Microscopic particle densities

Neutron wavelength Macroscopic number density Microscopic particle density operator Nucleus cross-section Nucleus coherent cross-section Nucleus incoherent cross-section... 

For coherent scattering, the scattering function depends on the pair correlation function, G(r, t), as indicated in Eq. 8b. This correlation function can be expressed in terms of the microscopic particle density which specifies the position of a particle by a delta function... 

Gas-phase reactions play a fundamental role in nature, for example atmospheric chemistry [1, 2, 3, 4 and 5] and interstellar chemistry [6], as well as in many teclmical processes, for example combustion and exliaust fiime cleansing [7, 8 and 9], Apart from such practical aspects the study of gas-phase reactions has provided the basis for our understanding of chemical reaction mechanisms on a microscopic level. The typically small particle densities in the gas phase mean that reactions occur in well defined elementary steps, usually not involving more than three particles. 

Syntactic Cellular Polymers. Syntactic cellular polymer is produced by dispersing rigid, foamed, microscopic particles in a fluid polymer and then stabilizing the system. The particles are generally spheres or microhalloons of phenoHc resin, urea—formaldehyde resin, glass, or siUca, ranging 30—120 lm dia. Commercial microhalloons have densities of approximately 144 kg/m (9 lbs/fT). The fluid polymers used are the usual coating resins, eg, epoxy resin, polyesters, and urea—formaldehyde resin. 

Spin densities determine many properties of radical species, and have an important effect on the chemical reactivity within the family of the most reactive substances containing free radicals. Momentum densities represent an alternative description of a microscopic many-particle system with emphasis placed on aspects different from those in the more conventional position space particle density model. In particular, momentum densities provide a description of molecules that, in some sense, turns the usual position space electron density model inside out , by reversing the relative emphasis of the peripheral and core regions of atomic neighborhoods. 

A variety of assays have been developed to quantify phagocytic activity. These include direct microscopic visualization (2,3), spectrophotometric evaluation of phagocytized paraffin droplets containing dye (4), scintillation counting of radiolabeled bacteria (5), fluorometric (6), and flow cytometric analysis of fluorescent particles (7-13). The flow cytometric assay offers the advantage of rapid analysis of thousands of cells and quantification of the internalized particle density for each analyzed cell. The assay may be performed with purified leukocyte preparations (7-13) or anficoagulated whole blood (14,15). 

Luminous matter has revealed dark matter, but the new substance remains obscure. What is it made from Is it perhaps composed of known forms of matter Only partly Is dark matter made up of microscopic particles If the answer is affirmative, we may suppose that this unknown form of energy penetrates and permeates the galaxies, the Solar System and even our own bodies, just as neutrinos pass through us every second without affecting us in any way. And like the neutrinos, these unknown particles would hardly interact at all with ordinary matter made from atoms. To absorb its own neutrinos, a star with the same density as the Sun would have to measure a billion solar radii in diameter. Luminous and radiating matter is a mere glimmer to dark matter. 

Nv) = nv particles. Define now microscopic, local density of the particle... 

Of special interest in the recent years was the kinetics of defect radiation-induced aggregation in a form of colloids-, in alkali halides MeX irradiated at high temperatures and high doses bubbles filled with X2 gas and metal particles with several nanometers in size were observed [58] more than once. Several theoretical formalisms were developed for describing this phenomenon, which could be classified as three general categories (i) macroscopic theory [59-62], which is based on the rate equations for macroscopic defect concentrations (ii) mesoscopic theory [63-65] operating with space-dependent local concentrations of point defects, and lastly (iii) discussed in Section 7.1 microscopic theory based on the hierarchy of equations for many-particle densities (in principle, it is infinite and contains complete information about all kinds of spatial correlation within different clusters of defects). 

Because particles may be hard and smooth in one case and rough and spongy in another, one must express densities with great care. Density is universally defined as weight per unit volume. Three types of densities—true density, particle density, and bulk density—can be defined, depending on the volume of particles containing microscopic cracks, internal pores, and capillary spaces. 

Consider physically small volume v. Due to discreteness of the matter distribution in space a number of particles Ny in a given volume is a random variable Ny = 0,1,2 — However, on the average each volume contains Ny) = nv particles. Define now microscopic, local density of the particle... 

The formulations of the population balance equation based on the continuum mechanical approach can be split into two categories, the macroscopic- and the microscopic population balance equation formulations. The macroscopic approach consists in describing the evolution in time and space of several groups or classes of the dispersed phase properties. The microscopic approach considers a continuum representation of a particle density function. 

Ottersen OP (1989) Postembedding immunogold labelling of fixed glutamate an electron microscopic analysis of the relationship between gold particle density and antigen concentration. J Chem Neuroanat 2 57-66. 

This procedure resulted in the median ion mass profile shown in Figure 1 as a smooth curve. Also shown are three points of median charged particle masses from the electron-microscope study of Adams (17). A particle density of 1.5 g cm was used to convert Adams reported particle diameter to masses. There is good agreement within a factor of 2 between the charged particle masses and the median ion mass profile. The greatest uncertainties in the median ion mass are seen to occur at distances between 1 and 2 cm from the burner where the mass increases by more than two orders of magnitude. 

Chapter 8 briefly introduced the concept of supercritical fluids in the context of undersea thermal vents. The supercritical point for water occurs at a temperature of 705°F (374°C) and a pressure of 222.3 bar (atmosphere). Above this temperature, no pressure can condense water to its liquid state. For carbon dioxide (CO2), the critical temperature (88.0°F or 31.1°C) and critical pressure (73.8 bar) are much lower. Above the supercritical point, CO2 behaves as a liquidlike gas liquidlike densities, gaslike viscosities. The solubility properties of supercritical CO2 are mnable by varying temperature and/or pressure. Density and dielectric constant increase with increasing pressure and decreasing temperature. Water and ionic substances are insoluble in supercritical CO2. The ability of supercritical CO2 to dissolve and extract relatively non-polar substances has been known for decades. The range may be extended by adding polar solvents such as methanol or acetone. The addition of surfactants helps to disperse microscopic particles to form colloidal suspensions. Carbon dioxide is nonflammable, nontoxic, and inexpensive. 

This equation is an example of a macroscopic reaction-transport equation that can be obtained in the long-time large-scale limit of mesoscopic equations. Recall that the mesoscopic approach is based on the idea that one can introduce mean-field equations for the particle density involving a detailed description of the movement of particles on the microscopic level. At the same time, random fluctuations around the mean behavior can be neglected due to a large number of individual particles. For example, we can obtain (3.1) from the mesoscopic integro-differential equation... 

This equation states that the particle density at time n + 1 is the sum of the densities at intermediate points x - z at time n multiplied by the probability of transition from X - z to X. This is a mesoscopic description. Although it only deals with the mean density of particles p(x,n), it involves a detailed description of the movement of particles on the microscopic level. Equation (3.13) is the same as the Kolmogorov forward equation (3.12). The solution to (3.13) can be rewritten as aconvolution... 

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