Superconductor penetration depth from σ(T)
==========================================

This chapter is a worked example of the canonical superconductor μSR
workflow: extract the temperature dependence of the vortex-lattice
depolarisation rate :math:`\sigma(T)` from transverse-field (TF) μSR data
in the mixed state, convert it to the magnetic penetration depth
:math:`\lambda_L`, and read the pairing physics from the shape of the
curve. The data are real. They come from the **A high-Tc cuprate** example
of the WiMDA muon-school corpus — a bismuth-strontium-calcium-copper-oxide
(BiSCCO, Bi-2212) sample measured on the ISIS **MUSR** spectrometer at
400 G and 200 G transverse field over 10–125 K. Every screenshot below is
generated by driving Asymmetry over the corpus ``.nxs`` runs, and every
fitted number is graded against the reference WiMDA ``.fit``/``.dat``
outputs shipped with that example.

The same workflow reappears, with material-specific twists, in two further
corpus examples cross-referenced at the end: the iron-pnictide LiFeAs, where
the linewidth is reported as a field width :math:`B_\mathrm{rms}(T)`, and the
noncentrosymmetric superconductor Re₆Zr, where the TF superfluid density sits
alongside a zero-field time-reversal-symmetry-breaking signal.

Physical motivation
-------------------

In the vortex state of a type-II superconductor, between :math:`H_{c1}`
and :math:`H_{c2}`, an applied transverse field penetrates the sample as a
lattice of flux tubes. A muon stops at a random position relative to that
lattice and samples the inhomogeneous internal field, drawn from the
lattice's field distribution :math:`p(B)`. The distribution is asymmetric:
a sharp peak at the saddle-point field between three vortices, with a long
tail toward the high-field cores (see Brandt, Phys. Rev. B **37**, 2349
(1988)). The TF-μSR line is well approximated by a Gaussian relaxation of
the time-domain asymmetry,

.. math::

   G(t) = \exp\!\left(-\tfrac{1}{2}\sigma^2 t^2\right),

whose second moment is set by the width of :math:`p(B)`,
:math:`\sigma = \gamma_\mu \langle \Delta B^2 \rangle^{1/2}`, with
:math:`\gamma_\mu / 2\pi = 135.5\;\mathrm{MHz/T}`. For a triangular vortex
lattice in the London limit :math:`\sigma` is fixed by the penetration
depth alone, so a single measured number returns :math:`\lambda_L`. The
WiMDA teaching example uses the compact guide relation

.. math::

   \sigma(T \to 0)\;[\mu\mathrm{s}^{-1}] = \frac{75780}{\lambda_L^2\;[\mathrm{nm}]},

so that :math:`\lambda_L = \sqrt{75780/\sigma}`.

The temperature dependence carries the physics. The superfluid density
:math:`\rho_s(T) \propto \sigma(T) \propto 1/\lambda^2(T)` reflects the
superconducting gap structure: a single isotropic s-wave gap freezes
:math:`\rho_s(T)` exponentially as :math:`T \to 0`; a d-wave gap with line
nodes gives a linear-in-:math:`T` decrease at low :math:`T`; a two-gap
(s+s) system shows a shoulder marking the smaller gap's freezing scale.

.. dropdown:: Derivation — the London-limit σ ↔ λ relation and its conventions

   For an ideal triangular flux-line lattice deep in the mixed state
   (:math:`H_{c1} \ll B \ll H_{c2}`, :math:`\lambda \gg \xi`) the second
   moment of :math:`p(B)` reduces to the Brandt form

   .. math::

      \sigma = 0.0609\,\gamma_\mu\,\frac{\Phi_0}{\lambda_L^2},

   with :math:`\Phi_0` the flux quantum. Asymmetry implements this as
   :func:`asymmetry.core.fitting.sc.constants.sigma_to_lambda_nm`, using
   :math:`\gamma_\mu` in rad s⁻¹ T⁻¹. The teaching guide's coefficient
   (75780, with :math:`\sigma` in μs⁻¹ and :math:`\lambda` in nm) and
   Bernhard *et al.*'s equivalent :math:`\sigma = 7.086\times10^{4}\,
   \lambda_{ab}^{-2}` (Phys. Rev. B **52**, 10488 (1995)) differ by about
   7 %: they are the same Brandt relation written in different conventions.
   The absolute :math:`\lambda_L` therefore carries a convention-level
   ambiguity of order 10 %, while the *shape* of :math:`\lambda_L(T)` — flat
   at low :math:`T`, divergent near :math:`T_c` — is convention-independent
   and is what fixes the gap structure.

