1. Introduction

Intermediate state (IS) is defined as a thermodynamically equilibrium state in which a type-I superconductor is split for domains of superconducting (S) and normal (N) phases [1–3]. For completeness of description, we begin with a brief overview of properties of the Meissner state, which will be necessary for discussion of the IS properties.

## 1.1. Meissner state in cylindrical specimens

Consider a specimen of a type-I superconductor at temperature T < Tc in a free space (vacuum) subjected to a uniform magnetic field H < Hcrð Þ T , where Tc is critical temperature at zero field and Hcrð Þ T is critical field of the S/N transition at given T. (We use notation Hcr instead of commonly used Hc because the latter is reserved for thermodynamic critical field, which can be different from Hcr). Assume that the specimen is a long cylinder with a circular base of radius R ≫ λ (λ is the penetration depth) and H is parallel to the cylinder as shown in Figure 1a. A demagnetizing factor η [2, 4] of such a specimen is zero, which means that outside it B ¼ Hi ¼ H (we use CGS units) all the way down to the sample surface.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Hi � B � 4πm, (2)

Intermediate State in Type-I Superconductors http://dx.doi.org/10.5772/intechopen.75742 91

dF~<sup>M</sup> ¼ �SVdT � <sup>M</sup> � <sup>d</sup>H, (3)

� �

� �

M � dH, (4)

<sup>n</sup> is the total free energy of

� �

<sup>s</sup>0) is the specimen magnetic energy

<sup>n</sup> nor the condensation

<sup>n</sup> ¼ Fn<sup>0</sup> and

where m is magnetization, which in superconductors is a macroscopic average of the magnetic moment per unit volume m ¼ M=V and, as it was mentioned above, Hi ¼ H due to geometry

Thermodynamics of our and any other singly connected superconductor can be described

where S is entropy per unit volume and a small variation of V due to changing magnetic field

Integrating Eq. (3) at constant temperature, we arrive at another well-known and very important formula for the total free energy of the singly connected superconductors in magnetic field

<sup>M</sup> � <sup>d</sup><sup>H</sup> <sup>¼</sup> <sup>F</sup>~<sup>M</sup>

field. Note that, since M in the N state is zero (because μ of the N phase is unity), the total free

<sup>s</sup><sup>0</sup> ¼ Fs0, where Fn<sup>0</sup> and Fs<sup>0</sup> are Helmholtz free energies F Tð Þ ; V; B in the normal and

<sup>c</sup>=8<sup>π</sup> � �<sup>V</sup> depends on the specimen shape, and therefore Eq. (4) is valid for singly

<sup>0</sup> M � dH, representing energy of interaction of the external field H with the specimen

� �

connected specimens of any shape. Secondly, Eq. (4) explicitly shows that the extra total free

magnetic moment M induced by this field. More specifically, EM is kinetic energy of electrons (Cooper pares) carrying the induced currents [1]. And thirdly, Eq. (4) shows that the source of EM is condensation energy. Finiteness of the later makes transition to the N state a mandatory property of any superconductor [2]. At the S/N transition, the magnetic energy of any specimen equals to its condensation energy, or area under M Hð Þ curve plotted as 4πM=VHc vs.

This as-called rule of 1/2 (or in general case Eq. (4)), represents the energy balance or the first law of thermodynamics for singly connected superconductors at constant temperature; compliance with this rule/equation is a necessary condition for discussion of equilibrium proper-

<sup>s</sup><sup>0</sup> is the total free energy of the S state in zero field, <sup>F</sup>~<sup>M</sup>

energy for this state does not depend on the field. This means that F~<sup>M</sup>

Importance of Eq. (4) is associated, firstly, with the fact that neither F~<sup>M</sup>

� �

<sup>n</sup> � <sup>H</sup><sup>2</sup> c 8π V � ð<sup>H</sup> 0

<sup>c</sup> <sup>=</sup>8<sup>π</sup> � �<sup>V</sup> is the condensation energy, where Hc is thermodynamic critical

using total free energy <sup>F</sup>~Mð Þ <sup>T</sup>; <sup>V</sup>; <sup>H</sup> , which differential dF~Mð Þ <sup>T</sup>; <sup>V</sup>; <sup>H</sup> is [2].

of our specimen.

is neglected.

