Relativistic collisions - different frames, threshold energy

unscientific · May 27, 2013

Homework Statement

(a) Show that E² - p²c² is a Lorentz invariant, where E is the total energy, p is the momentum, c is speed of light.

(b) In the lab frame, an utlra-relativistic electron with γ = 100 collides head-on with a photon of energy E. Find the threshold energy E required to create an electron-positron pair.

(c) In another head-on collision with the same setup, the path of the photon is reversed. Find the photon's energy in the electron's rest frame.

(d) Find the back-scattered energy of the photon in the same frame.

(e) Find the KE of the electron in the same frame.

(f) Find the back-scattered energy of photon in LAB frame.

Printed exam question 9, parts (a) to (f), on relativistic collisions: showing E^2 - p^2c^2 is Lorentz invariant, electron-photon threshold energy, back-scattered photon energy, and electron kinetic energy.

Homework Equations

The Attempt at a Solution

Handwritten working for part (b): an electron moving right toward a photon moving left, with energies E1 and E, shown in the lab frame; a dashed line separates it from a pair created at rest with energy 2E0, and the four-momentum equations and final result E = 3E0²/(2E1 + 2p1c) are shown.

Handwritten working for a relativistic Doppler and energy-momentum problem: f' = f0 sqrt((1+β)/(1−β)), E = 3/2 E0, then a two-frame sketch with four-momenta p^μ = (5/2 E0/c, 3/2 E0/c) on the left and p^μ = ((E0+E')/c, E'/c) on the right, the equation (5/2 E0/c)² − (3/2 E0/c)² = ((E0+E')/c)² − (E'/c)², and the result E' = 15/4 E0.

Handwritten working for part (f) of a relativistic collision problem: four-momentum vectors in two frames, showing electron and photon energies, momenta and the relation E3 squared minus p3c squared equals E0 squared, ending with E2 approximately 150 E0.

unscientific · May 29, 2013

Any help would be very appreciated!

clamtrox · May 29, 2013

unscientific said:

Any help would be very appreciated!

Help would be more forthcoming if you said where it's needed.

unscientific · May 30, 2013

clamtrox said:

Help would be more forthcoming if you said where it's needed.

Part (d): Is it alright to assume the electron remains stationary throughout the collision (Like a tennis ball rebounding against the wall)

Part(e): I'm not sure if this is a trick question, but in the elctron's frame, shouldn't it's own KE = 0?

unscientific · May 31, 2013

bumpp

unscientific · Jun 2, 2013

anyone?

Relativistic collisions - different frames, threshold energy

Homework Statement

Homework Equations

The Attempt at a Solution

Discussion

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