**Antisymmetrized Molecular Dynamics and Nuclear Structure**

Gaotsiwe J. Rampho and Sofianos A. Sofianos *Department of Physics, University of South Africa South Africa*

#### **1. Introduction**

132 Molecular Dynamics – Theoretical Developments and Applications in Nanotechnology and Energy

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Ligand of Zinc in Acutolysin A Using Combined Quantum Mechanical and Molecular Mechanical Simulation. *Journal of Physical Chemistry B,* Vol. 113, No.8, pp. One of the aims of nuclear physics studies is to establish a complete theoretical description of the structure of nuclear systems. The correct theoretical description of nuclear structure is expected to help explain and accurately predict different properties of and process in nuclei (Donnelly & Raskin, 1986). Fundamental to a complete description of nuclear structure are the wave function describing nuclear systems, the Hamiltonian describing interactions in the nucleus and electromagnetic form factors describing charge and currents distributions in the nucleus. None of these components is completely understood and, therefore, none can be completely determined for a given nuclear system, yet. As a result, theoretical models of these components, based on different approximations that are guided by experimental observations, are usually employed in the description of nuclear systems. The quality of such models is often judged by their ability to explain existing experimental observations. Parallel to the theoretical developments, the developments in experimental technologies has led not only to the availability of more precise experimental data, but also to data in kinematical regions previously not accessible. The availability of precise experimental data in a wide kinematical region allow for more accurate quantitative testing and, therefore, development of realistic theoretical models that, in turn, generate more accurate predictions of experimental outcomes (Golak et al., 2005).

Besides the understanding of static and dynamical properties of nuclear matter, studies in nuclear physics are aimed at constructing a comprehensive description of properties of nucleon-nucleon interactions. Few-nucleon systems provide unique favourable environment for such investigations (Rampho, 2010). In theoretical investigations, the interaction models in few-nucleon systems can be treated realistically and the resulting dynamical equations can be solved directly. The formulation and solution of dynamical equations for many-body systems is, on the other hand, quite challenging. Progress towards a better understanding of the nuclear force has been made over the years. Based on the accumulated experimental nucleon-nucleon scattering data different phenomenological nucleon-nucleon interaction models have been suggested. The models are constructed by fitting the models to existing nucleon-nucleon scattering data as well as some properties of the 2H nucleus (Cottingham et al., 1973; Lagaris & Pandharipande, 1981; Machleidt et al., 1987; Nagels et al., 1978; Wiringa et al., 1984). These interaction models are known as modern or realistic nucleon-nucleon potentials and are able to explain most of the static properties of light nuclei. Since the exact form of the short-range behavior of the nucleon-nucleon interaction is not completely determined, yet, the short-range part of many of these potential models is

of variational spatial, spin and isospin functions to represent single-particle wave functions (Doté & Horiuchi, 2000; Doté et al., 2006). The resulting wave functions are suitable for investigating systems with tensor forces. The other approach is to use a linear combination of several Slater determinants (Kanada-En'yo et al., 1995; 2003; Kimura, 2004) to represent the total wave function of the system. In the approach of reference (Doté et al., 1997) orthogonal single-particle wave functions characteristic of the Hartree-Fock orbitals are constructed from the AMD wave function. The AMD+Hartree-Fock wave function is also more flexible than the original AMD wave function. These modifications to the AMD wave function introduced significant improvements in the description of ground-state properties, mean-field and cluster structure of nuclear systems. It should be noted that most of the studies indicated employed phenomenological nucleon-nucleon potentials of Gaussian radial form. The main reason for this is that the expectation values of this type of potentials can be evaluated analytically.

Antisymmetrized Molecular Dynamics and Nuclear Structure 135

To extend the application of the AMD approach to realistic potentials further proposals were made. In Refs. (Togashi & Kat*o*, 2007; Togashi et al., 2009) the *G*-matrix approach was used to incorporate two-body correlations in the wave function. The resulting Bruckner-AMD wave function can be constructed with realistic nuclear potentials. In Ref. (Watanabe et al., 2009; Watanabe & Oryu, 2006) Jacobi coordinates are employed to construct the wave function. Many-body correlations are included, more variational parameters are considered and the effects of the center-of-mass are completely and explicitly removed. Most realistic potentials are of non-Gaussian form and are, in many cases, expanded as the sum-of-Gaussian for application in the AMD approach. In Ref. (Rampho, 2011) a numerical technique of evaluating expectation values of non-Gaussian potentials is introduced. The technique approximates the expectation value of the potential operator with a rapidly converging series of Talmi integrals. This technique is further elaborated on in this work. Ground-state properties of selected few-nucleon systems are determined using the Argonne V4' nucleon-nucleon potential.

In section 2 the construction of the AMD wave function is outlined while the variational technique used to determine the variational parameters is briefly discussed in section 3. Application of AMD to selected light nuclei is presented in section 4 while section 5 is devoted to the charge form factors of ground state in three- and four-nucleon systems. Conclusions are

Consider a nuclear system consisting of *A* nucleons. The wave function Ψ describing the system depends on the position *ri*, spin *σ<sup>i</sup>* and isospin *τ<sup>i</sup>* (*i* = 1, 2, 3, . . . , *A*), vectors of the nucleons. In what follows the collective vector, *υi*, is used to represent the set {*siriσ<sup>i</sup> τ<sup>i</sup>* } where *si* is a complex variational parameter. For a systems of fermions, like nuclei, the total wave function is required to be antisymmetric with respect to the interchange of any two particles in the system. One way of constructing such a wave function is by the use of Slater

> *ψ*1(*υ*1) *ψ*2(*υ*1) *ψ*3(*υ*1) ··· *ψA*(*υ*1) *ψ*1(*υ*2) *ψ*2(*υ*2) *ψ*3(*υ*2) ··· *ψA*(*υ*2) *ψ*1(*υ*3) *ψ*2(*υ*3) *ψ*3(*υ*3) ··· *ψA*(*υ*3)

> *ψ*1(*υA*) *ψ*2(*υA*) *ψ*3(*υA*) ··· *ψA*(*υA*)

. ... .

. .  

. (1)

given in section 6.

**2. The AMD wave function**

determinants of single-particle wave functions *ψi*(*υj*) as

<sup>√</sup>*A*!

 

. . . . . . . .

<sup>Ψ</sup>*t*(*<sup>υ</sup>*1,*<sup>υ</sup>*2,*<sup>υ</sup>*3,...,*<sup>υ</sup>A*) = <sup>1</sup>

often musked by introducing short-distance cut-off factors. Knowledge of the short-range behavior of the nucleon-nucleon force may reveal some information about the limits and boundaries between hadronic degrees of freedom and quark degrees of freedom. The suggestion that the short-range behavior of the strong nuclear interaction could manifest itself through short-range nucleon-nucleon correlations and momentum distributions in a nucleus (Schroeder et al., 1979) refocused the need for a better understanding of, and therefore more intensive investigations into, these concepts. Hence, there are continued experimental (Egiyan et al., 2007; 2006; Jones et al., 2000; Ulmer et al., 2002) and theoretical (Alvioli et al., 2008; Frankfurt et al., 1993; Piasetzky et al., 2006) investigations of these concepts.

Over the years a variety of methods have been developed and refined in the study of properties of nuclei. Very accurate wave functions for bound and scattering states in few-nucleon systems can now be constructed using realistic Hamiltonian for the systems. A demonstration of the level of accuracy that can now be achieved in describing ground state properties of the four-nucleon system using seven different state-of-the-art methods is given in reference (Kamada et al., 2001). The use of some of these methods in the study of properties of few-nucleon systems is shown in references (Carlson & Schiavilla, 1998; Golak et al., 2005). The application of these methods to systems consisting of more than four particles is still a challenge. For investigations of light to medium nuclei microscopic models have been employed in the study of static and dynamical properties of nuclei (Aichelin, 1991; Boffi et al., 1968; Kanada-En'yo et al., 2003; Neff et al., 2005; Ono & Horiuchi, 2004). These methods are continuously developing. Following the Time-Dependent Cluster Model (Caurier et al., 1982), microscopic simulation models were developed (Feldmeier, 1990; Horiuchi, 1991) for the study of fermionic systems. These models combine Fermi-Dirac statistics with elementary quantum mechanics to treat the motion of particles in a system (Feldmeier, 1990). However, the models are not fully quantum mechanical and do not assume a shell structure for the system. In this work the antisymmetrized molecular dynamics (AMD) approach is employed. The AMD total wave function is constructed as Slater determinant of single particle shifted Gaussian wave functions. The shift parameters of the Gaussian functions are complex variational parameters which are treated as generalised coordinates of the system. The width parameters are taken as free real parameters and are chosen to be the same for all the Gaussian. The equations of motion for the variational parameters are determined from the time-dependent variational principle. The equations are then solved by using the frictional cooling technique (Ono et al., 1992) to determine the variational parameters.

In the past, the AMD approach was employed in the study of dynamics of heavy-ion collisions (Ono et al., 1992) and elastic proton-nucleus scattering (Engel et al., 1995; Tanaka et al., 1995). Clustering in nuclei as well as angular distributions of scattered protons in proton-nucleus scattering can be well explained with the AMD model (Tanaka et al., 1995). Some properties and/or processes in physical systems are governed by conservation laws related to parity and total angular momentum of the system. Such systems are best described in terms of wave functions that have definite parity and angular momentum. The AMD wave function does not have definite parity nor does it possess definite total angular momentum. As a result, for applications in realistic investigations of properties of physical systems the AMD wave function requires some improvement. A number of modifications have since been introduced in the AMD formalism. In reference (Kanada-En'yo et al., 1995) numerical technique are used to project the AMD wave function onto the eigenstates of parity and total angular momentum. The resulting wave functions have definite parity and total angular momentum. In addition, a number of different techniques were considered for constructing a more flexible total wave function. One way was to use linear combinations

2 Will-be-set-by-IN-TECH

often musked by introducing short-distance cut-off factors. Knowledge of the short-range behavior of the nucleon-nucleon force may reveal some information about the limits and boundaries between hadronic degrees of freedom and quark degrees of freedom. The suggestion that the short-range behavior of the strong nuclear interaction could manifest itself through short-range nucleon-nucleon correlations and momentum distributions in a nucleus (Schroeder et al., 1979) refocused the need for a better understanding of, and therefore more intensive investigations into, these concepts. Hence, there are continued experimental (Egiyan et al., 2007; 2006; Jones et al., 2000; Ulmer et al., 2002) and theoretical (Alvioli et al.,

Over the years a variety of methods have been developed and refined in the study of properties of nuclei. Very accurate wave functions for bound and scattering states in few-nucleon systems can now be constructed using realistic Hamiltonian for the systems. A demonstration of the level of accuracy that can now be achieved in describing ground state properties of the four-nucleon system using seven different state-of-the-art methods is given in reference (Kamada et al., 2001). The use of some of these methods in the study of properties of few-nucleon systems is shown in references (Carlson & Schiavilla, 1998; Golak et al., 2005). The application of these methods to systems consisting of more than four particles is still a challenge. For investigations of light to medium nuclei microscopic models have been employed in the study of static and dynamical properties of nuclei (Aichelin, 1991; Boffi et al., 1968; Kanada-En'yo et al., 2003; Neff et al., 2005; Ono & Horiuchi, 2004). These methods are continuously developing. Following the Time-Dependent Cluster Model (Caurier et al., 1982), microscopic simulation models were developed (Feldmeier, 1990; Horiuchi, 1991) for the study of fermionic systems. These models combine Fermi-Dirac statistics with elementary quantum mechanics to treat the motion of particles in a system (Feldmeier, 1990). However, the models are not fully quantum mechanical and do not assume a shell structure for the system. In this work the antisymmetrized molecular dynamics (AMD) approach is employed. The AMD total wave function is constructed as Slater determinant of single particle shifted Gaussian wave functions. The shift parameters of the Gaussian functions are complex variational parameters which are treated as generalised coordinates of the system. The width parameters are taken as free real parameters and are chosen to be the same for all the Gaussian. The equations of motion for the variational parameters are determined from the time-dependent variational principle. The equations are then solved by using the frictional

2008; Frankfurt et al., 1993; Piasetzky et al., 2006) investigations of these concepts.

cooling technique (Ono et al., 1992) to determine the variational parameters.

