Nuclear Reactions in the Crusts of Accreting Neutron Stars

Unless otherwise stated, we use $\dot{m}=0.3{\dot{m}}_{\mathrm{Edd}}$ in the rest frame at the surface, with the local Eddington accretion rate ${\dot{m}}_{\mathrm{Edd}}=8.8\times {10}^{4}\,{\rm{g}}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$, and $g=1.85\times {10}^{14}\,\mathrm{cm}\,{{\rm{s}}}^{-2}$. This accretion rate is in the range for mixed H/He bursts powered by the rp-process, as well as superbursts, and is therefore appropriate for the initial compositions explored in this work. The calculation starts at a density of $\rho =1.4\,\times {10}^{9}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$. The temperature is treated as a free parameter and set to T = 0.5 GK throughout the crust. This corresponds closely to the temperature profile used in Gupta et al. (2007). This approach is suitable for identifying the typical nuclear reactions, independent of specific temperature profiles that vary from system to system and with time, and depends on a number of additional parameters outside of our model (see discussion below). Temperature is not expected to dramatically alter reaction sequences as kT ≪ μ_e everywhere and kT ≪ μ_n everywhere except for a very narrow layer at neutron drip. Pycnonuclear fusion reaction rates are not temperature sensitive either. The one nuclear process that is strongly temperature dependent is the strength of nuclear Urca cooling in the outer crust (Schatz et al. 2014). Choosing a relatively high constant temperature allows us to clearly identify critical Urca cooling pairs with their intrinsic strengths, which may play a role in limiting crustal heating.

Nuclear masses are among the most important input parameters. We use the Atomic Mass Evaluation AME12 (Wang et al. 2012) for experimental masses closer to stability. For the majority of nuclei for which masses are experimentally unknown, we employ the FRDM (Möller et al. 1995) mass model. We do not mix experimental and theoretical masses to calculate reaction Q-values of interest here, such as electron capture thresholds or neutron separation energies.

In general, mass models for isolated nuclei such as the FRDM are not applicable in the inner crust, where interactions with the free neutrons result in significant modifications of masses and nuclear structure. However, the goal of this work is to calculate the nuclear reactions in the outer crust and the transition from the outer crust into the inner crust. We stop the calculations in the outermost region of the inner crust below $\rho =2.7\times {10}^{12}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ before such modifications become significant. While the neutron mass fraction at the end of our calculation is X_n = 0.52, the neutron Fermi energy is 1.4 MeV, and the free neutron density is only $9\times {10}^{-4}$ fm⁻³, less than 1% of the neutron density inside a nucleus. For these conditions, the Mackie & Baym (1977) mass model, which includes the impact of free neutrons on the surface energy but neglects shell structure, shows an average correction of neutron separation energies of only 200 keV (maximum 300 keV), well within mass model uncertainties. Interactions with the free neutron gas in the inner crust also affect the single-particle level structure of nuclei. In particular, with increasing free neutron density the shell structure of spherical nuclei is expected to be modified, and for sufficiently large densities it effectively disappears (Negele & Vautherin 1973). This would affect the shell correction term in the FRDM, which turns out to be important in our work. Figure 5 in Negele & Vautherin (1973) shows that modifications of single-particle levels are expected to only set in at baryon densities in excess of n_b = $2\times {10}^{36}$, above the maximum n_b = $1.6\times {10}^{36}$ in our calculations.

Electron capture rates and β⁻-decay rates are determined from strength functions calculated in a model based on wave functions in a deformed folded-Yukawa single-particle potential with residual pairing and Gamow–Teller interactions. They are solved for in a quasi-particle random-phase approximation (QRPA). The original theory was based on a deformed oscillator single-particle potential (Krumlinde & Möller 1984). To obtain greater global predictive power, a folded-Yukawa single-particle model has been used later instead (Möller & Randrup 1990; Möller et al. 1997). That is the model used here, and we refer to it as QRPA-fY. Only allowed Gamow–Teller transitions are considered. Parent nuclei are assumed to be in their ground state, which is a reasonable assumption for the low temperatures (T < 1 GK) encountered in neutron star crusts. Weak interaction thresholds are corrected for lattice energy changes following Chamel & Haensel (2008). The weak reaction rates are then calculated for each time step using nuclear masses and a fast phase-space approximation (Becerril Reyes et al. 2006; Gupta et al. 2007). Neutron emission is determined individually for each transition from the parent ground state to a daughter state with excitation energy E_x. We make the simplifying assumption that the highest number of emitted neutrons that is energetically possible will occur in all cases, similar to the approach of calculating branchings for β-delayed neutron emission in Möller et al. (1997). To take into account Pauli blocking due to the positive neutron chemical potential, we impose the additional condition of (E_x − S_n)/N > μ_n, with neutron separation energy S_n and number of emitted neutrons N.

Neutron capture rates were computed with the TALYS statistical model code as part of a systematic effort to create a reaction rate database for nucleosynthesis studies, using the same atomic masses used to calculate the weak interaction rates (Xu et al. 2013). These neutron capture rates and the rates of the reverse reactions were corrected to account for plasma screening of photons and neutron degeneracy following Shternin et al. (2012). Pycnonuclear fusion rates were calculated from the S-factors of Beard et al. (2010) and Afanasjev et al. (2012) using the formalism described in Yakovlev et al. (2006) in the uniformly mixed multicomponent plasma approximation. We implement a total of 4844 pycnonuclear fusion rates from Be to Si.

We begin by discussing in detail the reaction sequences for an initial composition of pure ⁵⁶Fe. Our calculation follows the compositional change in an accreted fluid element as density and therefore electron chemical potential μ_e are slowly rising. The evolution of the main composition as a function of depth is shown in Figure 2, and the major compositional transitions are listed in Table 1.

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Table 1.  Major Compositional Transitions for Initial ⁵⁶Fe

Transition	Pa	ρb	μec	Xnd
56Fe → 56Cr	$3.4\times {10}^{27}$	$4.9\times {10}^{9}$	6.2	<10−25
56Cr → 56Ti	$1.7\times {10}^{28}$	$1.8\times {10}^{10}$	9.6	<10−25
56Ti → 56Ca	$1.1\times {10}^{29}$	$8.1\times {10}^{10}$	15.6	<10−25
56Ca → 56Ar,54Ar,58Ca	$5.5\times {10}^{29}$	$2.9\times {10}^{11}$	23.3	$1.2\times {10}^{-18}$
56Ar,54Ar,58Ca → 56Ar	$8.3\times {10}^{29}$	$4.2\times {10}^{11}$	25.9	$7.2\times {10}^{-20}$
56Ar → 40Mg,62Ar	$1.8\times {10}^{30}$	$7.8\times {10}^{11}$	31.6	$5.4\times {10}^{-8}$
40Mg,62Ar → 40Mg,48Si	$2.3\times {10}^{30}$	$1.1\times {10}^{12}$	33.5	0.13
40Mg,48Si → 40Mg	$4.2\times {10}^{30}$	$2.8\times {10}^{12}$	37.1	0.54

Notes.

^aPressure in dynes cm⁻². ^bMass density in g cm⁻³. ^cElectron chemical potential (without rest mass) in MeV. ^dNeutron abundance.

