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1. You may answer as many parts of question 1 as you wish. All work you do will be assessed and the marks totalled but the maximum total credit for this question will be 20 marks. (a) (i) How many lattice points are there per unit cell in simple cubic, body- centred cubic and face-centred cubic lattices? (ii) What is the spacing between second neighbour lattice points (in units of the lattice parameter a)inthe cubic and face centred cubic lattices? (b) x rays of wavelength are diffracted from crystal as shown schematically in the figure below 0 is the X rays incident angle. Determine the Bragg law for the diffracted beam. Explain the difference between how the two diffracted (-rays shown in the figure interfere when the Bragg law is fulfilled and when it is not fulfilled X rays Crystal planes sin0 CRYSTAL STRUCTURE 2A) Write the Miller indices of the planes drawn in the figure below: Z Z Z y y y X X x Z Z x y y 3. SCHRODINGER EQUATION: FREE PARTICLES IN A ID BOX There are 8 free electrons in 1D box of length I 0x< The Schrodinger equation for a free particle in box is given by: 12 d²w 2m dx2 Ev(x) (a) Define the left- -hand side term, § and E in this Schrodinger equation. What is the main difference between classical approach and quantum mechanical one, as far as the energy of free particle in 2 1D box is concerned? (b) Write down the boundary conditions for 'P(x) that are compatible with the physical problem discussed Find the only solution of this Schrodinger equation (i.e., the normalized wave function 'P(x)) that is compatible with the boundary conditions for '(P(x). Plot P(x) for the first quantum numbers. n-l, and 3. (c) Whatis the physical interpretation of 'P(x)? (d) Calculate the energy eigenvalues En and write down the principal quantum number n and the spin quantum number ms that is. the pair of quantum numbers (n, ms), for all 8 electrons. At which positions (x takes values between and L) is the probability density maximum for states with principal quantum number n=4? (e) Free particles in 1D box constitute the simplest possible model for electrons in metals. What the main simplifying element considered in this model? How will the Schrodinger equation change in more realistic model? 4.NEARLY FREE ELECTRON MODEL AND THE HALL EFFECT Nearly free electron model (a) Write down the two solutions of the Schrodinger equation for the case when Bragg reflection condition Ak =G is fulfilled (here is the wave vector and Gis vector in the reciprocal space) Explain the difference between them as far as their physical interpretation as probability distributions for the electrons is concerned What the solution of the Schrodinger equation when Bragg reflection condition Ak =G is not fulfilled. (b) Explain qualitatively the origin of the energy gap in the nearly free electron model. Sketch graph of the energy versus wave vector for nearly free electron model. Explain how this differs from the free electron model.

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