How do you observe "silent" quantum vibrations? In the theory of quantum vibrations (aka phonons) it is useful to divide up the vibrational normal modes of a crystal based on their representation within the symmetry group of the crystal.  The representations signify how the phonon will transform under symmetry operations like reflection, rotation, inversion, etc. For example, a particular phonon might have the $A_{1u}$ representation in a cubic crystal group $O_h$, and the subscript $u$ would tell you the phonon is antisymmetric under inversion.
Based on the representation, one can usually assign representations of phonons as infrared- or Raman-active based on their symmetry. In a nutshell, the former requires something that is antisymmetric under inversion, while the latter requires inversion symmetry. This assignment is useful in actual experiments that use infrared absorption or Raman scattering to predict which phonons should be visible.
However, not all representations can be classified as infrared- or Raman-active. In crystals without inversion symmetry, some representations are both infrared- and Raman-active, while others are neither and are classified as silent modes (see Chapter 8.8 of Group theory by Dresselhaus page 160).
My question is following: is there a general way using light to observe silent phonons? If there is no such method using light, how can these silent modes be observed?
I do want to emphasize the word "general" in my question, as it might be possible to observe some silent modes in special cases. What I am interested in is a systematic method for routinely observing all these so-called silent modes.
 A: One of the techniques, which allow you to probe silent modes is the Hyper-Raman scattering. This method is quite similar to the usual Raman one, but, it involves a three-photon process: two photons with energy $\omega _i$ are exciting the system, and one photon with energy $2\omega _i \pm \omega _{phonon}$ is emitted. It is inherently nonlinear and involves a quadratic term in the expansion of the induced polarization of the crystal.
For the detailed description of the experiment/theory, please, check [HYPER-RAMAN SCATTERING BY VIBRATIONAL EXCITATIONS IN CRYSTALS, GLASSES AND LIQUIDS by V.N. DENISOV, B.N. MAVRIN and YB. PODOBEDOV in PHYSICS REPORTS (1987)].
However, if you are lucky enough you could try to observe just the second-order Raman scattering at twice the energy of expected $A_{u}$ mode, since (e.g. for mmm point group) $A_{u} \otimes A_{u}$ = $A_{g}$
A: The reference Mr.Eight recommended (thanks!) gives detailed analysis of the selection rules for HRS; for example:
"In HRS of centrosymmetrical systems only the odd modes (the “u” symmetry) may be active.
Since in the RS spectra of these systems the even modes are only active, the rule of mutual exclusion holds in the RS and HRS spectra of centrosymmetrical systems."
So one should not automatically suppose that all modes will be active in the hyper-Raman scattering.
