# Onset of Electron Captures and Shallow Heating in Magnetars

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Equation of State of Magnetar Crusts

#### 2.1. Main Equations

#### 2.2. Weakly Quantizing Magnetic Field

#### 2.3. Strongly Quantizing Magnetic Field

## 3. Initial Composition of Magnetar Crusts

#### 3.1. Interface between Adjacent Crustal Layers

#### 3.2. No Magnetic Field

#### 3.3. Strongly Quantizing Magnetic Field

- ${\gamma}_{e}^{1\to 2}>0$ and $\overline{F}({Z}_{1},{A}_{1};{Z}_{2},{A}_{2};{B}_{\u2605})>0$$${\gamma}_{e}=8\overline{F}{({Z}_{1},{A}_{1};{Z}_{2},{A}_{2};{B}_{\u2605})}^{3/2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{sinh}}^{3}\left(\frac{1}{3}\mathrm{arcsinh}\phantom{\rule{3.33333pt}{0ex}}\upsilon \right)\phantom{\rule{0.166667em}{0ex}},$$
- ${\gamma}_{e}^{1\to 2}>0$ and $\overline{F}({Z}_{1},{A}_{1};{Z}_{2},{A}_{2};{B}_{\u2605})<0$$${\gamma}_{e}=\left\{\begin{array}{cc}8|\overline{F}({Z}_{1},{A}_{1};{Z}_{2},{A}_{2};{B}_{\u2605}){|}^{3/2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{cosh}}^{3}\left(\frac{1}{3}\mathrm{arccosh}\phantom{\rule{3.33333pt}{0ex}}\upsilon \right)\hfill & \mathrm{if}\phantom{\rule{4pt}{0ex}}\upsilon \ge 1\phantom{\rule{0.166667em}{0ex}},\hfill \\ 8|\overline{F}({Z}_{1},{A}_{1};{Z}_{2},{A}_{2};{B}_{\u2605}){|}^{3/2}\phantom{\rule{0.166667em}{0ex}}{cos}^{3}\left(\frac{1}{3}arccos\upsilon \right)\hfill & \mathrm{if}\phantom{\rule{4pt}{0ex}}0\le \upsilon <1\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}\right.$$
- ${\gamma}_{e}^{1\to 2}<0$ and $\overline{F}({Z}_{1},{A}_{1};{Z}_{2},{A}_{2};{B}_{\u2605})<0$$${\gamma}_{e}=8{\left|\overline{F}({Z}_{1},{A}_{1};{Z}_{2},{A}_{2};{B}_{\u2605})\right|}^{3/2}\phantom{\rule{0.166667em}{0ex}}{cos}^{3}{\theta}_{k}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{if}\phantom{\rule{4pt}{0ex}}-1<\upsilon \le 0\phantom{\rule{0.166667em}{0ex}},$$$${\theta}_{k}\equiv \frac{1}{3}arccos\upsilon +\frac{2\pi k}{3}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{and}\phantom{\rule{4pt}{0ex}}k=0,2\phantom{\rule{0.166667em}{0ex}}.$$

#### 3.4. Intermediate Magnetic Fields

## 4. Magnetic Field Decay and Electron Captures

#### 4.1. Onset of Electron Captures

#### 4.2. No Magnetic Field

#### 4.3. Intermediate Magnetic Field

#### 4.4. Strongly Quantizing Magnetic Field

#### 4.5. Heat Released

#### 4.6. Neutron Delayed Emission

## 5. Results and Discussions

#### 5.1. Initial Composition of the Outer Crust

#### 5.2. Heating

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Notes

1 | http://www.astro.ulb.ac.be/bruslib/, accessed on 9 June 2022. |

2 | https://compose.obspm.fr, accessed on 9 June 2022. |

3 | https://www-nds.iaea.org/relnsd/NdsEnsdf/QueryForm.html, accessed on 9 June 2022. |

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**Figure 1.**Relative errors (in %) of the approximate analytical expansions (14) and (16) for the baryon number density $\overline{n}$ (dashed line) and electron pressure ${P}_{e}$ (solid line), respectively, as a function of the magnetic field strength ${B}_{\u2605}=B/{B}_{\mathrm{rel}}$ with ${\gamma}_{e}=10$. The errors are obtained by taking the difference between the approximate and exact results and dividing by the exact result. All electrons are confined to the lowest Landau–Rabi level at ${B}_{\u2605}=({\gamma}_{e}^{2}-1)/2=49.5$.

