13. A sphere is tangent to all six edges of a regular tetrahedron. If the edge length of the tetrahedron is $a$, then the volume of the sphere is $\qquad$
13. $\frac{\sqrt{2} \pi a^{3}}{24}$.

We can transform the regular tetrahedron into a regular tetrahedron inscribed in a cube. In this case, the edge length of the cube is $\frac{\sqrt{2} a}{2}$, thus, the radius of the sphere is $\frac{\sqrt{2} a}{4}$.