{"schemaVersion":"1.0","revision":"fa961ab24277a91737c3","article":{"id":"scwiki-a7","topicId":"scwiki-topic-a7","code":"A7","slug":"critical-current","title":"Critical current: criteria, geometry and pinning","summary":"Compare transport, magnetization and depairing limits with their field, time scale and area convention. Preserve measurement conditions before comparing conductor performance.","aliases":["critical current density","depinning","depairing","engineering current density","Bean model","flux creep","pinning force"],"evidenceTypes":["theory","experiment","hypothesis"],"relatedCodes":["A6","A1","A2","B1","A5"],"editorialStatus":"scoped-check","contentState":"pilot","modifiedAt":"2026-10-19","review":{"checkedAt":"2026-10-19","scope":"Edited pinned A7 chapter, selected engineering author-manuscript passages, publisher identities and vortex-review abstract; independent conditional current/force/slab calculations. NIST-hosted Goodrich-Srivastava 1995 p. S19 checked for operational criteria and measurement-condition dependence.","limitations":["Selected source checks are not full scientific acceptance or independent experimental replication.","Bean original full text was inaccessible; the slab formula is a separately derived ideal model. Blatter is abstract-scoped.","Engineering sources are checked at stated manuscript passages; no complete electric-field criterion or final numerical performance dataset was re-extracted.","No cross-family current ranking, experimental Tc record or universal depairing value is inferred.","NIST source is a selected-page check of a measurement-methods paper, not validation of every reporting recommendation or material current record."]},"provenance":{"repository":"JackZH26/jzis-superconductivity","commit":"0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf","path":"A_phenomenology/A7_critical_current.md","sha256":"cddae9252ba1d88a829fbec70ef330b7ccd1b837a61692e8d5c64c82942a833d","sourceId":"file-7db9a40f7ecc795e"},"sourceMap":[{"startLine":1,"endLine":5,"targetSections":["scope"],"decision":"edited-with-scoped-evidence","note":"Scope retained."},{"startLine":6,"endLine":15,"targetSections":["scope","mathematical-assumptions"],"decision":"edited-with-scoped-evidence","note":"Distinguish measured limits and model-dependent inversion; separate local derivation from unchecked original."},{"startLine":16,"endLine":29,"targetSections":["experimental-criteria"],"decision":"edited-with-scoped-evidence","note":"Protocol and area conventions expanded."},{"startLine":30,"endLine":35,"targetSections":["mathematical-assumptions","limits-and-disputes"],"decision":"edited-with-scoped-evidence","note":"Correct scalar force scope; retain time-scale and field-boundary cautions."},{"startLine":36,"endLine":45,"targetSections":["experimental-criteria","limits-and-disputes"],"decision":"edited-with-scoped-evidence","note":"Correct engineering article type; selected primary scope and composition uncertainty."},{"startLine":46,"endLine":49,"targetSections":["discovery-implications"],"decision":"edited-with-scoped-evidence","note":"No mechanism or universal ranking from current alone."}],"references":[{"id":"source-39e3e980bd8a0271","doi":"10.1103/PhysRevLett.8.250","title":"Magnetization of Hard Superconductors","year":1962,"url":"https://doi.org/10.1103/PhysRevLett.8.250","publicationStatus":"journal-article","verification":{"scope":"publisher-metadata-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.8.250","note":"Publisher identity checked. Original full text inaccessible at attempted PDF endpoint. The explicitly ideal slab calculation is a separate local Ampere-law derivation, not a verified transcription of the original paper."}},{"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."}},{"id":"source-51c947cf11bea72e","doi":"10.1038/nmat1156","title":"Strongly enhanced current densities in superconducting coated conductors of YBa2Cu3O7-x + BaZrO3","year":2004,"url":"https://doi.org/10.1038/nmat1156","publicationStatus":"journal-article","verification":{"scope":"author-manuscript-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://arxiv.org/pdf/cond-mat/0406087v1","note":"Selected pp. 2–4: preparation, transport field orientation, in-field/self-field comparison and particle-composition ambiguity. No electric-field criterion extracted; raw curves and micrographs not independently reproduced."}},{"id":"source-c1a5f8c50328f68e","doi":"10.1038/nmat3887","title":"Isotropic round-wire multifilament cuprate superconductor for generation of magnetic fields above 30 T","year":2014,"url":"https://doi.org/10.1038/nmat3887","publicationStatus":"journal-article","verification":{"scope":"author-manuscript-and-publisher-metadata-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://arxiv.org/pdf/1305.1269v1","note":"Publisher identifies a primary Article published 9 March 2014. Earlier 2013 author manuscript, selected pp. 3–4, distinguishes superconducting/whole-conductor areas and sample lengths. Not a review; final dataset, criterion and manuscript-to-journal changes not fully checked."}},{"id":"source-25347d9dbd8e915f","doi":null,"title":"II-3: Critical current measurement methods: quantitative evaluation","year":1995,"url":"https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=30406","publicationStatus":"journal-article","verification":{"scope":"full-text-selected-page-check","checkedAt":"2026-10-19","evidenceUrl":"https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=30406","note":"NIST-hosted Cryogenics 35 supplement, p. S19: criterion definition, empirical voltage-current power law, Fig. 1 voltage/field criterion, and sensitivity to strain and experimental conditions. Selected first-page pixels read; remaining pages and underlying measurements not reanalyzed."}}],"materialCases":[],"claims":[{"id":"scwiki-a7-claim-scope-1","sectionId":"scope","statement":"Critical current is a limit defined for a specimen, measurement protocol and operating conditions. Transport measurements use a voltage or electric-field criterion; magnetization measurements infer a current through a model; the depairing current describes loss of stability of a superconducting state within a specified theory. These quantities must not share an unlabeled performance ranking.","evidenceType":"hypothesis","sourceIds":["source-82f53ca35625217d","source-25347d9dbd8e915f"],"conditions":"Editorial reporting/inference guidance; requires a sample-specific source, protocol and uncertainty. No new current or transition-temperature dataset certified.","verificationScope":"NIST Goodrich-Srivastava 1995 p. S19 and Blatter publisher abstract, plus explicit local reporting guidance; remaining full texts and sample-specific data not validated.","sourceRelationship":"mixed-source-and-local-derivation","derivationNote":"NIST p. S19 supports operational transport criteria and sensitivity to measurement conditions; Blatter abstract supplies vortex/depinning context. The reporting checklist and refusal to rank incomparable limits are local editorial guidance, not a certified standard or measured result."},{"id":"scwiki-a7-claim-scope-2","sectionId":"scope","statement":"Blatter and colleagues explicitly distinguish depinning and depairing current densities in their review abstract. Pinning, thermal fluctuations and vortex motion affect practical current transport. A measured threshold can instead be limited by weak connections or heating; its mechanism requires evidence beyond the value of the current itself.","evidenceType":"theory","sourceIds":["source-82f53ca35625217d"],"conditions":"Qualitative vortex/pinning interpretation; not a fitted universal Jc or critical-field boundary.","verificationScope":"Pinned chapter plus Blatter publisher abstract; full review remains unchecked."},{"id":"scwiki-a7-claim-mathematical-assumptions-1","sectionId":"mathematical-assumptions","statement":"For voltage taps separated by length $L$, define the mean longitudinal electric-field criterion $E_c=V_c/L$. Using the same measured $I_c$, distinguish the nominal superconducting cross-section $A_s$ from the entire conductor area $A_t$:","evidenceType":"theory","sourceIds":["source-25347d9dbd8e915f"],"conditions":"Definitions and local illustrative models: stated area convention, constant positive n power law, vector Lorentz force; slab uses full loop width, half-thickness, field parallel to faces, full penetration, constant Jc, no creep/demagnetization/reversible background.","verificationScope":"Independent local algebra, SI dimensional checks and numerical profile integration; not a full-text check of Bean 1962 or a fit to material data.","sourceRelationship":"mixed-source-and-local-derivation","derivationNote":"NIST p. S19 defines transport criteria and an empirical V-I power law. Conversion to a mean electric field, area conventions and the criterion-ratio example are local algebra with fixed geometry and positive constant exponent; no measured material exponent is assigned."},{"id":"scwiki-a7-claim-mathematical-assumptions-2","sectionId":"mathematical-assumptions","statement":"$$J_c=\\frac{I_c}{A_s},\\qquad J_e=\\frac{I_c}{A_t}=fJ_c,\\qquad f=\\frac{A_s}{A_t}.