Difference between revisions of "Crystal structures"

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Entry by [[Emily Redston]], AP 225, Fall 2011
 
Entry by [[Emily Redston]], AP 225, Fall 2011
  
A crystalline material is one in which the atoms are situated in a repeating or periodic array over large atomic distances; that is, long-range order exists such that, upon solidification, the atoms will position themselves in a repetitive three-dimensional pattern. If this long-range atomic order is absent, an [[amorphous]] material will form. The <b>crystal structure</b> of a material is the manner in which atoms, ions, or molecules are spatially arranged. There is an extremely large number of different crystal structures all having long-range order; these vary from relatively simple structures for metals to exceedingly complex ones, as displayed by some of the ceramic and polymeric materials. When describing crystalline structures, atoms are thought of as being solid spheres having well-defined diameters. This is an atomic [[hard sphere] model in which spheres representing nearest-neighbors atoms touch one another. The term [[lattice]] is oftentimes used to describe a three-dimensional array of points coinciding with atom positions.  
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Crystalline materials are characterized by repeating (or periodic) arrays of atoms over large atomic distances. Typically we think about these materials in the context of solids, though there are also [[liquid crystals]]. [[Crystallization]] is the process by which crystal structures form; atoms will position themselves in a repetitive three-dimensional structure upon solidification. The system is said to display [[long-range order]]. If a solid does not have long-range order, it is considered [[amorphous]]. The <b>crystal structure</b> of a material is the unique way in which atoms, ions, or molecules are spatially arranged. There is an extremely large number of different crystal structures that all have long-range order. Some of these are relatively simple structures (e.g. for metals) while others can be exceedingly complex (e.g. for ceramic and polymeric materials). When describing crystal structures, it is useful to think about the atoms as solid spheres having well-defined radii. We call this the atomic [[hard sphere]] model, where spheres representing nearest neighbor atoms touch one another. The term [[lattice]] is used to describe the three-dimensional array of points that coincide with the atom positions.  
  
The atomic order in crystallline order in crystalline solids indicates that small groups of atoms form a repetitive pattern. Thus, in describing crystal structures, it is often convenient to sub-divide the structure in small repeat entities called [[unit cells]]. A unit cell is choisen to represent the [[symmetry]] of the crystal structure, wherein all the atom positions in the crystal may be generated by translation of the unit cell integral distances along each of its edges. Thus, the unit cell is the basis structural unit of the crystal structure and defines the crystal the crystal structure by virtue of its geometry and the atom positions within.  
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In describing crystal structures, it is often convenient to sub-divide the structure into small repeat entities called unit cells. We typically choose a [[unit cell]] such that it represents the [[symmetry]] of the crystal structure (the [[conventional unit cell]]). The unit cell is a small structure that repeats itself by translation through the crystal; if the unit cells are stacked together in three-dimensions, they describe the bulk arrangement of the atoms in the crystal. Thus the unit cell is the basic structural unit of the crystal structure. The unit cell is described by its [[lattice parameters]], which are the lengths of the cell edges and the angles between them. Using these parameters, we can define 14 [[Bravais lattices]] in three-dimensional space that describe the geometric arrangement of the lattice points. The 32 crystallographic [[point groups]] describe the rotation and mirror symmetries of the unit cell. To describe the full symmetry of a crystal (including translational symmetry operations), we have to consider the 230 [[space groups]].  
  
Briefly, since they are relevant to hard sphere colloidal packing, I will go over two essential structures: face-centered cubic and hexagonal close packed. For hard-sphere models, you typically end up with relatively large numbers of nearest neighbors and dense atomic packings due to the minimal restrictions as to the number and position of nearest-neighbor atoms.
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[[Image:Closepacking.png‎  |thumb| 400px | left | Figure 1 – The hcp lattice (left) and the fcc lattice (right). The outline of each respective Bravais lattice is shown in red. The letters indicate which layers are the same. There are two "A" layers in the hcp matrix, where all the spheres are in the same position. All three layers in the fcc stack are different. Note the fcc stacking may be converted to the hcp stacking by translation of the upper-most sphere, as shown by the dashed outline. (http://en.wikipedia.org/wiki/Close-packing_of_equal_spheres)]]
  
The face-centered cubic system (F) has lattice points on the faces of the cube, that each gives exactly one half contribution, in addition to the corner lattice points, giving a total of 4 lattice points per unit cell (1⁄8 × 8 from the corners plus 1⁄2 × 6 from the faces).
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Many crystal structures are based on the close-packing of atoms, which we describe using hard spheres. Figure 1 shows two of the most common crystal structures: [[face-centered cubic]] (fcc) and [[hexagonal close-packed]] (hcp). These structures are especially relevant to colloidal packing, and are important because they have the highest possible packing density. Both structures begin from the hexagonal sheet of atoms we see in Figure 1, but they differ in how we stack these sheets on top of one another. Let's call the positions of this first layer A. We can see that there are two possible positions to fit another sheet on top of this one called B (place atoms on downward-facing triangular spaces) and C (place atoms on upward-facing triangular spaces).
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[[Image:Hcp2.gif  |thumb| 200px | right | Figure 2 – An animation of hcp lattice generation. (http://en.wikipedia.org/wiki/Close-packing_of_equal_spheres)]]
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<center>fcc = ABCABCA... stacking</center>
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<center>hcp = ABABABA... stacking</center>
  
In mineralogy and crystallography, crystal structure is a unique arrangement of atoms or molecules in a crystalline liquid or solid. A crystal structure is composed of a pattern, a set of atoms arranged in a particular way, and a lattice exhibiting long-range order and symmetry. Patterns are located upon the points of a lattice, which is an array of points repeating periodically in three dimensions. The points can be thought of as forming identical tiny boxes, called unit cells, that fill the space of the lattice. The lengths of the edges of a unit cell and the angles between them are called the lattice parameters. The symmetry properties of the crystal are embodied in its space group.
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Figure 2 shows a nice animation of how hcp is generated. In both fcc and hcp, each atom has twelve nearest neighbors, so we say that the [[coordination number]] is 12. The [[atomic packing factor]] (APF) of both structures is 0.74.  
 
