Ab initio study of the photoabsorption of {}^{4}He

Ab initio study of the photoabsorption of He

W. Horiuchi RIKEN Nishina Center, Wako 351-0198, Japan    Y. Suzuki Department of Physics, Niigata University, Niigata 950-2181, Japan RIKEN Nishina Center, Wako 351-0198, Japan    K. Arai Division of General Education, Nagaoka National College of Technology, Nagaoka 940-8532, Japan

There are some discrepancies in the low energy data on the photoabsorption cross section of He. We calculate the cross section with realistic nuclear forces and explicitly correlated Gaussian functions. Final state interactions and two- and three-body decay channels are taken into account. The cross section is evaluated in two methods: With the complex scaling method the total absorption cross section is obtained up to the rest energy of a pion, and with the microscopic -matrix method both cross sections He()H and He()He are calculated below 40 MeV. Both methods give virtually the same result. The cross section rises sharply from the H+ threshold, reaching a giant resonance peak at 26–27 MeV. Our calculation reproduces almost all the data above 30 MeV. We stress the importance of H+ and He+ cluster configurations on the cross section as well as the effect of the one-pion exchange potential on the photonuclear sum rule.

25.20.Dc, 25.40.Lw, 27.10.+h, 21.60.De

I Introduction

Nuclear strength or response functions for electroweak interactions provide us with important information on the resonant and continuum structure of the nuclear system as well as the detailed property of the underlying interactions. In this paper we focus on the photoabsorption of He. The experimental study of and reactions on He has a long history over the last half century. See Refs. shima (); nilsson (); quaglioni04 () and references therein. Unfortunately the experimental data presented so far are in serious disagreement, and thus a measurement of the photoabsorption cross section is still actively performed with different techniques in order to resolve this enigma nakayama (); tornow ().

Calculations of the cross section on He have been performed in several methods focusing on e.g., the peak position of the giant electric dipole () resonance, charge symmetry breaking effects, and sum rules efros (); wachter (); gazitb (). The photoabsorption cross section has extensively been calculated in the Lorentz integral transform (LIT) method LIT (), among others, which does not require calculating continuum wave functions. In the LIT the cross section is obtained by inverting the integral transform of the strength function, which is calculable using square-integrable () functions. The calculations were done with Malfliet-Tjon central force quaglioni04 (), the realistic Argonne 18 potential gazit (); bacca (), and an interaction based on chiral effective field theory quaglioni ().

In the calculations with the realistic interactions some singular nature of them, especially the short-range repulsion, has been appropriately replaced with the effective one that adapts to the model space of the respective approaches, that is, the hyperspherical harmonics method gazit (); bacca () and the no-core shell model quaglioni (). All of these calculations show the cross section that disagrees with the data shima () especially in the low excitation energy near the H+ threshold. The resonance peak obtained theoretically appears at about 27 MeV consistently with the experiments nilsson (); nakayama (); tornow (), but in a marked difference from that of Ref. shima ().

We have recently reported that all the observed levels of He below 26 MeV are well reproduced in a four-body calculation using bare realistic nuclear interactions dgvr (); inversion (). It is found that using the realistic interaction is vital to reproduce the He spectrum as well as the well-developed + (H+ and He+) cluster states with positive and negative parities. In this calculation the wave functions of the states are approximated in a combination of explicitly correlated Gaussians boys (); singer () reinforced with a global vector representation for the angular motion varga (); svm (). Furthermore this approach has very recently been applied to successfully describe four-nucleon scattering and reactions arai11 (); fbaoyama () with the aid of a microscopic -matrix method (MRM) desc (). It is found that the tensor force plays a crucial role in accounting for the astrophysical factors of the radiative capture reaction HHe as well as the nucleon transfer reactions, HH and HHe arai11 ().

The aim of this paper is to examine the issue of the photoabsorption cross section of He. Because four-body bound state problems with realistic nucleon-nucleon () interactions can be accurately solved with the correlated Gaussians, it is interesting to apply that approach to a calculation of the photoabsorption cross section. For this purpose we have to convert the continuum problem to such a bound-state like problem that can be treated in the basis functions. Differently from the previous theoretical calculations quaglioni04 (); gazit (); bacca (); quaglioni (), we employ a complex scaling method (CSM) ho (); moiseyev (); CSM () for avoiding a construction of the continuum wave functions. One of the advantages of the CSM is that the cross section can be directly obtained without recourse to a sophisticated inversion technique as used in the LIT or an artificial energy averaging procedure. We will pay special attention to the following points:

  1. To use a realistic interaction as it is

  2. To include couplings with final decay channels explicitly

  3. To perform calculations in both MRM and CSM as a cross-check.

Here point (1) indicates that the interaction is not changed to an effective force by some transformation. This looks sound and appealing because the cross section may depend on the -state probability of He wachter () and hence the effect of the tensor force on the cross section could be seen directly. In point (2) we make use of the flexibility of the correlated Gaussians to include such important configurations that have H+, He+, and ++ partitions. Thanks to this treatment the effects of final-state interactions are expected to be fully taken into account. Point (3) is probably most significant in our approach. We mean by this point that the photoabsorption cross section is calculated in two independent methods. In the MRM we calculate the cross sections for the radiative capture reactions, HHe and HeHe, and these cross sections are converted to the photoabsorption cross section using a formula due to the detailed balance. In the CSM we make use of the fact that the final continuum states of He, if rotated on the complex coordinate plane, can be expanded in the functions. Consistency of the two results, if attained, serves strong evidence for that the obtained cross section is reliable. We hope to shed light on resolving the controversy from our theoretical input.

In Sec. II we present our theoretical prescriptions to calculate the photoabsorption cross section. The two approaches, the CSM and the MRM, are explained in this section with emphasis on the method of how discretized states are employed for the continuum problem. We give the basic inputs of our calculation in Sec. III. The detail of our correlated basis functions is given in Sec. III.2, and various configurations needed to take into account the final state interactions as well as two- and three-body decay channels are explained in Sec. III.4. We show results on the photoabsorption cross section in Sec. IV. The strength function and the transition densities calculated from the continuum discretized states are presented in Sec. IV.1. A comparison of CSM and MRM cross sections is made in Sec. IV.2. The photonuclear sum rules are examined in Sec. IV.3. The calculated photoabsorption cross sections are compared to experiment in Sec. IV.4. Finally we draw conclusions of this work in Sec. V.

Ii Formulation of photoabsorption cross section calculation

ii.1 Basic formula

The photoabsorption takes place mainly through the transition, which can be treated by the perturbation theory. The wavelength of the photon energy (MeV) is about (fm), so that it is long enough compared to the radius of He even when is close to the rest energy of a pion. The photoabsorption cross section can be calculated by the formula ring ()


where is the strength function for the transition


The symbol denotes the operator, and and are the wave functions of the ground state with energy and the final state with the excitation energy of He, respectively. The recoil energy of He is ignored, so that is equal to the nuclear excitation energy . The symbol indicates a summation over and all possible final states . The final state of He is actually a continuum state lying above the H+ threshold, and it is normalized according to . The sum or integral for the final states in can be taken using the closure relation, leading to a well-known expression for the strength function


where a positive infinitesimal ensures the outgoing wave after the excitation of He. A method of calculation of in the CSM is presented in Sec. II.2.

A partial photoabsorption cross section for the two-body final state comprising nuclei, A and B, can be calculated in another way. With use of the detailed balance the cross section is related to that of its inverse process, the radiative capture cross section  thompson09 (), induced by the transition, at the incident energy ,


where is the A+B threshold energy. Here and are the angular momenta of the nuclei, and , and is the angular momentum of the ground state of He. The wave number is where is the reduced mass of the two nuclei and is the photon wave number . The photoabsorption cross section is equal to a sum of and provided that three- and four-body breakup contributions are negligible. A calculation of the radiative capture cross section will be performed in the MRM as explained in Sec. II.3.

The fact that we have two independent methods of calculating is quite important to assess their validity.

ii.2 Complex scaling method

The quantity of Eq. (3) is evaluated using the CSM, which makes a continuum state that has an outgoing wave in the asymptotic region damp at large distances, thus enabling us to avoid an explicit construction of the continuum state. In the CSM the single particle coordinate and momentum are subject to a rotation by an angle :


Applying this transformation in Eq. (3) leads to


where is the complex scaled resolvent




A key point in the CSM is that within a suitable range of positive the eigenvalue problem


can be solved in a set of basis functions


We are interested in with . With the solution of Eq. (9), an expression for reads myo (); threebody ()






Note that the energy of the bound state of in principle remains the same against the scaling angle . Also is to be understood as a solution of Eq. (9) for corresponding to the ground-state energy threebody (). This stability condition will be met when the basis functions are chosen sufficiently.

