of the three terms, the most rapid energy dependence occurs in T F because it contains the effect of the compound resonant states.For completeness we copy the expressions for T D and T F from Eq. (2.40) of [2]: where The identities used in the KKM to cast the T -matrix into a sum of its average and fluctuating parts in Eq. ( 8) cannot be used when doorway space D is treated explicitly.A desired separation of the T -matrix into its average and fluctuating parts is accomplished below by expressing the T -matrix completely in terms of scalar matrix elements.Despite this departure from the KKM, the approximations used in the derivation below are consistent with those of the KKM, so that the results below ought to be consistent with those of the KKM.To proceed, we expand the T F in terms of eigenvectors3 and (complex) eigenvalues of the operator expressions appearing in T F : The expansion yields where the composite partial widths G cλ , are defined in terms of partial widths where d| is a bi-orthogonal counterpart of |d .Next, we energy-average the T F,cc in Eq. ( 24) over an intermediate energy-averaging interval, I int , that is smaller than a doorway state width Γ d (in order to preserve intermediate structure) but is much larger than the width of fine compound resonances Γ λ (in order to smooth out the fine structure), or Since T P and T D in Eq. (18) are assumed not to vary appreciably over the energy-averaging interval I int , the effect of this energy-averaging on these terms is neglected.Similarly, the effect of energy-averaging the composite partial widths G cλ over the same energy-averaging interval I int is assumed to be small.Therefore, only the narrow compound resonance poles, E λ , of Eq. ( 24) need to be energy-averaged.
Energy-averaging with a Lorentzian weight of width I int can be performed by contour integration 4 that, for a function T (E) that is regular in the upper half-plane, yields Application of this analytical result to Eq. ( 24), along with the approximations just described, yields An expression for the fluctuating part of the T -matrix can now be determined from its definition: Inserting Eqs. ( 24) and (30) into Eq.( 32), and displaying the channel indices, yields where Now that we derived an expression for the fluctuating part, T fluct F , we turn our attention to the intermediate energy-averaged T -matrix.For the purposes of this work, it suffices to state a formal expression for the intermediate structure Hamiltonian, given in Eq (2.89) of [2], which for energy-averaging with a Lorentzian weight of width I int , becomes Since this H int is constructed so that its T -matrix is approximately equal to the intermediate energyaverage of the total T -matrix, namely, we arrive at a desired separation of the T -matrix: This separation of the T -matrix into its intermediate energy-average plus a fluctuating part is analogous to the KKM separation into optical plus its fluctuating part.It is this separation that makes it possible to retrace the steps in the derivation of the KKM cross section.Doing so yields an expression for the intermediate-structure cross section arising from intermediate energy-averaging of |T fluct F | 2 in Eq. (33): This expression is identical in form to the KKM expression in Eq. ( 16), the only difference being that Satchler's penetrability matrix P is computed for H int instead of H opt . 4See for example Eq. (2.19) of [2] for more details.

KKM Revisited
The derivation of the fluctuating T -matrix for the intermediate structure in Sect. 3 could be used to derive an alternative to Eq. ( 13).Writing the fluctuating T -matrix as in Eq. (31) but using the energyaveraging interval of the gross structure, I, yields Energy-averaging of T Q cc in Eq. ( 6) using a Lorentzian weight of width I, and performing the subtraction above, yields where This expression is simpler than the one in Eq. ( 13) because it bypasses computation of eigenvectors |q and eigenvalues E q in Eq. ( 12).The expression for T fluct Q above is very similar to that in Eq. ( 23) of [1] where it was used to argue that its energy average ought to vanish.This suggests that the expression for T fluct Q obtained in a simplified derivation could be used to derive the KKM expression for cross section in Eq. ( 16).

Model and Results
A computer model for studying the validity of approximations used in the KKM derivation was constructed in [5] in order to verify numerically that T fluct cc I /T opt cc 1. Eigenvalues and eigenvectors in Eqs. ( 5) and (12) were assumed not to vary appreciably over the energy-averaging range in computation of T fluct cc I .We remove this assumption in order to quantify the effect of the energy dependence of eigenvectors and eigenvalues over the energy-averaging interval.
Input parameters for this computation were otherwise mostly identical to those in [5].T fluct cc was energy-averaged with a Lorentzian weight of half-width 0.5 MeV at an incoming energy E = 20 MeV.The Lorentzian energy average was performed over 100 equidistant points between 18 and 22 MeV.The eigenvalue Eqs. ( 5) and ( 12) were solved at each of 100 energy points spanning the energyaveraging region.
Forty s-wave channels and 1,600 compound levels were used in this computation.The random interaction was defined on N R = 20 equidistant radial points between the origin and 7 fm.The variance of the coupling strength was set to 0.5 MeV fm 3/2 .For simplicity, we set H PP to be kinetic energy, so that T P = 0, and therefore T  1.A closeness of the two results suggests that energy dependence of eigenvectors and eigenvalues may be neglected without a significant loss in accuracy.Table 1.The average value and the square-root-variance for the two histograms plotted in Fig. 1.We observe a slightly larger average of the histogram for E-dependent eigenvectors and eigenvalues.
Average Ratio SQRT(Variance) E-independent 0.0037 0.0053 E-dependent 0.0042 0.0049 opt cc could be computed as T opt cc ≈ T Q,cc I .A histogram of T fluct cc I /T opt cc for 40 × 40 channel pairs, with and without using energy dependent eigenvectors and eigenvalues, is shown in Figure 1.The average values and square-root variances of the two histograms are displayed in Table

Fig. 1 .
Fig. 1.A histogram of ratios T fluct cc (E) I /T opt cc (E), computed with and without accounting for energy dependence of eigenvalues and eigenvectors.Both histograms corroborate a central result of the KKM theory T fluct cc (E) I /T opt cc (E) 1.A variance between the histograms is an estimate of the error caused by neglecting the energy-dependence.