Two Special Cases

In liquefied rare gases (LRG) the ejected electron has a long thermalization distance, because the subexcitation electrons can only be thermalized by elastic collisions, a very inefficient process predicated by the small mass ratio of the electron to that of the rare gas atom. Thus, even at a minimum of LET (for a -1-MeV electron), the thermalization distance exceeds the interionization distance on the track, determined by the LET and the W value, by an order of magnitude or more (Mozumder, 1995). Therefore, isolated spurs are never seen in LRG, and even at the minimum LET the track model is better described with a cylindrical symmetry. This matter is of great consequence to the theoretical understanding of free-ion yields in LRG (see Sect. 9.6). 

Recently, a biexcitonic quenching mechanism has been proposed to explain the variation of scintillation intensity in liquid argon (LAr) with the LET and quality of incident radiation (Hitachi et al., 1992). According to this, quenching occurs mainly in the track core due to high-energy deposition density. This 

(1961), The Radiation Chemistry of Water and Aqueous Solutions, ch. 5. Van Nostrand, Princeton, N.J. 

Chapter 3 Structure of Charged Particle Tracks in Condensed Media 

Lamerton, L. F. (1963), in Proc. Second Intern. Cong. Radiation Res. (Harrogate, England), p. 1, Yearbook Medical Publishing, Chicago. 

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The derivation of the D matrix for a given contour is based on first deriving the adiabatic-to-diabatic transformation matrix, A, as a function of s and then obtaining its value at the end of the arbitrary closed contours (when s becomes io). Since A is a real unitary matrix it can be expressed in terms of cosine and sine functions of given angles. First, we shall consider briefly the two special cases with M = 2 and 3. 

Returning now to the rigid-body rotational Hamiltonian shown above, there are two special cases for which exact eigenfunctions and energy levels can be found using the general properties of angular momentum operators. 

Multicomponent Diffusion. In multicomponent systems, the binary diffusion coefficient has to be replaced by an effective or mean diffusivity Although its rigorous computation from the binary coefficients is difficult, it may be estimated by one of several methods (27—29). Any degree of counterdiffusion, including the two special cases "equimolar counterdiffusion" and "no counterdiffusion" treated above, may arise in multicomponent gas absorption. The influence of bulk flow of material through the films is corrected for by the film factor concept (28). It is based on a slightly different form of equation 13 ... 

The 6/3-amino group of 6-APA may be alkylated either with diazoalkanes <67LA(702)163) or by the reduction of Schiff bases (Scheme 50) (65JCS3616). Two special cases of N-alkylation are also shown in Scheme 50 the formation of an imidazolidinone ring upon treating ampicillin with acetone (66JOC897), and the formation of a 6/3-amidinopenicillanic acid from 6-APA (77MI51105). 

Confidence limits are partial integrations over a probability density function. There are two special cases failure with time and failure with demand. 

There are two special cases for which equations 2.6-7 and 2.6-8 are easily solved to fold a prior distribution with the update distribution to obtain a posterior distribution with the sarai rm as the prior distribution. These distributions are the Bayes conjugates shown in Table 2.6-1. 

One of the first solutions to the problem of stresses around an elliptical hole in an infinite anisotropic plate was given by Lekhnitskii [6-7]. A more recent and comprehensive summary of the problem and many others is Savin s monograph [6-8]. Numerous results by Lekhnitskii are shown in his books [6-9 and 6-10]. Two special cases are of particular interest. 

We now consider two special cases of the symmetric matrix A] namely, (1) the entries of A are allowed to take on only integer values, and (2) the entries of A are all binary valued. 

The most straightforward preparation of A-(l-hydroxyalkyl)amides (or carbamates) involves addition of primary or secondary amides (carbamates) to aldehydes or ketones. This is an equilibrium process in which formation of the adduct is usually disfavored, except for two special cases ... 

The time of effect duration (TED) is a ftmction of the elimination half-life. Two special cases are most important, the 50% effect bisection time (TED50) and the duration of 90% of the effect (TED90). 

In two special cases, (i) ftN large so that all available sites (N0) are rapidly exhausted and (ii) ftN very small so that kB — feT(a) >> kNN0, the same approximation is applicable... 

Both net and multiplet effects must normally be considered except in two special cases (i) when = 0 and only multiplet effects are observed and (ii) when ai = 0 in which case there is no CIDNP to observe. In addition, if there is no coupling between a given nucleus or nuclei and any other nuclei in the product, the n.m.r. spectrum will be a single peak, which of necessity can show only net polarization. 

