Phase transition

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This diagram shows the nomenclature for the different phase transitions. Phase change - en.svg
This diagram shows the nomenclature for the different phase transitions.

In physics, chemistry, and other related fields like biology, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of matter: solid, liquid, and gas, and in rare cases, plasma. A phase of a thermodynamic system and the states of matter have uniform physical properties. During a phase transition of a given medium, certain properties of the medium change as a result of the change of external conditions, such as temperature or pressure. This can be a discontinuous change; for example, a liquid may become gas upon heating to its boiling point, resulting in an abrupt change in volume. The identification of the external conditions at which a transformation occurs defines the phase transition point.

Contents

Types of phase transition

States of matter

A simplified phase diagram for water, showing whether solid ice, liquid water, or gaseous water vapor is the most stable at different combinations of temperature and pressure. Phase diagram of water simplified.svg
A simplified phase diagram for water, showing whether solid ice, liquid water, or gaseous water vapor is the most stable at different combinations of temperature and pressure.

Phase transitions commonly refer to when a substance transforms between one of the four states of matter to another. At the phase transition point for a substance, for instance the boiling point, the two phases involved - liquid and vapor, have identical free energies and therefore are equally likely to exist. Below the boiling point, the liquid is the more stable state of the two, whereas above the boiling point the gaseous form is the more stable.

Common transitions between the solid, liquid, and gaseous phases of a single component, due to the effects of temperature and/or pressure are identified in the following table:

Phase transitions of matter (
)
To
From
 Solid Liquid Gas Plasma
Solid
 Melting Sublimation
Liquid Freezing
Vaporization
Gas Deposition Condensation
Ionization
Plasma Recombination

For a single component, the most stable phase at different temperatures and pressures can be shown on a phase diagram. Such a diagram usually depicts states in equilibrium. A phase transition usually occurs when the pressure or temperature changes and the system crosses from one region to another, like water turning from liquid to solid as soon as the temperature drops below the freezing point. In exception to the usual case, it is sometimes possible to change the state of a system diabatically (as opposed to adiabatically) in such a way that it can be brought past a phase transition point without undergoing a phase transition. The resulting state is metastable, i.e., less stable than the phase to which the transition would have occurred, but not unstable either. This occurs in superheating and supercooling, for example. Metastable states do not appear on usual phase diagrams.

Structural

A phase diagram showing the allotropes of iron, distinguishing between several different crystal structures including ferrite (a-iron) and austenite (g-iron). Pure iron phase diagram (EN).svg
A phase diagram showing the allotropes of iron, distinguishing between several different crystal structures including ferrite (α-iron) and austenite (γ-iron).

Phase transitions can also occur when a solid changes to a different structure without changing its chemical makeup. In elements, this is known as allotropy, whereas in compounds it is known as polymorphism. The change from one crystal structure to another, from a crystalline solid to an amorphous solid, or from one amorphous structure to another (polyamorphs) are all examples of solid to solid phase transitions.

The martensitic transformation occurs as one of the many phase transformations in carbon steel and stands as a model for displacive phase transformations. Order-disorder transitions such as in alpha-titanium aluminides. As with states of matter, there is also a metastable to equilibrium phase transformation for structural phase transitions. A metastable polymorph which forms rapidly due to lower surface energy will transform to an equilibrium phase given sufficient thermal input to overcome an energetic barrier.

Magnetic

A phase diagram showing different magnetic structures in the same crystal structure of Manganese monosilicide. MnSi magnetic phase diagram.png
A phase diagram showing different magnetic structures in the same crystal structure of Manganese monosilicide.

Phase transitions can also describe the change between different kinds of magnetic ordering. The most well-known is the transition between the ferromagnetic and paramagnetic phases of magnetic materials, which occurs at what is called the Curie point. Another example is the transition between differently ordered, commensurate or incommensurate, magnetic structures, such as in cerium antimonide. A simplified but highly useful model of magnetic phase transitions is provided by the Ising Model

Mixtures

A binary phase diagram showing the most stable chemical compounds of titanium and nickel at different mixing ratios and temperatures. Binary phase diagram of NiTI (phase and temperature).JPG
A binary phase diagram showing the most stable chemical compounds of titanium and nickel at different mixing ratios and temperatures.

