Four-nucleon scattering with a correlated Gaussian basis method

Four-nucleon scattering with a correlated Gaussian basis method

S. Aoyama, K. Arai, Y. Suzuki, P. Descouvemont, D. Baye
Abstract

Elastic-scattering phase shifts for four-nucleon systems are studied in an - type cluster model in order to clarify the role of the tensor force and to investigate cluster distortions in low energy + and + scattering. In the present method, the description of the cluster wave function is extended from a simple (0) harmonic-oscillator shell model to a few-body model with a realistic interaction, in which the wave function of the subsystems are determined with the Stochastic Variational Method. In order to calculate the matrix elements of the four-body system, we have developed a Triple Global Vector Representation method for the correlated Gaussian basis functions. To compare effects of the cluster distortion with realistic and effective interactions, we employ the AV8 potential as a realistic interaction and the Minnesota potential as an effective interaction. Especially for , the calculated phase shifts show that the + and + channels are strongly coupled to the + channel for the case of the realistic interaction. On the contrary, the coupling of these channels plays a relatively minor role for the case of the effective interaction. This difference between both potentials originates from the tensor term in the realistic interaction. Furthermore, the tensor interaction makes the energy splitting of the negative parity states of He consistent with experiments. No such splitting is however reproduced with the effective interaction.

1 Introduction

The microscopic cluster model is one of the successful models to study the structure and reactions of light nuclei [1]. In the conventional cluster model, one assumes that the nucleus is composed of several simple clusters with which are described by (0) harmonic-oscillator shell model functions, and use an effective - interaction which is appropriate for such a model space. However, it is well known that the ground states of the typical clusters , , and He have non-negligible admixtures of -wave component due to the tensor interaction. Since the conventional cluster model does not directly treat the -wave component, the strong attraction of the nucleon-nucleon interaction due to the tensor term is assumed to be renormalized into the central term of the effective interaction.

Recently, - structure calculations [2] have been successfully developed: Stochastic Variational Method (SVM) [3, 4, 5, 6], Global Vector Representation method (GVR) [7, 8], Green’s function Monte Carlo method [9], no core shell model [10], correlated hyperspherical harmonics method [11], unitary correlation operator method [12], and so on. Although the application of - reaction calculations with a realistic interaction are restricted so much in comparison with structure calculations, it has been intensively applied to the four-nucleon systems + and + [13, 14, 15, 16, 17, 18, 19]. Especially + scattering states, which couple to + and + channels, have attracted much attention, because the + radiative capture is one of the mechanisms making He through electro-magnetic transitions [20, 21] and also have posed intriguing puzzles for analyzing powers [22, 23, 24, 25, 26], which are motivated by the famous problem in the three-nucleon system.

Furthermore, the + elastic-scattering phase shifts are interesting because the astrophysical S-factor of the (,)He reaction is not explained by any calculation using an effective interaction that contains no tensor term, and is expected to be contributed by the -wave components of the clusters through transitions [27, 28].

Also, thanks to recent developments of the microscopic cluster model, the simple model using the (0) harmonic-oscillator wave function with an effective interaction is not mandatory any more, at least, in light nuclei. We can use a kind of - cluster model which employs more realistic cluster wave functions with realistic interactions. Therefore, it is interesting to see the difference between the - reaction calculations with a tensor term and the conventional cluster model calculations without a tensor term in few-body systems.