**Figure 2.**Same as Figure 1 for ${\gamma}_{e}=50$. All electrons are confined to the lowest Landau–Rabi level at ${B}_{\u2605}=({\gamma}_{e}^{2}-1)/2=1249.5$.

**Figure 3.**

**Top panel**: Exact threshold density (in cgs units) for the onset of electron captures by ${}^{56}$Fe as a function of the magnetic field strength ${B}_{\u2605}=B/{B}_{\mathrm{rel}}$ up to the onset of the strongly quantizing regime.

**Bottom panel**: Relative error (in %) of the approximate analytical expression.

**Figure 4.**Same as Figure 3 in the strongly quantizing regime.

**Figure 5.**

**Top panel**: Exact threshold pressure (in cgs units) for the onset of electron captures by ${}^{56}$Fe as a function of the magnetic field strength ${B}_{\u2605}=B/{B}_{\mathrm{rel}}$ up to the onset of the strongly quantizing regime.

**Bottom panel**: Relative error (in %) of the approximate analytical expression.

**Figure 6.**Same as Figure 5 in the strongly quantizing regime.

**Figure 7.**

**Top panel**: Exact heat released per nucleon from electron captures by ${}^{56}$Fe as a function of the magnetic field strength ${B}_{\u2605}=B/{B}_{\mathrm{rel}}$ up to the onset of the strongly quantizing regime.

**Bottom panel**: Relative error (in %) of the approximate analytical expression.

**Figure 8.**Same as Figure 7 in the strongly quantizing regime.

**Figure 9.**

**Top panel**: Exact threshold density (in cgs units) for the onset of electron captures by ${}^{122}$Zr as a function of the magnetic field strength ${B}_{\u2605}=B/{B}_{\mathrm{rel}}$ up to the onset of the strongly quantizing regime.

**Bottom panel**: Relative error (in %) of the approximate analytical expression.

**Figure 10.**Same as Figure 9 in the strongly quantizing regime.

**Figure 11.**

**Top panel**: Exact threshold pressure (in cgs units) for the onset of electron captures by ${}^{122}$Zr as a function of the magnetic field strength ${B}_{\u2605}=B/{B}_{\mathrm{rel}}$ up to the onset of the strongly quantizing regime.

**Bottom panel**: Relative error (in %) of the approximate analytical expression.

**Figure 12.**Same as Figure 11 in the strongly quantizing regime.

**Figure 13.**

**Top panel**: Exact heat released per nucleon from electron captures by ${}^{122}$Zr as a function of the magnetic field strength ${B}_{\u2605}=B/{B}_{\mathrm{rel}}$ up to the onset of the strongly quantizing regime.

**Bottom panel**: Relative error (in %) of the approximate analytical expression.

**Figure 14.**Same as Figure 13 in the strongly quantizing regime.

**Table 1.**Magnetic field strength ${B}_{\u2605}=B/{B}_{\mathrm{rel}}$ for the appearance (+) or the disappearance (−) of a nuclide in the outer crust of a magnetar.

Nuclide | ${\mathit{B}}_{\u2605}$ |
---|---|

${}^{66}$Ni(−) | 67 |

${}^{88}$Sr(+) | 858 |

${}^{126}$Ru(+) | 1023 |

${}^{80}$Ni(−) | 1072 |

${}^{128}$Pd(+) | 1249 |

${}^{78}$Ni(−) | 1416 |

${}^{64}$Ni(−) | 1669 |

${}^{124}$Zr(+) | 1872 |

${}^{121}$Y(−) | 1907 |

${}^{132}$Sn(+) | 1986 |

${}^{80}$Ni(−) | 2087 |

**Table 2.**Sequence of equilibrium nuclides with increasing depth (from top to bottom) in the outer crust of a magnetar for different magnetic field strengths. The first row indicates values of ${B}_{\u2605}=B/{B}_{\mathrm{rel}}$ associated with a change of composition. Results are valid up to ${B}_{\u2605}=2087$.