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The initial reaction sequence up to μ_e = 15.6 MeV and $\rho \,=8.1\times {10}^{10}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ takes 2.3 × 10¹⁰ s and is characterized by a series of three, two-step, electron capture (EC) reactions ⁵⁶Fe(2EC)⁵⁶Cr, ⁵⁶Cr(2EC)⁵⁶Ti, and ⁵⁶Ti(2EC)⁵⁶Ca described already in Haensel & Zdunik (1990) (see Figure 3). These electron captures proceed in steps of two because of the odd–even staggering of the electron capture thresholds. During the last sequence, (γ, n) reactions release small amounts of neutrons that get recaptured but do not appreciably change the reaction flows. For most EC transitions, the inverse process, β⁻-decay, is blocked, as the decay feeds primarily excited daughter states, which reduces the energy of the emitted electrons, resulting in effective Fermi blocking. The exception is ⁵⁶Ti(EC)⁵⁶Sc, where β⁻-decay of ⁵⁶Sc does occur, leading to a ⁵⁶Ti–⁵⁶Sc EC/β⁻ Urca cycle (Schatz et al. 2014; Meisel et al. 2015b). However, as discussed in Meisel et al. (2015b), the cycle is weak because of the fast ⁵⁶Sc(EC)⁵⁶Ca reaction for the nuclear physics inputs used here, and thus it does not affect nuclear energy generation.

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At μ_e = 23.3 MeV and $\rho =2.9\times {10}^{11}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$, the destruction of ⁵⁶Ca by electron capture occurs. However, this step proceeds entirely differently owing to the rising significance of free neutrons (see Figure 3). These neutrons are released in the second step of the two-step electron capture sequence, which proceeds as ⁵⁶Ca(EC)⁵⁶K(EC, 2n)⁵⁴Ar. The neutron separation energy of ⁵⁶Ar is sufficiently low for most of the EC transitions from ⁵⁶K to proceed to neutron-unbound states, leading to the emission of neutrons. The released neutrons are recaptured by the most abundant nucleus, which is still ⁵⁶Ca, leading to a neutron capture sequence to ⁵⁸Ca. The reaction path therefore splits into two branches leading to ⁵⁴Ar and ⁵⁸Ca, respectively. However, branchings between electron capture and neutron capture at ⁵⁷Ca and ⁵⁶K divert some of the reaction flow to ⁵⁶Ar via ⁵⁷Ca(EC)⁵⁷K(EC, n)⁵⁶Ar and ⁵⁶K(n, γ)⁵⁷K(EC, n)⁵⁶Ar, respectively. The result is a three-nuclide composition, dominated by ⁵⁶Ar, but with admixtures of ⁵⁸Ca and ⁵⁴Ar at about 0.2% mass fraction each.

This admixture is, however, short-lived, as at ${\mu }_{e}\,=25.9\,\mathrm{MeV}$ and $\rho =4.2\times {10}^{11}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$, ⁵⁸Ca and ⁵⁴Ar are converted into ⁵⁶Ar (Figure 4). The destruction of ⁵⁸Ca proceeds via ⁵⁸Ca(EC, 1n)⁵⁷K(EC, 1n, 2n, 3n), resulting in a range of Ar isotopes, which, together with the already-existing ⁵⁴Ar, are quickly transformed into ⁵⁶Ar by neutron capture. At this point, the crust is rather pure and mainly composed of ⁵⁶Ar.

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At μ_e = 31.6 MeV and $\rho =7.8\times {10}^{11}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$, ⁵⁶Ar is destroyed by the first previously defined SEC (Gupta et al. 2008; see also Figure 5). This reaction sequence occurs when the neutron emission following an electron capture leads to a nucleus with $| {Q}_{\mathrm{EC}}| \ll {\mu }_{e}$, which therefore immediately captures electrons again, and so on. In this particular case, an SEC leading from ⁵⁶Ar all the way to ⁴⁰Mg is established. The detailed reaction sequence is shown in Figure 5 and is characterized by electron captures with the emission of mostly four to five neutrons. The released neutrons are recaptured by ⁵⁶Ar, which is still the most abundant nuclide. This leads again to a split of the reaction path into the SEC from ⁵⁶Ar to ⁴⁰Mg and a sequence of neutron captures from ⁵⁶Ar to ⁶²Ar. In the initial phase of the SEC, there is a significant abundance buildup of ⁵⁰S produced by neutron capture from the SEC path, and to a lesser extent of ⁴²Si. However, with only a slight rise of μ_e, electron capture quickly destroys these isotopes, and they are converted into ⁴⁰Mg as well. The end result is a layer that consists primarily of ⁶²Ar (80% mass fraction) and ⁴⁰Mg (20% mass fraction). There is a small admixture of ⁵⁹Cl (10⁻⁵ mass fraction). The free neutron abundance is significantly increased to 4.8 × 10⁻⁵.

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At μ_e = 33.5 MeV and $\rho =1.1\times {10}^{12}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ ⁶²Ar is destroyed by an SEC and converted into ⁴⁸Si and ⁴⁰Mg (Figure 6). The initial reaction sequence proceeds via ⁶²Ar(EC, 3n)⁵⁹Cl(2n, γ)⁶¹Cl(EC, 11n)⁵⁰S (Figure 6). At ⁵⁰S, four destruction paths carry significant flow: (EC, 3n), (EC, 4n)⁴⁶P(n, γ), (EC, 5n)⁴⁵P(2n, γ), and (4n, γ)⁵⁴S(EC, 7n), all leading to ⁴⁷P, which then undergoes an (EC, 5n) reaction, leading to ⁴²Si (N = 28). At this point the neutron abundance has reached Y_n = 0.23 (Figure 7), and this considerable neutron density results in a significant neutron capture branch that drives reaction flow out of N = 28 into ⁴⁸Si. However, ⁴²Si(EC, n) is not negligible and results in an additional buildup of ⁴⁰Mg via ⁴²Si(EC, n)⁴¹Al(EC, 3n and 4n)^37,38 Mg(2–3n, γ)⁴⁰Mg. As a consequence, ⁴⁰Mg increases significantly in abundance. After the destruction of ⁶²Ar is complete, the layer consists of ⁴⁶Si (44% mass fraction), ⁴⁰Mg (32% mass fraction), and neutrons (23% mass fraction).

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The destruction of ⁶²Ar coincides with the onset of a weak reaction flow through the first pycnonuclear fusion reaction, ⁴⁰Mg+⁴⁰Mg → ⁸⁰Cr (Figure 6). The fusion reaction is immediately followed by a rapid SEC sequence leading back to ⁴⁰Mg and establishing a pycnonuclear fusion-SEC cycle (Figure 6). The net effect of the cycle is a ⁴⁰Mg+⁴⁰ Mg → ⁴⁰Mg+40n reaction resulting in the conversion of ⁴⁰Mg into neutrons with increasing depth. ⁴⁶Si is the only significant bottleneck in the cycle besides ⁴⁰Mg and maintains a significant, roughly constant abundance while reactions produce and destroy the isotope.

At ${\mu }_{e}=37.1\,\mathrm{MeV}$ and $\rho =1.7\times {10}^{12}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$, ⁴⁶Si begins to be depleted significantly. However, because of its location in the fusion-SEC cycle, its abundance is initially not dropping to zero but is merely reduced to about 10⁻⁴. At this depth, ⁴⁰Mg electron capture begins to initiate an SEC sequence toward lighter nuclei (Figure 8). This SEC sequence ends at ²⁵N, where the pycnonuclear fusion reaction ²⁵N+⁴⁰Mg ${\to }^{65}{\rm{K}}$ (not visible in Figure 6 because the flow is too weak) dominates over further electron capture. The resulting ⁶⁵K is immediately destroyed by a reaction sequence that merges into the SEC of the ⁴⁰Mg+⁴⁰Mg main pycnonuclear fusion-SEC cycle. An additional branching occurs at ²⁸O, where EC is comparable to ²⁸O+⁴⁰Mg → ⁶⁸Ca fusion. Again, the resulting ⁶⁸Ca does not accumulate but merges into the main SEC. EC on ⁴⁰Mg therefore effectively leads to a branching of the reaction flow into two pycnonuclear fusion subcycles. The effect of all these cycles is the same ⁴⁰Mg+⁴⁰Mg → ⁴⁰Mg+40n net reaction, converting one ⁴⁰Mg nucleus into neutrons on each full loop. The first significant depletion of ⁴⁰Mg (Figure 2) marks therefore the onset of significant pycnonuclear fusion.

