Recent developments for the optical model of nuclei

Recent developments for the optical model of nuclei

W. H. Dickhoff, R. J. Charity
Department of Physics, Washington University, St. Louis, MO 63130, USA
Department of Chemistry, Washington University, St. Louis, MO 63130, USA
Abstract

A brief overview of various approaches to the optical-model description of nuclei is presented. A survey of some of the formal aspects is given which links the Feshbach formulation for either the hole or particle Green’s function to the time-ordered quantity of many-body theory. The link between the reducible self-energy and the elastic nucleon-nucleus scattering amplitude is also presented using the development of Villars. A brief summary of the essential elements of the multiple-scattering approach is also included. Several ingredients contained in the time-ordered Green’s function are summarized for the formal framework of the dispersive optical model (DOM). Empirical approaches to the optical potential are reviewed with emphasis on the latest global parametrizations for nucleons and composites. Various calculations that start from an underlying realistic nucleon-nucleon interaction are discussed with emphasis on more recent work. The efficacy of the DOM is illustrated in relating nuclear structure and nuclear reaction information. Its use as an intermediate between experimental data and theoretical calculations is advocated. Applications of the use of optical models are pointed out in the context of the description of nuclear reactions other than elastic nucleon-nucleus scattering.

1 Introduction

The concept of an optical potential with both real and imaginary components was introduced in 1949 to describe the elastic scattering of neutrons. This quantum-mechanical approach allows the incident particle waves to be scattered, transmitted, and absorbed by the potential. The reference to “optical” comes from the use of complex refractive indices to explain similar phenomenona for light rays. The optical potential is an effective interaction which is used not just for elastic scattering, but as an ingredient in predicting the cross sections and angular distributions of many direct reaction processes and therefore plays an important role in nuclear physics. Reviews of the history and early use of the optical model by Feshbach and Hodgson can be found in Refs. [1, 2, 3, 4, 5]. However, since the last of these reviews was in 2002, it is timely to survey some of the more recent advances.

The new era of radioactive-beam experiments has clearly brought into focus the need to synthesize our approach to nuclear reactions and nuclear structure. Strongly-interaction probes of such exotic nuclei involve interactions that are not yet well constrained by experimental data. THerefore, there is an ongoing need to better understand the potentials that protons and neutrons in such systems experience. One important aspect is covered by the positive-energy optical potentials that govern elastic nucleon scattering while there is a clear need to establish the link with bound-state potentials which are hardly distinct from the positive-energy ones when the drip lines are approached.

The number of works that are encompassed by the title of “optical model” is immense and any review must be selective. The topics and studies present in this review are biased by the interests of the authors. We start with several theoretical derivations of the optical potential and its connection to the self-energy in many-body formulations in Sec. 2. In Sec. 2.1 the analysis of Capuzzi and Mahaux [6] will be employed to link the Feshbach approach to the many-body perspective provided by the time-ordered Green’s function. We complement this in Sec. 2.2 with a derivation by Villars [7] that formalizes the work of Ref. [8] that demonstrates the equivalence of the nucleon-nucleus -matrix to the on-shell reducible nucleon self-energy associated with the time-ordered Green’s function. This approach is important as it can be extended to the description of more complicated reactions involving composite projectiles or ejectiles. In Sec. 2.3 we briefly summarize some elements of the multiple-scattering approach but refrain from a detailed presentation which is available in Ref. [9]. We will take particular interest in the empirical dispersive optical model (DOM) where the real and imaginary potentials are connected by dispersion relations thereby enforcing causality. This approach allows one to exploit the formal properties of the time-ordered Green’s function including its link to the nucleon self-energy through the Dyson equation [10, 11]. As the dispersion relations require the optical potentials to be defined at both positive and negative energy, the formalism makes connections between nuclear reactions and nuclear structure needed for a better understanding of rare isotopes. Some ingredients relevant for the discussion of the DOM are summarized in Sec. 2.4.

