Answer to the Question 09/98
MISSING MOMENTUMThe question was:
Consider the creation of a phonon by the scattering of a neutron from a crystal. Let the initial and final momentum of the neutron be p and p'. The neutron interacts only weakly with matter, and is essentially a free particle. The phonon which is produced has a pseudomomentum K=p-p'.
(a) Prove that momentum of a phonon vanishes.
(b) Where does the missing momentum go?
(6/99) The problem has been solved by Groshaus Javier (e-mail jgros@physics.technion.ac.il). This problem was discussed in great detail by Rudolf Pierls in his book Surprises in Theoretical Physics, ch.4.2 (Princeton University Press, NJ, 1979). Short solution of the problem is presented below.
The solution:
In a phonon instantaneous displacement of an atom is proportional to cos(Q x - w t) where Q=K/{hbar}, x is the position of the atom, w the relevant frequency, and t is time. Thus the velocity of an atom (and, consequently, its momentum) will also oscillate sinusoidally and on the average will be zero. Moreover, if we sum over all positions of atoms on the lattice, and Q is such that the phonon forms a standing wave the summation over all atoms vanishes identically at every moment! Thus a phonon has vanishing momentum.
The above argument is not valid for Q=0 mode. Thus, the momentum of the neutron is transferred to the entire lattice. This means that the entire solid starts moving with velocity K/M, where M is the mass of the entire body.
The actual situation is a little more complicated, since the neutron is somewhat localized and therefore creates not a single phonon but rather a localized packet of phonons. Read the book of Pierls about this question as well as other related questions.
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