Determine number of nucleons which can be packed within a nucleus of given size.

Question

Forces caused by binding energy of quarks (sub-nucleonic particles) hold protons and neutrons together in atom’s nucleus (core). Binding energy of a quark pair is in excess of 300 MeV (million electron volts). My question; Is there a mathematical way that computes maximum number N (exact) of protons and nutrons (together) which reside within a sphere? It is given that radius of a proton and a neutron are r units respectively and radius of nucleus, the sphere is R units.  Typical value of r (best estimate) is 0.84 fm (femto meters = 10^-15 meters) and value of R is 15 fm for Uranium nucleus.

Answer 1

My answer:

I found a systematic way to calculate exact number of nucleons that can occupy volume of a given sphere of radius R units.

Concept:

1.    Divide sphere of radius R in to K spherical shells.

2.    Each inner shell’s diameter is 4r smaller than the next outside sphere. K = R/4r

3.    Next we must calculate how many nucleons can be packed in each shell.

4.    We flatten surface of each shell into an elliptical disc thickness 2r..

5.    The shape of a flat shell is an ellipse at top, semi-major axis pi X radius of the shell and semi-minor

axis radius of the shell.

6.    Now we can calculate exact number of nucleons because perimeter of ellipse of each shell is known

. Perimeter = 2pi X [ ½ (a^2 + b^2)]^ ½ , here a is semi-major axis & b semi-minor axis. Distance

between centers of two nucleons next to each other is 2r.

7.    We add nucleons for all K shells.

What we did, transformed the volume of sphere into volume of a cone whose base is an ellipse with semi-major axis pi X R and semi-minor axis R. Upper bound on nucleon count occurs when apex of the cone has 4 nucleons packed. In this age of computing one can write a program, computation of nucleon count for given values of R and r by a code that implement algorithm correspond to steps described. When I discussed this solution with Marcus Stidham, Center Director at Mathnasium, he commented that distance between nucleons protons and neutrons is also effected by strong charge forces of quarks. .

Answer 2

A rough estimate would be to divide the volume of the nucleus by the volume of a single nucleon. This method does not take into account the fact that no matter how you stack spheres, there will always be some unused space. Nor does it take into account that the packing space must be spherical. Taking those details into account would require some trial and error to determine a packing algorithm.