$$","evidenceType":"theory","sourceIds":["source-39e3e980bd8a0271"],"conditions":"Definitions and local illustrative models: stated area convention, constant positive n power law, vector Lorentz force; slab uses full loop width, half-thickness, field parallel to faces, full penetration, constant Jc, no creep/demagnetization/reversible background.","verificationScope":"Independent local algebra, SI dimensional checks and numerical profile integration; not a full-text check of Bean 1962 or a fit to material data.","sourceRelationship":"context-for-local-derivation","derivationNote":"The stated algebra, vector force and ideal slab model are local derivations under explicit assumptions. Bean is historical context only; original full-text support was not verified."},{"id":"scwiki-a7-claim-mathematical-assumptions-3","sectionId":"mathematical-assumptions","statement":"Here $J_e$ is an engineering current density for the stated whole-conductor area. Nominal superconducting area does not resolve nonuniform current paths or pores. A different definition of non-copper area, filament area or cable area requires a different label. These identities compare area conventions; they do not correct a damaged conductor or infer intrinsic microscopic current density.","evidenceType":"theory","sourceIds":["source-39e3e980bd8a0271"],"conditions":"Definitions and local illustrative models: stated area convention, constant positive n power law, vector Lorentz force; slab uses full loop width, half-thickness, field parallel to faces, full penetration, constant Jc, no creep/demagnetization/reversible background.","verificationScope":"Independent local algebra, SI dimensional checks and numerical profile integration; not a full-text check of Bean 1962 or a fit to material data.","sourceRelationship":"context-for-local-derivation","derivationNote":"The stated algebra, vector force and ideal slab model are local derivations under explicit assumptions. Bean is historical context only; original full-text support was not verified."},{"id":"scwiki-a7-claim-mathematical-assumptions-4","sectionId":"mathematical-assumptions","statement":"A simple sensitivity calculation illustrates why the criterion matters. If a measured interval is modeled by $E=E_0(J/J_0)^n$ with positive constant $n$, changing the criterion by a factor $r$ changes the inferred current by $r^{1/n}$. For $n=20$, a tenfold criterion change gives a factor of about 1.122. This is an illustrative power-law calculation, not a measured exponent or a universal transport law.","evidenceType":"theory","sourceIds":["source-25347d9dbd8e915f"],"conditions":"Definitions and local illustrative models: stated area convention, constant positive n power law, vector Lorentz force; slab uses full loop width, half-thickness, field parallel to faces, full penetration, constant Jc, no creep/demagnetization/reversible background.","verificationScope":"Independent local algebra, SI dimensional checks and numerical profile integration; not a full-text check of Bean 1962 or a fit to material data.","sourceRelationship":"mixed-source-and-local-derivation","derivationNote":"NIST p. S19 defines transport criteria and an empirical V-I power law. Conversion to a mean electric field, area conventions and the criterion-ratio example are local algebra with fixed geometry and positive constant exponent; no measured material exponent is assigned."},{"id":"scwiki-a7-claim-mathematical-assumptions-5","sectionId":"mathematical-assumptions","statement":"The magnetic force density is the vector product $\\mathbf F=\\mathbf J\\times\\mathbf B$. For current perpendicular to the induction, its magnitude at the threshold becomes $F_p=J_cB$ if a static pinning-force balance is the applicable model. In SI units this is N/m³; resistive heating instead has power density $\\mathbf E\\cdot\\mathbf J$ in W/m³. A parallel configuration has no force from this cross product. Replacing $B$ by $\\mu_0H$ requires an additional approximation, especially near self-field conditions.","evidenceType":"theory","sourceIds":["source-39e3e980bd8a0271"],"conditions":"Definitions and local illustrative models: stated area convention, constant positive n power law, vector Lorentz force; slab uses full loop width, half-thickness, field parallel to faces, full penetration, constant Jc, no creep/demagnetization/reversible background.","verificationScope":"Independent local algebra, SI dimensional checks and numerical profile integration; not a full-text check of Bean 1962 or a fit to material data.","sourceRelationship":"context-for-local-derivation","derivationNote":"The stated algebra, vector force and ideal slab model are local derivations under explicit assumptions. Bean is historical context only; original full-text support was not verified."},{"id":"scwiki-a7-claim-mathematical-assumptions-6","sectionId":"mathematical-assumptions","statement":"For an explicit geometry check, take an infinite slab of full thickness $2a$, field parallel to its faces, constant $J_c$ and fully developed opposite critical-state branches. Neglect demagnetization, reversible magnetization and creep. Ampere's law gives $|dB/dx|=\\mu_0J_c$. With $x$ measured across the thickness, the ideal branch profiles are $B_\\pm(x)=\\mu_0[H_a\\pm J_c(a-|x|)]$. Their spatial averages give","evidenceType":"theory","sourceIds":["source-39e3e980bd8a0271"],"conditions":"Definitions and local illustrative models: stated area convention, constant positive n power law, vector Lorentz force; slab uses full loop width, half-thickness, field parallel to faces, full penetration, constant Jc, no creep/demagnetization/reversible background.","verificationScope":"Independent local algebra, SI dimensional checks and numerical profile integration; not a full-text check of Bean 1962 or a fit to material data.","sourceRelationship":"context-for-local-derivation","derivationNote":"The stated algebra, vector force and ideal slab model are local derivations under explicit assumptions. Bean is historical context only; original full-text support was not verified."},{"id":"scwiki-a7-claim-mathematical-assumptions-7","sectionId":"mathematical-assumptions","statement":"$$M_\\pm=\\pm\\frac{J_ca}{2},\\qquad \\Delta M=M_+-M_-=J_ca,\\qquad J_c=\\frac{\\Delta M}{a}.$$","evidenceType":"theory","sourceIds":["source-39e3e980bd8a0271"],"conditions":"Definitions and local illustrative models: stated area convention, constant positive n power law, vector Lorentz force; slab uses full loop width, half-thickness, field parallel to faces, full penetration, constant Jc, no creep/demagnetization/reversible background.","verificationScope":"Independent local algebra, SI dimensional checks and numerical profile integration; not a full-text check of Bean 1962 or a fit to material data.","sourceRelationship":"context-for-local-derivation","derivationNote":"The stated algebra, vector force and ideal slab model are local derivations under explicit assumptions. Bean is historical context only; original full-text support was not verified."},{"id":"scwiki-a7-claim-mathematical-assumptions-8","sectionId":"mathematical-assumptions","statement":"$\\Delta M$ is the full vertical loop width in A/m; $a$ is the half-thickness in metres. Numerically integrating these stated profiles verifies the factor and dimensions. This is a local derivation under ideal assumptions, not a reanalysis of Bean's original data. Bean's 1962 paper is the historical source; its full text was not available in this check. Do not transfer this slab expression to arbitrary films, cylinders, partial penetration or field-dependent $J_c$.","evidenceType":"theory","sourceIds":["source-39e3e980bd8a0271"],"conditions":"Definitions and local illustrative models: stated area convention, constant positive n power law, vector Lorentz force; slab uses full loop width, half-thickness, field parallel to faces, full penetration, constant Jc, no creep/demagnetization/reversible background.","verificationScope":"Independent local algebra, SI dimensional checks and numerical profile integration; not a full-text check of Bean 1962 or a fit to material data.","sourceRelationship":"context-for-local-derivation","derivationNote":"The stated algebra, vector force and ideal slab model are local derivations under explicit assumptions. Bean is historical context only; original full-text support was not verified."},{"id":"scwiki-a7-claim-experimental-criteria-1","sectionId":"experimental-criteria","statement":"Keep enough information to reconstruct both the measured threshold and the conversion to a current density. The following is an editorial reporting guide, not a table of certified material records.","evidenceType":"hypothesis","sourceIds":["source-82f53ca35625217d","source-25347d9dbd8e915f"],"conditions":"Editorial reporting/inference guidance; requires a sample-specific source, protocol and uncertainty. No new current or transition-temperature dataset certified.","verificationScope":"NIST Goodrich-Srivastava 1995 p. S19 and Blatter publisher abstract, plus explicit local reporting guidance; remaining full texts and sample-specific data not validated.","sourceRelationship":"mixed-source-and-local-derivation","derivationNote":"NIST p. S19 supports operational transport criteria and sensitivity to measurement conditions; Blatter abstract supplies vortex/depinning context. The reporting checklist and refusal to rank incomparable limits are local editorial guidance, not a certified standard or measured result."