# Phase Transition

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## phase transition

[′fāz tran‚zish·ən]## Phase Transition

(also phase transformation), in a broad sense, a transition of a substance from one phase to another upon a change in external conditions, such as temperature, pressure, and magnetic or electric fields; in a narrow sense, an abrupt change in physical properties upon a continuous change in external parameters. The difference between the two definitions of “phase transition” is made clear by the following example. In the narrow sense, the transition of a substance from the gaseous to the plasma state (*see*PLASMA) is not a phase transition, since the gas is gradually ionized; however, in the broad sense, it is a phase transition. In this article, “phase transition” is discussed in the narrow sense.

The value of the temperature, pressure, or other physical quantity at which a phase transition occurs is called a transition point.

A distinction is made between two orders of phase transitions. In a first-order phase transition, the thermodynamic properties of a substance, such as the density or the component concentration, change abruptly. In this case, a completely specified amount of heat, called the heat of transformation, is released or absorbed per unit mass. In a second-order phase transition, some physical quantity that is equal to zero on one side of a transition point gradually increases from zero with increasing distance on the other side of the transition point. In this case, the density and concentrations change continuously, and heat is neither released nor absorbed.

Phase transitions are widespread in nature. First-order phase transitions include evaporation and condensation, melting and solidification, sublimation and condensation to the solid phase, and certain structural transformations in solids, such as the formation of martensite in an iron-carbon alloy. In antiferromagnets with one axis of sublattice magnetization, a first-order phase transition occurs in an applied magnetic field that is parallel to the axis. At a certain field strength, the sublattice magnetic moments are turned in a direction perpendicular to the field; that is, the sublattices flip. In pure superconductors, a magnetic field causes a first-order phase transition from the superconducting to the normal state.

At a temperature of absolute zero and a fixed volume, the phase with the least energy is the thermodynamically stable phase. In this case, a first-order phase transition occurs at the pressure and applied-field values where the energies of two different phases become equal. If the pressure *p*, rather than the volume *V*, is fixed, the Gibbs free energy Φ, or *G*, is the least energy in a state of thermodynamic equilibrium, and phases with the same values of Φ are in equilibrium with one another at a transition point.

At low pressures, many substances crystallize into loosely packed structures. For example, crystalline hydrogen consists of molecules located at relatively large distances from one another. The structure of graphite is a series of relatively widely spaced sheets of carbon atoms. At sufficiently high pressures, large values of the Gibbs free energy correspond to such friable structures. Under such conditions, stable close-packed phases correspond to smaller values of Φ. Therefore, at high pressures, graphite transforms into diamond, and crystalline molecular hydrogen should transform into atomic—that is, metallic—hydrogen. At standard pressure, the quantum liquids ^{3}He and ^{4}He remain liquid down to the lowest temperatures attained (*T* ~ 0.001°K). The reason for this behavior lies in the weak interaction of the particles and the large amplitude of the particle vibrations at temperatures close to absolute zero; such vibrations are called zero-point vibrations (*see*UNCERTAINTY PRINCIPLE). However, an increase in pressure (to 20 atmospheres at *T* ≈ 0°K) results in the solidification of liquid helium. At nonzero temperatures and at a fixed pressure and temperature, the phase with the least Gibbs free energy—that is, the minimum energy minus the work of the pressure forces and the quantity of heat imparted to the system—is still the stable phase.

The existence of a metastable equilibrium region in the vicinity of a phase-transition curve is characteristic of a first-order phase transition. For example, a liquid may be heated to a temperature above the boiling point or supercooled to below the freezing point. Metastable states exist for a long time because the formation of a new phase with a lower value of Φ, which is thermodynamically more favorable, is initiated when the nucleation of the new phase commences. A gain in the value of Φ when a nucleus forms is proportional to the volume of the nucleus, and a loss is proportional to the surface area of the nucleus, that is, to the value of its surface energy. Small nuclei that form increase Φ; therefore, such nuclei are highly likely to decrease and vanish. However, nuclei that attain some critical size grow, and the entire substance undergoes a transition to a new phase. The formation of a nucleus of critical size is a very unlikely process and seldom occurs. The probability that nuclei of critical size will form is increased if foreign inclusions of macroscopic size, such as dust particles in a liquid, are present in the substance. Near a critical point, both the difference between phases in equilibrium and the surface energy decrease, and nuclei of large size and fantastic shape readily form, thus affecting the properties of the substance (*see*CRITICAL PHENOMENA).