Since the last review there are significantly more data available on elastic scattering, total and absorption cross sections. Of particular importance is the availability of more neutron data with separated isotopes and, in addition, a few elastic-scattering experiments in inverse kinematics with radioactive beams. With the new data have come better global empirical parametrizations. These will be discussed in Sec. 3 for scattering of nucleons and light projectiles.

Since the concept of the optical model was first formulated, there has been considerable interest in calculating it microscopically from the underlying nucleon-nucleon (NN) interaction. Some recent attempts of such studies are presented in Sec. 4 with examples obtained from a number of approaches. The nuclear-matter approach pioneered in Ref. [12] is summarized and illustrated in Sec. 4.1 as it has played an important role in practical applications. The method accounts for the short-range repulsion of NN interactions in nuclear matter including the effect of the Pauli principle and utilizes suitable local-density approximations. An example of a recent nuclear-matter study relying on modern chiral NN interactions emphasizing the isovector properties of optical potentials, is discussed in Sec. 4.2. Recent applications of the multiple-scattering approach are summarized in Sec. 4.3.

Calculations based on the Green’s-function method are presented in Sec. 4.4. We emphasize and illustrate the influence of short-range correlations in Sec. 4.4.1. Such calculations attempt to properly account for the short-range repulsion of the NN interaction by summing ladder diagrams in the nucleus under consideration and are therefore quite distinct from nuclear-matter approaches. We devote attention to the effect of long-range correlations in Sec. 4.4.2. The emphasis in this analysis is on the role of low-lying collective states in determining the nucleon self-energy in an energy domain that includes the physics of giant resonances. In Sec. 4.4.3 a recent analysis is presented that combines the structure calculations of rare isotopes employing ab initio many-body methods with optical potentials in an attempt to extract matter radii. Recent work employing the coupled-cluster method to determine the Green’s function and thereby the optical potential, is presented in Sec. 4.4.4

Applications of the DOM provide an important tool to establish a connection between experimental data and theory. We discuss various incarnations of the developments of the DOM in Sec. 5. A brief section on applications of optical potentials is presented in Sec. 6. This includes transfer reactions (Sec. 6.1), heavy-ion knockout reactions (Sec. 6.2), electron and proton induced knockout reactions (Sec. 6.3), and in Sec. 6.4 some other applications of DOM potentials to these reactions. Finally, we draw some conclusions in Sec. 7.

2 Formal background

In this section we illustrate several different approaches to the optical potential.

2.1 Hole and particle Green’ functions and the Feshbach formulation and its relation to the time-ordered Green’s function

We start by taking a slightly different route in introducing the optical potential by following some of the development discussed in Ref. [6], which provides a more complete formal discussion. This paper starts by introducing the hole Green’s function for complex

(1)

where represents the nondegenerate ground state of the -particle system belonging to the Hamiltonian . We employ a coordinate-space notation but implicitly refer to a complete set of single-particle (sp) quantum numbers relevant for the system under study. By inserting a complete set of states for the -particle system on can write

(2)

Overlap functions for bound states are given by , whereas those in the continuum are given by indicating the relevant channel by and the energy by . Excitation energies in the system are given with respect to the -body ground state . Each channel has an appropriate threshold indicated by , which is the experimental threshold with respect to the ground-state energy of the -body system. The hole spectral density is given by

(3)

which allows one to write the hole Green’s function as

(4)

For complex , one may define the “hole Hamiltonian” operator by using the operator form of as follows

(5)

For the propagator it is useful to adopt the limit

(6)

for real. This hole Hamiltonian has eigenstates corresponding to the discrete overlap states

(7)

As was introduced in Refs. [13, 14] for nucleon scattering, one can decompose any vector in the Hilbert space for particles with two hermitian projection operators

(8)

with . It is then required that the hole projection operator has the property

(9)

for all . The one-body density matrix provides the following link between overlap states and

(10)

It is possible to show that [6]

(11)

Using standard manipulation, the hole Hamiltonian can be written in terms of the projection operators given in Eq. (8) according to

(12)

which leads to the corresponding eigenvalue equation

(13)

where in the coordinate representation

(14)

and

(15)

Note that Eq. (13) is equivalent to Eq. (7).

