Potassium dihydrogen phosphate, KH PO (KDP)

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Condensed Matter, Materials & Soft Matter · Crystal structure, bonding and diffraction

Solid State Physics, Ch Symmetry and the classification of crystals

Problem

Potassium dihydrogen phosphate, KH2_2PO4_4 (KDP), crystallises in the tetragonal point group 4ˉ2m\bar42m. Its eight symmetry operations are: the identity; a four-fold roto-inversion 4ˉ\bar4 about the cc axis, applied once and three times (4ˉ\bar4, 4ˉ3\bar4^3); the proper two-fold rotation about cc that equals 4ˉ2\bar4^2; two-fold rotations about the aa and bb axes; and two mirror planes containing cc and bisecting the angle between aa and bb. ((a)) Is 4ˉ2m\bar42m centrosymmetric? Using the rule that all noncentrosymmetric point groups except 432432 are piezoelectric, decide whether KDP can be piezoelectric. ((b)) Does 4ˉ2m\bar42m consist of proper rotations only? Decide whether KDP crystals can be grown in two distinguishable (left- and right-handed) forms. ((c)) The ten polar point groups are 1,2,m,mm2,3,3m,4,4mm,6,6mm1,2,m,mm2,3,3m,4,4mm,6,6mm. Is 4ˉ2m\bar42m one of them? Decide whether KDP can be pyroelectric. ((d)) The gyration tensor gijg_{ij} is symmetric and axial, g′=(det⁡R) RgRTg'=(\det R)\,RgR^{\mathsf T}. Taking xx, yy, zz along aa, bb, cc, with 4ˉ\bar4 sending (x,y,z)(x,y,z) to (y,−x,−z)(y,-x,-z), find which components of gijg_{ij} survive every operation of 4ˉ2m\bar42m. For light travelling along a unit direction n^\hat{\mathbf n} the optical rotation is proportional to n^igijn^j\hat n_i g_{ij}\hat n_j. Can KDP be optically active, and for light travelling along which of aa, bb and cc?

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