F~M � �

sH <sup>¼</sup> <sup>F</sup>~<sup>M</sup> � � <sup>s</sup><sup>0</sup> � ð<sup>H</sup> 0

superconducting states at zero field, respectively.

H=Hc, when M is aligned to H, is 1/2.

ties of superconductors [1].

energy (above the free energy of the ground state F~<sup>M</sup>

[1, 2]:

where F~<sup>M</sup> � �

F~M � �

energy H<sup>2</sup>

EM ¼ � <sup>Ð</sup> <sup>H</sup>

the N state, and H<sup>2</sup>

Figure 1. Cross-sectional view of specimens (shown in gray) with demagnetizing factor η ¼ 0 (a), η ¼ 1=2 (b), and η ¼ 1 (c) in a weak magnetic field H. In (a) and (b) the specimen (a cylinder) is in the Meissner state; in (c) the specimen (an infinite slab) is in the intermediate state starting from any H exceeding zero.

Here B is magnetic induction or magnetic flux density [5] or merely B-field [6]. B is an average microscopic magnetic field available for measurements [2]. Hi is magnetic field strength, also referred as magnetic and magnetizing force [4], Maxwell field [7], thermodynamic field [8], magnetic field [5], H-field [6], and others. And H is applied field set by a magnet power supply (for simplicity we will ignore a small contribution of Earth magnetism); it is the field away from the specimen or the field which would be in a space occupied by the specimen if the latter is absent. Away from the specimen, Hi is identical to H, but it can be not so near and inside the specimen. Everywhere outside the specimen, B ¼ Hi because magnetic permeability of the free space, as well as permeability of the N phase in superconductors, μ � B=Hi is unity by definition.

Our cylindrical specimen is in the Meissner state, implying that inside it B ¼ 0 and Hi ¼ H due to continuity of the tangential component of this field [9]. A jump of induction at the specimen surface ΔB ¼ H means that there is a surface current I, in which linear density g � I=l ¼ ΔBc=4π ¼ cH=4π, where l is length of the specimen and c is speed of light. This surface current is regarded as a screening current protecting the specimen interior from the external field. Taking into account direction of g (¼ n � Hc=4π, where n is the unit vector normal to the surface and directed outward), we arrive at a well-familiar formula for the specimen magnetic moment M:

$$M = -gl\frac{A}{c} = \left(\frac{cH}{4\pi}l\right)\frac{A}{c} = -\frac{H}{4\pi}V,\tag{1}$$

where A and V are the base area and volume of the specimen, respectively.

The same result follows from definition of the field strength:

$$H\_i \equiv B - 4\pi m\_i \tag{2}$$

where m is magnetization, which in superconductors is a macroscopic average of the magnetic moment per unit volume m ¼ M=V and, as it was mentioned above, Hi ¼ H due to geometry of our specimen.

Thermodynamics of our and any other singly connected superconductor can be described using total free energy <sup>F</sup>~Mð Þ <sup>T</sup>; <sup>V</sup>; <sup>H</sup> , which differential dF~Mð Þ <sup>T</sup>; <sup>V</sup>; <sup>H</sup> is [2].

$$d\ddot{F}\_M = -SVdT - \mathbf{M} \cdot d\mathbf{H},\tag{3}$$

where S is entropy per unit volume and a small variation of V due to changing magnetic field is neglected.

