Fermi gas

From Wikipedia, the free encyclopedia

An ideal Fermi gas is a phase of matter which is an ensemble of a large number of non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons and neutrons, and in general, particles with half-integer spin. These statistics determine the energy distribution of fermions in a Fermi gas in thermal equilibrium, and is characterized by their number density, temperature, and the set of available energy states. The model is named after the Italian physicist Enrico Fermi.

This physical model can be accurately applied to many systems with a large number of fermions. Some key examples are the behaviour of charge carriers in a metal, nucleons in an atomic nucleus, neutrons in a neutron star, and electrons in a white dwarf.

Description

Illustration of the energy states: Energy-occupation diagram for a system with 7 energy levels, the energy E_i is degenerate D_i times and has an occupancy given by N_i, with i=1,2,3,4,5,6,7. By the Pauli exclusion principle, up to D_i fermions can occupy a level of energy E_i of the system.

An ideal Fermi gas or free Fermi gas is a physical model assuming a collection of non-interacting fermions in a constant potential well. It is the quantum mechanical version of an ideal gas, for the case of fermionic particles.

By the Pauli exclusion principle, no quantum state can be occupied by more than one fermion with an identical set of quantum numbers. Thus a non-interacting Fermi gas, unlike a Bose gas, concentrates a small number of particles per energy. Thus a Fermi gas is prohibited from condensing into a Bose–Einstein condensate, although weakly-interacting Fermi gases might. The total energy of the Fermi gas at absolute zero is larger than the sum of the single-particle ground states because the Pauli principle implies a sort of interaction or pressure that keeps fermions separated and moving. For this reason, the pressure of a Fermi gas is non-zero even at zero temperature, in contrast to that of a classical ideal gas. For example, this so-called degeneracy pressure stabilizes a neutron star (a Fermi gas of neutrons) or a white dwarf star (a Fermi gas of electrons) against the inward pull of gravity, which would ostensibly collapse the star into a black hole. Only when a star is sufficiently massive to overcome the degeneracy pressure can it collapse into a singularity.

It is possible to define a Fermi temperature below which the gas can be considered degenerate (its pressure derives almost exclusively from the Pauli principle). This temperature depends on the mass of the fermions and the density of energy states.

The main assumption of the free electron model to describe the delocalized electrons in a metal can be derived from the Fermi gas. Since interactions are neglected due to screening effect, the problem of treating the equilibrium properties and dynamics of an ideal Fermi gas reduces to the study of the behaviour of single independent particles. In these systems the Fermi temperature is generally many thousands of kelvins, so in human applications the electron gas can be considered degenerate. The maximum energy of the fermions at zero temperature is called the Fermi energy. The Fermi energy surface in momentum space is known as the Fermi surface.

The nearly free electron model adapts the Fermi gas model to consider the crystal structure of metals and semiconductors. Where electrons in a crystal lattice are substituted by Bloch waves with a corresponding crystal momentum. As such, periodic systems are still relatively tractable and the model forms the starting point for more advanced theories that deal with interactions, e.g., using the perturbation theory.

Illustration of the Fermi energy for a one-dimensional well

The one-dimensional infinite square well of length L is a model for a one-dimensional box. It is a standard model-system in quantum mechanics for which the solution for a single particle is well known. The levels are labelled by a single quantum number n and the energies are given by

${\displaystyle E_{n}=E_{0}+{\frac {\hbar ^{2}\pi ^{2}}{2mL^{2}}}n^{2}.\,}$

where ${\displaystyle E_{0}}$ is the potential energy level inside the box, ${\displaystyle m}$ the mass of a single fermion, and ${\displaystyle \hbar }$ is the reduced Planck constant.

Suppose now that instead of one particle in this box we have N particles in the box and that these particles are fermions with spin-½. Then not more than two particles can have the same energy, i.e., two particles can have the energy of ${\textstyle E_{1}}$, two other particles can have energy ${\textstyle E_{2}}$ and so forth. The reason that two particles can have the same energy is that a particle can have a spin of ½ (spin up) or a spin of −½ (spin down), leading to two states for each energy level. In the configuration for which the total energy is lowest (the ground state), all the energy levels up to n = N/2 are occupied and all the higher levels are empty.

Defining the reference for the Fermi energy to be ${\displaystyle E_{0}}$, the Fermi energy is therefore given by

${\displaystyle E_{\mathrm {F} }^{({\text{1D}})}=E_{N^{*}/2}-E_{0}={\frac {\hbar ^{2}\pi ^{2}}{2mL^{2}}}(N^{*}/2)^{2},}$

where N* is equal to (N − 1) when the number of particles N is odd, and its equal to (N) for an even number of particles.

Three-dimensional case

A model of the atomic nucleus showing it as a compact bundle of the two types of nucleons: protons (red) and neutrons (blue). As a first approximation, the nucleus can be treated as composed of non-interacting proton and neutron gases.

