# Forbidden mechanism

In spectroscopy, a forbidden mechanism (forbidden transition or forbidden line) is a spectral line associated with absorption or emission of light by atomic nuclei, atoms, or molecules which undergo a transition that is not allowed by a particular selection rule but is allowed if the approximation associated with that rule is not made.[1] For example, in a situation where, according to usual approximations (such as the electric-dipole approximation for the interaction with light), the process cannot happen, but at a higher level of approximation (e.g. magnetic dipole, or electric quadrupole) the process is allowed but at a much lower rate.

An example is phosphorescent glow in the dark materials,[2] which absorb light and form an excited state whose decay involves a spin flip, and is therefore forbidden by electric dipole transitions. The result is emission of light slowly over minutes or hours.

Although the transitions are nominally forbidden, there is a small probability of their spontaneous occurrence, should an atomic nucleus, atom or molecule be raised to an excited state. More precisely, there is a certain probability that such an excited entity will make a forbidden transition to a lower energy state per unit time; by definition, this probability is much lower than that for any transition permitted or allowed by the selection rules. Therefore, if a state can de-excite via a permitted transition (or otherwise, e.g. via collisions) it will almost certainly do so before any transition occurs via a forbidden route. Nevertheless, most forbidden transitions are only relatively unlikely: states that can only decay in this way (so-called meta-stable states) usually have lifetimes on the order milliseconds to seconds, compared to less than a microsecond for decay via permitted transitions. In some radioactive decay systems, multiple levels of forbiddenness can stretch life times by many orders of magnitude for each additional unit by which the system changes beyond what is most allowed under the selection rules. Such excited states can last years, or even for many billions of years (too long to have been measured).

### Gamma decay

The most common mechanism for suppression of the rate of gamma decay of excited atomic nuclei, and thus make possible the existence of a metastable isomer for the nucleus, is lack of a decay route for the excited state that will change nuclear angular momentum (along any given direction) by the most common (allowed) amount of 1 quantum unit ${\displaystyle \hbar }$ of spin angular momentum. Such a change is necessary to emit a gamma-ray photon, which has a spin of 1 unit in this system. Integral changes of 2, 3, 4, and more units in angular momentum are possible (the emitted photons carry off the additional angular momentum), but changes of more than 1 unit are known as forbidden transitions. Each degree of forbiddenness (additional unit of spin change larger than 1, that the emitted gamma ray must carry) inhibits decay rate by about 5 orders of magnitude.[3] The highest known spin change of 8 units occurs in the decay of Ta-180m, which suppresses its decay by a factor of 1035 from that associated with 1 unit, so that instead of a natural gamma decay half life of 10−12 seconds, it has a half life of more than 1023 seconds, or at least 3 x 1015 years, and thus has yet to be observed to decay.

Although gamma decays with nuclear angular momentum changes of 2, 3, 4, etc., are forbidden, they are only relatively forbidden, and do proceed, but with a slower rate than the normal allowed change of 1 unit. However, gamma emission is absolutely forbidden when the nucleus begins in a zero-spin state, as such an emission would not conserve angular momentum. These transitions cannot occur by gamma decay, but must proceed by another route, such as beta decay in some cases, or internal conversion where beta decay is not favored.

### Beta decay

Beta decay is classified according to the L-value of the emitted radiation. Unlike gamma decay, beta decay may proceed from a nucleus with a spin of zero and even parity to a nucleus also with a spin of zero and even parity (Fermi transition). This is possible because the electron and neutrino emitted may be of opposing spin (giving a radiation total angular momentum of zero), thus preserving angular momentum of the initial state even if the nucleus remains at spin-zero before and after emission. This type of emission is super-allowed meaning that it is the most rapid type of beta decay in nuclei that are susceptible to a change in proton/neutron ratios that accompanies a beta decay process.

The next possible total angular momentum of the electron and neutrino emitted in beta decay is a combined spin of 1 (electron and neutrino spinning in the same direction), and is allowed. This type of emission (Gamow-Teller transition) changes nuclear spin by 1 to compensate. States involving higher angular momenta of the emitted radiation (2, 3, 4, etc.) are forbidden and are ranked in degree of forbiddenness by their increasing angular momentum.

