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Chapter 5
Magnetic Materials
In the previous chapter, we have considered currents that were able to move on a mesoscopicscale (i.e. on the order of µm). Such currents are called free currents. These are due to themovement of free charges.
However, in some media found in nature, one cannot neglect the influence of bound currents,that is to say currents which displacements occur on a maximum scale of the order of the nm.Indeed, although bound, these currents can significantly contribute to the mesoscopically-averagedcurrent densities.
Once again, Maxwell’s equations stated in chapter 1 are still valid if one considers both freeand bound charges and currents. Like in chapter 3, however, we will here see that it is often morepractical to express Maxwell’s equations under the form of what are called “Maxwell’s equationsin material media” where the electric and magnetic field are replaced by the electric displacement
field−→D and by the magnetizing field
−→H .
In this chapter, we will first examine the general expression and consequences of Maxwell’sequations. We will then look more specifically into non dielectric materials (εr = 1). Some media,however, are both dielectrics and magnetic materials, in which case the results obtained in chapter3 will have to be combined with those given in the present chapter in order to achieve a completedescription of the material behavior in electromagnetic field. We will end by giving an overview ofthe microscopic phenomena which can engender a magnetic behavior in materials.
5.1 Maxwell’s equations in magnetic materials
5.1.1 Magnetization vector
5.1.1.a Definition
Under the influence of an external magnetic field, bound charges can be set into motion ona nanometer scale. Thus appear nanoscopic current loops, i.e. nanoscopic magnetic dipoles.Materials capable of magnetizing themselves in such a way are called magnetic materials.
Definition : In a magnetic material, the magnetization vector−→M is the mesoscopically-
averaged 1 magnetic dipole moment per unit volume :
−→M =
d−→M
dV(5.1)
It is expressed in A.m−1.
1. See section 1.7.
108
5.1. MAXWELL’S EQUATIONS IN MAGNETIC MATERIALS 109
This quantity is directly related to the magnetic dipoles studied in section 4.3.
Indeed, let us consider an elemental volume dV containing a large number of magnetic dipoles
{−→Mk}k, then :
−→M =
1
dV
∑
k
−→Mk
=1
dVdN〈
−→Mk〉
=d−→M
dV
where dN is the number of dipoles contained in dV and 〈−→Mk〉 the average magnetic dipole moment
inside the volume dV .
♣ Example : Lattice of current loops
Let us illustrate the mechanism at play in mag-netic materials on a simple - and simplistic - exam-ple. We will consider the opposite 3D array of cur-rent loops, placed at abscissae xn = na, and throughwhich flows a current In = nI0.
Let us calculate the magnetization vector. Inthis case :
−→M =
d−→M
dV=
nI0a2
a3−→uz
=I0xn
a2−→uz
=I0x
a2−→uz
Note that the averaging for the calculation of−→M is done on a scale ≫ a.
The effective current - the magnetization current - can be calculated at point xn by summingthe branches of adjacent loops :
−→J M =
dI
dS−→uy = −
I0 × number of wires in dS
dS−→uy
= −I0
dSa2
dS−→uy = −
I0
a2−→uy
On this simple example, one can verify the relation−→J M =
−→rot
−→M which we will now prove in
the general case.
5.1.1.b Magnetization current density
In this paragraph, we would like to derive general formulae for the magnetization current den-sity. We will therefore consider a magnetic material of volume (V ) and surface (Σ), as schematizedfigure 5.1.
110 CHAPTER 5. Magnetic Materials
Figure 5.1: Two equivalent ways of dealing with a magnetic material : either consider the magne-tization or think in terms of magnetization current densities (in volume and on the surface).
At point M , the vector potential created by a single magnetic dipole−→M placed at point Q is
(equation 4.19):
−→A (M) =
µ0
4π
−→M∧
−−→QM
QM3
The vector potential created by the entire magnetic material can then be written as :
−→A (M) =
µ0
4π
˚
(V )
−→M∧
−−→QM
QM3
dN
dVdV
with dN the average number of magnetic dipoles per unit volume. Hence :
−→A (M) =
µ0
4π
˚
(V )
−→M ∧
−−→QM
QM3dV (5.2)
We have seen in section 3.1.1.b that :−−→QM
QM3= −
−−→gradM
(
1
QM
)
=−−→gradQ
(
1
QM
)
And we know from vector analysis that−−→curl (α
−→A ) = α
−−→curl
−→A +
−−→grad α ∧
−→A
which gives :−→M ∧
−−→QM
QM3=−→M ∧
−−→gradQ
(
1
QM
)
=
−−→curl
−→M
QM−−−→curl
( −→M
QM
)
So that :
−→A (M) =
µ0
4π
˚
(V )
−−→curl
−→M
QMdV −
µ0
4π
˚
(V )
−→rot
( −→M
QM
)
dV
=µ0
4π
˚
(V )
−−→curl
−→M
QMdV −
µ0
4π
‹
(Σ)
−→n ∧−→M
QMdS
−→A (M) =
µ0
4π
˚
(V )
−−→curl
−→M
QMdV +
µ0
4π
‹
(Σ)
−→M ∧ −→n
QMdS
5.1. MAXWELL’S EQUATIONS IN MAGNETIC MATERIALS 111
As can be seen from the last expression, the vector potential created by the magnetized material
is equivalent to the one created by the combination of a volume current density−→J M and a surface
current density−→J s
M :
−→J M =
−−→curl
−→M (5.3)
−→J s
M =−→M ∧ −→n (5.4)
Figure 5.2: Schematization of surface currents in a magnet. The bulk current density is notrepresented.
⋆ A few comments on these expressions:• If the magnetization is non uniform, its curl gives the net current density appearing in thematerial.
• These currents are real currents. They are called “magnetization currents” to distinguishthem from “free” currents, but both represent a physical reality. In a magnet, for instance,there can be no free currents circulating through the sample, but the magnetization induces
surface currents (see figure 5.2) : if the magnetization−→M is constant,
−→J M =
−→0 , whereas
−→J s
M =−→M ∧ −→n =M−→uz ∧
−→ur = −M−→uθ.
• As stated on numerous occasions, Maxwell’s equations hold for magnetic materials. However,the volume current density which has to be considered is the total volume current density
which also includes the magnetization current density :−→J tot =
−→J free +
−→J M .
5.1.2 Maxwell’s equations in material media
5.1.2.a Charges and Current distributions
Taking into account the contributions of both free and bound charges, one has the total chargedensity :
ρ = ρfree + ρP
ρ = ρfree − div−→P (5.5)
and the total current density :
−→J =
−→J free +
−→J P +
−→J M
−→J =
−→J free +
∂−→P
∂t+−−→curl
−→M (5.6)
By substituting these total charge and current densities into Maxwell-Gauss and Maxwell-Ampere equations, one obtains :
div−→E =
ρ
ε0−div−→P
ε0
div−→D = ρfree
112 CHAPTER 5. Magnetic Materials
and :
−−→curl
−→B = µ0
−→J + ε0µ0
∂−→E
∂t
= µ0−→J free + µ0
−−→curl
−→M + µ0
∂−→P
∂t+ ε0µ0
∂−→E
∂t
−−→curl
(−→B
µ0−−→M
)
=−→J free +
∂−→D
∂t
5.1.2.b H-vector
This leads us to define the H-vector.
Definition : The magnetic excitation vector−→H is defined by :
−→H =
−→B
µ0−−→M (5.7)
It is expressed in A.m−1.
The H-vector represents the properties of bound charges appearing when a magnetic materialis magnetized.
The terminology for the magnetic-field related vectors is sometimes confused. The importantthing to remember are the equations and their meaning. The actual name given to the differentvectors does not really matter as long as you do not get confused. Here are however the namesthat can be found in the literature. In bold are the names we will adopt from now on:
Symbol Unit Possible name
M A.m−1 Magnetization
Magnetic polarization (Weber)B T Magnetic field (Feynman)
Magnetic induction (Slater, Pauli, Feynman)Magnetic flux intensity (Jackson, Weber)
H A.m−1 H-vector
Magnetic excitationMagnetic field (Slater, Jackson, Feynman)Magnetic field intensity (Pauli)Magnetic intensity (Slater, Weber)Magnetizing force (Weber)
5.1.2.c Maxwell’s equations
Maxwell’s equations in material media can therefore be written :
Maxwell-Gauss : div−→D =
−→∇ .−→D = ρfree (5.8)
Conservation of Magnetic Flux : div−→B =
−→∇ .−→B = 0 (5.9)
Maxwell-Faraday :−−→curl
−→E =
−→∇ ∧
−→E = −
∂−→B
∂t(5.10)
Maxwell-Ampere :−−→curl
−→H =
−→∇ ∧
−→H =
−→J free +
∂−→D
∂t(5.11)
5.1. MAXWELL’S EQUATIONS IN MAGNETIC MATERIALS 113
⋆ A few comments on these equations :• Note that in these expressions, the charge and current densities are those corresponding to
the free charges. The bound charges and current densities are included in−→D and
−→H .
• One can either choose to write Maxwell’s equations in the general case (equations 1.4 to 1.7)
with ρ and−→J being the total charge and current densities OR use the above expressions
where only the free charge and current densities appear. Please do not mix both expressions...
