Superconducting penetration depth models

The Fit Parameters trending panel showing synthetic MgB₂ σ(T) data (Tc = 36 K) fitted with the two-gap ``SC_TwoGap_SS`` superconductor model (the smooth trend curve). The σ(T) curve maps onto the London penetration depth via ρ_s(T) ∝ 1/λ²(T) and the relation σ ∝ 1/λ², making μSR one of the few bulk probes of the superfluid density in unconventional superconductors (Sonier et al., Rev. Mod. Phys. 72, 769 (2000)).

For an end-to-end walk-through that exercises every model on this page, see Superconductor penetration depth from σ(T).

This page documents the superconducting sigma(T) models used for transverse-field (TF) μSR vortex-state analysis and interleaves the core theory with the API for each model family.

Scope and terminology

In this context, lambda denotes the London penetration depth \(\lambda_L(T)\), not the time-domain relaxation-rate symbol Lambda used in asymmetry models for \(A(t)\).

The superconducting workflow is:

1. Fit each TF-muSR run in the time domain to extract a Gaussian rate \(\sigma(T_i)\).

2. Fit the resulting \(\sigma(T)\) trend with superconducting gap models.

3. Interpret the fitted model in terms of \(\rho_s(T)\) and optionally convert to \(\lambda_L(T)\).

Core equations

In the London limit for a triangular vortex lattice,

\[\sigma_{sc}(T) \propto \lambda_L^{-2}(T) \propto \rho_s(T),\]

with normalised superfluid density

\[\rho_s(T) = \left[\frac{\lambda(0)}{\lambda(T)}\right]^2.\]

The code returns sigma(T) through either

\[\sigma(T) = \sigma_0\,\rho_s(T) + \sigma_{bg}\]

or the quadrature convention

\[\sigma^2(T) = \sigma_{sc}^2\,\rho_s^2(T) + \sigma_{nm}^2.\]

The quadrature form is motivated when superconducting and non-superconducting linewidth channels are treated as statistically independent Gaussian contributions to the field-distribution second moment [3].

The gap is factorised as

\[\Delta(T,\mathbf{k}) = \Delta_0\,\delta(T/T_c)\,g(\mathbf{k}),\]

with the normalised superfluid density \(\rho_s(T)\) obtained from the semiclassical response integral and the reduced-gap temperature law \(\delta(T/T_c)\). The derivation-level detail of both — the response tensor and the per-symmetry gap-amplitude constants used by each model — is collapsed below; the practical starting point for most users is the model-selection table that follows.

Model selection at a glance

Physical Appropriateness Guide

Model	Node Structure	Typical Low-T Trend	Use When
SC_SWave	Fully gapped	Activated/exponential	Baseline nodeless BCS scenario
SC_DWave	Line nodes	Approximately linear (clean)	Cuprate-like nodal behaviour is expected
SC_AnisotropicS_Cos4	Nodeless for \(|a|<1\)	Weaker than nodal d-wave	Fourfold anisotropy without committed nodal symmetry
SC_SPlusG	Strong anisotropy, can be near-nodal	Intermediate between simple s and d	Anisotropic singlet systems not captured by pure d-wave
SC_NonmonotonicD	d-wave with shifted angular maximum	Can show sub-linear curvature	Electron-doped cuprate phenomenology
SC_TwoGap_SS	Two nodeless gaps	Curvature from two energy scales	MgB2-style multiband superconductivity [2]
SC_TwoGap_SD	Mixed nodal and nodeless	Interpolates between s and d signatures	One band appears nodal and another appears nodeless

Mathematical detail: the normalised superfluid-density integral

The semiclassical response tensor form used in the literature is [1]

\[R_{ij}(T) = \frac{e^2}{4\pi^3\hbar c} \int_{FS} dS_k\,\frac{v_{Fi}v_{Fj}}{|v_F|} \left[ 1 + 2\int_{\Delta(\mathbf{k},T)}^{\infty} \frac{\partial f}{\partial E} \frac{E\,dE}{\sqrt{E^2-\Delta^2(\mathbf{k},T)}} \right].\]

With \(\rho_{ii}(T)=R_{ii}(T)/R_{ii}(0)\), the practical fitting form is

\[\rho_s(T)=1+2\left\langle \int_{\Delta(T,\mathbf{k})}^{\infty} \frac{\partial f}{\partial E} \frac{E\,dE}{\sqrt{E^2-\Delta^2(T,\mathbf{k})}} \right\rangle_{FS}.\]

The implementation evaluates the energy and angular integrals with Gauss-Legendre quadrature and enforces stable limits \(\rho_s(0)=1\), \(\rho_s(T\ge T_c)=0\).

