Under reaction–theoretical aspects, the DSCE mechanism amounts to a two-step reaction

As depicted in Figure 1, DSCE reactions are a sequence of two single charge exchange events, each of them mediated by the two–body NN–isovector interaction ${\mathcal{T}}_{NN}$, the latter acting by one–body operators on the projectile and the target nucleus, respectively. For a reaction $\alpha =a(Z_a,N_a)+A(Z_A,N_A)\to \beta =b(Z_a\pm 2,N_a\mp 2)+B(Z_A\mp 2,N_A\pm 2)$ the reaction amplitude is written down readily as a quantum mechanical second order reaction matrix element [6]: Initial (ISI) and final state (FSI) interactions are taken into account by optical potentials and the distorted waves ${\chi }_{\alpha ,\beta }^{(\pm )}$, depending on the center–of–mass (c.m.) momenta ${\mathbf{k}}_{\alpha ,\beta }$ and obeying outgoing and incoming spherical wave boundary conditions, respectively. The available c.m. energy is ${\omega }_{\alpha }=\sqrt{s_{\alpha }}$, Denoting projectile and target nucleus by a and A, respectively, the available energy available in the center–of–mass rest frame $s_{\alpha }={(T_{lab}+M_a+M_A)}^2-T_{lab}(T_{lab}+2M_a)$. The perturbative approach of Eq. (1) is justified by the weak coupling of the DSCE channels to the elastic scattering channel. However, coupling to non–elastic channels cannot be excluded a priori and should be checked for each reaction [2].

The many–body Green’s function of the combined projectile–target system is denoted by ${\mathcal{G}}_{aA}^{(+)}$. In operator form it is given by the resolvent of the projectile plus target many–body Schrödinger equation. Since DCE reactions are peripheral, grazing collisions, the $a+A$ system remains in each reaction step separable into projectile and target nucleus, allowing to express the total Hamiltonian by the sum of the Hamiltonians $H_a$ and $H_A$ of the isolated nuclei plus the Hamiltonian $H_{aA}=H_{rel}+V_{aA}$. $H_{rel}=T_{cm}+U_{aA}$ accounts for the kinetic energy of relative motion ($T_{cm}$) and elastic ion–ion interactions ($U_{aA}$) which for the assumed reaction scenario are well approximated by an optical model Hamiltonian $H_{opt}$. The residual projectile–target interactions are approximated the nucleon–nucleon (NN) T–matrix, $V_{aA}\approx {\mathcal{T}}_{NN}$, thus including the full Lippmann–Schwinger scattering series of NN–interactions. In a fully microscopic approach as in [6], $U_{aA}$ is calculated by in a folding approach by using ${\mathcal{T}}_{NN}$ and nuclear ground state, typically derived self–consistently by functional methods like Hartree–Fock–Bogolyubov (HFB) theory [4].

As shown in Figure 2 the proper treatment of time ordering is taken care of by using the retarded propagator: where the first and second term correspond to the ladder–graph and the so–called Z–graph, respectively. $\eta \to 0+$ denotes an infinitesimal, ensuring outgoing spherical wave boundary conditions by approaching zero from positive values.

The many–body wave function ${\Psi }_{aA}^{(+)}$ with outgoing spherical scattering waves is expanded into the basis of eigenstates of the nuclear Hamiltonians, $(H_a-E_c)|c\rangle =0$ and $(H_A-E_C)|C\rangle =0$, respectively. Hence, we obtain the representation: leading to the channel propagators which describe relative motion in a given partition $|\gamma \rangle =|cC\rangle$. The energy $M_{\gamma }=M_c+M_C+{\epsilon }_{\gamma }$ is given by the nuclear masses $M_{c,C}$ and the sum of excitation energies ${\epsilon }_{\gamma }={\epsilon }_c+{\epsilon }_C$.