Examples of second-order phase transitions, which take place below a specific temperature in each case, include the occurrence of a magnetic dipole moment in a magnetic substance upon a transition from the paramagnetic to the ferromagnetic state, the occurrence of antiferromagnetic ordering upon a transition from the paramagnetic to the antiferromagnetic state, the occurrence of superconductivity in metals and alloys, the occurrence of superfluidity in ^{4}He and ^{3}He, the ordering of alloys, and the spontaneous polarization of a substance upon a transition from the paraelectric to the ferroelectric phase.

In 1937, L. D. Landau proposed a general interpretation of all second-order phase transitions as points where symmetry changes. According to Landau’s interpretation, a system has higher symmetry above a transition point than below it. For example, above the transition point, the directions of the elementary magnetic moments—that is, the spins—of the particles in a magnetic substance are randomly distributed. Therefore, the simultaneous reversal of all the spins does not alter the physical properties of the system. Below the transition point, the spins have a preferred orientation, so that the simultaneous reversal of the spins changes the direction of the system’s magnetic dipole moment. Another example is a binary alloy in which the atoms of the two species *A* and *B* occupy the points of a simple cubic lattice. The disordered state is characterized by a random distribution of the *A* and *B* atoms over the lattice points, so that a slip of the lattice by a distance equal to the lattice constant does not change the properties of the lattice. Below the transition point, the atoms of the alloy are arranged in an ordered manner: . . . *ABAB* . . . . The slip of such a lattice by a distance equal to the lattice constant leads to the replacement of all *A* atoms by *B* atoms or vice versa. As a result of the establishment of order in the arrangement of the atoms, the symmetry of the lattice is lowered. Symmetry itself appears and disappears abruptly. However, the quantity that characterizes asymmetry, which is called the order parameter, may change continuously. During a second-order phase transition, the order parameter is equal to zero above and at the transition point. Similar behavior is exhibited by, for example, the magnetic dipole moment of a ferromagnet, the electric polarization of a ferroelectric, the density of the superfluid component in liquid ^{4}He, and the probability of finding an *A* atom at the appropriate lattice point of a binary alloy.

The absence of discontinuities in density, concentration, and heat of transformation is characteristic of second-order phase transitions. However, precisely the same pattern is observed at a critical point on the curve of a first-order phase transition. The similarity turns out to be very profound. In the vicinity of a critical point, the state of a substance may be characterized by a quantity that serves as the order parameter. For example, in the case of the critical point on a vaporization curve, this quantity is the deviation of the density from the mean value. As we move along the critical isochor from the high-temperature region, the vapor is homogeneous, and the density deviation is equal to zero. Below the critical temperature, the substance stratifies into two phases; in each phase, the deviation of the density from the critical value is nonzero. Since the phases differ little from one another in the vicinity of the transition point of a second-order phase transition, the formation of large nuclei of one phase in the other phase, known as fluctuations of concentration, is possible, just as it is in the vicinity of the critical point. Many critical effects in second-order phase transitions are associated with this situation, including the unlimited increase in the magnetic susceptibility of ferromagnets and in the dielectric constant of ferroelectrics (the increase in compressibility in the vicinity of a liquid-vapor critical point is analogous to such increases), an unlimited increase in heat capacity, the anomalous scattering of neutrons in ferromagnets, and the anomalous scattering of electromagnetic waves, for example, of light waves in a liquid or vapor (*see*OPALESCENCE, CRITICAL) and of X rays in solids. Dynamic phenomena also change substantially; this is associated with the very slow dissipation of fluctuations. For example, a Rayleigh line is narrowed near a liquid-vapor critical point, and spin diffusion is inhibited in the vicinity of the Curie point of ferromagnets and the Néel point of antiferromagnets (*see*SPIN WAVE). The mean size, or correlation radius *R*, of fluctuations increases as the transition point of a second-order phase transition is approached and becomes infinitely large at the transition point.