In such a case where sharp resonances exist, the angle has to be rotated to cover their resonance poles on the complex energy plane moiseyev (); CSM (). A choice of is made by examining the stability of with respect to the angle. One of the advantages of the CSM is that one needs no artificial energy smoothing procedure but obtains the continuous cross section naturally.

ii.3 Microscopic -matrix method

The calculation of involves the matrix element of between the scattering state initiated through the A+B entrance channel and the final state, i.e., the ground state of He. See, e.g., Ref. arai02 (). The scattering problem is solved in the MRM. As is discussed in detail for the four-nucleon scattering arai10 (); fbaoyama (), an accurate solution for the scattering problem with realistic potentials in general requires a full account of couplings of various channels. In the present study we include the following two-body channels: H()+, He()+, (1)+(1), (0)+(0), and (0)+(0). Here, for example, H() stands for not only the ground state of H but also its excited states. The latter are actually unbound, and these configurations together with the ground-state wave function are obtained by diagonalizing the intrinsic Hamiltonian for the ++ system in basis functions. Similarly (0), (0), and (0) stand for the two-nucleon pseudo states with the isospin .

The total wave function may be expressed in terms of a combination of various components, , with


where e.g. is the basis size for the nucleus A, is the intrinsic wave function of its th state with the angular momentum and the parity , and is the relative motion function between the two nuclei. The angular momenta of the two nuclei are coupled to the channel spin , which is further coupled with the partial wave for the relative motion to the total angular momentum . The index denotes a set of (, , , ). The parity of the total wave function is .

In the MRM the configuration space is divided into two regions, internal and external, by a channel radius. The total wave function in the internal region, , is constructed by expanding in terms of with a suitable set of , while the total wave function in the external region, , is represented by expressing with Coulomb or Whitakker functions depending on whether the channel is open or not. The scattering wave function and the -matrix are determined by solving a Schrödinger equation


in the internal region together with the continuity condition at the channel radius. Here is the Bloch operator. See Ref. desc () for detail.

In the MRM the ground-state wave function of He is approximated in combinations of the multi-channel configurations.

Iii Model

iii.1 Hamiltonian

The Hamiltonian we use reads


The kinetic energy of the center of mass motion is subtracted and the two-nucleon interaction consists of nuclear and Coulomb parts. As the potential we employ Argonne 8 (AV8AV8p () and G3RS tamagaki () potentials that contain central, tensor and spin-orbit components. The and terms in the G3RS potential are omitted. The potential of AV8 type contains eight pieces: =, where and are the radial form factor and the operator characterizing each piece of the potential. The operators are defined as =1, ====,  ===, where is the tensor operator, and is the spin-orbit operator. For the sake of later convenience, we define by


The AV8 potential is more repulsive at short distances and has a stronger tensor component than the G3RS potential. Due to this property one has to perform calculations of high accuracy particularly when the AV8 potential is used, in order to be safe from those problems of the CSM that are raised by Witała and Glöckle witala (). To reproduce the two- and three-body threshold energies is vital for a realistic calculation of . To this end we add a three-nucleon force (3NF) , and adopt a purely phenomenological potential hiyama () that is determined to fit the inelastic electron form factor from the ground state to the first excited state of He as well as the binding energies of H, He and He.

iii.2 Gaussian basis functions

Basis functions defined here can apply to any number of nucleons. The basis function we use for -nucleon system takes a general form in coupling scheme


where is the antisymmetrizer. We define spin functions by a successive coupling of each spin function


Since taking all possible intermediate spins (, ) forms a complete set for a given , any spin function can be expanded in terms of the functions (19). Similarly the isospin function can also be expanded using a set of isospin functions . In the MRM calculation we use a particle basis that in general contains a mixing of the total isospin , which is caused by the Coulomb potential.

There is no complete set that is flexible enough to describe the spatial part . For example, harmonic-oscillator functions are quite inconvenient to describe spatially extended configurations. We use an expansion in terms of correlated Gaussians varga (); svm (). As demonstrated in Ref. kamada (), the Gaussian basis leads to accurate solutions for few-body bound states interacting with the realistic potentials.

Two types of Gaussians are used. One is a basis expressed in a partial wave expansion




Here the coordinates are a set of relative coordinates. The angular part is represented by successively coupling the partial wave associated with each coordinate. The values of and as well as the intermediate angular momenta are variational parameters. The angular momentum is limited to in the present calculation. This basis is employed to construct the internal wave function of the MRM calculation.