Integrating Eq. (3) at constant temperature, we arrive at another well-known and very important formula for the total free energy of the singly connected superconductors in magnetic field [1, 2]:

$$\left(\tilde{\boldsymbol{F}}\_{M}\right)\_{sH} = \left(\tilde{\boldsymbol{F}}\_{M}\right)\_{s0} - \int\_{0}^{H} \mathbf{M} \cdot d\mathbf{H} = \left(\tilde{\boldsymbol{F}}\_{M}\right)\_{n} - \frac{H\_{c}^{2}}{8\pi}V - \int\_{0}^{H} \mathbf{M} \cdot d\mathbf{H},\tag{4}$$

where F~<sup>M</sup> � � <sup>s</sup><sup>0</sup> is the total free energy of the S state in zero field, <sup>F</sup>~<sup>M</sup> � � <sup>n</sup> is the total free energy of the N state, and H<sup>2</sup> <sup>c</sup> <sup>=</sup>8<sup>π</sup> � �<sup>V</sup> is the condensation energy, where Hc is thermodynamic critical field. Note that, since M in the N state is zero (because μ of the N phase is unity), the total free energy for this state does not depend on the field. This means that F~<sup>M</sup> � � <sup>n</sup> ¼ Fn<sup>0</sup> and F~M � � <sup>s</sup><sup>0</sup> ¼ Fs0, where Fn<sup>0</sup> and Fs<sup>0</sup> are Helmholtz free energies F Tð Þ ; V; B in the normal and superconducting states at zero field, respectively.

Here B is magnetic induction or magnetic flux density [5] or merely B-field [6]. B is an average microscopic magnetic field available for measurements [2]. Hi is magnetic field strength, also referred as magnetic and magnetizing force [4], Maxwell field [7], thermodynamic field [8], magnetic field [5], H-field [6], and others. And H is applied field set by a magnet power supply (for simplicity we will ignore a small contribution of Earth magnetism); it is the field away from the specimen or the field which would be in a space occupied by the specimen if the latter is absent. Away from the specimen, Hi is identical to H, but it can be not so near and inside the specimen. Everywhere outside the specimen, B ¼ Hi because magnetic permeability of the free space, as well as permeability of the N phase in superconductors, μ � B=Hi is unity by

Figure 1. Cross-sectional view of specimens (shown in gray) with demagnetizing factor η ¼ 0 (a), η ¼ 1=2 (b), and η ¼ 1 (c) in a weak magnetic field H. In (a) and (b) the specimen (a cylinder) is in the Meissner state; in (c) the specimen (an

Our cylindrical specimen is in the Meissner state, implying that inside it B ¼ 0 and Hi ¼ H due to continuity of the tangential component of this field [9]. A jump of induction at the specimen surface ΔB ¼ H means that there is a surface current I, in which linear density g � I=l ¼ ΔBc=4π ¼ cH=4π, where l is length of the specimen and c is speed of light. This surface current is regarded as a screening current protecting the specimen interior from the external field. Taking into account direction of g (¼ n � Hc=4π, where n is the unit vector normal to the surface and directed outward), we arrive at a well-familiar formula for the specimen magnetic

> <sup>c</sup> <sup>¼</sup> cH 4π l A

<sup>c</sup> ¼ � <sup>H</sup> 4π

V, (1)

<sup>M</sup> ¼ �gl <sup>A</sup>

The same result follows from definition of the field strength:

infinite slab) is in the intermediate state starting from any H exceeding zero.

where A and V are the base area and volume of the specimen, respectively.

definition.

90 Superfluids and Superconductors

moment M:

Importance of Eq. (4) is associated, firstly, with the fact that neither F~<sup>M</sup> � � <sup>n</sup> nor the condensation energy H<sup>2</sup> <sup>c</sup>=8<sup>π</sup> � �<sup>V</sup> depends on the specimen shape, and therefore Eq. (4) is valid for singly connected specimens of any shape. Secondly, Eq. (4) explicitly shows that the extra total free energy (above the free energy of the ground state F~<sup>M</sup> � � <sup>s</sup>0) is the specimen magnetic energy EM ¼ � <sup>Ð</sup> <sup>H</sup> <sup>0</sup> M � dH, representing energy of interaction of the external field H with the specimen magnetic moment M induced by this field. More specifically, EM is kinetic energy of electrons (Cooper pares) carrying the induced currents [1]. And thirdly, Eq. (4) shows that the source of EM is condensation energy. Finiteness of the later makes transition to the N state a mandatory property of any superconductor [2]. At the S/N transition, the magnetic energy of any specimen equals to its condensation energy, or area under M Hð Þ curve plotted as 4πM=VHc vs. H=Hc, when M is aligned to H, is 1/2.