It is possible to employ the original Hamiltonian to separate the kinetic and interaction contributions to in Eq. (14) written as

(16)

It is then possible to introduce the hole self-energy operator in the one-body space by subtracting the corresponding kinetic-energy operator as in

(17)

The first term in Eq. (12) diverges at large . This can be avoided by deriving other, related, Schrödinger-type equations. Indeed, by multiplying Eq. (13) on the left by any energy-independent, but possibly nonlocal, operator one obtains

(18)

where

(19)

The divergence can now be eliminated by setting in Eq. (19) which then becomes

(20)

Note that this Hamiltonian does not have a scattering eigenstate because it reflects the action of a removal operator on the localized ground state. This form of the Hamiltonian is completely analogous to the one introduced by Feshbach [13] to develop the optical potential for elastic nucleon scattering (see below), hence the subscript. Additional forms of the hole Hamiltonian are discussed in Ref. [6].

From the hole Hamiltonians considered above, it is possible to calculate the hole spectral function with access to overlap functions and their normalization. This reflects their analytic properties in the complex-energy plane which exhibits poles on the real axis for negative energy and a left-hand cut from to the threshold energy associated with nucleon emission from the -system.

Particle one-body Hamiltonians can be introduced in complete analogy with the hole version. The main difference consists is the presence of elastic overlap states as eigenstates

(21)

where is the asymptotic kinetic energy of the incident nucleon . The particle Hamiltonian therefore has eigenstates corresponding to discrete and continuum overlap states

(22)
(23)

For complex , the particle Green’s function is defined as

(24)

and can be written employing complete sets of -system states as

(25)

The particle Green’s function may therefore have discrete poles at negative energy on the real axis and a cut chosen to be just below the real-energy axis for .

The particle Hamiltonian, as well as the hole one in Eq. (17), can be decomposed as follows

(26)

with the kinetic-energy operator and the particle self-energy. For scattering eigenstates of the particle Hamiltonian, one has

(27)

with the index referring to the asymptotic boundary condition. The relevant Lippmann-Schwinger equation is given by

(28)

with the usual plane-wave eigenstates of the kinetic energy (with suitable modifications for protons).

For the -body problem, consider the elastic-scattering eigenstate in which the target is in the ground state

(29)

with

(30)

The corresponding Lippmann-Schwinger equation is given by

(31)

The elastic overlap in coordinate space is given by

(32)

It is now possible to demonstrate [6] that

(33)

thereby clarifying the importance of the particle self-energy as the scattering potential that generates the elastic scattering wave.

Following similar steps as for the hole Green’s function, a Feshbach Hamiltonian for the particle Green’s function can be generated employing the relevant projection operators. As this particle Hamiltonian also has a divergence at large , one may proceed as before to eliminate it, although in the present case, the relevant auxiliary operator is given by

(34)

The corresponding Hamiltonian then reads

(35)

The equivalence of this derivation to the original Feshbach optical potential [13] is demonstrated in Ref. [6].

The Green’s-function theory of the optical potential introduced in Ref. [8] identifies the optical potential with the self-energy associated with the time-ordered Green’s function. Unlike the separate hole and particle self-energies, this self-energy contains information on sp properties of both the ()- and ()-system, simultaneously. Indeed, the one-body Hamiltonian

(36)

has , , and as eigenstates.