5.1.3 Consequences of Maxwell’s equations
The conservation of magnetic flux and Maxwell-Faraday equations are unchanged so that :
‹
(S)
−→B.−→dS = 0 (equation 1.44)
and Faraday’s law still holds :
˛
(C)
−→E .−→dl = −
∂φ
∂t= e (equation 1.46)
Maxwell-Gauss equation is the same than what we saw in chapter 3, so that Gauss theorem forthe electric displacement field is still valid :
‹
(S)
−→D.−→dS = Qfree,int (equation 3.12)
5.1.3.a Generalized Ampere’s theorem
However Maxwell-Ampere’s law has been modified and Ampere’s theorem can therefore berephrased 2.
Let us consider a surface (S) delimited by a closed countour (C) which contains a free current
volume density−→J free, and a total free current Ifree. The generalized Ampere’s law can be
expressed :˛
(C)
−→H.−→dl = Ifree,(S) +
¨
(S)
∂−→D
∂t.−→dS (5.12)
♦ Proof :
˛
(C)
−→H.−→dl =
¨
(S)
−−→curl
−→H.−→dS
=
¨
(S)
(
−→J free +
∂−→D
∂t
)
.−→dS = Ifree,(S) +
¨
(S)
∂−→D
∂t.−→dS
In the quasi-steady state regime :
˛
(C)
−→H.−→dl = Ifree,(S) (5.13)
2. Once again, the original formulation seen in section 1.4 holds, provided the charge and current distributionsused are the total ones.
114 CHAPTER 5. Magnetic Materials
5.1.3.b Discontinuity equations
The discontinuity equations are obtained in a similar manner as in sections 1.5.2 and 1.5.3. Fortwo media (1) and (2) separated by an interface characterized by a unit vector −→n 1→2, with a free
surface charge density σfree and a free surface current density−→J s,free :
div−→D = ρfree →
−−→D2,n −
−−→D1,n = σfree
−→n 1→2 (5.14)
−→rot
−→E = −
∂−→B
∂t→
−−→E2,t −
−−→E1,t =
−→0 (5.15)
div−→B = 0 →
−−→B2,n −
−−→B1,n =
−→0 (5.16)
−→rot
−→H = µ0
−→J free +
∂−→D
∂t→
−−→H2,t −
−−→H1,t =
−→J s,free ∧
−→n 1→2 (5.17)
⋆ A few comments on these equations :• The bound surface charge density can still be expressed (equation 3.19) :
−−→P2,n −
−−→P1,n = −σP
−→n 1→2
• The bound surface current density can be retrieved :
−→J s
M =−→M ∧ −→n 1→2
♦ Proof :
−−→H2,t −
−−→H1,t =
−→J s,free ∧
−→n 1→2
−−→B2,t
µ0−
−−→B1,t
µ0−−−→M2,t +
−−→M1,t =
(−→J s,tot −
−→J s
M
)
∧ −→n 1→2
−−−→M2,t +
−−→M1,t = −
−→J s
M ∧ −→n 1→2
5.2 Linear magnetic materials
From this point on, we will concentrate on non-dielectric magnetic materials (εr = 1 ; µr 6= 1).
5.2.1 Constitutive relations
The above equations describe the relations between−→E ,−→H and
−→B . Unless
−→H can be expressed
as a function of−→E and
−→B , Maxwell’s equations cannot be solved. The connection between these
quantities are called the constitutive relations :
−→H =
−→H[−→E ,−→B]
We will now examine particular cases for these constitutive relations.
5.2. LINEAR MAGNETIC MATERIALS 115
5.2.2 Categorizing the magnetic media
5.2.2.a Linear magnetic materials
Definition : A magnetic medium is called linear if there is a magnetic tensor 3 such that:
−→B = [µ(ω)]
−→H (5.18)
⋆ A few comments :
– This property means that the different components of−→H ,
−→E and
−→B are related by linear
partial differential equations with constant coefficients.– The coefficient can however be a function of the frequency ω. In that case, it is useful tointroduce the complex notation.
– In linear magnetic materials, one assumes that the response of the material to an appliedfield is linear 4. This excludes the case of materials presenting a spontaneous magnetization.Outside of these materials, the linear response is reasonable as long as the fields are not toolarge.
5.2.2.b Linear Isotropic magnetic materials
Definition : A magnetic medium is called linear isotropic if there is a scalar magnetic
permeability µ(ω) such that :−→B = µ(ω)
−→B (5.19)
µr =µµ0
is then called the relative permeability. One also defines themagnetic susceptibility
χm such that:µ = µ0 (1 + χm) (5.20)
CAUTION : Electric and magnetic susceptibilities are often both noted χ. It is therefore impor-tant to know which one you are dealing with...
⋆ A few comments :– The relative permeability is a priori a complex number. We will see the physical meaning ofthe real and imaginary parts in a later chapter.
– In isotropic materials, the relative permeability does not depend on the directions.– In linear isotropic materials, the magnetic field and the magnetization are colinear.
5.2.2.c Linear Homogeneous Isotropic magnetic materials
Definition : A magnetic medium is called linear homogeneous isotropic (lhi) if it islinear, isotropic and if the magnetic permeability µ does not depend on the pointM of the material.The magnetic permeability is then characteristic of the material.
3. i.e. a matrix.4. This is more generally called the linear response theory.
116 CHAPTER 5. Magnetic Materials
Mu-metal (used for magnetic field shielding) 50’000Iron 5’000Ferrite 500Platinum 1.000265Teflon 1Air 1Bismuth 0.999834Superconductors 0
Table 5.1: Indicative relative permeability for a few materials.
5.2.3 Useful relations in lhi magnetic materials
Combining the previously seen relations for lhi magnetic materials, one obtains :
−→B = µ0
(−→H +
−→M)
= µ−→H = µ0 (1 + χm)
−→H = µ0µr
−→H (5.21)
−→B =
µ
χm
−→M =
µ0µr
µr − 1
−→M =
µ0 (1 + χm)
χm
−→M (5.22)
−→M =
χm
µ0 (1 + χm)
−→B =
µr − 1
µ0µr
−→B (5.23)
−→M = χm
−→H = (µr − 1)
−→H (5.24)
χm = µr − 1 (5.25)
Using:
−→J total =
−→J free +
−→J P +
−→J M
−−→curl
−→B = µ0
−→J total
−−→curl
−→H =
−→J free
and−−→curl
−→M =
−→J M
one could find relations between−→J total,
−→J free and
−→J M like what we have done in section 3.2.3 for
the charge densities. However, these would be true ONLY if−→J P =
−→0 , that is to say if
−→E =
−→0
or in the steady state regime... Otherwise, things are more complicated, due to the polarizationcurrent density.
⋆ Comment on the sign of χm:• If χm = 0, the material is non-magnetic.• If χm > 0, the material is either paramagnetic, ferromagnetic, ferrimagnetic or anti-
ferromagnetic. In those systems, the magnetic field within the material is increased due toa magnetization parallel to the applied field.
• If χm < 0, the material is diamagnetic. In those systems, the magnetic field within thematerial is weakened due to a magnetization anti-parallel to the applied field.
• Superconductors are perfectly diamagnetic: the magnetization completely compensatesthe applied field, so that the effective magnetic field within the material is exactly zero.
We will detail the different kind of magnetic behaviors in section 5.6.
5.2. LINEAR MAGNETIC MATERIALS 117
5.2.4 Snell-Descartes relations for a dielectric material
Figure 5.3: Evolution of the magnetic field at a boundary between two magnetic materials, in theabsence of free currents.
The discontinuity equations applied at the boundary between to lhi magnetic materials, in theabsence of free currents, gives, with the notations of figure 5.3 :
−−→B2,n −
−−→B1,n =
−→0 → B2 cosα2 = B1 cosα1
−−→H2,t −
−−→H1,t =
−→0 →
B2
µ2sinα2 =
B1
µ1sinα1
so that :tanα1
µ1=tanα2
µ2
When a magnetic field crosses from a region of low µ to a region of high µ, the field lines tend tomove apart, very much like what happens in optics when light crosses to a material with a higherrefractive index.
Hence, as seen figure 5.4, magnetic materials tend to bend the magnetic field lines. For high µr
materials, the field lines can be considered to be parallel to the incoming surfaces of the magnet.
Figure 5.4: Evolution of the magnetic field in magnetic rods, depending on their permeability µr.
118 CHAPTER 5. Magnetic Materials
♣ Application: Choke coil inductor - Choke coil inductors (as represented figure 5.2.4 are widelyused in electronic devices. They consist in a wire through which flows a current I, winded arounda high magnetic permeability material (iron, or ferrite for example). They enable to achieve highinductances while limiting magnetic field leakage 5.
Figure 5.5: Choke coil inductor. Picture taken from http://www.indiamart.com.
♦ Proof : Let us suppose that the magnetic material has a sufficiently high µr to confine themagnetic field in its core. The field lines are therefore circular and the ferrite is a flux tube. Theconservation of magnetic flux gives that the magnetic field is constant in all the material.
