Glassy quantum nuclear pasta in neutron star crusts

We conduct a comprehensive survey of the shape parameter space of the nuclear pasta phases in neutron star crusts by conducting three-dimensional $\mathrm{Hartree-Fock}+\mathrm{BCS}$ calculations. Spaghetti, waffles, lasagna, bicontinuous phases and cylindrical holes occupy local minima in the resulting constant-pressure Gibbs energy surfaces, implying multiple geometries coexist at a given depth. Notably, the bicontinuous phase, in which both the neutron gas and nuclear matter extend continuously in all dimensions appears over a large depth range. Our results support the idea that nuclear pasta is a glassy system. At a characteristic temperature, of order ${10}^8\text{–}{10}^9\mathrm{K}$, different phases may become frozen into domains whose sizes we estimate to be 1–50 times the lattice spacing and over which the local density and electron fraction can vary. Above this temperature, very little long-range order exists and matter is an amorphous solid. Electron scattering off domain boundaries may contribute to the disorder resistivity of the pasta phases. Annealing of the domains may occur during cooling; repopulating of local minima during crustal heating might lead to temperature-dependent transport properties in the deep crust layers. We identify four regions distinguished by whether pasta is the true ground state, and whether the pasta structure allows delocalization of protons. The whole pasta region can occupy up to 70% of the crust by mass and 25% by thickness, and the layer in which protons are delocalized could occupy 45% of the crust mass and 15% of its thickness.

Figure 1

The pure neutron matter EOS of the NRAPR Skyrme parametrization used in this work (red dashed line) compared to the region predicted by ab initio calculations of PNM [70, 71, 72]. Given the importance of the PNM EOS to crust properties, it is important to consider the consistency of the interaction used to model the crust and our best theoretical knowledge of PNM.

Figure 2

The red and blue dashed lines are the bounds of the total nucleon number in the cell $A_{\mathrm{cell}}$ and the average proton fraction $y_{\mathrm{p}}$ from a compressible liquid drop model using the NRAPR interaction [57] and varying the surface energy parameters over a reasonable range [73]. Vertical dotted lines indicate the transition to cylindrical nuclear pasta, slabs, cylindrical holes and spherical holes with increasing density. The transitions are accompanied by discontinuities in the cell size. The blue diamonds are the values of $A_{\mathrm{cell}}$ we choose to perform our 3DHF calculations at. The red points indicate the $\beta$-equilibrium proton fractions we at each density from our 3DHF calculations, and the “error bars” through them indicate the range of proton fractions we performed calculations at.

Figure 3

Energy-deformation surfaces at an average density of $n_{\mathrm{b}}=0.06{\mathrm{fm}}^{-3}$ for cells containing $A_{\mathrm{cell}}=614$ nucleons and a proton fraction of 0.023 (a), 0.026 (b), and 0.029 (c). Many local minima (dark blue troughs) appear in the landscapes; the pasta configurations appearing in those minima can be seen in Fig. 5.

Figure 4

Gibbs energy-deformation surfaces (a)–(c) and average baryon density surfaces (d)–(f) at a constant pressure of $0.291\mathrm{MeV}{\mathrm{fm}}^{-3}$ for cells containing $A_{\mathrm{cell}}=614$ nucleons. Results are shown for proton fraction of 0.023 (a), d), 0.026 (b), (e), and 0.029 (c), (f). The local minima differ in baryon density by of order 5%. The pasta configurations appearing in those minima can be seen in Fig. 5.

Figure 5

The top row (a)–(c) shows the Gibbs free-energy surfaces from Fig. 4 at a constant pressure of $0.291\mathrm{MeV}{\mathrm{fm}}^{-3}$ (densities around 0.06 ${\mathrm{fm}}^{-3}$) with red arrows indicating paths between local minima that pass over the smallest energy barriers. Underneath is the corresponding one-dimensional plot of the Gibbs energy along those paths (d). Selected $(\beta ,\gamma )$ coordinates are shown along the one-dimensional plots. 3D surfaces at constant neutron density in the cell show the shape of the nuclear cluster in the cell at the local minima indicated. The other apparent local minima can transition exothermically to one of the highlighted minima by $\beta$ decay or electron capture.