Mathematical detail: gap-amplitude approximations per symmetry

The implementation does not apply one temperature-amplitude law to every symmetry.

For isotropic s-wave-derived channels, the reduced gap uses the Carrington-Manzano interpolation [2]

\[\delta_{CM}(t)=\tanh\left(1.82[1.018(1/t-1)]^{0.51}\right),\quad t=T/T_c,\]

while models with tabulated symmetry-dependent weak-coupling shape factors use the generalised Gross-style form discussed in the review literature [1]

\[\delta_{gen}(t)=\tanh\left[ \frac{\pi}{\Delta_0(0)/(k_B T_c)}\sqrt{a\left(\frac{1}{t}-1\right)} \right],\quad t=T/T_c.\]

This reduced form is algebraically equivalent to the review expression once energies are written in units of \(k_B T_c\).

The current code paths are:

• Isotropic s-wave, alpha model, and s-wave channels in two-gap models: use \(\Delta_0(0)/(k_B T_c)=1.764\), following Carrington-Manzano.

• d-wave and nonmonotonic d-wave channels: default \(\Delta_0(0)/(k_B T_c)=2.14\), generalised form with \(a=4/3\).

• extended s-wave channel with \(g(\phi)=\cos(2\phi)\) or \(|\cos(2\phi)|\): default \(\Delta_0(0)/(k_B T_c)=2.14\), generalised form with \(a=4/3\).

• s+g channel: default \(\Delta_0(0)/(k_B T_c)=2.77\), generalised form with \(a=2\).

• Anisotropic s-wave and p-wave examples: model-dependent gap ratio; use the generalised form only when the user supplies a positive shape_factor_a. Otherwise these channels fall back to the Carrington-Manzano interpolation.

In the GUI, shape_factor_a may be left at 0 to indicate that no symmetry-specific weak-coupling value is being supplied. A positive fixed value uses the generalised amplitude law directly, while a positive free value allows the fit to determine \(a\).

Superconducting gap models

Isotropic s-wave

\[g(\phi)=1,\qquad \Delta_0(0)/(k_B T_c)\approx 1.764.\]

Use when low-T behaviour is exponentially activated and no nodal signatures are required [1].

asymmetry.core.fitting.sc.models.rho_s_wave(T: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, *, Tc: float, gap_ratio: float = 1.764, gap_mev: float | None = None, n_phi: int = 64, n_energy: int = 48) → ndarray[tuple[int, ...], dtype[float64]]

Return \(\rho_s(T)\) for isotropic s-wave gap, \(g(\phi)=1\).

Parameters:

• T – Temperature in K.

• Tc – Critical temperature in K.

• gap_ratio – Dimensionless ratio \(\Delta_0/(k_B T_c)\).

• gap_mev – Optional \(\Delta_0\) in meV. Overrides gap_ratio when given.

• n_phi – Number of angular quadrature points for Fermi-surface averaging.

• n_energy – Number of Gauss-Legendre nodes for the energy integral.

Returns:

Normalized superfluid density \(\rho_s(T)\).

Return type:

numpy.ndarray

asymmetry.core.fitting.sc.models.sc_s_wave(T: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, sigma_0: float, Tc: float, gap_ratio: float = 1.764, sigma_bg: float = 0.0, gap_mev: float | None = None) → ndarray[tuple[int, ...], dtype[float64]]

Additive isotropic s-wave model for measured \(\sigma(T)\).

\[\sigma(T) = \sigma_0\,\rho_s(T) + \sigma_{bg}\]

d_{x^2-y^2} d-wave

\[g(\phi)=\cos(2\phi),\qquad \Delta_0(0)/(k_B T_c)\approx 2.14.\]

Use when line nodes are physically expected (for example cuprate-like systems) and low-T data are inconsistent with activated behaviour [1].

asymmetry.core.fitting.sc.models.rho_d_wave(T: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, *, Tc: float, gap_ratio: float = 2.14, gap_mev: float | None = None, n_phi: int = 64, n_energy: int = 48) → ndarray[tuple[int, ...], dtype[float64]]

Return \(\rho_s(T)\) for d-wave \(g(\phi)=\cos(2\phi)\).

