Spontaneous emission

From Wikipedia, the free encyclopedia

Spontaneous emission is the process in which a quantum mechanical system (such as an atom, molecule or subatomic particle) transitions from an excited energy state to a lower energy state (e.g., its ground state) and emits a quantum in the form of a photon. Spontaneous emission is ultimately responsible for most of the light we see all around us; it is so ubiquitous that there are many names given to what is essentially the same process. If atoms (or molecules) are excited by some means other than heating, the spontaneous emission is called luminescence. For example, fireflies are luminescent. And there are different forms of luminescence depending on how excited atoms are produced (electroluminescence, chemiluminescence etc.). If the excitation is affected by the absorption of radiation the spontaneous emission is called fluorescence. Sometimes molecules have a metastable level and continue to fluoresce long after the exciting radiation is turned off; this is called phosphorescence. Figurines that glow in the dark are phosphorescent. Lasers start via spontaneous emission, then during continuous operation work by stimulated emission.

Spontaneous emission cannot be explained by classical electromagnetic theory and is fundamentally a quantum process. The first person to derive the rate of spontaneous emission accurately from first principles was Dirac in his quantum theory of radiation,^[1] the precursor to the theory which he later coined quantum electrodynamics.^[2] Contemporary physicists, when asked to give a physical explanation for spontaneous emission, generally invoke the zero-point energy of the electromagnetic field.^[3][4] In 1963 the Jaynes-Cummings model^[5] was developed describing the system of a two-level atom interacting with a quantized field mode (i.e. the vacuum) within an optical cavity. It gave the nonintuitive prediction that the rate of spontaneous emission could be controlled depending on the boundary conditions of the surrounding vacuum field. These experiments gave rise to cavity quantum electrodynamics (CQED), the study of effects of mirrors and cavities on radiative corrections.

Introduction

If a light source ('the atom') is in an excited state with energy ${\displaystyle E_{2}}$, it may spontaneously decay to a lower lying level (e.g., the ground state) with energy ${\displaystyle E_{1}}$, releasing the difference in energy between the two states as a photon. The photon will have angular frequency ${\displaystyle \omega }$ and an energy ${\displaystyle \hbar \omega }$:

${\displaystyle E_{2}-E_{1}=\hbar \omega ,}$

where ${\displaystyle \hbar }$ is the reduced Planck constant. Note: ${\displaystyle \hbar \omega =h\nu }$, where ${\displaystyle h}$ is the Planck constant and ${\displaystyle \nu }$ is the linear frequency. The phase of the photon in spontaneous emission is random as is the direction in which the photon propagates. This is not true for stimulated emission. An energy level diagram illustrating the process of spontaneous emission is shown below:


If the number of light sources in the excited state at time ${\displaystyle t}$ is given by ${\displaystyle N(t)}$, the rate at which ${\displaystyle N}$ decays is:

${\displaystyle {\frac {\partial N(t)}{\partial t}}=-A_{21}N(t),}$

where ${\displaystyle A_{21}}$ is the rate of spontaneous emission. In the rate-equation ${\displaystyle A_{21}}$ is a proportionality constant for this particular transition in this particular light source. The constant is referred to as the Einstein A coefficient, and has units ${\displaystyle s^{-1}}$.^[6] The above equation can be solved to give:

${\displaystyle N(t)=N(0)e^{-A_{21}t}=N(0)e^{-\Gamma _{rad}t},}$

where ${\displaystyle N(0)}$ is the initial number of light sources in the excited state, ${\displaystyle t}$ is the time and ${\displaystyle \Gamma _{rad}}$ is the radiative decay rate of the transition. The number of excited states ${\displaystyle N}$ thus decays exponentially with time, similar to radioactive decay. After one lifetime, the number of excited states decays to 36.8% of its original value (${\displaystyle {\frac {1}{e}}}$-time). The radiative decay rate ${\displaystyle \Gamma _{rad}}$ is inversely proportional to the lifetime ${\displaystyle \tau _{21}}$:

${\displaystyle A_{21}=\Gamma _{21}={\frac {1}{\tau _{21}}}.}$

Theory

Spontaneous transitions were not explainable within the framework of the Schrödinger equation, in which the electronic energy levels were quantized, but the electromagnetic field was not. Given that the eigenstates of an atom are properly diagonalized, the overlap of the wavefunctions between the excited state and the ground state of the atom is zero. Thus, in the absence of a quantized electromagnetic field, the excited state atom cannot decay to the ground state. In order to explain spontaneous transitions, quantum mechanics must be extended to a quantum field theory, wherein the electromagnetic field is quantized at every point in space. The quantum field theory of electrons and electromagnetic fields is known as quantum electrodynamics.