The data
--------

The **A high-Tc cuprate** example ships 48 ISIS muon runs. The
authoritative deliverable is the MUSR 400 G temperature scan (runs
1276–1289, 10–125 K), a single Gaussian-relaxed precession fitted run by
run to give :math:`\sigma(T)`; a companion MUSR 200 G scan (runs
1290–1303) supports the field comparison, and an EMU 150 G scan is present
but its reference fits are unreliable and are not used here. Run 1276
(125 K) sits above :math:`T_c` and serves as the normal-state reference.
The workflow runs entirely in the GUI: fit one TF run, inspect the vortex
line shape in the frequency domain, trend :math:`\sigma(T)` through
:math:`T_c`, and compare the two fields.

Step 1 — Fit a TF run for σ
---------------------------

.. figure:: /_generated/corpus_screenshots/corpus_bscco_tf_fit.png
   :alt: Converged Oscillatory × Gaussian fit on the BiSCCO 10 K, 400 G run
   :width: 100%

   The **Fit** dock (**Single** tab) after fitting the base-temperature
   400 G run (1277, 10 K). The composite model
   ``Oscillatory * Gaussian + Constant`` —
   ``A_1*cos(2*pi*frequency*t)*exp(-(sigma*t)^2) + A_bg`` in the **MODEL**
   box — carries the vortex signal: the oscillation at the 400 G Larmor
   frequency (``f`` = 5.24 MHz, shifted below the applied field by the
   diamagnetic response) under a Gaussian envelope that collapses the
   precession within ~2 μs. The **PARAMETERS** table reports
   ``σ`` = 1.164 μs⁻¹, and **FIT RESULTS** shows "Fit converged" with
   :math:`\chi^2_\nu = 1.09`.

For each temperature the per-run recipe is the same. Open the **Fit** dock,
build the composite ``Oscillatory * Gaussian + Constant`` in the function
builder (see :doc:`/reference/composite_models`), and record the Gaussian
envelope's :math:`\sigma` — the second moment of :math:`p(B)` — with its
uncertainty. The ``Oscillatory`` component carries the average field's
Larmor frequency; the additive ``Constant`` absorbs the non-precessing
baseline. The fitted ``σ`` = 1.164 μs⁻¹ reproduces the WiMDA reference
value of 1.1467(75) μs⁻¹ for this run to within about 1.5 %, and the
agreement holds to a few per cent across the superconducting range of the
scan. (Asymmetry delivers the loader's forward/backward asymmetry rather
than WiMDA's full eight-group grouping, which is the source of the small
residual offset.)

Step 2 — Inspect the vortex line shape
--------------------------------------

The guide asks the analyst to compare the time-domain fit with the
frequency-domain line shape, first by FFT and then by maximum entropy.
Both views make the vortex broadening directly visible.

.. figure:: /_generated/corpus_screenshots/corpus_bscco_vortex_fft.png
   :alt: FFT of the BiSCCO 10 K, 400 G run showing the broad vortex line
   :width: 100%

   The **Frequency** domain (**FFT** under **FREQUENCY DOMAIN**) of the
   10 K run. With **X Units:** ``Frequency (MHz)`` and **Reference:**
   400.00 G, the broad, asymmetric vortex line sits on the 400 G Larmor
   frequency near 5.4 MHz — the frequency-domain image of the mixed-state
   :math:`p(B)`. Its width is the same second moment the time-domain
   Gaussian :math:`\sigma` measures; above :math:`T_c` this line collapses
   to a narrow nuclear-dipolar Gaussian.

