applications of group theory to...
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© K. S. Suslick, 2013
Applications of Group Theory to Spectroscopy
Vibrational Spectroscopy
Raman & IR Apparatus and Concept
Selection Rules (Allowedness)
Symmetry of Vibrational Modes
Normal mode analysis
Raman, Resonance Raman, CARS
Electron Energy Loss Spectroscopy (EELS)
(Rotational Spectroscopy: not to be covered in class)
© K. S. Suslick, 2013
Comparison between IR and Raman
light scattering is a 2 photon event!
“virtuallevels”
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Comparison between IR and Ramanlight scattering is a 2 photon event:
© K. S. Suslick, 2013
2 Kinds of Light Scattering
John William Strutt, 3rd Baron Rayleigh, Nobel Prize
for Physics in 1904, for Ar discovery. 1842-1919.
Sir ChandrasekharaVenkata Rāman,
1888-1970. Nobel in Physics 1930.
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Infrared Spectroscopy : Absorbance of IR light.
Dispersive (dual beam)
FTIR: Fourier TransformInfraRed
Infrared Vibrational Spectroscopy
Michelsoninterferometer
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An FTIR interferogram converts wavelength into mirror position (i.e. energy into time). The central peak is at the ZPD position ("Zero Path Difference") where the maximum amount of light passes through the interferometer to the detector.
FTIR Interferogram
The interferogram is then fast Fourier transformed (FFT) back into the wavelength regime: i.e., the IR absorbance spectrum.
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• Speed: all frequencies measured simultaneously, FT-IR scans in sec. rather than several minutes. Felgett Advantage.
• Sensitivity: Sensitivity dramatically improved with FT-IR. The detectors are much more sensitive. Optical throughput is much higher (Jacquinot Advantage, better S/N).Fast scans enable co-addition of several scans (signal averaging).
• Internally Calibrated: HeNe laser used as internal wavelength calibration (Connes Advantage).
FTIR Advantages
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Raman: Scattering of visible light with loss (or gain)of frequency corresponding to vibrational quanta.
Raman Spectroscopy
laser beam
sample
grating
resolution slit
CCD
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Monochromator
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Raman Microscope
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Raman Sample Handling
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Raman Spectroscopy
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Molecular Vibrations: 3N-6 (5) Rule
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Infrared Absorption Selection Rule
IR Selection Rule: For any IR absorption,
the vibrational excitation must change
the dipole moment of the molecule
in order for radiation absorption to occur.
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Infrared Absorption Selection Rule
The Dipole Moment (M) must change during vibration
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Group Theory View of Selection Rules
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All f2(x) will/must include an A1g (totally symmetric) function.
Group Theory View of Selection Rules
© K. S. Suslick, 2013
Infrared Absorption Selection Rule
The ground vibrational state is alwaystotally symmetric (A1g): i.e., the atoms aren’t “moving” (sorta…)
And dipole moments are just vectors:i.e., they transform as x, y and z.
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The Dipole Moment (M) must change during vibration
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Infrared Absorption Selection Rule
In GT terms: must contain an A1g to be non-zero:
i.e., ( Γ x A1g = Γ )
must be x, y, or z symmetryfor allowed IR band!
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The Dipole Moment (M) must change during vibration
© K. S. Suslick, 2013
Infrared absorption:(1 photon process)
Raman Scattering:(2 photon process)
The polarizability tensor goes as the quadratics of xyz.
polarizabilitytensor
dipole moment vector
Raman vs. Infrared Selection Rules
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Infrared Absorbance & Dipole Moment Changes
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Vibrational Selection Rules
Selection Rules:
IR active modes must have IrrReps that go as x, y, z.
Raman active modes must go as quadratics (xy, xz, yz, x2, y2, z2)(Raman is a 2-photon process: photon in, scattered photon out)
IRActive
RamanActive
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Vibrational Selection Rules
Selection Rule Summary:
IR active modes must have IrrReps that go as x, y, z.
Raman active modes must go as quadratics (xy, xz, yz, x2, y2, z2)
IRActive
RamanActive
© K. S. Suslick, 2013
The Centrosymmetric Rule
If a molecule has an inversion center(i.e., is “centrosymmetric”), thenno vibration can be both IR and Raman allowed (active, absorbing).
x, y, z symmetries must have
i = -1 (for IR activity)
Quadratic symmetries must have
i = +1 (for Raman activity)
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The Centrosymmetric Rule: CO2
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Normal Modes of H2O
= A1 + B2 => z & y
= A1 => z
Accounting Rule: stretches/bends behave as double headed arrows, counted individually only if they stay in the same place after symmetry operation.