In quantum electrodynamics (or QED), the electromagnetic field has a ground state, the QED vacuum, which can mix with the excited stationary states of the atom.^[2] As a result of this interaction, the "stationary state" of the atom is no longer a true eigenstate of the combined system of the atom plus electromagnetic field. In particular, the electron transition from the excited state to the electronic ground state mixes with the transition of the electromagnetic field from the ground state to an excited state, a field state with one photon in it. Spontaneous emission in free space depends upon vacuum fluctuations to get started.^[7][8]

Although there is only one electronic transition from the excited state to ground state, there are many ways in which the electromagnetic field may go from the ground state to a one-photon state. That is, the electromagnetic field has infinitely more degrees of freedom, corresponding to the different directions in which the photon can be emitted. Equivalently, one might say that the phase space offered by the electromagnetic field is infinitely larger than that offered by the atom. This infinite degree of freedom for the emission of the photon results in the apparent irreversible decay, i.e., spontaneous emission.

In the presence of electromagnetic vacuum modes, the combined atom-vacuum system is explained by the superposition of the wavefunctions of the excited state atom with no photon and the ground state atom with a single emitted photon:

${\displaystyle |\psi (t)\rangle =a(t)e^{-i\omega _{0}t}|e;0\rangle +\sum _{k,s}b_{ks}(t)e^{-i\omega _{k}t}|g;1_{ks}\rangle }$

where ${\displaystyle |e;0\rangle }$ and ${\displaystyle a(t)}$ are the atomic excited state-electromagnetic vacuum wavefunction and its probability amplitude, ${\displaystyle |g;1_{ks}\rangle }$ and ${\displaystyle b_{ks}(t)}$ are the ground state atom with a single photon (of mode ${\displaystyle ks}$) wavefunction and its probability amplitude, ${\displaystyle \omega _{0}}$ is the atomic transition frequency, and ${\displaystyle \omega _{k}=c|k|}$ is the frequency of the photon. The sum is over ${\displaystyle k}$ and ${\displaystyle s}$, which are the wavenumber and polarization of the emitted photon, respectively. As mentioned above, the emitted photon has a chance to be emitted with different wavenumbers and polarizations, and the resulting wavefunction is a superposition of these possibilities. To calculate the probability of the atom at the ground state (${\displaystyle |b(t)|^{2}}$), one needs to solve the time evolution of the wavefunction with an appropriate Hamiltonian.^[1] To solve for the transition amplitude, one needs to average over (integrate over) all the vacuum modes, since one must consider the probabilities that the emitted photon occupies various parts of phase space equally. The "spontaneously" emitted photon has infinite different modes to propagate into, thus the probability of the atom re-absorbing the photon and returning to the original state is negligible, making the atomic decay practically irreversible. Such irreversible time evolution of the atom-vacuum system is responsible for the apparent spontaneous decay of an excited atom. If one were to keep track of all the vacuum modes, the combined atom-vacuum system would undergo unitary time evolution, making the decay process reversible. Cavity quantum electrodynamics is one such system where the vacuum modes are modified resulting in the reversible decay process, see also Quantum revival. The theory of the spontaneous emission under the QED framework was first calculated by Weisskopf and Wigner.

