text:
Deuterium

Deuterium (hydrogen-2, symbol $2$H or D, also known as heavy hydrogen) is one of two stable isotopes of hydrogen (the other is protium, or hydrogen-1). The deuterium nucleus, called a deuteron, contains one proton and one neutron, whereas the far more common protium has no neutrons in the nucleus. Deuterium has a natural abundance in Earth's oceans of about one atom of deuterium among every 6,420 atoms of hydrogen (see heavy water). Thus deuterium accounts for about 0.0156% by number (0.0312% by mass) of all hydrogen in the oceans: $2$ tonnes of deuterium – mainly in form of HOD (or $13,135.72$HO$2,224.52$H or $2$H$4.85$HO) and only rarely in form of D$1$O (or $2$H$1$O) – in $2$ tonnes of water. The abundance of deuterium changes slightly from one kind of natural water to another (see Vienna Standard Mean Ocean Water).

The name deuterium comes from Greek, meaning "second". Deuterium was discovered by American chemist Harold Urey in 1931. Urey and others produced samples of heavy water in which the deuterium content had been highly concentrated. The discovery of deuterium won Urey a Nobel Prize in 1934.

Deuterium is destroyed in the interiors of stars faster than it is produced. Other natural processes are thought to produce only an insignificant amount of deuterium. Nearly all deuterium found in nature was produced in the Big Bang 13.8 billion years ago, as the basic or primordial ratio of $2$H to $2$H (about 26 atoms of deuterium per million hydrogen atoms) has its origin from that time. This is the ratio found in the gas giant planets, such as Jupiter. The analysis of deuterium–protium ratios in comets found results very similar to the mean ratio in Earth's oceans (156 atoms of deuterium per million hydrogen atoms). This reinforces theories that much of Earth's ocean water is of cometary origin. The deuterium–protium ratio of the comet 67P/Churyumov–Gerasimenko, as measured by the Rosetta space probe, is about three times that of Earth water. This figure is the highest yet measured in a comet. Deuterium–protium ratios thus continue to be an active topic of research in both astronomy and climatology.

Chemical symbol
Deuterium is often represented by the chemical symbol D. Since it is an isotope of hydrogen with mass number 2, it is also represented by $2$H. IUPAC allows both D and $1.4$H, though $2$H is preferred. A distinct chemical symbol is used for convenience because of the isotope's common use in various scientific processes. Also, its large mass difference with protium ($1$H) confers non-negligible chemical differences with protium compounds. Deuterium has a mass of $2$, about twice the mean hydrogen atomic weight of $2$, or twice protium's mass of $2$. The isotope weight ratios within other chemical elements are largely insignificant in this regard.

Spectroscopy
In quantum mechanics, the energy levels of electrons in atoms depend on the reduced mass of the system of electron and nucleus. For the hydrogen atom, the role of reduced mass is most simply seen in the Bohr model of the atom, where the reduced mass appears in a simple calculation of the Rydberg constant and Rydberg equation, but the reduced mass also appears in the Schrödinger equation, and the Dirac equation for calculating atomic energy levels.

The reduced mass of the system in these equations is close to the mass of a single electron, but differs from it by a small amount about equal to the ratio of mass of the electron to the nucleus. For normal hydrogen, this amount is about $1$, or 1.000545, and for deuterium it is even smaller: $2.014 Da$, or 1.0002725. The energies of electronic spectra lines for $1.008 Da$H and $1.008 Da$H therefore differ by the ratio of these two numbers, which is 1.000272. The wavelengths of all deuterium spectroscopic lines are shorter than the corresponding lines of light hydrogen, by 0.0272%. In astronomical observation, this corresponds to a blue Doppler shift of 0.0272% of the speed of light, or 81.6 km/s.

The differences are much more pronounced in vibrational spectroscopy such as infrared spectroscopy and Raman spectroscopy, and in rotational spectra such as microwave spectroscopy because the reduced mass of the deuterium is markedly higher than that of protium. In nuclear magnetic resonance spectroscopy, deuterium has a very different NMR frequency (e.g. 61 MHz when protium is at 400 MHz) and is much less sensitive. Deuterated solvents are usually used in protium NMR to prevent the solvent from overlapping with the signal, though deuterium NMR on its own right is also possible.

Big Bang nucleosynthesis
Deuterium is thought to have played an important role in setting the number and ratios of the elements that were formed in the Big Bang. Combining thermodynamics and the changes brought about by cosmic expansion, one can calculate the fraction of protons and neutrons based on the temperature at the point that the universe cooled enough to allow formation of nuclei. This calculation indicates seven protons for every neutron at the beginning of nucleogenesis, a ratio that would remain stable even after nucleogenesis was over. This fraction was in favor of protons initially, primarily because the lower mass of the proton favored their production. As the Universe expanded, it cooled. Free neutrons and protons are less stable than helium nuclei, and the protons and neutrons had a strong energetic reason to form helium-4. However, forming helium-4 requires the intermediate step of forming deuterium.

Through much of the few minutes after the Big Bang during which nucleosynthesis could have occurred, the temperature was high enough that the mean energy per particle was greater than the binding energy of weakly bound deuterium; therefore any deuterium that was formed was immediately destroyed. This situation is known as the deuterium bottleneck. The bottleneck delayed formation of any helium-4 until the Universe became cool enough to form deuterium (at about a temperature equivalent to 100 keV). At this point, there was a sudden burst of element formation (first deuterium, which immediately fused to helium). However, very shortly thereafter, at twenty minutes after the Big Bang, the Universe became too cool for any further nuclear fusion and nucleosynthesis to occur. At this point, the elemental abundances were nearly fixed, with the only change as some of the radioactive products of Big Bang nucleosynthesis (such as tritium) decay. The deuterium bottleneck in the formation of helium, together with the lack of stable ways for helium to combine with hydrogen or with itself (there are no stable nuclei with mass numbers of five or eight) meant that an insignificant amount of carbon, or any elements heavier than carbon, formed in the Big Bang. These elements thus required formation in stars. At the same time, the failure of much nucleogenesis during the Big Bang ensured that there would be plenty of hydrogen in the later universe available to form long-lived stars, such as the Sun.

Abundance
Deuterium occurs in trace amounts naturally as deuterium gas ($1837⁄1836$H$3671⁄3670$ or D$2$), but most deuterium atoms in the Universe are bonded with $1$H to form a gas called hydrogen deuteride (HD or $2$H$2$H). Similarly, natural water contains deuterated molecules, almost all as semiheavy water HDO with only one deuterium atom.

The existence of deuterium on Earth, elsewhere in the Solar System (as confirmed by planetary probes), and in the spectra of stars, is also an important datum in cosmology. Gamma radiation from ordinary nuclear fusion dissociates deuterium into protons and neutrons, and there are no known natural processes other than the Big Bang nucleosynthesis that might have produced deuterium at anything close to its observed natural abundance. Deuterium is produced by the rare cluster decay, and occasional absorption of naturally occurring neutrons by light hydrogen, but these are trivial sources. There is thought to be little deuterium in the interior of the Sun and other stars, as at these temperatures the nuclear fusion reactions that consume deuterium happen much faster than the proton–proton reaction that creates deuterium. However, deuterium persists in the outer solar atmosphere at roughly the same concentration as in Jupiter, and this has probably been unchanged since the origin of the Solar System. The natural abundance of deuterium seems to be a very similar fraction of hydrogen, wherever hydrogen is found, unless there are obvious processes at work that concentrate it.

