Meissner effect, diamagnetism, and classical physics—a reviewHanno Essén and Miguel C. N. Fiolhais Citation: American Journal of Physics 80, 164 (2012); doi: 10.1119/1.3662027 View online: http://dx.doi.org/10.1119/1.3662027 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/80/2?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in Imaging flux distributions around superconductors: Geometrical susceptibility in the Meissner state J. Appl. Phys. 114, 203904 (2013); 10.1063/1.4834519 Sn doped ( Cu 0.5 Tl 0.5 ) Ba 2 Ca 2 Cu 3 − y Sn y O 10 − δ superconductors: Effect on the diamagnetism andphonon modes J. Appl. Phys. 106, 083901 (2009); 10.1063/1.3238516 Finite-size-induced stability of a permanent magnet levitating over a superconductor in the Meissner state Appl. Phys. Lett. 91, 142503 (2007); 10.1063/1.2794408 Magnetic scattering: Pair-breaking effect and anomalous diamagnetism above H c2 and T c AIP Conf. Proc. 483, 153 (1999); 10.1063/1.59589 The emergence of resistive state in the Meissner phase of superconductors upon an increase in temperaturebelow the critical point Low Temp. Phys. 24, 537 (1998); 10.1063/1.593632

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Meissner effect, diamagnetism, and classical physics—a review

Hanno EssenDepartment of Mechanics, Royal Institute of Technology (KTH), Stockholm SE-100 44, Sweden

Miguel C. N. FiolhaisLIP-Coimbra, Department of Physics, University of Coimbra, Coimbra 3004-516, Portugal

(Received 8 September 2011; accepted 28 October 2011)

We review the literature on what classical physics says about the Meissner effect and the London

equations. We discuss the relevance of the Bohr-van Leeuwen theorem for the perfect diamagnetism

of superconductors and conclude that the theorem is based on invalid assumptions. We also point

out results in the literature that show how magnetic flux expulsion from a sample cooled to

superconductivity can be understood as an approach to the magnetostatic energy minimum. These

results have been published several times but many textbooks on magnetism still claim that there is

no classical diamagnetism, and virtually all books on superconductivity repeat Meissner’s 1933

statement that flux expulsion has no classical explanation. VC 2012 American Association of Physics Teachers.

[DOI: 10.1119/1.3662027]

I. INTRODUCTION

It is now a century since superconductivity was discoveredby Kammerlingh-Onnes in Leiden in 1911. From the begin-ning, there was considerable interest from theoretical physi-cists. Unfortunately progress has been slow and one cansafely say that the phenomenon is still not completely under-stood, at least not in a fundamental reductionist sense. It istherefore important that the things that can be understood arecorrectly presented in textbooks. To the contrary, textbooksoften repeat two myths which have become so ingrained inthe minds of physicists that they hamper progress. These are:

• There is no classical diamagnetism (Bohr,1 van Leeuwen2).• There is no classical explanation of flux expulsion from a

superconductor (Meissner3).

Although both statements have been disproved multipletimes in the scientific literature, these myths continue to bespread. We hope this review will improve the situation.

Meissner found it natural that a weak magnetic field couldnot penetrate a Type-I superconductor. The fact that this isalready in conflict with classical physics, according to theBohr-van Leeuwen theorem is rarely mentioned. On theother hand, Meissner found it remarkable that a normal metalwith a magnetic field inside will expel this field when cooledto superconductivity. This was claimed to have no classicalexplanation, and consequently superconductors were notconsidered perfect conductors.

We will not present any new results in this paper. Instead,we will review some of the strong evidence in the archivalliterature demonstrating that the above two statements arefalse. We begin with the Bohr-van Leeuwen theorem andthen turn to the classical explanation of flux expulsion. Afterdiscussing the arguments for the traditional point of view,we will demonstrate why they are wrong.

