Linked polyhedra Flashcards
For a polyhedra with formula MmXx, how many vertices does it have?
n = Σ N_X[j]
Where N_X[j] is the number of j-connected vertices
For a polyhedra with formula MmXx, how can you determine m and x if you know the connectedness of the vertices?
x/m = Σ N_X[j] / j
Where N_X[j] is the number of j-connected vertices.
How can you get MX by connecting identical tetrahedra?
Tetrahedra, n = 4
MX, x/m = 1
Three options:
1 2-connected, 3 6-connected.
2 3-connected, 2-6 connected
4 4-connected (real)
How can you get B2O3 by connecting tetrahedra?
Tetrahedra, n = 4
M2X3, x/m = 3/2
Three options:
1 1-connected, 3 6-connected
2 2-connected, 2 4-connected
1 2-connected, 3 3-connected (real)
How can you get MX2 by connecting tetrahedra?
Tetrahedra, n = 4
MX2, x/m = 2
Three options:
1 1-connected, 3 3-connected
4 2-connected (real)
1 1-connected, 1 2-connected, 2 4-connected
What does Pauling’s 5th rule say?
The number of essentially different kinds of constituents in a crystal tend to be small.
What is Pauling’s 5th rule called?
The rule of parsimony
How can you get MX by connected octahedra?
Octahedra, n = 6
MX, x/m = 1
Two options:
2 4-connected, 4 8-connected
6 6-connected (real)
How can you get M2X3 by connected octahedra?
Octahedra, n = 6
M2X3, x/m = 3/2
Five options: 1 1-connected, 5 10-connected 2 2-connected, 4 8-connected 3 3-connected, 3 6-connected 6 4-connected (real) 1 2-connected, 2 4-connected, 3 6-connected
How can you get MX2 by connected octahedra?
Octahedra, n = 6
MX2, x/m = 2
4 options: 1 1-connected, 5 5-connected 2 2-connected, 4 4-connected 6 3-connected (real) 1 2-connected, 3 3-connected, 2 4-connected
How can you get MX3 by connected octahedra?
Octahedra, n = 6
MX3, x/m = 3
3 options:
1 1-connected, 2 2-connected, 3 3-connected
2 1-connected, 4 4-connected
6 2-connected
In this case, there are examples of all, but most simple compounds form latter.
How can you get MX2 by connected cubes?
Cube, n = 8
MX2, x/m = 2
5 options: 1 1-connected, 7-connected 2 2-connected, 6 6-connected 3 3-connected, 5 5-connected 8 4-connected (real, CaF2) 2 2-connected, 2 4-connected, 4 8-connected
How can you get MX4 by connected dodecahedra?
Dodecahedra, n = 8
MX4, x/m = 4
3 options:
2 1-connected, 6 3-connected
8 2-connected
1 1-connected, 4 2-connected, 3 3-connected
How can you get MX3 by connected MX9?
MX9, n = 9
MX3, x/m = 3
7 options:
2 1-connected, 7 7-connected
4 2-connected, 5 5-connected
9 3-connected
2 1-connected, 3 6-connected, 4 8-connected
1 1-connected, 2 2-connected, 6 6-connected
1 2-connected, 6 3-connected, 2 4-connected
2 2-connected, 3 3-connected, 4 4-connected
What is the range of bond angles for undistorted tetrahedra linked by vertices?
102.1 to 180°
What is the range of M-M distances for undistorted tetrahedra linked by vertices, as a multiple of polyhedron edge length?
0.95 to 1.22
What is the range of bond angles for undistorted tetrahedra linked by edges?
66 to 70.5°
What is the range of M-M distances for undistorted tetrahedra linked by vertices, as a multiple of polyhedron edge length?
0.66 to 0.71
What is the range of bond angles for undistorted tetrahedra linked by faces?
No range, only 38.9°
What is the range of M-M distances for undistorted tetrahedra linked by faces, as a multiple of polyhedron edge length?
No range, only 0.41.
What is the range of bond angles for undistorted octahedra linked by vertices?
131.8 to 180°
What is the range of M-M distances for undistorted octahedra linked by vertices, as a multiple of polyhedron edge length?
1.29 to 1.41
What is the range of bond angles for undistorted octahedra linked by edges?
No range, only 90°
What is the range of M-M distances for undistorted octahedra linked by edges, as a multiple of polyhedron edge length?
No range, only 1.00.
What is the range of bond angles for undistorted octahedra linked by faces?
No range, only 70.5°
What is the range of M-M distances for undistorted octahedra linked by faces, as a multiple of polyhedron edge length?
No range, only 0.82.
