Quantum concentration

Find the number density of positrons resulting from pair production by y-rays in thermal equilibrium in oxygen at a temperature of 109 K and a density of 1000 gmcm-3, using the twin conditions that the gas is electrically neutral and that the chemical potentials of positrons and electrons are equal and opposite. (At this temperature, the electrons can be taken as non-relativistic.) The quantum concentration for positrons and electrons is 8.1 x 1028 T93/2 cm-3, the electron mass is 511 keV and kT = 86.2 T9 keV. 

The equation of state described in Sect. 7.3 is valid when the gas can be described as classical, i.e. when the average separation of particles is large compared with the de Broglie wavelength5. As the average separation becomes small, the lightest particles will experience a breakdown of classical physics. This occurs when the particle density n significantly exceeds the critical quantum concentration ... 

This section will concentrate on the motions of atoms within molecules— internal molecular motions —as comprehended by the revolutionary quantum ideas of the 20th century. Necessarily, limitations of space prevent many topics from being treated in the detail they deserve. Some of these are treated in more detail in... 

In the present section, we concentrate on coherent preparation by irradiation with a properly chosen laser pulse during a given time interval. The quantum state at time t may be chosen to be the vibrational ground... 

Optical metiiods, in both bulb and beam expermrents, have been employed to detemiine tlie relative populations of individual internal quantum states of products of chemical reactions. Most connnonly, such methods employ a transition to an excited electronic, rather than vibrational, level of tlie molecule. Molecular electronic transitions occur in the visible and ultraviolet, and detection of emission in these spectral regions can be accomplished much more sensitively than in the infrared, where vibrational transitions occur. In addition to their use in the study of collisional reaction dynamics, laser spectroscopic methods have been widely applied for the measurement of temperature and species concentrations in many different kinds of reaction media, including combustion media [31] and atmospheric chemistry [32]. 

This chapter concentrates on describing molecular simulation methods which have a counectiou with the statistical mechanical description of condensed matter, and hence relate to theoretical approaches to understanding phenomena such as phase equilibria, rare events, and quantum mechanical effects. 

A substantial sunnner school proceedings, concentrating on modem teclmiques for studying rare events and quantum mechanical phenomena. 

Nonradiative reiaxation and quenching processes wiii aiso affect the quantum yieid of fluorescence, ( )p = /cj /(/cj + Rsiative measurements of fluorescence quantum yieid at different quencher concentrations are easiiy made in steady state measurements absoiute measurements (to detemrine /cpjj ) are most easiiy obtained by comparisons of steady state fluorescence intensity with a fluorescence standard. The usefuiness of this situation for transient studies... 

In quantum theory, physical systems move in vector spaces that are, unlike those in classical physics, essentially complex. This difference has had considerable impact on the status, interpretation, and mathematics of the theory. These aspects will be discussed in this chapter within the general context of simple molecular systems, while concentrating at the same time on instances in which the electronic states of the molecule are exactly or neatly degenerate. It is hoped... 

This section attempts a brief review of several areas of research on the significance of phases, mainly for quantum phenomena in molecular systems. Evidently, due to limitation of space, one cannot do justice to the breadth of the subject and numerous important works will go unmentioned. It is hoped that the several cited papers (some of which have been chosen from quite recent publications) will lead the reader to other, related and earlier, publications. It is essential to state at the outset that the overall phase of the wave function is arbitrary and only the relative phases of its components are observable in any meaningful sense. Throughout, we concentrate on the relative phases of the components. (In a coordinate representation of the state function, the phases of the components are none other than the coordinate-dependent parts of the phase, so it is also true that this part is susceptible to measurement. Similar statements can be made in momentum, energy, etc., representations.)... 

For larger systems, various approximate schemes have been developed, called mixed methods as they treat parts of the system using different levels of theory. Of interest to us here are quantuin-seiniclassical methods, which use full quantum mechanics to treat the electrons, but use approximations based on trajectories in a classical phase space to describe the nuclear motion. The prefix quantum may be dropped, and we will talk of seiniclassical methods. There are a number of different approaches, but here we shall concentrate on the few that are suitable for direct dynamics molecular simulations. An overview of other methods is given in the introduction of [21]. 

