Related to the tetrahedron are two spheres which have received much attention. The "twelve-point sphere," discovered by P. M. E. Prouhet (1817-1867) in 1863, is somewhat analogous to the nine-point circle of a triangle. If the perpendiculars from the vertices to the opposite faces of a tetrahedron be concurrent, then a sphere passes through the four feet of the perpendiculars, and consequently through the centre of gravity of each of the four faces, and through the mid-points of the segments of the perpendiculars between the vertices and their common point of intersection. This theorem has been generalized for any tetrahedron; a sphere can be drawn through the four feet of the perpendiculars, and consequently through the mid-points of the lines from the vertices to the centre of the hyperboloid having these perpendiculars as generators, and through the orthogonal projections of these points on the opposite faces.

I would imagine the form you are looking for is the second of these three . . . the surface of the sphere should pass precisely through the midpoint of the tetrahedral vertices. If the tetrahedron vertex length was 1. . . it appears that the radius of the sphere would be approaching 0.61 (PHI) of the tetrahedron vertex length. . . . which would provide the energetic exuberance you have noted . . .

(see equation (15) at mathworld.wolfram.com/Tetrahedron.html)

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