Specifically, when L > 0 the decay is referred to as forbidden. Nuclear selection rules require L-values greater than two to be accompanied by changes in both nuclear spin (J) and parity (π). The selection rules for the Lth forbidden transitions are

${\displaystyle \Delta J=L-1,L,L+1;\Delta \pi =(-1)^{L},}$

where Δπ = 1 or −1 corresponds to no parity change or parity change, respectively. As noted, the special case of a Fermi 0+ → 0+ transition (which in gamma decay is absolutely forbidden) is referred to as super-allowed for beta decay, and proceeds very quickly if beta decay is possible. The following table lists the ΔJ and Δπ values for the first few values of L:

Forbiddenness ΔJ Δπ
Superallowed 0+ → 0+ no
Allowed 0, 1 no
First forbidden 0, 1, 2 yes
Second forbidden 1, 2, 3 no
Third forbidden 2, 3, 4 yes

As with gamma decay, each degree of increasing forbiddenness increases the half life of the beta decay process involved by a factor of about 4 to 5 orders of magnitude.[4]

Double beta decay has been observed in the laboratory, e.g. in 82
Se
.[5] Geochemical experiments have also found this rare type of forbidden decay in several isotopes.[6] with mean half lives over 1018 yr .

## In solid-state physics

Forbidden transitions in rare earth atoms such as erbium and neodymium make them useful as dopants for solid-state lasing media [7]. In such media, the atoms are held in a matrix which keeps them from de-exciting by collision, and the long half life of their excited states makes them easy to optically pump to create a large population of excited atoms. Neodymium doped glass derives its unusual coloration from forbidden f-f transitions within the neodymium atom, and is used in extremely high power solid state lasers. Bulk semiconductor transitions can also be forbidden by symmetry, which change the functional form of the absorption spectrum, as can be shown in a Tauc plot.

## In astrophysics and atomic physics

Forbidden emission lines have been observed in extremely low-density gases and plasmas, either in outer space or in the extreme upper atmosphere of the Earth.[8] In space environments, densities may be only a few atoms per cubic centimetre, making atomic collisions unlikely. Under such conditions, once an atom or molecule has been excited for any reason into a meta-stable state, then it is almost certain to decay by emitting a forbidden-line photon. Since meta-stable states are rather common, forbidden transitions account for a significant percentage of the photons emitted by the ultra-low density gas in space. Forbidden transitions in highly charged ions resulting in the emission of visible, vacuum-ultraviolet, soft x-ray and x-ray photons are routinely observed in certain laboratory devices such as electron beam ion traps [9] and ion storage rings, where in both cases residual gas densities are sufficiently low for forbidden line emission to occur before atoms are collisionally de-excited. Using laser spectroscopy techniques, forbidden transitions are used to stabilize atomic clocks and quantum clocks that have the highest accuracies currently available.

Forbidden lines of nitrogen ([N II] at 654.8 and 658.4 nm), sulfur ([S II] at 671.6 and 673.1 nm), and oxygen ([O II] at 372.7 nm, and [O III] at 495.9 and 500.7 nm) are commonly observed in astrophysical plasmas. These lines are important to the energy balance of planetary nebulae and H II regions. The forbidden 21-cm hydrogen line is particularly important for radio astronomy as it allows very cold neutral hydrogen gas to be seen. Also, the presence of [O I] and [S II] forbidden lines in the spectra of T-tauri stars implies low gas density.

### Notation

Forbidden line transitions are noted by placing square brackets around the atomic or molecular species in question, e.g. [O III] or [S II].[8]

## References

1. ^ Philip R. Bunker; Per Jensen (2006). Molecular Symmetry and Spectroscopy. NRC Research Press. p. 414. ISBN 978-0-660-19628-2.
2. ^ Lisensky, George C.; Patel, Manish N.; Reich, Megan L. (1996). "Experiments with Glow-in-the-Dark Toys: Kinetics of Doped ZnS Phosphorescence". Journal of Chemical Education. 73 (11): 1048. Bibcode:1996JChEd..73.1048L. doi:10.1021/ed073p1048. ISSN 0021-9584.
3. ^
4. ^ Beta decay types
5. ^ Elliott, S. R.; Hahn, A. A.; Moe; M. K. (1987). "Direct evidence for two-neutrino double-beta decay in 82Se". Physical Review Letters. 59 (18): 2020–2023. Bibcode:1987PhRvL..59.2020E. doi:10.1103/PhysRevLett.59.2020.
6. ^ Barabash, A. S. (2011). "Experiment double beta decay: Historical review of 75 years of research". Physics of Atomic Nuclei. 74 (4): 603–613. arXiv:1104.2714. Bibcode:2011PAN....74..603B. doi:10.1134/S1063778811030070.
7. ^ Kolesov, R., et al. "Optical detection of a single rare-earth ion in a crystal." Nature Communications 3 (2012): 1029.
8. ^ a b "Заборонені лінії" (PDF). Астрономічний енциклопедичний словник [Encyclopedic Dictionary of Astronomy] (in Ukrainian). За загальною редакцією І. А. Климишина та А. О. Корсунь. Львів: ЛНУ—ГАО НАНУ. 2003. p. 161. ISBN 966-613-263-X. Lay summary.
9. ^ Mäckel, V. and Klawitter, R. and Brenner, G. and Crespo López-Urrutia, J. R. and Ullrich, J. (2011). "Laser Spectroscopy on Forbidden Transitions in Trapped Highly Charged Ar13+ Ions". Physical Review Letters. American Physical Society. 107 (14): 143002. Bibcode:2011PhRvL.107n3002M. doi:10.1103/PhysRevLett.107.143002.CS1 maint: Uses authors parameter (link)