In spectroscopy one can frequently find that atoms or molecules in the excited states dissipate their energy in the absence of any external source of photons. This is not spontaneous emission, but is actually nonradiative relaxation of the atoms or molecules caused by the fluctuation of the surrounding molecules present inside the bulk.^{[clarification needed]}

Rate of spontaneous emission

The rate of spontaneous emission (i.e., the radiative rate) can be described by Fermi's golden rule.^[9] The rate of emission depends on two factors: an 'atomic part', which describes the internal structure of the light source and a 'field part', which describes the density of electromagnetic modes of the environment. The atomic part describes the strength of a transition between two states in terms of transition moments. In a homogeneous medium, such as free space, the rate of spontaneous emission in the dipole approximation is given by:

${\displaystyle \Gamma _{rad}(\omega )={\frac {\omega ^{3}n|\mu _{12}|^{2}}{3\pi \varepsilon _{0}\hbar c^{3}}}={\frac {4\alpha \omega ^{3}n|\langle 1|\mathbf {r} |2\rangle |^{2}}{3c^{2}}}}$

${\displaystyle {\frac {|\mu _{12}|^{2}}{\pi \varepsilon _{0}\hbar c}}=4\alpha |\langle 1|\mathbf {r} |2\rangle |^{2}}$

where ${\displaystyle \omega }$ is the emission frequency, ${\displaystyle n}$ is the index of refraction, ${\displaystyle \mu _{12}}$ is the transition dipole moment, ${\displaystyle \varepsilon _{0}}$ is the vacuum permittivity, ${\displaystyle \hbar }$ is the reduced Planck constant, ${\displaystyle c}$ is the vacuum speed of light, and ${\displaystyle \alpha }$ is the fine structure constant.^{[clarification needed]} (This approximation breaks down in the case of inner shell electrons in high-Z atoms.) The above equation clearly shows that the rate of spontaneous emission in free space increases proportionally to ${\displaystyle \omega ^{3}}$.

In contrast with atoms, which have a discrete emission spectrum, quantum dots can be tuned continuously by changing their size. This property has been used to check the ${\displaystyle \omega ^{3}}$-frequency dependence of the spontaneous emission rate as described by Fermi's golden rule.^[10]

Radiative and nonradiative decay: the quantum efficiency

In the rate-equation above, it is assumed that decay of the number of excited states ${\displaystyle N}$ only occurs under emission of light. In this case one speaks of full radiative decay and this means that the quantum efficiency is 100%. Besides radiative decay, which occurs under the emission of light, there is a second decay mechanism; nonradiative decay. To determine the total decay rate ${\displaystyle \Gamma _{tot}}$, radiative and nonradiative rates should be summed:

${\displaystyle \Gamma _{tot}=\Gamma _{rad}+\Gamma _{nrad}}$

where ${\displaystyle \Gamma _{tot}}$ is the total decay rate, ${\displaystyle \Gamma _{rad}}$ is the radiative decay rate and ${\displaystyle \Gamma _{nrad}}$ the nonradiative decay rate. The quantum efficiency (QE) is defined as the fraction of emission processes in which emission of light is involved:

${\displaystyle QE={\frac {\Gamma _{rad}}{\Gamma _{nrad}+\Gamma _{rad}}}.}$

In nonradiative relaxation, the energy is released as phonons, more commonly known as heat. Nonradiative relaxation occurs when the energy difference between the levels is very small, and these typically occur on a much faster time scale than radiative transitions. For many materials (for instance, semiconductors), electrons move quickly from a high energy level to a meta-stable level via small nonradiative transitions and then make the final move down to the bottom level via an optical or radiative transition. This final transition is the transition over the bandgap in semiconductors. Large nonradiative transitions do not occur frequently because the crystal structure generally cannot support large vibrations without destroying bonds (which generally doesn't happen for relaxation). Meta-stable states form a very important feature that is exploited in the construction of lasers. Specifically, since electrons decay slowly from them, they can be deliberately piled up in this state without too much loss and then stimulated emission can be used to boost an optical signal.

Accretion (astrophysics)

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Atacama Large Millimeter Array image of HL Tauri, a protoplanetary disk

In astrophysics, accretion is the accumulation of particles into a massive object by gravitationally attracting more matter, typically gaseous matter, in an accretion disk.^[1][2] Most astronomical objects, such as galaxies, stars, and planets, are formed by accretion processes.

Overview

The idea proposed in the 19th century that Earth and the other terrestrial planets formed from meteoric material was developed in a quantitative way in 1969 by Viktor Safronov. He calculated, in detail, the different stages of terrestrial planet formation.^[3][4] Since then, the theory has been further developed using intensive numerical simulations to study planetesimal accumulation.

