Module: BSk

The functions are listed below in alphabetical order. Here, please find those functions regrouped by topic:

Pairing

• neutron_pairing_field()

• symmetric_pairing_field()

• neutron_ref_pairing_field()

• proton_ref_pairing_field()

Move

• eF_n()

• E_minigap_n()

• mu_q()

Statistics

• energy_per_nucleon()

• pressure_n()

• derivative_epsilon_rho_n()

• derivative_pressure_rho_n()

• speed_of_sound_n()

Effective mass

• effMn()

• effMp()

• isoscalarM()

• isovectorM()

Mean field

• U_q()

• B_q()

Density energy functional

See also:

• epsilon_np()

• epsilon_rho_np()

• epsilon_tau_np()

• epsilon_delta_rho_np()

• epsilon_pi_np()

• v_pi()

Auxiliary

• C0_rho()

• C1_rho()

• C0_tau()

• C1_tau()

• I()

• Lambda()

Todo

Describe properties

Todo

Give references

Module: BSk

The module that contains Brussels-Montreal parametrization with formulas for energy density, effective masses, etc. Both for uniform system and general expressions.

BSk31 Skyrme Parameters

The following table lists all Skyrme parameters used in the BSk31 functional:

BSk31 Skyrme Parameters

Parameter	Value	Unit
t₀	-2302.01	MeV fm³
t₁	762.99	MeV fm⁵
t₂	0.0	MeV fm⁵
t₃	13797.83	MeV fm⁽³⁺³ᵅ⁾
t₄	-500.0	MeV fm⁽⁵⁺³ᵝ⁾
t₅	-40.0	MeV fm⁽⁵⁺³ᵞ⁾
x₀	0.676655	dimensionless
x₁	2.658109	dimensionless
x₂t₂	-422.29	MeV fm⁵
x₃	0.83982	dimensionless
x₄	5.0	dimensionless
x₅	-12.0	dimensionless
α	1/5	dimensionless
β	1/12	dimensionless
γ	1/4	dimensionless

Individual Parameters

T0   =-2302.01

Skyrme parameter \(t_0\) [MeV fm ³]

T1   =762.99

Skyrme parameter \(t_1\) [MeV fm ⁵]

T2   =0.0

Skyrme parameter \(t_2\) [MeV fm ⁵]

T3   =13797.83

Skyrme parameter \(t_3\) [MeV fm ^(3+3*ALPHA)]

T4   =-500.

Skyrme parameter \(t_4\) [MeV fm ^(5+3*BETA)]

T5   =-40.

Skyrme parameter \(t_5\) [MeV fm ^(5+3*GAMMA)]

X0   =0.676655

Skyrme parameter \(x_0\) [1]

X1   =2.658109

Skyrme parameter \(x_1\) [1]

T2X2 =-422.29

Skyrme parameter \(x_2t_2\) [1][MeV fm ⁵]

X3   =0.83982

Skyrme parameter \(x_3\) [1]

X4   =5.

Skyrme parameter \(x_4\) [1]

X5   =-12.

Skyrme parameter \(x_5\) [1]

ALPHA =(1./5.)

[1]

BETA  =(1./12.)

[1]

GAMMA =(1./4.)

[1]

Note

these are not important at the moment

YW   =2.

[1]

FNP  =1.00

[1]

FNM  =1.06

[1]

FPP  =1.00

[1]

FPM  =1.04

[1]

KAPPAN =-36630.4

[MeV fm ⁸]

KAPPAP =-45207.2

[MeV fm ⁸]

List of functions

libnest.bsk.B_q(rho_n, rho_p, q)

Returns the mean field potential B_q (coming from variation over kinetic density, or effective mass).

\[B_q = \frac{\hbar^2}{2 M^*_q} = \frac{\hbar^2}{2 M_q} + C^\tau_0\rho + C^\tau_1 (\rho_{q} - \rho_{q'})\]

Parameters:

• rho_n (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

• rho_p (float) – proton density \(\rho_p\) [fm ^-3]; sum of both spin components

• q (string) – nucleon type choice (‘p’ - proton, or ‘n’ - neutron)

Returns:

effective mass of a proton \(M_{p}^{*}\) [MeV fm ²]

Return type:

float

libnest.bsk.C0_rho(rho_n, rho_p)

Calculates the energy functional \(C^{\rho}_0\) coefficient [BDEN02].

