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Lecture 4!
ü Review on atom/ion size!ü Crystal structure (Chap 4 of Nesseʼs
book)!
42 Si4+
15 C4+
Size of atoms!
Hefferan and O’Brien, 2010; Earth Materials
Force balance!
Crystal structure (Chap. 4)!
1. How are mineral structures illustrated?!
In halite structure each Na+ ion is surrounded by 6 Cl- anions and each Cl- anion is surrounded by 6 Na+ cations. The Cl ions are located at the corners of a regular octahedron as are the Na ions: The ionic radius of Na+ is 99 picometers (0.99Å) and the radius of Cl- is 181 pm (1.81Å)!
Atoms arrangements in molecules
Halite
Cl
Cl
Cl
Cl
Na
Space lattice and unit cell of NaCl: On a model of the structure of halite we can identify all the points where a Cl ion is in contact with a Na ion directly above. If we remove the ions we are left with a space lattice (scaffolding) composed of identipoints as shown on the right. The smallest unit in this lattice which can, by translation, build up the entire lattice is known as the unit cell. In this case, the unit cell is a cube with 8 identipoints at the corners and 6 identipoints in the middle of each face.
From Bloss (1971) Crystallography and crystal chemistry (Fig 6.6)
Identipoint at corner of unit cell
Identipoint at center of face of unit cell
“Ball and stick” model of a unit cell of halite with Na ions at the corners of the cube and in the middle of each cube face. Ions shown reduced in size in order to show their spatial distribution. a = 5.64 Å (dimension of unit cell).
Silicate Mineral Structures!The basic building block of silicate structures is the silica tetrahedron [SiO4]4-.
Silicate minerals are classified on the basis of the linkages of these tetrahedra.!“Ball and stick” model of tetrahedron “Polyhedral” model of tetrahedron
Si has a coordination number of 4
Crystal structure!
2. What determines how elements combine and pack together?!
Coordination number!
ü Defined originally in 1893 by Alfred Werner!
ü Radius ratio = Rcation/Ranion!ü Total number of neighbors of a central
atom in a molecule!
Nobel Prize in 1913
Radius Ratio: RC/RA = 1.0 Sphere stacking!
Cannonball Pile!
Coordination Polyhedra!How many spheres can be packed around, i.e., touching, a
single sphere if the spheres are all the same size?!
Two dimple types: Type 1 point NE Type 2 point NW They are equivalent since you could rotate the whole structure 60o and exchange them
Answer is 12
1 2
1
1 2
2
Closest Packing of spheres of uniform radius!Add next layer (red atoms) The atoms in this second
layer (red atoms) can only settle in one dimple type. In this case red atoms fill all “type 2” dimples.
Once the first red atom
settles in, the other red atoms must also fill “type 2” dimples.
What happens if we add a
third layer of atoms?
Closest Packing of spheres!
Third layer (yellow atoms) directly above the first Hexagonal closest packed structure (HCP) in which the coordination number is 12
Closest Packing of spheres!
Hexagonal closest packing arrangement viewed for a different perspective.
Closest Packing!
Third layer: Not directly above Blue layer atoms are now in a unique position above voids between atoms in layers A and B cubic closest packed structure (CCP)
Closest Packing of spheres!
Amount of space left in between spheres?
Carl Friedrich Gauss proved that the highest average density that can be achieved by a regular lattice arrangement is!
Coordination number = 12!6 coplanar atom 3 above the plane 3 below the plane
What happens when RC/RA decreases?
The center cation becomes too small for the 12-
cordinated site. It would “rattle around” in this site producing an energetically unstable structure. To compensate for the lower RC/RA ratio, the coordination number drops to the next favorable number (8). It will do this even if the cation is slightly too large for the next lower site.
The eight anions (O2- in silicate and oxide minerals and S2- in sulfides) are located at the corners of a cube and the cation (red) is located at the center of the cube. This arrangement is known as Body-Centered Cubic (BCC)
The next smaller crystal site has a coordination number of 8
If the RC/RA ratio decreases even further the cation would become too small and “rattle around” in its cubic “cage”. The coordination number then drops to ?
Answer = 6
This figure shows a central atom (red) surrounded by 6 anions that define an octahedron so this type of coordination is commonly referred to as “octahedral”. The cation is in the center of an octahedron of closest-packed oxygen atoms
Octahedral coordination
As RC/RA continues to decrease the cation will move to the next lower coordination (4) known as tetrahedral because the cation is in the center of a tetrahedron of closest-packed oxygen atoms
Tetrahedral coordination
1y0.5
As RC/RA continues to decrease the cation will move to the next lower coordination (3) called triangular coordination. The cation moves from the center of the tetrahedron to the center of an coplanar array of 3 oxygen atoms.
Finally, if RC/RA decreases even more the cation will move to the next lower coordination: (2). The cation moves directly between 2 neighboring oxygen atoms.
Ionic packing and Pauling’s Rules!
Pauling’s First Rule: A coordination polyhedron of anions is formed about each cation, the cation to anion distance equaling the sum of their ionic radii and their radius ratio (RC/RA) determining the nature of the coordination polyhedron and the coordination number of the cation.!
General Rule: Only those cations will be accepted into the anion dominated structure that are sufficiently large so as not to “rattle” around in the interstices, i.e., ions should touch tangentially (minimizes the potential energy of the crystal).
anion
cation
Bond length
+ 1/6
+ 1/6
+ 1/6
+ 1/6
Na
Na
Na
Na Cl-
“The electrostatic valence 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 the 6 Na ions) Charge of Cl anion = -1 These charges are equal in magnitude so the structure is stable
The strength of an electrostatic bond is the valence charge divided by the coordination number, e.g. 1/6 for Na in NaCl crystal structure
Pauling’s Second rule - Ionic bond strength!
Pauling’s Third Rule!The sharing of edges, and particularly of faces, of adjacent polyhedra tend to decrease the stability of an ionic structure
Fig 9-18 of Bloss, Crystallography and Crystal Chemistry. © MSA
stable Less stable unstable
This rule is particularly valid when the cation is highly charged, e.g., Si
ü In a crystal structure containing several cations, those of high valence and small coordination number tend not to share
polyhedral elements. ü Rule of Parsimony: the number of different
kinds of constituents in a crystal tends to
be small.
Pauling’s fourth and fifth rules!
Common CN with O2-!
