Class nucmass_ldrop_skin (o2scl)

O2scl : Class List

class nucmass_ldrop_skin : public o2scl::nucmass_ldrop

More advanced liquid drop model.

In addition to the physics in nucmass_ldrop, this includes corrections for

• finite temperature

• neutron skin

• an isospin-dependent surface energy

• decrease in the Coulomb energy from external protons

Bulk energy

The central densities and radii, \( n_n, n_p, R_n, R_p \) are all determined in the same way as nucmass_ldrop, except that now \( \delta \equiv I \zeta \), where \( \zeta \) is stored in doi . Note that this means \( N > Z~\mathrm{iff}~R_n>R_p \).

If new_skin_mode is false, then the bulk energy is also computed as in nucmass_ldrop. Otherwise, the number of nucleons in the core is computed with

\[\begin{split}\begin{eqnarray*} A_{\mathrm{core}} = Z (n_n+n_p)/n_p~\mathrm{for}~N\geq Z \\ A_{\mathrm{core}} = N (n_n+n_p)/n_p~\mathrm{for}~Z>N \\ \end{eqnarray*}\end{split}\]

and \( A_{\mathrm{skin}} = A - A_{\mathrm{core}} \). The core contribution to the bulk energy is

\[ E_{\mathrm{core}}/A = \left(\frac{A_{\mathrm{core}}}{A}\right) \frac{\hbar c}{n_{L} } \left[\varepsilon(n_n,n_p) - n_n m_n - n_p m_p \right] \]

then the skin contribution is

\[ E_{\mathrm{skin}}/A = \left(\frac{A_{\mathrm{skin}}}{A}\right) \frac{\hbar c}{n_{L} } \left[\varepsilon(n_n,0) - n_n m_n \right]~\mathrm{for}~N>Z \]

and

\[ E_{\mathrm{skin}}/A = \left(\frac{A_{\mathrm{skin}}}{A}\right) \frac{\hbar c}{n_{L} } \left[\varepsilon(0,n_p) - n_p m_p \right]~\mathrm{for}~Z>N \]

Surface energy

If full_surface is false, then the surface energy is just that from nucmass_ldrop , with an extra factor for the surface symmetry energy

\[ E_{\mathrm{surf}} = \frac{\sigma}{n_L} \left(\frac{36 \pi n_L}{A} \right)^{1/3} \left( 1- \sigma_{\delta} \delta^2 \right) \]

where \( \sigma_{\delta} \) is unitless and stored in ss.

If full_surface is true, then the surface energy is modified by a cubic dependence for the medium and contains finite temperature corrections.

Coulomb energy

The Coulomb energy density (see also Ravenhall et al. (1983)) is

\[ \varepsilon = 2 \pi e^2 R_p^2 n_p^2 f_d(\chi_p) \]

where the function \( f_d(\chi) \) is

\[ f_d(\chi_p) = \frac{1}{(d+2)} \left[ \frac{2}{(d-2)} \left( 1 - \frac{d}{2} \chi_p^{(1-2/d)} \right) + \chi_p \right] \]

This class takes \( d=3 \) .

Todos and Future

Todo

In class nucmass_ldrop_skin:

• This is based on LPRL, but it’s a little different in Lattimer and Swesty. I should document what the difference is.

• The testing could be updated.

Idea for Future:

Add translational energy?

Idea for Future:

Remove excluded volume correction and compute nuclear mass relative to the gas rather than relative to the vacuum.

Idea for Future:

In principle, Tc should be self-consistently determined from the EOS.

Idea for Future:

Does this work if the nucleus is “inside-out”?

References

Designed in [Steiner08] and [Souza09] based in part on [Lattimer85] and [Lattimer91].

Note

The input parameter T should be given in units of inverse Fermis — this is a bit unusual since the binding energy is returned in MeV, but we keep it for now.

Subclassed by o2scl::nucmass_ldrop_pair

Input parameters for temperature dependence

double pp

Exponent (default 1.25)

double a0

Coefficient (default 0.935)

double a2

Coefficient (default -5.1)

double a4

Coefficient (default -1.1)

bool rel_vacuum

If true, define the nuclear mass relative to the vacuum (default true)

double Tchalf

The critical temperature of isospin-symmetric matter in \( fm^{-1} \) (default \( 20.085/(\hbar c)\).)

virtual double drip_binding_energy_d(double Z, double N, double npout, double nnout, double chi, double T)

Return the free binding energy of a \nucleus in a many-body environment.

Public Functions

nucmass_ldrop_skin()

inline virtual const char *type()

Return the type, "nucmass_ldrop_skin".

virtual int fit_fun(size_t nv, const ubvector &x)

Fix parameters from an array for fitting.

virtual int guess_fun(size_t nv, ubvector &x)

Fill array with guess from present values for fitting.

Public Members

bool full_surface

If true, properly fix the surface for the pure neutron matter limit (default true)

bool new_skin_mode

If true, separately compute the skin for the bulk energy (default false)

double doi

Ratio of \( \delta/I \) (default 0.8).

double ss

Surface symmetry energy (default 0.5)