$G_{\alpha \gamma }^{(+)}$ is expanded into the bi–orthogonal set of optical model distorted waves (DW) being solution of the wave equation defined by $H_{opt}$ [4,5,6,8]. This leads to the reduced propagators with the kinetic energy $T_{\gamma }\left(k_{\gamma }\right)=\sqrt{k_{\gamma }^2+E_c^2}+\sqrt{k_{\gamma }^2+E_C^2}-(E_c+E_C)\sim \frac{k_{\gamma }^2}{2m_{\gamma }}$ at the off–shell momentum $k_{\gamma }$. $1/m_{\gamma }=1/E_c+1/E_C$ is the reduced mass. This allows to rewrite the DSCE reaction amplitude as The non–Hermitian nature of $H_{opt}$ has led to two distinct half off–shell first order DWBA amplitudes [1,5,6]: where ${\tilde{\chi }}_{\gamma }^{(+)}$ is the dual distorted wave obeying $\langle {\tilde{\chi }}_{\gamma }^{(+)}|{\chi }_{{\gamma }^{\prime }}^{(+)}\rangle ={\left(2\pi \right)}^3\delta ({\mathbf{k}}_{\gamma }-{\mathbf{k}}_{\gamma }^{\prime }){\delta }_{\gamma {\gamma }^{\prime }}$.

The structure of the DSCE transition form factor becomes especially transparent when considered in momentum space. In momentum representation, the isospin–reduced, anti–symmetrized, energy dependent isovector T–matrix, acting between a nucleon in $\left\{a\right\}$ and a nucleon in $\left\{A\right\}$, is given by the Fourier–Bessel transform [4,8] In the ion–ion rest frame the relative distance between a nucleon $i\in \left\{a\right\}$ and $k\in \left\{A\right\}$${\mathbf{x}}_{ik}={\mathbf{r}}_k-{\mathbf{r}}_i+{\mathbf{r}}_{\lambda }$ is given by the intrinsic nuclear coordinates ${\mathbf{r}}_{i,k}$ and the ion–ion relative coordinate ${\mathbf{r}}_{\lambda }$, $\lambda =\alpha ,\beta$, the latter describing the distance between the center–of-mass of the projectile–like and the target–like nucleus.

SCE and DCE reaction are rely on isovector interactions. Hence, we choose as residual interaction the isovector ($T=1$) part of the NN T–matrix, consisting of central rank—0 spin–scalar, spin–vector and non–central rank-2 spin–tensor interactions: where only the isospin–ladder parts ${\tau }_{i,k}^{\pm }$ are relevant for the SCE process. The (complex) form factors $V_{ST}$ and $V_{Tn}$, playing the role of variable coupling constants, are functions of the momentum transfer $\mathbf{p}$ and of the energies of the interacting nucleons. $Y_{2M}\left(\widehat{\mathbf{p}}\right)$ is a rank–2 spherical harmonics, above being contracted with a rank–2 spin–tensor. The spin-scalar parts excite Fermi–type (F) transitions, the spin–vector and spin–tensor components induce Gamow–Teller–type modes, mutually in projectile and target.

In momentum representation, the SCE transitions are described by matrix elements of the scattering operators The plane wave factor accounts for the spatial multipole structure, resulting in the Bessel–Fourier transforms of the one–body transition densities, constructed from the $n{p}^{-1}$ r $p{n}^{-1}$ particle–hole configuration amplitudes and the corresponding single particle wave functions [4]. Odd indices $i=1,3\dots$ and even indices $k=2,4\dots$ will be used to denote states in the a– and the A–system, respectively.

The DSCE amplitude of Eq. (1) is the standard form for a DW matrix element in second order perturbation theory in ${\mathcal{T}}_{NN}$. In momentum representation, the DSCE reaction amplitude is retrieved, however, in a rather different form Remarkably, the spectroscopic and reaction contribution appear in separated form. Nuclear spectroscopy is contained in the DSCE nuclear matrix element given by the form factor where the reduced channel propagator $g_{\alpha \gamma }^{(+)}\left(k_{\gamma }\right)$ depends on the energies of the intermediate states. The NME has the structure of second order perturbation theory. The formal similarity to the NME $2\nu$–DBD is easily recognized.

Initial state (ISI) and final state (FSI) interactions are contained now completely in the reaction kernel which is determined by the distortion coefficients [4]

In the energy denominators of $g_{\alpha \gamma }^{(+)}$ contained in ${\mathcal{N}}_{\alpha \beta }$ we add and subtract a state–independent auxiliary energy ${\omega }_{\gamma }$ and perform a Taylor–series expansion in $\xi =1-\frac{M_{\gamma }}{{\omega }_{\gamma }}$. Since in leading order the propagator becomes independent of the intermediate states, the summations over the channel states $c,C$ can be performed and we obtain the DSCE NME in closure approximation Initial and final nuclear states are connected directly by the effective DSCE rank–2 isotensor interaction The DSCE transition operator ${\mathcal{R}}_{NN}$ is composed of a combination of two–body–operators, the one acting in projectile and the other in the target nucleus. Hence, in the combined projectile–target system ${\mathcal{R}}_{NN}$ is an effective four–body interaction.