˛
−→H.−→dl = 2πRH = NI
B = µrH =µrNI
2πR
φB = N
¨
−→B.−→dS =
µrN2IS
2πR= LI
L =µrN
2S
2πR
Moreover, in series with resistors and capacitors, this inductance gives rise to a resonant circuit
which quality factor is given by Q = 1R
√
LC, meaning that a higher inductance renders the circuit
more selective and hence less prone to electronic noise.
5.3 Forces and momentum
We have seen in section 4.3.3.a that the force exerted by an external magnetic field−→B 0 on a
rigid dipole−→M can be expressed as :
−→F =
−→∇(−→M.−→B 0
)
A magnetic material submitted to an external magnetic field−→B 0 is therefore submitted to a volume
density of external force 6:
d−→F
dV=−→∇(−→M.−→B)
(5.26)
5. These device therefore limit the electromagnetic interferences which are an important issue in a society whereelectronic devices are numerous.
6. This expression does not take into account the internal forces within the magnetic material.
5.4. ENERGY IN MATERIAL MEDIA 119
where−→B is the magnetic field as seen by each one of the dipoles.
The corresponding angular momentum can be shown to be :
−→ΓO =
−→M ∧
−→B (5.27)
5.4 Energy in material media
We had seen in section 1.2.2.b, that :
∂Uem
∂t= −
˚
(V )
−→J .−→E dV −
‹
(S)
−→R.−→dS
with−→J the total current density, Uem =
˝
(V )
(
ε0E2
2 + B2
2µ0
)
dV the total electromagnetic energy
contained in a volume V and−→R =
−→E∧−→B
µ0
the Poynting vector.However, this expression - valid in all cases, including material media - involves the total current
density. In some case it is useful to derive a similar expression containing only the free currentdensity.
−→J free.
−→E =
−−→curl
−→H.−→E −
∂−→D
∂t.−→E
=(−−→curl
−→H.−→E −
−→H.−−→curl
−→E)
+−→H.−−→curl
−→E −
∂−→D
∂t.−→E
= −div(−→E ∧
−→H)
+−→H.
(
−∂−→B
∂t
)
−∂−→D
∂t.−→E
Hence the local energy conservation equation :
div(−→E ∧
−→H)
+−→H.
(
∂−→B
∂t
)
+∂−→D
∂t.−→E = −
−→J free.
−→E (5.28)
In the case of a lhi material :
−−→J free.
−→E = div
(−→E ∧
−→H)
+
−→B
µ.
(
∂−→B
∂t
)
+∂ε−→E
∂t.−→E
= div(−→E ∧
−→H)
+∂
∂t
(
B2
2µ+
εE2
2
)
One can then write :∂U ′em∂t
= −
˚
(V )
−→J free.
−→E dV −
‹
(S)
−→R′.−→dS (5.29)
with
U ′em =
˚
(V )
(
εE2
2+
B2
2µ
)
dV (5.30)
the total electromagnetic energy contained in a volume V,−→J free the free current density, and
−→R′ =
−→E ∧
−→H a version of the Poynting vector for material media. Note that the material properties
are taken into account in ε and µ.
⋆ A few comments :
120 CHAPTER 5. Magnetic Materials
– Energetics in dielectric or magnetic materials is a difficult and tricky problem. The discussionon this matter has here been kept to a minimum. For a more complete discussion on thesubtleties of this issue, see, for example, J.D. Jackson, Classical Electrodynamics, chapters 4,5 and 6.
– Let us note however that U ′em corresponds to the total electromagnetic energy density inthe medium. It corresponds to the total energy required to produce the magnetization inthe medium. It includes the energy for establishing the magnetization in a given field andthe energy for the creation of the magnetic moment and the energy for maintaining themagnetization.
– It can be compared to the magnetic energy −−→M.−→B seen in equation 5.26, which corresponds
only to the energy for establishing the magnetization in a given field.
5.5 Examples
5.5.1 Example #1 : Uniformly magnetized sphere
Let us consider a uniformly magnetized sphere
of magnetization−→M =M−→uz. Through equation 5.2,
we can express the vector potential and use an anal-ogy with a uniformly charged sphere and the calcu-lation of the corresponding electric field.
−→A (M) =
µ0
4π
˚
(V )
−→M ∧
−−→QM
QM3dV
=µ0
4π
−→M ∧
˚
(V )
−−→QM
QM3dV
Let us now make the analogy with the electric cre-ated by a uniformly charged sphere (of charge den-
sity ρ). In that case :
−→E =
¨
ρ−−→QM
4πε0QM3dV
which was solved via Gauss theorem :
−→E int =
rρ
3ε0
−→ur if r < R
−→E ext =
R3ρ
3ε0r2−→ur if r > R
Reintroducing these expressions in the initial problem yields :
−→A int =
µ0
4π
−→M ∧
4πε0ρ
rρ
3ε0
−→ur if r < R
−→A int =
µ0r
3
−→M ∧ −→ur
−→A ext =
µ0
4π
−→M ∧
4πε0ρ
R3ρ
3ε0r2−→ur if r > R
−→A ext =
µ0R3
3r2−→M ∧ −→ur
5.5. EXAMPLES 121
Since−→M = cos θ−→ur + sin θ−→uθ :
−→A int =
µ0rM sin θ
3−→uφ if r < R
−→A ext =
µ0R3M sin θ
3r2−→uφ if r > R
and using the expression of the curl in spherical coordinates :
−→B int =
1
r sin2 θ
∂
∂θ
(
µ0rM sin θ
3
)
−→ur −1
r
∂
∂r
(
µ0r2M sin θ
3
)
−→uθ if r < R
=2µ0M cos θ
3−→ur −
2µ0M sin θ
3−→uθ
=2µ0
3(M cos θ−→ur −M sin θ−→uθ)
=2µ0
3
−→M
−→H int =
−→B int
µ0−−→M = −
−→M
3if r < R
−→B ext =
1
r sin θ
∂
∂θ
(
µ0R3M sin2 θ
3r2
)
−→ur −1
r
∂
∂r
(
µ0R3M sin θ
3r
)
−→uθ if r > R
=2µ0R
3M cos θ
3r3−→ur +
µ0R3M sin θ
3r3−→uθ
=µ0R
3
3r3(2M cos θ−→ur +M sin θ−→uθ)
=µ0
4π
(
3(−→m.−→ur)−→ur −
−→m
r3
)
with −→m =4
3πR3−→M
−→H ext =
Bext
µ0if r > R
Inside the material−→H ‖ −
−→M . This can be generalized : The H-vector always points in the
direction opposite to that of the magnetization.
Outside the material, the sphere behaves like a magnetic moment −→m = 43πR
3−→M .Figure 5.6 gives the corresponding field lines.
Figure 5.6: Field lines for a uniformly polarized sphere. Taken from http://web.mit.edu/6.013_
book/www/chapter8/8.5.html.
122 CHAPTER 5. Magnetic Materials
5.5.2 Example #2: Quincke Tube
Figure 5.7: Experimental setup for Quincke’s Tube experiment.
Let us consider a magnetic liquid material of magnetization−→M , of mass per unit volume ρ, and
of magnetic susceptibility χ. The liquid surface is placed in the air gap of an electromagnet which
creates a magnetic field−→B = B0(z)
−→ux.−→B can be taken to be uniform in the air gap but decreases
rapidly outside the gap (see figure 5.7). When an uniform field is created in the electromagnet airgap, one observes that the liquid rises in the smaller tube (of radius r) from point A to point A′,while the liquid falls in the larger tube (of radius R) from point B to point B′. Explain.
First, let us examine the value of the magnetic field within the liquid. At the interface betweenair and the magnetic liquid, the tangential component of the H-vector is conserved :
−→H =
−→B0
µ0=
−→B liq
µ0µr
where−→B liq is the magnetic field within the magnetic liquid.
Moreover, due to volume conservation :
h′ = hr2
R2
When the magnetic liquid rises, the magnetic force exerted by the electromagnet on the liquidmust exactly compensate the weight of the liquid that is lifted up the smaller tube. The energyrequired for the creation of a magnetization within the liquid in the air gap can be understood asthe replacement of air by the liquid in the air gap :
Uem =1
2BliqHliqπr
2h−1
2BairHairπr
2h
=1
2πr2h
(
µH2 − µ0H2)
=1
2πr2hµ0H
2χm
Hence :
−→F magn = −
−−→grad (−Uem) =
1
2πr2µ0H
2χm−→uz
−→F grav = −mg−→uz = −πr
2(h+ h′)ρg−→uz
= −πr2h
(
1 +r2
R2
)
ρg−→uz ≃ −πr2hρg−→uz
5.6. CATEGORIZING MAGNETIC MATERIALS 123
Here, we have moreover neglected the weight of the displaced air and we have used the fact thatr ≪ R. Then, the condition for equilibrium gives :
πr2hρg =1
2πr2µ0H
2χm
h =µ0H
2χm
2ρg
h =B2
0χm
2µ0ρg
For a paramagnetic liquid with ρ = 103 kg.m−3 and χm = 10−4, one can compute h = 4 mm.For a diamagnetic liquid, since, as we will see χ < 0, the liquid lowers in the smaller tube.
