Class nucmass_ldrop_skin (o2scl)¶
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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
Nuclear radii and volume fractions
The nuclear, neutron and proton radii are determined by
\[\begin{split}\begin{eqnarray*} R &=& \left( \frac{3 A}{4 \pi n_L} \right)^{1/3} \nonumber \\ R_n &=& \left( \frac{3 N}{4 \pi n_n} \right)^{1/3} \nonumber \\ R_p &=& \left( \frac{3 Z}{4 \pi n_p} \right)^{1/3} \end{eqnarray*}\end{split}\]where the densities \( n_L, n_n \) and \( n_p \) are determined in the same way as in nucmass_ldrop, except that now \( \delta \equiv I \zeta \), where \( \zeta \) is stored in doi .The volume fraction occupied by protons, \( \chi_p \) is
\[ \chi_p = \left(\frac{R_p}{R_{\mathrm{WS}}}\right)^3 \]and similarly for neutrons. We also define \( \chi \) as\[ \chi = \left(\frac{R}{R_{\mathrm{WS}}}\right)^3 \, . \]We need to use charge neutrality\[ \frac{4 \pi}{3} R_p^3 \left(n_p - n_{p,\mathrm{out}}\right) + \frac{4 \pi}{3} R_{\mathrm{WS}}^3 n_{p,\mathrm{out}} = \frac{4 \pi}{3} R_{\mathrm{WS}}^3 n_{e,\mathrm{out}} \]thus\[ \chi_p \left(n_p - n_{p,\mathrm{out}}\right) + n_{p,\mathrm{out}} = n_{e,\mathrm{out}} \, . \]Bulk energy
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\[ E_{\mathrm{core}}/A = \left(\frac{A_{\mathrm{core}}}{A}\right) \frac{1}{n_{L} } \left[\varepsilon(n_n,n_p) - n_n m_n - n_p m_p \right] \]then the skin contribution is\[ E_{\mathrm{skin}}/A = \left(\frac{A_{\mathrm{skin}}}{A}\right) \frac{1}{n_{L} } \left[\varepsilon(n_n,0) - n_n m_n \right] \quad\mathrm{for}\quad N>Z \]and\[ E_{\mathrm{skin}}/A = \left(\frac{A_{\mathrm{skin}}}{A}\right) \frac{1}{n_{L} } \left[\varepsilon(0,n_p) - n_p m_p \right] \quad\mathrm{for}\quad Z>N \]Surface energy
If full_surface is false, then the surface energy per baryon is just that from nucmass_ldrop, with extra factors for dimension and the surface symmetry energy
\[ E_{\mathrm{surf}}/A = \frac{\sigma d}{3 n_L} \left(\frac{36 \pi n_L}{A} \right)^{1/3} \left( 1- \sigma_{\delta} \delta^2 \right) \]where \( \sigma_{\delta} \) is unitless and stored in ss.If full_surface is true, then the following temperature- and isospin-dependent surface energy is used. Taking \( x \equiv n_p /(n_n+n_p) \), the new surface tension is
\[ \sigma(x,T) = \left[ \frac{16+b}{x^{-3}+b+(1-x)^{-3}} \right] \left[\frac{1-T^2/T_c(x)^2}{1+a(x) T^2/T_c(x)^2}\right]^{p} \]where\[ a(x) = a_0 + a_2 y^2 + a_4 y^4 \, , \]\[ T_c(x) = T_c(x=1/2) \left( 1-c y^2 - d y^4\right)^{1/2} \]and \( y=1/2-x \). The value \( p \) is stored in pp and the value \( T_c(x=1/2) \) is stored in Tchalf. Currently, the values of \( c=3.313 \) and \( d=7.362 \) are fixed and cannot be changed. The value of \( b \) is determined from\[ b=-16+\frac{96}{\sigma_{\delta}} \]which is chosen to ensure that the surface energy is identical to the expression used when full_surface is false for small values of \( \delta \).This surface energy comes from [Steiner08], which was originally based on the expression in [Lattimer85].