Attaching the modified, state–independent propagator to the reaction kernel and including also the $k_{\gamma }$ integration we obtain The closure approach renders Eq. (12) into an intriguing simple form: where terms of first and higher order in $\xi$ are contained in $Res\left(\xi \right)$.

An instructive – albeit unrealistic – limiting case is to neglect all elastic interactions leading to the plane wave approximation (PWA). In PWA the distortion coefficients, Eq. (15), achieve a particular simple form, ${\tilde{D}}_{\alpha \gamma }=\delta ({\mathbf{q}}_{\alpha \gamma }+{\mathbf{p}}_1)$ and $D_{\gamma \beta }=\delta ({\mathbf{q}}_{\gamma \beta }+{\mathbf{p}}_2)$, respectively, where ${\mathbf{q}}_{\alpha \gamma }={\mathbf{k}}_{\alpha }-{\mathbf{k}}_{\gamma }$ and ${\mathbf{q}}_{\gamma \beta }={\mathbf{k}}_{\gamma }-{\mathbf{k}}_{\beta }$ are the half off–shell momentum transfers in the first and second SCE interaction. The total on–shell momentum transfer is ${\mathbf{q}}_{\alpha \beta }={\mathbf{q}}_{\alpha \gamma }+{\mathbf{q}}_{\gamma \beta }$.

Changing the momentum coordinates to ${\mathbf{p}}_{1,2}\mapsto \mathbf{P}\pm \mathbf{q}/2$, and introducing the mean on–shell channel momentum ${\mathbf{P}}_{\alpha \beta }=({\mathbf{k}}_{\alpha }+{\mathbf{k}}_{\beta })/2$, the product of PWA distortion factors and the $k_{\gamma }$–integral are easily evaluated, resulting in the PWA reaction kernel Both the delta–distribution and the functional structure of the propagator impose constraints on the momenta $\mathbf{P}$ and $\mathbf{q}$.1 The above result indicates also that $\mathbf{P}$, i.e. the sum of the momenta ${\mathbf{p}}_1$ and ${\mathbf{p}}_2$, is closely related to the physical momentum transfer ${\mathbf{q}}_{\alpha \beta }$.

In order to gain insight in the effects introduced by adding ISI/FSI interactions we use a simple but realistic approach: The product of distorted waves is factorized into a plane wave part and a reduced, in general complex–valued, amplitude: For strongly absorbing systems like colliding heavy ions the reduced DW–amplitudes are strongly suppressed in the overlap region, as is easily verified in Wentzel–Kramers–Brillouin (WKB) or eikonal theory. That region, centered at the origin, corresponds to an avoided volume in the sense that the spatial density distributions of the incoming and outgoing distorted waves vanishes. This kind of localized absorption of probability implies that the (quantum mechanical) survival probability of the initial system in that region is approaching zero. As a consequence, reactions will be restricted to regions outside of the ion–ion overlap volume, characterized by a strong probability flux into other reaction channels.

The shape and extension of that void is imprinted in the reduced amplitude. Describing the shape and size of the probability voids by the absorption form factors $|h_{ij}\left(\mathbf{r}\right)|$ we obtain ${\eta }_{ij}\left(\mathbf{r}\right)=1-h_{ij}\left(\mathbf{r}\right)$. For heavy ion reactions $|h_{ij}\left(\mathbf{r}\right)|$ is unity up to the absorption radius $r=R_{abs}$ and decreases rapidly at larger radii [4],in many aspects resembling a Heaviside distribution $h_{ij}\left(\mathbf{r}\right)\sim \Theta (R_{abs}^2-r^2)$. In Ref. [4] it was shown that $R_{abs}$ is directly related to the total ion–ion reaction cross section, varying with energy and increasing with the mass numbers of the interacting nuclei.