\[C^0_\rho(\rho) = \frac{3}{8} t_0 + \frac{3}{48} t_3 \rho^\alpha\]

Parameters:

• rho_n (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

• rho_p (float) – proton density \(\rho_p\) [fm ^-3]; sum of both spin components

Returns:

\(C^{\rho}_0\) coefficient

Return type:

float

libnest.bsk.C0_tau(rho_n, rho_p)

Calculates the energy functional \(C^{\tau}_0\) coefficient [BDEN02].

\[C^0_\tau(\rho) = \frac{3}{16} t_1 + \frac{1}{4} t_2 \left(\frac{5}{4} + x_2\right) + \frac{3}{16} t_4 \rho^\beta + \frac{1}{4} t_5 \left(\frac{5}{4} + x_5 \right) \rho^\gamma\]

Parameters:

• rho_n (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

• rho_p (float) – proton density \(\rho_p\) [fm ^-3]; sum of both spin components

Returns:

\(C^{\tau}_0\) coefficient

Return type:

float

libnest.bsk.C1_rho(rho_n, rho_p)

Calculates the energy functional \(C^{\rho}_1\) coefficient [BDEN02].

\[C^1_\rho(\rho) = -\frac{1}{4} t_0 \left(\frac{1}{2} + x_0\right) - \frac{1}{24} t_3 \left(\frac{1}{2} + x_3 \right) \rho^\alpha\]

Parameters:

• rho_n (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

• rho_p (float) – proton density \(\rho_p\) [fm ^-3]; sum of both spin components

Returns:

\(C^{\rho}_1\) coefficient

Return type:

float

libnest.bsk.C1_tau(rho_n, rho_p)

Calculates the energy functional \(C^{\tau}_1\) coefficient [BDEN02].

\[C^1_\tau(\rho) = -\frac{1}{8} t_1 \left( \frac{1}{2} + x_1 \right) + \frac{1}{8} t_2 \left(\frac{1}{2} + x_2\right) - \frac{1}{8} t_4 \rho^\beta \left( \frac{1}{2} + x_4 \right) + \frac{1}{8} t_5 \left(\frac{1}{2} + x_5 \right) \rho^\gamma\]

Parameters:

• rho_n (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

• rho_p (float) – proton density \(\rho_p\) [fm ^-3]; sum of both spin components

Returns:

\(C^{\tau}_1\) coefficient

Return type:

float

libnest.bsk.I(rho_n, rho_p, q)

Approximates the solution to the integral present in the original \(\nu^{\pi}\) formula for single-particle energies below 6.5 MeV. Check [PkecakCMWlazlowski21] and earlier papers of Chamel.

\[I_q(\rho_n,\rho_p) = \sqrt{\mu_q(\rho_n,\rho_p) } \left( 2 \ln{\frac{2 \mu_q(\rho_n,\rho_p) }{\Delta_q(\rho_n,\rho_p) }} + \Lambda\left( \frac{\varepsilon_\Lambda}{\mu_q(\rho_n,\rho_p) } \right) \right)\]

Parameters:

• rho_n (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

• rho_p (float) – proton density \(\rho_p\) [fm ^-3]; sum of both spin components

• q (string) – nucleon type choice (‘p’ - proton, or ‘n’ - neutron)

Returns:

analytic integral solution

Return type:

float

See also

Lambda() mu_q() neutron_ref_pairing_field() proton_ref_pairing_field()

libnest.bsk.Lambda(x)

Function \(\Lambda\) used in I(). Check [PkecakCMWlazlowski21] and earlier papers of Chamel.

\[\Lambda(x) = \ln(16x) + 2\sqrt{1+x} - 2 \ln\left({1+\sqrt{1+x}}\right) - 4.\]

Parameters:

x (float) – variable

Returns:

Lambda

Return type:

float

See also

I()

libnest.bsk.U_q(rho_n, rho_p, q)

Returns the mean field potential from density \(\rho\) variation.

rho_q is either rho_n or rho_p.

Parameters:

• rho_n (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

• rho_p (float) – proton density \(\rho_p\) [fm ^-3]; sum of both spin components

• q (string) – nucleon type choice (‘p’ - proton, or ‘n’ - neutron)

Returns:

Mean field potential \(U_q\) [MeV]

Return type:

float

libnest.bsk.derivative_energy_per_nucleon_n(rho)

First derivative of energy per nucleon (for neutrons only).