It is convenient also for later practical applications employing this framework to introduce the average Fermi energy

(37)

The spectral function is the sum of the hole and particle spectral functions

(38)
(39)
(40)

which can be utilized to write the time-ordered Green’s function according to

(41)

which is analytic for all complex values of . It exhibits singularities on the real axis with a left-hand cut along the real axis from to and right-hand cut from 0 to . It is therefore clear that

(42)

For complex , one may write

(43)

With similar limits as for real , one can write the one-body Hamiltonian as

(44)

The self-energy or mass operator is in general complex and its perturbation expansion can be analyzed in the standard manner as shown for example in Ref. [10].

2.2 Alternative derivation linking the nucleon self-energy to the optical potential

While it is intuitively clear that the sp propagator contains information related to the elastic scattering of particles from a target ground state, it is appropriate to provide an additional derivation for this observation. Historically, the first attempt for such a connection was presented in Ref. [8]. We will follow the presentation of [7] and [15] because it has the merit that it can be extended to composite projectiles and ejectiles. We will not deal with the complications associated with treating the recoil of the nucleus [16]. Neither will we attempt to discuss this problem in terms of the more correct wave-packet formulation. The initial and final state of the target nucleus are denoted by , representing an even-even nucleus with total angular momentum zero. As before, . The states describing the projectile before and after the reaction will have the same magnitude of momentum and correspond to plane waves with momentum and , respectively. An extension to charged-particle scattering is possible by replacing the momentum states by Coulomb ones. We suppress spin (isospin) for convenience and represent the initial state of the combined projectile-target system by

(45)

with energy

(46)

For the final state we write similarly

(47)

with

(48)

Note that these states are not eigenstates of the Hamiltonian in the wave-packet sense, except in the distant past or future, respectively. This becomes clear by evaluating

(49)

and

(50)

The first term on the right in Eqs. (49) and (50) comes from the commutator with the kinetic-energy operator and the other terms can be obtained from the interaction, involving two addition and one removal operator for and two removal and one addition operator for , if one is restricted to two-body interactions. From Eq. (49) we infer

(51)

and similarly from Eq. (50)

(52)

The stationary scattering states are eigenstates of and obey

(53)

Inverting Eq. (51), while adding a solution of the homogeneous equation (53), yields

(54)

where the signals the outgoing-wave boundary condition and it can be shown that is properly normalized. Similarly, one finds

(55)

The -matrix element for the transition is then  [17]

(56)

Inserting Eqs. (54) and (55) judiciously in this expression, one finds

(57)

Using Eq. (50) one may show

(58)

With this result, the second term of Eq. (57) can be rewritten as

(59)

where Eq. (54) has been employed as well. Inserting Eq. (59) into (57) yields, after further manipulation,

(60)

Using Eqs. (51) and (52), the second term on the right side can be shown to be proportional to , and therefore vanishes for all . We may therefore identify the transition matrix from Eq. (60) according to

(61)

A similar derivation yields the equivalent relation

(62)

noting that the condition must hold.

It remains to be shown how to relate these expressions to the sp propagator. For this purpose we use Eq. (54) to transform Eq. (61) into

(63)

The second term in this relation can be rewritten by utilizing again Eq. (49) yielding

(64)

where the denominator never vanishes. This observation follows by inserting a complete set of states on the right and noting that the ground-state energy can never be equal to the sum of the energies of the two fragments. We can then write

(65)

It is now straightforward but somewhat tedious to show that

(66)

and furthermore, using the Dyson equation in terms of the reducible self-energy (see Sec. 2.4), that

(67)

which completes the proof of the relation between the elastic-scattering amplitude and the sp propagator. From now on we will use both the self-energy notation as well as to denote the same quantity in the context of elastic nucleon-nucleus scattering. We will also employ the symbol for the two-body quantity in free space which should be clear from the context of the discussion.

