Degenerate matter in physics is a collection of free, non-interacting
particles with a pressure and other physical characteristics determined by
quantum mechanical effects. It is the analogue of an ideal gas in classical
mechanics. The degenerate state of matter, in the sense of deviant from an
ideal gas, arises at extraordinarily high density or at extremely low
temperatures in laboratories. It occurs for matter particles such as electrons,
neutrons, protons, and fermions in general and is referred to as
electron-degenerate matter, neutron-degenerate matter, etc. In a
mixture of particles, such as ions and electrons in white dwarfs or metals, the
electrons may be degenerate, while the ions are not.
In a quantum mechanical description, free particles limited to a finite
volume may take only a discrete set of energies, called quantum states. The
Pauli exclusion principle prevents identical fermions from occupying the
same quantum state. At lowest total energy, all the lowest energy quantum
states are filled. This state is referred to as full degeneracy. The
pressure remains nonzero even near absolute zero temperature. Adding
particles or reducing the volume forces the particles into higher-energy quantum
states. This requires a compression force, and is made manifest as a
resisting pressure. The key feature is that this degeneracy pressure does not
depend on the temperature and only on the density of the fermions. It keeps
dense stars in equilibrium independent of the thermal structure of the star.
Degenerate matter is also called a Fermi gas or a degenerate gas. A degenerate
state with velocities of the fermions close to the speed of light is called
relativistic degenerate matter. Degenerate matter was first described
for a mixture of ions and electrons in 1926 by Ralph H. Fowler, showing that at
densities observed in white dwarfs the electrons have a pressure much higher
than the partial pressure of the ions. Concept
Imagine that a plasma is cooled and compressed repeatedly. Eventually, it
will not be possible to compress the plasma any further, because the Pauli
exclusion principle states that two fermions cannot share the same quantum
state. When in this state, since there is no extra space for any particles, we
can also say that a particle's location is extremely defined. Therefore, since
ΔpΔx ≥ ħ/2 where Δp is the uncertainty in the particle's momentum and Δx is the
uncertainty in position, then we must say that their momentum is extremely
uncertain since the particles are located in a very confined space.
Therefore, even though the plasma is cold, the particles must be moving very
fast on average. This leads to the conclusion that in order to compress an
object into a very small space, tremendous force is required to control
its particles' momentum. Unlike a classical ideal gas, whose
pressure is proportional to its temperature, the pressure exerted by
degenerate matter depends only weakly on its temperature. In particular, the
pressure remains nonzero even at absolute zero temperature. At relatively
low densities, the pressure of a fully degenerate gas is given by P = K(n3,
where K depends on the properties of the particles making up the gas. At very
high densities, where most of the particles are forced into quantum states
with relativistic energies, the pressure is given by P = K′(n3, where K′ again
depends on the properties of the particles making up the gas.
All matter experiences both normal thermal pressure and degeneracy
pressure, but in commonly encountered gases, thermal pressure dominates so
much that degeneracy pressure can be ignored. Likewise, degenerate matter
still has normal thermal pressure, but at extremely high densities the
degeneracy pressure usually dominates. Exotic examples of degenerate matter
include neutronium, strange matter, metallic hydrogen and white dwarf
matter. Degeneracy pressure contributes to the pressure of conventional solids,
but these are not usually considered to be degenerate matter because a
significant contribution to their pressure is provided by electrical
repulsion of atomic nuclei and the screening of nuclei from each other by
electrons. In metals it is useful to treat the conduction electrons alone as
a degenerate, free electron gas while the majority of the electrons are
regarded as occupying bound quantum states. This contrasts with degenerate
matter that forms the body of a white dwarf, where all the electrons would be
treated as occupying free particle momentum states.
Degenerate gases Degenerate gases are gases composed of
fermions that have a particular configuration that usually forms at high
densities. Fermions are particles with half-integer spin. Their behavior is
regulated by a set of quantum mechanical rules called the Fermi–Dirac statistics.
One particular rule is the Pauli exclusion principle, which states that
there can be only one fermion occupying each quantum state, which also applies
to electrons that are not bound to a nucleus but merely confined to a fixed
volume, such as in the deep interior of a star. Such particles as electrons,
protons, neutrons, and neutrinos are all fermions and obey Fermi–Dirac
statistics. A fermion gas in which all energy states
below some energy level are filled is called a fully degenerate fermion gas.