In the past, the AMD approach was employed in the study of dynamics of heavy-ion collisions (Ono et al., 1992) and elastic proton-nucleus scattering (Engel et al., 1995; Tanaka et al., 1995). Clustering in nuclei as well as angular distributions of scattered protons in proton-nucleus scattering can be well explained with the AMD model (Tanaka et al., 1995). Some properties and/or processes in physical systems are governed by conservation laws related to parity and total angular momentum of the system. Such systems are best described in terms of wave functions that have definite parity and angular momentum. The AMD wave function does not have definite parity nor does it possess definite total angular momentum. As a result, for applications in realistic investigations of properties of physical systems the AMD wave function requires some improvement. A number of modifications have since been introduced in the AMD formalism. In reference (Kanada-En'yo et al., 1995) numerical technique are used to project the AMD wave function onto the eigenstates of parity and total angular momentum. The resulting wave functions have definite parity and total angular momentum. In addition, a number of different techniques were considered for constructing a more flexible total wave function. One way was to use linear combinations

of variational spatial, spin and isospin functions to represent single-particle wave functions (Doté & Horiuchi, 2000; Doté et al., 2006). The resulting wave functions are suitable for investigating systems with tensor forces. The other approach is to use a linear combination of several Slater determinants (Kanada-En'yo et al., 1995; 2003; Kimura, 2004) to represent the total wave function of the system. In the approach of reference (Doté et al., 1997) orthogonal single-particle wave functions characteristic of the Hartree-Fock orbitals are constructed from the AMD wave function. The AMD+Hartree-Fock wave function is also more flexible than the original AMD wave function. These modifications to the AMD wave function introduced significant improvements in the description of ground-state properties, mean-field and cluster structure of nuclear systems. It should be noted that most of the studies indicated employed phenomenological nucleon-nucleon potentials of Gaussian radial form. The main reason for this is that the expectation values of this type of potentials can be evaluated analytically.

To extend the application of the AMD approach to realistic potentials further proposals were made. In Refs. (Togashi & Kat*o*, 2007; Togashi et al., 2009) the *G*-matrix approach was used to incorporate two-body correlations in the wave function. The resulting Bruckner-AMD wave function can be constructed with realistic nuclear potentials. In Ref. (Watanabe et al., 2009; Watanabe & Oryu, 2006) Jacobi coordinates are employed to construct the wave function. Many-body correlations are included, more variational parameters are considered and the effects of the center-of-mass are completely and explicitly removed. Most realistic potentials are of non-Gaussian form and are, in many cases, expanded as the sum-of-Gaussian for application in the AMD approach. In Ref. (Rampho, 2011) a numerical technique of evaluating expectation values of non-Gaussian potentials is introduced. The technique approximates the expectation value of the potential operator with a rapidly converging series of Talmi integrals. This technique is further elaborated on in this work. Ground-state properties of selected few-nucleon systems are determined using the Argonne V4' nucleon-nucleon potential.

In section 2 the construction of the AMD wave function is outlined while the variational technique used to determine the variational parameters is briefly discussed in section 3. Application of AMD to selected light nuclei is presented in section 4 while section 5 is devoted to the charge form factors of ground state in three- and four-nucleon systems. Conclusions are given in section 6.

#### **2. The AMD wave function**

Consider a nuclear system consisting of *A* nucleons. The wave function Ψ describing the system depends on the position *ri*, spin *σ<sup>i</sup>* and isospin *τ<sup>i</sup>* (*i* = 1, 2, 3, . . . , *A*), vectors of the nucleons. In what follows the collective vector, *υi*, is used to represent the set {*siriσ<sup>i</sup> τ<sup>i</sup>* } where *si* is a complex variational parameter. For a systems of fermions, like nuclei, the total wave function is required to be antisymmetric with respect to the interchange of any two particles in the system. One way of constructing such a wave function is by the use of Slater determinants of single-particle wave functions *ψi*(*υj*) as

$$\Psi\_{t}(\vec{v}\_{1},\vec{v}\_{2},\vec{v}\_{3},\ldots,\vec{v}\_{A}) = \frac{1}{\sqrt{A!}} \begin{vmatrix} \psi\_{1}(\vec{v}\_{1}) \ \psi\_{2}(\vec{v}\_{1}) \ \psi\_{3}(\vec{v}\_{1}) & \cdots & \psi\_{A}(\vec{v}\_{1}) \\ \psi\_{1}(\vec{v}\_{2}) \ \psi\_{2}(\vec{v}\_{2}) \ \psi\_{3}(\vec{v}\_{2}) & \cdots & \psi\_{A}(\vec{v}\_{2}) \\ \psi\_{1}(\vec{v}\_{3}) \ \psi\_{2}(\vec{v}\_{3}) \ \psi\_{3}(\vec{v}\_{3}) & \cdots & \psi\_{A}(\vec{v}\_{3}) \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \psi\_{1}(\vec{v}\_{A}) \ \psi\_{2}(\vec{v}\_{A}) \ \psi\_{3}(\vec{v}\_{A}) & \cdots & \psi\_{A}(\vec{v}\_{A}) \end{vmatrix} . \tag{1}$$

The functions *ψi*(*υj*) do not form an orthogonal basis. However, an orthonormal basis can be constructed from these functions (Doté et al., 1997; Togashi & Kat*o*, 2007; Togashi et al., 2009). The ground state wave function of the system is represented by Ψ*t*( S) where S ≡ {*s*1,*s*2,*s*3,...,*sA*}.

The single-particle wave function *ψ*(*υ*) is assumed to be separable in the form

$$
\psi(\vec{v}) = \phi(\vec{s}, \vec{r}) \chi(\vec{v}) \,\xi(\vec{r}) \tag{2}
$$

A more realistic wave function is constructed from the AMD wave function (1) by projecting on to states with definite parity *π* and total angular momentum *J*. The parity projected wave

Antisymmetrized Molecular Dynamics and Nuclear Structure 137

where *P<sup>π</sup>* is the parity projection operator. A wave function with a definite parity *π*, total angular momentum *J*, and angular momentum projections *MK*, is constructed from the AMD

operator. The angular momentum projection operator is defined by (Peierls & Yoccoz, 1957)

S. To establish the time evolution of such a wave function, the form of the time dependence of the variational parameters, and therefore the equations of motion, is determined. The equations of motion of the parameters are derived from the time-dependent variational

where Ψ˙ = dΨ/d*t* and Ψ<sup>∗</sup> the complex conjugate of the wave function. In this definition of the variational principle the wave function Ψ is not normalized. Therefore the equations of motion for the variational parameters will not depend on either the normalization or the phase of the wave function. The variation of the wave function in Eq. (10) with respect to time

> *∂* Ψ *∂si* + d ˜si ∗ d*t*

The equations of motion of the variational parameters can then be obtained by expressing

*∂ ∂sj* ∗ 

<sup>−</sup> d ˜sj ∗ d*t*

 d ˜si d*t*

Ψ(

*MK*(Ω) is the angular momentum projection operator, *<sup>P</sup><sup>π</sup>* the parity projection

dΩ *DJ*<sup>∗</sup>

*MK*(Ω) is the Wigner *<sup>D</sup>*-function and *<sup>R</sup>*ˆ(Ω) the rotation operator with <sup>Ω</sup> ≡ {*α*, *<sup>β</sup>*, *<sup>γ</sup>*}

S) (7)

Ψ*AMD* (8)

*MK*(Ω) *<sup>R</sup>*ˆ(Ω) (9)

d*t* = 0 (10)

. (12)

d*t* = 0 (13)

S) depends on the set of complex variational parameters

�Ψ|Ψ�

*δ*Ψ(*t*1) = *δ*Ψ(*t*2) = *δ*Ψ∗(*t*1) = *δ*Ψ∗(*t*2) = 0 (11)

*∂* Ψ *∂si* ∗ 

ln�Ψ|Ψ� − *E*

*δs*(*t*1) = *δs*(*t*2) = *δs*∗(*t*1) = *δs*∗(*t*2) = 0 (14)

<sup>Ψ</sup>*<sup>π</sup>* <sup>=</sup> <sup>1</sup> 2 <sup>1</sup> <sup>±</sup> *<sup>P</sup><sup>π</sup>*

Ψ*J<sup>π</sup> MK* <sup>=</sup> <sup>1</sup> 2 *PJ MK*(Ω) <sup>1</sup> <sup>±</sup> *<sup>P</sup><sup>π</sup>*

*MK*(Ω) = <sup>2</sup> *<sup>J</sup>* <sup>+</sup> <sup>1</sup>

8 *π*<sup>2</sup>

*PJ*

representing Euler rotation angles.

The wave function of the system Ψ(

principle (Kramer & Saraceno, 1981)

Eq. (10) in terms of the parameters as

*δ t*<sup>2</sup> *t*1

 i ¯*h* <sup>2</sup> ∑ *j*

*δ t*<sup>2</sup> *t*1

 i¯*h* 2 Ψ Ψ˙ − Ψ˙ Ψ �Ψ|<sup>Ψ</sup> � <sup>−</sup> �<sup>Ψ</sup> <sup>|</sup>*H*<sup>|</sup> <sup>Ψ</sup>�

d Ψ <sup>d</sup> *<sup>t</sup>* <sup>=</sup> ∑ *i*

> d ˜sj d*t*

*∂ ∂sj*

**3. Equations of motion**

with the constraints

can be cast in the form

with the constraints

function has the form

wave function as

where *P<sup>J</sup>*

where *D<sup>J</sup>*

where *φ*(*s*,*r*), *χ*(*σ*), and *ξ*(*τ*) are the spatial, spin and isospin wave functions, respectively. The intrinsic spin state *χ*(*σ*) of a fermion, with spin *s* and third component of the spin *ms*, is denoted by

*χ*(*σ*) = |*s*, *ms* � = ⎧ ⎨ ⎩ � � � 1 <sup>2</sup> , <sup>+</sup><sup>1</sup> 2 � = | ↑ � � � � 1 <sup>2</sup> , <sup>−</sup><sup>1</sup> 2 � = | ↓ � (3)

where | ↑ � represents the *spin-up* and | ↓ � the *spin-down* state. These two spin states are orthogonal to each other. To improve the quality of the wave function the spins of the fermions may be allowed to vary. This is done by representing the fermion spin as a general spinor. Such a spinor is expressed as a linear combination of the spin-up and spin-down states using complex variational parameters (Kanada-En'yo et al., 2003). The isospin state, *ξ*(*τ*), in the case of a nucleon, with isospin *t* and third component of the isospin *mt*, is given by

$$\xi(\vec{\tau}) = \begin{pmatrix} t \ \prime \ m\_{\ell} \end{pmatrix} = \begin{cases} \left| \frac{1}{2} \ \prime + \frac{1}{2} \right\rangle = \left| p \right\rangle \\\\ \left| \frac{1}{2} \ \prime - \frac{1}{2} \right\rangle = \left| n \right\rangle \end{cases} \tag{4}$$

where | *p* � refers to a proton while | *n* � refers to a neutron state. These states are time-independent also.