Stars form by the gravitational collapse of interstellar gas. Prior to collapse, this gas is mostly in the form of molecular clouds, such as the Orion Nebula. As the cloud collapses, losing potential energy, it heats up, gaining kinetic energy, and the conservation of angular momentum ensures that the cloud forms a flatted disk—the accretion disk.

Accretion of galaxies

A few hundred thousand years after the Big Bang, the Universe cooled to the point where atoms could form. As the Universe continued to expand and cool, the atoms lost enough kinetic energy, and dark matter coalesced sufficiently, to form protogalaxies. As further accretion occurred, galaxies formed.^[5] Indirect evidence is widespread.^[5] Galaxies grow through mergers and smooth gas accretion. Accretion also occurs inside galaxies, forming stars.

Accretion of stars

The visible-light (left) and infrared (right) views of the Trifid Nebula, a giant star-forming cloud of gas and dust located 5,400 light-years (1,700 pc) away in the constellation Sagittarius

Stars are thought to form inside giant clouds of cold molecular hydrogen—giant molecular clouds of roughly 300,000 M_☉ and 65 light-years (20 pc) in diameter.^[6][7] Over millions of years, giant molecular clouds are prone to collapse and fragmentation.^[8] These fragments then form small, dense cores, which in turn collapse into stars.^[7] The cores range in mass from a fraction to several times that of the Sun and are called protostellar (protosolar) nebulae.^[6] They possess diameters of 2,000–20,000 astronomical units (0.01–0.1 pc) and a particle number density of roughly 10,000 to 100,000/cm³ (160,000 to 1,600,000/cu in). Compare it with the particle number density of the air at the sea level—2.8×10¹⁹/cm³ (4.6×10²⁰/cu in).^[7][9]

The initial collapse of a solar-mass protostellar nebula takes around 100,000 years.^[6][7] Every nebula begins with a certain amount of angular momentum. Gas in the central part of the nebula, with relatively low angular momentum, undergoes fast compression and forms a hot hydrostatic (non-contracting) core containing a small fraction of the mass of the original nebula. This core forms the seed of what will become a star.^[6] As the collapse continues, conservation of angular momentum dictates that the rotation of the infalling envelope accelerates, which eventually forms a disk.

Infrared image of the molecular outflow from an otherwise hidden newborn star HH 46/47

As the infall of material from the disk continues, the envelope eventually becomes thin and transparent and the young stellar object (YSO) becomes observable, initially in far-infrared light and later in the visible.^[9] Around this time the protostar begins to fuse deuterium. If the protostar is sufficiently massive (above 80 M_J), hydrogen fusion follows. Otherwise, if its mass is too low, the object becomes a brown dwarf.^[10] This birth of a new star occurs approximately 100,000 years after the collapse begins.^[6] Objects at this stage are known as Class I protostars, which are also called young T Tauri stars, evolved protostars, or young stellar objects. By this time, the forming star has already accreted much of its mass; the total mass of the disk and remaining envelope does not exceed 10–20% of the mass of the central YSO.^[9]

When the lower-mass star in a binary system enters an expansion phase, its outer atmosphere may fall onto the compact star, forming an accretion disk

At the next stage, the envelope completely disappears, having been gathered up by the disk, and the protostar becomes a classical T Tauri star.^[11] The latter have accretion disks and continue to accrete hot gas, which manifests itself by strong emission lines in their spectrum. The former do not possess accretion disks. Classical T Tauri stars evolve into weakly lined T Tauri stars.^[12] This happens after about 1 million years.^[6] The mass of the disk around a classical T Tauri star is about 1–3% of the stellar mass, and it is accreted at a rate of 10⁻⁷ to 10⁻⁹ M_☉ per year.^[13] A pair of bipolar jets is usually present as well. The accretion explains all peculiar properties of classical T Tauri stars: strong flux in the emission lines (up to 100% of the intrinsic luminosity of the star), magnetic activity, photometric variability and jets.^[14] The emission lines actually form as the accreted gas hits the "surface" of the star, which happens around its magnetic poles.^[14] The jets are byproducts of accretion: they carry away excessive angular momentum. The classical T Tauri stage lasts about 10 million years.^[6] The disk eventually disappears due to accretion onto the central star, planet formation, ejection by jets, and photoevaporation by ultraviolet radiation from the central star and nearby stars.^[15] As a result, the young star becomes a weakly lined T Tauri star, which, over hundreds of millions of years, evolves into an ordinary Sun-like star, dependent on its initial mass.

