Spectroscopy Notes

Ṗ 2πρ(ω) N X X X

cells 0 0 2

nn

| |d · |ni| ×

= λ (β)|hn

elec n i k,s

j

∂ω∂Ω V ω

} k

0

i,j

k,s nn

i Z −i(ω −ω

i(ω +ω )t )t

× 0 0

dt n e + (n + 1)e

k k

nn nn

k,s k,s

where we see exactly how the first contribution describes an absorption process with

E > E , while the latter an emission one. Higher order contributions, which also

0

n n

include the diamagnetic term, would include 2-photons processes.

An additional contribution to the linear response would come from the delocalized

electrons in metals, which would show a continuous excitation spectrum, defining the

Drude behaviour developed in the previous sections. This is clean until the plasma

frequency (proportional to the conduction band width) is reached, where the interband

transitions become possible and the latter developments taint the Drude one with char-

acteristic peaks. Since we are in linear response, excited electrons will decay instantly

without showing a conduction behaviour at ω < ω . Phonons excitations will overlap

p

with the stronger ones given by the delocalized electrons, being hidden in the macro-

scopic measurements, while characterizing the low frequency behaviour of insulators.

6 Photon Scattering: Classical Approach

Second order effects for the interaction of light with matter cover two phenomena:

elastic and inelastic scattering, and two-photon absorption. We will focus our study

only on the former.

We can distinguish multiple scattering processes. The elastic one is known as

Rayleigh scattering, and is analogous to the Bragg effect for neutrons. Diving into

'

inelastic processes, we find the Brillouin scattering characterised by ,

}ω }ω

i f

obtained when photons are scattered by the variation of density in the crystals, in-

duced by phonons, polarons or magnons. Rayleigh scattering could also be induced

by density waves, but it is associated to incoherent thermal excitations. When the po-

larization variation is associated to roto-vibrational modes between two neighbouring

27 6

atoms in the crystal, we’re dealing with Raman scattering, for which = .

}ω }ω

i f

The main difference with Brilluoin scattering is the nature of the phonons involved,

for the latter we are dealing with long-range oscillations, i.e., low energy phonons,

while for Raman the oscillations are localized for the molecule (high energy phonons).

Increasing the photon energy, we find Thomson and Compton scattering, respectively

for X-rays and γ-rays (with free charges like electrons).

Using a classical approach, we can compute the scattering effects for high energy

photons (the low energies ones are absorbed) through the study of the creation of

dipole moments in the crystal induced by the external electric field. For metals, the

electric field doesn’t penetrate enough to observe these effects, in fact, the instanta-

neous de-excitement of the electrons (difficult to transform their acquired energy into

heat) will diffuse the radiation back to the vacuum (summed up to obtain an almost

perfect specular reflection), at least until the interband excitation dominion is reached

(excited states live longer). The high energy photons which pass through are in gen-

eral absorbed to create dipoles (deformation), and, due to the energetic delay between

the electron dispersion and the phonon or roto-vibrational one (which characterize the

energy conversion time), it can be remitted with an energy equal, greater or lower than

the initial one. Classically, the dipole generated and the external field will be linked

by a tensor known as generalized polarizability

1 ˆ

β̂(t)E)E + ...

(

d = χ̂(E, t)E = α̂(t)E + 2

which will depend, in its more general form, on the external (local) field and the crystal

structure. The expansion in the local external field permits to separate the linear and

non-linear effects, in particular, the former is characterized by the polarizability tensor

α̂, while the latter higher order tensors take the general name of iper-polarizability of

order n, where n is equal to 1 for the quadratic contribution. The explicit depen-

dence on time is given by the evolution of the crystal geometry due to thermal effects,

in particular vibrations and rotations for molecules. If we suppose zero interaction

between these two degrees of freedom (no centrifugal effects), we can separate their

contribution to first order such as

∂ α̂ ∂ α̂

' , α̂ =

α̂(t) α̂ + α̂ (t) + α̂ (t) α̂ =

0 V R V R

∂Q ∂R

R Q

eq eq

Given the classical behaviour of the oscillations, we can write the time dependency as

sinusoidal functions, i.e.

α̂ (t) = α̂ sin(ω t) α̂ (t) = α̂ sin(2ω t)

V V V R R R

where the double nature of the rotational frequency is due to a degeneracy in the

dipole orientation: since the probe wavelengths are way longer than the interatomic

distances, the deformation given by a rotation starting from zero angle will be the

same as the one starting from 180.