This model has line nodes and therefore stronger low-temperature variation than isotropic s-wave, typically close to linear-in-T in clean limits [1]. The reduced gap amplitude uses the generalized weak-coupling form with the d-wave shape factor \(a=4/3\).

asymmetry.core.fitting.sc.models.sc_d_wave(T: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, sigma_0: float, Tc: float, gap_ratio: float = 2.14, sigma_bg: float = 0.0, gap_mev: float | None = None) → ndarray[tuple[int, ...], dtype[float64]]

Additive d-wave model for measured \(\sigma(T)\).

Anisotropic s-wave (cos4)

\[g(\phi)=1+a\cos(4\phi).\]

Use when an s-wave baseline is too rigid but a strict nodal d-wave model is not yet justified. For \(|a|<1\) the model remains nodeless. If shape_factor_a is left at 0, the model uses the Carrington-Manzano temperature dependence; if shape_factor_a > 0, it switches to the generalised weak-coupling reduced-gap law with that value.

asymmetry.core.fitting.sc.models.rho_anisotropic_s_cos4(T: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, *, Tc: float, gap_ratio: float = 1.764, a_anis: float = 0.2, shape_factor_a: float = 0.0, gap_mev: float | None = None, n_phi: int = 64, n_energy: int = 48) → ndarray[tuple[int, ...], dtype[float64]]

Return \(\rho_s(T)\) for anisotropic s-wave 1 + a*cos(4phi).

a_anis controls anisotropy. For abs(a_anis) < 1 the gap remains nodeless; accidental nodes may appear when abs(a_anis) >= 1. If shape_factor_a > 0, use the generalized weak-coupling reduced-gap law with that value; otherwise fall back to the Carrington-Manzano interpolation.

asymmetry.core.fitting.sc.models.sc_anisotropic_s_cos4(T: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, sigma_0: float, Tc: float, gap_ratio: float = 1.764, a_anis: float = 0.2, shape_factor_a: float = 0.0, sigma_bg: float = 0.0, gap_mev: float | None = None) → ndarray[tuple[int, ...], dtype[float64]]

Additive anisotropic-s model for measured \(\sigma(T)\).

s+g anisotropic singlet

\[g(\theta,\phi)=\frac{1-\sin^4\theta\cos(4\phi)}{2}.\]

Use when strong anisotropy is indicated but pure \(d_{x^2-y^2}\) does not capture the observed shape [1].

asymmetry.core.fitting.sc.models.rho_s_plus_g(T: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, *, Tc: float, gap_ratio: float = 2.77, gap_mev: float | None = None, n_theta: int = 24, n_phi: int = 48, n_energy: int = 48) → ndarray[tuple[int, ...], dtype[float64]]

Return \(\rho_s(T)\) for s+g-wave anisotropic singlet gap.

\[g(\theta,\phi)=\frac{1-\sin^4\theta\cos(4\phi)}{2}.\]

The default gap_ratio=2.77 follows the weak-coupling tabulation used in Ref. [1]. The reduced gap amplitude uses the generalized weak-coupling form with the s+g shape factor \(a=2\).

Parameters:

• T – Temperature in K.

• Tc – Critical temperature in K.

• gap_ratio – Dimensionless ratio \(\Delta_0/(k_B T_c)\).

• gap_mev – Optional \(\Delta_0\) in meV. Overrides gap_ratio when given.

• n_theta – Number of polar-angle quadrature points.

• n_phi – Number of azimuthal-angle quadrature points.