All the constituent particles are described by the same form of the spatial wave function. The spatial component, *φ*(*s*,*r*), is parametrized as a normalized Gaussian wave packet (Ono et al., 1992)

$$\phi(\vec{s}, \vec{r}) = \left(\frac{2a}{\pi}\right)^{3/4} \exp\left[-a\left(\vec{r} - \frac{\vec{s}}{\sqrt{a}}\right)^2 + \frac{\vec{s}^2}{2}\right].\tag{5}$$

The phase *s*2/2 is included mainly to simplify the structure of the elements of the resulting overlap matrix, � *φ<sup>i</sup>* | *φ<sup>j</sup>* �. The width parameter *α* is treated as a real constant and assumed to be the same for both protons and neutrons in the nucleus. It can be argued that the assumption is still valid if the masses of the constituents are not significantly different, like in nuclear systems. When the momentum operator of a fermion is denoted by *p*, then the real component Re[*s*] and the imaginary component Im[*s*] of the parameter*s* are given by

$$\operatorname{Re}[\vec{s}] = \sqrt{\mathfrak{a}} \frac{\langle \phi(\vec{r}) \,|\, \vec{r} \,|\, \phi(\vec{r}) \rangle}{\langle \phi(\vec{r}) \,|\, \phi(\vec{r}) \rangle} \qquad \text{and} \qquad \operatorname{Im}[\vec{s}] = \frac{1}{2\hbar \sqrt{\mathfrak{a}}} \frac{\langle \phi(\vec{r}) \,|\, \vec{p} \,|\, \phi(\vec{r}) \rangle}{\langle \phi(\vec{r}) \,|\, \phi(\vec{r}) \rangle} \,. \tag{6}$$

The single-particle wave function in Eq. (5) satisfies the minimum uncertainty Δ*r* Δ*p* = *h*¯ /2 (Ono & Horiuchi, 2004) which helps in the choice of the value for the width parameter *α*. Following Ref. (Ono et al., 1992), the value of *α* is chosen so as to generate reasonable description of the ground state properties of light nuclei. The expectation values of relevant operators calculated with this form of the wave function can be determined in analytical form. A more realistic wave function is constructed from the AMD wave function (1) by projecting on to states with definite parity *π* and total angular momentum *J*. The parity projected wave function has the form

$$\Psi^{\pi} = \frac{1}{2} \left[ 1 \pm P^{\pi} \right] \Psi(\vec{\mathbf{S}}) \tag{7}$$

where *P<sup>π</sup>* is the parity projection operator. A wave function with a definite parity *π*, total angular momentum *J*, and angular momentum projections *MK*, is constructed from the AMD wave function as

$$\Psi\_{MK}^{f\pi} = \frac{1}{2} \operatorname{P}\_{MK}^{J}(\Omega) \left[1 \pm P^{\pi}\right] \Psi\_{AMD} \tag{8}$$

where *P<sup>J</sup> MK*(Ω) is the angular momentum projection operator, *<sup>P</sup><sup>π</sup>* the parity projection operator. The angular momentum projection operator is defined by (Peierls & Yoccoz, 1957)

$$P\_{MK}^{I}(\Omega) = \frac{2J+1}{8\pi^2} \int d\Omega \, D\_{MK}^{I\*}(\Omega) \, \not{R}(\Omega) \tag{9}$$

where *D<sup>J</sup> MK*(Ω) is the Wigner *<sup>D</sup>*-function and *<sup>R</sup>*ˆ(Ω) the rotation operator with <sup>Ω</sup> ≡ {*α*, *<sup>β</sup>*, *<sup>γ</sup>*} representing Euler rotation angles.

#### **3. Equations of motion**

4 Will-be-set-by-IN-TECH

The functions *ψi*(*υj*) do not form an orthogonal basis. However, an orthonormal basis can be constructed from these functions (Doté et al., 1997; Togashi & Kat*o*, 2007; Togashi et al.,

where *φ*(*s*,*r*), *χ*(*σ*), and *ξ*(*τ*) are the spatial, spin and isospin wave functions, respectively. The intrinsic spin state *χ*(*σ*) of a fermion, with spin *s* and third component of the spin *ms*, is

> ⎧ ⎨ ⎩

where | ↑ � represents the *spin-up* and | ↓ � the *spin-down* state. These two spin states are orthogonal to each other. To improve the quality of the wave function the spins of the fermions may be allowed to vary. This is done by representing the fermion spin as a general spinor. Such a spinor is expressed as a linear combination of the spin-up and spin-down states using complex variational parameters (Kanada-En'yo et al., 2003). The isospin state, *ξ*(*τ*), in the

> ⎧ ⎨ ⎩

where | *p* � refers to a proton while | *n* � refers to a neutron state. These states are

All the constituent particles are described by the same form of the spatial wave function. The spatial component, *φ*(*s*,*r*), is parametrized as a normalized Gaussian wave packet (Ono et al.,

The phase *s*2/2 is included mainly to simplify the structure of the elements of the resulting overlap matrix, � *φ<sup>i</sup>* | *φ<sup>j</sup>* �. The width parameter *α* is treated as a real constant and assumed to be the same for both protons and neutrons in the nucleus. It can be argued that the assumption is still valid if the masses of the constituents are not significantly different, like in nuclear systems. When the momentum operator of a fermion is denoted by *p*, then the real component

� � � 1 <sup>2</sup> , <sup>+</sup><sup>1</sup> 2 �

� � � 1 <sup>2</sup> , <sup>−</sup><sup>1</sup> 2 �

case of a nucleon, with isospin *t* and third component of the isospin *mt*, is given by

� � � 1 <sup>2</sup> , <sup>+</sup><sup>1</sup> 2 �

� � � 1 <sup>2</sup> , <sup>−</sup><sup>1</sup> 2 �

*ψ*(*υ*) = *φ*(*s*,*r*) *χ*(*σ*) *ξ*(*τ*) (2)

= | ↑ �

= | ↓ �

= | *p* �

= | *n* �

�<sup>2</sup> <sup>+</sup> *<sup>s</sup>*<sup>2</sup> 2

2 ¯*h* √*α* �

� *φ*(*r*)|*p* | *φ*(*r*)�

� *<sup>φ</sup>*(*<sup>r</sup>*)<sup>|</sup> *<sup>φ</sup>*(*<sup>r</sup>*)� . (6)

. (5)

S) where

S ≡

(3)

(4)

2009). The ground state wave function of the system is represented by Ψ*t*(

The single-particle wave function *ψ*(*υ*) is assumed to be separable in the form

*χ*(*σ*) = |*s*, *ms* � =

*ξ*(*τ*) = | *t* , *mt* � =

*φ*(*s*,*r*) =

Re[*<sup>s</sup>*] = <sup>√</sup>*<sup>α</sup>* � *<sup>φ</sup>*(*<sup>r</sup>*)<sup>|</sup>*<sup>r</sup>* <sup>|</sup> *<sup>φ</sup>*(*<sup>r</sup>*)�

� 2*α π*

�3/4

Re[*s*] and the imaginary component Im[*s*] of the parameter*s* are given by

exp � − *α* � *<sup>r</sup>* <sup>−</sup> *<sup>s</sup>* √*α*

� *<sup>φ</sup>*(*<sup>r</sup>*)<sup>|</sup> *<sup>φ</sup>*(*<sup>r</sup>*)� and Im[*<sup>s</sup>*] = <sup>1</sup>

The single-particle wave function in Eq. (5) satisfies the minimum uncertainty Δ*r* Δ*p* = *h*¯ /2 (Ono & Horiuchi, 2004) which helps in the choice of the value for the width parameter *α*. Following Ref. (Ono et al., 1992), the value of *α* is chosen so as to generate reasonable description of the ground state properties of light nuclei. The expectation values of relevant operators calculated with this form of the wave function can be determined in analytical form.

{*s*1,*s*2,*s*3,...,*sA*}.

denoted by

time-independent also.

1992)

The wave function of the system Ψ( S) depends on the set of complex variational parameters S. To establish the time evolution of such a wave function, the form of the time dependence of the variational parameters, and therefore the equations of motion, is determined. The equations of motion of the parameters are derived from the time-dependent variational principle (Kramer & Saraceno, 1981)

$$\delta \int\_{t\_1}^{t\_2} \left[ \frac{\mathrm{i}\hbar}{2} \frac{\langle \Psi \left| \dot{\Psi} \right\rangle - \langle \dot{\Psi} \left| \Psi \right\rangle}{\langle \Psi \left| \Psi \right\rangle} - \frac{\langle \Psi \left| H \right| \Psi \rangle}{\langle \Psi \left| \Psi \right\rangle} \right] \mathrm{d}t = 0 \tag{10}$$

with the constraints

$$
\delta \Psi(t\_1) = \delta \Psi(t\_2) = \delta \Psi^\*(t\_1) = \delta \Psi^\*(t\_2) = 0 \tag{11}
$$

where Ψ˙ = dΨ/d*t* and Ψ<sup>∗</sup> the complex conjugate of the wave function. In this definition of the variational principle the wave function Ψ is not normalized. Therefore the equations of motion for the variational parameters will not depend on either the normalization or the phase of the wave function. The variation of the wave function in Eq. (10) with respect to time can be cast in the form

$$\frac{\mathbf{d}\,\Psi}{\mathbf{d}\,t} = \sum\_{i} \left[ \frac{\mathbf{d}\,\mathbf{\tilde{s}}\_{i}}{\mathbf{d}t} \frac{\partial}{\partial\,\mathbf{\tilde{s}}\_{i}} + \frac{\mathbf{d}\,\mathbf{\tilde{s}}\_{i}^{\*}}{\mathbf{d}t} \frac{\partial}{\partial\,\mathbf{\tilde{s}}\_{i}^{\*}} \right]. \tag{12}$$

The equations of motion of the variational parameters can then be obtained by expressing Eq. (10) in terms of the parameters as

$$\delta \int\_{t\_1}^{t\_2} \left[ \frac{\mathbf{i} \,\hbar}{2} \sum\_{\mathbf{j}} \left( \frac{\mathbf{d} \,\tilde{\mathbf{s}}\_{\mathbf{j}}}{\mathbf{d}t} \frac{\partial}{\partial \,\tilde{s}\_{\mathbf{j}}} - \frac{\mathbf{d} \,\tilde{\mathbf{s}}\_{\mathbf{j}}^{\*}}{\mathbf{d}t} \frac{\partial}{\partial \,\tilde{s}\_{\mathbf{j}}^{\*}} \right) \ln \langle \mathbf{\varPsi} | \mathbf{\varPsi} \rangle - E \right] \mathbf{d}t = \mathbf{0} \tag{13}$$

with the constraints

$$
\delta\vec{s}(t\_1) = \delta\vec{s}(t\_2) = \delta\vec{s}^\*(t\_1) = \delta\vec{s}^\*(t\_2) = 0\tag{14}
$$

where

$$E(\vec{\mathbf{S}}, \vec{\mathbf{S}}^{\*}) = \frac{\langle \Psi(\vec{\mathbf{S}}) | H | \Psi(\vec{\mathbf{S}}) \rangle}{\langle \Psi(\vec{\mathbf{S}}) | \Psi(\vec{\mathbf{S}}) \rangle} \tag{15}$$

are solved with the constraint ∑*<sup>i</sup> si* = 0. The zero-point oscillation of the center-of-mass need

Antisymmetrized Molecular Dynamics and Nuclear Structure 139

To evaluate the variational energy functional and solve for the variation parameters, the Hamiltonian *H* of the nucleus is required as input. Considering only two-body interactions in

where *mi* is the mass of nucleon *i*, *VNN*(*r*) the nucleon-nucleon *NN* potential, *VC*(*r*) the Coulomb potential and*r* the relative position vector of the interacting nucleons. In this work the AV4� *NN* potential with the *VC*1(*r*) Coulomb component is used (Wiringa & Pieper, 2002). In operator form the general Argonne nucleon-nucleon potential can be written in the form

*m*

1 , *σ<sup>i</sup>* ·*σ<sup>j</sup>*

1 ∑ *S*=0 *Vc*

where the subscripts *S* (*T*) indicate the coupled spin (isospin) of the interacting nucleon pair.

