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6. Spectroscopy Fundamentals

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6. Spectroscopy Fundamentals 6. Spectroscopy fundamentals This chapter provides a broad overview of the spectroscopic principles required in order to perform quantitative spectroscopy of atmospheres and to couple the details of the atmospheric spectroscopy with the radiative transfer processes and also with the assessment of spectroscopic measurements of atmospheres. 6.1 Einstein A and B coefficients These provide the relationship between spontaneous emission by atoms or molecules that are in quantum states above the ground state, including the thermal emission by atmospheres (Einstein A coefficient) and induced absorption and emission, (Einstein B coefficient). The blackbody flux Figure 6.1. A 2 quantum- density (eq. 4.7, converted to emitted photons) level system at equilibrium 2πσcd3 σ where, by definition, the is Rd()σσ= , and the radiation density (eq. 4.8) n cT2σ / number of transitions in e −1 absorption equals the 8πhcσ 3dσ number in emission. is ρ(σ )dσ = c /T . e 2σ −1 Consider a macroscopic two quantum-level system, perhaps a volume of a simple gas, at equilibrium as shown in Figure 6.1. At equilibrium, the number of molecules in each of the two states does not change macroscopically, although individual molecules may change state at any time. The populations in each state are determined by Boltzmann statistics: −c2σ /T N1 = N0e . (6.1) At equilibrium, dN12//0 dT== dN dT , implying NBNBNA001110110ρσ()=+ ρσ () . (6.2) Applying the Boltzmann factor N0 cT2σ / ρσ()BeBBA01==+ ρσ () 01 ρσ () 10 10 , (6.3) N1 or A 8πσhc 3 ρσ()==10 . (6.4) cT22σσ// cT Be01−− B 10 e 1 Therefore BB01= 10 (6.5a) 3 A10= 8.πσ hc B 10 (6.5b) This establishes the ratios among the absorption and emission rates for given radiation densities. Absolute values are given in Section 6.2. 1 3 3 Note the very strong wavenumber/frequency σ (or ν ) dependence of A10; emission at short wavelengths is very fast, and the upper states have very short lifetimes (and broad line shapes (Section 7.2). 3 If the upper and lower state degeneracies are different, A10 = 8πhcσ B10 as before, but g B = g B 0 01 1 10 . 6.2 Rotational spectroscopy 6.2.1 Diatomic molecules Figure 6.2. Schematic of a diatomic molecule with internuclear distance r and atomic masses m1 and m2. The moment of inertia I for rotation about the center of mass is given by I = m r2 = m r2 + m r2. ∑ i i 1 1 2 2 (6.6) since rm2 rm1 r1 = ;r2 = , m1 + m2 m1 + m2 (6.7) 2222 rmm12 rmm 1 222 mm 12 Irr=+=22 ≡µ , (6.8) ()()mm12++ mm 12 mm 12 + where µ is the reduced mass for rotation (reduced masses for other degrees of freedom of motion are also possible; reduced masses for modes of vibration are particularly notable). Note the common case of “hydrogenic” reduced mass: For HX molecules (X=F, Cl, Br, I, O, …) molecules, the reduced mass for rotation is small, similar to the mass of the hydrogen atom. 2 In general, for axis i of a three-dimensional object, I= r dm , where r2 is the distance iii∫ from the axis and dmi the differential in mass. 2 The moment of inertia tensor has diagonal elements, e.g., Imyz=+()22 and off- xx∑i i i i diagonal elements, e.g., II mxy. The principal axes are the choice of axes xy== yx−∑ i i i i that diagonalize the moment of inertia tensor for an object: IIIxy=== xz yz 0. The rotational energy EI= ω2 /2, classically, and the rotational angular momentum L2 LI= ω, or, E = . However, quantum mechanics says that L must be quantized in 2I h nh(/2) nh π 2 units of = , where h is Planck’s constant: LE==; 2π 22π I The Hamiltonian expression for rotation about one axis is Jˆ 2Ψ J (J +1)2Ψ = EΨ = , (6.9) 2I 2I h22h with solution at energies EBJJ=+( 1), where B == . B is the rotational 28IIπ 2 constant. Nonlinear polyatomic molecules have 2 or 3 rotational constants, depending on the symmetry of the molecule. Expressing B in cm-1: B (H35Cl) 10.59 ← hydrogenic B (HF) 20.94 ← hydrogenic B (OH) 18.87 ← hydrogenic B (O2) 1.446 ← not hydrogenic The spectroscopic consequences of having small, hydrogenic, moments of inertia, with corresponding large rotational constants, are that such molecules (or the corresponding rotational degrees of freedom in polyatomic molecules having one or more hydrogenic reduced masses about rotational axes) 1. have smaller rotational partition functions, and thus distribute the rotational spectra into fewer, stronger transitions; 2. have Boltzmann population distributions that extend to higher energies. This corresponds to the far infrared (submillimeter, terahertz) spectral region, where intensities also are stronger because of the σ3 factor in the blackbody expression. This spectral