The Joint Institute for Nuclear Astrophysics

This project is funded by the NSF through grants This project is funded by the NSF through grants PHY0216783 (JINA), the NSCL, and by Michigan State PHY0216783 (JINA), the NSCL, and by Michigan State University. University.

Comparison between the two compilations of electron capture rates:

We start by comparing the EC rates at the lowest values of temperature and density (T9=0.01 and Log10(Ye) = 1.0). Then we compare rates for higher values of density, while keeping T9 constant.

Below are shown the ratios for T9=0.01 and

Log10(Ye) = 1.0, 8.0, 9.0 and 10.0.

The color code for these plots is as follows: R -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 R1.0

Thick black borders denote stable nuclei.

Comparison between the two compilations of electron capture rates:

We start by comparing the EC rates at the lowest values of temperature and density (T9=0.01 and Log10(Ye) = 1.0). Then we compare rates for higher values of density, while keeping T9 constant.

Below are shown the ratios for T9=0.01 and

Log10(Ye) = 1.0, 8.0, 9.0 and 10.0.

The color code for these plots is as follows: R -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 R1.0

Thick black borders denote stable nuclei.

EC in crusts of accreting neutron stars (models of crust evolution)

Even at the low accretion rates of ~10-10 Msolar yr-1, a neutron star can accrete enough material from the secondary to replace its entire crust with ashes of H/He burning not in NSE. The rising electron chemical potential with density as the ashes are pushed deeper in the crust will switch on energetically unfavorable EC transitions. This lowers the nuclear charge and generates heat.

Threshold effects at low temperature cannot be captured accurately in tables. Our analytic implementation of the electron capture phase space is fast enough to be used in real-time inside a reaction network.

This implementation is not susceptible to low-temperature inaccuracies due to the Fermi-Dirac distribution shape (as Gauss-Laguerre quadrature schemes are in the grid region T9 = 0.01-0.1 when compared to trapezoidal rule schemes). Inaccuracies are only introduced because we evaluate the distortion of the electron wave function at only one “effective” electron energy. Most approximations ignore the Coulomb Correction altogether by setting it to unity over the integration range. By retaining an effective correction we retain the effects of a varying nuclear radius on the phase space (important for a rate compilation sufficiently global over the nuclear chart).

EC in crusts of accreting neutron stars (models of crust evolution)

Even at the low accretion rates of ~10-10 Msolar yr-1, a neutron star can accrete enough material from the secondary to replace its entire crust with ashes of H/He burning not in NSE. The rising electron chemical potential with density as the ashes are pushed deeper in the crust will switch on energetically unfavorable EC transitions. This lowers the nuclear charge and generates heat.

Threshold effects at low temperature cannot be captured accurately in tables. Our analytic implementation of the electron capture phase space is fast enough to be used in real-time inside a reaction network.

This implementation is not susceptible to low-temperature inaccuracies due to the Fermi-Dirac distribution shape (as Gauss-Laguerre quadrature schemes are in the grid region T9 = 0.01-0.1 when compared to trapezoidal rule schemes). Inaccuracies are only introduced because we evaluate the distortion of the electron wave function at only one “effective” electron energy. Most approximations ignore the Coulomb Correction altogether by setting it to unity over the integration range. By retaining an effective correction we retain the effects of a varying nuclear radius on the phase space (important for a rate compilation sufficiently global over the nuclear chart).

Where we are now:

EC rates successfully implemented in Neutron Star Crust simulation. (Gupta, Brown, Schatz, Möller, Kratz. TBP).

Where we are headed:

Calculating rates with excited states in parent nuclei.Using these rates in high temperature (T9), high density (Log10(Ye)) conditions in core – collapse supernova simulations.

Where we are now:

EC rates successfully implemented in Neutron Star Crust simulation. (Gupta, Brown, Schatz, Möller, Kratz. TBP).

Where we are headed:

Calculating rates with excited states in parent nuclei.Using these rates in high temperature (T9), high density (Log10(Ye)) conditions in core – collapse supernova simulations.

Analytic formulation of rate from state I in parent to state J in daughter:

R=(ln2*f)/(ft_value for transition from I to J) Phase space factor: f(T9 ,u,w,q) = Geff * FWhere: u, w, q are electron chemical potential, capture threshold and capture q-value respectively, in mec2 units (electron rest mass).

Rising “u” in NS crust allows electrons to overcome unfavorable capture thresholds.

For pre-threshold captures important in NS crusts:F(u < w,w = 1) = (w4+2qw3+q2w2)(T9 /5.93)f1(z)+(4w3+6qw2+2q2w)(T9 /5.93)2f2(z) + (12w2+12qw+2q2)(T9 /5.93)3f3(z)+(24w+12q) (T9 /5.93)4f4(z)+24(T9 /5.93)5f5(z)

For pre-threshold when q < -1 :F(u < w,w = - q) = 2w2 (T9 /5.93)3f3(z)+12w (T9 /5.93)4f4(z)+24(T9 /5.93)5f5(z)

Analytic formulation of rate from state I in parent to state J in daughter:

R=(ln2*f)/(ft_value for transition from I to J) Phase space factor: f(T9 ,u,w,q) = Geff * FWhere: u, w, q are electron chemical potential, capture threshold and capture q-value respectively, in mec2 units (electron rest mass).

Rising “u” in NS crust allows electrons to overcome unfavorable capture thresholds.

