Atomic Structure

In ionic crystals, the ions may be considered to behave as rigid spheres. As many minerals, especially the silicate minerals, can be considered to be ionic crystals, even though this may not be completely true; this is the ionic approximation.

Coordination Polyhedra Consider coordination of anions about a central cation Halite Na Cl Cl Cl Cl [6]NaCl

Coordination Polyhedra Could do the opposite, but conventionally choose the cation Can predict the coordination by considering the radius ratio: RC/RA Cations are generally smaller than anions so begin with maximum ratio = 1.0 Na [6]Cl Na Na Cl Na

Coordination Polyhedra Radius Ratio: RC/RA = 1.0 (commonly native elements) Equal sized spheres “Closest Packed” Hexagonal array: 6 nearest neighbors in the plane Note dimples in which next layer atoms will settle Two dimple types: Type 1 Type 2 They are equivalent since you could rotate the whole structure 60o and exchange them 2 1 1

Closest Packing Add next layer (red) 1 Red atoms can only settle in one dimple type Both types are identical and red atoms could settle in either Once first red atom settles in, one can only fill other dimples of that type In this case filled all type 2 dimples Type 1 dimples remain 1

Closest Packing A Third layer ?? Third layer dimples are now different! Call layer 1 A sites Layer 2 = B sites (no matter which choice of dimples is occupied) Layer 3 can now occupy A-type site (directly above yellow atoms) or C-type site (above voids in both A and B layers) A

Closest Packing Third layer: If occupy A-type site the layer ordering becomes A-B-A-B and creates a hexagonal closest packed structure (HCP) Coordination number (nearest or touching neighbors) = 12 6 coplanar 3 above the plane 3 below the plane

Closest Packing Third layer: If occupy A-type site the layer ordering becomes A-B-A-B and creates a hexagonal closest packed structure (HCP) Note top layer atoms are directly above bottom layer atoms

Closest Packing Unit-cell: smallest identical unit in a Third layer: Unit cell Unit-cell: smallest identical unit in a crystal structure, when repeated, generates the entire structure

Closest Packing Third layer: View from top shows hexagonal unit cell

A single close-packed layer

A single close-packed layer hexagonal unit cell

A single close-packed layer B positions for the next layer

A single close-packed layer C positions for the next layer

Two close-packed layers Sequence : AB

Two close-packed layers Sequence : AC For 2 layers AB is the same as AC

Two close-packed layers Sequence : AB Two choices for the third layer : above the A position ………...

Two close-packed layers Sequence : AB Two choices for the third layer : above the A position, or above the C position

Hexagonal close-packing : ABABAB…... z x y

Closest Packing Alternatively we could place the third layer in the C-type site (above voids in both A and B layers)

Closest Packing Third layer: If occupy C-type site the layer ordering is A-B-C-A-B-C and creates a cubic closest packed structure (CCP) Blue layer atoms are now in a unique position above voids between atoms in layers A and B

Closest Packing Third layer: If occupy C-type site the layer ordering is A-B-C-A-B-C and creates a cubic closest packed structure (CCP) Blue layer atoms are now in a unique position above voids between atoms in layers A and B

Closest Packing Third layer: If occupy C-type site the layer ordering is A-B-C-A-B-C and creates a cubic closest packed structure (CCP) Blue layer atoms are now in a unique position above voids between atoms in layers A and B

Closest Packing A-layer C-layer B-layer A-layer The atoms are slightly shrunken to aid in visualizing the structure A-layer C-layer B-layer A-layer

Closest Packing Rotating toward a top view

Closest Packing You are looking at a top yellow layer A with a blue layer C below, then a red layer B and a yellow layer A again at the bottom

Cubic close packing : a smaller section of the ABC stacking

What happens when RC/RA decreases? The center cation becomes too small for the [12] site and it drops to the next lower coordination number (next smaller site). It will do this even if it is slightly too large for the next lower site. It is as though it is better to fit a slightly large cation into a smaller site than to have one rattle about in a site that is too large.

The next smaller crystal site is: Body-Centered Cubic (BCC) with cation (red) in the center of a cube Coordination number is now 8 (corners of cube)

A central cation will remain in [8]-coordination with decreasing RC/RA until it again reaches the limiting situation in which all atoms mutually touch. Then a hard-sphere cation would “rattle” in the position, and it would shift to the next lower coordination (next smaller site). What is the RC/RA of that limiting condition?? Set = 1 arbitrary since will deal with ratios Diagonal length then = 2

A central cation will remain in [8]-coordination with decreasing RC/RA until it again reaches the limiting situation in which all atoms mutually touch. Then a hard-sphere cation would “rattle” in the position, and it would shift to the next lower coordination (next smaller site). What is the RC/RA of that limiting condition??

A central cation will remain in [8]-coordination with decreasing RC/RA until it again reaches the limiting situation in which all atoms mutually touch. Central Plane What is the RC/RA of that limiting condition?? 1.732 = dC + dA If dA = 1 then dC = 0.732 dC/dA = RC/RA = 0.732/1 = 0.732

The limits for [8]-coordination are thus between 1 The limits for [8]-coordination are thus between 1.0 (when it would by CubicClosePacking or HexagonalClosePacking) and 0.732 Note: BodyCenteredCubic is not a closest-packed oxygen arrangement.