• n_energy – Number of Gauss-Legendre nodes for the energy integral.

asymmetry.core.fitting.sc.models.sc_s_plus_g(T: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, sigma_0: float, Tc: float, gap_ratio: float = 2.77, sigma_bg: float = 0.0, gap_mev: float | None = None) → ndarray[tuple[int, ...], dtype[float64]]

Additive s+g model for measured \(\sigma(T)\).

\[\sigma(T) = \sigma_0\,\rho_{s+g}(T) + \sigma_{bg}.\]

Nonmonotonic d-wave

\[g(\phi)=\beta\cos(2\phi)+(1-\beta)\cos(6\phi).\]

Use when monotonic d-wave cannot reproduce curvature and an electron-doped cuprate-like nonmonotonic form is physically plausible [1].

asymmetry.core.fitting.sc.models.rho_nonmonotonic_d(T: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, *, Tc: float, gap_ratio: float = 2.14, beta_nm: float = 0.8, gap_mev: float | None = None, n_phi: int = 64, n_energy: int = 48) → ndarray[tuple[int, ...], dtype[float64]]

Return \(\rho_s(T)\) for nonmonotonic d-wave.

\[g(\phi)=\beta\cos(2\phi)+(1-\beta)\cos(6\phi).\]

This form is commonly used as a phenomenological extension when a monotonic \(\cos(2\phi)\) d-wave is insufficient [1]. The temperature-dependent gap amplitude uses the same d-wave weak-coupling shape factor \(a=4/3\) as rho_d_wave().

asymmetry.core.fitting.sc.models.sc_nonmonotonic_d(T: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, sigma_0: float, Tc: float, gap_ratio: float = 2.14, beta_nm: float = 0.8, sigma_bg: float = 0.0, gap_mev: float | None = None) → ndarray[tuple[int, ...], dtype[float64]]

Additive nonmonotonic-d model for measured \(\sigma(T)\).

Extended s-wave

The extended-s channel is implemented from the \(\cos(2\phi)\) basis (\(\cos(2\phi)\) or \(|\cos(2\phi)|\)) and uses the generalised weak-coupling reduced-gap form with \(a=4/3\), consistent with the same angular basis used for the d-wave tabulation in Ref. [1].

asymmetry.core.fitting.sc.models.rho_extended_s(T: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, *, Tc: float, gap_ratio: float = 2.14, signed: bool = False, gap_mev: float | None = None, n_phi: int = 64, n_energy: int = 48) → ndarray[tuple[int, ...], dtype[float64]]

Return \(\rho_s(T)\) for extended s-wave based on cos(2phi).

Set signed=True to preserve sign of \(\cos(2\phi)\). The default uses magnitude because the quasiparticle excitation energy depends on \(|\Delta|\). The reduced gap amplitude uses the generalized weak-coupling form with \(a=4/3\), consistent with the \(\cos(2\phi)\) basis tabulated in Ref. [1].

asymmetry.core.fitting.sc.models.sc_extended_s(T: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, sigma_0: float, Tc: float, gap_ratio: float = 2.14, signed_gap: float = 0.0, sigma_bg: float = 0.0, gap_mev: float | None = None) → ndarray[tuple[int, ...], dtype[float64]]

Additive extended-s model for measured \(\sigma(T)\).

p-wave examples

These are phenomenological alternatives for anisotropic or unconventional scenarios when domain-specific theory suggests them. If shape_factor_a is left at 0, the model uses the Carrington-Manzano temperature dependence; if shape_factor_a > 0, it switches to the generalised weak-coupling reduced-gap law with that value.

asymmetry.core.fitting.sc.models.rho_p_wave_axial(T: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, *, Tc: float, gap_ratio: float = 2.0, shape_factor_a: float = 0.0, gap_mev: float | None = None, n_phi: int = 64, n_energy: int = 48) → ndarray[tuple[int, ...], dtype[float64]]

Return \(\rho_s(T)\) for 2D axial p-wave, \(g(\phi)=\cos\phi\).