The microscopic -matrix method (MRM) with a cluster model (GCM or RGM) has been applied to studies of many nuclei [29, 30, 31, 32]. It is now used in descriptions of collisions [16]. We have also applied the MRM to the + scattering problem with more realistic cluster wave functions by using a realistic interaction [13]. The Gaussian basis functions for the expansion of the cluster wave functions are chosen by a technique of the SVM [5]. In the MRM, as will be shown later, the relative wave function between clusters ( and ) is connected to the boundary condition at a channel radius. The problem is how to calculate the matrix elements. In this paper, we develop a method called the Triple Global Vector Representation method (TGVR), by which we calculate the matrix elements in a unified way. Although we restricted ourselves to four nucleon systems in the present paper, the formulation of the TGVR itself can be applied to more than four-body systems as in the previous studies of the Global Vector Representation methods (GVR) [8]. Furthermore, for scattering problems, the TGVR can deal with more complicated systems than the double (or single) global vector which was given in the previous papers [7, 8], because we need three representative orbital angular momenta, the total internal orbital momenta of both clusters and the orbital momentum of their relative motion, in order to reasonably describe the scattering states. In other words, the first global vector represents the angular momentum of cluster , the second global vector represents the angular momentum of cluster , and the third global vector represents the relative angular momentum between the clusters.

In this paper, we will investigate the effect of the distortion of clusters on the + elastic-scattering by comparing the phase shifts calculated with a realistic and an effective interaction. In section 2, we explain the MRM in brief. In section 3, the correlated Gaussian (CG) method with the TGVR, which has newly been developed for the present analysis, will be presented. In section 4, we will explain how to calculate the matrix elements with TGVR basis functions. The typical matrix elements are also given in the appendix. In section 5, we will present and discuss the calculated scattering phase shifts in detail. Finally, summary and conclusions are given in section 6.

2 Microscopic R-matrix method

In the present study we calculate + and + (and +) elastic scattering phase shifts with the microscopic -matrix method. Though the method is well documented in e.g. Refs. [29, 30, 32], we briefly explain it below in order to present definitions and equations needed in the subsequent sections. Since our interest is on low-energy scattering, we consider only two-body channels. A channel is specified by the two nuclei (clusters) , their angular momenta, , the channel spin that is a resultant of the coupling of and , and the orbital angular momentum for the relative motion of and . The wave function of channel with the total angular momentum , its projection , and the parity takes the form

 ΨJMπα=\@fontswitchA[[ΦaIaΦbIb]Iχα(\boldmathρα)]JM, (2.1)

where and are respectively antisymmetrized intrinsic wave functions of and , and is an operator that antisymmetrizes between the clusters. The square bracket denotes the angular momentum coupling. The coordinate in the relative motion function is the relative distance vector of the clusters. The channel spin and the relative angular momentum in are coupled to give the total angular momentum . The relative-motion functions also depend on and . For simplicity, this dependence is not displayed explicitly in the notation for as well as for some other quantities below.

The configuration space is divided into two regions, internal and external, by the channel radius . In the internal region (), the total wave function may be expressed in terms of a combination of various s

 ΨJMπint = ∑αΨJMπα (2.2) = ∑α∑nfαn\@fontswitchAuαn(ρα)ϕJMπα,

with

 ϕJMπα=1√(1+δIaIbδab)(1+δab){[[ΦaIaΦbIb]IYℓ(ˆ\boldmathρα)]JM +(−1)Aa+Ia+Ib−I+ℓ[[ΦbIbΦaIa]IYℓ(ˆ\boldmathρα)]JMδab}, (2.3)

where is the number of nucleons in cluster , is unity if and are identical clusters and zero otherwise, and is unity if the clusters are in identical states and zero otherwise. In the second line of Eq. (2.2), the relative motion functions of Eq. (2) are expanded in terms of some basis functions as

 χαm(\boldmathρα)=∑nfαnuαn(ρα)Yℓm(ˆ\boldmathρα). (2.4)

In what follows we take

 uαn(ρα)=ρℓαexp(−12λnρ2α) (2.5)

with a suitable set of s.

