Section 6 Scattering (lecture 10) Previously: Quantum theory of atoms / molecules Quantum Mechanics Valence Atomic and Molecular Spectroscopy Scattering The scattering process Elastic (Rayleigh) and inelastic () scattering Selection rules for Similarities and differences with dipole allowed absorption
6.1 Scattering In addition to being absorbed and emitted by atoms and molecules, photons may also be scattered (approx. 1 in 10 7 in a transparent medium). This is not due to defects or dust but a molecular effect which provides another way to study energy levels. This scattering may be: Elastic and leave the molecule in the same state (Rayleigh Scattering) or Inelastic and leave the molecule in a different quantum state ( Scattering) 6.2 Rayleigh Scattering Lord Rayleigh calculated that a dipole scatterer << l scatters with an intensity: no. of scatterers wavelength 2 0 4 2 polarizability 2 n.b., distance scatterer - observer Nobel Prize 1904 (physics) 4 Nobel Prize 1930 (physics) 5 times more effective for 400nm than 600nm Hence the sky is blue! (and sunsets red)
6.3 Inelastic () Scattering Energy exchange between the photon and molecule leads to inelastic scatter. Virtual state The strongest scattering is Rayleigh scatter n 0 Stokes Rayleigh Anti-Stokes In Scattering the scattered photon has different energy (frequency, wavelength) than the incident photon: Stokes lines are those in which the photon has lost energy to the molecule n 0 n t n 0 Anti-Stokes lines are those in which the photon has gained energy from the molecule n 0 + n t n 0 n t n n 0 + n t Since molecular energy levels are quantised this produces discrete lines from which we can gain info on the molecule itself.
6.4 Scattering selection rules Scattering is not an oscillating dipole phenomenon! (no TDM) The presence of an electric field E induces a polarization in an atom/ molecule given by ind polarizability If the field is oscillating (e.g., photon) n ind 0 In atoms the polarizability is isotropic, and the atom acts like an antenna and reradiates at the incident frequency Rayleigh Scattering only In molecules the polarizability may be anisotropic, and depends on the rotational and vibrational coordinates. This can also give rise to Scattering. Gross Selection Rule: To be active a molecule must have anisotropic polarizability [Less restrictive than the need for a dipole moment, symmetric molecules can be active]
6.5 Rotational 6.5.1 Linear Molecules: The polarizability tensor is anisotropic ( ) As a molecule rotates the polarizability presented to the E field changes: the induced dipole is modulated by rotation results in rotational transitions n 0 Stokes Rayleigh Anti-Stokes J + 2 J J 2 Specific Selection Rule: Effective two-photon process and J Rayleigh Stokes lines Anti-Stokes lines Even non-polar molecules (O 2, N 2, CO 2 ) exhibit rotational Spectra
6.5.1 Rotational spectra J Assuming a rigid rotor: F(J) = BJ(J+1) Stokes lines are observed at: J J J n n n 0 0 and Anti- Stokes lines at: J J J - 2 n n n 0 0 n.b. 1 st Anti-Stokes line is J = 2 i.e., a gap of 6B between n 0 and 1 st lines of each branch lines in each branch of equal spacing = 4B
6.5.1 Example Rotational spectra Stokes H 2 Anti-Stokes 3:1 intensity alternation observed due to nuclear spin-statistics (3 times as many ortho-h 2 levels (odd J) as para-h 2 (even J)) Spectrum allowed because all transitions connect levels of the same symmetry. Likewise the 14 N 2 spectrum shows 2:1 aternations For the same reason, alternate lies are completely missing in the spectra of 16 O 2 and C 16 O 2.(if the level doesn t exist one can t see transitions to and from it) In deducing B from spacings, beware the possibility of missing lines in the spectrum.
6.6 Vibrational Gross Selection Rule: The polarizability must change during the vibration q 0 In practice this means the normal mode must transform with the same symmetry as the quadratic forms (x 2, xy, etc.) 6.6.1 Diatomics: Even homonuclear diatomics satisfy the gross selection rule and exhibit spectra Specific Selection Rule: Dv = ± 1 (+ Stokes, Anti-Stokes) n.b. Anti-Stokes rarely observed because v > 0 weakly populated 6.6.2 Polyatomics: Need to check each normal mode against the gross selection rule: H 2 O
CO 2 : D h g u IR Inactive IR Inactive u IR Inactive
6.7 The Rule of Mutual Exclusion In the case of CO 2 it is not coincidence that those modes which are active are IR inactive and vice versa. This is an example of the rule of mutual exclusion which states: In a centrosymmetric molecule (i.e., one with a centre of inversion symmetry) a vibrational mode may be either IR active or active but not both. acetylene D h Infra Red Infra Red
6.8 Vibration-Rotation In the same way that rotational transitions accompany vibrational absorptions so rotational structure is observed in high resolution spectra. Vibrational / Rotational spectrum of CO. The Q-branch identifies the vibrational spacing (we -2wexe)