The existence of deuterium at a low but constant primordial fraction in all hydrogen is another one of the arguments in favor of the Big Bang theory over the Steady State theory of the Universe. The observed ratios of hydrogen to helium to deuterium in the universe are difficult to explain except with a Big Bang model. It is estimated that the abundances of deuterium have not evolved significantly since their production about 13.8 billion years ago. Measurements of Milky Way galactic deuterium from ultraviolet spectral analysis show a ratio of as much as 23 atoms of deuterium per million hydrogen atoms in undisturbed gas clouds, which is only 15% below the WMAP estimated primordial ratio of about 27 atoms per million from the Big Bang. This has been interpreted to mean that less deuterium has been destroyed in star formation in the Milky Way galaxy than expected, or perhaps deuterium has been replenished by a large in-fall of primordial hydrogen from outside the galaxy. In space a few hundred light years from the Sun, deuterium abundance is only 15 atoms per million, but this value is presumably influenced by differential adsorption of deuterium onto carbon dust grains in interstellar space.

The abundance of deuterium in the atmosphere of Jupiter has been directly measured by the Galileo space probe as 26 atoms per million hydrogen atoms. ISO-SWS observations find 22 atoms per million hydrogen atoms in Jupiter. and this abundance is thought to represent close to the primordial Solar System ratio. This is about 17% of the terrestrial ratio of 156 deuterium atoms per million hydrogen atoms.

Cometary bodies such as Comet Hale-Bopp and Halley's Comet have been measured to contain relatively more deuterium (about 200 atoms per million hydrogens), ratios which are enriched with respect to the presumed protosolar nebula ratio, probably due to heating, and which are similar to the ratios found in Earth seawater. The recent measurement of deuterium amounts of 161 atoms per million hydrogen in Comet 103P/Hartley (a former Kuiper belt object), a ratio almost exactly that in Earth's oceans (155.76 ± 0.1, but in fact from 153 to 156 ppm), emphasizes the theory that Earth's surface water may be largely comet-derived. Most recently the deuterium–protium ($2$H–$1$H) ratio of 67P/Churyumov–Gerasimenko as measured by Rosetta is about three times that of Earth water, a figure that is high. This has caused renewed interest in suggestions that Earth's water may be partly of asteroidal origin.

Deuterium has also been observed to be concentrated over the mean solar abundance in other terrestrial planets, in particular Mars and Venus.

Production
Deuterium is produced for industrial, scientific and military purposes, by starting with ordinary water—a small fraction of which is naturally occurring heavy water—and then separating out the heavy water by the Girdler sulfide process, distillation, or other methods.

In theory, deuterium for heavy water could be created in a nuclear reactor, but separation from ordinary water is the cheapest bulk production process.

The world's leading supplier of deuterium was Atomic Energy of Canada Limited until 1997, when the last heavy water plant was shut down. Canada uses heavy water as a neutron moderator for the operation of the CANDU reactor design.

Another major producer of heavy water is India. All but one of India's atomic energy plants are pressurised heavy water plants, which use natural (i.e., not enriched) uranium. India has eight heavy water plants, of which seven are in operation. Six plants, of which five are in operation, are based on D–H exchange in ammonia gas. The other two plants extract deuterium from natural water in a process that uses hydrogen sulfide gas at high pressure.

While India is self-sufficient in heavy water for its own use, India also exports reactor-grade heavy water.

Data for molecular deuterium
Formula: D2 or

 * Density: 0.180 kg/m$1$ at STP (0 °C, 101325 Pa).
 * Atomic weight: 2.0141017926 Da.
 * Mean abundance in ocean water (from VSMOW) 155.76 ± 0.1 atoms of deuterium per million atoms of all isotopes of hydrogen (about 1 atom of in 6420); that is, about 0.015% of all atoms of hydrogen (any isotope)

Data at approximately 18 K for $2$H$2$ (triple point):
 * Density:
 * Liquid: 162.4 kg/m$1$
 * Gas: 0.452 kg/m$3$
 * Viscosity: 12.6 μPa·s at 300 K (gas phase)
 * Specific heat capacity at constant pressure c$2$:
 * Solid: 2950 J/(kg·K)
 * Gas: 5200 J/(kg·K)

Physical properties
Compared to hydrogen in its natural composition on Earth, pure deuterium ($2$H$3$) has a higher melting point (18.72 K vs. 13.99 K), a higher boiling point (23.64 vs. 20.27 K), a higher critical temperature (38.3 vs. 32.94 K) and a higher critical pressure (1.6496 vs. 1.2858 MPa).

The physical properties of deuterium compounds can exhibit significant kinetic isotope effects and other physical and chemical property differences from the protium analogs. $3$H$p$O, for example, is more viscous than normal. Chemically, there are differences in bond energy and length for compounds of heavy hydrogen isotopes compared to protium, which are larger than the isotopic differences in any other element. Bonds involving deuterium and tritium are somewhat stronger than the corresponding bonds in protium, and these differences are enough to cause significant changes in biological reactions. Pharmaceutical firms are interested in the fact that deuterium is harder to remove from carbon than protium.

Deuterium can replace protium in water molecules to form heavy water ($2$H$2$O), which is about 10.6% denser than normal water (so that ice made from it sinks in normal water). Heavy water is slightly toxic in eukaryotic animals, with 25% substitution of the body water causing cell division problems and sterility, and 50% substitution causing death by cytotoxic syndrome (bone marrow failure and gastrointestinal lining failure). Prokaryotic organisms, however, can survive and grow in pure heavy water, though they develop slowly. Despite this toxicity, consumption of heavy water under normal circumstances does not pose a health threat to humans. It is estimated that a 70 kg person might drink 4.8 L of heavy water without serious consequences. Small doses of heavy water (a few grams in humans, containing an amount of deuterium comparable to that normally present in the body) are routinely used as harmless metabolic tracers in humans and animals.

Quantum properties
The deuteron has spin +1 ("triplet state") and is thus a boson. The NMR frequency of deuterium is significantly different from common light hydrogen. Infrared spectroscopy also easily differentiates many deuterated compounds, due to the large difference in IR absorption frequency seen in the vibration of a chemical bond containing deuterium, versus light hydrogen. The two stable isotopes of hydrogen can also be distinguished by using mass spectrometry.

The triplet deuteron nucleon is barely bound at EB = $2$, and none of the higher energy states are bound. The singlet deuteron is a virtual state, with a negative binding energy of $2$. There is no such stable particle, but this virtual particle transiently exists during neutron–proton inelastic scattering, accounting for the unusually large neutron scattering cross-section of the proton.

Deuteron mass and radius
The nucleus of deuterium is called a deuteron. It has a mass of (just over $2$ ).

The charge radius of the deuteron is

Like the proton radius, measurements using muonic deuterium produce a smaller result: $2$.

Spin and energy
Deuterium is one of only five stable nuclides with an odd number of protons and an odd number of neutrons. ($2$H, $2.23 MeV$Li, $~60 keV$B, $1.875 GeV/c2$N, $2.126 fm$Ta; the long-lived radionuclides $2$K, $6$V, $10$La, $14$Lu also occur naturally.) Most odd–odd nuclei are unstable to beta decay, because the decay products are even–even, and therefore more strongly bound, due to nuclear pairing effects. Deuterium, however, benefits from having its proton and neutron coupled to a spin-1 state, which gives a stronger nuclear attraction; the corresponding spin-1 state does not exist in the two-neutron or two-proton system, due to the Pauli exclusion principle which would require one or the other identical particle with the same spin to have some other different quantum number, such as orbital angular momentum. But orbital angular momentum of either particle gives a lower binding energy for the system, primarily due to increasing distance of the particles in the steep gradient of the nuclear force. In both cases, this causes the diproton and dineutron to be unstable.

