1 IR Spectroscopy 2 Planetary Spectra

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1 IR Spectroscopy consider only the absorption due to gas. The giant planets and Titan all have simi- lar optical and near-IR spectra, because they “Time is nature’s way of keeping everything are all dominated by methane (CH ) vibra- from happening at once.” - Woody Allen 4 tional bands. Methane, therefore, plays a The absorption of visible, infrared and large roll in establishing the pressure lev- far infrared radiation can excite molecules els where most of the solar insolation is ab- through electronic, vibrational and rotational sorbed. The visible to near-IR spectra of transitions. In fact one can define the state Venus and Mars indicate CO2 features, as of a molecule in terms of its electronic, vi- does that of Earth. But Earth distinguishes brational and rotational states; a given elec- itself with water, oxygen, and in the UV, tronic state is subdivided into energy states ozone signatures, which sets it apart from corresponding to the vibrational levels, each other planets. of which is further subdivided into rotational Consider the infrared spectra of planetary levels, with each taking progressively less en- atmospheres. The thermal emission of the ergy to excite. As a result their spectral outer planets and Titan is dominated in part signatures end up at different spectroscopic by stratospheric emission features due largely regions: signatures due to electronic tran- to photochemically produced species, created sitions appear generally at UV and visible in part from the dissociation of methane. wavelengths while vibrational features extend Two primary coolants are ethane (C2H6) and from optical to middle IR wavelengths and acetylene (C2H2). Titan’s atmosphere is also signatures due to pure rotational transitions cooled by emission from hydrogen cyanide end up in the far infrared (HCN), which is produced from the dissoci- This is a very quick, somewhat qualitative, ation of its main constituent, N2. Another review of nature of molecular spectroscopy. component of Titan’s infrared spectrum is This review will give you enough material to the pressure-induced absorption of N2, H2 calculate the radiative transfer effects of gases and CH4, which creates a greenhouse effect once others have derived the line parameters, that warms the surface by 21 K. In contrast, but not, of course, to derive line parameters the spectra of the inner rocky planets exhibit yourselves. absorption by CO2 and, for Earth, a host of other constituents such as water, nitrous oxide (N2O), and ozone. Earth is quite the 2 Planetary Spectra weirdo. We’ll get back to that later.

Let’s ignore, for now, the eﬀects of scatter- ing by the gas (Raleigh) and particulates and

1 of ~J, where J is an integer (the quantum number) and ~ is h/2π, such that:

~2 E (J)= J(J + 1). (1) R 2I The energy difference between two succes- sive levels (~2/I) is generally on the order of 10−4 to 10−3 eV. At room temperature, the translational thermal energy of molecules −2 (3/2kBT ) is 2.5×10 eV. Therefore collisions in planetary atmospheres transfer the necessary energy of excitation, and broad range of energy states are occupied. In fact, for temperatures typical of planetary atmospheres, the rotational state populations obey the Boltzmann distribution and are thus in LTE. As we shall see later on, fluorescence more readily occurs for vibrational and elec- Figure 1: Molecular electronic, vibra- tronic transitions, which require higher en- tion, and rotation transitions. ergy collisions to excite. Since rotational transitions occur from 2.1 Rotation Spectra. the interaction of the electric dipole of the molecule and the electric field of the incident Molecular rotation can be envisioned classi- radiation, diatomic molecules of identical nu- cally as the rotation of a semi-rigid molecule clei, like N2 and O2, do not exhibit pure ro- about it’s center of mass. In classical me- tation spectra. These molecules lack electric chanics, the rotational energy, ER, depends dipole moments. That said these molecules on angular momentum, L, as: can be perturbed sufficiently to attain tempo-

2 rary dipole moments that allow for collision 1 2 L ER = Iω = induced spectra. 2 2I Note that the separation between spec- 1 ~ where ω is the angular velocity and I the mo- tral lines, ∆( λ ) = /2πcI, is constant with ment of inertia1. Quantization of angular mo- wavenumber Thus the value of I and, for mentum associates L with a quantized value diatomic molecules, the internuclear separation, R, can be measured from the spacing 1For example: for a diatomic molecule of reduced between spectral lines. mass µ=m1m2/m1 +m2, and internuclear separation R: I = µR2

2 2.2 Vibration-Rotation Spec- tra.