Phase transitions involving solutions and mixtures are more complicated than transitions involving a single compound. While chemically pure compounds exhibit a single temperature melting point between solid and liquid phases, mixtures can either have a single melting point, known as congruent melting, or they have different liquidus and solidus temperatures resulting in a temperature span where solid and liquid coexist in equilibrium. This is often the case in solid solutions, where the two components are isostructural.

There are also a number of phase transitions involving three phases: a eutectic transformation, in which a two-component single-phase liquid is cooled and transforms into two solid phases. The same process, but beginning with a solid instead of a liquid is called a eutectoid transformation. A peritectic transformation, in which a two-component single-phase solid is heated and transforms into a solid phase and a liquid phase. A peritectoid reaction is a peritectoid rection, except involving only solid phases. A monotectic reaction consists of change from a liquid and to a combination of a solid and a second liquid, where the two liquids display a miscibility gap. [1]

Separation into multiple phases can occur via spinodal decomposition, in which a single phase is cooled and separates into two different compositions.

Non-equilibrium mixtures can occur, such as in supersaturation.

Other examples

A small piece of rapidly melting solid argon shows two concurrent phase changes. The transition from solid to liquid, and gas to liquid (shown by the white condensed water vapour). Argon ice 1.jpg
A small piece of rapidly melting solid argon shows two concurrent phase changes. The transition from solid to liquid, and gas to liquid (shown by the white condensed water vapour).

Other phase changes include:

Phase transitions occur when the thermodynamic free energy of a system is non-analytic for some choice of thermodynamic variables (cf. phases). This condition generally stems from the interactions of a large number of particles in a system, and does not appear in systems that are small. Phase transitions can occur for non-thermodynamic systems, where temperature is not a parameter. Examples include: quantum phase transitions, dynamic phase transitions, and topological (structural) phase transitions. In these types of systems other parameters take the place of temperature. For instance, connection probability replaces temperature for percolating networks.

Classifications

Ehrenfest classification

Paul Ehrenfest classified phase transitions based on the behavior of the thermodynamic free energy as a function of other thermodynamic variables. [5] Under this scheme, phase transitions were labeled by the lowest derivative of the free energy that is discontinuous at the transition. First-order phase transitions exhibit a discontinuity in the first derivative of the free energy with respect to some thermodynamic variable. [6] The various solid/liquid/gas transitions are classified as first-order transitions because they involve a discontinuous change in density, which is the (inverse of the) first derivative of the free energy with respect to pressure. Second-order phase transitions are continuous in the first derivative (the order parameter, which is the first derivative of the free energy with respect to the external field, is continuous across the transition) but exhibit discontinuity in a second derivative of the free energy. [6] These include the ferromagnetic phase transition in materials such as iron, where the magnetization, which is the first derivative of the free energy with respect to the applied magnetic field strength, increases continuously from zero as the temperature is lowered below the Curie temperature. The magnetic susceptibility, the second derivative of the free energy with the field, changes discontinuously. Under the Ehrenfest classification scheme, there could in principle be third, fourth, and higher-order phase transitions. For example, the Gross–Witten–Wadia phase transition in 2-d lattice quantum chromodynamics is a third-order phase transition. [7] [8] The Curie points of many ferromagnetics is also a third-order transition, as shown by their specific heat having a sudden change in slope. [9] [10]

In practice, only the first- and second-order phase transitions are typically observed. The second-order phase transition was for a while controversial, as it seems to require two sheets of the Gibbs free energy to osculate exactly, which is so unlikely as to never occur in practice. Cornelis Gorter replied the criticism by pointing out that the Gibbs free energy surface might have two sheets on one side, but only one sheet on the other side, creating a forked appearance. [11] ( [9] pp. 146--150)

The Ehrenfest classification implicitly allows for continuous phase transformations, where the bonding character of a material changes, but there is no discontinuity in any free energy derivative. An example of this occurs at the supercritical liquid–gas boundaries.

