Atom polarizabilities, confined atoms

The linear response function [3], R(r, r ) = (hp(r)/hv(r ))N, is used to study the effect of varying v(r) at constant N. If the system is acted upon by a weak electric field, polarizability (a) may be used as a measure of the corresponding response. A minimum polarizability principle [17] may be stated as, the natural direction of evolution of any system is towards a state of minimum polarizability. Another important principle is that of maximum entropy [18] which states that, the most probable distribution is associated with the maximum value of the Shannon entropy of the information theory. Attempts have been made to provide formal proofs of these principles [19-21], The application of these concepts and related principles vis-a-vis their validity has been studied in the contexts of molecular vibrations and internal rotations [22], chemical reactions [23], hydrogen bonded complexes [24], electronic excitations [25], ion-atom collision [26], atom-field interaction [27], chaotic ionization [28], conservation of orbital symmetry [29], atomic shell structure [30], solvent effects [31], confined systems [32], electric field effects [33], and toxicity [34], In the present chapter, will restrict ourselves to mostly the work done by us. For an elegant review which showcases the contributions from active researchers in the field, see [4], Atomic units are used throughout this chapter unless otherwise specified. 

The independent work of Michels et al. [11] at the end of 1937 considered the hydrogen atom in an impenetrable sphere as a physical model for hydrogen under high pressure and studied the effects of confinement on the polarizability of hydrogen. Usually this paper is referred to as the first work on the confined atom problem, where the Dirichlet boundary conditions were used for quantum-mechanical problems. This work was followed by the work of Sommerfeld et al. [12,13] with the introduction of confluent... 

Isotope superlattices of nonpolar semiconductors gave an insight on how the coherent optical phonon wavepackets are created [49]. High-order coherent confined optical phonons were observed in 70Ge/74Ge isotope superlattices. Comparison with the calculated spectrum based on a planar force-constant model and a bond polarizability approach indicated that the coherent phonon amplitudes are determined solely by the degree of the atomic displacement, and that only the Raman active odd-number-order modes are observable. 

As we have seen, an atom under pressure changes its electron structure drastically and consequently, its chemical reactivity is also modified. In this direction we can use the significant chemical concepts such as the electronegativity and hardness, which have foundations in the density functional theory [9]. The intuition tells us that the polarizability of an atom must be reduced when it is confined, because the electron density has less possibility to be extended. Furthermore, it is known that the polarizability is related directly with the softness of a system [14], Thus, we expect atoms to be harder than usual when they are confined by rigid walls. Estimates of the electronegativity, x and die hardness, tj, can be obtained from [9]... 

Figure 5 Variation of dipole polarizability against (a) confinement radius, (b) pressure (atm) due to confinement for different Debye screening parameters for a hydrogen atom. Reprinted with permission from Claude Bertout, Editor-in-Chief, A A (Ref. [172]).

Table 3 The magnetizability and polarizability of some polyaromatic hydrocarbons as obtained using high-level ab-initio methods. Polarizabilities reported in atomic units, and magnetizabilities in ppm cgs. The molecules are confined to the xy plane, the x direction along the longest side of the...

Here we consider [25] the properties of H at the centre of a spherical box of radius R, using a numerical approach to obtain the energies and polarizabilities. We also develop some model wave functions, simple expressions for the energies and polarizability, deduce the critical radius R for which E = 0, and extend the analysis to the confined helium atom with effective screening. 

Table 1 The energies of hydrogen atom confined to a box of radius R, obtained from numerical calculations, model wave functions, and the simple expression in Equation (3.39). For each state and R, the first row is from numerical calculation, the second row in brackets is from model wave function, and the third row in brackets is from the simple expression in Equation (3.39). The last column is the dipole polarizability for the ground state...

The calculated values [25] of a for the hydrogen atom, for some values of R are given in Table 1. It is interesting to observe that the polarizability decreases rapidly with decreasing radius of confinement. This is expected since the smaller domain of confinement decreases the flexibility of the electron. 

D. Baye, K.D. Sen, Confined hydrogen atom by the Lagrange-mesh method Energies, mean radii, and dynamic polarizabilities, Phys. Rev. E 78 (2008) 026701(7p). 

Seventy years ago Michels et al. [1] proposed a model consisting of a hydrogen atom confined at the centre of an impenetrable spherical box, in order to study how the pressure and polarizability evolve as a function of the compression. The model of confinement in boxes of different sizes and geometrical forms has become very popular and it is widely used in a variety of quantum systems. 

In a second work [100], they studied the polarization of the confined helium atom via the Kirkwood approach [57], finding a decreasing tendency for polarizability as the pressure grows. 

We have confined ourselves to spherically symmetric atoms. It is interesting to note that polyatomic molecules with spherically symmetric polarizability (e.g. methane, CH4 and carbon tetra-fiuoride, CF4) can be treated in the same manner. 

Since ions show stronger interactions with EM fields than neutral atoms, which experience only a weak force because of their polarizability, they can be stored more effectively in EM traps. Therefore trapping of ions was achieved long before neutral particles were flapped [1215,1216]. Two different techniques have been developed to store ions within a small volume in the radio frequency (RF) quadrupole trap [1216,1217,1242] the ions are confined within a hyperbolic electric dc field superimposed by a RF field, while in the Penning trap [1220] a dc magnetic field with a superimposed electric field of hyperbolic geometry is used to flap the ions. 

The first, most primitive, model is the infinite barrier model (IBM). Here the electronic motion is confined by a spherical potential hole with infinitely high barriers. Once the electronic wave functions (spherical Bessel functions) and eigenvalues are known, one can proceed and calculate the dynamic polarizability a co). From this quantity the collective excitations are determined in a straightforward manner (see below). The theoretical prediction [50], shown in Figure 1.2, matches the experimental data (indicated by dots) rather well from very small to mesoscopic particle sizes. The result obtained shows that the IBM, which models the kinetic repulsion of the occupied 4d-shell of atomic Xe, works surprisingly well. This repulsion causes an enhanced electronic density, leading to the blue-shift of the surface-plasmon line. 

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