Lecture 18 Long Wavelength Spectroscopy

Transcription

1 Lecture 18 Long Wavelength Spectroscopy 1. Introduction. The Carriers of the Spectra 3. Molecular Structure 4. Emission and Absorption References Herzberg, Molecular Spectra & Molecular Structure (c. 1950, 3 volumes) Herzberg, The Spectra & Structure of Simple Free Radicals, (Cornell, 1971), esp. Parts I & II Townes & Schawlow, "Microwave Spectroscopy" (Dover 1975) Astro texts: Lightman & Rybicki, Ch. 10 Shu I, Chs Dopita & Sutherland, Ch. 1 Palla & Stahler, Ch. 5

2 1. Introduction to Long Wavelength Spectroscopy This course has so far dealt mainly with warm diffuse gas. Perhaps it was fitting that the very last discussion of the warm and very diffuse IGM ended with the question of how the first molecules formed and how molecular hydrogen cooled the cosmic gas enough to form the first stars. The argument has been made that the theory of primordial star formation is simpler than recent star formation because of the absence of dust and magnetic fields. But this misses the important point that theory by itself is nothing without tests and guidance from observations. We therefore switch our focus to the nearby clouds that are now making stars in nearby galaxies (the main topic of the second half of the course) and the central role of long wavelength astronomy.

3 Signatures of the Cool ISM Recall that much of Lecture 1 introduced long wavelength observations of the ISM and noted the primary roles of dust and molecules. COBE mm spectrum of the Galactic plane. Major IR Space Observatories IRAS 1983 COBE 1991 ISO 1996 SPITZER 003 Why doesn t the COBE spectrum show molecular features?

4 Molecular Emission Unlike atoms, molecules can produce many long-wavelength vibrational & rotational transitions by virtue of having (extra degrees of freedom from) more than one nucleus. The energies of such transitions cover a wide range. Some rough rules are: Vibrational transitions occur in the NIR-MIR e.g., H 1-0 S(1) at.1 µm and CO 1-0 at 4.6 µm Rotational transitions occur in the MIR FIR-mm band e.g., H 0-0 S(0) at 8. µm & CO J=1-0 at.6 mm Optical and UV transitions are spread over many lines,

5 Discovery of Interstellar Molecules Lecture 1 discussed the discovery of interstellar molecules. First optical detections: CH Dunham et al. (1937) CN Swings & Rosenfeld (1937) CH + McKellar (1940) Radio detections (pioneered by Townes & collaborators): OH Weinreb et al. (1963) NH 3 & H O Cheung et al. 1968, 1969 H CO Snyder at al. (1969) CO Wilson, Jefferts, & Penzias (1970). UV detection H Carruthers (1970)

6 Phillips & Keene Proc IEEE Schematic and now dated sub-mm spectrum of a molecular cloud showing fine structure & molecular emission lines (only the lowest rotational transition is shown). Not all of the suggested species have been detected. See real ISO spectra on the next pages.

7 ρ Oph Molecular Cloud E. van Dishoeck, ARAA

8 ISO NIR-MIR spectrum of a shock in Orion E. van Dishoeck ARAA

9 . Carriers of the Spectra Under their cool or not too warm conditions, interstellar clouds emit mainly in molecular rotational & atomic fine structure transitions. As we shall see, other small energy transitions also play important roles. a. Rotational Transitions The signature example is the rigid rotor whose energy levels are given by h E J = J ( J + 1) B B = I For a diatomic molecule, I = µ 1 r 1. With typical separations in Angstroms, the masses of the molecule are determining. e.g., B(CO) = k B (.77K) B(H ) = k B (51 K) More about rotational transitions below.

10 b. Fine structure transitions Although neutral atoms may not have fine structure (because either the spin or orbital angular momentum is zero), some of their ions will have these transitions, as shown in this small table (wavelengths in microns). Atom I II III C ( 3 P,1,0 ) N ( 4 S 3/ ) O ( 3 P 0,1, ) NB Two numbers appear for 3 P configurations & one for P.

11 c. Hyperfine Transitions Clouds with molecules also are partially atomic, so the classic H I 1-cm transition remains relevant. In addition, molecules with non-zero nuclear spin add hyperfine structure to molecular spectra, the primary example being the main nitrogen isotope 14 N with I = 1. d. Λ-Doubling A form of molecular spin-orbit coupling. Λ is the projection of the orbital angular momentum on the symmetry axis of a molecule. The splitting in OH has a convenient wavelength of ~ 18 cm. e. Inversion Splitting - Pyramidal molecules may have a well spaced tunneling inversion, e.g., it is ~ 1 cm. In NH. As will be discussed later, it provides an excellent thermometer for molecular regions. f. Radio-Recombination Lines Transitions between atomic levels of high principal quantum number (Rydberg states) range form cm to mm wavelengths (for n varying from > 50 to < 30). The small A-values may be compensated by a tendency to mase.

12 Erot 3. Molecular Structure The fundamental idea is the Born-Oppenheimer Approximation. Because of their large mass, the motion of nuclei is much slower than the electrons. E E + E + E ev, ro t E vib Averaging over the fast electronic motion gives the nuclear potential energy as a function of the nuclear separations. This might be called a frozen nucleus approximation. vib el ev, E el V(R) 1 10 ev R Figure: Schematic potential for a diatomic model, with asymptotic R -6 van der Waals interaction & harmonic oscillator potential at minimum

13 Going Beyond Born & Oppenheimer Within the Born Oppenheimer picture, the nuclei can vibrate and rotate about the equilibrium positions. Beyond that, the most important interactions are magnetic interactions between their various orbital & spin angular momenta, some to be discussed below. In summary, molecular dynamics is reduced to two famous problems in classical (& quantum) physics: I. Rigid Body Motion II. Normal Modes of Oscillation The first relates to the rotational motion of astrophysically interesting molecules with wavelengths in the FIR-mm bands and the second to vibrational motions observable at NIR wavelengths.

