Pierre Descouvemont

Three-alpha continuum states

Keywords: numbers: 21.60.Gx, 25.70.Ef, 24.10.Cn

published or not. The documents may come

Introduction

The 12 C spectroscopy below the 3α threshold is well known since many years (see Ref. [1] and references therein). The 0 + ground state and the 2 + first excited state (E x = 4.44 MeV) have been investigated in many experimental and theoretical works, and their properties are now well established. The situation, however, is very different above the 3α threshold. In the continuum region, the second 0 + state at E x = 7.65 MeV plays a very important role in nuclear physics and in astrophysics. This state cannot be reproduced in standard [2,3] or in no-core [4] shell model calculations. The 0 + 2 state is known to have a well-developed cluster structure and has been successfully described by α cluster models [5,6,7,8,9]. Fermionic molecular dynamics (FMD) calculations [10] also reproduce the 0 + 2 resonance provided that α-cluster structures are introduced in the FMD basis.

Historically, the interest for the 0 + 2 state started with the prediction of Hoyle [11] that a resonance with zero angular momentum should exist just above the α+ 8 Be threshold to explain the 12 C abundance in the Universe. Soon after this suggestion, the 0 + 2 resonance, called the "Hoyle state", was observed experimentally 375 keV above the 3α threshold [12,13]. In astrophysics, the determination of the triple α reaction rate at typical He-burning temperatures [14] is directly related to the properties of the Hoyle state (energy, α and γ widths). Although recently challenged by CDCC calculations [15], the 3α reaction rate is believed to be well understood.

In nuclear physics the interest for the 0 + 2 structure was recently revived by its interpretation as a dilute cluster gas state of three weakly interacting α particles [16]. The analysis of electron inelastic scattering data [17], as well as cluster calculations [9] suggest a large radius of the 0 + 2 state, which reduces the overlap between the α particles. This leads to the idea of a new form of nuclear matter, in analogy with the Bose-Einstein condensates [16].

In the theoretical viewpoint, it seems well established that the Hoyle state presents an α+ 8 Be cluster structure [9,18]. This form of α clustering is expected in light nuclei close to the α breakup threshold [19], and has been observed in many nuclei (see references in Ref. [7]). The specificity of the 12 C nucleus is that 8 Be, the other constituent of the α+ 8 Be system, presents itself an α cluster structure. Based on this interpretation of the 0 + 2 state as a bandhead of a K = 0 + rotational band, microscopic cluster models [8,9,18] predict the existence of a 2 + member at an energy of about 2 ∼ 3 MeV above the 3α threshold. This result is supported by the FMD [10] and by a non-microscopic 3α model [20]. This state is expected to be broad (Γ ∼ 1 MeV) and may overlap with other resonances.

Historically, the experimental search for a 2 + broad resonance is recent. Due to its specific structure, the observation of such resonance is extremely difficult. Some evidence for a 2 + broad level was reported by Itoh et al. [21] near E α ≈ 9 -10 MeV, though not widely accepted. More recently, the situation was clarified, and a possible 2 + resonance at E α = 9.6±0.1 MeV with Γ = 0.6 MeV was observed by Freer et al. [22]. These properties are in fair agreement with predictions of microscopic cluster models [9,18].

The theoretical description of the α + α system is well mastered in microscopic as well as in non-microscopic models. Microscopic theories [START_REF] Wildermuth | A Unified Theory of the Nucleus[END_REF] are based on a nucleonnucleon interaction and take an exact account of the Pauli principle. In non-microscopic approaches [START_REF] Buck | [END_REF], the structure of the α particle is neglected, and the Pauli principle is approximated by an appropriate choice of the α+α potential. However, the extension of these models, which reproduce the properties of the α + α phase shifts in a wide energy range, is not yet satisfactory for the 3α system. Microscopic approaches fail to describe accurately the α + α and α + α + α systems with the same nucleon-nucleon interaction [25]. With a NN potential that reproduces the α + α phase shifts, the 12 C ground-state is overbound. This was interpreted in Ref. [25] as a need for a three-body force. The situation is even more complex in non-microscopic theories, based on an α + α potential. Although highly accurate potentials exist in the literature, their application to 12 C is still unclear, even for the ground state (see Refs. [26,27]).

The aim of the present work is to investigate the 3α continuum in a non-microscopic approach. In the literature, this problem is usually tackled by using approximate methods to describe resonances. This can be done, for example, with the Complex Scaling Method (CSM, see Ref. [28]) or with the Analytic Continuation in the Coupling Constant (ACCC, see Ref. [START_REF] Kukulin | Theory of Resonances, Principles and Applications[END_REF]) method. Here, we go beyond these approximations and also determine the triple α phase shifts. Recently, we have extended the hyperspherical formalism [START_REF] Lin | [END_REF] to three-body continuum states by implementing the R-matrix theory with a Lagrange basis [31]. Although the hyperspherical theory is known to converge slowly in the continuum [32,33,31], it provides genuine 3-body phase shifts. The model was applied in Ref. [31] to the 3-body α + n + n phase shifts, and suggests the existence of broad 0 + and 1 -resonances in the low-energy spectrum of 6 He. The determination of phase shifts provides a coherent approach to continuum properties. It complements approximate methods by providing the wave functions not only at resonance energies, but also off resonances. Of course, resonances predicted by approximate methods should be reflected in the phase shifts. A combined approach therefore offers a rather complete picture of the continuum.

In Section 2, we briefly review the current literature on the subject. Section 3 is devoted to the models used here. We give the main properties of the hyperspherical method associated with the R-matrix theory. We also give an outline of the CSM, used as a complement to the 3α phase shifts. Results are shown in Section 4, where we use two different α + α potentials, and compare resonance properties derived from the CSM with values extracted from the 3-body phase shifts. Conclusions and outlook are presented in Section 5.

Brief overview of the literature

Theoretical works

The literature concerning 12 C is very abundant, and it is of course impossible to review all papers on the subject. Here we essentially focus on works aimed at investigating states above the 3α threshold. In a theoretical point of view, the Hamiltonian is given by

H = 3 i=1 T i + 3 i>j=1 V αα (|r i -r j |) + V ααα (r 1 , r 2 , r 3 ), (1)
where T i is the kinetic energy of α particle i with coordinate r i , and V αα an α + α potential. A phenomenological 3α potential V ααα is in general included in this Hamiltonian [34,20]. Although 3-body continuum calculations become feasible with microscopic cluster models [35] where the internal structure of the particles is taken into account, their application is currently limited to lighter systems such as α + n + n. Here, we focus on non-microscopic approaches where the α internal structure is neglected. Essentially three types of α + α potentials are available:

• Local deep potentials. In this variant, the potentials present additional (unphysical) bound states, which simulate the Pauli forbidden states [36]. Pauli forbidden states stem from antisymmetrization effects. They show up naturally in microscopic cluster theories where the antisymmetrization between all nucleons is exactly taken into account, and where the internal wave functions of the clusters are defined in the shell model with a common oscillator parameter b. In that case, Pauli forbidden states are expressed as harmonic-oscillator functions. For the α + α system, the Buck potential [START_REF] Buck | [END_REF] is known to accurately reproduce the phase shifts up to 20 MeV. The use of deep potentials in two-body problems is very simple, as forbidden states are eliminated automatically. For multi-body systems, however, the situation is more complicated as these two-body forbidden states introduce spurious states. Consequently, the total wave function must be orthogonalized to the two-body forbidden states ϕ k (r). This is achieved by adding, in the Hamiltonian (1), the projector [37]

P = Λ n k=1 | ϕ k (r) ϕ k (r) |, (2)
for each α + α pair. In Eq. ( 2), r is the α -α coordinate, and Λ is a constant chosen large enough (typically Λ ∼ 10 4 -10 5 MeV) to move the forbidden states in the high-energy region. This projector makes the α + α interaction non local. The number of forbidden states n depends on the system and on the angular momentum (2n + = 8 for α + α). For the 3α system, the use of the pseudo-potential method provides controversial results [38,26,39]. As the forbidden states ϕ k (r) stem from microscopic arguments, it seems natural to take harmonic-oscillator functions with an appropriate oscillator parameter (for example, deduced from the radius of the α particle). This procedure provides an overbinding of the 12 C ground state [26,25]. On the other hand, an alternative is to use the deep states generated by the potentials as forbidden states, which leads to an underbinding of the 12 C ground state [38,40]. This paradox was clarified by Matsumura et al. [27] who showed that removing the forbidden states of the potential is inconsistent with microscopic arguments.

• Local shallow potentials. These potentials do not contain forbidden states. A typical example is the Ali-Bodmer potential [41], which presents a repulsive core at low angular momenta. The Ali-Bodmer potential provides a good fit of the α + α phase shifts. In fact, it was shown that deep and shallow potentials are related to each other by a supersymmetry transform [42], and are therefore phase equivalent.

In shallow potentials, microscopic effects are simulated by a repulsive core at short distances. This variant is of course easier to use for multi-α systems as spurious states do not show up and there is no need for the projector (2). It has been adopted by Alvarez-Rodriguez et al. [34] to investigate low-lying 12 C resonances.

According to the work of Matsumura et al. [27], it is however not clear whether this potential is well adapted to multi-particle systems or not.

• Non-local potentials. The α + α potentials derived from microscopic cluster models are non local [43]. Recently some progress has been made to implement non-local potentials in three-body calculations [44,45], but they are currently limited to bound states. Their use in variational calculations is possible, but owing to the many problems encountered in the calculation of 3-body phase shifts, they are quite difficult to adapt to scattering theories. c Non-microscopic model with a deep α + α potential and a repulsive 3α force.

d Non-microscopic model with the Ali-Bodmer α + α potential and an attractive 3α force.