II. THE BOHR-VAN LEEUWEN THEOREM AND

CLASSICAL DIAMAGNETISM

The Hamiltonian H, for a system of charged particlesinteracting via a (scalar) potential energy U is

Hðrk; pkÞ ¼XN

j¼1

p2j

2mjþ UðrkÞ: (1)

The particles have masses mj, position vectors rj, andmomenta pj. The presence of an external magnetic field withvector potential A(r) alters the Hamiltonian to

Hðrk; pkÞ ¼XN

j¼1

pj �ej

c AðrjÞ� �2

2mjþ UðrkÞ; (2)

where ej are the charges of masses mj. Using this equationone can show that statistical mechanics predicts that theenergy—the thermal average of the Hamiltonian—does notdepend on the external field. Hence, the system exhibits nei-ther a paramagnetic nor a diamagnetic response.

In his 1911 doctoral dissertation, Niels Bohr1 used theabove equation and concluded there is no magnetic responseof a metal according to classical physics. In 1919 HendrikaJohanna van Leeuwen independently came to the same con-clusion in her Leiden thesis. When her work was publishedin 1921, Miss van Leeuwen2 noted that similar conclusionshad been reached by Bohr.

Many books on magnetism refer to the above results asthe Bohr-van Leeuwen theorem, or simply the van Leeuwentheorem. The theorem is often summarized as stating there isno classical magnetism. Since this is obvious in the case ofspin or atomic angular momenta—the quantum phenomenaresponsible for paramagnetism—the more interestingconclusion is that classical statistical mechanics and electro-magnetism cannot explain diamagnetism. A very thoroughtreatment of the Bohr-van Leeuwen theorem can be foundin Van Vleck.4 Other books, such as Mohn,5 Getzlaff,6

Aharoni,7 and Levy,8 mention the theorem more or lessbriefly. Weaknesses in classical derivations of diamagnetismin modern textbooks have been pointed out by O’Dell andZia.9 A discussion of the theorem can also be found in theFeynman lectures10 (Vol. 2, Sec. 34-6), which points out thata constant external field will not do any work on a system ofcharges. Therefore, the energy of this system cannot dependon the external field. In addition, the fact that the diamag-netic response of most materials is very small lends empiri-cal support to the theorem.

A. The inadequacy of the basic assumptions

If we accept the popular version of the Bohr-van Leeuwentheorem as true, then one must conclude that the perfect

164 Am. J. Phys. 80 (2), February 2012 http://aapt.org/ajp VC 2012 American Association of Physics Teachers 164

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diamagnetism of superconductors cannot have a classical ex-planation. A study of the proof of the theorem, however,reveals that it is only valid under assumptions that do not nec-essarily hold. The assumed Hamiltonian of Eq. (2) onlyincludes the vector potential of the external magnetic field.But it has been known since 1920 that the best Hamiltonianfor a system of classical charged particles is the DarwinHamiltonian.11 In this Hamiltonian one takes into account theinternal magnetic fields generated by the moving charged par-ticles of the system itself, plus any corresponding interactions.

The simplest way to see that the total energy of a systemof charged particles depends on the external field is to notethat the total energy includes a magnetic energy12

EB ¼1

8p

ðB2ðrÞdV ¼ 1

8p

ððBe þ BiÞ2dV; (3)

where Be is the external magnetic field and Bi is the internalfield. Using the Biot-Savart law this field is given, to firstorder in v/c, by

BiðrÞ ¼XN

j¼1

BjðrÞ ¼XN

j¼1

ej

c

vj � ðr � rjÞjr � rjj3

: (4)

Equation (3) makes it obvious that to minimize the energythe internal field will be, as much as possible, equal in mag-nitude and opposite in direction to the external field. Thisgives diamagnetism.

Charles Galton Darwin was first to derive an approximateLagrangian for a system of charged particles (neglectingradiation) that is correct to order (v/c)2.11,13–16 In a time inde-pendent external magnetic field, there is then a conservedDarwin energy given by

ED ¼XN

j¼1

mj

2v2

j þej

cvj �

1

2Aiðrj; rk;vkÞ þ AeðrjÞ

� �� �

þ UðrkÞ þ Ee: (5)

In this equation, vj are velocity vectors, Ae is the externalvector potential, Ee is the (constant) energy of the externalmagnetic field, and Ai is the internal vector potential givenby

