2 Elimination of the Compound Nucleus States

Because the model space in direct reaction theory is not the whole physical range, we need to define a division of the full Hilbert space by means of projection operators. Following Feshbach[7] we define P as the projection operator onto the model space, including the entrance channels, and Q as projecting on to the remaining space. Such operators must obey P²=P, Q²=Q, PQ=QP=0 and P+Q=I, where I is the identity operator. With these operators we divide the physical wave function $\Psi$ of the system as $\Psi = \Psi_P + \Psi_Q$ where $\Psi_P = P\Psi$ and $\Psi_Q = Q \Psi$. The $\Psi_P$ component, includes the elastic channel and just those channels `directly' related to it that we choose to include in our direct reaction model. The $\Psi_P$ contains the same reaction channels as the model wave function $\Psi_{\rm model}$, but the wave functions are not identical since the model Hamiltonian is obtained by some energy-averaging procedure to be discussed below. The physical Hamiltonian H governs the full wave function $\Psi$ at energy E by the Schrödinger equation $(H - E)\Psi=0$. This equation is now separated into two coupled equations for $\Psi_P$ and $\Psi_Q$:

$$(E - H_{PP}) \Psi_P = H_{PQ} \Psi_Q$$ (2)

$$(E - H_{QQ}) \Psi_Q = H_{QP} \Psi_P$$ (3)

where $H_{PP}\equiv PHP$, $H_{PQ}\equiv PHQ$ and so on. The Eq. (2) has an incoming wave boundary condition in the elastic channel, and there are outgoing waves in all other channels of this and Eq. (3) too. We may therefore formally solve Eq. (3) as

$$\Psi_Q = (E + i\varepsilon - H_{QQ}) ^{-1} H_{QP} \Psi_P$$ (4)

and substitute this into Eq. (2) to obtain a formally exact uncoupled equation for $\Psi_P$:

$$\left (E -H_{PP} - H_{PQ} \left ( E + i\varepsilon - H_{QQ} \right )^{-1} H_{QP} \right ) \Psi_P = 0 \ .$$

2.1 Optical Operator

The Feshbach procedure therefore gives an effective Hamiltonian $H_{\rm eff}$ for the direct-reaction model space $P\Psi$:

$$H_{\rm eff}(E) = H_{PP} + H_{PQ} \left ( E + i\varepsilon - H_{QQ} \right )^{-1} H_{QP}\ .$$ (5)

This is an exact expression, and describes precisely the effect on the model space all variations and resonances (for example) in the eliminated space. The effective Hamiltonian however, is non-local and energy-dependent even when the potential interactions in H are local and energy-independent. The contributions of distinct compound-nucleus state to the effective Hamiltonian may be seen by expanding over a complete set of such states:

$$H_{QQ} \vert Q_\lambda (E') \rangle = E' \vert Q_\lambda (E') \rangle$$

where $\lambda$ distinguishes among degenerate states. Then the second term on the r.h.s of Eq. (5) becomes

$$H_{PQ} \left ( E + i\varepsilon - H_{QQ} \right )^{-1} H_{QP}$$

$$\sum_\lambda \int dE' {{H_{PQ} \vert Q_\lambda (E') \rangle\langle Q_\lambda (E')\vert H_{QP} } \over {(E - E') + i\varepsilon }} \ .$$ = (6)

This term, from coupling to the Q channels, has Hermitian and anti-Hermitian parts,

$$H_{PQ}\left (E+i\varepsilon-H_{QQ} \right )^{-1} H_{QP}={\cal H}_R-i{\cal H}_I$$ (7)

where

$${\cal H}_R \sum_\lambda\! \int \!\! dE' {{H_{PQ} \vert Q_\lambda (E') \rangle (E-E') \langle Q_\lambda (E')\vert H_{QP} } \over {(E - E')^2 + \varepsilon^2 }}$$ = (8)

$${\cal H}_I \sum_\lambda H_{PQ} \vert Q_\lambda (E) \rangle \rho(E) \langle Q_\lambda (E)\vert H_{QP}$$ =

$$\times \int {{\varepsilon dE'}\over {(E - E')^2 + \varepsilon^2 }}$$ (9)

$$\pi \sum_\lambda H_{PQ} \vert Q_\lambda (E) \rangle \rho(E) \langle Q_\lambda (E)\vert H_{QP}$$ = (10)

with $\rho(E)$ the density of states of H_QQ at energy E. The anti-Hermitian part ${\cal H}_I$ is positive definite, and arises because the compound-nucleus $\Psi_Q$ channels act, asymmetrically, only to remove flux from the the model-space channels that are in $\Psi_P$.

