Theory for Powder Raman

The Powder Raman calculation predicts non-resonant Raman spectra from Raman tensors calculated by a solid-state quantum mechanical program. The calculation follows the Placzek approximation, in which the intensity of a first-order Stokes line is determined by the derivative of the polarisability with respect to the normal-mode coordinate. For a mode \(m\), incident laser frequency \(\nu_L\), scattered frequency \(\nu_S = \nu_L - \nu_m\), incident polarisation \(\mathbf{e}_L\), and detected polarisation \(\mathbf{e}_S\), the scattering strength is proportional to

(75)\[S_m = \frac{\nu_S^4}{c^4} \frac{n(\nu_m)+1}{8\pi^2\nu_m} \left|\mathbf{e}_S^{T}\tensorbf{R}^{(m)}\mathbf{e}_L\right|^2\]

where \(n(\nu_m)\) is the Bose-Einstein occupation factor. PDielec reports Raman intensities in arbitrary units, so the most important outputs are the relative mode strengths and the changes introduced by particle shape, local fields, polarisation geometry and linewidth.

Raman Tensor

Every Raman output reader returns the bulk dielectric-derivative tensor

(76)\[\tensorbf{R}^{(m)}_\epsilon = \sqrt{V_{cell}} \frac{\partial \tensorbs{\varepsilon}_\infty}{\partial Q_m}\]

where \(V_{cell}\) is in \(\text{Å}^3\) and \(Q_m\) is in \(\text{Å}\) times the square root of amu. The tensor units are \((\text{Å}/\mathrm{amu})^{1/2}\). The reader already includes \(\sqrt{V_{cell}}\); the powder and crystal calculations apply no additional cell-volume factor. A source polarizability-volume derivative \(R_\alpha=\partial[V_{cell}(\varepsilon-I)/(4\pi)]/\partial Q_m\) is converted on read using \(R_\epsilon=4\pi R_\alpha/\sqrt{V_{cell}}\). Activities in \(\text{Å}^4/\mathrm{amu}\) are display quantities, obtained by multiplying the internal activities by \(V_{cell}/(16\pi^2)\).

For particles whose Raman tensor does not depend on shape, the powder average can be written in terms of rotational invariants of \(\tensorbf{R}\). For one mode,

(77)\[\begin{split}\alpha &= \frac{1}{3}\left(R_{xx}+R_{yy}+R_{zz}\right) \\ \tensorbf{\gamma} &= \frac{1}{2}\left(\tensorbf{R}+\tensorbf{R}^{T}\right) - \alpha\tensorbf{1} \\ \tensorbf{\kappa} &= \frac{1}{2}\left(\tensorbf{R}-\tensorbf{R}^{T}\right)\end{split}\]

and

(78)\[\begin{split}\alpha^2 &= \alpha\alpha^* \\ \gamma^2 &= \frac{3}{2}\sum_{ij}\gamma_{ij}\gamma_{ij}^* \\ \kappa^2 &= \frac{3}{2}\sum_{ij}\kappa_{ij}\kappa_{ij}^*\end{split}\]

The usual parallel (VV), crossed (VH), and unpolarised powder strengths are then

(79)\[\begin{split}I_{VV} &\propto 45\alpha^2 + 4\gamma^2 \\ I_{VH} &\propto 3\gamma^2 + 5\kappa^2 \\ I_{total} &\propto 45\alpha^2 + 7\gamma^2 + 5\kappa^2\end{split}\]

For non-chiral, non-magnetic Raman scattering the antisymmetric term is normally zero.

Local Field Correction

For powder infrared calculations PDielec treats each crystallite as a small inclusion in a non-absorbing matrix. The powder Raman calculation uses the same macroscopic picture when local-field effects are requested. The particle is assumed to be much smaller than the laser wavelength, so the field inside the particle is uniform. For an inclusion with optical permittivity \(\tensorbs{\varepsilon}_i\), a matrix permittivity \(\varepsilon_e\), and a depolarisation tensor \(\tensorbf{L}\), the internal field is

(80)\[\mathbf{E}_i = \tensorbf{N}\mathbf{E}_e\]

with

(81)\[\tensorbf{N} = \left[ \tensorbf{1} + \frac{1}{\varepsilon_e} \tensorbf{L} \left(\tensorbs{\varepsilon}_i-\varepsilon_e\tensorbf{1}\right) \right]^{-1}\]

For a sphere \(\tensorbf{L}\) has diagonal elements of \(1/3\). Other ellipsoidal shapes use the same depolarisation tensor as the powder infrared effective-medium theory.

