San José State University

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Thayer Watkins
Silicon Valley
USA

 The Density of Nuclear Matter

The best estimate of the root-mean-square (rms) radius of the proton is 0.895±0.018 fermi; i.e. 8.95×10−16 meters. That means its volume is 3.00×10−45 m³. The mass of a proton is 1.672623×10−27 kilograms (kg). Therefore the density of nuclear matter is 1.8×1018 kg/m³. Since the mass of the Earth is only 6×1024 kg it would take only 3.33×106 cubic meters of nuclear material to equal the mass of the Earth. This would be a cubic block of nuclear material of 150 meters on each side.

The best estimate of the rms radius of an alpha particle is 1.68 fermi. This means its volume is 1.99×10−44 m³. Its mass is 6.64424×10−27 kg. Therefore its mass density is 3.345×1017 kg/m³. This is about 18.6 percent of the density of a proton.

The rms radius of a Uranium nucleus (U238) is about 8 fermi. Thus its volume is 2.145×10−42 m³. With a mass of 3.92×10−25 kg its density is 1.83×1017 kg/m³, which is about 10 percent of the mass density of a proton. This suggests that quite a bit of space may exist between the nucleons of the Uranium nucleus.

With 238 nucleons in the U238 nucleus the volume per nucleon is about 9×10−45 m³ per nucleon. This is the volume of a cube with 2.08×10−15 which is just 2.08 fermi on each side. This also means that the separation distance between nucleons is 2.08 fermi. This is approximately the same as the 2.25 fermi separation distance between the nucleons in a deuteron. This suggests that the nucleons in a nucleus are close packed like spheres.

It is plausible, even likely, that the nucleons in a nucleus are arranged in alpha particles when possible and any additional nucleons are arranged in pairs. For U238 this would mean 46 alpha particles and 73 neutron pairs for a total of 119 substructures. This gives an average volume per substructure of 1.8×10−44 m³. This is a cubic volume which is 2.62 fermi on each side and hence the average separation distance between substructures would be 2.62 fermi. Since the rms radius of an alpha particle is 1.68 fermi the separation distance of the centers of two alpha particles is 3.32 fermi. The rms radius of a neutron is about 1.1 fermi. This makes the separation distance of the centers of the neutrons in a neutron pair on the order of 2.2 fermi. The average of 46 alphas with a separation distance of 3.32 fermi and 73 neutron pairs with a separation distance of 2.2 fermi is 2.63 fermi. This also indicates the close packing of the substructures in a nucleus.