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2-2: Size of Nuclei
[Measurement of Size of Nuclei]
It had already been known from the analyses with the Ratherford atomic model that the nuclear size is smaller than roughly speaking. How can we more precisely measure the size of individual nucleus?
When we intend to measure the size of an object, we irradiate light on the object and observe the reflected light. The reflection occurs because of the interaction between the light and object.
Similarly, in order to measure the nuclear radius, it appears convenient to bombard some kinds of particles to the nucleus and observe their reaction. For this purpose, alpha particles, nucleons, electrons, photons etc. could be candidates as projectiles. Among them, electrons must be the most convenient. Electrons can interact with protons because they are both electrically charged. But electrons do not so strongly interact with neutron, so that the reflection of electrons, i.e. electron scattering, can be used only for the measurement of the charge distribution (or the proton distribution) in a nucleus.
The proton distribution in a nucleus may not always be equal to the mass distribution including both of protons and neutrons. However it would be possible to get a rough estimation of the nuclear mass distribution and the nuclear radius.
This type of precise measurements have been carried out and a lot of detailed data of the nuclear charge distribution for various nuclei have been obtained. A part is shown in the following figure.
[Experimental Data of Nuclear Charge Distributions]
The data obtained by the electron scattering. As an example, the red curve shows the proton distribution in Ca whose radius is shown by R.

[Nuclear Radius]
Most nuclei are nearly spherical. Namely, many protons and neutrons collect to compose a spherical cluster, i.e. a nucleus. (Precisely speaking, some of them may not be exactly spherical but slightly deformed. We will later discuss the nuclear deformation.)
According to the experimental results shown above, the charge distribution in a nucleus, , is well represented by a function

This is sometimes called the Woods-Saxon type function. In this function, the parameters , and are adjusted to fit the numerical results to the experimental data.
Adjusting these parameters, we can well reproduce the experimental charge distribution. The value of the parameter R is the radius of the nucleus.
For example, the red-colored R shows the radius of the Ca nucleus.
In this way, precise investigations of nuclear radii have carried out for a wide range of nuclides. Not only electron scattering but also various methods have been used for this purpose and then it has been clarified that the nuclear radius is generally given by the following formula:

where is the mass number of the nucleus.

[Nuclear Density and Its Saturation]
From the above formula for the nuclear radius, we can easily get the volume of a nucleus, V, as

Therefore the density of the nucleus is

This means that the nuclear density is almost constant independently of kind of nucleus. This is called the saturation of nuclear density. It is one of the remarkable properties of atomic nuclei. From this property, we can see that the mean value of the internucleon distance d is
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