In Table 1, we present recent theoretical results on 3α resonances. We limit the discussion to 0 + and 2 + states. In Ref. [34], the authors use the Ali-Bodmer potential [41] with an attractive 3α interaction to fit the experimental ground-state energy on the first 0 + state of the model. The continuum is treated with the CSM method. Although the rotation angle is rather small, several additional resonances, not known experimentally, are found. In particular, a narrow 2 + state is obtained at E cm = 1.38 MeV. The use of a larger rotation angle may still provide more resonances. As shown by Matsumura et al. [27], the treatment of forbidden states in the 3α system is a delicate problem. The authors recommend to use deep α + α potentials with forbidden states consistent with the underlying microscopic model. From recent works on the 3α system, the 3-body potential seems to be repulsive [20,44], which is not consistent with the choice adopted in Ref. [34]. The repulsive nature of the 3α interaction is supported by microscopic theories where, starting from a nucleon-nucleon force fitting the α + α phase shifts, the 3α ground state is overbound [9,18,25]. Kurokawa and Katō [20] use a similar 3α model complemented by the CSM method. In contrast with Ref. [34], the α + α potential is deep [47], and a repulsive 3α interaction must be used to compensate the ground state overbinding. Concerning 0 + resonances, Kurokawa and Katō find a state near 4 MeV, in agreement with Ref. [34]. However, an additional broad resonance is found near 1.7 MeV (Γ = 1.48 MeV). This state could not be seen in Ref. [34] owing to the small rotation angle used in the CSM. For the 2 + resonances, the results are very different. This may be related to the different α + α potentials used in these references.

Funaki et al. [46] and Arai [18] investigate the 3α continuum in a microscopic threecluster model with different basis functions. A microscopic approach avoids the choice of an α + α potential since the model is based on a nucleon-nucleon force. However, NN interactions fitting the α + α phase shifts provide 12 C low lying states too strongly bound. The properties of the 0 + 2 Hoyle state are well reproduced, in agreement with the interpretation of this resonance as a cluster state. Funaki et al. [46] use the ACCC method to derive resonance properties. This method is known to require a high accuracy of the basis [48] and becomes unstable for broad resonances. The authors only find a 2 + 2 resonance near 2.6 MeV.

Arai [18] uses the CSM and finds a broad 0 + state near 5 MeV. The properties of this resonance agree reasonably well within the different models. The two additional 2 + resonances nearly correspond to the semi-microscopic results of Ref. [20]. Notice that Arai treats resonant states in two ways: with the CSM and with an R-matrix calculation of the α+ 8 Be phase shifts. Although the latter do not exactly correspond to three-body phase shifts ( 8 Be is considered as a bound system), they provide a more direct insight on the continuum properties. In particular, the 2 + 2 state is shown to have an important reduced α width and, consequently, should correspond to the 2 + member of a rotational band based on the 0 + 2 Hoyle state.

Experimental works

For a long time, only a broad 0 + 3 state near E cm ≈ 3 MeV (Γ ≈ 3 MeV) was reported in data compilations [1]. The presence of broad 0 + states above the 3α threshold was confirmed by β-decay experiments [49,50] from 12 B and 12 N, but precise values for the energies and widths could not be derived. It was, however, suggested that broad states may overlap with each other and make the data analysis rather complicated.

More recently, some advances have been performed in the search for a 2 + 2 state. The interest for this state is twofold. According to most theoretical predictions, the Hoyle state should be considered as the bandhead of a 0 + rotational band. Then, the existence of a 2 + resonance would be a strong support to cluster theories. On the other hand, this resonance should affect the triple-α reaction rate at high temperatures (T ≥ 10 9 K). A first evidence for a 2 + broad state was reported by Itoh et al. [21,51] near E x ≈ 9-10 MeV. More recently, Freer et al. [22] performed a 12 C(p,p') experiment to various 12 C states, and conclude on a possible 2 + state at 9.6 ± 0.1 MeV, with a width of Γ = 0.6 ± 0.1 MeV. These properties agree nicely with microscopic predictions [9,18]. Diget et al. [49] populate 12 C continuum states by β decay of 12 B and 12 N and investigate breakup channels. They conclude on the possible existence of a 2 + state in the range E x = 10.5 -11 MeV.

Outline of the theory

The hyperspherical formalism

For 3-body systems, the Schrödinger equation associated with Eq. ( 1) can be solved with the hyperspherical approach [START_REF] Lin | [END_REF]. The hyperradius ρ and hyperangle α are deduced from the scaled Jacobi coordinates

x = √ 2(r 1 -r 2 ), y = 32 12 r 3 - r 1 + r 2 2 , (3)
as

ρ = x 2 + y 2 , α = arctan(y/x). (4)
Of course, two other choices can be made for x and y, with permutations of the α particles. The basis functions associated with these different choices are related to each other by a unitary transform involving the Raynal-Revai coefficients [52]. In this coordinate system, the 3-body kinetic energy involves the operator K 2 which generalizes the concept of angular momentum in 2-body systems. It commutes with 2

x and 2 y and their common eigenfunctions Y JM γK (Ω 5 ) are known analytically (see Ref. [53] for details). The eigenvalue of K 2 is K(K + 4) where the integer K is the hypermomentum quantum number. In these definitions, Ω 5 = (Ω x , Ω y , α) and γ stands for γ = ( x , y ) where ( x , y ) are the angular momenta associated with (x, y). The total angular momentum J results from the coupling of x and y .

The wave function in partial wave Jπ is then expanded over hyperspherical harmonics as

Ψ JMπ (ρ, Ω 5 ) = ρ -5/2 γK χ Jπ γK (ρ) Y JM γK (Ω 5 ), (5)
where the hyperradial functions χ Jπ γK (ρ) have to be determined. The Schrödinger equation is replaced by a system of coupled differential equations

- 2 2m N d 2 dρ 2 - (K + 3/2)(K + 5/2) ρ 2 -E χ Jπ γK (ρ) + K γ V Jπ Kγ,K γ (ρ) χ Jπ γ K (ρ) = 0, ( 6
)
where the matrix elements of the 2-body potentials are defined as

V Jπ Kγ,K γ (ρ) = Y JM γK (Ω 5 )| 3 i>j=1 V αα (r j -r i )|Y JM γ K (Ω 5 ) . (7)
In Eq. ( 7), the integrals over Ω x and Ω y are performed analytically, whereas a numerical quadrature is used for the integral over the hyperangle α. With the Raynal-Revai coefficients [52] the evaluation of the potential matrix elements ( 7) is rather easy. In Eq. ( 5), a truncation must be done in the summation over K; the maximum K value is denoted as K max . The number of γK components increases rapidly when K max increases. In Ref. [54], we have solved the system (6) with the Lagrange-mesh method [55]. Each hyperradial wave function is expanded as

χ Jπ γK (ρ) = N i=1 c Jπ γKi ϕ i (ρ), (8)
where ϕ i (ρ) are N Lagrange functions and c Jπ γKi variational coefficients. The use of these basis functions greatly simplifies the calculations, since matrix elements computed at the Gauss approximation do not need any numerical integral. In spite of its simplicity the method is very accurate (see Ref. [55] for futher detail).

The 3α wave function must be symmetrized under α exchanges. In other words, the wave function (5) must satisfy

SΨ JMπ = 1 3 (1 + S 13 + S 23 )Ψ JMπ = Ψ JMπ , (9)
where S ij is the exchange operator between particles i and j. Symmetry between particles 1 and 2 is automatic, since only even partial waves x are included in expansion (5). Eq. ( 9) can be easily solved with the help of the Raynal-Revai coefficients. For a given K value, operator S is diagonalized, which provides eigenvalues 1 and 0. Only the former eigenstates are kept in the basis.

The R-matrix theory for 3-body continuum states

The treatment of three-body continuum states (E > 0), with exact three-body asymptotic conditions, is recent [56,31]. In the R-matrix method [57,58], the configuration space is divided into two regions: the internal region (with radius a) where the nuclear force must be taken into account, and the external region where the potentials have reached their asymptotic (Coulomb) behaviour. Consequently the solutions of the system (6) are written as,

χ Jπ γK,int (ρ) = N i=1 c Jπ γKi ϕ i (ρ), for ρ < a. (10)
In the external region, potentials ( 7) are assumed to be given by their asymptotic form. As discussed in Refs. [59,31], the Coulomb part of potential (7) tends to

V Jπ,Coul Kγ,K γ (ρ) → z Jπ Kγ,K γ e 2 ρ , (11)
where matrix z Jπ represents 3-body effective charges. They are computed from the 2-body Coulomb potentials. An example is given in Table 2 for J = 0 + , and for K ≤ 8.

In most cases the non-diagonal elements are small compared to the diagonal terms. Neglecting non-diagonal terms provides uncoupled solutions of Eq. ( 6). Accordingly, the external solution reads, for an entrance channel γ ,

χ Jπ γK,ext (ρ) = C Jπ γK H - γK (kρ)δ γγ δ KK -U Jπ γK,γ K H + γK (kρ) , (12)
where k = 2m N E/ 2 is the three-body wave number (m N is the nucleon mass). In Eq. ( 12), C Jπ γK is a normalization coefficient, U Jπ is the three-body collision matrix and the incoming and outgoing functions H ± γK (x) are defined as

H ± γK (x) = G K+ 3 2 (η γK , x) ± iF K+ 3 2 (η γK , x), (13)
where the Sommerfeld parameters η γK are given by

η γK = z Jπ γK,γK m N e 2 2 k , (14)
and therefore depend on the channel. According to the R-matrix formalism, matrix elements between basis functions ϕ i (ρ) must be computed over the internal region. Then the use of the Bloch operator [58] makes the kinetic energy Hermitian, and ensures the continuity of the derivative of the wave function at ρ = a. The matching between the internal and external solutions provides the collision matrix U Jπ . A strong test of the calculation is that the collision matrix should not depend on the channel radius a and on the number N of basis functions, provided that they are large enough to meet the R-matrix requirements.

As shown in Ref. [31], an important issue in 3-body scattering states is the long range of the three-body potential (7). At large distances, the nuclear part of this potential behaves as

V Jπ,Nuc Kγ,K γ (ρ) -→ ρ→∞ v Jπ 0,Kγ,K γ ρ 3 , (15)
even with short-range two-body interactions. This property arises from the definition of the hyperspherical coordinates. Even for large ρ values, two particles can still be close to each other and strongly interact. Constants v Jπ 0,Kγ,K γ can be quite large (examples are given in Ref. [31]). For this reason, the R-matrix radius takes unusually large values (typically a ∼ 800 -1000 fm for the 3α system) to ensure that potential (15) is negligible compared with the centrifugal and Coulomb terms (for comparison, typical values in two-body scattering is a ≈ 10 fm [58]).

