Fowler equation

The total electron current generated in this process is given by the Fowler equation ... 

Another method for finding the W value, called the Fowler equation approach (Inokuti, 1975), is based on three assumptions, some of which can be relaxed. These are (1) that the incident particle is an electron (2) that there is only one ionization potential and (3) that the ionization efficiency is unity— that is, any energy loss E > I results in an ionization with a primary of energy... 

An early theory of the IT value was proffered by Spencer and Fano [44], based on the degradation spectrum. Another method, the Fowler equation, was employed by Inokuti [47] for electron irradiation, based on the approximation that there is only one ionization potential and that the ionization efficiency is unity. These restrictions can be relaxed. The main result of Inokuti s analysis may be given as follows. 

A note on a form of the Emden-Fowler Equation (with B.N. Mehta). J. Math. Anal. Appl. 36, 611-621 (1971). 

Assuming lateral interactions between the adsorbed species we may derive [9] the "Stern-Langmuir-Fowler" equation... 

It is curious that he never conuuented on the failure to fit the analytic theory even though that treatment—with the quadratic fonn of the coexistence curve—was presented in great detail in it Statistical Thermodynamics (Fowler and Guggenlieim, 1939). The paper does not discuss any of the other critical exponents, except to fit the vanishing of the surface tension a at the critical point to an equation... 

One important direetion of study has been to use empirieal adsorption data, together with the preassumed model for loeal adsorption, and attempt to extraet information about the form of x(e) [13,14]. The ehoiee of the model for loeal adsorption, whieh is an important input here, has been eustomarily treated quite easually, assuming that it has rather limited influenee on the form and properties of the evaluated EADFs. Usually, one of so many existing equations developed for adsorption on uniform surfaees is used as the loeal adsorption isotherm. The most often used forms of 0 p, T,e) are the Langmuir [6] and the Fowler-Guggenheim [15] equations for loealized adsorption. Ross and Olivier [4] extensively used the equation for mobile adsorption, whieh results from the two-dimensional version of the van der Waals theory of fluids. The most radieal solution has been... 

The field-induced tunneling across a potential barrier can be described using the Fowler-Nordheim equation [79] ... 

However, the values for the current, which are obtained using the Fowler-Nordheim equation with the actual device parameters, are several orders of magnitude [81, 82) higher than the values for the measured current in real devices. 

For the same the single layer devices based on Alq3, Peyghambarian et al. [83] found that the 1/V characteristics can also be described by an SCL current flow in the low cu ire lit regime. However, in the low current regime the 1/V characteristics can be qualitatively modeled by the Fowler-Nordheim equation (even if, quantitatively, the real device current differs from the calculated by seven orders of magnitude) [83] and thermionic emission ]78]. 

Even with the simpler Frumkin or Fowler-Guggenheim approach (Eqs. 6.50 and 6.52), treating the coadsorption and surface reaction of different adsorbates leads immediately to mathematically intractable expressions and to the introduction of new parameters, whereas equation... 

This isotherm is consistent with the modified electrochemical Langmuir isotherm, the Nemst equation and the potential-work function equivalence. For intermediate 0j and Pj values the isotherm of Eq. (6.58) is well approximated both by the Fowler-Guggenheim and by the Temkin isotherms. 

The Fowler-Guggenheim-Jovanovic model [3] assumes (as it was the earlier case also) the occurrence of intermolecular interactions among the molecules adsorbed as a monolayer but is based on the Jovanovic isotherm. The single-component isotherm is represented by the equation ... 

The most spectacular peak profiles, which suggest self-associative interactions, were obtained for 5-phenyl-1-pentanol on the Whatman No. 1 and No. 3 chromatographic papers (see Figure 2.15 and Figure 2.16). Very similar band profiles can be obtained using the mass-transfer model (Eqnation 2.21), coupled with the Fowler-Guggenheim isotherm of adsorption (Equation 2.4), or with the multilayer isotherm (Equation 2.7). 

The characteristics of the network also enter into the considerations in selecting the problem formulation. If the network is acyclic, are the formulations involving cycle equations still appropriate Epp and Fowler (E2) suggested that if the pressures at two connected vertices are given, a cycle equation can always be written for a pseudo-loop which contains a fictitious edge linking these two vertices. In this way formulation D may be modified to accommodate networks that are not completely cyclic. However, for other types of specifications, formulations A and C are clearly more appropriate. The choice of a formulation is thus closely intertwined with the nature of problem specification which is the next topic of our discussion. 

The change of rate equation has been attributed to wall effects which were shown by Fowler and Beaver127, to be important under these conditions. 

The Frumkin Equation (also referred to as the Frumkin-Fowler-Guggenheim, FFG, equation) has been specifically developped to take lateral interactions at the surface into account. In the FFG equation, the term 6 / (1-6) in (4.10b) is multiplied by the factor exp(-2 a0) which reflects the extent of lateral interactions... 

Hiickel s application of this approach to the aromatic compounds gave new confidence to those physicists and chemists following up on the Hund-Mulliken analysis. It was regarded by many people as the simplest of the quantum mechanical valence-bond methods based on the Schrodinger equation. 66 Hiickel s was part of a series of applications of the method of linear combination of atom wave functions (atomic orbitals), a method that Felix Bloch had extended from H2+ to metals in 1928 and that Fowler s student, Lennard-Jones, had further developed for diatomic molecules in 1929. Now Hiickel extended the method to polyatomic molecules.67... 

Field emission is the emission of electrons from a solid under an intense electric field, usually at ambient temperatures. It occurs by the quantum mechanical tunneling of electrons through a potential barrier (Fig. 13.1). This leads to an exponential dependence of emission current density J on the local electric field, as given by the Fowler Nordheim equation,... 

CO adsorbed Pd/Al Oj, 37 69-70 diffuse reflectance spectroscopy, 34 14 Fowler-Nordheim equation, 21 72 Fraction exposed, of total metal atoms definition, 36 72-73... 

The adsorption isotherm was modeled in order to deduce a numerical value for A. Following the model of Frumkin and Fowler reported in [21], is given by the following set of equations ... 

The reaction rate coefficients in the above equations may be related to reaction rates per pair of particles 2/, in nuclear physics (e.g., Fowler et al., 1975 Harris et al., 1983) by k = Xj/(1 + 5/ ), where 8 = 0 except for i= , for which 5/ = 1. That is, for Reactions 2-145 and 2-147 in which two identical particles collide to react, the definition of k is half of defined by nuclear physicists and for reactions in which different particles collide, the definition of k is the same as Xij. The reaction rate coefficients depend on temperature in a complicated way (Table 2-3) and may be calculated as the average value of the product of relative velocity times cross section. The concentrations of the intermediate species can be derived as follows. From Equation 2-155, 145 [ H] = ki4e[ H]pH]. That is. 

Fig. 1.38. The Fowler Nordheim equation for field emission. The relevant dimensions are much smaller than the tip radius. Therefore, a one-dimensional model is adequate. Neglecting the image-force effect, the potential U(z) outside the metal surface is linear with respect to distance z- The relevant parameters are work function of the material, tj), and the field intensity near the surface, F.

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