Two or more particles

For the parameters used to obtain the results in Fig. 3, X 0.6 so the mean free path is comparable to the cell length. If X -C 1, the correspondence between the analytical expression for D in Eq. (43) and the simulation results breaks down. Figure 4a plots the deviation of the simulated values of D from Do as a function of X. For small X values there is a strong discrepancy, which may be attributed to correlations that are not accounted for in Do, which assumes that collisions are uncorrelated in the time x. For very small mean free paths, there is a high probability that two or more particles will occupy the same collision volume at different time steps, an effect that is not accounted for in the geometric series approximation that leads to Do. The origins of such corrections have been studied [19-22]. 

For systems of two or more particles orbital and spin angular momenta are added separately by the rules of vector addition. Integral projections of the shorter vector on the direction of the longer vector, are added to the long vector, to give the possible eigenvalues of total angular momentum. 

It is noted that if ei = e2 the anti-symmetric wave function vanishes, ipa = 0. Two identical particles with half-spin can hence not be in the same non-degenerate energy state. This conclusion reflects Pauli s principle. Particles with integral spin are not subject to the exclusion principle (ips 0) and two or more particles may occur in the same energy state. 

Effect of particle size. If the rate of reaction, ( rA)ohs, is measured for two or more particle sizes (values of Le), while other conditions are kept constant, two extremes of behavior may be observed. 

The term interaction is sometimes used almost synonymously with cooperativity. In this book we reserve the term interaction to mean direct interaction energy between two (or more) particles. Indeed, sometimes interaction, in the above sense, is the sole source of cooperativity, in which case the two terms may be used interchangeably. However, in most cases of interest in biochemistry, interaction in the above sense is almost negligible, such as in two oxygen molecules in hemoglobin. Cooperativity in such systems is achieved by indirect routes of communication between the ligands. 

Many experiments (see Section I.B) require the energy analysis and detection of two or more particles with time correlation in other words, coincidence counting techniques must be used. Coincidence methods have long been used in nuclear physics because of the convenient fast detectors that have long been available. The more recent availability of fast, high-gain electron multipliers has created the possibility of coincidence measurements in electron spectroscopy. Various aspects of coincidence measurements have been discussed elsewhere.100 102... 

Coalescence is when two or more particles, droplets or bubbles fuse together to form a single larger unit, reducing the total surface area. 

There were other, less theoretical but no less persuasive objections. Some substances, such as ammonium chloride, dissociate in the vapor phase. That is, a single particle of vapor turns into two or more particles. Two or more particles occupy two or more times the volume that one particle does. That wreaks havoc with measurements of gas volumes and provides empirical evidence that fails to obey Gay-Lussac s law, making apparent nonsense of Avogadro s hypothesis. It was not until the phenomenon of dissociation was understood, and interpreted in terms of reaction kinetics, that this objection could be countered. Similar objections were raised against Dalton s laws of combining proportions, which work only for compounds of fixed composition. Metallic alloys and salt solutions, to take two of the most obvious exceptions, do seem to share some of the characteristics of chemical compounds, but they do not fit Daltons laws. The simplest way to avoid that objection was to say that only those substances that did fit Dalton s laws were true chemical compounds, but that is a circular argument that did not convince critics. 

When using polyelectrolytes even more complicated effects can arise. The addition of large polyelectrolytes to a suspension of opposite charged particles is likely to lead to bridging with again one polymer chain attached to two or more particles, in this way forming a necklace . 

Flocculation processes are complicated phenomena because of the varieties of both particle morphology and chemical reactions they encompass.34 A few concepts of a general nature have emerged, however, and they will be the focus of this chapter. From the perspective of kinetics, perhaps the most important of these broad generalizations is the distinction that can be made between transport-controlled and reaction-controlled flocculation, parallel to the classification of adsorption processes described in Section 4.5. Flocculation kinetics are said to exhibit transport control if the rate-limiting step is the movement of two (or more) particles toward one another prior to their close encounter and subsequent combination into a larger particle. Reaction control occurs if it is particle combination instead of particle movement (toward collision) that limits the rate of flocculation. 

Attraction of Two Spheres—An expression for the attraction between two or more particles, based on collision theory rather than attraction due to the motion of spheres, may be developed by the methods of dimensional analysis. Let the force of attraction between two particles of diameters d and d2 be F and assume that when the particles are close enough so that the gaseous film enveloping each particle coalesces over a region about the point of contact, then the whole attraction is due to a free surface energy a. If the average surface of contact of the particles is denoted by Sc then... 

Optical absorption and recombination processes involve two or more particles and so may include correlation effects. Electron-hole pairs form excitons in a crystal, with the result that the absorption and emission spectra are not described by the one-particle density of states distributions. Although excitons can exist in an amorphous material (see Chapter 3), they are not detected in the optical spectra and the absorption is described by the convolution of the one-particle densities of valence and conduction band states. The correlation effects in... 

The purpose of adding a polymer is not always to stabilize colloidal particles in other cases one aims at imparting an effective attraction which will bring the particles together in a floe or a network. The Interparticle attraction is then caused by the formation of so-called polymer bridges, strands of pol rmer that are adsorbed simultaneously onto two or more particles, in this way bridging the gap between them. In these applications kinetic aspects, which have not received much attention in the present chapter, may be crucial. We mention a number of examples of this type of destabilization. 

Primary Coincidence. Two or more particles in the measurement zone give rise to two or more overlapping pulses. Depending on their proximity and electrical resolution, these pulses may not be resolved, leading to loss of count. 

The optical and spectroscopic data of the local nanoparticle structures investigated showed that SERS is a local and time-dependent phenomenon, because (1) only few particles are Raman active particles, (2) strongest enhancements in SERS are obtained from particle agglomerates, (3) typically the Raman radiation is emitted from irregular stmctures like the necks between two or more particles agglomerated, (4) a time-dependent behavior characterized by intensity fluctuations was observed. 

It is important to note that the interference pattern is built up from single, separate particles. There is no interference between two or more particles during their evolution in the apparatus. Single particle interference is evidenced in our case by two independent arguments. 

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