• Simple Cubic Coordination Number
• Coordination Number Of Hcp
• What Is Coordination Number Of Fcc
• Coordination Number Of Fcc

Explore granular search interfaces into more than 40 specialized FCC databases such as radio call signs and equipment authorization. Number of equidistant neighbors from the atom in the center of the unit cell. BCC coordination number is therefore 8, as each cube corner atom is the nearest neighbor. HCP HCP is a closed-packed structure and therefore, by the same argument as that used for FCC, it has a coordination number of 12 (provided the c/a ratio shown in fig. The coordination number for BCC, SCC, and FCC are 8, 6 and 12 respectively as shown in the diagram. Answer verified by Toppr 26 Views Upvote (0).

The term 'closest packed structures' refers to the most tightly packed or space-efficient composition of crystal structures (lattices). Imagine an atom in a crystal lattice as a sphere. While cubes may easily be stacked to fill up all empty space, unfilled space will always exist in the packing of spheres. To maximize the efficiency of packing and minimize the volume of unfilled space, the spheres must be arranged as close as possible to each other. These arrangements are called closest packed structures.

The packing of spheres can describe the solid structures of crystals. In a crystal structure, the centers of atoms, ions, or molecules lie on the lattice points. Atoms are assumed to be spherical to explain the bonding and structures of metallic crystals. These spherical particles can be packed into different arrangements. In closest packed structures, the arrangement of the spheres are densely packed in order to take up the greatest amount of space possible.

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. The hole formed between three spheres is called a trigonal hole because it resembles a triangle. In the example below, two out of the the six trigonal holes have been highlighted green.

Once the first layer of spheres is laid down, a second layer may be placed on top of it. 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.

Closest Pack Crystal Structures

Hexagonal Closest Packed (HCP)

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.' The atoms in a hexagonal closest packed structure efficiently occupy 74% of space while 26% is empty space.

Cubic Closest Packed (CCP)

The arrangement in a cubic closest packing also efficiently fills up 74% of space. 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.'

Coordination Number and Number of Atoms Per Unit Cell

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.

• The hexagonal closest packed (hcp) has a coordination number of 12 and contains 6 atoms per unit cell.
• The face-centered cubic (fcc) has a coordination number of 12 and contains 4 atoms per unit cell.
• The body-centered cubic (bcc) has a coordination number of 8 and contains 2 atoms per unit cell.
• The simple cubic has a coordination number of 6 and contains 1 atom per unit cell.

Note

*For the hexagonal close-packed structure the derivation is similar. 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. So the total number of atoms in the cell is 3 + (1/2)×2 + (1/6)×6×2 = 6, however this results in 2 per primitive unit cell.

Simple Cubic Coordination Number

References

1. Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, 1993.
2. Krishna, P. and Verma, A. R. Closed Packed Structures, Chester, UK: International Union of Crystallography, 1981.
3. Petrucci, Ralph H., William S. Harwood, F. Geoffrey Herring, and Jeffry D. Madura. 'Crystal Structures' General Chemistry: Principles & Modern Applications, ninth Edition. New Jersey: Pearson Education, Inc., 2007. 501-508.

Contributors and Attributions

• Brittanie Harbick, Laura Suh, Jenny Fong

Perhaps the most common crystal structure is Face-Centered Cubic (FCC). The crystal structure is based on the Bravais lattice of the same name, with a single atom at each lattice point on the cube’s corners and faces. FCC is one of the most stable crystal structures and has the highest packing density. In some textbooks, FCC may also be abbreviated as CCP, which stands for Cubic Close-Packed. The face-centered cubic cell belongs to space group #225 or , Strukturbericht A₁, and Pearson symbol cF4 . 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. FCC has 4 atoms per unit cell, lattice constant a = 2R√2, Coordination Number CN = 12, and Atomic Packing Factor APF = 74%. FCC is a close-packed structure with ABC-ABC stacking.

Don’t worry, I’ll explain what those numbers mean and why they’re important later in the article. For now, let’s talk about which materials actually exist as face-centered cubic.

Common Examples of Face-Centered Cubic Materials

Since FCC is one of the most common crystal structures, there are many materials to choose from!

Aluminum, calcium, nickel, copper, strontium, rhodium, palladium, silver, ytterbium, iridium, platinum, gold, lead, actinium, and thorium all have an FCC structure. This list is not comprehensive; FCC can also be found in high pressure/temperature phases (like lanthanum), solidified gases (like xenon), or in alloys like steel or cobalt-based superalloys.

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.

One reason that FCC has its properties is because of its high coordination number.

Face-Centered Cubic Coordination Number

Coordination Number (CN) is the number of nearest neighbors that each atom has.