Since for a given reaction $h_{ij}$ is fixed essentially by the entrance channel, we may use universal form factors, $h_{ij}\left(\mathbf{r}\right)\equiv H_S\left(\mathbf{r}\right)$ and ${\tilde{\eta }}_{ij}\equiv {\tilde{\ddot{H}}}_S\left(\mathbf{r}\right)$. Thus, we obtain Hence, with ISI/FSI interactions the distortion factors are given by subtracting the interactions occurring in the avoided volume from the maximal possible interaction probability density described by the PWA delta-distribution.

In DWA the DCE reaction kernel is found to be given by the PWA–kernel with a unit interaction probability density, which is corrected for the overcounting of reaction probability by subtracting the interaction probability densities related to the avoided volume: The two–step character of the DSCE reaction amplitude is reflected by the first and second order kernels $K_{\alpha \beta }^{\left(1\right)}$ and $K_{\alpha \beta }^{\left(2\right)}$, respectively. With the $\{\mathbf{P},\mathbf{q}\}$ coordinates, the first order absorption kernel is found as Performing the $k_{\gamma }$–integration the result is Considering that $H_S\left(\mathbf{Q}\right)$ has a pronounced maximum at $Q=0$ we may replace Using $H_S=e^{2i\Phi }{F}_S$ and ${\tilde{H}}_S{e}^{2i\tilde{\Phi }}{F}_S$, we obtain with $\varphi =\Phi +\tilde{\Phi }$ where the phases may depend on momentum.

The kernel of second order in the absorption form factors is The product of form factors is evaluated easily by going back to the definition of the form factors $F_S$ as Fourier–Bessel transforms and interchanging the order of momentum and radial integrations. The detailed description in Appendix B leads to the conclusion that to a good approximation the kernel separates into a product of P– and q–dependent form factors where Collecting results, we obtain the total reaction kernel A phase factor was extracted from the last term, ${\overline{F}}_S^{\left(2\right)}\equiv e^{-2i\varphi }{F}_{\alpha \beta }^{\left(2\right)}$. The factorized $\mathbf{q}$ and $\mathbf{P}$ dependence is still maintained.

Finally, the P and q integrations are performed and as the main result of this section, we obtain the DSCE reaction amplitude where we have introduced the distortion form factor accounting for the interaction effects induced by the kernels ${\mathcal{K}}_{\alpha \beta }^{(1,2)}$ and being used in the form

Further insight is gained by expressing the propagator by the Fourier transform of the coordinate space propagator where $k_{+}=\sqrt{{\omega }_{\alpha }^2-{\omega }_{\gamma }^2}$ and $k_{-}=-i\sqrt{{\omega }_{\alpha }^2+{\omega }_{\gamma }^2}$ denote the poles of $g_{\alpha \beta }^{(+)}\left(k\right)$ in the upper and lower complex half–planes, respectively. In Figure 3 the propagator is displayed as encountered in the reaction${ }^{40}$Ca${(}^{18}$O${,}^{18}$Ne${)}^{40}$Ar at $T_{lab}=270$ MeV.

The DSCE–NME defined by deserves a more detailed consideration because care has to be taken to avoiding overcounting. This means to exclude that the same nucleon takes part in the first end the second SCE events. Most elegantly, this is achieved by using the coordinates $\{\mathbf{P},\mathbf{q}\}$ and center and relative spatial coordinates $\{{\mathbf{r}}_{\mu },{\mathbf{x}}_{\mu }\}$, $\mu =\left(13\right)$ or $\mu =\left(24\right)$, respectively. Hence, we replace ${\mathbf{p}}_{1,2}=\mathbf{P}\pm \mathbf{q}/2$ and $r_{1,3}={\mathbf{r}}_{13}/2\pm {\mathbf{x}}_{13}$ and accordingly $r_{2,4}={\mathbf{r}}_{24}/2\pm {\mathbf{x}}_{24}$. Furthermore, for the sake of focusing on the essential features and a more transparent formulation, the momentum dependence of T–matrices is approximated by ${\mathbf{p}}_{1,2}\sim {\mathbf{q}}_{\alpha \beta }/2$ which is eligible considering that ${\mathbf{p}}_1+{\mathbf{p}}_2\simeq {\mathbf{q}}_{\alpha \beta }$. Thus, we introduce Then, the exclusion principle is treated properly by using which vanishes for $i=j$ and changes sign under exchanges $1\leftrightarrow 3$ and $2\leftrightarrow 4$, respectively.

