Question B1 Palladium oxide is a semiconductor with a primitive tetragonal lattice, with unit cell parameters a = 3.02 Å and c = 5.31 Å. The motif consists of Pd atoms at (0, ½, ½) and (½, 0, 0) and O atoms at (0, 0, 1) and (0, 0, 1). a) Draw a plan (2D projection) of the unit cell, roughly to scale, looking down the a axis and labelling the b and c axes. Show the Pd atoms as large circles and the O atoms as small circles. Label each atom with its a fractional coordinate. b) State the coordination number and the coordination geometry (e.g., octahedral) of the Pd and O atoms. c) At the gamma point, the three vibrational modes lowest in frequency in fact have zero frequency. (i) Explain briefly what is meant by the gamma point in this context. (ii) Explain briefly why these three modes have zero frequency. The two vibrational modes that are next-lowest in frequency are also degenerate (i.e., have the same frequency). Each of them involves one Pd atom moving in one direction while the remaining three atoms in the unit cell move in the opposite direction. (iii) On two separate diagrams, sketch the motion of the atoms in these two modes. (Hint: low-frequency modes will be those that change the nearest-neighbour distances as little as possible.) d) At the wavevector k = (0, 1, 0), the two lowest-frequency modes involve only one Pd atom moving, while the remaining three atoms are stationary. These modes are similar in frequency but not degenerate. On two separate diagrams, sketch the motion of the atoms in these two modes. Explain which has the higher frequency.

Question B1 Palladium oxide is a semiconductor with a primitive tetragonal lattice, with unit cell parameters a = 3.02 Å and c = 5.31 Å. The motif consists of Pd atoms at (0, ½, ½) and (½, 0, 0) and O atoms at (0, 0, 1) and (0, 0, 1). a) Draw a plan (2D projection) of the unit cell, roughly to scale, looking down the a axis and labelling the b and c axes. Show the Pd atoms as large circles and the O atoms as small circles. Label each atom with its a fractional coordinate. b) State the coordination number and the coordination geometry (e.g., octahedral) of the Pd and O atoms. c) At the gamma point, the three vibrational modes lowest in frequency in fact have zero frequency. (i) Explain briefly what is meant by the gamma point in this context. (ii) Explain briefly why these three modes have zero frequency. The two vibrational modes that are next-lowest in frequency are also degenerate (i.e., have the same frequency). Each of them involves one Pd atom moving in one direction while the remaining three atoms in the unit cell move in the opposite direction. (iii) On two separate diagrams, sketch the motion of the atoms in these two modes. (Hint: low-frequency modes will be those that change the nearest-neighbour distances as little as possible.) d) At the wavevector k = (0, 1, 0), the two lowest-frequency modes involve only one Pd atom moving, while the remaining three atoms are stationary. These modes are similar in frequency but not degenerate. On two separate diagrams, sketch the motion of the atoms in these two modes. Explain which has the higher frequency.

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