Landau theory

In physics, the Landau theory is a theory for describing phase transitions . It is named after the Russian physicist Lev Landau . This theory is based on a polynomial expansion of the free energy as a function of a parameter, the so-called order parameter, in the vicinity of the phase transition.

This theory is applied to phase transitions that are characterized by the loss of certain elements of symmetry. The shape of the Landau potential is determined by the symmetry of the phases and can therefore be determined by group theoretic methods. Indeed, land theory is the first application of group theory in thermodynamics .

The basic principles of this theory were presented by Landau in 1937. Subsequently, this general theory was applied by various working groups to special cases, which are therefore referred to by slightly different names: Landau-Ginzburg theory of superconductors , Landau-Devonshire theory of ferroelectrics, etc.

General properties

The Landau theory is a "local" theory. It was intended as an approximation in the area around the phase transition point, i.e. for small values ​​of the order parameter. However, it happens that the scope of this theory encompasses a much wider area.

The Landau theory is a “ phenomenological ” theory: using thermodynamic methods it is able to describe all phenomena that occur in connection with a phase transition in a uniform model, but it does not make any statements about the microscopic causes of this phase transition . In practice, the expansion coefficients of the land auto theory are determined by experiment.

In addition, this theory is a “ mean-field” theory : the underlying microscopic interactions are not considered individually, but rather averaged over them. Therefore, this theory cannot take into account the fluctuations of the order parameter around its equilibrium value. However, these can play an important role in the vicinity of the phase transition.

Concepts

Symmetry breaking

→ Main article : Spontaneous symmetry breaking

The properties of a body are closely related to its symmetry , which in many cases can be described by a corresponding space group . In the case of a second order phase transition, the symmetry of the system changes and thus its properties. Thus, among other things, additional variables such as magnetization, dielectric polarization or deformation can arise spontaneously.

In contrast to a first-order phase transition, the state of the system changes continuously with a second-order phase transition. At the point of the phase transition, the states of the high and low temperature phases coincide. It follows that one space group must be a subgroup of the other. In most cases, the higher symmetry phase corresponds to the high temperature phase and the lower symmetry corresponds to the low temperature phase. However, this is not a thermodynamic law and therefore allows exceptions, for example at the lower Curie point of the Seignette salt .

Order parameter Q

→ Main article : order parameters

In the highly symmetrical phase, according to the postulates of thermodynamics, it is possible to characterize the entire system by specifying a small number of state variables (such as pressure and temperature ). Some symmetry properties disappear at the phase transition. Specifying pressure and temperature is no longer sufficient to characterize the state. To: an additional variable, therefore you need order parameter Q . The order parameter is a priori an abstract quantity . It describes the process that is causally responsible for the phase transition. In many cases, it can therefore be identified using a specific microscopic process. The order parameter is generally a tensor quantity . The order parameter is defined in such a way that it has the value zero in the more symmetrical phase and a value other than zero in the more symmetrical phase. In addition, its symmetry behavior is important for the theory.

In order for the Landau potential to lead to a second order phase transition, the three Landau conditions and the Landau-Ginzburg criterion must be met:

• 1. Landing condition: The symmetry group of the phase with broken symmetry R ₁ must be a subgroup of the phase with full symmetry R ₀ .
• 2. Landing condition: The symmetry breaking is described by a single representation of R ₀ , the active representation, which must not be the 1 representation of R ₀ .
• 3. Landing condition: The symmetrical third power of the active representation must not contain the 1 representation of R ₀ .
• Landau-Ginzburg criterion: The antisymmetric square of the active representation must not contain a representation that transforms like the component of a vector.

Overall, the conditions mean that powers of odd order do not appear in the Landau potential. The fourth criterion severely restricts the possible locations in the Brillouin zone at which the phase transition can take place.

The landau potential

To describe the phase transition, the order parameter is taken into account as an additional variable in the free enthalpy . It must be noted that in a certain sense it is not equivalent to and : while pressure and temperature can be specified as desired, the equilibrium value of must be determined from the condition that the free enthalpy should assume a minimum. ${\displaystyle Q}$ ${\displaystyle G(P,T)}$ ${\displaystyle Q}$ ${\displaystyle P}$ ${\displaystyle T}$ ${\displaystyle Q}$

In the vicinity of the phase transition, the order parameter assumes small values. Therefore the free energy can be expanded into a number of powers of . First and third order terms in are not taken into account, since otherwise the high temperature phase or the phase transition point would not be thermodynamically stable states. The landing conditions also result from this requirement (see above). The free enthalpy thus has the following form: ${\displaystyle Q}$ ${\displaystyle Q}$

${\displaystyle G(T,P,Q)=G_{0}(T,P)+{\frac {A(T,P)}{2}}\,Q^{2}+{\frac {B(T,P)}{4}}\,Q^{4}}$

where the development coefficients can in principle depend on pressure and temperature. The value of the order parameter is determined from

1. ${\displaystyle {\frac {\partial G(P,T,Q)}{\partial Q}}=0=AQ+BQ^{3}=Q(A+BQ^{2})}$

and

2. ${\displaystyle {\frac {\partial ^{2}G(P,T,Q)}{\partial Q^{2}}}=A+3BQ^{2}>0}$

The possible solutions to these equations are summarized below, along with the conditions for the coefficients and the meaning of the corresponding phase:

Order parameters	Coefficients	temperature	phase
${\displaystyle Q=0}$	${\displaystyle A>0}$	${\displaystyle T>Tc}$	High temperature phase
${\displaystyle Q=0}$	${\displaystyle A=0}$	${\displaystyle T=Tc}$	Phase transition point
${\displaystyle Q^{2}=-{\frac {A}{B}}}$	${\displaystyle A<0,B>0}$	${\displaystyle T<Tc}$	Low temperature phase

where is the phase transition temperature. Within the framework of the Landau theory, assumptions are made for the expansion coefficients and the simplest assumptions that meet this requirement: ${\displaystyle T_{c}}$ ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle \ A(T,P)=a(T-T_{c})}$ and ${\displaystyle \ B(T,P)=B>0}$

where is constant and greater than zero. If you insert this into the free enthalpy, the result is the Landau potential: ${\displaystyle a}$

${\displaystyle G(T,P,Q)=G_{0}(T,P)+{\frac {a}{2}}(T-T_{c})Q^{2}+{\frac {B}{4}}Q^{4}}$

For the order parameter applies in the high temperature and in the low temperature phase . ${\displaystyle Q=0}$ ${\displaystyle Q^{2}={\frac {a}{B}}(T_{c}-T)}$

The phase transition

To investigate the behavior of the system at the phase transition, the Landau potential is treated like a normal thermodynamic potential. The following applies to entropy : ${\displaystyle S}$

${\displaystyle S=-{\frac {\partial G}{\partial T}}=-{\frac {\partial G_{0}}{\partial T}}-{\frac {\partial \;\left({\frac {a}{2}}(T-T_{c})Q^{2}+{\frac {B}{4}}Q^{4}\right)}{\partial T}}=S_{0}-aQ^{2}}$

where the entropy of the system has no phase transition. Substituting for the equilibrium values ​​results in: ${\displaystyle S_{0}=-{\frac {\partial G_{0}}{\partial T}}}$ ${\displaystyle Q}$

Temperature range	phase	entropy
${\displaystyle T>T_{c}}$	High temperature phase	${\displaystyle \ S_{0}}$
${\displaystyle T=T_{c}}$	Phase transition point	${\displaystyle \ S_{0}}$
${\displaystyle T<T_{c}}$	Low temperature phase	${\displaystyle \ S_{0}-{\frac {a^{2}}{B}}(T_{c}-T)}$

The entropy remains constant in the phase transition. It is lower in the low-temperature phase than the entropy of the high-temperature phase extrapolated into the low-temperature phase. The specific heat capacity results from: ${\displaystyle C_{p}}$

${\displaystyle C_{p}=T\left({\frac {\partial S}{\partial T}}\right)_{P}={\begin{cases}C_{p0}&{\text{Hochtemperaturphase}}\\C_{p0}+{\frac {a^{2}T_{c}}{2B}}&{\text{Tieftemperaturphase}}\end{cases}}}$

Here, too, the specific heat capacity of the system is without a phase transition. The specific heat capacity has a jump. Since and are positive quantities, the heat capacity is higher in the low-temperature phase than in the high-temperature phase. ${\displaystyle C_{p0}=T\left({\frac {\partial S_{0}}{\partial T}}\right)_{P}}$ ${\displaystyle T_{c}}$ ${\displaystyle a}$ ${\displaystyle B}$

The fact that the 1st derivative of the Landau potential is continuous but the 2nd derivative is discontinuous means that the Landau potential in this form does indeed describe a 2nd order phase transition. In turn, this also shows that a phase transition associated with a change in the symmetry of the system must be of at least 2nd order. ${\displaystyle T_{c}}$

The order parameter susceptibility

The order parameter susceptibility is also important for the experimental investigation . Its inverse is the 2nd derivative of the Landau potential according to the order parameter: ${\displaystyle \chi }$

${\displaystyle {\frac {1}{\chi }}={\frac {\partial ^{2}G}{\partial Q^{2}}}=A+3BQ^{2}={\begin{cases}a(T-T_{c})&{\text{Hochtemperaturphase}}\;(Q=0)\\-2a(T-T_{c})&{\text{Tieftemperaturphase}}\;(Q^{2}=-{\frac {A}{B}})\end{cases}}}$

The order parameter susceptibility therefore has the form in both phases:

${\displaystyle \chi ={\frac {C}{T-Tc}}}$

In the context of land theory, a Curie-Weiss law with the Curie constants (high temperature) and (low temperature phase) follows for the order parameter susceptibility . ${\displaystyle C=1/a}$ ${\displaystyle C=-1/2a}$

Applications

Superconductivity: Landau-Ginzburg theory

→ Main article : Landau-Ginzburg theory

Individual evidence

1. ↑ Wadhawan2000, page = 131
2. ↑ Landau LD, Zh. Eksp. Teor. Fiz. 7, pp. 19–32 (1937) ( Memento of the original dated December 14, 2015 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (PDF; 380 kB) 
3. ↑ Wadhawan2000, page = 154

literature

• EKH Salje: Phase Transitions in Ferroelastic and Co-elastic Crystals . Cambridge University Press, 1993 (Salje 1993).
• VK Wadhawan: Introduction to ferroic materials . Gordon and Breach Science Publishers, 2000 (Wadhawan 2000).
• W. Gebhardt, U. Krey (1980): Phase transitions and critical phenomena - An introduction , Vieweg, ISBN 3-528-08422-7
• LD Landau, EM Lifshitz: Textbook of theoretical physics V . Statistical Physics. Akademie Verlag, Berlin 1970.

Web links

• Notes de cours sur la théorie de Landau (PDF; 2.7 MB).
• A Landau Primer for Ferroelectrics .
• Landau LD, Zh. Eksp. Teor. Fiz. 7, pp. 19–32 (1937) (PDF; 380 kB)