The centosymmetric crystals are suitable for the third order NLO properties. But why it is not exihibt the SHG?
Even-order nonlinear optical effects are forbidden in materials with inversion symmetry.
For SHG to occur you need an "even" response of the polarization P to the field E, i.e. the polarization will have the same direction for a certain field E and an opposite (in direction) field -E. It is equivalent to say that the function P(E) must have a component in E^2 in its Taylor expansion. For (obvious) symmetry reasons this is not possible if your lattice is centrosymmetric.
There are several approaches to answer this question. Centrosymmetric materials such as Si has the property that it has a centre of inversion so that the potential satisfy:
V(r) = V(-r)
or is symmetric with respect to its centre of inversion. A charge that is driven by an alternating electric field such as those coming from a laser will experience the same shift when oscillating to the left and right. Now, the nonlinear polarization (total dipole per volume) produced by SHG is given by
P_shg = /chi_shg E(omega) E(omega)
Now because the external field is oscillating then it changes sign and the polarization should also changes sign
-P_shg = /chi_shg -E(omega) -E(omega) = /chi_shg E(omega E(omega) = P_shg
but -P_shg = P_shg this is only possible if /chi = 0
Indeed it can be shown using group theory that the third rank tensor /chi _shg for centrosymmetric material such as Si which is Oh is indeed zero.
The other approach is by starting from a symmetric potential and find the solution of the Lorentz force equation. Due to anharmonic effects the solution should be found using the standard Reyleigh Schrödinger perturbation theory where it can be show that /chi_shg should be zero.
For THG this is not the case because P_thg is proportional to E(omega)^3 not E(omega)^2 as in the previous cases. So a change in the E(omega) sign does not imply that the fourth rank tensor /chi_thg should be zero. This can be shown also by grozp theory where the fourth rank tensor related to THG for centrosymmetric material is indeed not zero contrary ro the third rank tensor for SHG.
More on this can be found from our publications in JOSA B, 2014, 2015, 2016 please check my publication list.
Another approach is using quantum mechanics. SHG by dipoles is forbidden from within the bulk of centrosymmetric materials due to parity symmetry. This is related to forbidden transition from symmetric to symmetric states when the electron is relaxating back from their virtual/real excited states to the round states. For SHG the final transition is from levels with the same state which are forbidden by parity symmetry this is not the case for THG.
Pay attention however, that SHG from dipoles at the surface of centrosymmetric materials is not forbidden due to symmetry breaking. Also Quadrupole contribution to SHG from within the bulk is also allowed since it is a fourth rank tensor effect
P_shgquadrupole = X_shgquadrupole E(omega) nabla E(omega)
I hope this will help you.
But some Researchers mentioned that by introducing some non-centrosymmetric properties in organic molecule by hydrogen bonding , there is possibility of SHG Studies in Organic Molecules too.
Donor-acceptor type dyads, which are best seen as centrosymmetric, turn non-centrosymmetric, upon strong ICT that operates from D to A and lead to a change in dipole moment, which is one of the prerequisites for a molecule to be NLO active (Ouder and Chemla). Such molecules would therefore show large first-hyperpolarizability, beta (SHG).@Priya Dharshini
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