Figure 6

Illustration of our schematic model for pasta domains in local Gibbs energy minima at densities around $0.06{\mathrm{fm}}^{-3}$ where calculations present four physically distinct local minima. To visualize the phases better we have plotted a region of four by four unit cells, but it is important to remember we only simulate one single unit cell. The proton fractions corresponding to the minima we found are shown above each minima. The fictive temperature $T_{\mathrm{f}}$ is given by the height of the energy barriers, and the energy differences between he local minima are indicated as $\mathrm{\Delta }G$ The relative abundances of the phases at a temperature equal to the fictive temperature is shown below the visualizations of the phases. In the simplified model we take a single barrier height to be the fictive temperature. This is determined as the average of the various barriers in our calculation. These abundances will be frozen in as the temperature drops further unless processes such as quantum tunneling anneal the matter.

Figure 7

Top row: Gibbs free-energy surfaces at a pressure of 0.094 MeV ${\mathrm{fm}}^{-3}$ corresponding to a baryon density of $\approx 0.035{\mathrm{fm}}^{-3}$ at proton fractions of 0.018 (a), 0.022 (b), and 0.026 (c), for cells containing $A_{\mathrm{cell}}=454$ nucleons. Below is the Gibbs free-energy variation along one dimensional paths passing through the energy minimum (d). Selected $(\beta ,\gamma )$ coordinates are shown along the one-dimensional plots. Visualizations of the minimum energy nuclear shapes are shown, obtained by plotting a surface of constant neutron density corresponding to the average neutron density in the cell. In all cases the minimum is a spherical nucleus.

Figure 8

Top row: Gibbs free-energy surfaces at a pressure of $0.120\mathrm{MeV}{\mathrm{fm}}^{-3}$, corresponding to a baryon density of $\approx 0.04{\mathrm{fm}}^{-3}$, at proton fractions of 0.018 (a), 0.022 (b), and 0.026 (c), for cells containing $A_{\mathrm{cell}}=454$ nucleons. Below is the Gibbs free-energy variation along one dimensional paths passing through the energy minimum (d). Selected $(\beta ,\gamma )$ coordinates are shown along the one-dimensional plots. Visualizations of the minimum energy nuclear shapes are shown, obtained by plotting a surface of constant neutron density corresponding to the average neutron density in the cell. The minimum energy nuclear shape is roughly spherical, but deformed nuclear and cylindrical nuclear shapes appear as local minima. The relative abundances of the phases at a temperature equal to the fictive temperature is shown next to the visualizations of the phases.

Figure 9

Top row: Gibbs free-energy surfaces at a pressure of $0.150\mathrm{MeV}{\mathrm{fm}}^{-3}$, corresponding to a baryon density of $\approx 0.045{\mathrm{fm}}^{-3}$, at proton fractions of 0.018 (a), 0.022 (b), and 0.026 (c), for cells containing $A_{\mathrm{cell}}=454$ nucleons. Below is the Gibbs free-energy variation along one dimensional paths passing through the energy minimum (d). Selected $(\beta ,\gamma )$ coordinates are shown along the one-dimensional plots. Visualizations of the minimum energy nuclear shapes are shown, obtained by plotting a surface of constant neutron density corresponding to the average neutron density in the cell. The minimum energy nuclear shape is roughly spherical, but deformed nuclear and cylindrical nuclear shapes appear as local minima. The relative abundances of the phases at a temperature equal to the fictive temperature is shown below the visualizations of the phases.

Figure 10

Average baryon density surfaces (bottom) at a constant pressure of $0.15\mathrm{MeV}{\mathrm{fm}}^{-3}$, corresponding to a baryon density of $\approx 0.045{\mathrm{fm}}^{-3}$, for cells containing $A_{\mathrm{cell}}=454$ nucleons. Results are shown for proton fraction of 0.018 (a), 0.022 (b), and 0.026 (c). The local minima differ in baron density by of order 5%.

Figure 11

Top row: Gibbs free-energy surfaces at a pressure of $0.184\mathrm{MeV}{\mathrm{fm}}^{-3}$, corresponding to a baryon density of $\approx 0.05{\mathrm{fm}}^{-3}$, at proton fractions of 0.021 (a), 0.023 (b), and 0.025 (c), for cells containing $A_{\mathrm{cell}}=956$ nucleons. Below is the Gibbs free-energy variation along one-dimensional paths passing through the energy minimum (d). Selected $(\beta ,\gamma )$ coordinates are shown along the one-dimensional plots. Visualizations of the minimum energy nuclear shapes are shown, obtained by plotting a surface of constant neutron density corresponding to the average neutron density in the cell. The minimum energy nuclear shape is a nuclear “waffle,” with an isolated nuclear phase and spaghetti phase as local minima at a different proton fraction. The relative abundances of the phases at a temperature equal to the fictive temperature is shown below the visualizations of the phases.