.. figure:: /_generated/corpus_screenshots/corpus_bscco_maxent.png
   :alt: Maximum-entropy spectra of the BiSCCO 10 K vortex line versus the 125 K normal-state line
   :width: 100%

   The maximum-entropy (**MaxEnt**) reconstruction, the guide's
   FFT-versus-MaxEnt comparison. The 10 K vortex distribution (filled,
   :math:`\chi^2/N = 1.04`) spreads the same unit spectral area that the
   125 K normal-state line concentrates into a single narrow peak — the
   vortex lattice broadening :math:`p(B)`. The 10 K peak (5.40 MHz) sits
   just below the normal-state line (5.42 MHz): the diamagnetic shift of
   the mixed state. On this real forward/backward asymmetry the estimator
   converges out of the box, seeding the four MUSR quadrant-group phases
   from the data — the **Seed phases from data** option, on by default.

.. note::

   A maximum-entropy line shape is a regularised *estimate* of
   :math:`p(B)`, not a calibrated linewidth. Treat the MaxEnt and FFT views
   as qualitative pictures of the broadening; the quantitative
   :math:`\sigma` comes from the time-domain Gaussian fit of Step 1.

Step 3 — Trend σ(T) through T_c
-------------------------------

.. figure:: /_generated/corpus_screenshots/corpus_bscco_sigma_t.png
   :alt: BiSCCO vortex depolarisation rate σ(T) from 10 to 125 K in the trend panel
   :width: 100%

   The **Fit Parameters** trend panel with the per-run 400 G
   :math:`\sigma(T)` series loaded and ``σ (µs⁻¹)`` on the y-axis. The
   points reproduce the 14-row WiMDA reference trend: :math:`\sigma` falls
   from ≈ 1.16 μs⁻¹ at 10 K to a small residual ≈ 0.06 μs⁻¹ above
   :math:`T_c \approx 107\;\mathrm{K}`. The base-temperature value maps to
   :math:`\lambda_L = \sqrt{75780/\sigma} \approx 255\;\mathrm{nm}` — noted
   on the plot as indicative only (see below).

Loading each run's fit result into the trend panel and selecting
``σ (µs⁻¹)`` from **Y parameters** plots the depolarisation rate against
temperature. The curve has the canonical mixed-state form: a low-:math:`T`
plateau where the superfluid density is fully developed, a monotonic fall
on warming as :math:`\rho_s` melts, and a collapse to the nuclear-dipolar
background as superconductivity is lost near 107 K. The lowest-:math:`T`
value, :math:`\sigma(10\;\mathrm{K}) \approx 1.16\;\mu\mathrm{s}^{-1}`,
gives :math:`\lambda_L \approx 255\;\mathrm{nm}` through the guide relation,
consistent with the reference σ (257 nm) and the literature order of
magnitude for optimally-doped Bi-2212 (:math:`\lambda_{ab} \approx
260\;\mathrm{nm}`).

.. warning::

   **A single :math:`\lambda_L` is only indicative for Bi-2212 at these
   fields.** Bi-2212 is extremely anisotropic, and above a crossover field
   :math:`B^* \approx 500\;\mathrm{G}` its flux lines break into weakly
   correlated pancake vortices. All three teaching fields (150, 200 and
   400 G) sit below :math:`B^*`, so the extended-flux-line interpretation of
   :math:`\sigma` as :math:`\lambda_{ab}` is not physically robust here (see
   Bernhard *et al.*, Phys. Rev. B **52**, 10488 (1995), which states that
   meaningful :math:`\lambda_{ab}` values "cannot be extracted from
   :math:`\sigma`" in the usual way for this material). Quote
   :math:`\lambda_L \approx 255\;\mathrm{nm}` as an order-of-magnitude guide
   from :math:`\lambda_L = \sqrt{75780/\sigma}`, and grade the
   :math:`\sigma(T)` curve itself rather than a single derived
   :math:`\lambda_L`.

