Scattering Cross Section

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Scattering Cross Section

Scattering.png


dσdΩ=(number of particles scattered/seconddΩ)(number of incoming particles/secondcm2)=dNLdΩ=differential scattering cross section


where dΩ=sinθdθdϕ


σ=πθ=02πϕ=0(dσdΩ) sinθdθdϕ=NLtotal scattering cross section

Since this is just a ratio of detected particles to total particles, this gives the cross section as a relative probablity of a scattering, or reaction, to occur.

Luminosity

Luminosity is the quantity that measures the ability of a particle accelerator to produce the required number of interactions and is the proportional to the number of events per second dR/dt and the total cross section σp:

dN=dRdt=L×σp

In order to compute a luminosity for fixed target experiment, it is necessary to take into account the properties of the incoming beam and the stationary target.


L=Φρibeamρ

where

Φflux, or incoming particles per second


ρtarget density


length of target


For the differential cross-section:

dσdΩ=dNLdΩ=dNΦρdΩ

Transforming Cross Section Between Frames

Cross Section as a Function of Momentum and Solid Angle

Transforming the cross section between two different frames of reference has the condition that the quantity must be equal in both frames. This is due to the fact that

σ=NL=constant number


This makes the total cross section a Lorentz invariant in that it is not effected by any relativistic transformations.


This implies that the number of particles going into the solid-angle element d ΩLab and having a momentum between pLab and pLab+dpLabbe the same as the number going into the corresponding solid-angle element CM and having a corresponding momentum between pCM and pCM+dpCM




Expressing this in terms of the solid angle components,


As shown earlier,


Thus,


Simplify our expression for the cross section gives:


We can use the fact that


To give




Using Chain Rule

We can use the chain rule to find the transformation term on the right hand side:


Starting with the term:


Similarly,



Using the conversion of cartesian to spherical coordinates we know:


and the fact that as was shown earlier, that



This allows us to express the term:



Again, similarly



To find the middle component in the chain rule expansion,

which gives,




We can use the relativistic definition of the total Energy,





Then using the fact that


Final Expression

Using the values found above, our expression becomes:



This gives,

Cross Section as a Function of Energy, Momentum, and Solid Angle





We can use the chain rule to find the transformation term on the right hand side:


Using the expression found earlier,


This gives,


where it was already shown that since pz is the only cartesian component affected by the Lorentz shift


Our final expression

By symmetry,

Cross Section as a Function of Energy, and Solid Angle




Using the chain rule


Using previous results