Vibrational spectroscopy, particularly Raman spectroscopy, is an essential technique for probing the vibrational modes of materials, useful for identifying molecular compositions and applied widely in industry and research. The document details the theoretical foundations of vibrational modes and interaction of light with molecules, emphasizing the process of Raman scattering and its practical applications in chemical analysis and material characterization. Key concepts include the quantization of vibrational energy levels and the measurement of vibrational modes, which facilitate substance identification and quality assurance.

1. Vibrational spectroscopy
Prof David Lidzey
University of Sheffield

2. Vibrational Spectroscopy
• 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.

3. 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
Thus
Find a general solution
where
F = -kQ
F = m
d2
Q
dt2
m
d2
Q
dt2
+kQ = 0
Q(t)= Acos(wvibt)
wvib =
k
m

4. Insert potential into time independent Schrodinger equation:
To find quantized solutions
V(x) =
1
2
kx2
k =
d2
V
dx2
æ
è
ç
ö
ø
÷
From classical to quantum
V(x) = V(0) + x
dV
dx
æ
è
ç
ö
ø
÷+
1
2
x2 d2
V
dx2
æ
è
ç
ö
ø
÷ +
1
3!
x3 d3
V
dx3
æ
è
ç
ö
ø
÷ + ...
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) = -
2
2m
Ñ2
+V(x)
é
ë
ê
ù
û
úY(x)
En = wvib (n+
1
2
)

5. 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
values.
Selection rules dictate that harmonic
Oscillator transitions are only allowed for
Dn = ± 1
En = (n+
1
2
) wvib
0
1
2
3
4
5
6
7
8
Displacement (x)
Energy
Potential energy V
wvib
V(x) =
1
2
kx2

6. Molecules have many different vibrational modes
O C O
O C O
Asymmetric stretch mode
O
C
O
Bending mode
CO O
Symmetric stretch mode
(100)
(010)
(020)
(030)
(001)
Symmetric
stretch
mode
Bending
mode
Asymmetric
stretch
mode
171 meV
82 meV
290 meV
Example: CO2
Mode frequency dependent on mass of
Atoms, bond stiffness and type of
vibration involved (stretching, rocking,
breathing etc)

7. Light-molecule interactions
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
P =aE
E = E0 cos(w0t)
P =aE0 cos(w0t)

8. When a molecular bond undergoes vibration at its characteristic frequency
wvib, the atoms undergo a displacement dQ around their equilibrium position
Q0
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)
Q0
Q0+dQ
a =a0 +
¶a
¶Q
dQ+...
a =a0 +
¶a
¶Q
Q0 cos(wvibt)

9. 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)+
¶a
¶Q
Q0E0 cos(w0t)cos(wvibt)
cos(a)cos(b) =
1
2
cos(a-b)+cos(a+b)[ ]
P =a0E0 cos(w0t)+
¶a
¶Q
Q0E0
2
cos((w0 -wvib )t)+cos((w0 +wvib )t)[ ]
w0 w0 +wvibw0 -wvib

10. Results in a processes called Raman scattering
• Raman-spectroscopy is a form of inelastic
light-scattering.
• Photon interacts with a molecule in its ground
vibronic state or an excited vibronic state.
• Molecule makes a brief transition to a virtual
energy state.
• (Virtual state is an ‘imaginary’ intermediate
state. Lifetime of such states dictated by
uncertainty principle)
• The “scattered” (emitted) photon can be of
lower energy (Stokes shifted) than the
incoming photon, leaving the molecule in an
excited vibrational state.
Ground state
0
1
2
3
4
Virtual state
hn hn '

11. Anti-stokes scattering.
• Can also have a transition from a
vibrationally excited state to the
virtual state.
• 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
fluorescence.
Ground state
0
1
2
3
4
Virtual state
hn hn '

12. 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
happen?
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.
¶a
¶Q
¹ 0
Q0-DQ
Q0 Q0+DQ
Max compression Equilibrium Max elongation

13. 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
Laser light.
Raman scattered cross section given
By
Where
and
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
nvib
nvib

14. Example: acetone
394 492 532
789
899
1068
1220
1353
1427 1711
1746
(C=O stretch)
(CC2 symmetric stretch)
(CH3 deformation)
(CH3
rock)
(identification based on Harris et al, Journal of molecular spectroscopy, 43 (1972) 117)

15. Example: silicon
Raman map of silicon, showing strain
Around a laser drilled hole.

16. Kishan Dholakia and colleagues:
University of St. Andrews
Raman used in
chemical analysis
Quality assurance and
Substance identification
Detecting counterfeits
Mapping drug dispersion
in pharmaceuticals

17. Coupling electronic and vibronic transitions
• We have seen that we can directly measure
the vibrational modes of a material using
Raman spectroscopy.
• Molecules typically vibrate as the make
transitions between electronic states.
• So how does the vibration of a bond affect the
fluorescence of a molecule?

18. Molecular transitions
The ground state and the excited states
of molecules can be represented by
harmonic oscillators with quantized
vibrational modes.
Electronic transitions are allowed between
these modes.
Mass of an electron is very different from
the nuclei. Thus electronic transitions occur
in a stationary nuclear framework (Franck
Condon Principle).
We plot electronic transitions as vertical
lines, representing the same nuclear
distribution in ground and excited states.
Nuclear Displacement
Energy
hn

19. 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.
Stokes shift

20. Example: Absorption and PL of diphenyl anthracene
DE ~162 meV
(~1309 cm-1)
Probably a C-C
Stretch mode.
DE

21. The effects of disorder
See strongly broadened transitions
caused by inhomogeneous broadening.
Polymers can be very disordered materials
F8BT
S0 S1 S1 S0S0 S2

22. Vibrational spectroscopy
• 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.