Exercises

  1. Consider a simple cubic (SC) unit cell and a simple hexagonal (SH) unit cell.  For both of these cells, Z = 1 and a = b = c = 2r, where r is the radius of the host atoms which occupy the eight lattice points.  Derive an exact mathematical expression for the ratio of cell volumes SC:SH, and report this irrational number to six significant figures.

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  2. Derive an exact mathematical expression for the packing efficiency of a simple cubic unit cell, and report this irrational number to six significant figures.

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  3. Derive an exact mathematical expression for the packing efficiency of a body centered cubic unit cell, and report this irrational number to six significant figures.

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  4. Derive an exact mathematical expression for the packing efficiency of a face centered cubic unit cell, and report this irrational number to six significant figures.

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  5. Prove that the inter-edge angles of the standard reduced cell in the FCC lattice are all 60o.

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  6. Derive an exact mathematical expression for the closest packed interlayer spacing (CPIS), and report this irrational number to six significant figures. [Hint: CPIS is the height of the trigonal pyramid which is a regular tetrahedron, all six edges of which are of length 2r.)

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  7. Derive an exact mathematical expression for the packing efficiency of a simple hexagonal cell, and report this irrational number to six significant figures.

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  8. Derive an exact mathematical expression for the packing efficiency of a hexagonal closest packed (AB) unit cell, and report this irrational number to six significant figures.

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  9. Derive an exact mathematical expression for the packing efficiency in the hexagonal unit cell (Z = 3) of the CCP (ABC) lattice, and report this irrational number to six significant figures.

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  10. Derive an exact mathematical expression for the length of the longest edge of the trigonal antiprism described in the discussion of CCP Coordination, and report this irrational number to six significant figures.
  11. How many distinctly different closest packed lattices are possible by stacking hexagonal closest packed planes in 7-layer repeat patterns?

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  12. Derive an exact mathematical expression for the size of the sphere which just fills the tetrahedral hole in a closest packed bilayer, and report this irrational number to six significant figures.  This radius will be a function of r, the radius of the host atoms composing the bilayer.

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  13. Derive an exact mathematical expression for the radius of the sphere which just fills the octahedral hole in a closest packed bilayer, and report this irrational number to six significant figures.  This radius will be a function of r, the radius of the host atoms composing the bilayer.

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  14. Prove that in a closest packed lattice, there are twice as many tetrahedral as octahedral holes.