The first example of a phase transition which did not fit into the Ehrenfest classification was the exact solution of the Ising model, discovered in 1944 by Lars Onsager. The exact specific heat differed from the earlier mean-field approximations, which had predicted that it has a simple discontinuity at critical temperature. Instead, the exact specific heat had a logarithmic divergence at the critical temperature. [12] In the following decades, the Ehrenfest classification was replaced by a simplified classification scheme that is able to incorporate such transitions.

Modern classifications

In the modern classification scheme, phase transitions are divided into two broad categories, named similarly to the Ehrenfest classes: [5]

First-order phase transitions are those that involve a latent heat. During such a transition, a system either absorbs or releases a fixed (and typically large) amount of energy per volume. During this process, the temperature of the system will stay constant as heat is added: the system is in a "mixed-phase regime" in which some parts of the system have completed the transition and others have not. [13] [14]

Familiar examples are the melting of ice or the boiling of water (the water does not instantly turn into vapor, but forms a turbulent mixture of liquid water and vapor bubbles). Yoseph Imry and Michael Wortis showed that quenched disorder can broaden a first-order transition. That is, the transformation is completed over a finite range of temperatures, but phenomena like supercooling and superheating survive and hysteresis is observed on thermal cycling. [15] [16] [17]

Second-order phase transitions are also called "continuous phase transitions". They are characterized by a divergent susceptibility, an infinite correlation length, and a power law decay of correlations near criticality. Examples of second-order phase transitions are the ferromagnetic transition, superconducting transition (for a Type-I superconductor the phase transition is second-order at zero external field and for a Type-II superconductor the phase transition is second-order for both normal-state–mixed-state and mixed-state–superconducting-state transitions) and the superfluid transition. In contrast to viscosity, thermal expansion and heat capacity of amorphous materials show a relatively sudden change at the glass transition temperature [18] which enables accurate detection using differential scanning calorimetry measurements. Lev Landau gave a phenomenological theory of second-order phase transitions.

Apart from isolated, simple phase transitions, there exist transition lines as well as multicritical points, when varying external parameters like the magnetic field or composition.

Several transitions are known as infinite-order phase transitions. They are continuous but break no symmetries. The most famous example is the Kosterlitz–Thouless transition in the two-dimensional XY model. Many quantum phase transitions, e.g., in two-dimensional electron gases, belong to this class.

The liquid–glass transition is observed in many polymers and other liquids that can be supercooled far below the melting point of the crystalline phase. This is atypical in several respects. It is not a transition between thermodynamic ground states: it is widely believed that the true ground state is always crystalline. Glass is a quenched disorder state, and its entropy, density, and so on, depend on the thermal history. Therefore, the glass transition is primarily a dynamic phenomenon: on cooling a liquid, internal degrees of freedom successively fall out of equilibrium. Some theoretical methods predict an underlying phase transition in the hypothetical limit of infinitely long relaxation times. [19] [20] No direct experimental evidence supports the existence of these transitions.

