Quantum states, size-dependent

An explanation for these size-dependent optical properties, tenned quantum confinement , was first outlined by Bms and co-workers in the early 1980s, [156, 158, 159, 160 and 161] and has fonned the basis for nearly all subsequent discussions of these systems. Though recent work has modified and elaborated on this simple model, its basic predictions are surjDrisingly accurate. The energy of the lowest-lying exciton state is given by the following simple fonnula ... 

Another reason for the size-dependent properties in granular metals is not manifestation of quantum effects, but the rise of the surface state fraction. If one considers a metal granule as a sphere with diameter D and thickness of the surface layer hs, it is clear that the interface portion depends on the granule size and its relative contents are described by the following equation ... 

Specifically, this volume focuses on the synthesis, processing, and structural tailoring of nanocrystalline and nanoporous materials. Nanocrystalline materials possess unique hybrid properties characteristic of neither the molecular nor the bulk solid-state limits and may be confined in nanometersized domains in one, two, or three dimensions for unusual size-dependent behavior. Nanoporous materials, characterized by well-defined pores or cavities in the nanometer size regime and controlled pore diameter and structure, give rise to unique molecular sieving capabilities and ultrahigh internal surface areas. Nanoporous structures also act as hosts and templates for the fabrication of quantum dots and quantum wires. 

Both CIS and TDHF have the correct size dependence and can be applied to large molecules and solids (we will shortly substantiate what is meant by the correct size dependence ) [42-51], It is this property and their relatively low computer cost that render these methods unique significance in the subject area of this book despite their obvious weaknesses as quantitative excited-state theories. They can usually provide an adequate zeroth-order description of excitons in solids [50], Adapting the TDHF or CIS equations (or any methods with correct size dependence, for that matter) to infinitely extended, periodic insulators is rather straightforward. First, we recognize that a canonical HF orbital of a periodic system is characterized by a quantum number k (wave vector), which is proportional to the electron s linear momentum kh. In a one-dimensional extended system, the orbital is... 

Nanocrystals of metal and semiconductors with diameters in the range 1 to 50 nm form a class of materials with unusual properties which are size-dependent. Excellent electrical conductivity that primarily characterizes a metallic state, becomes a rare entity in small nanocrystals (< 2 nm) due to quantum confinement of the electronic states. Similarly, magnetic metals lose much of the coercivity with diminishing size. On the other hand, chemical properties such as reactivity may show up better at smaller sizes due to a greater number of surface bonding sites and other electronic effects. Considering the importance of nanocrystals in tech-... 

The theory of energy transfer considered in this subsection was used to interpret the experiments with PbSe quantum dots (58) on the size-dependent energy relaxation in a quantum dot. In this paper it was shown that smaller dots have faster relaxation. In the theoretical paper by Hong et al. (59) it was assumed that the above energy transfer from a quantum dot exciton to surface states of the dot is a dominant channel of the electronic energy relaxation. Hong et al. considered in their calculations a spherical quantum dot of radius R and the transfer rate was obtained from the calculation of the power dissipation W on the surface of the quantum dot by the relation... 

Quantum mechanics makes it clear that no atom has a fixed size. Electron orbitals extend from the nucleus to a greater or lesser extent, depending upon the chemical and physical environment in the locality of the atomic nucleus. Indeed, recent research on Bose-Einstein and Fermi condensation reveals that a collection of millions of atoms can enter an identical quantum state at temperatures just above 0 K and behave as a single atom, with a single wavefunction that spreads over the whole collection. 

The zeolite matrix allows the preparation of dispersions with distinctly different and narrow particle size distributions in a range where size dependent electronic properties can be studied (quantum size particles). Quantum-mechanical calculations suggest that the energy level of the first excited state of the exciton increases with decreasing particle size of the semiconductors in correspondence with the experimentally observed blue-shift of the optical absorption edge (refs. 8-10). 

The electron s wave function (iK atomic orbital) is a mathematical description of the electron s wavelike behavior in an atom. Each wave function is associated with one of the atom s allowed energy states. The probability density of finding the electron at a particular location is represented by An electron density diagram and a radial probability distribution plot show how the electron occupies the space near the nucleus for a particular energy level. Three features of an atomic orbital are described by quantum numbers size (n), shape (/), and orientation (m/). Orbitals with the same n and / values constitute a sublevel sublevels with the same n value constitute an energy level. A sublevel with / = 0 has a spherical (s) orbital a sublevel with / = 1 has three, two-lobed (p) orbitals and a sublevel with / = 2 has five, multi-lobed (d) orbitals. In the special case of the H atom, the energy levels depend on the n value only. 

Apart from their interesting structures, molecular clusters are intensively studied because of their physical properties. It has been shown for transition-metal clusters that small uniform particles of a certain size show interesting electronic, size-dependent effects (quantum size effects). In Au55(PR3)i2Cl6 a few electrons are trapped in a metallic state and single electrons tunnel between the cluster units. (SET, single electron tunneling).Such effects have yet to be observed in main-... 

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