This as-called rule of 1/2 (or in general case Eq. (4)), represents the energy balance or the first law of thermodynamics for singly connected superconductors at constant temperature; compliance with this rule/equation is a necessary condition for discussion of equilibrium properties of superconductors [1].

Using Eq. (1) for M of our cylindrical sample, we rewrite Eq. (4) as

$$
\tilde{F}\_M(H) = F\_n - \frac{H\_c^2}{8\pi} V + \frac{H^2}{8\pi} V. \tag{5}
$$

of Hi [2, 4]. Indeed, our cylindrical specimen in perpendicular field in the Meissner state represents a uniformly magnetized (B ¼ const ¼ 0) prolate ellipsoid with η = 1/2 [2, 4]. Inside of any uniform ellipsoid, Hi is also uniform, and when H is parallel to an axis of ellipsoid with respect to which the demagnetizing factor is η, the fields Hi, B, and H are connected with each

Hence, the field Hi inside our specimen in the Meissner state is H=ð Þ 1 � η , and therefore Hi on

Therefore near the "poles" of our specimen the field is zero, whereas near "equator" it is twice as big as the applied field. This implies that the external field near "equator" reaches the critical value Hc at H ¼ Hcð Þ¼ 1 � η Hc=2. When H is increased beyond this value, the field must enter the specimen destroying superconductivity. However, contrarily to the previous (parallel) case, superconductivity cannot be destroyed completely because there is still plenty

Indeed, the specimen magnetic moment M � ð Þ B � Hi V=4π ¼ �HV=4πð Þ¼� 1 � η HV=2π;

At the first sight, one might expect that at H > Hcð Þ 1 � η , the field will gradually enter the specimen, thus destroying superconductivity over the field range from Hcð Þ 1 � η to Hc. The superconducting cylinder in such case would stay resistanceless with gradually changing volume of the S core as shown in Figure 3. However, this scenario is problematic because as soon as the field enters the specimen, the density of the field lines near the "equator" decreases and hence the field inside the convex blue region in Figure 3 becomes smaller than Hc. Then this region should go back to the S state.<sup>2</sup> This means that when <sup>H</sup> <sup>&</sup>gt; Hcð Þ <sup>1</sup> � <sup>η</sup> , the ellipsoidal specimen splits into S and N regions, as it was suggested for the first time by Gorter and

Historically impossibility of configuration like that shown in Figure 3 was explained basing on a paradigm of instability of the N phase against transforming to the S phase at Hi < Hc (see, e.g. [8]). However, this (the N phase at Hi < H) does take place in specimens in the IS, but only at Hi in the upper part of the IS field range. At the lower edge of this range (at

� �

<sup>M</sup> � <sup>d</sup><sup>H</sup> <sup>¼</sup> VH<sup>2</sup>

c 16π < VH<sup>2</sup> c

the external side of the specimen surface (the external field) is

where θ is the angle between the normal and the applied field H.

EM ¼ � <sup>ð</sup><sup>0</sup>:5Hc 0

Derivation of Eq. (6) can be found in [2]; Maxwell using it in [4] refers to Poisson.

H ¼ ð Þ 1 � η Hc ) B in the first N domain and therefore Hi throughout the specimen is always Hc .

of condensation energy left in the specimen.

therefore magnetic energy EM at H ¼ Hc=2 is

Hence, as seen from Eq. (4), F~<sup>M</sup> < F~<sup>M</sup>

superconducting.

Casimir [10].