In the external region (), the total wave function takes the form

 ΨJMπext=∑αgα(ρα)ϕJMπα. (2.6)

Note that the antisymmetrization between the clusters is dropped in the external region under the condition that the channel radius is large enough. The function of Eq. (2.6) is a solution of the equation

 [−ℏ22μα(d2dρ2α+2ραddρα−ℓ(ℓ+1)ρ2α)+ZaZbe2ρα]gα(ρα)=Eαgα(ρα), (2.7)

where is the reduced mass for the relative motion in channel , and are the charges of and , and is the energy for the relative motion, where is the total energy, and and are the internal energies for the clusters and , respectively. For the scattering initiated through the channel , the asymptotic form of for the open channel is

 gα(ρα)=v−1/2αρ−1α[Iα(kαρα)δαα0−SJπαα0Oα(kαρα)], (2.8)

where , and is an element of the -matrix (or collision matrix) to be determined. Here and are the incoming and outgoing waves defined by

 Iα(kαρα)=Oα(kαρα)∗=Gℓ(ηα,kαρα)−iFℓ(ηα,kαρα), (2.9)

with the regular and irregular Coulomb functions and . The Sommerfeld parameter is . For a closed channel , the asymptotic form of is given by the Whittaker function

 gα(ρα)∝ρ−1αW−ηα,ℓ+1/2(2kαρα). (2.10)

The matrix elements are determined by solving a Schrödinger equation with a microscopic Hamiltonian involving the nucleons,

 (H+\@fontswitchL−E)ΨJMπint=\@fontswitchLΨJMπext, (2.11)

with the Bloch operator

 \@fontswitchL=∑αℏ22μαa|ϕJMπα⟩δ(ρα−a)(∂∂ρα−bαρα)ρα⟨ϕJMπα|, (2.12)

where the channel radius is chosen to be the same for all channels, and the are arbitrary constants. Here, we choose for the open channels and for the closed channels. The results do not depend on the choices for but these values simplify the calculations. Notice that the projector on in Eq. (2.12) is not essential in a microscopic calculation and can be dropped since the various channels are orthogonal at the channel radius.

The Bloch operator ensures that the logarithmic derivative of the wave function is continuous at the channel radius. In addition, must be equal to at . Projecting the Schrödinger equation on a basis state, one obtains

 ∑αnCα′n′,αnfαn=⟨ΦJMπα′n′|\@fontswitchL|ΨJMπext⟩ (2.13)

with

 (2.14)

and

 ΦJMπαn=uαn(ρα)ϕJMπα. (2.15)

Here indicates that the integration with respect to is to be carried out in the internal region. Actually is obtained by calculating the matrix element in the entire space and subtracting the corresponding external matrix element that is easily obtained because no intercluster antisymmetrization is needed. The -matrix and -matrix are defined by

 \@fontswitchRα′α≡ℏ2a2(kα′μα′μαkα)12∑n′nuα′n′(a)(C−1)α′n′,αnuαn(a), (2.16) \@fontswitchZα′α≡Iα(kαa)δα′α−\@fontswitchRα′αkαaI′α(kαa). (2.17)

The -matrix is finally obtained as

 SJπ=(\@fontswitchZ∗)−1\@fontswitchZ. (2.18)

In this paper we focus on the elastic phase shifts that are defined by the diagonal elements of the -matrix,

 SJπαα=ηJπαe2iδJπα. (2.19)

We study four-nucleon scattering involving the +, + and + channels in the energy region around and below the + threshold. In Table 1 we list all possible labels of physical channels for , , and , assuming . Here “physical” means that the channels involve the cluster bound states that appear in the external region as well. Non-physical channels involving excited pseudo states will also be included in most calculations. Note that the + channel must satisfy the condition of even (see Eq. (2.3)). The channel spin or 2 can couple with only even , but with only odd . It is noted that the relative motion for the + scattering can have only when is equal to and .

Because one of our purposes in this investigation is to understand the role of the tensor force played in the four-nucleon dynamics, we want to compare the phase shifts obtained with two Hamiltonians that differ in the type of interactions. One is a realistic interaction called the AV8 potential [33] that includes central, tensor and spin-orbit components. We also add an effective three-nucleon force (TNF) in order to reproduce reasonably the binding energies of , and He [34], which makes reasonable thresholds. In the present calculation, the TNF is included in all calculations for AV8. Another is an effective central interaction called the Minnesota (MN) potential [35], which reproduce reasonably the binding energies of , and He, though it has central terms alone (with an exchange parameter ). The Coulomb potential is included for both potentials.