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At a slightly larger depth, at μ_e = 37.2 MeV, ρ = 2.4 × 10¹² g cm⁻³, P = 3.9 × 10³⁰ dynes cm⁻², and y = 2.1 × 10¹⁶ g cm^–2 abundance builds up at the edge of our network, and we stop the calculation. The neutron mass fraction has reached 46%. The calculations indicate, though, that the next step is the conversion of N = 28 ⁴⁰Mg into ⁴⁴Mg as a consequence of the increasing neutron density. Deeper fusion-SEC cycles develop then, starting on ⁴⁴Mg. While ⁴⁴Mg is not magic, it is the preferred nucleus at these higher neutron densities, because of the jump in neutron binding from Na to Mg predicted by the FRDM mass model leading to an extension of the neutron drip line by eight isotopes (see, e.g., Figure 8).

For X-ray bursters that do not exhibit superburst burning, the ashes of the rp-process are the appropriate initial composition for the crust processes. We use here the composition calculated by the X-ray burst model of Schatz et al. (2001), which has ignition conditions that correspond to systems with high accretion rate and low metallicity, resulting in a relatively large amount of hydrogen (mass fraction X = 0.66) at ignition (see Figure 9). While such bursts would be rare in nature, the model serves as a useful tool to explore the consequences of a maximally extended rp-process that reaches the Sn-Sb-Te cycle. We ignore elements lighter than neon, assuming that they are destroyed by residual helium burning and other thermonuclear fusion processes near the surface.

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The initial compositional evolution is characterized by sequences of electron capture reactions along chains of constant mass number (Figure 10). In this shallow region, the original composition as a function of mass number is preserved and simply pushed to more neutron-rich nuclei. In some cases, β⁻ decay is not completely blocked, creating a local nuclear Urca cycle (Schatz et al. 2014) where both EC and β⁻ decay occur between a pair of nuclei. This can lead to significant neutrino cooling at the depth where the cycle forms, especially at high temperatures. Such Urca cycles do not occur for all EC transitions. They require a strong ground-state-to-ground-state transition (or a transition to a very low-lying state with excitation energy E_x ≪ kT) and an effective blocking of the subsequent EC reaction that would otherwise drain the cycle. EC–β-decay pairs therefore occur predominantly in odd-A chains, though there are a few exceptions, for example, in the A = 96 and A = 98 chains. The complete set of relevant Urca pairs can be identified in Figure 11 using the β⁻ flow as an indicator of the Urca cycling strength. Table 2 lists the most important Urca cycling pairs.

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Table 2.  Strongest Urca Pairs for Extreme rp-process Ashes

Urca Pair	ρ (g cm−3)	Relative Flowa
${}^{105}\mathrm{Zr}$–${}^{105}{\rm{Y}}$	$3.15\times {10}^{10}$	1.00
${}^{103}\mathrm{Sr}$–${}^{103}\mathrm{Rb}$	$6.40\times {10}^{10}$	0.42
${}^{103}\mathrm{Zr}$–${}^{103}{\rm{Y}}$	$2.19\times {10}^{10}$	0.31
${}^{105}{\rm{Y}}$–${}^{105}\mathrm{Sr}$	$4.40\times {10}^{10}$	0.14
${}^{105}\mathrm{Nb}$–${}^{105}\mathrm{Zr}$	$1.64\times {10}^{10}$	0.08
${}^{98}\mathrm{Kr}$–${}^{98}\mathrm{Br}$	$1.16\times {10}^{11}$	0.08
${}^{96}\mathrm{Kr}$–${}^{96}\mathrm{Br}$	$8.38\times {10}^{10}$	0.06
${}^{31}\mathrm{Mg}$–${}^{31}\mathrm{Na}$	$8.72\times {10}^{10}$	0.06
${}^{93}\mathrm{Sr}$–${}^{93}\mathrm{Rb}$	$1.10\times {10}^{10}$	0.04
${}^{93}\mathrm{Rb}$–${}^{93}\mathrm{Kr}$	$1.60\times {10}^{10}$	0.03
${}^{99}\mathrm{Sr}$–${}^{99}\mathrm{Rb}$	$3.80\times {10}^{10}$	0.02
${}^{97}\mathrm{Rb}$–${}^{97}\mathrm{Kr}$	$3.58\times {10}^{10}$	0.02
${}^{99}\mathrm{Rb}$–${}^{99}\mathrm{Kr}$	$4.99\times {10}^{10}$	0.02
${}^{97}\mathrm{Kr}$–${}^{97}\mathrm{Br}$	$6.40\times {10}^{10}$	0.01
${}^{29}\mathrm{Mg}$–${}^{29}\mathrm{Na}$	$5.58\times {10}^{10}$	0.01
${}^{55}\mathrm{Ti}$–${}^{55}\mathrm{Sc}$	$3.94\times {10}^{10}$	0.01
${}^{31}\mathrm{Al}$–${}^{31}\mathrm{Mg}$	$3.96\times {10}^{10}$	0.01
${}^{93}{\rm{Y}}$–${}^{93}\mathrm{Sr}$	$2.03\times {10}^{9}$	0.01
${}^{21}{\rm{O}}$–${}^{21}{\rm{N}}$	$1.22\times {10}^{11}$	0.01
${}^{103}\mathrm{Rb}$–${}^{103}\mathrm{Kr}$	$7.74\times {10}^{10}$	0.01

Note.

^aTime-integrated reaction flow relative to the strongest Urca pair.

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At around ${\mu }_{{\rm{e}}}=10.3\,\mathrm{MeV}$ and $\rho =2.36\times {10}^{10}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ the first neutrons are created. These neutrons are immediately recaptured by nuclei in other mass chains. The further evolution is therefore characterized by a combination of EC reactions that drive the composition more neutron-rich and neutron capture reactions that deplete some mass chains and enhance others. Figure 10 shows the reaction sequences up to the point where neutrons first appear. The first neutrons are created by ⁸⁸Rb(EC, n)⁸⁷Kr, which competes with ⁸⁸Rb(EC)⁸⁸Kr. The reason that neutron emission can occur relatively close to stability is that EC on ⁸⁸Rb is predicted to occur through a relatively high-lying excited state in ⁸⁸Kr at E_x = 6.9 MeV. This is a consequence of the proximity of the N = 50 neutron shell closure (⁸⁸Rb has N = 51), resulting in spherically shaped nuclei and EC strength distributions that are concentrated in a few states (Schatz et al. 2014). This highly excited state is sufficiently close to ${S}_{{\rm{n}}}{(}^{88}\mathrm{Kr})=7.0\,\mathrm{MeV}$ to lead to a significant neutron emission branch. The released neutrons are readily recaptured by nuclei in other mass chains, primarily at higher mass numbers, where neutron capture rates tend to be higher. In this case, neutron capture is dominated by ¹⁰⁵Zr, with smaller capture branches on ¹⁰⁵Y, ^130,104,106 Zr, and ¹⁰⁸Mo, which are the most abundant high-A nuclei present at this time (Figure 10). Because of the rapid recapture of the released neutrons, the free neutron abundance remains negligibly small. The chief result of the early neutron release is therefore not the appearance of free neutrons, but changes in the composition as function of mass number (see discussion in Section 4.1.2).