5.6 Categorizing magnetic materials
In this section, we will discuss the origins and properties of some magnetic materials 7. Thesecan be evidenced by the experiment schematized figure 5.8: if a piece of material is suspendedbetween the poles of an electromagnet, different behaviors can be observed. Let us first note thatthe magnetic field is here strongest near the South pole, due to the convergence of field lines.Ferromagnets (section 5.6.4) will be strongly attracted towards the high-field region, whereas allother magnetic materials will experience a force of much smaller intensity. Paramagnets (section5.6.3) and anti-ferromagnets (section 5.6.5) will be weakly attracted towards the high-field region,whereas diamagnetic materials (section 5.6.2) will be repelled towards the low-field region.
Figure 5.8: Schematic representation of an experiment designed to evidence the magnetic propertiesof materials. Taken from R.P. Feynman, The Feynman Lectures.
5.6.1 Microscopic origin of magnetism
Magnetic effects are either due to the permanent magnetic moments carried by individualatoms or to the movement of electrons or nuclei. They can therefore only be rigorously explainedby quantum mechanics. However, semiclassical models can give us a flavor of what is going on inthese materials. We will here limit ourselves to this level of explanation 8.
7. All magnetic materials will not be treated. For instance, we will not talk about ferrimagnetic materials,helimagnets or spinels.
8. For a quantum mechanical treatment of magnetism - of which you have had an introduction in your QuantumMechanics course - see, for example, N.W. Ashcroft and N.D. Mermin, Solid State Physics, chapter 31.
124 CHAPTER 5. Magnetic Materials
Magnetic
moment
alig
nment
Typical
valueofχm
Magnetiz
atio
nEvolutio
nof
χm
with
temperatu
re
Comments
Examples
Diamagnetis
mAnti-p
arallelto
anextern
almagn
eticfield
χm
<0,
|χm|≃10−9-
10−5,
forsu-
percon
ductors
χm=1
χm=−
µ0nZe2
6m
〈r2〉
Bism
uth,
Mercu
ry,Silver,
Argo
n,Heliu
m,
Water,
N2 ,most
organic
compounds
Paramagnetis
mParallel
toan
exter-
nalmagn
eticfield
χm
>0,
χm
≃10−5-
10−3
Atlow
temperatu
re,onehas
Curie’s
law:
χm=
CT
Tungsten
,Sodium,
Magnesiu
m,Aluminum
Ferromagnetis
mSpontan
eousmagn
e-tization
;mom
ents
alignparallel
toone
anoth
er
χm
>0
For
T>
TCuriepara-
magnetic
behavior
;For
T<
TCuriefer-
romagnetic
behavior
Gadolinium
(TCurie
=292
K),
Cobalt
(TCurie
=1388
K),Iron
(TCurie=1043
K),
NdFeB
(TCurie=583
K)
Antife
rromagnetis
mMom
ents
alignanti-
parallel
toonean-
other
χm
>0
For
T>
θparam
ag-netic
behavior
;For
T<
θantiferrom
ag-netic
behavior
Chrom
ium,NiO
Table5.2:
Princip
alcharacteristics
ofsom
emagn
eticmaterials.
5.6. CATEGORIZING MAGNETIC MATERIALS 125
We have seen in section 4.3 how a semi-classical model could account for the relation betweenthe electronic orbital moment
−→L and the corresponding magnetic moment
−−→ML:
−−→ML = −
e
2m
−→L = γ
−→L (5.31)
with γ the gyromagnetic ratio.In the same way, one can derive from quantum mechanics the relation between the electronic
spin−→S and the corresponding magnetic moment
−−→MS :
−−→MS = −
e
m
−→S (5.32)
Note that the contribution of the spin moment is twice as important as the contribution of theorbital moment. Hence, depending on the authorized combinations of orbital and spin moments 9,one obtains the electronic magnetic moment:
−→M = −g
e
2m
−→J total (5.33)
where g is a dimensionless factor, of the order of 1, called the Lande factor. It is characteristicof the considered wavefunction. The direction of the magnetic moment is opposite to the directionof the angular momentum.
The nucleus can also contribute to the global magnetic moment in a similar manner :
−→Mnucl = +g
e
2mp
−→J total (5.34)
where mp is the proton mass and g the nuclear Lande factor. Since the magnetic moment isinversely proportional to the mass, a proton contributes 1000 times less than an electron to theaverage moment, all other things being equal.
The principal categories of magnetic materials are summarized table 5.2.
5.6.2 Diamagnetism
In numerous compounds, the electron spins motion are exactly counter-balanced by their orbital
motion, so that there is no average magnetic moment. In that case, an external magnetic field−→B
induces some small extra currents, which tend to oppose the applied field. Diamagnetic compoundsusually are made out of atoms where all electrons are paired, so that the net spin of each molecule
is zero. The resulting magnetic moment is hence of opposite direction than that of−→B 10. Hence
the diamagnetism.Let us consider once again the semi-classical model of electrons orbiting around the nucleus.
Since the atom bears no average magnetic moment in the absence of magnetic field, the only non-zero contribution to M is due to the Lorentz force acting on the electrons. The angular velocityof such motion is ω = γB = eB
2m . Then the current associated to the motion of Z electrons can bewritten :
I = (−Ze)ω
2π
9. That is one application of the moments composition seen in Quantum Mechanics.10. Diamagnetic material therefore tend to expel the magnetic field. This property can be used to make objects
levitate. This has been done by Andrey Geim - more famous for the discovery of Graphene, Nobel Prize 2010 -.He has explained it in numerous papers, among which “Everyone’s magnetism”, Physics Today, p. 36, sept 1998.The videos of the levitating frog/cricket/strawberry/tomato/... can be found at http://www.ru.nl/hfml/research/levitation/diamagnetic/. He, and M.V. Berry, have won the IgNobel for this work...
126 CHAPTER 5. Magnetic Materials
Hence:
M = IS−→n = −Ze2
4πmBS−→n
Since S is the surface of the orbit in the (x, y) plane :
S = π(
〈x2〉+ 〈y2〉)
S = π
(
2
3〈r2〉
)
with 〈r2〉 = 〈x2〉 + 〈y2〉 + 〈z2〉 the mean distance of the electrons to the nucleus. Hence, themagnetic susceptibility for one atom is :
χm =M
H=
µ0M
H
= −µ0Ze2
6m〈r2〉
and for n atoms per unit volume:
χm = −µ0nZe2
6m〈r2〉 (5.35)
⋆ Comment on the diamagnetism of superconductors : Superconductors are considered to be perfect
diamagnetic materials (χm = −1). This actually means that−→B = µ0 (1 + χm)
−→H =
−→0 . In other
words, a superconductor expels the magnetic field from its volume, thanks to non-dissipative so-called supercurrents that are the consequence of the pairing of electrons into Cooper pairs 11. Thisdiamagnetism explains the Meissner effect and originates from purely quantum mechanical effectsthat have nothing to do with the simplistic semi-classical model developed above.
5.6.3 Paramagnetism
In paramagnetic compounds, the atoms usually have at least one unpaired electron, i.e. a netspin larger than zero. In all cases, the molecules possess a non-zero magnetic moment. Whenan external magnetic field is applied, this moment tends to align with the field, with a statisticdistribution similar to what we have studied for the orientation polarization section 3.5.2. In thesematerial, the diamagnetic effect is always present but is negligible compared to paramagnetic effect.Let us examine a semi-classical model for paramagnetism due to Leon Brillouin.
5.6.3.a Case of a single spin
We will examine the case of a single spin and extend the obtained result in the general case.The hypotheses are twofold:
1. Each electron has a magnetic moment associated with its spin. The projection of the spin
along the direction of the applied magnetic field−→B can only take the values
−→M or −
−→M. The
corresponding potential energies are Um = −−→M.−→B .
2. Interactions between magnetic moments are neglected. Only thermal agitation and the cor-responding chocs between molecules can change the directions of the magnetic moments.
If the magnetic field is along the Oz axis (−→B = B−→uz), the projection Mz of the magnetic
moment of a molecule can only take the valuesMz =M orMz = −M. Each value corresponds
11. See section 5.7.
5.6. CATEGORIZING MAGNETIC MATERIALS 127
to a number of molecules : N+ = A exp−−MBkBT
and N− = A exp−MBkBT
. We moreover have the
total number of molecules N = N+ +N− = 2Acosh(
MBkBT
)
.
