• russdot
In summary, the conversation discusses the calculation of mean and root mean square (rms) radius for a particular charge distribution. The equations for these values are given, with the use of density and wave functions. It is noted that the mean radius is not commonly used, but the equation for the rms radius is correct. The conversation ends with a confirmation that the mean value equation is also correct.
russdot
Hello,
Given a particular charge distribution p(r) = p_0*exp(-r$$^{2}$$/a$$^{2}$$), I was wondering if the proper way to calculate the mean radius <r> would be $$\int$$p(r)*r*p(r) dV ?
Which would make <r$$^{2}$$>$$^{1/2}$$ = ($$\int$$p(r)*r$$^{2}$$*p(r) dV)$$^{1/2}$$, correct?

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The mean radius is not generally used.

Great, thanks!
I'm assuming if the rms equation is correct, then the mean value equation is also correct..

I have always used:

$$<r^2> = \int \rho (r)r^2 d\vec{r}$$

since the wave function(s): $$\psi (r)^* \cdot \psi (r) = \rho (r)$$

If the density is normalised to unity: $$\int \rho (r) d\vec{r} = 1$$

Otherwise:
$$<r^2> = \int \rho (r)r^2 d\vec{r} / \int \rho (r) d\vec{r}$$

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## 1. What is the mean radius of a nucleus?

The mean radius of a nucleus is the average distance from the center of the nucleus to the outer edge. It is typically measured in femtometers (fm) or meters (m).

## 2. How is the mean radius of a nucleus calculated?

The mean radius of a nucleus is calculated using the root-mean-square (r.m.s.) formula, which takes into account the distribution of protons and neutrons within the nucleus. This formula is based on the average distance from the center of mass of the nucleus to each individual nucleon (proton or neutron).

## 3. What factors affect the mean radius of a nucleus?

The mean radius of a nucleus can be affected by the number of protons and neutrons within the nucleus, as well as the nuclear force that holds these particles together. Additionally, the shape of the nucleus can also impact its mean radius.

## 4. How does the mean radius of a nucleus relate to nuclear stability?

The mean radius of a nucleus is directly related to its nuclear stability. Generally, larger nuclei have a smaller mean radius and are less stable due to the repulsive forces between protons. Smaller nuclei have a larger mean radius and are more stable due to the stronger nuclear force between protons and neutrons.

## 5. Can the mean radius of a nucleus change?

Yes, the mean radius of a nucleus can change due to nuclear reactions or changes in the number of protons and neutrons within the nucleus. These changes can also affect the stability of the nucleus.

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