Aiðrj; rk;vkÞ ¼XN

k 6¼j

ek

rkj

vk þ ðvk � ekjÞekj

2c; (6)

where rkj¼ |rj – rk| and ekj¼ (rj – rk)/rkj. Although there is acorresponding Hamiltonian, it cannot be written in closedform. When the Darwin magnetic interactions are taken intoaccount, the Bohr-van Leeuwen theorem is no longer validbecause the magnetic fields of the moving charges will con-tribute to the total magnetic energy. The fact that this inva-lidates the Bohr-van Leeuwen theorem for superconductorswas stated explicitly by Pfleiderer17 in a letter to Nature in1966. A more recent discussion of classical diamagnetismand the Darwin Hamiltonian is given by Essen.18

B. What about Larmor’s theorem?

Larmor’s theorem states that a spherically symmetric sys-tem of charged particles will start to rotate if an externalmagnetic field is turned on (see Landau and Lifshitz,13 x45).

The rotation of such a system produces a circulating currentand thus a magnetic field. Simple calculations show that thisfield is of opposite direction to the external field and so thesystem is diamagnetic. This idea was first used by Langevinto derive diamagnetism. But from the point of view of theBohr-van Leeuwen theorem this seems strange. In Feyn-man’s lectures10 the Bohr-van Leeuwen theorem is thereforenot considered to be valid for systems that can rotate. Asnoted above, the problem is simply that the Bohr-van Leeu-wen theorem does not take into account the magnetic fieldproduced by the particles of the system itself. When this in-ternal field is accounted for, diamagnetism follows naturally,whether for systems of atoms from the Larmor’s theorem19

or for superconductors from the Darwin formalism.20

C. The Shanghai experiment—measuring a diamagneticcurrent?

If it is true that the classical Hamiltonian for a system ofcharged particles predicts diamagnetism, then why is the phe-nomenon so weak and insignificant in most cases? Is thereany evidence for classical diamagnetism for systems otherthan superconductors? As stressed by Mahajan,21 plasmas aretypically diamagnetic. However, plasmas are not usually inthermal equilibrium so it is difficult to reach any definite con-clusions from them. An experiment on an electron gas in ther-mal equilibrium, performed by Xinyong Fu and Zitao Fu22 inShanghai, is therefore of considerable interest.

Two electrodes of Ag-O-Cs side-by-side in a vacuum tubeemit electrons at room temperature because of their lowwork function. If a magnetic field is imposed on this system,an asymmetry arises and electrons flow from one electrodeto the other. For a field strength of about 4 gauss a steadycurrent of �10�14 A is measured at room temperature. Thecurrent grows with increasing field strength and changesdirection as the polarity is reversed. The authors interpretthis result as if the magnetic field acts as a Maxwell demonthat can violate the second law of thermodynamics.22

In view of statistical mechanics based on the Darwin Hamil-tonian,18 it is more natural to interpret this result as a dia-magnetic response of the system. The current—just as thesuper-current of the Meissner effect—is due to a diamagneticthermal equilibrium. This means that no useful work can beextracted from the system.

III. ON THE ALLEGED INCONSISTENCY OF

MAGNETIC FLUX EXPULSION WITH

CLASSICAL PHYSICS

Meissner and Ochsenfeld3,23 discovered the Meissnereffect in 1933. To their surprise a magnetic field was notonly unable to penetrate a superconductor, it was alsoexpelled from the interior of a conductor as it was cooledbelow its critical temperature. The first effect—ideal dia-magnetism—seemed natural to them even though it violatesthe Bohr-van Leeuwen theorem. As already discussed, byviolating this theorem ideal diamagnetism should have beenconsidered a non-classical effect. The flux expulsion, on theother hand, was explicitly proclaimed by Meissner to haveno classical explanation. Meissner does not give any argu-ments or references to support this statement, but accordingto Dahl24 the theoretical basis was Lippmann’s theorem onthe conservation of magnetic flux through an ideally con-ducting current loop. Later, Forrest23 and others have argued

165 Am. J. Phys., Vol. 80, No. 2, February 2012 H. Essen and M. C. N. Fiolhais 165

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that the magnetohydrodynamic theorem on frozen-in fluxlines also supports this notion. In this section, we discussthese arguments and point out that they do not rule out aclassical explanation of the Meissner effect.