1 ∑ *T*=0

*ij*

*V<sup>τ</sup>* = 2 

*V<sup>σ</sup>* = 2 

*V*(*rij*) = ∑

*vm*(*rij*) <sup>O</sup>*<sup>m</sup>*

<sup>⊗</sup>

*ST*(*rij*) <sup>Ω</sup>(*S*)

and Ω(*T*)

*ij* <sup>−</sup> *<sup>V</sup>τ*(*rij*) *<sup>P</sup><sup>σ</sup>*

where the dependence on *r* is understood. These radial form factors are very difficult to

1 , *τ<sup>i</sup>* ·*τ<sup>j</sup>*

*ij* <sup>Ω</sup>(*T*)

*ij* <sup>=</sup> <sup>1</sup> 2 

*VNN*(*rij*) + *VC*(*rij*)

*ij* two-body nucleon operators. The Argonne V4'

*ij* (23)

. (24)

*ij* (25)

*ij*

*ij*) is the spin (isospin)

*ij* . (27)

(26)

<sup>1</sup> + (−1)*T*+<sup>1</sup> *<sup>P</sup><sup>τ</sup>*

*ij <sup>P</sup><sup>τ</sup>*

*ij* (*P<sup>τ</sup>*

*ij* <sup>−</sup> *<sup>V</sup>στ*(*rij*) *<sup>P</sup><sup>σ</sup>*

*Vc* = *vc* − *v<sup>τ</sup>* − *v<sup>σ</sup>* + *vστ* (28)

*Vστ* = 4 *vστ* (31)

*<sup>v</sup><sup>τ</sup>* <sup>−</sup> *<sup>v</sup>στ* (29)

*<sup>v</sup><sup>σ</sup>* <sup>−</sup> *<sup>v</sup>στ* (30)

(22)

be subtracted from the expectation value of the Hamiltonian of the system.

*h*¯ 2 2 *mi* ∇2 *<sup>i</sup>* + 1 <sup>2</sup> ∑ *i*�=*j*

potential employed in this work consists of the first four operators

*V*(*rij*) =

<sup>O</sup>1→<sup>4</sup> *ij* <sup>≡</sup>

<sup>1</sup> + (−1)*S*+<sup>1</sup> *<sup>P</sup><sup>σ</sup>*

*V*(*rij*) = *Vc*(*rij*) + *Vσ*(*rij*) *P<sup>τ</sup>*

of the spin and isospin projection operators, respectively, where *P<sup>σ</sup>*

The relation between the form factors in equations (23) and (27) are

approximate satisfactorily with less than twenty Gaussian functions.

**4. Nuclear ground state properties**

the nucleus the nuclear Hamiltonian has the form

where *vm*(*rij*) are radial form factors and <sup>O</sup>*<sup>m</sup>*

Using the transformations

Ω(*S*) *ij* <sup>=</sup> <sup>1</sup> 2 

exchange operator, one obtains

The V4 potential model can also be cast in the form

*<sup>H</sup>* = − ∑ *i*

is the energy functional of the system. Minimizing the action in Eq. (13) with the constraints (14) results in the equations

$$\frac{\mathrm{i}\,\mathrm{i}\,\mathrm{\hbar}}{2}\sum\_{\mathrm{i}}\left(\frac{\partial^{2}\ln\langle\mathbf{V}|\Psi\rangle}{\partial\,\vec{s}\_{\mathrm{i}}\,\partial\,\vec{s}\_{\mathrm{j}}^{\ast}}\right)\frac{\mathrm{d}\,\vec{s}\_{\mathrm{i}}}{\mathrm{d}\,t}=\frac{\mathrm{\partial}E}{\mathrm{\partial}\,\vec{s}\_{\mathrm{j}}^{\ast}}\qquad\mathrm{and}\qquad-\frac{\mathrm{i}\,\mathrm{\hbar}}{2}\sum\_{\mathrm{j}}\left(\frac{\partial^{2}\ln\langle\mathbf{V}|\Psi\rangle}{\partial\,\vec{s}\_{\mathrm{i}}\,\partial\,\vec{s}\_{\mathrm{j}}^{\ast}}\right)\frac{\mathrm{d}\,\vec{s}\_{\mathrm{j}}^{\ast}}{\mathrm{d}\,t}=\frac{\mathrm{\partial}E}{\mathrm{\partial}\,\vec{s}\_{\mathrm{i}}}\tag{16}$$

of the variational parameters. Defining a Hermitian and positive definite matrix C with elements

$$\mathcal{L}\_{ij} = \frac{\partial^2 \ln \langle \Psi | \Psi \rangle}{\partial \vec{s}\_i \partial \vec{s}\_j^\*} \,' \, \tag{17}$$

the equations of motion can be compactly expressed in the form (Kramer & Saraceno, 1981)

$$
\frac{1}{2} \begin{bmatrix} 0 & \mathcal{L} \\\\ -\mathcal{L}^\* & 0 \end{bmatrix} \begin{bmatrix} \frac{\mathbf{d} \cdot \vec{s}^\*}{\mathbf{d} \cdot t} \\\\ \frac{\mathbf{d} \cdot \vec{s}}{\mathbf{d} \cdot t} \end{bmatrix} = \begin{bmatrix} \frac{\partial E}{\partial \vec{s}^\*} \\\\ \frac{\partial E}{\partial \vec{s}} \end{bmatrix} . \tag{18}
$$

The solution to these equations provide the time evolution of the variational parameters and therefore of the wave function of the system.

To determine the parameters�*si* and the variational energy *E* of the system Eq. (18) are modified by multiplying the right-hand-side by a complex constant *μ* = *a* + i*b* where *a* and *b* are arbitrary real numbers. The coefficient *μ* introduce friction in the equations. The resulting equations can be solved numerically. To show that solving the modified equation of motion minimizes the energy the matrix C is replaced by a simpler positive definite matrix, the unit matrix *I* (Ono & Horiuchi, 2004). Then the equations of motion for the variational parameters takes the form

$$\frac{\mathrm{d}\vec{s}\_{\mathrm{i}}}{\mathrm{d}\,t} = \frac{2\mu}{\mathrm{i}\,\hbar} \frac{\partial}{\partial \vec{s}\_{\mathrm{i}}^{\*}} \qquad \text{and} \qquad \frac{\mathrm{d}\vec{s}^{\*}}{\mathrm{d}\,t} = -\frac{2\mu^{\*}}{\mathrm{i}\,\hbar} \frac{\partial}{\partial \vec{s}\_{\mathrm{i}}} \,. \tag{19}$$

These equations have a form similar to that of Hamilton's classical equations of motion for canonically conjugate variables. The variation of the energy functional with time leads to

$$\frac{\partial \mathbf{E}}{\partial \mathbf{d}} = \sum\_{\mathbf{i}} \left[ \frac{\partial E}{\partial \vec{s}\_{\mathbf{i}}} \frac{\mathbf{d} \, \mathbf{\tilde{s}}\_{\mathbf{i}}}{\mathbf{d} \, \mathbf{t}} + \frac{\partial E}{\partial \vec{s}\_{\mathbf{i}}^{\*}} \frac{\mathbf{d} \, \mathbf{\tilde{s}}\_{\mathbf{i}}^{\*}}{\mathbf{d} \, \mathbf{t}} \right] = \frac{4b}{\hbar} \sum\_{\mathbf{i}} \frac{\partial E}{\partial \vec{s}\_{\mathbf{i}}} \frac{\partial E}{\partial \vec{s}\_{\mathbf{i}}^{\*}}.\tag{20}$$

The same result can be obtained using the positive definite matrix C (Ono & Horiuchi, 2004). Since the two terms in Eq. (20) are positive at all times, it is evident that

$$\frac{\text{d}\,\text{E}}{\text{d}\,t} < 0 \quad \text{when} \quad b < 0 \,\,\text{A} \tag{21}$$

so that choosing *b* < 0 results in the energy functional decreasing with time during the variation of the parameters. Hence the parameter *μ* is referred to as the coefficient of friction and this technique of lowering the energy of a system is called *frictional cooling*. The equations are solved with the constraint ∑*<sup>i</sup> si* = 0. The zero-point oscillation of the center-of-mass need be subtracted from the expectation value of the Hamiltonian of the system.