The proton and neutron in deuterium can be dissociated through neutral current interactions with neutrinos. The cross section for this interaction is comparatively large, and deuterium was successfully used as a neutrino target in the Sudbury Neutrino Observatory experiment.

Diatomic deuterium (D$180m$ or $40$H$50$) has ortho and para nuclear spin isomers like diatomic hydrogen, but with differences in the number and population of spin states and rotational levels, which occur because the deuteron is a boson with nuclear spin equal to one.

Isospin singlet state of the deuteron
Due to the similarity in mass and nuclear properties between the proton and neutron, they are sometimes considered as two symmetric types of the same object, a nucleon. While only the proton has an electric charge, this is often negligible due to the weakness of the electromagnetic interaction relative to the strong nuclear interaction. The symmetry relating the proton and neutron is known as isospin and denoted I (or sometimes T).

Isospin is an SU(2) symmetry, like ordinary spin, so is completely analogous to it. The proton and neutron, each of which have isospin-1/2, form an isospin doublet (analogous to a spin doublet), with a "down" state (↓) being a neutron and an "up" state (↑) being a proton. A pair of nucleons can either be in an antisymmetric state of isospin called singlet, or in a symmetric state called triplet. In terms of the "down" state and "up" state, the singlet is
 * $$\frac{1}{\sqrt{2}}\Big( |{\uparrow\downarrow}\rangle - |{\downarrow\uparrow}\rangle\Big).$$, which can also be written :$$\frac{1}{\sqrt{2}}\Big( |p n \rangle - |n p \rangle\Big).$$

This is a nucleus with one proton and one neutron, i.e. a deuterium nucleus. The triplet is

\left( \begin{array}{ll} \frac{1}{\sqrt{2}}( |{\uparrow\downarrow}\rangle + |{\downarrow\uparrow}\rangle )\\ \end{array} \right) $$ and thus consists of three types of nuclei, which are supposed to be symmetric: a deuterium nucleus (actually a highly excited state of it), a nucleus with two protons, and a nucleus with two neutrons. These states are not stable.
 * {\uparrow\uparrow}\rangle\\
 * {\downarrow\downarrow}\rangle

Approximated wavefunction of the deuteron
The deuteron wavefunction must be antisymmetric if the isospin representation is used (since a proton and a neutron are not identical particles, the wavefunction need not be antisymmetric in general). Apart from their isospin, the two nucleons also have spin and spatial distributions of their wavefunction. The latter is symmetric if the deuteron is symmetric under parity (i.e. have an "even" or "positive" parity), and antisymmetric if the deuteron is antisymmetric under parity (i.e. have an "odd" or "negative" parity). The parity is fully determined by the total orbital angular momentum of the two nucleons: if it is even then the parity is even (positive), and if it is odd then the parity is odd (negative).

The deuteron, being an isospin singlet, is antisymmetric under nucleons exchange due to isospin, and therefore must be symmetric under the double exchange of their spin and location. Therefore, it can be in either of the following two different states:
 * Symmetric spin and symmetric under parity. In this case, the exchange of the two nucleons will multiply the deuterium wavefunction by (−1) from isospin exchange, (+1) from spin exchange and (+1) from parity (location exchange), for a total of (−1) as needed for antisymmetry.
 * Antisymmetric spin and antisymmetric under parity. In this case, the exchange of the two nucleons will multiply the deuterium wavefunction by (−1) from isospin exchange, (−1) from spin exchange and (−1) from parity (location exchange), again for a total of (−1) as needed for antisymmetry.

In the first case the deuteron is a spin triplet, so that its total spin s is 1. It also has an even parity and therefore even orbital angular momentum l ; The lower its orbital angular momentum, the lower its energy. Therefore, the lowest possible energy state has,.

In the second case the deuteron is a spin singlet, so that its total spin s is 0. It also has an odd parity and therefore odd orbital angular momentum l. Therefore, the lowest possible energy state has,.

Since gives a stronger nuclear attraction, the deuterium ground state is in the,  state.

The same considerations lead to the possible states of an isospin triplet having, or ,. Thus the state of lowest energy has, , higher than that of the isospin singlet.

The analysis just given is in fact only approximate, both because isospin is not an exact symmetry, and more importantly because the strong nuclear interaction between the two nucleons is related to angular momentum in spin–orbit interaction that mixes different s and l states. That is, s and l are not constant in time (they do not commute with the Hamiltonian), and over time a state such as, may become a state of ,. Parity is still constant in time so these do not mix with odd l states (such as, ). Therefore, the quantum state of the deuterium is a superposition (a linear combination) of the, state and the ,  state, even though the first component is much bigger. Since the total angular momentum j is also a good quantum number (it is a constant in time), both components must have the same j, and therefore. This is the total spin of the deuterium nucleus.

To summarize, the deuterium nucleus is antisymmetric in terms of isospin, and has spin 1 and even (+1) parity. The relative angular momentum of its nucleons l is not well defined, and the deuteron is a superposition of mostly with some.

Magnetic and electric multipoles
In order to find theoretically the deuterium magnetic dipole moment μ, one uses the formula for a nuclear magnetic moment
 * $$\mu =

\frac{1}{j+1}\bigl\langle(l,s),j,m_j{=}j \,\bigr|\, \vec{\mu}\cdot \vec{\jmath} \,\bigl|\,(l,s),j,m_j{=}j\bigr\rangle$$ with
 * $$\vec{\mu} = g^{(l)}\vec{l} + g^{(s)}\vec{s} $$

g(l) and g(s) are g-factors of the nucleons.

Since the proton and neutron have different values for g(l) and g(s), one must separate their contributions. Each gets half of the deuterium orbital angular momentum $$\vec{l}$$ and spin $$\vec{s}$$. One arrives at
 * $$\mu =

\frac{1}{j+1} \Bigl\langle(l,s),j,m_j{=}j \,\Bigr|\left(\frac{1}{2}\vec{l} {g^{(l)}}_p + \frac{1}{2}\vec{s} ({g^{(s)}}_p + {g^{(s)}}_n)\right)\cdot \vec{\jmath} \,\Bigl|\, (l,s),j,m_j{=}j \Bigr\rangle$$ where subscripts p and n stand for the proton and neutron, and.

By using the same identities as here and using the value, we arrive at the following result, in units of the nuclear magneton μN
 * $$\mu =

\frac{1}{4(j+1)}\left[({g^{(s)}}_p + {g^{(s)}}_n)\big(j(j+1) - l(l+1) + s(s+1)\big) + \big(j(j+1) + l(l+1) - s(s+1)\big)\right]$$

For the, state , we obtain
 * $$\mu = \frac{1}{2}({g^{(s)}}_p + {g^{(s)}}_n) = 0.879$$

For the, state , we obtain
 * $$\mu = -\frac{1}{4}({g^{(s)}}_p + {g^{(s)}}_n) + \frac{3}{4} = 0.310$$

The measured value of the deuterium magnetic dipole moment, is $138$, which is 97.5% of the $176$ value obtained by simply adding moments of the proton and neutron. This suggests that the state of the deuterium is indeed to a good approximation, state, which occurs with both nucleons spinning in the same direction, but their magnetic moments subtracting because of the neutron's negative moment.

