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Why does second order nonlinear polarization occur only in crystal materials with a non-centrosymmetric crystal structure? (Nonlinear effects at crystal surfaces are an exception). Why does third order polarization occur in basically all media? The answer, found here did not help (at least for now).

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It's because centrosymmetric crystals prohibit a nonzero second-order nonlinear polarization by the centrosymmetry (I will call it parity, sorry), $$ P: (x,y,z) \to (-x,-y,-z).$$ Work with the formula $$ P_i = A_{ij} E_{j} + B_{ijk} E_{j} E_{k} $$ where I renamed the coefficients $A,B$ relatively to the usual conventions. Under the parity symmetry $P$, the vectors and tensors map as $$ P_i\to -P_i, \qquad E_i\to -E_i $$ because they have an odd number of indices and they're polar vectors, not axial vectors. Centrosymmetric crystals have the internal constants $A,B$ constant under the parity symmetry $$A_{ij} \to A_{ij}, \qquad B_{ijk} \to B_{ijk}. $$ It's important that there's no sign in the transformation of $B$, either (despite the odd number of three indices): we literally get the same crystal back by the parity symmetry so there can't be any changes of its material constants, not even the tensor-valued ones.

When the transformation laws are applied, the original equation transforms to $\to$ $$ -P_i = -A_{ij} E_{j} + B_{ijk} E_{j} E_{k} $$ which contradicts the original formula unless $B_{ijk}=0$. So the nonlinear term must vanish because of the transformation rules under parity.

Only odd powers of $E_i$ might appear on the right hand side with a nonzero coefficient. In particular, terms that are cubic in $E_i$ are exactly as allowed by the symmetry as the terms that are linear in $E_i$; for example, $E^2\cdot E_i$ has pretty much identical transformation rules under any parity-like symmetry as $E_i$, so if $E_i$ is allowed, so must be the $E^2\cdot E_i$ terms. In fact, the lowest order (linear) terms might be set to zero by some special tuning but it's virtually impossible to get rid of the cubic terms which have a higher number of independent tensor components.

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