Electron confinement

Estimate the minimum uncertainty in (a) the position of a marble of mass 1.0 g given that its speed is known to within 1.0 mm s 1 and (b) the speed of an electron confined to within the diameter of a typical atom (200. pm). 

The uncertainty principle is negligible for macroscopic objects. Electronic devices, however, are being manufactured on a smaller and smaller scale, and the properties of nanoparticles, particles with sizes that range from a few to several hundred nanometers, may be different from those of larger particles as a result of quantum mechanical phenomena, (a) Calculate the minimum uncertainty in the speed of an electron confined in a nanoparticle of diameter 200. nm and compare that uncertainty with the uncertainty in speed of an electron confined to a wire of length 1.00 mm. (b) Calculate the minimum uncertainty in the speed of a I.i+ ion confined in a nanoparticle that has a diameter of 200. nm and is composed of a lithium compound through which the lithium ions can move at elevated temperatures (ionic conductor), (c) Which could be measured more accurately in a nanoparticle, the speed of an electron or the speed of a Li+ ion ... 

Mills, G., Gordon, M.S. and Metiu, H. (2003) Oxygen adsorption on Au clusters and a rough Au(lll) surface The role of surface flatness, electron confinement, excess electrons, and band gap. Journal of Chemical Physics, 118, 4198-4205. 

The chance to observe an electronic confinement in a OD nanoparticle depends on several conditions ... 

In this Section we want to present one of the fingerprints of noble-metal cluster formation, that is the development of a well-defined absorption band in the visible or near UV spectrum which is called the surface plasma resonance (SPR) absorption. SPR is typical of s-type metals like noble and alkali metals and it is due to a collective excitation of the delocalized conduction electrons confined within the cluster volume [15]. The theory developed by G. Mie in 1908 [22], for spherical non-interacting nanoparticles of radius R embedded in a non-absorbing medium with dielectric constant s i (i.e. with a refractive index n = Sm ) gives the extinction cross-section a(o),R) in the dipolar approximation as ... 

A more general relation between potential and electronic pressure for a density-functional treatment of a metal-metal interface has been given.74) For two metals, 1 and 2, in contact, equilibrium with respect to electron transfer requires that the electrochemical potential of the electron be the same in each. Ignoring the contribution of chemical or short-range forces, this means that —e + (h2/ m)x (3n/7r)2/3 should be the same for both metals. In the Sommerfeld model for a metal38 (uniformly distributed electrons confined to the interior of the metal by a step-function potential), there is no surface potential, so the difference of outer potentials, which is the contact potential, is given by... 

Sommerfeld suggested that the potential in a metal crystal could be assumed constant. This assumption implies that the forces acting on an electron cancel to zero and that the electrons in a metal can be described like a non-interacting gas of electrons, confined to a box that represents the metal. The only restriction on electronic motion would be the Pauli principle. The electronic energy in a three-dimensional rectangular box is known as... 

Sastre, G., Viruela, P.M. and Corma, A. (1997). Quantum chemistry calculations on the effect of electron confinement upon the frontier molecular jorbitals of ethylene and benzene in sodalite. Implications on reactivity. Chem. Phys. Lett. 264, 565-572... 

Zhang, L.Z., Sun, W. and Cheng, P. (2003). Spectroscopic and theoretical studies of quantum and electronic confinement effects in nanostructured materials. Molecules 8, 207-222... 

Zicovich-Wilson, C.M., Corma, A. and Viruela, P. (1994). Electronic confinement of molecules in microscopic pores. A new concept which contributes to explain the catalytic activity of zeolites. J. Phys. Chem. 98, 10863-10870... 

The external potential [1] is responsible for keeping the electrons confined to a region of space. For the case of an isolated molecule, the external potential is the potential generated by its nuclei. When one considers the interaction between a molecule and another species, then the external potential is the one generated by the nuclei of both species, and it acts on all the electrons. However, when they are very far apart from each other, since the electrons of both species are localized in, basically, separated regions, then the external potential of each species may be assumed to be the one generated by its own nuclei, and by the nuclei and the electrons of the other species. 