H2O Bend: θ | 1 1 1 1 |
Both O-H Stretches: O-H | 2 0 0 2 |
For H2O (because the H’s are light),stretching frequencies are much higher than bending frequencies.
i.e., very little mixing of stretches and bends; we can consider stretches and bends separately.
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Normal Modes of H2O
= A1 + B2 => z & y
= A1 => zH2O Bend: θ | 1 1 1 1 |
Both O-H Stretches: O-H | 2 0 0 2 |
O
H H
A1: 3657 cm-1 B2: 3756 cm-1
O
H H
O
H H
A1: 1595 cm-1
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Symmetry Effects on Infrared Active Vibrations
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Non-coordinated: Td
Unidentate: C3v
Bidendate: C2v
M' & M same atom, chelated: "η2-"M' & M different, bridging: "μ-"
Sulfate (SO42- ) can be ligated in several ways:
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Symmetry Effects on Infrared Active Vibrations
S-O | 4 1 2 | All S-O Stretches
Accounting Rule: stretches are double headed arrows,counted individually only if they stay on the same bond.
© K. S. Suslick, 2013
Irreducible Representations (IrR):
Order = 1+2+3 = 6 = h
# of A1 = 1/h ( 1*1*4 + 2*1*1 + 3*1*2) = 1/6 (12) = 2 A1
# of A2 = 1/h ( 1*1*4 + 2*1*1 + 3*-1*2) = 1/6 (0) = 0 A2
# of E = 1/h ( 1*2*4 + 2*-1*1 + 3*0*2) = 1/6 (6) = 1 E
1
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Symmetry Effects on Infrared Active Vibrations
ICBST: For Td symmetry, only 1 IR Active stretch,
For C2v symmetry, 4 IR Active stretches.
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Thus, ΓS-O = 2 A1 + E for C3v (unidentate)
A1 and E are both IR active ( i.e., z and (xy) )
Therefore: 3 IR bands expected for C3v unidentate SO42-
© K. S. Suslick, 2013
For S-O stretches (1000 – 1200 cm-1):“Descent in Symmetry”
1 IR Band Non-coordinated (Td)3 IR bands Unidentate (C3v)4 IR bands Bidentate (C2v)
Symmetry Effects on Infrared Active Vibrations
SO4-2 can be ligated in several ways:
Non-coordinated: Td
Unidentated: C3v
Bidentate: C2v
M & M’ same atom:chelated, “2-”
M & M’ different:bridging, “-”
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Symmetry Effects on Infrared Active Vibrations
SO4-2 can be ligated in several ways:
Non-coordinated: Td
Unidentated: C3v
Bidentate: C2v
M & M’ same atom:chelated, “2-”
M & M’ different:bridging, “-”
There exist "correlation tables" for this kind of "descent in symmetry", which is easier than re-figuring out the local symmetry point group and allowed IR or Raman transitions.
e.g., http://www.staff.ncl.ac.uk/j.p.goss/symmetry/Correlation.html
© K. S. Suslick, 2013
GT & Vibrational Spectroscopy: Cisplatin
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GT & Vibrational Spectroscopy: Cisplatin
… etc. for B1 & B2
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GT & Vibrational Spectroscopy: Cisplatin
symmetricstretch
anti-symmetricstretch
(w.r.t. C2 or σyz)
But actually thereis a little bit of
Pd motion, too.
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GT & Vibrational Spectroscopy: Cisplatin
Raman Active IR Active
ΓM-Cl 2 0 0 0 0 2 0 2
h = 8
= A1g + B2u
z y
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GT & Vibrational Spectroscopy: Pd(NH3)2Cl2
2 bands in IR
1 band in IR
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IR Stretches for Metal Carbonyls# IRbands
1
3
1
4
2
3
CO
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IR Stretches for Metal Carbonyls
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Polyatomic Vibrations: Normal Modes
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Naming Normal Modes
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GT & Cartesian Coordinate Analysis
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GT & Cartesian Coordinate Analysis
Raman IR IR
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GT & Cartesian Coordinate Analysis
But what are these vibrations (i.e., 3 Ag, Au, and 2 Bu) ?
