Spontaneous Emission, Stimulated Emission, and Absorption
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1 Chapter Six Spontaneous Emission, Stimulated Emission, and Absorption In this chapter, we review the general principles governing absorption and emission of radiation by absorbers with quantized energy levels. The absorbers in question can be atoms, ions, molecules, dust grains, or any objects with energy levels. 6.1 Emission and Absorption of Photons If an absorber X is in a level and there is radiation present with photons having an energy equal to E u E, where E and E u are the energies of levels (for lower ) and u (for upper ), the absorber can absorb a photon and undergo an upward transition: absorption : X + hν X u, hν = E u E. (6.1) Suppose that we have number density n of absorbers X in level. The rate per volume at which the absorbers absorb photons will obviously be proportional to both the density of photons of the appropriate energy and the number density n, so we can write the rate of change of n due to photoabsorption by level as dnu dn = = n B u u ν, ν = E u E, (6.2) u u h populate level u depopulate level where u ν is the radiation energy density per unit frequency, and the proportionality constant B u is the Einstein B coefficient 1 for the transition u. An absorber X in an excited level u can decay to a lower level with emission of a photon. There are two ways this can happen: spontaneous emission : X u X + hν ν = (E u E )/h, (6.3) stimulated emission : X u + hν X + 2hν ν = (E u E )/h. (6.4) 1 Einstein was the first to discuss the statistical mechanics of the interaction of absorbers with the quantized radiation field.
2 54 CHAPTER 6 Spontaneous emission is a random process, independent of the presence of a radiation field, with a probability per unit time A u the Einstein A coefficient. Stimulated emission occurs if photons of the identical frequency, polarization, and direction of propagation are already present, and the rate of stimulated emission is proportional to the density of these photons. Thus the total rate of depopulation of level u due to emission of photons can be written dn u = dnu u = n u (A u + B u u ν ), (6.5) where the coefficient B u is the Einstein B coefficient for the downward transition u. Thus we now have three coefficients characterizing radiative transitions between levels u and : A u, B u and B u. We will now see that they are not independent of one another. In thermal equilibrium, the radiation field becomes the blackbody radiation field, with intensity given by the blackbody spectrum with specific energy density B ν = 2hν3 c 2 1 e hν/kt 1, (6.6) (u ν ) LTE = 4π c B ν(t ) = 8πhν3 c 3 1 e hν/kt 1. (6.7) If we place absorbers X into a blackbody radiation field, then the net rate of change of level u is dn u = dnu dnu + u u = n B u 8πhν 3 c 3 1 e hν/kt 1 n u 8πhν A 3 1 u + B u c 3. (6.8) e hν/kt 1 If the absorbers are allowed to come to equilibrium with the radiation field, levels and u must be populated according to n u /n l = (g u /g )e (E E u)/kt, with dn u / = 0. From Eq. (6.8) it is easy to show 2 that B u and B u must be related to A u by B u = c3 8πhν 3 A u, (6.9) B u = g u g B u = g u g c 3 8πhν 3 A u. (6.10) 2 Hint: consider the two limits T 0 and T. Equation (6.8), with dn u/ = 0, must be valid in both limits.