The difference between this energy level and the lowest energy level is known as
the Fermi energy. The electron gas in ordinary metals and in the interior of
white dwarf stars constitute two examples of a degenerate electron gas.
Most stars are supported against their own gravitation by normal thermal gas
pressure. White dwarf stars are supported by the degeneracy pressure of
the electron gas in their interior, while for neutron stars the degenerate
particles are neutrons. = Electron degeneracy=
In an ordinary fermion gas in which thermal effects dominate, most of the
available electron energy levels are unfilled and the electrons are free to
move to these states. As particle density is increased, electrons
progressively fill the lower energy states and additional electrons are
forced to occupy states of higher energy even at low temperatures. Degenerate
gases strongly resist further compression because the electrons cannot
move to already filled lower energy levels due to the Pauli exclusion
principle. Since electrons cannot give up energy by moving to lower energy
states, no thermal energy can be extracted. The momentum of the fermions
in the fermion gas nevertheless generates pressure, termed degeneracy
pressure. Under high densities the matter becomes
a degenerate gas when the electrons are all stripped from their parent atoms. In
the core of a star, once hydrogen burning in nuclear fusion reactions
stops, it becomes a collection of positively charged ions, largely helium
and carbon nuclei, floating in a sea of electrons, which have been stripped from
the nuclei. Degenerate gas is an almost perfect conductor of heat and does not
obey the ordinary gas laws. White dwarfs are luminous not because they are
generating any energy but rather because they have trapped a large amount of heat
which is gradually radiated away. Normal gas exerts higher pressure when it is
heated and expands, but the pressure in a degenerate gas does not depend on the
temperature. When gas becomes super-compressed, particles position
right up against each other to produce degenerate gas that behaves more like a
solid. In degenerate gases the kinetic energies of electrons are quite high and
the rate of collision between electrons and other particles is quite low,
therefore degenerate electrons can travel great distances at velocities
that approach the speed of light. Instead of temperature, the pressure in
a degenerate gas depends only on the speed of the degenerate particles;
however, adding heat does not increase the speed. Pressure is only increased by
the mass of the particles, which increases the gravitational force
pulling the particles closer together. Therefore, the phenomenon is the
opposite of that normally found in matter where if the mass of the matter
is increased, the object becomes bigger. In degenerate gas, when the mass is
increased, the pressure is increased, and the particles become spaced closer
together, so the object becomes smaller. Degenerate gas can be compressed to very
high densities, typical values being in the range of 10,000 kilograms per cubic
centimeter. There is an upper limit to the mass of
an electron-degenerate object, the Chandrasekhar limit, beyond which
electron degeneracy pressure cannot support the object against collapse. The
limit is approximately 1.44solar masses for objects with compositions similar to
the sun. The mass cutoff changes with the chemical composition of the object,
as this affects the ratio of mass to number of electrons present. Celestial
objects below this limit are white dwarf stars, formed by the collapse of the
cores of stars that run out of fuel. During collapse, an electron-degenerate
gas forms in the core, providing sufficient degeneracy pressure as it is
compressed to resist further collapse. Above this mass limit, a neutron star or
a black hole may be formed instead. = Proton degeneracy=
Sufficiently dense matter containing protons experiences proton degeneracy
pressure, in a manner similar to the electron degeneracy pressure in
electron-degenerate matter: protons confined to a sufficiently small volume
have a large uncertainty in their momentum due to the Heisenberg
uncertainty principle. Because protons are much more massive than electrons,
the same momentum represents a much smaller velocity for protons than for
electrons. As a result, in matter with approximately equal numbers of protons
and electrons, proton degeneracy pressure is much smaller than electron
degeneracy pressure, and proton degeneracy is usually modeled as a
correction to the equations of state of electron-degenerate matter.
