Scattering, Frame Transformations

Problem:
Show that in a nuclear reaction of the type below

the nuclear disintegration energy (Q value) is given by

,

where the kinetic energies Ei are << Mic2 and M3 represents the light particle.

Solution:

wpe2.jpg (55081 bytes)

Problem:
(a) Show that in elastic collisions between two particles m1 and m2, with one particle initially at rest, the fractional energy loss of the moving particle is given by

Image412.gif (1333 bytes).

where q is the scattering angle in the center of mass frame.
(b) What is the maximum amount of energy lost by a 1 eV electron in an elastic collision with a helium atom?

Solution:

wpe3.jpg (55817 bytes)

Problem:
A fixed force center scatters a particle of mass m and initial velocity u0 according to the force law F(r)=k/r3.  Determine the differential scattering cross section.
Solution:

wpe7.jpg (63625 bytes)

Problem:

A particle of mass m moves under a central repulsive force F(r)=km/r3.   At its distance of closest approach r0 it has speed v0.

(a)  Find the orbital equation r(q) for the particle motion, evaluating constants in terms of r0 and v0.
(b)  Find the impact parameter and the total angular deflection, assuming the particle approaches from large r.
(c)  Sketch the particle trajectory, indicating the impact parameter and total deflection calculated in part (b).

Solution:

wpe4.jpg (59428 bytes)

Problem:

An excited, very heavy, non-relativistic particle traveling at speed v0, in the z-direction decays by emitting a light particle of mass m (also non-relativistic) at a precise speed vc, at an angle qc, in the rest frame of the emitter.

(a)  At what corresponding angle qL does an emitted particle of a corresponding speed vL emerge in the laboratory frame?

(b)  If the angular distribution of such events is isotropic in the rest frame of the emitter, i.e. if all angles of emission qc are equally probable in the rest frame of the emitter, what is the angular distribution in the laboratory frame?
[ Translation: if the number of emitted particles emitted at angles between qc , and qc+dqc in the projectile rest frame is independent of qc, what fraction of the total number of emitted particles is emitted between qL and qL+ dqL in the laboratory frame? ]
Give a formula for the laboratory frame angular distribution in terms of the isotropic distribution a constant in the rest frame of the emitter; and in terms of vL, vc, v0, and qL.

(c)  Show that when v0>vc there is a maximum angle of emission qL, which is given by sinqLmax=vc/v0.
[NOTE that you can derive the answer to part (c) even if you have not been able to answer parts (a) and (b).]

(d)  Make a qualitative sketch What happens to this cross section near

Solution: