Using polarized Raman spectroscopy to study the stress gradient in mineral systems with anomalous birefringence

Abstract

Polarized Raman spectroscopy was applied to garnet hosts which exhibit anomalous birefringence around inclusions of zircon and quartz to elucidate the spatial distribution of the anisotropic strain fields in the vicinity of the host-inclusion boundary. We show that there is a direct relationship between the stress-induced birefringence and the Raman scattering generated by the fully symmetric phonon modes (the A_1g modes in cubic crystals). Our experimental results coupled with selected finite element models show that the ratio between the measured Raman peak intensity collected in cross and parallel polarized scattering geometries of totally symmetric modes represents a useful tool to constrain the radial stress profile in the host around the inclusions. Further, we demonstrate how group-theoretical considerations and tensor analysis of the morphic effect (external-field-induced change of the symmetry) on the phonons and the optical properties of the host can help to derive useful information on the symmetry of the stress field. Finally, we show experimentally that, under the same amount of applied stress, this approach is more sensitive than the commonly used approach of measuring differences in phonon frequencies and provides better opportunities to map the spatial variations of strain. This approach is an alternative technique to study structural phenomena associated with anomalous birefringence in host crystals surrounding stressed inclusions and could be applied to other systems in which similar optical effects are observed.

Appendix

As discussed in the main text, each Raman-active phonon mode m in a crystal will have its own Raman polarizability tensor $\alpha^{m}$ and the intensity of the Raman scattering peak from the mode depends on Eq. (2) which is repeated here:

$$I^{m} \propto \left| {e_{k}^{i} \alpha^{m}_{kl} e_{l}^{s} } \right|^{2} .$$

(13)

Thus the intensity of the Raman scattering peak from the mode depends on the orientation of the electric-field unit vectors $e^{i}$ and $e^{s}$ of both the incident and scattered light. Experimentally, the directions of the electric-field vectors of the incident and scattered light are defined by the polarisers placed in the incident and scattered beam paths, so $e^{i}$ and $e^{s}$ (in the crystal) depend on the orientation of the crystal with respect to the polarisers in the spectrometer system. Written out in the terms of the individual components of the tensors, Eq. (13) becomes:

$$I^{m} \propto \left( {e_{1}^{i} \alpha^{m}_{11} e_{1}^{s} + e_{1}^{i} \alpha^{m}_{12} e_{2}^{s} + e_{1}^{i} \alpha^{m}_{13} e_{3}^{s} + e_{2}^{i} \alpha^{m}_{21} e_{1}^{s} + e_{2}^{i} \alpha^{m}_{22} e_{2}^{s} + e_{2}^{i} \alpha^{m}_{23} e_{3}^{s} + e_{3}^{i} \alpha^{m}_{31} e_{1}^{s} + e_{3}^{i} \alpha^{m}_{32} e_{2}^{s} + e_{3}^{i} \alpha^{m}_{33} e_{3}^{s} } \right)^{2} .$$

(14)

With $\alpha^{m}_{kl} = \alpha^{m}_{lk}$ for first-order Raman scattering.

Symmetry constraints on the Raman polarizability tensor for A _1g modes

Which components of the Raman polarizability tensor are zero, and the constraints on the values of the non-zero components, are defined by the symmetry of the mode. As shown in Table 2, for the A_1g modes in cubic crystals cubic crystals with point symmetry $m\bar{3}m$, $\alpha^{{A_{1g} }}_{11} = \alpha^{{A_{1g} }}_{22} = \alpha^{{A_{1g} }}_{33}$, and $\alpha^{{A_{1g} }}_{12} = \alpha^{{A_{1g} }}_{13} = \alpha^{{A_{1g} }}_{23} = 0$, the same as the constraints on the $\varvec{B}$ tensor for optical birefringence. This is the reason why the behaviour of the intensities of A_1g modes in cubic crystals under stress follows that of the optical birefringence. And the expression for the intensity for A_1g mode in cubic crystals becomes:

$$I^{{A_{1g} }} \propto \left( {e_{1}^{i} \alpha^{{A_{1g} }}_{11} e_{1}^{s} + e_{2}^{i} \alpha^{{A_{1g} }}_{22} e_{2}^{s} + e_{3}^{i} \alpha^{{A_{1g} }}_{33} e_{3}^{s} } \right)^{2} = \left( {\alpha_{11}^{{A_{1g} }} } \right)^{2} \left( {e_{1}^{i} e_{1}^{s} + e_{2}^{i} e_{2}^{s} + e_{3}^{i} e_{3}^{s} } \right)^{2} .$$

(15)

This equation shows that the symmetry of the Raman polarizability tensor of the A_1g modes in cubic crystals means that the intensity is independent of the direction of propagation of light through the crystal, and depends only on the relative orientation of the two polarisers to one another. We can now choose any convenient coordinate system for these polarisers and the experiment. For this example, we choose the incident beam to be along the z-axis, and the polarisation direction to be along the x-axis. Thus $e^{i}$ = [1 0 0]. If one makes measurements in HH polarisation, the polarisation of the scattered beam is parallel to the incident beam, so $e^{i}$ = $e^{s}$, and Eq. (14) for the intensity becomes:

$$I^{m} \propto \left( {\alpha^{m}_{11} e_{1}^{i} e_{1}^{s} } \right)^{2} = \left( {\alpha_{11}^{m} } \right)^{2} .$$

(16)

By contrast, when the polarisations of the incident and scattered radiation are ‘crossed’, (i.e. mutually perpendicular and denoted VH), then $e^{s}$ = [0 1 0] and the intensity given by 14 is zero. Thus, in cubic crystals, A_1g modes can be observed by polarised Raman spectroscopy with HH polarisation, but are not observed with VH polarisation.