Perey–Buck Ansatz in Nonlocal Nuclear Potentials

• Perey–Buck ansatz is a phenomenological representation of nonlocal nucleon–nucleus interactions that factorizes the optical potential into a midpoint radial strength and a Gaussian displacement function.
• It underlies the Perey effect by dampening the wave function in the nuclear interior and shifting its tail outward compared to local potential treatments.
• The ansatz serves as a benchmark in optical-model phenomenology and has driven the development of advanced numerical methods for direct-reaction and bound-state calculations.

Searching arXiv for papers on Perey-Buck ansatz and related nonlocal optical potentials. The Perey–Buck ansatz is a phenomenological representation of nonlocal nucleon–nucleus interactions in which the optical potential depends on both $\mathbf r$ and $\mathbf r'$ and is factorized into a smooth radial strength evaluated at a midpoint coordinate and a Gaussian function of the displacement $\mathbf r-\mathbf r'$ (Sailaubek et al., 2019, Tian et al., 2018). In its standard coordinate-space form,

$$U(\mathbf r,\mathbf r')=U_N\!\left(\tfrac12|\mathbf r+\mathbf r'|\right)\, \frac{1}{\pi^{3/2}\beta^3}\exp\!\left[-\left(\frac{|\mathbf r-\mathbf r'|}{\beta}\right)^2\right],$$

with $\beta$ the nonlocality range parameter (Stahulak et al., 4 Sep 2025). Historically, this ansatz underlies the “Perey effect,” namely the reduction of the wave function in the nuclear interior when a nonlocal problem is compared with a local one, and it remains a standard reference point for optical-model phenomenology, direct-reaction theory, and modern microscopic reinterpretations of nonlocality (Titus et al., 2014, Arellano et al., 2022).

1. Formal structure of the ansatz

In the conventional Perey–Buck form, the nonlocal kernel is separable into a radial factor depending on the average coordinate and a Gaussian nonlocality factor depending on the relative coordinate (Tian et al., 2018). The Gaussian is normalized as

$$H(\mathbf x)=\pi^{-3/2}\beta^{-3}e^{-(\mathbf x/\beta)^2},$$

or equivalently

$$H_{P\!B}(s)=\frac{1}{\pi^{3/2}\beta^3}e^{-s^2/\beta^2},$$

with momentum-space transform

$\tilde H_{P\!B}(K)=e^{-\beta^2K^2/4}$

(Gómez-Ramos et al., 2019, Arellano et al., 2022).

A standard Woods–Saxon realization writes the local central and spin-orbit form factors as

$$V_L(r)=V_L f_0(r), \qquad V_{\rm so}(r)=2V_{\rm so}\frac{1}{r}\frac{d}{dr}f_{\rm so}(r),$$

with

$$f_i(r)=\left[1+\exp\left(\frac{r-R_i}{a_i}\right)\right]^{-1}, \qquad R_i=r_iA^{1/3},$$

while the nonlocal central term is replaced by a Perey–Buck-type kernel (Tian et al., 2018). A commonly used form is

$\mathbf r'$0

followed by the usual approximation in which the variable $\mathbf r'$1 entering the Woods–Saxon factor is replaced by $\mathbf r'$2 (Sailaubek et al., 2019).

After partial-wave projection, the ansatz generates a radial nonlocal kernel

$\mathbf r'$3

with

$\mathbf r'$4

or, in the notation used in bound-state ANC calculations,

$\mathbf r'$5

(Tian et al., 2018, Sailaubek et al., 2019). This projected kernel is the operative object in radial integro-differential equations and basis-matrix formulations.

A persistent implementation detail is that many practical calculations take only the central part of the potential to be nonlocal, while the spin-orbit and Coulomb terms remain local (Tian et al., 2018). The same pattern appears in bound-state ANC calculations, where the explicit nonlocal kernel contains only the central Woods–Saxon term, with the spin-orbit structure retained separately (Sailaubek et al., 2019). This suggests that, in much of the phenomenological literature, “Perey–Buck nonlocality” denotes a nonlocal central interaction rather than a fully nonlocal optical potential in every operator channel.

