Unsold approximation

Using an average energy or Unsold approximation [87] an expression of the type... 

Robinson (1967) has used the Unsold approximation for the energy levels to express the polarizabilities in terms of the electrostatic moments of the ground-state electron distribution. The expressions have been applied to X-ray charge densities by Zyss, Baert, and coworkers (Fkyerat et al. 1995 F. Hamzaoui, F. Baert and J. Zyss, private communication). A detailed description of the derivation and the approximations involved is beyond the scope of this treatise. However, it should be mentioned that the severe approximations are made that all excited-state energy levels are equal, and that the exciting light frequency is equal to zero. 

Due to the excitation energies of both S and Q in the denominator the local integrations do not lead directly to monomer factors, but by invoking the Unsold approximation [97] it is possible to split the denominator into a product ... 

The last integral can be estimated rather well using Unsold approximation [10] that gives the following expressions for the (hyper)polarizabilities ... 

An attempt was made by Unsold (33) to evaluate to the second-order the interaction of a proton and a hydrogen atom. He found, neglecting the resonance phenomenon, that the second-order perturbation energy is given approximately by the expression... 

A form of the curve of growth more relevant to stellar (as opposed to interstellar) absorption lines is derived from work by E. A. Milne, A. S. Eddington, M. Min-naert, D. H. Menzel and A. Unsold. In the Milne-Eddington model of a stellar photosphere, the continuum source function (equated to the Planck function in the LTE approximation) increases linearly with continuum optical depth rA and there is a selective absorption i]K, in the line, where rj(Av), the ratio of selective to continuous absorption, is a constant independent of depth given by... 

This is the Menzel-Minnaert-Unsold interpolation formula (often used assuming Roo = 1). It gives a better approximation to stellar absorption-line profiles (which are definitely not flat-bottomed) than does the exponential formula the shape of the corresponding curve of growth is much the same, but its use leads to a b-parameter that is about 25 per cent higher for the same observational data. Denoting the central value of p by po, the Doppler part of the curve is given by... 

At the end of the project, Schott had created a new material for precision optics applications, had sold some telescope mirror blanks, but had also quite an amount of unsold material of different quality in stock. It was now up to the sales people to find other applications for Zerodur than telescope mirror substrates, thereby creating new markets for this new material. Relatively soon two new applications were found which will be described in more detail in Sect. 4.6 one was mirror substrates for reflective optics in microlithography, for which castings of approximately 500 mm in diameter were needed, and the other was housings for laser gyroscopes, for which continuous bars of cross sections of about 140 x 40 mm were the best delivery... 

The difference with the computed value —1.4496 4Rh is therefore 0.00215 4Rh or 1.5 per thousand. In relation to the ionisation potential it is a little bit bigger, namely 4.8 per thousand. By comparison it is noted that in the first approximar tion, computed by Unsold, the error is still 17%, and by the fourth approximation computed Kellner still 3%. In contrast the value obtained by Slater agrees with mine. 

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