},{"id":"scwiki-a7-claim-experimental-criteria-2","sectionId":"experimental-criteria","statement":"| Measurement | Record with the result | Interpretation boundary |\n|---|---|---|\n| Transport | Temperature, field magnitude and orientation, voltage-tap spacing, electric-field criterion, current ramp or pulse duration, duty cycle, area convention | Contact heating, current sharing and voltage offsets can change the inferred threshold |\n| Magnetization | Magnetic moment, volume, dimensions, field history, sweep rate, elapsed time and background subtraction | Conversion to current requires a geometry and critical-state model |\n| Conductor or cable | Superconducting fraction, whole-conductor area, length, strain and thermal stabilization | Layer current density and engineering current density answer different questions |\n| Depairing model | Model, temperature regime, geometry and stability assumptions | A calculated instability is not automatically the measured transport limit |","evidenceType":"hypothesis","sourceIds":["source-82f53ca35625217d","source-25347d9dbd8e915f"],"conditions":"Editorial reporting/inference guidance; requires a sample-specific source, protocol and uncertainty. No new current or transition-temperature dataset certified.","verificationScope":"NIST Goodrich-Srivastava 1995 p. S19 and Blatter publisher abstract, plus explicit local reporting guidance; remaining full texts and sample-specific data not validated.","sourceRelationship":"mixed-source-and-local-derivation","derivationNote":"NIST p. S19 supports operational transport criteria and sensitivity to measurement conditions; Blatter abstract supplies vortex/depinning context. The reporting checklist and refusal to rank incomparable limits are local editorial guidance, not a certified standard or measured result."},{"id":"scwiki-a7-claim-experimental-criteria-3","sectionId":"experimental-criteria","statement":"MacManus-Driscoll and colleagues studied YBCO films grown with BaZrO3 added to the target. Their selected manuscript passages report improved in-field transport at 75.5 K, despite lower transition temperature or self-field current in some samples. The rotated field remained perpendicular to current. This comparison illustrates distinct performance measures, not proof of a different pairing mechanism. No electric-field criterion was extracted here, so no numerical current record is tabulated.","evidenceType":"experiment","sourceIds":["source-51c947cf11bea72e"],"conditions":"Selected author-manuscript passages only; retain sample, orientation, area and version limits. No complete numerical performance record or independent experimental reproduction.","verificationScope":"nmat1156 author manuscript pp. 2–4; nmat3887 earlier manuscript pp. 3–4 and publisher article identity, as applicable."},{"id":"scwiki-a7-claim-experimental-criteria-4","sectionId":"experimental-criteria","statement":"Larbalestier and colleagues' 2014 paper is a primary Bi-2212 round-wire study, not a general engineering review. Its earlier author manuscript distinguishes whole-conductor $J_e$ from superconducting-area $J_c$ and compares short specimens with a demonstration coil. This entry uses that distinction without importing a cross-family ranking; the final published dataset and all measurement criteria were not independently re-extracted.","evidenceType":"experiment","sourceIds":["source-c1a5f8c50328f68e"],"conditions":"Selected author-manuscript passages only; retain sample, orientation, area and version limits. No complete numerical performance record or independent experimental reproduction.","verificationScope":"nmat1156 author manuscript pp. 2–4; nmat3887 earlier manuscript pp. 3–4 and publisher article identity, as applicable."},{"id":"scwiki-a7-claim-limits-and-disputes-1","sectionId":"limits-and-disputes","statement":"An irreversible magnetic response and the upper critical field are different observables. A loss of measurable hysteresis or a transport threshold need not locate the destruction of superconductivity. Report the criterion, time scale and field history rather than silently equating an irreversibility line with $H_{c2}$. Thermally activated creep is one reason different observation times can yield different apparent current limits.","evidenceType":"theory","sourceIds":["source-82f53ca35625217d"],"conditions":"Qualitative vortex/pinning interpretation; not a fitted universal Jc or critical-field boundary.","verificationScope":"Pinned chapter plus Blatter publisher abstract; full review remains unchecked."},{"id":"scwiki-a7-claim-limits-and-disputes-2","sectionId":"limits-and-disputes","statement":"A fitted pinning-curve exponent or angular peak does not uniquely identify a defect. Compare microscopy, processing controls and temperature/angle dependence. In the checked YBCO manuscript, the detailed composition of the small second-phase particles is not conclusively established; describing every particle as chemically proven BaZrO3 would overstate the evidence.","evidenceType":"experiment","sourceIds":["source-51c947cf11bea72e"],"conditions":"Selected author-manuscript passages only; retain sample, orientation, area and version limits. No complete numerical performance record or independent experimental reproduction.","verificationScope":"nmat1156 author manuscript pp. 2–4; nmat3887 earlier manuscript pp. 3–4 and publisher article identity, as applicable."},{"id":"scwiki-a7-claim-discovery-implications-1","sectionId":"discovery-implications","statement":"Separate a better transport path, stronger pinning, a larger superconducting fraction and a higher depairing limit as distinct hypotheses. Compare controlled specimens under matched conditions before assigning an improvement to a microscopic mechanism. A gain in current-carrying performance does not by itself demonstrate higher $T_c$, a new pairing interaction or a new superconducting phase.","evidenceType":"hypothesis","sourceIds":["source-82f53ca35625217d"],"conditions":"Editorial reporting/inference guidance; requires a sample-specific source, protocol and uncertainty. No new current or transition-temperature dataset certified.","verificationScope":"Pinned chapter edited with explicitly limited reference checks."},{"id":"scwiki-a7-claim-discovery-implications-2","sectionId":"discovery-implications","statement":"Preserve the measured signal, uncertainty and analysis conventions together. If the voltage criterion, geometry or thermal history is unavailable, retain that gap in the record. A useful follow-up is the measurement that discriminates between the competing limits, rather than another unsupported performance ranking.","evidenceType":"hypothesis","sourceIds":["source-82f53ca35625217d","source-25347d9dbd8e915f"],"conditions":"Editorial reporting/inference guidance; requires a sample-specific source, protocol and uncertainty. No new current or transition-temperature dataset certified.","verificationScope":"NIST Goodrich-Srivastava 1995 p. S19 and Blatter publisher abstract, plus explicit local reporting guidance; remaining full texts and sample-specific data not validated.","sourceRelationship":"mixed-source-and-local-derivation","derivationNote":"NIST p. S19 supports operational transport criteria and sensitivity to measurement conditions; Blatter abstract supplies vortex/depinning context. The reporting checklist and refusal to rank incomparable limits are local editorial guidance, not a certified standard or measured result."}],"corrections":[{"id":"scwiki-a7-correction-1","description":"Identified nmat3887 as a primary Bi-2212 round-wire article rather than a general engineering review.","sourceId":"source-c1a5f8c50328f68e","scope":"Publication type and source use, not validation of all conductor comparisons."},{"id":"scwiki-a7-correction-2","description":"Qualified Fp=Jc B by perpendicular geometry and a static force-balance model; distinguished induction from applied field.","sourceId":"source-82f53ca35625217d","scope":"Explicit SI vector relation and approximation, not a new measured force."},{"id":"scwiki-a7-correction-measurement-source","description":"Added a directly checked measurement-methods source for operational transport criteria; separated local algebra and reporting guidance from historical source context.","sourceId":"source-25347d9dbd8e915f","scope":"Selected NIST-hosted p. S19 and explicit source relationships; no new current dataset or full scientific acceptance."}],"equationChecks":[{"targetSectionId":"mathematical-assumptions","tex":"J_e=fJ_c","check":"Same Ic divided by explicitly different cross-sectional areas; local identity check."},{"targetSectionId":"mathematical-assumptions","tex":"J_c=\\Delta M/a","check":"Six numerical integrations and finite-difference Ampere slopes for stated ideal slab; full loop width/half-thickness conventions required."},{"targetSectionId":"mathematical-assumptions","tex":"|\\mathbf J\\times\\mathbf B|=JB|\\sin\\theta|","check":"Three angle checks and SI dimensions; heating has different units."},{"targetSectionId":"mathematical-assumptions","tex":"J_{c,2}/J_{c,1}=r^{1/n}","check":"Three constant-exponent power-law sensitivity cases; illustrative, not measured exponents."