As RC/RA continues to decrease below the 0 As RC/RA continues to decrease below the 0.732 the cation will move to the next lower coordination: [6], or octahedral. The cation is in the center of an octahedron of closest-packed oxygen atoms. (view of face of octahedron)

As RC/RA continues to decrease below the 0 As RC/RA continues to decrease below the 0.732 the cation will move to the next lower coordination: [6], or octahedral. The cation is in the center of an octahedron of closest-packed oxygen atoms. (view of apex of octahedron)

Interstitial sites in close-packed structures i. e Interstitial sites in close-packed structures i.e. the spaces between the spheres There are two types : tetrahedral sites and octahedral sites I : Tetrahedral sites : the small spaces between 4 close-packed spheres

Interstitial sites in close-packed structures II : Octahedral sites : the larger spaces between 6 close-packed spheres

As RC/RA continues to decrease below the 0 As RC/RA continues to decrease below the 0.732 the cation will move to the next lower coordination: [6], or octahedral. The cation is in the center of an octahedron of closest-packed oxygen atoms What is the RC/RA of that limiting condition?? 1.414 = dC + dA If dA = 1 then dC = 0.414 dC/dA = RC/RA = 0.414/1 = 0.414

As RC/RA continues to decrease below the 0 As RC/RA continues to decrease below the 0.414 the cation will move to the next lower coordination: [4], or tetrahedral. The cation is in the center of an tetrahedron of closest-packed oxygen atoms

As RC/RA continues to decrease below the 0 As RC/RA continues to decrease below the 0.414 the cation will move to the next lower coordination: [4], or tetrahedral. The cation is in the center of an tetrahedron of closest-packed oxygen atoms What is the RC/RA of the limiting condition?? Center-to-corner distance of a tetrahedron with edges of 1.0 = 0.6124 RC = 0.612 - 0.5 = 0.1124 RC/RA = 0.1124/0.5 = 0.225

As RC/RA continues to decrease below the 0 As RC/RA continues to decrease below the 0.22 the cation will move to the next lower coordination: [3]. The cation moves from the center of the tetrahedron to the center of an coplanar tetrahedral face of 3 oxygen atoms What is the RC/RA of the limiting condition?? cos 60 = 0.5/y y = 0.577 RC = 0.577 - 0.5 = 0.077 RC/RA = 0.077/0.5 = 0.155

If RC/RA decreases below the 0 If RC/RA decreases below the 0.15, the cation will move to the next lower coordination: [2]. The cation moves directly between 2 neighboring oxygen atoms

The nickel arsenide (NiAs) structure The structure can be described as a hexagonal close packed As ions, with Ni occupying all of the octahedral sites

The wurtzite (ZnS) structure The structure can be described as a hexagonal close packed S ions, with Zn2+ ions occupying half of the tetrahedral sites

The sodium chloride (NaCl) structure The structure can be described as a cubic close packed Cl- ions, with Na+ ions in all the octahedral sites

The sphalerite (ZnS) structure The structure can be described as a cubic close packed S ions, with Zn2+ ions in half of the tetrahedral sites

Atomic and Ionic Radii Can't absolutely determine: e- cloud is nebulous & based on probability of encountering an e- In crystalline solids the center-to-center distance = bond length = sum of ionic radii How get ionic radius of X and Y in XY compound??

Atomic Radii a a Need one pure element Native Cu. Atomic radius = 1/2 bond length a a

Ionic Radii Ionic radius of an element varies with Charge Coordination number Bond-type Structure type Effective ionic radii = empirically determined from highly accurate structure data = Average ionic radii for one valence and one coordination number but includes different bond-types and different structure types

Ionic radii versus coordination number

Pauling’s Rules for Ionic Crystals Deal with the energy state of the crystal structure 1st Rule The cation-anion distance =  radii Can use RC/RA to determine the coordination number of the cation This is our previous discussion on coordination polyhedra

Pauling’s Rules for Ionic Crystals 2nd Rule: The electrostatic valency principle First note that the strength of an electrostatic bond = valence / CN Na+ in NaCl is in [6] coordination For Na+ the strength = +1 divided by 6 = + 1/6 Cl Cl Cl Cl Na

Pauling’s Rules for Ionic Crystals 2nd Rule: the electrostatic valency principle An ionic structure will be stable to the extent that the sum of the strengths of electrostatic bonds that reach an anion from adjacent cations = the charge of that anion 6 ( + 1/6 ) = +1 (sum from Na’s) charge of Cl = -1 These charges are equal in magnitude so the structure is stable + 1/6 Na + 1/6 Na Na Cl- + 1/6 + 1/6 Na

Pauling’s Rules 3rd Rule: The sharing of edges, and particularly of faces, of adjacent polyhedra tend to decrease the stability of an ionic structure

If the edge of the edge of the tetrahedron = 1, B = 0.71 C = 0.58 D = base of tetrahedral poyhedron at maximum separation of the central ion

Pauling’s Rules 4th Rule: In a crystal with different cations, those of high valence and small CN tend not to share polyhedral elements An extension of Rule 3 Si4+ in [4] coordination is very unlikely to share edges or faces

Pauling’s Rules 5th Rule: The number of different kinds of constituents in a crystal tends to be small

Representing crystal structures as polyhedral models In many cases it is more convenient to represent crystal structures as polyhedral models, than as ball-and-stick or space-filling models, especially when the structure is complex. e.g. the NaCl structure can be described as an array of edge-shared octahedra space-filling models ball-and-stick Polyhedral presentation

e.g. the sphalerite structure of ZnS can be described as an array of corner sharing tetrahedra

The spinel AB2O4 structure