Summary and Terms

Summary

We've spent the last two chapters exploring the ways in which two atoms can interact and bond with one another. In this chapter, we've increased the complexity a notch, moving into assemblies of multiple atoms and further into crystals, periodic arrays of many atoms.

In our simple 2D models , we noticed that when multiple atoms group up, some arrangements are preferred over others, usually the ones where as many bonds between atoms as possible are formed. You may have also seen that certain arrangements that don't have the maximum number of bonds will stay in these suboptimal arrangements unless we give them enough energy to rearrange - a concept known as metastability. From this simulation we identified the concept of a 2D-crystal, and introduced the concepts of a bravais lattice, unit cell, and motif, which together define a crystal in 2D or 3D.

Next, we moved into three dimensions, where the underlying rules for defining a crystal are the same but take on increased complexity due to adding a dimension. Bravais lattices and unit cells are still just as important, there are just more of them to choose from. Broadly speaking, a unit cell in 3D is defined by three side lengths $( a, b, c )$ and three angles $(\alpha, \beta, \gamma )$, but in this class we'll only consider those with orthogonal basis vectors ($(\alpha = \beta = \gamma = 90^\circ)$

Navigating a 3D unit cell is a non-trivial affair, but with some simple vector math we're able to define any point in a cell in terms of the three lattice vectors relative to the origin. Furthermore, we defined crystallographic directions $[uvw]$ and crystallographic planes $(hkl)$, vectors and stacks of parallel planes respectively which help us navigate and describe unit cells and the crystals that they represent. We also described how equivalent directions and planes fall into families based on the symmetry of the unit cell in which they are defined.

We use all of these tools to understand the structure of crystals, the most approachable of which are often common metals. We defined two ways of visualizing a unit cell, the ball-and-stick and space-filling models, which each provide us with a useful but by nature not completely accurate view of a unit cell. These two models have distinct strengths and weaknesses. For instance, it's generally easier to determine coordination number using the ball-and-stick model, while visualizing the interstitial sites between lattices atoms is easier with the space-filling model. We also explored how using a unit cell's lattice parameter(s), we can define the density of atoms within the cell and the atomic packing factor.

Interstitial sites, the open spaces between atoms in a unit cell, are very important for considering phenomena such as alloying and diffusion, and we spent some time exploring how to determine their volume based on the space-filling model. There are several types of interstitial site, and they are named based on the polyhedron that their nearest neighbors make around them (i.e. a tetrahedral site is enclosed by four atoms in a tetrahedron). Interstitial sites in a unit cell comprise a sublattice - they repeat regularly just like the atoms on lattice points. The kind of sublattice (FCC, BCC, SC, etc.) depends on the site and may or may not be the same as the lattice of the parent crystal structure.

Crystals aren't restricted just to metals. Ceramics and polymers can also form crystals which typically contain at least two types of atoms. For ionic solids, we can use Pauling's rules to determine the type of structure that will form based on the compound's stoichiometry and the size of the ions involved. One of the most important predictions we get out of these rules is the coordination geometry of ions in the crystal - that is, how many nearest neighbors an ion of a given type will have. Many of the crystal structures expressed look just like the ones we saw in metals but with two atom types instead of one and some occupied interstitial sites.

While materials for some specialty applications can approach perfect, unbroken crystallinity, most of the time this is not the case. Single crystals like these have high long and short-range order and their properties are anisotropic - they depend on which crystalline direction we interact with. Polycrystalline materials are comprised of many crystallites, which can vary widely in size. They have predictable short range order (within a single crystallite) but their long range order is much lower than a single crystal. Finally, amorphous materials are not crystalline at all - they have only short range order on the scale of a bond length, but completely random arrangements beyond this length scale.

Terms

Metastable State: A state where a system is in a local energy minimum, but not the global energy minimum. In our simulations we saw this as atoms that could have three bonds if they moved, but stay at two bonds because they can't overcome the activation barrier for movement. Question 5.3.1.4

Crystal: A periodic, ordered arrangement of atoms or molecules. Section 5.4.1

Motif: The objects that get translated to form a crystal, such as groups of atoms. A motif is associated with a lattice point, but all elements in a motif are not necessarily located on lattice points. Section 5.4.2

Unit Cell: A box that contains the motif and can be translated (but not rotated) to fill 3-D or 2-D space. Defined by lattice vectors, which tell us the size and shape of the unit cell (the side lengths of the box and the angles between its axes). Section 5.4.2

Bravais Lattice: an infinite arrangement of symmetry-equivalent points in space which define the periodicity of a crystal. In 2-dimensions there are 5 bravais lattices, while in 3-dimensions there are 14. Section 5.4.3