Parameters:

rho_n (float) – rho_n (float): neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

Returns:

derivative of energy per neutron, \(\frac{\delta e_n}{\delta \rho}\) [MeV]

Return type:

float

See also

energy_per_nucleon_n()

libnest.bsk.effMn(rho_n, rho_p)

Returns the effective mass of a neutron in nuclear medium.

\[M_n^* = \frac{\hbar^2}{2 M_n} \frac{1}{B_n(\rho_n,\rho_p)}\]

Parameters:

• rho_n (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

• rho_p (float) – proton density \(\rho_p\) [fm ^-3]; sum of both spin components

Returns:

effective mass of a neutron \(M_{n}^{*}\) [MeV]

Return type:

float

See also

B_q()

libnest.bsk.effMp(rho_n, rho_p)

Returns the effective mass of a proton in nuclear medium.

\[M_p^* = \frac{\hbar^2}{2 M_p} \frac{1}{B_p(\rho_n,\rho_p)}\]

Parameters:

• rho_n (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

• rho_p (float) – proton density \(\rho_p\) [fm ^-3]; sum of both spin components

Returns:

effective mass of a proton \(M_{p}^{*}\) [MeV]

Return type:

float

See also

B_q()

libnest.bsk.energy_per_nucleon(rho_n, rho_p)

Returns the energy per nucleon on infinite nuclear matter of given density of protons and neutrons, rho_p and rho_n, respectively, in MeV. Formula (A13) from https://journals.aps.org/prc/pdf/10.1103/PhysRevC.80.065804 [CGP09].

Parameters:

• rho_n (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

• rho_p (float) – proton density \(\rho_p\) [fm ^-3]; sum of both spin components

Returns:

energy per nucleon \(e_n\) [MeV]

Return type:

float

libnest.bsk.energy_per_nucleon_n(rho)

Energy per nucleon on infinite neutron matter of density \(\rho_n\). Derived from formula (A13) from https://journals.aps.org/prc/pdf/10.1103/PhysRevC.80.065804 [CGP09].

Parameters:

rho_n (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

Returns:

energy per neutron \(e_n\) [MeV]

Return type:

float

libnest.bsk.epsilon_delta_rho_np(rho_n, rho_p, rho_grad_n_square, rho_grad_p_square, rho_grad_square)

Energy functional \(\epsilon_{\Delta \rho}\) for particle matter, related to the interaction of the nucleons with the background fluctuations.

\[\begin{equation*} \begin{split} \varepsilon_{\Delta\rho}&(\rho_n,\vec\nabla\rho_n,\rho_p,\vec\nabla\rho_p) = +\frac{3}{16} t_1 \left[ \left( 1 + \frac{1}{2}x_1\right) \left( \nabla(\rho_n+\rho_p) \right)^2 -\left(\frac{1}{2} + x_1 \right) \sum_q \left( \nabla\rho_q \right)^2 \right] \\ &- \frac{1}{16}t_2 \left[ \left( 1 + \frac{1}{2} x_2\right) \left( \nabla(\rho_n+\rho_p) \right)^2 + \left( \frac{1}{2} + x_2\right) \sum_{q} \left( \nabla\rho_q\right)^2 \right] \\ &+\frac{3}{16} t_4 (\rho_n+\rho_p)^\beta \left[ \left(1 + \frac{1}{2}x_4\right) \left( \nabla(\rho_n+\rho_p) \right)^2 - \left(\frac{1}{2} + x_4 \right) \sum_q \left( \nabla\rho_q \right)^2 \right] \\ &+ \frac{\beta}{8}t_4 (\rho_n+\rho_p)^{\beta-1} \left[ \left(1 + \frac{1}{2}x_4\right) (\rho_n+\rho_p)\left(\nabla(\rho_n+\rho_p)\right)^2 - \left(\frac{1}{2} + x_4 \right) \nabla(\rho_n+\rho_p)\cdot\left(\sum_q \rho_q\nabla\rho_q\right) \right] \\ &- \frac{1}{16}t_5 (\rho_n+\rho_p)^\gamma \left[ \left( 1 + \frac{1}{2} x_5\right) \left( \nabla(\rho_n+\rho_p)\right)^2 +\left(\frac{1}{2} + x_5\right) \sum_{q} \left( \nabla\rho_q\right)^2 \right]. \end{split} \end{equation*} \]

Parameters:

• rho_n (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

• rho_p (float) – proton density \(\rho_p\) [fm ^-3]; sum of both spin components

• rho_grad_n_square (float) – neutron density gradient squared \(\nabla \rho\) [fm ^-8]

• rho_grad_p_square (float) – proton density gradient squared \(\nabla \rho\) [fm ^-8]

• rho_grad_square (float) – total density gradient squared \(\nabla \rho\) [fm ^-8]

Returns:

energy functional \(\epsilon_{\Delta \rho}\)

Return type:

float

libnest.bsk.epsilon_derivative_n(rho)

Calculates the derivative of energy density \(\frac{\delta \epsilon}{\delta \rho}\) in neutron matter.