#### **4. Nuclear ground state properties**

6 Will-be-set-by-IN-TECH

�Ψ(� <sup>S</sup>)|Ψ(�

and <sup>−</sup> i ¯*<sup>h</sup>*

of the variational parameters. Defining a Hermitian and positive definite matrix C with

<sup>C</sup>*ij* <sup>=</sup> *<sup>∂</sup>*<sup>2</sup> ln�Ψ|Ψ� *∂*�*si ∂*�*s*<sup>∗</sup> *j*

the equations of motion can be compactly expressed in the form (Kramer & Saraceno, 1981)

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

The solution to these equations provide the time evolution of the variational parameters and

To determine the parameters�*si* and the variational energy *E* of the system Eq. (18) are modified by multiplying the right-hand-side by a complex constant *μ* = *a* + i*b* where *a* and *b* are arbitrary real numbers. The coefficient *μ* introduce friction in the equations. The resulting equations can be solved numerically. To show that solving the modified equation of motion minimizes the energy the matrix C is replaced by a simpler positive definite matrix, the unit matrix *I* (Ono & Horiuchi, 2004). Then the equations of motion for the variational parameters

and <sup>d</sup>�*s*<sup>∗</sup>

d ˜s∗ i d *t*

� <sup>=</sup> <sup>4</sup>*<sup>b</sup> <sup>h</sup>*¯ ∑ *i*

These equations have a form similar to that of Hamilton's classical equations of motion for canonically conjugate variables. The variation of the energy functional with time leads to

> *∂ E ∂*�*s*∗ *i*

The same result can be obtained using the positive definite matrix C (Ono & Horiuchi, 2004).

so that choosing *b* < 0 results in the energy functional decreasing with time during the variation of the parameters. Hence the parameter *μ* is referred to as the coefficient of friction and this technique of lowering the energy of a system is called *frictional cooling*. The equations

d�*s*∗ d *t* d�*s* d *t*

⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣

*∂ E ∂*�*s*∗ ⎤ ⎥ ⎥ ⎥ ⎥

*∂ E ∂*�*s*

<sup>d</sup> *<sup>t</sup>* <sup>=</sup> <sup>−</sup>2*μ*<sup>∗</sup> i ¯*h*

*∂ E ∂*�*si*

*∂ E ∂*�*si*

< 0 when *b* < 0 , (21)

*∂ E ∂*�*s*∗ *i*

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

is the energy functional of the system. Minimizing the action in Eq. (13) with the constraints

<sup>S</sup>)<sup>|</sup> *<sup>H</sup>* <sup>|</sup>Ψ(�

S)�

�

*<sup>∂</sup>*<sup>2</sup> ln�Ψ|Ψ� *∂*�*si ∂*�*s*<sup>∗</sup> *j*

<sup>2</sup> ∑ *j*

<sup>S</sup>)� (15)

� d ˜s<sup>∗</sup> j <sup>d</sup>*<sup>t</sup>* <sup>=</sup> *<sup>∂</sup><sup>E</sup> ∂*�*si*

, (17)

<sup>⎦</sup> . (18)

. (19)

. (20)

(16)

<sup>S</sup>∗) = �Ψ(�

*E*(� S,�

where

i ¯*h* <sup>2</sup> ∑ *i*

elements

takes the form

(14) results in the equations

*<sup>∂</sup>*<sup>2</sup> ln�Ψ|Ψ� *∂*�*si ∂*�*s*<sup>∗</sup> *j*

� d ˜si <sup>d</sup>*<sup>t</sup>* <sup>=</sup> *<sup>∂</sup><sup>E</sup> ∂*�*s*∗ *j*

> i ¯*h* 2

therefore of the wave function of the system.

d�*si* <sup>d</sup> *<sup>t</sup>* <sup>=</sup> <sup>2</sup>*<sup>μ</sup>* i ¯*h*

d E <sup>d</sup> *<sup>t</sup>* <sup>=</sup> ∑ *i*

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

0 C

−C<sup>∗</sup> 0

*∂ E ∂*�*s*∗ *i*

� *∂ E ∂*�*si*

d ˜si <sup>d</sup> *<sup>t</sup>* <sup>+</sup>

Since the two terms in Eq. (20) are positive at all times, it is evident that d E d *t*

�

To evaluate the variational energy functional and solve for the variation parameters, the Hamiltonian *H* of the nucleus is required as input. Considering only two-body interactions in the nucleus the nuclear Hamiltonian has the form

$$H = -\sum\_{i} \frac{\hbar^2}{2m\_i} \nabla\_i^2 + \frac{1}{2} \sum\_{i \neq j} \left[ V\_{NN}(\vec{r}\_{ij}) + V\_{\mathbb{C}}(\vec{r}\_{ij}) \right] \tag{22}$$

where *mi* is the mass of nucleon *i*, *VNN*(*r*) the nucleon-nucleon *NN* potential, *VC*(*r*) the Coulomb potential and*r* the relative position vector of the interacting nucleons. In this work the AV4� *NN* potential with the *VC*1(*r*) Coulomb component is used (Wiringa & Pieper, 2002). In operator form the general Argonne nucleon-nucleon potential can be written in the form

$$V(\vec{r}\_{ij}) = \sum\_{m} v\_{m}(r\_{ij}) \, \mathcal{O}^{m}\_{ij} \tag{23}$$

where *vm*(*rij*) are radial form factors and <sup>O</sup>*<sup>m</sup> ij* two-body nucleon operators. The Argonne V4' potential employed in this work consists of the first four operators

$$\mathcal{O}\_{ij}^{1 \to 4} \equiv \left\{ 1, \vec{\sigma}\_{\dot{l}} \cdot \vec{\sigma}\_{\dot{l}} \right\} \otimes \left\{ 1, \vec{\tau}\_{\dot{l}} \cdot \vec{\tau}\_{\dot{l}} \right\}. \tag{24}$$

The V4 potential model can also be cast in the form

$$V(\vec{r}\_{ij}) = \sum\_{T=0}^{1} \sum\_{S=0}^{1} V\_{ST}^{c}(r\_{ij}) \, \Omega\_{ij}^{(S)} \, \Omega\_{ij}^{(T)} \tag{25}$$

where the subscripts *S* (*T*) indicate the coupled spin (isospin) of the interacting nucleon pair. Using the transformations

$$
\Omega\_{ij}^{(S)} = \frac{1}{2} \left[ 1 + (-1)^{S+1} P\_{ij}^{\sigma} \right] \qquad \text{and} \qquad \Omega\_{ij}^{(T)} = \frac{1}{2} \left[ 1 + (-1)^{T+1} P\_{ij}^{\tau} \right] \tag{26}
$$

of the spin and isospin projection operators, respectively, where *P<sup>σ</sup> ij* (*P<sup>τ</sup> ij*) is the spin (isospin) exchange operator, one obtains

$$V(\vec{r}\_{\text{ij}}) = V\_{\mathfrak{c}}(r\_{\text{ij}}) + V\_{\mathfrak{c}}(r\_{\text{ij}}) \, P\_{\text{ij}}^{\mathfrak{r}} - V\_{\mathfrak{r}}(r\_{\text{ij}}) \, P\_{\text{ij}}^{\mathfrak{r}} - V\_{\mathfrak{c}\mathfrak{r}}(r\_{\text{ij}}) \, P\_{\text{ij}}^{\mathfrak{r}} \, P\_{\text{ij}}^{\mathfrak{r}} \, . \tag{27}$$

The relation between the form factors in equations (23) and (27) are

$$V\_{\mathcal{L}} = v\_{\mathcal{L}} - v\_{\tau} - v\_{\mathcal{U}} + v\_{\mathcal{U}\tau} \tag{28}$$

$$V\_{\tau} = 2\left[v\_{\tau} - v\_{\sigma\tau}\right] \tag{29}$$

$$V\_{\sigma} = 2\left[v\_{\sigma} - v\_{\sigma\tau}\right] \tag{30}$$

$$V\_{\sigma\tau} = 4\,\upsilon\_{\sigma\tau} \tag{31}$$

where the dependence on *r* is understood. These radial form factors are very difficult to approximate satisfactorily with less than twenty Gaussian functions.

The variational energy *<sup>E</sup>*0, root-mean-square (rms) radius �*r*2� and magnetic moment *<sup>μ</sup>* of the nuclei are calculated using the parity projected AMD wave function, Ψ*π*. These three quantities are given by

$$E\_0 = \frac{\langle \Psi^\pi | H | \Psi^\pi \rangle}{\langle \Psi^\pi | \Psi^\pi \rangle} \tag{32}$$

of the series in (37) were considered. The integral in *Iα*(*m*) is evaluated numerically using a Gaussian quadrature. The value of *α* was chosen to satisfactorily reproduce the experimental binding energy of the three-nucleon systems. The variational energies are corrected by subtracting contributions from spurious center-of-mass. The results are presented along with corresponding experimental data in Table 1. As can be observed from this table the theoretical prediction of the experimental binding energies of the three-nucleon systems is satisfactory, as expected. However, the experimental binding energy of the 2H nucleus is overestimated by about 60 %. The binding energy of the 4He nucleus is underestimated by 10 % whereas that of the 6Li and 8Be nuclei are underestimated by 14 %. The 8Be nucleus is very unstable

Antisymmetrized Molecular Dynamics and Nuclear Structure 141

The AMD generates reasonable predictions of the experimental values for the rms radii of the nuclei. The deviation of the predicted values from the experimental values for the nuclei range from 0 %, for the 3He nucleus, to 11 %, for the 6Li nucleus. The theoretical radii of the 6Li and the 8Be nuclei are almost the same. In general the theoretical results overestimate the experimental values of the rms radii for all the nuclei. The calculated magnetic moments of the three-nucleon systems are different from the experimental values by 6 % and 10 %. For the three-nucleon systems the theoretical moments equal the magnetic moment of the unlike nucleon in the system. This reason also explains why the calculated magnetic moments of the 2H and 6Li nuclei, which overestimate the respective experimental values by 3 % and 7 %, are equal. The calculated magnetic moments of the 4He and 8Be nuclei are both equal to zero. These results are consistent with the theoretical expectation when only the dominant spherical ground state wave function of the system is used in the calculations (Phillips, 1977). The results for the parity and angular momentum projected wave function are given in Table 2 for the three-nucleon and the four-nucleon systems. The parity projection is done before the variation of the parameters and the angular momentum projection is done after the optimisation of the parameters. The rotation increases the theoretical energies of the three-nucleon systems by 10 % and decreases that of the four-nucleon system by 10 %, relative to the un-rotated wave function. Since the nuclear systems are described with spherical wave functions, spatial rotations are not expected to introduce significant modifications to the results presented in Table 1. The binding energy found for the 3H system overestimates

*<sup>E</sup>*<sup>0</sup> (MeV) *<sup>r</sup>*<sup>2</sup> (fm) *<sup>μ</sup>* (*μN*)

*<sup>A</sup>*X*<sup>π</sup>* AMD EXP AMD EXP AMD EXP 2H<sup>+</sup> -3.53 -2.23 2.16 1.96 0.880 0.857 3H<sup>+</sup> -8.12 -8.48 1.76 1.60 2.793 2.979 3He<sup>+</sup> -7.78 -7.72 1.76 1.77 -1.913 -2.128

6Li<sup>+</sup> -27.54 -31.99 2.70 2.4 0.880 0.822

Table 1. The ground state energies, rms radii and magnetic moments of selected light nuclei.

4He<sup>+</sup> -25.50 -28.3 1.53 1.47 0.000

8Be<sup>+</sup> -49.38 -55.99 2.75 0.000

The experimental data are taken from reference (Suzuki et al., 2008).