The other is an explicitly correlated Gaussian with a global vector representation varga (); svm (); dgvr (); fbaoyama ()


where is an positive definite symmetric matrix and is an -dimensional column vector. Both and are variational parameters. The tilde symbol denotes a transpose, that is, and . The latter specifies the global vector responsible for the rotation. The basis function (22) will be used in the CSM calculation. Actually a choice of the angular part of Eq. (22) is here restricted to . With the two global vectors any states but can be constructed with a suitable choice of and .

Apparently the basis function (22) includes correlations among the nucleons through the non-vanishing off diagonal elements of . Contrary to this, the basis function (20) takes a product form of a function depending on each coordinate, so that the correlation is usually accounted for by including the so-called rearrangement channels that are described with different coordinate sets kamimura (). A great advantage of Eq. (22) is that it keeps its functional form under the coordinate transformation. Hence one needs no such rearrangement channels but can use just one particular coordinate set, which enables us to calculate Hamiltonian matrix elements in a unified way. See Refs. dgvr (); fbaoyama () for details.

The variational parameters are determined by the stochastic variational method varga (); svm (). It is confirmed that both types of basis functions produce accurate results for the ground-state properties of H, He, and He dgvr (). Table 1 lists the properties of H and He obtained using the basis (22). Included and values are the same as those used in Refs. dgvr (); inversion (). Both potentials of AV8+3NF and G3RS+3NF reproduce the binding energy and the root-mean square radius of He satisfactorily. The G3RS+3NF potential gives a slightly larger radius and a smaller -state probability than the AV8+3NF potential.

H He H He
 (MeV) 8.41 28.43 8.35 28.56
 (fm) 1.70 1.45 1.74 1.47
 (fm) 2.41 2.45
91.25 85.56 92.85 88.33
8.68 14.07 7.10 11.42
0.07 0.37 0.05 0.25
Table 1: Ground-state properties of H and He calculated with the correlated Gaussians (22) using the AV8 and G3RS potentials together with 3NF. Here , , and denote the energy, the root-mean-square radius of proton distribution and the root-mean-square relative distance of protons, respectively, and stands for the probability (in %) of finding the component with the total orbital angular momentum and the spin . The experimental energy of He is  MeV and the point proton radius is 1.457(14) fm mueller ().

iii.3 Two- and three-body decay channels

As is well-known, the electric dipole operator


is an isovector, where is the center of mass coordinate of He, and is the Jacobi coordinate: , , . This operator excites the ground state of He to those states that have = in so far as a small isospin admixture in the ground state of He is ignored. Moreover those excited states should mainly have component, because the ground state of He is dominated by the component. See Table 1. Excited states with or 2 components will be weakly populated by the transition through the minor components (12–14%) of the He ground state.

According to the -matrix phenomenology as quoted in Ref. tilley (), two levels with are identified. Their excitation energies and widths in MeV are respectively =(23.64,  6.20), (25.95,  12.66). We have recently studied the level structure of He and succeeded to reproduce all the known levels below 26 MeV inversion (). With including the 3NF, two states are predicted at about 23 and 27 MeV in case of the AV potential. They are however not clearly identified as resonances in a recent microscopic scattering calculation fbaoyama (). In Sec. IV.1, we will show that three states with strong strength are obtained below 35 MeV in a diagonalization using the basis and will discuss the properties of those states.

Low-lying excited states with decay to H+ and He+ channels with wave. Possible channel spins that the H+ or He+ continuum state takes are and  fbaoyama (). A main component of the continuum state is found to be while that of the continuum state is . Thus it is expected that the excitation of He is followed mainly by the H+ and He+ decays in the channel, which agrees with the result of a resonating group method calculation including the H+, He+, and + physical channels wachter ().