Many variations are possible around the basic flash calculation. Pressure and V/F can be specified and T calculated, and so on. Details can be found in King7. However, two special cases are of particular interest. If it is necessary to calculate the bubble point, then V/F = 0 in Equation 4.55, which simplifies to ... 

Operation at constant reflux ratio is better than operation with constant distillate composition for high-yield batch separations. However, operation with constant distillate composition might be necessary if high product purity is required. In fact, it is not necessary to operate in one of these two special cases of constant reflux ratio or constant distillate composition. Given the appropriate control scheme, the reflux ratio can be varied through the batch... 

Two special cases of radiative heat transfer which are applicable to heat transfer in freeze drying are illustrated by Figure 32. Heat transfer between body 1 and body 2 are illustrated for case I (Fig. 32a), where body 1 is of much greater area than body 2, Ai A2, and surrounds body 2. The freeze-drying example is heat exchange between the top of the vial (body 2) and the freeze dryer shelf... 

Figure 32 Two special cases of radiation heat transfer with importance to freeze-drying applications, (a) Case I Body 1 is the surroundings. (b) Case II Bodies 1 and 2 are of equal areas, and e = 1 for body 2.

Before we move on to a discussion of the methods of solution, we may note the two special cases of this general formulation. The special case of... 

This can be considered as the average energy of a specific atom over time or the average energy of all the atoms at a specific time. It is useful to consider two special cases. At low temperature, hv > kT and the average energy is approximately 1/2hv. At high temperature, hv[Pg.246]

Vapor flow through pipes is modeled using two special cases adiabatic and isothermal behavior. The adiabatic case corresponds to rapid vapor flow through an insulated pipe. The isothermal case corresponds to flow through an uninsulated pipe maintained at a constant temperature an underwater pipeline is an excellent example. Real vapor flows behave somewhere between the adiabatic and isothermal cases. Unfortunately, the real case is difficult to model and no generalized and useful equations are available. 

Corresponding equations for the two special cases of gas-film mass-transfer control and surface-reaction-rate control may be obtained from these results (they may also be derived individually). The results for the latter case are of the same form as those for reaction-rate control in the SCM (see Table 9.1, for a sphere) with R0 replacing (constant) R (and (variable) R replacing rc in the development). The footnote in Example 9-2 does not apply here (explain why). 

It is necessary to distinguish among three rate quantities. We use the symbol NA to represent the flux of A, in mol m-2 s-1, through gas and/or liquid film if reaction takes place in the liquid film, NA includes the effect of reaction (loss of A). We use the symbol (—rA), in mol m-2 s 1, to represent the intensive rate of reaction per unit interfacial area. Dimensionally, (—rA) corresponds to NA, but (— rA) and NA are equal only in the two special cases (1) and (2) above. In case (3), they are not equal, because reaction occurs in the bulk liquid (in which there is no flux) as well as in the liquid film. In this case, furthermore, we need to distinguish between the flux of A into the liquid film at the gas-liquid interface, NA(z = 0), and the flux from the liquid film to the bulk liquid, Na(z = 1), where z is the relative distance into the film from the interface these two fluxes differ because of the loss of A by reaction in the liquid film. The third rate quantity is ( rA)int in mol irT3 s-1, the intrinsic rate of reaction per unit volume of liquid in the bulk liquid. ( rA) and (- rA)int are related as shown in equation 9.2-17 below. 

The concept of reduction potential is introduced in Chapter 6. When the reduction potentials of two species differ by 0.1 V or more, the resulting redox reaction will proceed rapidly and stoichiometrically so that it may be used as the basis for a titrimetric procedure. The end point of a redox titration may be observed by following the potential of the titrand with an indicator electrode or with a visual indicator. In two special cases, the reagent (potassium permanganate and iodine) is self-indicating (vide infra). 

A change of state according to equation 6.27 is called a polytropic change. Two special cases are the isothermal change and the adiabatic... 

Two special cases of the fontuilu (32.8) which occur frequently are... 

These observations are explained by the conversion of CO to CO2 on sites active for CO conversion, as the intrinsically produced CO makes its way out of the catalyst bead by diffusion. The greater the diffusion limitation, the more the CO conversion. Weisz gave the mathematical solution for two special cases of these phenomena and demonstrated that the above interpretation was correct. 

Two special cases of Eq. 3-108a,b are of interest. When a chain-transfer agent is absent, the term in [S] diappears and... 

There are two special cases that are worth special mention. For the case that the interaction term is the arithmetic mean, then... 

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