Characteristic properties

Phase coexistence

A disorder-broadened first-order transition occurs over a finite range of temperatures where the fraction of the low-temperature equilibrium phase grows from zero to one (100%) as the temperature is lowered. This continuous variation of the coexisting fractions with temperature raised interesting possibilities. On cooling, some liquids vitrify into a glass rather than transform to the equilibrium crystal phase. This happens if the cooling rate is faster than a critical cooling rate, and is attributed to the molecular motions becoming so slow that the molecules cannot rearrange into the crystal positions. [21] This slowing down happens below a glass-formation temperature Tg, which may depend on the applied pressure. [18] [22] If the first-order freezing transition occurs over a range of temperatures, and Tg falls within this range, then there is an interesting possibility that the transition is arrested when it is partial and incomplete. Extending these ideas to first-order magnetic transitions being arrested at low temperatures, resulted in the observation of incomplete magnetic transitions, with two magnetic phases coexisting, down to the lowest temperature. First reported in the case of a ferromagnetic to anti-ferromagnetic transition, [23] such persistent phase coexistence has now been reported across a variety of first-order magnetic transitions. These include colossal-magnetoresistance manganite materials, [24] [25] magnetocaloric materials, [26] magnetic shape memory materials, [27] and other materials. [28] The interesting feature of these observations of Tg falling within the temperature range over which the transition occurs is that the first-order magnetic transition is influenced by magnetic field, just like the structural transition is influenced by pressure. The relative ease with which magnetic fields can be controlled, in contrast to pressure, raises the possibility that one can study the interplay between Tg and Tc in an exhaustive way. Phase coexistence across first-order magnetic transitions will then enable the resolution of outstanding issues in understanding glasses.

Critical points

In any system containing liquid and gaseous phases, there exists a special combination of pressure and temperature, known as the critical point, at which the transition between liquid and gas becomes a second-order transition. Near the critical point, the fluid is sufficiently hot and compressed that the distinction between the liquid and gaseous phases is almost non-existent. This is associated with the phenomenon of critical opalescence, a milky appearance of the liquid due to density fluctuations at all possible wavelengths (including those of visible light).

Symmetry

Phase transitions often involve a symmetry breaking process. For instance, the cooling of a fluid into a crystalline solid breaks continuous translation symmetry: each point in the fluid has the same properties, but each point in a crystal does not have the same properties (unless the points are chosen from the lattice points of the crystal lattice). Typically, the high-temperature phase contains more symmetries than the low-temperature phase due to spontaneous symmetry breaking, with the exception of certain accidental symmetries (e.g. the formation of heavy virtual particles, which only occurs at low temperatures). [29]

Order parameters

An order parameter is a measure of the degree of order across the boundaries in a phase transition system; it normally ranges between zero in one phase (usually above the critical point) and nonzero in the other. [30] At the critical point, the order parameter susceptibility will usually diverge.

An example of an order parameter is the net magnetization in a ferromagnetic system undergoing a phase transition. For liquid/gas transitions, the order parameter is the difference of the densities.

From a theoretical perspective, order parameters arise from symmetry breaking. When this happens, one needs to introduce one or more extra variables to describe the state of the system. For example, in the ferromagnetic phase, one must provide the net magnetization, whose direction was spontaneously chosen when the system cooled below the Curie point. However, note that order parameters can also be defined for non-symmetry-breaking transitions.[ citation needed ]

Some phase transitions, such as superconducting and ferromagnetic, can have order parameters for more than one degree of freedom. In such phases, the order parameter may take the form of a complex number, a vector, or even a tensor, the magnitude of which goes to zero at the phase transition.[ citation needed ]

There also exist dual descriptions of phase transitions in terms of disorder parameters. These indicate the presence of line-like excitations such as vortex- or defect lines.

Relevance in cosmology

Symmetry-breaking phase transitions play an important role in cosmology. As the universe expanded and cooled, the vacuum underwent a series of symmetry-breaking phase transitions. For example, the electroweak transition broke the SU(2)×U(1) symmetry of the electroweak field into the U(1) symmetry of the present-day electromagnetic field. This transition is important to explain the asymmetry between the amount of matter and antimatter in the present-day universe, according to electroweak baryogenesis theory.

Progressive phase transitions in an expanding universe are implicated in the development of order in the universe, as is illustrated by the work of Eric Chaisson [31] and David Layzer. [32]

See also relational order theories and order and disorder.

Critical exponents and universality classes

Continuous phase transitions are easier to study than first-order transitions due to the absence of latent heat, and they have been discovered to have many interesting properties. The phenomena associated with continuous phase transitions are called critical phenomena, due to their association with critical points.