The mesoscopically averaged magnetic moment is :
〈M〉 =N+M−N−M
N
= M2Asinh
(
MBkBT
)
2Acosh(
MBkBT
)
〈M〉 = Mtanh
(
MB
kBT
)
And the magnetization can then be expressed as :
−→M = n〈M〉−→uz
−→M = nMtanh
(
MB
kBT
)
−→uz
where n is the molecular density. Let us now examine the order of magnitude of the involvedenergies :
MB
kBT=
e~B
2mkBT
≃10−19 × 10−34 × 1
10−31 × 10−23 × 300MB
kBT≃ 0.03≪ 1
We can therefore rightfully linearize the tanh is the previous expression :
−→M ≃
nM2B
kBT−→uz
=χm
µ0 (1 + χm)
−→B ≃
χm
µ0for χm ≪ 1
Hence :
χm =nM2µ0
kBT
5.6.3.b Extension to the case of a general magnetic moment
These results can then be extended to general case of a magnetic moment−→J . Then the possible
projections on the Oz axis are given by Quantum Mechanics: mJgµB where µB = ~e2m is the Bohr
magneton, g the Lande factor and mJ the eigenvalue of the projection of−→J on the Oz axis
(−J ≤ mJ ≤ J). The calculation then gives the magnetization:
−→M = ngµBJBJ(x) (5.36)
where BJ(x) is the Brillouin function (see figure 5.9) defined by:
BJ(x) =2J + 1
2Jcotanh
(
2J + 1
2Jx
)
−1
2Jcotanh
(
1
2Jx
)
(5.37)
128 CHAPTER 5. Magnetic Materials
and x is given by:
x =JgµBB
kBT
In the case where µBB ≪ kBT , we can linearize the previous expression to obtain Curie’s law
which states that the magnetic permeability is inversely proportional to the temperature :
χm =µ0nµ
2eff
3kBT=
C
T(5.38)
with C the so-called Curie’s constant:
µeff = gµB
√
J(J + 1) (5.39)
Note that in the high-field limit, the magnetization reaches a plateau of value:
M =Ms = ngµB (5.40)
which corresponds to the maximum magnetization reached when all magnetic moments are alignedwith the field.
Figure 5.9: The Brillouin function for for J = 12 to J = 5
2 . Taken from http://moxbee.blogspot.
fr/.
5.6.4 Ferromagnetism
Some materials, like iron, nickel or cobalt, display, below a characteristic temperature calledthe Curie temperature TCurie a spontaneous magnetization, even in the absence of externalmagnetic field. Although we will not review the detailed microscopic mechanisms that give rise tothis phenomenon 12, let us review its main features.
5.6.4.a Phenomenological description
• For temperatures larger than TCurie, ferromagnets behave like paramagnetic materials :
χm =C
T − TCurie
for T > TCurie (5.41)
12. For more details, see, for example, R.P. Feynman, The Feynman Lectures, chapters 36 and 37 and referencetherein.
5.6. CATEGORIZING MAGNETIC MATERIALS 129
• For temperatures smaller than TCurie, ferromagnets have a non-defined χm - since the magne-tization is spontaneous and does not depend on the applied magnetic field -. The idea is thatinteractions between magnetic moments are sufficiently strong - unlike in the case of param-agnets - to induce a magnetization. Once one magnetic moment has a certain directions, allneighboring moments tend to align with it.
5.6.4.b Calculation of the magnetization of a ferromagnet
In order to compute the magnetization of a ferromagnet, we will extend the semi-classical modelstudied in the case of paramagnets (section 5.6.3) in order to take into account interactions betweenmagnetic moments.
We will suppose that a moment−→Mi has an interaction energy with other moments:
Ep = −J∑
k
−→Mi.
−−→Mk = −
−→Mi.
(
J∑
k
−−→Mk
)
(5.42)
where J is a constant which positive sign models the fact that the magnetic moments tend to alignwith one another to minimize their energy.
All is as if an effective, virtual, magnetic field−→B eq = J
∑
k
−−→Mk acts on the magnetic moment
−→Mi. In the framework of the mean field theory developed by Pierre Weiss 13, we will suppose that:
−→B eq = λ
−→M (5.43)
with λ > 0.By applying the result found in section 5.6.3 in the case of the virtual field
−→B eq, one finds :
−→M = nMtanh
(
MBeq
kBT
)
−→uz (5.44)
−→M = nMtanh
(
MλM
kBT
)
−→uz (5.45)
This equation can be solved numerically or graphically (see figure 5.10) by plotting the inter-section of tanhx (with x = λMM
kBT) and M
nM= kBT
nM2λx = px. Then :
• For T > TCurie =λnM2
kB
, p > 1 and the only solution is M = 0.• For T < TCurie, p < 1 and there are two solutions : M = 0 which is an unstable solution ;andM 6= 0 which is the more stable solution, thermodynamically speaking. This correspondsto a spontaneous magnetization and hence to the ferromagnetic state.
• The transition between those two cases is achieved when T = TCurie. This critical tempera-ture defines a phase transition between a paramagnetic phase and a ferromagnetic phase.
13. The mean field theory was actually developed by Pierre Weiss and Pierre Curie to explain phase transitions.See Pierre Weiss, “L’hypothese du champ moleculaire et la propriete ferromagnetique”, J. Phys.Theor. Appl., 6,pp.661-690, 1907.
130 CHAPTER 5. Magnetic Materials
Figure 5.10: Graphic solution of the self-consistent equation for magnetization in ferromagnets.
5.6.4.c The magnetization curve
The magnetization curve of a ferromagnet is illustrated figure 5.11 in the case of soft iron. Thedifferent steps can be understood as follows :
– Let us start by a non-magnetized ferromagnet. As soon as the applied H-field is applied(curve a), the magnetization - given by B - increases.
– The magnetization eventually saturates at a value Msat = nM representing the maximumsaturation achieved when all magnetic moments are aligned.
– When the H-field is decreased (curve b), there is a competition between the applied H-fieldand the interactions between the magnetic moments inside the material. The magnetizationslowly decreases. At zero H-field, the magnetization is non-zero.
– When the polarity of the H-field is reversed, the magnetization eventually reverses until itreaches −Msat.
– Then, again, if the applied H-field is increased (curve c), the magnetization will progressivelyreverse.
Figure 5.11: Magnetization curve for soft iron. Taken from R.P. Feynman, The Feynman Lectures.
5.7. ADDITIONAL READING 131
5.6.5 Anti-ferromagnetism
• For temperatures larger than θ, the Neel temperature, antiferromagnets behave like para-magnetic materials :
χm =C
T − θfor T > θ (5.46)
• For temperatures smaller than θ, antiferromagnets tend to have their magnetic momentsorganize into a periodic array of parallel/antiparallel moments. This gives rise to a zeromagnetization at low field. More generally χm = M
Hdecreases with the temperature : at low
temperature, the thermal agitation is not strong enough to act against the anti-alignment ofmagnetic moments.
5.7 Additional reading
The following article 14, reviews Meissner effect in superconductors (the expulsion of the mag-netic field within the bulk of these materials) in link with classical electromagnetism.
14. Essen and Fiolhais, Am. J. Phys., 80, 164 2012.
Meissnereffect,diamagnetism,andclassicalphysics—areview
HannoEssen
DepartmentofMechanics,RoyalInstituteofTechnology(KTH),Stockholm
SE-10044,Sweden
MiguelC
.N.Fiolhais
LIP-Coimbra,DepartmentofPhysics,UniversityofCoimbra,Coimbra
3004-516,Portugal
(Received
8September
2011;accepted28October
2011)
Wereview
theliterature
onwhat
classicalphysics
saysabouttheMeissner
effect
andtheLondon
equations.Wediscuss
therelevance
oftheBohr-van
Leeuwen
theorem
fortheperfectdiamagnetism
ofsuperconductors
andconcludethat
thetheorem
isbased
oninvalid
assumptions.Wealso
point
outresultsin
theliterature
that
show
how
magnetic
fluxexpulsionfrom
asample
cooledto
superconductivitycanbeunderstoodas
anapproachto
themagnetostatic
energyminim
um.These
resultshavebeenpublished
severaltimes
butmanytextbooksonmagnetism
stillclaim
thatthereis
noclassicaldiamagnetism,andvirtually
allbooksonsuperconductivityrepeatMeissner’s
1933
statem
entthatfluxexpulsionhas
noclassicalexplanation.V
C2012AmericanAssociationofPhysicsTeachers.
[DOI:10.1119/1.3662027]
I.IN
TRODUCTIO
N
Itisnowacentury
since
superconductivitywas
discovered
byKam
merlingh-O
nnes
inLeiden
in1911.From
thebegin-
ning,therewas
considerable
interestfrom
theoreticalphysi-
cists.
Unfortunately
progress
has
been
slow
and
onecan
safely
saythatthephenomenonisstillnotcompletely
under-
stood,at
leastnotin
afundam
entalreductionistsense.Itis
therefore
importantthatthethingsthatcanbeunderstoodare
correctlypresentedin
textbooks.Tothecontrary,textbooks
often
repeattwomythswhichhavebecomeso
ingrained
inthemindsofphysiciststhatthey
ham
per
progress.Theseare:
•Thereisnoclassicaldiamagnetism
(Bohr,1van
Leeuwen
2).
•Thereis
noclassicalexplanationoffluxexpulsionfrom
asuperconductor(M
eissner
3).
Althoughboth
statem
ents
havebeendisproved
multiple
times
inthescientificliterature,thesemythscontinueto
be
spread.Wehopethisreviewwillim
provethesituation.
Meissner
founditnaturalthataweakmagnetic
fieldcould
notpenetrate
aType-Isuperconductor.Thefact
that
this
isalreadyin
conflictwithclassicalphysics,accordingto
the
Bohr-van
Leeuwen
theorem
israrely
mentioned.On
the
other
hand,Meissner
founditremarkablethatanorm
almetal
withamagnetic
fieldinsidewillexpelthisfieldwhen
cooled
tosuperconductivity.This
was
claimed
tohavenoclassical
explanation,and
consequently
superconductors
were
not
considered
perfectconductors.