• Osterbrock, D.E., Astrophysics of gaseous nebulae and active galactic nuclei, University Science Books, 1989, ISBN 0-935702-22-9.
• Heinrich Beyer, Heinrich F. Beyer, H.-Jürgen Kluge, H.-J. Kluge, Viatcheslav Petrovich Shevelʹko, X-Ray Radiation of Highly Charged Ions, Springer Science & Business Media, 1997, ISBN 978-3-540-63185-9.
• Gillaspy, John, editor, Trapping Highly Charged Ions: Fundamentals and Applications, Edited by John Gillaspy. Published by Nova Science Publishers, Inc., Huntington, NY, 1999, ISBN 1-56072-725-X.
• Wolfgang Quint, Manuel Vogel, editors, Fundamental Physics in Particle Traps, Springer Tracts in Modern Physics, Volume 256 2014, ISBN 978-3-642-45200-0.
Aluminium(II) oxide

Aluminium(II) oxide or aluminium monoxide is a compound of aluminium and oxygen with the chemical formula AlO. It has been detected in the gas phase after explosion of aluminized grenades in the upper atmosphere and in stellar absorption spectra.

Carbon monosulfide

Carbon monosulfide is a chemical compound with the formula CS. This diatomic molecule is the sulfur analogue of carbon monoxide, and is unstable as a solid or a liquid, but it has been observed as a gas both in the laboratory and in the interstellar medium. The molecule resembles carbon monoxide with a triple bond between carbon and sulfur. The molecule is not intrinsically unstable, but it tends to polymerize. This tendency reflects the greater stability of C−S single bonds.

Polymers with the formula (CS)n have been reported. Also, CS has been observed as a ligand in certain transition metal complexes.

Circumstellar envelope

A circumstellar envelope (CSE) is a part of a star that has a roughly spherical shape and is not gravitationally bound to the star core. Usually circumstellar envelopes are formed from the dense stellar wind, or they are present before the formation of the star. Circumstellar envelopes of the old stars (Mira variables and OH/IR stars) eventually evolve into protoplanetary nebulae, and circumstellar envelopes of the young stellar objects evolve into circumstellar discs.

Cyclopropenone

Cyclopropenone is an organic compound with molecular formula C3H2O consisting of a cyclopropene carbon framework with a ketone functional group. It is a colorless, volatile liquid that boils near room temperature. Neat cyclopropenone polymerizes upon standing at room temperature. The chemical properties of the compound are dominated by the strong polarization of the carbonyl group, which gives a partial positive charge with aromatic stabilization on the ring and a partial negative charge on oxygen. It is an aromatic compound.

Dicarbon monoxide

Dicarbon monoxide (C2O) is a molecule that contains two carbon atoms and one oxygen atom. It is a linear molecule that, because of its simplicity, is of interest in a variety of areas. It is, however, so extremely reactive that it is not encountered in everyday life. It is classified as a cumulene and an oxocarbon.

Fulminic acid

Fulminic acid is a chemical compound with a molecular formula HCNO. Its silver salt was discovered in 1800 by Edward Charles Howard and later investigated in 1824 by Justus von Liebig. It is an organic acid and an isomer of isocyanic acid, whose silver salt was discovered one year later by Friedrich Woehler. The free acid was first isolated in 1966.Fulminic acid and its salts (fulminates), for instance mercury fulminate, are very dangerous, and are often used as detonators for other explosive materials, and are examples of primary explosives. The vapors also are toxic.

Intergalactic dust

Intergalactic dust is cosmic dust in between galaxies in intergalactic space. Evidence for intergalactic dust has been suggested as early as 1949, and study of it grew throughout the late 20th century. There are large variations in the distribution of intergalactic dust. The dust may affect intergalactic distance measurements, such as to supernova and quasars in other galaxies.Intergalactic dust can form intergalactic dust clouds, known to exist around some galaxies since the 1960s. By the 1980s, at least four intergalactic dust clouds had been discovered within several megaparsec (Mpc) of the Milky Way galaxy, exemplified by the Okroy cloud.In February 2014, NASA announced a greatly upgraded database for tracking polycyclic aromatic hydrocarbons (PAHs) in the universe. According to scientists, more than 20% of the carbon in the universe may be associated with PAHs, possible starting materials for the formation of life. PAHs seem to have been formed as early as two billion years after the big bang, are widespread throughout the universe, and are associated with new stars and exoplanets.