3 SPONTANEOUS EMISSION, STIMULATED EMISSION, AND ABSORPTION 55 Thus the strength of stimulated emission (B u ) and absorption (B u ) are both determined by A u and the ratio g u /g l. Rather than discussing absorption and stimulated emission in terms of the radiation energy density u ν, it is helpful to characterize the intensity of the radiation field by a dimensionless quantity, the photon occupation number n γ : n γ n γ c2 2hν 3 I ν, (6.11) c2 2hν 3 Īν = c3 8πhν 3 u ν, (6.12) where the bar denotes averaging over directions. With this definition of n γ, we can rewrite Eqs. (6.2 and 6.5) as simply dn dnu = n u A u (1 + n γ ), (6.13) u = n gu A u n γ. (6.14) u g If the radiation field depends on frequency in the vicinity of the transition frequency ν u, then n γ needs to be averaged over the emission profile in (6.13) and over the absorption profile in (6.14). From Eq. (6.13) we immediately see that the photon occupation number n γ determines the relative importance of stimulated and spontaneous emission: stimulated emission is unimportant when n γ 1, but should otherwise be included in analyses of level excitation. 6.2 Absorption Cross Section Having determined the rate at which photons are absorbed by an absorber exposed to electromagnetic radiation, it is useful to recast this in terms of an absorption cross section. The photon density per unit frequency is just u ν /hν. Let σ u (ν) be the cross section for absorption of photons of frequency ν with resulting u transition. The absorption rate is then dnu = n u dν σ u (ν)c u ν hν n c u ν dν σ u (ν), (6.15) hν where we have assumed that u ν (and hν) do not vary appreciably over the line profile of σ u. Thus B u = c hν dν σ u (ν), (6.16)
4 56 CHAPTER 6 and, using Eq. (6.10), we obtain the integral over the absorption cross section: dν σ u (ν) = g u c 2 g 8πνu 2 A u. (6.17) Thus we may relate the monochromatic absorption cross section σ u (ν) to a normalized line profile φ ν : σ u (ν) = g u c 2 g 8πνu 2 A u φ ν with φ ν dν = 1. (6.18) The frequency dependence of the normalized line profile φ ν is discussed in the following. 6.3 Oscillator Strength Earlier we characterized the strength of radiative transitions by the Einstein A coefficient, A u. Equivalently, we can characterize the strength of an absorption transition u by the oscillator strength f u, defined by the relation f u m ec πe 2 σ u (ν)dν. (6.19) From Eqs. (6.17 and 6.19), we see that the Einstein A coefficient for spontaneous decay is related to the absorption oscillator strength of the upward transition by A u = 8π2 e 2 ν 2 u m e c 3 g f u = cm2 s 1 g g u λ 2 f u. (6.20) u g u The oscillator strength f u for a downward transition u is negative, and is defined by g f u = g u f u. (6.21) The rate of stimulated emission is proportional to the downward oscillator strength, so it is natural that it should be negative, as it results in depopulation of the upper level. With this definition, the transitions for a one-electron atom in an initial state i obey the Thomas-Reich-Kuhn sum rule: f ij = 1, (6.22) j where the sum over final states j includes transitions to bound states and also to the continuum (i.e., photoionization). If the initial state i is not the ground state, the
5 SPONTANEOUS EMISSION, STIMULATED EMISSION, AND ABSORPTION 57 sum includes downward transitions with f ij < 0. For multielectron atoms or ions, the sum rule (6.22) generalizes to f ij = N, (6.23) j where N is the number of electrons, and the sum is over all transitions out of initial state i. The absorption cross section σ u (ν) is related to the oscillator strength by σ u (ν) = πe2 m e c f uφ ν with φ ν dν = 1. (6.24) 6.4 Intrinsic Line Profile The intrinsic line profile is characterized by a normalized profile function φ intr. σ intr. (ν) = πe2 m e c f u φ intr. ν φ intr. ν dν = 1 ν :. (6.25) The intrinsic line profile of an absorption line is normally described by the Lorentz line profile function: φ intr. ν = 4γ u 16π 2 (ν ν u ) 2 + γ 2 u, (6.26) where ν u (E u E )/h. The Lorentz profile in Eq. (6.26) provides an accurate (but not exact) 3 approximation to the actual line profile. The Lorentz line profile has a full wih at half maximum (FWHM) ( ν) intr. FWHM = γ u 2π. (6.27) The intrinsic wih of the absorption line reflects the uncertainty in the energies of levels u and due to the finite lifetimes of these levels 4 against transitions to all other levels, including both radiative and collisional transitions. If the primary process for depopulating levels u and is spontaneous decay (as is often the case in the ISM), then γ u γ u = A uj + A j. (6.28) E j<e u E j<e 3 The line profile is more accurately given by the Kramers-Heisenberg formula; Lee (2003) discusses application of this formula to the Lyman α line. 4 The Heisenberg uncertainty principle E t h implies that an energy level u has a wih E u h/τ u, where τ u is the level lifetime.
