How to calculate a coulomb barrier with a prolate deformed nucleus

Trying to calculate a coulomb barrier for a prolate deformed nucleus with a spherical projectile has consumed my attention since Friday; I could not find anyone in the literature who did this without using fancy quantum mechanics. Anyone can look in an E.M. textbook to learn how to calculate a coulomb potential for two spherical charged balls, but once you perturb the spherical geometry of one of the balls it’s no longer a simple problem.

Disregarding the classical physics aspects of the problem and only caring about the nuclear chemistry part, the formula you would use to calculate the coulomb barrier will vary slightly depending on how heavy the two nuclei are (ie. where you lie in the periodic table). Since I deal with the heaviest elements humans can make, I finally decided to use Swiatecki’s coulomb barrier equation but modified his coulomb barrier parameter, the original formula is shown below.

z is Swiatecki’s coulomb barrier parameter, Zt atomic number of the target nucleus, Zp atomic number of the projectile, At and Ap are the mass numbers of the target nucleus and projectile nucleus respectively.

To calculate the coulomb barrier parameter with a prolate deformed target nucleus, to a first order approximation, I used the following expansion around the At term.

Where b2t is the quadrupole deformation parameter(degree of non-sphericalness), which one can look up for any target atom of interest. The x term is the angle of contact and ranges from 0 to pi/2. An angle of 0 corresponds to an equatorial touching, while pi/2 corresponds to polar touching, shown respectively in the next figure.

From staring at the equation all day, this is most likely only physically valid at an angle of zero and at an angle of pi/2, which were the only two conformations I was interested in anyways. The above seems to hold up well against authors who used a quantum mechanical approach, so it does hold some physical merit.

Note1: All units are in MeV
Note2: After solving for the coulomb barrier parameter you have to plug this back into Swiatecki’s coulomb barrier equation, to determine the actual coulomb barrier.
Note3: You can download Swiatecki’s paper from here: Fusion by Diffusion II.
Note4: I do realize I posted this right after making fun of p-chemist in the previous entry.

Mitch

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