The first fusion reactions are initiated relatively early at $\rho =1.2\times {10}^{11}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ and ${\mu }_{{\rm{e}}}=17.5\,\mathrm{MeV}$(Figure 11). Previously, ²⁰O produced by EC processes from the initially abundant ²⁰Ne has been partially converted by neutron capture into ²¹O. As soon as ²¹O undergoes an EC transition to ²¹N, ²¹N+²¹O and ²¹N+²¹N fusion reactions occur. Slightly deeper at $\rho =1.3\times {10}^{11}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ and μ_e = 18.0 MeV, two EC transitions on the remaining ²⁰O produce ²⁰C, triggering ²⁰C+²⁰O, ²⁰C+²¹N, and ²⁰C+²⁰C fusion reactions. Other fusion reaction combinations, including reactions involving ²²O produced by neutron capture from ²¹O, also occur but are an order of magnitude weaker. At somewhat higher $\rho =1.92\,\times {10}^{11}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ and ${\mu }_{{\rm{e}}}=20.4\,\mathrm{MeV}$, the pycnonuclear fusion of oxygen becomes possible. ²⁴O, produced by electron captures from the initially present ²⁴Mg, with some contribution from ²¹O neutron captures, is destroyed via ²⁴O+²⁴O. With this reaction, all oxygen is destroyed, leaving neon as the lightest element present in the crust.

At $\rho =3.78\times {10}^{11}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ and ${\mu }_{{\rm{e}}}=25.2\,\mathrm{MeV}$ nuclei in most reaction chains have reached the neutron drip line (Figure 11). The free neutron abundance is still low, with Y_n = $1.1\times {10}^{-13}$. This is sufficient, however, to drive the composition into an (n, γ)–(γ, n) equilibrium within each isotopic chain. Note that the neutron Fermi energy ${E}_{{\rm{F}}{\rm{n}}}\ll {kT}$ at this stage. Overall, neutron capture reactions have significantly altered the composition as a function of mass number. In particular, abundances in most odd-A chains have been drastically reduced; the only remaining odd-A nucleus with a significant abundance is ⁸⁹Cu (Section 4.1.2).

Beyond $\rho =3.78\times {10}^{11}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ and ${\mu }_{{\rm{e}}}=25.2\,\mathrm{MeV}$, SEC chains begin to play a role and rapidly convert nuclei along the neutron drip line into lighter species until a particularly strongly bound nucleus with a large EC threshold is reached (Figure 12). The associated release of neutrons leads to a drastic increase of the free neutron abundance, marking the location of neutron drip.

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Neutron captures also drive the abundance in the neon isotopic chain, predominantly originating from initial ²⁸Si in the burst ashes, into ³²Ne and ³⁴Ne. At $\rho =7.7\times {10}^{11}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$, pycnonuclear fusion reactions set in and destroy ³²Ne and ³⁴Ne. The most important reactions are ³⁴Ne+³⁴Ne, ³²Ne+³⁴Ne, ³⁴Ne+²⁴O, and ³⁴Ne+²⁰ C. ²⁴O is produced via ³²Ne(EC, n)³¹Na(EC, 8n)²³O(n, γ)²⁴O. ²⁰C is produced from ²⁴O via ²⁴O(EC, 2n)²²N(n, γ)²³N(EC, 5n)¹⁸C(2n, γ)²⁰C.

All these processes are essentially completed at $\rho \,=1.28\times {10}^{12}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ and ${\mu }_{{\rm{e}}}=33.6\,\mathrm{MeV}$, at which point the composition is concentrated in a few nuclei at N = 82 (¹¹⁶Se, abundance Y = $4.4\times {10}^{-4}$), at N = 50 (⁷⁰Ca Y = $1.0\,\times {10}^{-2}$, near N = 28; ⁴⁶Si, Y = $2.6\times {10}^{-4}$), and at N = 28 (⁴⁰Mg, Y = $8.4\times {10}^{-5}$). The neutron abundance has reached Y_n = 0.21 (Figure 7), and E_{F n} has reached 0.62 MeV, exceeding kT ≈ 40 keV, resulting in degenerate neutrons. The abundance accumulated at the three locations where the neutron drip line intersects the neutron numbers N = 28, 50, and 82 can be mapped to different mass ranges in the initial composition. N = 82 is mostly produced from initial A ≥ 106 nuclei, with some contribution from A = 102–105. The initial A ≥ 106 abundance is $7.5\times {10}^{-4}$, already larger than the final N = 82 abundance. The main branch points that govern leakage to lighter nuclei for A ≥ 106 material are ¹⁰⁶Kr and ¹⁰⁴Se. At ¹⁰⁶Kr, neutron capture moves material toward N = 82, while EC feeds ¹⁰⁴Se via an (EC)(γ, n)(EC, n) sequence. At ¹⁰⁴Se, (EC, n) moves material ultimately to N = 50, while neutron capture feeds N = 82. The final N = 82 abundance may be increased by a small contribution from A = 102–105. The key branchings are ¹⁰²Se and again ¹⁰⁴Se, where in each case the EC branch moves the abundance toward N = 50.

A ≤ 56 nuclei are mostly converted into nuclei near the N = 28 region. A = 56 is the borderline case, and the reaction sequence is similar to the pure ⁵⁶Fe case discussed in Section 3.1. An isolated exception is the initial ²⁸Si abundance, which in part ends up in the N = 50 region owing to fusion reactions: ²⁸Si is converted into neon isotopes via ECs. At ²⁸Ne neutron capture competes with ²⁸Ne(EC, n). The ²⁸Ne(EC, n) branch leads to fluorine and then oxygen, which then fuses into nuclei in the sulfur region. However, a significant fraction of the initial ²⁸Si abundance is processed through the ²⁸Ne(n, γ) branch, leading to neutron-rich neon isotopes, which then fuse into calcium and ultimately end up in ⁷⁰Ca (N = 50). Initial elements lighter than silicon fuse into nuclei below calcium and are therefore converted into N ≈ 28 nuclei. Initial elements heavier than silicon but lighter than iron end up as magnesium or silicon isotopes, which do not fuse until much greater depths, when nuclei are sufficiently neutron-rich for SECs to prevent accumulation at N = 50.

We continue the simulation beyond $\rho =1.25\times {10}^{12}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ and ${\mu }_{{\rm{e}}}=33.6\,\mathrm{MeV}$ to $\rho =2.5\times {10}^{12}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ and ${\mu }_{{\rm{e}}}\,=37\,\mathrm{MeV}$, at which point the increasing neutron density drives the composition toward the edge of our reaction network (Figure 13). At $\rho =2\times {10}^{12}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ and ${\mu }_{{\rm{e}}}=35.1\,\mathrm{MeV}$N = 50 ⁷⁰Ca is destroyed by an SEC and converted into nuclei in the N = 28 region, leaving only the N = 28 and N = 82 regions with significant abundance. We also see the onset of significant ⁴⁰Mg+⁴⁰Mg fusion, resulting in a similar fusion-SEC cycle to that discussed in Section 3.1.

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Figure 14 shows the calculated time-integrated nuclear energy production. The various drops indicate significant cooling from nuclear Urca pairs in the outer crust. Indeed, the location of the top three pairs listed in Table 2 coincides with the major drops visible in Figure 14 around $\rho =2.2\times {10}^{10}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$, $\rho =3.2\times {10}^{10}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$, and $\rho =6.4\times {10}^{10}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$.