To avoid huge basis sizes, propagation techniques are necessary. We follow the method of Ref. [31], where the R-matrix is computed at some channel radius a 0 with the basis (10). Typical values of a 0 are a 0 ≈ 30-40 fm with N ≈ 40-50 basis functions. At this radius the nuclear interaction is far from being negligible, and the asymptotic form ( 12) is not valid. Consequently the R-matrix determined at a 0 is propagated up to ρ = a with the Numerov algorithm. This technique is based on an analytical approximation of the nuclear potential in the region

a 0 ≤ ρ ≤ a V Jπ,Nuc γK,γ K (ρ) ≈ ∞ k=0 v k ρ 2k+3 , (16)
where coefficients v k are computed from integrals of the two-body potentials. They depend on the different quantum numbers. Expansion ( 16) is a generalization of the asymptotic behaviour (15). In practice the summation over index k is truncated at some k max value, which can be quite large in the 3α system to achieve a good accuracry (k max ≈ 30 -40). This formalism is limited to local potentials, and is not valid for non-local interactions, where expansion ( 16) must be adapted. Consequently current calculations are restricted to local α + α interactions. This means that deep potentials cannot be used since the associated projector ( 2) is non-local. Only shallow potentials, such as the Ali-Bodmer potential, can be currently considered to derive three-body phase shifts. Further developments are still necessary to deal with non-local interactions in the Rmatrix formalism.

The collision matrix may involve many channels, in particular for large K max values, and is in general analyzed through its eigenvalues. In addition, the Coulomb contribution should be removed from the eigenphases. As explained in Ref. [31] this is achieved by evaluating the pure Coulomb collision matrix U Jπ C (the nuclear potential is set to zero). Then the nuclear collision matrix U Jπ N is defined as

U Jπ = U Jπ C 1/2 U Jπ N U Jπ C 1/2 . ( 17
)
Matrix U Jπ N is easily shown to be symmetric and unitary. Its eigenvalues can therefore be written as

u Jπ N,i = exp(2iδ Jπ i ), (18)
where the eigenphases δ Jπ i are real. The number of eigenphases is of course equal to the number of channels γK.

The Complex Scaling method

The R-matrix theory provides 3-body wave functions at any scattering energy. The calculation of the phase shifts is obviously the optimal way to investigate the continuum, and in particular broad resonances. In practice, however, dealing with the continuum is an heavy task, especially for three-body systems. As mentioned before, only shallow local potentials can be currently employed. For these reasons a number of approximate methods have been developed. The common idea is to derive resonance properties (energy and width) from bound-state calculations, which only requires slight modifications of well mastered techniques. In the CSM [60,61,28,62], the space radial coordinate r and the momentum p are transformed as

U (θ)r = r exp(iθ), U (θ)p = p exp(-iθ), ( 19
)
where θ is the scaling angle. Under this transformation the Schrödinger equation reads

H(θ)Ψ(θ) = E(θ)Ψ(θ), (20)
with

H(θ) = U (θ)HU(θ) -1 . ( 21
)
The diagonalization of H(θ) with square-integrable functions can be done in various bases, such as Gaussian [38] or Lagrange basis [54]. The ABC theorem [60,61] demonstrates that this diagonalization provides continuum states along straight lines in the complex plane, rotated by an angle 2θ from the positive real axis. On the contrary, resonant states are not modified under variations of θ. The resonance energy E R and width Γ are determined from

E(θ) = E R -iΓ/2, (22)
and are independent (at the numerical accuracy of the calculation) of θ. The main interest for the CSM is its simplicity. Standard three-body codes can be easily adapted by introducing transformation (17), and by diagonalizing a complex matrix. Of course broad resonances require large rotation angles (tan 2θ ≥ Γ/2E R ). A limitation of the CSM is that the potential must be analytic. In addition, only the total width of a resonance can be determined. In coupled-channel calculations the partial widths in the different channels cannot be computed individually. Further detail, and in particular applications to three-body systems in nuclear physics, can be found in Ref. [62].

Results

Conditions of the calculations

The calculations are performed with two α + α potentials widely used in the literature: the shallow Ali-Bodmer (AB) potential [41] ( -dependent potential d), and the deep potential of Buck et al. (BFW) [START_REF] Buck | [END_REF]. Both interactions reproduce fairly well the α + α phase shifts up to E ≈ 20 MeV, i.e. below the proton threshold in 8 Be. As the BFW potential introduces spurious 3α states in the 12 C spectrum, the projector (2) is used to remove α + α Pauli forbidden states. According to Ref. [27], the two-body forbidden states are defined as harmonic-oscillator orbitals. The oscillator parameter b is chosen as in Ref. [44], i.e. b = 1.395 fm. The constant Λ is taken as 10 4 MeV, but all results are insensitive to that choice provided it is large enough (Λ 10 3 MeV).

As discussed in Section 2, both potentials have been used by several authors to investigate 12 C bound states. It is well known that a 3α potential must be introduced to reproduce the ground-state energy. According to Ref. [34], we choose the three-body interaction as

V ααα (ρ) = v 3 exp(-(ρ/ρ 3 ) 2 ), (23)
with the range ρ 3 = 6 fm. This phenomenological potential only depends on the hyperradius, and can be easily introduced in the hyperspherical equations (6). The amplitude v 3 is adjusted, for each potential and angular momentum, on the experimental energies. The values are given in Table 3, which confirms that the 3α potential must be attractive for the AB potential, and repulsive for the BFW potential. For the AB potential, v 3 strongly depends on angular momentum, whereas this dependence is weak for the BFW potential. Calculations with the CSM method are performed with Lagrange-Laguerre basis functions [54] (typically N = 50), and a scaling parameter h = 0.3 fm. The Rmatrix calculations (see Ref. [31] for detail) are performed with 50 Lagrange-Legendre functions with a channel radius a 0 = 40 fm. The R-matrix is then propagated up to a = 1000 fm, where the wave functions have reached their asymptotic behaviour (12). The sensitivity with respect to these parameters is quite low (less than 1 • in the phase shifts). As continuum states are known to converge slowly in the hyperspherical formalism [32,56,31], large K max values must be used in the expansion (5). However, our aim is not to derive fully converged values, but to investigate qualitatively the 12 C spectrum above the 3α threshold. For J = 0 + , we use K max = 44, which represents 144 values of quantum numbers γK. For J = 2 + , the number of channels is of course larger, and the K max value has to be reduced. We use K max = 24, which represents 120 channels.

Continuum states with the AB potential

We start the analysis without three-body potential (v 3 = 0). In Fig. 1, we show the 0 + eigenphases (18) and the corresponding complex energies (22). Here and in the following, bound states (Re(E) < 0) are not shown in the figures, but are given in Table 4. We analyze the convergence with respect to K max (K max = 36, 40 and 44 are presented). The first three eigenphases (i = 1, 2, 3 in Eq. ( 18)) present a narrow resonance between 1 and 2 MeV. These resonances are less and less narrow when energy increases. Their energy and width deduced from the phase shifts are very close to those obtained with the CSM calculations, but the phase shifts also provide information off resonances. The bound-state and resonance energies for K max = 44 are displayed in Table 4 which shows that the lowest 0 + state is found at -1.60 MeV. The underbinding of 12 C with the AB potential is well known [5,40], and can be solved by using an attractive three-body interaction.

The J = 0 + eigenphases and complex energies with the 3α potentials of Table 3 are displayed in Fig. 2, for K max = 44. To check the stability of physical resonances with respect to the scaling angle, we use two values: θ = 0.25 and θ = 0.30. Several states are stable and can also be observed in the eigenphases. The second 0 + state is located at -0.12 MeV (see Table 4) but, as for v 3 = 0, several additional resonances between 0.5 and 2 MeV are observed in the eigenphases, and confirmed by the CSM method. A further narrow state near 3 MeV (Γ ≈ 0.4 MeV) is obtained. Table 4. Energies E R and widths Γ (in MeV) with the AB potential. The results for J = 2 + are presented in Fig. 3. Without the 3α potential the lowest 2 + state is unbound (see Table 4), and would show up as a sharp resonance in the phase shifts. The CSM analysis shows that several resonances are insensitive to the θ value, and can be observed in the phase shifts. With v 3 = -11.9 MeV, a narrow state is found near 4 MeV. The present analysis may help in understanding the recent results of Ogata et al. on the 3α process within the CDCC framework [15]. Although the models are different, both provide three-body wave functions which should tend to the exact solution of the Schrödinger equation. Ogata et al. find a triple-α reaction rate about 10 20 times larger than previously accepted at a temperature T = 10 7 K. This huge factor, inconsistent with stellar models [63], might be explained by additional 0 + resonances. The α + α potential used by Ogata et al. is very similar to the Ali-Bodmer potential, and is also expected to give rise to additional resonances in the CDCC approach. This would explain the unexpectedly large factor obtained in the CDCC 3α reaction rate.

J = 0 + J = 2 + v 3 = 0 v 3 = -22.0 MeV v 3 = 0 v 3 = -

Continuum states with the BFW potential

As explained in Section 3.2, the analysis of continuum states with a deep potential is limited to the CSM. The reason is that the projector over forbidden states (2) introduces non-locality in the three-body equations, even at large ρ values.

