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The spinel structure shown by MgAl2O4and by other mixed oxides of di- andtervalent metals has an elementary cell containing 32 O ions in almost perfect
cubic closest packing.
Based upon the 1/8 of the FCC unit cell shown above, eight of the 64 tetrahedralinterstices are filled by the divalent Mg and sixteen of the 32 octahedral intersticesare filled with the trivalent Al ions.
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AmBnXpCompounds.
These compounds contain more than one cation.
A common structure for crystal of this type is perovskite (CaTiO3) arrangement.
Here the Ca and O ions taken together form an FCC arrangement with the Ti ionsin the octahedral voids.
The coordination number of Ca is 12, Ti is 6 and O is 2.
The atomic coordinates in the primitive cell are Ti (0, 0, 0), Ca (1/2, 1/2. 1/2) and O(0, 0, 1/2), (0, 1/2, 0) and (1/2, 0, 0).
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In addition to that of calcium fluorite, a common structure of AX2compounds is thatof rutile (TiO2).
The lattice is primitive tetragonal with c/a of approximately 0.65.
There are titanium atoms at (0, 0, 0) and (1/2, 1/2, 1/2) and oxygen atoms at (u, u,0) and (u+ 1/2, 1/2-u, 1/2) where u is approximately 0.30.
Thus there are two formula units per cell.
Each Ti atom is surrounded by six O atoms but the octahedron of O atoms is notquite regular.
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AmXpCompounds.
In these compounds the charge on the ions are not the same so that m and p willnot both equal 1.
If all of the tetrahedral interstices in the FCC structure are filled with atoms of adifferent kind from those at the lattice points, the calcium fluoride (CaF2) structureis obtained.
The lattice is FCC with atoms one kind (Ca) at (0, 0, 0) and the other lattice pointsfilled with F atoms at (1/4, 1/4, 1/4) and (1/4, 3/4, 1/4) and equivalent positions.
There are four formula units per unit cell.
The coordination number of Ca is 8 with the nearest neighbors being atoms of Farranged at the corners of a cube.
The F atoms lie at the lattice points of a primitive cubic lattice in which each F atomhas tetrahedral coordination with four Ca atoms.
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A structure related to the one above is the -ZnS or wurtzite structure:
Here alternate tetrahedral interstices in the HCP structure are filled.
The lattice is hexagonal with atoms of one kind at (0, 0, 0) and (2/3, 1/3, 1/2) andthose of the other at (0, 0, 3/8) and (2/3, 1/3,7/8).
Each atom is tetrahedrally coordinated with four of the opposite kind.
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The cubic form of -ZnS has the sphalerite structure:
The lattice is FCC with one atom of each kind associated with each lattice point,i.e. A at (0, 0, 0) and X at (1/4, 1/4, 1/4).
Each atom lies at the lattice points of an FCC lattice and each lies in the secondlargest interstice of the FCC lattice of the other.
The coordination number of the atoms is 4 with the nearest neighbors at a distanceof 3ao/4 arranged at the vertices of a regular tetrahedron.
The bonds that make up this structure have a strong covalent character.
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CsCl, CsBr, CsI as well as many intermetallic compounds such as CuZn have thecesium chloride structure:
This structure has a primitive cubic lattice with one atom of each kind associatedwith each lattice point, i.e. A at (0, 0, 0) and X at (1/2, 1/2, 1/2).
The coordination number is 8 for each atom; the nearest neighbors are at 3ao/2
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AX Compounds.
A third of all compounds of the type AX crystallize in the sodium chloride structure:
The lattice is face centered cubic with 2 different atoms associate with each latticepoint, i.e. A at (0, 0, 0) and X at (0, 0, 1/2)
Each of the two atoms lies upon an FCC lattice, and each lies at the largestinterstice of the others FCC lattice.
Each atom therefore has coordination number of 6, the neighbors being at thevertices of a regular octahedron.
There are four formula units per conventional unit cell but there is clearly no trace
of a molecule of AX in this structure.When the metallic ion is of variable valence, crystals with this structure often formwith some ion positions unoccupied.
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Ionic Crystal Structures.
Overview
As discussed above, the metallic structures contain holes or interstices.
The geometry of the important interstices for each of these crystal structures canbe summarized as follows:
The structures of a great many compounds may be formed by placing atoms ofdifferent intrinsic sizes and charges into these holes.
These structures are then formed from a combination of cations and anions suchthat overall charge neutrality is maintained.
Ionic crystals (many of the ceramics are among them) are usually described interms of the structure of the anion arrangement, followed by a specification of theinterstitial location of the cations.
Sometimes this scheme is reversed for convenience.
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If made up of equal spheres, this structure has largest holes at coordinates (1/2,1/4, 0) and equivalent positions.
Where O = (1/2, 1/2, 0) and X = (1/2, 1/4, 0).
There are 12 such positions per unit cell, i.e. 6 per lattice point. The largest sphere
fitting such an interstice has a radius r = R(5/3) 1) = 0.288R.
It has four nearest neighbors equidistant from it but the tetrahedron which theseneighbors form is not regular.
The second largest interstice is at (1/2, 1/2, 0) and equivalent positions.
There are 6 such sites per unit cell and so 3 per lattice point.
Each can accommodate a sphere of radius r = 0.15R.