The intrinsic wave function of cluster is described with a combination of basis functions with different and values

 ΦkIkMIk=Nk∑\@fontswitchA[ψ(space)Lkψ(spin)Sk]IkMIkψ(isospin)TkMTk, (2.20)

where , and denote the space, spin and isospin parts of the cluster wave function. In the case of the AV8 potential, the (or ) wave function is approximated with thirty Gaussian basis functions that include , and and . The deuteron wave function is also approximated with Gaussian basis functions, four terms both in the - and -waves, respectively. The falloff parameters of the Gaussian functions are selected using the SVM [5] and the expansion coefficients are determined by diagonalizing the intrinsic cluster Hamiltonian. A similar procedure is applied to the case of the MN potential.

The calculated energies , root-mean-square (rms) radii and state probabilities are given in the fourth to sixth columns in Table 2. We use the truncated basis in order to obtain the phase shifts in reasonable computer times, they slightly deviate from more elaborate calculations, which are given in the last three columns. Fortunately, except for the small shift of the threshold energy, the phase shifts are not very sensitive to the details of the cluster wave functions because they are determined by the change of the relative motion function of the clusters. The values in parenthesis for He are the number of basis functions in the major multi-channel calculation. The energy of He calculated in Table 2 with the multi-channel calculation is thus not optimized but found to be very close to that of the more extensive calculation. It is noted that the calculated value for the deuteron is smaller than in other calculations. This is due to the restricted choice of the length parameters of the basis functions, which permits us to use a relatively small channel radius of fm. We have checked that the phase shifts for a single channel calculation do not change even when more extended deuteron wave functions are employed.

3 Correlated Gaussian function with triple global vectors

As explained in the previous section, the calculation of the -matrix reduces to that of the Hamiltonian and overlap matrix elements with the functions defined by (2) and (2.20), and it is conveniently performed by transforming that wave function into an -coupled form,

 \@fontswitchA[[[ψ(space)Laψ(space)Lb]Labχα(% \boldmathρα)]L[ψ(spin)Saψ(spin)Sb]S]JM. (3.1)

The transformation can be done as

 \@fontswitchA[[[ψ(space)Laψ(spin)Sa]Ia[ψ(space)Lbψ(spin)Sb]Ib]Iχα(% \boldmathρα)]JMψ(isospin)TaMTaψ(isospin)TbMTb (3.2) = ∑LabLS⎡⎢⎣LaSaIaLbSbIbLabSI⎤⎥⎦(−1)Lab+J−I−LU(SLabJℓ;IL) ×\@fontswitchA[ψ(space)LaLb(Lab)ℓL[ψ(spin)Saψ(spin)Sb]S]JMψ(isospin)TaMTaψ(isospin)TbMTb

with

 ψ(space)LaLb(Lab)ℓL=[[ψ(space)Laψ(space)Lb]Labχα(\boldmathρα)]L, (3.3)

where and are Racah and 9 coefficients in unitary form [5].

The evaluation of the matrix element can be done in the spatial, spin, and isospin parts separately. The spin and isospin parts are obtained straightforwardly. In the following we concentrate on the spatial matrix element. The spatial part (3.3) of the total wave function is given as a product of the cluster intrinsic parts and their relative motion part. The coordinates used to describe the + channel are depicted in Fig. 1(a) with , whereas the coordinates suitable for the + and + channels are shown in Fig. 1(b) with . These coordinate sets are often called H-type and K-type. Therefore the calculation of the spatial matrix element requires a coordinate transformation involving the angular momenta and . Moreover the permutation operator in causes a complicated coordinate transformation. All these complexities are treated elegantly by introducing a correlated Gaussian [36, 6, 5], provided each part of is given in terms of (a combination of) Gaussian functions as in the present case. In what follows we will demonstrate how it is performed. Because the formulation with the correlated Gaussian is not restricted to four nucleons but can be applied to a system including more particles, the number of nucleons is assumed to be in this and next sections as well as in Appendices B and C unless otherwise mentioned.