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A more typical estimate of the final composition of mixed H/He bursts is provided by calculations using the 1D multizone code KEPLER. We use the model described in more detail in Cyburt et al. (2016), which was shown to reproduce the observed light-curve features of GS 1826-24 reasonably well (Heger et al. 2007). The final composition entering the crust is calculated by averaging over the deeper layers in the accreted material after a sequence of about 14 bursts, excluding the bottom layers that are produced by the atypical first burst (see Cyburt et al. 2016, for details). Figure 15 shows the initial composition as a function of mass number. The main difference from the extreme rp-process discussed in Section 3.2 is the reduced amount of heavier nuclei beyond A = 72. This is due to increased CNO hydrogen burning in between bursts, which leads to a lower hydrogen abundance at ignition, and a lower ignition depth, which leads to lower peak temperature and a less extended αp-process. Both of these effects result in a lower hydrogen-to-seed ratio and a shorter rp-process (see Equation (13) in Schatz et al. 1999).

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The evolution of the composition with increasing depth is overall very similar to the extreme rp-process ashes case, though the different mass chains have different relative abundances owing to the different initial composition (Figure 16). The main global difference is a shift of the appearance of significant amounts of free neutrons toward higher densities by about 50%, which is more in line with the calculation for pure ⁵⁶Fe ashes (Figure 7). This is due to the much reduced abundance of heavy (A > 72) nuclei that tend to reach the neutron drip line at shallower depth and lead to an early release of neutrons in the case of the extreme rp-process ashes. The deeper onset of neutron drip has some consequences for the further evolution of the composition. In particular, when the heavier mass chains reach the neutron drip line, the neutron abundance is lower, and the equilibrium nucleus is therefore closer to stability where electron capture thresholds are lower. SEC cascades therefore set in earlier, compared to the case of extreme rp-process ashes, where nuclei are pushed to more neutron-rich isotopes that require higher μ_e to capture electrons (Figure 17). This, together with the much lower initial abundance of A > 103 nuclei, leads to a negligible production of N = 82 nuclei. The final composition after SECs have converted the composition into nuclei at or near closed shells or with large single-particle energy level gaps is ⁷⁰Ca (N = 50, Y = $8.5\times {10}^{-3}$), ⁴⁶Si (N = 32, Y = $1.24\times {10}^{-3}$), and ⁴⁰Mg (N = 28, Y = $6.4\times {10}^{-3}$) with a neutron abundance of Y_n = 0.092.

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Again we can map these final abundances to the distribution by mass number of the initial composition. As in the case of the extreme rp-process ashes, A ≈ 56 is the approximate dividing line between material ending up in N = 50 and near N = 28. However, because of the lower free neutron abundance, nuclei tend to be less neutron-rich and have lower EC thresholds. Therefore, in the case of the KEPLER X-ray burst ashes, there is some leakage from the A = 58–60 mass chains toward the N = 28 region, primarily due to branch points where neutron capture competes with EC such as ⁵⁹Ca(EC),⁶⁰Ca(EC, n), and ⁶²Ca(EC, 3n). While the abundance that remains in the Ca isotopic chain is ultimately converted into N = 50 ⁷⁰Ca, any leakage to lighter elements feeds the N = 28 region. As in the case of the extreme rp-process ashes, the initial ²⁸Si abundance is fed into the Ca isotopic chain and converted to ⁷⁰Ca. Indeed, the sum of the initial abundances of ²⁸Si and A > 60 is $9.5\times {10}^{-3}$ mol g⁻¹, which exceeds the produced N = 50 abundance slightly, indicating that most of the other nuclei end up near N = 28.

At depths beyond $\rho =1.10\times {10}^{12}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ ($y=1.27\,\times {10}^{16}\,{\rm{g}}\,{\mathrm{cm}}^{-2}$) the evolution is essentially the same as in the case of the extreme rp-process ashes (see Section 3.2). The nuclear energy release is overall similar (Figure 14), though there is somewhat stronger nuclear Urca cooling and a slightly higher nuclear energy generation. The dominant Urca pairs are listed in Table 3. Compared to the extreme rp-process ashes, there are two additional important Urca pairs in lighter mass chains, ³³Al–³³Mg and ⁶⁵Fe–⁶⁵Mn. The lack of shallower cooling from heavier nuclei is more than offset by the very strong cooling from ³¹Mg–³¹Na at around $\rho =1\times {10}^{11}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ (Figure 14). This is due to the much larger initial abundance of A = 31 nuclei ($9.1\times {10}^{-5}$ versus $4.9\times {10}^{-6}$).

Table 3.  Strongest Urca Pairs for KEPLER X-ray Burst Ashes

Urca Pair	ρ (g cm−3)	Relative Flowa
${}^{31}\mathrm{Mg}$–${}^{31}\mathrm{Na}$	$8.60\times {10}^{10}$	1.00
${}^{33}\mathrm{Al}$–${}^{33}\mathrm{Mg}$	$5.62\times {10}^{10}$	0.47
${}^{55}\mathrm{Ti}$–${}^{55}\mathrm{Sc}$	$3.72\times {10}^{10}$	0.40
${}^{31}\mathrm{Al}$–${}^{31}\mathrm{Mg}$	$3.75\times {10}^{10}$	0.14
${}^{65}\mathrm{Fe}$–${}^{65}\mathrm{Mn}$	$2.59\times {10}^{10}$	0.11
${}^{59}\mathrm{Cr}$–${}^{59}{\rm{V}}$	$2.42\times {10}^{10}$	0.09
${}^{29}\mathrm{Mg}$–${}^{29}\mathrm{Na}$	$5.32\times {10}^{10}$	0.07
${}^{57}\mathrm{Cr}$–${}^{57}{\rm{V}}$	$1.36\times {10}^{10}$	0.05
${}^{57}{\rm{V}}$–${}^{57}\mathrm{Ti}$	$2.62\times {10}^{10}$	0.04
${}^{63}\mathrm{Fe}$–${}^{63}\mathrm{Mn}$	$1.61\times {10}^{10}$	0.04
${}^{59}\mathrm{Mn}$–${}^{59}\mathrm{Cr}$	$1.08\times {10}^{10}$	0.03
${}^{63}\mathrm{Cr}$–${}^{63}{\rm{V}}$	$4.56\times {10}^{10}$	0.03
${}^{65}\mathrm{Ni}$–${}^{65}\mathrm{Co}$	$5.27\times {10}^{9}$	0.03
${}^{65}\mathrm{Co}$–${}^{65}\mathrm{Fe}$	$1.21\times {10}^{10}$	0.02
${}^{59}{\rm{V}}$–${}^{59}\mathrm{Ti}$	$3.39\times {10}^{10}$	0.02
${}^{61}\mathrm{Fe}$–${}^{61}\mathrm{Mn}$	$9.09\times {10}^{9}$	0.02
${}^{59}\mathrm{Fe}$–${}^{59}\mathrm{Mn}$	$3.48\times {10}^{9}$	0.01
${}^{57}\mathrm{Ti}$–${}^{57}\mathrm{Sc}$	$5.78\times {10}^{10}$	0.01
${}^{55}{\rm{V}}$–${}^{55}\mathrm{Ti}$	$9.90\times {10}^{9}$	0.01

Note.

^aTime-integrated reaction flow relative to strongest Urca pair.

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In some X-ray bursting systems, rare superbursts may further modify the composition at a depth around $\rho =1\times {10}^{9}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ (Cumming et al. 2006). For systems that regularly exhibit superbursts, the ashes of superburst burning are the appropriate initial composition for nuclear processes in the crust. We use the final composition produced in a superburst model calculated with the KEPLER code (Keek & Heger 2011). This composition is shown in Figure 18. The sequence of EC and neutron captures leading to the neutron drip line is shown in Figure 19 and, for the mass chains with significant abundance, is very similar to the result with the KEPLER X-ray burst ashes. Figure 20 shows the reaction sequences and final composition when the composition is consolidated to a few nuclei that are particularly strongly bound owing to shell effects. One difference from the calculation with the KEPLER X-ray burst ashes is that this point is reached at a slightly higher density ($\rho =1.23\times {10}^{12}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ instead of $\rho =1.1\times {10}^{12}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$). The reason is that Y_e is smaller owing to the higher neutron abundance, and therefore a higher density is required to achieve ${\mu }_{{\rm{e}}}=33.7\,\mathrm{MeV}$, needed to destroy ⁶²Ar.