Step 4 — Compare the two fields
-------------------------------

.. figure:: /_generated/corpus_screenshots/corpus_bscco_field_compare.png
   :alt: BiSCCO σ(T) at 400 G overlaid on 200 G in the trend panel
   :width: 100%

   The trend panel overlaying the 400 G and 200 G :math:`\sigma(T)` scans
   as two coloured series (armed through ``select_series``, the equivalent
   of Shift-clicking the second series pill). Both are genuine per-run TF
   Gaussian fits. The 200 G plateau (≈ 0.9 μs⁻¹) sits *below* the 400 G
   plateau (≈ 1.15 μs⁻¹): the field dependence of :math:`\sigma` in Bi-2212
   is the pancake-vortex physics of the warning above, not fit scatter. The
   10 K, 200 G run (1291) is excluded — its reference fit is a documented
   negative-:math:`\sigma` pathology.

The guide's field-comparison task is exactly this overlay. For a
conventional, weakly-anisotropic superconductor deep in the mixed state,
:math:`\sigma` is field-independent between :math:`H_{c1}` and
:math:`H_{c2}`, and the two scans would coincide. That they do not is the
signature of dimensional crossover: below :math:`B^*` the pancake vortices
in adjacent CuO₂ planes decouple, the lattice order softens, and the
measured second moment shrinks. The overlay is a plot-only comparison;
**Export TSV** writes both series, while GLE export and any trend
**Model Fit** apply to the active series alone.

Fitting a gap model
-------------------

Where the material and the field permit a clean single-:math:`\lambda`
interpretation — which Bi-2212 at 150–400 G does not — the next step is to
fit the superfluid density :math:`\sigma(T)` (background-subtracted) to a
gap model and read off the pairing symmetry. Asymmetry's parametric-model
registry ships the standard family, each documented in the component-info
dialog:

- ``SC_SWave`` — a single isotropic, fully-gapped BCS order parameter;
  :math:`\sigma(T)` saturates exponentially at low :math:`T`.
- ``SC_DWave`` — a nodal d-wave gap; the line nodes give :math:`\sigma(T)`
  a linear low-:math:`T` slope rather than activated saturation.
- ``SC_TwoGap_SS`` — two nodeless gaps for multiband superconductors (the
  MgB₂-type case), with a weight partitioning the superfluid density
  between the two channels.
- ``SC_TwoGap_SD`` — a mixed s + d interpretation for cases where neither
  pure form fits the full temperature range.

The ``_Q`` variants (``SC_SWave_Q``, ``SC_DWave_Q``) combine the
superconducting and normal linewidth contributions in quadrature rather
than linearly, appropriate when the two sources are independent. Fit these
in the trend panel by selecting ``σ (µs⁻¹)`` and clicking **Model Fit**, or
export the trend (**Export TSV**) and fit outside the GUI with
``scipy.optimize.curve_fit``. In every case, include a few points above
:math:`T_c` so the high-:math:`T` tail anchors both :math:`T_c` and the
nuclear-dipolar background :math:`\sigma_{bg}`, and subtract
:math:`\sigma_{bg}` in quadrature before converting :math:`\sigma \to
\lambda`.

Cross-reference — LiFeAs, the field width B_rms(T)
--------------------------------------------------

The **LiFeAs** corpus example runs the same vortex-lattice workflow on the
"111" iron-arsenide superconductor, but reports the linewidth as a field
width :math:`B_\mathrm{rms} = \sigma_\mathrm{VL}/\gamma_\mu` (mT) rather
than a rate, following Pratt *et al.*, Phys. Rev. B **79**, 052508 (2009).
The superconducting broadening adds in quadrature to a
temperature-independent nuclear width, :math:`\sigma^2 = \sigma_\mathrm{VL}^2
+ \sigma_n^2`, and the powder London limit gives
:math:`B_\mathrm{rms} = \sqrt{0.00371}\,\Phi_0/(3^{1/4}\lambda_{ab})^2`.

.. figure:: /_generated/corpus_screenshots/corpus_lifeas_brms_t.png
   :alt: LiFeAs vortex-lattice field width B_rms(T) for two samples in the trend panel
   :width: 100%

   :math:`B_\mathrm{rms}(T)` at :math:`B_0 = 40\;\mathrm{mT}` for two LiFeAs
   samples overlaid in the trend panel: Sample 1 (real Asymmetry
   two-Gaussian fits, :math:`T_c = 16\;\mathrm{K}`, plateau ≈ 1.9 mT)
   against Sample 2 (digitised from the paper's Fig. 1, :math:`T_c \approx
   12\;\mathrm{K}`, ≈ 1.2 mT). The dotted lines mark the London-limit
   plateaux for the printed :math:`\lambda_{ab} = 195(2)\;\mathrm{nm}` and
   :math:`244(2)\;\mathrm{nm}`. Points near :math:`T_c`, where the
   signal/background split degenerates, are ringed grey and drop out of the
   trend.

