Degenerate matter^[1] is a highly dense state of fermionic matter in which the Pauli exclusion principle exerts significant pressure in addition to, or in lieu of thermal pressure. The description applies to matter composed of electrons, protons, neutrons or other fermions. The term is mainly used in astrophysics to refer to dense stellar objects where gravitational pressure is so extreme that quantum mechanical effects are significant. This type of matter is naturally found in stars in their final evolutionary states, such as white dwarfs and neutron stars, where thermal pressure alone is not enough to avoid gravitational collapse.

The term degenerative understandably implies that symptoms will get worse with age. However, the term does not refer to the symptoms, but rather describes the process of the disc degenerating over time. See Degenerative Disc Disease Progression over Time. Despite what the name suggests, degenerative disc disease is not a disease, but a condition in which natural, age-related wear-and-tear on a. Definition of degenerate (Entry 1 of 3) 1 a: having declined or become less specialized (as in nature, character, structure, or function) from an ancestral or former state the last degenerate member of a noble family — W.

Degenerate matter is usually modelled as an ideal Fermi gas, an ensemble of non-interacting fermions. In a quantum mechanical description, 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 (when the thermal energy of the particles is negligible), all the lowest energy quantum states are filled. This state is referred to as full degeneracy. This degeneracy pressure remains non-zero even at absolute zero temperature.^[2][3] Adding particles or reducing the volume forces the particles into higher-energy quantum states. In this situation, a compression force is required, and is made manifest as a resisting pressure. The key feature is that this degeneracy pressure does not depend on the temperature but only on the density of the fermions. Degeneracy pressure keeps dense stars in equilibrium, independent of the thermal structure of the star.

A degenerate mass whose fermions have velocities close to the speed of light (particle energy larger than its rest mass energy) is called relativistic degenerate matter.

The concept of degenerate stars, stellar objects composed of degenerate matter, was originally developed in a joint effort between Arthur Eddington, Ralph Fowler and Arthur Milne. Eddington had suggested that the atoms in Sirius B were almost completely ionised and closely packed. Fowler described white dwarfs as composed of a gas of particles that became degenerate at low temperature. Milne proposed that degenerate matter is found in most of the nuclei of stars, not only in compact stars.^[4][5]

Concept[edit]

If a plasma is cooled and under increasing pressure, it will eventually not be possible to compress the plasma any further. This constraint is due to the Pauli exclusion principle, which states that two fermions cannot share the same quantum state. When in this highly compressed state, since there is no extra space for any particles, a particle's location is extremely defined. Since the locations of the particles of a highly compressed plasma have very low uncertainty, their momentum is extremely uncertain. The Heisenberg uncertainty principle states

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where Δp is the uncertainty in the particle's momentum and Δx is the uncertainty in position (and ħ is the reduced Planck constant). Therefore, even though the plasma is cold, such particles must on average be moving very fast. Large kinetic energies lead 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ('Math extension cannot connect to Restbase.') from server '/mathoid/local/v1/':): {displaystyle P=k_{rm B}frac{NT}{V} },

where P is pressure, k_B is Boltzmann's constant, N is the number of particles—typically atoms or molecules—, T is temperature, and V is the volume, 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 can be derived by treating the system as an ideal Fermi gas, in this way

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ('Math extension cannot connect to Restbase.') from server '/mathoid/local/v1/':): {displaystyle P=frac{(3pi^2)^{2/3}hbar^2}{5m}left(frac{N}{V}right)^{5/3}},

where m is the mass of the individual 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ('Math extension cannot connect to Restbase.') from server '/mathoid/local/v1/':): {displaystyle P=Kleft(frac{N}{V}right)^{4/3}},

where K is another proportionality constant depending on the properties of the particles making up the gas.^[6]

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, the degeneracy pressure dominates to the point that temperature has a negligible effect on the total pressure. The adjacent figure shows how the pressure of a Fermi gas saturates as it is cooled down, relative to a classical ideal gas.

