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Einstein coefficients
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Einstein coefficients
In physics, one thinks of atomic spectral lines from two viewpoints.
• An emission line is formed when an electron makes a transition
from a particular discrete energy level E2 of an atom, to a lower
energy level E1, emitting a photon of a particular energy and
wavelength. A spectrum of many such photons will show an
emission spike at the wavelength associated with these photons.
• An absorption line is formed when an electron makes a transition
from a lower, E1, to a higher discrete energy state, E2, with a photon
being absorbed in the process. These absorbed photons generally
come from background continuum radiation and a spectrum will
show a drop in the continuum radiation at the wavelength associated
with the absorbed photons.
Emission lines and absorption lines compared to
a continuous spectrum.
The two states must be bound states in which the electron is bound to the atom, so the transition is sometimes
referred to as a "bound–bound" transition, as opposed to a transition in which the electron is ejected out of the atom
completely ("bound–free" transition) into a continuum state, leaving an ionized atom, and generating continuum
radiation.
A photon with an energy equal to the difference E2 - E1 between the energy levels is released or absorbed in the
process. The frequency ν at which the spectral line occurs is related to the photon energy by Bohr's frequency
condition
where h denotes Planck's constant.[1][2][3][4][5][6]
Emission and absorption coefficients
An atomic spectral line refers to emission and absorption events in a gas in which
is the density of atoms in the
upper energy state for the line, and
is the density of atoms in the lower energy state for the line.
The emission of atomic line radiation at frequency ν may be described by an emission coefficient with units of
energy/time/volume/solid angle. ε dt dV dΩ is then the energy emitted by a volume element
in time
into
solid angle
where
. For atomic line radiation:
is the Einstein coefficient for spontaneous emission, which is fixed by the intrinsic properties of the
relevant atom for the two relevant energy levels.
The absorption of atomic line radiation may be described by an absorption coefficient with units of 1/length. The
expression κ' dx gives the fraction of intensity absorbed for a light beam at frequency ν while traveling distance dx.
The absorption coefficient is given by:
where
coefficient
and
are the Einstein coefficients for photo absorption and induced emission respectively. Like the
, these are also fixed by the intrinsic properties of the relevant atom for the two relevant energy
levels. For thermodynamics and for the application of Kirchhoff's law, it is necessary that the total absorption be
expressed as the algebraic sum of two components, described respectively by
and
, which may be
regarded as positive and negative absorption, which are, respectively, the direct photon absorption, and what is
commonly called stimulated or induced emission.[7][8][9]
Einstein coefficients
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The above equations have ignored the influence of the spectral line shape. To be accurate, the above equations need
to be multiplied by the (normalized) spectral line shape, in which case the units will change to include a 1/Hz term.
For conditions of thermodynamic equilibrium, together the number densities
and
, the Einstein coefficients,
and the spectral energy density provide sufficient information to determine the absorption and emission rates.
Equilibrium conditions
The number densities
and
are set by the physical state of the gas in which the spectral line occurs, including
the local spectral radiance (or, in some presentations, the local spectral radiant energy density). When that state is
either one of strict thermodynamic equilibrium, or one of so-called 'local thermodynamic equilibrium',[10][11][12] then
the distribution of atomic states of excitation (which includes
and
) determines the rates of atomic emissions
and absorptions to be such that Kirchhoff's law of equality of radiative absorptivity and emissivity holds. In strict
thermodynamic equilibrium, the radiation field is said to be black-body radiation, and is described by Planck's law.
For local thermodynamic equilibrium, the radiation field does not have to be a black-body field, but the rate of
interatomic collisions must vastly exceed the rates of absorption and emission of quanta of light, so that the
interatomic collisions entirely dominate the distribution of states of atomic excitation. Circumstances occur in which
local thermodynamic equilibrium does not prevail, because the strong radiative effects overwhelm the tendency to
the Maxwell-Boltzmann distribution of molecular velocities. For example, in the atmosphere of the sun, the great
strength of the radiation dominates. In the upper atmosphere of the earth, at altitudes over 100 km, the rarity of
intermolecular collisions is decisive.
In the cases of thermodynamic equilibrium and of local thermodynamic equilibrium, the number densities of the
atoms, both excited and unexcited, may be calculated from the Maxwell–Boltzmann distribution, but for other cases,
(e.g. lasers) the calculation is more complicated.
Einstein coefficients
In 1916, Albert Einstein proposed that there are three processes occurring in the formation of an atomic spectral line.
