Thomson formula

The scattering of X-radiation by matter was first observed by Sagnac [SAG 97b] in 1897. The basic relation expressing the intensity scattered by an electron was laid down the following year by Thomson [THO 98dj. 

Consider a free electron located inside a parallel X-ray beam with intensity Iq. This beam constitntes a plane wave traveling along the x-axis, encountering a free electron located in O. The electron, subjected to the acceleration  

Therefore, we obtain a wave traveling through P with the same frequency as the incident wave and with amplitude  

The unit surface located in P is observed from O with a solid angle equal to 1/r. Therefore, the intensity in question, with respect to the unit of solid angle is 

I = Igte. Likewise, the intensity along the Oy-axis is given by = Igr cos 20. 

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A quantitative test of the Gibbs-Thomson formula will accordingly involve the following measurements we first determine the surface tension of a solution for a number of different concentrations, plot the [Pg.41]

In view of the fundamental importance of the Gibbs-Thomson formula, and the magnitude of the discrepancies between the figures calculated from it and the experimental results, it is of obvious interest to inquire to What causes the deviations may be due. The first point to be noticed is that the complex substances which exhibit them most markedly form, at least at higher concentrations, colloidal and not true solutions. It is, therefore, very probable that they may form gelatinous or semi-solid skins on the adsorbent surface, in which the concentration may be very great. There is a considerable amount of evidence to support this view. Thus Lewis finds that, if the thickness of the surface layer be taken as equal to the radius of molecular attraction, say 2 X io 7 cms., and the concentration calculated from the observed adsorption, it is found, for instance, for methyl orange, to be about 39%, whereas the solubility of the substance is only about 078%. The surface layer, therefore, cannot possibly consist of a more concentrated solution of the dye, which is the only case that can be dealt with theoretically, but must be formed of a semi-solid deposit. 

This equation, called the Thomson formula, is given mainly in old textbooks on electrochemistry as an approximate equation which expresses the relation between the EMF of a cell and the heat of reaction. As will be seen from the explanation of electrolysis below, the same formula may be also used for a rough estimate of the decomposition voltage of a compound, if no other data, except the reaction heat, are available. It is worth mentioning that the Thompson formula is precisely valid only with such systems the electromotive force of which does not change with temperature. 

This is the Thomson formula for scattering from a free charge. The differential Thomson cross section a is given by the angle-independent part of Eq. (3.25.17) ... 

The general formula for scattered intensities (away from X-ray absorption edges) in the Thomson formula is... 

This relation is called the Thomson formula [THO 98d]. It has a specific use the scattering power of a given object can be defined as the nmnber of free and independent electrons this object would have to be replaced with, in order to obtain the same scattered intensity. 

This is called the Thomson formula for the scattering of x-rays by a single electron. 

Direct Ionization by Electron Impact Thomson Formula... 

When the transferred energy exceeds the ionization potential, Ae > 7, direct ionization takes place. Thus, integration of (2-10) over Ae > 7 gives an expression for the ionization cross section by direct electron impact, known as the Thomson formula ... 

The Thomson formula can be rewritten taking into account the kinetic energy v of the valence electron (Smirnov, 2001) ... 

The Thomson formula (2-13) agrees with (2-12), assuming the valence electron is at rest and Sv = 0. An interesting variation of the Thomson formula (2-13) can be obtained assuming a Coulomb interaction of the valence electron with the rest of the atom and taking... 

All the modifications of the Thomson formula can be combined using the generalized function /(e/7) common for all atoms ... 

Here Zv is the number of valence electrons in an atom. For the simplest Thomson formula (2-11), the generalized function can be expressed as... 

Ionization by Direct Electron Impact, Thomson Formula. Compare cross sections of the direct ionization by electron impact calculated using different modifications of the Thomson formula (2-11), (2-13), and (2-14). Calculate the maximum values of the cross sections and corresponding values of electron energy for each of the Thomson formula modifications. 

As to the total cross-section of scattering by free electron irradiated by unpolarized light, it is described by the same Thomson formula (5.7) that is easy to check by integrating (5.8) over 20 (from 0 to Jt) and over (p (from 0 to 27t). 

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