While degeneracy pressure usually dominates at extremely high densities, it is the ratio between degenerate pressure and thermal pressure which determines degeneracy. Given a sufficiently drastic increase in temperature (such as during a red giant star's helium flash), matter can become non-degenerate without reducing its density.

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. The free electron model of metals derives their physical properties by considering the conduction electrons alone as a degenerate gas, while the majority of the electrons are regarded as occupying bound quantum states. This solid state contrasts with degenerate matter that forms the body of a white dwarf, where most of the electrons would be treated as occupying free particle momentum states.

Exotic examples of degenerate matter include neutron degenerate matter, strange matter, metallic hydrogen and white dwarf matter.

Degenerate gases[edit]

Degenerate gases are gases composed of fermions such as electrons, protons, and neutrons rather than molecules of ordinary matter. The electron gas in ordinary metals and in the interior of white dwarfs are two examples. Following the Pauli exclusion principle, there can be only one fermion occupying each quantum state. In a degenerate gas, all quantum states are filled up to the Fermi energy. Most stars are supported against their own gravitation by normal thermal gas pressure, while in white dwarf stars the supporting force comes from the degeneracy pressure of the electron gas in their interior. In neutron stars, the degenerate particles are neutrons.

A fermion gas in which all quantum states below a given 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.

Electron degeneracy[edit]

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 of most of the electrons, because they are stuck in fully occupied quantum states. Pressure is increased only 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 particles become spaced closer together due to gravity (and the pressure is increased), 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^[7]solar masses for objects with typical compositions expected for white dwarf stars (carbon and oxygen with two baryons per electron). This mass cutoff is appropriate only for a star supported by ideal electron degeneracy pressure under Newtonian gravity; in general relativity and with realistic Coulomb corrections, the corresponding mass limit is around 1.38 solar masses.^[8] The limit may also change with the chemical composition of the object, as it affects the ratio of mass to number of electrons present. The object's rotation, which counteracts the gravitational force, also changes the limit for any particular object. Celestial objects below this limit are white dwarf stars, formed by the gradual shrinking of the cores of stars that run out of fuel. During this shrinking, 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 (primarily supported by neutron degeneracy pressure) or a black hole may be formed instead.

Neutron degeneracy[edit]

Neutron degeneracy is analogous to electron degeneracy and is demonstrated in neutron stars, which are partially supported by the pressure from a degenerate neutron gas.^[9] The collapse happens when the core of a white dwarf exceeds approximately 1.4 solar masses, which is the Chandrasekhar limit, above which the collapse is not halted by the pressure of 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 electron capture). The result 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 dwarfs, this phenomenon 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. The result is 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 theoretical limit for non-relativistic objects supported by ideal neutron degeneracy pressure is only 0.75 solar masses;^[10] however, with more realistic models including baryon interaction, 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 dense forms of degenerate matter.^[a]

Proton degeneracy[edit]

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. However, 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.

Quark degeneracy[edit]

At densities greater than those supported by neutron degeneracy, quark matter is expected to occur.^[11] Several variations of this hypothesis 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. Quark stars are considered to be an intermediate category between neutron stars and black holes.^[12]

See also[edit]

Notes[edit]

1. ^Possible “denser forms of matter” include quark matter, preon stars, etc., if those forms of matter actually exist, and if they have suitable properties. The main issue being whether the hypothetical material’s equation of state shows a degree of compressibility, or ‘stiffness’, compatible with the stellar remnant model.

Citations[edit]

