Developer Guide

Tables to Download

Feb. 6 2022, first release

Tables with electrons and photons:

Baryons-only tables without derivatives of F:

Electron-photon EOS:

Electrons and photons

Nuclear masses:

Nuclear masses

Table Format

The table is an HDF5 file and the datasets are given below. Several quantities are stored in o2scl:tensor_grid objects, which are stored as HDF5 groups with the name of the group given below. The contents of the rank 3 tensor are stored in the dataset named data in each group.

Grid

n_nB: number of points in baryon density grid
n_Ye: number of points in electron fraction grid
n_T: number of points in temperature grid
nB_grid: array containing baryon density grid (in \(\mathrm{fm}^{-3}\))
Ye_grid: array containing electron fraction grid
T_grid: array containing temperature grid (in \(\mathrm{MeV}\))

Physical quantities

These are stored as double-precision numbers.

hc: \(\hbar c\) (in \(\mathrm{MeV~fm}\))
alpha_em: \(\alpha_{\mathrm{EM}}\), the fine structure constant
m_neut: the mass of the neutron (in \(\mathrm{MeV}\))
m_prot: the mass of the proton in MeV (in \(\mathrm{MeV}\))

Flags

These are stored as integers.

baryons_only: 1 (true) if the thermodynamic quantities with baryons only are provided (always 1)
with_leptons: 1 (true) if the thermodynamic quantities with leptons are provided, and 0 (false) otherwise.
derivs_computed: 1 (true) if the baryon part of the pressure and the nucleon chemical potentials are included.
alg_mode: an integer representing the algorithm used to compute the table
include_muons: 0 (false) if muons are not included

Quantities for the solver

These are all o2scl:tensor_grid objects. The contents of the tensor are stored in an HDF5 dataset named “data” in row-major order (the first index is the baryon density, the second is the electron fraction, and the third is the temperature).

flag: A flag indicating if each point is complete, empty, or if a guess has been stored (10 means the point is complete).
log_xn: \(\log_{10}(x_n)\) where \(x_n\equiv n_n^{\prime}/n_B\).
log_xp: \(\log_{10}(x_p)\) where \(x_p\equiv n_p^{\prime}/n_B\).
A: The average nuclear mass number
Z: The average nuclear charge number

Also, if alg_mode is either 2, 3, or 4, then the following tensors are included:

A_min: the smallest value of A in the distribution, not including n, p, d, t, \(^{3}\mathrm{He}\), \(^{4}\mathrm{Li}\), and \(\alpha\).
A_max: largest value of A in the distribution, not including n, p, d, t, \(^{3}\mathrm{He}\), \(^{4}\mathrm{Li}\), and \(\alpha\).
NmZ_min: the smallest value of \(N-Z\) in the distribution, not including n, p, d, t, \(^{3}\mathrm{He}\), \(^{4}\mathrm{Li}\), and \(\alpha\).
NmZ_max: the largest value of \(N-Z\) in the distribution, not including n, p, d, t, \(^{3}\mathrm{He}\), \(^{4}\mathrm{Li}\), and \(\alpha\).

Composition

These are all o2scl:tensor_grid objects and included for all tables.

Xn: the baryon number fraction of neutrons (this includes only the neutrons in the low-density phase, i.e. the neutrons not inside nuclei)
Xp: the baryon number fraction of protons (this includes only the protons in the low-density phase, i.e. the protons not inside nuclei)
Xalpha: the baryon number fraction of alpha particles
Xd: the baryon number fraction of deuterons
Xt: the baryon number fraction of tritons
XHe3: the baryon number fraction of \(^{3}\mathrm{He}\),
XLi4: the baryon number fraction of \(^{4}\mathrm{Li}\)
Xnuclei: the baryon number fraction of nuclei

The sum of these eight quantities should be close to 1 for all points in the table. Note that Xn and Xp do not store the local density of neutrons in the gas phase, those quantities are given in Du et al. (2022): \(n_n^{\prime}\equiv n_n/\xi\) and \(n_n^{\prime}\equiv n_p/\xi\). These densities are not stored in the tables, but are accessible, for example, in the point-nuclei command. At high densities or temperatures when nuclei are not present, \(\xi=1\).