Before starting the discussion on confined atoms, we shall briefly describe the simplest standard confined quantum mechanical system in three dimensions (3-D), namely the particle-in-a-(spherical)-box (PIAB) model [1], The analysis of this system is useful in order to understand the main characteristics of a confined system. Let us note that all other spherically confined systems with impenetrable walls located at a certain radius, Rc, transform into the PIAB model in the limit of Rc —> 0. For the sake of simplicity, we present the model in one-dimension (1-D). In atomic units (a.u.) (me=l, qc 1, and h = 1), the Schrodinger equation for an electron confined in one-dimensional box is... 

Model calculations for the Cs suboxides in comparison with elemental Cs have shown that the decrease in the work function that corresponds to an increase in the Fermi level with respect to the vacuum level can be explained semi-quantitatively with the assumption of a void metal [65], The Coulomb repulsion of the conduction electrons by the cluster centers results in an electronic confinement and a raising of the Fermi energy due to a quantum size effect. 

On a somewhat larger scale, there has been considerable activity in the area of nanocrystals, quantum dots, and systems in the tens of nanometers scale. Interesting questions have arisen regarding electronic properties such as the semiconductor energy band gap dependence on nanocrystal size and the nature of the electronic states in these small systems. Application [31] of the approaches described here, with the appropriate boundary conditions [32] to assure that electron confinement effects are properly addressed, have been successful. Questions regarding excitations, such as exdtons and vibrational properties, are among the many that will require considerable scrutiny. It is likely that there will be important input from quantum chemistry as well as condensed matter physics. 

Because of their importance as basic primary centers, we will now discuss the optical bands associated with the F centers in alkali halide crystals. The simplest approximation is to consider the F center - that is, an electron trapped in a vacancy (see Figure 6.12) - as an electron confined inside a rigid cubic box of dimension 2a, where a is the anion-cation distance (the Cr -Na+ distance in NaCl). Solving for the energy levels of such an electron is a common problem in quantum mechanics. The energy levels are given by... 

Borysiuk J, Bozek R, Strupinski W et al (2010) Graphene growth on C and Si-face of 4 H-SiC - TEM and AFM studies. Mater Sci Forum 645-648 577-580 Berger C, Song Z, Li X et al (2006) Electronic confinement and coherence in patterned epitaxial graphene. Science 312 1191-1196... 

The hydrogen atom has a single electron confined to the neighbourhood of the nucleus by a potential field V, given by — e jr. The solution of the appropriate Schrodinger equation becomes possible if the equation is expressed in polar coordinates r, 0 and independent equations each containing only one variable ... 

For transitions between different atomic or molecular electronic states, the wavelengths usually lie in the ultraviolet typically, A 103 A. For vibrational and rotational transitions, A is even larger. The size of atoms and molecules is about 1 A. Hence A is usually much greater than the size of the molecule as far as electrons confined to move within the molecule are concerned, the spatial variation of the radiation s electric field is negligible zjA 0. With this further approximation, we have... 

The uncertainty principle is negligible for. macroscopic objects. Electronic devices, however, are being manufactured on a smaller and smaller scale so that the properties of nanoparticles, particles whose sizes range from a few to several hundred nanometers, may be different from those of larger particles due to quantum mechanical phenomena, (a) Calculate the minimum uncertainty in the speed of an electron confined in a nanoparticle with a diameter of... 

Consider an electron, confined to a ring of radius 0.1 nm (about the size of an atom). Calculate the energies of the five lowest states, and express your answer in electron volts. 

In other words, the electrons confined to their ground state are only slightly disturbed by the incident photons Hence the NLO effects should be classified into the second category where short-range forces play a decisive role We therefore make the assumption that, in the NLO effects, the electron motion may be regarded as confined to small regions In other words, any NLO susceptibility (or second-order susceptibility) in crystals is a localized effect arising from the action of incident photons on the electrons in certain orbitals of atomic clusters ... 

Figure 1 Energy spectrum of the low-lying states of four electrons confined in a quasi-one-dimensional Gaussian potential with (D, a>z,a>xy) = (4.0, 0.1, 20.0) for different-size basis sets. Energy levels of different spin multiplicities are indicated by different colors (See the caption to Figure 2). The number in the round brackets specifies the total number of basis functions and the parameter v p specifies the extended polyad quantum number (See the text for details). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this book.)...

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