Spectroscopy Notes

Spectroscopic methods and nanophotonics

Notes from the University La Sapienza course held by Prof. Stefano Lupi

Tomarchio Luca

July 23, 2019

Contents

1. Scattering Matrix and Cross Section
2. Linear Response Theory
3. Neutron Scattering
   • Study of a Perfect Gas
   • Study of the Harmonic Crystal
   • Zero Order Coherent Scattering Section
   • First Order Coherent Scattering Section
   • Incoherent Response
   • Study of an Amorphous Solid or Liquid
4. Electromagnetic Field Quantization
5. Matter-Radiation Interaction
   • Optical Properties of the Drude Model
     • Free Electrons Contribution
   • Ionic and Covalent Solids
     • Localized Electrons Contribution
6. Photon Scattering: Classical Approach
   • Rule of Mutual Exclusion
7. Photon Scattering: Quantistic Approach
   • Derivation of the Hydrodynamic Regime
   • Crystal Light Scattering
8. Near Field Spectroscopy
   • Near-Field Scanning Optical Microscopy (SNOM)
   • Aperture-less Mode: AFM NFS
9. Time-Resolved Spectroscopy
10. Biological Applications of IR Spectroscopy

Spectroscopy interaction

The interaction between a probe and a condensed matter system, or target, is the study of spectroscopy. It is principally based upon the linear response theory, which implies that the interaction isn’t strong enough to modify the dispersion relations or the properties that play a role in the interaction, i.e., the output will be a ”photograph” of the unperturbed system. We distinguish two types of spectroscopy techniques: frequency and time resolved. The former are the ones that give information about the Fourier transforms of the response functions, while the latter, possible since the invention of ultra-rapid lasers, are able to highlight the time dependence of the functions and give information about outside equilibrium properties (phase transitions and fluctuations). The response of a condensed system can be exploited as an excitation of single particles or collective behaviour, which can even describe biologic or non-physics systems.

Scattering matrix and cross section

The fundamental idea behind these processes is the interaction between two initially independent systems, the probe and the target, both described through a Hamiltonian. Which one is the probe is just a matter of choice, the entire point of view is relative. The change of the probe quantum state gives information on the medium, hence, the main idea is to connect the two systems for a finite time Δt = t - t, adding an interaction Hamiltonian V(t), which is zero for times outside the interval. In general, the probe used will depend on the system we want to study, and its energy has to be slightly higher than the characteristic energies we want to excite. Furthermore, for direct images of the object, spatial resolution has to be kept in mind (diffraction limit). We define the total Hamiltonian as:

H = H_t + H_p + V(t)

and assume the two systems to be isolated from the environment, obtaining two conservation laws:

p_f = p_i - p_t

and

E_f = E_i - E_t

with an additional angular momentum relation, such that the probe state is completely characterized. The initial and final occupation, |p_i⟩ and |p_f⟩, are considered to be pure states, while the states of the target are distributed through a Boltzmann statistic based on temperature. The probe state, if we want to compute all the collective effects of the probe, must be the state of the entire flux of multiple components incident on the medium, and interacting with it and then for the time of propagation throughout its length. Since the interaction term is not infinitesimally small, the Schrödinger equation can only be approached by a Dirac representation, used to exploit the Dyson formula and the scattering operator:

S(t_i, t_f) = T exp[-i/ħ ∫_ti^t_fV(t')dt']

It's now easy to define the transition probability per unit time between two states as:

P_f←i = |⟨f|S(t_i, t_f)|i⟩|²/T

where the states of the two systems are independent due to the null interaction outside the time interval and the scattering matrix now starts from order one. This search of a unit time response permits to obtain results which are independent on the material dimension, especially when a continuous probe is sent through the target. The nature of the total state will depend on the statistic of both the target and the probe. This will be given in an interaction picture, hence, it can’t be calculated exactly, but, due to the adiabatic switching on only for a finite time, at the two time extremes the interaction picture state will converge to the Heisenberg one. This process, for systems of permanent interaction (electrons in a metal), should be considered to start:

t_i = -∞

and finish at:

t_f = +∞

maintaining a convergence proved by the Gell-Mann and Low theorem. For our case, we can send the time interval to infinity by introducing a switching on and off factor to the interaction term, without changing the physics. In reality, thanks to the thermal distribution, we have to consider all the possible initial states of the target to obtain the total transition rate. By introducing a density of final states for the probe, we define the number of states in an infinitesimal interval of solid angles dΩ (from the axial propagation of the output probe) and energies:

ρ(E_f)dE_fdΩ

where the density should also have a dependency on the direction of the output probe. It is then straightforward to write:

dP/dΩdE = (1/T)Σ_i⟨i|S(t_i, t_f)|f⟩|²e^-βE_i

By considering just a pure probe initial state and an isolated system, we have obtained a completely general relation which can be re-expressed into a total differential scattering cross section:

dσ/dΩdω = Σ_diff + Σ_abs + Σ_refl

with φ incident flux with dimensions [φ] = l^-2t^-1, and N number of interacting particles in the target. The total (integrated) cross section is measured in barns, where 1 barn = 10^-24 cm². This general cross section absorbs all the processes, i.e., diffusion, absorption and reflection, and describes their global probability of happening for a single probe element incident to a single target one (mean field response). The net difference between this approach and the Green function one are the properties we want to study. For the latter we want to analyze the intrinsic properties of a material and how they respond to a perturbation, while for the former we require the probabilities of transitions and how the target structure affects the probe.

Linear response theory

The theory is developed to obtain analytical expressions of the response properties of a system given a ”not too great” external perturbation. If the latter statement is true, the interaction can be translated in the Hamiltonian as a coupling between a property of the system A and the external perturbation α. For bulk quantities:

v(t) = ∫α(r, t)A(r)dr

Since we are not interested in thermal averages, we want to directly use the scattering matrix to obtain our formulation in real space. Given the cross section, we can write:

ΣΣ|⟨f|S|i⟩|² = ∑⟨p_f|⟨t_f|S|t_i⟩|p_i⟩⟨p_i|⟨t_i|S|t_f⟩|p_f⟩

If we can find, in the scattering matrix S, contributions which sum takes the linear form:

W = ∫drdtφ_p(r, t)φ_t(r, t)

substituting in the former relation:

⟨p_f|φ_p⟩⟨p_f|φ_t⟩⟨t_i|φ_t⟩∫dr₁dr₂dt₁dt₂(r₁, t₁)|p₂(r₂, t₂)|p₁(r₁, t₁)|t_i⟩⟨t_i|

where the result is obtained considering an independence of the two operators, possible only for linear approximations, since the two systems are considered adiabatic, meaning that they don’t evolve during the interaction with respect to the other system’s parameters. The time evolution of operators in the interaction representation can be simplified for the same reason, taking the form:

φ_p(r, t) = e^iH_ptφ_p(r, 0)e^-iH_tt

Such a linear approximation means that we need not a long flux of incident particles, but a single propagating layer is enough (not a single particle since we want to analyze the entire volume). The exponential operators will apply to the border Heisenberg kets, obtaining:

⟨p_f|φ_p(r, t)|p_i⟩ = ⟨p_f|φ_p(r, 0)|p_i⟩e^iωt

Since the contribution of φ depends only on the difference t - t₁, i.e., the system is automatically stationary, its evolution doesn’t depend on a specific time reference:

⟨t⟩ = ∫e^iωt

making trivial the limit to infinite times. A further simplification is obtained by writing the initial and final states of the probe as plane waves:

⟨p_f|φ_p(r, 0)|p_i⟩ = (1/V)∫A(q, r)e^-ikf·re^ikr·rdr

Since φ is an observable, the contribution A(q) arising from the remaining term will become A(-q). Furthermore, by considering a total translational invariance of the probe even inside the medium (possible again due to linearity, which doesn’t change the picture of the input), we can write:

φ_p(r, 0) = e^-ip·rφ_p(0, 0)e^ip·r

In the end, the cross section relation takes the form:

∂σ/∂Ω∂ω = ρ(ω)∫∫e^-iq·re^iωt|A(q)|²dtdr

For continuous homogeneous media, the dependence on one of the spatial variables will disappear, obtaining a simpler form:

