Balian-Bloch theory

DEFORMED HARMONIC OSCILLATORS FOR METAL CLUSTERS AND BALIAN-BLOCH THEORY... 

Comparing the predictions of the 3D q-HO model and of Balian—Bloch theory... 

Deformed harmonic oscillators for metal clusters and Balian-Bloch theory 409... 

The semiclassical theories of Gutzwiller (1967, 1971, 1980), Balian and Bloch (1972), and Berry and Tabor (1976, 1977), which are based on Feynman s path-integral formulation of quantum mechanics (Feynman and Hibbs 1965), provide a solid theoretical justification. 

In this paper we confront the results provided by the 3D g-HO model to the restrictions imposed by the theory of Balian and Bloch. It turns out that agreement between the predictions of the two theories can be established by introducing in the Hamiltonian of the 3D g-HO an additional term, which is characterized by the same symmetry as the original Hamiltonian. Furthermore, it can be seen that this additional term occius naturally through a variational procedure, analogous to the one used for obtaining the Morse oscillator [25] from the usual harmonic oscillator. 

In Section 2 of this paper a brief account of the 3D q -HO is given, while in Section 3 the appUcation of the periodic orbit theory of Balian and Bloch to metal clusters is briefly described. The predictions of the 3D g-HO model axe compared to the restrictions imposed by the theory of Balian and Bloch in Section 4, while in Section 5 a modified Hamiltonian for the 3D q-HO is introduced, allowing for full agreement with the theory of Balian and... 

In the theory of Balian and Bloch a cluster of N alkali atoms can be studied [24] by considering the N valence electrons moving in a sphere with... 

As we have seen in the previous section, a hallmark of the theory of Balian and Bloch is that the plot of vs. i, where are the magic numbers and i the index counting them (i = 1, 2, 3,. ..), should be a straight line having a slope of 0.61 in the case of alkali meteils, while in the case of A1 clusters the slope should be 0.32 [4]. 

The discrepancy between the 3D -HO model and the theory of Balian and Bloch can be lifted by considering the Hamiltonian... 

We therefore conclude that the addition of the second term in the Hamiltonian of the 3D q-H.0, leading to the VFO, makes the predictions of the 3D 9-HO model compatible with those of the theory of Balian and Bloch. 

In the theory of Balian and Bloch, as mentioned above, a phase shift by a half unit of the running index i should be observed in the plot of the magic numbers vs. i when passing from a supersheU to the next [4,24]. In Fig. 1(a) such a shift is seen quite clearly aroimd Ni = 1000 in the cases of e = 0.006, 0.007, 0.008, whUe no clear shift of this type is seen in Fig. 1(b). 

Furthermore, the theory of Balian and Bloch, as mentioned above, predicts that within each supersheU the minima of the shell energy, Eshell, should appear at equidistant p>ositions (i.e, they should exhibit a periodicity) when plotted vs. The change in the periodicity when passing... 

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