The two-body decay to + is suppressed due to the isospin conservation. Above the ++ threshold at 26.07 MeV, this three-body decay becomes possible where the decaying pair is in the state. In fact the cross section to this three-body decay is observed experimentally shima ().

iii.4 Square-integrable basis with

The accuracy of the CSM calculation crucially depends on how sufficiently the basis functions for are prepared for solving the eigenvalue problem (9). We attempt at constructing the basis paying attention to two points: the sum rule of strength and the decay channels as discussed in Sec. III.3. As the operator (23) suggests, we will construct the basis with by choosing the following three operators and acting them on the basis functions that constitute the ground state of He: (i) a single-particle excitation built with , (ii) a + (H+ and He+) two-body disintegration due to , (iii) a ++ three-body disintegration due to . See Fig. 1. The basis (i) is useful for satisfying the sum rule, and the bases (ii) and (iii) take care of the two- and three-body decay asymptotics. These cluster configurations will be better described using the relevant coordinates rather than the single-particle coordinate. It should be noted that the classification label does not necessarily indicate strictly exclusive meanings because the basis functions belonging to the different classes have some overlap among others because of their non-orthogonality.

Figure 1: (Color online) Three patterns for the dipole excitations for He. Thick solid lines denote the coordinates on which the spatial part of the operator acts.

We will slightly truncate the ground-state wave functions of H, He, and He when they are needed to construct the above configurations, (i) and (ii). With this truncation a full calculation presented in Sec. IV will be possible without excessive computer time. As shown in Table 1, the ground states of these nuclei contain a small amount (less than 0.5%) of component, so that we omit this component and reconstruct the ground-state wave functions using only with =0, 2 and in Eq. (22). The energy loss is found to be small compared to the accurate energy of Table 1. E.g., in the case of AV8+3NF, the loss is 0.23 MeV for H in 64 basis dimension and 1.53 MeV for He in 200 basis dimension. The truncated ground-state wave function is denoted for and for He.

Note, however, that we use the accurate wave function of Table 1 for the He ground state in computing with Eq. (11).

iii.4.1 Single-particle (sp) excitation

As is well-known, applying the operator on a ground state leads to a coherent state that exhausts all the strength from the ground state. The coherent state is however not an eigenstate of the Hamiltonian. In analogy to this, the basis of type (i) is constructed as follows


where is the space-spin part of the th basis function of . We include all the basis functions and all possible for the four-nucleon isospin state with =10. The truncated basis consists of either or in the notation of Eq. (18). The former contains no global vector, while the latter contains one global vector. Since is rewritten as with , the basis (24) contains at most two global vectors and reduces to the correlated Gaussian (22). For example, the basis with the latter case can be reduced, after the angular momentum recoupling, to the standard form with


Each component of is included as an independent basis function in what follows.

iii.4.2 + two-body disintegration

In this basis the nucleon couples with the ground and pseudo states of the system. Their relative motion carries -wave excitations, and it is described in a combination of several Gaussians. The basis function takes the following form


where is the space-spin part of the th basis function of . The value of takes and , and takes any of , and that, with , can add up to the angular momentum 1. The parameter is taken in a geometric progression as in fm. As in the basis of the single-particle excitation the space-spin part is again expressed in the correlated Gaussians (22) with at most two global vectors, where one of the global vectors is = with . All the basis states with different values of and are included independently.

iii.4.3 ++ three-body disintegration

In this basis the relative motion between and is wave but the system is excited to the + configuration with -wave relative motion. Here does not necessarily mean its ground state but include pseudo states with the angular momentum . The spatial part is however taken from the basis functions of the deuteron ground state. The three-body basis function takes the following form




where is the (pseudo) deuteron wave function mentioned above. Both of and take and . All possible sets of and values that satisfy the angular momentum addition rule are included in the calculation. Both and are again given in a geometric progression, in fm. Note that = with . After recoupling the orbital and spin angular momenta, the basis (27) leads to the following space-spin parts: with =0 or 2, and all possible values of are allowed. These are included independently. Note that the matrix of becomes diagonal.

The basis dimension included is 7400 (7760) for AV8 (G3RS)+3NF, 1200 (1560) from (i), 3000 from (ii), and 3200 from (iii), respectively.

Iv Results

iv.1 Discretized strength of electric dipole transition

Continuum states with are discretized by diagonalizing the Hamiltonian in the basis functions defined in Sec. III. These discretized states provide us with an approximate distribution of the strength. Figure 2 displays the reduced transition probability


as a function of the discretized energy . The calculations were performed in each basis set of (i)–(iii) as well as a full basis that includes all of them. The distribution of depends rather weakly on the potentials.

