Some Nuclear Properties — Question 4

Question 4

A uniform sphere relates the sharp radius to a measured rms radius

Assume a spherical nucleus with constant charge density inside radius R. The mean-square charge radius is $\langle r^2\rangle=(1/Q)\int r^2\,dq$. A measurement gives $r_{\mathrm{rms}}=4.00\,\mathrm{fm}$.

1. Derive the relation between $r_{\mathrm{rms}}$ and R.

2. Infer R and estimate A using $R=1.20A^{1/3}\,\mathrm{fm}$.

Question 4 - Solution

A uniform sphere relates the sharp radius to a measured rms radius

(a) Method and calculation

$$Q=4\pi\rho_qR^3/3,\quad\int r^2dq=4\pi\rho_q\int_0^Rr^4dr=4\pi\rho_qR^5/5.$$$$\boxed{\langle r^2\rangle=3R^2/5,\quad r_{\mathrm{rms}}=\sqrt{3/5}\,R}.$$

(b) Interpretation and further calculation

$$\boxed{R=\sqrt{5/3}(4.00)=5.164\,\mathrm{fm}}.$$$$\boxed{A\simeq(R/1.20\,\mathrm{fm})^3=79.69\approx80}.$$

Physical check

The result is a model-dependent estimate of mass number, not isotope identification. An rms radius and a sharp-edge radius are not interchangeable.