Accretion of planets

Artist's impression of a protoplanetary disk showing a young star at its center

Self-accretion of cosmic dust accelerates the growth of the particles into boulder-sized planetesimals. The more massive planetesimals accrete some smaller ones, while others shatter in collisions. Accretion disks are common around smaller stars, or stellar remnants in a close binary, or black holes surrounded by material, such as those at the centers of galaxies. Some dynamics in the disk, such as dynamical friction, are necessary to allow orbiting gas to lose angular momentum and fall onto the central massive object. Occasionally, this can result in stellar surface fusion (see Bondi accretion).

In the formation of terrestrial planets or planetary cores, several stages can be considered. First, when gas and dust grains collide, they agglomerate by microphysical processes like van der Waals forces and electromagnetic forces, forming micrometer-sized particles; during this stage, accumulation mechanisms are largely non-gravitational in nature.^[16] However, planetesimal formation in the centimeter-to-meter range is not well understood, and no convincing explanation is offered as to why such grains would accumulate rather than simply rebound.^[16]:341 In particular, it is still not clear how these objects grow to become 0.1–1 km (0.06–0.6 mi) sized planetesimals;^[3][17] this problem is known as the "meter size barrier":^[18] As dust particles grow by coagulation, they acquire increasingly large relative velocities with respect to other particles in their vicinity, as well as a systematic inward drift velocity, that leads to destructive collisions, and thereby limit the growth of the aggregates to some maximum size.^[19] Ward (1996) suggests that when slow moving grains collide, the very low, yet non-zero, gravity of colliding grains impedes their escape.^[16]:341 It is also thought that grain fragmentation plays an important role replenishing small grains and keeping the disk thick, but also in maintaining a relatively high abundance of solids of all sizes.^[19]

A number of mechanisms have been proposed for crossing the 'meter-sized' barrier. Local concentrations of pebbles may form, which then gravitationally collapse into planetesimals the size of large asteroids. These concentrations can occur passively due to the structure of the gas disk, for example, between eddies, at pressure bumps, at the edge of a gap created by a giant planet, or at the boundaries of turbulent regions of the disk.^[20] Or, the particles may take an active role in their concentration via a feedback mechanism referred to as a streaming instability. In a streaming instability the interaction between the solids and the gas in the protoplanetary disk results in the growth of local concentrations, as new particles accumulate in the wake of small concentrations, causing them to grow into massive filaments.^[20] Alternatively, if the grains that form due to the agglomeration of dust are highly porous their growth may continue until they become large enough to collapse due to their own gravity. The low density of these objects allows them to remain strongly coupled with the gas, thereby avoiding high velocity collisions which could result in their erosion or fragmentation.^[21]

Grains eventually stick together to form mountain-size (or larger) bodies called planetesimals. Collisions and gravitational interactions between planetesimals combine to produce Moon-size planetary embryos (protoplanets) over roughly 0.1–1 million years. Finally, the planetary embryos collide to form planets over 10–100 million years.^[17] The planetesimals are massive enough that mutual gravitational interactions are significant enough to be taken into account when computing their evolution.^[3] Growth is aided by orbital decay of smaller bodies due to gas drag, which prevents them from being stranded between orbits of the embryos.^[22][23] Further collisions and accumulation lead to terrestrial planets or the core of giant planets.

If the planetesimals formed via the gravitational collapse of local concentrations of pebbles their growth into planetary embryos and the cores of giant planets is dominated by the further accretions of pebbles. Pebble accretion is aided by the gas drag felt by objects as they accelerate toward a massive body. Gas drag slows the pebbles below the escape velocity of the massive body causing them to spiral toward and to be accreted by it. Pebble accretion may accelerate the formation of planets by a factor of 1000 compared to the accretion of planetesimals, allowing giant planets to form before the dissipation of the gas disk.^[24][25] Yet, core growth via pebble accretion appears incompatible with the final masses and compositions of Uranus and Neptune.^[26]

The formation of terrestrial planets differs from that of giant gas planets, also called Jovian planets. The particles that make up the terrestrial planets are made from metal and rock that condense in the inner Solar System. However, Jovian planets begin as large, icy planetesimals, which then capture hydrogen and helium gas from the solar nebula.^[27] Differentiation between these two classes of planetesimals arise due to the frost line of the solar nebula.^[28]

Accretion of asteroids

Chondrules in a chondrite meteorite. A millimeter scale is shown.

