{"schemaVersion":"1.0","revision":"fa961ab24277a91737c3","article":{"id":"scwiki-a2","topicId":"scwiki-topic-a2","code":"A2","slug":"meissner-effect","title":"The Meissner effect and magnetic evidence","summary":"Distinguish equilibrium flux expulsion from shielding, then interpret penetration depth and magnetic volume estimates with their geometric assumptions.","aliases":["Meissner Ochsenfeld","magnetic shielding","London penetration depth","diamagnetism"],"evidenceTypes":["experiment","theory","hypothesis"],"relatedCodes":["A1","A6","A7","A3","A5","F7"],"editorialStatus":"scoped-check","contentState":"pilot","modifiedAt":"2026-10-19","review":{"checkedAt":"2026-10-18","scope":"Selected historical experiment reproduction and London original scan, modern demagnetization manuscript and Hardy abstract; independent London boundary-value, demagnetization and sheet-energy checks.","limitations":["Scoped passage checks and model calculations are not full scientific acceptance or independent experimental replication.","London pp. 72–73 and Prozorov–Kogan pp. 1–2 only; arbitrary-shape simulation, microscopic/nonlocal theories and full historical manuscripts not exhaustively checked.","Hardy remains abstract-scoped. Bozovic LSCO is a bibliographic pointer; no scaling dataset or experimental Tc was re-extracted.","Shielding, demagnetization and sheet inductance formulas require their stated geometry, SI normalization and response regime."]},"provenance":{"repository":"JackZH26/jzis-superconductivity","commit":"0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf","path":"A_phenomenology/A2_meissner_effect.md","sha256":"6f9cc20701142991b8034214d36e80f59cf2a70b125e0e4f85c505b85e6c45f6","sourceId":"file-95fa6ac71c9a09ec"},"sourceMap":[{"startLine":1,"endLine":5,"targetSections":["scope"],"decision":"edited-with-scoped-evidence","note":"Magnetic evidence scope retained."},{"startLine":6,"endLine":11,"targetSections":["scope","limits-and-disputes"],"decision":"edited-with-scoped-evidence","note":"Selected historical primary observations distinguish wall and cavity; London magnetic-state postulate separated from perfect conductivity."},{"startLine":12,"endLine":21,"targetSections":["mathematical-assumptions","experimental-criteria"],"decision":"edited-with-scoped-evidence","note":"Modern SI London limits, explicit local model checks and Hardy abstract scope."},{"startLine":22,"endLine":31,"targetSections":["mathematical-assumptions","experimental-criteria","limits-and-disputes"],"decision":"edited-with-scoped-evidence","note":"Demagnetization now has a specific modern primary model source, rather than a general historical citation."},{"startLine":32,"endLine":35,"targetSections":["mathematical-assumptions","limits-and-disputes"],"decision":"edited-with-scoped-evidence","note":"Sheet normalization clarified; quantum-geometric decomposition deferred to dedicated F7, no unchecked Tian claim repeated."},{"startLine":36,"endLine":43,"targetSections":["experimental-criteria","limits-and-disputes"],"decision":"edited-with-scoped-evidence","note":"Bibliography repaired; exact checked source scope retained."},{"startLine":44,"endLine":46,"targetSections":["discovery-implications"],"decision":"edited-with-scoped-evidence","note":"Do not infer mechanism or volume without a model and controls."}],"corrections":[{"id":"scwiki-a2-correction-1","description":"Repaired corrupted historical title and multiline chemical-formula metadata in the Hardy reference.","sourceId":"source-512705f3e64d88d7","scope":"Bibliographic presentation; not a new scientific result."},{"id":"scwiki-a2-correction-2","description":"Replaced broad London attribution for geometry corrections with a specifically checked modern demagnetization source; separated theoretical/reporting statements from experiments.","sourceId":"source-37f27e50dbdcf3a5","scope":"Claim mapping and SI/model boundary."},{"id":"scwiki-a2-correction-3","description":"Specified Hardy sample composition and probe; changed unsupported ambient-pressure/equilibrium labels to unknown at the checked abstract scope.","sourceId":"source-b7bc48c845a47b53","scope":"Evidence conditions; no transition temperature added."}],"equationChecks":[{"targetSectionId":"mathematical-assumptions","tex":"B(x)/B_0=\\cosh(x/\\lambda)/\\cosh(a/\\lambda)","check":"Independent finite-difference boundary-value solutions at five thickness ratios and two grid sizes agree and converge; parallel infinite slab only."},{"targetSectionId":"mathematical-assumptions","tex":"\\chi_{app}=\\chi/(1+N\\chi)","check":"Four SI ellipsoid perfect-diamagnet cases satisfy B=mu0(Hint+M)=0; apparent magnitude can exceed one."},{"targetSectionId":"mathematical-assumptions","tex":"L_k=\\mu_0\\lambda^2/d","check":"Carrier density/charge/mass convention and sheet kinetic energy agree for uniform thin-film current; not an arbitrary film response fit."}],"references":[{"id":"source-512705f3e64d88d7","doi":"10.1007/BF01504252","title":"Ein neuer Effekt bei Eintritt der Supraleitfähigkeit","year":1933,"url":"https://doi.org/10.1007/BF01504252","publicationStatus":"journal-article","verification":{"scope":"primary-reproduction-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://courses.physics.ucsd.edu/2025/Fall/physics139/meissnerpapertranslated.pdf","note":"Selected first two PDF pages: German original with English translation by A. M. Forrest, external field redistribution in tin/lead and retained field in a lead-tube cavity. Original apparatus, uncertainties and complete translation not independently reproduced."}},{"id":"source-0590b237a1d87de4","doi":"10.1098/rspa.1935.0048","title":"The electromagnetic equations of the supraconductor","year":1935,"url":"https://doi.org/10.1098/rspa.1935.0048","publicationStatus":"journal-article","verification":{"scope":"primary-scan-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://w2agz.com/Library/Classic%20Papers%20in%20Superconductivity/London%20%26%20London%2C%20Proc%20Roy%20Soc%20London%20A149%2C%2071%20%281935%29.pdf","note":"Original pp. 72–73 visually checked, Eqs. (2)–(6): initial-field integration constant, screening equation and additional magnetic-state postulate. Public equations use modern SI conventions and stated local assumptions; not an exhaustive check of the historical paper or later electrodynamics."}},{"id":"source-b7bc48c845a47b53","doi":"10.1103/PhysRevLett.70.3999","title":"Precision measurements of the temperature dependence of lambda in YBa2Cu3O6.95: Strong evidence for nodes in the gap function","year":1993,"url":"https://doi.org/10.1103/PhysRevLett.70.3999","publicationStatus":"journal-article","verification":{"scope":"primary-abstract-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.70.3999","note":"Publisher abstract: single-crystal YBa2Cu3O6.95, 900 MHz resonator, penetration-depth change and linear term around 3–25 K. Full curves, fitting and defect interpretation not independently reproduced; not a phase measurement."}},{"id":"source-b96ffab89d45664d","doi":"10.1038/nature19061","title":"Dependence of the critical temperature in overdoped copper oxides on superfluid density","year":2016,"url":"https://doi.org/10.1038/nature19061","publicationStatus":"journal-article","verification":{"scope":"publisher-and-author-metadata-only","checkedAt":"2026-10-18","evidenceUrl":"https://www.bnl.gov/cmpmsd/mbe/publications/","note":"Author laboratory publication list and publisher record agree on DOI, title, Nature 536, 309–311 (2016). Scaling data and full primary results were not reanalyzed; this reference is a bibliographic pointer only."}},{"id":"source-37f27e50dbdcf3a5","doi":"10.1103/PhysRevApplied.10.014030","title":"Effective Demagnetizing Factors of Diamagnetic Samples of Various Shapes","year":2018,"url":"https://doi.org/10.1103/PhysRevApplied.10.014030","publicationStatus":"journal-article","verification":{"scope":"author-manuscript-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://arxiv.org/pdf/1712.06037v2","note":"Selected pp. 1–2, section II Eqs. (1)–(7): SI susceptibility, ellipsoidal demagnetization and distinction from integral effective factors for other shapes. Full three-dimensional numerical simulations and all shape approximations not reproduced."}},{"id":"source-82f53ca35625217d","doi":"10.1103/RevModPhys.66.1125","title":"Vortices in high-temperature superconductors","year":1994,"url":"https://doi.org/10.1103/RevModPhys.66.1125","publicationStatus":"journal-article","verification":{"scope":"review-abstract-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.66.1125","note":"Publisher abstract distinguishes depinning and depairing currents and discusses thermal fluctuations, pinning and creep. Full review derivations and quantitative phase boundaries were not checked."