The complex energies for J = 0 + and 2 + are presented in Fig. 4 for two scaling angles (see also Table 5). Without a three-body force, the low-lying states are overbound. This result is well known [5,26]. In these conditions, narrow resonances show up near E = 1.45 MeV (J = 0 + ) and E = 1.84 MeV (J = 2 + ). Introducing a 3α potential moves the 0 + 2 Hoyle state slightly above the experimental energy (E = 0.38 MeV) and gives rise to 0 + broad resonances near 2.3 and 4.8 MeV. These states are quite stable when changing the scaling angle (see Fig. 4). A similar behaviour is obtained for J = 2 + . With v 3 = 25.1 MeV, we obtain a broad resonance (Γ = 0.8 MeV) at E R = 3.6 MeV. Our results with the BFW potential are qualitatively in agreement with those of Kurokawa and Katō [20] (see Table 1), who use a deep potential derived from a folding model. Our 0 + 3 and 2 + 2 broad resonances are at slightly higher energies. The present calculations suggest that the description of the 3α continuum is very sensitive to the type of potential. The deep BFW potential provides results consistent with experiment, in contrast with shallow potentials, such as the AB interaction, which predict several narrow states, not supported by experiment.

Conclusion

The theoretical study of the 3α continuum represents a typical example of open problem. Our goal is of course not to provide a definite solution to the 3α problem, but to draw attention on the strong sensitivity of continuum states on the α + α potential. Although the physics of 12 C above the α threshold is an important issue in nuclear physics and in nuclear astrophysics, it remains unclear. A rigorous treatment of the continuum requires the calculation of the phase shifts which provides, not only resonance properties, but also information on the 3α process off resonances. Until now, threebody continuum calculations were essentially limited to the α + n + n system, where the phase shifts suggest broad resonances at low energies. In the 3α system, the phase shifts computed with the AB potential present several narrow resonances, which do not have experimental counterparts. The existence of such narrow states could be an explanation for the huge triple α reaction rate of Ogata et al. [15], who use an α + α potential very similar to the AB potential.

Of course R-matrix calculations can only be performed by neglecting non-diagonal terms of the Coulomb interaction at large distances (see the discussion in Section 3.2). However, the consistency between the phase shift analysis and the CSM results, as well as the stability of the phase shifts against variations of different parameters (R-matrix radii a 0 and a, number of basis functions) indicate that this approximation should not affect the conclusions.

An important output of our work is the sensitivity of the 3α continuum properties with respect to the choice of the α + α potential. This result is not surprising as the bound-state spectrum is also dependent on the potential (see for example Ref. [5]).

Calculations with the deep BFW potential are currently limited to the CSM method, but provide resonance energies which significantly differ from the AB interaction. In addition to the 0 + 1 , 0 + 2 and 2 + 1 states, only broad resonances beyond 3 MeV are found. The 2 + 2 resonance might correspond to the new state observed in recent experiments [22,49].

Improvements of the present study could be done by using non-local potentials. Recent works propose non-local α + α interactions derived either from the RGM kernels [44] or from a phenomenological approach [45]. In particular the potential of Ref. [45] reproduces simultaneously the experimental energies of the 0 + 1 and 0 + 2 states in a 3α description of 12 C. However performing continuum calculations with non-local potentials strongly increases the complexity of the model, in particular for the R-matrix theory. An alternative approach is to use microscopic cluster models, which do not rely on α + α potentials, but on a nucleon-nucleon interaction. Calculations of the α+ 8 Be have been performed [9,18], but the determination of the 3α phase shifts in this framework is still a challenge for the future.

Figure 1 .Figure 2 .

 12 Figure1. Three-α phase shifts (upper panel) and complex energies (lower panel, θ = 0.25) for the Ali-Bodmer potential (J = 0 + ) without 3-body potential (v 3 = 0). In the phase shifts, dotted, dashed and solid lines correspond to K max = 36, 40 and 44, respectively.

Figure 3 .

 3 Figure 3. Continuum states for J = 2 + with the Ali-Bodmer potential. Upper panel: three-α phase shifts (solid lines: v 3 = 0, dotted lines: v 3 = -11.9 MeV). Lower panel: complex energies for v 3 = 0 and v 3 = -11.9 MeV with two scaling angles (θ = 0.25 and 0.30).

Figure 4 .

 4 Figure 4. Complex energies with the BFW potential for J = 2 + with and without a 3α potential.

Table 1 .

 1 Theoretical properties (energy E R , width Γ, in MeV) of 0 + and 2 + resonances in various models. J π Ref. [46] a Ref. [18] b Ref. [20] c Ref. [34] d

	0 +	4.7, 1.0	1.66, 1.48 3.95, 1.0
			4.58, 1.1
	2 + 2.6 ± 0.3, 1.0 ± 0.3 2.1, 0.8	2.28, 1.1	1.38, 0.132
		4.9, 0.9	5.14, 1.9	4.48, 1.086
				6.49, 2.250

a Microscopic model with the ACCC method. b Microscopic model with the CSM method and α+ 8 Be phase shifts.

Table 2 .

 2 Effective charges z Jπ Kγ,K γ [see Eq. (11)] for J = 0 + ( x = y , x = y ).

	K, x

Table 3 .

 3 Amplitude v 3 (in MeV) of the 3α potential[START_REF] Wildermuth | A Unified Theory of the Nucleus[END_REF] for the AB and BFW α + α interactions.

	J π AB	BFW
	0 + -22.0 23.3
	2 + -11.9 25.2

Table 5 .

 5 Energies E R and widths Γ (in MeV) with the BFW potential.

		J = 0 +	J = 2 +
	v 3 = 0	v 3 = 23.3 MeV	v 3 = 0	v 3 = 25.1 MeV
	-20.62	-7.18	-17.17	-3.01
	-1.26	1.31, 0.02	1.84, 0.0017	3.6, 0.8
	1.45, 0.07	2.3, 1.1	3 .5, 0.2
	2.7, 1.3	4 .8, 1.5
	4.5, 1.5

Acknowledgments

This text presents research results of the IAP programme P6/23 initiated by the Belgianstate Federal Services for Scientific, Technical and Cultural Affairs.


O Moreno

E Moya De Guerra

P Sarriguren

J M Udías

Parity violating elastic electron scattering and nuclear structure

Keywords: numbers: 25.30.Bf, 21.10.Gv, 21.60.Jz

published or not. The documents may come

Introduction

Knowledge of the neutron density distributions in atomic nuclei is of fundamental importance to understand basic aspects of the nuclear structure, but contrary to what happens with proton distributions, the experimental information available on neutrons is still clearly insufficient. The neutron distributions, and in particular the neutron radii of heavy nuclei are essential nuclear-structure observables that still remain elusive. Improving our knowledge of the neutron density distributions in nuclei is one of the strongest challenges in present nuclear structure activities.

An accurate measurement of the neutron densities in heavy nuclei would have significant implications for nuclear structure. First of all, one should bear in mind that because of the uncertainties in the neutron density, modern effective nuclear forces are typically constructed without any constraint on them. However, a correct parametrization of the isovector channel of the effective nuclear force is essential for the description of phenomena in exotic neutron-rich nuclei such as halos [1] or neutron skins [2]. Once these interactions are constrained to reproduce a neutron radius in a stable nucleus such as 208 Pb, they can make improved predictions for a variety of unstable nuclei. The isospin dependence of the energy functional in nuclear matter would be also constrained by this information. In particular, it would lead to an improved neutron equation of state with important consequences in astrophysics [3] (including the structure of neutron stars). Precise neutron density distributions in nuclei are also required to make progress in atomic parity non-conservation (PNC) experiments [4]. As the accuracy of these experiments improves, they will need more and more precise information on neutron densities because the parity violating interaction is basically proportional to the overlap between the electrons in the atomic orbits and the neutrons [5].

Electron-nucleus scattering has been in the past an excellent tool for studying the nuclear structure. Much reliable information on electromagnetic form factors and charge density distributions has been accumulated for stable nuclei [6] and it is expected that the new facilities in GSI [START_REF]Technical Proposal for the Design, Construction, Commissioning, and Operation of the ELISe Setup[END_REF] and RIKEN [START_REF] Suda | [END_REF] will provide a good opportunity to extend the study of the charge density to unstable nuclei as well. Unfortunately, a measurement of the neutron density distribution to a precision and detail comparable to that of the proton one is hardly possible. Progress on our knowledge of the neutron densities has been limited by the use of hadronic probes that are subject to large and controversial uncertainties because of the not well known reaction mechanism. Neutron density measurements are usually performed using probes having different sensitivities to protons and neutrons. The methods used include hadron scattering, antiprotonic atoms, as well as excitations of the giant-dipole resonance (GDR) and the spin-dipole resonance (SDR). The latter have been especially used to determine neutron skin radii.

Neutron radii were extracted firstly from Coulomb energy differences [9] and from neutron pickup reactions (p, d) and (d, t) [10], but since these reactions are mainly sensitive to the tail of the neutron density, model assumptions were needed for the interior density. Hadron scattering and in particular proton-nucleus elastic scattering at intermediate energies [11], is a well-established method for probing nuclear matter density distributions of stable nuclei. Although the reaction is surface-dominated, it is sensitive to both the surface and the interior neutron density. This reaction produces large cross sections, but suffers from the complexity of the strong interaction. Therefore, it is difficult to describe the reaction mechanism and to perform a model-independent analysis of the data. In general, present hadron scattering interpretation involves a strong model dependence of the results. Difference between neutron and proton density distributions at large radial distances, on the nuclear surface, has been determined with antiprotonic atoms [12], where a slow antiproton is captured into an atom. These differences have also been determined by exciting the GDR [13]. This has been done in inelastic alpha scattering to the GDR, where the cross section of the process depends strongly on the ratio (R n -R p )/R. Unfortunately, the cross section of the GDR excitation is very small relative to those of other overlapping resonances. Similarly, the cross sections of the isovector SDR has been used to extract the neutron skin thickness [14]. This resonance is excited in (p, n), or ( 3 He, t) charge exchange reactions using inverse kinematics.

To summarize, there have been many measurements of neutron densities with strongly interacting probes. Unfortunately, the measured neutron densities are still model-dependent and the error bars are much larger than those of proton densities due to the uncertainties inherent to the strong interaction. As a result, no existing measurement of neutron densities or radii has an established accuracy of 1%. As an example, in 208 Pb, while electron scattering experiments have determined the charge radius to better than 0.001 fm, realistic estimates place the uncertainty in the neutron radius at about 0.2 fm.

