Geometry of a cube Flashcards
How do you calculate size lengths or diagonal of a cubic cell?
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}
How can you calculate the number of atoms in a cubic closed packed unit cell?
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
How can you calculate the volume of the atoms in a ccp unit cell?
4 atoms in unit cell
Atomic radius=r
Volume of sphere is 4/3pir3
So Total volume is 16/3pir3
How do you calculate the volume of the ccp unit cell?
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
How can you find the packing efficiency of the ccp unit cell
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
What is the density of a solid?
Density=mass/ volume
How do you find the mass of the unit cell?
Mass= number of atomsxRAM/ Na
What are interstitial sites?
These are the holes in the lattice
What are the two types of interstitial sites of a close packed structure?
Octahedral sites and tetrahedral sites
Why is the holes in the lattice important?
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
Describe the interstitial sites in a ccp
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
What do octahedral sites look like in ccp?
They lie between a triangle of the atoms above and a triangle of the atoms below
8 faces
What do tetrahedral sites look like in ccp?
The tetrahedral sites like between a triangle of atoms in one row and a single atom in the row below
4 faces
Describe the interstitial sites in hcp
Octahedral sites share faces and tetrahedral sites share faces
What determines coordinate geometry?
The shape of the polyhedron formed by neighbouring atoms
How many octahedral and tetrahedral sites are there per atom?
For each atom there is 1 octahdedral site and 2 tetrahedral sites
What is a cell projection diagram
This is a 2D representation of the cell and is used when unit cells are more difficult to draw
How do you describe the position of atoms in a cell projection diagram?
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)
How many octahedral and tetrahedral sites would a unit cell have?
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
Describe the relative size of anions and cations
Generally, anions are larger than cations
In a binary solid, describe the position of the anions and cations
How would you describe the structure?
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
In what situation would anions be in the interstitial holes?
This is when the symmetry for this layout is high because the highest symmetry is favoured
How do you describe the structure of a binary solid?
Describe structure in terms of coordination number and geometry of cation and anion
4 coordinate= tetrahedral
6 coordinate= octahedral/ trigonal prismatic
8 coordinate= cubic
Describe the difference between octahedral and trigonal prismatic
Both occur with 6 coordinate geometry
With octahedral, triangular faces point in different directions
With trigonal prismatic, triangular faces point in the same direction
Describe the structure of NiAs
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
What is the NiAs hcp arrangement an analogue of?
Analogue of ccp NaCl with metal atoms occupying octahedral states
What is there no equivalent analogue NiAs hcp arrangement for fluorite or antifluorite structures?
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
What is a polymorph?
This is solid material that can have more than one lattice structure
Give an example of a polymorph and describe the two structures
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
What is a binary solid that is not close packed?
CsCl
This is not a body centred cubic
It starts from primitive cubic packing
Describe the interstitial sites of a primitive cubic structure
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)
What is a ternary solid and give an example
A solid made up of more than 2 types of ions
CaTiO3
Describe the unit cell of a ternary solid
More than one unit cell can be defined
The two unit cells give the same structure when repeated in three dimensions
How do you calculate the empirical formula from the structure of a solid?
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
