San José State University
Department of Economics
Thayer Watkins
Silicon Valley
& Tornado Alley

The Forces Between Protons at Nuclear Distances

Two protons are subject to two forces; the nuclear force and the electrostatic force. When they are very close together the nuclear force is dominant and when they are far apart it is the electrostatic force which is dominant. Somewhere two protons experience zero net force because the two opposing forces are equal. The following materical is to make quantitative estimates of the forces.

The electrostatic (Coulomb) force is well know. For the charges of the protons the formula for the force reduces to

F = α/s²
where s is the separation distance of the proton centers and α is equal to 2.3103×10-28

The nuclear force is generally not explicitly specified. Here the formula

F = −He-s/s0/s²

is used. The logic for the form is given in Nuclear Force Formula and the estimates based upon the dimensions of the deuteron are explained in Estimates. The value of s0 is 1.522 fermi (1 fermi = 10-15 meters.) and H is equal to 3.4×10-26.

The graph below shows the two forces plotted against separation distances ranging from 5 fermi to 20 fermi.

The plotted values are in newton-meters. One newton meter is an enormous force for a particle having mass of only 1.6×10-27 kg. The net force is shown below.

The point of zero net force is at about 7.6 fermi. That is just about exactly 5 times s0. The separation distance of the nucleon centers in a deuteron is about 3.2 fermi.

Nuclear Dimensions

The radius of a spherical nucleus containing A nucleons is given by the formula

r = r0A
where r0 is equal to 1.25 fermi.

From this formula the diameter of a nucleus would reach 7.6 fermi when A is equal to about 28. This corresponds to silicon. When the radius of the nucleus is 7.6 fermi it means that the protons on the edge of one side of the nucleus are being repelled by all of the protons on the other side to the center of the nucleus. The radius of a nucleus reaches level 7.6 fermi for mass number of 225. This corresponds to elements like Actinium and Uranium. This is where radioactivity beomes a general property of the nuclides.

(To be continued.)

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