Degenerate matter
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encyclopedia
Degenerate matter is matter
which has such extraordinarily high density
that the dominant contribution to its pressure
rises from the Pauli
exclusion principle.[1]
The pressure maintained by a body of degenerate matter is called the
degeneracy pressure, and arises because the Pauli principle
prevents the constituent particles from occupying identical quantum
states. Any attempt to force them close enough
together that they are not clearly separated by position must place
them in different energy levels. Therefore, reducing the volume
requires forcing many of the particles into higher-energy quantum
states. This requires additional compression force, and is made
manifest as a resisting pressure.
Concept
Imagine that a plasma
is cooled and compressed repeatedly. Eventually, we will not be able
to compress the plasma any further, because the exclusion principle
states that two particles 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 (according to the Heisenberg uncertainty
principle) 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 molecules are located in a very
confined space. Therefore, even though the plasma is cold, the
molecules must be moving very fast on average. This leads to the
conclusion that if you want to compress an object into a very small
space, you must use tremendous force to control its particles'
momentum.
Unlike a classical ideal
gas, whose pressure is proportional to its temperature
(P = nkT / V, where P is pressure, V
is the volume, n is the number of particles—typically atoms
or molecules—k is Boltzmann's
constant, and T is 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(n / V)5
/ 3, 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'(n / V)4
/ 3, where K' again depends on the properties of the
particles making up the gas.[2]
All matter experiences both normal thermal pressure and degeneracy
pressure, but in commonly encountered gasses, thermal pressure
dominates so much that degeneracy pressure can be neglected.
Likewise, degenerate matter still has normal thermal pressure, but at
high densities the degeneracy pressure dominates. Thus, increasing
the temperature of degenerate matter has a minor effect on total
pressure until the temperature rises so high that thermal pressure
again dominates total pressure.
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 which usually forms at high
densities. Fermions
are subatomic particles with half-integer
spin.
Their behaviour is regulated by a set of quantum mechanical rules
called the Fermi-Dirac
statistics. One particular rule is the Pauli
exclusion principle that 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 the energy states below a critical value,
designated Fermi
energy, are filled is called a fully degenerate
fermion gas. 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
gas pressure. White
dwarf stars are supported by the degeneracy pressure
of the electron gas in their interior. For white dwarfs the
degenerate particles are the electrons while for neutron
stars the degenerate particles are neutrons.
Electron degeneracy
In ordinary gas, most of the electron energy levels (n-spheres)
are unfilled and the electrons are free to move about. As particle
density is increased electrons progressively fill the lower energy
states and additional electrons are forced to occupy states of higher
energy. Degenerate gases strongly resist further compression because
the electrons cannot move to lower energy levels which are already
filled. The Pauli Exclusion Principle causes this. Even though
thermal energy may be extracted from the gas, it still may not cool
down, since electrons cannot give up energy by moving to a lower
energy state. This increases the pressure of the fermion gas termed
degeneracy pressure. In a degenerate gas, the average pressure
opposes the force of gravity and limits its compression.
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. 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 1.44 solar
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
which 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 (supported by neutron degeneracy pressure) 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 supported by the pressure
from a degenerate neutron gas. This happens when a stellar core above
1.44 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 (via inverse beta
decay, also termed "neutralization" and
electron
capture). 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 (such as quark matter) if these forms exist and
have suitable properties (mainly related to degree of
compressibility, or "stiffness", described by the equations
of state).
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 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
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
causes the matter to collapse into a point of zero volume. As far as
is known today, no degeneracy state can exist within the
Schwarzschild
radius of a black
hole, thus all its energy (mass) will be located in an
infinitely dense singularity.