6 58 CHAPTER 6 In the case of a resonance line, where is the ground state, the second sum vanishes. It is convenient to describe line wihs in terms of the line-of-sight velocities that would produce Doppler shifts of the same amount. Thus the intrinsic wih of an absorption line can be given in terms of velocity: ( v) intr. FWHM ν u FWHM = c ( ν)intr. = λ uγ u 2π = km s λu γ u 7618 cm s 1, (6.29) where λ u γ u = 7618 cm s 1 is the value for H Lyman α. The intrinsic line wih can also be written in terms of the energy and oscillator strength of the transition: ( v) intr hν FWHM g α3 f u c = I H g u hν I H g g u f u km s 1, (6.30) where α e 2 / hc = 1/ is the fine-structure constant, I H = (1/2)α 2 m e c 2 = ev is the ionization energy of H, and the inequality is because γ u A u [see Eq. (6.28)]. From (6.29), we see that optical and ultraviolet absorption lines, for which hν/i H < 1 and f u < 1, will have ( v) intr. FWHM < 0.1 km s 1. For example, H Lyman α (λ = 1215 Å) has hν/i H = 3/4, f lu = , g /g u = 2/6, and ( v) intr. FWHM = km s 1. H Lyman α has a relatively large energy (0.75I H ) and relatively large oscillator strength (0.4162); most other optical and ultraviolet permitted lines have even smaller ( v) intr. FWHM. Because ( v) intr. FWHM hν, x-ray transitions can have considerably larger intrinsic line wihs. For example, the 6.68 kev Fe 24+ 1s2p 1s 2 line has an intrinsic linewih ( v) intr. FWHM = 13.5 km s Doppler Broadening: The Voigt Line Profile Atoms and ions are generally in motion, and the velocity distribution is often approximated by a Gaussian, this being of course the correct form if the velocities are entirely due to thermal motions: p v = 1 1 e (v v0) 2 /2σ v = π 2 2π σ v b e (v v0) /b 2, (6.31) where p v dv is the probability of the velocity along the line of sight being in the interval [v, v + dv], σ v is the one-dimensional velocity dispersion, and the broadening parameter b 2σ v. The wih of the velocity distribution is also sometimes specified in terms of the FWHM; for a Gaussian distribution of velocities, this is just ( v) FWHM = 8 ln 2 σ v = 2 ln 2 b. (6.32)
7 SPONTANEOUS EMISSION, STIMULATED EMISSION, AND ABSORPTION 59 If the velocity dispersion is entirely due to thermal motion with kinetic temperature T = 10 4 T 4 K, then 1/2 1/2 kt T4 σ v = = 9.12 km s 1, (6.33) M M/amu 1/2 1/2 2kT T4 b = = km s 1, (6.34) M M/amu 1/2 1/2 (8 ln 2) kt ( v) therm T4 FWHM = =21.47 km s 1. (6.35) M M/amu The intrinsic absorption line profile φ intr. ν must be convolved with the velocity distribution of the absorbers to obtain the line profile 4γ u φ ν = dv p v (v) 16π 2 [ν (1 v/c)ν u ] 2, (6.36) + γu 2 where p v dv is the probability of the absorber having radial velocity in the interval (v, v + dv). If the absorbers have a Maxwellian (i.e., Gaussian) one-dimensional velocity distribution p v (Eq. 6.31), then the absorption line will have a so-called Voigt line profile: φ Voigt ν 1 dv e v 2 /2σ 2 v 2π σ v 4γ u 16π 2 [ν (1 v/c)ν u ] 2 + γ 2 u. (6.37) Unfortunately, the Voigt line profile cannot be obtained analytically except for limiting cases. 5 However, if, as is generally the case, the one-dimensional velocity dispersion σ v ( v) intr. FWHM, the central core of the line profile is well-approximated by treating the intrinsic line profile as a δ-function, so that the central core of the line has a Maxwellian profile: φ ν 1 π 1 ν ul c b exp v 2 /b 2, b 2 σ v. (6.38) We will discuss the Voigt profile further in Chapter Transition from Doppler Core to Damping Wings Near line-center, the line profile is well-approximated by the Doppler core profile, which for a Gaussian velocity distribution gives σ π e2 m e c f u λ u e v2 /b 2, (6.39) b 5 Accurate approximation formulae have been developed for the Voigt profile see Armstrong (1967).