}}],"materialCases":[{"formula":"YBa2Cu3O6.95","sample":"single crystals in Hardy et al. publisher abstract","recordType":"experiment","tcK":null,"temperatureNote":"measurement interval is not an extracted transition temperature","criterion":"penetration-depth-change","pressureRegime":"unknown","pressureGPa":null,"equilibrium":"unknown","probe":"900 MHz resonator; penetration-depth change over approximately 3–25 K","id":"scwiki-a2-material-1","sourceId":"source-b7bc48c845a47b53","reportedTemperatureK":null,"temperatureKind":"unassigned","aliases":["cuprate","cuprates"]}],"claims":[{"id":"scwiki-a2-claim-scope-1","sectionId":"scope","statement":"The Meissner effect describes magnetic flux exclusion from superconducting material in the appropriate equilibrium, low-field state. It adds information beyond persistent current or zero resistance. The London brothers explicitly distinguished their magnetic-state condition from an acceleration-only description that can retain the initial field. Their historical equations are used here in a modern SI convention, within a stated local model.","evidenceType":"theory","sourceIds":["source-0590b237a1d87de4"],"conditions":"Homogeneous isotropic vortex-free local London response, quasistatic weak field; slab full thickness 2a and parallel field. Ellipsoid uses scalar SI susceptibility along principal axis; thin-film sheet inductance requires d much less than lambda and uniform current.","verificationScope":"London original pp. 72–73, Prozorov–Kogan selected pp. 1–2; explicitly separate independent numerical/local deductions and qualitative vortex abstract."},{"id":"scwiki-a2-claim-scope-2","sectionId":"scope","statement":"The 1933 Meissner–Ochsenfeld report compares the external field around tin and lead before and after cooling through the transition. It also describes field retained in the cavity of a hollow lead tube. A field in that cavity is not a measurement of field penetrating the superconducting wall. Selected pages of a reproduction containing the original German and an English translation were checked; the historical experiment was not independently reproduced.","evidenceType":"experiment","sourceIds":["source-512705f3e64d88d7"],"conditions":"Selected historical first two reproduction pages, or Hardy YBa2Cu3O6.95 single crystals at 900 MHz and approximately 3–25 K; no inferred Tc or independent replication.","verificationScope":"Primary reproduction/abstract passages only; no raw-data reanalysis."},{"id":"scwiki-a2-claim-scope-3","sectionId":"scope","statement":"Separate zero-field-cooled shielding, field-cooled magnetization and hysteresis. A type-II specimen can contain vortices, and pinning can prevent equilibration during cooling or field sweeps. Thus a protocol-dependent signal should not be labeled complete equilibrium flux expulsion. The vortex review supplies qualitative context; it is not a calibration of a particular sample's volume fraction.","evidenceType":"theory","sourceIds":["source-82f53ca35625217d"],"conditions":"Homogeneous isotropic vortex-free local London response, quasistatic weak field; slab full thickness 2a and parallel field. Ellipsoid uses scalar SI susceptibility along principal axis; thin-film sheet inductance requires d much less than lambda and uniform current.","verificationScope":"London original pp. 72–73, Prozorov–Kogan selected pp. 1–2; explicitly separate independent numerical/local deductions and qualitative vortex abstract."},{"id":"scwiki-a2-claim-mathematical-assumptions-1","sectionId":"mathematical-assumptions","statement":"In a homogeneous, isotropic, vortex-free local London model, with displacement current neglected, combine the magnetic London relation with Ampere's law and $\\nabla\\cdot\\mathbf B=0$:","evidenceType":"theory","sourceIds":["source-0590b237a1d87de4"],"conditions":"Homogeneous isotropic vortex-free local London response, quasistatic weak field; slab full thickness 2a and parallel field. Ellipsoid uses scalar SI susceptibility along principal axis; thin-film sheet inductance requires d much less than lambda and uniform current.","verificationScope":"London original pp. 72–73, Prozorov–Kogan selected pp. 1–2; explicitly separate independent numerical/local deductions and qualitative vortex abstract.","sourceRelationship":"mixed-source-and-local-derivation","derivationNote":"Modern SI local-model deduction using the magnetic London relation, Ampere law and div B=0; historical source uses different electromagnetic units."},{"id":"scwiki-a2-claim-mathematical-assumptions-2","sectionId":"mathematical-assumptions","statement":"$$\\nabla\\times\\mathbf j=-\\frac{\\mathbf B}{\\mu_0\\lambda^2},\\qquad \\nabla\\times\\mathbf B=\\mu_0\\mathbf j,\\qquad \\nabla^2\\mathbf B=\\frac{\\mathbf B}{\\lambda^2}.$$","evidenceType":"theory","sourceIds":["source-0590b237a1d87de4"],"conditions":"Homogeneous isotropic vortex-free local London response, quasistatic weak field; slab full thickness 2a and parallel field. Ellipsoid uses scalar SI susceptibility along principal axis; thin-film sheet inductance requires d much less than lambda and uniform current.","verificationScope":"London original pp. 72–73, Prozorov–Kogan selected pp. 1–2; explicitly separate independent numerical/local deductions and qualitative vortex abstract."},{"id":"scwiki-a2-claim-mathematical-assumptions-3","sectionId":"mathematical-assumptions","statement":"For a half-space $x>0$ with parallel surface field $B(0)=B_0$ and bounded field at depth, the solution is $B(x)=B_0e^{-x/\\lambda}$. The surface boundary and exclusion of a growing solution matter; the differential equation alone does not specify an arbitrary specimen's field map. These are modern local-model deductions, not a claim that London theory covers every superconducting state.","evidenceType":"theory","sourceIds":["source-0590b237a1d87de4"],"conditions":"Homogeneous isotropic vortex-free local London response, quasistatic weak field; slab full thickness 2a and parallel field. Ellipsoid uses scalar SI susceptibility along principal axis; thin-film sheet inductance requires d much less than lambda and uniform current.","verificationScope":"London original pp. 72–73, Prozorov–Kogan selected pp. 1–2; explicitly separate independent numerical/local deductions and qualitative vortex abstract.","sourceRelationship":"mixed-source-and-local-derivation","derivationNote":"Local boundary-value solution of the screening equation; the historical source supports exponential decay, while the stated modern boundary conditions specify this solution."},{"id":"scwiki-a2-claim-mathematical-assumptions-4","sectionId":"mathematical-assumptions","statement":"For an infinite slab $-a<x<a$ with equal parallel fields on its two faces, no vortices and constant $\\lambda$, the same boundary-value problem gives","evidenceType":"theory","sourceIds":["source-0590b237a1d87de4"],"conditions":"Homogeneous isotropic vortex-free local London response, quasistatic weak field; slab full thickness 2a and parallel field. Ellipsoid uses scalar SI susceptibility along principal axis; thin-film sheet inductance requires d much less than lambda and uniform current.","verificationScope":"London original pp. 72–73, Prozorov–Kogan selected pp. 1–2; explicitly separate independent numerical/local deductions and qualitative vortex abstract."},{"id":"scwiki-a2-claim-mathematical-assumptions-5","sectionId":"mathematical-assumptions","statement":"$$\\frac{B(x)}{B_0}=\\frac{\\cosh(x/\\lambda)}{\\cosh(a/\\lambda)},\\qquad \\chi_{\\parallel}=\\frac{\\langle B\\rangle}{B_0}-1=\\frac{\\lambda}{a}\\tanh\\!\\left(\\frac{a}{\\lambda}\\right)-1.$$","evidenceType":"theory","sourceIds":["source-0590b237a1d87de4"],"conditions":"Homogeneous isotropic vortex-free local London response, quasistatic weak field; slab full thickness 2a and parallel field. Ellipsoid uses scalar SI susceptibility along principal axis; thin-film sheet inductance requires d much less than lambda and uniform current.","verificationScope":"London original pp. 72–73, Prozorov–Kogan selected pp. 1–2; explicitly separate independent numerical/local deductions and qualitative vortex abstract."},{"id":"scwiki-a2-claim-mathematical-assumptions-6","sectionId":"mathematical-assumptions","statement":"Here $2a$ is the full thickness and $B_0=\\mu_0H_a$. A local finite-difference solution independently checks the field profile and its spatial average. A thick slab tends toward $\\chi_{\\parallel}=-1$; a very thin slab in this parallel geometry has much weaker shielding even when the entire slab obeys the superconducting model. This is not a formula for a perpendicular-field thin film or a porous specimen.","evidenceType":"theory","sourceIds":["source-0590b237a1d87de4"],"conditions":"Homogeneous isotropic vortex-free local London response, quasistatic weak field; slab full thickness 2a and parallel field. Ellipsoid uses scalar SI susceptibility along principal axis; thin-film sheet inductance requires d much less than lambda and uniform current.","verificationScope":"London original pp. 72–73, Prozorov–Kogan selected pp. 1–2; explicitly separate independent numerical/local deductions and qualitative vortex abstract."