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

[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?

 

[Bill_K]

The nuclear force has a strong repulsive core that tends to keeps nucleons a constant distance apart. See http://webs.mn.catholic.edu.au/physics/emery/hsc_quanta_quarks.htmfor 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.)

 

[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?

 

[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.

 

[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)

http://books.google.com/books?id=R3...ge&q=neutron proton scattering length&f=false

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.

 

[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?

 

[Vanadium 50]

There is no analytic expression derived from QFT for the nuclear force? Even not for a single proton/neutron?

Nope. The best you can get are semi-empirical formulas. But you can't start from QCD and get an exact solution for the nucleon-nucleon force. It's highly non perturbative, so you would have to go to the lattice.

 

[Aidyan]

Ok, thanks so far. The problem is that these empirical data hide the real physical information. In fact, I would like to understand from what kind of physical constants depends the nucleus size? For the Coulomb potential it is easy, but when it comes to the strong force interaction it isn't clear to me. I suppose the speed of light, Planck's constant, etc. (pion mass?) Did someone elaborate on this?