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Body Centered Cubic Structure.
This structure is shown by the alkali metals (Na) at room temperature, thetransition metals (Cr) and by Fe, Ti and Zr in certain temperature ranges.
The lattice is body centered cubic with one atom at each lattice point, (0, 0, 0) and(1/2, 1/2, 1/2).
Each atom has 8 neighbors at a separation of 3ao/2.
The nearest neighbors of one atom are not nearest neighbors of one another.
If the structure is made up of equal spheres, these have radius R = 3ao/4.
The 8 directions are closest packed.
The packing factor is 3/8 = 0.68
There are no closest packed planes.
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There are two such interstices per cell and each has octahedral coordination.
The second largest interstices lie at (0, 0, 3/8), (0, 0, 5/8), (2/3, 1/3, 1/8) and (2/3,1/3, 7/8).
These have r = 0.225R and there are 4 per unit cell each with tetrahedralcoordination.
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If each sphere has 12 nearest neighbors, then the axial ratio c/a must equal (8/3)or 1.633.
The packing factor is the 0.74 as in the FCC structure.
If the atomic centers are projected onto a plane parallel to (0001), then thestructure can be regarded as made up by stacking rafts of spheres in the sequence
ABABAB .... orACACAC ....
If c/a is equal to the ideal value for sphere packing:
There are 6 closest packed direction of the type in the basal plane.
These are the only closest packed directions.
The largest interstices have the coordinates (1/3, 2/3, 1/4) and (1/3, 2/3, 3/4).
The largest sphere which can be inserted without disturbing the spheres of radiusR packed in contact has r = 0.414R.
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This structure is exhibited by the early transition metals (Ti), the divalent metals
including Mg, Zn and Cd and by most of the rare earths. The hexagonal primitiveunit cell contains two atoms with coordinates (0, 0, 0) and (2/3, 1/3, 1/3).
There are thus two atoms associated with each lattice point. Packing togetherequal spheres can reproduce this structure:
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The largest sphere which can be placed in this position without disturbing the
arrangement of spheres at the lattice points without disturbing the arrangement ofspheres at the lattice points has r = (2 1)R; this sphere would have octagonalcoordination with 6 nearest neighbors.
The second largest interstice occurs at points with coordinates (1/4, 1/4, 1/4) andequivalent positions.
The largest sphere which can be placed here has radius r = (1.5-1)R andpossesses tetrahedral coordination.
There are 8 such points in the unit cell and hence 2 per lattice points.
Hexagonal close packed structure.
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A given crystal can be considered as made up by stacking, one above the other,successive planar rafts of closest packed spheres so that proceeding in the [111]
direction the centers of the atoms in adjacent rafts follow the sequenceABCABCABC ...
The same crystal structure but in a different orientation would be described by thesequence of rafts:ACBACBACB ...
When the structure is regarded as being made up of spheres in contact, the size ofthe interstices (holes) between the spheres is important because many othercrystal structures contain at least one set of atoms in an FCC arrangement,
The largest interstices occurs at positions in the unit cell with coordinates (1/2, 1/2,1/2) and equivalent positions, i.e. (0, 1/2, 0), (0, 0, 1/2) and (1/2, 0, 0).
There are thus four of these per unit cell, hence one per lattice point.
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If the atoms are spheres of R = ao/22, the appearance of a {111} plane is shownby the full circles in the following figure.
Each sphere is in contact with six equidistant spheres with centers in the plane.
Since this arrangement is the closest packing of circles in a plane, the {111} planesare termed the closest packed.
There are 8 such planes in the lattice if distinction is made between parallelnormals of opposite sense e.g., and
Each then contains 6 closest packed directions.The centers of the atoms in a given (111) plane occupy points such as A in theabove figure.
If the positions of the centers of the atoms in adjacent (111) planes are projectedonto the given (111) plane they occupy positions such as those marked B or C inthe above figure.
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Each atom possesses 12 nearest neighbors at a distance of ao/2. Thus, thecoordination number is 12.
This structure is the one obtained if equal spheres are placed in contact at thelattice points of a face-centered cubic lattice.
If the radius of the spheres is R, then R = ao/22
The proportion of space filled by the spheres, i.e., the packing fraction, is a 2/6or 74%
Along directions in the lattice, rows of spheres are in contact along the linejoining their centers.
Such directions are termed closest packed.
There are 12 such directions in all if account is taken of the change in sign.The atomic centers lie at the lattice points so in the {111} planes, which lie normalto the triad axes, the atoms centers form a tri-equiangular net of points.
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Metallic crystal structures
Overview
The vast majority of the elements are metallic
Most metals crystallize on cooling from the melt.
The metallic elements with the notable exceptions of manganese, gallium, indium,tin, mercury, uranium and plutonium all possess one of three structures: FCC,HCP or BCC.
Face Centered Cubic Structure.
The noble metals (Au), the metals of higher valence (Al), the later transition metals(Co) and the inert gases (Ar) all possess this structure.
The lattice is face centered cubic with one atom at each lattice point.
The coordinates of the atoms in the conventional cell are thus (0, 0, 0), (1 /2, 1/2,
0), (1/2, 0, 1/2) and (0, 1/2, 1/2).
There are four atoms per unit cell.