2. Local-equivalent representation and the Perey factor

The ansatz is closely associated with the construction of a local-equivalent interaction. In leading order, the local-equivalent potential satisfies a transcendental relation of the form

$\mathbf r'$6

for two-body scattering (Gómez-Ramos et al., 2019). In a related microscopic reformulation, the same basic idea appears as

$\mathbf r'$7

which is used to map an energy-dependent local optical potential onto an energy-independent nonlocal one (Stahulak et al., 4 Sep 2025).

Beyond the leading local-energy approximation, the localized Schrödinger equation acquires derivative terms,

$\mathbf r'$8

and one introduces a factorization

$\mathbf r'$9

to remove the first-derivative term and recover an ordinary local equation for $\mathbf r-\mathbf r'$0 (Gómez-Ramos et al., 2019). In that context, one form of the Perey factor is

$\mathbf r-\mathbf r'$1

with

$\mathbf r-\mathbf r'$2

(Gómez-Ramos et al., 2019).

The more familiar practical correction factor used in reaction calculations is

$\mathbf r-\mathbf r'$3

obtained from the more complete expression

$\mathbf r-\mathbf r'$4

after neglecting the $\mathbf r-\mathbf r'$5 term (Titus et al., 2014). The same logic appears in the transfer-reaction thesis, where the full expression includes the surface-sensitive $\mathbf r-\mathbf r'$6 contribution and the standard Perey correction factor follows from dropping it (Titus, 2016).

A central limitation is therefore intrinsic to the approximation: the neglected $\mathbf r-\mathbf r'$7 term contributes mainly in the surface region (Titus et al., 2014, Titus, 2016). This is why the Perey factor captures the gross interior damping but does not fully reproduce surface and peripheral behavior. It is also why several later studies treat the nonlocal equation directly instead of relying on a local-equivalent reduction (Rawitscher, 2012, Tian et al., 2018).

3. Nonlocal Schrödinger equations and exact numerical treatments

The basic nonlocal radial equation used in Perey–Buck calculations is

$\mathbf r-\mathbf r'$8

for scattering or bound states (Tian et al., 2018). In three dimensions one may write

$\mathbf r-\mathbf r'$9

or the equivalent radial Lippmann–Schwinger form (Rawitscher, 2012).

Several exact or near-exact numerical strategies have been developed for such kernels. One approach solves the integro-differential equation iteratively by recasting it as a sequence of inhomogeneous differential equations (Tian et al., 2018). Another uses a Chebyshev spectral expansion combined with singular value decomposition of the discretized nonlocal kernel; for the Perey–Buck application, this method reached an accuracy “between $$U(\mathbf r,\mathbf r')=U_N\!\left(\tfrac12|\mathbf r+\mathbf r'|\right)\, \frac{1}{\pi^{3/2}\beta^3}\exp\!\left[-\left(\frac{|\mathbf r-\mathbf r'|}{\beta}\right)^2\right],$$0 to $$U(\mathbf r,\mathbf r')=U_N\!\left(\tfrac12|\mathbf r+\mathbf r'|\right)\, \frac{1}{\pi^{3/2}\beta^3}\exp\!\left[-\left(\frac{|\mathbf r-\mathbf r'|}{\beta}\right)^2\right],$$1,” depending on the number of polynomials employed (Rawitscher, 2012). The same study also introduced a Sturmian-function expansion supplemented by iterative correction, reporting that for $$U(\mathbf r,\mathbf r')=U_N\!\left(\tfrac12|\mathbf r+\mathbf r'|\right)\, \frac{1}{\pi^{3/2}\beta^3}\exp\!\left[-\left(\frac{|\mathbf r-\mathbf r'|}{\beta}\right)^2\right],$$2 an accuracy of $$U(\mathbf r,\mathbf r')=U_N\!\left(\tfrac12|\mathbf r+\mathbf r'|\right)\, \frac{1}{\pi^{3/2}\beta^3}\exp\!\left[-\left(\frac{|\mathbf r-\mathbf r'|}{\beta}\right)^2\right],$$3 is obtained without iterations and, after one iteration, $$U(\mathbf r,\mathbf r')=U_N\!\left(\tfrac12|\mathbf r+\mathbf r'|\right)\, \frac{1}{\pi^{3/2}\beta^3}\exp\!\left[-\left(\frac{|\mathbf r-\mathbf r'|}{\beta}\right)^2\right],$$4 (Rawitscher, 2012).