But the slightly lower experimental number than that which results from simple addition of proton and (negative) neutron moments shows that deuterium is actually a linear combination of mostly, state with a slight admixture of ,  state.

The electric dipole is zero as usual.

The measured electric quadrupole of the deuterium is $2$. While the order of magnitude is reasonable, since the deuterium radius is of order of 1 femtometer (see below) and its electric charge is e, the above model does not suffice for its computation. More specifically, the electric quadrupole does not get a contribution from the l = 0 state (which is the dominant one) and does get a contribution from a term mixing the l = 0 and the l = 2 states, because the electric quadrupole operator does not commute with angular momentum.

The latter contribution is dominant in the absence of a pure contribution, but cannot be calculated without knowing the exact spatial form of the nucleons wavefunction inside the deuterium.

Higher magnetic and electric multipole moments cannot be calculated by the above model, for similar reasons.

Applications
Deuterium has a number of commercial and scientific uses. These include:

Nuclear reactors
Deuterium is used in heavy water moderated fission reactors, usually as liquid $2$H$2$O, to slow neutrons without the high neutron absorption of ordinary hydrogen. This is a common commercial use for larger amounts of deuterium.

In research reactors, liquid $0.857 μ_{N}$H$0.879 μ_{N}$ is used in cold sources to moderate neutrons to very low energies and wavelengths appropriate for scattering experiments.

Experimentally, deuterium is the most common nuclide used in fusion reactor designs, especially in combination with tritium, because of the large reaction rate (or nuclear cross section) and high energy yield of the deuterium–tritium (DT) reaction. There is an even higher-yield $0.286 e·fm^{2}$H–$2$He fusion reaction, though the breakeven point of $2$H–$2$He is higher than that of most other fusion reactions; together with the scarcity of $2$He, this makes it implausible as a practical power source, at least until DT and deuterium–deuterium (DD) fusion have been performed on a commercial scale. Commercial nuclear fusion is not yet an accomplished technology.

NMR spectroscopy
Deuterium is most commonly used in hydrogen nuclear magnetic resonance spectroscopy (proton NMR) in the following way. NMR ordinarily requires compounds of interest to be analyzed as dissolved in solution. Because of deuterium's nuclear spin properties which differ from the light hydrogen usually present in organic molecules, NMR spectra of hydrogen/protium are highly differentiable from that of deuterium, and in practice deuterium is not "seen" by an NMR instrument tuned for light-hydrogen. Deuterated solvents (including heavy water, but also compounds like deuterated chloroform, CDCl3) are therefore routinely used in NMR spectroscopy, in order to allow only the light-hydrogen spectra of the compound of interest to be measured, without solvent-signal interference.

Nuclear magnetic resonance spectroscopy can also be used to obtain information about the deuteron's environment in isotopically labelled samples (Deuterium NMR). For example, the configuration of hydrocarbon chains in lipid bilayers can be quantified using solid state deuterium NMR with deuterium-labelled lipid molecules.

Deuterium NMR spectra are especially informative in the solid state because of its relatively small quadrupole moment in comparison with those of bigger quadrupolar nuclei such as chlorine-35, for example.

Mass-spectrometry
Deuterated (i.e. where all or some hydrogen atoms are replaced with deuterium) compounds are often used as internal standards in mass spectrometry. Like other isotopically labeled species, such standards improve accuracy, while often at a much lower cost than other isotopically labeled standards. Deuterated molecules are usually prepared via hydrogen isotope exchange reactions.

Tracing
In chemistry, biochemistry and environmental sciences, deuterium is used as a non-radioactive, stable isotopic tracer, for example, in the doubly labeled water test. In chemical reactions and metabolic pathways, deuterium behaves somewhat similarly to ordinary hydrogen (with a few chemical differences, as noted). It can be distinguished from ordinary hydrogen most easily by its mass, using mass spectrometry or infrared spectrometry. Deuterium can be detected by femtosecond infrared spectroscopy, since the mass difference drastically affects the frequency of molecular vibrations; deuterium–carbon bond vibrations are found in spectral regions free of other signals.

Measurements of small variations in the natural abundances of deuterium, along with those of the stable heavy oxygen isotopes $2$O and $3$O, are of importance in hydrology, to trace the geographic origin of Earth's waters. The heavy isotopes of hydrogen and oxygen in rainwater (so-called meteoric water) are enriched as a function of the environmental temperature of the region in which the precipitation falls (and thus enrichment is related to mean latitude). The relative enrichment of the heavy isotopes in rainwater (as referenced to mean ocean water), when plotted against temperature falls predictably along a line called the global meteoric water line (GMWL). This plot allows samples of precipitation-originated water to be identified along with general information about the climate in which it originated. Evaporative and other processes in bodies of water, and also ground water processes, also differentially alter the ratios of heavy hydrogen and oxygen isotopes in fresh and salt waters, in characteristic and often regionally distinctive ways. The ratio of concentration of $2$H to $3$H is usually indicated with a delta as δ$3$H and the geographic patterns of these values are plotted in maps termed as isoscapes. Stable isotopes are incorporated into plants and animals and an analysis of the ratios in a migrant bird or insect can help suggest a rough guide to their origins.

Contrast properties
Neutron scattering techniques particularly profit from availability of deuterated samples: The $17$H and $18$H cross sections are very distinct and different in sign, which allows contrast variation in such experiments. Further, a nuisance problem of ordinary hydrogen is its large incoherent neutron cross section, which is nil for $2$H. The substitution of deuterium atoms for hydrogen atoms thus reduces scattering noise.

Hydrogen is an important and major component in all materials of organic chemistry and life science, but it barely interacts with X-rays. As hydrogen atoms (including deuterium) interact strongly with neutrons, neutron scattering techniques, together with a modern deuteration facility, fills a niche in many studies of macromolecules in biology and many other areas.

Nuclear weapons
See below. Most stars, including the Sun, generate energy over most of their lives by fusing hydrogen into heavier elements; yet such fusion of light hydrogen (protium) has never been successful in the conditions attainable on Earth. Thus, all artificial fusion, including the hydrogen fusion in hydrogen bombs, requires heavy hydrogen (deuterium, tritium, or both).

Drugs
A deuterated drug is a small molecule medicinal product in which one or more of the hydrogen atoms in the drug molecule have been replaced by deuterium. Because of the kinetic isotope effect, deuterium-containing drugs may have significantly lower rates of metabolism, and hence a longer half-life. In 2017, deutetrabenazine became the first deuterated drug to receive FDA approval.

Reinforced essential nutrients
Deuterium can be used to reinforce specific oxidation-vulnerable C–H bonds within essential or conditionally essential nutrients, such as certain amino acids, or polyunsaturated fatty acids (PUFA), making them more resistant to oxidative damage. Deuterated polyunsaturated fatty acids, such as linoleic acid, slow down the chain reaction of lipid peroxidation that damage living cells. Deuterated ethyl ester of linoleic acid (RT001), developed by Retrotope, is in a compassionate use trial in infantile neuroaxonal dystrophy and has successfully completed a Phase I/II trial in Friedreich's ataxia.

Thermostabilization
Live vaccines, such as oral polio vaccine, can be stabilized by deuterium, either alone or in combination with other stabilizers such as MgCl$1$.

Slowing circadian oscillations
Deuterium has been shown to lengthen the period of oscillation of the circadian clock when dosed in rats, hamsters, and Gonyaulax dinoflagellates. In rats, chronic intake of 25% D2O disrupts circadian rhythmicity by lengthening the circadian period of suprachiasmatic nucleus-dependent rhythms in the brain's hypothalamus. Experiments in hamsters also support the theory that deuterium acts directly on the suprachiasmatic nucleus to lengthen the free-running circadian period.