Wewillnotpresentanynew
resultsin
thispaper.Instead,
wewillreview
someofthestrongevidence
inthearchival
literature
dem
onstratingthat
theabovetwostatem
ents
are
false.
Webegin
withtheBohr-van
Leeuwen
theorem
and
then
turn
totheclassicalexplanationoffluxexpulsion.After
discussingthearguments
forthetraditionalpointofview,
wewilldem
onstratewhythey
arewrong.
II.THEBOHR-V
ANLEEUW
ENTHEOREM
AND
CLASSIC
ALDIA
MAGNETISM
TheHam
iltonian
H,forasystem
ofcharged
particles
interactingviaa(scalar)potentialenergyUis
Hðr
k;p
kÞ¼
X
N j¼1
p2 j
2m
j
þUðr
kÞ:
(1)
The
particles
have
masses
mj,
position
vectors
r j,and
momenta
pj.Thepresence
ofan
external
magnetic
fieldwith
vectorpotentialA(r)alterstheHam
iltonianto
Hðr
k;p
kÞ¼
X
N j¼1
pjÿ
e j cAðr
j�
�
2
2m
j
þUðr
kÞ;
(2)
wheree j
arethecharges
ofmassesmj.Usingthis
equation
onecan
show
that
statisticalmechanicspredicts
that
the
energy—thethermal
averageoftheHam
iltonian—does
not
dependontheexternal
field.Hence,thesystem
exhibitsnei-
ther
aparam
agneticnoradiamagneticresponse.
Inhis
1911doctoraldissertation,Niels
Bohr1
usedthe
aboveequationandconcluded
thereisnomagnetic
response
ofametal
accordingto
classicalphysics.In
1919Hendrika
Johannavan
Leeuwen
independentlycameto
thesamecon-
clusionin
her
Leiden
thesis.When
her
work
was
published
in1921,Missvan
Leeuwen
2notedthat
similar
conclusions
had
beenreached
byBohr.
Manybooksonmagnetism
referto
theaboveresultsas
theBohr-van
Leeuwen
theorem,orsimply
thevan
Leeuwen
theorem.Thetheorem
isoften
summarized
asstatingthereis
noclassicalmagnetism.Since
this
isobviousin
thecase
of
spin
oratomic
angularmomenta—thequantum
phenomena
responsible
for
param
agnetism—the
more
interesting
conclusionisthat
classicalstatisticalmechanicsandelectro-
magnetism
cannotexplain
diamagnetism.A
verythorough
treatm
entoftheBohr-van
Leeuwen
theorem
canbefound
inVan
Vleck.4
Other
books,
such
asMohn,5
Getzlaff,6
Aharoni,7and
Levy,8
mention
thetheorem
more
orless
briefly.Weaknessesin
classicalderivationsofdiamagnetism
inmoderntextbookshavebeenpointedoutbyO’D
elland
Zia.9A
discussionofthetheorem
canalso
befoundin
the
Feynman
lectures1
0(V
ol.2,Sec.34-6),whichpointsoutthat
aconstantexternal
fieldwillnotdoanywork
onasystem
of
charges.Therefore,theenergyofthissystem
cannotdepend
ontheexternalfield.In
addition,thefact
that
thediamag-
netic
response
ofmostmaterialsisverysm
alllendsem
piri-
calsupportto
thetheorem.
A.Theinadequacy
ofthebasicassumptions
IfweacceptthepopularversionoftheBohr-van
Leeuwen
theorem
astrue,
then
onemust
concludethat
theperfect
164
Am.J.Phys.80(2),February2012
http://aapt.org/ajp
VC2012American
AssociationofPhysics
Teachers
164
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art
icle
is c
opyrighte
d a
s indic
ate
d in t
he a
rtic
le.
Reuse
of A
AP
T c
onte
nt
is s
ubje
ct
to t
he t
erm
s a
t: h
ttp:/
/scitation.a
ip.o
rg/t
erm
sconditio
ns.
Dow
nlo
aded t
o I
P:
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diamagnetism
ofsuperconductors
cannothaveaclassicalex-
planation.A
studyoftheproofofthetheorem,however,
revealsthatitisonly
valid
under
assumptionsthatdonotnec-
essarily
hold.The
assumed
Ham
iltonian
ofEq.(2)only
includes
thevectorpotential
oftheexternal
magnetic
field.
Butithas
beenknownsince
1920that
thebestHam
iltonian
forasystem
ofclassicalcharged
particles
istheDarwin
Ham
iltonian.11In
thisHam
iltonianonetakes
into
accountthe
internalmagneticfieldsgenerated
bythemovingcharged
par-
ticles
ofthesystem
itself,plusanycorrespondinginteractions.
Thesimplestway
toseethat
thetotalenergyofasystem
ofcharged
particles
dependsontheexternal
fieldis
tonote
thatthetotalenergyincludes
amagneticenergy12
EB¼
1 8p
ð
B2ðrÞdV¼
1 8p
ð
ðBeþBiÞ2dV;
(3)
whereBeistheexternal
magnetic
fieldandBiistheinternal
field.UsingtheBiot-Savartlaw
this
fieldis
given,to
first
order
inv/c,by
BiðrÞ
¼X
N j¼1
BjðrÞ
¼X
N j¼1
e j c
vj�ðr
ÿr jÞ
jrÿr jj3
:(4)
Equation(3)makes
itobviousthat
tominim
izetheenergy
theinternal
fieldwillbe,as
much
aspossible,equal
inmag-
nitudeandopposite
indirectionto
theexternal
field.This
gives
diamagnetism.
CharlesGaltonDarwin
was
firstto
derivean
approxim
ate
Lagrangian
forasystem
ofcharged
particles(neglecting
radiation)thatiscorrectto
order
(v/c)2.11,13–16In
atimeinde-
pendentexternal
magnetic
field,thereis
then
aconserved
Darwin
energygiven
by
ED¼
X
N j¼1
mj
2v2 jþe j cvj�
1 2Aiðr j;r k;v
kÞþAeðr
jÞ
��
��
þUðr
kÞþEe:
(5)
Inthis
equation,vjarevelocity
vectors,Aeis
theexternal
vectorpotential,Eeis
the(constant)
energyoftheexternal
magnetic
field,andAiis
theinternal
vectorpotential
given
by
Aiðr j;r k;v
kÞ¼
X
N k6¼j
e k r kj
vkþðv
k�e
kjÞe
kj
2c
;(6)
wherer kj¼|rj–r k|andekj¼(rj–r k)/r kj.Althoughthereisa
correspondingHam
iltonian,it
cannotbewritten
inclosed
form
.When
theDarwin
magnetic
interactionsaretaken
into
account,theBohr-van
Leeuwen
theorem
isnolonger
valid
because
themagnetic
fieldsofthemovingcharges
willcon-
tribute
tothetotalmagnetic
energy.Thefact
that
this
inva-
lidates
theBohr-van
Leeuwen
theorem
forsuperconductors
was
stated
explicitlybyPfleiderer
17in
aletter
toNature
in1966.A
more
recentdiscussionofclassicaldiamagnetism
andtheDarwin
Ham
iltonianisgiven
byEssen.18
B.WhataboutLarm
or’stheorem?
Larmor’stheorem
states
that
aspherically
symmetricsys-
tem
ofcharged
particles
willstartto
rotate
ifan
external
magnetic
fieldisturned
on(see
Landau
andLifshitz,13x45).
Therotationofsuch
asystem
producesacirculatingcurrent
andthusamagnetic
field.Sim
ple
calculationsshow
that
this
fieldis
ofopposite
directionto
theexternal
fieldandso
the
system
isdiamagnetic.Thisidea
was
firstusedbyLangevin
toderivediamagnetism.Butfrom
thepointofview
ofthe
Bohr-van
Leeuwen
theorem
this
seem
sstrange.
InFeyn-
man’s
lectures1
0theBohr-vanLeeuwen
theorem
istherefore
notconsidered
tobevalid
forsystem
sthat
canrotate.As
notedabove,
theproblem
issimply
that
theBohr-van
Leeu-
wen
theorem
does
nottakeinto
accountthemagnetic
field
producedbytheparticles
ofthesystem
itself.When
thisin-
ternal
fieldisaccountedfor,diamagnetism
followsnaturally,
whether
forsystem
sofatomsfrom
theLarmor’stheorem
19
orforsuperconductorsfrom
theDarwin
form
alism.20
C.TheShanghaiexperim
ent—
measuringadiamagnetic
current?
Ifitis
truethat
theclassicalHam
iltonianforasystem
of
charged
particles
predictsdiamagnetism,then
whyisthephe-
nomenonso
weakandinsignificantin
most
cases?
Isthere
anyevidence
forclassicaldiamagnetism
forsystem
sother
than
superconductors?Asstressed
byMahajan,21plasm
asare
typically
diamagnetic.However,plasm
asarenotusually
inthermalequilibrium
soitisdifficultto
reachanydefinitecon-
clusionsfrom
them
.Anexperim
entonan
electrongas
inther-
malequilibrium,perform
edbyXinyongFuandZitao
Fu22in
Shanghai,istherefore
ofconsiderableinterest.
