I have a question about symmetry energy in semi-empirical mass formula, According to semi-empirical mass formula as follows: E=a_{v}A-a_{s}A^{2/3}-a_{c}Z(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?
(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.
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.
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.
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 X^{2}. 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.