2.3 Multiple scattering

The formal approach sketched in Sec. 2.1 demonstrates that it is possible to formulate the effective one-body problem to describe elastic scattering using a projection operator onto the elastic channel and the complementary one that projects on all others. However, using the NN interaction to generate the energy-independent term of Eq. (35) is recognized as an inappropriate starting point as realistic NN interactions contain strong repulsion that is only tamed by an all-order summation that is contained in the energy-dependent term. With this recognition, it was suggested one could include important multiple-scattering events from the very start either in terms of the NN -matrix [18, 19] or the -matrix that only allows intermediate states in accord with the Pauli principle. A comprehensive overview of this approach is given in Ref. [9] together with an extensive bibliography. Only a brief discussion is therefore provided here in order to make the transition to schemes that allow for practical calculations.

It is assumed that two-particle interactions between projectile and target nucleons dominate. With the projectile tagged 0 and the target particles , one can write

(68)

The NN -matrix associated with can be obtained from the Lippmann-Schwinger equation as

(69)

with representing propagation in free space while generating the correct boundary condition. Replacing the NN interaction in the optical potential by the -matrix yields a workable approach to the optical-model potential. While this formulation is expected to work at higher energies, it is possible to make corrections for the Pauli principle and mean-field propagation in which extend the approach to lower energies. The most utilized approximation is the -matrix discussed in Sec.4.1 for nuclear matter.

A full-folding approach is an important implementation of the spectator expansion of multiple-scattering theory. The -particle Hamiltonian for this problem can be written as

(70)

Using the Green’s function

(71)

the full -matrix for the elastic scattering from the target’s ground state of nucleons can be written as

(72)

where includes all pairwise interactions of the projectile with all target nucleons. Assuming the elastic-scattering channel to be described in terms of an optical potential , one can write

(73)

with

(74)

employing the relevant projection operators, e.g. . The full elastic-scattering -matrix can then be written as

(75)

A spectator expansion then involves an ordering of the scattering process in a sequence of active projectile-target interactions. In first order, the interaction between projectile and target is written in standard notation as

(76)

where

(77)

with reduced amplitudes

(78)

For elastic scattering, only contributes to the optical potential. The task is then to find suitable approximations and solutions of Eqs. (77) and (78). The effective interaction can be obtained from the free -matrix by correcting the propagator for the binding associated with the nucleon that the projectile interacts with as well as allowing for its possible mean-field corrections. Introducing initial and final momentum and , as well as

(79)

the latter is associated with the integration over the internal momenta of the target nucleons. Convolution of the effective interaction with the target’s ground state then yields a nonlocal optical potential which can be written in momentum space as [9]

(80)

where is evaluated in the NN rest frame, imposes Lorentz invariant flux in transforming from the NN to the nucleon- system, and represents the one-body density matrix. In the spectator model, the free -matrix is used at an energy associated with the beam energy and the binding of the struck particle. When -matrix interactions are employed, the coupling with the integration variable can be included [9].

As noted above, an extensive review was published about this approach in Ref. [9] with many citations to original work. We will only discuss some recent applications of this approach in Sec. 4.3.

2.4 Nucleon self-energy approach: dispersive optical model

We introduce here some relevant results from Green’s-function theory as it pertains to the application of the dispersive optical model, promoted successfully by Mahaux [20]. Some of the material below is a condensed version of results recently reviewed in Ref. [11]. The aim of this section is to introduce relevant quantities related to experiment that can be obtained from the single-particle propagator. In addition, this section presents the pertinent one-body equations that need to be solved including their interpretation. The link between the particle and hole domain plays an important role in this formulation.

The nucleon propagator with respect to the -body ground state in the sp basis with good radial position, orbital angular momentum (parity) and total angular momentum while suppressing the projection of the total angular momentum and the isospin quantum numbers can be obtained from Eq. (42) and written as

(81)

where the continuum solutions in the systems are also implied in the completeness relations. The numerators of the particle and hole components of the propagator represent the products of overlap functions associated with adding or removing a nucleon from the -body ground state. The Dyson equation can be obtained by analyzing the perturbation expansion or the equation of motion for the propagator [10]. It has the following form

(82)

with representing the noninteracting propagator containing only kinetic-energy contributions. The nucleon self-energy contains all linked-diagram contributions that are irreducible with respect to propagation represented by .