The molecular moment of inertia, I, is not however a ﬁxed entity. Instead it changes with rotation with the extension of the internuclear distance. In addition it changes with the vibration of the molecule. According to classical mechanics, the vibration of a semi-rigid system can be decom- posed into a set of normal modes. The number of vibrations possible for a molecule generally increase with the number of atoms, N, it contains. As a rule of thumb, for molecules of N>2, there are 3N-6 independent modes for a non-linear molecule and 3N-5 for a linear molecule. If the vibrations are small in amplitude as occurs for the lowest energy states, the oscil- lations can be approximated as simple harmonic, for which the quantized states are:

EV = hνk(v +1/2) (2) Figure 2: Vibration modes of some com- where ν is the mode frequency and v is k k mon molecules. Note that the ν1 and ν2 an integer - the vibrational quantum num- modes of CH4 are inactive. ber. The mode is denoted by the vibrational constant, k. Another term is also used, the vibrational constant, ωek=νk/c. The separa- plane of the molecule. The mode frequency −1 tion between states is constant for a simple ν1∼2127 cm , where the 1→0 transition harmonic oscillator and equal to hνk. The lines occur. The 2→1 transitions occur at lowest energy state is not zero, because of the same wavelength (roughly) and the 2→0 the uncertainty principal. For each particu- transitions occur at twice this frequency. We lar mode the frequency spacing between the can estimate the frequencies of the lines of vibrational states is constant and equal to the higher transitions, only these transitions de- mode frequency νk if the simple harmonic ap- viate further from the simple harmonic oscil- proximation applies. lator assumption. Consider CO. There is only only one vi- In fact since the rotation transitions ac- bration mode, that of stretching along the company vibration transitions, the energy

3 levels are written in terms of an anharmonic oscillator and the interaction between the rotation and vibration transitions, e.g.

2 E(v,J)= hc[ωe(v +1/2) − ωexe(v +1/2)

2 2 +BvJ(J + 1) − DvJ (J + 1) ], the second term is a higher order vibrational anharmonic term and the latter two terms are interaction terms (as indicated by the v subscript and the J quantum number). The selection rule of v= ±1 applies generally. But there are also transitions where v= ±2, ±3 and so on, a result of the anharmonic nature of most transitions. The rotational transitions provide a ﬁne structure to the spectrum of a particular vibration transition, with spectral lines having wavenumbers smaller than the band head, the P-branch, and longer than the band head, the R-branch. These are formed from the ′′ Figure 3: The R and P branches of transition of the lower rotational states J vibration-rotation transitions. to the higher state J ′ by:

∆J = J ′ −J ′′ = +1 R−branch the rotational selection rules are ∆J = 0, ±1, and the Q branch is allowed. If Λ= 0, then ∆J = J ′−J ′′ = −1 P −branch. ∆J = ±1. Generally Q branches do not occur in diatomic molecules. Q branches occur The R − branch transitions to a higher rota- in spherical top molecules like methane. tional energy level at the highest vibrational level, requiring higher energy level photons than P − branch. There is a further correction to the rota- 3 Line Strengths tional energies derived above. It turns out that the electron angular momenta about the The strengths of spectroscopic lines depends internuclear axis of a diatomic molecule are on the probability of the individual transi- comparable to the nuclear value. Only the tion, but also on the population of the states component of the angular momentum, Λ, on that are involved. Here we’re going to as- the spin axis remains constant. If Λ=6 0 then sume LTE conditions. We therefore return

4 to the Boltzmann distribution of states. The ratio of a particular rotational state’s population to the population of all rotational states combined is: n(J) (2J + 1)e−EJ /kB T = , n QR where:

2 ′ ′ ′ −(~ /2I)J (J +1)/kBT QR = (2J + 1)e XJ′ is the partition function. The term 2J +1 is the degeneracy of the rotational state J, a quantum mechanical result. If the spacing Figure 4: The populations of rotation of the states is small compared to the extent states control relative intensities of ro- of the rotational band, one can integrate that tational lines, which thus can serve as above expression in terms of dJ and one finds an atmospheric thermometer. 2 the approximation that QR ∼ kBT /(~ /2I). Note this ratio provides us with a probabil- expression for the population of rotational ity that the particular lower energy rotational states. state will be populated. We use this ratio, in- hνB − − stead of n1 when defining the line strength. i EJ /kB T hν/kT σ¯ν = (2J + 1)e (1 − e ). The line strength is defined in terms of the 4πQi cross section σν of an individual vibration- rotation line: Generally, data bases of line parameters give you the line intensity at a particular reference temperature, T0, somewhere near S = σνφ(ν)dν. Z room temperature. In order to determine the line intensity at a temperature of interest, T , The expression of the LTE cross section in you must take into account the different pop- terms of the Einstein coefficients is: ulation of states at T0 and T , and the temper- hν n¯ ature dependence of the stimulated emission σ¯ = B 1 (1 − exp(−hν/kT )), ν 4π 12 n term. The line intensity at T can then be expressed in terms of the line intensity at T0 where we have used the Einstein coefficients as: that are defined in terms of the intensity. Let’s generalize to a multi-level atom and Q (T )Q (T )e−E/kBT (1 − e−hνi/kBT ) σ¯ (T )=¯σ (T ) V 0 R 0 ν ν 0 −E/kBT0 −hνi/kBT0 use Bi rather than B12 and write in full the QV (T )QR(T )e (1 − e )