In molecular dynamics applications there is a growing interest in mixed quantum-classical models various kinds of which have been proposed in the current literature. We will concentrate on two of these models the adiabatic or time-dependent Born-Oppenheimer (BO) model, [8, 13], and the so-called QCMD model. Both models describe most atoms of the molecular system by the means of classical mechanics but an important, small portion of the system by the means of a wavefunction. In the BO model this wavefunction is adiabatically coupled to the classical motion while the QCMD model consists of a singularly perturbed Schrddinger equation nonlinearly coupled to classical Newtonian equations, 2.2. 

Various kinds of mixed quantum-classical models have been introduced in the literature. We will concentrate on the so-called quantum-classical molecular dynamics (QCMD) model, which consists of a Schrodinger equation coupled to classical Newtonian equations (cf. Sec. 2). 

Quantum chemists have devised efficient short-hand notation schemes to denote the basis set aseti in an ab initio calculation, although this does mean that a proliferation of abbrevia-liijii.s and acronyms are introduced. However, the codes are usually quite simple to under-sland. We shall concentrate on the notation used by Pople and co-workers in their Gaussian aerie-, of programs (see also the appendix to this chapter). 

The intensity of fluorescence therefore, increases with an increase in quantum efficiency, incident power of the excitation source, and the molar absorptivity and concentration of the fluorescing species. 

Standardizing the Method Equations 10.32 and 10.33 show that the intensity of fluorescent or phosphorescent emission is proportional to the concentration of the photoluminescent species, provided that the absorbance of radiation from the excitation source (A = ebC) is less than approximately 0.01. Quantitative methods are usually standardized using a set of external standards. Calibration curves are linear over as much as four to six orders of magnitude for fluorescence and two to four orders of magnitude for phosphorescence. Calibration curves become nonlinear for high concentrations of the photoluminescent species at which the intensity of emission is given by equation 10.31. Nonlinearity also may be observed at low concentrations due to the presence of fluorescent or phosphorescent contaminants. As discussed earlier, the quantum efficiency for emission is sensitive to temperature and sample matrix, both of which must be controlled if external standards are to be used. In addition, emission intensity depends on the molar absorptivity of the photoluminescent species, which is sensitive to the sample matrix. 

C is the concentration of limiting reactant in mol/L, c is the chemiluminescence quantum yield in ein/mol, and P is a photopic factor that is determined by the sensitivity of the human eye to the spectral distribution of the light. Because the human eye is most responsive to yellow light, where the photopic factor for a yellow fluorescer such as fluorescein can be as high as 0.85, blue or red formulations have inherently lower light capacities. 

High quantum yields are uncommon and, moreover, the quantum yield almost always decreases as the concentration of chemiluminescent reactant approaches practical levels. Thus even reactions with high inherent quantum yields at low concentrations, such as the firefly reaction, do not necessarily provide high light capacities (237). 

The theoretical limit of light capacity has been estimated for an ideal reaction that provides yellow light with a photopic factor of 0.85 in a quantum yield of one at 5 Af concentration as 173,000 (Im-h)/L, equivalent to the light output of a 40-W bulb burning continuously for two weeks (237). The most efficient formulation available, based on oxaUc ester chemiluminescence, produces about 0.5% of that limit, with a light capacity of 880 (Im-h)/L (237). 

In practice, o2one concentrations obtained by commercial uv devices ate low. This is because the low intensity, low pressure mercury lamps employed produce not only the 185-nm radiation responsible for o2one formation, but also the 254-nm radiation that destroys o2one, resulting in a quantum yield of - 0.5 compared to the theoretical yield of 2.0. Furthermore, the low efficiency (- 1%) of these lamps results in a low o2one production rate of 2 g/kWh (100). 

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