Bound-state treatments can also be cast as matrix diagonalization in a complex-ranged Gaussian basis. There the reduced radial wave function is expanded as

$$U(\mathbf r,\mathbf r')=U_N\!\left(\tfrac12|\mathbf r+\mathbf r'|\right)\, \frac{1}{\pi^{3/2}\beta^3}\exp\!\left[-\left(\frac{|\mathbf r-\mathbf r'|}{\beta}\right)^2\right],$$5

with basis functions

$$U(\mathbf r,\mathbf r')=U_N\!\left(\tfrac12|\mathbf r+\mathbf r'|\right)\, \frac{1}{\pi^{3/2}\beta^3}\exp\!\left[-\left(\frac{|\mathbf r-\mathbf r'|}{\beta}\right)^2\right],$$6

obtained from complex Gaussians (Sailaubek et al., 2019). The coefficients follow from the generalized eigenvalue problem

$$U(\mathbf r,\mathbf r')=U_N\!\left(\tfrac12|\mathbf r+\mathbf r'|\right)\, \frac{1}{\pi^{3/2}\beta^3}\exp\!\left[-\left(\frac{|\mathbf r-\mathbf r'|}{\beta}\right)^2\right],$$7

and the nonlocal matrix elements are

$$U(\mathbf r,\mathbf r')=U_N\!\left(\tfrac12|\mathbf r+\mathbf r'|\right)\, \frac{1}{\pi^{3/2}\beta^3}\exp\!\left[-\left(\frac{|\mathbf r-\mathbf r'|}{\beta}\right)^2\right],$$8

(Sailaubek et al., 2019). In the $$U(\mathbf r,\mathbf r')=U_N\!\left(\tfrac12|\mathbf r+\mathbf r'|\right)\, \frac{1}{\pi^{3/2}\beta^3}\exp\!\left[-\left(\frac{|\mathbf r-\mathbf r'|}{\beta}\right)^2\right],$$9 example, $\beta$0 was sufficient for convergence of the local-case CRGB wave function and Whittaker ratio, and the same framework was then applied to the Perey–Buck nonlocal kernel (Sailaubek et al., 2019).

These developments altered the methodological status of the ansatz. In earlier work, the Perey–Buck form was tightly linked to local-equivalent approximations; in later work, it became equally a benchmark kernel for direct numerical solution (Rawitscher, 2012, Titus et al., 2014). A plausible implication is that the historical importance of the ansatz now lies as much in providing a controlled nonlocal test problem as in providing a closed-form correction factor.

4. The Perey effect

The standard physical content associated with the ansatz is the “Perey effect.” In exact nonlocal calculations, the wave function is reduced in the nuclear interior relative to the local-equivalent case and is shifted outward in radius (Sailaubek et al., 2019, Rawitscher, 2012). For bound states, one formulation is explicit: “Being compared to the local equivalent case, the nonlocality reduces the amplitude of the wave function and slightly shifts its tail to farer distances in full agreement with the so-called Perey effect” (Sailaubek et al., 2019).

Because the bound-state wave function is normalized to unity, interior suppression is accompanied by an enhanced exterior tail. This is why exact nonlocal calculations with Perey–Buck kernels systematically produce larger asymptotic normalization coefficients than corresponding local potentials (Sailaubek et al., 2019). The same interior suppression underlies the practical correction factor

$\beta$1

with $\beta$2 asymptotically and $\beta$3 in the interaction region (Titus et al., 2014, Gómez-Ramos et al., 2019).

Comparisons of exact nonlocal and Perey-corrected local solutions show that the correction factor is not uniformly accurate. For bound states, the Perey-corrected wave function agrees well with the exact nonlocal one in the interior, but discrepancies appear in the surface and peripheral regions (Titus et al., 2014, Titus, 2016). For scattering states, the correction is overall adequate except for a few partial waves associated with grazing impact parameters, again reflecting the surface sensitivity of the neglected terms (Titus et al., 2014, Titus, 2016).

The same idea has been extended to three-body entrance channels in $\beta$4 reactions within CDCC. There the three-body analog is

$\beta$5

so that the Perey effect becomes a product of neutron and proton factors rather than a single deuteron factor (Gómez-Ramos et al., 2019). In practical terms, the modified overlap is reduced in the nuclear interior by about $\beta$6–$\beta$7 relative to asymptotic normalization (Gómez-Ramos et al., 2019). This suggests that the effect is not merely a two-body optical-model artifact but a transferable structural feature of nonlocal reductions.