Continuous phase transitions can be characterized by parameters known as critical exponents. The most important one is perhaps the exponent describing the divergence of the thermal correlation length by approaching the transition. For instance, let us examine the behavior of the heat capacity near such a transition. We vary the temperature T of the system while keeping all the other thermodynamic variables fixed and find that the transition occurs at some critical temperature Tc. When T is near Tc, the heat capacity C typically has a power law behavior:

The heat capacity of amorphous materials has such a behaviour near the glass transition temperature where the universal critical exponent α = 0.59 [33] A similar behavior, but with the exponent ν instead of α, applies for the correlation length.

The exponent ν is positive. This is different with α. Its actual value depends on the type of phase transition we are considering.

The critical exponents are not necessarily the same above and below the critical temperature. When a continuous symmetry is explicitly broken down to a discrete symmetry by irrelevant (in the renormalization group sense) anisotropies, then some exponents (such as , the exponent of the susceptibility) are not identical. [34]

For −1 <α< 0, the heat capacity has a "kink" at the transition temperature. This is the behavior of liquid helium at the lambda transition from a normal state to the superfluid state, for which experiments have found α = −0.013 ± 0.003. At least one experiment was performed in the zero-gravity conditions of an orbiting satellite to minimize pressure differences in the sample. [35] This experimental value of α agrees with theoretical predictions based on variational perturbation theory. [36]

For 0 <α< 1, the heat capacity diverges at the transition temperature (though, since α< 1, the enthalpy stays finite). An example of such behavior is the 3D ferromagnetic phase transition. In the three-dimensional Ising model for uniaxial magnets, detailed theoretical studies have yielded the exponent α ≈ +0.110.

Some model systems do not obey a power-law behavior. For example, mean field theory predicts a finite discontinuity of the heat capacity at the transition temperature, and the two-dimensional Ising model has a logarithmic divergence. However, these systems are limiting cases and an exception to the rule. Real phase transitions exhibit power-law behavior.

Several other critical exponents, β, γ, δ, ν, and η, are defined, examining the power law behavior of a measurable physical quantity near the phase transition. Exponents are related by scaling relations, such as

It can be shown that there are only two independent exponents, e.g. ν and η.

It is a remarkable fact that phase transitions arising in different systems often possess the same set of critical exponents. This phenomenon is known as universality. For example, the critical exponents at the liquid–gas critical point have been found to be independent of the chemical composition of the fluid.

More impressively, but understandably from above, they are an exact match for the critical exponents of the ferromagnetic phase transition in uniaxial magnets. Such systems are said to be in the same universality class. Universality is a prediction of the renormalization group theory of phase transitions, which states that the thermodynamic properties of a system near a phase transition depend only on a small number of features, such as dimensionality and symmetry, and are insensitive to the underlying microscopic properties of the system. Again, the divergence of the correlation length is the essential point.

Critical phenomena

There are also other critical phenomena; e.g., besides static functions there is also critical dynamics. As a consequence, at a phase transition one may observe critical slowing down or speeding up. Connected to the previous phenomenon is also the phenomenon of enhanced fluctuations before the phase transition, as a consequence of lower degree of stability of the initial phase of the system. The large static universality classes of a continuous phase transition split into smaller dynamic universality classes. In addition to the critical exponents, there are also universal relations for certain static or dynamic functions of the magnetic fields and temperature differences from the critical value.[ citation needed ]

Phase transitions in biological systems

Phase transitions play many important roles in biological systems. Examples include the lipid bilayer formation, the coil-globule transition in the process of protein folding and DNA melting, liquid crystal-like transitions in the process of DNA condensation, and cooperative ligand binding to DNA and proteins with the character of phase transition. [37]

In biological membranes, gel to liquid crystalline phase transitions play a critical role in physiological functioning of biomembranes. In gel phase, due to low fluidity of membrane lipid fatty-acyl chains, membrane proteins have restricted movement and thus are restrained in exercise of their physiological role. Plants depend critically on photosynthesis by chloroplast thylakoid membranes which are exposed cold environmental temperatures. Thylakoid membranes retain innate fluidity even at relatively low temperatures because of high degree of fatty-acyl disorder allowed by their high content of linolenic acid, 18-carbon chain with 3-double bonds. [38] Gel-to-liquid crystalline phase transition temperature of biological membranes can be determined by many techniques including calorimetry, fluorescence, spin label electron paramagnetic resonance and NMR by recording measurements of the concerned parameter by at series of sample temperatures. A simple method for its determination from 13-C NMR line intensities has also been proposed. [39]

It has been proposed that some biological systems might lie near critical points. Examples include neural networks in the salamander retina, [40] bird flocks [41] gene expression networks in Drosophila, [42] and protein folding. [43] However, it is not clear whether or not alternative reasons could explain some of the phenomena supporting arguments for criticality. [44] It has also been suggested that biological organisms share two key properties of phase transitions: the change of macroscopic behavior and the coherence of a system at a critical point. [45] Phase transitions are prominent feature of motor behavior in biological systems. [46] Spontaneous gait transitions, [47] as well as fatigue-induced motor task disengagements, [48] show typical critical behavior as an intimation of the sudden qualitative change of the previously stable motor behavioral pattern.

The characteristic feature of second order phase transitions is the appearance of fractals in some scale-free properties. It has long been known that protein globules are shaped by interactions with water. There are 20 amino acids that form side groups on protein peptide chains range from hydrophilic to hydrophobic, causing the former to lie near the globular surface, while the latter lie closer to the globular center. Twenty fractals were discovered in solvent associated surface areas of > 5000 protein segments. [49] The existence of these fractals proves that proteins function near critical points of second-order phase transitions.

In groups of organisms in stress (when approaching critical transitions), correlations tend to increase, while at the same time, fluctuations also increase. This effect is supported by many experiments and observations of groups of people, mice, trees, and grassy plants. [50]

Phase transitions in social systems

Phase transitions have been hypothesised to occur in social systems viewed as dynamical systems. A hypothesis proposed in the 1990s and 2000s in the context of peace and armed conflict is that when a conflict that is non-violent shifts to a phase of armed conflict, this is a phase transition from latent to manifest phases within the dynamical system. [51] : 49 

Experimental

A variety of methods are applied for studying the various effects. Selected examples are:

See also

Related Research Articles

<span class="mw-page-title-main">Condensed matter physics</span> Branch of physics

Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases, that arise from electromagnetic forces between atoms and electrons. More generally, the subject deals with condensed phases of matter: systems of many constituents with strong interactions among them. More exotic condensed phases include the superconducting phase exhibited by certain materials at extremely low cryogenic temperatures, the ferromagnetic and antiferromagnetic phases of spins on crystal lattices of atoms, the Bose–Einstein condensates found in ultracold atomic systems, and liquid crystals. Condensed matter physicists seek to understand the behavior of these phases by experiments to measure various material properties, and by applying the physical laws of quantum mechanics, electromagnetism, statistical mechanics, and other physics theories to develop mathematical models and predict the properties of extremely large groups of atoms.

<span class="mw-page-title-main">Spontaneous symmetry breaking</span> Symmetry breaking through the vacuum state

Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or the Lagrangian obey symmetries, but the lowest-energy vacuum solutions do not exhibit that same symmetry. When the system goes to one of those vacuum solutions, the symmetry is broken for perturbations around that vacuum even though the entire Lagrangian retains that symmetry.

In statistical mechanics, a universality class is a collection of mathematical models which share a single scale-invariant limit under the process of renormalization group flow. While the models within a class may differ dramatically at finite scales, their behavior will become increasingly similar as the limit scale is approached. In particular, asymptotic phenomena such as critical exponents will be the same for all models in the class.