The three-dimensional isotropic and non-relativistic case is known as the Fermi sphere.

Let us now consider a three-dimensional infinite square well, that is, a cubical box that has a side length L. The states are now labelled by three quantum numbers n_x, n_y, and n_z. The single particle energies are

${\displaystyle E_{n_{x},n_{y},n_{z}}=E_{0}+{\frac {\hbar ^{2}\pi ^{2}}{2mL^{2}}}\left(n_{x}^{2}+n_{y}^{2}+n_{z}^{2}\right)\,}$,

where n_x, n_y, n_z are positive integers. There are multiple states with the same energy, for example ${\displaystyle E_{211}=E_{121}=E_{112}}$. Now let's put N non-interacting fermions of spin ½ into this box. To calculate the Fermi energy, we look at the case where N is large.

If we introduce a vector ${\displaystyle \mathbf {n} =(n_{x},n_{y},n_{z})}$ then each quantum state corresponds to a point in 'n-space' with energy

${\displaystyle E_{\mathbf {n} }=E_{0}+{\frac {\hbar ^{2}\pi ^{2}}{2mL^{2}}}|\mathbf {n} |^{2}\,}$

With ${\displaystyle |\mathbf {n} |^{2}}$denoting the square of the usual Euclidean length ${\displaystyle |\mathbf {n} |={\sqrt {n_{x}^{2}+n_{y}^{2}+n_{z}^{2}}}}$. The number of states with energy less than E_F +  E₀ is equal to the number of states that lie within a sphere of radius ${\displaystyle |\mathbf {n} _{\mathrm {F} }|}$ in the region of n-space where n_x, n_y, n_z are positive. In the ground state this number equals the number of fermions in the system.

${\displaystyle N=2\times {\frac {1}{8}}\times {\frac {4}{3}}\pi n_{\mathrm {F} }^{3}\,}$

The free fermions that occupy the lowest energy states form a sphere in momentum space. The surface of this sphere is the Fermi surface.

The factor of two is found again because there are two spin states, while the factor of 1/8 is because only 1/8 of the sphere lies in the region where all n are positive. We find

${\displaystyle n_{\mathrm {F} }=\left({\frac {3N}{\pi }}\right)^{1/3}}$

so the Fermi energy is given by

${\displaystyle E_{\mathrm {F} }={\frac {\hbar ^{2}\pi ^{2}}{2mL^{2}}}n_{\mathrm {F} }^{2}={\frac {\hbar ^{2}\pi ^{2}}{2mL^{2}}}\left({\frac {3N}{\pi }}\right)^{2/3}}$

Which results in a relationship between the Fermi energy and the number of particles per volume (when we replace L² with V^2/3):

${\displaystyle E_{\mathrm {F} }={\frac {\hbar ^{2}}{2m}}\left({\frac {3\pi ^{2}N}{V}}\right)^{2/3}\,}$

The total energy of a Fermi sphere of ${\displaystyle N}$ fermions is given by

${\displaystyle E_{\rm {T}}=NE_{0}+\int _{0}^{N}E_{\mathrm {F} }\,\mathrm {d} N^{\prime }=\left({\frac {3}{5}}E_{\mathrm {F} }+E_{0}\right)N}$

Therefore, the average energy per particle is given by:

${\displaystyle E_{\mathrm {av} }=E_{0}+{\frac {3}{5}}E_{\mathrm {F} }}$

Related quantities

Using this definition of Fermi energy, various related quantities can be useful. The Fermi temperature is defined as:

${\displaystyle T_{\mathrm {F} }={\frac {E_{\mathrm {F} }}{k_{\rm {B}}}}}$

where ${\displaystyle k_{\rm {B}}}$ is the Boltzmann constant. The Fermi temperature can be thought of as the temperature at which thermal effects are comparable to quantum effects associated with Fermi statistics. The Fermi temperature for a metal is a couple of orders of magnitude above room temperature.

Other quantities defined in this context are Fermi momentum

${\displaystyle p_{\mathrm {F} }={\sqrt {2mE_{\mathrm {F} }}}}$,

and Fermi velocity

${\displaystyle v_{\mathrm {F} }={\frac {p_{\mathrm {F} }}{m}}}$.

These quantities are the momentum and group velocity, respectively, of a fermion at the Fermi surface. The Fermi momentum can also be described as ${\displaystyle p_{\mathrm {F} }=\hbar k_{\mathrm {F} }}$, where ${\displaystyle k_{\mathrm {F} }}$ is the radius of the Fermi sphere and is called the Fermi wave vector.

These quantities are not well-defined in cases where the Fermi surface is non-spherical.

