Bulk Parameters

The Bulk Parameters module computes saturation properties of nuclear matter from a tabulated equation of state (EoS) in \((n_B, y_Q)\) space, where \(n_B\) is the baryon number density and \(y_Q = n_Q / n_B\) is the charge fraction. The EoS is provided as a CSV file over a grid of temperatures.

The module locates the saturation point (where the pressure vanishes in symmetric nuclear matter, \(y_Q = 0.5\)) and extracts the following bulk parameters at that point:

  • Saturation density \(n_0\) (fm-3)

  • Binding energy \(\text{BE}\) (MeV)

  • Incompressibility \(K_0\) (MeV)

  • Skewness \(Q_0\) (MeV)

  • Kurtosis \(Z_0\) (MeV)

  • Symmetry energy \(J\) (MeV)

  • Slope of the symmetry energy \(L\) (MeV)

  • Curvature of the symmetry energy \(K_\text{sym}\) (MeV)

  • Skewness of the symmetry energy \(Q_\text{sym}\) (MeV)

  • Kurtosis of the symmetry energy \(Z_\text{sym}\) (MeV)

These parameters arise from the Taylor expansion of the energy per baryon around the saturation point:

\[E(n_B,\, y_Q) = \text{BE} + \frac{K_0}{2} x^2 + \frac{Q_0}{6} x^3 + \frac{Z_0}{24} x^4 + \left( J + L\, x + \frac{K_\text{sym}}{2} x^2 + \frac{Q_\text{sym}}{6} x^3 + \frac{Z_\text{sym}}{24} x^4 \right) \delta^2 + \mathcal{O}(x^5,\, \delta^4)\]

where \(x = (n_B - n_0)/(3\, n_0)\) and \(\delta = 1 - 2 y_Q\).

Two computation methods are available:

Fitting (method: fitting)

Non-linear least-squares fit of the full dataset to the Taylor expansion above using the Ceres solver with AutoDiff. All ten parameters are extracted simultaneously.

Derivative (method: derivative)

Numerical derivatives of the energy per baryon at \(n_0\). The parameters computed depend on the backend:

  • derivatives.method: gsl — GSL cubic spline; supports first and second derivatives only (n0, BE, K0, J, L). Setting isoscalar_expansion_order >= 3 with this backend raises a configuration error.

  • derivatives.method: finite_difference — central finite-difference stencils; supports up to third-order derivatives (adds Q0).

Symmetry energy when yQ grid stops at 0.5: if the EoS grid does not extend beyond :math: y_Q = 0.5, the module automatically falls back to a 2nd-order one-sided (backward) stencil using the :math: y_Q grid spacing as the step size so that the evaluation points land on or between grid nodes. J and L remain finite but are less accurate than with a full central stencil.

Z0, Ksym, Qsym, and Zsym require fourth-order derivatives and are not computable via the derivative method regardless of backend; their output columns are set to NaN.

This is available within the MUSES Calculation Engine. See the Quickstart for a minimal example of how to use the module.

To use it locally, see the Installation instructions.

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