In physics, critical phenomena is the collective name associated with the physics of critical points. Most of them stem from the divergence of the correlation length, but also the dynamics slows down. Critical phenomena include scaling relations among different quantities, power-law divergences of some quantities described by critical exponents, universality, fractal behaviour, and ergodicity breaking. Critical phenomena take place in second order phase transitions, although not exclusively.

In physics, a quantum phase transition (QPT) is a phase transition between different quantum phases. Contrary to classical phase transitions, quantum phase transitions can only be accessed by varying a physical parameter—such as magnetic field or pressure—at absolute zero temperature. The transition describes an abrupt change in the ground state of a many-body system due to its quantum fluctuations. Such a quantum phase transition can be a second-order phase transition. Quantum phase transitions can also be represented by the topological fermion condensation quantum phase transition, see e.g. strongly correlated quantum spin liquid. In case of three dimensional Fermi liquid, this transition transforms the Fermi surface into a Fermi volume. Such a transition can be a first-order phase transition, for it transforms two dimensional structure into three dimensional. As a result, the topological charge of Fermi liquid changes abruptly, since it takes only one of a discrete set of values.

<span class="mw-page-title-main">Magnon</span> Spin 1 quasiparticle; quantum of a spin wave

A magnon is a quasiparticle, a collective excitation of the spin structure of an electron in a crystal lattice. In the equivalent wave picture of quantum mechanics, a magnon can be viewed as a quantized spin wave. Magnons carry a fixed amount of energy and lattice momentum, and are spin-1, indicating they obey boson behavior.

<span class="mw-page-title-main">Lambda point</span>

The lambda point is the temperature at which normal fluid helium makes the transition to superfluid helium II. The lowest pressure at which He-I and He-II can coexist is the vapor−He-I−He-II triple point at 2.1768 K (−270.9732 °C) and 5.0418 kPa (0.049759 atm), which is the "saturated vapor pressure" at that temperature. The highest pressure at which He-I and He-II can coexist is the bcc−He-I−He-II triple point with a helium solid at 1.762 K (−271.388 °C), 29.725 atm (3,011.9 kPa).

Critical exponents describe the behavior of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e. they do not depend on the details of the physical system, but only on some of its general features. For instance, for ferromagnetic systems at thermal equilibrium, the critical exponents depend only on:

Multiferroics are defined as materials that exhibit more than one of the primary ferroic properties in the same phase:

A quantum critical point is a point in the phase diagram of a material where a continuous phase transition takes place at absolute zero. A quantum critical point is typically achieved by a continuous suppression of a nonzero temperature phase transition to zero temperature by the application of a pressure, field, or through doping. Conventional phase transitions occur at nonzero temperature when the growth of random thermal fluctuations leads to a change in the physical state of a system. Condensed matter physics research over the past few decades has revealed a new class of phase transitions called quantum phase transitions which take place at absolute zero. In the absence of the thermal fluctuations which trigger conventional phase transitions, quantum phase transitions are driven by the zero point quantum fluctuations associated with Heisenberg's uncertainty principle.

A Josephson junction (JJ) is a quantum mechanical device which is made of two superconducting electrodes separated by a barrier. A π Josephson junction is a Josephson junction in which the Josephson phase φ equals π in the ground state, i.e. when no external current or magnetic field is applied.

In materials science, heavy fermion materials are a specific type of intermetallic compound, containing elements with 4f or 5f electrons in unfilled electron bands. Electrons are one type of fermion, and when they are found in such materials, they are sometimes referred to as heavy electrons. Heavy fermion materials have a low-temperature specific heat whose linear term is up to 1000 times larger than the value expected from the free electron model. The properties of the heavy fermion compounds often derive from the partly filled f-orbitals of rare-earth or actinide ions, which behave like localized magnetic moments.

In condensed matter physics, a quantum spin liquid is a phase of matter that can be formed by interacting quantum spins in certain magnetic materials. Quantum spin liquids (QSL) are generally characterized by their long-range quantum entanglement, fractionalized excitations, and absence of ordinary magnetic order.

