(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Take the case of elastic scattering (A+B -> A+B); if particle A carries energy E_{A}', and scatters at an angle θ, in the CM (center of mass/momentum frame), what is its energy in the Breit* frame? Find the velocity of the Breit frame (magnitude and direction) relative to the CM.

*The Breit frame (or 'brick wall' frame) is the system in which A recoils with its momentum reversed (p_{after}= -p_{before}), as though it had bounced off a brick wall.

2. Relevant equations

(Note:boldletters are 3-vectors, letters with a subscript x, y or z are vector components, and letters with a super or subscript μ are contravariant and covariant 4-vectors respectively.)

p^{μ}= (E/c,p_{x},p_{y},p_{z})

Lorentz transformations for momentum and Energy

E/c = γ(E'/c - β*p_{x}')

p_{x}= γ(p_{x}' - β*E'/c)

p_{y}= p_{y}'

p_{z}= p_{z}'

(Where β = v/c and γ^{2}= 1/(1-β^{2})

Lorentz Invariant

p_{μ}*p^{μ}= E^{2}/c^{2}-p^{2}= m^{2}c^{2}

or in the more common form:

E^{2}=p^{2}c^{2}+ m^{2}c^{4}

Maybe more, I'm not sure.

3. The attempt at a solution

So in the CM frame the scattering angle is related to the relative masses/energies of the particles and since we "know" the energy of particle A in the CM we can say that the scattering angle, θ, is related to the total energy/mass of the system. This means I can write a set of 4 equations of the following form:

|p_{Af}'|*cos(θ) =p_{Afx}'

Which might be useful...

In the CM I can say that the total momentum 4-vector

p^{μ}_{TOT}' = ((E_{A}' + E_{B}')/c,0)

and in the Breit frame

p^{μ}_{TOT}= ((E_{A}+ E_{B})/c,p_{A})

I think if I were to take the inner product of each of the total momentum 4-vectors with itself I could set the results equal to each other... Maybe mass comes in handy since B initially only has rest mass in the Breit frame?

Somehow I need to get γ so I can do the second part of the question...

I feel like I'm close to some pivotal revelation, just not seeing it. There are so many unknowns I'm not sure how to start reducing the number of unknowns I'm dealing with.

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# Homework Help: Relativistic elastic scattering in CM and Breit frames

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