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Nuclear Rotations

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 angular momentum is always h²[I(I+1)]]^{½}.
Thus there is an equipartition of angular momenta among the various modes of rotation.

Order of Magnitude Estimates of the Rate of Rotation of a Alpha Particle

An alpha particle consists of two protons and two neutrons. Neutrons have slightly more mass
than protons. 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-alpha particle.

The diameter of the alpha particle is approximately 3.6 fermi. The charge radius of a proton
is 0.877 fermi. Deducting two proton radii from the diameter of the alpha particle gives a separation
distance of the particle centers of 1.846 fermi and thus an orbit radius of 0.923 fermi.

The mass of an alpha particle is 6.64466×10^{-27} kg. Thus the moment of inertia
of a alpha particle for rotation about an axis perpendicular to its longitudinal axis is

J = (6.64466×10^{-27})(0.923×10^{-15})²
J = 5.6608×10^{-57} kg m²

The minimum rate of rotation is thus

ω = √2(1.05457×10^{-34})/(5.6608×10^{-57}) = 2.635×10^{22} radians per second

The number of complete rotations per second is 4.194×10^{21}.

Thus the order of magnitude of the rotations of a alpha particle is about 10^{20} per second.