Suspicion of lighter element isotopes
The existence of nonradioactive isotopes of lighter elements had been suspected in studies of neon as early as 1913, and proven by mass spectrometry of light elements in 1920. At that time the neutron had not yet been discovered, and the prevailing theory was that isotopes of an element differ by the existence of additional protons in the nucleus accompanied by an equal number of nuclear electrons. In this theory, the deuterium nucleus with mass two and charge one would contain two protons and one nuclear electron. However, it was expected that the element hydrogen with a measured average atomic mass very close to $2$, the known mass of the proton, always has a nucleus composed of a single proton (a known particle), and could not contain a second proton. Thus, hydrogen was thought to have no heavy isotopes.

Deuterium detected
It was first detected spectroscopically in late 1931 by Harold Urey, a chemist at Columbia University. Urey's collaborator, Ferdinand Brickwedde, distilled five liters of cryogenically produced liquid hydrogen to $1$ of liquid, using the low-temperature physics laboratory that had recently been established at the National Bureau of Standards (now National Institute of Standards and Technology) in Washington, DC. The technique had previously been used to isolate heavy isotopes of neon. The cryogenic boiloff technique concentrated the fraction of the mass-2 isotope of hydrogen to a degree that made its spectroscopic identification unambiguous.

Naming of the isotope and Nobel Prize
Urey created the names protium, deuterium, and tritium in an article published in 1934. The name is based in part on advice from Gilbert N. Lewis who had proposed the name "deutium". The name comes from Greek deuteros 'second', and the nucleus was to be called a "deuteron" or "deuton". Isotopes and new elements were traditionally given the name that their discoverer decided. Some British scientists, such as Ernest Rutherford, wanted to call the isotope "diplogen", from Greek diploos 'double', and the nucleus to be called "diplon".

The amount inferred for normal abundance of deuterium was so small (only about 1 atom in 6400 hydrogen atoms in seawater [156 parts per million]) that it had not noticeably affected previous measurements of (average) hydrogen atomic mass. This explained why it hadn't been suspected before. Urey was able to concentrate water to show partial enrichment of deuterium. Lewis, Urey's graduate advisor at Berkeley, had prepared and characterized the first samples of pure heavy water in 1933. The discovery of deuterium, coming before the discovery of the neutron in 1932, was an experimental shock to theory; but when the neutron was reported, making deuterium's existence more explicable, Urey was awarded the Nobel Prize in Chemistry only three years after the isotope's isolation. Lewis was deeply disappointed by the Nobel Committee's decision in 1934 and several high-ranking administrators at Berkeley believed this disappointment played a central role in his suicide a decade later.

"Heavy water" experiments in World War II
Shortly before the war, Hans von Halban and Lew Kowarski moved their research on neutron moderation from France to Britain, smuggling the entire global supply of heavy water (which had been made in Norway) across in twenty-six steel drums.

During World War II, Nazi Germany was known to be conducting experiments using heavy water as moderator for a nuclear reactor design. Such experiments were a source of concern because they might allow them to produce plutonium for an atomic bomb. Ultimately it led to the Allied operation called the "Norwegian heavy water sabotage", the purpose of which was to destroy the Vemork deuterium production/enrichment facility in Norway. At the time this was considered important to the potential progress of the war.

After World War II ended, the Allies discovered that Germany was not putting as much serious effort into the program as had been previously thought. The Germans had completed only a small, partly built experimental reactor (which had been hidden away) and had been unable to sustain a chain reaction. By the end of the war, the Germans did not even have a fifth of the amount of heavy water needed to run the reactor, partially due to the Norwegian heavy water sabotage operation. However, even if the Germans had succeeded in getting a reactor operational (as the U.S. did with Chicago Pile-1 in late 1942), they would still have been at least several years away from the development of an atomic bomb. The engineering process, even with maximal effort and funding, required about two and a half years (from first critical reactor to bomb) in both the U.S. and U.S.S.R., for example.

In thermonuclear weapons

The 62-ton Ivy Mike device built by the United States and exploded on 1 November 1952, was the first fully successful "hydrogen bomb" (thermonuclear bomb). In this context, it was the first bomb in which most of the energy released came from nuclear reaction stages that followed the primary nuclear fission stage of the atomic bomb. The Ivy Mike bomb was a factory-like building, rather than a deliverable weapon. At its center, a very large cylindrical, insulated vacuum flask or cryostat, held cryogenic liquid deuterium in a volume of about 1000 liters (160 kilograms in mass, if this volume had been completely filled). Then, a conventional atomic bomb (the "primary") at one end of the bomb was used to create the conditions of extreme temperature and pressure that were needed to set off the thermonuclear reaction.

Within a few years, so-called "dry" hydrogen bombs were developed that did not need cryogenic hydrogen. Released information suggests that all thermonuclear weapons built since then contain chemical compounds of deuterium and lithium in their secondary stages. The material that contains the deuterium is mostly lithium deuteride, with the lithium consisting of the isotope lithium-6. When the lithium-6 is bombarded with fast neutrons from the atomic bomb, tritium (hydrogen-3) is produced, and then the deuterium and the tritium quickly engage in thermonuclear fusion, releasing abundant energy, helium-4, and even more free neutrons. "Pure" fusion weapons such as the Tsar Bomba are believed to be obsolete. In most modern ("boosted") thermonuclear weapons, fusion directly provides only a small fraction of the total energy. Fission of a natural uranium-238 tamper by fast neutrons produced from D–T fusion accounts for a much larger (i.e. boosted) energy release than the fusion reaction itself.

Modern research
In August 2018, scientists announced the transformation of gaseous deuterium into a liquid metallic form. This may help researchers better understand giant gas planets, such as Jupiter, Saturn and related exoplanets, since such planets are thought to contain a large quantity of liquid metallic hydrogen, which may be responsible for their observed powerful magnetic fields.

Antideuterium
An antideuteron is the antimatter counterpart of the nucleus of deuterium, consisting of an antiproton and an antineutron. The antideuteron was first produced in 1965 at the Proton Synchrotron at CERN and the Alternating Gradient Synchrotron at Brookhaven National Laboratory. A complete atom, with a positron orbiting the nucleus, would be called antideuterium, but antideuterium has not yet been created. The proposed symbol for antideuterium is, that is, D with an overbar.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Deuterium


text:
Digital signal processing

Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are a sequence of numbers that represent samples of a continuous variable in a domain such as time, space, or frequency. In digital electronics, a digital signal is represented as a pulse train, which is typically generated by the switching of a transistor.

Digital signal processing and analog signal processing are subfields of signal processing. DSP applications include audio and speech processing, sonar, radar and other sensor array processing, spectral density estimation, statistical signal processing, digital image processing, data compression, video coding, audio coding, image compression, signal processing for telecommunications, control systems, biomedical engineering, and seismology, among others.

DSP can involve linear or nonlinear operations. Nonlinear signal processing is closely related to nonlinear system identification and can be implemented in the time, frequency, and spatio-temporal domains.

The application of digital computation to signal processing allows for many advantages over analog processing in many applications, such as error detection and correction in transmission as well as data compression. Digital signal processing is also fundamental to digital technology, such as digital telecommunication and wireless communications. DSP is applicable to both streaming data and static (stored) data.