8 60 CHAPTER 6 where the velocity v (ν 0 ν)c/ν 0. For very large frequency shifts, the profile can be approximated by just the damping wings: σ π e2 f u λ u 1 γ u λ u b 2 m e c b 4π 3/2 b v 2. (6.40) For what frequency shift, expressed as a velocity, do we make the transition from the Doppler core to the damping wings? The condition for z v/b is obtained by equating (6.39) and (6.40): e z2 = 4π 3/2 b z 2 γ u λ u 7618 cm s 1 = 2924 b 6 z 2, (6.41) γ u λ u where b 6 b/10 km s 1. The solution to this transcendental equation is 7618 cm s z ln b 6, (6.42) γ u λ u provided that the quantity in square brackets is not very large or very small. Therefore, for a strong permitted line (such as Lyman α), the damping wings dominate for velocity shifts z > 3.2, or v > 32b 6 km s Selection Rules for Radiative Transitions Some energy levels are connected by strong radiative transitions; in other cases, radiative transitions between the levels may be extremely slow. The strong transitions always satisfy what are referred to as the selection rules for electric dipole transitions. Here, we summarize the selection rules for the strong electric dipole transitions, and we also give the selection rules for intersystem and forbidden transitions that do not satisfy the electric dipole selection rules but nevertheless are strong enough to be astrophysically important. We will use the ion N II as an example; the first nine energy levels of N II are shown in Fig Allowed = Electric Dipole Transitions The strongest transitions are electric dipole transitions. These are transitions satisfying the following selection rules: 1. Parity must change. 2. L = 0, ±1. 3. J = 0, ±1, but J = 0 0 is forbidden. 4. Only one single-electron wave function n changes, with = ±1.
9 SPONTANEOUS EMISSION, STIMULATED EMISSION, AND ABSORPTION 61 Figure 6.1 First nine energy levels of N II. Forbidden transitions are indicated by broken lines, and allowed transitions by solid lines; forbidden decays are not shown from levels that have permitted decay channels. Fine-structure splitting is not to scale. Hyperfine splitting is not shown. 5. S = 0: Spin does not change. An allowed transition is denoted without square brackets, for example, N II Å 3 P 0 3 D o 1. This is a transition between the = 1s 2 2s 2 2p 2 3 P 0 and u = 1s 2 2s 2 2p3s 3 D o 1 levels of N II, with a wavelength λ u = Å. The transition has A u = s 1. This decay is very fast the lifetime of the 3 D o 1 level against this decay is only 1/A u = 4.6 ns!
10 62 CHAPTER Spin-Forbidden or Intersystem Transitions These are transitions that fulfill the electric dipole selection rules 1 to 4 but have S = 0. These transitions are considerably weaker than allowed transitions. Such transitions are sometimes referred to as semiforbidden, or intercombination, or intersystem transitions; the latter is the terminology that we will use here. An intersystem transition is denoted with a single right bracket for example, N II] Å 3 P 2 5 S o 2, a transition between = 1s 2 2s 2 2p 2 3 P 2 and u = 1s 2 2s2p 3 5 S o 2, with wavelength λ u = Å and A u = s Forbidden Transitions Forbidden transitions are those that fail to fulfill at least one of the selection rules 1 to 4. The transition probabilities vary widely, depending on the values of the electric quadrupole or magnetic dipole matrix elements between the upper and lower states. A forbidden transition is denoted with two square brackets for example, [N II] Å 3 P 1 1 D 2, a transition between = 1s 2 2s 2 2p 2 3 P 1 and u = 1s 2 2s 2 2p 2 1 D 2, with λ u = Å and A u = s 1. This fails rule 1 (parity is unchanged) and it fails rule 4 (single electron wave functions are unchanged). This is an example of a magnetic dipole transition. Another example of a forbidden transition is the electric quadrupole transition [N II] Å 1 D 2 1 S 0, between = 1s 2 2s 2 2p 2 1 D 2 and u = 1s 2 2s 2 2p 2 1 S 0, with λ u = Å and A u = 1.17 s 1. This fails rules 1 (parity is unchanged) and 4 (single electron wave functions are unchanged) and it fails rules 2 and 3 ( L= 2 and J = 2), yet its transition probability is three orders of magnitude larger than the magnetic dipole transition [N II] Å! We see then that there is a hierarchy in the transition probabilities: very roughly speaking, intersystem lines are 10 6 times weaker than permitted transitions, and forbidden lines are times weaker than intersystem transitions. Despite being very weak, forbidden transitions are important in astrophysics for the simple reason that every atom and ion has excited states that can only decay via forbidden transitions. At high densities, such excited states would be depopulated by collisions, but at the very low densities of interstellar space, collisions are sufficiently infrequent that there is time for forbidden radiative transitions to take place.
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