∂σ/∂Ω∂ω = ρ(ω)∫e^-iq·re^iωt|A(q)|²dt

where the time interval can be extended to infinity by inserting the switching on factor to the second power. We see that the cross section is linked to the Fourier transform of the correlation function of the operator φ between two space-time locations, which defines the selection rules of the process. This function simply highlights the influence of an initial state on the evolution of the operator along time and space. The response of a system is therefore studied using these correlations, for instance, neutrons couple with the density operator, and with them is possible to study the variation of the density in response to a density perturbation in another space-time location. The only difficulty is the fact that experimentally we obtain the correlation in the Fourier space, and a perfect inversion is impossible because of the limited interval of ω and q exchanged with the probe. If we have neutrons with average energy of 25 meV, the maximum exchanged momentum is 10^-8 cm^-1, which is great enough to analyze the entire BZ, but the same is not valid for electromagnetic radiation. Furthermore, if we analyze only a small solid angle dΩ around a propagation direction, we would find only a limited vectorial interval for q. In summary, our knowledge of the response will depend on the number of degrees of freedom our experiment is able to measure. If we were able to measure the entire solid angle, we would also find retro-diffused neutrons, exploiting a momentum range of 2π/a, i.e., the entire BZ, such that we can analyze responses with a spatial limit of λ = 10^-8 cm. By decreasing the solid angle measured, we can isolate just the macroscopic fluctuations, like sound. The same results are found for the energy integration over ω.

We conclude the section by exploiting the detailed balance principle. It gives the reversed probability of transition P_i→f, which will follow the same relations, with the exception of a different thermal occupation, in particular:

P_f→i = e^-βħωP_i→f

All the processes which admit a reversed transition will have a probability for it to happen (Stokes and anti-Stokes peaks are an example).

Neutron scattering

Spectroscopy using neutrons can give a lot of structural information for different media, ranging from crystals to molecules, liquids and magnetic structures. The main difficulties with this method are related to the generation of neutrons, which can be done in two ways: by nuclear fission or by spallation. Neutrons obtained through the former method are very energetic (in the MeV range) and can resolve spatial distances of the order of a thousand of an Å, and are useless for the study of condensed matter. Therefore, these particles are thermalized, to thermal energies 300 K = 25 meV λ = 1 Å, impacting them to a target of the order of the atomic nucleus (graphite, in general). The interaction of neutrons with matter is difficult to exploit, but, for thermal neutrons, we can write the Fermi pseudo-potential as:

V(r, t) = (2πħ/μ)Σδ(r - R_l(t))b_l

where b is the scattering parameter for each atom in the medium. The ”pseudo” word is due to the wavelength of the neutrons, which is too long to see the single core elements, interacting instead with the total core as a homogeneous element. The scattering parameter is in general complex, but in absence of absorption, it is just real. Furthermore, it depends on the mass number (neutrons + protons, i.e., isotopic state) and from the nuclear total spin. In the linear response theory, we get:

W = (1/N)∂σ/∂Ω∂ω = (8π²k_i/k_f)∫∫dtdr₁e^{-iq(r₁-r₂)}e^iωt⟨[b_lδ(r₁ - R_l(t))][b_lδ(r₂ - R_l(0))]⟩

The scattering parameter can’t be put outside the bracket since it will depend on the state of the system. Furthermore, it will have statistical fluctuation such that:

b = b̄ + δb_l

In general, since there are no isotopic aggregations (random distributions of isotopes throughout the medium), the scattering parameters product will be independent from the density operators, obtaining:

C(r₁, r₂, t) = Σ⟨(b̄ + δb_l)(b̄ + δb_l)[δ(r₁ - R_l(t))][δ(r₂ - R_l(0))]⟩

from which we obtain two contributions, known as coherent (collective modes) and incoherent (single particle mode) responses, respectively:

C(r₁, r₂, t) = Σ⟨ρ(r₁, t)ρ(r₂, 0)⟩ + Σ⟨δ(r₁ - R_l(t))δ(r₂ - R_l(0))⟩

The density-density correlation can be written as a function known as Van Hove function:

G(r₁, r₂, t) = G_s(r₁, r₂, t) + G_d(r₁, r₂, t)

where the two contributions (obtained in a classical regime) are the self and distinct parts. The latter can be expressed through a pair distribution function g:

G_d(r₁, r₂, t) = Σ⟨δ(r₁ - R_l(t))δ(r₂ - R_l(0))⟩/⟨ρ(r₁, t)⟩