Thermodynamic quantities

Degeneracy pressure

As we showed above, a Fermi gas has a non-zero total energy even at zero temperature. By using the first law of thermodynamics, we can associate a pressure to this internal energy, that is

${\displaystyle P=-{\frac {\partial E_{\rm {T}}}{\partial V}}={\frac {2}{5}}{\frac {N}{V}}E_{\mathrm {F} }={\frac {(3\pi ^{2})^{2/3}\hbar ^{2}}{5m}}\left({\frac {N}{V}}\right)^{5/3},}$

where the expression is only valid for temperatures much smaller than the Fermi temperature. This pressure is known as the degeneracy pressure. In this sense, systems composed of fermions are also referred as degenerate matter.

Standard stars avoid collapse by balancing thermal pressure (plasma and radiation) against gravitational forces. At the end of the star lifetime, when thermal processes are weaker, some stars may become white dwarfs, which are only sustained against gravity by electron degeneracy pressure. Using the Fermi gas as a model, it is possible to calculate the Chandrasekhar limit, i.e. the maximum mass any star may acquire (without significant thermally generated pressure) before collapsing into a black hole or a neutron star. The latter, is a star mainly composed of neutrons, where the collapse is also avoided by neutron degeneracy pressure.

For the case of metals, the electron degeneracy pressure contributes to the compressibility or bulk modulus of the material.

Chemical potential

Assuming that the concentration of fermions does not change with temperature, then the total chemical potential µ (Fermi level) of the three-dimensional ideal Fermi gas is related to the zero temperature Fermi energy E_F by a Sommerfeld expansion (assuming ${\displaystyle k_{\rm {B}}T\ll E_{\mathrm {F} }}$):

${\displaystyle \mu (T)=E_{0}+E_{\mathrm {F} }\left[1-{\frac {\pi ^{2}}{12}}\left({\frac {k_{\rm {B}}T}{E_{\mathrm {F} }}}\right)^{2}-{\frac {\pi ^{4}}{80}}\left({\frac {k_{\rm {B}}T}{E_{\mathrm {F} }}}\right)^{4}+\cdots \right]}$,

where T is the temperature.

Hence, the internal chemical potential, µ-E₀, is approximately equal to the Fermi energy at temperatures that are much lower than the characteristic Fermi temperature T_F. This characteristic temperature is on the order of 10⁵ K for a metal, hence at room temperature (300 K), the Fermi energy and internal chemical potential are essentially equivalent.

Density of states

Figure 3: Density of states (DOS) of a Fermi gas in 3-dimensions

For the three dimensional case, with fermions of spin-½, we can obtain the number of particles as a function of the energy ${\textstyle N(E)}$ by substituting the Fermi energy by a variable energy ${\textstyle (E-E_{0})}$:

${\displaystyle N(E)={\frac {V}{3\pi ^{2}}}\left[{\frac {2m}{\hbar ^{2}}}(E-E_{0})\right]^{3/2}}$,

which we can use to obtain the density of states (number of energy states per energy per volume) ${\displaystyle g(E)}$. It can be calculated by differentiating the number of particles with respect to the energy:

${\displaystyle g(E)={\frac {1}{V}}{\frac {\partial N(E)}{\partial E}}={\frac {1}{2\pi ^{2}}}\left({\frac {2m}{\hbar ^{2}}}\right)^{3/2}{\sqrt {E-E_{0}}}}$.

Arbitrary-dimensional case

Using a volume integral on ${\textstyle d}$ dimensions, we can find the state density:

${\displaystyle g^{(d)}(E)=g_{s}\int {\frac {\mathrm {d} ^{d}\mathbf {k} }{(2\pi )^{d}}}\delta \left(E-E_{0}-{\frac {\hbar ^{2}|\mathbf {k} |^{2}}{2m}}\right)=g_{s}\ {\frac {d}{2}}\left({\frac {m}{2\pi \hbar ^{2}}}\right)^{d/2}{\frac {(E-E_{0})^{d/2-1}}{\Gamma (d/2+1)}}}$

By then looking for the number density of particles, we can extract the Fermi energy:

${\displaystyle \rho ={\frac {N}{V}}=\int _{E_{0}}^{E_{0}+E_{\mathrm {F} }^{(d)}}g^{(d)}(E)\,\mathrm {d} E}$

To get:

${\displaystyle E_{\mathrm {F} }^{(d)}={\frac {2\pi \hbar ^{2}}{m}}\left({\tfrac {1}{g_{s}}}\Gamma \left({\tfrac {d}{2}}+1\right){\frac {N}{V}}\right)^{2/d}}$

where ${\textstyle V}$ is the corresponding d-dimensional volume, ${\textstyle g_{s}}$ is the dimension for the internal Hilbert space. For the case of spin-½, every energy is twice-degenerate, so in this case ${\textstyle g_{s}=2}$.

A particular result is obtained for ${\displaystyle d=2}$, where the density of states becomes a constant (does not depend on the energy):

${\displaystyle g^{(2\mathrm {D} )}(E)={\frac {g_{s}}{2}}{\frac {m}{\pi \hbar ^{2}}}}$.