Figure 12

Top row: Gibbs free-energy surfaces at a pressure of 0.234 MeV ${\mathrm{fm}}^{-3}$, corresponding to a baryon density of $\approx 0.054{\mathrm{fm}}^{-3}$, at proton fractions of 0.022 (a), 0.024 (b), and 0.026 (c), for cells containing $A_{\mathrm{cell}}=1166$ nucleons. Below is the Gibbs free-energy variation along one dimensional paths passing through the energy minimum (d). Selected $(\beta ,\gamma )$ coordinates are shown along the one-dimensional plots. Visualizations of the minimum energy nuclear shapes are shown, obtained by plotting a surface of constant neutron density corresponding to the average neutron density in the cell. The minimum energy nuclear shape is roughly spherical, followed by cylindrical and waffle shapes. The relative abundances of the phases at a temperature equal to the fictive temperature is shown below the visualizations of the phases.

Figure 13

Top row: Gibbs free-energy surfaces at a pressure of 0.30 MeV ${\mathrm{fm}}^{-3}$, corresponding to a baryon density of $\approx 0.066{\mathrm{fm}}^{-3}$, at proton fractions of 0.022 (a), 0.025 (b), and 0.028 (c), for cells containing $A_{\mathrm{cell}}=784$ nucleons. Below is the Gibbs free-energy variation along one dimensional paths passing through the energy minimum (d). Selected $(\beta ,\gamma )$ coordinates are shown along the one-dimensional plots. Visualizations of the minimum energy nuclear shapes are shown, obtained by plotting a surface of constant neutron density corresponding to the average neutron density in the cell. The minimum energy nuclear configuration is the bicontinuous cubic-P phase, followed by planar and cylindrical geometries. The relative abundances of the phases at a temperature equal to the fictive temperature is shown below the visualizations of the phases.

Figure 14

Top row: Gibbs free-energy surfaces at a pressure of 0.32 MeV ${\mathrm{fm}}^{-3}$, corresponding to a baryon density of $\approx 0.07{\mathrm{fm}}^{-3}$, at proton fractions of 0.03 (a), 0.034 (b), and 0.038 (c), for cells containing $A_{\mathrm{cell}}=532$ nucleons. Below is the Gibbs free-energy variation along one dimensional paths passing through the energy minimum. Selected $(\beta ,\gamma )$ coordinates are shown along the one-dimensional plots. Visualizations of the minimum energy nuclear shapes are shown, obtained by plotting a surface of constant neutron density corresponding to the average neutron density in the cell. The minimum energy nuclear configuration is planar, followed by waffle and bicontinuous cubic-P phases. The relative abundances of the phases at a temperature equal to the fictive temperature is shown below the visualizations of the phases.

Figure 15

Top row: Gibbs free-energy surfaces at a pressure of 0.34 MeV ${\mathrm{fm}}^{-3}$, corresponding to a baryon density of $\approx 0.076{\mathrm{fm}}^{-3}$, at proton fractions of 0.034 (a), 0.038 (b), and 0.042 (c), for cells containing $A_{\mathrm{cell}}=532$ nucleons. Below is the Gibbs free-energy variation along one dimensional paths passing through the energy minimum (d). Selected $(\beta ,\gamma )$ coordinates are shown along the one-dimensional plots. Visualizations of the minimum energy nuclear shapes are shown, obtained by plotting a surface of constant neutron density corresponding to the average neutron density in the cell. All three minima are variations of the bicontinuous cubic-P phase, albeit at different proton fractions. The relative abundances of the phases at a temperature equal to the fictive temperature is shown below the visualizations of the phases.

Figure 16

Average baryon density surfaces (bottom) at a constant pressure of $0.524\mathrm{MeV}{\mathrm{fm}}^{-3}$, corresponding to a baryon density of $\approx 0.076{\mathrm{fm}}^{-3}$, for cells containing $A_{\mathrm{cell}}=532$ nucleons. Results are shown for proton fraction of 0.034 (a), 0.038 (b), and 0.042 (c). The local minima differ in baron density by of order 5%.

Figure 17

Top row: Gibbs free-energy surfaces at a pressure of 0.45 MeV ${\mathrm{fm}}^{-3}$, corresponding to a baryon density of $\approx 0.082{\mathrm{fm}}^{-3}$, at proton fractions of 0.024 (a), 0.030 (b), and 0.036 (c), for cells containing $A_{\mathrm{cell}}=332$ nucleons. Below is the Gibbs free-energy variation along one dimensional paths passing through the energy minimum (d). Selected $(\beta ,\gamma )$ coordinates are shown along the one-dimensional plots. Visualizations of the minimum energy nuclear shapes are shown, obtained by plotting a surface of constant neutron density corresponding to the average neutron density in the cell. The first minimum is a combination of both a cylindrical high density and low density region (spaghetti and antispaghetti), whereas the second minimum is cylindrical. The relative abundances of the phases at a temperature equal to the fictive temperature is shown below the visualizations of the phases. In the simplified model we take a single barrier height to be the fictive temperature. These abundances will be frozen in as the temperature drops further unless quantum tunneling processes anneal the matter.