The Kibble–Zurek mechanism (KZM) describes the non-equilibrium dynamics and the formation of topological defects in a system which is driven through a continuous phase transition at finite rate. It is named after Tom W. B. Kibble, who pioneered the study of domain structure formation through cosmological phase transitions in the early universe, and Wojciech H. Zurek, who related the number of defects it creates to the critical exponents of the transition and to its rate—to how quickly the critical point is traversed.

<span class="mw-page-title-main">Arrott plot</span>

In condensed matter physics, an Arrott plot is a plot of the square of the magnetization of a substance, against the ratio of the applied magnetic field to magnetization at one fixed temperature(s). Arrott plots are an easy way of determining the presence of ferromagnetic order in a material. They are named after American physicist Anthony Arrott who introduced them as a technique for studying magnetism in 1957.

In theoretical physics, the curvature renormalization group (CRG) method is an analytical approach to determine the phase boundaries and the critical behavior of topological systems. Topological phases are phases of matter that appear in certain quantum mechanical systems at zero temperature because of a robust degeneracy in the ground-state wave function. They are called topological because they can be described by different (discrete) values of a nonlocal topological invariant. This is to contrast with non-topological phases of matter that can be described by different values of a local order parameter. States with different values of the topological invariant cannot change into each other without a phase transition. The topological invariant is constructed from a curvature function that can be calculated from the bulk Hamiltonian of the system. At the phase transition, the curvature function diverges, and the topological invariant correspondingly jumps abruptly from one value to another. The CRG method works by detecting the divergence in the curvature function, and thus determining the boundaries between different topological phases. Furthermore, from the divergence of the curvature function, it extracts scaling laws that describe the critical behavior, i.e. how different quantities behave as the topological phase transition is approached. The CRG method has been successfully applied to a variety of static, periodically driven, weakly and strongly interacting systems to classify the nature of the corresponding topological phase transitions.

<span class="mw-page-title-main">Amnon Aharony</span> Physicist at Ben Gurion University in Israel

Amnon Aharony is an Israeli Professor (Emeritus) of Physics in the School of Physics and Astronomy at Tel Aviv University, Israel and in the Physics Department of Ben Gurion University of the Negev, Israel. After years of research on statistical physics, his current research focuses on condensed matter theory, especially in mesoscopic physics and spintronics. He is a member of the Israel Academy of Sciences and Humanities, a Foreign Honorary Member of the American Academy of Arts and Sciences and of several other academies. He also received several prizes, including the Rothschild Prize in Physical Sciences, and the Gunnar Randers Research Prize, awarded every other year by the King of Norway.

Shang-keng Ma was a Chinese theoretical physicist, known for his work on the theory of critical phenomena and random systems. He is known as the co-author with Bertrand Halperin and Pierre Hohenberg of a 1972 paper that "generalized the renormalization group theory to dynamical critical phenomena." Ma is also known as the co-author with Yoseph Imry of a 1975 paper and with Amnon Aharony and Imry of a 1976 paper that established the foundation of the random field Ising model (RFIM)

<span class="mw-page-title-main">Carlo Di Castro</span> Italian theoretical physicist

Carlo Di Castro is an Italian theoretical physicist in the field of statistical mechanics, superconductivity, and condensed matter physics. He is a patriarch of Italian theoretical condensed matter physics, founder of the “Rome Group”, member of the Accademia dei Lincei, and emeritus professor of Sapienza University of Rome. In 1969, Di Castro, in co-authorship with Giovanni Jona-Lasinio, introduced the revolutionary renormalization group approach into the study of critical phenomena, providing a first example of complexity in physical systems.

References

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  2. ↑ Rybin, M.V.; et al. (2015). "Phase diagram for the transition from photonic crystals to dielectric metamaterials". Nature Communications. 6: 10102. arXiv: 1507.08901 . Bibcode:2015NatCo...610102R. doi:10.1038/ncomms10102. PMC   4686770 . PMID   26626302.
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