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The final composition shown in Figure 20 has been determined at the same μ_e as Figure 17 and demonstrates that indeed the same nuclei are populated. Because this composition is reached at a greater depth, it is much closer to the onset of pycnonuclear fusion of ⁴⁰Mg. However, the relative population of ⁴⁰Mg (Y = $9.1\times {10}^{-3}$), ⁴⁶Si (Y = $8.7\times {10}^{-3}$), and ⁷⁰Ca (Y = $3.2\times {10}^{-4}$) is different because of the different initial composition. Most of the abundance is concentrated around N = 28 with only a smaller N = 50 contribution from ⁷⁰Ca. Because the initial composition has only a small amount of A > 60 nuclei (Y = $8\times {10}^{-8}$ mol g⁻¹), the only contribution to N = 50 comes from parts of the initial A = 60 and A = 28 abundances (see discussion in Section 3.2). Again the competition of neutron capture and EC at ⁶⁰Ca and ²⁸Ne, respectively, is critical in determining the relative distribution of N = 28 and N = 50 nuclei in the inner crust.

In crusts with initial superburst ashes, Urca cooling is comparable to the case of extreme rp-process ashes (Figure 14); however, cooling comes almost exclusively from the ⁵⁵Sc–⁵⁵Ti pair. All other Urca cooling pairs are at least a factor of 50 weaker. Heating is significantly higher than for either of the X-ray burst ash cases.

The reaction sequences obtained here with a full reaction network differ substantially from previous work, especially when compared to models that consider only a single species at a given time such as Haensel & Zdunik (1990, 2008). One important difference are the EC/β Urca pairs already discussed in Schatz et al. (2014), which can occur when both the parent and the daughter nucleus of an EC are present and the β⁻ decay is not fully blocked.

In the case of the initial ⁵⁶Fe ashes, we agree with Haensel & Zdunik (1990) that the composition reaches the neutron drip line with the destruction of ⁵⁶Ar at around $\rho =7.8\,\times {10}^{11}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$. However, taking into account the finite time needed for the transition and the change in neutron density during the transition, we find that the reaction flow branches into an EC sequence and a neutron capture sequence unlike Haensel & Zdunik (1990). This leads to the appearance of more than one species in the composition. Also, unlike Haensel & Zdunik (1990), we confirm that beyond neutron drip, nuclei are converted rapidly via the SECs found in Gupta et al. (2008) into much lighter nuclei. For example, ⁵⁶Ar is converted into ⁴⁰Mg in a single step so that ⁴⁰Mg is already produced at the time of ⁵⁶Ar destruction at $\rho =7.8\times {10}^{11}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ and μ_e = 31.6 MeV. This is in contrast to Haensel & Zdunik (1990), where, after ⁵⁶Ar destruction at $\rho =6.1\times {10}^{11}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$, several EC reactions at stepwise increasing μ_e have to occur before ⁴⁰Mg is produced at $\rho =1.1\times {10}^{12}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$. In addition, we find that the reaction sequence branches at ⁴²Si, leading to the additional production of ⁴⁸Si via neutron captures, resulting in a two-component composition of ⁴⁰Mg and ⁴⁸Si.

The onset of pycnonuclear fusion also differs. In Haensel & Zdunik (1990), the first fusion is ³⁴Ne+³⁴Ne, triggered by EC on ⁴⁰Mg at $\rho =1.46\times {10}^{12}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ and μ_e = 34.3 MeV. We find that, due to our mass model that includes shell effects, the threshold for ⁴⁰Mg(EC) is higher so that ⁴⁰Mg(EC) occurs deeper at $\rho =1.8\times {10}^{12}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$. As our calculation allows for the presence of multiple nuclear species, we find that the lighter nuclides produced by the ensuing SEC chains preferably fuse with the still abundant ⁴⁰Mg, rather than with themselves, leading to the occurrence of fusion between unlike nuclides. Furthermore, the SEC chains on ⁴⁰Mg are faster, leading to lighter nuclides before fusion sets in. A branching at ²⁸O, where EC and fusion compete, leads to the creation of two major species undergoing fusion, resulting in two major fusion reactions, ²⁵N + ⁴⁰Mg$\,{\to }^{65}{\rm{K}}$ and ²⁸O+⁴⁰Mg → ⁶⁸Ca. In addition, at the larger depth where these reactions are triggered, ⁴⁰Mg+⁴⁰Mg → ⁸⁰Cr becomes significant as well. In summary, instead of a single fusion reaction between like species, three fusion reactions occur simultaneously, two of them between very different species.

A fundamental difference here is that we do not find a large abundance buildup of the fusion reaction products as found in Haensel & Zdunik (1990, 2008). Instead, the reaction product is immediately recycled via an SEC. Fusion reactions therefore lead to fusion-SEC cycles. In the case of ⁴⁰Mg, for example, the resulting net reaction of all fusion-SEC cycles is ⁴⁰Mg+⁴⁰Mg → ⁴⁰Mg+40n. The fusion-SEC cycles slowly convert ⁴⁰Mg into neutrons, until the increasing neutron density shifts the composition to more neutron-rich nuclei. A single fusion reaction effectively only destroys a single ⁴⁰Mg nucleus instead of two, allowing for more fusion reactions at a shallower depth. A fusion-induced cycle for the destruction of ⁴⁰Mg has been described in Steiner (2012), who used a quasi-statistical equilibrium model that allows for the presence of multiple species. He found a cycle that starts at ⁴⁰Mg with an SEC to ²²C, followed by ²²C+²²C → ⁴⁴Mg(γ, 4n)⁴⁰Mg. However, taking into account the finite speed of the nuclear reactions, we find that ⁴⁰Mg and the lighter nuclides produced by an SEC coexist, leading to asymmetric fusion reactions. Our model also tracks individual reaction channels and can therefore resolve branchings between competing reactions. This also broadens the range of fusion reactions.

The reaction sequences starting with broader initial composition distributions are of similar type, characterized by four phases: EC chains without neutrons, EC chains with neutron-induced reactions, SECs at neutron drip, and pycnonuclear fusion. As soon as neutrons are released, the evolution in a given isobaric EC chain starts to depend on what happens in other chains. Initial abundances of lighter species such as ²⁰Ne, ²⁴Mg, or ²⁸Si are transformed into even lighter nuclei, which then undergo pycnonuclear fusion prior to neutron drip. This has already been suggested by Horowitz et al. (2008). We confirm that these reactions occur at a depth around $\rho =1\times {10}^{11}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ , but the types of fusion reactions differ significantly from previous predictions. The most dominant fusion reactions around this depth are ²¹N+²¹O, ²¹N+²¹N, ²⁰C+²⁰O, ²⁰C+²¹N, ²⁰C+²⁰C, and ²⁴O+²⁴O, but many weaker reactions occur. Ne fusion sets in at slightly higher $\rho \,=4\times {10}^{11}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ via ³⁴Ne+³⁴Ne, ³²Ne+³⁴ Ne, ³⁴Ne+²⁴O, and ³⁴Ne+²⁰C. At greater depths we find the pycnonuclear fusion-SEC cycles:

$$\begin{eqnarray*}&&{}^{40}\mathrm{Mg}+{}^{40}\mathrm{Mg}\to \,{}^{80}\mathrm{Cr}\to {}^{40}\mathrm{Mg}+40{\rm{n}}\\ &&{}^{25}{\rm{N}}+{}^{40}\mathrm{Mg}\to \,{}^{65}{\rm{K}}\to {}^{40}\mathrm{Mg}+25{\rm{n}}\\ &&{}^{28}{\rm{O}}+{}^{40}\mathrm{Mg}\to \,{}^{68}\mathrm{Ca}\to {}^{40}\mathrm{Mg}+28{\rm{n}}.\end{eqnarray*}$$

Here the fusion reaction rates determine how rapidly nuclei are converted into free neutrons.

