Isovector dipole excitations in the relativistic RPA

Recent theoretical models based on quantum hadrodynamics have been successfully applied in the description of various nuclear phenomena. Mean field approximation of quantum hadrodynamics offers accurate quantitative predictions of bulk properties from stable to the exotic nuclei with large neutron/proton excess [6,10,9]. Collective excitations have been described within the relativistic random phase approximation [5,3] and the time-dependent relativistic mean field model [7,8] in nuclei over the whole periodic table, describing its properties that vary smoothly with the mass number. In the present analysis we want to explore to what extent the random phase approximation in the relativistic mean field theory can answer the question of collectivity in the low-lying isovector dipole excitations. Within the relativistic mean-field theory (RMFT), the nucleus is described as a system of Dirac nucleons interacting through the exchange of virtual mesons and photons [6]. The Lagrangian density reads

$${\cal L} = \bar\psi\left(i\gamma\cdot\partial-m\right)\psi ~+~\frac{1}{2}(\partial\sigma)^2-U(\sigma )$$

$$-~\frac{1}{4}\Omega_{\mu\nu}\Omega^{\mu\nu} +\frac{1}{2}m^2_\omeg... ...frac{1}{2}m^2_\rho\vec\rho^{\,2} ~-~\frac{1}{4}{\rm F}_{\mu\nu}{\rm F}^{\mu\nu}$$

$$-~g_\sigma\bar\psi\sigma\psi~-~ g_\omega\bar\psi\gamma\cdot\omega... ...\cdot\vec\rho\vec\tau\psi~-~ e\bar\psi\gamma\cdot A \frac{(1-\tau_3)}{2}\psi\;.$$ (1)

Here the Dirac spinor $\psi$ correspond to the nucleon with mass $m$, while $m_\sigma$, $m_\omega$, and $m_\rho$ are the masses of the $\sigma$-meson, the $\omega$-meson, and the $\rho$-meson. The parameters $g_\sigma$, $g_\omega$ and $g_\rho$, are the coupling constants for the mesons to the nucleon. $U(\sigma)$ denotes the nonlinear $\sigma$ self-interaction

$$U(\sigma)~=~\frac{1}{2}m^2_\sigma\sigma^2+\frac{1}{3}g_2\sigma^3+ \frac{1}{4}g_3\sigma^4,$$ (2)

and $\Omega^{\mu\nu}$, $\vec R^{\mu\nu}$, and $F^{\mu\nu}$ are field tensors. The non-linear self-interaction of the $\sigma$ field is very important for a quantitative description of ground and excited states in finite nuclei. Within this model, the Dirac-Hartree single-particle spectrum is constructed by expanding the fields in the harmonic oscillator basis [4] for the spherical nuclei. The relativistic random phase approximation (RRPA) correspond to the small amplitude limit of the time-dependent relativistic mean field theory. We solve the RRPA equations given in the matrix form [2];

$$\left( \begin{array}{cc} A^J & B^J \\ B^{^\ast J} & A^{^\as... ...M}_{\tilde p h} \\ Y^{\nu,JM}_{\tilde p h} \end{array}\right)$$ (3)

where $\tilde p$ and h denotes particle(antiparticle) and hole single-particle states within the RMFT. The matrix elements are determined by the single-particle energies and residual interaction,

$$A^J_{j_{\tilde p} j_h, j_{\tilde q} j_i} = \left(\epsilon_{j... ...e q}}\delta_{j_h j_i} +V^J_{j_{\tilde p} j_i j_h j_{\tilde q}}$$ (4)

$$B^J_{j_{\tilde p} j_h, j_{\tilde q} j_i} = (-1)^{j_{\tilde q} - j_i + J}V^J_{j_{\tilde p} j_{\tilde q} j_h j_i}.$$ (5)

Relativistic two-body interaction is obtained from the same Lagrangian (1) with the parameter set NL3 that is used for construction of the ground state basis [4] of single-particle energies and corresponding wave functions. The vacuum polarization is not taken into account, but this effects are included implicitly because the parameters in (1) are fitted to the experimental data in a few finite closed shell nuclei. However, the parameters of the interaction are fixed in further calculations and there are no additional adjustments or special parameters for fitting to the properties of particular excitation. Configuration space that include only pairs formed by excitations of particle from the Fermi sea to a higher state above the Fermi level is not sufficient for successful description of giant resonances. Recent investigations indicate that one should also include the transitions from the Fermi sea to the negative-energy Dirac sea in order to obtain decoupling of the spurious state and preserve the current conservation [2]. Solution of the RRPA eigenvalue problem( ) is applied to evaluate the electric dipole response,

$$B(EJ,\omega_{\nu}) = \frac{1}{2J+1} \left\vert \sum_{j_{\ti... ...t{Q}_J \vert\vert j_{\tilde p} \rangle \, \right\vert^2 \quad.$$ (6)

For dipole excitations J=1, while $\nu$ denotes the particular RRPA-eigenvalue. The presence of the spurious state corresponding to the center of mass motion, should be carefully treated. In the case of ideal accuracy, the spurious state is degenerate with the ground state, while because of the truncation effect, one can expect low energy spurious peak. A large dimension of the configuration space is suggested to ensure that the spurious component is eliminated from the physical states [2]. To avoid the coupling of the spurious state to electric transition operator, the following isovector dipole operator is applied,

$$\hat{Q}_{1 \mu}^{T=1} \ = \frac{N}{N+Z}\sum^{Z}_{p=1} r_{p}Y_{1 \mu} - \frac{Z}{N+Z}\sum^{N}_{n=1} r_{n}Y_{1 \mu.}$$ (7)

When considering isovector dipole modes, the contributions from Dirac states do not contribute significantly, in contrast to the large effect to the RRPA strength distributions in the isoscalar case, where contribution of scalar $\sigma$-meson dominates if compared with the role of the vector $\omega$ and $\rho$-mesons [28]. The isovector dipole strength distribution for the $^{208}$Pb is shown in the Fig.1, separately for calculations without and with Dirac states contributions due to exchange of scalar and vector mesons. The exact position of GDR energy is independent from the contributions of antiparticle-particle excitations, but without it, only 2/3 of dipole strength in the main peak is exhausted. Therefore in the further analysis, we apply the full self-consistent RRPA including in addition to the ordinary p-h excitations, the antiparticle-hole excitations from the occupied Fermi to the empty Dirac sea.

Nils Paar, 2001.