5 The expression for the line intensity is thus:

−E/kB T −hνi/kB T T0 m e (1 − e ) σ¯ν (T )=¯σν(T0)h i , T e−E/kB T0 (1 − e−hνi/kB T0 )

where m depends, as shown above, on the Figure 5: The shape, qualitatively molecular geometry. Note that this simple speaking, of the R (right) and P (left) expression for the ratio of rotational partition branches of a diatomic molecule’s vi- functions does not apply well for all molecules bration spectrum. and conditions, e.g. it is way off for water at the temperatures (500-2000 K) characteristic of extrasolar “hot Jupiter” planets. Here we see that the ratio of the line intensities depends on the ratio of the fractional number of particles occupying the lower energy state, and the ratio of the stimulated 4 Practical Application emission, which acts as a negative absorption. Note that both of these effects depend Molecular line bases give you the wavelength on temperature. The degeneracy of the state of the intensity, the lowest energy state, E doesn’t depend on temperature, so its ratio ′′ − = E , in units of cm 1, the line intensity is unity. −1 σ¯ν (T0) = S in units of cm /(molecule - The partition functions of the vibration cm−2), and the pressure broadening coeffi- transitions are usually close to one. Often cient. With this information you can calcu- these values do not need to be considered. late the absorption coefficient for vibration- One exception to the rule is the ν9 vibrational rotation transitions. These units are defined band of C H . Another exception is wa- 2 6 in terms of wavenumber, νi, which is related ter. The values of QV R(T) = QV (T) QR(T) to the frequency of the transition, f and the were derived by Vidler & Tennyson (J. Chem. speed of light, c, as νi = f/c. Also note that Phys. 113, 9766, 2000). since E is the wavenumber units it must be As we saw above, the rotational partition multiplied by hc to get the energy. Second function, QR(T ), depends roughly on the first note the units of intensity. The absorption power of temperature. A better approxima- coefficient depends on Si φ(νi) where φ(ν) tion is the following: is the line profile function, which is in units of 1/cm−1, that is inverse wavenumber. The combined value of S φ(ν ) is in units of in- QR(T ) ∼ T linear molecules i i verse column abundance, which are the final units of the absorption coefficient. 3/2 QR(T ) ∼ T non−linear molecules

6 5 Appendix which oscillates about an equilibrium posi- tion with a frequency ν of So where do equations 1 and 2 come from? 1 Well there is not enough time to go into the ν = k/m, 2π whole story, but this will give you an idea. In p 1926 Erwin Schrodinger formulated an equa- where m is the mass. tion, call aptly the Schrodinger Equation that In quantum mechanics, the solutions the is essentially an equation of motion, like New- time-independent Schrodinger Equation for ton’s equation, but for quantum mechanics - this potential is: the physics of the microscopic states in nature. His equation can be written as: En =(n +1/2)hν, ~2 δ2Ψ(x, t) δΨ(x, t) where ν is given above. So the the ﬁrst eigen- − + V (x)Ψ(x, t)= i~ 2m δx2 δt value is: 1 E = hν If the potential energy function is indepen- 0 2 dent of time, we can simplify the equation The ﬁrst eigenfunction is: using a separation of variables. First we write −µ2/2 Ψ(x, t) = ψ(x)φ(t). Then we plug this into ψ0 = A0e , the equation and rearrange it to see if we can get all of the x dependent variables on one where (km)1/4 side and the t dependent on the other. In- µ = x deed we can. Then the common variable, ~1/2 call it G, is not dependent on either x or t You can check this by plugging it into the and we end up with two equations, which time-independent Schrodinger Equation. are the 2 sides of the equation that are set equal G. From this exercise we get the time- independent Schrodinger Equation: ~2 δ2ψ(x) − + V (x)ψ(x)= Eψ(x), 2m δx2 and φ(t)= e−iEt/~. Now consider a simple harmonic oscillator. The potential function for this system in a classical world is: k V (x)= x2, 2 7 Figure 6: A Sample of the HITRAN data base.