},{"id":"scwiki-a2-claim-mathematical-assumptions-7","sectionId":"mathematical-assumptions","statement":"Demagnetization is a separate shape effect. For an ellipsoid along a principal axis, with uniform scalar susceptibility in SI, let $H_{\\mathrm{int}}=H_a-NM$ and $M=\\chi H_{\\mathrm{int}}$. Then","evidenceType":"theory","sourceIds":["source-0590b237a1d87de4"],"conditions":"Homogeneous isotropic vortex-free local London response, quasistatic weak field; slab full thickness 2a and parallel field. Ellipsoid uses scalar SI susceptibility along principal axis; thin-film sheet inductance requires d much less than lambda and uniform current.","verificationScope":"London original pp. 72–73, Prozorov–Kogan selected pp. 1–2; explicitly separate independent numerical/local deductions and qualitative vortex abstract."},{"id":"scwiki-a2-claim-mathematical-assumptions-8","sectionId":"mathematical-assumptions","statement":"$$\\chi_{\\mathrm{app}}=\\frac{M}{H_a}=\\frac{\\chi}{1+N\\chi}.$$","evidenceType":"theory","sourceIds":["source-0590b237a1d87de4"],"conditions":"Homogeneous isotropic vortex-free local London response, quasistatic weak field; slab full thickness 2a and parallel field. Ellipsoid uses scalar SI susceptibility along principal axis; thin-film sheet inductance requires d much less than lambda and uniform current.","verificationScope":"London original pp. 72–73, Prozorov–Kogan selected pp. 1–2; explicitly separate independent numerical/local deductions and qualitative vortex abstract."},{"id":"scwiki-a2-claim-mathematical-assumptions-9","sectionId":"mathematical-assumptions","statement":"For the perfect-diamagnet limit $\\chi=-1$, this gives $\\chi_{\\mathrm{app}}=-1/(1-N)$, for $N<1$. A sphere has $N=1/3$, so an uncorrected magnitude of 1.5 is consistent with this ideal geometry, not a 150% superconducting volume. The Prozorov–Kogan manuscript defines the convention and explains why nonellipsoidal specimens need additional treatment. Its numerical arbitrary-shape results are not reproduced here.","evidenceType":"theory","sourceIds":["source-37f27e50dbdcf3a5"],"conditions":"Homogeneous isotropic vortex-free local London response, quasistatic weak field; slab full thickness 2a and parallel field. Ellipsoid uses scalar SI susceptibility along principal axis; thin-film sheet inductance requires d much less than lambda and uniform current.","verificationScope":"London original pp. 72–73, Prozorov–Kogan selected pp. 1–2; explicitly separate independent numerical/local deductions and qualitative vortex abstract."},{"id":"scwiki-a2-claim-mathematical-assumptions-10","sectionId":"mathematical-assumptions","statement":"In a single-component carrier convention, $\\lambda^{-2}=\\mu_0n_sq^2/m^*$. Count charge, mass and density consistently: pair density is not electron density. For a uniform thin film with thickness $d\\ll\\lambda$ and negligible dissipative response, the sheet kinetic inductance is $L_k=\\mu_0\\lambda^2/d$. Equating magnetic-model current response and carrier kinetic energy checks this normalization. Inferring a bulk penetration depth from a sheet response therefore requires thickness and a response model; multiband or nonlocal systems need their corresponding treatment.","evidenceType":"theory","sourceIds":["source-0590b237a1d87de4"],"conditions":"Homogeneous isotropic vortex-free local London response, quasistatic weak field; slab full thickness 2a and parallel field. Ellipsoid uses scalar SI susceptibility along principal axis; thin-film sheet inductance requires d much less than lambda and uniform current.","verificationScope":"London original pp. 72–73, Prozorov–Kogan selected pp. 1–2; explicitly separate independent numerical/local deductions and qualitative vortex abstract.","sourceRelationship":"context-for-local-derivation","derivationNote":"Local sheet kinetic-energy deduction with consistent carrier density, mass and charge. Historical London references provide model context, not a direct thin-film measurement."},{"id":"scwiki-a2-claim-experimental-criteria-1","sectionId":"experimental-criteria","statement":"Report specimen dimensions, field orientation and amplitude, cooling and sweep history, magnetic moment normalization, background subtraction and the assumed geometry. For AC or microwave probes, include frequency, amplitude and the conversion from the measured response. These are reporting requirements; a successful model inversion is not an independent demonstration of its assumptions.","evidenceType":"hypothesis","sourceIds":["source-37f27e50dbdcf3a5"],"conditions":"Editorial reporting/inference boundary; sample-specific protocol, geometry, uncertainty and source required.","verificationScope":"Pinned chapter edited with source checks stated individually; no new experimental dataset certified."},{"id":"scwiki-a2-claim-experimental-criteria-2","sectionId":"experimental-criteria","statement":"Hardy and colleagues report a linear contribution to the penetration-depth change of YBa2Cu3O6.95 single crystals over approximately 3–25 K with a 900 MHz resonator. This abstract-scoped observation constrains low-energy excitations. It does not directly measure gap phase or uniquely identify the pairing representation; sample disorder and electrodynamics belong in the interpretation. No transition temperature is extracted here.","evidenceType":"experiment","sourceIds":["source-b7bc48c845a47b53"],"conditions":"Selected historical first two reproduction pages, or Hardy YBa2Cu3O6.95 single crystals at 900 MHz and approximately 3–25 K; no inferred Tc or independent replication.","verificationScope":"Primary reproduction/abstract passages only; no raw-data reanalysis."},{"id":"scwiki-a2-claim-limits-and-disputes-1","sectionId":"limits-and-disputes","statement":"Shielding and expulsion can differ because of field history, geometry, connectivity and pinning. A small signal alone does not distinguish a small superconducting region from a measurement or geometry limitation. Conversely, a connected superconducting shell can screen a nonsuperconducting interior; a shielding measurement alone cannot establish the interior's volume fraction. The historical hollow-tube observation makes the distinction between material and enclosed space especially important.","evidenceType":"hypothesis","sourceIds":["source-512705f3e64d88d7"],"conditions":"Editorial reporting/inference boundary; sample-specific protocol, geometry, uncertainty and source required.","verificationScope":"Pinned chapter edited with source checks stated individually; no new experimental dataset certified."},{"id":"scwiki-a2-claim-limits-and-disputes-2","sectionId":"limits-and-disputes","statement":"Božović and colleagues' overdoped LSCO study is retained as a reference for superfluid-density measurements, but this check confirms its bibliographic identity rather than reanalyzing its scaling data. Do not infer a pairing mediator from a correlation alone. Likewise, a measured total stiffness cannot be decomposed into conventional and geometric contributions merely by changing its label. Follow the dedicated geometric-superfluid-weight entry for that separate model question.","evidenceType":"hypothesis","sourceIds":["source-b96ffab89d45664d"],"conditions":"Editorial reporting/inference boundary; sample-specific protocol, geometry, uncertainty and source required.","verificationScope":"Pinned chapter edited with source checks stated individually; no new experimental dataset certified."},{"id":"scwiki-a2-claim-discovery-implications-1","sectionId":"discovery-implications","statement":"Build a magnetic evidence package around discriminating measurements: cooling history for trapping, dimensions and orientation for demagnetization, thickness for sheet response, and transport or heat capacity on comparable specimens for complementary evidence. Preserve disagreements between probes until sample and analysis differences are tested.","evidenceType":"hypothesis","sourceIds":["source-37f27e50dbdcf3a5"],"conditions":"Editorial reporting/inference boundary; sample-specific protocol, geometry, uncertainty and source required.","verificationScope":"Pinned chapter edited with source checks stated individually; no new experimental dataset certified."},{"id":"scwiki-a2-claim-discovery-implications-2","sectionId":"discovery-implications","statement":"Separate measured magnetic response, the fitted electrodynamic parameters and the microscopic interpretation. This keeps magnetic evidence useful for discovering superconductivity without turning one model-dependent inversion into an unqualified material property or mechanism assignment.","evidenceType":"hypothesis","sourceIds":["source-37f27e50dbdcf3a5"],"conditions":"Editorial reporting/inference boundary; sample-specific protocol, geometry, uncertainty and source required.","verificationScope":"Pinned chapter edited with source checks stated individually; no new experimental dataset certified."