At this point it is of advantage to incorporate again the propagator into the nuclear matrix element such that the q–integration can be performed: Thus, the second order dynamics originally introduced as a reaction dynamical effect has been converted to a spectroscopic property. The DSCE reaction amplitude is obtained in the compact and intuitive form The plane wave matrix element describes the unit DCE interaction probability without ISI/FSI, thus representing the maximal DCE transition strength. The second component corresponds to a fictitious DCE process occurring in the avoided ion–ion overlap volume. As a result, the heavy ion DCE amplitude is determined by the subtraction of the reaction dynamical forbidden matrix element from the unconstrained plane wave matrix element of unit interaction probability.

From Eq. (39) it is found that the q–dependence is separated, being completely contained in product of the two sine–functions. The q–integral is easily performed by substituting $\mathbf{k}=\mathbf{q}/2+{\mathbf{P}}_{\alpha \beta }$. We also introduce ${\mathbf{x}}_{\pm }={\mathbf{x}}_{13}\pm {\mathbf{x}}_{24}$. The addition theorems of trigonometric functions yield: where A closer look shows that the k–integral serves to project on the monopole component of the integrand. By symmetry reasons only the cosine–terms will contribute, leading to Contour integration leads to the correlation function From the functional properties of the reduced propagators $g_{\alpha \beta }^{(+)}\left(k_0{x}_{\pm }\right)$, we conclude that the regions ${\mathbf{x}}_{\pm }={\mathbf{x}}_{13}\pm {\mathbf{x}}_{24}\sim 0$ are weighted most strongly. Since ${\mathbf{x}}_{13}$ and ${\mathbf{x}}_{24}$ denote the distances between two SCE vertices in the interacting nuclei, respectively, we find that the relative motion has induced a pronounced correlation corresponding to constraining the vertices to appear at ${\mathbf{x}}_{13}\sim \pm {\mathbf{x}}_{24}$. Thus, we encounter a configuration of two coupled isospin–dipoles, combined to an isospin quadrupole where the two dipoles are separated by $\Delta x=|{\mathbf{x}}_{13}|=|{\mathbf{x}}_{24}|$, with a fluctuation $\delta x\sim \hslash c/k_0$. Remembering that the pole momentum $k_0$ depends on the ion masses, the nuclear excitation energies, and the kinetic energy, the magnitude of the fluctuation will vary such that increasing any of those conditions will decrease the quantal fluctuations and vice versa. The interference of the two kind of dipoles $x_{+}$ and $x_{-}$, respectively, is determined mainly by the monopole component given by a Riccati–Bessel function $j_0\left(x\right)$ The quadrupole and higher order contributions - by symmetry reasons of only even multipolarity – are suppressed because the functions $j_{2n}\left(x\right)$, $n\ge 1$, will gain strength only at larger distances considerably beyond the effective range defined by the propagator.

Another look to the correlated DSCE–NME reveals that also the total momentum transfer variable $\mathbf{K}$ induces a correlation exists. As seen above, $\mathbf{K}$ imposes a constraint on the center coordinates of the nucleon pairs, ${\mathbf{r}}_{13}\sim {\mathbf{r}}_{24}$ within an interval of the order $\delta r\sim 1/K$. In the plane wave part of the NME, where $K=\frac{1}{2} q_{\alpha \beta }$, the correlation length varies with the scattering angle. As seen in Eq. (41), the distortion part of the NME reaction amplitude is given by folding ${\mathcal{B}}_{\alpha \beta }$ with the momentum distribution of the DCE distortion form factor. Using Eq. (48), the P–integration reduces to evaluate a Bessel–Fourier integral, mapping the distortion amplitude from momentum space to r–space:

The closure approximation has led to effective two–body interactions ${\mathcal{R}}_{NN}$ acting in the projectile and target nuclei, and corresponding in total to a nucleus-nucleus four–body interaction. As indicated in Fig. 2 by the ladder diagram, originally the interactions carry an operator structure defined by the t–channel formalism as appropriate for a nuclear reaction. Nuclear matrix elements, however, are defined within a given nucleus which requires a transformation of the operators into a s–cannel formalism. In Ref. [8] that problem was solved on the level of matrix elements. Here, we address that question from the operator side by appropriate reordering of operators and applications of angular momentum recoupling techniques.