If shape_factor_a > 0, use the generalized weak-coupling reduced-gap law with that value; otherwise fall back to the Carrington-Manzano interpolation.

asymmetry.core.fitting.sc.models.sc_p_wave_axial(T: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, sigma_0: float, Tc: float, gap_ratio: float = 2.0, shape_factor_a: float = 0.0, sigma_bg: float = 0.0, gap_mev: float | None = None) → ndarray[tuple[int, ...], dtype[float64]]

Additive 2D axial p-wave model for measured \(\sigma(T)\).

asymmetry.core.fitting.sc.models.rho_p_wave_polar_3d(T: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, *, Tc: float, gap_ratio: float = 2.0, shape_factor_a: float = 0.0, gap_mev: float | None = None, n_theta: int = 24, n_phi: int = 48, n_energy: int = 48) → ndarray[tuple[int, ...], dtype[float64]]

Return \(\rho_s(T)\) for 3D polar p-wave line-node example.

Uses \(g(\theta)=\sin\theta\) with spherical angular averaging. If shape_factor_a > 0, use the generalized weak-coupling reduced-gap law with that value; otherwise fall back to the Carrington-Manzano interpolation.

MgB2-style two-gap weighted sums

The standard multiband weighted-sum form is [2]

\[\rho_{tot}(T)=w\rho_1(T)+(1-w)\rho_2(T),\qquad 0\le w\le 1.\]

For SC_TwoGap_SS both channels are isotropic s-wave. For SC_TwoGap_SD one channel is isotropic s-wave and one is d-wave.

asymmetry.core.fitting.sc.models.sc_two_gap_ss(T: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, sigma_0: float, Tc: float, gap_ratio_1: float = 1.2, gap_ratio_2: float = 2.2, weight: float = 0.5, sigma_bg: float = 0.0) → ndarray[tuple[int, ...], dtype[float64]]

Two-gap isotropic model, weighted superfluid density sum.

\[ \begin{align}\begin{aligned} \rho_s(T) = w\rho_1(T) + (1-w)\rho_2(T),\quad 0\le w\le 1.\\This is the standard MgB2-style alpha-model decomposition for multiband superconductors [2].\end{aligned}\end{align} \]

asymmetry.core.fitting.sc.models.sc_two_gap_sd(T: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, sigma_0: float, Tc: float, gap_ratio_s: float = 1.764, gap_ratio_d: float = 2.14, weight: float = 0.5, sigma_bg: float = 0.0) → ndarray[tuple[int, ...], dtype[float64]]

Two-gap mixed-symmetry model (s + d weighted sum).

Alpha-model and quadrature conventions

SC_AlphaModel rescales the weak-coupling s-wave ratio, \(\Delta_0/(k_B T_c)=\alpha_{sc}\times 1.764\).

SC_SWave_Q and SC_DWave_Q use

\[\sigma(T)=\sqrt{(\sigma_{sc}\rho_s(T))^2+\sigma_{nm}^2}.\]

Use quadrature models when the non-superconducting linewidth contribution is best represented as an independent Gaussian channel that combines at the second-moment level rather than by direct addition.

asymmetry.core.fitting.sc.models.sc_alpha_model(T: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, sigma_0: float, Tc: float, alpha_sc: float = 1.0, sigma_bg: float = 0.0) → ndarray[tuple[int, ...], dtype[float64]]

Single-gap alpha model using isotropic BCS kernel.

alpha_sc rescales the weak-coupling s-wave ratio 1.764.

asymmetry.core.fitting.sc.models.sc_s_wave_q(T: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, sigma_sc: float, sigma_nm: float, Tc: float, gap_ratio: float = 1.764, gap_mev: float | None = None) → ndarray[tuple[int, ...], dtype[float64]]

Quadrature s-wave model for measured \(\sigma(T)\).

Use this convention when independent Gaussian broadening channels combine at the second-moment level, motivating \(\sigma^2=(\sigma_{sc}\rho_s)^2+\sigma_{nm}^2\).

asymmetry.core.fitting.sc.models.sc_d_wave_q(T: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, sigma_sc: float, sigma_nm: float, Tc: float, gap_ratio: float = 2.14, gap_mev: float | None = None) → ndarray[tuple[int, ...], dtype[float64]]

Quadrature d-wave model for measured \(\sigma(T)\).

asymmetry.core.fitting.sc.models.sc_s_plus_g_q(T: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, sigma_sc: float, sigma_nm: float, Tc: float, gap_ratio: float = 2.77, gap_mev: float | None = None) → ndarray[tuple[int, ...], dtype[float64]]

Quadrature s+g model for measured \(\sigma(T)\).