A. Lippmann’s theorem

Gabriel Lippmann (1845–1921), winner of the 1908 NobelPrize in Physics, published a theorem in 1889 stating that themagnetic flux through an ideally conducting current loop isconserved. In 1919, when superconductivity had been dis-covered, Lippmann25 again published this result in three dif-ferent French journals (see Sauer26). Although the idea offlux conservation is considered highly fundamental,27 nowa-days references to Lippman’s theorem are hard to find. Butat the time Lippman’s theorem was quite influential, andDahl24 explains how this was one of the results that made theMeissner effect seem surprising—and non-classical—at thetime of its discovery.

The proof of Lippmann’s theorem follows by noting thatthe self inductance L of a closed circuit, or loop, is related tothe magnetic flux U from the current in the loop via

U ¼ cL _q; (7)

where c is the speed of light and _q is the current through thecircuit, an overdot denoting a time derivative (see Landauand Lifshitz,28 Vol. 8, x33). The equation of motion for a sin-gle loop electric circuit is

L€qþ R _qþ C�1q ¼ EðtÞ; (8)

where R is the resistance, C is the capacitance, and EðtÞ isthe emf driving the current. If there is no resistance, no ca-pacitance, and no emf, this equation becomes

L€q ¼ 0: (9)

Therefore, if the self inductance L is constant, Eqs. (7) and(9) tell us that U¼ constant.

B. Lippmann’s theorem and superconductors

Although Lippmann’s theorem is correct, its relevance forthe prevention of flux expulsion is not clear. For supercon-ductors there are two points to consider, the assumption ofzero emf, and the fact that constant flux does not imply con-stant magnetic field. We consider these points one at a time.

Consider a superconducting sphere of radius r in a con-stant external magnetic field Be. When the sphere expels thisfield by generating (surface) currents that produce Bi¼�Be

in its interior, the total magnetic energy is reduced. The mag-netic energy change is

DEB ¼ �34pr3

3

� �B2

e

8p; (10)

or three times the initial interior magnetic energy.29 Thisenergy is thus available for producing the emf required togenerate currents in the sphere’s interior. The assumption ofLippmann’s theorem—that the emf is zero—is therefore notfulfilled.

When a steady current flows through a fixed metal wire boththe flux and the magnetic field distribution are constant. On theother hand, when current flows in a loop in a conducting

medium, a constant flux through the loop does not imply a con-stant magnetic field because the loop can change in size, shape,or location. Normally current loops are subject to forces thatincrease their radius (see e.g., Landau and Lifshitz,28 Vol. 8,x34, Problem 4, or Essen,30 Sec. 4.1). In view of this,Lippmann’s theorem does not automatically imply that themagnetic field must be constant, even if the emf is zero.

Other authors have reached similar conclusions. Mei andLiang31 carefully considered the electromagnetics of super-conductors in 1991 and write, “Thus Meissner’s experimentshould be viewed through its time history instead of as astrictly dc event. In that case classical electromagnetic theorywill be consistent with the Meissner effect.”

C. Frozen-in field lines

The simplest derivation of the frozen-in field result for aconducting medium begins with Ohm’s law j¼rE. If r !1 we must have E¼ 0 to prevent infinite current. Faraday’slaw then gives @B=@t ¼ �c r� Eð Þ ¼ 0 which tells usthe magnetic field B is constant (Forrest,23 Alfven andFalthammar32).

When dealing with a conducting fluid, the equations ofmagnetohydrodynamics and the limit of infinite conductivityin Ohm’s law give rise to32

@B

@t¼ r� ðv� BÞ: (11)

This result tells us the magnetic field convects with the fluidbut does not dissipate. In addition, Eq. (11) can be used toderive the following two statements that are often referred toas the frozen-in field theorem:33

1. the magnetic flux through any closed curve moving withthe fluid is constant, and

2. a magnetic field line moving with the fluid remains amagnetic field line for all time.

The first of these is a just a restatement of Lippmann’s the-orem for a current loop in a fluid.