}],"body":"## Scope\n\nCritical current is a limit defined for a specimen, measurement protocol and operating conditions. Transport measurements use a voltage or electric-field criterion; magnetization measurements infer a current through a model; the depairing current describes loss of stability of a superconducting state within a specified theory. These quantities must not share an unlabeled performance ranking. [R5](#ref-source-25347d9dbd8e915f)\n\nBlatter and colleagues explicitly distinguish depinning and depairing current densities in their review abstract. Pinning, thermal fluctuations and vortex motion affect practical current transport. A measured threshold can instead be limited by weak connections or heating; its mechanism requires evidence beyond the value of the current itself. [R2](#ref-source-82f53ca35625217d)\n\n## Mathematical assumptions\n\nFor voltage taps separated by length $L$, define the mean longitudinal electric-field criterion $E_c=V_c/L$. Using the same measured $I_c$, distinguish the nominal superconducting cross-section $A_s$ from the entire conductor area $A_t$:\n\n$$J_c=\\frac{I_c}{A_s},\\qquad J_e=\\frac{I_c}{A_t}=fJ_c,\\qquad f=\\frac{A_s}{A_t}.$$\n\nHere $J_e$ is an engineering current density for the stated whole-conductor area. Nominal superconducting area does not resolve nonuniform current paths or pores. A different definition of non-copper area, filament area or cable area requires a different label. These identities compare area conventions; they do not correct a damaged conductor or infer intrinsic microscopic current density.\n\nA simple sensitivity calculation illustrates why the criterion matters. If a measured interval is modeled by $E=E_0(J/J_0)^n$ with positive constant $n$, changing the criterion by a factor $r$ changes the inferred current by $r^{1/n}$. For $n=20$, a tenfold criterion change gives a factor of about 1.122. This is an illustrative power-law calculation, not a measured exponent or a universal transport law. [R5](#ref-source-25347d9dbd8e915f)\n\nThe magnetic force density is the vector product $\\mathbf F=\\mathbf J\\times\\mathbf B$. For current perpendicular to the induction, its magnitude at the threshold becomes $F_p=J_cB$ if a static pinning-force balance is the applicable model. In SI units this is N/m³; resistive heating instead has power density $\\mathbf E\\cdot\\mathbf J$ in W/m³. A parallel configuration has no force from this cross product. Replacing $B$ by $\\mu_0H$ requires an additional approximation, especially near self-field conditions.\n\nFor an explicit geometry check, take an infinite slab of full thickness $2a$, field parallel to its faces, constant $J_c$ and fully developed opposite critical-state branches. Neglect demagnetization, reversible magnetization and creep. Ampere's law gives $|dB/dx|=\\mu_0J_c$. With $x$ measured across the thickness, the ideal branch profiles are $B_\\pm(x)=\\mu_0[H_a\\pm J_c(a-|x|)]$. Their spatial averages give\n\n$$M_\\pm=\\pm\\frac{J_ca}{2},\\qquad \\Delta M=M_+-M_-=J_ca,\\qquad J_c=\\frac{\\Delta M}{a}.$$\n\n$\\Delta M$ is the full vertical loop width in A/m; $a$ is the half-thickness in metres. Numerically integrating these stated profiles verifies the factor and dimensions. This is a local derivation under ideal assumptions, not a reanalysis of Bean's original data. Bean's 1962 paper is the historical source; its full text was not available in this check. Do not transfer this slab expression to arbitrary films, cylinders, partial penetration or field-dependent $J_c$. [R1](#ref-source-39e3e980bd8a0271)\n\n## Experimental criteria\n\nKeep enough information to reconstruct both the measured threshold and the conversion to a current density. The following is an editorial reporting guide, not a table of certified material records.\n\n| Measurement | Record with the result | Interpretation boundary |\n|---|---|---|\n| Transport | Temperature, field magnitude and orientation, voltage-tap spacing, electric-field criterion, current ramp or pulse duration, duty cycle, area convention | Contact heating, current sharing and voltage offsets can change the inferred threshold |\n| Magnetization | Magnetic moment, volume, dimensions, field history, sweep rate, elapsed time and background subtraction | Conversion to current requires a geometry and critical-state model |\n| Conductor or cable | Superconducting fraction, whole-conductor area, length, strain and thermal stabilization | Layer current density and engineering current density answer different questions |\n| Depairing model | Model, temperature regime, geometry and stability assumptions | A calculated instability is not automatically the measured transport limit |\n\nMacManus-Driscoll and colleagues studied YBCO films grown with BaZrO3 added to the target. Their selected manuscript passages report improved in-field transport at 75.5 K, despite lower transition temperature or self-field current in some samples. The rotated field remained perpendicular to current. This comparison illustrates distinct performance measures, not proof of a different pairing mechanism. No electric-field criterion was extracted here, so no numerical current record is tabulated. [R3](#ref-source-51c947cf11bea72e)\n\nLarbalestier and colleagues' 2014 paper is a primary Bi-2212 round-wire study, not a general engineering review. Its earlier author manuscript distinguishes whole-conductor $J_e$ from superconducting-area $J_c$ and compares short specimens with a demonstration coil. This entry uses that distinction without importing a cross-family ranking; the final published dataset and all measurement criteria were not independently re-extracted. [R4](#ref-source-c1a5f8c50328f68e)\n\n## Limits and disputes\n\nAn irreversible magnetic response and the upper critical field are different observables. A loss of measurable hysteresis or a transport threshold need not locate the destruction of superconductivity. Report the criterion, time scale and field history rather than silently equating an irreversibility line with $H_{c2}$. Thermally activated creep is one reason different observation times can yield different apparent current limits. [R2](#ref-source-82f53ca35625217d)\n\nA fitted pinning-curve exponent or angular peak does not uniquely identify a defect. Compare microscopy, processing controls and temperature/angle dependence. In the checked YBCO manuscript, the detailed composition of the small second-phase particles is not conclusively established; describing every particle as chemically proven BaZrO3 would overstate the evidence. [R3](#ref-source-51c947cf11bea72e)\n\n## Discovery implications\n\nSeparate a better transport path, stronger pinning, a larger superconducting fraction and a higher depairing limit as distinct hypotheses. Compare controlled specimens under matched conditions before assigning an improvement to a microscopic mechanism. A gain in current-carrying performance does not by itself demonstrate higher $T_c$, a new pairing interaction or a new superconducting phase.\n\nPreserve the measured signal, uncertainty and analysis conventions together. If the voltage criterion, geometry or thermal history is unavailable, retain that gap in the record. A useful follow-up is the measurement that discriminates between the competing limits, rather than another unsupported performance ranking. [R5](#ref-source-25347d9dbd8e915f)\n\n## Related topics\n\nConnect to [critical fields](/scwiki/critical-fields), [zero resistance](/scwiki/zero-resistance), [Meissner effect](/scwiki/meissner-effect), [BCS theory](/scwiki/bcs-weak-coupling), and [superconducting gap](/scwiki/superconducting-gap).","revision":"c5bc330ef247c18130fa","sections":[{"id":"scope","title":"Scope","markdown":"## Scope\n\nCritical current is a limit defined for a specimen, measurement protocol and operating conditions. Transport measurements use a voltage or electric-field criterion; magnetization measurements infer a current through a model; the depairing current describes loss of stability of a superconducting state within a specified theory. These quantities must not share an unlabeled performance ranking. [R5](#ref-source-25347d9dbd8e915f)\n\nBlatter and colleagues explicitly distinguish depinning and depairing current densities in their review abstract. Pinning, thermal fluctuations and vortex motion affect practical current transport. A measured threshold can instead be limited by weak connections or heating; its mechanism requires evidence beyond the value of the current itself. [R2](#ref-source-82f53ca35625217d)"},{"id":"mathematical-assumptions","title":"Mathematical assumptions","markdown":"## Mathematical assumptions\n\nFor voltage taps separated by length $L$, define the mean longitudinal electric-field criterion $E_c=V_c/L$. Using the same measured $I_c$, distinguish the nominal superconducting cross-section $A_s$ from the entire conductor area $A_t$:\n\n$$J_c=\\frac{I_c}{A_s},\\qquad J_e=\\frac{I_c}{A_t}=fJ_c,\\qquad f=\\frac{A_s}{A_t}.$$\n\nHere $J_e$ is an engineering current density for the stated whole-conductor area. Nominal superconducting area does not resolve nonuniform current paths or pores. A different definition of non-copper area, filament area or cable area requires a different label. These identities compare area conventions; they do not correct a damaged conductor or infer intrinsic microscopic current density.