Ketenimine

Ketenimines are a group of organic compounds sharing a common functional group with the general structure R1R2C=C=NR3. A ketenimine is a cumulated alkene and imine and is related to an allene and a ketene.

The parent compound is ketenimine or CH2CNH. The most recent work by Bane et al. investigates the rovibrational structure of the ν8 and ν12 bands in the high-resolution FTIR spectrum, complementing the earlier analysis of the pure rotational spectrum. This pair of Coriolis coupled bands provide a rare example where intensity sharing between bands yields sufficient intensity for an otherwise invisible band (ν12).

Methoxy group

A methoxy group is the functional group consisting of a methyl group bound to oxygen. This alkoxy group has the formula O–CH3. On a benzene ring, the Hammett equation classifies a methoxy substituent as an electron-donating group.

Methyl formate

Methyl formate, also called methyl methanoate, is the methyl ester of formic acid. The simplest example of an ester, it is a colorless liquid with an ethereal odour, high vapor pressure, and low surface tension. It is a precursor to many other compounds of commercial interest.

Methylidynephosphane

Methylidynephosphane (phosphaethyne) is a chemical compound which was the first phosphaalkyne compound discovered, containing the unusual C≡P carbon-phosphorus triple bond.

Octatetraynyl radical (C8H) is an organic free radical with eight carbon atoms linked in a chain with alternating single bonds and triple bonds.

In 2007 negatively charged octatetraynyl was detected in Galactic molecular source TMC-1, making it the second type of anion to be found in the interstellar medium (after Hexatriynyl radical) and the largest such molecule detected to date.

Phosphorus mononitride

Phosphorus mononitride is an inorganic compound with the chemical formula PN. Containing only phosphorus and nitrogen, this material is classified as a binary nitride.

It is the first identified phosphorus compound in the interstellar medium.It is an important molecule in interstellar medium and the atmospheres of Jupiter and Saturn.

Photodissociation region

Photodissociation regions (or photon-dominated regions, or PDRs) are predominantly neutral regions of the interstellar medium in which far ultraviolet photons strongly influence the gas chemistry and act as the most important source of heat. They occur in any region of interstellar gas that is dense and cold enough to remain neutral, but that has too low a column density to prevent the penetration of far-UV photons from distant, massive stars. A typical and well-studied example is the gas at the boundary of a giant molecular cloud. PDRs are also associated with HII regions, reflection nebulae, active galactic nuclei, and Planetary nebulae. All the atomic gas and most of the molecular gas in the galaxy is found in PDRs.

Propionaldehyde

Propionaldehyde or propanal is the organic compound with the formula CH3CH2CHO. It is a saturated 3-carbon aldehyde and is a structural isomer of acetone. It is a colorless liquid with a slightly irritating, fruity odor.

Propynal

Propynal is an organic compound with molecular formula HC2CHO. It is the simplest chemical compound containing both alkyne and aldehyde functional groups. It is a colorless liquid with explosive properties.The compound exhibits reactions expected for an electrophilic alkynyl aldehyde. It is a dienophile and a good Michael acceptor. Grignard reagents add to the carbonyl center.

Silicon monosulfide

Silicon monosulfide is a chemical compound of silicon and sulfur. The chemical formula is SiS. Molecular SiS has been detected at high temperature in the gas phase. The gas phase molecule has an Si-S bondlength of 192.93 pm, this compares to the normal single bond length of 216 pm, and is shorter than the Si=S bond length of around 201 pm reported in an organosilanethione. Historically a pale yellow-red amorphous solid compound has been reported. The behavior of silicon can be contrasted with germanium which forms a stable solid monosulfide.

Titanium oxide

Titanium oxide may refer to:

Titanium dioxide (titanium(IV) oxide), TiO2

Titanium(II) oxide (titanium monoxide), TiO, a non-stoichiometric oxide

Titanium(III) oxide (dititanium trioxide), Ti2O3

Ti3O

Ti2O

δ-TiOx (x= 0.68 - 0.75)

TinO2n−1 where n ranges from 3–9 inclusive, e.g. Ti3O5, Ti4O7, etc.

Triatomic molecule

Triatomic molecules are molecules composed of three atoms, of either the same or different chemical elements. Examples include H2O, CO2 (pictured) and HCN.

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