region is thus particularly useful for the measurements of gases that are important in stratospheric chemistry, including OH, H2O, HO2, H2O2, HCl, HF, and HBr. These are not hydrides. In hydrides, the hydrogen atom has a formal negative charge (e.g., NaH, LiH, LiAlH4). ˆ 2 J Ψ iM =EΨ has the set of solutions Ψ=Pe(cosθ )φ , which are spherical harmonics, 2I JM where J = 0, 1, 2, …. and |M| ≤ J. There are thus 2J + 1 “states” of same energy for each J. Their degeneracy (in the absence of magnetic and electric fields m and E which may 3 perturb these states from energy degeneracy when the molecules have magnetic or electric moments, Section 6.3) is 2J + 1 (Figure 6.3). This shows why we need to modify our statement for the Einstein coefficients from BBnm= mn to gBnnm= gB mmn. The microscopic probabilities are the same but the summed quantities are not. Detailed balance requires the above relationship, e.g., Figure 6.3. Illustration of the degeneracies of rotational energies 13.×BB01=× 10 Degeneracy in general is given by for the lowest several rotational 2F +1, where F is the total angular momentum of the states of a diatomic molecule state (including rotation, orbital, spin, electronic, nuclear). 2πσσhc23 d Reminders: Blackbody radiancy Rdσ = σ ehcσ / kT −1 3 8πσhc -3 -1 -2 The radiation density ρσ()= erg cm /cm = erg cm . ehcσ / kT −1 hcσ -3 -1 Since ρσ()= F () σ (F = flux), Fh()σρσσ= ()/ in cm s . c Dipole moments, transition moments, line strengths, and intensities Spectroscopic transitions are effected by the interaction of electromagnetic radiation with electromagnetic moments in atoms and molecules. Important Figure 6.4. Left: a simpler arrangement of masses and charges, transitions giving rise to the corresponding to a (hypothetical) purely ionically-bound molecule. spectra of atmospheres Right: The more general case of an arbitrarily shaped molecule arise from electric dipole and charge distribution. moments and (generally much weaker) magnetic dipole moments. Electric quadrupole and even higher moments can also contribute in limited cases (electric quadrupole- allowed transitions are significant in the near infrared A band of O2). The electric dipole moment of a molecule has between one and three components depending on the symmetry and arrangement of electric charges in the molecule (Figure 6.4). The components of the dipole moment are determined by taking integrals over the product of charge density and distance from the center of mass: µ = e r dr, r = x, y, z. (6.10) r ∫ r 4 Figure 6.5. An electric dipole moment of 1 Debye corresponds to unit plus and minus charges separated by 1 Å (10-10 m). The intensity of a transition is proportional to µ 2 : The square of the quantum-mechanical matrix element which governs intensity is 2 ΨΨΩµµd =2 × (direction cosine matrix element)2, (6.11) ∫ 12 0 where the direction cosine matrix elements describe the relative intensities for transition among different quantum states. Their squares are the line strength factors. (Additionally the Boltzmann statistics, Chapter 4, must be included for a complete description of line intensities.) The standard unit of the electric dipole moment µe is the Debye (D, 1 Debye = 1 esu-Å, as illustrated in Figure 6.5). Magnetic dipole moments arise from circulating electronic charges in the atom or molecule, take on quantized values, and generally interact much more weakly with radiation. The standard unit of the magnetic dipole moment µB is the Bohr magneton. To gauge the relative ranges of strength of electric dipole allowed and magnetic dipole allowed transitions, (1 Debye)2 = 10-36 erg cm3 and (1 Bohr magneton)2 -41 3 = 8.60×10 erg cm . For several important atmospheric molecules, as examples, µe (HCl) = 1.1 D, µe (OH) = 1.7 D, µe (O2) = 0, but µB = 2 Bohr magnetons (quantized to almost exactly 2, with tiny corrections, including a relativistic correction). For simple diatomic molecules, with electric dipole moments but without additional electronic or nuclear moments, we identify µe with µ0 and find intensities proportional to 2 J +1 2 J +1 µ = µ 2 ; µ = µ 2 . J→J+1 0 2J +1 J←J+1 0 2J + 3 (6.11) The usual selection rules, determining by symmetries of the quantum mechanic matrix elements as in Eq. 6.11 for rotational transitions of polar diatomics are ΔJ =±1 (from the symmetry of ΨΨµ ). Then, the rotational lines occur at energies of ΔE = 2B, 4B, 6B ∫ ab (Figure 6.6). 5 Figure 6.6. The lowest several rotational energy levels for polar diatomic molecules. More generally, the rotational energy levels can be expanded as: 2 2 3 3 E = BJ(J +1) − DJ (J +1) + HJ (J +1) + .... Departures from the simple BJ(1) J + are due to centrifugal distortion.
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