What Determines the r_0 Constant in Nuclear Radius Calculations?

[mateomy]

I've been noticing that for expressions finding the nuclear radius there is a constant term $r_0$, that I can't seem to find an explanation for. The full term is
$$R=r_{0}A^{1/3}$$
That the atomic radius is roughly the 3rd root of the number of nucleons. But where is the $r_0$ coming from? I'm using Cottingham and Wong as sources but both of them fail to mention it's derivation. Plus, they both give ${slightly}$ different values, without looking I think Wong assigns it a value of 1.7 fm and Cottingham 1.1 fm. Can anyone explain this to me?


Thanks.

 

[TSny]

Nuclei are sort of fuzzy and don't have definite radii (they're not even necessarily spherical). The formula gives an approximate value of the radius and the value of r_o in the formula is determined from various experiments. See http://en.wikipedia.org/wiki/Nuclear_size.

 

[tia89]

This is I think an expression found by experimental or at least phenomenological means. As there $A$ is the number of nucleons, you would have that $r_0$ is the size of a single nucleon (and has to be determined experimentally of course). It is needed in this expression formally to give the right dimension to $R$.

Also notice that (something I think is interesting) if you compute the volume of the nucleus (interpreting really $r_0$ as the radius of the single nucleon), you will have that the total volume is $A$ times the volume of a nucleon, which is clearly consistent with what one would imagine.

 

[tia89]

the formula is determined from various experiments. See http://en.wikipedia.org/wiki/Nuclear_size.

Looking in the Wikipedia article to the values given for the experimental sizes of neutron and proton, indeed $r_0$ is a constant determined empirically and seems to me a sort of mean value of the two.

 

[paranormal]

The constant term r_0 in the expression for nuclear radius is known as the "nuclear radius parameter" and it represents the average distance between neighboring nucleons in the nucleus. It is a fundamental constant in nuclear physics and its value is determined experimentally. The slightly different values given by different sources may be due to different experimental methods used to measure it.

The origin of the r_0 parameter can be traced back to the liquid drop model of the nucleus, which describes the nucleus as a drop of incompressible nuclear matter. In this model, the r_0 parameter is related to the nuclear binding energy and surface tension.

It is important to note that the expression for nuclear radius, R=r_{0}A^{1/3}, is an empirical formula and it may not hold true for all nuclei. The actual size and shape of a nucleus can vary depending on its composition and other factors. In addition, the r_0 parameter may also be influenced by nuclear interactions and other effects.

Overall, the r_0 parameter is a key component in understanding the size and structure of atomic nuclei. Further research and experiments are needed to fully understand its origin and variations in different nuclei.