1

2

ð Þ 1 � η Hi þ ηB ¼ H: (6)

Intermediate State in Type-I Superconductors http://dx.doi.org/10.5772/intechopen.75742 93

Hi ¼ Hsinθ=ð Þ 1 � η , (7)

<sup>8</sup><sup>π</sup> : (8)

<sup>n</sup>, and therefore the specimen must remain

other as.1

Now a question arises; up to what fields Eq. (1) is valid? Vast majority of superconductors are of type II, for which Eq. (1) holds up to a low critical field Hc<sup>1</sup> < Hcr and Hcr ¼ Hc2, which is an upper critical field. However there is a relatively small group of mostly pure elementary materials, for which Eq. (1) (or the Meissner condition B ¼ 0) holds in the entire field range of the superconducting state, i.e., up to Hcr. Those are type-I superconductors. An example of M Hð Þ dependences for a typical type-I superconductor with η ¼ 0 is shown in Figure 2.

S/N transition takes place when free energies of the S and N states are equal, i.e., <sup>F</sup>~Mð Þ¼ Hcr F~M <sup>n</sup>. For our type-I cylindrical sample, as seen from Eq. (5), this implies that Hcr ¼ Hc and therefore the S/N transition in specimens with η ¼ 0 must be discontinuous, i.e., thermodynamic phase transition of the first order, in full agreement with experimental results, e.g., with those shown in Figure 2.

#### 1.2. Intermediate state

Now, we turn our cylinder perpendicular to the applied field. In a weak field, the specimen is in the Meissner state (inside it B ¼ 0), but the pattern of field lines looks now as shown in Figure 1b. The external field near the specimen is tangential to its surface, which follows from always valid conditions of continuity of the normal component of B and of tangential component

Figure 2. Experimental data for magnetic moment of a pure indium film 2.79 μ m thick measured in parallel applied field H at indicated temperatures. Errors up and down indicate that the measurements were conducted at increasing and decreasing field, respectively.

of Hi [2, 4]. Indeed, our cylindrical specimen in perpendicular field in the Meissner state represents a uniformly magnetized (B ¼ const ¼ 0) prolate ellipsoid with η = 1/2 [2, 4]. Inside of any uniform ellipsoid, Hi is also uniform, and when H is parallel to an axis of ellipsoid with respect to which the demagnetizing factor is η, the fields Hi, B, and H are connected with each other as.1

$$(1 - \eta)H\_i + \eta B = H.\tag{6}$$

Hence, the field Hi inside our specimen in the Meissner state is H=ð Þ 1 � η , and therefore Hi on the external side of the specimen surface (the external field) is

$$H\_i = H \sin \theta / (1 - \eta),\tag{7}$$

where θ is the angle between the normal and the applied field H.

Using Eq. (1) for M of our cylindrical sample, we rewrite Eq. (4) as

F~M 

those shown in Figure 2.

92 Superfluids and Superconductors

1.2. Intermediate state

decreasing field, respectively.

<sup>F</sup>~Mð Þ¼ <sup>H</sup> Fn � <sup>H</sup><sup>2</sup>

c 8π V þ H2 8π

Now a question arises; up to what fields Eq. (1) is valid? Vast majority of superconductors are of type II, for which Eq. (1) holds up to a low critical field Hc<sup>1</sup> < Hcr and Hcr ¼ Hc2, which is an upper critical field. However there is a relatively small group of mostly pure elementary materials, for which Eq. (1) (or the Meissner condition B ¼ 0) holds in the entire field range of the superconducting state, i.e., up to Hcr. Those are type-I superconductors. An example of M Hð Þ dependences for a typical type-I superconductor with η ¼ 0 is shown in Figure 2.

