Bethe surface

B1.6.2.4 THE BETHE SURFACE BINARY VERSUS DIPOLE COLLISIONS... 

A succinct picture of the nature of high-energy electron scattering is provided by the Bethe surface [4], a tlnee-dimensional plot of the generalized oscillator strength as a fiinction of the logaritlnn of the square of the... 

Figure Bl.6.6 The Bethe surface. The sharp ridge corresponds to scattering from a single stationary target electron the broadened ridge to scattering from the electrons in an atom or molecule.

Although the actual form of f(w, q) is different from formula (4.27), the latter leads to reasonable results when we use it to calculate the cross sections of inelastic collisions and the ionization losses.120 As one of the reasons for using approximation (4.27), one can consider the fact that the data concerning the Bethe surfaces for molecules are very scant, while there is extensive information about the optical oscillator strengths of molecules both in the discrete and in the continuous regions of the spectrum (see Refs. 119 and 121). 

Hammerschmidt, S., Bethe, G., Remane, P. H. and Chhatwal, G. S. (1999). Identification of pneumococcal surface protein A as a lactoferrin-binding protein of Streptococcus pneumoniae, Infect. Immun., 67, 1683-1687. 

The blame for this should be placed on the use of tree-like models. These tree-like, Bethe networks of pores are characterized by a finite ratio of surface pores to those in the bulk (equal to (Z - 2) / (Z -... 

For the catalyst particle modeled by the network this ratio corresponds to the ratio of the geometric area to overall surface area, which for most catalysts is essentially zero as is also the case for three-dimensional networks. Accounting for the activity of the surface pores in Bethe networks tends to smooth out all the abrupt changes in the activity that would be otherwise observed at the percolation threshold. 

The model in Fig. 3.2 is sufficient to predict the general features of N E), but much more detailed calculations are needed to obtain an accurate density of states distribution. Present theories are not yet as accurate as the corresponding results for the crystalline band structure. The lack of structural periodicity complicates the calculations, which are instead based on specific structural models containing a cluster of atoms. A small cluster gives a tractable numerical computation, but a large fraction of the atoms are at the edge of the cluster and so are not properly representative of the real structure. Large clusters reduce the problem of surface atoms, but rapidly become intractable to calculate. There are various ways to terminate a cluster which ease the problem. For example, a periodic array of clusters can be constructed or a cluster can be terminated with a Bethe lattice. Both approaches are chosen for their ease of calculation, but correspond to structures which deviate from the actual a-Si H network. 

Even Davisson and Germer s first work on the reflection of slow electrons by crystal lattices made it clear that the facts could not be accurately represented by equations (3) and (5) on the contrary, definite deviations from Bragg s law of reflection occur. These were first explained by Patterson as being due to a diminution of the distance between the lattice planes at the surface. Bethe has shown, however, that better agreement with experiment is obtained by expressing the action of the crystal on the electrons by means of a mean lattice potential V. Schrodinger s equation for the de Broglie waves with an internal lattice potential is then... 

One of us has presented a statistical theory for strong electrolytes according to this point of view. The model of order-disorder is similar to that applied by Bethe in his theory of binary alloys. This model has been applied to adsorption phenomena of gases on surfaces, where it has explained the abnormal cases of isobaric adsorption, and it reproduces the isotherm of Langmuir. It has also been used for condensation phenomena, where we obtain from it the relations between physical quantities at the critical point. 

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