May 28, 1969
TO: John Blethen
FROM: Linus Pauling
SUBJECT: Calculations on Nuclear Structure
I suggest that during the next month or two you make an attack on the structure of polyhelionic nuclei, somewhat along the lines of the one that you carried out nearly two years ago.
Let us assume an interaction energy between pairs of helions with the form of the Morse function. The three parameters of the Morse function are to me evaluated by comparison of calculated and observed energy levels for the lighter polyhelionic nuclei, from carbon-12 on. For carbon 12, for example, the normal state energy has to be corrected for the zero-state vibrational energy, with the frequencies calculated by a normal coordinate treatment. It is possible that the first excited state, 2+, at 4.433, is a rotational state around an axis of symmetry in the plane of the triangle, and that the level 14.05 is the next rotational state around this axis, 4+. These rotational states would give the value of the distance between helions corresponding to the minimum in the Morse function. Presumably the rotational levels should be corrected for centrifugal stretching of the molecule, as determined by the curvature of the Morse function.
The 0+ level at 7.656 Mev may correspond to the breathing vibration.
Similar calculations may be carried out for 0 16.
In each case there is a reported 3- level, which may correspond to rotation around a three fold axis.
A possible complication that has occurred to me, as a result of some preliminary calculations is that
For example, the 3- level for 0 16 lies about 30 percent below that for carbon 12, corresponding to a larger moment of inertia. Our simple picture would make the moment of inertia the same, namely that for a triangle of helions. If there is a good bit of resonance of the sort described above, it would decrease the moment of inertia for carbon 12 and increase that for 0 16, perhaps in a way to account for the difference.
Similar calculations should be made for neon 20, with the assumption that the structure of the normal state is that of the trigonal bipyramid. The first excited state is 2+, and there is a 4» state higher up (perhaps at 5.631). These rotational states the first excited vibrational states, symmetric and antisymmetric, along the symmetry axis occur.
I suggest leaving magnesium 24 for consideration later. There is no doubt in my mind that this has a structure of orthorhombic, and that it is not a regular octahedron. When we do discuss the structure we might find that the octahedral structure.
Similar calculations should be made for silicon 28, with the configuration of the pentagonal bipyramid. The first excited state, 2+,and the second excited state, probably 4+, corresponds to rotation around an axis perpendicular to the five fold axis.
I have found that the helion-helion distances do not come out with nearly the same values for these different nuclei when the rotational states are calculated for a rigid oscillator. Perhaps there will be greater constancy when centrifugal stretching is taken into consideration.
The goal in this work would be to find a single Morse function that would account in a moderately satisfactory way for a number of the energy levels of the polyhelionic nuclei, and that would agree also with the observed energies of formation in the normal state from separate helions.
As a further possible application I would suggest that the 0+ excited state for calcium at 3.35 Mev (the first excited state) may correspond to a structure in which there are two interpenetrating pentagonal bipyramids, with the normal state being one in which there is a central helion surrounded by 9 helions. The rotational levels of the prolate structure (the two inter- , penetrating pentagonal bipyramids) should lie rather close, corresponding to the large moment of inertia.
For the lighter elements we might be able to predict energies of certain unusual excited states, such as with 3 helions in a row for carbon 12 and 4 helions in a row for oxygen 16. Another possibility is a rhomb (a planar structure) for oxygen 16. There are some sequences of energy levels reported for some of these lighter atoms in which the spacing is quite small, corresponding to a very moment of inertia, such as might be explained by these structures. The assignment of structures such as these to the excited levels would be made-with greater confidence if we could predict the values of the energy levels from the general treatment that we may hope to develop.
Linus Pauling
PS Later on, I think, we should discuss the structure of polyhelionic nuclei in the neighborhood of iron 52, in which presumably there are 2 helions in the core of a structure with prolate deformation.
May
28, 1969
