Electron statistics

StatSoft, Inc. (1999) Electronic Statistics Textbook. Tulsa, OK StatSoft. WEB http //www.statsoft.com/ textbook/stathome.html. 

De Dominies L, Fantoni R (2006) Effects of electrons statistic on carbon nanotuhes hyperpolarizability frequency dependence determined with sum over states method. J Raman Spectrosc 37 669... 

Electronic statistical weight = 2 and rotational statistical weights ... 

Several pharmaceutical companies (Merck, Sharp and Dohme Smith, Kline and French and Schering Corp.) have applied the IBM 101 electronic statistical machine to literature searching (8, p. 232 41 49, p. 240). The number of columns which may be searched simultaneously depends upon the number of punches appearing in a single column, but as many as 60 columns may be examined at one pass of the cards. The 101 is a versatile machine which cuts down the searching time as compared with the simple sorter. A similar machine called ILAS was developed for searching coded steroid structures (49, p. 447) in the U. S. Patent OflBce. 

Finally, it may be noted that the basic aim of this paper is not solely to demonstrate the influence of quantum confinement on the photoemission from non-parabolic semiconductors but also to formulate the appropriate electron statistics in the most generalized form. Transport and other phenomena in semiconductors having different band structures and the derivation of the expressions of many important electronic properties are based on the temperature-dependent electron statistics in such materials. 

In semiconductors, there is a special valence energy band under the conduction band. With pure (intrinsic) semiconductors, the energy levels in between are forbidden levels, and at room temperature very few electrons statistically have sufficient energy to cross the forbidden band and reach the conduction band that is, the conductivity is low. For example, the energy gap is 0.7 eV for germanium, 1.1 eV for silicon, and 5.2 eV for diamond (an insulator). 

The Gibbs energy functions and the electronic statistical multiplicities for the rare-earth monoxides. Chandrasekharaiah and Gingerich (1989). 

The factor 4 comes from the two electron statistical weights for the two levels of spin, and m is the real mass of the electron. 

Goodman, Statistical properties of laser sparkle patterns . Tech. Report N°2303-l, Stanford Electronics Laboratories.(1963). 

Electrons, protons and neutrons and all other particles that have s = are known as fennions. Other particles are restricted to s = 0 or 1 and are known as bosons. There are thus profound differences in the quantum-mechanical properties of fennions and bosons, which have important implications in fields ranging from statistical mechanics to spectroscopic selection mles. It can be shown that the spin quantum number S associated with an even number of fennions must be integral, while that for an odd number of them must be half-integral. The resulting composite particles behave collectively like bosons and fennions, respectively, so the wavefunction synnnetry properties associated with bosons can be relevant in chemical physics. One prominent example is the treatment of nuclei, which are typically considered as composite particles rather than interacting protons and neutrons. Nuclei with even atomic number tlierefore behave like individual bosons and those with odd atomic number as fennions, a distinction that plays an important role in rotational spectroscopy of polyatomic molecules. 

Another important accomplislnnent of the free electron model concerns tire heat capacity of a metal. At low temperatures, the heat capacity of a metal goes linearly with the temperature and vanishes at absolute zero. This behaviour is in contrast with classical statistical mechanics. According to classical theories, the equipartition theory predicts that a free particle should have a heat capacity of where is the Boltzmann constant. An ideal gas has a heat capacity consistent with tliis value. The electrical conductivity of a metal suggests that the conduction electrons behave like free particles and might also have a heat capacity of 3/fg,... 

The resolution of this issue is based on the application of the Pauli exclusion principle and Femii-Dirac statistics. From the free electron model, the total electronic energy, U, can be written as... 

These limitations lead to electron spin multiplicity restrictions and to differing nuclear spin statistical weights for the rotational levels. Writing the electronic wavefunction as the product of an orbital fiinction and a spin fiinction there are restrictions on how these functions can be combined. The restrictions are imposed by the fact that the complete function has to be of synnnetry... 

In many materials, the relaxations between the layers oscillate. For example, if the first-to-second layer spacing is reduced by a few percent, the second-to-third layer spacing would be increased, but by a smaller amount, as illustrated in figure Al,7,31b). These oscillatory relaxations have been measured with FEED [4, 5] and ion scattering [6, 7] to extend to at least the fifth atomic layer into the material. The oscillatory nature of the relaxations results from oscillations in the electron density perpendicular to the surface, which are called Eriedel oscillations [8]. The Eriedel oscillations arise from Eenni-Dirac statistics and impart oscillatory forces to the ion cores. 

In this chapter, the foundations of equilibrium statistical mechanics are introduced and applied to ideal and weakly interacting systems. The coimection between statistical mechanics and thennodynamics is made by introducing ensemble methods. The role of mechanics, both quantum and classical, is described. In particular, the concept and use of the density of states is utilized. Applications are made to ideal quantum and classical gases, ideal gas of diatomic molecules, photons and the black body radiation, phonons in a hannonic solid, conduction electrons in metals and the Bose—Einstein condensation. Introductory aspects of the density... 

Andersen H C and Chandler D 1970 Mode expansion in equilibrium statistical mechanics I. General theory and application to electron gas J. Chem. Phys. 53 547... 

Comcidence experiments have been connnon in nuclear physics since the 1930s.The widely used coincidence circuit of Rossi [9] allowed experimenters to detennine, within tire resolution time of the electronics of the day, whether two events were coincident in time. The early circuits were capable of submicrosecond resolution, but lacked the flexibility of today s equipment. The most important distinction between modem comcidence methods and those of the earlier days is the availability of semiconductor memories that allow one to now record precisely the time relations between all particles detected in an experiment. We shall see the importance of tliis in the evaluation of the statistical uncertainty of the results. 

It is beyond the scope of these introductory notes to treat individual problems in fine detail, but it is interesting to close the discussion by considering certain, geometric phase related, symmetry effects associated with systems of identical particles. The following account summarizes results from Mead and Truhlar [10] for three such particles. We know, for example, that the fermion statistics for H atoms require that the vibrational-rotational states on the ground electronic energy surface of NH3 must be antisymmetric with respect to binary exchange... 

Final state analysis is where dynamical methods of evolving states meet the concepts of stationary states. By their definition, final states are relatively long lived. Therefore experiment often selects a single stationary state or a statistical mixture of stationary states. Since END evolution includes the possibility of electronic excitations, we analyze reaction products in terms of rovibronic states. 

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