Au is easy. By inspection:
(remember, center of mass must be preserved or it becomes a translation/rotation!)
© K. S. Suslick, 2013
GT & Cartesian Coordinate Analysis
For the 3 Ag and 2 Bu, let us use Bond Stretch Vectors and Anglesas a starting basis set:
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GT & Cartesian Coordinate Analysis
For the 3 Ag and 2 Bu, let us use Bond Stretch Vectors and Anglesas a starting basis set:
But in all cases, the extent of mixing (“coupling”) is NOT available from GT.
© K. S. Suslick, 2013
Alternative Method: Projection Operators
So again, the 2 Bu vibrations are a combination of the N-F asymmetric stretch and the asymmetric bend (θ1 – θ2).
E C2 i σBu = 1 -1 -1 1
form a projection operator:
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Extent of Mode Mixing
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Short, Short Cut
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A Short, Short Cut
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A Short, Short Cut
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Vibrational Group Frequencies
© K. S. Suslick, 2013
Vibrational Group Frequencies
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Complications: Coupling of Normal Modes
If two normal modes have the same symmetry, then they can mix (“Fermi Coupling”)
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Harmonic vs. Anharmonic Oscillators
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Harmonic vs. Anharmonic OscillatorsSecond kind of anharmonicity: mixtures of normal modes.
e.g., combinations, overtones, difference bands.
Even if symmetry allowed, Anharmonic bands are usually weak.
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Normal Coordinate Analysis
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Normal Coordinate Analysis
Molecular Vibrations: The Theory of Infrared and Raman Vibrational, by E. B. Wilson, J. C. Decius, P. C. GrossInfrared Spectra of Inorganic and Coordination Compounds, by K. Nakamoto
F11 = force along L1F12 = force between L1 and L2F13 = force between L1 and θF22 = force along L2F33 = force along θ ….
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Normal Coordinate Analysis
(GF = distance x force = energy)
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Normal Coordinate Analysis
GF matrices seldom are solvable due to lack of sufficient νso we make simplifications or we calculate potentials from ab initio or DFT:
© K. S. Suslick, 2013
Example: GVFF
But even 16 forceconstants out-numbers the numberof observablesconsiderably!
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Applications of Group Theory to Spectroscopy
Vibrational Spectroscopy
Raman & IR Apparatus and Concept
Selection Rules (Allowedness)
Symmetry of Vibrational Modes
Return to vibrations and normal mode analysis
Raman, Resonance Raman, CARS
Electron Energy Loss Spectroscopy (EELS)
Rotational Spectroscopy
© K. S. Suslick, 2013
Complications: Anharmonicity
The Harmonic Oscillator
Hook’s Law
ICBST: For a HO, only Δn = ±1 is IR allowed.
all same energy
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Complications: Anharmonicity
The Harmonic Oscillator
relevant toelectronic spectra, too.
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Complications: Anharmonicity
Bonds can never break for any HO!
For any non-HO, |Δn| > ±1 are allowed (but weak).
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Complications: Anharmonicity
Morse Potential hard to solve Schroedinger Eqn.Easier to do a Cubic Perturbation:
k = force constant (Hook’s Law)l = cubic correction
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Consequences of Anharmonicity:Coupling of Normal Modes
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Consequences of Anharmonicity:Coupling of Normal Modes
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Even if symmetry allowed, coupled bands are usually weak.
Consequences of Anharmonicity:Coupling of Normal Modes
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Consequences of Anharmonicity:Coupling of Normal Modes
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The Force Constant
Force Constant, k, measures the curvature of the bottom of the potential energy well.k is a measure of the stiffness of internal motions.
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How much force to break?280 J / 0.5 m = 560 Nt ~ 100 Lbf
Sinclair’s Monofilament (a la Larry Niven)aka Shigawire (Dune by Frank Herbert), monomolecular filament (Johnny Mnemonic
by William Gibson) and The Space Elevator (of Arthur C. Clarke)
What’s the ultimate strength of a monomolecular chain of macroscopic length?
How much energy?
~ 280 J
Strength of 1 cm2 rope?1014 nm2/cm2 / 1 nm2 / chain
~ 1016 Lbf(= 5x106 Empire State Buildings)
© K. S. Suslick, 2013
Isotope Effects on Vibrations
Remember the problem in polyatomicsof not enough frequencies vs.too many force constants?
Isotopic labeling gives you more observables,but same force constants.