5. Applications to nuclear observables

The ansatz has been used extensively in direct radiative capture. In a potential-model treatment of $\beta$8Ca$\beta$9Ca, $$H(\mathbf x)=\pi^{-3/2}\beta^{-3}e^{-(\mathbf x/\beta)^2},$$0Li$$H(\mathbf x)=\pi^{-3/2}\beta^{-3}e^{-(\mathbf x/\beta)^2},$$1Li, and $$H(\mathbf x)=\pi^{-3/2}\beta^{-3}e^{-(\mathbf x/\beta)^2},$$2C$$H(\mathbf x)=\pi^{-3/2}\beta^{-3}e^{-(\mathbf x/\beta)^2},$$3N, explicit Perey–Buck nonlocality changed the calculated direct-capture cross sections by around $$H(\mathbf x)=\pi^{-3/2}\beta^{-3}e^{-(\mathbf x/\beta)^2},$$4 (Tian et al., 2018). The effect was reaction dependent: it was about $$H(\mathbf x)=\pi^{-3/2}\beta^{-3}e^{-(\mathbf x/\beta)^2},$$5 for $$H(\mathbf x)=\pi^{-3/2}\beta^{-3}e^{-(\mathbf x/\beta)^2},$$6Ca$$H(\mathbf x)=\pi^{-3/2}\beta^{-3}e^{-(\mathbf x/\beta)^2},$$7Ca, about $$H(\mathbf x)=\pi^{-3/2}\beta^{-3}e^{-(\mathbf x/\beta)^2},$$8 for $$H(\mathbf x)=\pi^{-3/2}\beta^{-3}e^{-(\mathbf x/\beta)^2},$$9Li$$H_{P\!B}(s)=\frac{1}{\pi^{3/2}\beta^3}e^{-s^2/\beta^2},$$0Li below $$H_{P\!B}(s)=\frac{1}{\pi^{3/2}\beta^3}e^{-s^2/\beta^2},$$1 MeV, and up to about $$H_{P\!B}(s)=\frac{1}{\pi^{3/2}\beta^3}e^{-s^2/\beta^2},$$2 in the resonant part of the $$H_{P\!B}(s)=\frac{1}{\pi^{3/2}\beta^3}e^{-s^2/\beta^2},$$3C$$H_{P\!B}(s)=\frac{1}{\pi^{3/2}\beta^3}e^{-s^2/\beta^2},$$4N astrophysical $$H_{P\!B}(s)=\frac{1}{\pi^{3/2}\beta^3}e^{-s^2/\beta^2},$$5 factor around $$H_{P\!B}(s)=\frac{1}{\pi^{3/2}\beta^3}e^{-s^2/\beta^2},$$6 MeV (Tian et al., 2018). The same work showed that nonlocality in bound and scattering states can interfere constructively or destructively depending on the reaction (Tian et al., 2018).

In bound-state ANC calculations, the ansatz has been embedded directly in a complex-ranged Gaussian basis. For the mirror systems $$H_{P\!B}(s)=\frac{1}{\pi^{3/2}\beta^3}e^{-s^2/\beta^2},$$7–$$H_{P\!B}(s)=\frac{1}{\pi^{3/2}\beta^3}e^{-s^2/\beta^2},$$8, $$H_{P\!B}(s)=\frac{1}{\pi^{3/2}\beta^3}e^{-s^2/\beta^2},$$9–$\tilde H_{P\!B}(K)=e^{-\beta^2K^2/4}$0, and $\tilde H_{P\!B}(K)=e^{-\beta^2K^2/4}$1–$\tilde H_{P\!B}(K)=e^{-\beta^2K^2/4}$2, nonlocal Perey–Buck-type potentials yielded systematically larger single-particle ANCs than local potentials (Sailaubek et al., 2019). Representative values are

$\tilde H_{P\!B}(K)=e^{-\beta^2K^2/4}$3

$\tilde H_{P\!B}(K)=e^{-\beta^2K^2/4}$4

$\tilde H_{P\!B}(K)=e^{-\beta^2K^2/4}$5

(Sailaubek et al., 2019).

