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Concept# Atomic form factor

Summary

In physics, the atomic form factor, or atomic scattering factor, is a measure of the scattering amplitude of a wave by an isolated atom. The atomic form factor depends on the type of scattering, which in turn depends on the nature of the incident radiation, typically X-ray, electron or neutron. The common feature of all form factors is that they involve a Fourier transform of a spatial density distribution of the scattering object from real space to momentum space (also known as reciprocal space). For an object with spatial density distribution, \rho(\mathbf{r}), the form factor, f(\mathbf{Q}), is defined as
:f(\mathbf{Q})=\int \rho(\mathbf{r}) e^{i\mathbf{Q} \cdot \mathbf{r}}\mathrm{d}^3\mathbf{r},
where \rho(\mathbf{r}) is the spatial density of the scatterer about its center of mass (\mathbf{r}=0), and \mathbf{Q} is the momentum transfer. As a result of the nature of the Fourier transform, the broader

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2011The exact formalism from B. Zotter to compute beam coupling impedances has been fully developed only in the case of an infinitely long circular beam pipe. For other two dimensional geometries, some form factors are known only in the ultrarelativistic case and under certain assumptions of conductivity and frequency of the pipe material. We present here a new and exact formalism to compute the beam coupling impedances in the case of a collimator-like geometry where the jaws are made of two infinite plates of any linear material. It is shown that the impedances can be computed theoretically without any assumptions on the beam speed, material conductivity or frequency range. The final formula involves coefficients in the form of integrals that can be calculated numerically. This way we obtain new generalized form factors between the circular and the flat chamber cases, which eventually reduce to the so-called Yokoya factors under certain conditions.

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