Module: Definitions
Todo
Describe properties
Todo
Give references
Module: Definitions
Handy functions from different fields of physics
List of functions
- libnest.definitions.E_minigap_delta_n(delta, rho_n)
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:
delta (float) – pairing field for neutrons \(\Delta_n\) [MeV]
rho_n (float) – neutron density \(\rho_n\) [fm -3]; sum of both spin components
- Returns:
energy of minigap \(E_{mg}\) [MeV]
- Return type:
float
See also
- libnest.definitions.E_minigap_rho_n(rho_n)
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.
- Parameters:
rho_n (float) – neutron density \(\rho_n\) [fm -3]; sum of both spin components
- Returns:
energy of minigap \(E_{mg}\) [MeV]
- Return type:
float
- libnest.definitions.Meff_hydro(rho_in, rho_out, R)
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.
- Parameters:
rho_in (float) – density of nucleus \(\rho_{\mathrm{in}}\) [fm -3]
rho_out (float) – density of superlufid neutrons \(\rho_{\mathrm{out}}\) [fm -3]
- Returns:
effective mass \(M_{\mathrm{eff}}\) in units neutron mass
- Return type:
float
- libnest.definitions.eF_n(kF)
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.
- Parameters:
kF (float) – wavevector \(k_F\)
- Returns:
Fermi energy \(\epsilon_F\) [MeV]
- Return type:
float
- libnest.definitions.kf2rho(kF)
Returns rho based on wavevector kF.
It uses the relation for a uniform Fermi system and yields:
\[\rho = \frac{k_F^3}{3 \pi^2}.\]If we have a two-component mixture of protons and neutrons, both having two spin components (up and down), then \(\rho\) states for the density of one isospin, and one spin component only.
- Parameters:
kF (float) – density \(\rho\) for a single component
- Returns:
wavevector \(k_F\) [fm -1]
- Return type:
float
See also
- libnest.definitions.mu_q(rho_n, rho_p, q)
Calculates the chemical potential \(\mu\) defined with the wavevector \(k_F\) [GCP09].
\[\mu_q = \frac{\hbar^2 k_F^2}{2M_q}\]- 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:
chemical potential \(\mu\) [MeV]
- Return type:
float
- libnest.definitions.rho2kf(rho)
Returns wavevector kF based on density rho.
It uses the relation for a uniform Fermi system and yields:
\[k_F = (3 \pi^2 \rho )^{1/3}.\]If we have a two-component mixture of protons and neutrons, both having two spin components (up and down), then \(\rho\) states for the density of one isospin, and one spin component only.
- Parameters:
rho (float) – density \(\rho\) for a single component
- Returns:
wavevector \(k_F\) [fm -1]
- Return type:
float
See also
- libnest.definitions.rho2tau(rho)
Returns kinetic density \(\tau\) for uniform Fermi system of density \(\rho\).
\[\tau = \frac{3}{5} \left(3 \pi\right)^{2/3} \rho^{5/3} \]- Parameters:
rho (float) – density \(\rho\) [fm -3] for a single component
- Returns:
kinetic density \(\tau\) [fm -3]
- Return type:
float
- libnest.definitions.rhoEta(rho_n, rho_p)
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.- 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:
pair of total density \(\rho\) [fm -3], and density difference \(\eta\) [fm -3]
- Return type:
float
- libnest.definitions.superfluidFraction(j, rho, vsf)
Returns superfluid fraction: how much of the matter is superfluid.
\[\eta_\mathrm{sf} = \frac{\hbar c}{M} \frac{{ j}}{\rho} \frac{1}{v_\mathrm{sf}}\]- Parameters:
rho (float) – density \(\rho\) [fm -3]
j (float) – a three-component current vector \(\vec j\) [fm -4]
vsf (float) – \(v_{\mathrm{sf}}\) [c]
- Returns:
superfluid fraction: a number between 0 and 1 [1]
- Return type:
float
- libnest.definitions.vLandau(delta, kF)
Returns Landau velocity.
Landau velocity shows at which velocity the superfluid medium starts to be excited: phonons appear.
\[v_L = \frac{\Delta}{\hbar k_F} c\]- Parameters:
delta (float) – \(\Delta\) [MeV]
kF (float) – \(k_F\) [MeV]
- Returns:
Landau velocity \(v_L\) in units of speed of light [c]
- Return type:
float
- libnest.definitions.v_NV(B, j, rho, A)
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:
B (float) – mean field potential coming from kinetic energy variation B [MeV fm 2]
j (float) – a three-component current vector \(\vec j\) [fm -4]
rho (float) – density \(\rho\) [fm -3]
A (float) – mean field potential coming from current variation A [MeV fm]
- Returns:
mass velocity \(v_{\mathrm{NV}}\) in units of percentage of speed of light [c]
- Return type:
float
- libnest.definitions.vcritical(delta, kF)
Returns critical velocity for superfluid.
At this velocity the system is no longer superfluid: the Cooper pairs break.
\[v_L = \frac{e}{2} \frac{\Delta}{\hbar k_F} c \]- Parameters:
delta (float) – \(\Delta\) [MeV]
kF (float) – \(k_F\) [MeV]
- Returns:
Landau velocity \(v_c\) in units of speed of light [c]
- Return type:
float
- libnest.definitions.vsf(r)
Calculates the velocity based on gradient of the pairing field gradient.
\[v_\mathrm{sf} = \frac{\hbar c}{M} \frac{1}{2r}\]- Parameters:
r (float) – distance from the center of a vortex in femtometers [fm]
- Returns:
velocity in units of percentage of speed of light [c]
- Return type:
float
- libnest.definitions.vsf_NV(B, vsf, A)
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:
B (float) – mean field potential coming from kinetic energy variation B [MeV fm 2]
vsf (float) – \(v_{\mathrm{sf}}\) [c]
A (float) – mean field potential coming from current variation A [MeV fm]
- Returns:
velocity \(v_\mathrm{SF}^{NV}\) in units of percentage of speed of light [c]
- Return type:
float
- libnest.definitions.xiBCS(kF, delta=None)
Calculates the coherence length from BCS theory. If no delta argument is provided, it is assumed that we consider pure neutron matter.
\[\xi_\mathrm{BCS} = \frac{(\hbar c)^2 k_F}{\pi \Delta Mc^2}\]- Parameters:
k_F (float) – Fermi momentum [fm -1]
delta (float) – pairing field [MeV]
- Returns:
coherence length [fm]
- Return type:
float