Radial potential well

Fig. 4.70. A stacked ring ion guide. Rings are connected to an RF drive voltage in alteration of the phase. Normally tens of pairs are used to build one ion guide. Ions entering from the left experience a deep radial potential well with comparatively steep walls toward the inner ring surfaces when traveling though the apertures. Reproduced from Ref. [242] with permission. Elsevier Science Publishers, 2007.

The metal cluster will be modeled as an infinitely deep spherical potential well with the represented by an infinitely high spherical barrier. Let us place this barrier in the center of the spherical cluster to simplify the calculations. The simple Schrodinger equation, containing only the interaction of the electrons with the static potential and the kinetic energy term and neglecting any electron-electron interaction, can then be solved analytically, the solutions for the radial wave functions being linear combinations of spherical Bessel and Neumann functions. 

Consider now the solutions of the spherical potential well with a barrier at the center. Figure 14 shows how the energies of the subshells vary as a function of the ratio between the radius of the C o barrier Rc and the outer radius of the metal layer R ui- The subshells are labeled with n and /, where n is the principal quantum number used in nuclear physics denoting the number of extrema in the radial wave function, and / is the angular momentum quantum number. 

The results of Table 28.1 show that for xenon, the 4/-electron is localized in the outward potential well far from the nucleus. Therefore, its radial orbital overlaps with those of inward electrons only a little. For this reason all integrals are very small and are of the same order for all LS values of the configuration considered. However, for barium we have a completely different picture, illustrating the collapse phenomenon. Only for the term 1P does the electron remain not collapsed for the rest of the terms the values of radial integrals have increased by 2-5 orders, evidencing... 

Figure 21 Relativistic Dirac-Fock calculated data [29] of the valence ns radial electron probability densities Pns 2 for free Mg, Ca, Sr and Baas well as for the atoms encaged in C60, marked Mg, Ca, d>Sr and Ba, respectively. The domain of the C60 potential well is encompassed by the dashed vertical lines in this figure.

Figure 25 Radial orbitals (r) of free Ca and Ca in Ca Cgo (d>Ca) [20]. Note the collapse of the 4d orbital in a dipole 3p 4d excited state of Ca into a hollow space of CgQ but significant electron density transfer of the 3p - 3d excited orbital in Ca into the Cf,o potential well.

Fig. 5. Schematic representation (after [13]) of the numerical calculation of the spatial part of the matrix element Mspace in the p + p—> d c u, reaction. The top part shows the potential well of depth Vo and nuclear radius R of deuterium with binding energy of —2.22 MeV. The next part shows the radius dependence of the deuterium radial wave function Xd(r)- The wave-function extends far outside the nuclear radius with appreciable amplitude due to the loose binding of deuterium ground state. The p-p wave-function XppM which comprise the U = 0 initial state has small amplitude inside the final nuclear radius. The radial part of the integrand entering into the calculation of Mspac is a convolution of both Xd and Xpp in the second and third panels and is given with the hatched shading in the bottom panel. It has the major contribution far outside the nuclear radius...

Centrifugal barrier effects have their origin in the balance between the repulsive term in the radial Schrodinger equation, which varies as 1/r2, and the attractive electrostatic potential experienced by an electron in a many-electron atom, whose variation with radius differs from atom to atom because of screening effects. In order to understand them properly, it is necessary to appreciate the different properties of short and of long range potential wells in quantum mechanics. 

In fig. 5.4, we show the double-well radial potential as computed by Griffin et al. [208] for elements with Z 56. Note that the radial scale used in plotting this figure is highly nonlinear on a linear scale, the double-well or double-valley potential looks rather similar to that of fig. 5.8. Fig. 5.4 is plotted in this way to show how very small changes in the wells can precipitate a very large change in the radius of the 4/ wavefunction from a radius of about 13 atomic units in Ba I, it collapses... 

Fig. 5.4. The double-well radial potential for 4/ elements close to the point at which orbital collapse occurs. Note the highly nonlinear scales on both axes, designed to show as much detail as possible for both wells (after D.C. Griffin et al. [208]).

Fig. 5.5. The effective radial potential for 3d electrons, showing the knee at the edge of the inner well (after D.C. Griffin et at [209]).

Fig. 5.8. Comparison between the effective radial potential for / electrons in H and the double-well radial potential in a heavy element by Goppert-Mayer-Fermi theory (not to scale).

Numerical procedures provide accurate solutions of the SCF problem, but are perhaps not enough to understand why a phenomenon appears. The double well occurs for rather simple reasons which we now endeavour to make clear. Consider first the effective radial potential of H for i = 3, which is shown in the lower inset of fig. 5.8. This potential has a hard centre , due to the repulsive centrifugal force, which varies as 1/r2, and therefore dominates at small r. On the other hand, it also has an attractive well at large r, which is due to the Coulomb term. At one point of intermediate radius rc, the two terms cancel each other exactly. 

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