Parameters:

rho (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

Returns:

derivative of energy density \(\frac{\delta \epsilon}{\delta \rho}\)

Return type:

float

See also

derivative_energy_per_nucleon_n() energy_per_nucleon()

libnest.bsk.epsilon_np(rho_n, rho_p, rho_grad_n, rho_grad_p, tau_n, tau_p, jsum2, jdiff2, nu_n, nu_p, kappa_n, kappa_p)

Calculates the total energy functional \(\epsilon\), with a limit for single-particle energies \(\epsilon_{\Lambda}\) = 6.5 MeV.

\[\varepsilon(\rho_n,\vec\nabla\rho_n, \tilde{\rho}_n,\tau_n,{ j}_n,\rho_p,\vec\nabla\rho_p,\tilde{\rho}_p,\tau_p,{ j}_p) = \nonumber \\ = \frac{\hbar^2}{2 M_n} \tau_n + \frac{\hbar^2}{2 M_p} \tau_p + \varepsilon_\rho(\rho_n,\rho_p) + \varepsilon_\tau(\rho_n,\tau_n,{ j}_n,\rho_p,\tau_p,{ j}_p) \nonumber \\ + \varepsilon_{\Delta\rho}(\rho_n,\vec\nabla\rho_n,\rho_p,\vec\nabla\rho_p) + \varepsilon_\pi(\rho_n,\vec\nabla\rho_n,\tilde{\rho}_n,\rho_p,\vec\nabla\rho_p,\tilde{\rho}_p), \]

Parameters:

• rho_n (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

• rho_p (float) – proton density \(\rho_p\) [fm ^-3]; sum of both spin components

• rho_grad_n (float) – neutron density gradient \(\nabla \rho\) [fm ^-4]

• rho_grad_p (float) – proton density gradient \(\nabla \rho\) [fm ^-4]

• tau_n (float) – kinetic density \(\tau\) [fm ^-5]

• tau_n – kinetic density \(\tau\) [fm ^-5]

• jsum2 (float) – sum of momentum density/current vectors \(j\) [fm ^-3]

• jdiff2 (float) – difference of momentum density/current vectors \(j\) [fm ^-3]

• nu_n (float) – neutron anomalous density \(\nu\) [fm ^-3]

• nu_p (float) – proton anomalous density \(\nu\) [fm ^-3]

• kappa_n (float)

• kappa_p (float) – what is kappa? (no Eq.9 in Ref.41)

TO DO: write eq for jsum2 and jdiff2

Returns

float: nergy functional \(\epsilon\)

libnest.bsk.epsilon_pi_np(rho_n, rho_p, rho_grad_n, rho_grad_p, nu_n, nu_p, kappa_n, kappa_p)

Energy functional \(\epsilon_{\pi}\) for particle matter, related to pairng energy density.

\[\varepsilon_\pi(\rho_n,\vec\nabla\rho_n,\tilde{\rho}_n,\rho_p,\vec\nabla\rho_p,\tilde{\rho}_p) =\frac{1}{4} f^\pm_n \left( v^{\pi n}(\rho_n,\rho_p) + \kappa_n|\nabla\rho_n|^2 \right) \tilde{\rho_n}^2 +\frac{1}{4} f^\pm_p \left( v^{\pi p}(\rho_n,\rho_p) + \kappa_p|\nabla\rho_p|^2 \right) \tilde{\rho_p}^2,\]

Parameters:

• rho_n (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

• rho_p (float) – proton density \(\rho_p\) [fm ^-3]; sum of both spin components

• rho_grad_n (float) – neutron density gradient \(\nabla \rho_n\) [fm ^-4]

• rho_grad_p (float) – proton density gradient \(\nabla \rho_p\) [fm ^-4]

• nu_n (float) – neutron anomalous density \(\nu_n\) [fm ^-3]