How many edges does an octahedron have?
12 edges
How many faces does an octahedron have?
8 faces
How many vertices does an octahedron have?
6 vertices
How many edges does a tetrahedron have?
6 edges
How many faces does a tetrahedron have?
4 faces
How many vertices does a tetrahedron have?
4 vertices
How many polyhedra can be shared at a vertex without sharing edges and faces?
Up to 4 tetrahedra, up to 2 octahedra.
How many polyhedra can be shared at a vertex with shared edges and faces?
Up to 8 regular tetrahedra, up to 6 regular octahedra.
What is the relation between having 6 regular octahedra shared at a vertex (with shared edges and faces) and packing of a structure?
With 6 regular octahedra shared, this corresponds to closest packing with all octahedral holes filled.
What is the Niggli formula for octahedra sharing 2 vertices?
MX4+2/2
Which configurations can octahedra sharing 2 vertices form?
Cis and trans, and for each there can also be straight connections and rotated connections.
What is the Niggli formula for octahedra sharing 4 vertices?
MX2+4/2
Which configurations can octahedra sharing 4 vertices form?
Triple chain, cis-layer, trans-layer and network.
What is the Niggli formula for octahedra sharing 6 vertices?
MX6/2
Which structures has the Niggli formula MX6/2
Octahedra sharing all vertices
Without rotation: ReO3
With full rotation: RhF3
Which configurations can octahedra sharing two edges take?
Cis, trans and ‘cis, trans’
Which configurations can octahedra sharing three edges take?
They both form layers, either in a ccp structure or a hcp structure.
Give example of structures with octahedra sharing 3 edges with vertex connectivity 2 or less.
YCl3 (ccp) and BiI3 (hcp)
How many different ways can an octahedron share two edges with vertex connectivity 2 or less?
Two ways:
- Edges directly opposite on base (trans)
- Edge on base, and one of the edges going up from edge on opposite side (cis)
How many different ways can an octahedron share three edges with vertex connectivity 2 or less?
One way:
1) One edge bottom part, one edge on base and one edge on top part. None can share vertex.
How many different ways can an octahedron share two edges with vertex connectivity more than 2?
Two ways:
1) Two edges adjacent on base
2) One edge on base, and one coming up from that edge.
How many different ways can an octahedron share three edges with vertex connectivity more than 2?
Four ways:
1) All edges connected at base
2) Two edges connected at base with one going upwards opposite
3) Two edges connected at base with one going upwards from one vertex
4) One edge at base with one going up and one going down from each vertex
How many different ways can an octahedron share four edges with vertex connectivity more than 2, but without face sharing?
Six ways:
1) Two edges on opposite side of base with two edges going down and meets at summit.
2) Two edges meeting at each summit from opposite sides
3) All edges around base
4) Two edges meeting at each summit, rotated by 90°.
5) Two edges connected at base with two edges going up from each edge and meets at summit
6) One edge along base, then down to bottom vertex, up again and up to top vertex
How many different ways can an octahedron share six edges with vertex connectivity more than 2, but without face sharing?
One way:
Two edges to meet at top vertex, two edges opposite on base and two edges to meet at bottom vertex.
Which configuration can octahedra with four shared edges, but without face sharing, take?
1) Double chain (e.g. NH4CdCl3)
- Connected with two edges on opposite sides at base, and two edges meeting at bottom vertex
2) Layer (e.g. NH4HgCl3)
- Connected all around base
3) Network (TiO2, anatase)
- Two edges meeting at each extreme vertex, rotated 90° around each other.
Which configurations can octahedra with six shared edges, but without face sharing, take?
Two types of layered:
1) CdCl2-type (cubic closest packing of halogen atoms)
2) CdI2-type (hexagonal closest packing of halogen atoms)
Which configurations can octahedra with twelve shared edges, but without face sharing, take?
Network. NaCl-type.
How are octahedra connected in corundum?
One face shared, plus and additional 3 edges.
How are octahedra connected in rutile?
In a network, with a trans chain of shared edges forming the chains, and with corner sharing connecting layers.
How are octahedra connected in α-PbO2
In a network, with a cis chain of shared edges forming the chains and with corner sharing connecting layers.
How are octahedra connected in brookite (TiO2)?
Nigli formula: MX6/3
A cis-trans chain of shared edges, and corner sharing.
How are tetrahedra connected in zinc blende?
Corner sharing. All vertices are four connected. Tetrahedra are oriented same way in all layers.
How are tetrahedra connected in wurtzite?
Corner sharing. All vertices are four connected. Tetrahedra rotated relative to each other from layer to layer.
