Multiple scattering for particles in matter

Multiple scattering of charged particles in silicon layer and solid state detector

In addition to inelastic collisions with the atomic electrons, particles passing through matter suffer repeated elastic Coulomb scattering from nuclei although with a smaller probability.

Considering that usually nuclei have mass greater than the incoming particle, the energy transfer is negligible but each scattering centre adds a small deviation to the incoming particle’s trajectory also. Even if this deflection is small the sum of all the contribution adds a random component to the particle’s path which proceeds with a zig-zag path (see Figure 1.). As result, incoming beam after a thickness of material shown a divergence greater than the initial.

Multiple scattering in silicon wafer

  Fig 1. Effect of Multiple Coulomb Scattering.

Three situations can be considered:

1 - Single scattering. When the thickness is extremely small and the probability to have more than one interaction is negligible. This situation is well described by the Rutherford formula:

rutherford scattering formula ( light scattering)

2 - Plural scattering. When the number of Coulomb scattering increases but remains under few tens of interactions. This is the most difficult case to deal with, several works have been done by different authors (see [1] for further information).

3 - Multiple scattering. When the thickness increases and the number of interactions become high the angular dispersion can be modelled as Gaussian.

Referring to multiple scattering, that is the most common situation, naming Θ the solid angle into which is concentrated the 98% of the beam after a thicknessXof material, if we define Θ0= Θ/√2 as the projection of Θ on a plane, the angular dispersion can be calculated by the relation:

Coulomb scattering (particle light scattering)

where p is the momentum and Xo is the radiation length. This last quantity is characteristic of the material and can be found tabulated by Y.S. Tsai [2] or can be used the approximated formula

radiation length formula


  1.  E. Keil, E.Zeitler, and W. Zinn, Zeitschrift f r Naturforschung A, vol. 15A, no. 1031, 1960.

    Yung-Su Tsai, Pair production and bremsstrahlung of charged leptons, Reviews of Modern Physics, vol. 46, no. 815, 1974

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