• nu_p (float) – proton anomalous density \(\nu_p\) [fm ^-3]

• kappa_n – \(\kappa_n\) (float)

• kappa_p – \(\kappa_p\) (float)

Returns:

energy functional \(\epsilon_{\pi}\) [MeV]

Return type:

float

See also

v_pi()

libnest.bsk.epsilon_rho_np(rho_n, rho_p)

Energy functional \(\epsilon_{\rho}\) for particle matter, related to the interaction of the nucleons with the background matter density.

\[\varepsilon_\rho(\rho_n,\rho_p) = C^\rho_0(\rho_n+\rho_p) \left( \rho_n + \rho_p \right)^2 + C^\rho_1(\rho_n+\rho_p) \left( \rho_n - \rho_p \right)^2\]

Parameters:

• rho_n (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

• rho_p (float) – proton density \(\rho_p\) [fm ^-3]; sum of both spin components

Returns:

energy functional \(\epsilon_{\rho}\)

Return type:

float

libnest.bsk.epsilon_tau_np(rho_n, rho_p, tau_n, tau_p, jsum2, jdiff2)

Energy functional \(\epsilon_{\tau}\) for particle matter, related to the density-dependent effective mass. It gives rise to current couplings.

\[\varepsilon_\tau(\rho_n,\tau_n,{ j}_n,\rho_p,\tau_p,{ j}_p) = C^\tau_0(\rho_n+\rho_p) \left[ (\rho_n + \rho_p)(\tau_n + \tau_p) - ({ j}_n + { j}_p)^2 \right] + C^\tau_1(\rho_n+\rho_p) \left[ (\rho_n - \rho_p)(\tau_n - \tau_p) - ({ j}_n - { j}_p)^2 \right].\]

Parameters:

• rho_n (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

• rho_p (float) – proton density \(\rho_p\) [fm ^-3]; sum of both spin components

• tau_n (float) – kinetic density \(\tau\) [fm ^-5]

• tau_n – kinetic density \(\tau\) [fm ^-5]

• jsum2 (float) – sum of momentum density/current vectors \(j\) [fm ^-3]

• jdiff2 (float) – difference of momentum density/current vectors \(j\) [fm ^-3]

Returns:

energy functional \(\epsilon_{\tau}\)

Return type:

float

libnest.bsk.isoscalarM(rho_n, rho_p)

Calculates effective isoscalar mass M_s for a given uniform system with neutron and proton densities rho_n and rho_p respectively.

\[M^*_s = 2 \left( 1/M_n^* + 1/M_p^* \right)^{-1}\]

Parameters:

• rho_n (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

• rho_p (float) – proton density \(\rho_p\) [fm ^-3]; sum of both spin components

Returns:

effective isoscalar mass \(M_{s}^{*}\) [MeV]

Return type:

float

See also

effMn() effMp()

libnest.bsk.isovectorM(rho_n, rho_p)

Calculated effective isovector mass \(M_v\) for a given uniform system with neutron and proton densities rho_n, rho_p respectively.

\[M^{*}_v = M^{*}_s M_p^{*} \frac{ 2 \rho_p-\rho}{2 M_p^*\rho_p-\rho}\]

Parameters:

• rho_n (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

• rho_p (float) – proton density \(\rho_p\) [fm ^-3]; sum of both spin components

Returns:

effective isovector mass \(M_{v}^{*}\) [MeV]

Return type:

float

See also

effMn() effMp() isoscalarM()

libnest.bsk.neutron_pairing_field(rho_n)

Returns the pairing field for uniform pure neutron nuclear matter. For kF larger than 1.38 fm ^-1 it returns (numerical) zero.

\[\Delta_{\\mathrm{NeuM}}(k_F) = \\frac{3.37968 k_F^2}{k_F^2+0.556092^2} \\frac{(k_F-1.38236)^2}{(k_F-1.38236)^2+0.327517^2},\]

Parameters:

rho_n (float) – neutron density \(\\rho_n\) [fm ^-3]; sum of both spin components

Returns:

pairing field for neutron matter \(\\Delta_{\\mathrm{NeuM}}\) [MeV]

Return type:

float

See also

symmetric_pairing_field() neutron_ref_pairing_field() proton_ref_pairing_field()

libnest.bsk.neutron_ref_pairing_field(rho_n, rho_p)