This situation has led to a further consideration of leptonic probes to investigate neutron distributions. Elastic magnetic electron scattering from odd-N nuclei is sensitive to the neutron magnetic moment and information about the odd neutron density can be extracted. However, information about the whole neutron density cannot be directly measured since most of the neutrons in a heavy nucleus are coupled to spin zero and make no contribution to the magnetization. An exciting new possibility could be the direct measurements of the neutron density form factors from the asymmetry in parityviolating (PV) elastic polarized electron scattering [15]. The electroweak experiments can be both accurate and model independent and the data can be interpreted with as much confidence as electromagnetic (EM) scattering. PV electron-nucleus scattering arises from the interference of EM and weak neutral amplitudes and it is a clean and powerful tool for measuring the spatial distribution of neutrons in nuclei with unprecedented accuracy.

Parity violating electron scattering

Introduction

The first motivation for studying PV electron scattering from nuclei was to use it as a tool to extract information on the weak neutral current (WNC) and thereby to test the validity of the Standard Model (SM) in the low-energy regime [16,17]. This possibility lies in the fact that the PV asymmetry acquires a very simple, model-independent expression in terms of basic coupling constants because nuclear structure effects cancel out when certain conditions are met. Later on, the emphasis shifted to test nucleon and nuclear structure with such probes. The first experiments in this line were aimed to probe strange quark contributions to the form factors of the nucleon. The SAMPLE experiment [18] measured the strange magnetic form factor at low momentum transfer. The HAPPEX experiment made the most precise measurement yet of strange quarks in the proton [19]. The results from these experiments suggest that strange quarks do not make large contributions to the nucleon electric or magnetic form factors. The contributions from strange quarks are less than 1 percent of the proton charge distribution and no more than 4 percent of its magnetic moment. They are actually compatible with zero [20].

The present interest on PV electron scattering is focused on one hand on the study of the isospin mixing in nuclei. This is so because understanding the sensitivity of PV electron scattering to the nuclear isospin mixing is crucial to determine the precision up to which the SM constants can be deduced and to what extent strangeness effects in the WNC can be studied. On the other hand, the focus is made on the study of neutron densities. This is because the Z 0 couples predominantly to neutrons. The coupling of the Z 0 with the proton depends on the small factor (1 -4 sin 2 θ W ), while the coupling with the neutron is more than ten times larger. This situation is opposite to the EM charge coupling where electrons couple to protons and neutrons with strengths 1 and 0 respectively. This property of the weak interaction can be exploited to provide information about the spatial distribution of neutrons in the nuclear ground state. Indeed, this idea was proposed as part of the original study made in Ref. [15], namely that a measurement of the PV-asymmetry in elastic electron-scattering can provide a direct measurement of the Fourier Transform of the neutron density. In fact, this measurement is now being carried out in the Parity Radius Experiment (PREX) [21] at Jefferson Laboratory which has the goal of measuring the neutron radius of 208 Pb to a 1% precision, using PV elastic electron scattering. This is an electroweak alternative to the hadronic program and promises to measure the neutron radius of 208 Pb accurately and model independently. Another advantage of the electroweak program is related to the possibility of calibrating proton-nucleus scattering to reproduce the neutron density in a stable nucleus. This is analogous to use beta decay to calibrate the charge-exchange reactions as a tool to extract the Gamow-Teller strength. Then, proton scattering could be used in a wide variety of other nuclei, including radioactive beams with hydrogen targets in inverse kinematics. In fact, the nuclear matter distribution in 6 He and 8 He has been already determined at GSI by using small angle proton scattering in inverse kinematics [22].

It has also been proposed [23] that PV electron scattering can be used to determine the type (skin or halo) of neutron distribution in neutron-rich stable nuclei. In particular, asymmetries for skin-type neutron distributions are larger than those of halo-type neutron distributions.

Formalism

Polarized electron scattering from unpolarized nuclei can be used to study parity violation, since both electromagnetic and weak interactions contribute to the process via γ and Z 0 exchange, respectively. The PV asymmetry is given by [15]

A = dσ + -dσ - dσ + + dσ -, (1)
where dσ + (dσ -) is the cross section for electrons longitudinally polarized parallel (antiparallel) to their momentum and the asymmetry is proportional to the interference between the γ and Z 0 amplitude. For a J + = 0 + target in Plane Wave Born Approximation (PWBA) the asymmetry A can be written as

A = a F W (q) F ch (q) , (2)
with a a linear function of Q 2 proportional to the ratio between neutron and proton numbers:

a = G F Q 2 4πα √ 2 N Z , (3)
where G F and α are the Fermi and fine-structure coupling constants, respectively, and

Q 2 = -q 2 µ = 4ǫ 2 sin 2 (θ e /2
) is the four-momentum transfer in the scattering process. F W (q) and F ch (q) are the elastic weak and EM charge form factors, respectively, containing the dependence on the nuclear structure. These form factors are monopole Coulomb-type and are defined as follows,

F ch (q) = G Ep (q)F 0 p + N Z G En (q)F 0 n , (4)
and

F W (q) = G Ep (q) F 0 n - Z N (1 -4 sin 2 θ W )F 0 p (5) -G En (q) F 0 n (1 -4 sin 2 θ W ) - Z N F 0 p ,
where G Ep and G En are the charge form factors of the proton and the neutron (G Ep (q = 0) = 1 and G En (q = 0) = 0) and θ W is the Weinberg angle. The monopole form factors for point nucleons are

F 0 p (q) = 1 Z d 3 rj 0 (qr)ρ p (r), F 0 n (q) = 1 N d 3 rj 0 (qr)ρ n (r) . (6)
In these expressions the neutron and proton densities are normalized to the numbers of their corresponding type of nucleons and F 0 p,n (q = 0) = 1. Therefore, we see that the asymmetry, aside from a , depends mainly on the ratio of neutron to proton nuclear form factor (F 0 n (q)/F 0 p (q)):

A = a   F 0 n (q) F 0 p (q) -Z N (1 -4 sin 2 θ W ) - G En (q) G Ep (q) F 0 n (q) F 0 p (q) (1 -4 sin 2 θ W ) -Z N 1 + N Z G En (q) G Ep (q) F 0 n (q) F 0 p (q)   .(7)
At the q-values that are of our concern here, the charge form factor of the neutron (G En ) can be neglected and we can write

A = a F 0 n (q) F 0 p (q) - Z N (1 -4 sin 2 θ W ) . (8)
In a first approximation the radial distribution of mass can be assumed to follow that of charge so that the radial dependence of the neutron density is that of the proton density and they are both scaled by the N/Z factor (ρ n (r) = ρ p (r) N/Z). In this case F 0 n = F 0 p for all q and one gets:

A 0 = a 1 -Z N (1 -4 sin 2 θ W ) , (9)
or

A 0 ≈ a for Z/N < 1. (10)
For N = Z nuclei Eq. ( 9) leads to the usual expression:

A 0 N =Z = a 4 sin 2 θ W , (11)
as obtained in the exact SU(2) isospin limit in Refs. [15,24]. In nuclei, isospin is not an exact symmetry, but still at q = 0, F 0 n = F 0 p . For q > 0 the assumption F 0 n = F 0 p amounts to assume that the nuclear ground state is a good zero-isospin eigenstate. Then only isoscalar matrix elements contribute and the WNC and EM form factors become proportional. Since isospin is not an exact symmetry in nuclei the actual PV asymmetry deviates from the linear Q 2 dependence by a correction Γ, where

A = A 0 [1 + Γ(q)] . (12)
The deviation Γ accounts not only for the effects of nuclear isospin mixing but also for the strangeness content in the PV asymmetry, Γ = Γ I + Γ s [24]. The last measurements [20] give a strangeness density ρ s consistent with zero at the q values of interest here, and therefore we will discuss only Γ = Γ I in the next section. Since F 0 p is well known from standard (parity conserving) electron scattering, a precise information on F 0 n (hence on ρ n ) can be obtained by accurate measurements of A at different q values. In the region 0 < q 5R -1 , that requires q values of the order of A -1/3 GeV at most, the elastic cross section is sufficiently large to measure A accurately [24].

We note that in general a direct measurement of the q dependence of F 0 n /F 0 p is obtained from the difference between the exact asymmetry A and the approximated one A 0 as:

F 0 n (q) F 0 p (q) = 1 + [A -A 0 ] a . (13)
This equation provides the means to obtain the experimental q-dependent ratio (F 0 n (q)/F 0 p (q)) exp from the experimental asymmetry values, A = A exp , substituting A 0 and a by the known linear functions of Q 2 in Eq. ( 3) and Eq. ( 9).

As already mentioned there are several corrections to the asymmetry expressed by Eq. ( 8) that could interfere the neutron radius measurement. The corrections have been studied in Ref. [25], concluding that they should not be an issue for the interpretation of the experiment because they are either negligible or well under control. These corrections include effects from strange quarks, neutron electric form factors, parity admixtures in the ground state, dispersion corrections, meson exchange currents, radiative corrections, contributions from excited states, and target impurities. By far the largest correction to the asymmetry in heavy nuclei comes from Coulomb distortions of the electron wave function. Distortion effects can be included exactly [24,25,26] by solving numerically the Dirac equation for an electron moving in the Coulomb potential (vector) plus the small weak neutral potential (axial-vector). The axial-vector potential is given by

W W (r) = α sin 2 (2θ W ) 2π m Z ∞ 0 ρ W (r ′ ) r ′ r e -m Z |r-r ′ | -e -m Z (r+r ′ ) dr ′ (14)
where m Z is the mass of the Z 0 boson and ρ W (r) is the weak density (inverse Fourier transform of the weak form factor of Eq. ( 5)). For massless electrons, one has positive and negative helicity states that scatter from a V + A and V -A potentials. Thus, one obtains the cross sections dσ + (dσ -) entering in Eq. ( 1) from the phase shifts which result from the numerical solution of the Dirac equation, involving a partial wave expansion. For heavy nuclei, distortion corrections can be large but they can be calculated with an accuracy significantly better than the 3% experimental error expected for the asymmetry. When Coulomb distortion effects are taken into account we can still define a distorted wave (DW) neutron to proton form factor ratio,

F 0 n (q) F 0 p (q) DW = 1 + A DW -A 0 a ( 15
)
which is the actual ratio to be compared to experiment.