S/N transition takes place when free energies of the S and N states are equal, i.e., <sup>F</sup>~Mð Þ¼ Hcr

Now, we turn our cylinder perpendicular to the applied field. In a weak field, the specimen is in the Meissner state (inside it B ¼ 0), but the pattern of field lines looks now as shown in Figure 1b. The external field near the specimen is tangential to its surface, which follows from always valid conditions of continuity of the normal component of B and of tangential component

Figure 2. Experimental data for magnetic moment of a pure indium film 2.79 μ m thick measured in parallel applied field H at indicated temperatures. Errors up and down indicate that the measurements were conducted at increasing and

<sup>n</sup>. For our type-I cylindrical sample, as seen from Eq. (5), this implies that Hcr ¼ Hc and therefore the S/N transition in specimens with η ¼ 0 must be discontinuous, i.e., thermodynamic phase transition of the first order, in full agreement with experimental results, e.g., with

V: (5)

Therefore near the "poles" of our specimen the field is zero, whereas near "equator" it is twice as big as the applied field. This implies that the external field near "equator" reaches the critical value Hc at H ¼ Hcð Þ¼ 1 � η Hc=2. When H is increased beyond this value, the field must enter the specimen destroying superconductivity. However, contrarily to the previous (parallel) case, superconductivity cannot be destroyed completely because there is still plenty of condensation energy left in the specimen.

Indeed, the specimen magnetic moment M � ð Þ B � Hi V=4π ¼ �HV=4πð Þ¼� 1 � η HV=2π; therefore magnetic energy EM at H ¼ Hc=2 is

$$E\_M = -\int\_0^{0.5H\_c} \mathbf{M} \cdot d\mathbf{H} = \frac{V H\_c^2}{16\pi} < \frac{V H\_c^2}{8\pi}.\tag{8}$$

Hence, as seen from Eq. (4), F~<sup>M</sup> < F~<sup>M</sup> � � <sup>n</sup>, and therefore the specimen must remain superconducting.

At the first sight, one might expect that at H > Hcð Þ 1 � η , the field will gradually enter the specimen, thus destroying superconductivity over the field range from Hcð Þ 1 � η to Hc. The superconducting cylinder in such case would stay resistanceless with gradually changing volume of the S core as shown in Figure 3. However, this scenario is problematic because as soon as the field enters the specimen, the density of the field lines near the "equator" decreases and hence the field inside the convex blue region in Figure 3 becomes smaller than Hc. Then this region should go back to the S state.<sup>2</sup> This means that when <sup>H</sup> <sup>&</sup>gt; Hcð Þ <sup>1</sup> � <sup>η</sup> , the ellipsoidal specimen splits into S and N regions, as it was suggested for the first time by Gorter and Casimir [10].

<sup>1</sup> Derivation of Eq. (6) can be found in [2]; Maxwell using it in [4] refers to Poisson.

<sup>2</sup> Historically impossibility of configuration like that shown in Figure 3 was explained basing on a paradigm of instability of the N phase against transforming to the S phase at Hi < Hc (see, e.g. [8]). However, this (the N phase at Hi < H) does take place in specimens in the IS, but only at Hi in the upper part of the IS field range. At the lower edge of this range (at H ¼ ð Þ 1 � η Hc ) B in the first N domain and therefore Hi throughout the specimen is always Hc .

Figure 3. Cross section of the cylindrical sample in case if superconducting phase (S, colored in gray) is gradually replaced by the normal (N, colored in blue) phase filled by the field.

After Peierls [11] this inhomogeneous state in type-I superconductors is named the intermediate state. Properties of the IS were (and in some extent still are) one of the longest-standing challenges of physics of superconductivity. Below we will expose the main theoretical ideas and key experimental achievements addressing these properties. Comprehensive reviews of the experimental and theoretical works on the IS published before 1970 are available in [1, 12, 13]; for references to more recent publications, we recommend papers by Brandt and Das [14] and Clem et al. [15].

