# Critical fields: thermodynamic, orbital and Pauli scales

Distinguish Hc, Hc1 and Hc2, compare orbital and spin limits, and keep experimental field criteria attached to their conditions.

## Scope

A critical field needs a physical definition. The thermodynamic field $H_c$ represents the zero-field condensation-energy density, $f_n-f_s=\mu_0H_c^2/2$. This energy scale exists for both type-I and type-II superconductors; it is not generally the field at which a type-II sample becomes normal. In the ideal bulk type-II description, $H_{c1}$ marks the equilibrium onset of the mixed state and $H_{c2}$ its upper boundary. The vortex construction distinguishes those boundaries from one thermodynamic energy scale. [R1](#ref-source-89f5503f5e470385)

Here $H$ is magnetic field strength in amperes per metre, while $B$ and $\mu_0H$ are expressed in tesla. Inside matter, $B=\mu_0(H+M)$; replacing $B$ by $\mu_0H$ requires a stated magnetization approximation. An applied-field value, an internal field and a flux density must not silently share the same symbol. The formulas below use SI conventions, rather than copying Gaussian-unit factors from historical papers.

## Mathematical assumptions

For a bulk, isotropic, single-component Ginzburg–Landau model near $T_c$, neglecting paramagnetic pair breaking, linearization gives the orbital result

$$
B_{c2}^{\mathrm{orb}}=\frac{\Phi_0}{2\pi\xi^2},\qquad
\Phi_0=\frac{h}{2e},\qquad
\kappa=\frac{\lambda}{\xi}.
$$

The usual $\mu_0H_{c2}^{\mathrm{orb}}$ notation uses $B\simeq\mu_0H$ near this continuous boundary when normal-state magnetic response is negligible. In the same model, $H_{c2}^{\mathrm{orb}}=\sqrt{2}\kappa H_c$. The GL coherence length $\xi$ and penetration depth $\lambda$ are evaluated at the same temperature. They are not interchangeable with every microscopic pair-size or fitted length. The isotropic GL type-II boundary is $\kappa=1/\sqrt{2}$. [R1](#ref-source-89f5503f5e470385)

A useful independent check follows from the linearized kinetic operator: the lowest Landau-level energy of a pair with charge magnitude $2e$ is $\hbar(2e)B/(2m^\ast)$. Equating it to $|\alpha|$, with $\xi^2=\hbar^2/(2m^\ast|\alpha|)$, gives the flux-over-area expression above. This is a model derivation, not a measured field or an all-temperature interpolation.

For a separate, purely paramagnetic comparison at zero temperature, take an isotropic weak-coupling singlet with $g=2$, negligible spin susceptibility in the paired state, no orbital cost and no competing modulated state. Equating its condensation energy to the normal-state polarization-energy gain gives

$$
B_P=\frac{\Delta_0}{\sqrt{2}\mu_B},\qquad
\frac{B_P}{T_c}=\frac{\pi e^{-\gamma}k_B}{\sqrt{2}\mu_B}
\simeq 1.86\;\mathrm{T\,K^{-1}}.
$$

This is the Clogston–Chandrasekhar reference scale. The commonly quoted $1.84\;\mathrm{T\,K^{-1}}$ uses rounded weak-coupling gap constants; neither coefficient is an exact universal experimental bound. The numerical value here uses the same modern weak-coupling gap convention as the BCS entry. The energy comparison is supported by the cited perspective; the original Clogston paper's identity is checked, but its full text was not available in this review. [R4](#ref-source-d32df19524b2a1c8) [R5](#ref-source-b717fb1db41e9f29) [R6](#ref-source-ba10e800a1246bf6)

## Experimental criteria

Report the probe and the rule used to locate a field line. Resistance onset, a stated percentage of the normal-state resistance and a voltage-resolution-limited zero need not coincide. Neither a magnetic irreversibility line nor a first flux-entry signal should automatically be renamed $H_{c2}$ or equilibrium $H_{c1}$. Demagnetizing geometry, surface barriers, vortex motion, pinning and inhomogeneous current paths can affect the observed signal.

Record temperature, field orientation and angular uncertainty, sample geometry, pressure where relevant, current density, sweep direction and rate, and the normal-state baseline. Compare transport with an appropriate bulk thermodynamic or magnetic measurement on comparable specimens. Preserve hysteresis and transition widths instead of choosing the most favorable boundary. These are measurement-reporting requirements from the pinned chapter, not a claim that each possible artifact is present in a particular sample.