2.2 Energy Averaging

In direct reaction calculations, the precise compound nuclear resonances are not needed in all their fluctuations, but only the average effect of these and similar channels. This is most easily accomplished by averaging $H_{\rm eff}(E)$ over small energy intervals, giving $\overline H_{\rm eff}$ as

$$\overline H_{\rm eff}(E) = \int dE' f(E-E') H_{\rm eff}(E')$$ (11)

where f(E-E') is some distribution function of unit integral and width of the order $\Delta E$. If $\Delta E$ is significantly larger than the average spacing of the compound nucleus levels ( $\rho(E)\Delta E \gg 1$), then the resulting $\overline H_{\rm eff}(E)$ has Hermitian and anti-Hermitian components that vary rather slowly with energy E.

2.3 Optical Model

In order to formulate an Optical Model, we further assume that the energy-averaged effective Hamiltonian $\overline H_{\rm eff}$ can be approximated by a local potential that depends only on the coordinate degrees of freedom that are explicitly treated in the model wave function. That is, we approximate $\overline H_{\rm eff}(E) \approx H_{\rm opt}$, which depends only on the collective and/or single-particle degrees of freedom that distinguish the particular N nuclei eigenstates $\phi_{pi}$ and $\phi_{ti}$. If the model space contains only the elastic channel (N=1), we thereby reduce the effective Hamiltonian to contain a local optical potential $U_i({\bf R}_i)$ that depends only on the radial separation of the pair of interacting nuclei. This gives a single-channel Hamiltonian operator

$$H_{\rm opt}({\bf R}_i) = H_i \equiv T_i + h_{pi} + h_{ti} + U_i^{(1)}({\bf R}_i)$$ (12)

for the pair i of the interacting nuclei. The ⁽¹⁾ superscript indicates the size of the model space. If there are more than 1 channel ($N\ge 2$), then the same optical model Hamiltonian may be written with different partitioning of the kinetic and internal energies that are appropriate for the different mass partitions. Thus, there will be a way of writing the optical channel Hamiltonian for each channel:

$$\overline H_{\rm opt}(E) = H_i \equiv T_i({\bf R}_i) + h_{pi} + h_{ti} + V_i$$ (13)

each with some effective potential V_i. This last term can always be separated into diagonal and off-diagonal parts as $V_i = U_i^{(N)}({\bf R}_i) + {\cal V}_i$, where ${\cal V}_i$ is the term that couples together the different channels. The separation is often made unique by requiring that ${\cal V}_i$ has zero diagonal matrix element. The optical potentials (their sum labelled V_i in general) give rise to the elastic scattering cross section, and the Optical Model procedure uses this causality in reverse, to determine them as those local potentials which fit elastic scattering. We typically look for optical potentials that vary only smoothly and slowly with energy, as appropriate to averaging over some energy scale $\Delta E$, and is most often found just for the one channel case (N=1). Note that in the coupled channels case (N>1) the diagonal potentials $U_i^{(N)}({\bf R}_i)$ do not by themselves reproduce the elastic scattering without the work of the off-diagonal couplings ${\cal V}_i$. We therefore call the diagonal U_i^(N) the bare potentials, because, even though they are optical potentials which include the effects of $Q\Psi$ channels not in the model space, they do not include the dressing effects of the inter-channel couplings within the model space $P\Psi$. Only the potential in the one-channel model space U_i⁽¹⁾ is supposed to reproduce the elastic scattering by itself. Because $H_{\rm eff}(E)$ has Hermitian and anti-Hermitian parts, the optical potentials will have real and imaginary terms, and because ${\cal H}_I$ is positive definite the imaginary parts will be negative and absorptive.