Connection between the Strutinsky level density and the semiclassical level density

We establish an analytical link between the level density obtained by means of the Strutinsky averaging method and the semiclassical level density. This link occurs only in the so-called asymptotic limit. It turns out that the Strutinsky method amounts to an approximation to the semiclassical method. This approximation contains an unavoidable remainder that constitutes an intrinsic noise in comparison to the semiclassical method. Thus, the problem of the dependency of the Strutinsky procedure on the two free smoothing parameters of the averaging is intimately connected to this noise. However, we demonstrate that the noise of the method is small in the average density of states and in the average energy, whereas it might be non-negligible in the shell correction itself. To improve this method, we give a rule that consists simply of minimizing the relative error for the average energy.

Figure 1

Strutinsky density of states ${\overline{g}}_{M,\gamma }(\lambda )$ of the harmonic oscillator as a function of the Fermi level λ. Three values of the smoothing parameter γ are considered; the order of the curvature correction is $M=0$. The shell gap is fixed arbitrarily at $\hslash \omega =10$ MeV.

Figure 2

(Top) Strutinsky density of states ${\overline{g}}_{M,\gamma }(\lambda )$ (curve a) and semiclassical density of states $g_{\mathrm{sc}}(\lambda )$ (curve b) of the cubic box as a functions of the Fermi level $\lambda$. The Strutinsky level density is calculated with the order $M=0$, and the smoothing parameter $\gamma =25E_0$ (for the definition of $E_0$ see Sec. 3c). The two level densities are practically indistinguishable. (Bottom) Relative error of the Strutinsky density of states defined by ($\mathrm{b}-\mathrm{a})/\mathrm{b}$ compared to the theoretical value (see text).

Figure 3

(Top) Strutinsky smooth energy ${\overline{E}}_{M,\gamma }(\lambda )$ (curve a) and semiclassical energy $E_{\mathrm{sc}}(\lambda )$ (curve b) of the cubic box as a functions of the Fermi level λ. The two free parameters are given in the graph. As in Fig. 2 (top), the two smooth energies are indistinguishable. (Bottom) Relative error of the Strutinsky smooth energy compared to the theoretical value (see also text).

Figure 4

(Top) The new understanding of the plateau condition is that the relative variations of the Strutinsky energy ${\overline{E}}_{M,\gamma }(\lambda )$ are so small compared to ${\overline{E}}_{M,\gamma }(\lambda )$. This means that, at the scale of the plateau's value, the fluctuations are negligible. (Bottom) Same as Fig. 4 (top), but at a reduced scale. Here, the fluctuations of the plateau are obvious.

Figure 5

(Top) Minimization of the mean relative error (see text). The minimum occurs approximately at $\gamma \approx 28.5E_0$. (Bottom) Comparison between the semiclassical energy and the different values of the Strutinsky smooth energy used in Fig. 5 (top). Note that the optimization [see Fig. 5 (top)] is made on the smooth energy and not on the shell correction itself as in the old method of the plateau.

Figure 6

Semiclassical and Strutinsky level densities for a finite well (here the Woods-Saxon potential). The main differences occur at the top of the well, near the singularity of the semiclassical density.