Calculating the Ratio (semimajor-axis/semiminor-axis) from the Deformation

As chemists we view the nucleus as some arbitrary positively charged sphere that keeps the electrons bound to the atom. As a nuclear chemist we know the only nuclei that are perfectly spherical are nuclei that contain a magic number(2,8,20,28,50,82,126) of protons, or neutrons, or both. The other nuclei, not associated with magic numbers, will be deformed by some degree. The measurement of the deformity is called the deformation parameter. The deformation parameter is typically interchangeable with what we discuss as the quadrupole deformation, although higher order deformations (hexadecapole, hexacontatetrapole) can and will be calculated by others. A quadrupole deformation can have two different effects on a nucleus. 1) It can cause a prolate deformation, making the spherical nucleus actually appear more like a football, positive quadrupole deformations. 2) It can cause an oblate deformation, making the spherical nucleus actually appear more like a doorknob, negative quadrupole deformations. Both are shown below

Prolate

Oblate

All that is nice, but lets get to the point. Often in the literature, authors will list the deformation of a nucleus and will leave it at that. I often find myself wanting to know physically what does it all mean. Simply put, the deformation is a fancy equation nuclear scientists use that relates the length of the semimajor-axis to the length of the semiminor-axis. Think of a 2d-ellipse, turn it into 3-dimensions (ellipsoid) and you have the generic picture I’m trying to convey. In order to translate a deformation into reality (i.e. the ratio of the longest side to the shortest side) you can use the following equation I painfully solved yesterday during night-shift. I purposefully didn’t simplify it and left it in the form of the quadratic formula in case any curious readers wanted to play with it, or attempt to back-track to the original logic behind it. Why they do not give these equations in any text-book, or journal article that I have come across since joining this field is beyond me. Below are the equations.

For a prolate nucleus use the following:

For an oblate nucleus use the following:

Where beta is the quadrupole deformation.

Have fun with it, if anyone ever needs it.

Mitch

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