Twoelectrodes
ofAg-O
-Csside-by-sidein
avacuum
tube
emit
electronsat
room
temperature
because
oftheirlow
work
function.Ifamagnetic
fieldisim
posedonthissystem
,an
asymmetry
arises
andelectronsflow
from
oneelectrode
totheother.Forafieldstrength
ofabout4gauss
asteady
currentof�10ÿ14A
ismeasuredat
room
temperature.The
currentgrowswith
increasing
field
strength
and
changes
directionas
thepolarity
isreversed.Theauthors
interpret
thisresultas
ifthemagnetic
fieldactsas
aMaxwelldem
on
that
can
violate
the
second
law
of
thermodynam
ics.22
Inview
ofstatisticalmechanicsbased
ontheDarwin
Ham
il-
tonian,18itis
more
naturalto
interpretthis
resultas
adia-
magnetic
response
ofthesystem
.Thecurrent—
just
asthe
super-currentoftheMeissner
effect—isdueto
adiamagnetic
thermal
equilibrium.Thismeansthat
nousefulwork
canbe
extractedfrom
thesystem
.
III.
ONTHEALLEGEDIN
CONSISTENCYOF
MAGNETIC
FLUXEXPULSIO
NWIT
HCLASSIC
ALPHYSIC
S
Meissner
and
Ochsenfeld
3,23
discovered
the
Meissner
effect
in1933.Totheirsurprise
amagnetic
fieldwas
not
only
unable
topenetrate
asuperconductor,
itwas
also
expelledfrom
theinteriorofaconductoras
itwas
cooled
below
itscritical
temperature.Thefirsteffect—
idealdia-
magnetism—seem
ednaturalto
them
even
thoughitviolates
theBohr-van
Leeuwen
theorem.Asalreadydiscussed,by
violatingthistheorem
idealdiamagnetism
should
havebeen
consideredanon-classical
effect.Thefluxexpulsion,onthe
other
hand,was
explicitlyproclaimed
byMeissner
tohave
noclassicalexplanation.Meissner
does
notgiveanyargu-
mentsorreferencesto
supportthisstatem
ent,butaccording
toDahl24thetheoreticalbasis
was
Lippmann’s
theorem
on
theconservationofmagnetic
fluxthroughan
ideallycon-
ductingcurrentloop.Later,Forrest23andothershaveargued
165
Am.J.Phys.,Vol.80,No.2,February2012
H.Essen
andM.C.N.Fiolhais
165
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that
themagnetohydrodynam
ictheorem
on
frozen-in
flux
lines
also
supportsthis
notion.In
this
section,wediscuss
thesearguments
andpointoutthat
they
donotrule
outa
classicalexplanationoftheMeissner
effect.
A.Lippmann’stheorem
GabrielLippmann(1845–1921),winner
ofthe1908Nobel
Prize
inPhysics,published
atheorem
in1889statingthatthe
magnetic
fluxthroughan
ideallyconductingcurrentloopis
conserved.In
1919,when
superconductivityhad
beendis-
covered,Lippmann25againpublished
thisresultin
threedif-
ferentFrench
journals(see
Sauer
26).
Althoughtheidea
of
fluxconservationisconsidered
highly
fundam
ental,27nowa-
daysreferencesto
Lippman’s
theorem
arehardto
find.But
atthetimeLippman’s
theorem
was
quiteinfluential,and
Dahl24explainshowthiswas
oneoftheresultsthatmadethe
Meissner
effect
seem
surprising—andnon-classical—at
the
timeofitsdiscovery.
TheproofofLippmann’s
theorem
followsbynotingthat
theselfinductance
Lofaclosedcircuit,orloop,isrelatedto
themagneticfluxUfrom
thecurrentin
theloopvia
U¼
cL_ q;
(7)
wherecisthespeedoflightand_ qisthecurrentthroughthe
circuit,an
overdotdenotingatimederivative(see
Landau
andLifshitz,28Vol.8,x33).Theequationofmotionforasin-
gleloopelectriccircuitis
L€ qþR_ qþCÿ1q¼
EðtÞ;
(8)
whereRis
theresistance,C
isthecapacitance,andEðtÞis
theem
fdrivingthecurrent.If
thereis
noresistance,noca-
pacitance,andnoem
f,thisequationbecomes
L€ q¼
0:
(9)
Therefore,iftheselfinductance
Lis
constant,Eqs.(7)and
(9)tellusthatU¼constant.
B.Lippmann’stheorem
andsuperconductors
AlthoughLippmann’stheorem
iscorrect,itsrelevance
for
thepreventionoffluxexpulsionis
notclear.Forsupercon-
ductors
therearetwopoints
toconsider,theassumptionof
zero
emf,andthefact
that
constantfluxdoes
notim
ply
con-
stantmagneticfield.Weconsider
thesepointsoneatatime.
Consider
asuperconductingsphereofradiusrin
acon-
stantexternal
magnetic
fieldBe.When
thesphereexpelsthis
fieldbygenerating(surface)
currentsthat
produce
Bi¼ÿBe
initsinterior,thetotalmagneticenergyisreduced.Themag-
neticenergychangeis
DEB¼
ÿ3
4pr3 3
��
B2 e
8p;
(10)
orthreetimes
theinitialinteriormagnetic
energy.29This
energyis
thusavailable
forproducingtheem
frequired
togeneratecurrentsin
thesphere’sinterior.Theassumptionof
Lippmann’s
theorem—that
theem
fiszero—istherefore
not
fulfilled.
When
asteadycurrentflowsthroughafixed
metalwireboth
thefluxandthemagneticfielddistributionareconstant.Onthe
other
hand,when
currentflowsin
aloop
inaconducting
medium,aconstantfluxthroughtheloopdoes
notim
ply
acon-
stantmagneticfieldbecause
theloopcanchangein
size,shape,
orlocation.Norm
ally
currentloopsaresubject
toforces
that
increase
theirradius(see
e.g.,Landau
andLifshitz,28Vol.8,
x34,Problem
4,orEssen,30
Sec.4.1).
Inview
ofthis,
Lippmann’s
theorem
does
notautomatically
imply
that
the
magneticfieldmustbeconstant,even
iftheem
fiszero.
Other
authors
havereached
similar
conclusions.Mei
and
Liang31carefullyconsidered
theelectromagneticsofsuper-
conductors
in1991andwrite,“T
husMeissner’s
experim
ent
should
beviewed
throughitstimehistory
insteadofas
astrictly
dcevent.In
thatcase
classicalelectromagnetictheory
willbeconsistentwiththeMeissner
effect.”
C.Frozen-infieldlines
Thesimplest
derivationofthefrozen-infieldresultfora
conductingmedium
beginswithOhm’s
law
j¼rE.If
r!
1wemusthaveE¼0to
preventinfinitecurrent.Faraday’s
law
then
gives
@B=@t¼
ÿcr
�E
ðÞ¼
0which
tells
us
the
magnetic
field
Bis
constant(Forrest,23Alfven
and
Faltham
mar
32).
When
dealingwithaconductingfluid,theequationsof
magnetohydrodynam
icsandthelimitofinfiniteconductivity
inOhm’slawgiverise
to32
@B @t¼
r�ðv
�BÞ:
(11)
Thisresulttellsusthemagnetic
fieldconvectswiththefluid
butdoes
notdissipate.
Inaddition,Eq.(11)canbeusedto
derivethefollowingtwostatem
entsthatareoften
referred
toas
thefrozen-infieldtheorem:33
1.themagnetic
fluxthroughanyclosedcurvemovingwith
thefluid
isconstant,and
2.amagnetic
fieldlinemovingwiththefluid
remainsa
magneticfieldlineforalltime.
Thefirstoftheseisajustarestatem
entofLippmann’sthe-
orem
foracurrentloopin
afluid.
D.Magneticfieldlines
insuperconductors
Asshownin
theprevioussection,Ohm’s
law
withinfinite
conductivityim
plies
therecanbenochangein
themagnetic
fieldsince
thiswould
giverise
toan
infinitecurrentdensity.
Itis,however,notphysicallymeaningfulto
takethelimit
r!
1in
Ohm’s
law.In
amedium
ofzero
resistivityone
must
instead
use
the
equation
ofmotion
forthe
charge
carriers.Onecanthen
derive2
9,34
dj
dt¼
e2n
mEþ
e mcj�B;
(12)
forthetimerate
ofchangeofthecurrentdensity.Thetime
derivativehereis
aconvective,
ormaterial,timederivative.
Ohm’s
law
simply
does
notapply,andtherefore
itcannot
imply
thatthemagneticfielddoes
notchange.
Thesecondstatem
entofthefrozen-infieldtheorem
can
bequestioned
onthesamegrounds.Justas
withLippmann’s
theorem,theconclusionthat
themagnetic
fieldremainscon-
stantdoes
notfollow,even
ifthestatem
entisassumed
valid.
Themagnetic
fieldisdetermined
bythedensity
ofmagnetic
fieldlines
andconstancy
ofthisdensity
requires
thatthecon-
ductingfluid
isincompressible.Onecaneasily
imaginethat
166
Am.J.Phys.,Vol.80,No.2,February2012
H.Essen
andM.C.N.Fiolhais
166
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thefluid
ofsuperconductingelectronsis
compressible,and
that
magnetic
pressure
33pushes
thefluid
(withitsmagnetic
fieldlines)to
thesurfaceofthematerial.Corroboratingthis
pointofview,AlfvenandFaltham
mar
32(Sec.5.4.2)state
that“inlowdensity
plasm
astheconceptoffrozen-inlines
of
forceisquestionable.”