1. ^Academic Press dictionary of science and technology. Morris, Christopher G., Academic Press. San Diego: Academic Press. 1992. pp. 662. ISBN0122004000. OCLC22952145.CS1 maint: others (link)
2. ^see http://apod.nasa.gov/apod/ap100228.html
3. ^Andrew G. Truscott, Kevin E. Strecker, William I. McAlexander, Guthrie Partridge, and Randall G. Hulet, 'Observation of Fermi Pressure in a Gas of Trapped Atoms', Science, 2 March 2001
4. ^Fowler, R. H. (1926-12-10). 'On Dense Matter'. Monthly Notices of the Royal Astronomical Society. 87 (2): 114–122. Bibcode:1926MNRAS..87..114F. doi:10.1093/mnras/87.2.114. ISSN0035-8711.
5. ^David., Leverington (1995). A History of Astronomy : from 1890 to the Present. London: Springer London. ISBN1447121244. OCLC840277483.
6. ^Stellar Structure and Evolution section 15.3 – R Kippenhahn & A. Weigert, 1990, 3rd printing 1994. ISBN0-387-58013-1
7. ^ENCYCLOPAEDIA BRITANNICA
8. ^Rotondo, M. et al. 2010, Phys. Rev. D, 84, 084007, https://arxiv.org/abs/1012.0154
9. ^Potekhin, A. Y. (2011). 'The Physics of Neutron Stars'. Physics-Uspekhi. 53 (12): 1235–1256. arXiv:1102.5735. Bibcode:2010PhyU...53.1235Y. doi:10.3367/UFNe.0180.201012c.1279. S2CID119231427.
10. ^Oppenheimer, J.R.; Volkoff, G.M. (1939). 'On massive neutron cores'. Physical Review. American Physical Society. 55 (374): 374–381. doi:10.1103/PhysRev.55.374.
11. ^Annala, Eemeli; Gorda, Tyler; Kurkela, Aleksi; Nättilä, Joonas; Vuorinen, Aleksi (2020-06-01). 'Evidence for quark-matter cores in massive neutron stars'. Nature Physics. 16 (9): 907–910. doi:10.1038/s41567-020-0914-9. ISSN1745-2481.
12. ^Cain, Fraser (2016-07-25). 'What are Quark Stars?'. Universe Today. Retrieved 2021-01-15.

References[edit]

• Cohen-Tanoudji, Claude (2011). Advances in Atomic Physics. World Scientific. p. 791. ISBN978-981-277-496-5. Archived from the original on 2012-05-11. Retrieved 2012-01-31.

External links[edit]

Overview

What is degenerative disk disease?

Disk degeneration is a normal part of aging. It’s also known as degenerative disk disease (DDD).

The degeneration develops over time. It affects the rubber-like disks between vertebrae — the small bones that make up the spinal column (backbone).

The disks act like cushions between those bones. When the cushions wear away, the bones can start to rub together. This contact can cause pain and other problems, such as:

• Adult scoliosis, where the spine curves.
• Herniated disk, also called a bulged, slipped or ruptured disk.
• Spinal stenosis, when the spaces around your spine narrow.
• Spondylolisthesis, when vertebrae move in and out of place.

How common is intervertebral disk degeneration?

Almost everyone has some disk degeneration after age 40, even if they don’t develop symptoms. It can lead to back pain in about 5% of adults.

Are certain people more likely to get DDD?

Certain people have a higher chance of developing disk degeneration:

• People who are very overweight.
• People who experience trauma to the spine.
• Professional drivers (for example, taxi and truck drivers).
• Gymnasts.
• Smokers.

Symptoms and Causes

What causes DDD?

A healthy back contains a number of rubbery cushions called disks. Each disk sits between a set of vertebrae, the bones that stack up to make the spinal column. Together, the discs allow a person to bend, twist and move freely.

As we age, our disks begin to wear away, for several reasons:

• Activities or sports can cause small tears in the discs over the years.
• Discs dry out or get weak over time.
• Injury can cause discs to break down faster.

Because discs are primarily composed of collagen and have a relatively poor blood supply, they do not heal like other parts of the body.

What are the symptoms of DDD?

When disks wear down too much, the vertebrae rub together. The grinding of the bones can cause:

• Pain.
• Stiffness.
• Tingling or numbness.
• Trouble with movement.
• Weakness in the legs or foot drop (can’t raise the front part of one or both feet).

What Is A Degenerate Triangle

What does degenerative disk pain feel like?

