Question

Al is metal, silvery in colour, with a resistivity of 2.65 μΩ.cm at room temperature. it has the so-called face centred cubic (FCC) structure, in which the fractional coordinates of the atoms within the (conventional cubic, side length of a 4.046 A) unit cell are: (0,0,0), (12%,0), (12,0,%), (0,%,%) The intermetallic compound AbAu is bright purple in colour, has a melting point of 1060°C, is rather brittle, and has a resistivity of p-8 u2.cm.1 the repeating unit is also cubic with a side length we shall denote a), and has atoms at fractional coordinates: 4, 74, 74), (74,74,74), (74, 74, 74), (74, 74,4 74, 74,74),74,74,74 Au: (0,0,0), (%,%,0), (14,0,%), (0,%,%) 74,74,74, 74, 74,74 Fig shows simulated powder diffraction pattern profiles for three different samples using Cu Ka (X-ray) radiation with a wavelength of 1.54 Å. Powder diffraction is reminiscent of the polycrystalline diffraction patterm shown in lectures: the sample comprises many small crystals in random orientations and the resultant diffraction pattern only varies in one dimension 1.0 0.8 0.6 0.4 0.2 0.0 (A)2 20 25 35 40 45 26(degrees) (B)10 0.8 0.6 0.4 0.2 0.0 35 45 2θ (degrees) 0.8 2 0.6 0.2 0.0 35 50 20(degrees) Fig. 2: Simulated powder diffraction patterns. Note: Consistent with the standard experimental geometry , the horizontal axis is given in units of 20 rather than θ.) For questions (a)-(d), use the quantum free electron gas model a. Again assuming that each Al atom contributes three electrons to the electron gas, calculate the Fermi energy, EF, at 0 K for aluminium. Compare your answer with the measured Fermi level for aluminium of 11.7 eV and comment b. Using the measured Fermi level of .7 eV for aluminium, calculate the corresponding Fermi velocity. Assuming that Ef at 0 K is the same as Ef at 300 K, further calculate the average time between electronic collisions and the mean free path of electrons. Compare this with the nearest-neighbour atom distance and comment. In the quantum free electron gas model, the number of occupied states with energy between E and E+dE is given by C. Show that the probability for any given electron to have energy between E and E+dE is given by VE dE d. Making use of your answer to part (c), what is the probability that a randomly chosen "free" electron in aluminium will have (kinetic) energy between 12.0 eV and 12.05 eV at room temperature (i.e. T-300 K)? Note: an accurate answer requires the evaluation of an integral that you cannot do analytically Show your approximations and justifying why these are reasonable

Answer 1

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