Section 6 Raman Scattering (lecture 10)

Section 6

Raman Scattering

(lecture 10)

Quantum theory of atoms / molecules Previously: Quantum Mechanics Valence Atomic and Molecular Spectroscopy

Raman Scattering

 _{The scattering process}

 _{Elastic (Rayleigh) and inelastic (Raman) scattering}  _{Selection rules for Raman}

6.1 Scattering

In addition to being absorbed and emitted by atoms and molecules, photons may also be scattered (approx. 1 in 107 _{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 (Raman Scattering)

6.2 Rayleigh Scattering

Lord Rayleigh calculated that a dipole scatterer << l scatters with an intensity:





2 2 0 4 2

 

_



  polarizability no. of scatterers distance scatterer - observer wavelength 4



 5 times more effective_{for 400nm than 600nm} Hence the sky is blue!

(and sunsets red)

n.b.,

Nobel Prize 1904 (physics)

Nobel Prize 1930 (physics)

6.3 Inelastic (Raman) Scattering

Energy exchange between the photon and molecule leads to inelastic scatter.

n₀– n_t

In Raman 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

Anti-Stokes lines are those in which the photon

has gained energy from the molecule

n₀+ n_t

The strongest scattering is Rayleigh scatter

St ok es An ti-St ok es Ra yl ei gh n₀+ n_t n₀ n n₀– n_t n₀

Since molecular energy levels are quantised this produces discrete lines from which we can gain info on the molecule itself.

6.4 Raman Scattering selection rules

Scattering is not an oscillating dipole phenomenon! (no TDM)

ind







The presence of an electric field E induces a polarization in an atom/ molecule given by

polarizability

If the field is oscillating (e.g., photon)



_ind





₀



n



In atoms the polarizability is isotropic, and the atom acts like an antenna and re-radiates 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 Raman Scattering.

Gross Selection Rule:

To be Raman active a molecule must have anisotropic polarizability

6.5 Rotational Raman

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

St ok es An ti-St ok es Ra yl ei gh n₀ J J + 2 J – 2

Effective two-photon process and

Specific Selection Rule:

  

J

Rayleigh

Stokes lines

Anti-Stokes lines

6.5.1 Rotational Raman spectra

  

J

Assuming a rigid rotor: F(J) = BJ(J+1)

 Stokes lines are observed at:



  









0 0

n n

 

J

 

J

 

n

J



and Anti- Stokes lines at:

   









0 0

n n

 

J



J

 

n

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 n.b. 1st _{Anti-Stokes line is J = 2}

6.5.1 Example Rotational Raman spectra

H₂

Stokes

Anti-Stokes

3:1 intensity alternation observed due to nuclear spin-statistics (3 times as many

ortho-H₂ levels (odd J) as para-H₂ (even J))

Spectrum allowed because all transitions connect levels of the same symmetry.

For the same reason, alternate lies are completely missing in the Raman spectra of

16_O

2and C16O2.(if the level doesn’t exist one can’t see transitions to and from it)

Likewise the 14_N

2 Raman spectrum shows 2:1 aternations

6.6 Vibrational Raman

Gross Selection Rule: The polarizability must change during the vibration

Even homonuclear diatomics satisfy the gross selection rule and exhibit Raman spectra

Specific Selection Rule: Dv = ± 1 (+ Stokes, – Anti-Stokes)

n.b. Anti-Stokes rarely observed because v > 0 weakly populated

6.6.1 Diatomics:

6.6.2 Polyatomics:

Need to check each normal mode against the gross selection rule:

Raman Active Raman Active Raman Active H₂O 0 q         

In practice this means the normal mode must transform with the same symmetry as the quadratic forms (x2_{, xy, etc.)}

CO

₂

:

D

_h Raman Active Raman Inactive Raman Inactive IR Active IR Active IR Inactive g 



u 



u



6.7 The Rule of Mutual Exclusion

In the case of CO₂ it is not coincidence that those modes which are Raman 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 Raman active but not both.

acetylene

D_h

Raman Raman Infra Red

Infra Red Raman

6.8 Vibration-Rotation Raman

In the same way that rotational transitions accompany vibrational absorptions so rotational structure is observed in high resolution Raman spectra.

Vibrational / Rotational Raman spectrum of CO.

The Q-branch identifies the vibrational spacing (w_e -2w_ex_e)