Figure 18

Top row: Gibbs free-energy surfaces at a pressure of 0.53 MeV ${\mathrm{fm}}^{-3}$, corresponding to a baryon density of $\approx 0.088{\mathrm{fm}}^{-3}$, at proton fractions of 0.024 (a), 0.030 (b), and 0.036 (c), for cells containing $A_{\mathrm{cell}}=332$ nucleons. Below is the Gibbs free-energy variation along one dimensional paths passing through the energy minimum (d). Selected $(\beta ,\gamma )$ coordinates are shown along the one-dimensional plots. Visualizations of the minimum energy nuclear shapes are shown, obtained by plotting a surface of constant neutron density corresponding to the average neutron density in the cell. The first minimum is planar, and the second minimum is a combination of both a cylindrical high density and low density region (spaghetti and antispaghetti). The relative abundances of the phases at a temperature equal to the fictive temperature is shown below the visualizations of the phases.

Figure 19

Overview of all nuclear geometries that emerge from our calculations. To visualize the phases better we have plotted a region of four unit cells squared, but it is important to remember we only calculate one single unit cell. The colors are just for clarity of visualization and have no physical meaning. The left column gives the approximate baryon densities the configurations to the right cover. The minimum energy configurations are the leftmost pictures in each row, with the energy of the minimum increasing rightwards. We display only a representative selection of configurations. Horizontal lines divide the graphic into the four distinct regions of pasta: from low to high density, the region where pasta first appears as a local minimum at higher energy than the isolated nuclear phase; the region where pasta shapes become the absolute minimum, but isolated nuclei remain as higher energy local minima; the region where only pasta shapes exist; and the region where only pasta shapes exist, and protons have become delocalized in all dimensions.

Figure 20

(a) The pressure of matter in the crust as calculated by the compressible liquid drop model (dashed line), and the pressure of each of the configurations examined in our 3DHF calculations (blue diamonds). (b) The chemical potential from the CLDM and our 3DHF calculations. Vertical dotted lines divide the four regions of pasta P1-P4 as described in the text. The mass coordinate $y$ calculated from the CLDM pressure is given in the right vertical scale in panel (a), and the radial coordinate $r$ calculated from the CLDM chemical potential is given in the right vertical scale in panel (b).

Figure 21

The fictive temperature $T_{\mathrm{f}}$ as a function of mass coordinate in the crust $y$ (the fraction of the mass of the crust above a given point) assuming that the relevant degrees of freedom are individual nucleons (red points), or a collection of nucleons of order the proton number $y_{\mathrm{p}}{A}_{\mathrm{cell}}$ (blue diamonds). The bars span the range of of barrier heights from our calculations. Vertical dotted lines divide the four regions of pasta P1–P4 as described in the text.

Figure 22

The long-range order of the pasta phases $\mathrm{\Delta }L$ calculated at the fictive temperature, assuming individual nucleons are the relevant degrees of freedom, as a function of mass parameter. We give the length scale in absolute terms (a) and in units of cell size (b). The horizontal dashed lines indicate the point where the order drops below 10 fm—approximately the characteristic width of nuclear clusters (a)—and below one cell size (b). Spaghettilike configurations are shown as red pints, and lasagnalike configurations are shown by blue diamonds. At high $y$ the Lasagna order becomes much less than the cell size, too low to appear on the plots; at that point the formalism used to calculate the disorder breaks down. Below the dashed lines, matter is completely disordered. These are upper limits on the long-range order: if we assume that clusters of nucleons are the relevant degrees of freedom, then the fictive temperature increases (see Fig. 21) and the long-range order drops. Vertical dotted lines divide the four regions of pasta P1–P4 as described in the text.

Figure 23

The lattice energy per nucleon in the unit cell for minimum energy configurations as a function of mass coordinate $y$. The lattice energy drives the stability of the phases; it peaks at around $y=0.4$ (a baryon density of $\approx 0.06{\mathrm{fm}}^{-3}$), where isolated nuclei disappear from the minima in the energy landscape, and then decreases roughly exponentially with density thereafter.