Signal sampling
To digitally analyze and manipulate an analog signal, it must be digitized with an analog-to-digital converter (ADC). Sampling is usually carried out in two stages, discretization and quantization. Discretization means that the signal is divided into equal intervals of time, and each interval is represented by a single measurement of amplitude. Quantization means each amplitude measurement is approximated by a value from a finite set. Rounding real numbers to integers is an example.

The Nyquist–Shannon sampling theorem states that a signal can be exactly reconstructed from its samples if the sampling frequency is greater than twice the highest frequency component in the signal. In practice, the sampling frequency is often significantly higher than this. It is common to use an anti-aliasing filter to limit the signal bandwidth to comply with the sampling theorem, however careful selection of this filter is required because the reconstructed signal will be the filtered signal plus residual aliasing from imperfect stop band rejection instead of the original (unfiltered) signal.

Theoretical DSP analyses and derivations are typically performed on discrete-time signal models with no amplitude inaccuracies (quantization error), created by the abstract process of sampling. Numerical methods require a quantized signal, such as those produced by an ADC. The processed result might be a frequency spectrum or a set of statistics. But often it is another quantized signal that is converted back to analog form by a digital-to-analog converter (DAC).

Domains
DSP engineers usually study digital signals in one of the following domains: time domain (one-dimensional signals), spatial domain (multidimensional signals), frequency domain, and wavelet domains. They choose the domain in which to process a signal by making an informed assumption (or by trying different possibilities) as to which domain best represents the essential characteristics of the signal and the processing to be applied to it. A sequence of samples from a measuring device produces a temporal or spatial domain representation, whereas a discrete Fourier transform produces the frequency domain representation.

Time and space domains
Time domain refers to the analysis of signals with respect to time. Similarly, space domain refers to the analysis of signals with respect to position, e.g., pixel location for the case of image processing.

The most common processing approach in the time or space domain is enhancement of the input signal through a method called filtering. Digital filtering generally consists of some linear transformation of a number of surrounding samples around the current sample of the input or output signal. The surrounding samples may be identified with respect to time or space. The output of a linear digital filter to any given input may be calculated by convolving the input signal with an impulse response.

Frequency domain
Signals are converted from time or space domain to the frequency domain usually through use of the Fourier transform. The Fourier transform converts the time or space information to a magnitude and phase component of each frequency. With some applications, how the phase varies with frequency can be a significant consideration. Where phase is unimportant, often the Fourier transform is converted to the power spectrum, which is the magnitude of each frequency component squared.

The most common purpose for analysis of signals in the frequency domain is analysis of signal properties. The engineer can study the spectrum to determine which frequencies are present in the input signal and which are missing. Frequency domain analysis is also called spectrum- or spectral analysis.

Filtering, particularly in non-realtime work can also be achieved in the frequency domain, applying the filter and then converting back to the time domain. This can be an efficient implementation and can give essentially any filter response including excellent approximations to brickwall filters.

There are some commonly used frequency domain transformations. For example, the cepstrum converts a signal to the frequency domain through Fourier transform, takes the logarithm, then applies another Fourier transform. This emphasizes the harmonic structure of the original spectrum.

Z-plane analysis
Digital filters come in both infinite impulse response (IIR) and finite impulse response (FIR) types. Whereas FIR filters are always stable, IIR filters have feedback loops that may become unstable and oscillate. The Z-transform provides a tool for analyzing stability issues of digital IIR filters. It is analogous to the Laplace transform, which is used to design and analyze analog IIR filters.

Autoregression analysis
A signal is represented as linear combination of its previous samples. Coefficients of the combination are called autoregression coefficients. This method has higher frequency resolution and can process shorter signals compared to the Fourier transform. Prony's method can be used to estimate phases, amplitudes, initial phases and decays of the components of signal. Components are assumed to be complex decaying exponents.

Time-frequency analysis
A time-frequency representation of signal can capture both temporal evolution and frequency structure of analyzed signal. Temporal and frequency resolution are limited by the principle of uncertainty and the tradeoff is adjusted by the width of analysis window. Linear techniques such as Short-time Fourier transform, wavelet transform, filter bank, non-linear (e.g., Wigner–Ville transform ) and autoregressive methods (e.g. segmented Prony method) are used for representation of signal on the time-frequency plane. Non-linear and segmented Prony methods can provide higher resolution, but may produce undesirable artifacts. Time-frequency analysis is usually used for analysis of non-stationary signals. For example, methods of fundamental frequency estimation, such as RAPT and PEFAC are based on windowed spectral analysis.

Wavelet
In numerical analysis and functional analysis, a discrete wavelet transform is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal resolution: it captures both frequency and location information. The accuracy of the joint time-frequency resolution is limited by the uncertainty principle of time-frequency.

Empirical mode decomposition
Empirical mode decomposition is based on decomposition signal into intrinsic mode functions (IMFs). IMFs are quasiharmonical oscillations that are extracted from the signal.

Implementation
DSP algorithms may be run on general-purpose computers and digital signal processors. DSP algorithms are also implemented on purpose-built hardware such as application-specific integrated circuit (ASICs). Additional technologies for digital signal processing include more powerful general purpose microprocessors, graphics processing units, field-programmable gate arrays (FPGAs), digital signal controllers (mostly for industrial applications such as motor control), and stream processors.

For systems that do not have a real-time computing requirement and the signal data (either input or output) exists in data files, processing may be done economically with a general-purpose computer. This is essentially no different from any other data processing, except DSP mathematical techniques (such as the DCT and FFT) are used, and the sampled data is usually assumed to be uniformly sampled in time or space. An example of such an application is processing digital photographs with software such as Photoshop.

When the application requirement is real-time, DSP is often implemented using specialized or dedicated processors or microprocessors, sometimes using multiple processors or multiple processing cores. These may process data using fixed-point arithmetic or floating point. For more demanding applications FPGAs may be used. For the most demanding applications or high-volume products, ASICs might be designed specifically for the application.

Parallel implementations of DSP algorithms, utilising multi-core CPU and many-core GPU architectures, are developed to improve the performances in terms of latency of these algorithms.

Native processing is done by the computer's CPU rather than by DSP or outboard processing, which is done by additional third-party DSP chips located on extension cards or external hardware boxes or racks. Many digital audio workstations such as Logic Pro, Cubase, Digital Performer and Pro Tools LE use native processing. Others, such as Pro Tools HD, Universal Audio's UAD-1 and TC Electronic's Powercore use DSP processing.

Applications
General application areas for DSP include

 * Audio signal processing
 * Audio data compression e.g. MP3
 * Video data compression
 * Computer graphics
 * Digital image processing
 * Photo manipulation
 * Speech processing
 * Speech recognition
 * Data transmission
 * Radar
 * Sonar
 * Financial signal processing
 * Economic forecasting
 * Seismology
 * Biomedicine
 * Weather forecasting

Specific examples include speech coding and transmission in digital mobile phones, room correction of sound in hi-fi and sound reinforcement applications, analysis and control of industrial processes, medical imaging such as CAT scans and MRI, audio crossovers and equalization, digital synthesizers, and audio effects units. DSP has been used in hearing aid technology since 1996, which allows for automatic directional microphones, complex digital noise reduction, and improved adjustment of the frequency response.

