Spherical potential well

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. 

Fig. 14. Energy levels calculated for an infinitely deep spherical potential well of radius with an infinitely high central potential barrier with a radius the zigzag line...

In the s-wave-tip model (Tersoff and Hamann, 1983, 1985), the tip was also modeled as a protruded piece of Sommerfeld metal, with a radius of curvature R, see Fig. 1.25. The solutions of the Schrodinger equation for a spherical potential well of radius R were taken as tip wavefunctions. Among the numerous solutions of this macroscopic quantum-mechanical problem, Tersoff and Hamann assumed that only the s-wave solution was important. Under such assumptions, the tunneling current has an extremely simple form. At low bias, the tunneling current is proportional to the Fermi-level LDOS at the center of curvature of the tip Pq. 

Fig. 1.25. The s-wave-tip model. The tip was modeled as a spherical potential well of radius R. The distance of nearest approach is d. The center of curvature of tip is To, at a distance (R + d) from the sample surface. Only the 5-wave solution of the spherical-potential-well problem is taken as the tip wavefunction. In the interpretation of the images of the reconstructions on Au(llO), the parameters used are R = 9 A, d = 6 A. The center of curvature of the tip is 15 A from the Au surface. (After Tersoff and Hamann, 1983.)...

The original 5-wave-tip model described the tip as a macroscopic spherical potential well, for example, with r 9 A. It describes the protruded end of a free-electron-metal tip. Another incarnation of the 5-wave-tip model is the Na-atom-tip model. It assumes that the tip is an alkali metal atom, for example, a Na atom, weakly adsorbed on a metal surface (Lang, 1986 see Section 6.3). Similar to the original 5-wave model, the Na-atom-tip model predicts a very low intrinsic lateral resolution. 

In the case of metal clusters, for example, valence electrons show the shell structure which is characteristic of the system consisting of a finite number of fermions confined in a spherical potential well [2]. This electronic shell structure, in turn, motivated some theorists to study clusters as atomlike building blocks of materials [3]. The electronic structure of the metallofullerenes La C60 [4] and K C60 [5] was investigated from this viewpoint. This theorists dream of using clusters as atomlike building blocks was first realized by the macroscopic production of C60 and simultaneous discovery of crystalline solid C60, where C60 fullerenes form a close-packed crystalline lattice [6]. 

This relation applies generally and specifically to spheres and cylinders. Empirical relations have been developed to convert the measured lifetime (x) to this mean free path ( ). In case of small pores (<2 nm) an earlier version is adequate [43, 44], The model is based on the assumptions of spherical potential wells with infinite depth and radius r that traps the positronium. 

This species is essentially an electron stabilised by the surrounding water molecules. It has been the subject of detailed theoretical studies(35), but can be considered as an electron in a spherical potential well consisting of solvent molecules. Specific short-range solvation effects are thought to be important as well as long-range polarization forces. 

In the period 1940-1946, Ogg (132) developed the first quantitative theory for the solvated electron states in liquid ammonia. The Ogg description relied primarily on the picture of a particle in a box. A spherical cavity of radius R is assumed around the electron, and the ammonia molecules create an effective spherical potential well with an infinitely high repulsive barrier to the electron. It is this latter feature that does not satisfactorily represent the relatively weakly bound states of the excess electron (9,103). However, the idea of a potential cavity formed the basis of subsequent theoretical treatments. Indeed, as Brodsky and Tsarevsky (9) have recently pointed out, the simple approach used by Ogg for the excess electron in ammonia forms the basis of the modem theory (157) of localized excess-electron states in the nonpolar, rare-gas systems. [The similarities between the current treatments of trapped H atoms and excess electrons in the rare-gas solids has also recently been reviewed by Edwards (59).]... 

Extensions to the spherical jellium model have been made to incorporate deviations from sphericality. Clemenger [15] replaced the Woods-Saxon potential with a perturbed harmonic oscillator model, which enables the spherical potential well to undergo prolate and oblate distortions. The expansion of a potential field in terms of spherical harmonics has been used in crystal field theory, and these ideas have been extended to the nuclear configuration in a cluster in the structural jellium model [16]. 

The rate of this reaction is expressed by a pseudo-first order reaction as X k[X]. In diamagnetic substances, however, Ps feels strong repulsive force based on the Pauli s exclusion principle. Hence in crystals Ps will exist in interstitial sites as a Bloch wave, but in substances having density fluctuation it is trapped in open space. Describing the Ps state in the interstitial or in the trapped states by a spherical potential well with infinite height, its zero-point energy is given as. 

Once o-Ps is conflned in a hole it stays there colliding many times on the wall until an electron, having anti-parallel spin to the e" spin, in the wall meets e" and is annihilated (pick-off annihilation). Theoretically the rate of this pick-off annihilation is proportional to the overlap integral of the e wave function with those of external electrons. In a simple but useful model a spherical potential well is assumed for the hole and the external electrons are dealt with as an electron layer pasted over the wall with a thickness A R. The o-Ps lifetime is then given as (3) ... 

To treat the quantum size effect, an electron or hole inside a spherical particle is modeled as a particle in a sphere. " " The electron is considered to be confined to a spherical potential well of radius a with potential energy V(r) = 0 for r < u and V(r) = oo for r > u. Solving the Schrodinger equation yields the wave functions" ... 

Generally, three to four lifetime components are resolved in polymers, and their attribution is as follows. The shortest lifetime component ri with intensity h is attributed to contributions from free positron annihilation (inclusive of p-Ps lifetime). The intermediate lifetime component Z2 with intensity 12 is considered to be due to the annihilation of positrons trapped at defects present in the crystalline regions, or those trapped at the crystalline-amorphous interface boundaries. The longest-lived component T3 with intensity 1, is due to pick-off annihilation of the o-Ps in the free volume cavities present mainly in the amorphous regions of the polymer [42,43]. The simple model of a Ps atom in a spherical potential well of radius R leads to a correlation between o-Ps hfetime and R [70,128-130] ... 

Fig. 1. Proton tunnelling between the two spherical potential wells of constant depth -Vq and of radius a, which are located on equivalent positions of the two identical molecules A and B.

We have considered a typical numerical example by taking realistic values Vq 0.75 eV, a = 0,2 aQ —0.1 A, corresponding to an energy Eq of the ground state of each spherical potential well Eq = -0.080 eV. 

This simple 3D modle represents an improvement with respect to the usual linear models and has the advantage to be solved analytically. This theory will be improved by taking the random rotational motion of the molecules into account. As the spherical potential wells for the protons are not at the centers of the donor and acceptor molecules A and B, the distance between these wells depends on the relative position of the centers of the molecules together with their orientations. 

FRIES - The rate constant k for proton tunnelling given by Eqs. (1 ) and nir is inversely proportional to the relative diffusion constant D of the spherical potential wells. As the temperature decreases, so does D and thus k increases. This effect is easily interpreted by the fact that the reactants, i.e. the wells, stay longer in close contact where the proton can effectively tunnel. 

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