Geometry of a cube

Question 1

How do you calculate size lengths or diagonal of a cubic cell?

Answer 1

Use Pythagoras theorem
Side =a
Face diagonal=b
Body diagonal =c

b2= a2 + a2
So b= {2a}
C2= a2+ b2= a2+ a2 +a2
So c= {3a}

Question 2

How can you calculate the number of atoms in a cubic closed packed unit cell?

Answer 2

The atoms on the faces contribute 1/2 to the unit cell
The atoms on the vertices contribute 1/8

8x 1/8 + 6x 1/2 =4
So there are 4 atoms in the unit cell

Question 3

How can you calculate the volume of the atoms in a ccp unit cell?

Answer 3

4 atoms in unit cell
Atomic radius=r
Volume of sphere is 4/3pir3
So Total volume is 16/3pir3

Question 4

How do you calculate the volume of the ccp unit cell?

Answer 4

Face diagonal is 4r
Find side l of cubic cell I in terms of r 
l2 + l2 =(4r)2
l= {8}r
Volume of cube is l3
l3= ({8}r)3

Draw a triangle using side lengths and a diagonal line across the face

Question 5

How can you find the packing efficiency of the ccp unit cell

Answer 5

Packing efficiency= volume of atoms/ volume of cell

A crystal will not have a greater packing efficient than 74%
The lower the coordination number the lower the packing efficiency

Question 6

What is the density of a solid?

Answer 6

Density=mass/ volume

Question 7

How do you find the mass of the unit cell?

Answer 7

Mass= number of atomsxRAM/ Na

Question 8

What are interstitial sites?

Answer 8

These are the holes in the lattice

Question 9

What are the two types of interstitial sites of a close packed structure?

Answer 9

Octahedral sites and tetrahedral sites

Question 10

Why is the holes in the lattice important?

Answer 10

The free space is very important for determining the structures of compounds that are based on sphere packings with other atoms in the holes
The number of faces determines the environment

Question 11

Describe the interstitial sites in a ccp

Answer 11

In ccp, octahedral sites share edges and tetrahedral sites share edges
There are 4 octahedral sites and 8 tetrahedron holes inside a unit cell
The T sites are smaller

Question 12

What do octahedral sites look like in ccp?

Answer 12

They lie between a triangle of the atoms above and a triangle of the atoms below
8 faces

Question 13

What do tetrahedral sites look like in ccp?

Answer 13

The tetrahedral sites like between a triangle of atoms in one row and a single atom in the row below
4 faces

Question 14

Describe the interstitial sites in hcp

Answer 14

Octahedral sites share faces and tetrahedral sites share faces

Question 15

What determines coordinate geometry?

Answer 15

The shape of the polyhedron formed by neighbouring atoms

Question 16

How many octahedral and tetrahedral sites are there per atom?

Answer 16

For each atom there is 1 octahdedral site and 2 tetrahedral sites

Question 17

What is a cell projection diagram

Answer 17

This is a 2D representation of the cell and is used when unit cells are more difficult to draw

Question 18

How do you describe the position of atoms in a cell projection diagram?

Answer 18

X and y are normal
You give the position with reference to Z
Z is given as a number where 0 is the bottom of the cell and 1 is the top of the cell (1/2 is half way)

Question 19

How many octahedral and tetrahedral sites would a unit cell have?

Answer 19

There are 4 atoms in a unit cell and each atom has 1 octahedral site and 2 tetrahedral sites
Overall there were 4 octahedral sites and 8 tetrahedral sites

Question 20

Describe the relative size of anions and cations

Answer 20

Generally, anions are larger than cations

Question 21

In a binary solid, describe the position of the anions and cations
How would you describe the structure?

Answer 21

Usually a close packed layer of the anions with cations occupying the interstitial holes

Describe structure in terms of coordination number of cation and anion

Question 22

In what situation would anions be in the interstitial holes?

Answer 22

This is when the symmetry for this layout is high because the highest symmetry is favoured

Question 23

How do you describe the structure of a binary solid?

Answer 23

Describe structure in terms of coordination number and geometry of cation and anion
4 coordinate= tetrahedral
6 coordinate= octahedral/ trigonal prismatic
8 coordinate= cubic

Question 24

Describe the difference between octahedral and trigonal prismatic

Answer 24

Both occur with 6 coordinate geometry

With octahedral, triangular faces point in different directions
With trigonal prismatic, triangular faces point in the same direction

Question 25

Describe the structure of NiAs

Answer 25

Structure is based on hexagonal close packed arrangement of arsenic atoms
Nickel atoms are in the octahedral sites
Both have 6 nearest neighbours
See diagrams

Question 26

What is the NiAs hcp arrangement an analogue of?

Answer 26

Analogue of ccp NaCl with metal atoms occupying octahedral states

Question 27

What is there no equivalent analogue NiAs hcp arrangement for fluorite or antifluorite structures?

Answer 27

This is because one set of ions occupied the tetrahedral sites and tetrahedral sites in hcp are closer together
This means if they were occupied this would lead to repulsion and destabilisation of the lattice

Question 28

What is a polymorph?

Answer 28

This is solid material that can have more than one lattice structure

Question 29

Give an example of a polymorph and describe the two structures

Answer 29

ZnS
- ccp of S2- ions with Zn2+ ions in half of the tetrahedral sites
-hcp arrangement of S2- ions with Zn2+ in half of the tetrahedral sites
Both have coordination 4
Cations are as far away from each other as possible

Question 30

What is a binary solid that is not close packed?

Answer 30

CsCl
This is not a body centred cubic
It starts from primitive cubic packing

Question 31

Describe the interstitial sites of a primitive cubic structure

Answer 31

Primitive cubic structure does not have tetrahedral or octahedral sites
All of the interstitial sites are equivalent
They lie at the centre of the cube and so have cubic geometry (8 coordinate)

Question 32

What is a ternary solid and give an example

Answer 32

A solid made up of more than 2 types of ions

CaTiO3

Question 33

Describe the unit cell of a ternary solid

Answer 33

More than one unit cell can be defined

The two unit cells give the same structure when repeated in three dimensions

Question 34

How do you calculate the empirical formula from the structure of a solid?

Answer 34

1) count how many of the same ion are on the face, edge, corner and body
2) multiply by the corresponding factor: body- in 1 cell- factor 1, face- in 2 cells- factor 1/2, edge- in 4 cells- factor 1/4, corner- in 8 cells- factor 1/8
3) do this for all ions and write as ratio