}],"body":"## Scope\n\nThe Meissner effect describes magnetic flux exclusion from superconducting material in the appropriate equilibrium, low-field state. It adds information beyond persistent current or zero resistance. The London brothers explicitly distinguished their magnetic-state condition from an acceleration-only description that can retain the initial field. Their historical equations are used here in a modern SI convention, within a stated local model. [R2](#ref-source-0590b237a1d87de4)\n\nThe 1933 Meissner–Ochsenfeld report compares the external field around tin and lead before and after cooling through the transition. It also describes field retained in the cavity of a hollow lead tube. A field in that cavity is not a measurement of field penetrating the superconducting wall. Selected pages of a reproduction containing the original German and an English translation were checked; the historical experiment was not independently reproduced. [R1](#ref-source-512705f3e64d88d7)\n\nSeparate zero-field-cooled shielding, field-cooled magnetization and hysteresis. A type-II specimen can contain vortices, and pinning can prevent equilibration during cooling or field sweeps. Thus a protocol-dependent signal should not be labeled complete equilibrium flux expulsion. The vortex review supplies qualitative context; it is not a calibration of a particular sample's volume fraction. [R6](#ref-source-82f53ca35625217d)\n\n## Mathematical assumptions\n\nIn a homogeneous, isotropic, vortex-free local London model, with displacement current neglected, combine the magnetic London relation with Ampere's law and $\\nabla\\cdot\\mathbf B=0$:\n\n$$\\nabla\\times\\mathbf j=-\\frac{\\mathbf B}{\\mu_0\\lambda^2},\\qquad \\nabla\\times\\mathbf B=\\mu_0\\mathbf j,\\qquad \\nabla^2\\mathbf B=\\frac{\\mathbf B}{\\lambda^2}.$$\n\nFor a half-space $x>0$ with parallel surface field $B(0)=B_0$ and bounded field at depth, the solution is $B(x)=B_0e^{-x/\\lambda}$. The surface boundary and exclusion of a growing solution matter; the differential equation alone does not specify an arbitrary specimen's field map. These are modern local-model deductions, not a claim that London theory covers every superconducting state. [R2](#ref-source-0590b237a1d87de4)\n\nFor an infinite slab $-a<x<a$ with equal parallel fields on its two faces, no vortices and constant $\\lambda$, the same boundary-value problem gives\n\n$$\\frac{B(x)}{B_0}=\\frac{\\cosh(x/\\lambda)}{\\cosh(a/\\lambda)},\\qquad \\chi_{\\parallel}=\\frac{\\langle B\\rangle}{B_0}-1=\\frac{\\lambda}{a}\\tanh\\!\\left(\\frac{a}{\\lambda}\\right)-1.$$\n\nHere $2a$ is the full thickness and $B_0=\\mu_0H_a$. A local finite-difference solution independently checks the field profile and its spatial average. A thick slab tends toward $\\chi_{\\parallel}=-1$; a very thin slab in this parallel geometry has much weaker shielding even when the entire slab obeys the superconducting model. This is not a formula for a perpendicular-field thin film or a porous specimen.\n\nDemagnetization is a separate shape effect. For an ellipsoid along a principal axis, with uniform scalar susceptibility in SI, let $H_{\\mathrm{int}}=H_a-NM$ and $M=\\chi H_{\\mathrm{int}}$. Then\n\n$$\\chi_{\\mathrm{app}}=\\frac{M}{H_a}=\\frac{\\chi}{1+N\\chi}.$$\n\nFor the perfect-diamagnet limit $\\chi=-1$, this gives $\\chi_{\\mathrm{app}}=-1/(1-N)$, for $N<1$. A sphere has $N=1/3$, so an uncorrected magnitude of 1.5 is consistent with this ideal geometry, not a 150% superconducting volume. The Prozorov–Kogan manuscript defines the convention and explains why nonellipsoidal specimens need additional treatment. Its numerical arbitrary-shape results are not reproduced here. [R5](#ref-source-37f27e50dbdcf3a5)\n\nIn a single-component carrier convention, $\\lambda^{-2}=\\mu_0n_sq^2/m^*$. Count charge, mass and density consistently: pair density is not electron density. For a uniform thin film with thickness $d\\ll\\lambda$ and negligible dissipative response, the sheet kinetic inductance is $L_k=\\mu_0\\lambda^2/d$. Equating magnetic-model current response and carrier kinetic energy checks this normalization. Inferring a bulk penetration depth from a sheet response therefore requires thickness and a response model; multiband or nonlocal systems need their corresponding treatment.\n\n## Experimental criteria\n\nReport specimen dimensions, field orientation and amplitude, cooling and sweep history, magnetic moment normalization, background subtraction and the assumed geometry. For AC or microwave probes, include frequency, amplitude and the conversion from the measured response. These are reporting requirements; a successful model inversion is not an independent demonstration of its assumptions.\n\nHardy and colleagues report a linear contribution to the penetration-depth change of YBa2Cu3O6.95 single crystals over approximately 3–25 K with a 900 MHz resonator. This abstract-scoped observation constrains low-energy excitations. It does not directly measure gap phase or uniquely identify the pairing representation; sample disorder and electrodynamics belong in the interpretation. No transition temperature is extracted here. [R3](#ref-source-b7bc48c845a47b53)\n\n## Limits and disputes\n\nShielding and expulsion can differ because of field history, geometry, connectivity and pinning. A small signal alone does not distinguish a small superconducting region from a measurement or geometry limitation. Conversely, a connected superconducting shell can screen a nonsuperconducting interior; a shielding measurement alone cannot establish the interior's volume fraction. The historical hollow-tube observation makes the distinction between material and enclosed space especially important. [R1](#ref-source-512705f3e64d88d7)\n\nBožović and colleagues' overdoped LSCO study is retained as a reference for superfluid-density measurements, but this check confirms its bibliographic identity rather than reanalyzing its scaling data. Do not infer a pairing mediator from a correlation alone. Likewise, a measured total stiffness cannot be decomposed into conventional and geometric contributions merely by changing its label. Follow the dedicated geometric-superfluid-weight entry for that separate model question. [R4](#ref-source-b96ffab89d45664d)\n\n## Discovery implications\n\nBuild a magnetic evidence package around discriminating measurements: cooling history for trapping, dimensions and orientation for demagnetization, thickness for sheet response, and transport or heat capacity on comparable specimens for complementary evidence. Preserve disagreements between probes until sample and analysis differences are tested.\n\nSeparate measured magnetic response, the fitted electrodynamic parameters and the microscopic interpretation. This keeps magnetic evidence useful for discovering superconductivity without turning one model-dependent inversion into an unqualified material property or mechanism assignment.\n\n## Related topics\n\nCompare [zero resistance](/scwiki/zero-resistance), [critical fields](/scwiki/critical-fields), [critical current](/scwiki/critical-current), [specific heat](/scwiki/specific-heat-jump), [superconducting gap](/scwiki/superconducting-gap), and [geometric superfluid weight](/scwiki/geometric-superfluid-weight).","revision":"beae8ed476bfb412c85f","sections":[{"id":"scope","title":"Scope","markdown":"## Scope\n\nThe Meissner effect describes magnetic flux exclusion from superconducting material in the appropriate equilibrium, low-field state. It adds information beyond persistent current or zero resistance. The London brothers explicitly distinguished their magnetic-state condition from an acceleration-only description that can retain the initial field. Their historical equations are used here in a modern SI convention, within a stated local model. [R2](#ref-source-0590b237a1d87de4)\n\nThe 1933 Meissner–Ochsenfeld report compares the external field around tin and lead before and after cooling through the transition. It also describes field retained in the cavity of a hollow lead tube. A field in that cavity is not a measurement of field penetrating the superconducting wall. Selected pages of a reproduction containing the original German and an English translation were checked; the historical experiment was not independently reproduced. [R1](#ref-source-512705f3e64d88d7)\n\nSeparate zero-field-cooled shielding, field-cooled magnetization and hysteresis. A type-II specimen can contain vortices, and pinning can prevent equilibration during cooling or field sweeps. Thus a protocol-dependent signal should not be labeled complete equilibrium flux expulsion. The vortex review supplies qualitative context; it is not a calibration of a particular sample's volume fraction. [R6](#ref-source-82f53ca35625217d)"},{"id":"mathematical-assumptions","title":"Mathematical assumptions","markdown":"## Mathematical assumptions\n\nIn a homogeneous, isotropic, vortex-free local London model, with displacement current neglected, combine the magnetic London relation with Ampere's law and $\\nabla\\cdot\\mathbf B=0$:\n\n$$\\nabla\\times\\mathbf j=-\\frac{\\mathbf B}{\\mu_0\\lambda^2},\\qquad \\nabla\\times\\mathbf B=\\mu_0\\mathbf j,\\qquad \\nabla^2\\mathbf B=\\frac{\\mathbf B}{\\lambda^2}.