\n\nA simple sensitivity calculation illustrates why the criterion matters. If a measured interval is modeled by $E=E_0(J/J_0)^n$ with positive constant $n$, changing the criterion by a factor $r$ changes the inferred current by $r^{1/n}$. For $n=20$, a tenfold criterion change gives a factor of about 1.122. This is an illustrative power-law calculation, not a measured exponent or a universal transport law. [R5](#ref-source-25347d9dbd8e915f)\n\nThe magnetic force density is the vector product $\\mathbf F=\\mathbf J\\times\\mathbf B$. For current perpendicular to the induction, its magnitude at the threshold becomes $F_p=J_cB$ if a static pinning-force balance is the applicable model. In SI units this is N/m³; resistive heating instead has power density $\\mathbf E\\cdot\\mathbf J$ in W/m³. A parallel configuration has no force from this cross product. Replacing $B$ by $\\mu_0H$ requires an additional approximation, especially near self-field conditions.\n\nFor an explicit geometry check, take an infinite slab of full thickness $2a$, field parallel to its faces, constant $J_c$ and fully developed opposite critical-state branches. Neglect demagnetization, reversible magnetization and creep. Ampere's law gives $|dB/dx|=\\mu_0J_c$. With $x$ measured across the thickness, the ideal branch profiles are $B_\\pm(x)=\\mu_0[H_a\\pm J_c(a-|x|)]$. Their spatial averages give\n\n$$M_\\pm=\\pm\\frac{J_ca}{2},\\qquad \\Delta M=M_+-M_-=J_ca,\\qquad J_c=\\frac{\\Delta M}{a}.$$\n\n$\\Delta M$ is the full vertical loop width in A/m; $a$ is the half-thickness in metres. Numerically integrating these stated profiles verifies the factor and dimensions. This is a local derivation under ideal assumptions, not a reanalysis of Bean's original data. Bean's 1962 paper is the historical source; its full text was not available in this check. Do not transfer this slab expression to arbitrary films, cylinders, partial penetration or field-dependent $J_c$. [R1](#ref-source-39e3e980bd8a0271)"},{"id":"experimental-criteria","title":"Experimental criteria","markdown":"## Experimental criteria\n\nKeep enough information to reconstruct both the measured threshold and the conversion to a current density. The following is an editorial reporting guide, not a table of certified material records.\n\n| Measurement | Record with the result | Interpretation boundary |\n|---|---|---|\n| Transport | Temperature, field magnitude and orientation, voltage-tap spacing, electric-field criterion, current ramp or pulse duration, duty cycle, area convention | Contact heating, current sharing and voltage offsets can change the inferred threshold |\n| Magnetization | Magnetic moment, volume, dimensions, field history, sweep rate, elapsed time and background subtraction | Conversion to current requires a geometry and critical-state model |\n| Conductor or cable | Superconducting fraction, whole-conductor area, length, strain and thermal stabilization | Layer current density and engineering current density answer different questions |\n| Depairing model | Model, temperature regime, geometry and stability assumptions | A calculated instability is not automatically the measured transport limit |\n\nMacManus-Driscoll and colleagues studied YBCO films grown with BaZrO3 added to the target. Their selected manuscript passages report improved in-field transport at 75.5 K, despite lower transition temperature or self-field current in some samples. The rotated field remained perpendicular to current. This comparison illustrates distinct performance measures, not proof of a different pairing mechanism. No electric-field criterion was extracted here, so no numerical current record is tabulated. [R3](#ref-source-51c947cf11bea72e)\n\nLarbalestier and colleagues' 2014 paper is a primary Bi-2212 round-wire study, not a general engineering review. Its earlier author manuscript distinguishes whole-conductor $J_e$ from superconducting-area $J_c$ and compares short specimens with a demonstration coil. This entry uses that distinction without importing a cross-family ranking; the final published dataset and all measurement criteria were not independently re-extracted. [R4](#ref-source-c1a5f8c50328f68e)"},{"id":"limits-and-disputes","title":"Limits and disputes","markdown":"## Limits and disputes\n\nAn irreversible magnetic response and the upper critical field are different observables. A loss of measurable hysteresis or a transport threshold need not locate the destruction of superconductivity. Report the criterion, time scale and field history rather than silently equating an irreversibility line with $H_{c2}$. Thermally activated creep is one reason different observation times can yield different apparent current limits. [R2](#ref-source-82f53ca35625217d)\n\nA fitted pinning-curve exponent or angular peak does not uniquely identify a defect. Compare microscopy, processing controls and temperature/angle dependence. In the checked YBCO manuscript, the detailed composition of the small second-phase particles is not conclusively established; describing every particle as chemically proven BaZrO3 would overstate the evidence. [R3](#ref-source-51c947cf11bea72e)"},{"id":"discovery-implications","title":"Discovery implications","markdown":"## Discovery implications\n\nSeparate a better transport path, stronger pinning, a larger superconducting fraction and a higher depairing limit as distinct hypotheses. Compare controlled specimens under matched conditions before assigning an improvement to a microscopic mechanism. A gain in current-carrying performance does not by itself demonstrate higher $T_c$, a new pairing interaction or a new superconducting phase.\n\nPreserve the measured signal, uncertainty and analysis conventions together. If the voltage criterion, geometry or thermal history is unavailable, retain that gap in the record. A useful follow-up is the measurement that discriminates between the competing limits, rather than another unsupported performance ranking. [R5](#ref-source-25347d9dbd8e915f)"},{"id":"related-topics","title":"Related topics","markdown":"## Related topics\n\nConnect to [critical fields](/scwiki/critical-fields), [zero resistance](/scwiki/zero-resistance), [Meissner effect](/scwiki/meissner-effect), [BCS theory](/scwiki/bcs-weak-coupling), and [superconducting gap](/scwiki/superconducting-gap)."},{"id":"material-cases","title":"Material cases","markdown":"## Material cases\n\nNo sample-specific material records are included in this article. Model results are not experimental transition temperatures."}],"markdown":"# Critical current: criteria, geometry and pinning\n\nCompare transport, magnetization and depairing limits with their field, time scale and area convention. Preserve measurement conditions before comparing conductor performance.\n\n## Scope\n\nCritical current is a limit defined for a specimen, measurement protocol and operating conditions. Transport measurements use a voltage or electric-field criterion; magnetization measurements infer a current through a model; the depairing current describes loss of stability of a superconducting state within a specified theory. These quantities must not share an unlabeled performance ranking. [R5](#ref-source-25347d9dbd8e915f)\n\nBlatter and colleagues explicitly distinguish depinning and depairing current densities in their review abstract. Pinning, thermal fluctuations and vortex motion affect practical current transport. A measured threshold can instead be limited by weak connections or heating; its mechanism requires evidence beyond the value of the current itself. [R2](#ref-source-82f53ca35625217d)\n\n## Mathematical assumptions\n\nFor voltage taps separated by length $L$, define the mean longitudinal electric-field criterion $E_c=V_c/L$. Using the same measured $I_c$, distinguish the nominal superconducting cross-section $A_s$ from the entire conductor area $A_t$:\n\n$$J_c=\\frac{I_c}{A_s},\\qquad J_e=\\frac{I_c}{A_t}=fJ_c,\\qquad f=\\frac{A_s}{A_t}.$$\n\nHere $J_e$ is an engineering current density for the stated whole-conductor area. Nominal superconducting area does not resolve nonuniform current paths or pores. A different definition of non-copper area, filament area or cable area requires a different label. These identities compare area conventions; they do not correct a damaged conductor or infer intrinsic microscopic current density.\n\nA simple sensitivity calculation illustrates why the criterion matters. If a measured interval is modeled by $E=E_0(J/J_0)^n$ with positive constant $n$, changing the criterion by a factor $r$ changes the inferred current by $r^{1/n}$. For $n=20$, a tenfold criterion change gives a factor of about 1.122. This is an illustrative power-law calculation, not a measured exponent or a universal transport law. [R5](#ref-source-25347d9dbd8e915f)\n\nThe magnetic force density is the vector product $\\mathbf F=\\mathbf J\\times\\mathbf B$. For current perpendicular to the induction, its magnitude at the threshold becomes $F_p=J_cB$ if a static pinning-force balance is the applicable model. In SI units this is N/m³; resistive heating instead has power density $\\mathbf E\\cdot\\mathbf J$ in W/m³. A parallel configuration has no force from this cross product. Replacing $B$ by $\\mu_0H$ requires an additional approximation, especially near self-field conditions.\n\nFor an explicit geometry check, take an infinite slab of full thickness $2a$, field parallel to its faces, constant $J_c$ and fully developed opposite critical-state branches. Neglect demagnetization, reversible magnetization and creep. Ampere's law gives $|dB/dx|=\\mu_0J_c$. With $x$ measured across the thickness, the ideal branch profiles are $B_\\pm(x)=\\mu_0[H_a\\pm J_c(a-|x|)]$. Their spatial averages give\n\n$$M_\\pm=\\pm\\frac{J_ca}{2},\\qquad \\Delta M=M_+-M_-=J_ca,\\qquad J_c=\\frac{\\Delta M}{a}.$$\n\n$\\Delta M$ is the full vertical loop width in A/m; $a$ is the half-thickness in metres. Numerically integrating these stated profiles verifies the factor and dimensions. This is a local derivation under ideal assumptions, not a reanalysis of Bean's original data. Bean's 1962 paper is the historical source; its full text was not available in this check. Do not transfer this slab expression to arbitrary films, cylinders, partial penetration or field-dependent $J_c$. [R1](#ref-source-39e3e980bd8a0271)\n\n## Experimental criteria\n\nKeep enough information to reconstruct both the measured threshold and the conversion to a current density. The following is an editorial reporting guide, not a table of certified material records.\n\n| Measurement | Record with the result | Interpretation boundary |\n|---|---|---|\n| Transport | Temperature, field magnitude and orientation, voltage-tap spacing, electric-field criterion, current ramp or pulse duration, duty cycle, area convention | Contact heating, current sharing and voltage offsets can change the inferred threshold |\n| Magnetization | Magnetic moment, volume, dimensions, field history, sweep rate, elapsed time and background subtraction | Conversion to current requires a geometry and critical-state model |\n| Conductor or cable | Superconducting fraction, whole-conductor area, length, strain and thermal stabilization | Layer current density and engineering current density answer different questions |\n| Depairing model | Model, temperature regime, geometry and stability assumptions | A calculated instability is not automatically the measured transport limit |\n\nMacManus-Driscoll and colleagues studied YBCO films grown with BaZrO3 added to the target. Their selected manuscript passages report improved in-field transport at 75.5 K, despite lower transition temperature or self-field current in some samples. The rotated field remained perpendicular to current. This comparison illustrates distinct performance measures, not proof of a different pairing mechanism. No electric-field criterion was extracted here, so no numerical current record is tabulated. [R3](#ref-source-51c947cf11bea72e)\n\nLarbalestier and colleagues' 2014 paper is a primary Bi-2212 round-wire study, not a general engineering review. Its earlier author manuscript distinguishes whole-conductor $J_e$ from superconducting-area $J_c$ and compares short specimens with a demonstration coil. This entry uses that distinction without importing a cross-family ranking; the final published dataset and all measurement criteria were not independently re-extracted. [R4](#ref-source-c1a5f8c50328f68e)\n\n## Limits and disputes\n\nAn irreversible magnetic response and the upper critical field are different observables. A loss of measurable hysteresis or a transport threshold need not locate the destruction of superconductivity. Report the criterion, time scale and field history rather than silently equating an irreversibility line with $H_{c2}$. Thermally activated creep is one reason different observation times can yield different apparent current limits. [R2](#ref-source-82f53ca35625217d)\n\nA fitted pinning-curve exponent or angular peak does not uniquely identify a defect. Compare microscopy, processing controls and temperature/angle dependence. In the checked YBCO manuscript, the detailed composition of the small second-phase particles is not conclusively established; describing every particle as chemically proven BaZrO3 would overstate the evidence. [R3](#ref-source-51c947cf11bea72e)\n\n## Discovery implications\n\nSeparate a better transport path, stronger pinning, a larger superconducting fraction and a higher depairing limit as distinct hypotheses. Compare controlled specimens under matched conditions before assigning an improvement to a microscopic mechanism. A gain in current-carrying performance does not by itself demonstrate higher $T_c$, a new pairing interaction or a new superconducting phase.\n\nPreserve the measured signal, uncertainty and analysis conventions together. If the voltage criterion, geometry or thermal history is unavailable, retain that gap in the record. A useful follow-up is the measurement that discriminates between the competing limits, rather than another unsupported performance ranking. [R5](#ref-source-25347d9dbd8e915f)\n\n## Related topics\n\nConnect to [critical fields](/scwiki/critical-fields), [zero resistance](/scwiki/zero-resistance), [Meissner effect](/scwiki/meissner-effect), [BCS theory](/scwiki/bcs-weak-coupling), and [superconducting gap](/scwiki/superconducting-gap).\n\n## Material cases\n\nNo sample-specific material records are included in this article. Model results are not experimental transition temperatures.\n\n## Sources\n\n### Ref source-39e3e980bd8a0271\n\n[R1] Magnetization of Hard Superconductors (1962). [DOI 10.1103/PhysRevLett.8.250](https://doi.org/10.1103/PhysRevLett.8.250). Publication: journal-article. Check: publisher-metadata-scoped-check. Checked: 2026-10-18. Publisher identity checked. Original full text inaccessible at attempted PDF endpoint. The explicitly ideal slab calculation is a separate local Ampere-law derivation, not a verified transcription of the original paper.\n### Ref source-82f53ca35625217d\n\n[R2] 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### Ref source-51c947cf11bea72e\n\n[R3] Strongly enhanced current densities in superconducting coated conductors of YBa2Cu3O7-x + BaZrO3 (2004). [DOI 10.1038/nmat1156](https://doi.org/10.1038/nmat1156). Publication: journal-article. Check: author-manuscript-scoped-check. Checked: 2026-10-18. Selected pp. 2–4: preparation, transport field orientation, in-field/self-field comparison and particle-composition ambiguity. No electric-field criterion extracted; raw curves and micrographs not independently reproduced.\n### Ref source-c1a5f8c50328f68e\n\n[R4] Isotropic round-wire multifilament cuprate superconductor for generation of magnetic fields above 30 T (2014). [DOI 10.1038/nmat3887](https://doi.org/10.1038/nmat3887). Publication: journal-article. Check: author-manuscript-and-publisher-metadata-scoped-check. Checked: 2026-10-18. Publisher identifies a primary Article published 9 March 2014. Earlier 2013 author manuscript, selected pp. 3–4, distinguishes superconducting/whole-conductor areas and sample lengths. Not a review; final dataset, criterion and manuscript-to-journal changes not fully checked.\n### Ref source-25347d9dbd8e915f\n\n[R5] II-3: Critical current measurement methods: quantitative evaluation (1995). [Source publication](https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=30406). Publication: journal-article. Check: full-text-selected-page-check. Checked: 2026-10-19. NIST-hosted Cryogenics 35 supplement, p. S19: criterion definition, empirical voltage-current power law, Fig. 1 voltage/field criterion, and sensitivity to strain and experimental conditions. Selected first-page pixels read; remaining pages and underlying measurements not reanalyzed.\n\n## Claim evidence and conditions\n\n### scwiki-a7-claim-scope-1\n\nCritical current is a limit defined for a specimen, measurement protocol and operating conditions. Transport measurements use a voltage or electric-field criterion; magnetization measurements infer a current through a model; the depairing current describes loss of stability of a superconducting state within a specified theory. These quantities must not share an unlabeled performance ranking.\n\nEvidence: hypothesis. Sources: source-82f53ca35625217d, source-25347d9dbd8e915f.\n\nSource relationship: mixed-source-and-local-derivation. NIST p. S19 supports operational transport criteria and sensitivity to measurement conditions; Blatter abstract supplies vortex/depinning context. The reporting checklist and refusal to rank incomparable limits are local editorial guidance, not a certified standard or measured result.\n\nConditions: Editorial reporting/inference guidance; requires a sample-specific source, protocol and uncertainty. No new current or transition-temperature dataset certified.\n\nVerification scope: NIST Goodrich-Srivastava 1995 p. S19 and Blatter publisher abstract, plus explicit local reporting guidance; remaining full texts and sample-specific data not validated.\n\n### scwiki-a7-claim-scope-2\n\nBlatter and colleagues explicitly distinguish depinning and depairing current densities in their review abstract. Pinning, thermal fluctuations and vortex motion affect practical current transport. A measured threshold can instead be limited by weak connections or heating; its mechanism requires evidence beyond the value of the current itself.\n\nEvidence: theory. Sources: source-82f53ca35625217d.\n\nConditions: Qualitative vortex/pinning interpretation; not a fitted universal Jc or critical-field boundary.\n\nVerification scope: Pinned chapter plus Blatter publisher abstract; full review remains unchecked.