(constant vs. isotope)
(changes vs.isotope)µ
C-H vs. C-D
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© K. S. Suslick, 2013
Isotope Effects on Vibrations
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Correlations between k and Physical Properties
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© K. S. Suslick, 2013
Correlations between k and Physical Properties
© K. S. Suslick, 2013
Correlations between k and Physical Properties
Fo
rce
Co
nst
ant
Badger’s Rule
CO adsorbed on Pt and Ir
as function of electrode potentials.
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© K. S. Suslick, 2013
Correlations between k and Physical Properties
Force Constant
Badger’s Rule applied to Fe-O stretches in heme and non-heme oxygenases
X-ray crystallography
© K. S. Suslick, 2013
Correlations between k and Physical Properties
Force Constant
Lippincott’s Rule for diatomics
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© K. S. Suslick, 2013
Effects of Metal Ion Coordination on
© K. S. Suslick, 2013
Comparison between IR and Raman
light scattering is a 2 photon event:
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Infrared absorption:(1 photon process)
Raman Scattering:(2 photon process)
The polarizability tensor goes as the quadratics of xyz.
polarizabilitytensor
dipole moment vector
Raman vs. Infrared Selection Rules
© K. S. Suslick, 2013
2 Kinds of Light Scattering
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© K. S. Suslick, 2013
Raman Spectroscopy: Stokes v. Anti-Stokes
© K. S. Suslick, 2013
Raman Spectroscopy: Stokes v. Anti-Stokes
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© K. S. Suslick, 2013
Raman Spectroscopy
© K. S. Suslick, 2013
Raman Selection Rule
The polarizability (i.e, induced dipole created by the electromagneticradiation field) must change during the excited vibration.
Consider N2
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© K. S. Suslick, 2013
Raman Selection Rule
The polarizability goes as the quadraticswithin any point group.
And the polarizability is frequency dependent (more later).
Iraman½
© K. S. Suslick, 2013
Raman Scattering Intensities
where:
spongeanalogy:
reason sky is blue!
...
2
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© K. S. Suslick, 2013
Raman Scattering Intensities:Kramers-Heisenberg-Dirac Equation (!)
The polarizability is a function of photon frequency:
© K. S. Suslick, 2013
Raman Scattering Intensities:Kramers-Heisenberg-Dirac Equation (!)
The polarizability is a function of photon frequency:
Thus the polarizability, and therefore Iraman increase as:
1. Increases in strength of coupling of electronic-excited stateswith both ground and vibrationally excited states.
2. Closeness of incident radiation to actual electronic transition: "Resonant enhancement"i.e., "Resonance Raman Spectroscopy"
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© K. S. Suslick, 2013
Type Iscat / I incident
Fluorescence 1 – 0.01Rayleigh 10-4
Raman Fundamentals (Stokes) 10-7
Raman Overtones (Stokes 10-9
Raman Anti-Stokes Fundamentals 10-9
Resonance Raman 10-2 to 10-4
Surf. Enhanced Raman >1 (multiple scatters)
Relative Intensities of Raman Scattering
© K. S. Suslick, 2013
Raman Scattering Cross-Section
(ex) = target area presented by a
molecule for scattering
scattered flux/unit solid angle
indident flux/unit solid angle
d
d
dd
d
Process Cross-Section of (cm2)
absorption UV 10-18
absorption IR 10-21
emission Fluorescence 10-19
scattering Rayleigh 10-26
scattering Raman 10-29
scattering RR 10-24
scattering SERRS 10-15
scattering SERS 10-16 Table adapted from Aroca, Surface EnhancedVibrational Spectroscopy, 2006
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Resonance Raman Scattering
Advantages:1. Affected only by chromophore
(usually the metal center)2. Sensitive to small structural changes3. Unaffected by medium
(e.g., H2O, powders, glasses, …)4. High sensitivity (μM)
Disadvantages:1. Affected only by the chromophore!2. Fluorescence often interfers (impurities, …)3. Thermal and photo- stability can be problematic.4. Analysis usually by empirical correlations.
© K. S. Suslick, 2013
Uses of Resonance Raman Scattering
M1. Materials available only in low concentrations (> µM)
Especially biological chromophores.Probes changes only near the chromophore.Often hard to assign bands except emipirically
(e.g., fingerprinting oxidation states).
2. Assignment of Electronic Spectraif one already knows the vibration modes, then the coupling between the electronic absorbance bands and the vibrational modescan assign electronic states.