$$\n\nFor a half-space $x>0$ with parallel surface field $B(0)=B_0$ and bounded field at depth, the solution is $B(x)=B_0e^{-x/\\lambda}$. The surface boundary and exclusion of a growing solution matter; the differential equation alone does not specify an arbitrary specimen's field map. These are modern local-model deductions, not a claim that London theory covers every superconducting state. [R2](#ref-source-0590b237a1d87de4)\n\nFor an infinite slab $-a<x<a$ with equal parallel fields on its two faces, no vortices and constant $\\lambda$, the same boundary-value problem gives\n\n$$\\frac{B(x)}{B_0}=\\frac{\\cosh(x/\\lambda)}{\\cosh(a/\\lambda)},\\qquad \\chi_{\\parallel}=\\frac{\\langle B\\rangle}{B_0}-1=\\frac{\\lambda}{a}\\tanh\\!\\left(\\frac{a}{\\lambda}\\right)-1.$$\n\nHere $2a$ is the full thickness and $B_0=\\mu_0H_a$. A local finite-difference solution independently checks the field profile and its spatial average. A thick slab tends toward $\\chi_{\\parallel}=-1$; a very thin slab in this parallel geometry has much weaker shielding even when the entire slab obeys the superconducting model. This is not a formula for a perpendicular-field thin film or a porous specimen.\n\nDemagnetization is a separate shape effect. For an ellipsoid along a principal axis, with uniform scalar susceptibility in SI, let $H_{\\mathrm{int}}=H_a-NM$ and $M=\\chi H_{\\mathrm{int}}$. Then\n\n$$\\chi_{\\mathrm{app}}=\\frac{M}{H_a}=\\frac{\\chi}{1+N\\chi}.$$\n\nFor the perfect-diamagnet limit $\\chi=-1$, this gives $\\chi_{\\mathrm{app}}=-1/(1-N)$, for $N<1$. A sphere has $N=1/3$, so an uncorrected magnitude of 1.5 is consistent with this ideal geometry, not a 150% superconducting volume. The Prozorov–Kogan manuscript defines the convention and explains why nonellipsoidal specimens need additional treatment. Its numerical arbitrary-shape results are not reproduced here. [R5](#ref-source-37f27e50dbdcf3a5)\n\nIn a single-component carrier convention, $\\lambda^{-2}=\\mu_0n_sq^2/m^*$. Count charge, mass and density consistently: pair density is not electron density. For a uniform thin film with thickness $d\\ll\\lambda$ and negligible dissipative response, the sheet kinetic inductance is $L_k=\\mu_0\\lambda^2/d$. Equating magnetic-model current response and carrier kinetic energy checks this normalization. Inferring a bulk penetration depth from a sheet response therefore requires thickness and a response model; multiband or nonlocal systems need their corresponding treatment."},{"id":"experimental-criteria","title":"Experimental criteria","markdown":"## Experimental criteria\n\nReport specimen dimensions, field orientation and amplitude, cooling and sweep history, magnetic moment normalization, background subtraction and the assumed geometry. For AC or microwave probes, include frequency, amplitude and the conversion from the measured response. These are reporting requirements; a successful model inversion is not an independent demonstration of its assumptions.\n\nHardy and colleagues report a linear contribution to the penetration-depth change of YBa2Cu3O6.95 single crystals over approximately 3–25 K with a 900 MHz resonator. This abstract-scoped observation constrains low-energy excitations. It does not directly measure gap phase or uniquely identify the pairing representation; sample disorder and electrodynamics belong in the interpretation. No transition temperature is extracted here. [R3](#ref-source-b7bc48c845a47b53)"},{"id":"limits-and-disputes","title":"Limits and disputes","markdown":"## Limits and disputes\n\nShielding and expulsion can differ because of field history, geometry, connectivity and pinning. A small signal alone does not distinguish a small superconducting region from a measurement or geometry limitation. Conversely, a connected superconducting shell can screen a nonsuperconducting interior; a shielding measurement alone cannot establish the interior's volume fraction. The historical hollow-tube observation makes the distinction between material and enclosed space especially important. [R1](#ref-source-512705f3e64d88d7)\n\nBožović and colleagues' overdoped LSCO study is retained as a reference for superfluid-density measurements, but this check confirms its bibliographic identity rather than reanalyzing its scaling data. Do not infer a pairing mediator from a correlation alone. Likewise, a measured total stiffness cannot be decomposed into conventional and geometric contributions merely by changing its label. Follow the dedicated geometric-superfluid-weight entry for that separate model question. [R4](#ref-source-b96ffab89d45664d)"},{"id":"discovery-implications","title":"Discovery implications","markdown":"## Discovery implications\n\nBuild a magnetic evidence package around discriminating measurements: cooling history for trapping, dimensions and orientation for demagnetization, thickness for sheet response, and transport or heat capacity on comparable specimens for complementary evidence. Preserve disagreements between probes until sample and analysis differences are tested.\n\nSeparate measured magnetic response, the fitted electrodynamic parameters and the microscopic interpretation. This keeps magnetic evidence useful for discovering superconductivity without turning one model-dependent inversion into an unqualified material property or mechanism assignment."},{"id":"related-topics","title":"Related topics","markdown":"## Related topics\n\nCompare [zero resistance](/scwiki/zero-resistance), [critical fields](/scwiki/critical-fields), [critical current](/scwiki/critical-current), [specific heat](/scwiki/specific-heat-jump), [superconducting gap](/scwiki/superconducting-gap), and [geometric superfluid weight](/scwiki/geometric-superfluid-weight)."},{"id":"material-cases","title":"Material cases","markdown":"## Material cases\n\nValues describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.\n\n| Material and sample | Reported temperature | Criterion | Probe | Pressure | State | Evidence |\n|---|---|---|---|---|---|---|\n| YBa2Cu3O6.95 (aliases: cuprate, cuprates); single crystals in Hardy et al. publisher abstract | measurement interval is not an extracted transition temperature | unassigned; penetration-depth-change | 900 MHz resonator; penetration-depth change over approximately 3–25 K | unknown; unknown GPa | unknown | experiment; [source](#ref-source-b7bc48c845a47b53) |"}],"markdown":"# The Meissner effect and magnetic evidence\n\nDistinguish equilibrium flux expulsion from shielding, then interpret penetration depth and magnetic volume estimates with their geometric assumptions.\n\n## Scope\n\nThe Meissner effect describes magnetic flux exclusion from superconducting material in the appropriate equilibrium, low-field state. It adds information beyond persistent current or zero resistance. The London brothers explicitly distinguished their magnetic-state condition from an acceleration-only description that can retain the initial field. Their historical equations are used here in a modern SI convention, within a stated local model. [R2](#ref-source-0590b237a1d87de4)\n\nThe 1933 Meissner–Ochsenfeld report compares the external field around tin and lead before and after cooling through the transition. It also describes field retained in the cavity of a hollow lead tube. A field in that cavity is not a measurement of field penetrating the superconducting wall. Selected pages of a reproduction containing the original German and an English translation were checked; the historical experiment was not independently reproduced. [R1](#ref-source-512705f3e64d88d7)\n\nSeparate zero-field-cooled shielding, field-cooled magnetization and hysteresis. A type-II specimen can contain vortices, and pinning can prevent equilibration during cooling or field sweeps. Thus a protocol-dependent signal should not be labeled complete equilibrium flux expulsion. The vortex review supplies qualitative context; it is not a calibration of a particular sample's volume fraction. [R6](#ref-source-82f53ca35625217d)\n\n## Mathematical assumptions\n\nIn a homogeneous, isotropic, vortex-free local London model, with displacement current neglected, combine the magnetic London relation with Ampere's law and $\\nabla\\cdot\\mathbf B=0$:\n\n$$\\nabla\\times\\mathbf j=-\\frac{\\mathbf B}{\\mu_0\\lambda^2},\\qquad \\nabla\\times\\mathbf B=\\mu_0\\mathbf j,\\qquad \\nabla^2\\mathbf B=\\frac{\\mathbf B}{\\lambda^2}.