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Development and brief application of Raman selection rules. Alex Kitt. General Selection Rules:. Only deal with transitions to zone center Based on thermal Occupancy, only Stoke’s Only to first order. Raman Introduction and limitations. Classical Treatment. Things to note:

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Presentation Transcript
raman introduction and limitations
General Selection Rules:

Only deal with transitions to zone center

Based on thermal Occupancy, only Stoke’s

Only to first order

Raman Introduction and limitations
classical treatment
Classical Treatment

Things to note:

  • First term Raman, second term Stokes, third term anti-Stoke’s
  • Only have Raman effects if the polarizability it dependent on the normal mode
  • Relationship between derivative of polarizability and orientation of normal coordinate matters
quantum
To notice:

Effect is due to coupling between photons and phonons through the electrons

Vibration states are harmonic oscillator

Symmetry could help…

Quantum

The familiar E1 transition matrix for radiative emissions:

But, we can expand the electric dipole moment in a power series in the normal coordinates

The final term gives us the matrix element that is pertinent for Raman scattering

geometric groups of solids
Geometric groups of solids
  • Point group-Collections of all rotational (proper and improper) operations that do not change the molecule-32 possible
  • Space group-Include translational properties to allow a discussion of infinite lattices-230 types of space groups
  • Factor group-Space group modulo the primitive unit cell
    • Factor groups are isomorphic to point groups
irreducible representations
Any symmetry operation can be represented by a reducible matrix

Reducing the set of matrices provides the symmetry species and normal coordinates

Character table holds a lot of information for us

Irreducible representations
example character table
Example Character Table
  • The product of Cartesian coordinates are also classified to their symmetry species
selection rules
Selection Rules

According to the Kramer, Heisenberg, Dirac equation the operator above transforms like the product of Cartesian coordinates

  • For transitions from the ground, symmetric, state the final state and the operator must be of the same symmetry species
which symmetry species exist
Which symmetry species exist?

ni is number of modes in the ith symmetry species

g is the order of the factor group

gρ is the order of the factor group

xρ is the character of the reducible matrix element

xρi is the character of the symmetery species

BA DeAngelis, RE Newnham, WB White. American Minearalogist 57, 255 (1972)

graphene with and without strain
Without strain Graphene is in space group 191 which has a factor group isomorphic to D6h

2 atoms/unit cell  3 optical phonon modes

E2g and B2g modes exist

Graphene with and without Strain

F Tuinstra and JL Koenig. Jour of Chem Phy, 55 3, 1126 (1970)

http://img.chem.ucl.ac.uk/

strain
Strain
  • Under uni-axial strain the symmetry group is broken along with the degeneracy

Huang et al. PNAS April 21, 2009, 106 (16)

references
References

BA DeAngelis, RE Newnham, WB White. American Minearalogist 57, 255 (1972)

Huang et al. PNAS April 21, 2009, 106 (16)

McHale, Jeanne. Molecular Spectroscopy (1999)

F Tuinstra and JL Koenig. Jour of Chem Phy, 55 3, 1126 (1970)

M Cardona and G Guntherodt, Topics in Applied Physics: Light Scattering in Solids II

JR Ferraro, K Nakamoto, CW Brown. Introductory Raman Spectroscopy Second Edition

C Kittel, Introduction to Solid State Physics