In a face-centered cubic crystal, each atom has 12 nearest neighbors (NN). That’s the theoretical maximum number of NNs possible–each of those NNs contributes a bond, giving the crystal structure very high stability.

FCC has 6 next-nearest neighbors, and 24 next-next nearest neighbors.

Face-Centered Cubic Lattice Constants

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

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These are actually primitive vectors, which you can read about in the section below.

Face-Centered Cubic Atomic Packing Factor

The Atomic Packing Factor (APF) is essentially the density of the unit cell. 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.

APF is the

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.

If you count the portion of the atoms in the cell, 1/8th of each atom would count, and there are 8 corner atoms. , so there is one full corner atom. ½ of each face atom is inside the cell, and there are 6 face atoms, so .

Thus, there are 4 atoms per unit cell.

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 .

Now that we’ve written everything in terms of the radius of an atom, you can see that every face-centered cubic crystal will have the same packing factor regardless of the actual element.

As you can see, face-centered cubic crystals have 74% packing. This is the highest possible packing for spheres of the same size, although it’s possible to have higher packing if you used multiple kinds of atoms of different sizes.

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.

Primitive Face-Centered Cubic Unit Cell

The FCC cell that I have shown you is a conventional unit cell, not a primitive unit cell. This conventional cell has advantages because it is highly symmetric and easy for humans to understand.

However, when dealing with mathematical descriptions of crystals, it may be easier to describe the unit cell in the smallest form possible. The smallest possible unit cell is called the primitive cell. If you are interested in primitive cells, you can read all about them in this article.

The FCC primitive cell looks like this:

Here are the primitive vectors for the FCC unit cell.

Interstitial Sites in Face-Centered Cubic

Interstitial sites are the spaces inside a crystal where another kind of atom could fit. You can read all about interstitial sites in this article, but FCC has two types of interstitial sites: octahedral and tetrahedral. (Technically trigonal sites are also possible, but they are not practically useful).

FCC has 4 octahedral sites, which means that a small interstitial atom could fit in 4 positions such that it is equally surrounded by 6 FCC lattice atoms.

These octahedral interstitial atoms can be size.

Coordination Number Of Hcp

FCC also has 8 tetrahedral sites, which means that a small interstitial atom could fit in 8 positions such that it is equally surrounded by 4 FCC lattice atoms.

These tetrahedral atoms can be size.

Slip Systems in Face-Centered Cubic

Slip systems are the way that atoms slide past each other when deforming. Slip systems determine many mechanical properties of materials, and is the main reason why a material will be ductile or brittle.

To understand slip system directions, you will need to be familiar with Miller Indices notation, which you can read about in this article.

What Is Coordination Number Of Fcc

The FCC close-packed planes are {111}, so those are the slip planes. Within the {111} planes, the slip direction (close-packed direction) is <110>.

By combining the 4 close-packed planes , and with 6 close-packed directions
<>, <>, <>, <>, <>, and <> and dividing by two to eliminate duplicates, there are 12 independent slip systems.

12 independent slip systems is much larger than the 5 independent slip systems needed for ductility, so FCC metals are very ductile and have no ductile-to-brittle phase transition.

Coordination Number Of Fcc

Final Thoughts

The Face-Centered Cubic (FCC) crystal structure is one of the most common ways that atoms can be arranged in pure solids. FCC is close-packed, which means it has the maximum APF of 0.74. Because FCC has 12 independent slip systems, it is also very ductile.

Here is a summary chart of all FCC crystal properties:

References and Further Reading

You can check out this article if you want a fundamental explanation of atomic packing factor.

If you want a basic explanation of crystals and grains, check out this article.

If you would like a comprehensive analysis of the differences between FCC and BCC crystal structures, you may be interested in this article.

Finally, for a great reference for all things crystallography, check out The AFLOW Library of Crystallographic Prototypes, by Mehl et. al.

Single-Element Crystal Structures and the 14 Bravais Lattices

If you want to learn about specific crystal structures, here is a list of my articles about Bravais lattices and some related crystal structures for pure elements. Face-Centered Cubic is one of these 14 Bravais lattices and also occurs as a crystal structure.

Introduction to Bravais Lattices
1. Simple Cubic
2. Face-Centered Cubic
2a. Diamond Cubic
3. Body-Centered Cubic
4. Simple Hexagonal
4a. Hexagonal Close-Packed
4b. Double Hexagonal Close-Packed (La-type)
5. Rhombohedral
5a. Rhombohedral Close-Packed (Sm-type)
6. Simple Tetragonal
7. Body-Centered Tetragonal
7a. Diamond Tetragonal (White Tin)
8. Simple Orthorhombic
9. Base-Centered Orthorhombic
10. Face-Centered Orthorhombic
11. Body-Centered Orthorhombic
12. Simple Monoclinic
13. Base-Centered Monoclinic
14.Triclinic