4.1.1. Urca Cooling

4.1.2. Appearance of Free Neutrons

We find that free neutrons start to play a role long before the composition reaches the neutron drip line, the traditional point where free neutrons appear. This early release of neutrons stems from EC reactions that populate neutron-unbound excited states (E_x > S_n), which then decay by neutron emission. There are two basic mechanisms for EC to populate high-lying excited states. First, the EC threshold of a particular reaction may be increased by the excitation energy of the lowest-lying daughter state for an allowed transition. If this state is above the neutron separation energy, neutron emission will occur. Also, once the transition proceeds at threshold, μ_e can be higher than the threshold of the subsequent EC reaction, leading again to the population of excited daughter states that may be above the neutron separation energy. These effects can occur in even- and odd-A chains. An example is ⁸⁸Rb(EC, n)⁸⁷Kr discussed in Section 3.2, where the lowest-lying EC transition is predicted to go to a 6.9 MeV state in ⁸⁸Kr, close to ${S}_{{\rm{n}}}{(}^{88}\mathrm{Kr})=7.02\,\mathrm{MeV}$. This leads to neutron emission relatively close to stability. It will be important to explore how lower-lying forbidden transitions not included in the QRPA-fY calculations may reduce this effect.

The second mechanism to release neutrons prior to reaching the neutron drip line is the odd–even staggering of Q_EC in even-A chains, ${\rm{\Delta }}{Q}_{\mathrm{EC}}=| {Q}_{\mathrm{EC},\mathrm{even}-\mathrm{even}}-{Q}_{\mathrm{EC},\mathrm{odd}-\mathrm{odd}}|$. In these chains, an EC reaction on an even–even nucleus is immediately followed by an EC reaction on an odd–odd nucleus (Haensel & Zdunik 1990), where excited states up to ΔQ_EC can be populated (Gupta et al. 2007). Neutron emission is possible if ΔQ_EC > S_n. ΔQ_EC depends strongly on the mass model (Meisel et al. 2015b). For typical values of 3 MeV neutron release would only start closer to the drip line at S_n ≈ 3 MeV. However, the FRDM mass model predicts significantly larger ΔQ_EC in some cases.

The early release of neutrons does not lead to a buildup of a large free neutron abundance. Instead, the released neutrons are recaptured by other nuclei present at the same depth. This is a feature of the multicomponent composition of the outer crust. Nuclei with the largest abundance and largest neutron capture cross sections will dominate the neutron absorption. Interestingly, this tends to lead to the depletion of odd-A chains, starting as early as at $\rho =4\times {10}^{10}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ (see Figure 21). As the odd-A mass chains tend to have most of the nuclear Urca pairs, the early release of neutrons strongly limits Urca cooling in the deeper regions of the outer crust.

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The μ_e required for pre-drip line neutron release varies greatly from mass chain to mass chain. To illustrate this point, we provide a simple estimate for the minimum μ_e for neutron release in each mass chain, based on nuclear mass differences (${\mu }_{{\rm{e}}}\gt | {Q}_{\mathrm{EC}}| +{S}_{{\rm{n}}}$; Figure 22). EC transitions are assumed to proceed when ${\mu }_{{\rm{e}}}\gt | {Q}_{\mathrm{EC}}| +{E}_{{\rm{x}}0}$, with E_x0 being the daughter excitation energy of the lowest-lying EC transition. This simple estimate neglects lattice energy and finite temperature corrections, which depend on overall composition and astrophysical parameters and are included in the full network calculation. Clearly, the depth of early neutron release depends strongly on the mass chain and therefore on the initial composition created by thermonuclear burning on the neutron star surface. While transitions into excited states move the release of neutrons to shallower depths, on average by 6 MeV in μ_e (red solid line in Figure 22), the odd–even staggering of Q_EC alone leads to significant neutron release (red dashed line in Figure 22) prior to reaching the neutron drip line (blue line in Figure 22). All curves in Figure 22 show a pronounced variation in μ_e from mass chain to mass chain of up to about 10 MeV. Therefore, regardless of the detailed transition energies and odd–even staggering, there will be a transition region between the outer and inner crust where some mass chains release neutrons and others capture them. The characteristics of the compositional evolution in this region will depend sensitively on the composition of the X-ray burst ashes.

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4.1.3. Superthreshold Electron Capture Cascades

4.1.4. Shell Effects and the Composition beyond Neutron Drip

Heating and cooling by nuclear reactions in the crust link the nuclear processes identified in this work with observables. Figure 14 shows that the considerably different nuclear processes obtained with a full reaction network and a realistic initial multicomponent composition lead to differences in the heating and cooling of the neutron star crust compared to simplified single-component equilibrium calculations (Haensel & Zdunik 2008). In particular, for all types of realistic burst ashes, Urca cooling is significant at the 0.5 GK temperature investigated here and would likely lead to a cooler crust in a self-consistent model. As expected, the location and strength of Urca cooling depend sensitively on the initial composition.

There are also significant differences in heating between the models. This is shown in Figure 23, where we only integrate over segments of positive slope in Figure 14. This provides a lower limit of the heating, as we neglect any heating during a cooling episode. On one hand, all our calculations with a full reaction network show significantly more heating at shallower depths than the Haensel & Zdunik (2008, hereafter HZ08) estimate. At around $\rho =1.3\times {10}^{12}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$, the integrated difference has accumulated to about $0.5\,\mathrm{MeV}\,{{\rm{u}}}^{-1}$ (though this is a lower limit). This is in part due to our inclusion of transitions into excited states in the first step of the two-step electron capture sequences in even-A chains. These transitions not only reduce neutrino emission (as considered in HZ08) but also increase the electron capture energy thresholds and thus the total energy release in the sequence (Gupta et al. 2007). On the other hand, the calculations with realistic burst ashes and a full reaction network show remarkable similarity, despite the significant variations in initial compositions. Differences in the thermal structure for different realistic burst ashes will therefore predominantly arise from differences in Urca cooling, not from differences in heating.

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The much shallower onset of fusion reactions in the models with realistic ashes, around $\rho =(1.2-7.7)\times {10}^{11}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$, compared to $\rho =1.1\times {10}^{12}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ for pure ⁵⁶Fe ashes, contributes to an increased heat deposition at shallower depths. This is due to two effects. First, lighter nuclei in the initial composition tend to be converted more rapidly into the low-Z species that can undergo fusion reactions. Second, the SEC effect creates lighter nuclei earlier. Horowitz et al. (2008) pointed out the potential importance of fusion of lighter nuclei at shallower depths. Indeed, such reactions can deposit of the order of  = 0.7–0.9 $\mathrm{MeV}\,{{\rm{u}}}^{-1}$ of heat (Horowitz et al. 2008), provided that they would make up 100% of the composition. However, the mass fraction X of A ≤ 28 nuclei in the initial composition is only 0.7%, 5%, and 1% for superburst, KEPLER, and extreme burst ashes, respectively. The associated heating X is therefore rather small, 0.005–0.05 $\mathrm{MeV}\,{{\rm{u}}}^{-1}$, and comparable to electron capture heating in the more abundant mass chains.