Thermodynamic quantities

In this section, all quantities are stored as o2scl:tensor_grid objects.

Three quantities are included for all tables:

Fint: the baryon part of the free energy per baryon (in \(\mathrm{MeV}\))
Sint: the baryon part of the entropy per baryon
Eint: the baryon part of the internal energy per baryon (in \(\mathrm{MeV}\))

The relation \(F_{\mathrm{int}}=E_{\mathrm{int}}-T S_{\mathrm{int}}\) should hold at all points in the table.

If include_muons is 1, then Ymu, the muon fraction, is also included. If either include_muons or with_leptons is 1, then mue, the electron chemical potential is included. The electron chemical potential includes the electron rest mass and is in \(\mathrm{MeV}\).

If derivs_computed is 1, then the following quantities are also included:

Pint: the baryon part of the pressure (in \(\mathrm{MeV}/\mathrm{fm}^3\))
mun: the neutron chemical potential (in \(\mathrm{MeV}\))
mup: the proton chemical potential (in \(\mathrm{MeV}\))

The thermodynamic identity,

\[E_{\mathrm{int}} n_B = - P_{\mathrm{int}} + S_{\mathrm{int}} n_B T + \bar{n}_n \mu_{n} + \bar{n}_p \mu_{p}\]

where \(\bar{n}_n\equiv n_B(1-Y_e)\) and \(\bar{n}_p\equiv n_B Y_e\) should be satisfied for all of the points in the table.

The rest mass energy, \(m_n (1-Y_e) + m_p Y_e\), has been subtracted out from Fint and Eint. (See Rest mass contribution for more detail.) The neutron and proton rest mass have been subtracted out from the neutron and proton chemical potentials (indepedent of whether or not the model implies a relativistic dispersion relation for the nucleons). See also Chemical Potentials for some helpful information on chemical potentials and the associated thermodynamic identity. If with_leptons is 1, then the electron chemical potential is included (as described above) and the following four quantities are also included:

F: the total free energy per baryon (in \(\mathrm{MeV}\))
S: the total entropy per baryon
E: the total internal energy per baryon (in \(\mathrm{MeV}\))
P: the total pressure (in \(\mathrm{MeV}/\mathrm{fm}^3\))

The thermodynamic identity,

\[E n_B = - P + S n_B T + \bar{n}_n \mu_{n} + \bar{n}_p (\mu_{p}+\mu_{e})\]

should be satisfied for all of the points in the table. Note that F and E do not include the baryonic part of the rest mass energy but they do include the electron rest mass energy, \(m_e Y_e\).

String arrays

For compatibility with O₂scl, a set of two string arrays is also included. The first, oth_names contains the list: Xd, Xt, XHe3, XLi4, flag, log_xn, and log_xp. If alg_mode is 2 or larger, oth_names also contains A_min, A_max, NmZ_min, NmZ_max. The second, oth_units, contains a set of empty strings because none of the tensors referred to in the oth_names list have any units. The unsigned integer n_oth contains the size of the oth_names array.

Electron and photon table

The electron and photon table, contains five o2scl:tensor_grid objects which includes electrons, positrons, and photons,

F: the free energy per baryon (in \(\mathrm{MeV}\))
S: the entropy per baryon
E: the internal energy per baryon (in \(\mathrm{MeV}\))
P: the pressure (in \(\mathrm{MeV}/\mathrm{fm}^3\))
mue: the electron chemical potential (in \(\mathrm{MeV}\))

Nuclear masses table

The nuclear massses table, contains a o2scl:table object with the columns

Z: the proton number,
N: the neutron number,
g: the spin degeneracy,
m: the total mass (in \(\mathrm{MeV}\)),
be: the binding energy (in \(\mathrm{MeV}\)),
Sn: the neutron separation energy (in \(\mathrm{MeV}\)),
Sp: the proton separation energy (in \(\mathrm{MeV}\)),
mass_type: 1 for light nucleus, 2 for AME, 3 for FRDM, and 4 for extrapolated FRDM results, and
spin_type: 1 for light nucleus, 2 for Jexp from HFB fit, 3 for Jth from HFB fit, 4 for simple ansatz.