Prof David Lidzey
University of Sheffield
• A key experimental technique use to probe the vibrational
modes (normal modes) of a material.
• Raman spectroscopy is commonly used in chemistry to provide
a fingerprint by which molecules can be identified.
• Can be used to explore relative composition of a material (i.e.
relative concentration of a known compound in solution).
• Widely used in industry and quality assurance.
• Key technique in condensed matter research.
Simple harmonic motion
Atoms connected via chemical bonds are equivalent to masses
connected by springs. We can describe these using Hooke’s
law (Q is a displacement of an atom away from eqn position)
From Newton’s second law
Where m is the reduced mass
Find a general solution
F = -kQ
F = m
+kQ = 0
Insert potential into time independent Schrodinger equation:
To find quantized solutions
From classical to quantum
V(x) = V(0) + x
÷ + ...
If two nuclei are slightly displaced from equilibrium positions (x = R - Re),
can express their potential energy in a Taylor series:
Not interested in absolute potential, so set V(0) = 0.
At equilibrium, dV/dx = 0 (a potential minimum). Providing displacement is small, third
order term can be neglected. We can therefore write:
EY(x) = -
En = wvib (n+
This creates a ladder of vibrational modes
This is well-known case of a harmonic oscillator.
The energy of a quantum-mechanical harmonic
oscillator is quantized and limited to the
Selection rules dictate that harmonic
Oscillator transitions are only allowed for
Dn = ± 1
En = (n+
Potential energy V
Molecules have many different vibrational modes
O C O
O C O
Asymmetric stretch mode
Symmetric stretch mode
Mode frequency dependent on mass of
Atoms, bond stiffness and type of
vibration involved (stretching, rocking,
During the interaction between light and a molecule, the incident wave induces a
dipole P, given by
Where a is the polarizability of the molecule, and E is the strength of the EM wave.
(Polarisability is the tendency of an electron cloud to be distorted by a field)
The EM field of an incident wave at angular frequency wo can be expressed using.
So the time-dependent induced dipole moment is
E = E0 cos(w0t)
P =aE0 cos(w0t)
When a molecular bond undergoes vibration at its characteristic frequency
wvib, the atoms undergo a displacement dQ around their equilibrium position
For small displacements, we can express the change in the polarisability
using a Taylor series.
Here, a0 is the polarizability at the equilibrium position. Substituting, we have
dQ =Q0 cos(wvibt)
a =a0 +
a =a0 +
From our expression for P, we then find
Using the trig identity
It is easy to show
This tells us that dipole moments are created at 3 different frequencies:
P =a0E0 cos(w0t)+
P =a0E0 cos(w0t)+
cos((w0 -wvib )t)+cos((w0 +wvib )t)[ ]
w0 w0 +wvibw0 -wvib
Results in a processes called Raman scattering
• Raman-spectroscopy is a form of inelastic
• Photon interacts with a molecule in its ground
vibronic state or an excited vibronic state.
• Molecule makes a brief transition to a virtual
• (Virtual state is an ‘imaginary’ intermediate
state. Lifetime of such states dictated by
• The “scattered” (emitted) photon can be of
lower energy (Stokes shifted) than the
incoming photon, leaving the molecule in an
excited vibrational state.
hn hn '
• Can also have a transition from a
vibrationally excited state to the
• The molecule will then return to its
ground-state, with the scattered
photon carrying away more energy
than the incident photon.
• This is called anti-Stokes scattering.
• Raman scattering should not be
confused with the emission of
hn hn '
Raman ‘selection rules’.
A necessity for Raman scattering is that
i.e., as the bond vibrates, there is a change in its polarizability. Why does this
At max compression, electrons ‘feel’ effects of other nucleus, and are less
purturbed by EM field. At max elongation, electrons feel less interaction with
other atom, and are more perturbed by the EM field. We thus have a change
in polarisability as a function of displacement.
Max compression Equilibrium Max elongation
Raman spectroscopy: practicalities
Raman signal is often orders of magnitude
weaker than elastic scattering, so we need
A laser and rejection of stray light.
Use an ‘edge filter’ to reject the
Raman scattered cross section given
Can use shorter wavelengths (higher frequencies), but this can excite fluorescence
that often swamps the weak Raman signal.
Spectroscopists most often express wavenumber of vibrational mode in units of
cm-1 (which is a unit of energy). Typically goes from 200 to 4000 cm-1.
s µ(n0 -nvib )
n0 =1/ l0
nvib = 2pwvib /c
394 492 532
(CC2 symmetric stretch)
(identification based on Harris et al, Journal of molecular spectroscopy, 43 (1972) 117)
Raman map of silicon, showing strain
Around a laser drilled hole.
Kishan Dholakia and colleagues:
University of St. Andrews
Raman used in
Quality assurance and
Mapping drug dispersion
Coupling electronic and vibronic transitions
• We have seen that we can directly measure
the vibrational modes of a material using
• Molecules typically vibrate as the make
transitions between electronic states.
• So how does the vibration of a bond affect the
fluorescence of a molecule?
The ground state and the excited states
of molecules can be represented by
harmonic oscillators with quantized
Electronic transitions are allowed between
Mass of an electron is very different from
the nuclei. Thus electronic transitions occur
in a stationary nuclear framework (Franck
We plot electronic transitions as vertical
lines, representing the same nuclear
distribution in ground and excited states.
Molecular absorption and emission spectra contain ‘vibrational replicas’.
In ideal case, the excited and ground states have an identical harmonic
potential, and thus absorption spectrum is the mirror image of emission.
Stokes shift measure of energetic relaxation between ground and excited states.
Example: Absorption and PL of diphenyl anthracene
DE ~162 meV
Probably a C-C
The effects of disorder
See strongly broadened transitions
caused by inhomogeneous broadening.
Polymers can be very disordered materials
S0 S1 S1 S0S0 S2
• In many molecular systems, the harmonic potential results in
quantized vibrational modes.
• Raman spectroscopy allows you to identify and characterize
these vibrational modes.
• We can see fingerprinits of certain vibrational modes when we
measure absorption and fluorescence emission.
• Raman spectroscopy is highly useful in materials research and
is widely used as a routine characterization technique.