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Bioinorganic Applications of Resonance Raman
Bioinorganic systems with chromophores, e.g.,1. Heme proteins2. Photosynthetic reaction centers & antennae3. Rhodopsin & vision pigments (non metallic)4. Fe & Cu enzymes (often near UV)
Types of information obtained:a. Type of ligands (but selective, not all necessarily)
b. Oxidation state of metal (indirectly, marker bands)
c. Comparisons between proteins and synthetic analogs, or among proteins
d. Dynamics (down to ps) of photoinduced rxns.
© K. S. Suslick, 2013
Resonance Raman of Heme Proteins
π*
π
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Resonance Raman Scattering: Heme Proteins
© K. S. Suslick, 2013
Resonance Raman of Hb
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© K. S. Suslick, 2013
Resonance Raman Scattering: Heme Proteins
© K. S. Suslick, 2013
Low Frequency Raman Scattering:Lattice Vibrations
Whole molecules within a crystal will vibrate and partially rotatewith respect to one another: lattice vibrations and librations.a.k.a. “phonons”. Very low energy (big masses).
Very hard to do far-IR (<600 cm-1), not hard to do Raman.
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© K. S. Suslick, 2013
Surface Enhanced Raman Scattering: SERS
SERS is a surface sensitive technique that results in the enhancement of Raman scattering by molecules adsorbed on rough metal surfaces or nanoparticles.
The enhancement factor can be as much as 1012, which allows the technique to be sensitive enough to detect single molecules(through heroic efforts, multiple scatters per molecule over time).
© K. S. Suslick, 2013
How Does SERS Work?
SERS substrates commonly usedSilver (Ag), gold (Au) and copper (Cu)
The energy required to generate plasmons matches the Raman light source
Surface preparationsLargest enhancements for rough surfaces of 10 – 100 nm
metal island film roughened metal nanostructured surface
metal coatednanostructure
Base Support
metal coating
spin coatedlayer of n.p.
base support
metal coating
nanoparticle layer
Base Support
polymer layermetal nanoparticles
base support
embedded metal nanoparticle film
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© K. S. Suslick, 2013
Origins of SERS
A. Electromagnetic field enhancement mechanism• excitation of surface plasmon• tends to form spatially localized “hot areas” for adsorbates
• the magnitude of enhancement ~106- 107 times for single Ag np• ~108 for the gap between two coupled particles
B. Chemical enhancement • due to specific interactions, forming charge-transfer complexes• the magnitude of chemical enhancement ~10-100 times
Conductive Ag or Au nanofeatures concentrate oscillating electric field from light through plasmon resonance.
Even greater concentration of field at “hot spots” at small radii of curvature (sharp points)
© K. S. Suslick, 2013
CARS: Coherent Anti-Stokes Raman Spectroscopy
Raman = 2 photon process (1 laser + 1 scattered)
CARS = 4 photon Process (3 laser + 1 scattered)
non-linear optical effect, lasers must be very high flux (short pulsed)
1. When a - b = real vibration ,molecules are driven into v=1 state.
2. Then laser A photons (at a) drivev=1 molecules back to v=0, withemission of + a This is called
stimulated anti-Stokes Raman.
CARS is ~106 more efficient than Raman.
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© K. S. Suslick, 2013
EELS
Electron Energy Loss Spectroscopy
© K. S. Suslick, 2013
EELS
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© K. S. Suslick, 2013
High Resolution EELS (HREELS)
© K. S. Suslick, 2013
HREELS
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© K. S. Suslick, 2013
HREELS Selection Rule
For electron to lose/gain energy from scattering offof an adsorbate, it must interact with a surface chargeor dipole. Only adsorbate dipoles perpendicular to metallic surface will be felt by incoming e-.
adsorbate
metal surface
image dipole
No net surface dipole!
e-
© K. S. Suslick, 2013
HREELS of Adsorbates
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© K. S. Suslick, 2013
HREELS of Adsorbates
ethylene
© K. S. Suslick, 2013
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© K. S. Suslick, 2013
Rotational Spectra: Far IR-µwave, Rot. Ram.
© K. S. Suslick, 2013
Rotational Spectra: Selection Rules
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© K. S. Suslick, 2013
Rotational Spectra: Linear Rigid Rotor
© K. S. Suslick, 2013
Rotational Spectra Informational Content
Limitations: