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Even a crystal with a center of symmetry can have the properties of a fourth rank tensor, such as electrostriction or second-order electro-optical effects. Explain this with a tensor transformation equation?

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The angular frequency of vibrations in a one-dimensional monatomic crystal are given by where w (k) = 4K m sin k = ka 2 2π λ is the wavenumber of the vibration pattern in terms of its wavelength >, the spring constant between neighbouring atoms, and m the mass of each atom. (i) Show that the frequency does not change if one shifts 2π k→ k +. a (ii) Briefly explain the physical reason for this periodicity.