EMS172-2009-Homework6_Solutions

EMS172-2009-Homework6_Solutions - UNIVERSITY OF CALIFORNIA...

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EMS 172 Profs. Y. Takamura and A. Moulé, 2009 UNIVERSITY OF CALIFORNIA, DAVIS Department of Chemical Engineering and Materials Science EMS 172: Electronic, Magnetic, and Optical Properties of Materials Homework 6 Due Date: Tuesday, Nov. 17 th , 2009 in class Note: You must show all the steps you used to obtain your answer in order to receive full credit. 1. Spin Paramagnetism in Metals [11 pts] (a) Consider the two metals Au ([Xe] 4 f 14 5 d 10 6 s 1 ) and W ([Xe] 4 f 14 5 d 4 6 s 2 ) which reside on the same row of the period table. Looking at Table 14.1 from Hummel, despite the fact that both of these metals have unpaired electrons and therefore have a magnetic moment, the magnetic susceptibility for Au is negative and for W is positive. Why? For each, consider the contribution of spin para χ , dia (core electrons), and dia (valence electrons) – write the equation for each case . Your answer can be in terms of r (core electrons) and r (valence electrons) which are the average radii for the different electron orbitals, r (core electrons) < r (valence electrons), both ~ 1Å, and the number of core and valence electrons. Which term dominates for each metal? (b) In an Au-W alloy, we can sum the magnetic susceptibility of the two alloying elements. For delicate magnetic measurements, we need to create a material that has no net reaction to external magnetic fields, i.e. 0 = total . What ratio of Au to W must we use? According to Table 14.1 of Hummel, for Au, = -34.4x10 -6 while for W =77.7x10 -6 . In these metals the total magnetic susceptibility is due to the contributions of the spin paramagnetic susceptibility, the diamagnetic susceptibility from the core electrons, and the diamagnetic susceptibility from the valence electrons, according to the equations: V E Z F B spin para ) ( 0 2 μ = [1 pt] and mV r Z e dia 6 0 2 2 = [1 pt] For the diamagnetic susceptibility, we consider r and Z for the core and valence electrons separately. For Au and W, the diamagnetic susceptibility for the core electrons is the same, [1 pt] while the diamagnetic susceptibility for the valence electrons is larger for Au is larger than that of W because of the larger number of valence electrons and r (valence electrons). [1 pt] For the paramagnetic susceptibility, for Au, the valence band is nearly full, so Z(E F ) is small, and thus spin para is small [1 pt] . For W, the band is nearly half full so Z(E F ) is larger than in the case of Au, and thus spin para is larger than the case of Au. [1 pt] You can also consider that W has a half full 5d band, while Au has a half full 6s band, and in general the 6s bands are wide in energy, but narrow in Z(E), while the 5d bands are more narrow in energy and wide in Z(E) so that again Z(E F ) for W is larger than Z(E F ) for Au. For Au, the diamagnetic susceptibility dominates over the paramagnetic susceptibility so that the overall
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EMS172-2009-Homework6_Solutions - UNIVERSITY OF CALIFORNIA...

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