Degenerative disk pain:

• Can happen in the neck or lower back.
• May extend into the arms and hands or into the butt and legs.
• Can be mild, moderate or severe.
• May start and stop.
• Can get worse after certain activities such as bending, twisting or lifting.
• Can get worse over time.

Diagnosis and Tests

How is DDD diagnosed?

If you have symptoms of disk degeneration, you should tell a healthcare provider. The healthcare provider will:

• Review your medical history with you.
• Examine your body to see where it hurts.
• Ask you to describe what makes the pain worse or better.
• Ask you to rate your pain on a scale of zero to 10.

What tests might I get?

The healthcare provider might order some tests to take pictures of the bones and disks in your spine. The tests may include:

• CT scan.
• MRI.
• Spine X-ray.

Management and Treatment

Treatment for DDD usually starts with medications to control pain. It also involves physical therapy, or rehabilitation.

Common medications used to treat DDD include:

• Acetaminophen.
• Nonsteroidal anti-inflammatory drugs (NSAIDs), such as ibuprofen or meloxicam.
• Gabapentin, a membrane-stabilizer.
• Steroid injections into the disk area.

Physical therapy is also helpful to treat DDD. It can include many different strategies:

• Adjustments to the way you move (for example, how you lift a box) to lessen pain and avoid injury.
• Aquatic exercises, since being in the water takes pressure off your muscles and joints.
• Corrections to your posture.
• Joint mobilization or manipulation, which moves your joints in different directions to improve range of motion.
• Plans for exercising at home.
• Soft tissue mobilization, which puts deep pressure on muscles to stretch them and reduce tension.
• Strengthening your core, the muscles supporting your back.
• Stretches for flexibility.
• Traction, or gentle pulling of the arms and legs, often using ropes and weights.

How long will I need to do physical therapy?

Most people with DDD have rehabilitation one to three times a week for several weeks or months. Your rehabilitation needs will depend on:

• The severity of your symptoms.
• Your goals.
• Your insurance coverage.

You’ll “graduate” from physical therapy when you have reached your goals. Still, your healthcare provider will want you to exercise at home and stay active.

Will I need surgery for disk degeneration?

If more conservative options don’t work, some people choose to have surgery for disk degeneration. Options may include:

• Artificial disk replacement:Artificial disk replacement is also called total disk replacement. It involves surgery to remove the damaged disk (diskectomy). The surgeon then implants a manufactured device that looks and acts like a disk.
• Diskectomy and spinal fusion: A diskectomy surgically removes a damaged disk. Spinal fusion then joins vertebrae together for stability. To make the connection, the surgeon uses a bone graft. This piece of bone comes from elsewhere in your body or from a deceased donor. The graft fuses with your spine. The surgeon will also place screws, rods, hooks or plates into the bones of the spine. The hardware hold bones together so they fuse.

Prevention

How can I reduce my risk of disk degeneration?

Disk degeneration eventually happens to everyone. You can’t prevent it, but you can slow it down and take action to protect your vertebrae:

• Focus on good posture.
• Keep your core muscles strong.
• Maintain a healthy weight.
• Quit smoking.
• Stay active.
• Use good technique when you move, twist and lift to prevent injury.

Outlook / Prognosis

What is the outlook for people with DDD?

Without treatment, disk degeneration usually gets worse over time. But pain medications, physical therapy and sometimes surgery can reduce pain and improve movement.

Researchers continue to study DDD, particularly ways to delay and treat it.

Living With

What Is A Degenerate

What can I do to help ease the symptoms of DDD?

Several strategies can help you manage DDD pain:

• Do your physical therapy exercises at home exactly as you were shown.
• Keep your core muscles strong to support your back and neck.
• Take your pain medications as prescribed.
• Use good posture when sitting and standing.
• Use heat and cold on the area that hurts.

A note from Cleveland Clinic

What Is A Degenerate Orbital

Disk degeneration is a natural part of aging once you turn 40. Still, if you develop pain in your neck or back that does not respond to over-the-counter pain medications, talk to a healthcare provider. Medications and therapy can control the symptoms of disk degeneration and help you stay active.