Returns the reference pairing field for neutrons in uniform matter. This is an extrapolation between \(\Delta_{\\mathrm{SM}}\) and \(\Delta_{\\mathrm{NeuM}}\). In limits \(\\eta \\rightarrow 0\) reproduces symmetric matter and \(\\eta \\rightarrow 1\), the neutron matter.

\[\\Delta_n(\\rho_n,\\rho_p) = \\Delta_{\\mathrm{SM}}(\\rho_n+\\rho_p) \\left( 1 - |\\eta| \\right) + \Delta_{\\mathrm{NeuM}}(\\rho_n) \\eta \\frac{\\rho_n}{\\rho_n+\\rho_p}\]

Parameters:

• rho_n (float) – neutron density \(\\rho_n\) [fm ^-3]; sum of both spin components

• rho_p (float) – proton density \(\\rho_p\) [fm ^-3]; sum of both spin components

Returns:

pairing field for neutrons \(\\Delta_n\) [MeV]

Return type:

float

See also

neutron_pairing_field() symmetric_pairing_field() proton_ref_pairing_field()

libnest.bsk.numerical_derivative_energy_per_nucleon_n(rho_n)

First derivative of energy per nucleon, calculated numerically using np.gradient().

Args: rho_n (float): rho_n (float): neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

Returns

float: derivative of energy per neutron, \(\frac{\delta e_n}{\delta \rho}\) [MeV]

See also

energy_per_nucleon()

libnest.bsk.numerical_derivative_epsilon_n(rho_n)

Derivative of energy density \(\epsilon\) with respect to density, calculated numerically using np.gradient() for neutron matter and using the energy_per_nucleon() function, setting proton density \(\rho_p\) to 0.

Parameters:

rho_n (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

Returns:

Derivative of \(\epsilon\)

Return type:

float

See also

energy_per_nucleon()

libnest.bsk.numerical_derivative_pressure_n(rho_n)

Derivative of pressure \(P\) with respect to density for neutron matter, calculated numerically using np.gradient().

Parameters:

rho_n (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

Returns:

Derivative of \(P\)

Return type:

float

See also

numerical_pressure_n()

libnest.bsk.numerical_pressure_n(rho_n)

Pressure \(P\) in neutron matter, calculated using the np.gradient() function to calculate the derivative of data calculated by energy_per_nucleon().

Parameters:

rho_n (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

Returns:

Pressure \(P\)

Return type:

float

See also

energy_per_nucleon()

libnest.bsk.numerical_second_derivative_energy_per_nucleon_n(rho_n)

Second derivative of energy per nucleon, calculated numerically using np.gradient().

Args: rho_n (float): rho_n (float): neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

Returns

float: second derivative of energy per neutron, \(\frac{\delta^2 e_n}{\delta \rho^2}\) [MeV]

See also

energy_per_nucleon()

libnest.bsk.numerical_speed_of_sound_n(rho_n)

Velocity of sound in neutron matter dependent on total matter density equal to neuton density \(\rho_n\), given as a percentage of speed of light. Calculated based on numerical data.

Parameters:

rho_n (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

Returns:

velocity of sound \(v_s\)

Return type:

float

See also

numerical_derivative_pressure_n() numerical_derivative_epsilon_n()

libnest.bsk.pressure_derivative_n(rho)

Calculates the derivative of pressure \(\frac{\delta P}{\delta \rho}\) in neutron matter.

Parameters:

rho (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

Returns:

derivative of pressure \(\frac{\delta P}{\delta \rho}\)

Return type:

float

See also

derivative_energy_per_nucleon_n() second_derivative_energy_per_nucleon_n()

libnest.bsk.pressure_n(rho)

Pressure \(P\) in neutron matter.

Parameters:

rho (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

Returns:

Pressure \(P\)

Return type:

float

See also

derivative_energy_per_nucleon_n()

libnest.bsk.proton_ref_pairing_field(rho_n, rho_p)

Returns the reference pairing field for protons in uniform matter. This is an extrapolation between \(\Delta_{\\mathrm{SM}}\) and \(\Delta_{\\mathrm{NeuM}}\). In limits \(\\eta \\rightarrow 0\) reproduces symmetric matter and \(\\eta \\rightarrow 1\), the neutron matter.

\[\\Delta_p(\\rho_n,\\rho_p) = \\Delta_{\\mathrm{SM}}(\\rho_n+\\rho_p) \\left( 1 - |\\eta| \\right) - \Delta_{\\mathrm{NeuM}}(\\rho_n) \\eta \\frac{\\rho_p}{\\rho_n+\\rho_p}\]

Parameters:

• rho_n (float) – neutron density \(\\rho_n\) [fm ^-3]; sum of both spin components

• rho_p (float) – proton density \(\\rho_p\) [fm ^-3]; sum of both spin components

Returns:

pairing field for protons \(\\Delta_p\) [MeV]

Return type:

float

See also

neutron_pairing_field() symmetric_pairing_field() neutron_ref_pairing_field()

libnest.bsk.second_derivative_energy_per_nucleon_n(rho)

Second derivative of energy per nucleon (for neutrons only).