(Audi et al., 2003) and, therefore, very challenging to study experimentally.

$$
\langle r^2 \rangle = \frac{1}{A} \frac{\langle \Psi^\pi | \sum\_{i=1}^A \left[ \vec{r}\_i - \vec{R} \right]^2 | \Psi^\pi \rangle}{\langle \Psi^\pi | \Psi^\pi \rangle} \tag{33}
$$

$$\vec{\mu} = \frac{\langle \Psi^{\pi} | \sum\_{i=1}^{A} \left[ g\_{\ell} \vec{\ell}\_{i} + g\_{s} \vec{s}\_{i} \right] | \Psi^{\pi} \rangle}{\langle \Psi^{\pi} | \Psi^{\pi} \rangle} \tag{34}$$

where *R* is the center-of-mass of the nucleus,*<sup>i</sup>* (*si*) the orbital (spin) angular momentum and *g* (*gs*) the corresponding *g*-factor of a nucleon. The *g*-factors are constants with values (Wong, 1998)

$$\begin{array}{ll} \text{g } g\_{\ell} = \begin{cases} \text{1 for proton} \\ \text{0 for neutron} \end{cases} & \text{:} \qquad g\_{s} = \begin{cases} \text{ 5.585695 for proton} \\ -\text{3.826085 for neutron} \end{cases} \end{array} \tag{35}$$

The integrals in equations (32), (33) and (34) can be analytically evaluated (Rampho, 2010). As an illustration the evaluation of the variational energy is summarized. The variational energy is evaluated as

$$E\_0 = \sum\_{\mathbf{i}\mathbf{k}} \mathcal{T}\_{\mathbf{i}\mathbf{k}} \mathcal{B}\_{\mathbf{i}\mathbf{k}}^{-1} + \sum\_{\mathbf{i}\mathbf{j}\mathbf{k}\mathbf{l}} \mathcal{V}\_{\mathbf{i}\mathbf{j}\mathbf{k}\mathbf{l}} \mathcal{B}\_{\mathbf{i}\mathbf{i}} \mathcal{B}\_{\mathbf{j}\mathbf{j}} \left[\mathcal{P}\_{\mathbf{i}\mathbf{j}}^{\mathbf{d}} \mathcal{B}\_{\mathbf{i}\mathbf{l}}^{-1} \mathcal{B}\_{\mathbf{j}\mathbf{k}}^{-1} - \mathcal{P}\_{\mathbf{i}\mathbf{j}}^{\mathbf{e}} \mathcal{B}\_{\mathbf{i}\mathbf{k}}^{-1} \mathcal{B}\_{\mathbf{j}\mathbf{l}}^{-1}\right] \tag{36}$$

where the first sum generate the total kinetic energy while the second generate the total potential energy of the system. In this energy functional T*ik* and V*ijkl* represent the integrals involving the kinetic operator and potential functions, respectively, with *Bij* <sup>=</sup> �*ψi*|*ψj*� and <sup>P</sup><sup>d</sup> *ij* (Pe *ij*) resulting from direct (exchange) interactions. The evaluation of the expectation values of the potential can be evaluated analytically when the these radial components are given in terms of Gaussian functions (Tohsaki, 1992). Instead of expanding the potential in terms of Gaussians the corresponding expectation value is approximated by the series (Rampho, 2011)

$$\mathcal{V}\_{ijkl} = \exp\left(-\frac{\eta\_2^2}{4}\right) \sum\_{m=0}^{\infty} \left(\frac{\eta\_2^2}{4}\right)^m I\_{\mathbb{R}}(m) \tag{37}$$

where *η*<sup>2</sup> =*sl* −*sk* +*si* −*sj* and

$$I\_a(m) = \frac{2\,\mathrm{a}^{m+3/2}}{\Gamma(m+1)\Gamma(m+\frac{3}{2})} \int\_0^\infty r^{2\,m+2} V(r) \mathrm{e}^{-a\,r^2} \,\mathrm{d}r \tag{38}$$

the Talmi integral involving the gamma function Γ(*x*). If the range of the potential is not greater than 1/√<sup>2</sup> *<sup>α</sup>* then the series in Eq. (37) converges quite fast for any value of *<sup>η</sup>*<sup>2</sup> (Brink, 1965). Therefore, only the first few terms in the series tend to be significant. The Coulomb potential is treated with a Gaussian integral transform. In solving the cooling equations a random number generator is used to set up the initial values of the variational parameters. The required wave function is obtained when the energy functional for the system is independent of the variations of the parameters of the wave function. The parity projection of the wave functions is done before the variation of the parameters. Only the first five terms 8 Will-be-set-by-IN-TECH

The variational energy *<sup>E</sup>*0, root-mean-square (rms) radius �*r*2� and magnetic moment *<sup>μ</sup>* of the nuclei are calculated using the parity projected AMD wave function, Ψ*π*. These three

*<sup>i</sup>* + *gssi*

*R* is the center-of-mass of the nucleus,*<sup>i</sup>* (*si*) the orbital (spin) angular momentum and *g* (*gs*) the corresponding *g*-factor of a nucleon. The *g*-factors are constants with values (Wong,

The integrals in equations (32), (33) and (34) can be analytically evaluated (Rampho, 2010). As an illustration the evaluation of the variational energy is summarized. The variational energy

where the first sum generate the total kinetic energy while the second generate the total potential energy of the system. In this energy functional T*ik* and V*ijkl* represent the integrals involving the kinetic operator and potential functions, respectively, with *Bij* <sup>=</sup> �*ψi*|*ψj*� and <sup>P</sup><sup>d</sup>

*ij*) resulting from direct (exchange) interactions. The evaluation of the expectation values of the potential can be evaluated analytically when the these radial components are given in terms of Gaussian functions (Tohsaki, 1992). Instead of expanding the potential in terms of Gaussians the corresponding expectation value is approximated by the series (Rampho, 2011)

> ∞ ∑ *m*=0

> > 2 )

the Talmi integral involving the gamma function Γ(*x*). If the range of the potential is not greater than 1/√<sup>2</sup> *<sup>α</sup>* then the series in Eq. (37) converges quite fast for any value of *<sup>η</sup>*<sup>2</sup> (Brink, 1965). Therefore, only the first few terms in the series tend to be significant. The Coulomb potential is treated with a Gaussian integral transform. In solving the cooling equations a random number generator is used to set up the initial values of the variational parameters. The required wave function is obtained when the energy functional for the system is independent of the variations of the parameters of the wave function. The parity projection of the wave functions is done before the variation of the parameters. Only the first five terms

 ∞ 0

 *η*2 2 4

*<sup>m</sup>*

*r*<sup>2</sup> *<sup>m</sup>*+2*V*(*r*)e−*<sup>α</sup> <sup>r</sup>*<sup>2</sup>

 Pd *ij <sup>B</sup>*−<sup>1</sup> *il <sup>B</sup>*−<sup>1</sup>

V*ijkl Bli Bkj*

 <sup>Ψ</sup>*π*

�Ψ*π*|Ψ*π*� (32)

�Ψ*π*|Ψ*π*� (33)

�Ψ*π*|Ψ*π*� (34)

5.585695 for proton

*ij <sup>B</sup>*−<sup>1</sup> *ik <sup>B</sup>*−<sup>1</sup> *jl* 

*jk* − P<sup>e</sup>

<sup>−</sup>3.826085 for neutron (35)

*Iα*(*m*) (37)

d*r* (38)

(36)

*ij*

*<sup>E</sup>*<sup>0</sup> <sup>=</sup> �Ψ*π*<sup>|</sup> *<sup>H</sup>* <sup>|</sup>Ψ*π*�

�*r*2� <sup>=</sup> <sup>1</sup> *A* Ψ*<sup>π</sup>* <sup>∑</sup>*<sup>A</sup> i*=1 *ri* <sup>−</sup> *R* <sup>2</sup> <sup>Ψ</sup>*π*

*μ* = Ψ*<sup>π</sup>* <sup>∑</sup>*<sup>A</sup> i*=1 *g*

0 for neutron : *gs* <sup>=</sup>

quantities are given by

*g* =

*E*<sup>0</sup> = ∑ *ik*

where *η*<sup>2</sup> =*sl* −*sk* +*si* −*sj* and

1 for proton

<sup>T</sup>*ik <sup>B</sup>*−<sup>1</sup>

*ik* + ∑ *ijkl*

V*ijkl* = exp

*<sup>I</sup>α*(*m*) = <sup>2</sup> *<sup>α</sup>m*+3/2

 − *η*2 2 4

Γ(*m* + 1)Γ(*m* + <sup>3</sup>

where

is evaluated as

1998)

(Pe

of the series in (37) were considered. The integral in *Iα*(*m*) is evaluated numerically using a Gaussian quadrature. The value of *α* was chosen to satisfactorily reproduce the experimental binding energy of the three-nucleon systems. The variational energies are corrected by subtracting contributions from spurious center-of-mass. The results are presented along with corresponding experimental data in Table 1. As can be observed from this table the theoretical prediction of the experimental binding energies of the three-nucleon systems is satisfactory, as expected. However, the experimental binding energy of the 2H nucleus is overestimated by about 60 %. The binding energy of the 4He nucleus is underestimated by 10 % whereas that of the 6Li and 8Be nuclei are underestimated by 14 %. The 8Be nucleus is very unstable (Audi et al., 2003) and, therefore, very challenging to study experimentally.

The AMD generates reasonable predictions of the experimental values for the rms radii of the nuclei. The deviation of the predicted values from the experimental values for the nuclei range from 0 %, for the 3He nucleus, to 11 %, for the 6Li nucleus. The theoretical radii of the 6Li and the 8Be nuclei are almost the same. In general the theoretical results overestimate the experimental values of the rms radii for all the nuclei. The calculated magnetic moments of the three-nucleon systems are different from the experimental values by 6 % and 10 %. For the three-nucleon systems the theoretical moments equal the magnetic moment of the unlike nucleon in the system. This reason also explains why the calculated magnetic moments of the 2H and 6Li nuclei, which overestimate the respective experimental values by 3 % and 7 %, are equal. The calculated magnetic moments of the 4He and 8Be nuclei are both equal to zero. These results are consistent with the theoretical expectation when only the dominant spherical ground state wave function of the system is used in the calculations (Phillips, 1977). The results for the parity and angular momentum projected wave function are given in Table 2 for the three-nucleon and the four-nucleon systems. The parity projection is done before the variation of the parameters and the angular momentum projection is done after the optimisation of the parameters. The rotation increases the theoretical energies of the three-nucleon systems by 10 % and decreases that of the four-nucleon system by 10 %, relative to the un-rotated wave function. Since the nuclear systems are described with spherical wave functions, spatial rotations are not expected to introduce significant modifications to the results presented in Table 1. The binding energy found for the 3H system overestimates


Table 1. The ground state energies, rms radii and magnetic moments of selected light nuclei. The experimental data are taken from reference (Suzuki et al., 2008).


Table 2. The ground state energies, rms radii and magnetic moments of the three- and four-nucleon systems. The experimental values are taken from reference (Suzuki et al., 2008).

the experimental energy by 5 % and for the 3He system by 12 %. In contrast, the 4He results are lower than the experimental value by ∼ 19 %, a result which is in line with other calculations in the field using the same rank in the potential. The rms radii obtained for the 3H and 3He systems are lower than the experimental values by <sup>∼</sup> 16 % and <sup>∼</sup> 22 % less, respectively. Similar results are obtained for the 4He system where the calculated rms radius is underestimated by ∼ 21 %. In general, the AMD approach reproduces the experimental values for the magnetic moment of the nuclei quite satisfactorily.

Intrinsic density distribution in a nucleus is determined by rotating the arbitrary space-fixed coordinate axes on to the body-fixed principal axes of the nucleus. The rotation is generated by diagonalising the moment-of-inertia tensor of the nucleus which is defined by elements

$$I\_{\mu\nu} = \frac{\left\langle \Psi^{\pi} \left| \sum\_{i=1}^{A} \left[ \vec{r}\_{i} - \vec{R} \right]\_{\mu} \left[ \vec{r}\_{i} - \vec{R} \right]\_{\nu} \right| \Psi^{\pi} \right\rangle}{\left\langle \Psi^{\pi} \middle| \Psi^{\pi} \right\rangle} \tag{39}$$


momentum projected wave functions are employed.

nuclear charge operator. For a nucleus in an initial state

*<sup>F</sup>*ch(*<sup>q</sup>*) = <sup>1</sup>

*Ek*(*Q*2) <sup>−</sup> <sup>2</sup> *<sup>G</sup><sup>N</sup>*

*Z* Ψ*J<sup>π</sup> MK ρ*(*q*) <sup>Ψ</sup>*J<sup>π</sup> MK* 

 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Fig. 2. Density distributions of the 6Li nucleus (left) and the 8Be nucleus (right).

