The groups of all orthogonal matrices is defined It has a subgroup of matrices with determinant +1, This subgroup is called specail orthogonal group. $\newcommand{\U}[1]{\underline{#1}}$
Let $K'$ and $K''$ be two systems of coordinate axes obtained by application of a rotation $(n_1,\theta_1)$ followed by $(n_2,\theta_2)$ \begin{equation} K\stackrel{(n_1,\theta_1)}{\longrightarrow} K' \stackrel{(n_2,\theta_2)}{\longrightarrow} K'' \end{equation} Let $x,x',x''$ denote components of position vector of a point x. with respect to the three sets of coordinates. Thus \begin{equation} x'=R_{\hat{n}_1}(\theta_1) x \end{equation} and \begin{equation} x''=R_{n_2}(\theta_2)x' =R_{n_2}(\theta_2)R_{n_1}(\theta_1) x
The definition of cross section is formulated in probabilistic terms. This interpretation turns out to be useful for interpretation of the cross section as an area, and also for quantum mechanical problems.
The cross section is measured by measuring the intensity of beam, scattered from a thin foil, in the forward direction as a function of thickness of the foil.
The potential scattering can be thought of as a limiting case of two particle scattering when the target is very heavy compared to the incident particle. In this case motion of the target can be ignored. The motion of the incident particle is taken as that of particle getting scattered from a potential created by the target.
As an another view, scattering of a two particle in the centre of mass frame of reference is same as the the potential scattering of a particle having mass equal to the reduced mass.
In a scattering process a beam of particles, or of waves, is incident on a target. In most general situations, the final state may consist of 'anything', subject only to conservation laws such as energy, momentum etc..
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