Parameters:

rho_n (float) – rho_n (float): neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

Returns:

second derivative of energy per neutron, \(\frac{\delta^2 e_n}{\delta \rho^2}\) [MeV]

Return type:

float

See also

energy_per_nucleon_n() derivative_energy_per_nucleon_n()

libnest.bsk.speed_of_sound_n(rho_n)

Velocity of sound in neutron matter, dependent on total matter density equal to neuton density \(\rho_n\), given as a percentage of speed of light.

Parameters:

rho_n (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

Returns:

velocity of sound \(v_s\)

Return type:

float

See also

pressure_derivative_n() epsilon_derivative_n()

libnest.bsk.symmetric_pairing_field(rho_n, rho_p)

Returns the pairing field for uniform symmetric matter. For kF larger than 1.31 fm ^-1 it returns (numerical) zero.

\[\Delta_{\\mathrm{SM}}(k_F) = \\frac{11.5586 k_F^2}{k_F^2 + 0.489932^2}\\frac{(k_F - 1.3142)^2}{(k_F - 1.3142)^2 + 0.906146^2}.\]

Parameters:

• rho_n (float) – neutron density \(\\rho_n\) [fm ^-3]; sum of both spin components

• rho_p (float) – proton density \(\\rho_p\) [fm ^-3]; sum of both spin components

Returns:

pairing field for symmetric matter \(\\Delta_{\\mathrm{SM}}\) [MeV]

Return type:

float

See also

neutron_pairing_field() neutron_ref_pairing_field() proton_ref_pairing_field()

libnest.bsk.v_pi(rho_n, rho_p, q)

Calculates pairing strength \(\upsilon^{\pi}_q\) for neutrons or protons, for energies below 6.5 MeV.

Based on Equation 14 from Phys Rev C 104. Check [PkecakCMWlazlowski21] and earlier papers of Chamel.

Note

This function handles both neutron (q=’n’) and proton (q=’p’) cases with the same implementation. Consider splitting into separate functions for better code organization if species-specific modifications are needed.

\[v^{\pi q}(\rho_n,\rho_p) = - \frac{8 \pi^2}{I_q(\rho_n,\rho_p) } B_q(\rho_n,\rho_p)^{3/2}\]

Parameters:

• rho_n (float) – neutron density \(\rho_n\) [fm ^-3]; sum of both spin components

• rho_p (float) – proton density \(\rho_p\) [fm ^-3]; sum of both spin components

• q (string) – nucleon type choice (‘p’ - proton, or ‘n’ - neutron)

Returns:

pairing strength \(\upsilon^{pi}\) [fm ^-3]

Return type:

float

See also

I() effMn() effMp() Lambda() B_q()

References

[BDEN02] (1,2,3,4)

Michaël Bender, Jacek Dobaczewski, Jonathan Engel, and Witold Nazarewicz. Gamow-teller strength and the spin-isospin coupling constants of the skyrme energy functional. Physical Review C, 65(5):054322, 2002. URL: https://link.aps.org/doi/10.1103/PhysRevC.65.054322, doi:10.1103/PhysRevC.65.054322.

[CGP09] (1,2)

Nicolas Chamel, Stéphane Goriely, and John Michael Pearson. Further explorations of skyrme-hartree-fock-bogoliubov mass formulas. xi. stabilizing neutron stars against a ferromagnetic collapse. Physical Review C, 80(6):065804, 2009. URL: https://link.aps.org/doi/10.1103/PhysRevC.80.065804, doi:10.1103/PhysRevC.80.065804.

[PkecakCMWlazlowski21] (1,2,3)

Daniel Pęcak, Nicolas Chamel, Piotr Magierski, and Gabriel Wlazłowski. Properties of a quantum vortex in neutron matter at finite temperatures. Physical Review C, 104(5):055801, 2021.