This is useful when superconducting and non-superconducting linewidth channels are treated as independent Gaussian contributions that add in quadrature.

Field-dependent vortex-lattice line width (Brandt)

The models above fit the line width as a function of temperature at fixed field. The complementary measurement sweeps the applied field at fixed (low) temperature: in a type-II superconductor the vortex-lattice field-distribution second moment depends on field through Brandt’s Ginzburg-Landau result [5], and fitting it yields the absolute penetration depth \(\lambda\) and the upper critical field \(B_{c2}\) directly.

For an ideal triangular flux-line lattice (\(\kappa \gg 1\)) the Gaussian muon depolarisation rate is

\[\sigma(B_0) = \sigma_0(\lambda)\,\frac{(1-b)\,[1 + 1.21\,(1-\sqrt{b})^3]} {1 + 1.21},\qquad b = \frac{B_0}{B_{c2}},\]

where \(\sigma_0(\lambda)=\gamma_\mu\,C_B\,\Phi_0/\lambda^2\) is the field-independent London limit (the \(b\to 0\) maximum, the same scale used by asymmetry.core.fitting.sc.constants.lambda_nm_to_sigma_us()). The field factor is normalised so that \(\sigma(b\to0)=\sigma_0(\lambda)\) and \(\sigma(b\ge1)=0\). Equivalently, in the commonly cited single-crystal form, \(\sigma\,[\mu s^{-1}] \approx 4.85\times10^{4}\,(1-b)[1+1.21 (1-\sqrt{b})^3]\,\lambda^{-2}\,[\mathrm{nm}^{-2}]\) [6].

The field width is what the literature often reports as \(B_{\mathrm{rms}} = \sigma/\gamma_\mu\); fitting in \(\sigma\) (\(\mu s^{-1}\)) keeps the workflow identical to the time-domain Gaussian rate. An optional field-independent nuclear/background channel sigma_bg adds in quadrature, \(\sigma=\sqrt{\sigma_{VL}^2+\sigma_{bg}^2}\) (Pratt et al. Eq. (2) [6]).

These are field-scope components (they appear for field-trend fits, not temperature trends). x is the applied field in gauss; Bc2 is a parameter in tesla; lambda_ab is in nm.

SC_Brandt_VortexLattice

Single-crystal Brandt \(\sigma(B_0)\) for a type-II superconductor.

SC_Brandt_VortexLattice_Powder

Polycrystalline variant applying the \(3^{1/4}\) ab-plane powder average (so the line width is the single-crystal value divided by \(\sqrt{3}\), Pratt Eq. (3) [6]). Use this for powder samples — a single-crystal fit of powder data under-estimates \(\lambda_{ab}\) by a factor \(3^{1/4}=1.316\).

asymmetry.core.fitting.sc.models.brandt_field_width_sigma(B0: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, lambda_ab: float, Bc2: float, sigma_bg: float = 0.0, *, powder: bool = False) → ndarray[tuple[int, ...], dtype[float64]]

Brandt field-dependent vortex-lattice line width \(\sigma(B_0)\).

Returns the Gaussian muon depolarisation rate of a type-II superconductor as a function of applied transverse field, from which the magnetic penetration depth is extracted:

\[\sigma(B_0) = \sigma_0(\lambda)\,g(b), \qquad b = B_0/B_{c2},\]

where \(\sigma_0(\lambda)=\gamma_\mu\,C_B\,\Phi_0/\lambda^2\) is the field-independent London limit (the \(b\to 0\) maximum, supplied by asymmetry.core.fitting.sc.constants.lambda_nm_to_sigma_us()) and \(g(b)\) is the normalised Brandt factor (brandt_field_factor()). Equivalently, in the commonly cited single-crystal form,

\[\sigma(B_0)\,[\mu s^{-1}] \approx 4.85\times10^{4}\,(1-b) [1 + 1.21(1-\sqrt{b})^3]\,\lambda^{-2}\,[\mathrm{nm}^{-2}].\]

Parameters:

• B0 – Applied transverse field in gauss (the field-scope x convention).

• lambda_ab – Magnetic penetration depth in nm. For powder=True this is the ab-plane depth \(\lambda_{ab}\).