## Limits and disputes

Werthamer–Helfand–Hohenberg theory treats orbital and spin effects within a specified microscopic approximation, including spin–orbit impurity scattering. Its publisher abstract explicitly notes that such scattering counteracts paramagnetic limiting. A fitted WHH curve therefore needs its dirty-limit, spin and scattering assumptions; merely naming the fit does not establish them. This entry's WHH check is abstract-scoped. [R2](#ref-source-16926b3817cff55d)

Multiband structure also changes extrapolation. Gurevich's dirty two-gap model allows $H_{c2}(T)$ shapes controlled by different intraband diffusivities, including deviations from the one-gap slope estimate near $T_c$. That calculation is a counterexample to treating a one-band extrapolation coefficient as universal. It does not validate an arbitrary multiband fit or supply a field value for an unmeasured sample. [R3](#ref-source-7ba854cc57960377)

Exceeding the simple Pauli estimate does not uniquely identify triplet pairing. The spin response and spin–orbit energy scales may differ from the assumed reference model; the cited Ising-superconductivity perspective discusses a concrete route to a strongly anisotropic paramagnetic response. Orbital and spin limitations must be treated together when both matter. No material-specific critical-field record or pairing assignment is introduced here. [R5](#ref-source-b717fb1db41e9f29)

## Discovery implications

Use a field measurement to discriminate between declared models. First decide whether the observation concerns an equilibrium boundary, vortex dynamics or a transport threshold. Then test orientation and temperature dependence over a range where the model applies. Retain the fitting window, parameter uncertainty and alternative descriptions, particularly when extrapolating far below the measured temperatures.

A high critical field may motivate further study, but a useful discovery argument identifies which assumption failed and which new observable could distinguish the remaining explanations. Material comparisons should keep pressure, field direction and criterion attached to each value. A large fitted field alone establishes neither a microscopic pairing mediator nor topology. This is an editorial evaluation protocol; it is not a newly verified experimental result.

## Related topics

Read about [the Meissner effect](/scwiki/meissner-effect), [zero-resistance criteria](/scwiki/zero-resistance), [the BCS baseline](/scwiki/bcs-weak-coupling), and [multiband pairing](/scwiki/multiband-pairing).

## Material cases

No sample-specific material records are included in this article. Model results are not experimental transition temperatures.

## Sources

### Ref source-89f5503f5e470385

[R1] Nobel Lecture: Type-II superconductors and the vortex lattice (2004). [DOI 10.1103/RevModPhys.76.975](https://doi.org/10.1103/RevModPhys.76.975). Publication: journal-article. Check: primary-lecture-scoped-check. Checked: 2026-10-18. Selected pp. 975-977: GL type-II boundary, upper field Eq. (1), flux quantum Eq. (10), mixed-state boundary around Eq. (11). SI notation and the linearized GL derivation are editorial checks; original Gaussian-unit equations are not copied as SI. Material-specific claims and historical approximations outside these passages are not validated.
### Ref source-16926b3817cff55d

[R2] Temperature and Purity Dependence of the Superconducting Critical Field, Hc2. III. Electron Spin and Spin-Orbit Effects (1966). [DOI 10.1103/PhysRev.147.295](https://doi.org/10.1103/PhysRev.147.295). Publication: journal-article. Check: primary-abstract-scoped-check. Checked: 2026-10-18. Publisher abstract only: orbital/Pauli treatment and spin-orbit impurity scattering within the stated approximation. Full derivation, fit coefficients and specific experimental applications are not verified.
### Ref source-7ba854cc57960377

[R3] Enhancement of the upper critical field by nonmagnetic impurities in dirty two-gap superconductors (2003). [DOI 10.1103/PhysRevB.67.184515](https://doi.org/10.1103/PhysRevB.67.184515). Publication: journal-article. Check: primary-full-text-scoped-check. Checked: 2026-10-18. Abstract and introduction, pp. 184515-1 to -2: dirty two-band model, diffusivity-dependent field curves and limitations of one-gap extrapolation Eq. (1). Detailed Usadel solution and material field values are not independently reproduced.
### Ref source-d32df19524b2a1c8

[R4] Upper Limit for the Critical Field in Hard Superconductors (1962). [DOI 10.1103/PhysRevLett.9.266](https://doi.org/10.1103/PhysRevLett.9.266). Publication: journal-article. Check: publisher-metadata-only. Checked: 2026-10-18. Historical reference identity only. Full-text retrieval was unavailable; the local energy-balance check is supported separately by the cited perspective and declared BCS assumptions.
### Ref source-b717fb1db41e9f29

[R5] Ising superconductivity: A first-principles perspective (2023). [DOI 10.1063/5.0153345](https://doi.org/10.1063/5.0153345). Publication: journal-article. Check: review-full-text-scoped-check. Checked: 2026-10-18. Selected section II, pp. 240503-2 to -3: susceptibility/free-energy comparison and orientation-dependent spin response. No full validation of every Ising material, experimental field value or interpretation in the perspective.
### Ref source-ba10e800a1246bf6

[R6] Theory of Superconductivity (1957). [DOI 10.1103/PhysRev.108.1175](https://doi.org/10.1103/PhysRev.108.1175). Publication: journal-article. Check: primary-full-text-scoped-check. Checked: 2026-10-18. Selected weak-coupling gap passages previously checked for B1, pp. 1182-1183 and 1186. Modern gap constant used in an explicit local energy-balance calculation; this source check does not establish a universal critical field.

## Claim evidence and conditions

### scwiki-a6-claim-scope-1

A critical field needs a physical definition. The thermodynamic field $H_c$ represents the zero-field condensation-energy density, $f_n-f_s=\mu_0H_c^2/2$. This energy scale exists for both type-I and type-II superconductors; it is not generally the field at which a type-II sample becomes normal. In the ideal bulk type-II description, $H_{c1}$ marks the equilibrium onset of the mixed state and $H_{c2}$ its upper boundary. The vortex construction distinguishes those boundaries from one thermodynamic energy scale.

Evidence: theory. Sources: source-89f5503f5e470385.

Conditions: Bulk equilibrium field definitions; distinguish SI H, B and magnetization, and type-I versus type-II behavior.

Verification scope: Vortex-field distinction checked against Abrikosov pp. 975-977. Thermodynamic and SI notation are explicit editorial definitions; not an independent verification of every material.

### scwiki-a6-claim-scope-2

Here $H$ is magnetic field strength in amperes per metre, while $B$ and $\mu_0H$ are expressed in tesla. Inside matter, $B=\mu_0(H+M)$; replacing $B$ by $\mu_0H$ requires a stated magnetization approximation. An applied-field value, an internal field and a flux density must not silently share the same symbol. The formulas below use SI conventions, rather than copying Gaussian-unit factors from historical papers.

Evidence: theory. Sources: source-89f5503f5e470385.

Conditions: Bulk equilibrium field definitions; distinguish SI H, B and magnetization, and type-I versus type-II behavior.

Verification scope: Vortex-field distinction checked against Abrikosov pp. 975-977. Thermodynamic and SI notation are explicit editorial definitions; not an independent verification of every material.

### scwiki-a6-claim-mathematical-assumptions-1

For a bulk, isotropic, single-component Ginzburg–Landau model near $T_c$, neglecting paramagnetic pair breaking, linearization gives the orbital result

Evidence: theory. Sources: source-89f5503f5e470385.

Conditions: Orbital result: isotropic single-component GL near Tc, negligible paramagnetic and normal magnetic response. Pauli result separately assumes zero-temperature weak-coupling singlet, g=2, negligible paired-state spin susceptibility and no orbital or modulated-state competition.

Verification scope: Declared GL and energy-balance model calculation checked algebraically and numerically. Source-specific passage/access scope retained; not a measurement or independent microscopic-theory reproduction.

### scwiki-a6-claim-mathematical-assumptions-2

The usual $\mu_0H_{c2}^{\mathrm{orb}}$ notation uses $B\simeq\mu_0H$ near this continuous boundary when normal-state magnetic response is negligible. In the same model, $H_{c2}^{\mathrm{orb}}=\sqrt{2}\kappa H_c$. The GL coherence length $\xi$ and penetration depth $\lambda$ are evaluated at the same temperature. They are not interchangeable with every microscopic pair-size or fitted length. The isotropic GL type-II boundary is $\kappa=1/\sqrt{2}$.

Evidence: theory. Sources: source-89f5503f5e470385.

Conditions: Orbital result: isotropic single-component GL near Tc, negligible paramagnetic and normal magnetic response. Pauli result separately assumes zero-temperature weak-coupling singlet, g=2, negligible paired-state spin susceptibility and no orbital or modulated-state competition.

Verification scope: Declared GL and energy-balance model calculation checked algebraically and numerically. Source-specific passage/access scope retained; not a measurement or independent microscopic-theory reproduction.

### scwiki-a6-claim-mathematical-assumptions-3

A useful independent check follows from the linearized kinetic operator: the lowest Landau-level energy of a pair with charge magnitude $2e$ is $\hbar(2e)B/(2m^\ast)$. Equating it to $|\alpha|$, with $\xi^2=\hbar^2/(2m^\ast|\alpha|)$, gives the flux-over-area expression above. This is a model derivation, not a measured field or an all-temperature interpolation.

Evidence: theory. Sources: source-89f5503f5e470385.

Conditions: Orbital result: isotropic single-component GL near Tc, negligible paramagnetic and normal magnetic response. Pauli result separately assumes zero-temperature weak-coupling singlet, g=2, negligible paired-state spin susceptibility and no orbital or modulated-state competition.

Verification scope: Declared GL and energy-balance model calculation checked algebraically and numerically. Source-specific passage/access scope retained; not a measurement or independent microscopic-theory reproduction.

### scwiki-a6-claim-mathematical-assumptions-4

For a separate, purely paramagnetic comparison at zero temperature, take an isotropic weak-coupling singlet with $g=2$, negligible spin susceptibility in the paired state, no orbital cost and no competing modulated state. Equating its condensation energy to the normal-state polarization-energy gain gives

Evidence: theory. Sources: source-b717fb1db41e9f29, source-ba10e800a1246bf6.

Conditions: Orbital result: isotropic single-component GL near Tc, negligible paramagnetic and normal magnetic response. Pauli result separately assumes zero-temperature weak-coupling singlet, g=2, negligible paired-state spin susceptibility and no orbital or modulated-state competition.

Verification scope: Declared GL and energy-balance model calculation checked algebraically and numerically. Source-specific passage/access scope retained; not a measurement or independent microscopic-theory reproduction.

### scwiki-a6-claim-mathematical-assumptions-5

This is the Clogston–Chandrasekhar reference scale. The commonly quoted $1.84\;\mathrm{T\,K^{-1}}$ uses rounded weak-coupling gap constants; neither coefficient is an exact universal experimental bound. The numerical value here uses the same modern weak-coupling gap convention as the BCS entry. The energy comparison is supported by the cited perspective; the original Clogston paper's identity is checked, but its full text was not available in this review.

Evidence: theory. Sources: source-d32df19524b2a1c8, source-b717fb1db41e9f29, source-ba10e800a1246bf6.

Conditions: Orbital result: isotropic single-component GL near Tc, negligible paramagnetic and normal magnetic response. Pauli result separately assumes zero-temperature weak-coupling singlet, g=2, negligible paired-state spin susceptibility and no orbital or modulated-state competition.

Verification scope: Declared GL and energy-balance model calculation checked algebraically and numerically. Source-specific passage/access scope retained; not a measurement or independent microscopic-theory reproduction.

### scwiki-a6-claim-experimental-criteria-1

Report the probe and the rule used to locate a field line. Resistance onset, a stated percentage of the normal-state resistance and a voltage-resolution-limited zero need not coincide. Neither a magnetic irreversibility line nor a first flux-entry signal should automatically be renamed $H_{c2}$ or equilibrium $H_{c1}$. Demagnetizing geometry, surface barriers, vortex motion, pinning and inhomogeneous current paths can affect the observed signal.

Evidence: hypothesis. Sources: source-89f5503f5e470385.

Conditions: Probe-specific measurement protocol; geometry, temperature, orientation, pressure, current and sweep conditions must accompany each boundary. No sample is validated by this guidance.

Verification scope: Editorial reporting protocol adapted from pinned A6 chapter, not independently verified experimental findings. Cited theory motivates distinctions only; it does not certify each measurement artifact.

### scwiki-a6-claim-experimental-criteria-2

Record temperature, field orientation and angular uncertainty, sample geometry, pressure where relevant, current density, sweep direction and rate, and the normal-state baseline. Compare transport with an appropriate bulk thermodynamic or magnetic measurement on comparable specimens. Preserve hysteresis and transition widths instead of choosing the most favorable boundary. These are measurement-reporting requirements from the pinned chapter, not a claim that each possible artifact is present in a particular sample.

Evidence: hypothesis. Sources: source-89f5503f5e470385.

Conditions: Probe-specific measurement protocol; geometry, temperature, orientation, pressure, current and sweep conditions must accompany each boundary. No sample is validated by this guidance.

Verification scope: Editorial reporting protocol adapted from pinned A6 chapter, not independently verified experimental findings. Cited theory motivates distinctions only; it does not certify each measurement artifact.

### scwiki-a6-claim-limits-and-disputes-1

Werthamer–Helfand–Hohenberg theory treats orbital and spin effects within a specified microscopic approximation, including spin–orbit impurity scattering. Its publisher abstract explicitly notes that such scattering counteracts paramagnetic limiting. A fitted WHH curve therefore needs its dirty-limit, spin and scattering assumptions; merely naming the fit does not establish them. This entry's WHH check is abstract-scoped.

Evidence: theory. Sources: source-16926b3817cff55d.

Conditions: WHH abstract scope, dirty two-band model scope and specified spin response; no universal field extrapolation or pairing assignment follows.

Verification scope: WHH publisher abstract only; detailed fits and numerical theory remain unreviewed.

### scwiki-a6-claim-limits-and-disputes-2

Multiband structure also changes extrapolation. Gurevich's dirty two-gap model allows $H_{c2}(T)$ shapes controlled by different intraband diffusivities, including deviations from the one-gap slope estimate near $T_c$. That calculation is a counterexample to treating a one-band extrapolation coefficient as universal. It does not validate an arbitrary multiband fit or supply a field value for an unmeasured sample.

Evidence: theory. Sources: source-7ba854cc57960377.

Conditions: WHH abstract scope, dirty two-band model scope and specified spin response; no universal field extrapolation or pairing assignment follows.

Verification scope: Gurevich abstract and introduction pp. 184515-1 to -2; no numerical solution reproduced.

### scwiki-a6-claim-limits-and-disputes-3

Exceeding the simple Pauli estimate does not uniquely identify triplet pairing. The spin response and spin–orbit energy scales may differ from the assumed reference model; the cited Ising-superconductivity perspective discusses a concrete route to a strongly anisotropic paramagnetic response. Orbital and spin limitations must be treated together when both matter. No material-specific critical-field record or pairing assignment is introduced here.

Evidence: theory. Sources: source-b717fb1db41e9f29.

Conditions: WHH abstract scope, dirty two-band model scope and specified spin response; no universal field extrapolation or pairing assignment follows.

Verification scope: Wickramaratne-Mazin perspective section II, pp. 240503-2 to -3; model spin-response argument, no material-specific field or pairing validation.

### scwiki-a6-claim-discovery-implications-1

Use a field measurement to discriminate between declared models. First decide whether the observation concerns an equilibrium boundary, vortex dynamics or a transport threshold. Then test orientation and temperature dependence over a range where the model applies. Retain the fitting window, parameter uncertainty and alternative descriptions, particularly when extrapolating far below the measured temperatures.

Evidence: hypothesis. Sources: source-7ba854cc57960377.

Conditions: Editorial model-discrimination protocol; no measured critical-field values or established pairing mechanisms are asserted.

Verification scope: Editorial reporting protocol adapted from pinned A6 chapter, not independently verified experimental findings. Cited theory motivates distinctions only; it does not certify each measurement artifact.

### scwiki-a6-claim-discovery-implications-2

A high critical field may motivate further study, but a useful discovery argument identifies which assumption failed and which new observable could distinguish the remaining explanations. Material comparisons should keep pressure, field direction and criterion attached to each value. A large fitted field alone establishes neither a microscopic pairing mediator nor topology. This is an editorial evaluation protocol; it is not a newly verified experimental result.

Evidence: hypothesis. Sources: source-7ba854cc57960377.

Conditions: Editorial model-discrimination protocol; no measured critical-field values or established pairing mechanisms are asserted.

Verification scope: Editorial reporting protocol adapted from pinned A6 chapter, not independently verified experimental findings. Cited theory motivates distinctions only; it does not certify each measurement artifact.

## Revision and review

Article ID: scwiki-a6. Revision: 884c5be2585dc88338db. Updated: 2026-10-18T06:15:04.224300+00:00.

Review scope: Edited pinned A6 chapter with selected vortex/GL, dirty two-band and spin-susceptibility passages; explicit SI and model-equation checks. WHH is abstract-scoped and Clogston identity-only.

- This is not full scientific validation or independent reproduction of the microscopic theories.
- No sample-specific field values, transition criteria or pairing assignments have been verified for this article.
- The thermodynamic and model limits must not replace experimentally defined field lines without justification.
- Original Clogston/Chandrasekhar derivations and detailed WHH fits remain unreviewed; the Pauli coefficient is evaluated from stated model assumptions.

Source: [A_phenomenology/A6_critical_fields.md](https://github.com/JackZH26/jzis-superconductivity/blob/0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf/A_phenomenology/A6_critical_fields.md), commit 0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf, SHA-256 0a9dc38b486fa45e51d106da468ad665205b35a105a18151a062f7bd3262f352.