Infrared and Raman spectroscopy

Infrared spectroscopy and Raman scattering provide methods of mea­suring phonon frequencies that are complementary to neutron scatter­ing. We show that the use of electromagnetic radiation leads to con­straints that do not exist for neutron scattering, but which can be exploited. In this chapter we give an introductory description of the background theory and the experimental methods. We then outline some of the areas of application.

Introduction

Neutrons are complemented as probes of the dynamic behaviour of crystals by the quanta of electromagnetic radiation, photons. Having zero rest mass, pho­tons do not scatter from phonons in the same way that neutrons do; the differ­ences are highlighted by considering photon scattering from the viewpoint of neutron scattering.³⁵ We note from the start the numerical values of the impor­tant quantities. The photon angular frequency co and wave vector k are related by co = ck, where c is the velocity of light (3 x 10⁸ m s^_1). This linear relation contrasts with the quadratic relation for neutrons, equation (9.1). A change in energy of the photon due to the absorption or creation of a phonon will there­fore cause a linear change in its wave vector. If the change in frequency is 5 THz, say, for a typical optic mode, the corresponding change in wave vector Ak/2n will be | x 10~⁵ A^-1. This corresponds to a phonon wave vector that is very close to the Brillouin zone centre in a crystal of typical unit cell dimen­sions. These numbers demonstrate that photons will be scattered or absorbed only by phonons with very long wavelength.

We will consider first the case where the photon is absorbed to create a phonon of the same frequency and wave vector. Such an experiment will

involve shining a polychromatic beam of radiation on the sample, and measur­ing the frequencies at which absorption occurs. These will typically be in the infrared region of the electromagnetic spectrum, hence this technique is known as infrared spectroscopy.¹ Even when the conditions of energy and wave vec­tor conservation are met, there may be symmetry factors that prevent absorp­tion; in the language of scattering theory we would say that the cross section or scattering (structure) factor is zero, and in the language of spectroscopy we would say that the selection rules do not allow the absorption process.

We next consider a true scattering process, in which the photon is scattered with a change in frequency. In this case it is common to perform experiments using monochromatic radiation, usually lasers in the visible region of the electromagnetic spectrum. If we take an optic mode with a frequency co_Q that has a negligible dependence on wave vector, and consider the usual case where the light beam is scattered through an angle of 90°, the scattering equations are:

(0_f = (0_i±(0 o (10.1)

|Q|³⁶ = |k_;. ±k₇|² = kf +kj= c²(cof + a}) (10.2)

As with neutron scattering, photons can be scattered either by absorbing or creating a single phonon. This type of scattering is called Raman scattering,³⁷ named after one of its experimental discoverers.³⁸ Similar scattering pro­cesses can also occur that involve acoustic phonons; in this case the changes in the frequency of the light beam are so small that different experimental techniques are required. The scattering of photons by acoustic modes is called Brillouin scattering, in this case named not in honour of the experimental discovery of the effect but after the theoretical prediction³⁹ (Brillouin 1914, 1922).

We noted in the previous chapter that neutrons can be scattered by more than one phonon, but the conservation laws for multiphonon scattering are sufficiently slack that the scattering is not constrained to give peaks. Multiphonon scattering processes are also allowed in spectroscopy. However,

because the only phonons involved are those with small wave vector, the mul- tiphonon processes still lead to sharp peaks in the measured spectra. These are often called combination or overtone bands.

The interaction between photons and phonons, whether by scattering or absorption, can be treated in a number of ways. The macroscopic approach begins with Maxwell’s equations of electromagnetism as applied to a dielectric medium, and connects with the normal modes at the stage when the response of the crystal to the electric field of the photon is required. The microscopic approach is to consider scattering and absorption processes as involving transi­tions between different quantum states, and to calculate the appropriate matrix elements in a perturbation treatment. Both approaches require a considerable amount of background theory if one is to avoid merely quoting results; instead we adopt a more phenomenological approach to the understanding of the dif­ferent processes.