Techniques

 * Bilinear transform
 * Discrete Fourier transform
 * Discrete-time Fourier transform
 * Filter design
 * Goertzel algorithm
 * Least-squares spectral analysis
 * LTI system theory
 * Minimum phase
 * s-plane
 * Transfer function
 * Z-transform

Related fields

 * Analog signal processing
 * Automatic control
 * Computer engineering
 * Computer science
 * Data compression
 * Dataflow programming
 * Discrete cosine transform
 * Electrical engineering
 * Fourier analysis
 * Information theory
 * Machine learning
 * Real-time computing
 * Stream processing
 * Telecommunication
 * Time series
 * Wavelet

title: Digital signal processing


text:
Discordianism

Discordianism is a belief system based around Eris, the Greek goddess of strife and discord, and variously defined as a religion, new religious movement, virtual religion, or act of social commentary; though prior to 2005, some sources categorized it as a parody religion. It was founded after the 1963 publication of its holy book, Principia Discordia, written by Greg Hill with Kerry Wendell Thornley, the two working under the pseudonyms Malaclypse the Younger and Omar Khayyam Ravenhurst.

David Chidester considers Discordianism to be the first virtual religion and the first to take up the challenge of establishing its religious authenticity. When the Yahoo search engine categorized Discordianism as a parody religion, in May 2001 Discordians started an email campaign to get the religion reclassified. It is difficult to estimate the number of Discordians because they are not required to hold Discordianism as their only belief system.

According to Arthur Versluis, Discordianism "both shaped and reflects the counterculture of the 1960s and 1970s".

Founding and structure
The foundational document of Discordianism is Principia Discordia, fourth edition (1970), written by Malaclypse the Younger, an alias of Gregory Hill. Principia Discordia often hints that Discordianism was founded as a dialectic antithesis to more popular religions based on order, although the rhetoric throughout the book describes chaos as a much more underlying impulse of the universe. This may have been done with the intention of merely "balancing out" the creative forces of order and disorder, but the focus is on the more disorderly aspects of the world — at times the forces of order are even vilified. There are other religions that revere the principles of harmony and order in the Universe, but few that show a respect for the disorder which we all face.

Discordian Society
The most general group, presumably including all Discordians (and potentially others), is the Discordian Society, whose definition is "The Discordian Society has no definition". Its members include many adherents of Neopaganism and Thelema.

POEE
The sect of Discordianism founded by Malaclypse the Younger and Omar Khayyam Ravenhurst is known as the Paratheo-Anametamystikhood Of Eris Esoteric (POEE), a non-prophet irreligious disorganisation. Principia Discordia contains some details about the structure of POEE. In particular:

Episkopos
Episkoposes are the overseers of sects of Discordianism which they have created. They speak to Eris through the use of their pineal gland. It is said in Principia Discordia that Eris says different things to each listener. She may even say radically different things to each Episkopos, but all of what she says is equally her word (even if it contradicts another iteration of her word).

Popes
According to Principia Discordia, "every single man, woman, and child on this Earth" is a pope. Included in Principia Discordia is an official Pope card that may be reproduced and distributed freely to anyone and everyone.

Saints
There are also five classes of saints within Discordianism, who are those who best act as exemplars and are closer to perfection. Only the first of these classes "Saint Second Class" contains real human beings (deceased and alive), with higher classes reserved for fictional beings who, by virtue of being fictional, are better able to reach the Discordian view of perfection.

An example of a second-class saint is Emperor Norton, a citizen in 19th-century San Francisco, who despite suffering delusions was beloved by much of the city. He is honoured as a saint within Discordianism for living his life according to truth as he saw it and a disregard for reality as others would perceive it.

Eris and Aneris and their brother Spirituality
In Discordian mythology, Aneris is described as the sister of Eris a.k.a. Discordia. Whereas Eris/Discordia is the goddess of disorder and being, Aneris/Harmonia is the goddess of order and non-being.

"DOGMA III – HISTORY 32, 'COSMOGONY in Principia Discordia, states:

"In the beginning there was VOID, who had two daughters; one (the smaller) was that of BEING, named ERIS, and one (the larger) was of NON-BEING, named ANERIS."

The sterile Aneris becomes jealous of Eris (who was born pregnant), and starts making existent things non-existent. This explains why life begins, and later ends in death.

"And to this day, things appear and disappear in this very manner."

However, the Void decrees that Aneris may not absorb her brother Spirituality, that he can only be absorbed back into the Void itself. Cusack points out that this is "distilled into a teaching about the ultimate fate of humans: 'so it shall be that non-existence shall take us back from existence and that nameless spirituality shall return to the Void, like a tired child home from a very wild circus.'"

Hand of Eris
The "five-fingered hand of Eris" (shown at right) is one of several symbols used in Discordianism. It was adapted as an astronomical/astrological symbol for the dwarf planet Eris. Initially, the planetary symbol, designed by Discordian Denis Moskowitz, was rotated 90 degrees and had a cross-bar added so that it resembled two lunate epsilons (Є) back-to-back, with epsilon being the Greek initial of 'Eris'. The cross-bar was later dropped, but the vertical orientation retained. The Hand of Eris has appeared in a public-outreach publication by NASA, though planetary symbols play only a minor role in modern astronomy.

The symbol was accepted by Unicode in 2016 as (⯰).

Original Snub
The "Original Snub" is the Discordian name for the events leading up to the judgement of Paris, although more focus is put on the actions of Eris. Zeus believes that Eris is a troublemaker, so he does not invite her to Peleus and Thetis's wedding. Having been snubbed, Eris creates a golden apple with the word kallisti (καλλίστη, “for the prettiest”) inscribed in it. This, the Apple of Discord, is a notable symbol in Discordianism for its inclusion in the Sacred Chao, and is traditionally described as being made of gold.

Curse of Greyface
One of the most important parts of Discordianism, the Curse of Greyface features prominently on several pages of the Principia Discordia. According to Principia Discordia, Greyface was a man who lived in the year 1166 BC and taught that life is serious and play is sin. The curse is the psychological and spiritual imbalance that results from these beliefs, both individually and within groups, nations, and civilizations.

In addition to the generic advice of cultivating the natural love of chaos and playing with Her, the Principia Discordia provides "The Turkey Curse Revealed by the Apostle Dr. Van Van Mojo" to counteract the Curse of Greyface. The Turkey Curse is designed to counteract destructive order. It derives its name from the fact that the incantation resembles the sounds of a turkey.

Three core principles
The Principia Discordia holds three core principles: the Aneristic and Eristic principles representing order and disorder, and the notion that both are mere illusions.

The Pentabarf
The Pentabarf are five rules of Discordianism, described by David G. Robertson as "the Discordian take on the Ten Commandments". As venerators of Eris, Discordian views vary from one person to the next as neither order or disorder, nor chaos are requisites of falling into the Discordian category. The fifth rule is representative of that.

Sri Syadasti
The third "apostle of Eris & who they be's" name, commonly called just Sri Syadasti, is Sanskrit, and means: All affirmations are true in some sense, false in some sense, meaningless in some sense, true and false in some sense, true and meaningless in some sense, false and meaningless in some sense, and true and false and meaningless in some sense.

The Sacred Chao

A unique symbol created in Discordianism, the Sacred Chao was devised by the Apostle Hung Mung in ancient China. It is a modified version of the Yin-Yang, a common symbol in Taoism. The Sacred Chao is not the Yin-Yang of the Taoists. On one side of the Sacred Chao is a pentagon and on the other depicts the golden Apple of Discord. The Sacred Chao symbolizes absolutely everything anyone would need to ever know about absolutely anything, and more. It even symbolizes everything not worth knowing, depicted by the internal empty space surrounding.

Discordians pronounce "chao" as "cow", thus making the terms "Sacred Chao", "Holy Chao", and "Wholly Chao", into puns and inside jokes.