14 Rotational Motion The rotational motion of a molecule is determined by its moments of inertia & angular momentum. Classically, any object has three orthogonal principal moments of inertia (diagonal inertia tensor) and simple associated expressions for the rotational energy and angular momentum. This carries over to quantum mechanics, and it is customary to classify the rotational properties of molecules according to the values of the principle moments of inertia. The principle moments of inertia are designated Ia, Ib, and Ic in order of increasing magnitude

15 Linear Molecules & Symmetric Tops A molecule which is linear or has an axis of rotational symmetry is called a symmetric top Either I c = I b > I a or I c > I b = I a Linear molecules have a small I about the axis of the molecule so they are of the first variety & are called prolate symmetric tops Other molecules, e.g., benzene, have the largest moment of inertia about the symmetry axis & are called oblate symmetric tops Molecules which are spherically symmetric, e.g, methane have three equal moments of inertia and are called spherical tops Molecules I c I b I a are asymmetric tops

16 Symmetric Top Rotators Classically, the energy of rotation is Suppose I x = I y = I b. Since J = J x + J y + J z z z y y x x z z y y x x I J I J I J I I I E = + + = ω ω ω + = b c z b I I J I J E 1 1 The next step is to quantize these classical expressions.

17 Symmetric Top Rotators (cont d) Both the square & the projection on the symmetry axis of the angular momentum are good quantum numbers. NB The projection on a fixed axis is also conserved; it is usually denoted M z and enters into the Zeeman effect. J = J ( J + 1) h and J z = Kh J =0, 1,, K =0, 1,, J Hence the rotational energy is E = h I b J ( J or E = BJ ( J + 1) + h I c + 1) + ( C h I b B) K K on introducing the rotational constants B & C.

18 Some Rules of the Game for Symmetric Tops J can have any integral value As a projection of J, K has (J+1) values, +J, J-1, -J+1, -J The energy depends on K, so there are only J+1 distinct values, and the levels start at J=K For a prolate top (cigar) C-B > 0: levels increase with K For an oblate top (pancake) C-B < 0: levels decrease with K K =0 reduces to the simple rotational ladder of a linear molecule Note on asymmetric tops (all unequal moments): Only J and E are conserved. The states are labeled by J and (K -,K + ), which become conserved projections in the limit of prolate & oblate symmetric tops.

19 Energy Levels for Oblate & Prolate Symmetric Tops Prolate Oblate J K Prolate

20 4. Radiative Properties of Molecules a. Rotational Transitions The molecule should have a permanent dipole moment generate a rotational Spectrum. Molecules like H, C, O, CH4, CH do not have strong rotational spectra. For symmetric molecules like H, only quadrupole transitions occur with J =, e.g. λ( 0) = 8 µm etc. b. Selection Rules The general rules apply, albeit in new forms dictated by the notation of molecular spectroscopy. For a symmetric top, the dipole moment lies along the symmetry axis. The radiation field cannot exert a torque along this axis, so the selection rule for a pure rotational transition is K = 0 (and J = ±1).

21 c. Transition Probabilities for Ro-vibrational Transitions Convention: lower level has double, upper level has single prime v,j v,0 1st level in v band v,j V,0 1 st level in v band For dipole transitions, J = 0, ±1 allows three possibilities: P-transitions: J =J -1, i.e., 0-1, 1-, etc. R-transition: J =J +1, i.e., 1-0, -1, etc, Q-transitions J =J, i.e., 1-1, -, etc,

22 A d. A-value for Dipole Radiation From a Simple Rotor - the standard atomic formulae are slightly changed: 64π 3h ν ( ) c S( J ', J '') J ' ( v' J ' v'' J '') = J M J (v'' The upper level is (v J ) and the lower level is (v J ). S(J,J ) is the Honl-London factor. (v J M v J ) is the dipole matrix element = J ' 1 S( J ', J '') = J '( J ' + 1) J ''( J '' + 1) for J ' = J '' + 1 ( R) and J ' + 1 for J ' = The absorption cross section is σ (v' J ', v'' J '') = f (v' J ', v'' J '') = πe f (v' J ', v'' J '') ϕ( ν ) m c J ' + 1 λ(v'' J '', v' J ') 3(J '' + 1) λ(00,01) e '' J '' 1 ( P) A(v' J ', v'' J '') A(01,00) v' ')

23 Transition Energy Progressions For the pure rotational spectrum of a rigid rotor, the photon energy in the transition J =J J =J-1 E J = E( J ) E( J 1) = BJ generates the sequence B, 4B, 6B.... For the ro-vibrational spectrum, the transitions involve the fixed energy difference between the vibrational levels (and the electronic levels for an optical or UV transition) plus the difference in rotational energies: E( v' J ', v'' J '') = E( v', v'') vib + B' J '( J ' + 1) B'' J ''( J '' + 1) For no change in the electronic state, B & B are about the same. Substituting the appropriate changes for R and P transitions, J =J +1 & J =J -1 respectively, yields:

24 E( v' E( v' J J ', v'' ', v'' J J '') '') R P = E( v', v'') = E( v', v'') vib vib + BJ ' B( J ' + 1) The R-transitions generate a sequence of rovib transition energies just like a simple rigid rotor, with energies laddered on top of the reference energy (the difference between the first level in each vibrational energy potential). The P-transitions generate the opposite sequence of rovib transition energies with energies laddered below the reference energy -6B -4B -B 0 B B 6B P-transitions R-transitions