326 6 Special Theory of Relativity
6.68 A proton of momentumplarge compared with its rest massM, collides with
a proton inside a target nucleus with Fermi momentumpf. Find the available
kinetic energy in the collision, as compared with that for a free-nucleon target,
whenpandpfare (a) parallel (b) anti parallel (c) orthogonal.
6.69 An antiproton of momentum 5 GeV/c suffers a scattering. The angles of the
recoil proton and scattered antiproton are found to be 82◦and 2◦ 30 ′with
respect to the incident direction. Show that the event is consistent with an
elastic scattering of an antiproton with a free proton.
6.70 Show that ifEis the ultra-relativistic laboratory energy of electrons incident
on a nucleus of massM, the nucleus will acquire kinetic energy
EN=(E^2 /Mc^2 )(1−cosθ)/(1+E(1−cosθ)/Mc^2 )
whereθis the scattering angle.
6.71 A particle of massMmescatters elastically from an electron. If the inci-
dent particle’s momentum ispand the scattered electron’s relativistic energy
isEandφis the angle the electron makes with the incident particle, show that
M=P[{[E+me]/[E−me]}cos^2 φ−1]]^1 /^2
6.72 A neutrino of energy 2 GeV collides with an electron. Calculate the maximum
momentum transfer to the electron.
6.73 A particle of massm 1 collides elastically target particle of massm 2 at rela-
tivistic energy. Show that the maximum angle at whichm 1 is scattered in the
lab system is dependent only on the masses of particles providedm 1 >m 2
6.74 Show that if energyν(>mec^2 ) and momentumqare transferred to a free sta-
tionary electron the four-momentum transfer squared is given byq^2 =− 2 meν
6.75 A photon of energyEtravelling in the+xdirection collides elastically with
an electron of massmmoving in the opposite direction. After the collision,
the photon travels back along the –xdirection with the same energyE.
(a) Use the conservation of energy and momentum to demonstrate that the
initial and final electron momenta are equal and opposite and of magni-
tudeE/c.
(b) Hence show that the electron speed is given by
v/c=(1+(mc^2 /E)^2 )−^1 /^2
[adapted from the University of Manchester 2008]
6.2.4 Invariance Principle ................................
6.76 Use the invariance of scalar product of two four-vectors under Lorentz trans-
formation to obtain the expression for Compton scattering wavelength shift.
6.77 Show that for a high energy electron scattering at an angleθ,thevalueof
the squared four-momentum transfer is given approximately byQ^2 = 2 E^2