• Bc2 – Upper critical field \(B_{c2}\) in tesla.

• sigma_bg – Optional field-independent (e.g. nuclear) Gaussian rate in \(\mu s^{-1}\), added in quadrature \(\sigma=\sqrt{\sigma_{VL}^2+\sigma_{bg}^2}\) (Pratt et al., PRB 79, 052508 (2009), Eq. (2)). Default 0.

• powder – When True apply the polycrystalline ab-plane average, replacing \(\lambda_{ab}\) by \(3^{1/4}\lambda_{ab}\) so the line width is the single-crystal value divided by \(\sqrt{3}\) (Pratt Eq. (3)).

Returns:

\(\sigma(B_0)\) in \(\mu s^{-1}\).

Return type:

numpy.ndarray

References

5. 8. Brandt, Phys. Rev. B 37, 2349 (1988); 68, 054506 (2003).

asymmetry.core.fitting.sc.models.brandt_field_factor(b: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float) → ndarray[tuple[int, ...], dtype[float64]]

Return the normalised Brandt field-dependence factor \(g(b)\).

\[g(b) = \frac{(1-b)\,[1 + 1.21\,(1-\sqrt{b})^3]}{1 + 1.21}, \qquad b = B_0/B_{c2}.\]

This is Brandt’s Ginzburg-Landau result for the field dependence of the second moment of an ideal triangular vortex lattice (Brandt, Phys. Rev. B 68, 054506 (2003)), normalised so that \(g(0)=1\). For \(b \ge 1\) (at or above \(B_{c2}\) there is no vortex lattice) the factor is clamped to 0; negative b is clamped to 0 so the sqrt stays real.

Worked example: extract lambda from sigma(B0)

Fit a field sweep (line widths already extracted per run in the time domain) of a powder type-II superconductor, recovering \(\lambda_{ab}\) in nm:

import numpy as np
from asymmetry.core.fitting import Parameter, ParameterSet, ParameterCompositeModel
from asymmetry.core.fitting.parameter_models import fit_parameter_model

# Applied transverse field in gauss; sigma in us^-1 (one point per run).
B0 = np.array([100.0, 200.0, 800.0, 1600.0, 3200.0, 4000.0, 6000.0])
model = ParameterCompositeModel(["SC_Brandt_VortexLattice_Powder"])

params = ParameterSet([
    Parameter("lambda_ab", value=200.0, min=0.0),  # nm
    Parameter("Bc2", value=20.0, min=0.0),          # tesla
    Parameter("sigma_bg", value=0.0, min=0.0, fixed=True),
])

sigma = model.function(B0, lambda_ab=195.0, Bc2=25.0, sigma_bg=0.0)
sigma_err = np.full_like(B0, 0.02)
result = fit_parameter_model(B0, sigma, sigma_err, model, params)
# result.parameters -> lambda_ab ~ 195 nm; B_rms = sigma / gamma_mu.

Shared parameter semantics

sigma_0

Additive superconducting scale in \(\mu s^{-1}\).

sigma_bg

Additive background term in \(\mu s^{-1}\).

sigma_sc, sigma_nm

Superconducting and normal/nuclear terms in the quadrature convention.

Tc

Critical temperature in K.

lambda_ab

Magnetic (ab-plane) penetration depth in nm, fitted directly by the Brandt field-dependence models.

Bc2

Upper critical field in tesla, setting the reduced field \(b=B_0/B_{c2}\) in the Brandt vortex-lattice line width.

gap_ratio, gap_ratio_*

Dimensionless \(\Delta_0/(k_B T_c)\) values.

weight

Band weight constrained to \(0\le w\le 1\).

a_anis, beta_nm

Angular anisotropy controls for anisotropic-s and nonmonotonic-d models.

shape_factor_a

Optional weak-coupling shape-factor parameter for anisotropic-s and p-wave channels. In the GUI, 0 means “not supplied” and keeps the Carrington-Manzano fallback; any positive fixed or free value enables the generalised reduced-gap law.

alpha_sc

Scaling factor multiplying the weak-coupling s-wave ratio.

signed_gap

Extended-s convention control (signed versus magnitude form).

