Appearance of the single gyroid network phase in “nuclear pasta” matter

Nuclear matter under the conditions of a supernova explosion unfolds into a rich variety of spatially structured phases, called nuclear pasta. We investigate the role of periodic networklike structures with negatively curved interfaces in nuclear pasta structures, by static and dynamic Hartree-Fock simulations in periodic lattices. As the most prominent result, we identify for the first time the single gyroid network structure of cubic chiral $I{4}_{1}23$ symmetry, a well-known configuration in nanostructured soft-matter systems, both as a dynamical state and as a cooled static solution. Single gyroid structures form spontaneously in the course of the dynamical simulations. Most of them are isomeric states. The very small energy differences from the ground state indicate its relevance for structures in nuclear pasta.

Figure 1

Gyroidal pasta shape: The green structure on the left hand represents the density distribution of the gyroidal state of nuclear pasta matter computed with time-dependent Hartree-Fock calculations for an average density of $0.06 \mathrm{fm}{}^{-3}$ and a box length of $a=22$ fm. Shown is the Gibbs dividing surface with a corresponding threshold density. The solid volume represents densities above this value and the void represents densities below this value. The blue structure on the right-hand side shows the nodal approximation (4) of a single gyroid constant-mean-curvature surface at the same volume fraction. Also shown by orange bars is a gyroid network in the void phase of both the pasta shape and the nodal approximation, showing that they are indeed homotopic. Black frames are guides to the eye, of size $1.25a$, the cubic lattice parameter.

Figure 2

Distinction between the single gyroid (left) and double gyroid (right) geometries. The single gyroid consists of a single solid domain (gray) and a single void domain, separated by the CMC gyroid surface. The double gyroid consists in a solid domain (gray) represented by a sheet of finite width draped onto the gyroid minimal surface and two distinct void domains, each forming a networklike labyrinth. The interfaces between the solid and the void domain in the double gyroid are two distinct surfaces [brown (dark grey) and green (light grey)], modeled, e.g., by CMC surfaces with constant curvatures $\pm h_0$. A second form of the double gyroid, where solid and void are interchanged, also exists. (Images are from Ref. [34].)

Figure 3

Binding energies per nucleon $E/N$ for the metastable ground states for different volume fractions. G (gyroid, dots), P (primitive, squares), and S (slab, diamonds) denote the initial guiding potentials. The states shown in this plot remained topologically stable for the 9000 iterations. Higher values of $E/N$ correspond to more tightly bound and hence more favorable solutions.

Figure 4

Histograms of the resulting Euler characteristic $\chi$ for different box lengths of 20–24 fm of the dynamic, randomly initialized calculations. The shaded area marks the identified gyroidal structures.