Fundamentals of the Physics of Solids / 14-Magnetically Ordered Systems
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14.5 The General Description of Magnetic Phase Transitions |
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transition. In the previous section this transition was described using the Heisenberg Hamiltonian, and the temperature dependence of magnetization and susceptibility were determined on both sides of the transition point. In all our previous examples the transitions were of second order, that is magnetization and sublattice magnetization appeared gradually, not with an abrupt discontinuity. While this is not always the case, second-order magnetic phase transitions are very common, therefore we shall give a concise overview of the general theory of second-order transitions. We shall demonstrate that the characteristic physical quantities of the magnetic system can be derived without making specific assumptions about the Hamiltonian, and that the e ects of fluctuations may also be taken into account by going beyond the mean-field approximation.
14.5.1 The Landau Theory of Second-Order Phase Transitions
In 1937 L. D. Landau7 put forward a general phenomenological theory for the description of second-order phase transitions. He considered the sudden change of symmetry as one of the most important characteristics of such transitions. At high temperatures the system is in a disordered state that possesses high symmetry. At lower temperatures a more ordered – and consequently less symmetric – state occurs, therefore the higher symmetry of the previous phase is broken. The two phases are distinguished by the existence and nonexistence of some symmetries, therefore a sharp line can be drawn between the two. In Landau’s approach order can be characterized quantitatively by an order parameter, which is zero in the disordered phase and nonzero in the ordered phase. The physical meaning of the order parameter is determined by the character of the transition. In ferromagnetic ordering magnetization itself can serve as an order parameter, while in antiferromagnetic ordering the sublattice magnetization is a good candidate. In order–disorder transitions in alloys the concentration of individual components on the sublattices is a possible choice. When superfluidity and superconductivity are analyzed, the wavefunction of the condensate proves to be an appropriate order parameter. Below we shall demonstrate that, regardless of the physical meaning of the order parameter, general statements can be made about the variations of the free energy and other physical quantities during the phase transition.
Landau based the description of second-order phase transitions on the following assumptions:
1.The phase transition is continuous. This means that the order parameter ψ0 – which vanishes in the disordered phase but takes a nonzero value in the ordered phase below the critical temperature Tc – varies continuously in the phase transition. If the order parameter changed discontinuously to a finite value, the phase transition would be of first order.
7 See footnote on page 28.
490 14 Magnetically Ordered Systems
2.A free-energy-like quantity F can be introduced, which can be defined even for the nonequilibrium values ψ of the order parameter. This quantity has its minimum at the equilibrium value ψ0 of the order parameter, and this minimum value is just the equilibrium value of the free energy. Therefore F must have its minimum at ψ = 0 in the disordered phase and at a finite ψ0 in the ordered phase (i.e., below the critical temperature Tc).
3.F is an analytic function of ψ, that is, it can be expanded into a power series of ψ. Since the order parameter varies continuously in the vicinity of the transition point, only the first few terms need to be retained if the behavior of the system is studied in a small temperature range around Tc.
4.The expansion coe cients are analytical functions of temperature.
For simplicity we shall assume that the order parameter is a real scalar, although this is not always the case.8 It follows from the above assumptions that the expansion of F is
F = F0 |
+ A(T )ψ2 + 1 B(T )ψ4 |
+ . . . . |
(14.5.1) |
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The linear and cubic terms are missing for simple reasons. The linear term must be absent, otherwise ψ0, the order parameter at the minimum of the free energy, would be nonzero everywhere – with the possible exception of the point where the coe cient of the linear term vanishes –, and so there would be no order–disorder transition. The cubic term must also be absent, otherwise the position of the minimum would not change continuously with temperature – recall that the temperature dependence appears through the coe cients A and B – but ψ0 would jump abruptly from zero to a finite value, which would correspond to a first-order phase transition. Generally speaking, the requirement that the cubic term should vanish imposes severe restrictions on the possible transitions. In magnetic transitions the situation is simplified right from the start by the symmetry properties of the magnetic moment that permit only even powers of ψ. The free energy is plotted against ψ in Fig. 14.17; the minimum is at ψ = 0 on one graph, and at a finite value ψ0 on the other.