In what follows we show results of the asymmetries calculated from densities obtained within selfconsistent mean field approximation, with and without Coulomb distortion effects.

Selfconsistent mean field approach for quasiparticles

The nuclear structure calculation involved in the asymmetry, which basically concerns the proton and neutron form factors, is performed within a selfconsistent deformed mean field approach with pairing correlations. To generate the ground-state wave function we use a Skyrme density-dependent nucleon-nucleon interaction (SLy4 [27]). The Hartree-Fock (HF) equations are solved iteratively and at the end of each iteration we solve the BCS equations to generate the occupation probabilities in a consistent way. We obtain the single-particle levels, occupation numbers, and wave functions. The latter are expanded in a harmonic oscillator basis using 12 major shells. More details can be found in Refs. [24,28,29].

In this approach the collective isospin mixing effect of the Coulomb force is included non-perturbatively in the isospin-non-conserving HF mean field, along with other collective effects such as pairing and deformation. As a result, the HF+BCS ground state is made up of quasiparticles with rather complex admixtures of harmonic oscillator wave functions in many different major shells. So far, this approximation is known to provide optimal descriptions of charge form factors [30,31] of heavy nuclei and can be expected to be reliable for the theoretical description of the asymmetry.

Calculation of asymmetries

As a paradigmatic example of N > Z nuclei we show results for 208 Pb. In Fig. 1a we show the proton ρ p and neutron ρ n ground state densities of 208 Pb from a HF(SLy4) calculation. Also shown are the isoscalar (IS) (ρ IS = ρ p + ρ n ) and isovector (IV) (ρ IV = ρ n -ρ p ) densities.

In Fig. 1b we show the corresponding proton F 0 p and neutron F 0 n form factors in PWBA (see Eqs. ( 6)) together with R np (q) defined as:

R np (q) = F 0 n (q) F 0 p (q) -1 . (16)
In Fig. 2a we show the PV asymmetry in 208 Pb obtained from both PWBA (dashed line) and DWBA (solid line). The DWBA calculation is performed for an incident electron energy of 850 MeV. It is worth noticing how the PWBA singularities become smooth functions of q in the distorted case. In this kinematics, the expected asymmetry amounts to 8.4•10 -7 at q = 0.45 fm -1 , which is the momentum transfer chosen at PREX [21]. A word of caution should be made with regards to the effective transverse momentum that should be used when comparing DWBA or experimental data to PWBA calculations (see for instance Ref. [32] and references therein). A displacement in q of the DWBA peaks relative to the PWBA peaks is clearly seen in Fig. 2. The Coulomb distortion is also responsible for the anomalous behaviour of R np in DWBA as q → 0. Actually, the isospin mixing effect is best explored when comparing the DWBA asymmetry to that obtained also in DWBA but taking ρ n = ρ p N/Z (see figures 2 and 3 in Ref. [24]).

As an example of an N = Z nucleus we show results for 28 Si. For this nucleus the calculations are done with the same Skyrme interaction using constant pairing gaps ∆ p,n =1 MeV. Figs. 3 and4 contain similar results as Figs. 1 and2, but for 28 Si. The effects expected on the elastic parity-violating electron scattering at momentum transfers extending up to about 1.5 fm -1 from the isospin-mixing in the nuclear groundstate wave functions of a set of N = Z nuclei, including 28 Si, have been studied in Ref. [24]. The small differences between the proton and neutron density distributions yield both isoscalar and isovector ground-state Coulomb monopole matrix elements and produce modifications in the PV asymmetry from the model-independent result obtained in the absence of isospin-mixing.

Kinematic ranges where potential future measurements might be undertaken have been discussed in Ref. [24] by studying both the deviations in the PV asymmetry Γ and the experimental figure-of-merit. Isospin mixing will have a measurable effect on the asymmetries in the range 0.5 ≤ q ≤ 1.0 fm -1 .

Isospin mixing in N = Z nuclei

As seen in the previous section the PV asymmetry in N = Z nuclei is affected by the isospin mixing in the ground state. Isospin impurities are discussed in the literature in different schemes. In previous works [33] we evaluated isospin mixing by computing the mean value of the T 2 operator in the ground state. In Refs. [34,35] isospin mixing is evaluated in terms of the overlaps between spherical neutron and proton single particle wave functions with the same quantum numbers. The admixture of T = T 0 + 1 in the ground state with isospin T 0 is given in terms of those overlaps [34,35],

P (T = T 0 + 1) ≈ 1 T 0 + 1 nlj ′ N nlj C nlj , (17)
where N nlj is the number of protons in the orbit (nlj) and C nlj are the deviations from unity of the overlaps,

C nlj = 1 -R p nlj (r)R n nlj (r)r 2 dr . (18)
In a more general framework where deformation and pairing are included, as it is our case, a better formulation of the isospin mixing in the ground state wave function of N = Z nuclei is given as

P T = 1 -Ψ p |Ψ n , (19)
where Ψ p,n are the proton and neutron wave functions in the selfconsistent HF+BCS ground state. This expression takes into account all the possible overlaps between proton and neutron single particle wave functions, as can be seen in the explicit expressions for the BCS overlaps [36].

In Table 1 we show the isospin mixing probabilities (percentage) for various N = Z nuclei obtained in this work using Eq. 19. Our HF(SLy4)+BCS calculations for 12 C and 28 Si with ∆ p,n = 1 MeV give spherical ( 12 C) and oblate ( 28 Si, β = -0.25) solutions. For the doubly closed nuclei 40 Ca and 100 Sn the spherical ground states have zero pairing gaps. Also given in the table are the results obtained by Hamamoto et al. [34] (HF(Sk3) and HF(SG2)), by Satula et al. [37] before (BR) and after (AR) rediagonalization, as well as the values corresponding to the formulation in Bohr and Mottelson [START_REF] Bohr | Nuclear Structure[END_REF], p. 173. In these works all the nuclei are spherical and no pairing correlations are included. Our HF+BCS estimates are in accordance with other HF calculations [34,35] and are larger than the estimates given by the BM isospin mixing, obtained from a collective model description of the isovector giant monopole resonance.

Conclusions

The accurate knowledge of the neutron density in the nucleus is a challenge to the present understanding of the nuclear structure and its determination in a model independent way is still an open problem. In this work we have shown that parity violating elastic electron scattering is an attractive alternative to the use of hadronic probes as an instrument to get information on neutron density distributions. The reason for that is that, contrary to the strong force, the electroweak interaction is perfectly known and therefore information on the nuclear structure can be extracted in a clean way. On the other hand, while the standard parity conserving electron scattering is sensitive to the nuclear charge distribution, parity violating electron scattering is mainly sensitive to the neutron distributions. Thus, these experiments will allow to measure a nuclear weak-charge density distribution and finally to determine the neutron distribution with a highly improved accuracy as compared to hadronic probes, which could be calibrated afterwards to reproduce more precisely that information. Accurate measurements of the PV asymmetries in N = Z nuclei will also highlight the issue of isospin mixing that still remains an open problem too.

By a direct comparison to theory, these measurements will represent a critical test to the nuclear models and will have a deep impact on nuclear structure, as well as on atomic physics and astrophysics.

Figure 1 .

 1 Figure 1. Left panel: proton, neutron, IS and IV densities of 208 Pb in its ground state. Right panel: proton and neutron form factors together with R np , in PWBA.

Figure 2 .

 2 Figure 2. Left panel: Plane wave (PW) and distorted wave (DW) PV asymmetries of 208 Pb in its ground state. Right panel: for the same nucleus, the quantity R np in plane wave (PW) and distorted wave (DW).

Figure 3 .

 3 Figure 3. Same as in Fig. 1, but for the N = Z nucleus 28 Si.

Figure 4 .

 4 Figure 4. Same as in Fig.2, but for the N = Z nucleus28 Si.

Table 1 .

 1 Isospin mixing probabilities (percentage) for various N = Z nuclei obtained in this and in other works (see text).

		12 C	28 Si	40 Ca 100 Sn
	This work SLy4	0.041	0.28	0.58	3.9
	Hamamoto et al. Sk3 ∼0.05 Hamamoto et al. SG2 ∼0.05 Satula et al. SLy4 BR -Satula et al. SLy4 AR -Bohr and Mottelson 0.007 0.063 0.165 1.9 -∼0.6 ∼3.9 -∼0.6 ∼4.8 ∼0.35 ∼0.6 ∼3.9 ∼0.45 ∼0.8 ∼6.2

Acknowledgements

This work was supported by Ministerio de Ciencia e Innovación (Spain) under Contract FIS2008-01301. J.M.U. acknowledges support from INTAS Open Call grant No 05-1000008-8272, and Ministerio de Ciencia e Innovación (Spain) under grants FPA-2007-62616 and FPA-2006-07393, and UCM and Comunidad de Madrid under grant Grupo de Física Nuclear (910059).


A Covello
email: covello@na.infn.it

A Gargano

Realistic shell-model calculations: current status and open problems

The main steps involved in realistic shell-model calculations employing two-body low-momentum interactions are briefly reviewed. The practical value of this approach is exemplified by the results of recent calculations and some remaining open questions and directions for future research are discussed.

Introduction

A key challenge for nuclear physics is to develop a comprehensive understanding of the observed properties of nuclei starting from the forces among nucleons. Nowadays there are two main ways to try to achieve this goal, which are substantially complementary to each other. The first one is based on the so-called ab initio calculations in which nuclear properties, such as binding and excitation energies, are calculated directly from first principles of quantum mechanics using an appropriate computational scheme. To this category belong the Green's function Montecarlo Method (GFMC) [START_REF] Pieper | Proc. Int. School of Physics "E. Fermi[END_REF], the nocore shell-model (NCSM) [START_REF] Navrátil | [END_REF], and the coupled cluster method [3]. Clearly, all ab initio calculations need huge amount of computational resources and are therefore currently limited to light nuclei.

