hw1.solution - 1 Decorated cubic lattices Base-centered...

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1 Decorated cubic lattices Base-centered cubic is a Bravais lattice generated by { a ˆx , a ( ˆx + ˆy ) / 2 , ˆ z } . Side-centered cubic is not a Bravais lattice since it contains the vectors a ( ˆx + ˆ z ) / 2 and a ( ˆy + ˆ z ) / 2 but not the sum a ( ˆx / 2 + ˆy / 2 + ˆ z ). It is generated by a simple cubic lattice with basis { 0 , a ( ˆx + ˆ z ) / 2 , a ( ˆy + ˆ z ) / 2 } . Edge-centered cubic is not a Bravais lattice since it contains the vectors a ˆx / 2 and a ˆy / 2 but not the sum a ( ˆx + ˆy ) / 2. It is generated by a simple cubic lattice with basis { 0 , a ˆx / 2 , a ˆy / 2 , a ˆ z / 2 } . 2 Skipping alternate lattice points n i are all even . Displacements in the lattice are generated by { 2 ˆx , 2 ˆy , 2 ˆ z } . This is a simple cubic lattice with side 2 containing the origin. n i are all odd . This is a simple cubic lattice with side 2 containing the point (1,1,1). i n i is even . For a given lattice point, either all three coordinates are even, or exactly 1 coordinate is even, with the other two odd. Points of the first type form a simple cubic lattice with side 2. Points of the second type lie in the face centers of this lattice, forming an FCC lattice with side 2. i n i is odd . Either all three coordinates are odd, or exactly one coordinate is odd, with the other two even. Points of the first type form a SC of side two, and together with the second set this is again an FCC with side 2.
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