The second layer of spheres may be placed to cover the trigonal holes from the first layer. Holes now exist between the first layer the orange spheres and the second the lime spheres , but this time the holes are different. The triangular-shaped hole created over a orange sphere from the first layer is known as a tetrahedral hole. A hole from the second layer that also falls directly over a hole in the first layer is called an octahedral hole.
In a hexagonal closest packed structure, the third layer has the same arrangement of spheres as the first layer and covers all the tetrahedral holes. Since the structure repeats itself after every two layers, the stacking for hcp may be described as "a-b-a-b-a-b.
Similar to hexagonal closest packing, the second layer of spheres is placed on to of half of the depressions of the first layer. The third layer is completely different than that first two layers and is stacked in the depressions of the second layer, thus covering all of the octahedral holes. The spheres in the third layer are not in line with those in layer A, and the structure does not repeat until a fourth layer is added.
The fourth layer is the same as the first layer, so the arrangement of layers is "a-b-c-a-b-c. A unit cell is the smallest representation of an entire crystal. All crystal lattices are built of repeating unit cells. In a unit cell, an atom's coordination number is the number of atoms it is touching. Here the unit cell consist of three primitive unit cells is a hexagonal prism containing six atoms if the particles in the crystal are atoms.
Indeed, three are the atoms in the middle layer inside the prism ; in addition, for the top and bottom layers on the bases of the prism , the central atom is shared with the adjacent cell, and each of the six atoms at the vertices is shared with other five adjacent cells. Types of Holes From Close-Packing of Spheres When a single layer of spheres is arranged into the shape of a hexagon, gaps are left uncovered.
Cu is the prototype for FCC. The Face-Centered Cubic FCC unit cell can be imagined as a cube with an atom on each corner, and an atom on each face. It is one of the most common structures for metals.
Aluminum, calcium, nickel, copper, strontium, rhodium, palladium, silver, ytterbium, iridium, platinum, gold, lead, actinium, and thorium all have an FCC structure. FCC metals are usually very ductile and have no ductile-to-brittle phase transformation. If you are interested in the differences between FCC and BCC another common structure , you may be interested in this article.
In a face-centered cubic crystal, each atom has 12 nearest neighbors NN. The face-centered cubic lattice is a cube with an atom on each corner and each face. Using the hard sphere model, which imagines each atom as a discrete sphere, the FCC crystal has each atom touch along the face diagonal of the cube. That means that the face diagonal has a length of , so with a bit of geometry we find that the lattice parameter , or side length of the cube, has a length of.
If you wanted to describe the face-centered cubic crystal with math, you would describe the cell with the vectors. Since we use the hard sphere model, each point inside the cell is either part of an atom, or part of the void.
APF is basically the fraction of atoms to void. For a full article explaining APF, check out this link. The total volume of the unit cell is just the volume of a cube. The cube side length is a, so the volume is. Now we need to count how many atoms are in each unit cell. It may look like there are 14 atoms because there are 8 corners and 6 faces, but actually the cell only intersects portions of those atoms. The volume of a sphere is. We previously established that the volume of the whole cube is , and since , the volume of the cube is.
Since FCC has the maximum type of packing, it is a close-packed structure. The other common close-packed structure is hexagonal close-packed HCP , although there are also lesser-known types like the close-packed rhombohedral structure found in Samarium. The FCC cell that I have shown you is a conventional unit cell, not a primitive unit cell.
Do the atoms in the second layer have the bulk coordination? No - the fact that they are clearly exposed visible at the surface implies that they have a lower CN than they would in the bulk. What is the coordination number of these second layer atoms on the fcc surface? Rationale: The atoms in the second layer are only missing one atom from their complete coordination shell the atom that would have been directly above them i.
The surface is obtained by cutting the fcc metal in such a way that the surface plane intersects the x -, y - and z - axes at the same value - this exposes a surface with an atomic arrangement of 3-fold apparently 6-fold, hexagonal symmetry. This layer of surface atoms actually corresponds to one of the close-packed layers on which the fcc structure is based. The diagram below shows the conventional birds-eye view of the surface - emphasizing the hexagonal packing of the surface layer atoms.
Since this is the most efficient way of packing atoms within a single layer, they are said to be "close-packed". Rationale: Each surface atom has six nearest neighbours in the 1st layer, and another three in the layer immediately below ; a total of 9. Flat surfaces of single crystal samples correspond to a single Miller Index plane and, as we have seen, each individual surface has a well-defined atomic structure. It is these flat surfaces that are used in most surface science investigations, but it is worth a brief aside to consider what type of surfaces exist for an irregular shaped sample but one that is still based on a single crystal.
Such samples can exhibit facets corresponding to a range of different Miller Index planes. This is best illustrated by looking at the diagrams below. Figure: left an angled corner.
Hermann, Fritz-Haber-Institut, Berlin. Depending upon how an fcc single crystal is cleaved or cut, flat surfaces of macroscopic dimensions which exhibit a wide range of structural characteristics may be produced.
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