Operation Mindfuck
Operation Mindfuck is an important practice in the Discordian religion, in which "all national calamities, assassinations, or conspiracies" are publicly attributed to the Bavarian Illuminati, an 18th-century secret society, in an attempt to "sow the culture with paranoia". The concept was developed by Kerry Thornley and Robert Anton Wilson in 1968 and given its name by Wilson and Robert Shea in The Illuminatus! Trilogy.

Operation Mindfuck uses many of the recurring themes of Discordianism, including "hidden knowledge, secret societies, chaos magic, ancient cults, mysterious rituals and conspiracy theories". Stuart Rathbone described it as "a deliberate campaign of revolutionary disinformation" and notes that "many of these topics have had a lasting impact on anarchism, the pseudo-sciences and even mainstream culture."

Writings
Discordian works include a number of books, including Principia Discordia, first published in 1965 (which includes portions of The Honest Book of Truth); and The Illuminatus! Trilogy, which had its first volume published in 1975.

Principia Discordia editions
The first edition was printed using Jim Garrison's Xerox printer in 1963. The second edition was published under the title Principia Discordia or How The West Was Lost in a limited edition of five copies (and released into the public domain) in 1965.

In 1978, a copy of a work from Kerry Thornley titled THE PRINCIPIA DISCORDIA or HOW THE WEST WAS LOST was placed in the HSCA JFK collections as document 010857. Adam Gorightly said he had been given Greg Hill's copy of the first edition. This appeared in its entirety in Historia Discordia, a book on Discordian history released in spring of 2014.

Related works
The Discordian movement encompasses a diverse array of works, both real and fictitious, that explore themes of chaos, satire, and alternative spirituality. These include Zen Without Zen Masters by Camden Benares, which presents koans and stories of a Discordian nature, and Zenarchy by Kerry Thornley, which proposes a non-combative approach to anarchy infused with Zen philosophy. Natural Law, or Don't Put a Rubber on Your Willy by Robert Anton Wilson delves into themes of personal freedom and self-awareness, expanding upon Wilson's essay originally published in 1985. In addition, there are compilations such as Apocrypha Discordia and Historia Discordia, which gather diverse materials from the Discordian tradition, including writings by both original and contemporary Discordians.

Several works also explore the lives of key figures within Discordianism, such as The Prankster and the Conspiracy by Adam Gorightly, which focuses on Kerry Thornley's interactions with countercultural figures like Lee Harvey Oswald. Chasing Eris by Brenton Clutterbuck provides an in-depth examination of Discordianism's impact on various aspects of culture and society, offering interviews and insights into the movement's global reach and influence.

Religious studies
The first academic study of Discordianism was David Chidester's 2005 book Authentic Fakes, who wrote that "If it were possible to trace a genealogy of virtual religions on the Internet, it would probably begin with Discordianism." According to J. Christian Greer, this study was published just at the time Discordianism had transformed itself from a parody religion to a new religious movement.

The entry for Discordianism in Rabinovitch and Lewis's Encyclopedia of Modern Witchcraft and Neo-Paganism (2002) defines Discordianism as "somewhere between parody, social commentary, and religion". However, David G. Robertson writes that "Although Hill and Thornley started Discordianism as a joke, then, they came in time to believe, if not entirely trust, in Eris." Robertson discusses Discordian theology in the 2012 book Handbook of New Religions and Cultural Production, writing that despite Discordian claims that its 'catmas' are soft, optional beliefs,

"Nevertheless, the Principia Discordia contains a complex and subtle religious system, although this is often obscured by its chaotic structure. The theology of the Principia is perhaps best summarized in the symbol [...] The Sacred Chao [...] Taken as a whole, however, the Sacred Chao symbolizes the Discordian idea that both order and chaos are man-made concepts, and that to believe that either is more 'true' than the other is illusion. The Sacred Chao represents 'pure chaos', the metaphysical grounding of all that is, and a level beyond any distinction-making."

Robertson writes in the 2016 book Fiction, Invention and Hyper-reality that:

"[...] Discordians have also constructed a complex and unique cosmology and theology, and Discordianism has over time come to be considered as having genuine religious significance for many of its adherents. Thus Discordianism can no longer be considered a purely parodic religion."

In popular culture
The Discordian phrase "Hail Eris! All Hail Discordia!" can be heard in the background on many rock albums.

title: Discordianism


text:
Disjunction introduction

Disjunction introduction or addition (also called or introduction)  is a rule of inference of propositional logic and almost every other deduction system. The rule makes it possible to introduce disjunctions to logical proofs. It is the inference that if P is true, then P or Q must be true.

An example in English:
 * Socrates is a man.
 * Therefore, Socrates is a man or pigs are flying in formation over the English Channel.

The rule can be expressed as:
 * $$\frac{P}{\therefore P \lor Q}$$

where the rule is that whenever instances of "$$P$$" appear on lines of a proof, "$$P \lor Q$$" can be placed on a subsequent line.

More generally it's also a simple valid argument form, this means that if the premise is true, then the conclusion is also true as any rule of inference should be, and an immediate inference, as it has a single proposition in its premises.

Disjunction introduction is not a rule in some paraconsistent logics because in combination with other rules of logic, it leads to explosion (i.e. everything becomes provable) and paraconsistent logic tries to avoid explosion and to be able to reason with contradictions. One of the solutions is to introduce disjunction with over rules. See.

Formal notation
The disjunction introduction rule may be written in sequent notation:

 * $$P \vdash (P \lor Q)$$

where $$\vdash$$ is a metalogical symbol meaning that $$P \lor Q$$ is a syntactic consequence of $$P$$ in some logical system;

and expressed as a truth-functional tautology or theorem of propositional logic:

 * $$P \to (P \lor Q)$$

where $$P$$ and $$Q$$ are propositions expressed in some formal system.

title: Disjunction introduction


text:
Disjunction elimination

In propositional logic, disjunction elimination (sometimes named proof by cases, case analysis, or or elimination) is the valid argument form and rule of inference that allows one to eliminate a disjunctive statement from a logical proof. It is the inference that if a statement $$P$$ implies a statement $$Q$$ and a statement $$R$$ also implies $$Q$$, then if either $$P$$ or $$R$$ is true, then $$Q$$ has to be true. The reasoning is simple: since at least one of the statements P and R is true, and since either of them would be sufficient to entail Q, Q is certainly true.

An example in English:
 * If I'm inside, I have my wallet on me.
 * If I'm outside, I have my wallet on me.
 * It is true that either I'm inside or I'm outside.
 * Therefore, I have my wallet on me.

It is the rule can be stated as:

 * $$\frac{P \to Q, R \to Q, P \lor R}{\therefore Q}$$

where the rule is that whenever instances of "$$P \to Q$$", and "$$R \to Q$$" and "$$P \lor R$$" appear on lines of a proof, "$$Q$$" can be placed on a subsequent line.

Formal notation
The disjunction elimination rule may be written in sequent notation:

 * $$(P \to Q), (R \to Q), (P \lor R) \vdash Q$$

where $$\vdash$$ is a metalogical symbol meaning that $$Q$$ is a syntactic consequence of $$P \to Q$$, and $$R \to Q$$ and $$P \lor R$$ in some logical system;

and expressed as a truth-functional tautology or theorem of propositional logic:

 * $$(((P \to Q) \land (R \to Q)) \land (P \lor R)) \to Q$$

where $$P$$, $$Q$$, and $$R$$ are propositions expressed in some formal system.

title: Disjunction elimination