Lattice Points: sets of points positioned in the unit cell such that the each point's surroundings are identical. Section 5.4.8

Primitive Unit Cell: unit cells in which possess the smallest area (2D) or volume (3D) and still repeat the periodic pattern of the crystal. By definition, a primitive unit cell possesses only one lattice point. Section 5.4.8

Crystal System: The type of Bravais lattice that describes a crystal. Divided into cubic, tetragonal, orthorhombic, hexagonal, trigonal, monoclinic, and triclinic. Each crystal system has unique constraints on its lattice vectors. In this class we'll only work mathematically with systems that have orthogonal lattice vectors. Section 5.5.5

Common Lattices in Cubic, Tetragonal and Orthorhombic Systems

Primitive Lattice: denoted $P$, a lattice with spheres only at the corners of the unit cell. Only one lattice point per unit cell. Section 5.5.5

Body-Centered Lattice: denoted $I$, a lattice with spheres at the corners and one in the center of the unit cell. Two lattice points per unit cell. Section 5.5.5

Face Centered Lattice: denoted $F$, a lattice with spheres at the corners and one at the center of each face. Four lattice points per unit cell. Section 5.5.5

Base-Centered Lattice: denoted $B$, a lattice with spheres at the corners and one sphere at the center of two parallel faces. Two lattice points per unit cell. Section 5.5.5

Crystallographic Point Notation: A way of defining any location within a unit cell in terms of the cell's lattice vectors. The location of an atom or site is given by its $qrs$ coordinates, which describe the coefficients we would need to apply to the cell's lattice vectors to get us to that point. For example, a $qrs$ of $1/2$ $1/2$ $1/2$ describes the center of the unit cell because to get there we moved $1/2$a, $1/2$b, and $1/2$c to get there. Section 5.6.3

Crystallographic Direction: A vector given by $uvw$ coordinates that describes a specific direction in a crystal. $uvw$ coordinates are always normalized to the smallest possible integer values. Conventionally, square brackets $[uvw]$ are used to refer to specific directions. Section 5.7.2

Crystallographic Direction Family: crystallographic directions that are equivalent by symmetry belong to the same family of directions. Conventionally, we use angle brackets to refer to direction families $\langle uvw\rangle$. Section 5.9.2

Crystallographic Planes: Planes represented by $hkl$ coordinates, which describe the normalized reciprocal of the point where the plane intersects the three lattice vectors (see chapter 5.9 for an in-depth explanation). Conventionally, we use parentheses $(hkl)$ to refer to specific planes. Section 5.8.1

Crystallographic Plane Family: Groups of crystallographic planes that are equivalent by symmetry. Conventionally, we use curly brackets {$hkl$} to refer to families of planes. Section 5.9.2

Ball and Stick Model: A way of visualizing crystals where atoms are represented by small spheres and bonds are sticks that connect them. This model gives us an open view of the crystal and clearly shows coordination geometries, communicating to us how many bonds each atom. Section 5.10.2

Space Filling Model: A way of visualizing crystals in which we set the atomic radius of each atom such that it touches its nearest neighbors. This model is powerful in showing us the relative sizes of atoms within the structure and revealing open space in the crystal. Section 5.10.2

Atomic Coordination: The number of nearest neighbors a given atom has. Importantly, atoms on lattice points should have identical surroundings and thus the same coordination number. Section 5.10.4

Atomic Packing Factor: a measure of how much of the unit cell is occupied by atoms in the space-filling model. Essentially a measure of how closely the atoms are packed. Section 5.10.5

Interstitial Site: Lattice sites that exist between lattice atoms which are named based on the shape the surrounding atoms make. The space-filling model is typically used to define the size of these sites. Section 5.11.4

Pauling's Rules: A set of rules which leveraged knowledge of ion size, charge, and bonding to predict the structure of an ionic crystal. This is a powerful and simple theory which still helps scientists predict and understand both simple and relatively complex ionic solids. Section 5.12.3.

Polymorph: Different crystal structures observed in materials with the same composition. For materials consisting of a single element, these are known as allotropes. Section 5.12.9

Single Crystalline Material: This is a highly ordered structure in which atoms are located in predictable positions over the entire volume. Single crystalline materials are characterized by both short and long-range order. Section 5.13.3

Polycrystalline Material: Polycrystalline materials are those that have predictable short-range order (at the scale of the bond) and some degree of long-range order - but nothing that nears the degree of single crystalline materials. They are typically composed of many small crystallites which are not necessarily aligned with one another. Section 5.13.3

Amorphous Material: Materials characterized by a lock of long or short-range order beyond the scale of a single bond. Section 5.13.3