As discussed formally in Sec. 2.1, the solution of the Dyson equation generates all discrete poles corresponding to bound states explicitly given by Eq. (81) that can be reached by adding or removing a particle with quantum numbers . The hole spectral function is obtained from

(83)

for energies in the -continuum. For discrete energies as well as all continuum ones, overlap functions for the addition or removal of a particle are generated as well.

For discrete states in the system, one can show that the overlap function obeys a Schrödinger-like equation [10]. Introducing the notation

(84)

for the overlap function for the removal of a nucleon at with discrete quantum numbers and , one finds

(85)

where

(86)

and in coordinate space the radial momentum operator is given by . Discrete solutions to Eq. (85) exist in the domain where the self-energy has no imaginary part and these are normalized by utilizing the inhomogeneous term in the Dyson equation. For an eigenstate of the Schrödinger-like equation [Eq. (85)], the so-called quasihole state labeled by , the corresponding normalization or spectroscopic factor is given by [10]

(87)

Discrete solutions in the domain where the self-energy has no imaginary part can therefore be obtained by expressing Eq. (85) on a grid in coordinate space and performing the corresponding matrix diagonalization. Likewise, the solution of the Dyson equation [Eq. (82)] for continuum energies in the domain below the Fermi energy, can be formulated as a complex matrix inversion in coordinate space. This is advantageous in the case of a non-local self-energy representative of all microscopic approximations that include at least the HF approximation. Below the Fermi energy for the removal of a particle , the corresponding discretization is limited by the size of the nucleus as can be inferred from the removal amplitude given in Eq. (84), which demonstrates that only coordinates inside the nucleus need to be considered. Such a finite interval therefore presents no numerical difficulty.

The particle spectral function for a finite system can be generated by the calculation of the reducible self-energy . In some applications relevant for elucidating correlation effects, a momentum-space scattering code [21] to calculate can be employed. In an angular-momentum basis, iterating the irreducible self-energy to all orders, yields

(88)

where is the free propagator. The propagator is then obtained from an alternative form of the Dyson equation in the following form [10]

(89)

The on-shell matrix elements of the reducible self-energy in Eq. (88) are sufficient to describe all aspects of elastic scattering like differential, reaction, and total cross sections as well as polarization data [21]. The connection between the nucleon propagator and elastic-scattering data can therefore be made explicit by identifying the nucleon elastic-scattering -matrix with the reducible self-energy obtained by iterating the irreducible one to all orders with  [8, 10, 7, 15].

The spectral representation of the particle part of the propagator referring to the system, appropriate for a treatment of the continuum and possible open channels, is given by Eq. (25) in the present basis as

(90)

which is a generalizing of the discrete version of Eq. (81) using a momentum-space formulation instead of one in coordinate space. Overlap functions for bound states are given by , whereas those in the continuum are given by indicating the relevant channel by and the energy by . Excitation energies in the system are with respect to the -body ground-state . Each channel has an appropriate threshold indicated by which is the experimental threshold with respect to the ground-state energy of the -body system. The overlap function for the elastic channel can be explicitly calculated by solving the Dyson equation while it is also possible to obtain the complete spectral density for

(91)

In practice, this requires solving the scattering problem twice at each energy so that one may employ

(92)

with , and only the elastic-channel contribution to Eq. (91) is explicitly known. Equivalent expressions pertain to the hole part of the propagator  [20].

The calculations are performed in momentum space according to Eq. (88) to generate the off-shell reducible self-energy and thus the spectral density by employing Eqs. (89) and (92). Because the momentum-space spectral density contains a delta-function associated with the free propagator, it is convenient for visualization purposes to also consider the Fourier transform back to coordinate space