For a reciprocal dielectric, the effective bulk Raman tensor includes one incident and one scattered optical local-field factor:

(82)\[\tensorbf{R}^{(m)}_{eff} = \tensorbf{N}(\nu_S)^T \left[\tensorbf{R}^{(m)}_{\epsilon,0}+\Delta\tensorbf{R}^{(m)}_{\epsilon,EO}\right] \tensorbf{N}(\nu_L)\]

The transpose is ordinary, including for complex reciprocal permittivities. The current powder implementation uses the same optical permittivity at both frequencies. \(R_{eff}\) retains the bulk \(R_\epsilon\) normalization; it is not an extensive particle dipole-polarizability derivative.

Particle Frequencies

In ionic crystals the particle boundary conditions can also shift the vibrational frequencies. The phonon-induced polarisation is

(83)\[\mathbf{P}_{ph} = \frac{1}{V}\tensorbf{Z}^{mw}\mathbf{x}\]

where \(\tensorbf{Z}^{mw}\) is the mass-weighted Born charge tensor and \(\mathbf{x}\) is the vector of mass-weighted atomic displacements. The depolarisation field generated by this polarisation applies a restoring force to the ions. This gives an effective particle dynamical matrix

(84)\[\tensorbf{D}^{particle} = \tensorbf{D}^{TO} + \frac{1}{\epsilon_0\varepsilon_e V} \left(\tensorbf{Z}^{mw}\right)^T \tensorbf{N}_{bg}\tensorbf{L}\tensorbf{Z}^{mw}\]

where \(\tensorbf{D}^{TO}\) is the transverse-optic dynamical matrix and \(\tensorbf{N}_{bg}\) is the internal-field tensor constructed from the background permittivities. Diagonalising \(\tensorbf{D}^{particle}\) gives particle-mode frequencies and eigenvectors appropriate to the selected crystallite shape.

In the SI expression above Born charges are in coulombs and masses are in kilograms. The implementation uses charges in electrons, masses in electron-mass units, volume in \(\text{Bohr}^3\), and the prefactor \(4\pi/(\varepsilon_e V)\). The same electrostatic kernel \(K=N_{bg}L/\varepsilon_e\) determines the particle EO correction:

\[\Delta R^{(m)}_{\epsilon,EO,ij} =-8\pi\sum_l\widetilde\chi^{(2)}_{ijl}(K Z^{mw}u_m)_l .\]

Here \(\widetilde\chi^{(2)}\) is the cell-dependent internal reader value, not the raw susceptibility in pm/V. Its conversion is specified in Theory for Crystal Layer Raman. Both the mechanical Raman tensor and the mode charge must use the same particle eigenvector. No bulk propagation direction is needed for this ellipsoid boundary-value model. It describes a quasistatic particle in a lossless scalar host; it is not a bulk directional LO/TO powder average. Matrix=none bypasses particle and optical-field corrections; use a host with permittivity one to describe an isolated particle in vacuum.

Orientation Averaging

For a crystallite in orientation \(g\), the particle Raman tensor in the laboratory frame is

(85)\[\tensorbf{R}^{(m)}_{lab}(g) = g\tensorbf{R}^{(m)}_{eff}g^T\]

The powder strength is the orientational average over all rotations,

(86)\[\left<S_m\right>_{powder} = \int_{SO(3)} S_m(g)\,dg\]

For spherical particles this average reduces to the invariant expressions above. For more general particle tensors, PDielec samples the rotation group and averages the fixed laboratory polarisation geometry over many crystallite orientations.

If Born-charge or Hessian data are unavailable, PDielec retains the supplied TO frequencies and Raman tensors and omits the particle frequency/EO correction. Optical local fields still apply. For fixed relative shape/crystal axes in an isotropic host, the effective tensor rotates as one rank-two tensor, so the analytic invariants also supply the nonspherical fallback. Missing polar data therefore do not imply zero Raman scattering.

Both analytic and sampled paths exclude nonfinite, nonpositive and above-laser mode frequencies before evaluating the Stokes weight, as well as modes below the acoustic cutoff. This does not remove unstable frequencies from the reader.

The broadened Raman spectrum is formed as a sum over active modes,

(87)\[I(\Delta\nu) = \sum_m \left<S_m\right>_{powder} \frac{\sigma_m}{(\Delta\nu-\nu_m)^2+\sigma_m^2}\]

where \(\sigma_m\) is the Lorentzian half-width at half maximum.