67 | 858 | 1023 | 1072 | 1249 | 1416 | 1669 | 1872 | 1907 | 1986 | |
---|---|---|---|---|---|---|---|---|---|---|

${}^{56}$Fe | ${}^{56}$Fe | ${}^{56}$Fe | ${}^{56}$Fe | ${}^{56}$Fe | ${}^{56}$Fe | ${}^{56}$Fe | ${}^{56}$Fe | ${}^{56}$Fe | ${}^{56}$Fe | ${}^{56}$Fe |

${}^{62}$Ni | ${}^{62}$Ni | ${}^{62}$Ni | ${}^{62}$Ni | ${}^{62}$Ni | ${}^{62}$Ni | ${}^{62}$Ni | ${}^{62}$Ni | ${}^{62}$Ni | ${}^{62}$Ni | ${}^{62}$Ni |

${}^{64}$Ni | ${}^{64}$Ni | ${}^{64}$Ni | ${}^{64}$Ni | ${}^{64}$Ni | ${}^{64}$Ni | ${}^{64}$Ni | – | – | – | – |

${}^{66}$Ni | – | – | – | – | – | – | – | – | – | – |

– | – | ${}^{88}$Sr | ${}^{88}$Sr | ${}^{88}$Sr | ${}^{88}$Sr | ${}^{88}$Sr | ${}^{88}$Sr | ${}^{88}$Sr | ${}^{88}$Sr | ${}^{88}$Sr |

${}^{86}$Kr | ${}^{86}$Kr | ${}^{86}$Kr | ${}^{86}$Kr | ${}^{86}$Kr | ${}^{86}$Kr | ${}^{86}$Kr | ${}^{86}$Kr | ${}^{86}$Kr | ${}^{86}$Kr | ${}^{86}$Kr |

${}^{84}$Se | ${}^{84}$Se | ${}^{84}$Se | ${}^{84}$Se | ${}^{84}$Se | ${}^{84}$Se | ${}^{84}$Se | ${}^{84}$Se | ${}^{84}$Se | ${}^{84}$Se | ${}^{84}$Se |

${}^{82}$Ge | ${}^{82}$Ge | ${}^{82}$Ge | ${}^{82}$Ge | ${}^{82}$Ge | ${}^{82}$Ge | ${}^{82}$Ge | ${}^{82}$Ge | ${}^{82}$Ge | ${}^{82}$Ge | ${}^{82}$Ge |

– | – | – | – | – | – | – | – | – | – | ${}^{132}$Sn |

${}^{80}$Zn | ${}^{80}$Zn | ${}^{80}$Zn | ${}^{80}$Zn | ${}^{80}$Zn | ${}^{80}$Zn | ${}^{80}$Zn | ${}^{80}$Zn | ${}^{80}$Zn | ${}^{80}$Zn | ${}^{80}$Zn |

${}^{78}$Ni | ${}^{78}$Ni | ${}^{78}$Ni | ${}^{78}$Ni | ${}^{78}$Ni | ${}^{78}$Ni | – | – | – | – | – |

${}^{80}$Ni | ${}^{80}$Ni | ${}^{80}$Ni | ${}^{80}$Ni | – | – | – | – | – | – | – |

– | – | – | – | – | ${}^{128}$Pd | ${}^{128}$Pd | ${}^{128}$Pd | ${}^{128}$Pd | ${}^{128}$Pd | ${}^{128}$Pd |

– | – | – | ${}^{126}$Ru | ${}^{126}$Ru | ${}^{126}$Ru | ${}^{126}$Ru | ${}^{126}$Ru | ${}^{126}$Ru | ${}^{126}$Ru | ${}^{126}$Ru |

${}^{124}$Mo | ${}^{124}$Mo | ${}^{124}$Mo | ${}^{124}$Mo | ${}^{124}$Mo | ${}^{124}$Mo | ${}^{124}$Mo | ${}^{124}$Mo | ${}^{124}$Mo | ${}^{124}$Mo | ${}^{124}$Mo |

${}^{122}$Zr | ${}^{122}$Zr | ${}^{122}$Zr | ${}^{122}$Zr | ${}^{122}$Zr | ${}^{122}$Zr | ${}^{122}$Zr | ${}^{122}$Zr | ${}^{122}$Zr | ${}^{122}$Zr | ${}^{122}$Zr |

– | – | – | – | – | – | – | – | ${}^{124}$Zr | ${}^{124}$Zr | ${}^{124}$Zr |

${}^{121}$Y | ${}^{121}$Y | ${}^{121}$Y | ${}^{121}$Y | ${}^{121}$Y | ${}^{121}$Y | ${}^{121}$Y | ${}^{121}$Y | ${}^{121}$Y | – | – |

${}^{120}$Sr | ${}^{120}$Sr | ${}^{120}$Sr | ${}^{120}$Sr | ${}^{120}$Sr | ${}^{120}$Sr | ${}^{120}$Sr | ${}^{120}$Sr | ${}^{120}$Sr | ${}^{120}$Sr | ${}^{120}$Sr |

${}^{122}$Sr | ${}^{122}$Sr | ${}^{122}$Sr | ${}^{122}$Sr | ${}^{122}$Sr | ${}^{122}$Sr | ${}^{122}$Sr | ${}^{122}$Sr | ${}^{122}$Sr | ${}^{122}$Sr | ${}^{122}$Sr |

${}^{124}$Sr | ${}^{124}$Sr | ${}^{124}$Sr | ${}^{124}$Sr | ${}^{124}$Sr | ${}^{124}$Sr | ${}^{124}$Sr | ${}^{124}$Sr | ${}^{124}$Sr | ${}^{124}$Sr | ${}^{124}$Sr |

**Table 3.**Values of the nuclear parameter ${\gamma}_{e}^{1\to 2}$ from which the pressure and the densities at the boundary between adjacent layers in the outer crust of a magnetar can be calculated.

${\mathit{\gamma}}_{\mathit{e}}^{1\to 2}$ | Interface |
---|---|

1.8908 | ${}^{56}\mathrm{Fe}$–${}^{62}\mathrm{Ni}$ |

4.8972 | ${}^{62}\mathrm{Ni}$–${}^{64}\mathrm{Ni}$ |

8.6863 | ${}^{62}\mathrm{Ni}$–${}^{88}\mathrm{Sr}$ |

8.1312 | ${}^{64}\mathrm{Ni}$–${}^{66}\mathrm{Ni}$ |

9.3317 | ${}^{64}\mathrm{Ni}$–${}^{86}\mathrm{Kr}$ |

18.098 | ${}^{64}\mathrm{Ni}$–${}^{88}\mathrm{Sr}$ |

12.155 | ${}^{66}\mathrm{Ni}$–${}^{86}\mathrm{Kr}$ |

10.044 | ${}^{86}\mathrm{Kr}$–${}^{84}\mathrm{Se}$ |

5.5622 | ${}^{88}\mathrm{Sr}$–${}^{86}\mathrm{Kr}$ |

15.330 | ${}^{84}\mathrm{Se}$–${}^{82}\mathrm{Ge}$ |

20.519 | ${}^{82}\mathrm{Ge}$–${}^{80}\mathrm{Zn}$ |

38.310 | ${}^{82}\mathrm{Ge}$–${}^{132}\mathrm{Sn}$ |

25.926 | ${}^{80}\mathrm{Zn}$–${}^{78}\mathrm{Ni}$ |

37.083 | ${}^{80}\mathrm{Zn}$–${}^{128}\mathrm{Pd}$ |

$-33.289$ | ${}^{132}\mathrm{Sn}$–${}^{80}\mathrm{Zn}$ |

32.978 | ${}^{78}\mathrm{Ni}$–${}^{80}\mathrm{Ni}$ |

$-409.21$ | ${}^{78}\mathrm{Ni}$–${}^{128}\mathrm{Pd}$ |

48.055 | ${}^{78}\mathrm{Ni}$–${}^{126}\mathrm{Ru}$ |

45.122 | ${}^{80}\mathrm{Ni}$–${}^{124}\mathrm{Mo}$ |

218.54 | ${}^{80}\mathrm{Ni}$–${}^{126}\mathrm{Ru}$ |

30.039 | ${}^{128}\mathrm{Pd}$–${}^{126}\mathrm{Ru}$ |

32.010 | ${}^{126}\mathrm{Ru}$–${}^{124}\mathrm{Mo}$ |

37.441 | ${}^{124}\mathrm{Mo}$–${}^{122}\mathrm{Zr}$ |

40.352 | ${}^{122}\mathrm{Zr}$–${}^{121}\mathrm{Y}$ |

42.293 | ${}^{122}\mathrm{Zr}$–${}^{124}\mathrm{Zr}$ |

1.8541 | ${}^{124}\mathrm{Zr}$–${}^{121}\mathrm{Y}$ |

39.031 | ${}^{124}\mathrm{Zr}$–${}^{120}\mathrm{Sr}$ |

40.786 | ${}^{121}\mathrm{Y}$–${}^{120}\mathrm{Sr}$ |

44.857 | ${}^{120}\mathrm{Sr}$–${}^{122}\mathrm{Sr}$ |

47.747 | ${}^{122}\mathrm{Sr}$–${}^{124}\mathrm{Sr}$ |

**Table 4.**Values of the nuclear parameters ${\gamma}_{e}^{\beta}$ and ${\mathcal{Q}}^{\left(0\right)}$ (considering ground-state to ground-state transitions) from which the threshold density and pressure for the onset of electron captures, as well as the heat released can be calculated. The symbol (★) is used to distinguish reactions for which theoretical atomic masses were needed.

${\mathit{\gamma}}_{\mathit{e}}^{\mathit{\beta}}$ | Reaction | ${\mathcal{Q}}^{\left(0\right)}$ (MeV) |
---|---|---|

8.232 | ${}^{56}\mathrm{Fe}\to {}^{56}\mathrm{Cr}-2{e}^{-}+2{\nu}_{e}$ | 2.069 |

18.867 | ${}^{56}\mathrm{Cr}\to {}^{56}\mathrm{Ti}-2{e}^{-}+2{\nu}_{e}$ | 2.295 |

29.313 | ${}^{56}\mathrm{Ti}\to {}^{56}\mathrm{Ca}-2{e}^{-}+2{\nu}_{e}$ | 3.514 |

43.710 | ${}^{56}\mathrm{Ca}\to {}^{56}\mathrm{Ar}-2{e}^{-}+2{\nu}_{e}\phantom{\rule{4pt}{0ex}}(\u2605)$ | 2.045 |

11.415 | ${}^{62}\mathrm{Ni}\to {}^{62}\mathrm{Fe}-2{e}^{-}+2{\nu}_{e}$ | 2.776 |

21.262 | ${}^{62}\mathrm{Fe}\to {}^{62}\mathrm{Cr}-2{e}^{-}+2{\nu}_{e}$ | 2.725 |

31.174 | ${}^{62}\mathrm{Cr}\to {}^{62}\mathrm{Ti}-2{e}^{-}+2{\nu}_{e}$ | 2.442 |

41.470 | ${}^{62}\mathrm{Ti}\to {}^{62}\mathrm{Ca}-2{e}^{-}+2{\nu}_{e}\phantom{\rule{4pt}{0ex}}(\u2605)$ | 2.490 |

15.299 | ${}^{64}\mathrm{Ni}\to {}^{64}\mathrm{Fe}-2{e}^{-}+2{\nu}_{e}$ | 2.484 |

24.445 | ${}^{64}\mathrm{Fe}\to {}^{64}\mathrm{Cr}-2{e}^{-}+2{\nu}_{e}$ | 2.471 |

34.581 | ${}^{64}\mathrm{Cr}\to {}^{64}\mathrm{Ti}-2{e}^{-}+2{\nu}_{e}$ | 1.865 |

19.782 | ${}^{66}\mathrm{Ni}\to {}^{66}\mathrm{Fe}-2{e}^{-}+2{\nu}_{e}$ | 3.257 |

