Q: In beta decay of a nucleus initially at rest, the recoiling daughter nucleus and the emitted beta particle are sometimes observed not to move along the same straight line. How can this be explained in view of the law of conservation of linear momentum?
A: In beta decay, a nucleus emits a beta particle (electron or positron) and a neutrino (or antineutrino). The neutrino carries away some of the momentum. Since, initial linear momentum is $\vec 0$, conservation of linear momentum requires $\vec 0 = \overrightarrow {{p_R}} + \overrightarrow {{p_\beta }} + \overrightarrow {{p_\nu }} $.
If beta particle and neutrino move in straight line, $\overrightarrow {{p_\nu }} = k\overrightarrow {{p_\beta }} $ where k is a scalar and therefore $\overrightarrow {{p_R}} = - (1 + k)\overrightarrow {{p_\beta }} $ which means the daughter nucleus and the emitted beta particle move along the same straight line.
However, if beta particle and neutrino do not move in straight line, the daughter nucleus and the emitted beta particle also do not move along the same straight line.