Transfer reactions provide a more stringent test because they are surface dominated. In DWBA studies of $\tilde H_{P\!B}(K)=e^{-\beta^2K^2/4}$6 transfer using Perey–Buck nonlocality, first peaks differed by $\tilde H_{P\!B}(K)=e^{-\beta^2K^2/4}$7–$\tilde H_{P\!B}(K)=e^{-\beta^2K^2/4}$8 from local-equivalent calculations (Titus, 2016). The same thesis reported that with the nonlocal dispersive optical model the discrepancies grew to $\tilde H_{P\!B}(K)=e^{-\beta^2K^2/4}$9–$$V_L(r)=V_L f_0(r), \qquad V_{\rm so}(r)=2V_{\rm so}\frac{1}{r}\frac{d}{dr}f_{\rm so}(r),$$0, and that in nonlocal ADWA the disagreement was about $$V_L(r)=V_L f_0(r), \qquad V_{\rm so}(r)=2V_{\rm so}\frac{1}{r}\frac{d}{dr}f_{\rm so}(r),$$1 on average (Titus, 2016). A focused re-examination concluded that the Perey correction factor improves on a purely local treatment but is not sufficient if the desired accuracy is better than $$V_L(r)=V_L f_0(r), \qquad V_{\rm so}(r)=2V_{\rm so}\frac{1}{r}\frac{d}{dr}f_{\rm so}(r),$$2 (Titus et al., 2014).

Within CDCC, the three-body Perey-factor implementation produced smaller but still non-negligible effects. For energies typical of many $$V_L(r)=V_L f_0(r), \qquad V_{\rm so}(r)=2V_{\rm so}\frac{1}{r}\frac{d}{dr}f_{\rm so}(r),$$3 experiments the main peak changed by $$V_L(r)=V_L f_0(r), \qquad V_{\rm so}(r)=2V_{\rm so}\frac{1}{r}\frac{d}{dr}f_{\rm so}(r),$$4–$$V_L(r)=V_L f_0(r), \qquad V_{\rm so}(r)=2V_{\rm so}\frac{1}{r}\frac{d}{dr}f_{\rm so}(r),$$5, while at higher energies the effect reached about $$V_L(r)=V_L f_0(r), \qquad V_{\rm so}(r)=2V_{\rm so}\frac{1}{r}\frac{d}{dr}f_{\rm so}(r),$$6; for the nodal $$V_L(r)=V_L f_0(r), \qquad V_{\rm so}(r)=2V_{\rm so}\frac{1}{r}\frac{d}{dr}f_{\rm so}(r),$$7C$$V_L(r)=V_L f_0(r), \qquad V_{\rm so}(r)=2V_{\rm so}\frac{1}{r}\frac{d}{dr}f_{\rm so}(r),$$8C $$V_L(r)=V_L f_0(r), \qquad V_{\rm so}(r)=2V_{\rm so}\frac{1}{r}\frac{d}{dr}f_{\rm so}(r),$$9 state, the reported changes were about $$f_i(r)=\left[1+\exp\left(\frac{r-R_i}{a_i}\right)\right]^{-1}, \qquad R_i=r_iA^{1/3},$$0 at the first peak and about $$f_i(r)=\left[1+\exp\left(\frac{r-R_i}{a_i}\right)\right]^{-1}, \qquad R_i=r_iA^{1/3},$$1 at the second peak (Gómez-Ramos et al., 2019).

The ansatz has also been imported into decay theory through its local-equivalent potential. In WKB calculations of $$f_i(r)=\left[1+\exp\left(\frac{r-R_i}{a_i}\right)\right]^{-1}, \qquad R_i=r_iA^{1/3},$$2 and cluster decay, the Perey–Buck nonlocal interaction decreased all half-lives studied (Rojas-Gamboa et al., 2022). The reported reduction was roughly $$f_i(r)=\left[1+\exp\left(\frac{r-R_i}{a_i}\right)\right]^{-1}, \qquad R_i=r_iA^{1/3},$$3–$$f_i(r)=\left[1+\exp\left(\frac{r-R_i}{a_i}\right)\right]^{-1}, \qquad R_i=r_iA^{1/3},$$4 for many $$f_i(r)=\left[1+\exp\left(\frac{r-R_i}{a_i}\right)\right]^{-1}, \qquad R_i=r_iA^{1/3},$$5 decays and often $$f_i(r)=\left[1+\exp\left(\frac{r-R_i}{a_i}\right)\right]^{-1}, \qquad R_i=r_iA^{1/3},$$6–$$f_i(r)=\left[1+\exp\left(\frac{r-R_i}{a_i}\right)\right]^{-1}, \qquad R_i=r_iA^{1/3},$$7 for cluster decays, larger than in the Mumbai nonlocal model and relatively insensitive to $$f_i(r)=\left[1+\exp\left(\frac{r-R_i}{a_i}\right)\right]^{-1}, \qquad R_i=r_iA^{1/3},$$8 (Rojas-Gamboa et al., 2022). In that framework, the nonlocality range is scaled as

