Geometric approach to nuclear pasta phases

By use of the variational methods and differential geometry in the framework of the liquid drop model we formulate appropriate equilibrium equations for pasta phases with imposed periodicity. The extension of the Young-Laplace equation in the case of charged fluid is obtained. The $\beta$ equilibrium and virial theorem are also generalized. All equations are shown in gauge invariant form. For the first time, the pasta shape stability analysis is carried out. The proper stability condition in the form of the generalized Jacobi equation is derived. The presented formalism is tested on some particular cases.

Figure 1

Schematic view of rectangular cell with one proton cluster $\mathcal{P}$. The surface tension is associated with its boundary $\partial \mathcal{P}$.

Figure 2

The infinitesimal deformation of the proton cluster described by the normal component $\epsilon$.

Figure 3

Schematic view of the cell size variation.

Figure 4

(a) The axially symmetric deformation of cylindrical cluster. (b) shows the charge density profile for a fixed $\varphi ,z$.