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4.2 Nuclear processes and formation of accreted crusts

In this section we will describe the fate of X-ray burst ashes, produced at ≲ 107 g cm–3, and then sinking deeper and deeper under the weight of accreted plasma above them. We start at a few tens of meters below the surface, and we will end at a depth of ∼ 1 km, where the density ≳ 1013 g cm–3. Under conditions prevailing in accreting neutron star crusts, at 8 −3 ρ > 10 g cm matter is strongly degenerate, and is “relatively cold” (8 T ≲ 10 K, see Figures 25View Image and 26View Image), so that thermonuclear processes are strongly suppressed because interacting nuclei have to overcome a large Coulomb barrier. The structure of an accreted crust is shown in Figure 19View Image.
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Figure 19: Model of an accreting neutron star crust. The total mass of the star is M = 1.4M ⊙. Stable hydrogen burning takes place in the H-burning shell, and produces helium, which accumulates in the He-shell. Helium ignites at ρ ∼ 107 g cm–3, leading to a helium flash and explosive burning of all matter above the bottom of the He-layer into 56Ni, which captures electrons to become 56Fe. After ∼ 1 h, the cycle of accretion, burning of hydrogen and explosion triggered by a helium flash repeats again and the layer of iron from the previous burst is pushed down. Based on the unpublished results of calculations by P. Haensel and J.L. Zdunik. Accreted crust model of [186Jump To The Next Citation Point185Jump To The Next Citation Point]. The core model of [125Jump To The Next Citation Point].

In what follows we will use a simple model of the accreted crust formation, based on the one-component plasma approximation at T = 0 [185Jump To The Next Citation Point187Jump To The Next Citation Point]. The (initial) X-burst ashes are approximated by a one-component plasma with (Ai, Zi) nuclei.

At densities lower than the threshold for pycnonuclear fusion (which is very uncertain, see Yakovlev et al. [427]), 12 13 −3 ρpyc ∼ 10 –10 g cm, the number of nuclei in an element of matter does not change during the compression resulting from the increasing weight of accreted matter. Due to nucleon pairing, stable nuclei in dense matter have even N = A − Z and Z (even-even nuclides). In the outer crust, in which free neutrons are absent, the electron captures proceed in two steps,

(A,Z ) + e− −→ (A, Z − 1) + ν , (67 ) e
(A, Z − 1 ) + e− − → (A, Z − 2) + νe + Qj . (68 )
Electron captures lead to a systematic decrease in Z (therefore an increase in N = A − Z) with increasing density. The first capture, Equation (67View Equation), proceeds as soon as μe > E {A, Z − 1 } − E {A,Z }, in a quasi-equilibrium manner, with negligible energy release. It produces an odd-odd nucleus, which is strongly unstable in a dense medium, and captures a second electron in a nonequilibrium manner, Equation (68View Equation), with energy release Q j, where j is the label of the nonequilibrium process.
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Figure 20: Electron capture processes.
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Figure 21: Z and N of nuclei vs. matter density in an accreted crust, for different models of dense matter. Solid line: Ai = 106; dotted line: Ai = 56. Every change of N and Z, which takes place at a constant pressure, is accompanied by a jump in density: it is represented by small steep (but not perpendicular!) segments of the curves. These segments connect the top and the bottom density of a thin reaction shell. Arrows indicate positions of the neutron drip point. From Haensel & Zdunik [187Jump To The Next Citation Point].

After the neutron-drip point (11 − 3 ρ > ρND ≃ 6 × 10 g cm), electron captures trigger neutron emissions,

(A,Z ) + e− −→ (A, Z − 1) + νe, (69 )
(A, Z − 1) + e− − → (A − k,Z − 2) + k n + νe + Qj . (70 )
Due to the electron captures, the value of Z decreases with increasing density. In consequence, the Coulomb barrier prohibiting the nucleus-nucleus reaction lowers. This effect, combined with the decrease of the mean distance between the neighboring nuclei, and a simultaneous increase of energy of the quantum zero-point vibrations around the nuclear lattice sites, opens up a possibility for the pycnonuclear fusion associated with quantum-mechanical tunneling through the Coulomb barrier due to zero-point vibrations (see, e.g., Section 3.7 of the book by Shapiro & Teukolsky [373Jump To The Next Citation Point]). The pycnonuclear fusion timescale τpyc is a very sensitive function of Z. The chain of the reactions (69View Equation) and (70View Equation) leads to an abrupt decrease of τpyc typically by 7 to 10 orders of magnitude. In the one-component plasma approximation, the accreted crust is composed of spherical shells containing a single nuclide (A, Z). Pycnonuclear fusion switches on as soon as τpyc is smaller than the time of the travel of a piece of matter (due to accretion) through the considered shell of mass Mshell(A, Z), τacc ≡ Mshell∕M˙. The masses of the shells are on the order of 10 −5M ⊙. As a result, in the inner crust the chain of reactions (69View Equation and 70View Equation) in several cases is followed by the pycnonuclear reaction, occurring on a timescale much shorter than τacc. Introducing ′ Z = Z − 2, we then have
(A, Z ′) + (A,Z ′) −→ (2A, 2Z ′) + Qj,1, (71 )
(2A, 2Z ′) −→ (2A − k′,2Z ′) + k′ n + Qj,2, (72 )
... ... −→ ... ... + Qj,3 , (73 )
where dots in Equation (73View Equation) denote an actual nonequilibrium process, usually following reaction (72View Equation). The total heat deposition in matter, resulting from a chain of reactions involving a pycnonuclear fusion, Equations (71View Equation), (72View Equation) and (73View Equation), is Qj = Qj,1 + Qj,2 + Qj,3.

The composition of accreted neutron star crusts, obtained by Haensel & Zdunik [187Jump To The Next Citation Point], is shown in Figure 21View Image. These results describe crusts built of accreted and processed matter up to the density 13 − 3 5 × 10 g cm (slightly before the crust-core interface). At a constant accretion rate M˙ = M˙−9 × 10−9 M ⊙∕yr this will take ∼ 106 yr∕M˙−9. During that time, a shell of X-ray burst ashes will be compressed from ∼ 108 g cm −3 to ∼ 1013 g cm− 3.

Two different compositions of X-ray burst ashes at ≲ 108 g cm −3, Ai, Zi, were assumed. In the first case, Ai = 56, Zi = 26, which is a “standard composition”. In the second scenario Ai = 106, to imitate nuclear ashes obtained by Schatz et al. [364]. The value of Zi = 46 stems then from the condition of beta equilibrium at ρ = 108 g cm −3. As we see in Figure 21View Image, after the pycnonuclear fusion region is reached, both curves converge (as explained in Haensel & Zdunik [187Jump To The Next Citation Point], this results from Ai and Zi in two scenarios).

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