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The crystal structure of a material or the arrangement of atoms within a given type of crystal structure can be described in terms of its unit cell. The unit cell is a small box containing one or more atoms, a spatial arrangement of atoms. The unit cells stacked in three-dimensional space describe the bulk arrangement of atoms of the crystal. The crystal structure has a three-dimensional shape. The unit cell is given by its lattice parameters, which are the length of the cell edges and the angles between them, while the positions of the atoms inside the unit cell are described by the set of atomic positions (xi  , yi  , zi) measured from a lattice point.
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==References==
 
==References==
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[2] http://en.wikipedia.org/wiki/Close-packing_of_equal_spheres
 
[2] http://en.wikipedia.org/wiki/Close-packing_of_equal_spheres
  
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[3] Spaepen, Frans. ''Applied Physics 282: Solids: Structure and Defects''. Harvard University
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For a much more in-depth look into crystal structures, ''The Structure of Materials'' by Samuel M. Allen and Edwin L. Thomas is a great reference.
  
 
== Keyword in references: ==
 
== Keyword in references: ==
  
 
[[Phase Behavior and Structure of a New Colloidal Model System of Bowl-Shaped Particles]]
 
[[Phase Behavior and Structure of a New Colloidal Model System of Bowl-Shaped Particles]]

Latest revision as of 15:59, 10 December 2011

Entry by Emily Redston, AP 225, Fall 2011

Crystalline materials are characterized by repeating (or periodic) arrays of atoms over large atomic distances. Typically we think about these materials in the context of solids, though there are also liquid crystals. Crystallization is the process by which crystal structures form; atoms will position themselves in a repetitive three-dimensional structure upon solidification. The system is said to display long-range order. If a solid does not have long-range order, it is considered amorphous. The crystal structure of a material is the unique way in which atoms, ions, or molecules are spatially arranged. There is an extremely large number of different crystal structures that all have long-range order. Some of these are relatively simple structures (e.g. for metals) while others can be exceedingly complex (e.g. for ceramic and polymeric materials). When describing crystal structures, it is useful to think about the atoms as solid spheres having well-defined radii. We call this the atomic hard sphere model, where spheres representing nearest neighbor atoms touch one another. The term lattice is used to describe the three-dimensional array of points that coincide with the atom positions.

In describing crystal structures, it is often convenient to sub-divide the structure into small repeat entities called unit cells. We typically choose a unit cell such that it represents the symmetry of the crystal structure (the conventional unit cell). The unit cell is a small structure that repeats itself by translation through the crystal; if the unit cells are stacked together in three-dimensions, they describe the bulk arrangement of the atoms in the crystal. Thus the unit cell is the basic structural unit of the crystal structure. The unit cell is described by its lattice parameters, which are the lengths of the cell edges and the angles between them. Using these parameters, we can define 14 Bravais lattices in three-dimensional space that describe the geometric arrangement of the lattice points. The 32 crystallographic point groups describe the rotation and mirror symmetries of the unit cell. To describe the full symmetry of a crystal (including translational symmetry operations), we have to consider the 230 space groups.

Figure 1 – The hcp lattice (left) and the fcc lattice (right). The outline of each respective Bravais lattice is shown in red. The letters indicate which layers are the same. There are two "A" layers in the hcp matrix, where all the spheres are in the same position. All three layers in the fcc stack are different. Note the fcc stacking may be converted to the hcp stacking by translation of the upper-most sphere, as shown by the dashed outline. (http://en.wikipedia.org/wiki/Close-packing_of_equal_spheres)

Many crystal structures are based on the close-packing of atoms, which we describe using hard spheres. Figure 1 shows two of the most common crystal structures: face-centered cubic (fcc) and hexagonal close-packed (hcp). These structures are especially relevant to colloidal packing, and are important because they have the highest possible packing density. Both structures begin from the hexagonal sheet of atoms we see in Figure 1, but they differ in how we stack these sheets on top of one another. Let's call the positions of this first layer A. We can see that there are two possible positions to fit another sheet on top of this one called B (place atoms on downward-facing triangular spaces) and C (place atoms on upward-facing triangular spaces).

Figure 2 – An animation of hcp lattice generation. (http://en.wikipedia.org/wiki/Close-packing_of_equal_spheres)
fcc = ABCABCA... stacking
hcp = ABABABA... stacking

Figure 2 shows a nice animation of how hcp is generated. In both fcc and hcp, each atom has twelve nearest neighbors, so we say that the coordination number is 12. The atomic packing factor (APF) of both structures is 0.74.

References

[1] Callister, William D. Materials Science and Engineering: an Introduction. New York: John Wiley & Sons, 2007.

[2] http://en.wikipedia.org/wiki/Close-packing_of_equal_spheres

[3] Spaepen, Frans. Applied Physics 282: Solids: Structure and Defects. Harvard University

For a much more in-depth look into crystal structures, The Structure of Materials by Samuel M. Allen and Edwin L. Thomas is a great reference.

Keyword in references:

Phase Behavior and Structure of a New Colloidal Model System of Bowl-Shaped Particles