$13\mathrm{.}35*****$ A beam of particles is moving along an accelerator pipe in the $z$ direction. The particles are uniformly distributed in a cylindrical volume of length $L_0($in the $z$ direction$)$ and radius $R_0.$ The particles have momenta uniformly distributed with $p_z$ in an interval $p_0+-p_z$ and the transverse momentum inside a circle of radius $.$ To increase the particles' spatial density, the beam is focused by electric and magnetic fields, so that the radius shrinks to a smaller value $R.$ What does Liouville's theorem tell you about the spread in the transverse momentum and the subsequent behavior of the radius $R?($Assume that the focusing does not affect either $L_0$ or $p_2.)$