(a) What is the nuclear density of ^228 Th? (b) Do you...

(a) What is the nuclear density of ${ }^{228}$ Th? (b) Do you expect the nuclear density of an alpha particle to be greater than, less than, or the same as that of ${ }^{228}$Th? Explain. (c) Calculate the nuclear density of an alpha particle.

Key Concepts

Nuclear Density

Nuclear density is the average number of nucleons per unit volume within a nucleus. It is calculated by dividing the mass number (total number of protons and neutrons) by the nuclear volume, which is typically estimated using a radius proportional to A^(1/3). This density is remarkably constant across different nuclei due to the saturation of the nuclear force.

Nuclear Radius and Volume Scaling

The size of a nucleus is generally estimated with the formula R = r?·A^(1/3), where r? is an empirical constant (approximately 1.2 fm) and A is the mass number. Because the volume of a sphere scales as R³, substituting the A^(1/3) dependency shows that the nucleus’s volume is proportional to A, thereby implying that the nuclear density (A divided by the volume) remains almost constant.

Saturation Property of Nuclear Matter

The saturation property of nuclear matter implies that each nucleon interacts predominantly with its immediate neighbors, so nuclear forces do not add up over the entire nucleus. This behavior causes the average nucleon density to remain nearly the same for light nuclei (like an alpha particle) and heavy nuclei (like ²²?Th), despite their large differences in mass number.

Alpha Particle Characteristics

An alpha particle is a very light nucleus composed of two protons and two neutrons. Despite its small size, its radius can be estimated using the same scaling law as larger nuclei. As a result, when the density is calculated using R = r?·A^(1/3), the value turns out to be comparable to that of heavier nuclei, reflecting the nearly constant density of nuclear matter.

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