.. sidebar:: Data handling — pick the right detector pair

   .. figure:: /_generated/corpus_screenshots/corpus_lifeas_pair_select.png
      :alt: LiFeAs base-T run in default Forward/Back pairing versus the Up/Down transverse pair
      :width: 100%

   These are spin-rotated ("TF WED") GPS runs: the muon spin is rotated
   transverse, so the precession appears in the **Up/Down** detector pair.
   The loader's default Forward/Back pairing sees the two detectors in phase
   and the precession *cancels* (top), leaving noise; regrouping onto the
   Up/Down transverse pair recovers the clean 5.44 MHz vortex-damped signal
   (bottom). This is the essential grouping choice for the example, made in
   the **Grouping** dialog.

Cross-reference — Re₆Zr, superfluid density and a TRSB step
-----------------------------------------------------------

The **TRSB** corpus example (Re₆Zr, a noncentrosymmetric superconductor;
Singh, Hillier *et al.*, Phys. Rev. Lett. **112**, 107002 (2014)) carries
two observables. The transverse-field superfluid density behaves exactly as
above, while a separate zero-field measurement reveals broken time-reversal
symmetry.

.. figure:: /_generated/corpus_screenshots/corpus_trsb_sigma_sc_t.png
   :alt: Re6Zr TF superfluid-density σ_sc(T) melting through T_c in the trend panel
   :width: 100%

   The TF depolarisation rate :math:`\sigma_{sc}(T)` at 40 mT, falling from
   ≈ 0.45 μs⁻¹ at base temperature toward a ≈ 0.17 μs⁻¹ normal-state
   plateau through :math:`T_c = 6.75\;\mathrm{K}`. The low-:math:`T` plateau
   and the melt between ~3 and ~7 K are the s-wave superfluid-density
   signature, reproducing the WiMDA reference trend.

.. figure:: /_generated/corpus_screenshots/corpus_trsb_sigma_t_step.png
   :alt: Re6Zr zero-field Gaussian Kubo-Toyabe σ(T) showing the small TRSB step below T_c
   :width: 100%

   The zero-field Gaussian Kubo–Toyabe rate :math:`\sigma(T)`, framed on a
   tight 0.250–0.270 μs⁻¹ axis to resolve the small spontaneous rise below
   :math:`T_c = 6.75\;\mathrm{K}` (:math:`\Delta\sigma \approx
   0.006\text{–}0.01\;\mu\mathrm{s}^{-1}`). The onset of static spontaneous
   fields at :math:`T_c` — decoupled by a 10 mT longitudinal field — is the
   time-reversal-symmetry-breaking signature. Unlike the TF superfluid
   density, this small zero-field step is the headline physics of that
   example, and Asymmetry renders it legibly only because the axis is framed
   to the signal.

Interpretation
--------------

What the shape of :math:`\sigma(T)` (equivalently :math:`\rho_s(T)`) tells
you about the material:

- **Exponential low-:math:`T` saturation.** A single isotropic s-wave gap
  (``SC_SWave``); the :math:`\Delta/k_B T_c` value distinguishes
  weak-coupling (BCS, 1.764) from strong-coupling (Re₆Zr reaches 2.1).
- **Linear-in-:math:`T` low-:math:`T` decrease.** Line nodes in the gap
  (``SC_DWave``); the hallmark of a d-wave order parameter, as in the
  cuprates.
- **A shoulder or kink near :math:`T/T_c \approx 0.3\text{–}0.5`.**
  Multiband superconductivity with two gaps (``SC_TwoGap_SS``); the smaller
  gap freezes out at its own scale.
- **A field-dependent plateau.** As in Bi-2212 here: not a pairing-symmetry
  statement at all, but dimensional-crossover physics that invalidates a
  single-:math:`\lambda` reading.