= Neutron degeneracy= Neutron degeneracy is analogous to
electron degeneracy and is demonstrated in neutron stars, which are primarily
supported by the pressure from a degenerate neutron gas. This happens
when the core of a white dwarf star above the vicinity of 1.4 solar masses,
the Chandrasekhar limit, collapses and is not halted by the degenerate
electrons. As the star collapses, the Fermi energy of the electrons increases
to the point where it is energetically favorable for them to combine with
protons to produce neutrons. The result of this collapse is an extremely compact
star composed of nuclear matter, which is predominantly a degenerate neutron
gas, sometimes called neutronium, with a small admixture of degenerate proton and
electron gases. Neutrons in a degenerate neutron gas are
spaced much more closely than electrons in an electron-degenerate gas, because
the more massive neutron has a much shorter wavelength at a given energy. In
the case of neutron stars and white dwarf stars, this is compounded by the
fact that the pressures within neutron stars are much higher than those in
white dwarfs. The pressure increase is caused by the fact that the compactness
of a neutron star causes gravitational forces to be much higher than in a less
compact body with similar mass. This results in a star with a diameter on the
order of a thousandth that of a white dwarf.
There is an upper limit to the mass of a neutron-degenerate object, the
Tolman–Oppenheimer–Volkoff limit, which is analogous to the Chandrasekhar limit
for electron-degenerate objects. The precise limit is unknown, as it depends
on the equations of state of nuclear matter, for which a highly accurate
model is not yet available. Above this limit, a neutron star may collapse into
a black hole, or into other, denser forms of degenerate matter if these
forms exist and have suitable properties.
= Quark degeneracy= At densities greater than those
supported by neutron degeneracy, quark matter is expected to occur. Several
variations of this have been proposed that represent quark-degenerate states.
Strange matter is a degenerate gas of quarks that is often assumed to contain
strange quarks in addition to the usual up and down quarks. Color superconductor
materials are degenerate gases of quarks in which quarks pair up in a manner
similar to Cooper pairing in electrical superconductors. The equations of state
for the various proposed forms of quark-degenerate matter vary widely, and
are usually also poorly defined, due to the difficulty of modeling strong force
interactions. Quark-degenerate matter may occur in the
cores of neutron stars, depending on the equations of state of neutron-degenerate
matter. It may also occur in hypothetical quark stars, formed by the
collapse of objects above the Tolman–Oppenheimer–Volkoff mass limit
for neutron-degenerate objects. Whether quark-degenerate matter forms at all in
these situations depends on the equations of state of both
neutron-degenerate matter and quark-degenerate matter, both of which
are poorly known. = Preon degeneracy hypothesis=
Preons are subatomic particles proposed to be the constituents of quarks, which
become composite particles in preon-based models. If preons exist,
preon-degenerate matter might occur at densities greater than that which can be
supported by quark-degenerate matter. The expected properties of
preon-degenerate matter depend very strongly on the model chosen to describe
preons, and the existence of preons is not assumed by the majority of the
scientific community, due to conflicts between the preon models originally
proposed and experimental data from particle accelerators.
= Singularity= At densities greater than those
supported by any degeneracy, gravity overwhelms all other forces. To the best
of our current understanding, the body collapses to form a black hole. At the
same time, the material must be converted from fermions, subject to
degeneracy pressure, to bosons, which are not. Physics cannot currently
predict what sort of bosons those might be.
In the frame of reference that is co-moving with the collapsing matter,
general relativity models without quantum mechanics have all the matter
ending up in an infinitely dense singularity at the center of the event
horizon. It is a general result of quantum mechanics that no object can be
confined in a space smaller than its own wavelength, making such a singularity
impossible, but we do not have a theory that combines GR and QM sufficiently to
tell us what the structure inside a black hole might be.
In the frame of reference of an observer at infinity, the collapse viewed in
electromagnetic radiation asymptotically appears to approach the event horizon.
Observations in gravitational waves, which are planned at LIGO, would let
distant observers detect effects from matter falling inside the event horizon.
As a consequence of relativity, the extreme gravitational field and orbital
velocity experienced by infalling matter around a black hole would "slow" time
for that matter relative to a distant observer. In the slowed proper time
experienced by the infalling matter, the fall to the center of the black hole
would be quite short in duration. See also
Compact star White dwarf
Neutron star Quark star—QCD matter
Preon star—preon Pauli exclusion principle
Uncertainty principle Neutronium
Electron degeneracy pressure Nuclear matter
Gravitational time dilation List of plasma articles
Notes References
Cohen-Tanoudji, Claude. Advances in Atomic Physics. World Scientific. p.
791. ISBN 978-981-277-496-5. External links
Detailed mathematical explanation of degenerate gases
Mass-radius diagram of degenerate star types