Worksheet: Hole Geometries in Crystals

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In this worksheet, we will practice determining stoichiometries of atoms and ions in cubic lattices based on Bravais lattice type and hole occupancies.

Q1:

In a cubic ionic lattice, a cation of element M is located at each corner of the unit cell and an anion of element L is located at the center of each edge. What is the formula of this material?

  • A M L 2
  • B M L 2
  • C M L 4 3
  • D M L 2 3
  • E M L 3

Q2:

In the cubic unit cell of a manganese oxide, there is a manganese ion in the center of the cell and at each of the corners. Oxide ions are located at the center of each edge of the unit cell. What is the formula of this compound?

  • A M n O
  • B M n O 3 4
  • C M n O 3
  • D M n O 2 3
  • E M n O 2

Q3:

In an ionic lattice, cations of element M occupy all of the cubic holes in the simple cubic lattice formed by anions of element L. What is the formula of this material?

  • A M L 2 3
  • B M L 3 2
  • C M L
  • D M L 2
  • E M L 2

Q4:

In a face-centered cubic ionic lattice, cations of element A occupy one-eighth of the tetrahedral holes between anions of element L, while cations of element Z occupy half of the octahedral holes. What is the formula of this material?

  • A A Z L 2 4
  • B A Z L 4 3
  • C A Z L 4 2 3
  • D A Z L 4 3
  • E A Z L 2 4

Q5:

In a face-centered cubic ionic lattice, cations of element M occupy one-eighth of the tetrahedral holes and half of the octahedral holes between anions of element L. What is the formula of this material?

  • A M L 2 3
  • B M L 3
  • C M L 3 4
  • D M L 4 3
  • E M L 3 2

Q6:

The radius of a cation in a cubic ionic lattice is 70% smaller than the radius of the anion. The charges of the cation and anion are equal in magnitude. What type of hole is the cation likely to occupy?

  • AOctahedral
  • BTetrahedral
  • CBoth octahedral and tetrahedral
  • DBoth octahedral and cubic
  • ECubic

Q7:

N a C l and ZnS both have face-centered cubic structures. Why do the sodium ions in NaCl have a higher coordination number than the zinc ions in ZnS?

  • ASodium ions occupy both tetrahedral and octahedral holes, as they are more similar in size to chloride ions than zinc ions are to sulfide ions.
  • BSodium ions occupy octahedral holes, as they are less similar in size to chloride ions than zinc ions are to sulfide ions.
  • CSodium ions occupy tetrahedral holes, as they are more similar in size to chloride ions than zinc ions are to sulfide ions.
  • DSodium ions occupy octahedral holes, as they are more similar in size to chloride ions than zinc ions are to sulfide ions.
  • ESodium ions occupy tetrahedral holes, as they are less similar in size to chloride ions than zinc ions are to sulfide ions.

Q8:

What type of hole do lithium ions occupy in the cubic lattice of LiSe2?

  • ACubic
  • BOctahedral
  • CTetrahedral
  • DBoth octahedral and tetrahedral
  • EBoth cubic and tetrahedral

Q9:

In a hexagonal close-packed ionic lattice, cations of element M occupy two-thirds of the octahedral holes between anions of element L. What is the formula of this material?

  • A M L 3 2
  • B M L 2
  • C M L
  • D M L 2
  • E M L 2 3

Q10:

Fill in the blank: Tetrahedral holes have a coordination number of .

Q11:

How many tetrahedral holes would you expect to find in a close-packed crystal containing 6 atoms?

Q12:

How many octahedral holes would you expect to find in a closed-packed crystal containing 8 atoms?

Q13:

What is the size of a tetrahedral hole in a cubic close-packed system with a radius of 1.2 Å?

Q14:

In a close-packed arrangement of spheres with a uniform radius of 1.5 Å, what is the radius of the largest sphere that can occupy an octahedral hole?

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