Aage Bohr and Ben Mottleson found that nuclear rotations obey the I(I+1) rule; i.e., have energy levels E_I such that

E_I = [h²/(2J)]I(I+1)

where I is an integer, J is the moment of interia of the rotating nuclear shell and h is Planck's constant divided by 2π. Note what this implies about the quantification of angular momentum. If ω is the rate of rotation then

E_I = ½Jω² = [h²/(2J)]I(I+1)
and hence
ω² = [h²/J²]I(I+1)
and therefore
ω = {h/J][I(I+1)]^½

Thus angular momentum L is given by

L = Jω = h[I(I+1)]^½

This is in contrast to the Old Quantum Physics of Niels Bohr in which L would be hI.

Note that

ω = h[I(I+1)]^½/J

and thus the smaller the moment of inertia the faster a structure rotates. However the energy is always h²I(I+1)/(2J).

Order of Magnitude Estimates of
the Rate of Rotation of a Deuteron

A deuteron consists of a proton and a neutron. The neutron has slightly more mass than the proton. For purposes of this order of magnitude estimate the differences between the neutron and the proton will be ignored. It is thus the computation for a pseudo-deuteron.

The diameter of the deuteron is approximately 4.2 fermi. The charge radius of a proton is 0.877 fermi. Deducting two proton radii from the diameter of the deuteron gives a separation distance of the particle centers of 2.446 fermi and thus an orbit radius of 1.223 fermi.

The mass of a proton is 1.673×10^-27 kg. Thus the moment of inertia of a deuteron for rotation about an axis perpendicular to its longitudinal axis is

J = 2(1.673×10^-27)(1.223×10^-15)²
J = 5.005×10^-57 kg m²

The minimum rate of rotation is thus

ω = √2(1.05457×10^-34)/(5.005×10^-57) = 2.98×10²² radians per second

The number of complete rotations per second is 4.74×10²¹.

There could also be rotation about the longitudinal axis. The moment of inertia of a solid sphere of mass M is (2/5)MR² where R is the radius of the sphere. The moment of inertia of the pseudo-deteron about the longitudinal axis is then

J = 2(2/5)(1.673×10^-27(8.77×10^-16)² = 1.287×10^-56

And therefore the rate of rotation is

ω = √2h/J = 1.159×10²¹ radians per second
and
1.845×10²⁰ complete rotations per second

Thus the order of magnitude of the rotations of a deuteron is about 10²⁰ per second.