E.AclassicalderivationoftheLondonequations
In1981Edwards3
5published
amanuscriptwiththeabove
titlein
PhysicalReview
Letters.This
causedan
uproar
of
indignationandthejournal
laterpublished
threedifferent
criticisms
of
Edwards’
work.36–38
Incidentally,
all
of
Edwards’
criticsrestated
(orim
plied)thetextbookmyth
that
theMeissner
fluxexpulsiondoes
nothaveaclassicalexpla-
nation.In
addition,thejournal
Nature
published
astudyby
Taylor39pointingoutthefaultsofEdwards’
derivation.Of
course,
Edwardsisnottheonly
oneto
publish
anerroneous
derivationoftheLondonequations.A
much
earlierexam
ple
isthederivationbyMoore
40from
1976.Thefactthatvarious
derivationshavebeenwrong,ofcourse,does
notproveany-
thing—as
longas
acorrectderivationexists.
In1966Nature
published
aclassicalexplanationofthe
LondonequationsbyPfleiderer
17whichdid
notcause
any
comment.Indeedithas
notbeencitedasingle
time,
prob-
ably
because
itisverybrief
andcryptic.However,aclassical
derivationoftheLondonequationsandfluxexpulsioncan
befoundin
aclassictextbookbytheFrench
Nobel
laureate
PierreGillesdeGennes.41Because
thisderivationshould
be
more
recognized,werepeatithere.
F.deGennes’derivationoffluxexpulsion
Thetotalenergyoftherelevantelectronsin
thesupercon-
ductorisassumed
tohavethreecontributions:thecondensa-
tion
energy
associated
with
the
phase
transition
ES,the
energyofthemagneticfieldEB,andthekineticenergyofthe
movingsuperconductingelectronsEk.Thetotalenergyrele-
vantto
theproblem
isthustaken
tobe
E¼
ESþEBþEk:
(13)
Thecondensation
energy
isthen
assumed
tobeconstant
whiletheremainingtwocanvaryin
response
toexternal
fieldvariations.Thesuper-currentdensity
iswritten
as
jðrÞ
¼nðrÞevðrÞ;
(14)
wherenis
thenumber
density
ofsuperconductingelectrons
andvistheirvelocity,whichgives
akineticenergyof
Ek¼
ð
1 2nðrÞmv2ðrÞdV¼
ð
1 2
m
e2nðrÞj2ðrÞdV:
(15)
BymeansoftheMaxwellequationr
�B¼
4pj=candEq.
(3)forEB,thetotalenergy(13)becomes
E¼
ESþ
1 8p
ð
B2þk2ðr
�BÞ2
hi
dV;
(16)
wherewehaveassumed
that
nis
constantin
theregion
wherethereis
current,andtheLondonpenetrationdepth
isgiven
by
k¼
ffiffiffiffiffiffiffiffiffiffiffiffi
mc2
4pe2n
r
:(17)
Minim
izingtheenergyin
Eq.(16)withrespectto
Bthen
gives
theLondonequation
Bþk2r
�ðr
�BÞ¼
0;
(18)
inoneofitsequivalentform
s.Notice
thatthisderivationuti-
lizesnoquantum
concepts
anddoes
notcontain
Planck’s
constant.Itisthuscompletely
classical.A
similar
derivation
has
beenpublished
more
recentlybyBadıa-M
ajos.34
TheconclusionofdeGennes
isclearlystated
inhis1965
book41(emphasis
from
theoriginal):
“Thesuperconductor
findsanequilibrium
state
wherethesum
ofthekinetic
and
magnetic
energiesis
minimum,andthis
state,formacro-
scopic
samples,
correspondsto
theexpulsionofmagnetic
flux.”In
spiteofthis,most
textbookscontinueto
statethat
“fluxexpulsionhas
noclassicalexplanation”as
originally
stated
byMeissner
andOchsenfeld
3andrepeatedin
thein-
fluential
monographsbyLondon42andNobel
laureateMax
von
Laue.43
As
one
textbook
exam
ple,
Ashcroft
and
Mermin
44explain
that
“perfect
conductivityim
plies
atime-
independentmagneticfieldin
theinterior.”
G.Apurely
classicalderivationfrom
magnetostatics
Onemightobject
that
theelectronic
chargeeis
amicro-
scopic
constant,andthat
theLondonpenetrationdepth
kin
mostcasesisso
smallthat
itseem
sto
belongto
thedomain
ofmicrophysics.Soeven
ifquantum
concepts
donotenter
explicitly,theabovederivationdoes
havemicroscopic
ele-
ments.Itis
thusofinterest
that
from
apurely
macroscopic
pointofview
wecan
identify
thekinetic
energy
ofthe
conductionelectronssolely
withmagnetic
energy.45,46The
energythatshould
beminim
ized
would
then
beEq.(3)
EB¼
1 8p
ð
B2dV:
(19)
Unfortunately,this
willnotprovideanyinform
ationabout
thecurrents
that
arethesources
ofB.However,if
wecan
neglect
thecontributionfrom
fieldsat
asufficientlydistant
surface—
i.e.,when
radiationis
negligible—itis
possible
torewritethisexpressionas
E0 B¼
1 2c
ð
j�A
dV;
(20)
whereB¼
r�A.Theidea
isto
then
apply
thevariational
principleto
EjA¼
2E0 BÿEB,i.e.,theexpression
EjA¼
ð
1 cj�A
ÿ1 8p
r�A
ðÞ2
��
dV:
(21)
Heretheenergyis
written
interm
softhefieldA
andits
sourcej.TherearemanywaysofcombiningEqs.(19)and
(20)to
getan
energyexpression,butitturnsoutthatEq.(21)
istheonethatgives
thesimplestresults.
Startingfrom
Eq.(21)andaddingtheconstraintr
�j¼
0,
theintegration
issplitinto
interior,
surface,
and
exterior
regions.
Theresultis
atheorem
ofclassicalmagnetostatics
that
states,forasystem
ofperfect
conductors
themagnetic
fieldis
zero
intheinteriors
andallcurrentflowsontheir
surfaces(Fiolhaiset
al.29)This
theorem
isanalogousto
asimilar
resultin
electrostaticsforelectric
fieldsandcharge
densities
inconductors,sometim
esreferred
toas
Thomson’s
167
Am.J.Phys.,Vol.80,No.2,February2012
H.Essen
andM.C.N.Fiolhais
167
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theorem
(see
Stratton,47Sec.2.11).Readers
should
consult
thepaper
byFiolhaisetal.29fordetails.
Manyother
authors
havealso
reached
theconclusionthat
theequilibrium
stateofasuperconductorisin
fact
thestate
ofminim
um
magneticenergyofan
idealconductor.Someof
theseare,
inchronological
order,Cullwick,48Pfleiderer,17
Karlsson,49Badıa-M
ajos,34Kudinov,50andMahajan.21For
exam
ple,Cullwickwrites“T
hewell-knownMeissnereffect
inpure
superconductors
isshownto
bean
expectedrather
than
anunexpected
phenomenon…”
while
Kudinov50
explainsthat
it“isworthnotingin
this
connectionthat
the
expulsionofamagnetic
fieldto
theperiphery(theMeissner
effect)also
occurs
inaclassicalcollisionless
gas
ofcharged
particlesandthatthishappenssolely
because
thisstateisen-
ergetically
favorable.”
IV.CONCLUSIO
NS
Thereader
may
gettheim
pressionfrom
ourinvestigations
abovethat
weconsider
superconductivityto
beaclassical
phenomenon.Nothingcould
befurther
from
thetruth.As
implied
bytheGinzburg-Landautheory,51theBCStheory,52
andtheJosephsoneffect,53thephenomenonisquantum
me-
chanical
toalargeextent.Since
quantum
physics
mustlead
toclassicalphysics
insomemacroscopic
limit,itmust
be
possible
toderiveourclassicalresultfrom
aquantum
per-
spective.
Evans
and
Rickayzen54
did
indeed
derive
the
equivalence
ofzero
resistivityandtheMeissner
effect
quan-
tum
mechanically,butdid
notdiscuss
theclassicallimit.
What
wewantto
correctisthemis-statementthat
theMeiss-
ner
effect
proves
that
superconductors
are“notjust
perfect
conductors.”
Accordingto
basic
physics
andalargenumber
ofindependentinvestigators,thespecificphenomenon
of
fluxexpulsionfollowsnaturallyfrom
classicalphysics
and
thezero
resistance
property
ofthesuperconductor—
they
are
justperfectconductors.
Inconclusion,wehavecarefullyexam
ined
theevidence
fortheoftrepeatedstatem
ents
intextbooksthat
(1)thereis
noclassicaldiamagnetism
and(2)thereisnoclassicalexpla-
nationofmagnetic
fluxexpulsion.Wehavefoundthat
the
theoreticalarguments
forthesestatem
ents
arenotrigorous
andthat,forsuperconductors,theseparticularphenomena
arein
goodagreem
entwiththeclassicalphysics
ofideal
conductors.
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ct
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