Shell correction energies

Let us now invert the logic and define the Casimir energy as the energy resulting from the geometry-dependent part of the density of states (d.o.s.) - a concept that is closely related to the shell correction energy in nuclear physics ... 

The shell-correction energy is plotted in Figure 5a using data from Reference [55]. Two equally deep minima are obtained, one at Z = 108 and N = 162 for deformed nuclei with deformation parameters p2 0.22, p4 -0.07 and the other one at Z = 114 and N = 184 for spherical SHEs. Different results are obtained from self-consistent Hartree-Fock-Bogoliubov, HFB, calculations and relativistic mean-field models [56,57], They predict for spherical nuclei shells at Z = 114, 120 or 126 (dashed lines in Figure 5a) and N = 184 or 172. 

Fig. 5. Shell-correction energy (a) and partial half-lives for spontaneous fission (b) and a decay (c). See text for a detailed descriptions and for references.

Here (17) means a smoothed deformation dependence of the shell model energy as defined in Wagemans (1991) and dl7means the so-called shell correction energy. The qualitative behavior of the shell correction d 17 with deformation is easily understood. See Fig. 5.1. 

Upper end of the chart of nuclides showing nuclear half-lives. Theoretical shell correction energies (Smolanczuk 1997) are underlain in blue color. The shading indicates the height of the shell correction energy in steps of 1 MeV. The decay modes are as follows a decay (yellow), decay (red), spontaneous fission (green) (Courtesy M. Schadel)... 

By using the macroscopic-microscopic model, shell correction energies can be extracted from the experimental mass M and a macroscopic (spherical) mass ATld taken from a theoretical model. The shell correction is negative and it stabilizes the nucleus ... 

Upper panel Experimental shell correction energies (Esh) for even-even nuclides characterized by the difference N-Z = 48 compared to current theoretical predictions. Lower panel Experimental and calculated fission barriers for the same nuclides. The dashed line displays the macroscopic fission barrier as explained after Eq. (19.10)... 

In close analogy, an estimate of an experimental fission barrier can be obtained by superimposing the shell correction energy and the liquid drop barrier... 

Maps of shell correction energies obtained from self-consistent calculations plotted versus proton and neutron number. The darker the shading, the larger is the shell correction. The scale of Esh (upper right figure) is in MeV (Bender et al. 2001)... 

Fig. 8 Shell correction energies for the Sn region (top) and the superheavy region (bottom) calculated with various interactions. Increasing shell correction energies are colour coded from green (lowest) to red (highest). The shell stabilization closely traces the magic numbers around Sn while larger islands of stability are formed in the superheavy region. Adapted from [23]...

A Consistent Calculation of Atomic Energy Shell Corrections Strutinsky s Method in the Hartree-Fock-Roothaan Scheme... 

Equations (7) can be viewed as a formal Taylor-series expansion, around the averaged part of the one-particle density matrix, of the HF energy functional E[p] [16, 18], this defining a shell-correction series . In Eqn (13) the first-order term of this expansion is expressed in terms of the single-particle energies e,. 

With the determination of Yo the above-described averaging procedure can be performed, yielding the averaged value E p and the shell corrections S,E p and SjEjjp for the energy, as well as the averaged and fluctuating parts p and 5p of the one-particle density matrix. 

This process is repeated until self-consistency is reached, the final values p and n, being noted p and n, Using these values the first-order shell correction to the energy can be written ... 

In Section 3 we have formulated Strutinsky s shell-correction method in the framework of the analytic HFR scheme, for single open-shell atoms and molecules in their ground state. The consideration of two or many open-shell systems could be performed following the same pattern. Both the averaged part of the energy, E,jp, and its first-order shell-correction part, 8,E pr, have been derived in analytic form, and the self-consistent process for determining them has been described. 

Equation (3) incorporates relativistic effects, effects of target density, and corrections to account for binding of inner-shell electrons, as well as the mean excitation energy C/Z is determined from the shell corrections, S/2 is the density correction, Ifj accounts for the maximum energy that can be transferred in a single collision with a free electron, m/M is the ratio of the electron mass to the projectile mass, and mc is the electron rest energy. If the value in the bracket in Eq. (4) is set to unity, the maximum energy transfer for protons... 

Despite the fact that Bohr s stopping power theory is useful for heavy charged particles such as fission fragments, Rutherford s collision cross section on which it is based is not accurate unless both the incident particle velocity and that of the ejected electron are much greater than that of the atomic electrons. The quantum mechanical theory of Bethe, with energy and momentum transfers as kinematic variables, is based on the first Born approximation and certain other approximations [1,2]. This theory also requires high incident velocity. At relatively moderate velocities certain modifications, shell corrections, can be made to extend the validity of the approximation. Other corrections for relativistic effects and polarization screening (density effects) are easily made. Nevertheless, the Bethe-Born approximation... 

In previous papers [10,11] we have formulated a procedure for splitting the ground-state energy of a multifermionic system into an averaged, structure-less part, E, and a residual, shell-structure part, 8E. The latter originates from quantum interference effects of the one-particle motion in the confining potential [12] and has the form of a shell-correction expansion 5E = It was also shown [11] that the first-order corrective term,... 

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