Typical values

Metals

Under the free electron model, the electrons in a metal can be considered to form a Fermi gas. The number density ${\displaystyle N/V}$ of conduction electrons in metals ranges between approximately 10²⁸ and 10²⁹ electrons/m³, which is also the typical density of atoms in ordinary solid matter. This number density produces a Fermi energy of the order:

${\displaystyle E_{\mathrm {F} }={\frac {\hbar ^{2}}{2m_{e}}}\left(3\pi ^{2}\ 10^{28\ \sim \ 29}\ \mathrm {m} ^{-3}\right)^{2/3}\approx 2\ \sim \ 10\ \mathrm {eV} }$,

where m_e is the electron mass. This Fermi energy corresponds to a Fermi temperature of the order of 10⁶ kelvins, much higher than the temperature of the sun surface. Any metal will boil before reaching this temperature under atmospheric pressure. Thus for any practical purpose a metal, can be considered as a Fermi gas at zero temperature as a first approximation (normal temperatures are small compared to T_F).

White dwarfs

Stars known as white dwarfs have mass comparable to our Sun, but have about a hundredth of its radius. The high densities mean that the electrons are no longer bound to single nuclei and instead form a degenerate electron gas. The number density of electrons in a white dwarf is of the order of 10³⁶ electrons/m³. This means their Fermi energy is:

${\displaystyle E_{\mathrm {F} }={\frac {\hbar ^{2}}{2m_{e}}}\left({\frac {3\pi ^{2}(10^{36})}{1\ \mathrm {m} ^{3}}}\right)^{2/3}\approx 3\times 10^{5}\ \mathrm {eV} =0.3\ \mathrm {MeV} }$

Nucleus

Another typical example is that of the particles in a nucleus of an atom. The radius of the nucleus is roughly:

${\displaystyle R=\left(1.25\times 10^{-15}\mathrm {m} \right)\times A^{1/3}}$

where A is the number of nucleons.
The number density of nucleons in a nucleus is therefore:

${\displaystyle \rho ={\frac {A}{{\begin{matrix}{\frac {4}{3}}\end{matrix}}\pi R^{3}}}\approx 1.2\times 10^{44}\ \mathrm {m} ^{-3}}$

Now since the Fermi energy only applies to fermions of the same type, one must divide this density in two. This is because the presence of neutrons does not affect the Fermi energy of the protons in the nucleus, and vice versa.

So the Fermi energy of a nucleus is about:

${\displaystyle E_{\mathrm {F} }={\frac {\hbar ^{2}}{2m_{\rm {p}}}}\left({\frac {3\pi ^{2}(6\times 10^{43})}{1\ \mathrm {m} ^{3}}}\right)^{2/3}\approx 3\times 10^{7}\ \mathrm {eV} =30\ \mathrm {MeV} }$,

where m_p is the proton mass.

The radius of the nucleus admits deviations around the value mentioned above, so a typical value for the Fermi energy is usually given as 38 MeV.

Extensions to the model

Relativistic Fermi gas

Radius–mass relations for a model white dwarf, relativistic relation vs non-relativistic. The Chandrasekhar limit is indicated as M_Ch.

For the whole article, we have only discussed the case where particles have a parabolic relation between energy and momentum, as is the case in non-relativistic mechanics. For particles with energies close to their respective rest mass, we have to use the equations of special relativity. Where single-particle energy is given by:

${\displaystyle E={\sqrt {(pc)^{2}+(mc^{2})^{2}}}}$.

For this system, the Fermi energy is given by:

${\displaystyle E_{\mathrm {F} }={\sqrt {(p_{\mathrm {F} }c)^{2}+(mc^{2})^{2}}}-mc^{2}\approx p_{\mathrm {F} }c}$ ,

where the ${\displaystyle \approx }$ equality is only valid in the ultrarelativistic limit, and

${\displaystyle p_{\mathrm {F} }=\hbar \left({\frac {1}{g_{s}}}6\pi ^{2}{\frac {N}{V}}\right)^{1/3}}$.

The relativistic Fermi gas model is also used for the description of large white dwarfs which are close to the Chandresekhar limit. For the ultrarelativistic case, the degeneracy pressure is proportional to ${\displaystyle (N/V)^{4/3}}$.

Fermi liquid

In 1956, Lev Landau developed the Fermi liquid theory, where he treated the case of a Fermi liquid, i.e., a system with repulsive, not necessarily small, interactions between fermions. The theory shows that the thermodynamic properties of an ideal Fermi gas and a Fermi liquid do not differ that much. It can be shown that the Fermi liquid is equivalent to a Fermi gas composed of collective excitations or quasiparticles, each with a different effective mass and magnetic moment.

State of matter

From Wikipedia, the free encyclopedia

The four fundamental states of matter. Clockwise from top left, they are solid, liquid, plasma, and gas, represented by an ice sculpture, a drop of water, electrical arcing from a tesla coil, and the air around clouds, respectively.