Coulomb energy
The Coulomb energy density is
\[ \varepsilon_{\mathrm{Coul}} = 2 \pi \chi_p e^2 R_p^2 (n_p-n_{p,\mathrm{out}})^2 f_d(\chi_p) \]where the function \( f_d(\chi_p) \) is\[ f_d(\chi_p) = \frac{1}{(d+2)} \left[ \frac{2}{(d-2)} \left( 1 - \frac{d}{2} \chi_p^{(1-2/d)} \right) + \chi_p \right] \, . \]AWS, 12/23/24: The fit to nuclear data seems to be better if we use
\[ \varepsilon_{\mathrm{Coul}} = 2 \pi \chi e^2 R_p^2 (n_p-n_{p,\mathrm{out}})^2 f_d(\chi_p) \]instead. Thus\[ E_{\mathrm{Coul}}/A = \left(\frac{2 \pi e^2 R_p^2}{nL}\right) (n_p-n_{p,\mathrm{out}})^2 f_d(\chi_p) \]When \( d=3 \), \( f_3(\chi_p) \) reduces to\[ \frac{1}{5} \left[ 2 - 3 \chi_p^{1/3} + \chi_p \right] \, . \]Then, using the approximation \( \chi_p = \chi \) and the limit \( \chi_p \rightarrow 0 \) gives the expression used in nucmass_ldrop. The second term in square brackets above gives the Wigner-Seitz approximation to the so-called “lattice” contribution\[ \varepsilon_{\mathrm{L}} = -\frac{6 \pi e^2}{5} \chi_p^{4/3} e^2 R_p^2 (n_p-n_{p,\mathrm{out}})^2 \]Taking \( n_{p,\mathrm{out}}=0 \) and using\[ R_p \Rightarrow \left( \frac{3 Z}{4 \pi n_p}\right)^{1/3} \]we get\[ \varepsilon_{\mathrm{L}} = -\left(\frac{6 \cdot 3^{2/3} \pi^{1/3}} {5 \cdot 4^{2/3}}\right) e^2 Z^{2/3} \left(\chi_p n_p\right)^{4/3} \]and noting that charge equality implies \( \chi_p n_p = n_e \), gives\[ \varepsilon_{\mathrm{L}} = -1.4508 Z^{2/3} e^2 n_e^{4/3} \]which is approximately equal to the expression used in eos_crust .See [Ravenhall83] and [Steiner12].
Todos and Future
Todo
In class nucmass_ldrop_skin:
(future) Add translational energy?
(future) Remove excluded volume correction and compute nuclear mass relative to the gas rather than relative to the vacuum.
(future) In principle, Tc should be self-consistently determined from the EOS.
(future) Does this class work if the nucleus is “inside-out”?
References
Designed in [Steiner08] and [Souza09co] based in part on [Lattimer85] and [Lattimer91].
Note
The input parameter T should be given in units of inverse Fermis. This is a bit confusing, since the binding energy is returned in MeV.
Subclassed by o2scl::nucmass_ldrop_pair
Settings
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bool rel_vacuum¶
If true, define the nuclear mass relative to the vacuum (default true)
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bool full_surface¶
If true, properly fix the surface for the pure neutron matter limit (default true)
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bool new_skin_mode¶
If true, separately compute the skin for the bulk energy (default false)
Surface parameters
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double doi¶
Ratio of \( \delta/I \) (default 0.8).
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double ss¶
Surface symmetry energy coefficient (default 0.5)
Input parameters for temperature dependence
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double pp¶
Exponent (default 1.25)
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double a0¶
Coefficient (default 0.935)
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double a2¶
Coefficient (default -5.1)
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double a4¶
Coefficient (default -1.1)
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double Tchalf¶
The critical temperature of isospin-symmetric matter in \( \mathrm{fm}^{-1} \) (default \( 20.085/(\hbar c)\).)
Basic usage
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nucmass_ldrop_skin()¶
Default constructor.
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virtual double binding_energy_densmat(double Z, double N, double npout = 0.0, double nnout = 0.0, double ne = 0.0, double T = 0.0)¶
Return the free binding energy of a nucleus in a many-body environment.
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inline virtual const char *type()¶
Return the type,
"nucmass_ldrop_skin".