Forbidden mechanism

Last updated November 18, 2024

In spectroscopy, a forbidden mechanism (forbidden transition or forbidden line) is a spectral line associated with absorption or emission of photons 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 low rate.

Should an atomic nucleus, atom or molecule be raised to an excited state and should the transitions be nominally forbidden, then there is still a small probability of their spontaneous occurrence. 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.^{[ citation needed ]} Such excited states can last years, or even for many billions of years (too long to have been measured).

In radioactive decay

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 $\hbar$ 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 and ends 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 $L$ th forbidden transitions are

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

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 Δ $J$ and Δπ values for the first few values of $L$ :

Forbiddenness	Δ $J$	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]

Related Research Articles

Atoms are the basic particles of the chemical elements. An atom consists of a nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished from each other by the number of protons that are in their atoms. For example, any atom that contains 11 protons is sodium, and any atom that contains 29 protons is copper. Atoms with the same number of protons but a different number of neutrons are called isotopes of the same element.

Alpha decay or α-decay is a type of radioactive decay in which an atomic nucleus emits an alpha particle and thereby transforms or "decays" into a different atomic nucleus, with a mass number that is reduced by four and an atomic number that is reduced by two. An alpha particle is identical to the nucleus of a helium-4 atom, which consists of two protons and two neutrons. It has a charge of +2 e and a mass of 4 Da. For example, uranium-238 decays to form thorium-234.

In nuclear physics, beta decay (β-decay) is a type of radioactive decay in which an atomic nucleus emits a beta particle, transforming into an isobar of that nuclide. For example, beta decay of a neutron transforms it into a proton by the emission of an electron accompanied by an antineutrino; or, conversely a proton is converted into a neutron by the emission of a positron with a neutrino in what is called positron emission. Neither the beta particle nor its associated (anti-)neutrino exist within the nucleus prior to beta decay, but are created in the decay process. By this process, unstable atoms obtain a more stable ratio of protons to neutrons. The probability of a nuclide decaying due to beta and other forms of decay is determined by its nuclear binding energy. The binding energies of all existing nuclides form what is called the nuclear band or valley of stability. For either electron or positron emission to be energetically possible, the energy release or Q value must be positive.

A beta particle, also called beta ray or beta radiation, is a high-energy, high-speed electron or positron emitted by the radioactive decay of an atomic nucleus, known as beta decay. There are two forms of beta decay, β⁻ decay and β⁺ decay, which produce electrons and positrons, respectively.

The Mössbauer effect, or recoilless nuclear resonance fluorescence, is a physical phenomenon discovered by Rudolf Mössbauer in 1958. It involves the resonant and recoil-free emission and absorption of gamma radiation by atomic nuclei bound in a solid. Its main application is in Mössbauer spectroscopy.

A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The term is commonly used for the energy levels of the electrons in atoms, ions, or molecules, which are bound by the electric field of the nucleus, but can also refer to energy levels of nuclei or vibrational or rotational energy levels in molecules. The energy spectrum of a system with such discrete energy levels is said to be quantized.

Electron capture is a process in which the proton-rich nucleus of an electrically neutral atom absorbs an inner atomic electron, usually from the K or L electron shells. This process thereby changes a nuclear proton to a neutron and simultaneously causes the emission of an electron neutrino.

<span class="mw-page-title-main">Nuclear isomer</span> Metastable excited state of a nuclide

A nuclear isomer is a metastable state of an atomic nucleus, in which one or more nucleons (protons or neutrons) occupy excited state (higher energy) levels. "Metastable" describes nuclei whose excited states have half-lives 100 to 1000 times longer than the half-lives of the excited nuclear states that decay with a "prompt" half life (ordinarily on the order of 10⁻¹² seconds). The term "metastable" is usually restricted to isomers with half-lives of 10⁻⁹ seconds or longer. Some references recommend 5 × 10⁻⁹ seconds to distinguish the metastable half life from the normal "prompt" gamma-emission half-life. Occasionally the half-lives are far longer than this and can last minutes, hours, or years. For example, the ^180m
₇₃Ta nuclear isomer survives so long (at least 10¹⁵ years) that it has never been observed to decay spontaneously. The half-life of a nuclear isomer can even exceed that of the ground state of the same nuclide, as shown by ^180m
₇₃Ta as well as ^186m
₇₅Re, ^192m2
₇₇Ir, ^210m
₈₃Bi, ^212m
₈₄Po, ^242m
₉₅Am and multiple holmium isomers.

Radioactive decay is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is considered radioactive. Three of the most common types of decay are alpha, beta, and gamma decay. The weak force is the mechanism that is responsible for beta decay, while the other two are governed by the electromagnetic and nuclear forces.

In physics, atomic spectroscopy is the study of the electromagnetic radiation absorbed and emitted by atoms. Since unique elements have unique emission spectra, atomic spectroscopy is applied for determination of elemental compositions. It can be divided by atomization source or by the type of spectroscopy used. In the latter case, the main division is between optical and mass spectrometry. Mass spectrometry generally gives significantly better analytical performance, but is also significantly more complex. This complexity translates into higher purchase costs, higher operational costs, more operator training, and a greater number of components that can potentially fail. Because optical spectroscopy is often less expensive and has performance adequate for many tasks, it is far more common. Atomic absorption spectrometers are one of the most commonly sold and used analytical devices.