As expected, three types of basis functions play a distinctive and supplementary role in the strength distribution. The basis functions (i) produce strongly concentrated strength at about 27 MeV and another peak above 40 MeV. The component of these states is about 95%. With the + two-body configurations (ii), we obtain two peaks in the region of 20–30 MeV and one or two peaks at around 35 MeV. The two peaks at about 25 MeV may perhaps correspond to the levels with =1 at 23.64 and 25.95 MeV with very broad widths tilley (). Note, however, that a microscopic four-nucleon scattering calculation presents no conspicuous resonant phase shifts for and channels fbaoyama (). The three-body configurations (iii) give relatively small strength broadly in the excitation energy above 30 MeV. The three prominent peaks at around 25–35 MeV remain to exist in the full basis calculation. This implies that the low-lying strength mainly comes from the + configuration. We will return this issue in Sec. IV.4 The three discretized states are labeled by their excitation energies in what follows.

Figure 2: (Color online) Discretized strength of the transitions in He. See the text for the calculations classified by sp, +, ++, and Full.

Table 2 shows the properties of the three states that have strong strength. The expectation value of each piece of the Hamiltonian is a measure of its contribution to the energy. We see that the central (: ) and tensor (: ) terms are major contributors among the interaction pieces. The one-pion exchange potential (OPEP) consists of and terms, so that the tensor force of the OPEP is found to play a vital role. The value of in the table is obtained by the squared coefficient of the expansion


where is normalized. Note that no basis functions with are included in the present calculation as they are not expressible in the two global vectors. As expected, all of the three states dominantly consist of the component, which can be excited, by the operator, from the main component of the He ground state. We see a considerable admixture of the components especially with in the three states. This is understood from the role played by the tensor force that couples the and 2 states. In fact the states lose energy due to large kinetic energy contributions but gain energy owing to the coupling with the main component with through the tensor force. For example, in the case of state, the diagonal matrix elements of the kinetic energy, , are 196.5 (160.3), 198.6 (161.5), 199.3 (162.3) MeV for =1, 2, 3 with AV8 (G3RS)+3NF, while the tensor coupling matrix elements between and states, , are respectively 54.5 (40.2), 70.8 (52.1), 84.0 (61.9) MeV for =1, 2, 3 states.

23.96 27.05 33.02 24.08 27.25 33.43
4.46 1.38 4.60 4.48 1.31 4.88
51.21 54.78 43.71 44.34 48.37 49.65
6.42 6.37 4.44 0.14 0.24 0.31
3.41 3.68 1.61 3.07 3.38 2.94
2.17 2.15 1.65 3.81 3.75 3.43
23.83 24.04 16.09 20.45 20.81 18.46
0.22 0.22 0.14 0.41 0.41 0.37
30.60 30.51 22.71 20.60 20.64 18.80
4.79 4.77 3.55 2.33 2.33 2.13
6.76 6.73 4.96 2.37 2.38 2.15
0.74 0.86 0.55 0.72 0.85 0.85
0.42 0.45 0.32 0.41 0.45 0.42
87.18 84.58 82.70 90.12 88.47 79.73
4.76 7.47 7.59 3.18 4.89 13.86
0.16 0.25 0.22 0.09 0.13 0.36
0.89 0.74 4.56 0.85 0.76 0.95
2.17 1.99 1.40 1.89 1.79 1.41
4.85 4.97 3.53 3.86 3.95 3.69
Table 2: Properties of the three states that exhibit strong strength. The excitation energy and the expectation values are given in units of MeV. The value of is given in %. See Table 1 for the ground-state energy of He.

The transition density is defined as


which gives the transition matrix element through


Figure 3 displays the transition densities for the three states of Table 2 that give the large matrix elements. The dependence of the transition density on the interaction is rather weak except for the third state labeled by . The transition density extends to significantly large distances mainly due to the effect of the + configurations, so that for a reliable evaluation of the basis functions for must include configurations that reach far distances. The peak of appears at about 2 fm, which is much larger than the peak position (1.1 fm) of , where is the ground-state density of He. A comparison of the transition densities of the second () and third () states suggests that near  2-6 fm they exhibit a constructive pattern in the second state and a destructive pattern in the third state.

Figure 3: (Color online) Transition densities for the three discretized states listed in Table 2 that have strong strength.

iv.2 Test of CSM calculation

The strength function (11) calculated in the CSM using the full basis is plotted in Fig. 4 for some angles . Both AV8+3NF and G3RS+3NF potentials give similar results. With =10, shows some oscillations whose peaks appear at the energies of the discretized states shown in the full calculation of Fig. 2. To understand this behavior we note that the contribution of an eigenstate to is given by a Lorentz distribution