We introduce the rank–2 isotensor operator and defining the effective rank–2 isotensor interaction is rewritten as Since the isotensor is already in s–channel form, in the following we need to consider only the reduced DSCE interaction $R_{NN}$.

From the operator structure of the NN T–matrix, we derive immediately that $R_{NN}$ is a superposition of products of spin–scalar, spin–vector, and rank–2 spin–tensor interactions. For the rank–0 central interactions with $T=1$ we derive for example the structure: and corresponding expression are recovered for the tensor interaction and the mixed central–tensor terms. Hence, we find the vertex form factors They are playing the role of momentum dependent effective coupling constants, in addition varying also with the energy in the NN rest frame.

While for spin–scalar interactions the t– to s–channel transformation is trivial, the spin–dependent interactions require a considerably more involved treatment by explicit angular momentum recoupling. Since also the spacial degrees of freedoms have to be treated properly, we define the spin–scalar and spin–vector operators where ${\mathbf{1}}_{\sigma }$ is the spin–unity operator. $\mathbf{k}=\mathbf{p}$ for $i=1,3$ in the a–system and $\mathbf{k}=-{\mathbf{p}}^{\prime }$ for $i=2,4$ in the A–system.

The rank–0 central interactions lead to a superposition of Fermi–Fermi (FF), Gamow-Teller–Gamow-Teller (GG) and the mixed Fermi–Gamow–Teller transitions, FG and GF. Leaving aside from hereon the vertex form factors, the resulting operators are While the FF and FG/GF components are already in an appropriate form, the double Gamow–Teller operators (GG) require a more detailed treatment. The products of spin–vector components are recoupled by arranging the nucleon spin operators into intra–nuclear two–body operators of tensorial rank 0,1,and 2 which by contraction form a total rank–0 scalar operator in the overall projectile–target system: where the $S=0$ components are in fact scalar products, $R_1\left({\mathbf{p}}_1\right|i)·R_1\left({\mathbf{p}}_2\right|j)$.

An even richer spectrum of operators is obtained from the double–tensor (TT) term. Labeling for bookkeeping reasons the nucleon spin–operators by $S_i$, where $S_i=1$, defining $S_{ij}=2$, using $\widehat{J}=\sqrt{2J+1}$, and applying angular momentum recoupling techniques which result in a 9–j symbol, one finds The mixed central–tensor terms include the Fermi–Tensor components FT and TF With appropriate recoupling, the combined Gamow–Teller–Tensor (GT) components become with $L=2$. Accordingly,

The DSCE transition form factor is given by generalizes the concept of nuclear matrix elements to transition form factors of arbitrary momentum transfers. Nuclear matrix elements in the strict sense are recovered in the limiting case $p,p^{\prime }\to 0$.

A key question is what kind of spectroscopic information can be extracted from DSCE cross sections. For an answer, we have to have a closer look into the operator structure, given by multitude of terms. Already from the central spin–vector interactions two additional spin-tensor terms were obtained by recoupling. But most of the components are generated by the additional degrees of freedom provided by the rank–2 spin–tensor interactions. Counting the recoupling terms separately, more than 40 components are recognized. However, they are actually determined by four generic types of operators, namely the spin scalar–scalar terms, mixed scalar–vector and scalar–tensor, and rank–1 and rank–2 tensor terms. Moreover, by the plane wave parts, the scalar $R_0$ and vector ${\mathbf{R}}_1$ operators contain a rich spectrum of multipole operators by expanding the plane wave factors into partial waves as discussed in Appendices Appendix C and Appendix D, respectively. After reordering, each of the terms factorizes into a projectile and a target form factor but in general ${\mathcal{F}}_{\alpha \beta }$ as a whole will not be separable in a a– and a A–type form factor.

As an example, we consider ${\Sigma }_{FG}$ and introduce the partial form factor where the isospin matrix elements were left out for simplicity and the scalar product is evaluated explicitly in the spherical bi–orthogonal basis $\{{\mathbf{e}}_{\mu }^{*},{\mathbf{e}}_{\mu }\}$, $\mu =0,\pm 1$, see Appendix D. The form factors in the a– and the A–system, respectively, are defined as: Both form factors are given by products of spin–scalar and spin–vector operators, the latter expressed in spherical representation, following the rules discussed in the Appendix.