27.062 | ${}^{66}\mathrm{Fe}\to {}^{66}\mathrm{Cr}-2{e}^{-}+2{\nu}_{e}$ | 1.287 |

38.397 | ${}^{66}\mathrm{Cr}\to {}^{66}\mathrm{Ti}-2{e}^{-}+2{\nu}_{e}\phantom{\rule{4pt}{0ex}}(\u2605)$ | 3.540 |

15.938 | ${}^{86}\mathrm{Kr}\to {}^{86}\mathrm{Se}-2{e}^{-}+2{\nu}_{e}$ | 2.504 |

23.585 | ${}^{86}\mathrm{Se}\to {}^{86}\mathrm{Ge}-2{e}^{-}+2{\nu}_{e}$ | 1.979 |

30.980 | ${}^{86}\mathrm{Ge}\to {}^{86}\mathrm{Zn}-2{e}^{-}+2{\nu}_{e}\phantom{\rule{4pt}{0ex}}(\u2605)$ | 2.310 |

38.926 | ${}^{86}\mathrm{Zn}\to {}^{86}\mathrm{Ni}-2{e}^{-}+2{\nu}_{e}\phantom{\rule{4pt}{0ex}}(\u2605)$ | 2.320 |

20.754 | ${}^{84}\mathrm{Se}\to {}^{84}\mathrm{Ge}-2{e}^{-}+2{\nu}_{e}$ | 2.389 |

28.517 | ${}^{84}\mathrm{Ge}\to {}^{84}\mathrm{Zn}-2{e}^{-}+2{\nu}_{e}$ | 1.903 |

36.362 | ${}^{84}\mathrm{Zn}\to {}^{84}\mathrm{Ni}-2{e}^{-}+2{\nu}_{e}\phantom{\rule{4pt}{0ex}}(\u2605)$ | 2.390 |

25.431 | ${}^{82}\mathrm{Ge}\to {}^{82}\mathrm{Zn}-2{e}^{-}+2{\nu}_{e}$ | 1.868 |

34.256 | ${}^{82}\mathrm{Zn}\to {}^{82}\mathrm{Ni}-2{e}^{-}+2{\nu}_{e}\phantom{\rule{4pt}{0ex}}(\u2605)$ | 2.154 |

44.640 | ${}^{82}\mathrm{Ni}\to {}^{82}\mathrm{Fe}-2{e}^{-}+2{\nu}_{e}\phantom{\rule{4pt}{0ex}}(\u2605)$ | 2.080 |

31.233 | ${}^{80}\mathrm{Zn}\to {}^{80}\mathrm{Ni}-2{e}^{-}+2{\nu}_{e}$ | 1.879 |

39.435 | ${}^{78}\mathrm{Ni}\to {}^{78}\mathrm{Fe}-2{e}^{-}+2{\nu}_{e}\phantom{\rule{4pt}{0ex}}(\u2605)$ | 2.070 |

41.842 | ${}^{124}\mathrm{Mo}\to {}^{124}\mathrm{Zr}-2{e}^{-}+2{\nu}_{e}\phantom{\rule{4pt}{0ex}}(\u2605)$ | 2.920 |

42.722 | ${}^{122}\mathrm{Zr}\to {}^{122}\mathrm{Sr}-2{e}^{-}+2{\nu}_{e}\phantom{\rule{4pt}{0ex}}(\u2605)$ | 0.790 |

11.397 | ${}^{88}\mathrm{Sr}\to {}^{88}\mathrm{Kr}-2{e}^{-}+2{\nu}_{e}$ | 2.395 |

18.564 | ${}^{88}\mathrm{Kr}\to {}^{88}\mathrm{Se}-2{e}^{-}+2{\nu}_{e}$ | 2.144 |

26.761 | ${}^{88}\mathrm{Se}\to {}^{88}\mathrm{Ge}-2{e}^{-}+2{\nu}_{e}$ | 2.582 |

34.601 | ${}^{88}\mathrm{Ge}\to {}^{88}\mathrm{Zn}-2{e}^{-}+2{\nu}_{e}\phantom{\rule{4pt}{0ex}}(\u2605)$ | 2.740 |

