What is nuclear symmetry energy?

[fhqwgads2005]

in a nutshell

 

[humanino]

The strong force which is binding in nuclei is symmetric under the exchange neutron <-> proton (or down <-> up). There are other forces, for instance the electromagnetic repulsion between two protons, which are not binding and not symmetric. As the name suggests, the non-symmetric part is only a perturbation. So, using the asymmetry parameter
$$\alpha = \frac{N-Z}{A}$$
and the density $\rho$, we develop the energy density in the nuclear medium $E(\rho,\alpha)$ as a Taylor series
$$E(\rho,\alpha) = E(\rho,0) + S(\rho)\alpha^2 + O(\alpha^4) + \cdots$$
and expanding around the saturation density $\rho_s$
$$S(\rho) = \left.\frac{1}{2}\frac{\partial^2 E}{\partial\alpha^2}\right|_{\alpha=0,\rho=\rho_s}=a_v+\frac{p_0}{\rho_s^2}(\rho-\rho_s)+\cdots$$
The symmetry energy $a_v\approx 29 \pm 2$ MeV

source : The nuclear symmetry energy

 

[fhqwgads2005]

So, is it useful to think of it as the energy required to increase the asymmetry between N and Z in the nucleus, say, by electron capture?

 

[rahele]

why in the symmetry energy only squared asymmetry parameter are exist and there is not the first power of asymmetry parameter?

 

[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.

 

[bcrowell]

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.

Nice explanation. I would add that this topic is a little subtle because often we want to use these liquid-drop energies in the Strutinsky smearing technique, where we add in a quantum-mechanical shell correction. When we do that, we have two terms in the energy, classical and quantum-mechanical. We have to be careful not to double-count a particular energy in both the classical and the quantum-mechanical term (Strutinsky shell correction). As dauto correctly explains, the asymmetry energy is a quantum-mechanical effect arising from the exclusion principle. So you would think you should include it only in the quantum-mechanical term. However, the Strutinsky technique for, say, the neutrons, only adds a correction that represents the difference in binding energy between nucleus (N,Z) and the average of other nuclei (N+x,Z), where x is small, and this correction vanishes when the levels are uniformly spaced. So clearly the effect as described by dauto is not included in the Strutinsky correction, because it would occur even if the levels were uniformly spaced.