The relative and center of mass coordinates of the nucleons, , and the single-particle coordinates, , are mutually related by a linear transformation matrix and its inverse as follows:

 \boldmathxi=N∑j=1Uij\boldmathrj,     \boldmathri=N∑j=1(U−1)ij\boldmathxj. (3.4)

We use a matrix notation as much as possible in order to simplify formulas and expressions. Let denote an -dimensional column vector comprising all but the center of mass coordinate . Its transpose is a row vector and it is expressed as

 ˜\boldmathx=(\boldmathx1,\boldmathx2,...,\boldmathxN−1). (3.5)

The choice for is not unique but a set of Jacobi coordinates is conveniently employed. For the four-body system, the Jacobi set is identical to the K-type coordinate, and the corresponding matrix is given by

 UK=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝1−1001212−10131313−114141414⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (3.6)

The transformation matrix for the H-type coordinate reads

 UH=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝1−100001−11212−12−1214141414⎞⎟ ⎟ ⎟ ⎟ ⎟⎠. (3.7)

The K-type coordinate is obtained directly from the H-type coordinate by a transformation matrix

 UKU−1H=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝10000−12100232300001⎞⎟ ⎟ ⎟ ⎟ ⎟⎠=(UKH001), (3.8)

where is a 33 sub-matrix of .

Each coordinate set emphasizes particular correlations among the nucleons. As mentioned above, the H-coordinate is natural to describe the + channel, whereas the K-coordinate is suited for a description of the 3+ partition. It is of crucial importance to include both types of motion in order to fully describe the four-nucleon dynamics [2]. In order to develop a unified method that can incorporate both types of coordinates on an equal footing, we extend the explicitly correlated Gaussian function [37, 7] to include triple global vectors

 FL1L2(L12)L3LM(u1,u2,u3,A,% \boldmathx) =exp(−12˜% \boldmathxA\boldmathx)[[\@fontswitchYL1(˜u1\boldmathx)\@fontswitchYL2(˜u2\boldmathx)]L12\@fontswitchYL3(˜u3\boldmathx)]LM, (3.9)

where

 \@fontswitchYLiMi(˜ui\boldmathx)=|˜ui\boldmathx|LiYLiMi(ˆ˜ui\boldmathx) (3.10)

is a solid spherical harmonics and its argument, , what we call a global vector, is a vector defined through an -dimensional column vector and as

 ˜ui\boldmathx=N−1∑j=1(ui)j% \boldmathxj, (3.11)

where is the th element of . In Eq. (3.9) is an real and symmetric matrix, and it must be positive-definite for the function to have a finite norm, but otherwise may be arbitrary. Non-diagonal elements of can be nonzero.

The matrix and the vectors are parameters to characterize the “shape” of the correlated Gaussian function. The Gaussian function including describes a spherical motion of the system, while the global vectors are responsible for a rotational motion. The spatial function (3.3) is found to reduce to the general form (3.9). Suppose that stands for the H-type coordinate. Then a choice of =(1,0,0), =(0,1,0) and =(0,0,1) together with a diagonal matrix provides us with the basis function (3.3) employed to represent the configurations of the + channel. On the other hand, the K-type basis function looks like

 exp(−12˜\boldmathx′AK\boldmathx′)[[\@fontswitchYL1(\boldmathx′1)\@fontswitchYL2(\boldmathx′2)]L12\@fontswitchYL3(\boldmathx′3)]LM, (3.12)

where is the K-coordinate set (see Fig. 1(b)) and is a 33 diagonal matrix. Noting that is equal to , we observe that the basis function (3.12) is obtained from Eq. (3.9) by a particular choice of parameters, that is, =(1,0,0), =(0,,) and =(0,,), and the matrix is related to by

 A=(u1u2u3)AK⎛⎜⎝˜u1˜u2˜u3⎞⎟⎠=˜UKHAKUKH. (3.13)

Thus the form of the -function remains unchanged under the transformation of relative coordinates.

