Module: Definitions
Module: Definitions
Handy functions from different fields of physics
List of functions
- libnest.definitions.rho2kf(rho)[source]
Returns wavevector kF based on density rho.
It uses the relation for a uniform Fermi system and yields:
\[k_F = (3 \pi^2 \rho )^{1/3}.\]Here \(\rho\) is the density of a single nucleon species (neutrons or protons), summed over both spin components. For symmetric nuclear matter of total density \(\rho\) pass \(\rho/2\).
- Parameters:
rho (float) – density \(\rho_q\) of one nucleon species, both spin components [fm -3]
- Returns:
wavevector \(k_F\) [fm -1]
- Return type:
See also
- libnest.definitions.kf2rho(kF)[source]
Returns rho based on wavevector kF.
It uses the relation for a uniform Fermi system and yields:
\[\rho = \frac{k_F^3}{3 \pi^2}.\]The result is the density of a single nucleon species (neutrons or protons), summed over both spin components.
- Parameters:
kF (float) – Fermi wavevector \(k_F\) [fm -1]
- Returns:
density \(\rho_q\) of one nucleon species, both spin components [fm -3]
- Return type:
See also
- libnest.definitions.rho2tau(rho)[source]
Returns kinetic density \(\tau\) for uniform Fermi system of density \(\rho\).
\[\tau = \frac{3}{5} k_F^2 \rho = \frac{3}{5} \left(3 \pi^2\right)^{2/3} \rho^{5/3} \]
- libnest.definitions.rhoEta(rho_n, rho_p)[source]
Returns total density \(\rho\) and difference \(\eta\) of densities from neutron and proton densities.
\[\rho = \rho_n + \rho_p \eta = \rho_n - \rho_p\]Note
The function returns the absolute difference \(\eta = \rho_n - \rho_p\), not the relative asymmetry \(\eta/(\rho_n + \rho_p)\). This is consistent with usage in
neutron_ref_pairing_field()andproton_ref_pairing_field()where the ratio \(\eta/\rho\) is computed explicitly.
- libnest.definitions.xiBCS(kF, delta=None)[source]
Calculates the coherence length from BCS theory. If no delta argument is provided, it is assumed that we consider pure neutron matter.
\[\xi_\mathrm{BCS} = \frac{(\hbar c)^2 k_F}{\pi \Delta Mc^2}\]
- libnest.definitions.vsf(r)[source]
Calculates the velocity based on gradient of the pairing field gradient.
\[v_\mathrm{sf} = \frac{\hbar c}{M} \frac{1}{2r}\]
- libnest.definitions.vLandau(delta, kF)[source]
Returns Landau velocity.
Landau velocity shows at which velocity the superfluid medium starts to be excited: phonons appear.
\[v_L = \frac{\Delta}{\hbar k_F} c\]
- libnest.definitions.vcritical(delta, kF)[source]
Returns critical velocity for superfluid.
At this velocity the system is no longer superfluid: the Cooper pairs break.
\[v_L = \frac{e}{2} \frac{\Delta}{\hbar k_F} c \]
- libnest.definitions.superfluidFraction(j, rho, vsf)[source]
Returns superfluid fraction: how much of the matter is superfluid.
\[\eta_\mathrm{sf} = \frac{\hbar c}{M} \frac{{ j}}{\rho} \frac{1}{v_\mathrm{sf}}\]
- libnest.definitions.vsf_NV(B, vsf, A)[source]
Returns the velocity based on the gradient of the pairing field phase. However it is adjusted to the entrainment effects (definition by Nicolas Chamel Valentin Allard).
\[v_\mathrm{sf}^{NV} = \frac{\hbar^2}{2 M B}v_\mathrm{sf} + \frac{{ A}}{M}\]- Parameters:
- Returns:
velocity \(v_\mathrm{SF}^{NV}\) in units of percentage of speed of light [c]
- Return type:
- libnest.definitions.v_NV(B, j, rho, A)[source]
Returns the velocity (mass velocity). It is adjusted to the entrainment effects (definition by Nicolas Chamel Valentin Allard).
\[v^{\mathrm{NV}} = \hbar c \frac{\hbar^2}{2 M B} \frac{{ j}}{\rho} + \frac{{ A}}{M}\]- Parameters:
- Returns:
mass velocity \(v_{\mathrm{NV}}\) in units of percentage of speed of light [c]
- Return type:
- libnest.definitions.eF_n(kF)[source]
Returns Fermi energy for neutrons based on wavevector kF:
\[\epsilon_F = \frac{\hbar^2 k_F^2}{2 M_n},\]where \(M_n\) is mass of a neutron.
- libnest.definitions.Meff_hydro(rho_in, rho_out, R)[source]
Returns effective mass of a spherical nucleus of radius \(R\) of density \(\rho_{\mathrm{in}}\) immersed in superfluid neutrons of density \(\rho_{\mathrm{out}}\). The formula is from: [Mag04]
\[M_{\mathrm{eff}} = \frac{4}{3} \pi R^3 m_n \frac{(\rho_{\mathrm{in}}-\rho_{\mathrm{out}})^2}{\rho_{\mathrm{in}}+2\rho_{\mathrm{out}}},\]where \(m_n\) is mass of a neutron.
- libnest.definitions.E_minigap_delta_n(delta, rho_n)[source]
Returns the energy of minigap \(E_{mg}\) [MeV] for neutron matter given the pairing field and density.
The minigap energy can be approximated:
\[E_\mathrm{mg} = \frac{4}{3} \frac{|\Delta|^2}{\varepsilon_F},\]where \(\Delta\) is the pairing gap in the system, and \(\varepsilon_F\) is the Fermi energy.
- Parameters:
- Returns:
energy of minigap \(E_{mg}\) [MeV]
- Return type:
See also
- libnest.definitions.E_minigap_rho_n(rho_n)[source]
Returns the energy of minigap \(E_{mg}\) [MeV] for neutron matter, calculating the pairing field automatically from density.
This is a convenience wrapper around
E_minigap_delta_n()that computes the neutron pairing field using the BSk functional.The minigap energy can be approximated:
\[E_\mathrm{mg} = \frac{4}{3} \frac{|\Delta|^2}{\varepsilon_F},\]where \(\Delta\) is the pairing gap in the system, and \(\varepsilon_F\) is the Fermi energy.