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# Conservation of Momentum and Relativity

An electron moving in the positive x-direction is characterized by GAMMA = 1000 and BETA = 1 - 5 x 10^-7 which for this problem you can take as 1. A photon of energy h&#957;, traveling at 45 degrees with respect to the x axis collides with the electron. The collision results in the creation of an electron-positron pair plus an electron.

A. Find the threshold energy of the photon, that is, the minimum energy the photon must have for this reaction to occur. Express your answer in terms of mc^2 where m = the mass of the electron.

B. Find the direction of the Center of Momentum Frame at threshold.

#### Solution Preview

In the original frame S, the 4-momentum of the electron is

p1 = m*c*gamma*{1, beta, 0, 0} (1)

and the 4-vector of the photon is

p2 = h*nu/c*{1, cos45°, sin45°, 0} (2)

After the collision we have 3 electrons in the sense of the mass (that is we have 2 electrons and a positron, but the masses of all the three are the same mass of electron).

Let us go to the Center of Mass frame of the 3 electrons S'.
In this frame, the sum of all their energies would be the smallest, if ...

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