The previous expression has its minimum at ψ = 0 if the coe cients A and B are both positive. The minimum is at a finite ψ0 if A < 0 and B > 0. Therefore the following temperature dependence is required for the coe cients:
> 0 , |
if |
T > Tc , |
A(T ) |
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(14.5.2) |
< 0 , |
if |
T < Tc , |
while B(T ) has to be positive in both regions. Assuming that the coe cients are analytic, the simplest choice is
A(T ) = a(T − Tc) and B(T ) = B(Tc) = B > 0 . |
(14.5.3) |
8 Generalization to an n-component order parameter is straightforward.
14.5 The General Description of Magnetic Phase Transitions |
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Fig. 14.17. Free energy as a function of the order parameter in the vicinity of the critical temperature
Determined from the minimum condition for the free energy, the equilibrium value of the order parameter is
ψ02(T ) = |
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if |
(14.5.4) |
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0 , |
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T ≥ Tc , |
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A(T ) |
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T < Tc . |
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− |
B |
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The order parameter appears continuously in Tc indeed; its temperature dependence is
ψ0(T ) Tc − T . (14.5.5)
Using the terminology of critical phenomena: the critical exponent of the order parameter is β = 1/2 in the Landau theory.
This value is used in the determination of the equilibrium value of the free energy F in the ordered phase. F is smaller than the free energy F0 in the disordered phase:
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A2(T ) |
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a2 |
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F = F0 − |
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= F0 − |
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(T − Tc)2 . |
(14.5.6) |
2B |
2B |
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It follows immediately that the specific heat is not continuous but has a finite jump in the phase transition point:
C = |
a2 |
(14.5.7) |
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Tc . |
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B |
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If the order parameter is linearly coupled to some external field, e.g., magnetization to the external magnetic field, then the susceptibility associated with the response to the external field can also be determined from the Landau theory. In magnetic systems the energy contribution −M · B due to the magnetic field has to be taken into account. Choosing the order parameter as
492 14 Magnetically Ordered Systems
the component of magnetization along the direction of the field, the equilibrium value of the magnetization can be determined from the minimum of the free energy
F = F0 + A(T )M 2 + 21 BM 4 − μ0M H , |
(14.5.8) |
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which leads to |
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2A(T )M + 2BM 3 = μ0H . |
(14.5.9) |
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Keeping only the leading-order term above the critical temperature Tc, |
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M = |
μ0H |
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μ0H |
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(14.5.10) |
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2a(T − Tc) |
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2A(T ) |
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so the magnetic susceptibility is |
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χm = |
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μ0 |
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(14.5.11) |
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2a(T − Tc) |
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The same temperature dependence has appeared as in the Curie–Weiss law for ferromagnets. In antiferromagnets, where sublattice magnetization is the order parameter, susceptibility measured in a uniform field remains finite, however the staggered susceptibility shows the same kind of singularity at k = q0, the characteristic wave vector of the antiferromagnetic structure.
The di erential susceptibility in the ordered phase can be determined from the relation
2A(T ) |
∂M |
+ 6BM 2 |
∂M |
= μ0 , |
(14.5.12) |
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∂H |
∂H |
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which is derived from (14.5.9). Substituting the order parameter from (14.5.4),
χm = − |
μ0 |
= |
μ0 |
(14.5.13) |
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4A(T ) |
4a(Tc − T ) |
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Susceptibility diverges as the −1st power of |T − Tc| on both sides of the transition point, thus the critical exponent is γ = 1.
14.5.2 Determination of Possible Magnetic Structures
Up to now the order parameter has been assumed to be a real scalar, therefore only one secondand one fourth-order term appeared in the expansion of the free energy. As we shall see in Chapter 26, the analysis of superconductivity requires that a complex order parameter be used. However, even in magnetic systems the order parameter may be a multicomponent quantity. One could think that the vector character of magnetization requires a three-component order parameter. But the situation is not so simple. The number of components and the form of the secondand fourth-order terms in the free energy expansion are determined by the symmetries of the system.
The free energy of a crystal has to be invariant under all the symmetry operations that take the crystal into itself. To obtain such an invariant expression
14.5 The General Description of Magnetic Phase Transitions |
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for the free energy when the symmetries of the disordered phase are known, the order parameter has to be expanded in terms of the basis functions of the irreducible representations of the symmetry group of the disordered phase, and then secondand fourth-order invariants have to be constructed from the expansion coe cients. For simplicity, we shall assume that there are two irreducible representations: a twoand a three-dimensional one with normalized basis functions φ(1)i and φ(2)i . The expansion of the order parameter is then
ψ = (c1 |
φ(1) |
+ c2 |
φ(1)) + (d1 |
φ(2) |
+ d2 |
φ(2) |
+ d3 |
φ(2)) . |
(14.5.14) |
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Transferring the transformation properties of basis functions to the coe - cients, only those combinations may appear in the free energy that are invariant under all symmetries of the high-temperature phase. A single second-order invariant is associated with each irreducible representation, and these appear with independent coe cients in the expansion of the free energy:
F = F |
+ A (T )(c2 |
+ c2) + A (T )(d2 |
+ d2 |
+ d2) + . . . . |
(14.5.15) |
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We shall start from the disordered phase, where each coe cient Ai(T ) is positive, otherwise the minimum of F would not occur at vanishing order parameter. In the ordered phase those coe cients (ci, dj , or others if there are further irreducible representations) acquire a finite value that are associated with that particular irreducible representation for which the temperature-dependent co- e cients Ai disappears first (i.e., at the highest temperature). Therefore the dimension of the order parameter is determined by the dimension of the corresponding irreducible representation.
Recall that the irreducible representations of the crystal’s space group are specified by the wave vector k, which characterizes the behavior of the system under spatial translations, and more than one irreducible representation may be associated with a vector k. The coe cient Ai(T ) obviously depends on the wave vector k associated with the irreducible representation. To emphasize this dependence, the notation Aik (T ) will be used. According to the previous considerations, the transition temperature Tc and the wave vector k0 characterizing the behavior of the magnetic structure under translations are determined from the condition that it is the highest temperature where one of the coe cients Aik (T ) vanishes. We shall demonstrate that barring accidental solutions (which are not the consequences of symmetry considerations) k0 must be a high-symmetry point of the Brillouin zone.
This is because when the coe cients are continuous functions of k there is no reason why the linear term of the expansion around a general k0 in powers of k − k0 should be missing. Consequently, if the coe cient at wave vector k0 vanishes at a temperature Tc, i.e., Aik0 (Tc) = 0, then the linear expansion of the coe cients in both k − k0 and T − Tc shows that it vanishes at a higher temperature for some wave vector k that is di erent from k0 – in striking contradiction with our assumption. This argument does not apply to high-symmetry points of the Brillouin zone: symmetry considerations may
494 14 Magnetically Ordered Systems
imply that the coe cient has an extremum at such a k0. This explains why in the overwhelming majority of magnetic structures the wave vector k0, which describes the spatial modulation of the ordered magnetic moment, is a highsymmetry point of the Brillouin zone.
14.5.3 Spatial Inhomogeneities and the Correlation Length
The Landau theory leads to the same results as the mean-field theory since fluctuations are neglected in both. However, experiments show deviations from the Curie–Weiss law, and magnetization does not vanish in the Curie point with the exponent predicted by the mean-field theory, as illustrated in Fig. 14.12. This stems from the assumption that the order parameter is perfectly homogeneous in space. In reality, spatial inhomogeneities may become important in the vicinity of the phase transition point. Owing to thermal fluctuations, order is not perfectly homogeneous under Tc, and even above Tc the order parameter may acquire a nonzero value locally, over small regions. Nevertheless we cannot speak of a magnetically ordered state, since order can be observed only over a finite correlation length ξ – in other words, only shortrange order exists, and there is no correlation between more distant regions. To account for spatial inhomogeneities, we shall write the full free energy of the system in terms of a free-energy density f (r):
F = f (r) dr , (14.5.16)
where f (r) can be expressed in a form similar to (14.5.1) using a positiondependent local order parameter ψ(r). Because of the spatial variations, a new contribution may appear that is proportional to the square of the gradient of the order parameter and that corresponds to the energy increment due to spatial inhomogeneities:
f (r) = f0 + A(T )ψ2(r) + 12 B(T )ψ4(r) + 12 C(T ) ( ψ(r))2 + . . . . (14.5.17)
If the order parameter ψ fluctuates about ψ0, the order parameter of the homogeneous system satisfying the equation
2A(T )ψ0 + 2B(T )ψ03 = 0 , |
(14.5.18) |
then it can be written as ψ(r) = ψ0 + δψ(r), where δψ(r) is small and is subject to the constraint
δψ(r) dr = 0 . |
(14.5.19) |
Using this decomposition, the leading-order increment of the free energy due to fluctuations is
δF = |
dr |
21 C |
δψ(r) |
2 + A(T ) δψ(r) 2 |
+ 3B(T )ψ02 |
δψ(r) 2 |
. (14.5.20) |
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14.5 The General Description of Magnetic Phase Transitions |
495 |
Expanding fluctuations into a Fourier series,
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δψ(r) = |
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ψk eik·r , |
(14.5.21) |
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where the homogeneous term that corresponds to k = 0 is missing, thus the contribution of the fluctuations to the free energy is
δF = |
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21 Ck2 |
+ A(T ) + 3B(T )ψ02 |ψk|2 . |
(14.5.22) |
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Since the probability for such a fluctuation to occur is given by the Boltzmann factor
δF |
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P (δF ) = exp −kBT |
(14.5.23) |
the thermal average of the square of the component ψk is
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kBT V |
(14.5.24) |
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Ck2 + 2A(T ) + 6B(T )ψ02 |
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Above Tc, where ψ0 = 0 the amplitude of long-wavelength fluctuations increases as the temperature approaches the transition point. The same is true below Tc, where ψ02 = −A(T )/B, that is
|ψk |2 = |
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kBT V |
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(14.5.25) |
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Writing this expression in the form |
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χ |
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(14.5.26) |
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the correlation length ξ is defined as |
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C |
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ξ2 = |
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T > Tc , |
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2A(T ) |
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T < Tc . |
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C |
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(14.5.27) |
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As we shall see, this specifies the natural length scale over which the spatial correlations between fluctuations decay. On both sides of the transition point ξ2 diverges as 1/|T − Tc|, i.e., the correlation length itself is proportional to the inverse square root of |T − Tc|.
To see the significance of ξ, we have to determine the expectation value of the spatial correlation of fluctuations,
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Γ (r) = |
δψ(r)δψ(0) . |
496 14 Magnetically Ordered Systems
By recognizing that |ψk |2 is the Fourier transform of this expression, an inverse transformation gives
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In the general d-dimensional case (d ≥ 2), |
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Γ (r) |
e−r/ξ |
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Thus ξ is indeed the characteristic length of the exponential decay of correlations. Note that in the critical point, where the correlation length becomes infinitely large, the decay is no longer exponential but power-law-like.
14.5.4 Scaling Laws
Close to the critical point the behavior of the system can be characterized by a handful of parameters called critical exponents. In general, these specify the temperature and field dependence of certain thermodynamic quantities. In ferromagnetic transitions the quantities of particular interest are
C(T ) |T − Tc|−α , |
M (T ) (Tc − T )β , |
(14.5.31) |
χ(T ) |T − Tc|−γ , |
M (H) H1/δ . |
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The exponent ν that characterizes the divergence of the correlation length is defined as
ξ(T ) |T − Tc|−ν , |
(14.5.32) |
while the exponent η specifying the slow spatial decay of the correlation function of the order parameter in the critical point is defined as
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Γ (r) rd−2+η . |
(14.5.33) |
For those quantities that can be defined both above and below the critical temperature – such as specific heat, susceptibility, or correlation length –, the symbol used for the exponent measured in the range below Tc has an extra prime compared to the value above Tc.
According to the Landau theory, the above calculations yield α = 0, since specific heat does not have a singularity only a finite jump, and β = 1/2, γ = 1, δ = 3, ν = 1/2, η = 0. However, experiments do not confirm these results. As listed in Table 14.7, the critical exponents measured in ferromagnetic materials deviate substantially from the values predicted by the Landau theory. The table contains yet another critical exponent, x, which describes the temperature dependence of the excitation energies in the spectrum of spin waves (which will be discussed in the next chapter):
14.5 |
The General Description of Magnetic Phase Transitions |
497 |
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Table 14.7. Experimental values for some critical exponents |
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α |
β |
γ |
δ |
x |
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Fe |
−0.12 |
0.38 |
1.33 |
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0.37 |
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Co |
−0.10 |
0.42 |
1.21 |
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0.39 |
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Ni |
0.38 |
1.32 |
4.2 |
0.39 |
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D (Tc − T )x . |
(14.5.34) |
In the mean-field approximation x = 1/2.
Using strictly valid thermodynamic relations it can be showed that the critical exponents must satisfy certain inequalities. For example, the Rushbrook
(a), Gri ths (b, c), and Josephson (d, e) inequalities are9 |
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α + 2β + γ |
≥ 2 , |
(14.5.35-a) |
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β(δ + 1) |
≥ 2 |
− α , |
(14.5.35-b) |
γ |
≥ β(δ − 1) , |
(14.5.35-c) |
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dν |
≥ 2 |
− α , |
(14.5.35-d) |
dν |
≥ 2 |
− α . |
(14.5.35-e) |
For the values obtained in the Landau theory, equalities are satisfied instead of the rigorously derived inequalities. Even though the experimental values of the critical exponents deviate from the predictions of the Landau theory, the equalities are found to hold for them, too, within experimental error.
An analysis of the experimental data also reveals another important relationship for magnetization measured as a function of the reduced temperature di erence
t = |T − Tc|/Tc |
(14.5.36) |
and the magnetic field. Instead of studying directly the t- and H-dependence of magnetization, M/tβ can be plotted against H/tβδ. The measured values are then found to lie on a single curve in the vicinity of the critical point. Formulated mathematically,
M (t, H) = tβ f (H/tβδ) , |
(14.5.37) |
or alternatively
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t |
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(14.5.38) |
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9 G. S. Rushbrook, 1963, R. B. Griffiths, 1965, B. D. Josephson, 1967.
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To understand this scaling property, B. Widom (1965) assumed that for small values of t and H the free energy is a generalized homogeneous function of these variables. This means that the free energy – or more precisely, its singular part coming from critical fluctuations – satisfies the equation
Fsing(λat t, λaH H) = λFsing(t, H) . |
(14.5.39) |
This implies that specific heat, magnetization, and susceptibility show the same behavior:
λ2at C(λat t, λaH H) = λC(t, H) ,
λaH M (λat t, λaH H) = λM (t, H) , |
(14.5.40) |
λ2aH χm(λat t, λaH H) = λχm(t, H) .
It is readily established that each critical exponent can be expressed with at and aH :
α = 2 |
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2aH − 1 |
, δ = |
aH |
. (14.5.41) |
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This requires an appropriate choice of the scale parameter λ – for example λ = t−1/at for the specific heat. The above-mentioned scaling property follows directly from the equation for magnetization. It can be proved in much the same manner that the singular part of free energy also depends only on a suitably chosen combination of t and H:
Fsing(t, H) = |t|2−αf ± |
|t|βδ . |
(14.5.42) |
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Critical exponents are not independent of each other as they can be expressed with the two exponents in Widom’s homogeneous function. From the equations on β and δ
at = |
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aH = δ |
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(14.5.43) |
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Substituting them into the formulas for α and γ, it is readily seen that the Rushbrook and Gri ths inequalities are replaced by equalities:
α + 2β + γ = 2 , γ = β(δ − 1) . |
(14.5.44) |
In the phase transition point t = 0 and H = 0. If either t or H is nonzero, the system is no longer in the critical point, and so the correlation length becomes finite. The scaling hypothesis was understood through the insight that close to the critical point the behavior of the system is governed by the fluctuations on the only relevant scale: that of the correlation length. In 1966 L. P. Kadanoff assumed that the behavior of the system depended