# How to calculate the nuclear size from the strong nuclear force?

1. Mar 22, 2012

### Aidyan

I suppose that the evaluation of the size of a proton and of nucleons calculating the equilibrium between the repulsive electric force and the attracting strong nuclear force is a standard nuclear physics topic, but I couldn't find it. Can anyone help?

2. Mar 22, 2012

### Bill_K

The nuclear force has a strong repulsive core that tends to keeps nucleons a constant distance apart. See here for an illustration. Quoting:

The main properties of the strong nuclear force are:

At typical nucleon separation (1.3 x 10-15m) it is a very strong attractive force (104 N).

At much smaller separations between nucleons the force is very powerfully repulsive.

Beyond about 1.3 x 10-15m separation, the force quickly dies off to zero. Thus, the strong nuclear force is a very short-range force.

The much smaller Coulomb force between protons has a much larger range and becomes the only significant force between protons when their separation exceeds about 2.5 x 10-15m.

The strong nuclear force is not connected with charge. Proton-proton, proton-neutron and neutron-neutron forces are the same. (The force between protons, however, must always be modified by the Coulomb repulsion between them.)

3. Mar 22, 2012

### Aidyan

Ok, this is the qualitative description. But for the calculation, as I understand it, one has to obtain the force field from the Yukawa potential:

$V(r)=-g^2 \, \frac{e^{-mcr/\hbar}}{r}$

with m the mass of the strong force mediating pion, and then equate with the Coulomb force. Everyone tells $g$ is some constant without further clarification. What is its value?

4. Mar 22, 2012

### Bill_K

Nope. Did you look at the picture? The Yukawa potential has no repulsive core. The idea that the nuclear force was simply due to the exchange of pions was an early guess which turned out to be inaccurate.

5. Mar 22, 2012

### Bob S

Rutherford first discovered that the nuclear Coulomb force was not 1/r2 by scattering alpha particles off of thin foils (gold). Look in particular at the impact parameter b(θ) in the Coulomb scattering cross section, and the discussion on "calculating nuclear size" in

http://en.wikipedia.org/wiki/Rutherford_scattering

This section from Blatt and Weisskopf Theoretical Nuclear Physics (pages 86 - 94)

discusses low energy proton proton scattering (pages 86 - 94), including nuclear-Coulomb destructive interference observed in the differential cross section, showing that the nuclear force is attractive.

6. Mar 22, 2012

### Aidyan

Ok. But is the precise strong force potential inferred only by fitting the experimental data of cross sections? There is no analytic expression derived from QFT for the nuclear force? Even not for a single proton/neutron?

7. Mar 22, 2012