Most of the information accumulated about nuclear structure is derived from electron-nucleus scattering. In electron-nucleus scattering the electron transfers momentum *q* and energy *ω* to the target nucleus and the nucleus undergoes some transitions that are governed by selection rules related angular momentum and parity. For elastic scattering the initial and final states of the nucleus have the same angular momentum (de Forest Jr. & Walecka, 1966). This type of electron scattering is used to probe ground state charge and magnetisation distributions in nuclei. The ground state of the 3He nucleus has total angular momentum and parity

provide a good test of the quality of the wave function describing the system. In this section the charge form factors of 3He and 4He nuclei are calculated in the plane wave impulse approximation (PWIA) (Chew & Wick, 1952). In this approximation the nucleons inside the target nucleus are assumed non-interacting with one another during the interaction with the electron. This means that the electron interacts with independent nucleons inside the nucleus. Since the transitions are between states of definite angular momentum, the parity and angular

The charge distribution in the nucleus is inferred from the electric transitions in the nucleus due elastic electron-nucleus scattering. The charge form factor is the expectation value of the

> Ψ*J<sup>π</sup> MK* <sup>Ψ</sup>*J<sup>π</sup> MK*

where *Ze* is the total charge on and *ρ*(*q*) the charge operator of the nucleus with *q* being the momentum transferred to the nucleus by the electron. In the PWIA the nuclear charge operator is formed by the superposition of the individual nucleon charge operators and is

*Mk*(*Q*2) <sup>−</sup> *<sup>G</sup><sup>N</sup>*

<sup>√</sup><sup>1</sup> <sup>+</sup> *<sup>τ</sup>*

4 *m*<sup>2</sup> *N* <sup>Ψ</sup>*J<sup>π</sup> MK* 

*Ek* (*Q*2)

i*σ<sup>k</sup>* ·*q* ×*pk*

 exp

<sup>+</sup> whereas the 4He system has *<sup>J</sup><sup>π</sup>* <sup>=</sup> <sup>0</sup>+. Theoretical predictions of electromagnetic


 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

the charge form factor is given

i*q* ·*rk*

(41)

(40)



Y [fm]

5

Antisymmetrized Molecular Dynamics and Nuclear Structure 143


**5. Charge form factors**

*J<sup>π</sup>* = <sup>1</sup> 2

by

given by (Rampho, 2010)

*ρ*(*q*) =

*A* ∑ *k*=1  *q <sup>Q</sup> <sup>G</sup><sup>N</sup>*


3

Y [fm]

6

9

where { *μ*, *ν* }≡{ *x*, *y*, *z* }. The calculated intrinsic density using the parity projected wave function, in the body-fixed coordinate axes, are shown in Figure 1 for the 4He nucleus and in Figure 2 for the 6Li and 8Be nuclei. The 4He nuclei displays a spherical density distribution whereas the 6Li and 8Be nuclei display distinct cluster structures.

Fig. 1. Density distribution of the 4He nucleus.

Fig. 2. Density distributions of the 6Li nucleus (left) and the 8Be nucleus (right).

#### **5. Charge form factors**

10 Will-be-set-by-IN-TECH

*J<sup>π</sup>* AMD EXP AMD EXP AMD EXP

<sup>+</sup> -8.95 -8.48 1.33 1.60 2.769 2.979

<sup>+</sup> -8.61 -7.72 1.33 1.77 -1.847 -2.128

0<sup>+</sup> -23.04 -28.30 1.16 1.47 0.000

Table 2. The ground state energies, rms radii and magnetic moments of the three- and four-nucleon systems. The experimental values are taken from reference (Suzuki et al., 2008). the experimental energy by 5 % and for the 3He system by 12 %. In contrast, the 4He results are lower than the experimental value by ∼ 19 %, a result which is in line with other calculations in the field using the same rank in the potential. The rms radii obtained for the 3H and 3He systems are lower than the experimental values by <sup>∼</sup> 16 % and <sup>∼</sup> 22 % less, respectively. Similar results are obtained for the 4He system where the calculated rms radius is underestimated by ∼ 21 %. In general, the AMD approach reproduces the experimental

Intrinsic density distribution in a nucleus is determined by rotating the arbitrary space-fixed coordinate axes on to the body-fixed principal axes of the nucleus. The rotation is generated by diagonalising the moment-of-inertia tensor of the nucleus which is defined by elements

where { *μ*, *ν* }≡{ *x*, *y*, *z* }. The calculated intrinsic density using the parity projected wave function, in the body-fixed coordinate axes, are shown in Figure 1 for the 4He nucleus and in Figure 2 for the 6Li and 8Be nuclei. The 4He nuclei displays a spherical density distribution

> -4 0 4 X [fm]

�Ψ*π*|Ψ*π*� (39)

 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

values for the magnetic moment of the nuclei quite satisfactorily.

 Ψ*π* <sup>∑</sup>*<sup>A</sup> i*=1 *ri* <sup>−</sup> *R μ ri* <sup>−</sup> *R ν* Ψ*π*

whereas the 6Li and 8Be nuclei display distinct cluster structures.


Fig. 1. Density distribution of the 4He nucleus.


Y [fm]

3

*Iμν* =

*<sup>A</sup>*X 

3H <sup>1</sup> 2

3He <sup>1</sup> 2

4He

*<sup>E</sup>*<sup>0</sup> (MeV) �*r*2� (fm) *<sup>μ</sup>* (*μN*)

Most of the information accumulated about nuclear structure is derived from electron-nucleus scattering. In electron-nucleus scattering the electron transfers momentum *q* and energy *ω* to the target nucleus and the nucleus undergoes some transitions that are governed by selection rules related angular momentum and parity. For elastic scattering the initial and final states of the nucleus have the same angular momentum (de Forest Jr. & Walecka, 1966). This type of electron scattering is used to probe ground state charge and magnetisation distributions in nuclei. The ground state of the 3He nucleus has total angular momentum and parity *J<sup>π</sup>* = <sup>1</sup> 2 <sup>+</sup> whereas the 4He system has *<sup>J</sup><sup>π</sup>* <sup>=</sup> <sup>0</sup>+. Theoretical predictions of electromagnetic provide a good test of the quality of the wave function describing the system. In this section the charge form factors of 3He and 4He nuclei are calculated in the plane wave impulse approximation (PWIA) (Chew & Wick, 1952). In this approximation the nucleons inside the target nucleus are assumed non-interacting with one another during the interaction with the electron. This means that the electron interacts with independent nucleons inside the nucleus. Since the transitions are between states of definite angular momentum, the parity and angular momentum projected wave functions are employed.

The charge distribution in the nucleus is inferred from the electric transitions in the nucleus due elastic electron-nucleus scattering. The charge form factor is the expectation value of the nuclear charge operator. For a nucleus in an initial state <sup>Ψ</sup>*J<sup>π</sup> MK* the charge form factor is given by

$$F\_{\rm ch}(\vec{q}) = \frac{1}{Z} \frac{\langle \Psi\_{MK}^{J\pi} | \rho(\vec{q}) | \Psi\_{MK}^{J\pi} \rangle}{\langle \Psi\_{MK}^{J\pi} | \Psi\_{MK}^{J\pi} \rangle} \tag{40}$$

where *Ze* is the total charge on and *ρ*(*q*) the charge operator of the nucleus with *q* being the momentum transferred to the nucleus by the electron. In the PWIA the nuclear charge operator is formed by the superposition of the individual nucleon charge operators and is given by (Rampho, 2010)

$$\rho(\vec{q}) = \sum\_{k=1}^{A} \left[ \frac{q}{Q} \, \mathrm{G}\_{\mathrm{Ek}}^{N}(Q^{2}) - \frac{2 \, \mathrm{G}\_{\mathrm{Mk}}^{N}(Q^{2}) - \mathrm{G}\_{\mathrm{Ek}}^{N}(Q^{2})}{4 \, m\_{N}^{2} \sqrt{1+\tau}} \, \mathrm{i}\vec{\sigma}\_{k} \cdot \vec{q} \times \vec{p}\_{k} \right] \, \exp\left(\mathrm{i}\, \vec{q} \cdot \vec{r}\_{k}\right) \tag{41}$$

where *τ* = *Q*2/4 *m*<sup>2</sup> *<sup>N</sup>*, *<sup>Q</sup>*<sup>2</sup> <sup>=</sup> *<sup>q</sup>*<sup>2</sup> <sup>−</sup> *<sup>ω</sup>*2, *<sup>ω</sup>* <sup>=</sup> *q*<sup>2</sup> + *m*<sup>2</sup> *<sup>N</sup>* <sup>−</sup> *mN* and *<sup>G</sup><sup>N</sup> <sup>E</sup>* (*G<sup>N</sup> <sup>M</sup>*) the nucleon Sachs electric (magnetic) form factor. For the Sachs form factors the phenomenological parametrization derived in Ref. (Friedrich & Walcher, 2003) is adopted. The general multipole analysis of nuclear charge form factors is given by (Uberall, 1971)

$$F\_{\rm ch}(\overrightarrow{q}) = \sqrt{4\,\pi} \sum\_{L=0}^{\leq 2J} \langle JIL0|Jf \rangle \, F\_L^{\rho}(q) \, Y\_{L0}^\*(\hat{\mathbf{q}}) \tag{42}$$

rms radii, and the magnetic moments for selected few-nucleon systems. The nuclear Hamiltonian was constructed from the Argonne AV4 *NN* potential that includes also the Coulomb interaction. Comparison with the experimental data revealed that the reproduction of the ground state properties of light nuclei is quite satisfactory. The discrepancies observed can be attributed to reasons not entirely related to the AMD. These include i) the omission of mixed-symmetric states (for three-body) ii) the use of a limited rank for the Argonne AV18 potential, and iii) the omission of three-nucleon forces. As far as the magnetic moment is concerned, the inclusion of relativistic corrections to the magnetic moment operator, are also expected to contribute to the reduction of the discrepancy between theory and experiment. The technique used in the approximation of the variational energy can be easily extended to the three-body interactions. The implementation of this technique in nuclear three-body

Antisymmetrized Molecular Dynamics and Nuclear Structure 145

The angular momentum and parity projected AMD was used to calculate the ground state charge form factors for the 3He and 4He nuclei. In overall, the results obtained, within the AMD and PWIA approximation, reproduce the general behavior of the experimental form factors. For momentum transfer below the first diffraction minimum the reproduction of experimental form factors is fairly good. However, beyond the first diffraction minimum the results are lower than the data. The deviations of the theoretical results from experimental data can be minimize by employing improved wave functions. The wave functions can be constructed by using a more complete realistic Hamiltonian, three-body forces, and relativistic corrections in the electromagnetic operators. It should be noted that these results are consistent with other results obtained by competing theoretical models. In conclusion, the results indicate that the AMD method is a very promising method in calculating

Work is underway to construct wave functions for scattering processes applicable in nuclear breakup reactions (Rampho, 2010). In these constructs the AMD is combined with the Glauber multiple scattering to account for final state interactions. Wave functions for two-body and three-body scattering reactions can be treated accurately in the Faddeev formalism (Golak et al., 2005; Merkuriev et al., 1976). In the hyperspherical harmonics approach some progress in being made towards the construction of accurate wave functions for two-body scattering processes (Kievsky et al., 2008). However, these methods become involved for

Aichelin, J. (1991). "Quantum" molecular dynamics - a dynamical microscopic n-body

Alvioli, M., Ciofi degli Atti, C. & Morita, H. (2008). Proton-neutron and proton-proton

Amroun, A., Breton, V., Cavedon, J. M., Frois, B., Goutte, D., Juster, F. P., Leconte, P., Martino,

Audi, G., Bersillon, O., Blachot, J. & Wapstra, A. (2003). The NUBASE evaluation of nuclear

3He electromagnetic form factors, *Nucl. Phys.* A 579(3-4): 596–626.

approach to investigate fragment formation and the nuclear equation of state in

correlations in medium-weight nuclei and the role of the tensor force, *Phys. Rev. Lett.*

J., Mizuno, Y., Phan, X. H., Platchkov, S. K., Sick, I. & Williamson, S. (1994). 3H and

interactions will be considered future projects.

electromagnetic form factors of the general *A*-body nuclear system.

systems consisting of more than four constituent particles.

heavy ion collisions, *Phys. Rep.* 202(5): 233–360.

and decay properties, *Nucl. Phys.* A 729(1): 3–128.

