Relativistic collision and then quesion about cerenkov radiation

[jonjacson]

Homework Statement

a)

We have an incident photon beam into a fog chamber, and we observe a compton electron with a moment of 1,5 Mev/c emitted in a 10º angle to the incident beam. ¿Which is the energy of the incident and scattered photons?

b)

Find the minimum energy that should have the photon for the compton electron could be detected using cerenkov radiation (n=1,33).

Homework Equations

4-moment of the photon and the electron

Conservation laws of energy and momentum

And the relativistic relation:

$$E^2=p^2 c^2+ m^2c^4.$$

The Attempt at a Solution

I must apply conservation of moment and energy of the reaction $$\gamma$$ $$\epsilon$$ --->$$\gamma'$$ $$\epsilon'$$ :

$$p'_\gamma = p_e + p_\gamma -p'_e$$

and then my problem is to find every component of the 4vector, i am not sure if this is right:

I have some general doubts:

I have seen the expression of the 4vector energy momentum in three different ways:

-In the minkowsky space: $$\textbf{P}=(m\textbf{v}, icm)$$

-In another book i found: $$P=(mc\gamma,m\gamma\textbf{v})$$ then he uses this relationships: $$\textbf{p}=m\gamma\textbf{v}$$ and $$E=mc^2\gamma$$ to arrive at:

$$\textbf{P}=(E/c,\textbf{p})$$

And i don't know which should i apply, in one of them the energy is the P_0 component, in the other you have the imaginary number i, i will use the equation without the i component.

before the collision:

-I will supose that the electron is at rest so p=0 and :

$$P_e=(m_ec,0,0,0)$$

-For the photon, the mass is zero so :

$$E^2=p^2c^2$$ so the 4vector is:

$$P_\gamma=(E/c,E/c,0,0)$$

¿Is anything wrong until now?.

And after the collision, I must define two angles$$\theta$$ and $$\alpha$$, which are the angles between the electron and the photon to the incident beam.

So now we have:

-The 4vector of the electron is:

$$P'_e=(E'_e/c,p_e cos(\theta),p_e sin(\theta),o), p_e$$ is the moduli of the thridimensional momentum

-for the photon:

$$P'_\gamma=(E'_\gamma/c,p_\gamma cos(\alpha),p_\gamma sin(\alpha),0)$$Well , before applying conservation equations I would like to check if this is wrong ¿is it?

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b)

I think that i need understand part a) to solve part b).

 

[jonjacson]

Perhaps ¿Did i made anything wrong explaining the problem?