Assumptions and limitations
---------------------------

- **The London limit.** :math:`\sigma \to \lambda` is exact only for
  :math:`\lambda \gg \xi`. Where :math:`\xi` approaches :math:`\lambda` the
  Brandt expression acquires corrections (see Sonier *et al.*, Rev. Mod.
  Phys. **72**, 769 (2000), Appendix B).
- **The nuclear-dipolar background.** Even above :math:`T_c` the muon sees a
  non-zero :math:`\sigma` from nuclear dipoles (≈ 0.055 μs⁻¹ here). Fit it
  and subtract it in quadrature before converting :math:`\sigma \to
  \lambda`.
- **Anisotropy and the vortex regime.** Below :math:`H_{c1}` the field does
  not enter as a lattice; above :math:`H_{c2}` superconductivity is
  suppressed; and in extremely anisotropic layered materials a pancake-
  vortex crossover (Bi-2212 above) breaks the single-:math:`\lambda`
  picture even well inside the mixed state.
- **Convention on the absolute :math:`\lambda`.** The guide's coefficient
  (75780) and the Brandt/Bernhard form differ by ~7 %, so the absolute
  :math:`\lambda_L` is convention-dependent at the ~10 % level; the *shape*
  of :math:`\lambda_L(T)` is not.
- **Data range.** Without points above :math:`T_c` the fit cannot separate
  :math:`T_c` from :math:`\sigma_{bg}`; without points below
  :math:`T/T_c \approx 0.3` a small gap in a two-gap model is poorly
  constrained.

References
---------

.. rubric:: References

- E. H. Brandt, Phys. Rev. B **37**, 2349 (1988) — the vortex-lattice field
  distribution and its second moment.
- J. E. Sonier, J. H. Brewer, and R. F. Kiefl, Rev. Mod. Phys. **72**, 769
  (2000) — the canonical review of TF μSR in superconductors.
- C. Bernhard, Ch. Niedermayer, U. Binninger, A. Hofer, Ch. Wenger, J. L.
  Tallon, G. V. M. Williams, E. J. Ansaldo, J. I. Budnick, C. E. Stronach,
  D. R. Noakes, and M. A. Blankson-Mills, Phys. Rev. B **52**, 10488 (1995)
  — the σ ↔ λ relation and the pancake-vortex caveat for Bi-2212.
- 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) — LiFeAs :math:`B_\mathrm{rms}(T)` and
  :math:`\lambda_{ab}`.
- R. P. Singh, A. D. Hillier, B. Mazidian, J. Quintanilla, J. F. Annett, D.
  McK. Paul, G. Balakrishnan, and M. R. Lees, Phys. Rev. Lett. **112**,
  107002 (2014) — Re₆Zr TRSB and superfluid density.
- S. J. Blundell, R. De Renzi, T. Lancaster, and F. L. Pratt, *Muon
  Spectroscopy: An Introduction* (Oxford University Press, Oxford, 2022),
  Ch. 9 (superconductors), especially Ch. 9.5 (penetration depth and gap
  structure).
- A. Amato and E. Morenzoni, *Introduction to Muon Spin Spectroscopy:
  Applications to Solid State and Material Sciences*, Lecture Notes in
  Physics Vol. 961 (Springer, Cham, 2024), Ch. 6 — the vortex-state field
  distribution, multiband superconductivity, and unconventional pairing.

Cross-references
----------------

- :doc:`/reference/loading_data` — loading the ISIS ``.nxs`` and PSI
  ``.bin`` corpus runs.
- :doc:`/reference/detector_grouping` — the transverse (Up/Down) grouping
  choice for spin-rotated runs.
- :doc:`/reference/composite_models` — building the
  ``Oscillatory * Gaussian + Constant`` time-domain composite.
- :doc:`/reference/fourier_analysis` — inspecting the :math:`p(B)` line
  shape by FFT and maximum entropy.
- :doc:`/reference/parameter_trending` — the σ → λ → gap-model pipeline and
  multi-series overlays.
- :doc:`/reference/sc_penetration_depth` — the parametric ``SC_*`` gap-model
  reference page.