<span class="mw-page-title-main">Internal conversion</span> Process where an excited nucleus ejects an orbital electron from its atom

Internal conversion is an atomic decay process where an excited nucleus interacts electromagnetically with one of the orbital electrons of an atom. This causes the electron to be emitted (ejected) from the atom. Thus, in internal conversion, a high-energy electron is emitted from the excited atom, but not from the nucleus. For this reason, the high-speed electrons resulting from internal conversion are not called beta particles, since the latter come from beta decay, where they are newly created in the nuclear decay process.

In quantum mechanics, angular momentum coupling is the procedure of constructing eigenstates of total angular momentum out of eigenstates of separate angular momenta. For instance, the orbit and spin of a single particle can interact through spin–orbit interaction, in which case the complete physical picture must include spin–orbit coupling. Or two charged particles, each with a well-defined angular momentum, may interact by Coulomb forces, in which case coupling of the two one-particle angular momenta to a total angular momentum is a useful step in the solution of the two-particle Schrödinger equation. In both cases the separate angular momenta are no longer constants of motion, but the sum of the two angular momenta usually still is. Angular momentum coupling in atoms is of importance in atomic spectroscopy. Angular momentum coupling of electron spins is of importance in quantum chemistry. Also in the nuclear shell model angular momentum coupling is ubiquitous.

In physics and chemistry, a selection rule, or transition rule, formally constrains the possible transitions of a system from one quantum state to another. Selection rules have been derived for electromagnetic transitions in molecules, in atoms, in atomic nuclei, and so on. The selection rules may differ according to the technique used to observe the transition. The selection rule also plays a role in chemical reactions, where some are formally spin-forbidden reactions, that is, reactions where the spin state changes at least once from reactants to products.

<span class="mw-page-title-main">Double electron capture</span> Mode of radioactive decay

Double electron capture is a decay mode of an atomic nucleus. For a nuclide (A, Z) with a number of nucleons A and atomic number Z, double electron capture is only possible if the mass of the nuclide (A, Z−2) is lower.

In physics, induced gamma emission (IGE) refers to the process of fluorescent emission of gamma rays from excited nuclei, usually involving a specific nuclear isomer. It is analogous to conventional fluorescence, which is defined as the emission of a photon by an excited electron in an atom or molecule. In the case of IGE, nuclear isomers can store significant amounts of excitation energy for times long enough for them to serve as nuclear fluorescent materials. There are over 800 known nuclear isomers but almost all are too intrinsically radioactive to be considered for applications. As of 2006 there were two proposed nuclear isomers that appeared to be physically capable of IGE fluorescence in safe arrangements: tantalum-180m and hafnium-178m2.

Yrast is a technical term in nuclear physics that refers to a state of a nucleus with a minimum of energy for a given angular momentum. Yr is a Swedish adjective sharing the same root as the English whirl. Yrast is the superlative of yr and can be translated whirlingest, although it literally means "dizziest" or "most bewildered". The yrast levels are vital to understanding reactions, such as off-center heavy ion collisions, that result in high-spin states.

<span class="mw-page-title-main">Mössbauer spectroscopy</span> Spectroscopic technique

Mössbauer spectroscopy is a spectroscopic technique based on the Mössbauer effect. This effect, discovered by Rudolf Mössbauer in 1958, consists of the nearly recoil-free emission and absorption of nuclear gamma rays in solids. The consequent nuclear spectroscopy method is exquisitely sensitive to small changes in the chemical environment of certain nuclei.

A gamma ray, also known as gamma radiation (symbol $γ$ ), is a penetrating form of electromagnetic radiation arising from the radioactive decay of atomic nuclei. It consists of the shortest wavelength electromagnetic waves, typically shorter than those of X-rays. With frequencies above 30 exahertz (3×10¹⁹ Hz) and wavelengths less than 10 picometers (1×10⁻¹¹ m), gamma ray photons have the highest photon energy of any form of electromagnetic radiation. Paul Villard, a French chemist and physicist, discovered gamma radiation in 1900 while studying radiation emitted by radium. In 1903, Ernest Rutherford named this radiation gamma rays based on their relatively strong penetration of matter; in 1900, he had already named two less penetrating types of decay radiation (discovered by Henri Becquerel) alpha rays and beta rays in ascending order of penetrating power.

The Wu experiment was a particle and nuclear physics experiment conducted in 1956 by the Chinese American physicist Chien-Shiung Wu in collaboration with the Low Temperature Group of the US National Bureau of Standards. The experiment's purpose was to establish whether or not conservation of parity (P-conservation), which was previously established in the electromagnetic and strong interactions, also applied to weak interactions. If P-conservation were true, a mirrored version of the world (where left is right and right is left) would behave as the mirror image of the current world. If P-conservation were violated, then it would be possible to distinguish between a mirrored version of the world and the mirror image of the current world.

In nuclear physics, a beta decay transition is the change in state of an atomic nucleus undergoing beta decay. When undergoing beta decay, a nucleus emits a beta particle and a corresponding neutrino, transforming the original nuclide into one with the same mass number but differing atomic number.

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