The second line of attack consists of using the shell model with realistic two-body effective interactions, namely derived microscopically from the free nucleon-nucleon (NN) potential. In this kind of calculations only the valence nucleons are treated as active particles while the core polarization effects are taken into account perturbatively in the derivation of the effective interaction V eff . Of course, this approach allows one to perform calculations for medium-and heavy-mass nuclei which are far beyond the reach of ab initio calculations.

In this context, it should be mentioned that in recent years successful shell-model calculations, in particular for sdand pf-nuclei, have been performed [4,5,6] by employing modified versions of realistic effective interactions. Let us make it clear that this paper is concerned only with calculations employing two-body effective interactions derived from the free NN potential without any phenomenological adjustments, which we refer to as realistic shell-model calculations tout court.

The two main ingredients of realistic shell-model calculations are the NN potential V N N and the many-body methods for deriving V eff . As regards the latter, a main difficulty with its derivation is the strong short-range repulsion contained in modern NN potentials. As is well known, the traditional way to circumvent this problem is the Brueckner G-matrix method. However, a few years ago a new approach to the renormalization of V N N has been proposed [7,8], which has proved to be [9] an advantageous alternative to the use of the G-matrix . This consists in constructing a smooth low-momentum potential, V low-k , that can be used directly to derive V eff within the framework of a perturbative approach.

Since its introduction in 2002, we have used this approach to derive realistic lowmomentum effective interactions for shell-model calculations in various mass regions, which have provided an accurate description of nuclear structure properties. It should be noted that, having taken the single-nucleon energies from experiment, these calculations are essentially parameter-free.

While the realistic shell-model calculations performed thus far have all been based on a Hamiltonian containing only two-body interactions, the role of three-nucleon interactions has been deeply investigated within the framework of ab initio approaches, such as the GFMC and the NCSM. It is certainly a major open problem to investigate the contribution of three-body forces to the shell-model effective interaction.

In this paper, we shall try to point out some questions that in our opinion remain open in shell-model calculations with realistic effective interactions. This is done in the context of a brief description of the current status of the field, which serves the purpose to provide the appropriate frame for our discussion.

Theory and applications

The Schrödinger equation for a nucleus with A nucleons is written as

HΨ i = (H 0 + H 1 )Ψ i = E i Ψ i , (1)
where

H 0 = T + U (2)
and

H 1 = V N N -U, (3)
T being the kinetic energy and U an auxiliary potential introduced to define a convenient single-particle basis. The effective interaction V eff acting only within a reduced model space is then defined through the eigenvalue problem

P H eff P |Ψ α = P (H 0 + V eff )P |Ψ α = E α P Ψ α , ( 4
)
where the E α and the corresponding Ψ α are a subset of the eigenvalues and eigenfunctions of the original Hamiltonian defined in the complete Hilbert space. The P operator projects onto the chosen model space, which is defined in terms of the eigenvectors of the unperturbed Hamiltonian H 0 .

A well-established approach to the derivation of realistic effective interactions is provided by the Q-box folded-diagram expansion [9]. It is, however, outside the scope of this article to give any detailed discussion of this approach. Rather, our aim is to discuss synthetically its merits and limits. To this end, this section is divided into two parts. In the first one we examine in a critical way the main steps involved in the derivation of V eff . In the second one we give a brief survey of some applications to show the accuracy of realistic shell-model calculations in the description of nuclear structure properties.

Outline of theoretical framework

2.1.1. Choice of V N N Starting in the mid 1990s, several potential models have been constructed which predict almost identical deuteron observables and are phase-shift equivalent, namely they fit equally well (χ 2 /datum ≈ 1) the NN scattering data up to the inelastic threshold. These are the potentials constructed by the Nijmegen group, Nijm I and Nijm II [10], the Argonne V 18 potential [11], and the CD-Bonn potential [12].

More recently, new NN potentials have been derived within the framework of the chiral perturbation theory following the basic idea of Weinberg [13]. Efforts in this direction have resulted in the construction of next-to-next-to-next-to-leading order (N 3 LO, fourth order) potentials [14,15]. An interesting feature of this approach is that three-nucleon forces arise naturally at third order (N 2 LO).

We will not go further into this subject and refer the reader to [16] for a survey of the current status of the chiral potentials. However, a general comment may be in order here. Currently there are no real efforts to go beyond the one-boson-exchange (OBE) model in the derivation of V N N within the traditional meson theory, the main interest in the field being focused on chiral potentials. As regards the latter, there remain some open problems, the most important ones being the three-nucleon forces beyond N 2 LO [17] and the consistency of the Weinberg power counting [18].

Of course, the existence of several different phase-shift equivalent NN potentials raises the question of how much nuclear structure results may depend on the choice of the potential one starts with. However, the renormalization of the free NN potential, which is the subject of section 2.1.2, removes to a large extent the ambiguity in the choice of V N N . We have indeed verified [9] that low-momentum shell-model effective interactions derived from phase-shift equivalent NN potentials through the V low-k approach do not lead to significantly different results.

2.1.2.

Renormalization of V N N In recent years, a renormalization-group-based approach has been introduced [7,8] allowing one to handle the nonperturbative behaviour that characterizes all realistic NN potential models. In this approach, the high-momentum modes of V N N are integrated out down to a certain cutoff momentum Λ, which leads to a low-momentum potential that decouples high-and low-energy degrees of freedom while preserving the on-shell properties of the original V N N . By varying the cutoff, an entire class of low-momentum potentials can be associated with the initial Hamiltonian, which are all by construction softer than V N N . This greatly simplifies, or even makes feasible in some cases, nuclear structure calculations. It is a remarkable feature of this approach that with a cutoff in the range Λ ∼ 2.1 fm -1 low-momentum interactions derived from different NN potential become very similar to each other. This has recently led to a widespread use of these low-momentum potentials in various contexts, going from few-body systems to nuclear matter.

Different techniques have been developed to derive low-momentum potentials, the decimation of the initial potential being performed on the two-body systems in all cases. In our calculations, we have adopted a conventional effective interaction technique based on the Lee-Suzuki similarity transformation [19]. The cutoff Λ specifies the lowmomentum space P within which the V low-k is confined and its complement Q = 1 -P. Then, using a similarity transformation X a new Hamiltonian H = X -1 HX is defined which satisfies the decoupling equation QHP = 0.

(

The low-momentum potential is then given by

V low-k = PHP -PT P. (6)
The solution procedure for Eq. ( 5) can be found in [9]. Here we only mention that we adopt the iterative technique for non-degenerate model spaces proposed in [20] and employ a momentum-space discretization procedure with an adequate number of Gaussian mesh points.

As mentioned above, the V low-k is developed for the two-body system, which means that when used in calculations for A > 2 the low-energy observables are not the same as those produced by the original NN potential and depend, to a certain extent, on the value of the cutoff [21]. This dependence may be removed by complementing, for each value of Λ, the two-body V low-k potential with three-and higher-body components, which are in fact generated by the renormalization procedure when applied to an Abody system, even if the initial Hamiltonian contains only a two-body potential [22]. However, the three-body forces included in V low-k calculations to date are generally approximated by fitting the parameters entering the leading chiral three-nucleon forces. A consistent evolution of the three-body forces in the three-nucleon system has been only recently achieved in the work of [23]. There, calculations are performed for A ≤ 4 nuclei with an initial Hamiltonian including as two-and three-body forces the N 3 LO [14] and N 2 LO [24] interactions, respectively.

Note that the inclusion of forces beyond the two-body ones has been essentially limited to few-nucleon systems. As regards medium-and heavy-mass nuclei, only the NN potential is used at present in the derivation of the shell-model effective interaction. We shall come back to this point in section 3.

It is worth mentioning that in almost all our calculations the cutoff parameter Λ has been chosen in the vicinity of 2 fm -1 . This means that V low-k preserves the phase shifts up to E Lab ≈ 350 MeV, which is the inelastic threshold. We have found [25], however, that the shell-model results do not change significantly for moderate variations of the cutoff around this value. We have also verified [25] that allowing for limited changes (∼ 0.3 fm -1 ) in the value of Λ the V low-k 's extracted from different phase-shift equivalent potentials give essentially the same results. Further study is certainly needed to better clarify the effect of cutoff variations on shell-model effective interactions.

In concluding this section, it is worth insisting on the fact that the use of V low-k rather than the G matrix represents a substantial progress in the derivation of shellmodel effective interactions. The main advantages of V low-k may be summarized as follows: i) since it is a smooth NN potential, V low-k can be used directly in nuclear structure calculations within a perturbative approach; ii) it does not depend either on the starting energy or on the model space, as instead the case of the G matrix, which is defined in the nuclear medium; iii) the V low-k 's extracted from various phase-shift equivalent potentials are very similar to each other, thus suggesting the realization of a nearly unique low-momentum potential.

From the above, it is clear that V low-k is also a most valuable tool to tackle the tough problem of taking into account three-body forces in realistic shell-model calculations.

2.1.3. Derivation of the effective interaction As mentioned above, the calculation of the effective interaction within a chosen model space P is performed by using the Q-box folded-diagram expansion developed for two valence particles.

The effective interaction of Eq. ( 4) can be written as

V eff = Q + ∞ i=1 F i , ( 7
)
where the Q-box is defined as the sum of all irreducible and valence linked diagrams with V low-k replacing V N N in the H 1 vertices, and the F i 's represent Q-box i-folded diagrams. The latter can be expressed in terms of Q-box derivatives with respect to the energy variable ω, namely

F 1 = Q1 Q, F 2 = Q2 Q Q + Q1 Q1 Q, F 3 = Q3 Q Q Q + Q2 Q1 Q Q + Q2 Q Q1 Q + Q1 Q2 Q Q + Q1 Q1 Q1 Q, (8)
... ,

with Qm = 1 m! d m Q(ω) dω m ω=ω 0 , ( 9
)
ω 0 being the unperturbed energy of two valence particles in a degenerate model space, P H 0 P = ω 0 . As a first step, one has to calculate the Q-box. This calculation requires some approximations, since only diagrams up to a finite order in H 1 can be included and a truncation of the intermediate-state summation has to be made for each Q-box diagram. Then, for a given Q-box the folded-diagram series (7) can be summed up by the Lee-Suzuki iteration method [19].