For pre-threshold captures important in NS crusts:F(u < w,w = 1) = (w4+2qw3+q2w2)(T9 /5.93)f1(z)+(4w3+6qw2+2q2w)(T9 /5.93)2f2(z) + (12w2+12qw+2q2)(T9 /5.93)3f3(z)+(24w+12q) (T9 /5.93)4f4(z)+24(T9 /5.93)5f5(z)

For pre-threshold when q < -1 :F(u < w,w = - q) = 2w2 (T9 /5.93)3f3(z)+12w (T9 /5.93)4f4(z)+24(T9 /5.93)5f5(z)

Geff = Coulomb Correction = (p/w)FC (Z,A,w) evaluated at effective electron energy w=weff extracted from

weff 2 (weff +q)2 =5.93*F/(T9 f1(z)).

P = electron momentum, Z = nuclear charge of captor

FC (Z,A,w) = Distortion of electron wave function (for a given electron energy w) due to nuclear charge and finite size of nucleus.F = analytic expression above

Geff = Coulomb Correction = (p/w)FC (Z,A,w) evaluated at effective electron energy w=weff extracted from

weff 2 (weff +q)2 =5.93*F/(T9 f1(z)).

P = electron momentum, Z = nuclear charge of captor

FC (Z,A,w) = Distortion of electron wave function (for a given electron energy w) due to nuclear charge and finite size of nucleus.F = analytic expression above

Where fn(z) is a generalization of the Logarithm function: f1 (z)=ln(1+z)fn(z)=[ {(-1)k-1zk}/k n ] (k=1,…N<300 for convergence when n<6) and z=exp{-(5.93|w-u|)/ T9 } < 1

Where fn(z) is a generalization of the Logarithm function: f1 (z)=ln(1+z)fn(z)=[ {(-1)k-1zk}/k n ] (k=1,…N<300 for convergence when n<6) and z=exp{-(5.93|w-u|)/ T9 } < 1

Comparison of EC rates to those calculated by FFN

FULLER, FOWLER & NEWMAN (FFN) (Ap.J. Suppl.42,447 (1980))

GUPTA & MÖLLER(GM)

Spherical Independent Particle Model.

Experimental input (gs->gs or gs-> low-lying forbidden transitions) included.

Excited states of parent included (required for Core-Collapse Supernovae, not for Neutron Star Crusts)

Nilsson Model.Allowed (GT) only.

Ground state of parent only (thermal population effects in Neutron Star Crusts very mild).

No quenching of strength. Residual interactions using QRPA (Peter Möller).

Artist’s conception of neutron star EXO 0748-676 (blue sphere). It is part of a binary star system, and its neighboring star (yellow-red sphere) supplies the fuel for the thermonuclear bursts. (Image Credit: NASA)

)(

)(10 FFNECR

GMECRLogR

Results of comparisons

At the lowest density some differences arise due to low lying structure (experimental data vs. QRPA).

At higher values of Log10(Ye) (e.g. 8.0, 9.0, 10.0) a larger fraction of strength in daughter nuclei is accessed, and more n-rich nuclei can EC. Thus, the observed changes in the calculated rates may be due to:

In FFN deformation for neutron rich nuclei is not taken into account, but it is in GM. Therefore, calculated structure is very different. Significantly deformed neutron orbits can increase the number of states accessible by the electron chemical potential.

For rates in which the initial nucleus is even-even and the final is an odd-odd, g.s. g.s. may not be allowed via (GT) transition.

Gupta & Möller calculations do not include experimental input. We intend to compare the FFN and GM with available experimental level information.

Results of comparisons

At the lowest density some differences arise due to low lying structure (experimental data vs. QRPA).

At higher values of Log10(Ye) (e.g. 8.0, 9.0, 10.0) a larger fraction of strength in daughter nuclei is accessed, and more n-rich nuclei can EC. Thus, the observed changes in the calculated rates may be due to:

In FFN deformation for neutron rich nuclei is not taken into account, but it is in GM. Therefore, calculated structure is very different. Significantly deformed neutron orbits can increase the number of states accessible by the electron chemical potential.

For rates in which the initial nucleus is even-even and the final is an odd-odd, g.s. g.s. may not be allowed via (GT) transition.

Gupta & Möller calculations do not include experimental input. We intend to compare the FFN and GM with available experimental level information.

Z

N

Log10(Ye) = 1.0

UF(T9=0.01)=0.508 MeV

N

Z

Z

N

N

Z

Electron Capture Rates for Neutron Star Electron Capture Rates for Neutron Star Crusts.Crusts.

Ana D. Becerril ReyesAna D. Becerril Reyes1,21,2, Sanjib S. Gupta, Sanjib S. Gupta11 and Hendrik Schatz and Hendrik Schatz1,21,2..

Electron Capture Rates for Neutron Star Electron Capture Rates for Neutron Star Crusts.Crusts.

Ana D. Becerril ReyesAna D. Becerril Reyes1,21,2, Sanjib S. Gupta, Sanjib S. Gupta11 and Hendrik Schatz and Hendrik Schatz1,21,2..1 National Superconducting Cyclotron Laboratory, Michigan State University, 2 Dept. of Physics and Astronomy, Michigan State University

factor of 2

factor of 5

Log10(Ye) = 8.0

UF(T9=0.01)=2.447 MeV

Log10(Ye) = 9.0

UF(T9=0.01)=5.182 MeV

Log10(Ye) = 10.0

UF(T9=0.01)=11.119 MeV