Symmetry energy in nuclear physics

[rahele]

I have a question about symmetry energy in semi-empirical mass formula,
According to semi-empirical mass formula as follows:
E=a_vA-a_sA^2/3-a_cZ(Z-1)/A^1/3-a_sym(N-Z)²/A
why in the symmetry energy only squared parameter symmetry are exist and there is not the first power of asymmetry parameter?

 

[mfb]

(N-Z) would have nothing to do with symmetry, it would go from -infinity to +infinity (well, bounded by 0 neutrons and 0 protons of course). And the absolute proton and neutron numbers are in the total mass anyway (this is just the binding energy).

There could be |N-Z|, but experiments show this is not needed. And I don't see a physical reason for it.

 

[ChrisVer]

I think absolute values are not so favored... :) they miss nice functional properties. So we wouldn't search for a fitting in | | but in ( )^2 if we knew a priori that something is happening, and see how that works
Also, I guess, it's because it fits the experiments as mfb said.

 

[dauto]

Calculate the average potential energy of a brick in a brick wall of height N. Calculate the same for a wall of height Z. Keep the sum of the height A = Z + N fixed but allow their difference (N - Z) to be a free parameter. Find out the dependency of the total energy on that free parameter.

 

[lpetrich]

Semi-empirical mass formula - Wikipedia has a derivation of the form of the symmetry-energy term. The derivation treats protons and neutrons as separate but overlapping Fermi liquids that both extend over the nucleus.

E_kinetic = E_Fermi/A^2/3*(Z^5/3 + N^5/3)

One then sets Z = (A/2) + X and N = (A/2) - X and expands in X. The first term in X is a term in X².

The absolute-value function has a problem: it has a singularity at 0. Its first derivative is a step function and its second derivative a Dirac delta function.

The square function does not have that problem.