Scattering collisions/lorentz-invariant quantity

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High-energy electrons are crucial for probing the internal structure of protons and neutrons, as their energy allows them to penetrate these particles effectively. The minimum momentum required for the electrons can be calculated using the de Broglie wavelength formula, yielding a value of 1.2. In the laboratory frame, the energies of the electrons and target must be derived, although the participant expresses confusion about calculating velocities and final energies without given values. The total energy in the center-of-mass frame is a Lorentz-invariant quantity, which can be determined using the energies calculated earlier. The discussion highlights the need for a solid understanding of quantum mechanics and special relativity to address these calculations.
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hey guys, here's the question.

Physicts probe inside neutrons and protons by scattering electrons off them.
(i) Explain briefly why it is important that the electrons have high energy.

(ii) Assuming that the protons and neutrons were at rest, calculate the minimum momentum to which the electrons should be accelerated, in order to perform such experiments successfully.

(iii) Considering such scattering collisions in the laboratory frame of reference (where the target is stationary), write formulae for the energies of the electrons and target, and evaluate them in GeV (you may ignore the rest-mass of the electron).

(iv) The total energy in the centre-of-mass reference frame in such electron-nucleon collisions corresponds to the maximum possible mass of all particles in the final state and is a Lorentz-invariant quantity. Ignoring the rest-mass of the electron, calculate this quantity.

My workings:

(i) to ensure that they penetrate the protons and neutrons.

(ii) using wavelngth=planck's const/momentum gives 1.2.

(iii) can someone give me a hint for this part please. i don't understand how to get the velocity for the electron (it says ignore rest mass so using this in part (ii) won't help). also how are we supposed to know the final energies, as we're not told velocities etc.

(iv) are we supposed to use the energies calculated for the neutron is (iii) and somehow find the velocity so we can work out the mass?

thanks for any help.
 
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You have to proceed from the fact that the electron is a wave and that proton/neutron size is finite, R_n, for example. In order to "see" such a small size, the electron wave-length should be shorter than R_n (to be close to the case of "geometrical optics"). Then simple QM and SR formulae are sufficient to resolve this problem.

Bob.
 
Bob_for_short said:
You have to proceed from the fact that the electron is a wave and that proton/neutron size is finite, R_n, for example. In order to "see" such a small size, the electron wave-length should be shorter than R_n (to be close to the case of "geometrical optics"). Then simple QM and SR formulae are sufficient to resolve this problem.

Bob.

hi bob, thanks for the reply.
so the proton size is 1x10^{-15}, and so this helps me with (ii).

sorry i haven't done any relativity so i don't have a clue what it is on about (i'm a maths student doing a physics course). which formulae?

thanks
 
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