D. Magnetic field lines in superconductors

As shown in the previous section, Ohm’s law with infiniteconductivity implies there can be no change in the magneticfield since this would give rise to an infinite current density.It is, however, not physically meaningful to take the limitr ! 1 in Ohm’s law. In a medium of zero resistivity onemust instead use the equation of motion for the chargecarriers. One can then derive29,34

dj

dt¼ e2n

mEþ e

mcj � B; (12)

for the time rate of change of the current density. The timederivative here is a convective, or material, time derivative.Ohm’s law simply does not apply, and therefore it cannotimply that the magnetic field does not change.

The second statement of the frozen-in field theorem canbe questioned on the same grounds. Just as with Lippmann’stheorem, the conclusion that the magnetic field remains con-stant does not follow, even if the statement is assumed valid.The magnetic field is determined by the density of magneticfield lines and constancy of this density requires that the con-ducting fluid is incompressible. One can easily imagine that

166 Am. J. Phys., Vol. 80, No. 2, February 2012 H. Essen and M. C. N. Fiolhais 166

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the fluid of superconducting electrons is compressible, andthat magnetic pressure33 pushes the fluid (with its magneticfield lines) to the surface of the material. Corroborating thispoint of view, Alfven and Falthammar32 (Sec. 5.4.2) statethat “in low density plasmas the concept of frozen-in lines offorce is questionable.”

E. A classical derivation of the London equations

In 1981 Edwards35 published a manuscript with the abovetitle in Physical Review Letters. This caused an uproar ofindignation and the journal later published three differentcriticisms of Edwards’ work.36–38 Incidentally, all ofEdwards’ critics restated (or implied) the textbook myth thatthe Meissner flux expulsion does not have a classical expla-nation. In addition, the journal Nature published a study byTaylor39 pointing out the faults of Edwards’ derivation. Ofcourse, Edwards is not the only one to publish an erroneousderivation of the London equations. A much earlier exampleis the derivation by Moore40 from 1976. The fact that variousderivations have been wrong, of course, does not prove any-thing—as long as a correct derivation exists.

In 1966 Nature published a classical explanation of theLondon equations by Pfleiderer17 which did not cause anycomment. Indeed it has not been cited a single time, prob-ably because it is very brief and cryptic. However, a classicalderivation of the London equations and flux expulsion canbe found in a classic textbook by the French Nobel laureatePierre Gilles de Gennes.41 Because this derivation should bemore recognized, we repeat it here.

F. de Gennes’ derivation of flux expulsion

The total energy of the relevant electrons in the supercon-ductor is assumed to have three contributions: the condensa-tion energy associated with the phase transition ES, theenergy of the magnetic field EB, and the kinetic energy of themoving superconducting electrons Ek. The total energy rele-vant to the problem is thus taken to be

E ¼ ES þ EB þ Ek: (13)

The condensation energy is then assumed to be constantwhile the remaining two can vary in response to externalfield variations. The super-current density is written as

jðrÞ ¼ nðrÞevðrÞ; (14)

where n is the number density of superconducting electronsand v is their velocity, which gives a kinetic energy of

Ek ¼ð

1

2nðrÞmv2ðrÞdV ¼

ð1

2

m

e2nðrÞ j2ðrÞdV: (15)

By means of the Maxwell equation r� B ¼ 4pj=c and Eq.(3) for EB, the total energy (13) becomes

E ¼ ES þ1

8p

ðB2 þ k2ðr � BÞ2h i

dV; (16)

where we have assumed that n is constant in the regionwhere there is current, and the London penetration depth isgiven by

k ¼ffiffiffiffiffiffiffiffiffiffiffiffimc2

4pe2n

r: (17)

Minimizing the energy in Eq. (16) with respect to B thengives the London equation

Bþ k2r� ðr � BÞ ¼ 0; (18)

in one of its equivalent forms. Notice that this derivation uti-lizes no quantum concepts and does not contain Planck’sconstant. It is thus completely classical. A similar derivationhas been published more recently by Badıa-Majos.34

The conclusion of de Gennes is clearly stated in his 1965book41 (emphasis from the original): “The superconductorfinds an equilibrium state where the sum of the kinetic andmagnetic energies is minimum, and this state, for macro-scopic samples, corresponds to the expulsion of magneticflux.” In spite of this, most textbooks continue to state that“flux expulsion has no classical explanation” as originallystated by Meissner and Ochsenfeld3 and repeated in the in-fluential monographs by London42 and Nobel laureate Maxvon Laue.43 As one textbook example, Ashcroft andMermin44 explain that “perfect conductivity implies a time-independent magnetic field in the interior.”

