Average ground-state energy of finite Fermi systems

Semiclassical theories such as the Thomas-Fermi and Wigner-Kirkwood methods give a good description of the smooth average part of the total energy of a Fermi gas in some external potential when the chemical potential is varied. However, in systems with a fixed number of particles $N$, these methods overbind the actual average of the quantum energy as $N$ is varied. We describe a theory that accounts for this effect. Numerical illustrations are discussed for fermions trapped in a harmonic oscillator potential and in a hard-wall cavity, and for self-consistent calculations of atomic nuclei. In the latter case, the influence of deformations on the average behavior of the energy is also considered.

Figure 1

Accumulated level density (upper panel) and total energy (lower panel) with spin-degeneracy 2 for a spherical HO potential as a function of chemical potential μ. Staircase, solid, and dashed lines correspond to the quantum, semiclassical WK, and shell correction (quantum minus semiclassical) values, respectively.

Figure 2

(Upper panel) Quantum and WK values of the energy per particle for a spherical HO potential as a function of the number of particles, in units of $\hslash \omega$. The leading $N$ dependence given by Eq. (11) is subtracted from both curves. (Lower panel) The same as in the upper panel but for a strongly triaxially deformed HO potential. Notice that the WK curves are different in the spherical and deformed cases.

Figure 3

Quantum and WK values of the energy per particle in a triaxially deformed HO potential. Spin degeneracy is included.

Figure 4

(Upper panel) Normalized fluctuating part $\tilde{E}(N)=E(N)-\overline{E}(N)$ as a function of $N$ for a spherical cavity. $E_0=\hslash {}^2/2m{r}_0^2$, where $r_0$ is the radius of the sphere and $m$ is the mass of fermions. (Lower panel) Average $\langle \tilde{E}(N)\rangle {}_N$ of the fluctuating function in the upper panel (full line) compared to the theoretical prediction, Eq. (22) (dashed line).

Figure 5

(Upper panel) Fluctuating part $\tilde{E}(N)=E(N)-\overline{E}(N)$ as a function of $N$ for a 3D isotropic HO. (Lower panel) Average $\langle \tilde{E}(N)\rangle {}_N$ of the fluctuating function in the upper panel (full line) compared to the theoretical prediction, Eq. (26) (dashed line).

Figure 6

Difference between the minimized ground-state energy with respect to deformations and the spherical WK energy $\overline{E}(N)$ for fermions in a HO potential (squares joined by full line). For comparison, the dotted curve shows the energy difference for an isotropic HO (same curve as in the upper part of Fig. 5). The average of both curves is shown by a long-dashed line. The scaling $\hslash \omega =41(2N){}^{-1/3}$ MeV has been used in the calculations.

Figure 7

Energy per particle of atomic nuclei with an even number of protons and of neutrons along the periodic table, as obtained from self-consistent relativistic mean-field calculations. The deformed calculations are from Ref. [29]. For each value of the mass number $A$ we plot the most bound isotope according to the tabulation of Ref. [29] (except for $A=40$ [30]).

Figure 8

Energy difference between the deformed quantum solutions and the WK values of Fig. 7 for the self-consistent relativistic nuclear mean field. The result obtained by taking the difference of the experimental data [31] from the calculated WK energies is also shown for the purpose of illustration. The location of the magic neutron ($N$) and proton ($Z$) numbers is indicated.