## 2. Model of Peierls and London

The first successful theoretical model of the IS magnetic properties was developed in 1936 independently by Peierls [11] and London [16]. In this model properties of ellipsoidal samples are considered in an averaged limit, in which the nonuniform induction B is replaced by average B. This allowed to use Eq. (6) with demagnetizing factor η calculated for uniform ellipsoid. However Eq. (6) has two unknowns, B and Hi, both of which are needed to calculate the specimen magnetic moment. Basing on a paradigm that the N phase is unstable at Hi < Hc, Peierls and London postulated that inside the specimen in the IS (i.e., at 1ð Þ � η Hc < H < Hc),

$$H\_{\bar{l}} = H\_c.\tag{9}$$

Graphs of these functions for B and M are shown in Figure 4 in reduced coordinates. It is important that area under the graphs for 4πM Hð Þ=VHc vs. H=Hc is the same 1/2. Therefore this model meets the necessary thermodynamic condition of Eq. (4). The PL model fits well experimental data obtained for thick specimens, i.e., when the field inhomogeneities near the surfaces through which the flux enters and leaves the specimen are negligible. Overall, the PL model represents a global description of the IS in zero-order approximation [8]. Similar model for the mixed state in type-II superconductors is available in [17]. For type-I superconductors

Intermediate State in Type-I Superconductors http://dx.doi.org/10.5772/intechopen.75742 95

Magnetic flux structure of the IS was for the first time considered by Landau [18] for an infinite parallel-plane plate (slab) in perpendicular field, i.e., for the sample-field configuration shown in Figure 1c. In such a specimen the surface current (and hence the Meissner state) is absent because <sup>B</sup> <sup>¼</sup> <sup>H</sup>, and therefore <sup>g</sup> <sup>¼</sup> <sup>H</sup> � <sup>B</sup> <sup>c</sup>=4<sup>π</sup> <sup>¼</sup> 0 at any <sup>H</sup> from zero to Hcr. Due to that the IS starts at H right above zero, no matter how small is this field. Magnetic moment of this specimen (Landau considered thick plate) is M Hð Þ¼ �ð Þ Hc þ H V=4π; graphs for B and M are

Assuming that (i) the plate is split for regularly structured S and N laminae and (ii) the boundary of a cross section of the S laminae is the line of induction B with magnitude Hc at the S/N interface, Landau calculated shape of rounded corners of the S laminae near the sample surface. Landau's scenario for cross section of the S-lamina near the surface is shown in Figure 5a. To meet the second assumption, Landau splits a central field line for two branches (oba and ocd in Figure 5a) making a sharp (90�) turn at the splitting point (o). Hence, in this scenario the field fills all space outside the specimen, as it is supposed to be the case in magnetostatics. On the other hand, splitting the field line challenges the magnetostatics rules

Figure 4. Peierls and London model. Average magnetic induction (a) and magnetic moment (b) for specimens with demagnetizing factor η ¼ 1 (infinite slab in perpendicular field, green line), η ¼ 1=2 (long cylinder in perpendicular field,

blue line) and η ¼ 0 (long cylinder in parallel field, red line). NS designates the normal state (black line).

[4], and the sharp turn of the line may cost the system too much energy [2, 19].

this new model converts to the model of Peierls and London.

3. Landau laminar models

shown by the green lines in Figure 4a and b.

Eqs. (2), (6), and (9) constitute a complete system of equations. Solving it one finds B, Hi, and M:

$$H \le H\_c(1 - \eta) \begin{cases} B = 0 \\ H\_i = H/(1 - \eta) \\ M = HV/4\pi(1 - \eta) \end{cases} \tag{10}$$

$$H\_c(1 - \eta) \le H \le H\_c \begin{cases} \overline{B} = (H - H\_c(1 - \eta))/\eta \\ H\_i = H\_c \\ M = V(H - H\_c)/4\pi\eta \end{cases} . \tag{11}$$

Graphs of these functions for B and M are shown in Figure 4 in reduced coordinates. It is important that area under the graphs for 4πM Hð Þ=VHc vs. H=Hc is the same 1/2. Therefore this model meets the necessary thermodynamic condition of Eq. (4). The PL model fits well experimental data obtained for thick specimens, i.e., when the field inhomogeneities near the surfaces through which the flux enters and leaves the specimen are negligible. Overall, the PL model represents a global description of the IS in zero-order approximation [8]. Similar model for the mixed state in type-II superconductors is available in [17]. For type-I superconductors this new model converts to the model of Peierls and London.