$$\n\nFor a half-space $x>0$ with parallel surface field $B(0)=B_0$ and bounded field at depth, the solution is $B(x)=B_0e^{-x/\\lambda}$. The surface boundary and exclusion of a growing solution matter; the differential equation alone does not specify an arbitrary specimen's field map. These are modern local-model deductions, not a claim that London theory covers every superconducting state. [R2](#ref-source-0590b237a1d87de4)\n\nFor an infinite slab $-a<x<a$ with equal parallel fields on its two faces, no vortices and constant $\\lambda$, the same boundary-value problem gives\n\n$$\\frac{B(x)}{B_0}=\\frac{\\cosh(x/\\lambda)}{\\cosh(a/\\lambda)},\\qquad \\chi_{\\parallel}=\\frac{\\langle B\\rangle}{B_0}-1=\\frac{\\lambda}{a}\\tanh\\!\\left(\\frac{a}{\\lambda}\\right)-1.$$\n\nHere $2a$ is the full thickness and $B_0=\\mu_0H_a$. A local finite-difference solution independently checks the field profile and its spatial average. A thick slab tends toward $\\chi_{\\parallel}=-1$; a very thin slab in this parallel geometry has much weaker shielding even when the entire slab obeys the superconducting model. This is not a formula for a perpendicular-field thin film or a porous specimen.\n\nDemagnetization is a separate shape effect. For an ellipsoid along a principal axis, with uniform scalar susceptibility in SI, let $H_{\\mathrm{int}}=H_a-NM$ and $M=\\chi H_{\\mathrm{int}}$. Then\n\n$$\\chi_{\\mathrm{app}}=\\frac{M}{H_a}=\\frac{\\chi}{1+N\\chi}.$$\n\nFor the perfect-diamagnet limit $\\chi=-1$, this gives $\\chi_{\\mathrm{app}}=-1/(1-N)$, for $N<1$. A sphere has $N=1/3$, so an uncorrected magnitude of 1.5 is consistent with this ideal geometry, not a 150% superconducting volume. The Prozorov–Kogan manuscript defines the convention and explains why nonellipsoidal specimens need additional treatment. Its numerical arbitrary-shape results are not reproduced here. [R5](#ref-source-37f27e50dbdcf3a5)\n\nIn a single-component carrier convention, $\\lambda^{-2}=\\mu_0n_sq^2/m^*$. Count charge, mass and density consistently: pair density is not electron density. For a uniform thin film with thickness $d\\ll\\lambda$ and negligible dissipative response, the sheet kinetic inductance is $L_k=\\mu_0\\lambda^2/d$. Equating magnetic-model current response and carrier kinetic energy checks this normalization. Inferring a bulk penetration depth from a sheet response therefore requires thickness and a response model; multiband or nonlocal systems need their corresponding treatment.\n\n## Experimental criteria\n\nReport specimen dimensions, field orientation and amplitude, cooling and sweep history, magnetic moment normalization, background subtraction and the assumed geometry. For AC or microwave probes, include frequency, amplitude and the conversion from the measured response. These are reporting requirements; a successful model inversion is not an independent demonstration of its assumptions.\n\nHardy and colleagues report a linear contribution to the penetration-depth change of YBa2Cu3O6.95 single crystals over approximately 3–25 K with a 900 MHz resonator. This abstract-scoped observation constrains low-energy excitations. It does not directly measure gap phase or uniquely identify the pairing representation; sample disorder and electrodynamics belong in the interpretation. No transition temperature is extracted here. [R3](#ref-source-b7bc48c845a47b53)\n\n## Limits and disputes\n\nShielding and expulsion can differ because of field history, geometry, connectivity and pinning. A small signal alone does not distinguish a small superconducting region from a measurement or geometry limitation. Conversely, a connected superconducting shell can screen a nonsuperconducting interior; a shielding measurement alone cannot establish the interior's volume fraction. The historical hollow-tube observation makes the distinction between material and enclosed space especially important. [R1](#ref-source-512705f3e64d88d7)\n\nBožović and colleagues' overdoped LSCO study is retained as a reference for superfluid-density measurements, but this check confirms its bibliographic identity rather than reanalyzing its scaling data. Do not infer a pairing mediator from a correlation alone. Likewise, a measured total stiffness cannot be decomposed into conventional and geometric contributions merely by changing its label. Follow the dedicated geometric-superfluid-weight entry for that separate model question. [R4](#ref-source-b96ffab89d45664d)\n\n## Discovery implications\n\nBuild a magnetic evidence package around discriminating measurements: cooling history for trapping, dimensions and orientation for demagnetization, thickness for sheet response, and transport or heat capacity on comparable specimens for complementary evidence. Preserve disagreements between probes until sample and analysis differences are tested.\n\nSeparate measured magnetic response, the fitted electrodynamic parameters and the microscopic interpretation. This keeps magnetic evidence useful for discovering superconductivity without turning one model-dependent inversion into an unqualified material property or mechanism assignment.\n\n## Related topics\n\nCompare [zero resistance](/scwiki/zero-resistance), [critical fields](/scwiki/critical-fields), [critical current](/scwiki/critical-current), [specific heat](/scwiki/specific-heat-jump), [superconducting gap](/scwiki/superconducting-gap), and [geometric superfluid weight](/scwiki/geometric-superfluid-weight).\n\n## Material cases\n\nValues describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.\n\n| Material and sample | Reported temperature | Criterion | Probe | Pressure | State | Evidence |\n|---|---|---|---|---|---|---|\n| YBa2Cu3O6.95 (aliases: cuprate, cuprates); single crystals in Hardy et al. publisher abstract | measurement interval is not an extracted transition temperature | unassigned; penetration-depth-change | 900 MHz resonator; penetration-depth change over approximately 3–25 K | unknown; unknown GPa | unknown | experiment; [source](#ref-source-b7bc48c845a47b53) |\n\n## Sources\n\n### Ref source-512705f3e64d88d7\n\n[R1] Ein neuer Effekt bei Eintritt der Supraleitfähigkeit (1933). [DOI 10.1007/BF01504252](https://doi.org/10.1007/BF01504252). Publication: journal-article. Check: primary-reproduction-scoped-check. Checked: 2026-10-18. Selected first two PDF pages: German original with English translation by A. M. Forrest, external field redistribution in tin/lead and retained field in a lead-tube cavity. Original apparatus, uncertainties and complete translation not independently reproduced.\n### Ref source-0590b237a1d87de4\n\n[R2] The electromagnetic equations of the supraconductor (1935). [DOI 10.1098/rspa.1935.0048](https://doi.org/10.1098/rspa.1935.0048). Publication: journal-article. Check: primary-scan-scoped-check. Checked: 2026-10-18. Original pp. 72–73 visually checked, Eqs. (2)–(6): initial-field integration constant, screening equation and additional magnetic-state postulate. Public equations use modern SI conventions and stated local assumptions; not an exhaustive check of the historical paper or later electrodynamics.\n### Ref source-b7bc48c845a47b53\n\n[R3] Precision measurements of the temperature dependence of lambda in YBa2Cu3O6.95: Strong evidence for nodes in the gap function (1993). [DOI 10.1103/PhysRevLett.70.3999](https://doi.org/10.1103/PhysRevLett.70.3999). Publication: journal-article. Check: primary-abstract-scoped-check. Checked: 2026-10-18. Publisher abstract: single-crystal YBa2Cu3O6.95, 900 MHz resonator, penetration-depth change and linear term around 3–25 K. Full curves, fitting and defect interpretation not independently reproduced; not a phase measurement.\n### Ref source-b96ffab89d45664d\n\n[R4] Dependence of the critical temperature in overdoped copper oxides on superfluid density (2016). [DOI 10.1038/nature19061](https://doi.org/10.1038/nature19061). Publication: journal-article. Check: publisher-and-author-metadata-only. Checked: 2026-10-18. Author laboratory publication list and publisher record agree on DOI, title, Nature 536, 309–311 (2016). Scaling data and full primary results were not reanalyzed; this reference is a bibliographic pointer only.\n### Ref source-37f27e50dbdcf3a5\n\n[R5] Effective Demagnetizing Factors of Diamagnetic Samples of Various Shapes (2018). [DOI 10.1103/PhysRevApplied.10.014030](https://doi.org/10.1103/PhysRevApplied.10.014030). Publication: journal-article. Check: author-manuscript-scoped-check. Checked: 2026-10-18. Selected pp. 1–2, section II Eqs. (1)–(7): SI susceptibility, ellipsoidal demagnetization and distinction from integral effective factors for other shapes. Full three-dimensional numerical simulations and all shape approximations not reproduced.\n### Ref source-82f53ca35625217d\n\n[R6] Vortices in high-temperature superconductors (1994). [DOI 10.1103/RevModPhys.66.1125](https://doi.org/10.1103/RevModPhys.66.1125). Publication: journal-article. Check: review-abstract-scoped-check. Checked: 2026-10-18. Publisher abstract distinguishes depinning and depairing currents and discusses thermal fluctuations, pinning and creep. Full review derivations and quantitative phase boundaries were not checked.