Despite these differences in the distribution of heat deposition, the total heat deposited is remarkably similar for all our models, at least down to a depth where $\rho =1.6\times {10}^{12}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$. At that depth, total heat deposition is $1.1\,\mathrm{MeV}\,{{\rm{u}}}^{-1}$, $0.96\,\mathrm{MeV}\,{{\rm{u}}}^{-1}$, $0.88\,\mathrm{MeV}\,{{\rm{u}}}^{-1}$, $1.2\,\mathrm{MeV}\,{{\rm{u}}}^{-1}$, and $0.9\,\mathrm{MeV}\,{{\rm{u}}}^{-1}$ for HZ08, pure Fe ashes, extreme burst ashes, KEPLER burst ashes, and superburst ashes, respectively. Note however, that the latter three cases are lower limits, as some heat may be released in regions with net cooling. Our results confirm with a full reaction network the robustness of heating (but not Urca cooling) with to initial composition found in previous work using simplified approaches (Haensel & Zdunik 2008) or reaction networks without pycnonuclear fusion (Gupta et al. 2008).

We are now in the position to predict the impurity parameter ${Q}_{\mathrm{imp}}={\sum }_{i}{Y}_{i}{({Z}_{i}-\langle Z\rangle )}^{2}/{\sum }_{i}{Y}_{i}$ with average charge number $\langle Z\rangle$ and abundances Y_i (excluding neutrons) as a function of depth. Q_imp is important as it determines the thermal conductivity of the crust due to electron impurity scattering. Figure 24 shows impurity parameters for the various initial compositions as a function of density. The extreme X-ray burst ashes exhibit the broadest range of isotopes and has the largest Q_imp ≈ 80. The rp-process in the more typical KEPLER burst produces much fewer Z = 30–46 nuclei, resulting in a lower initial Q_imp ≈ 40. Superbursts drive the composition into nuclear statistical equilibrium, resulting in much less diverse ashes with a much smaller initial Q_imp ≈ 4. Down to a depth where $\rho =1\times {10}^{10}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$, Q_imp stays rather constant. At greater depth it begins to decrease substantially because heavier nuclei tend to electron capture more, reducing their Z faster, and because the early release of neutrons starts to eliminate abundance in some mass chains. At $\rho =1\times {10}^{11}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$, the extreme burst ashes case shows a drastic reduction in Q_imp, bringing it in line with the KEPLER ashes. This is due to the pycnonuclear fusion of oxygen produced via electron capture from the relatively large initial ²⁰Ne abundance. In addition, compared to the KEPLER ashes, the extreme burst ashes have relatively smaller initial abundances of ²⁴Mg and ²⁸Si, causing a much larger impact on Q_imp once lighter nuclei from the abundance ²⁰Ne start fusing. Between $\rho =2\times {10}^{11}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ and $\rho =7\times {10}^{11}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$, light-element fusion and SEC chains lead to a steady reduction in Q_imp for all cases.

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Interestingly, all initial compositions converge to a comparable Q_imp = 7–11 between $\rho =8\times {10}^{11}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ and $\rho =1.3\times {10}^{12}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ owing to shell effects that lock abundance in different locations. This includes even the pure ⁵⁶Fe ashes case, which turns into a multicomponent composition beyond neutron drip owing to the splitting of the reaction path discussed in Section 3.1. However, beyond $\rho =1.5\,\times {10}^{12}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ material trapped at the N = 50 spherical shell closure is destroyed, all compositions but the extreme burst ashes converge to a single nucleus, and Q_imp drops to less than 1. This is in line with previous predictions (Jones 2005; Gupta et al. 2008; Steiner 2012) that Q_imp is reduced to a value near 1 when transitioning from the outer to the inner crust, though we find that the transition is gradual and exhibits some variations. The exception is the extreme burst ashes case, the only case where material is also locked in at the N = 82 spherical shell closure owing to the heavy A ≥ 106 nuclei contained in the ashes. In this case, the conversion of N = 50 nuclei into lighter species, together with the unchanged heavy N = 82 nuclei, leads to the opposite behavior, an increase of Q_imp in the inner crust to values of around 20.

Our theoretical predictions of Q_imp can be compared with constraints extracted from observed cooling curves of transiently accreting neutron stars using crust cooling models. For KS 1731-260, the most recent analysis by Merritt et al. (2016) obtains ${Q}_{\mathrm{imp}}={4.4}_{-0.5}^{+2.2}$, in agreement with earlier results from Brown & Cumming (2009) (<4). For MXB 1659-29, Turlione et al. (2015) find Q_imp = 3.3–4, in agreement with earlier results from Brown & Cumming (2009). These results are also in line with work by Page & Reddy (2013), who use models with different Q_imp values for the outer and the inner crust and find Q_imp = 5 and 3 for KS 1731-260, Q_imp = 10 and 3 for MXB 1659-29, and Q_imp = 20 and 4 for XTE J1701-462 for the outer and inner crust, respectively. A significantly higher Q_imp = 40 has been found in an analysis of EXO 0748-676 (Degenaar et al. 2014). Turlione et al. (2015) obtain a good fit of the data from this source with Q_imp ≈ 1 but do not include the most recent data points considered in Degenaar et al. (2014).

Roggero & Reddy (2016) performed path integral Monte Carlo calculations of the electron–ion scattering with impurities at $\rho =1\times {10}^{10}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ and found that heat conductivity is reduced by about a factor of 2–4 compared to the simple approximation employed in current crust cooling models. If this result is indeed broadly applicable at higher densities, it would imply that the required impurity parameters to fit observations are reduced by about a factor of 2–4, and that thus a Q_imp ≈ 1–2 is needed to explain observations of KS 1731-260, MXB 1659-29, and XTE J1701-462.

Most crust cooling models used to analyze observational data employ a single Q_imp for the entire crust. These Q_imp values can be compared to our predictions at ρ > 10¹² g cm⁻³, where electron impurity scattering is expected to dominate heat transport (Brown & Cumming 2009). Only Page & Reddy (2013) provide values for Q_imp in the outer crust. However, a comparison is difficult, as we predict significant changes in Q_imp as a function of density and it is unclear how constraining the observational data are for the best-fit Q_imp values given.

The observational constraints of a small inner crust Q_imp in most sources overall agree with our predictions for KEPLER burst and superburst ashes, but they clearly disagree with our prediction for extreme rp-process ashes. Our finding of a high Q_imp = 20 for an initial composition from extreme rp-process ashes could in principle explain the high Q_imp = 40 inferred for EXO 0748-676 (Degenaar et al. 2014). An extreme rp-process would require particularly long bursts with burst durations of the order of 200 s. Indeed, EXO 0748-676 does exhibit such bursts (Boirin et al. 2007) during its outburst phase.

It has been suggested that chemical separation effects during the freezing of the crust at the ocean crust boundary, which are not included in our model, lead to a reduction in Q_imp in the outer crust (Horowitz et al. 2007, 2009; Medin & Cumming 2011, 2014; Mckinven et al. 2016). This is due to the different freeze-out properties of light and heavy nuclei. However, in the absence of nuclear reactions, once steady state is achieved, the solid crust composition must match on average the composition of the ashes entering the crust. Either chemical separation effects lead to time-dependent compositional changes in the ocean that counteract the chemical separation once steady state is achieved, or alternating layers of light and heavy nuclei form. In the latter case, Q_imp would indeed be significantly reduced, as each layer would have a higher purity than the average crust. However, if such layers form, what their thickness is and what the impact of layer boundaries are remain open questions.