Order parameters

Order parameters are used to describe the state of a physical system during a phase transition .

In the transition from a liquid to a crystalline solid phase , the system changes from high symmetry ( isotropy , homogeneity ) to a phase in which this symmetry is broken (only the lattice symmetry of the crystal remains ). Here, a parameter that is 0 in the liquid phase and has a finite value in the crystalline phase is a measure of the order of the system, which explains the term order parameter: a higher value corresponds to a stronger order, whereas 0 is disorder.

Even with phase transitions without symmetry breaking, the parameter with which the transition is described is called the order parameter. For example, the volume fraction of the liquid is a suitable order parameter to describe the transition from liquid to gaseous: in the gaseous state it is 0, in the liquid state it is just 1 (at sufficiently low pressure, i.e. not near the triple point ).

The description with order parameters can also be applied to systems that continuously change their order within a phase .

Depending on the type of phase transition, the order parameter can suddenly assume a new value and thus serve directly as an indication of the phase transition or it can change continuously. In a physical system there are often several effects that suggest order. From the physical quantities represented by these effects, one chooses the quantity from which the order parameter is calculated.

Vector order parameters are also possible. Their use makes sense in the event of changes in the order of the isotropy of the system. A direction is marked there in the orderly phase. The vector used then has this direction, and as the amount the thickness of the alignment of the individual components of the preferred direction.

Connection with symmetry

Order parameters are used in statistical physics , which investigates phase transitions with spontaneous symmetry breaking . There, the additional degree of freedom released by breaking the symmetry corresponds precisely to the order parameter.

An example of this is the spontaneous magnetization when a ferromagnet cools down . This occurs as an additional degree of freedom of the system during the phase transition from paramagnetism to ferromagnetism and changes - apart from small jumps - continuously from 0 to the final, complete magnetization of the ferromagnet.

An example of an order parameter that does not correspond to a new degree of freedom of the system after the phase transition is the use of the density to describe the liquid-gas transition.

Examples

Further examples of order parameters are:

• the charge carrier density at the transition from the insulator to the conductive state in the semiconductor .
• the wave function in the Landau-Ginzburg theory ; it can be complex and vary spatially. Their square indicates the density of the Cooper pairs . ${\displaystyle \psi }$ ${\displaystyle |\psi |^{2}\ }$

Continuous change

A striking example with several phases and continuous changes in order are liquid crystals . Their different phases , between which they can switch, are ordered differently, in that the areas with parallel alignment of the rod-shaped crystals are oriented differently. Within such a phase, slight deviations in the parallelism of the crystals can occur, which can be determined by an angle-dependent order parameter. However, there is also a disordered phase in which the crystals are oriented along random directions.

Sudden change

Transitions between phases that are different aggregate states change the order of the system by leaps and bounds. This effect is most pronounced (in relation to the density as an order parameter) in the sublimation of crystalline solids and their reversal, the resublimation : between the gas with a very low density and the solid with a significantly higher density , the system does not assume a state of medium density .
It also corresponds to the intuitive idea that a regular crystal lattice is more ordered than the randomly distributed molecules of a gas. In mathematical terms, the crystalline order has translational symmetry : a step with the distance between the atoms along the lattice leads back to an identical place (one atom surrounded by others at the same distance). In gas, on the other hand, this translation symmetry is broken; a fixed step size leads randomly to empty spaces as well as to other gas molecules.

literature

• Ludwig Bergmann, Thomas Dorfmüller, Clemens Schaefer: Textbook of Experimental Physics: Mechanics, Relativity, Warmth. 11th edition. Walter De Gruyter, Berlin 1998, ISBN 3-11-012870-5 .