Meteorites contain a record of accretion and impacts during all stages of asteroid origin and evolution; however, the mechanism of asteroid accretion and growth is not well understood.^[29] Evidence suggests the main growth of asteroids can result from gas-assisted accretion of chondrules, which are millimeter-sized spherules that form as molten (or partially molten) droplets in space before being accreted to their parent asteroids.^[29] In the inner Solar System, chondrules appear to have been crucial for initiating accretion.^[30] The tiny mass of asteroids may be partly due to inefficient chondrule formation beyond 2 AU, or less-efficient delivery of chondrules from near the protostar.^[30] Also, impacts controlled the formation and destruction of asteroids, and are thought to be a major factor in their geological evolution.^[30]

Chondrules, metal grains, and other components likely formed in the solar nebula. These accreted together to form parent asteroids. Some of these bodies subsequently melted, forming metallic cores and olivine-rich mantles; others were aqueously altered.^[30] After the asteroids had cooled, they were eroded by impacts for 4.5 billion years, or disrupted.^[31]

For accretion to occur, impact velocities must be less than about twice the escape velocity, which is about 140 m/s (460 ft/s) for a 100 km (60 mi) radius asteroid.^[30] Simple models for accretion in the asteroid belt generally assume micrometer-sized dust grains sticking together and settling to the midplane of the nebula to form a dense layer of dust, which, because of gravitational forces, was converted into a disk of kilometer-sized planetesimals. But, several arguments^[which?] suggest that asteroids may not have accreted this way.^[30]

Accretion of comets

The Helix Nebula has a cometary Oort cloud

Comets, or their precursors, formed in the outer Solar System, possibly millions of years before planet formation.^[32] How and when comets formed is debated, with distinct implications for Solar System formation, dynamics, and geology. Three-dimensional computer simulations indicate the major structural features observed on cometary nuclei can be explained by pairwise low velocity accretion of weak cometesimals.^[33][34] The currently favored formation mechanism is that of the nebular hypothesis, which states that comets are probably a remnant of the original planetesimal "building blocks" from which the planets grew.^[35][36][37]

Astronomers think that comets originate in both the Oort cloud and the scattered disk.^[38] The scattered disk was created when Neptune migrated outward into the proto-Kuiper belt, which at the time was much closer to the Sun, and left in its wake a population of dynamically stable objects that could never be affected by its orbit (the Kuiper belt proper), and a population whose perihelia are close enough that Neptune can still disturb them as it travels around the Sun (the scattered disk). Because the scattered disk is dynamically active and the Kuiper belt relatively dynamically stable, the scattered disk is now seen as the most likely point of origin for periodic comets.^[38] The classic Oort cloud theory states that the Oort cloud, a sphere measuring about 50,000 AU (0.24 pc) in radius, formed at the same time as the solar nebula and occasionally releases comets into the inner Solar System as a giant planet or star passes nearby and causes gravitational disruptions.^[39] Examples of such comet clouds may already have been seen in the Helix Nebula.^[40]

The Rosetta mission to comet 67P/Churyumov–Gerasimenko determined in 2015 that when Sun's heat penetrates the surface, it triggers evaporation (sublimation) of buried ice. While some of the resulting water vapour may escape from the nucleus, 80% of it recondenses in layers beneath the surface.^[41] This observation implies that the thin ice-rich layers exposed close to the surface may be a consequence of cometary activity and evolution, and that global layering does not necessarily occur early in the comet's formation history.^[41][42] While most scientists thought that all the evidence indicated that the structure of nuclei of comets is processed rubble piles of smaller ice planetesimals of a previous generation,^[43] the Rosetta mission dispelled the idea that comets are "rubble piles" of disparate material.^[44][45]