AMEDEE
The AMEDEE database provides predicted nuclear properties for nearly 7000 nuclei computed within the Hartree-Fock-Bogoliubov (HFB) framework using the Gogny D1S energy density functional. For each nucleus, the database gives access to the potential energy surface, ground-state properties (binding energy, deformation, chemical potentials, radii), and, for approximately 1700 even-even nuclei, low-lying collective spectroscopic properties derived from the five-dimensional collective Hamiltonian (5DCH) calculations. The database was previously hosted on this website.
← Browse the databaseTheoretical framework
The many-body problem is solved in the framework of the mean-field approximation accounting for pairing correlations, the Hartree-Fock-Bogoliubov (HFB) approximation. The self-consistent HFB equations are solved using an iterative method. They are deduced from the minimization of the total energy of the nucleus:
$$\delta \left( \langle \Phi | H - \lambda_Z Z - \lambda_N N - \mu_2 Q_{20} | \Phi \rangle \right) = 0$$
In this expression:
- $|\Phi\rangle$ is the HFB wave function.
- $\lambda_N$ and $\lambda_Z$ are the Lagrange parameters fixing the number of neutrons $N$ and protons $Z$.
- $\mu_2$ is the Lagrange parameter used to fix the quadrupole moment $q_{20}$, defined by $$q_{20} = \langle \Phi | Q_{20} | \Phi \rangle$$
with the operator
$$Q_{20} = \left(\frac{16\pi}{5}\right)^{1/2} r^2 Y_{20}$$
- $H$ is the nuclear Hamiltonian, which reads $$H = \sum_i T_i + \frac{1}{2} \sum_{i \neq j} V_{ij}$$
where $V_{ij}$ is the Gogny effective nucleon-nucleon interaction[^gogny] and $T_i$ is the kinetic energy term.
The HFB equations are solved in a harmonic oscillator basis within the axial symmetry hypothesis. The size of the basis is defined by the number of major shells $N_0$ used to expand the HFB wave function. This number depends on the number of nucleons contained in the nucleus. It is chosen so that the number of states in the basis is 8 times the maximum number of occupied states.
These states also depend on the nucleus constrained deformation and are defined by the quantum numbers of the deformed harmonic oscillator ($n_\perp$, $m$ and $n_z$), which obey the inequality:
$$\left(2n_\perp + m + 1\right) \hbar\omega_\perp + \left(n_z + \tfrac{1}{2}\right) \hbar\omega_z \leq \left(N_0 + 2\right) \hbar\omega_0$$
with
$$\left(\hbar\omega_0\right)^3 = \left(\hbar\omega_\perp\right)^2 \hbar\omega_z$$
where $\omega_\perp$ and $\omega_z$ are the two parameters of the axial oscillator.
These parameters should in principle be determined by minimizing the total energy of the system. This criterion is applied to determine, for each deformation, the optimal value of $\omega_0$. However, the ratio $q = \omega_\perp / \omega_z$ is not optimized but rather estimated through the relation:
$$q = \exp\left[\frac{1.5,\beta\cos(\gamma)}{2\beta + 1}\right]$$
with $\gamma = 0$ for $\beta > 0$ and $\gamma = \pi$ for $\beta \leq 0$,
based on the approximate deformation $\beta$ obtained when using the liquid-drop approximation of the nucleus. In this approach, both axial symmetry and parity are conserved.
The treatment of odd-$A$ and odd-odd nuclei is performed using the blocking procedure without breaking time-reversal symmetry. This procedure consists in blocking the unpaired nucleon in a fixed orbital during the minimization procedure. Several configurations are thus tested to determine the blocked quasi-particle yielding the minimum energy. For odd-$A$ nuclei, 11 configurations have been tested for each deformation, while 25 configurations are considered for odd-odd nuclei.
Technical details
The potential energy surfaces show the nucleus HFB energy, namely
$$E = \langle \Phi | H | \Phi \rangle$$
They are plotted as functions of the deformation parameter $\beta$:
$$\beta = \left(\frac{5\pi}{9}\right)^{1/2} \frac{q_{20}}{A R_0^2}$$
where $A = N + Z$ is the nucleus mass, $R_0 = 1{.}2, A^{1/3}$ its radius (expressed in fm) and $q_{20}$ is the mass quadrupole moment defined by
$$q_{20} = \langle \Phi | Q_{20} | \Phi \rangle$$
with the operator
$$Q_{20} = \left(\frac{16\pi}{5}\right)^{1/2} r^2 Y_{20}$$
When plotting only potential energy surfaces, dashed lines have been added corresponding to the approximate rotational energy correction for spins $I = 8$, $16$ and $24$. These are obtained by adding to the nucleus binding energy the rotational energy
$$E_\text{rot} = \frac{I(I+1)}{2,\mathfrak{I}_x}$$
given by the simplest rotational model, using however the moment of inertia $\mathfrak{I}_x$ calculated for every deformation $|\beta| \geq 0{.}15$.
The theoretical binding energies (defined as the minimum close to $\beta = 0$) as well as the experimental masses taken from the Audi-Wapstra mass tables are also indicated.
The chemical potentials are given by the Lagrange parameters $\lambda_Z$ and $\lambda_N$ for the protons and neutrons respectively.
The quadrupole collective masses $M_{20}$ and the moments of inertia $\mathfrak{I}_x$ have been calculated using the Inglis-Beliaev approximation. The determination of the zero point energies (ZPE) is also described in the same paper.
Concerning the proton and neutron pairing energies (p/n), they are defined by
$$E_P^{(p/n)} = \frac{1}{2} \operatorname{Tr}\left( \Delta^{(p/n)}, \kappa^{(p/n)} \right)$$
where $\Delta^{(p/n)}$ is the pairing field and $\kappa^{(p/n)}$ the pairing tensor obtained from the solution of the HFB equations.
The $\beta_2^{(p/n)}$ and $\beta_4^{(p/n)}$ parameters are deduced from the multipole moments $q_{20}$ and $q_{40}$ defined from the proton and neutron distributions. More precisely,
$$\beta_4^{(p/n)} = \frac{q_{40}^{(p/n)}}{A, R_0^4}$$
with
$$q_{40}^{(p/n)} = \langle \Phi^{(p/n)} | r^4 Y_{40} | \Phi^{(p/n)} \rangle$$
Finally, the proton and neutron radii are given by the square roots of the mean square radii $\langle r^2 \rangle^{(p/n)}$ defined by
$$\langle r^2 \rangle^{(p/n)} = \langle \Phi^{(p/n)} | r^2 | \Phi^{(p/n)} \rangle$$