Taking the external field to be harmonic E(t) = E sin(ωt), we can write the com-

0

plete linear relation as five different contributions

28

 

α̂ E

V 0 −

d = α̂ E sin(ωt) cos(ω ω )t

+ + cos(ω + ω )t +

0 V V

 

2

| {z } | {z } {z }

|

Rayleigh V ib. Raman−Stokes Anti−Stokes

 

α̂ E

R 0 −

cos(ω 2ω )t + cos(ω + 2ω )t

+ R R

 

2 | {z } | {z }

RRS RRAS

where the physical effect of scattering is obtained as the energy lost by an oscillating

dipole (acceleration), given by the Hertz law 2

∝ |

I d̈| sin θ

emitted

with θ angle between the dipole axis and the emitted light direction. The effects given

by the presence of two or more atoms are the development of inelastic processes: Stokes

and anti-Stokes are related to the emission of lower and higher energy with respect to

the absorbed one, exactly like neutron scattering.

Taking a step toward quantum mechanics, the energies for the roto-vibrational

modes take the form

1 −1 −1

∼ ∼

E = n + E = BJ(J+1) 4000 cm = 0.5 eV, B 40 cm = 5 meV

}ω }ω

V V R V

2 ±2.

with inelastic selection rules ∆J = 0,

Figure 4: Classically, Stokes and anti-Stokes peaks are equal, but on a quantum level

this is not true due to the occupation asymmetry given by thermal excita-

tions. Higher frequencies mean a greater disparity between Stokes (higher)

and anti-Stokes. Temperature measurements can therefore be computed

through the disparity of the peaks. The first excitations, starting from

the Rayleigh peak, are the rotational ones (Brillouin peaks overlap with

Rayleigh), which show plenty of possible energies due to the selection rules

and the multiple occupied initial states. Vibrational effects live at energies

hundreds of times bigger. 29

Optical instruments with resolving power high enough to see the first Raman scatter-

ings are easily obtained, since the requirements for a Helium-Neon laser are

E 1.25 eV

' ∼

RP = 300

−3

×

∆E 5 10 eV

where the frequency of the laser is associated to E and the energy difference to scan to

∆E. Modern monochromators can reach resolving power orders of magnitude higher.

6.1 Rule of Mutual Exclusion

In a molecule with a centre of symmetry it is seen that vibrations that are Raman active

are IR (absorption) inactive and vice-versa. This phenomenon is known as mutual

exclusion rule. For a complex molecule that has no symmetry except the identity, all

the normal modes (sinusoidal pattern) are active in Raman and IR spectra. However,

there exists modes that are neither Raman or IR active, which are called ”silent”, and

are present in molecules like benzene of ethylene. In general the strong bands in the

IR spectrum of a compound corresponds to weak bands in the Raman and vice versa.

This complimentary nature is due to the electrical characteristic of the vibration. If a

bond is strongly polarised, a small change in its length such as that occurring during

a vibration, will have only a small additional effect on polarisation, meaning that the

electronic states won’t change much from the application of the external field. Thus,

vibrations involving polar bonds (C-O , N-O , O-H) are comparatively weak Raman

scatterers. Such polarised bonds, however, carry their charges during the vibrational

motion, which results in a large net dipole moment change and produce strong IR

absorption band. Conversely, relatively neutral bonds (C-C , C-H , C=C) suffer large

changes in polarizability during a vibration, though this is less easy to visualise. But

the dipole moment is not similarly affected and vibrations that predominantly involve

this type of bond are strong Raman scatterers but weak in the IR. This phenomenon

is a powerful tool, in fact, by overlapping the two spectra, it is possible to acquire

information about the symmetries of the element. The main differences between the

two procedures are:

1. Raman is caused by polarizability variations due to roto-vibrational modes, while

IR is due to the direct changes in dipole moment.

2. Polarised molecules are prone to be IR active, but only slightly Raman active.

3. Water can’t be used as a solvent for IR spectra measurements due to its absorp-

tion capacity.

4. Raman gives an indication of covalent bondings, while IR of ionic ones.

5. Raman instrumentations are way more elaborate than their IR counterparts.

7 Photon Scattering: Quantistic Approach

The perturbative Hamiltonian in the dipole approximation takes the simple form V =

−d · E. The effects of this perturbation are the loss of energetic degeneracy and, at

30

second order, the renormalization of energies. For atoms, this is the Stark-Lo Surdo

effect, where the s and p states hybridize to form a new one where the electron can roam

freely. When the system shows a conventional unit cell, the dipole moment is better

P

understood as the polarization density operator, which takes the form q d .

i i

i∈U C

Taking a zero temperature approach, the effects of this perturbation can be ex-

ploited, for the electronic structure, through a non-degenerate perturbative theory,

where the new ground state takes the form

he |V |e i

X λ 0

0 0

i |e i

|e |e i he |V |e i

= + E = E +

0 λ 0 0 0

0 0

−

E E

0 λ

λ>0 − →

and is easily generalized for oscillating fields through the sub