G. A purely classical derivation from magnetostatics

One might object that the electronic charge e is a micro-scopic constant, and that the London penetration depth k inmost cases is so small that it seems to belong to the domainof microphysics. So even if quantum concepts do not enterexplicitly, the above derivation does have microscopic ele-ments. It is thus of interest that from a purely macroscopicpoint of view we can identify the kinetic energy of theconduction electrons solely with magnetic energy.45,46 Theenergy that should be minimized would then be Eq. (3)

EB ¼1

8p

ðB2dV: (19)

Unfortunately, this will not provide any information aboutthe currents that are the sources of B. However, if we canneglect the contribution from fields at a sufficiently distantsurface—i.e., when radiation is negligible—it is possible torewrite this expression as

E0B ¼1

2c

ðj � AdV; (20)

where B ¼ r� A. The idea is to then apply the variationalprinciple to EjA ¼ 2E0B � EB, i.e., the expression

EjA ¼ð

1

cj � A� 1

8pr� Að Þ2

� �dV: (21)

Here the energy is written in terms of the field A and itssource j. There are many ways of combining Eqs. (19) and(20) to get an energy expression, but it turns out that Eq. (21)is the one that gives the simplest results.

Starting from Eq. (21) and adding the constraint r � j ¼ 0,the integration is split into interior, surface, and exteriorregions. The result is a theorem of classical magnetostaticsthat states, for a system of perfect conductors the magneticfield is zero in the interiors and all current flows on theirsurfaces (Fiolhais et al.29) This theorem is analogous to asimilar result in electrostatics for electric fields and chargedensities in conductors, sometimes referred to as Thomson’s

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theorem (see Stratton,47 Sec. 2.11). Readers should consultthe paper by Fiolhais et al.29 for details.

Many other authors have also reached the conclusion thatthe equilibrium state of a superconductor is in fact the stateof minimum magnetic energy of an ideal conductor. Some ofthese are, in chronological order, Cullwick,48 Pfleiderer,17

Karlsson,49 Badıa-Majos,34 Kudinov,50 and Mahajan.21 Forexample, Cullwick writes “The well-known Meissner effectin pure superconductors is shown to be an expected ratherthan an unexpected phenomenon…” while Kudinov50

explains that it “is worth noting in this connection that theexpulsion of a magnetic field to the periphery (the Meissnereffect) also occurs in a classical collisionless gas of chargedparticles and that this happens solely because this state is en-ergetically favorable.”

IV. CONCLUSIONS

The reader may get the impression from our investigationsabove that we consider superconductivity to be a classicalphenomenon. Nothing could be further from the truth. Asimplied by the Ginzburg-Landau theory,51 the BCS theory,52

and the Josephson effect,53 the phenomenon is quantum me-chanical to a large extent. Since quantum physics must leadto classical physics in some macroscopic limit, it must bepossible to derive our classical result from a quantum per-spective. Evans and Rickayzen54 did indeed derive theequivalence of zero resistivity and the Meissner effect quan-tum mechanically, but did not discuss the classical limit.What we want to correct is the mis-statement that the Meiss-ner effect proves that superconductors are “not just perfectconductors.” According to basic physics and a large numberof independent investigators, the specific phenomenon offlux expulsion follows naturally from classical physics andthe zero resistance property of the superconductor—they arejust perfect conductors.

In conclusion, we have carefully examined the evidencefor the oft repeated statements in textbooks that (1) there isno classical diamagnetism and (2) there is no classical expla-nation of magnetic flux expulsion. We have found that thetheoretical arguments for these statements are not rigorousand that, for superconductors, these particular phenomenaare in good agreement with the classical physics of idealconductors.

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