## 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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