The three processes are referred to as spontaneous emission, stimulated emission, and absorption. With each is
associated an Einstein coefficient which is a measure of the probability of that particular process occurring. Einstein
considered the case of isotropic radiation of frequency ν, and spectral energy density ρ (ν).[2][13] Note that in some
treatments, the black-body spectral radiance is used rather than the spectral radiation energy density.[14][12]
Spontaneous emission
Spontaneous emission is the process by which an electron
"spontaneously" (i.e. without any outside influence) decays from a
higher energy level to a lower one. The process is described by the
Einstein coefficient A21 (s−1) which gives the probability per unit time
that an electron in state 2 with energy
will decay spontaneously to
state 1 with energy
, emitting a photon with an energy E2 − E1 =
hν. Due to the energy-time uncertainty principle, the transition actually
produces photons within a narrow range of frequencies called the
spectral linewidth. If
is the number density of atoms in state i then
the change in the number density of atoms in state 2 per unit time due
to spontaneous emission will be:
Schematic diagram of atomic spontaneous
emission
Einstein coefficients
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The same process results in increasing of the population of the state 1:
Stimulated emission
Stimulated emission (also known as induced emission) is the process
by which an electron is induced to jump from a higher energy level to a
lower one by the presence of electromagnetic radiation at (or near) the
frequency of the transition. From the thermodynamic viewpoint, this
process must be regarded as negative absorption. The process is
described by the Einstein coefficient
(J−1 m3 s−1), which gives the
probability per unit time per unit spectral energy density of the
radiation field that an electron in state 2 with energy
state 1 with energy
will decay to
, emitting a photon with an energy E2 − E1 =
hν. The change in the number density of atoms in state 1 per unit time
due to induced emission will be:
where
Schematic diagram of atomic stimulated emission
denotes the spectral energy density of the isotropic radiation field at the frequency of the transition (see
Planck's law).
Stimulated emission is one of the fundamental processes that led to the development of the laser. Laser radiation is,
however, very far from the present case of isotropic radiation.
Photo absorption
Absorption is the process by which a photon is absorbed by the atom,
causing an electron to jump from a lower energy level to a higher one.
The process is described by the Einstein coefficient
(J−1 m3 s−1),
which gives the probability per unit time per unit spectral energy
density of the radiation field that an electron in state 1 with energy
will absorb a photon with an energy E2 − E1 = hν and jump to state 2
with energy
. The change in the number density of atoms in state 1
per unit time due to absorption will be:
Schematic diagram of atomic absorption
Einstein coefficients
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Detailed balancing
The Einstein coefficients are fixed probabilities associated with each atom, and do not depend on the state of the gas
of which the atoms are a part. Therefore, any relationship that we can derive between the coefficients at, say,
thermodynamic equilibrium will be valid universally.
At thermodynamic equilibrium, we will have a simple balancing, in which the net change in the number of any
excited atoms is zero, being balanced by loss and gain due to all processes. With respect to bound-bound transitions,
we will have detailed balancing as well, which states that the net exchange between any two levels will be balanced.
This is because the probabilities of transition cannot be affected by the presence or absence of other excited atoms.
Detailed balance (valid only at equilibrium) requires that the change in time of the number of atoms in level 1 due to
the above three processes be zero:
Along with detailed balancing, at temperature T we may use our knowledge of the equilibrium energy distribution of
the atoms, as stated in the Maxwell–Boltzmann distribution, and the equilibrium distribution of the photons, as stated
in Planck's law of black body radiation to derive universal relationships between the Einstein coefficients.
From the Maxwell–Boltzmann distribution we have for the number of excited atomic species i:
where n is the total number density of the atomic species, excited and unexcited, k is Boltzmann's constant, T is the
temperature,
is the degeneracy (also called the multiplicity) of state i, and Z is the partition function. From
Planck's law of black-body radiation at temperature T we have for the spectral energy density at frequency ν
where:
where
is the speed of light and
is Planck's constant.
Substituting these expressions into the equation of detailed balancing and remembering that E2 − E1 = hν yields:
separating to:
The above equation must hold at any temperature, so
Therefore the three Einstein coefficients are interrelated by:
and
When this relation is inserted into the original equation, one can also find a relation between
involving Planck's law.
and
,
Einstein coefficients
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Oscillator strengths
The oscillator strength
is defined by the following relation to the cross section
for absorption:
where is the electron charge and
is the electron mass. This allows all three Einstein coefficients to be
expressed in terms of the single oscillator strength associated with the particular atomic spectral line:
References
[1] Bohr 1913
[2] Einstein 1916
[3]
[4]
[5]
[6]
[7]
[8]
Sommerfeld 1923, p. 43
Heisenberg 1925, p. 108
Brillouin 1970, p. 31
Jammer 1989, pp. 113, 115
Weinstein, M.A. (1960). On the validity of Kirchhoff's law for a freely radiating body, American Journal of Physics, 28: 123-25.
Burkhard, D.G., Lochhead, J.V.S., Penchina, C.M. (1972). On the validity of Kirchhoff's law in a nonequilibrium environment, American
Journal of Physics, 40: 1794-1798.