\n\n## Claim evidence and conditions\n\n### scwiki-a2-claim-scope-1\n\nThe Meissner effect describes magnetic flux exclusion from superconducting material in the appropriate equilibrium, low-field state. It adds information beyond persistent current or zero resistance. The London brothers explicitly distinguished their magnetic-state condition from an acceleration-only description that can retain the initial field. Their historical equations are used here in a modern SI convention, within a stated local model.\n\nEvidence: theory. Sources: source-0590b237a1d87de4.\n\nConditions: Homogeneous isotropic vortex-free local London response, quasistatic weak field; slab full thickness 2a and parallel field. Ellipsoid uses scalar SI susceptibility along principal axis; thin-film sheet inductance requires d much less than lambda and uniform current.\n\nVerification scope: London original pp. 72–73, Prozorov–Kogan selected pp. 1–2; explicitly separate independent numerical/local deductions and qualitative vortex abstract.\n\n### scwiki-a2-claim-scope-2\n\nThe 1933 Meissner–Ochsenfeld report compares the external field around tin and lead before and after cooling through the transition. It also describes field retained in the cavity of a hollow lead tube. A field in that cavity is not a measurement of field penetrating the superconducting wall. Selected pages of a reproduction containing the original German and an English translation were checked; the historical experiment was not independently reproduced.\n\nEvidence: experiment. Sources: source-512705f3e64d88d7.\n\nConditions: Selected historical first two reproduction pages, or Hardy YBa2Cu3O6.95 single crystals at 900 MHz and approximately 3–25 K; no inferred Tc or independent replication.\n\nVerification scope: Primary reproduction/abstract passages only; no raw-data reanalysis.\n\n### scwiki-a2-claim-scope-3\n\nSeparate zero-field-cooled shielding, field-cooled magnetization and hysteresis. A type-II specimen can contain vortices, and pinning can prevent equilibration during cooling or field sweeps. Thus a protocol-dependent signal should not be labeled complete equilibrium flux expulsion. The vortex review supplies qualitative context; it is not a calibration of a particular sample's volume fraction.\n\nEvidence: theory. Sources: source-82f53ca35625217d.\n\nConditions: Homogeneous isotropic vortex-free local London response, quasistatic weak field; slab full thickness 2a and parallel field. Ellipsoid uses scalar SI susceptibility along principal axis; thin-film sheet inductance requires d much less than lambda and uniform current.\n\nVerification scope: London original pp. 72–73, Prozorov–Kogan selected pp. 1–2; explicitly separate independent numerical/local deductions and qualitative vortex abstract.\n\n### scwiki-a2-claim-mathematical-assumptions-1\n\nIn a homogeneous, isotropic, vortex-free local London model, with displacement current neglected, combine the magnetic London relation with Ampere's law and $\\nabla\\cdot\\mathbf B=0$:\n\nEvidence: theory. Sources: source-0590b237a1d87de4.\n\nSource relationship: mixed-source-and-local-derivation. Modern SI local-model deduction using the magnetic London relation, Ampere law and div B=0; historical source uses different electromagnetic units.\n\nConditions: Homogeneous isotropic vortex-free local London response, quasistatic weak field; slab full thickness 2a and parallel field. Ellipsoid uses scalar SI susceptibility along principal axis; thin-film sheet inductance requires d much less than lambda and uniform current.\n\nVerification scope: London original pp. 72–73, Prozorov–Kogan selected pp. 1–2; explicitly separate independent numerical/local deductions and qualitative vortex abstract.\n\n### scwiki-a2-claim-mathematical-assumptions-2\n\n$$\\nabla\\times\\mathbf j=-\\frac{\\mathbf B}{\\mu_0\\lambda^2},\\qquad \\nabla\\times\\mathbf B=\\mu_0\\mathbf j,\\qquad \\nabla^2\\mathbf B=\\frac{\\mathbf B}{\\lambda^2}.$$\n\nEvidence: theory. Sources: source-0590b237a1d87de4.\n\nConditions: Homogeneous isotropic vortex-free local London response, quasistatic weak field; slab full thickness 2a and parallel field. Ellipsoid uses scalar SI susceptibility along principal axis; thin-film sheet inductance requires d much less than lambda and uniform current.\n\nVerification scope: London original pp. 72–73, Prozorov–Kogan selected pp. 1–2; explicitly separate independent numerical/local deductions and qualitative vortex abstract.\n\n### scwiki-a2-claim-mathematical-assumptions-3\n\nFor a half-space $x>0$ with parallel surface field $B(0)=B_0$ and bounded field at depth, the solution is $B(x)=B_0e^{-x/\\lambda}$. The surface boundary and exclusion of a growing solution matter; the differential equation alone does not specify an arbitrary specimen's field map. These are modern local-model deductions, not a claim that London theory covers every superconducting state.\n\nEvidence: theory. Sources: source-0590b237a1d87de4.\n\nSource relationship: mixed-source-and-local-derivation. Local boundary-value solution of the screening equation; the historical source supports exponential decay, while the stated modern boundary conditions specify this solution.\n\nConditions: Homogeneous isotropic vortex-free local London response, quasistatic weak field; slab full thickness 2a and parallel field. Ellipsoid uses scalar SI susceptibility along principal axis; thin-film sheet inductance requires d much less than lambda and uniform current.\n\nVerification scope: London original pp. 72–73, Prozorov–Kogan selected pp. 1–2; explicitly separate independent numerical/local deductions and qualitative vortex abstract.\n\n### scwiki-a2-claim-mathematical-assumptions-4\n\nFor an infinite slab $-a<x<a$ with equal parallel fields on its two faces, no vortices and constant $\\lambda$, the same boundary-value problem gives\n\nEvidence: theory. Sources: source-0590b237a1d87de4.\n\nConditions: Homogeneous isotropic vortex-free local London response, quasistatic weak field; slab full thickness 2a and parallel field. Ellipsoid uses scalar SI susceptibility along principal axis; thin-film sheet inductance requires d much less than lambda and uniform current.\n\nVerification scope: London original pp. 72–73, Prozorov–Kogan selected pp. 1–2; explicitly separate independent numerical/local deductions and qualitative vortex abstract.\n\n### scwiki-a2-claim-mathematical-assumptions-5\n\n$$\\frac{B(x)}{B_0}=\\frac{\\cosh(x/\\lambda)}{\\cosh(a/\\lambda)},\\qquad \\chi_{\\parallel}=\\frac{\\langle B\\rangle}{B_0}-1=\\frac{\\lambda}{a}\\tanh\\!\\left(\\frac{a}{\\lambda}\\right)-1.$$\n\nEvidence: theory. Sources: source-0590b237a1d87de4.\n\nConditions: Homogeneous isotropic vortex-free local London response, quasistatic weak field; slab full thickness 2a and parallel field. Ellipsoid uses scalar SI susceptibility along principal axis; thin-film sheet inductance requires d much less than lambda and uniform current.\n\nVerification scope: London original pp. 72–73, Prozorov–Kogan selected pp. 1–2; explicitly separate independent numerical/local deductions and qualitative vortex abstract.\n\n### scwiki-a2-claim-mathematical-assumptions-6\n\nHere $2a$ is the full thickness and $B_0=\\mu_0H_a$. A local finite-difference solution independently checks the field profile and its spatial average. A thick slab tends toward $\\chi_{\\parallel}=-1$; a very thin slab in this parallel geometry has much weaker shielding even when the entire slab obeys the superconducting model. This is not a formula for a perpendicular-field thin film or a porous specimen.\n\nEvidence: theory. Sources: source-0590b237a1d87de4.\n\nConditions: Homogeneous isotropic vortex-free local London response, quasistatic weak field; slab full thickness 2a and parallel field. Ellipsoid uses scalar SI susceptibility along principal axis; thin-film sheet inductance requires d much less than lambda and uniform current.\n\nVerification scope: London original pp. 72–73, Prozorov–Kogan selected pp. 1–2; explicitly separate independent numerical/local deductions and qualitative vortex abstract.\n\n### scwiki-a2-claim-mathematical-assumptions-7\n\nDemagnetization is a separate shape effect. For an ellipsoid along a principal axis, with uniform scalar susceptibility in SI, let $H_{\\mathrm{int}}=H_a-NM$ and $M=\\chi H_{\\mathrm{int}}$. Then\n\nEvidence: theory. Sources: source-0590b237a1d87de4.\n\nConditions: Homogeneous isotropic vortex-free local London response, quasistatic weak field; slab full thickness 2a and parallel field. Ellipsoid uses scalar SI susceptibility along principal axis; thin-film sheet inductance requires d much less than lambda and uniform current.