Note that is no longer diagonal. The choice of a different set of coordinates ends up only choosing appropriate parameters for , , , and .

It is also noted that the triple global vectors in Eq. (3.9) are a minimum number of vectors to provide all possible spatial functions with arbitrary and parity . A natural parity state with can be described by only one global vector, that is, using e.g., , , ,  [6, 38, 8]. To describe an unnatural parity state with except for case, we need at least two global vectors, say, , , ,  [37, 7]. The simplest choice for the state is to use three global vectors with  [37]. In this way, the basis function (3.9) can be versatile enough to describe bound states of not only four- but also more-particle systems with arbitrary and .

To assure the permutation symmetry of the wave function, we have to operate a permutation on . Since induces a linear transformation of the coordinate set, a new set of the permuted coordinates, , is related to the original coordinate set as with an matrix . As before, this permutation does not change the form of the -function:

 PFL1L2(L12)L3LM(u1,u2,u3,A,% \boldmathx) =FL1L2(L12)L3LM(u1,u2,u3,A,% \boldmathxP) =FL1L2(L12)L3LM(˜\@fontswitchPu1,˜\@fontswitchPu2,˜\@fontswitchPu3,˜\@fontswitchPA\@fontswitchP,\boldmathx). (3.14)

The fact that the functional form of remains unchanged under the permutation as well as the transformation of coordinates enables one to unify the method of calculating the matrix elements. This unique property is one of the most notable points in the present method.

4 Calculation of matrix elements

Calculations of matrix elements with the correlated Gaussian are greatly facilitated with the aid of the generating function  [6, 5]

 g(\boldmaths;A,\boldmathx)=exp(−12˜\boldmathxA\boldmathx+˜\boldmaths\boldmathx), (4.1)

with , where , is a 3-dimensional unit vector (), and is a scalar parameter. More explicitly

 ˜\boldmaths\boldmathx=N−1∑i=1% \boldmathsi⋅\boldmathxi=N−1∑i=13∑j=1λj(uj)i\boldmathej⋅\boldmathxi=3∑j=1λj\boldmathej⋅(˜uj\boldmathx). (4.2)

The correlated Gaussian is generated as follows:

 FL1L2(L12)L3LM(u1,u2,u3,A,% \boldmathx) =(3∏i=1BLiLi!∫d% \boldmathei)[[YL1(\boldmathe1)YL2(\boldmathe2)]L12YL3(\boldmathe3)]LM ×⎛⎝∂L1+L2+L3∂λL11∂λL22∂λL33g(\boldmaths;A,\boldmathx)⎞⎠∣∣∣λ1=λ2=λ3=0, (4.3)

where

 BL = (2L+1)!!4π. (4.4)

When is expanded in powers of , only the term of degree contributes in Eq. (4.3), and this term contains the th degree because and always appear simultaneously. In order for the term to contribute to the integration over , these vectors must couple to the angular momentum because of the orthonormality of the spherical harmonics , that is, they are uniquely coupled to the maximum possible angular momentum. The same rule applies to , and , as well.

We outline a method of calculating the matrix element for an operator

 ⟨FL4L5(L45)L6L′M′(u4,u5,u6,A′,\boldmathx)|\@fontswitchO|FL1L2(L12)L3LM(u1,u2,u3,A,\boldmathx)⟩. (4.5)

In what follows this matrix element is abbreviated as . Using Eq. (4.3) in Eq. (4.5) enables one to relate the matrix element to that between the generating functions:

 ⟨F