In physics, a state of matter is one of the distinct forms in which matter can exist. Four states of matter are observable in everyday life: solid, liquid, gas, and plasma. Many other states are known to exist, such as glass or liquid crystal, and some only exist under extreme conditions, such as Bose–Einstein condensates, neutron-degenerate matter, and quark-gluon plasma, which only occur, respectively, in situations of extreme cold, extreme density, and extremely high-energy. Some other states are believed to be possible but remain theoretical for now.

Historically, the distinction is made based on qualitative differences in properties. Matter in the solid state maintains a fixed volume and shape, with component particles (atoms, molecules or ions) close together and fixed into place. Matter in the liquid state maintains a fixed volume, but has a variable shape that adapts to fit its container. Its particles are still close together but move freely. Matter in the gaseous state has both variable volume and shape, adapting both to fit its container. Its particles are neither close together nor fixed in place. Matter in the plasma state has variable volume and shape, but as well as neutral atoms, it contains a significant number of ions and electrons, both of which can move around freely.

The term phase is sometimes used as a synonym for state of matter, but a system can contain several immiscible phases of the same state of matter.

The four fundamental states

Solid

A crystalline solid: atomic resolution image of strontium titanate. Brighter atoms are Sr and darker ones are Ti.

In a solid, constituent particles (ions, atoms, or molecules) are closely packed together. The forces between particles are so strong that the particles cannot move freely but can only vibrate. As a result, a solid has a stable, definite shape, and a definite volume. Solids can only change their shape by force, as when broken or cut.

In crystalline solids, the particles (atoms, molecules, or ions) are packed in a regularly ordered, repeating pattern. There are various different crystal structures, and the same substance can have more than one structure (or solid phase). For example, iron has a body-centred cubic structure at temperatures below 912 °C, and a face-centred cubic structure between 912 and 1394 °C. Ice has fifteen known crystal structures, or fifteen solid phases, which exist at various temperatures and pressures.

Glasses and other non-crystalline, amorphous solids without long-range order are not thermal equilibrium ground states; therefore they are described below as nonclassical states of matter.

Solids can be transformed into liquids by melting, and liquids can be transformed into solids by freezing. Solids can also change directly into gases through the process of sublimation, and gases can likewise change directly into solids through deposition.

Liquid

Structure of a classical monatomic liquid. Atoms have many nearest neighbors in contact, yet no long-range order is present.

A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. The volume is definite if the temperature and pressure are constant. When a solid is heated above its melting point, it becomes liquid, given that the pressure is higher than the triple point of the substance. Intermolecular (or interatomic or interionic) forces are still important, but the molecules have enough energy to move relative to each other and the structure is mobile. This means that the shape of a liquid is not definite but is determined by its container. The volume is usually greater than that of the corresponding solid, the best known exception being water, H₂O. The highest temperature at which a given liquid can exist is its critical temperature.

Gas

The spaces between gas molecules are very big. Gas molecules have very weak or no bonds at all. The molecules in "gas" can move freely and fast.

A gas is a compressible fluid. Not only will a gas conform to the shape of its container but it will also expand to fill the container.

In a gas, the molecules have enough kinetic energy so that the effect of intermolecular forces is small (or zero for an ideal gas), and the typical distance between neighboring molecules is much greater than the molecular size. A gas has no definite shape or volume, but occupies the entire container in which it is confined. A liquid may be converted to a gas by heating at constant pressure to the boiling point, or else by reducing the pressure at constant temperature.

At temperatures below its critical temperature, a gas is also called a vapor, and can be liquefied by compression alone without cooling. A vapor can exist in equilibrium with a liquid (or solid), in which case the gas pressure equals the vapor pressure of the liquid (or solid).

A supercritical fluid (SCF) is a gas whose temperature and pressure are above the critical temperature and critical pressure respectively. In this state, the distinction between liquid and gas disappears. A supercritical fluid has the physical properties of a gas, but its high density confers solvent properties in some cases, which leads to useful applications. For example, supercritical carbon dioxide is used to extract caffeine in the manufacture of decaffeinated coffee.

Plasma

In a plasma, electrons are ripped away from their nuclei, forming an electron "sea". This gives it the ability to conduct electricity.

Like a gas, plasma does not have definite shape or volume. Unlike gases, plasmas are electrically conductive, produce magnetic fields and electric currents, and respond strongly to electromagnetic forces. Positively charged nuclei swim in a "sea" of freely-moving disassociated electrons, similar to the way such charges exist in conductive metal, where this electron "sea" allows matter in the plasma state to conduct electricity.

A gas is usually converted to a plasma in one of two ways. e.g. Either from a huge voltage difference between two points, or by exposing it to extremely high temperatures. Heating matter to high temperatures causes electrons to leave the atoms, resulting in the presence of free electrons. This creates a so-called partially ionised plasma. At very high temperatures, such as those present in stars, it is assumed that essentially all electrons are "free", and that a very high-energy plasma is essentially bare nuclei swimming in a sea of electrons. This forms the so-called fully ionised plasma.

The plasma state is often misunderstood, and although not freely existing under normal conditions on Earth, it is quite commonly generated by either lightning, electric sparks, fluorescent lights, neon lights or in plasma televisions. Also plasma appears in some types of flame, the Sun's corona, and stars are all examples of illuminated matter in the plasma state.

Phase transitions

The transitions between the four fundamental states of matter.

A state of matter is also characterized by phase transitions. A phase transition indicates a change in structure and can be recognized by an abrupt change in properties. A distinct state of matter can be defined as any set of states distinguished from any other set of states by a phase transition. Water can be said to have several distinct solid states. The appearance of superconductivity is associated with a phase transition, so there are superconductive states. Likewise, ferromagnetic states are demarcated by phase transitions and have distinctive properties. When the change of state occurs in stages the intermediate steps are called mesophases. Such phases have been exploited by the introduction of liquid crystal technology.

The state or phase of a given set of matter can change depending on pressure and temperature conditions, transitioning to other phases as these conditions change to favor their existence; for example, solid transitions to liquid with an increase in temperature. Near absolute zero, a substance exists as a solid. As heat is added to this substance it melts into a liquid at its melting point, boils into a gas at its boiling point, and if heated high enough would enter a plasma state in which the electrons are so energized that they leave their parent atoms.

Forms of matter that are not composed of molecules and are organized by different forces can also be considered different states of matter. Superfluids (like Fermionic condensate) and the quark–gluon plasma are examples.

In a chemical equation, the state of matter of the chemicals may be shown as (s) for solid, (l) for liquid, and (g) for gas. An aqueous solution is denoted (aq). Matter in the plasma state is seldom used (if at all) in chemical equations, so there is no standard symbol to denote it. In the rare equations that plasma is used in plasma is symbolized as (p).

Non-classical states

Glass

Schematic representation of a random-network glassy form (left) and ordered crystalline lattice (right) of identical chemical composition.

Glass is a non-crystalline or amorphous solid material that exhibits a glass transition when heated towards the liquid state. Glasses can be made of quite different classes of materials: inorganic networks (such as window glass, made of silicate plus additives), metallic alloys, ionic melts, aqueous solutions, molecular liquids, and polymers. Thermodynamically, a glass is in a metastable state with respect to its crystalline counterpart. The conversion rate, however, is practically zero.

Crystals with some degree of disorder

A plastic crystal is a molecular solid with long-range positional order but with constituent molecules retaining rotational freedom; in an orientational glass this degree of freedom is frozen in a quenched disordered state.

Similarly, in a spin glass magnetic disorder is frozen.

Liquid crystal states

Liquid crystal states have properties intermediate between mobile liquids and ordered solids. Generally, they are able to flow like a liquid, but exhibiting long-range order. For example, the nematic phase consists of long rod-like molecules such as para-azoxyanisole, which is nematic in the temperature range 118–136 °C. In this state the molecules flow as in a liquid, but they all point in the same direction (within each domain) and cannot rotate freely. Like a crystalline solid, but unlike a liquid, liquid crystals react to polarized light.

Other types of liquid crystals are described in the main article on these states. Several types have technological importance, for example, in liquid crystal displays.

Magnetically ordered

Transition metal atoms often have magnetic moments due to the net spin of electrons that remain unpaired and do not form chemical bonds. In some solids the magnetic moments on different atoms are ordered and can form a ferromagnet, an antiferromagnet or a ferrimagnet.

In a ferromagnet—for instance, solid iron—the magnetic moment on each atom is aligned in the same direction (within a magnetic domain). If the domains are also aligned, the solid is a permanent magnet, which is magnetic even in the absence of an external magnetic field. The magnetization disappears when the magnet is heated to the Curie point, which for iron is 768 °C.

An antiferromagnet has two networks of equal and opposite magnetic moments, which cancel each other out so that the net magnetization is zero. For example, in nickel(II) oxide (NiO), half the nickel atoms have moments aligned in one direction and half in the opposite direction.

In a ferrimagnet, the two networks of magnetic moments are opposite but unequal, so that cancellation is incomplete and there is a non-zero net magnetization. An example is magnetite (Fe₃O₄), which contains Fe²⁺ and Fe³⁺ ions with different magnetic moments.

A quantum spin liquid (QSL) is a disordered state in a system of interacting quantum spins which preserves its disorder to very low temperatures, unlike other disordered states. It is not a liquid in physical sense, but a solid whose magnetic order is inherently disordered. The name "liquid" is due to an analogy with the molecular disorder in a conventional liquid. A QSL is neither a ferromagnet, where magnetic domains are parallel, nor an antiferromagnet, where the magnetic domains are antiparallel; instead, the magnetic domains are randomly oriented. This can be realized e.g. by geometrically frustrated magnetic moments that cannot point uniformly parallel or antiparallel. When cooling down and settling to a state, the domain must "choose" an orientation, but if the possible states are similar in energy, one will be chosen randomly. Consequently, despite strong short-range order, there is no long-range magnetic order.

Microphase-separated

SBS block copolymer in TEM

Copolymers can undergo microphase separation to form a diverse array of periodic nanostructures, as shown in the example of the styrene-butadiene-styrene block copolymer shown at right. Microphase separation can be understood by analogy to the phase separation between oil and water. Due to chemical incompatibility between the blocks, block copolymers undergo a similar phase separation. However, because the blocks are covalently bonded to each other, they cannot demix macroscopically as water and oil can, and so instead the blocks form nanometer-sized structures. Depending on the relative lengths of each block and the overall block topology of the polymer, many morphologies can be obtained, each its own phase of matter.

Ionic liquids also display microphase separation. The anion and cation are not necessarily compatible and would demix otherwise, but electric charge attraction prevents them from separating. Their anions and cations appear to diffuse within compartmentalized layers or micelles instead of freely as in a uniform liquid.

Low-temperature states

Superfluid

Liquid helium in a superfluid phase creeps up on the walls of the cup in a Rollin film, eventually dripping out from the cup.

Close to absolute zero, some liquids form a second liquid state described as superfluid because it has zero viscosity (or infinite fluidity; i.e., flowing without friction). This was discovered in 1937 for helium, which forms a superfluid below the lambda temperature of 2.17 K. In this state it will attempt to "climb" out of its container. It also has infinite thermal conductivity so that no temperature gradient can form in a superfluid. Placing a superfluid in a spinning container will result in quantized vortices.

These properties are explained by the theory that the common isotope helium-4 forms a Bose–Einstein condensate in the superfluid state. More recently, Fermionic condensate superfluids have been formed at even lower temperatures by the rare isotope helium-3 and by lithium-6.

Bose–Einstein condensate

Velocity in a gas of rubidium as it is cooled: the starting material is on the left, and Bose–Einstein condensate is on the right.

In 1924, Albert Einstein and Satyendra Nath Bose predicted the "Bose–Einstein condensate" (BEC), sometimes referred to as the fifth state of matter. In a BEC, matter stops behaving as independent particles, and collapses into a single quantum state that can be described with a single, uniform wavefunction.

In the gas phase, the Bose–Einstein condensate remained an unverified theoretical prediction for many years. In 1995, the research groups of Eric Cornell and Carl Wieman, of JILA at the University of Colorado at Boulder, produced the first such condensate experimentally. A Bose–Einstein condensate is "colder" than a solid. It may occur when atoms have very similar (or the same) quantum levels, at temperatures very close to absolute zero (−273.15 °C).

Fermionic condensate

A fermionic condensate is similar to the Bose–Einstein condensate but composed of fermions. The Pauli exclusion principle prevents fermions from entering the same quantum state, but a pair of fermions can behave as a boson, and multiple such pairs can then enter the same quantum state without restriction.

Rydberg molecule

One of the metastable states of strongly non-ideal plasma is Rydberg matter, which forms upon condensation of excited atoms. These atoms can also turn into ions and electrons if they reach a certain temperature. In April 2009, Nature reported the creation of Rydberg molecules from a Rydberg atom and a ground state atom, confirming that such a state of matter could exist. The experiment was performed using ultracold rubidium atoms.

Quantum Hall state

A quantum Hall state gives rise to quantized Hall voltage measured in the direction perpendicular to the current flow. A quantum spin Hall state is a theoretical phase that may pave the way for the development of electronic devices that dissipate less energy and generate less heat. This is a derivation of the Quantum Hall state of matter.

Photonic matter

Photonic matter is a phenomenon where photons interacting with a gas develop apparent mass, and can interact with each other, even forming photonic "molecules". The source of mass is the gas, which is massive. This is in contrast to photons moving in empty space, which have no rest mass, and cannot interact.

Dropleton

A "quantum fog" of electrons and holes that flow around each other and even ripple like a liquid, rather than existing as discrete pairs.

High-energy states

Degenerate matter

Under extremely high pressure, as in the cores of dead stars, ordinary matter undergoes a transition to a series of exotic states of matter collectively known as degenerate matter, which are supported mainly by quantum mechanical effects. In physics, "degenerate" refers to two states that have the same energy and are thus interchangeable. Degenerate matter is supported by the Pauli exclusion principle, which prevents two fermionic particles from occupying the same quantum state. Unlike regular plasma, degenerate plasma expands little when heated, because there are simply no momentum states left. Consequently, degenerate stars collapse into very high densities. More massive degenerate stars are smaller, because the gravitational force increases, but pressure does not increase proportionally.
Electron-degenerate matter is found inside white dwarf stars. Electrons remain bound to atoms but are able to transfer to adjacent atoms. Neutron-degenerate matter is found in neutron stars. Vast gravitational pressure compresses atoms so strongly that the electrons are forced to combine with protons via inverse beta-decay, resulting in a superdense conglomeration of neutrons. Normally free neutrons outside an atomic nucleus will decay with a half life of just under 15 minutes, but in a neutron star, the decay is overtaken by inverse decay. Cold degenerate matter is also present in planets such as Jupiter and in the even more massive brown dwarfs, which are expected to have a core with metallic hydrogen. Because of the degeneracy, more massive brown dwarfs are not significantly larger. In metals, the electrons can be modeled as a degenerate gas moving in a lattice of non-degenerate positive ions.

Quark matter

In regular cold matter, quarks, fundamental particles of nuclear matter, are confined by the strong force into hadrons that consist of 2–4 quarks, such as protons and neutrons. Quark matter or quantum chromodynanamical (QCD) matter is a group of phases where the strong force is overcome and quarks are deconfined and free to move. Quark matter phases occur at extremely high densities or temperatures, and there are no known ways to produce them in equilibrium in the laboratory; in ordinary conditions, any quark matter formed immediately undergoes radioactive decay.
Strange matter is a type of quark matter that is suspected to exist inside some neutron stars close to the Tolman–Oppenheimer–Volkoff limit (approximately 2–3 solar masses), although there is no direct evidence of its existence. In strange matter, part of the energy available manifests as strange quarks, a heavier analogue of the common down quark. It may be stable at lower energy states once formed, although this is not known.

Quark–gluon plasma is a very high-temperature phase in which quarks become free and able to move independently, rather than being perpetually bound into particles, in a sea of gluons, subatomic particles that transmit the strong force that binds quarks together. This is analogous to the liberation of electrons from atoms in a plasma. This state is briefly attainable in extremely high-energy heavy ion collisions in particle accelerators, and allows scientists to observe the properties of individual quarks, and not just theorize. Quark–gluon plasma was discovered at CERN in 2000. Unlike plasma, which flows like a gas, interactions within QGP are strong and it flows like a liquid.

At high densities but relatively low temperatures, quarks are theorized to form a quark liquid whose nature is presently unknown. It forms a distinct color-flavor locked (CFL) phase at even higher densities. This phase is superconductive for color charge. These phases may occur in neutron stars but they are presently theoretical.

Color-glass condensate

Color-glass condensate is a type of matter theorized to exist in atomic nuclei traveling near the speed of light. According to Einstein's theory of relativity, a high-energy nucleus appears length contracted, or compressed, along its direction of motion. As a result, the gluons inside the nucleus appear to a stationary observer as a "gluonic wall" traveling near the speed of light. At very high energies, the density of the gluons in this wall is seen to increase greatly. Unlike the quark–gluon plasma produced in the collision of such walls, the color-glass condensate describes the walls themselves, and is an intrinsic property of the particles that can only be observed under high-energy conditions such as those at RHIC and possibly at the Large Hadron Collider as well.

Very high energy states

Various theories predict new states of matter at very high energies. An unknown state has created the baryon asymmetry in the universe, but little is known about it. In string theory, a Hagedorn temperature is predicted for superstrings at about 10³⁰ K, where superstrings are copiously produced. At Planck temperature (10³² K), gravity becomes a significant force between individual particles. No current theory can describe these states and they cannot be produced with any foreseeable experiment. However, these states are important in cosmology because the universe may have passed through these states in the Big Bang.

The gravitational singularity predicted by general relativity to exist at the center of a black hole is not a phase of matter; it is not a material object at all (although the mass-energy of matter contributed to its creation) but rather a property of spacetime at a location. It could be argued, of course, that all particles are properties of spacetime at a location, leaving a half-note of controversy on the subject.

Other proposed states

Supersolid

A supersolid is a spatially ordered material (that is, a solid or crystal) with superfluid properties. Similar to a superfluid, a supersolid is able to move without friction but retains a rigid shape. Although a supersolid is a solid, it exhibits so many characteristic properties different from other solids that many argue it is another state of matter.

String-net liquid

In a string-net liquid, atoms have apparently unstable arrangement, like a liquid, but are still consistent in overall pattern, like a solid. When in a normal solid state, the atoms of matter align themselves in a grid pattern, so that the spin of any electron is the opposite of the spin of all electrons touching it. But in a string-net liquid, atoms are arranged in some pattern that requires some electrons to have neighbors with the same spin. This gives rise to curious properties, as well as supporting some unusual proposals about the fundamental conditions of the universe itself.

Superglass

A superglass is a phase of matter characterized, at the same time, by superfluidity and a frozen amorphous structure.

Dark matter

While dark matter is estimated to comprise 83% of the mass of matter in the universe, most of its properties remain a mystery due to the fact that it neither absorbs nor emits electromagnetic radiation, and there are many competing theories regarding what dark matter is actually made of. Thus, while it is hypothesized to exist and comprise the vast majority of matter in the universe, almost all of its properties are unknown and a matter of speculation, because it has only been observed through its gravitational effects.