Info helpers for trend components

Each model component shown in parameter-trend fitting (for example SC_SWave, SC_DWave, SC_SPlusG, SC_TwoGap_SS) has a concise physics summary in asymmetry.core.fitting.parameter_models.PARAMETER_MODEL_COMPONENTS via the description field. Parameter-level helper text is provided by asymmetry.core.fitting.get_param_info().

Assumptions and limitations

The current implementation is intended for practical experimental fitting and follows standard approximations:

• London-limit treatment.

• Clean-limit-like thermal kernel without explicit impurity self-energy.

• Isotropic angular weighting by default.

• No explicit field-dependent nonlinear Meissner corrections.

Treat fitted parameters accordingly when comparing strongly disordered, strongly anisotropic, or very low-field datasets.

Working with lambda(T) and lambda^{-2}(T)

The same \(\rho_s(T)\) can be mapped to penetration-depth observables:

\[\lambda^{-2}(T)=\rho_s(T)\lambda^{-2}(0),\qquad \lambda(T)=\frac{\lambda(0)}{\sqrt{\rho_s(T)}}.\]

asymmetry.core.fitting.sc.models.rho_to_lambda_inv_sq(rho: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, *, lambda_0_nm: float) → ndarray[tuple[int, ...], dtype[float64]]

Convert normalized \(\rho_s(T)\) to \(\lambda^{-2}(T)\) in nm^-2.

asymmetry.core.fitting.sc.models.rho_to_lambda(rho: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, *, lambda_0_nm: float) → ndarray[tuple[int, ...], dtype[float64]]

Convert normalized \(\rho_s(T)\) to \(\lambda(T)\) in nm.

asymmetry.core.fitting.sc.models.lambda_inv_sq_from_model(T: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, *, rho_function: Callable[[...], ndarray[tuple[int, ...], dtype[float64]]], lambda_0_nm: float, **kwargs: float) → ndarray[tuple[int, ...], dtype[float64]]

Evaluate \(\lambda^{-2}(T)\) from any rho_function callable.

asymmetry.core.fitting.sc.models.lambda_from_model(T: ndarray[tuple[int, ...], dtype[float64]] | list[float] | float, *, rho_function: Callable[[...], ndarray[tuple[int, ...], dtype[float64]]], lambda_0_nm: float, **kwargs: float) → ndarray[tuple[int, ...], dtype[float64]]

Evaluate \(\lambda(T)\) from any rho_function callable.

Example: Fit sigma(T) with an s-wave model

import numpy as np
from asymmetry.core.fitting import Parameter, ParameterSet, ParameterCompositeModel
from asymmetry.core.fitting.parameter_models import fit_parameter_model

T = np.linspace(1.0, 25.0, 20)
model = ParameterCompositeModel(["SC_SWave"])

params = ParameterSet([
    Parameter("sigma_0", value=1.0, min=0.0),
    Parameter("Tc", value=24.0, min=0.0),
    Parameter("gap_ratio", value=1.764, min=0.0),
    Parameter("sigma_bg", value=0.0, min=0.0),
])

y = model.function(T, sigma_0=1.2, Tc=24.0, gap_ratio=1.9, sigma_bg=0.03)
yerr = np.full_like(T, 0.02)
result = fit_parameter_model(T, y, yerr, model, params)

References

[1] R. Prozorov and R. W. Giannetta, Supercond. Sci. Technol. 19, R41 (2006).

[2] A. Carrington and F. Manzano, Physica C 385, 205 (2003).

[3] J. E. Sonier, J. H. Brewer, and R. F. Kiefl, Rev. Mod. Phys. 72, 769 (2000).

[4] R. Prozorov, M. A. Tanatar, R. T. Gordon, C. Martin, H. Kim, V. G. Kogan, N. Ni, M. E. Tillman, S. L. Bud’ko, and P. C. Canfield, Physica C 469, 582 (2009).

[5] E. H. Brandt, Phys. Rev. B 37, 2349 (1988); Phys. Rev. B 68, 054506 (2003).

[6] F. L. Pratt, P. J. Baker, S. J. Blundell, T. Lancaster, H. J. Lewtas, P. Adamson, M. J. Pitcher, D. R. Parker, and S. J. Clarke, Phys. Rev. B 79, 052508 (2009).