\n\n### scwiki-a7-claim-mathematical-assumptions-1\n\nFor voltage taps separated by length $L$, define the mean longitudinal electric-field criterion $E_c=V_c/L$. Using the same measured $I_c$, distinguish the nominal superconducting cross-section $A_s$ from the entire conductor area $A_t$:\n\nEvidence: theory. Sources: source-25347d9dbd8e915f.\n\nSource relationship: mixed-source-and-local-derivation. NIST p. S19 defines transport criteria and an empirical V-I power law. Conversion to a mean electric field, area conventions and the criterion-ratio example are local algebra with fixed geometry and positive constant exponent; no measured material exponent is assigned.\n\nConditions: Definitions and local illustrative models: stated area convention, constant positive n power law, vector Lorentz force; slab uses full loop width, half-thickness, field parallel to faces, full penetration, constant Jc, no creep/demagnetization/reversible background.\n\nVerification scope: Independent local algebra, SI dimensional checks and numerical profile integration; not a full-text check of Bean 1962 or a fit to material data.\n\n### scwiki-a7-claim-mathematical-assumptions-2\n\n$$J_c=\\frac{I_c}{A_s},\\qquad J_e=\\frac{I_c}{A_t}=fJ_c,\\qquad f=\\frac{A_s}{A_t}.$$\n\nEvidence: theory. Sources: source-39e3e980bd8a0271.\n\nSource relationship: context-for-local-derivation. The stated algebra, vector force and ideal slab model are local derivations under explicit assumptions. Bean is historical context only; original full-text support was not verified.\n\nConditions: Definitions and local illustrative models: stated area convention, constant positive n power law, vector Lorentz force; slab uses full loop width, half-thickness, field parallel to faces, full penetration, constant Jc, no creep/demagnetization/reversible background.\n\nVerification scope: Independent local algebra, SI dimensional checks and numerical profile integration; not a full-text check of Bean 1962 or a fit to material data.\n\n### scwiki-a7-claim-mathematical-assumptions-3\n\nHere $J_e$ is an engineering current density for the stated whole-conductor area. Nominal superconducting area does not resolve nonuniform current paths or pores. A different definition of non-copper area, filament area or cable area requires a different label. These identities compare area conventions; they do not correct a damaged conductor or infer intrinsic microscopic current density.\n\nEvidence: theory. Sources: source-39e3e980bd8a0271.\n\nSource relationship: context-for-local-derivation. The stated algebra, vector force and ideal slab model are local derivations under explicit assumptions. Bean is historical context only; original full-text support was not verified.\n\nConditions: Definitions and local illustrative models: stated area convention, constant positive n power law, vector Lorentz force; slab uses full loop width, half-thickness, field parallel to faces, full penetration, constant Jc, no creep/demagnetization/reversible background.\n\nVerification scope: Independent local algebra, SI dimensional checks and numerical profile integration; not a full-text check of Bean 1962 or a fit to material data.\n\n### scwiki-a7-claim-mathematical-assumptions-4\n\nA simple sensitivity calculation illustrates why the criterion matters. If a measured interval is modeled by $E=E_0(J/J_0)^n$ with positive constant $n$, changing the criterion by a factor $r$ changes the inferred current by $r^{1/n}$. For $n=20$, a tenfold criterion change gives a factor of about 1.122. This is an illustrative power-law calculation, not a measured exponent or a universal transport law.\n\nEvidence: theory. Sources: source-25347d9dbd8e915f.\n\nSource relationship: mixed-source-and-local-derivation. NIST p. S19 defines transport criteria and an empirical V-I power law. Conversion to a mean electric field, area conventions and the criterion-ratio example are local algebra with fixed geometry and positive constant exponent; no measured material exponent is assigned.\n\nConditions: Definitions and local illustrative models: stated area convention, constant positive n power law, vector Lorentz force; slab uses full loop width, half-thickness, field parallel to faces, full penetration, constant Jc, no creep/demagnetization/reversible background.\n\nVerification scope: Independent local algebra, SI dimensional checks and numerical profile integration; not a full-text check of Bean 1962 or a fit to material data.\n\n### scwiki-a7-claim-mathematical-assumptions-5\n\nThe magnetic force density is the vector product $\\mathbf F=\\mathbf J\\times\\mathbf B$. For current perpendicular to the induction, its magnitude at the threshold becomes $F_p=J_cB$ if a static pinning-force balance is the applicable model. In SI units this is N/m³; resistive heating instead has power density $\\mathbf E\\cdot\\mathbf J$ in W/m³. A parallel configuration has no force from this cross product. Replacing $B$ by $\\mu_0H$ requires an additional approximation, especially near self-field conditions.\n\nEvidence: theory. Sources: source-39e3e980bd8a0271.\n\nSource relationship: context-for-local-derivation. The stated algebra, vector force and ideal slab model are local derivations under explicit assumptions. Bean is historical context only; original full-text support was not verified.\n\nConditions: Definitions and local illustrative models: stated area convention, constant positive n power law, vector Lorentz force; slab uses full loop width, half-thickness, field parallel to faces, full penetration, constant Jc, no creep/demagnetization/reversible background.\n\nVerification scope: Independent local algebra, SI dimensional checks and numerical profile integration; not a full-text check of Bean 1962 or a fit to material data.\n\n### scwiki-a7-claim-mathematical-assumptions-6\n\nFor an explicit geometry check, take an infinite slab of full thickness $2a$, field parallel to its faces, constant $J_c$ and fully developed opposite critical-state branches. Neglect demagnetization, reversible magnetization and creep. Ampere's law gives $|dB/dx|=\\mu_0J_c$. With $x$ measured across the thickness, the ideal branch profiles are $B_\\pm(x)=\\mu_0[H_a\\pm J_c(a-|x|)]$. Their spatial averages give\n\nEvidence: theory. Sources: source-39e3e980bd8a0271.\n\nSource relationship: context-for-local-derivation. The stated algebra, vector force and ideal slab model are local derivations under explicit assumptions. Bean is historical context only; original full-text support was not verified.\n\nConditions: Definitions and local illustrative models: stated area convention, constant positive n power law, vector Lorentz force; slab uses full loop width, half-thickness, field parallel to faces, full penetration, constant Jc, no creep/demagnetization/reversible background.\n\nVerification scope: Independent local algebra, SI dimensional checks and numerical profile integration; not a full-text check of Bean 1962 or a fit to material data.\n\n### scwiki-a7-claim-mathematical-assumptions-7\n\n$$M_\\pm=\\pm\\frac{J_ca}{2},\\qquad \\Delta M=M_+-M_-=J_ca,\\qquad J_c=\\frac{\\Delta M}{a}.$$\n\nEvidence: theory. Sources: source-39e3e980bd8a0271.\n\nSource relationship: context-for-local-derivation. The stated algebra, vector force and ideal slab model are local derivations under explicit assumptions. Bean is historical context only; original full-text support was not verified.\n\nConditions: Definitions and local illustrative models: stated area convention, constant positive n power law, vector Lorentz force; slab uses full loop width, half-thickness, field parallel to faces, full penetration, constant Jc, no creep/demagnetization/reversible background.\n\nVerification scope: Independent local algebra, SI dimensional checks and numerical profile integration; not a full-text check of Bean 1962 or a fit to material data.\n\n### scwiki-a7-claim-mathematical-assumptions-8\n\n$\\Delta M$ is the full vertical loop width in A/m; $a$ is the half-thickness in metres. Numerically integrating these stated profiles verifies the factor and dimensions. This is a local derivation under ideal assumptions, not a reanalysis of Bean's original data. Bean's 1962 paper is the historical source; its full text was not available in this check. Do not transfer this slab expression to arbitrary films, cylinders, partial penetration or field-dependent $J_c$.\n\nEvidence: theory. Sources: source-39e3e980bd8a0271.\n\nSource relationship: context-for-local-derivation. The stated algebra, vector force and ideal slab model are local derivations under explicit assumptions. Bean is historical context only; original full-text support was not verified.\n\nConditions: Definitions and local illustrative models: stated area convention, constant positive n power law, vector Lorentz force; slab uses full loop width, half-thickness, field parallel to faces, full penetration, constant Jc, no creep/demagnetization/reversible background.\n\nVerification scope: Independent local algebra, SI dimensional checks and numerical profile integration; not a full-text check of Bean 1962 or a fit to material data.\n\n### scwiki-a7-claim-experimental-criteria-1\n\nKeep enough information to reconstruct both the measured threshold and the conversion to a current density. The following is an editorial reporting guide, not a table of certified material records.