$$f_i(r)=\left[1+\exp\left(\frac{r-R_i}{a_i}\right)\right]^{-1}, \qquad R_i=r_iA^{1/3},$$9

so that heavy cluster–daughter systems correspond to much smaller effective $\mathbf r'$00 than nucleon–nucleus systems (Rojas-Gamboa et al., 2022).

6. Microscopic reinterpretations, parameter systematics, and limitations

Although the Perey–Buck form is phenomenologically successful, it is not treated as an exact microscopic theorem. One direct-capture study states explicitly that it “is most widely used … although it may not represent the real structure of nonlocal potentials given by microscopic theories sufficiently well” (Tian et al., 2018). Another practical caveat is that local and nonlocal potentials fitted to the same limited observables are not necessarily phase equivalent (Tian et al., 2018).

Microscopic folding calculations in momentum space nevertheless provide quantitative support for a Perey–Buck-like nonlocality. A study of fully nonlocal microscopic optical potentials for $\mathbf r'$01Ca$\mathbf r'$02 identified an approximately separable momentum-space structure, termed $\mathbf r'$03,

$\mathbf r'$04

whose coordinate-space interpretation is a radial factor times a nonlocality form factor (Arellano et al., 2022). For the central channel, the extracted bell-shaped nonlocality factor was close to the Gaussian Perey–Buck form and yielded $\mathbf r'$05–$\mathbf r'$06 fm below about $\mathbf r'$07 MeV for both proton and neutron beams (Arellano et al., 2022). The same work found that the spin-orbit channel is also nonlocal and bell-shaped, but with a smaller range, about $\mathbf r'$08–$\mathbf r'$09 fm (Arellano et al., 2022). The microscopic conclusion was not that the kernel is exactly Gaussian, but that it has “close resemblance” to the Perey–Buck form (Arellano et al., 2022).

A later microscopic analysis using chiral effective field theory employed the Perey–Buck equivalence relation as a diagnostic of whether local energy dependence can be reinterpreted as spatial nonlocality (Stahulak et al., 4 Sep 2025). Its main conclusion was sharply differentiated by channel: the dominant source of energy dependence in the microscopic real optical potential arises from spatial nonlocalities, whereas the energy dependence of the microscopic imaginary optical potential is a genuine time nonlocality (Stahulak et al., 4 Sep 2025). For the real part, the extracted nonlocality scales were typically $\mathbf r'$10–$\mathbf r'$11 fm and the Perey–Buck mapping worked up to about $\mathbf r'$12 MeV; beyond that range, satisfactory fits were not obtained (Stahulak et al., 4 Sep 2025).

These microscopic studies also modify two common simplifications. First, the nonlocality range $\mathbf r'$13 is not strictly constant: it shows density, asymmetry, proton–neutron, and radial dependence in microscopic extractions (Stahulak et al., 4 Sep 2025). Second, the spin-orbit term need not be local microscopically, even though it is frequently kept local in phenomenological Perey–Buck implementations (Tian et al., 2018, Arellano et al., 2022). A plausible implication is that the traditional ansatz is best understood as a compact, empirically robust parameterization of the dominant central spatial nonlocality, rather than as a universal description of all nonlocal optical-structure effects.

The main misconception addressed by this literature is that the Perey factor is an exact substitute for nonlocal dynamics. The accumulated evidence does not support that view. Exact nonlocal treatments reproduce the same qualitative interior suppression, but they show surface, peripheral, and channel-dependent deviations large enough to matter in transfer and capture observables (Titus et al., 2014, Titus, 2016, Gómez-Ramos et al., 2019). The ansatz therefore occupies a dual position: it is both a historically central phenomenology of nonlocality and a benchmark against which more exact and more microscopic treatments are now measured.