36.695 | ${}^{126}\mathrm{Ru}\to {}^{126}\mathrm{Mo}-2{e}^{-}+2{\nu}_{e}\phantom{\rule{4pt}{0ex}}(\u2605)$ | 1.860 |

34.444 | ${}^{128}\mathrm{Pd}\to {}^{128}\mathrm{Ru}-2{e}^{-}+2{\nu}_{e}\phantom{\rule{4pt}{0ex}}(\u2605)$ | 2.290 |

28.662 | ${}^{132}\mathrm{Sn}\to {}^{132}\mathrm{Cd}-2{e}^{-}+2{\nu}_{e}$ | 1.987 |

33.237 | ${}^{132}\mathrm{Cd}\to {}^{132}\mathrm{Pd}-2{e}^{-}+2{\nu}_{e}\phantom{\rule{4pt}{0ex}}(\u2605)$ | 2.863 |

38.084 | ${}^{132}\mathrm{Pd}\to {}^{132}\mathrm{Ru}-2{e}^{-}+2{\nu}_{e}\phantom{\rule{4pt}{0ex}}(\u2605)$ | 2.940 |

**Table 5.**Same as in Table 4 but considering ground-state to excited state transitions (for which the excitation energy is experimentally known and ${E}_{\mathrm{exc}}>0$).

${\mathit{\gamma}}_{\mathit{e}}^{\mathit{\beta}}$ | Reaction | ${\mathcal{Q}}^{\left(0\right)}$ (MeV) |
---|---|---|

8.448 | ${}^{56}\mathrm{Fe}\to {}^{56}\mathrm{Cr}-2{e}^{-}+2{\nu}_{e}$ | 2.290 |

12.405 | ${}^{62}\mathrm{Ni}\to {}^{62}\mathrm{Fe}-2{e}^{-}+2{\nu}_{e}$ | 3.788 |

20.726 | ${}^{86}\mathrm{Kr}\to {}^{86}\mathrm{Se}-2{e}^{-}+2{\nu}_{e}$ | 7.397 |

31.260 | ${}^{82}\mathrm{Ge}\to {}^{82}\mathrm{Zn}-2{e}^{-}+2{\nu}_{e}$ | 7.825 |

15.764 | ${}^{88}\mathrm{Sr}\to {}^{88}\mathrm{Kr}-2{e}^{-}+2{\nu}_{e}$ | 6.858 |

22.290 | ${}^{88}\mathrm{Kr}\to {}^{88}\mathrm{Se}-2{e}^{-}+2{\nu}_{e}$ | 5.951 |

31.010 | ${}^{132}\mathrm{Sn}\to {}^{132}\mathrm{Cd}-2{e}^{-}+2{\nu}_{e}$ | 4.387 |

**Table 6.**Same as in Table 4 but considering reactions involving carbon and oxygen.

${\mathit{\gamma}}_{\mathit{e}}^{\mathit{\beta}}$ | Reaction | ${\mathcal{Q}}^{\left(0\right)}$ (MeV) |
---|---|---|

21.393 | ${}^{16}\mathrm{O}\to {}^{16}\mathrm{C}-2{e}^{-}+2{\nu}_{e}$ | 2.411 |

27.163 | ${}^{12}\mathrm{C}\to {}^{12}\mathrm{Be}-2{e}^{-}+2{\nu}_{e}$ | 1.661 |

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**MDPI and ACS Style**

Chamel, N.; Fantina, A.F.
Onset of Electron Captures and Shallow Heating in Magnetars. *Universe* **2022**, *8*, 328.
https://doi.org/10.3390/universe8060328

**AMA Style**

Chamel N, Fantina AF.
Onset of Electron Captures and Shallow Heating in Magnetars. *Universe*. 2022; 8(6):328.
https://doi.org/10.3390/universe8060328

**Chicago/Turabian Style**

Chamel, Nicolas, and Anthea Francesca Fantina.
2022. "Onset of Electron Captures and Shallow Heating in Magnetars" *Universe* 8, no. 6: 328.
https://doi.org/10.3390/universe8060328