[9] Baltes, H.P. (1976). On the validity of Kirchhoff's law of heat radiation for a body in a nonequilibrium environment, Chapter 1, pages 1-25 of
Progress in Optics XIII, edited by E. Wolf, North-Holland, ISSN 00796638.
[10] Milne, E.A. (1928). The effect of collisions on monochromatic radiative equilibrium, Monthly Notices of the Royal Astronomical Society,
88: 493-502. (http:/ / adsabs. harvard. edu/ cgi-bin/ nph-data_query?bibcode=1928MNRAS. . 88. . 493M& db_key=AST&
link_type=ABSTRACT& high=4c3363690619220)
[11] Chandrasekhar, S. (1950), p. 7.
[12] Mihalas, D., Weibel-Mihalas, B. (1984). Foundations of Radiation Hydrodynamics, Oxford University Press, New York, ISBN
0-19-503437-6. (http:/ / www. filestube. com/ 9c5b2744807c2c3d03e9/ details. html)
[13] Loudon, R. (2000), Section 1.5, pp. 16–19.
[14] Chandrasekhar, S. (1950), p. 354.
Cited bibliography
• Bohr, N. (1913). "On the constitution of atoms and molecules" (http://www.ffn.ub.es/luisnavarro/
nuevo_maletin/Bohr_1913.pdf). Philosophical Magazine 26: 1–25. doi:10.1080/14786441308634993.
• Brillouin, L. (1970). Relativity Reexamined. Academic Press. ISBN 978-0-12-134945-5.
• Chandrasekhar, S. (1950). Radiative Transfer, Oxford University Press, Oxford.
• Einstein, A. (1916). "Strahlungs-Emission und -Absorption nach der Quantentheorie". Verhandlungen der
Deutschen Physikalischen Gesellschaft 18: 318–323. Bibcode 1916DPhyG..18..318E. Also Einstein, A. (1916).
"Zur Quantentheorie der Strahlung". Mitteilungen der Physikalischen Gessellschaft Zürich 18: 47–62. And a
version nearly identical to the latter at Einstein, A. (1917). "Zur Quantentheorie der Strahlung". Physikalische
Zeitschrift 18: 121–128. Bibcode 1917PhyZ...18..121E. Translated in ter Haar, D. (1967). The Old Quantum
Theory. Pergamon. pp. 167–183. LCCN 66029628. Also in Boorse, H.A., Motz, L. (1966). The world of the atom,
edited with commentaries, Basic Books, Inc., New York, pp. 888–901. (http://astro1.panet.utoledo.edu/~ljc/
einstein_ab.pdf)
• Heisenberg, W. (1925). "Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen".
Zeitschrift für Physik 33: 879–893. Bibcode 1925ZPhy...33..879H. doi:10.1007/BF01328377. Translated as
Einstein coefficients
"Quantum-theoretical Re-interpretation of kinematic and mechanical relations" in van der Waerden, B.L. (1967).
Sources of Quantum Mechanics. North-Holland Publishing. pp. 261–276.
• Jammer, M. (1989). The Conceptual Development of Quantum Mechanics (second ed.). Tomash Publishers
American Institute of Physics. ISBN 0-88318-617-9.
• Loudon, R. (1973/2000). The Quantum Theory of Light, (first edition 1973), third edition 2000, Oxford University
Press, Oxford UK, ISBN 0-19-850177-3.
• Sommerfeld, A. (1923). Atomic Structure and Spectral Lines (http://books.google.com/books/about/
Atomic_structure_and_spectral_lines.html?id=u1UmAAAAMAAJ). Brose, H. L. (transl.) (from 3rd German
ed.). Methuen.
Other reading
• Condon, E.U. and Shortley, G.H. (1964). The Theory of Atomic Spectra. Cambridge University Press.
ISBN 0-521-09209-4.
• Rybicki, G.B. and Lightman, A.P. (1985). Radiative processes in Astrophysics. John Wiley & Sons, New York.
ISBN 0-471-82759-2.
• Shu, F.H. (1991). The Physics of Astrophysics - Volume 1 - Radiation. University Science Books, Mill Valley,
CA. ISBN 0-935702-64-4.
• Robert C. Hilborn (2002). "Einstein coefficients, cross sections, f values, dipole moments, and all that" (http://
arxiv.org/abs/physics/0202029). physics/0202029.
• Taylor, M.A. and Vilchez, J.M. (2009). "Tutorial: Exact solutions for the populations of the n-level ion". Pub.
Astron. Soc. Pac. 121, 885: 1257–1266.
External links
• Emission Spectra from various light sources (http://ioannis.virtualcomposer2000.com/spectroscope/amici.
html#colorphotos)
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