**7. References**

100(16): 162503.

where *Y*∗ *LM*(**q**ˆ) are the spherical harmonics, *L* the nuclear orbital angular momentum and �*JJL*0|*J J*� the Clebsch-Gordan coefficients and ˆ**q** = *q*/|*q*|. The summation is over even values of *L* only. The intrinsic charge form factor is corrected by dividing the calculated charge form factor by the contributions of the center-of-mass (Rampho, 2010). The integrals in Eq. (42) can be analytically evaluated.

Fig. 3. The charge form factor of 3He (left) and 4He (right) compared with the experimental data of (McCarthy et al., 1977) and the theoretical fit to the data (Amroun et al., 1994).

The results of the calculated ground-state charge form factors of the 3He and 4He nuclei are presented in Figure 3. In these figures the theoretical charge form factors are compared with experimental data from (Frosch et al., 1967; McCarthy et al., 1977). In the comparison phenomenological parametrization (Amroun et al., 1994) that fit experimental universal data of the form factors for electron-nucleus scattering are also shown. The charge form factors are normalized such that *F*ch(0) = 1. As can be seen in these figures, for low momentum transfers, up to the first diffraction minimum, the AMD gives a reasonable description, albeit it slightly overestimates the experimental data. Beyond the first diffraction minimum the results are lower than the data. The first diffraction minimum for the nuclei are consistent with, but not better than, the predictions of other theoretical models obtained with various nucleon-nucleon potentials in the PWIA (Kloet & Tjon, 1974). It should be noted that the overestimation of the position of the diffraction minimum indicates an underestimation of the nuclear charge radius.

#### **6. Conclusions**

To test the applicability of the AMD model in nuclear structure studies, the angular momentum and parity projected AMD wave function were used to calculate binding energies, 12 Will-be-set-by-IN-TECH

Sachs electric (magnetic) form factor. For the Sachs form factors the phenomenological parametrization derived in Ref. (Friedrich & Walcher, 2003) is adopted. The general multipole

�*JJL*0|*J J*� the Clebsch-Gordan coefficients and ˆ**q** = *q*/|*q*|. The summation is over even values of *L* only. The intrinsic charge form factor is corrected by dividing the calculated charge form factor by the contributions of the center-of-mass (Rampho, 2010). The integrals in Eq. (42) can

�*JJL*0|*J J*� *<sup>F</sup><sup>ρ</sup>*

*LM*(**q**ˆ) are the spherical harmonics, *L* the nuclear orbital angular momentum and

Fig. 3. The charge form factor of 3He (left) and 4He (right) compared with the experimental data of (McCarthy et al., 1977) and the theoretical fit to the data (Amroun et al., 1994).

The results of the calculated ground-state charge form factors of the 3He and 4He nuclei are presented in Figure 3. In these figures the theoretical charge form factors are compared with experimental data from (Frosch et al., 1967; McCarthy et al., 1977). In the comparison phenomenological parametrization (Amroun et al., 1994) that fit experimental universal data of the form factors for electron-nucleus scattering are also shown. The charge form factors are normalized such that *F*ch(0) = 1. As can be seen in these figures, for low momentum transfers, up to the first diffraction minimum, the AMD gives a reasonable description, albeit it slightly overestimates the experimental data. Beyond the first diffraction minimum the results are lower than the data. The first diffraction minimum for the nuclei are consistent with, but not better than, the predictions of other theoretical models obtained with various nucleon-nucleon potentials in the PWIA (Kloet & Tjon, 1974). It should be noted that the overestimation of the position of the diffraction minimum indicates an underestimation of the nuclear charge

To test the applicability of the AMD model in nuclear structure studies, the angular momentum and parity projected AMD wave function were used to calculate binding energies,

*Fch*(*q*2)

10<sup>0</sup>

10−<sup>1</sup>

10−<sup>2</sup>

10−<sup>3</sup>

10−<sup>4</sup>

*<sup>L</sup>*(*q*) *Y*<sup>∗</sup>

*q*<sup>2</sup> + *m*<sup>2</sup>

*<sup>N</sup>* <sup>−</sup> *mN* and *<sup>G</sup><sup>N</sup>*

*<sup>E</sup>* (*G<sup>N</sup>*

*<sup>L</sup>*0(**q**ˆ) (42)

AMD Fit Exp

*q*<sup>2</sup> [fm−<sup>2</sup>]

0 5 10 15 20

*<sup>M</sup>*) the nucleon

*<sup>N</sup>*, *<sup>Q</sup>*<sup>2</sup> <sup>=</sup> *<sup>q</sup>*<sup>2</sup> <sup>−</sup> *<sup>ω</sup>*2, *<sup>ω</sup>* <sup>=</sup>

analysis of nuclear charge form factors is given by (Uberall, 1971)

*<sup>F</sup>*ch(*<sup>q</sup>*) = <sup>√</sup>

AMD Fit Exp

*q*<sup>2</sup> [fm−<sup>2</sup>]

0 5 10 15 20

4 *π* ≤ 2*J* ∑ *L*=0

where *τ* = *Q*2/4 *m*<sup>2</sup>

be analytically evaluated.

where *Y*∗

10<sup>0</sup> 10−<sup>1</sup> 10−<sup>2</sup> 10−<sup>3</sup> 10−<sup>4</sup> 10−<sup>5</sup> 10−<sup>6</sup> 10−<sup>7</sup> 10−<sup>8</sup>

radius.

**6. Conclusions**

*Fch*(*q*2)

2

rms radii, and the magnetic moments for selected few-nucleon systems. The nuclear Hamiltonian was constructed from the Argonne AV4 *NN* potential that includes also the Coulomb interaction. Comparison with the experimental data revealed that the reproduction of the ground state properties of light nuclei is quite satisfactory. The discrepancies observed can be attributed to reasons not entirely related to the AMD. These include i) the omission of mixed-symmetric states (for three-body) ii) the use of a limited rank for the Argonne AV18 potential, and iii) the omission of three-nucleon forces. As far as the magnetic moment is concerned, the inclusion of relativistic corrections to the magnetic moment operator, are also expected to contribute to the reduction of the discrepancy between theory and experiment. The technique used in the approximation of the variational energy can be easily extended to the three-body interactions. The implementation of this technique in nuclear three-body interactions will be considered future projects.

The angular momentum and parity projected AMD was used to calculate the ground state charge form factors for the 3He and 4He nuclei. In overall, the results obtained, within the AMD and PWIA approximation, reproduce the general behavior of the experimental form factors. For momentum transfer below the first diffraction minimum the reproduction of experimental form factors is fairly good. However, beyond the first diffraction minimum the results are lower than the data. The deviations of the theoretical results from experimental data can be minimize by employing improved wave functions. The wave functions can be constructed by using a more complete realistic Hamiltonian, three-body forces, and relativistic corrections in the electromagnetic operators. It should be noted that these results are consistent with other results obtained by competing theoretical models. In conclusion, the results indicate that the AMD method is a very promising method in calculating electromagnetic form factors of the general *A*-body nuclear system.

Work is underway to construct wave functions for scattering processes applicable in nuclear breakup reactions (Rampho, 2010). In these constructs the AMD is combined with the Glauber multiple scattering to account for final state interactions. Wave functions for two-body and three-body scattering reactions can be treated accurately in the Faddeev formalism (Golak et al., 2005; Merkuriev et al., 1976). In the hyperspherical harmonics approach some progress in being made towards the construction of accurate wave functions for two-body scattering processes (Kievsky et al., 2008). However, these methods become involved for systems consisting of more than four constituent particles.

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Tomoaki Togashi1 and Kiyoshi Kato¯ <sup>2</sup>

*Japan*

**8**

<sup>1</sup>*Faculty of Advanced Life Science, Hokkaido University*

**Applications to Light Nuclei** 

<sup>2</sup>*Division of Physics, Graduate School of Science, Hokkaido University*

*Ab initio* calculation starting from the bare nuclear interactions is one of recent major theoretical subjects in nuclear physics. One hopes to understand the property of nuclei based on the nuclear forces against the background of the progress of the fundamental understanding of bare nuclear interactions and the study of unstable nuclei far from the beta stable line, which are expected to have structures different from stable nuclei. Nowadays, one can obtain the three-, four-, or few-body wave functions starting from the bare Hamiltonian by applying many kinds of few-body exact methods (Kamada et al., 2001). It is difficult to apply such methods directly to heavier nuclei in viewpoint of computational costs at this stage, therefore, in many of *ab initio* approaches, the effective interaction is constructed based on the bare interaction by applying, for example, the unitary correlated operator method (UCOM) (Neff & Feldmeier, 2004) and so on. As one of remarkable achievements in *ab initio* approaches, the quantum Monte Carlo method (Wiringa et al., 2000) has presented the appearance of the two-alpha (4He) cluster structure in 8Be starting from the bare interactions. That suggests that

**Antisymmetrized Molecular Dynamics with Bare** 

**Nuclear Interactions: Brueckner-AMD, and Its** 

one has the ability to discuss cluster structures based on the bare nuclear forces.

The existence of cluster states appearing around thresholds, which are weakly bound systems of subunits consisting of several nucleons to be strongly correlated each other, is predicted by the Ikeda diagram and the cluster models (Ikeda et al., 1972). The cluster states in some excited states of a light nucleus can be described as the states different from the mean-field-like structures. Cluster models in nuclear physics are able to describe these states successfully. In unstable nuclei, it is expected that mean-field-like and cluster-like structures coexist because their ground states often exist around the thresholds of decaying particles. The antisymmetrized molecular dynamics (AMD) (Kanada-En'yo et al., 1995) to be one of the recent developments of the cluster model has succeeded in understanding and predicting many properties of unstable nuclei. AMD can describe the nuclear structures without any assumption of configurations as the results of the energy variation in quantum mechanics. However, it is difficult for the bare nuclear interactions to be applied straightforwardly to the AMD framework because of the singularity and complexity in the interactions. Hence, AMD calculations have been performed with phenomenological interactions represented by

**1. Introduction**