Recent advances in the theory of second-order phase transitions and critical phenomena are based on similarity theory. In the approach used, it is assumed that if *R* is taken as the unit of length and if the mean value of the order parameter of a cell with an edge *R* is taken as the unit of the order parameter, the overall picture of fluctuations will depend on neither proximity to a transition point not the specific substance. All thermodynamic quantities are then power-law functions of *R*. The power-law indexes, which are called critical exponents, do not depend on the specific substance and are determined solely by the nature of the order parameter. For example, the exponents at the Curie point of an isotropic material, the magnetization vector of which is the order parameter, differ from the exponents at a liquid-vapor critical point or at the Curie point of a uniaxial magnetic substance, where the order parameter is a scalar.

In the vicinity of a transition point, an equation of state has the characteristic form of the law of corresponding states. For example, in the vicinity of a liquid-vapor critical point, the ratio (ρ – ρ_{c})/(ρ_{l} – ρ_{v}) depends only on (p – *p _{c}*)/(ρ

_{l}– ρ

_{v}) ×

*K*, where ρ is the density, ρ

_{T}_{c}is the critical density, ρ

_{l}is the density of the liquid, ρ

_{v}is the density of the vapor,

*p*is the pressure,

*p*is the critical pressure, and

_{c}*K*is the isothermal compressibility. If an appropriate scale is chosen, the form of the dependence is the same for all liquids (

_{T}*see*CRITICAL PHENOMENA).

Theoretical calculations of critical exponents and equations of state have been found to be in good agreement with experimental data. Approximate values of critical exponents are presented in Table 1.

Table 1. Critical exponents of thermodynamic and kinetic quantities | |
---|---|

Quantity | Exponent |

^{1} For example, the variation of density with pressure or of magnetization with magnetic field strength | |

Note: T_{c} is the critical temperature | |

T – T ..............._{c} | –3/2 |

Heat capacity ............... | 3/16 |

Susceptibility^{1} ............... | 2 |

Magnetic field ............... | –5/2 |

Magnetic moment ............... | –1/2 |

Rayleigh line width ............... | –3/2 |

The further development of the theory of second-order phase transitions is associated with the use of the methods of quantum field theory, particularly the renormalization group method. The method makes it possible in principle to obtain critical exponents to any required accuracy.

The division of phase transitions into two orders is somewhat arbitrary, since there are first-order phase transitions with small discontinuities in heat capacity and in other quantities and small heats of transformation in the presence of highly evolved fluctuations. A phase transition is a collective phenomenon that occurs at rigorously specified values of temperature and other quantities and only in a system that has in the limit an arbitrarily large number of particles.

### REFERENCES

Landau, L. D., and E. M. Lifshits.*Statisticheskaia fizika*, 2nd. ed. (

*Teoreticheskaia fizika*, vol. 5.) Moscow, 1964.

Landau, L. D., A. I. Akhiezer, and E. M. Lifshits.

*Kurs obshchei fiziki: Mekhanika i molekuliarnaia fizika*, 2nd ed. Moscow, 1969.

Brout, R.

*Fazovye perekhody*. Moscow, 1967. (Translated from English.)

Fisher, M.

*Priroda kriticheskogo sostoianiia*. Moscow, 1968. (Translated from English.)

Stanley, H.

*Fazovye perekhody i kriticheskie iavleniia*. Moscow, 1973. (Translated from English.)

Anisimov, M. A. “Issledovaniia kriticheskikh iavlenii v zhidko-stiakh.”

*Uspekhifizicheskikh nauk*, 1974, vol. 114, issue 2.

Patashinskii, A. Z., and V. L. Pokrovskii.

*Fluktuatsionnaia teoriia fazovykh perekhodov*. Moscow, 1975.

*Kvantovaia teoriia polia i fizika fazovykh perekhodov. (Novosti fundamental’noi fiziki*, fasc. 6.) Moscow, 1975. (Translated from English.)

Wilson, K., and J. Kogut.

*Renormalizatsionnaia gruppa i*∊-razlozhenie. (

*Novosti fundamental’noi fiziki*, fasc. 5.) Moscow, 1975. (Translated from English.)

V. L. POKROVSKII