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The Spacing of Particles in a Tetrahedral Arrangement

The alpha particle (the He4 nuclide) is the most important building block of nuclei. Scattering experiments indicate that its point group is C3v, which means that it has a tetrahedral arrangement of nucleons, neutrons and protons.

An alpha particle can be thought of as a pair of deuterons, neutron-proton pairs. The neutrons of the alpha particle form a pair and so do the protons. The neutron-neutron pair involves a spin pairing and a strong force bonding. The strong force between two neutrons is a repulsion which pushes the separation distance between the centers of the neutrons to the limit of their spin pairing. The same thing occurs for the protons except there is an electrostatic repulsion in addition to the strong force repulsion. The strong force between a neutron and a proton is an attraction and there has to be a rotation to maintain a separation between a neutron and a proton in a neutron-proton pair.

The arrangement in an alpha particle can be visualized as two neutron proton bonds, as shown below.

The arrangement of the two deuterons can be visualized in terms of two cylinders representing the bonds. One bond is positioned at a right angle to the other and the midpoints of the bonds separated, as shown below.

This visualization indicates that there are at least two parameters for a tetrahedral arrangement: the distance between the midpoints and the length of the bonds. The distance between the neutrons is necessarily smaller than the distance between the protons so the angle between the bonds of the deuterons is not exactly a right angle. Thus there are three parameters for a tetrahedral arrangement. However for the moment presume there is no difference between the neutron-neutron pair and the proton-proton pair. The coordinates for the simple arrangement in which the distance between a neutron and a proton is 2 and the distance between the midpoints of the neutron-proton bond is 1 is shown below.

Nucleon x y z
n1 0.5 1 0
p1 0.5 -1 0
n2 -0.5 0 1
p2 -0.5 0 -1

The coordinates of the center of this arrange are (0, 0, 0). The distance from this center to the nucleons is 1.118. This could be considered the radius of the arrangement. The distance between the neutrons is 1.732 and this also the distance between the protons. The distance between a neutron and a proton was presumed to be 2. Thus the ratio of the separation distance in the neutron-neutron and proton-proton bonds to the radius of the arrangement is 1.55, while the ratio for separation distance in neutron-proton bond to the radius is 1.79. This however is true only if the parameters of the arrangement are in a ratio of 2 to 1.

According to the article, "Precise Radii of Light Nuclei from Electron Scattering," by I. Sick published in Precision Physics of Simple Atoms and Molecules (edited by Savely G. Karshenboin), the root-mean-square (rms) charge radius of a Helium 4 nuclide is 1.68 fermi; and that of a proton is 0.895 fermi. Thus the distance from the centroid of the tetrahedron to the center of a proton is 0.785 fermi. The best estimate of the rms-charge diameter of a deuteron from page 70 of the above mentioned work is 4.260 fermi with a margin of error of ±0.02 fermi. Precision Physics of Simple Atoms and Molecules does not give an estimate for the radius of the neutron. Another source gives the rms-radius of the neutron as 1.11 fermi.

Thus the separation distance of the centers of the nucleons is

s = 4.260−0.895−1.113=2.252 fermi.

With the additional forces upon the nucleons the separation distance of a neutron and proton in an alpha particle would not necessarily be the same as in a deuteron. Nevertheless it is worth exploring the implications. The ratio of 2.252 f to 0.785 f is 2.869. It is not possible to get a separation distance of nucleons greater than twice the rms radius. With an angle of 120° between the radials from the centroid to the centers of the nucleons the separation distance of the nucleons would be √3·0.785=1.36 fermi.

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