\n\nVerification scope: London original pp. 72–73, Prozorov–Kogan selected pp. 1–2; explicitly separate independent numerical/local deductions and qualitative vortex abstract.\n\n### scwiki-a2-claim-mathematical-assumptions-8\n\n$$\\chi_{\\mathrm{app}}=\\frac{M}{H_a}=\\frac{\\chi}{1+N\\chi}.$$\n\nEvidence: theory. Sources: source-0590b237a1d87de4.\n\nConditions: Homogeneous isotropic vortex-free local London response, quasistatic weak field; slab full thickness 2a and parallel field. Ellipsoid uses scalar SI susceptibility along principal axis; thin-film sheet inductance requires d much less than lambda and uniform current.\n\nVerification scope: London original pp. 72–73, Prozorov–Kogan selected pp. 1–2; explicitly separate independent numerical/local deductions and qualitative vortex abstract.\n\n### scwiki-a2-claim-mathematical-assumptions-9\n\nFor the perfect-diamagnet limit $\\chi=-1$, this gives $\\chi_{\\mathrm{app}}=-1/(1-N)$, for $N<1$. A sphere has $N=1/3$, so an uncorrected magnitude of 1.5 is consistent with this ideal geometry, not a 150% superconducting volume. The Prozorov–Kogan manuscript defines the convention and explains why nonellipsoidal specimens need additional treatment. Its numerical arbitrary-shape results are not reproduced here.\n\nEvidence: theory. Sources: source-37f27e50dbdcf3a5.\n\nConditions: Homogeneous isotropic vortex-free local London response, quasistatic weak field; slab full thickness 2a and parallel field. Ellipsoid uses scalar SI susceptibility along principal axis; thin-film sheet inductance requires d much less than lambda and uniform current.\n\nVerification scope: London original pp. 72–73, Prozorov–Kogan selected pp. 1–2; explicitly separate independent numerical/local deductions and qualitative vortex abstract.\n\n### scwiki-a2-claim-mathematical-assumptions-10\n\nIn a single-component carrier convention, $\\lambda^{-2}=\\mu_0n_sq^2/m^*$. Count charge, mass and density consistently: pair density is not electron density. For a uniform thin film with thickness $d\\ll\\lambda$ and negligible dissipative response, the sheet kinetic inductance is $L_k=\\mu_0\\lambda^2/d$. Equating magnetic-model current response and carrier kinetic energy checks this normalization. Inferring a bulk penetration depth from a sheet response therefore requires thickness and a response model; multiband or nonlocal systems need their corresponding treatment.\n\nEvidence: theory. Sources: source-0590b237a1d87de4.\n\nSource relationship: context-for-local-derivation. Local sheet kinetic-energy deduction with consistent carrier density, mass and charge. Historical London references provide model context, not a direct thin-film measurement.\n\nConditions: Homogeneous isotropic vortex-free local London response, quasistatic weak field; slab full thickness 2a and parallel field. Ellipsoid uses scalar SI susceptibility along principal axis; thin-film sheet inductance requires d much less than lambda and uniform current.\n\nVerification scope: London original pp. 72–73, Prozorov–Kogan selected pp. 1–2; explicitly separate independent numerical/local deductions and qualitative vortex abstract.\n\n### scwiki-a2-claim-experimental-criteria-1\n\nReport specimen dimensions, field orientation and amplitude, cooling and sweep history, magnetic moment normalization, background subtraction and the assumed geometry. For AC or microwave probes, include frequency, amplitude and the conversion from the measured response. These are reporting requirements; a successful model inversion is not an independent demonstration of its assumptions.\n\nEvidence: hypothesis. Sources: source-37f27e50dbdcf3a5.\n\nConditions: Editorial reporting/inference boundary; sample-specific protocol, geometry, uncertainty and source required.\n\nVerification scope: Pinned chapter edited with source checks stated individually; no new experimental dataset certified.\n\n### scwiki-a2-claim-experimental-criteria-2\n\nHardy and colleagues report a linear contribution to the penetration-depth change of YBa2Cu3O6.95 single crystals over approximately 3–25 K with a 900 MHz resonator. This abstract-scoped observation constrains low-energy excitations. It does not directly measure gap phase or uniquely identify the pairing representation; sample disorder and electrodynamics belong in the interpretation. No transition temperature is extracted here.\n\nEvidence: experiment. Sources: source-b7bc48c845a47b53.\n\nConditions: Selected historical first two reproduction pages, or Hardy YBa2Cu3O6.95 single crystals at 900 MHz and approximately 3–25 K; no inferred Tc or independent replication.\n\nVerification scope: Primary reproduction/abstract passages only; no raw-data reanalysis.\n\n### scwiki-a2-claim-limits-and-disputes-1\n\nShielding and expulsion can differ because of field history, geometry, connectivity and pinning. A small signal alone does not distinguish a small superconducting region from a measurement or geometry limitation. Conversely, a connected superconducting shell can screen a nonsuperconducting interior; a shielding measurement alone cannot establish the interior's volume fraction. The historical hollow-tube observation makes the distinction between material and enclosed space especially important.\n\nEvidence: hypothesis. Sources: source-512705f3e64d88d7.\n\nConditions: Editorial reporting/inference boundary; sample-specific protocol, geometry, uncertainty and source required.\n\nVerification scope: Pinned chapter edited with source checks stated individually; no new experimental dataset certified.\n\n### scwiki-a2-claim-limits-and-disputes-2\n\nBožović and colleagues' overdoped LSCO study is retained as a reference for superfluid-density measurements, but this check confirms its bibliographic identity rather than reanalyzing its scaling data. Do not infer a pairing mediator from a correlation alone. Likewise, a measured total stiffness cannot be decomposed into conventional and geometric contributions merely by changing its label. Follow the dedicated geometric-superfluid-weight entry for that separate model question.\n\nEvidence: hypothesis. Sources: source-b96ffab89d45664d.\n\nConditions: Editorial reporting/inference boundary; sample-specific protocol, geometry, uncertainty and source required.\n\nVerification scope: Pinned chapter edited with source checks stated individually; no new experimental dataset certified.\n\n### scwiki-a2-claim-discovery-implications-1\n\nBuild a magnetic evidence package around discriminating measurements: cooling history for trapping, dimensions and orientation for demagnetization, thickness for sheet response, and transport or heat capacity on comparable specimens for complementary evidence. Preserve disagreements between probes until sample and analysis differences are tested.\n\nEvidence: hypothesis. Sources: source-37f27e50dbdcf3a5.\n\nConditions: Editorial reporting/inference boundary; sample-specific protocol, geometry, uncertainty and source required.\n\nVerification scope: Pinned chapter edited with source checks stated individually; no new experimental dataset certified.\n\n### scwiki-a2-claim-discovery-implications-2\n\nSeparate measured magnetic response, the fitted electrodynamic parameters and the microscopic interpretation. This keeps magnetic evidence useful for discovering superconductivity without turning one model-dependent inversion into an unqualified material property or mechanism assignment.\n\nEvidence: hypothesis. Sources: source-37f27e50dbdcf3a5.\n\nConditions: Editorial reporting/inference boundary; sample-specific protocol, geometry, uncertainty and source required.\n\nVerification scope: Pinned chapter edited with source checks stated individually; no new experimental dataset certified.\n\n## Revision and review\n\nArticle ID: scwiki-a2. Revision: beae8ed476bfb412c85f. Updated: 2026-10-19.\n\nReview scope: Selected historical experiment reproduction and London original scan, modern demagnetization manuscript and Hardy abstract; independent London boundary-value, demagnetization and sheet-energy checks.\n\n- Scoped passage checks and model calculations are not full scientific acceptance or independent experimental replication.\n- London pp. 72–73 and Prozorov–Kogan pp. 1–2 only; arbitrary-shape simulation, microscopic/nonlocal theories and full historical manuscripts not exhaustively checked.\n- Hardy remains abstract-scoped. Bozovic LSCO is a bibliographic pointer; no scaling dataset or experimental Tc was re-extracted.\n- Shielding, demagnetization and sheet inductance formulas require their stated geometry, SI normalization and response regime.\n\nSource: [A_phenomenology/A2_meissner_effect.md](https://github.com/JackZH26/jzis-superconductivity/blob/0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf/A_phenomenology/A2_meissner_effect.md), commit 0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf, SHA-256 6f9cc20701142991b8034214d36e80f59cf2a70b125e0e4f85c505b85e6c45f6.\n","wordCount":859}}