﻿ The Notion of Nuclear Rotations and Estimates of Rates
San José State University

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Thayer Watkins
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The Notion of Nuclear Rotations
and Estimates of Rates

What a profound notion is nuclear rotation. It is in contradiction to the Copenhagen Interpretation of quantum physics which holds that subatomic particles do not generally have a physical reality. Instead, according to that interpretation, they exist only as probability density distributions unless they are subjected to an observation that collapses their probability distribution to a point. Particles existing only as probability distributions cannot display structures and their rotation.

Nuclear rotation is also in contradiction to several of the standard models of nuclear structure such as the Independent Particle Model.

Nuclear rotation implies some sort of structure composed of nucleons. The structure may not be for the entire nucleus but it could be for a nuclear shell. The question is then what sort of structure can exist among the nucleons in a shell. One possibility and perhaps the only possibility is spin pairing. For more on this see What holds a nucleus together?.

One of the first studies of nuclear rotation was undertaken by Aage (pronounced oh-weh) Bohr and Ben Mottleson and published in Volume 2 of their Nuclear Structure. Bohr and Mottleson found that nuclear rotations obey the I(I+1) rule; i.e., have energy levels EI such that

#### EI = [h²/(2J)]I(I+1)

where I is an integer, J is the moment of inertia of the rotating nuclear shell and h is Planck's constant divided by 2π. Bohr and Mottleson cautiously stated that nuclei appear to rotate and satisfy the I(I+1) rule because AAge Bohr's father, Niels Bohr, was a major founder of the Copenhagen Interpretation that maintains that such rotations cannot exist.

Note what the Bohr-Mottleson I(I+1) rule implies about the quantification of angular momentum. If ω is the rate of rotation in radians per second then

#### EI = 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

where ω is the rotation rate in radians per second.

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. Most structures have a number of different modes of rotation, each with its own moment of inertia. Regardless of the differences in moments of inertia each mode of rotation will have the same angular momentum. 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 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. This is about 5 billion trillion times per second, an almost unbelievably fast rate of rotation.

This figure is so large that I looked for confirmation in the literature on nuclear physics. One possible source for confirmation is the book by Zdzislaw Szymanski of the University of Warsaw entitled Fast Nuclear Rotation. Incredibly in the 220 pages of text there is not one figure given for a rate of rotation in rotations per second. Rates of rotation are only given in terms of quantum numbers.

In the Guide to the Nuclear Science Wall Chart there is this statement

[…] scientists can create nuclei which have very high angular momentum. Nuclei respond to this rotation, which can be as fast as a hundred billion billion revolutions per second[…]

This is an order of magnitude smaller than the computed rotation rate for a deuteron but it is an equally almost unbelievably fast rate.

A similar computation for an alpha particle gives a rate of rotation of 4.194×1021 per second, the same order of magnitude as for a deuteron.

The question arises as to what such rates of rotation imply about the velocity of the subparticles of a structure in relation to the speed of light. That velocity is v=ωr, where r is the orbit radius for the particle. For the deuteron that is

#### v = (2.98×1022 radians per second)(1.223×10−15 m) = 3.64×107 m/sec which is, relative to the speed of light β = v/c ≅ 0.121

The factor 1/(1−(v/c)²)½ is then 1.007455, not a lot of correction needed.

Thus the computed rates of rotation are compatible with the Special Theory of Relativity.