Q 15 state blochs theorem and show that the wave

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q.1.5State Bloch’s theorem and show that the wave functionψkprq “ÿReik¨Rφpr´Rq(whereRare the points of the lattice andφpr´Rqis the ground-statewave function of an electron bound to the nucleus on siteR) is of theform required by Bloch’s theorem.[7 marks] 6
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1.6Consider barium titanate (BaTiO3), which is a ferroelectric material. Athigh temperature it has a simple cubic Bravais lattice, where Ba atomslie at the cube corners, Ti atoms at the cube centres and O atoms at thecentres of the cube faces. Sketch the unit cell and indicate the numberof atoms in the basis. How many acoustic and optical branches in thephonon dispersion curves for barium titanate would you expect?[6 marks] 1.7The figure below shows the measured phonon dispersion data for thefcc metal nickel.(a) The left hand panel shows dispersion from the centre of the zone tothe point (001), commonly designated by the X point. Why are theretwo sets of data in this panel? What is indicated by the abbreviateddesignations “L” and “T”? Are either of these curves degenerate?(b) Why does the curve labelled “T” fall below the “L” curve?(c) Make an estimate of the ratio of the sound speeds associated withthe “T” and “L” modes.(d) In the second panel from the right, connecting (000) to (011), thereare three curves. Why is this?(e) Phonon frequencies at (001) and (011) appear to be the same. Are(001) and (011) equivalent points in the Brillouin zone? 7
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[10 marks]
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