One of the most important questions arising in this kind of approach is the convergence of the effective interaction expansion, to which much attention has been paid over the years. Before folded diagrams were taken into account, both the order-byorder [26] and the intermediate state [27] convergence were questioned. Furthermore, it was shown [28] that the presence of an intruder state in the low-lying spectrum of the nucleus with two valence nucleons is an essential source of divergence. The problem of the intermediate state convergence was substantially solved by using NN potentials with a weak tensor force, while the introduction of folded diagrams and their summation through the Lee-Suzuki scheme was proved to be an important step to handle the nonconvergence of the order-by-order expansion of V eff , also when intruder states are present in the P -space [19,29]. As a matter of fact, if we know the Q-box and its derivatives, V eff is obtained after a small number of iterations. The convergence of the Q-box, however, still remains to some extent an open problem, although it has been shown that the differences in V eff 's calculated with Q-boxes to various orders, are attenuated when the folded diagrams are summed to all orders [30]. In particular, in [30] a comparative study between results obtained using a second-and third-order Q-box in the folded-diagram expansion has been performed. There it was found that both calculations give very similar results for nuclei with two valence nucleons, which, as it was also suggested in [29], should be traced to the cancellation of the higher-order Q-box diagrams by the folded diagrams. However, the final conclusion of [30] was that third-order calculations should be employed since third-order terms of the Q-box are not negligible. On the other hand, in almost all our calculations we have used effective interactions derived by including diagrams up to second order. It is worth mentioning that our choice is also supported by the results of [31]. There, a V low-k derived from the CD-Bonn NN potential has been employed and the sd-shell effective interaction has been calculated by summing core-polarization diagrams to all orders. The results of this calculation have turned out to be remarkably similar to those of a second-order one.

There are a few other points that we would like to discuss here. The effective interaction constructed within the above approach contains both one-and two-body terms. All the one-body contributions, once summed to the eigenvalues of H 0 , give the single-particle effective energies, which should represent the energies of the onevalence-nucleon system. In most of the realistic shell-model calculations to date, these single-particle energies are replaced by the experimental values, and only the two-body term of V eff is retained by employing the subtraction procedure described in [29].

The effective interaction is constructed for a two-valence-particle nucleus but is generally used also for more complex systems. This means that the three-or higher-body forces, which appear in the effective interaction for the latter, even if the initial potential is only a two-body one, are not taken into account. In [32] the lack of three-body forces, effective or original, was considered responsible for the failure to explain simultaneously binding energies and excitation spectra of several nuclei. In [33], using the two-level Lipkin model, it was shown that effective three-body forces are not negligible. On the basis of our calculations, however, we do not have at the moment a definite conclusion on the role of these forces. In the next section we shall comment on this point.

Finally, it is worth recalling that the perturbation in our starting Hamiltonian is H 1 = V low-k -U , and, as a consequence, the Q-box diagrams contain -U as well as V low-k vertices. However, in almost all calculations performed so far diagrams with the U potential and bubble insertions are not included. This assumption is based on the fact that these diagrams would cancel each other when using for U the Hartree-Fock self-consistent potential. The most common choice, however, is the harmonic oscillator potential, and the plausibility of the above approximation remains to be ascertained.

In some of our recent papers [34,35] all the V low-k and -U diagrams up to third order are taken into account.

From this discussion it seems that some questions inherent in the derivation of the effective interaction still remain to be settled, especially as regards the real advantages of an a priori more complete and consistent calculation of the effective interaction. We feel that further investigations aimed at finding specific answers to these questions are needed. However, these considerations cannot ignore the remarkable success achieved thus far by realistic shell-model calculations in explaining nuclear structure properties. This will be illustrated in section 2.2.

Practical value of realistic shell-model calculations

While the first efforts to derive the shell-model effective interaction from the free NN potential go back to more than forty years ago, the practical value of this approach has emerged only during the last decade or so. We shall here try to give a concise yet quantitative evaluation of the results produced by modern realistic shell-model calculations. These results concern spectroscopic properties of nuclei in various mass regions both near and far from stability, and have been obtained using second-order folded-diagram effective interactions derived from the CD-Bonn potential renormalized by constructing a V low-k with Λ = 2.1 -2.2 fm -1 . In particular, calculations have been performed in the regions around doubly magic 100 Sn, 132 Sn and 208 Pb involving some twenty nuclei altogether ( [9] and references therein, [36,37]). In practically all cases a very good agreement between the calculated and experimental spectra has been obtained. In general, the rms deviation does not exceed 100 keV. Also, the currently available experimental data on electromagnetic transition rates are well reproduced by the theory (see, for instance, [38]).

It is well outside the scope of this article to give even a brief survey of the results of the realistic shell-model calculations referred to above. However, to quantitatively illustrate their practical value, we feel it appropriate to just discuss in some detail the results for the two odd-odd nuclei 136 Sb and 134 I. In both cases the main motivation for our calculations [39,37] was to interpret new data of primary interest for the understanding of the shell structure of nuclei in the vicinity of 132 Sn. Note that 136 Sb with an N/Z ratio of 1.67 is at present the most exotic open-shell nucleus beyond 132 Sn for which information exists on excited states.

In the study [39] two new transitions, in addition to the one previously known, were observed in 136 Sb using γ-ray and conversion-electron spectroscopy, and a level scheme was constructed by comparison with the results of a realistic shell-model calculation performed within the theoretical framework described in the previous sections. As is seen in figure 1, the predicted energies are in very good agreement with measured values. The theory predicted also a B(E2; 6 -→ 4 -) value which comes quite close to the experimental one.

As regards the nucleus 134 I, five high-spin excited states up to an energy of about 3 MeV were very recently identified [40] through measurements of prompt γ rays from the spontaneous fission of 252 Cf. Angular correlations were performed, but were not sufficient to assign spins and parities. We have interpreted the observed states on the basis of a realistic shell-model calculation and made spin-parity assignments accordingly [37]. As is seen in figure 2, a very good agreement was found between the experimental and calculated levels. High-spin states in 134 I identified in [40] compared with the calculated level scheme for negative-parity yrast states.

More on open problems

In the previous sections we have already touched upon some interesting questions regarding realistic shell-model calculations. In this section, we give a more detailed discussion of what we feel are the issues worthy of further study in the near future.

The two examples given in section 2.2 provide good evidence of the predictive power of realistic shell-model calculations. However, as is the case for the majority of our calculations, they concern nuclei neighboring a double shell closure. It is, of course, a very interesting problem to find out what happens with increasing number of valence nucleons. From this viewpoint, of special interest are long isotopic or isotonic chains which provide the opportunity to investigate the effect of adding neutrons or protons to a doubly magic core over a large number of nuclei. To this aim, we have conducted a study of the N = 82 isotonic chain which has long been considered a benchmark for shell-model calculations employing effective interactions derived from free NN potentials [41,42]. The results of this study [9] have shown remarkable agreement with experiment for the low-energy spectra all over the isotonic chain. However, the ground-state energy, which is well reproduced for the two-valence-proton nucleus 134 Te, is increasingly overestimated with increasing number of protons. It should mentioned that a similar behavior was found in [START_REF] Engeland | Proc. 6th Int. Spring Seminar on Nuclear Physics (S. Agata sui due Golfi[END_REF] for the Sn isotopic chain. This study, however, differs from that of [9] in two respects: the short-range repulsion of V N N is taken care of by introducing the Brueckner G matrix and diagrams through third order in G are included in the Q-box. In [START_REF] Engeland | Proc. 6th Int. Spring Seminar on Nuclear Physics (S. Agata sui due Golfi[END_REF] it was shown that the calculated binding energies turned out to be in good agreement with experiment by adding a small repulsive monopole contribution to the effective two-body interaction.

These findings lend support to the suggestion that the overbinding of nuclei with more than two valence particles produced by realistic shell-model calculations may be traced to the lack of many-body forces, in particular the three-body ones. However, we have recently performed calculations for oxygen [START_REF] Coraggio | [END_REF] and calcium [34] isotopes, as well as for N = 82 isotones [35], including diagrams up to third order in V low-k . The results obtained seem to indicate that third-order diagrams may play a significant role in curing the above "overbinding problem".

In summary, we may say that a deeper comprehension of the role of both genuine and effective three-body forces and of contributions to the Q-box beyond second order certainly deserves further study. In this connection, it should be mentioned that an attempt to explain the overbinding in neutron-rich O isotopes in terms of genuine threebody forces is made in [45].

In the calculations for 136 Sb and 134 I mentioned above only particle-particle and particle-hole matrix elements come into play. We have, however, also performed several calculations for nuclei with two or more valence holes [46,47]. While the overall agreement between theory and experiment may be considered satisfactory in most cases, it is not in general of the same quality as that obtained for nuclei in which there is no hole-hole interaction. At present, we don't have a clear understanding of the origin of this dissimetry.

Final remarks

In this paper, we have tried to give an outline of the present status of shell-model calculations employing realistic two-body effective interactions and point out some problems that deserve further study. To place our discussion in a proper perspective, we have briefly surveyed the main steps involved in this kind of calculations, whose practical value has been exemplified by the results of two recent calculations in the neutron-rich 132 Sn region.

As regards open problems, we have focused on those issues that are raised by our own studies. This may have left out other interesting questions. This is certainly the case of the coupling with the scattering continuum which should play an important role in nuclei near the drip lines. We refer to [48] for a very recent study of this problem including references to earlier works. We hope, however, to have succeeded in highlighting the value of realistic shell-model calculations and the need for further efforts to clarify some remaining open questions.

Figure 1 .

 1 Figure 1. Experimental and calculated level scheme of 136 Sb.

Figure 2 .

 2 Figure 2.High-spin states in 134 I identified in[40] compared with the calculated level scheme for negative-parity yrast states.

Acknowlegments

The considerations made in this paper are essentialy the fruit of a long-standing collaboration with Luigi Coraggio, Nunzio Itaco and Tom Kuo. Frequent discussions with Luigi Coraggio and Nunzio Itaco have been of great help.