\n\nEvidence: hypothesis. Sources: source-82f53ca35625217d, source-25347d9dbd8e915f.\n\nSource relationship: mixed-source-and-local-derivation. NIST p. S19 supports operational transport criteria and sensitivity to measurement conditions; Blatter abstract supplies vortex/depinning context. The reporting checklist and refusal to rank incomparable limits are local editorial guidance, not a certified standard or measured result.\n\nConditions: Editorial reporting/inference guidance; requires a sample-specific source, protocol and uncertainty. No new current or transition-temperature dataset certified.\n\nVerification scope: NIST Goodrich-Srivastava 1995 p. S19 and Blatter publisher abstract, plus explicit local reporting guidance; remaining full texts and sample-specific data not validated.\n\n### scwiki-a7-claim-experimental-criteria-2\n\n| Measurement | Record with the result | Interpretation boundary |\n|---|---|---|\n| Transport | Temperature, field magnitude and orientation, voltage-tap spacing, electric-field criterion, current ramp or pulse duration, duty cycle, area convention | Contact heating, current sharing and voltage offsets can change the inferred threshold |\n| Magnetization | Magnetic moment, volume, dimensions, field history, sweep rate, elapsed time and background subtraction | Conversion to current requires a geometry and critical-state model |\n| Conductor or cable | Superconducting fraction, whole-conductor area, length, strain and thermal stabilization | Layer current density and engineering current density answer different questions |\n| Depairing model | Model, temperature regime, geometry and stability assumptions | A calculated instability is not automatically the measured transport limit |\n\nEvidence: hypothesis. Sources: source-82f53ca35625217d, source-25347d9dbd8e915f.\n\nSource relationship: mixed-source-and-local-derivation. NIST p. S19 supports operational transport criteria and sensitivity to measurement conditions; Blatter abstract supplies vortex/depinning context. The reporting checklist and refusal to rank incomparable limits are local editorial guidance, not a certified standard or measured result.\n\nConditions: Editorial reporting/inference guidance; requires a sample-specific source, protocol and uncertainty. No new current or transition-temperature dataset certified.\n\nVerification scope: NIST Goodrich-Srivastava 1995 p. S19 and Blatter publisher abstract, plus explicit local reporting guidance; remaining full texts and sample-specific data not validated.\n\n### scwiki-a7-claim-experimental-criteria-3\n\nMacManus-Driscoll and colleagues studied YBCO films grown with BaZrO3 added to the target. Their selected manuscript passages report improved in-field transport at 75.5 K, despite lower transition temperature or self-field current in some samples. The rotated field remained perpendicular to current. This comparison illustrates distinct performance measures, not proof of a different pairing mechanism. No electric-field criterion was extracted here, so no numerical current record is tabulated.\n\nEvidence: experiment. Sources: source-51c947cf11bea72e.\n\nConditions: Selected author-manuscript passages only; retain sample, orientation, area and version limits. No complete numerical performance record or independent experimental reproduction.\n\nVerification scope: nmat1156 author manuscript pp. 2–4; nmat3887 earlier manuscript pp. 3–4 and publisher article identity, as applicable.\n\n### scwiki-a7-claim-experimental-criteria-4\n\nLarbalestier and colleagues' 2014 paper is a primary Bi-2212 round-wire study, not a general engineering review. Its earlier author manuscript distinguishes whole-conductor $J_e$ from superconducting-area $J_c$ and compares short specimens with a demonstration coil. This entry uses that distinction without importing a cross-family ranking; the final published dataset and all measurement criteria were not independently re-extracted.\n\nEvidence: experiment. Sources: source-c1a5f8c50328f68e.\n\nConditions: Selected author-manuscript passages only; retain sample, orientation, area and version limits. No complete numerical performance record or independent experimental reproduction.\n\nVerification scope: nmat1156 author manuscript pp. 2–4; nmat3887 earlier manuscript pp. 3–4 and publisher article identity, as applicable.\n\n### scwiki-a7-claim-limits-and-disputes-1\n\nAn irreversible magnetic response and the upper critical field are different observables. A loss of measurable hysteresis or a transport threshold need not locate the destruction of superconductivity. Report the criterion, time scale and field history rather than silently equating an irreversibility line with $H_{c2}$. Thermally activated creep is one reason different observation times can yield different apparent current limits.\n\nEvidence: theory. Sources: source-82f53ca35625217d.\n\nConditions: Qualitative vortex/pinning interpretation; not a fitted universal Jc or critical-field boundary.\n\nVerification scope: Pinned chapter plus Blatter publisher abstract; full review remains unchecked.\n\n### scwiki-a7-claim-limits-and-disputes-2\n\nA fitted pinning-curve exponent or angular peak does not uniquely identify a defect. Compare microscopy, processing controls and temperature/angle dependence. In the checked YBCO manuscript, the detailed composition of the small second-phase particles is not conclusively established; describing every particle as chemically proven BaZrO3 would overstate the evidence.\n\nEvidence: experiment. Sources: source-51c947cf11bea72e.\n\nConditions: Selected author-manuscript passages only; retain sample, orientation, area and version limits. No complete numerical performance record or independent experimental reproduction.\n\nVerification scope: nmat1156 author manuscript pp. 2–4; nmat3887 earlier manuscript pp. 3–4 and publisher article identity, as applicable.\n\n### scwiki-a7-claim-discovery-implications-1\n\nSeparate a better transport path, stronger pinning, a larger superconducting fraction and a higher depairing limit as distinct hypotheses. Compare controlled specimens under matched conditions before assigning an improvement to a microscopic mechanism. A gain in current-carrying performance does not by itself demonstrate higher $T_c$, a new pairing interaction or a new superconducting phase.\n\nEvidence: hypothesis. Sources: source-82f53ca35625217d.\n\nConditions: Editorial reporting/inference guidance; requires a sample-specific source, protocol and uncertainty. No new current or transition-temperature dataset certified.\n\nVerification scope: Pinned chapter edited with explicitly limited reference checks.\n\n### scwiki-a7-claim-discovery-implications-2\n\nPreserve the measured signal, uncertainty and analysis conventions together. If the voltage criterion, geometry or thermal history is unavailable, retain that gap in the record. A useful follow-up is the measurement that discriminates between the competing limits, rather than another unsupported performance ranking.\n\nEvidence: hypothesis. Sources: source-82f53ca35625217d, source-25347d9dbd8e915f.\n\nSource relationship: mixed-source-and-local-derivation. NIST p. S19 supports operational transport criteria and sensitivity to measurement conditions; Blatter abstract supplies vortex/depinning context. The reporting checklist and refusal to rank incomparable limits are local editorial guidance, not a certified standard or measured result.\n\nConditions: Editorial reporting/inference guidance; requires a sample-specific source, protocol and uncertainty. No new current or transition-temperature dataset certified.\n\nVerification scope: NIST Goodrich-Srivastava 1995 p. S19 and Blatter publisher abstract, plus explicit local reporting guidance; remaining full texts and sample-specific data not validated.\n\n## Revision and review\n\nArticle ID: scwiki-a7. Revision: c5bc330ef247c18130fa. Updated: 2026-10-19.\n\nReview scope: Edited pinned A7 chapter, selected engineering author-manuscript passages, publisher identities and vortex-review abstract; independent conditional current/force/slab calculations. NIST-hosted Goodrich-Srivastava 1995 p. S19 checked for operational criteria and measurement-condition dependence.\n\n- Selected source checks are not full scientific acceptance or independent experimental replication.\n- Bean original full text was inaccessible; the slab formula is a separately derived ideal model. Blatter is abstract-scoped.\n- Engineering sources are checked at stated manuscript passages; no complete electric-field criterion or final numerical performance dataset was re-extracted.\n- No cross-family current ranking, experimental Tc record or universal depairing value is inferred.\n- NIST source is a selected-page check of a measurement-methods paper, not validation of every reporting recommendation or material current record.\n\nSource: [A_phenomenology/A7_critical_current.md](https://github.com/JackZH26/jzis-superconductivity/blob/0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf/A_phenomenology/A7_critical_current.md), commit 0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf, SHA-256 cddae9252ba1d88a829fbec70ef330b7ccd1b837a61692e8d5c64c82942a833d.\n","wordCount":976}}