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Homework #6 Solutions Spring 2014 2
1stZone: 222πa2ndZone: kf=2πa3rdZone: kf=522πa4thZone: kf=5262πa5thZone: kf=1022πa6thZone: kf=1022πa7thZone: kf=1022πaBased on this, our table for the first 13 electrons looks like: Brillouin Zone m kf/2πa1 2 3 4 5 6 7 1 .399 P E E E E E E 2 .564 P P E E E E E 3 .691 P P E E E E E 4 .798 F P P P E E E 5 .892 F P P P E E E 6 .977 F P P P E E E 7 1.055 F F P P P E E
8 1.128 F F F P P P E 9 1.197 F F F F P P E 10 1.262 F F F F P P P 11 1.323 F F F F P P P 12 1.382 F F F F P P P 13 1.438 F F F F P P P At m=12 electrons, the Fermi radius in units of 2πais 1.382. The shortest distance to the edge of the 7thBZ (i.e. the point at which the eighth BZ would begin to be filled) occurs along the diagonal of the reciprocal lattice (see Figure 9.15). At that point, the radius of the Fermi circle would be: kf=2πa2=1.4042πaThis Fermi wavevector is generated between m=12 electrons and m=13 electrons. Thus for m=12 electrons only zones less than or equal to 7 are occupied while for m=13 electrons we begin to fill the eight and higher zones. 2. a.