Working with Common Metallic Crystal Structures

Throughout the last few sections we've built up the fundamentals of crystals:

• Crystal structures emerge from regular bonding of atoms that persists in space to create a ordered and periodic structure.
• Crystal structures possess a unit cell - a geometrical entity that tiles to fill space.
• Within the unit cell there is a motif, which is repeated in space to formulate a crystal.
• One can navigate and describe the these structure using crystallographic points, directions, and planes.

The last three sections focused on simply navigating the Bravais lattices (note, we rarely mentioned atoms or molecules, we were exploring the lattice points). Now we'll "decorate" these Bravais lattices with atoms and molecules to produce real crystal structures. At this point, we'll be able to directly connect simple crystal structures with observed physical properties. We'll highlight structure-property relationships between crystal structure and density, mechanical anisotropy, chemical reactivity/catalytic activity, and so on.

Visualization Models

To visualize the atomic arrangements (and, importantly, the spaces in-between the atoms) we'll use two different structural models, each demonstrated in Figure 5.10.1 for the $\ce{CdI2}$ crystal - a crystal that has a trigonal unit cell and is used in phosphors.

• The ball-and-stick model Figure 5.10.1(a.), gives us an open view of the crystal and clearly shows coordination geometries, communicating to us how many bonds each atom. In this model it is often easier to see crystal symmetries.
• The space-filling model Figure 5.10.1(b.), 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 their packing structures. It's also for observing the open space in crystal structures, which is extremely important in alloying, processing, and the creation of compounds.

Each model has its advantages and disadvantages and we often use both to helps us explore crystal structures. It is, however, important to remember that atoms are not little round balls, and these visualizations possess inherent shortcomings - namely bonds are represented by little sticks and the atoms themselves are represented by spherically symmetric balls with hard shells. There are, of course, more advanced models used for more advanced purposes.

Common Metallic Structures

Let's being by decorating some of our simplest lattices with metal atoms. We'll start by decorating the cubic-P lattice with Po atoms to form the simplest crystal structure - which is perhaps not surprisingly called simple cubic (SC) structure. Then, we'll proceed to cubic-I lattices decorated with Fe (the body-centered cubic structure, BCC) and cubic-F decorated with Al (the face-centered cubic structure, FCC).

We'll also show a hexagonal-P lattice decorated with Ti atoms (the hexagonal close-packed structure, HCP). As this hexagonal structure does not have an orthogonal basis set, we won't do calculations on it. However, there are many structural similarities between HCP and FCC which are important to observe. HCP is a very common structure in metals.

Before continuing, we should describe a few important features of every crystal which we'll be able to derive from the crystal structures and lattice parameters:

• Atomic Coordination is the number of nearest neighbors an atom has. Remember, each lattice point has identical surroundings, so if an atom is located on a lattice point, all those points must have identical coordination. Coordination is usually easiest to see in ball-and-stick representations. Because coordination is important for bonding (electron transfer and sharing among neighbors) the coordination of atoms within a crystal influence electronic properties.
• The Number Atoms in Unit Cell is simply.... the number of atoms that are fully enclosed in the unit cell. Note - this can be tricky depending on the visualization because we often show full atoms at lattice points even though only a fraction of the atoms are positioned in the unit cell proper (the other fractions are in neighboring unit cells). We'll show examples, but for orthogonal unit cells (cubic, tetragonal, orthorhombic) with a single atomic species, you can calculated the number of atoms in the unit cell by counting the atomic sites and accounting for the atomic fractions located in the unit cell. We did this with lattices in the Section 5.10.3, but I'll reproduce it here :

where $N_{\text{Tot}}$ is the total number of spheres (or atoms, or ions, etc.) in cell, $ N_{\text{i}}$ is the number of atoms that are fully inside the cell, $N_{\text{f}}$ are the number of atoms on he faces of the cell, and $N_{\text{c}}$ are the number of atoms on the corner of the cell.

• The Lattice Parameter $a$ can be represented can be represented in terms of atomic radius $r$ when considering a space-filling structure. This is derived directly from our interpretation of the bond in an interatomic potential. The atomic radius is tabulated, and so frequently if we know the crystal structure, we can write the lattice parameters in terms of $r$. Examples below.
• The Atomic Packing Factor (APF) is a measure of the volume of the unit cell which is occupied in a space-filling model - or how closely the atoms are packed into the unit cell. The APF is calculated simply from computing how many atoms there are in the unit cell $N_{\text{Tot}}$ times the volume of each atom and dividing the product by the volume of the unit cell:

• Density is the mass per unit volume. It should be clear that we have (or can find) all the information to make this calculation: the number of atoms per unit cell, the atom's standard atomic weight ($A_{\text{r}}^{\circ}$), Avogadro's constant ($N_{\text{A}}$), and the volume of the unit cell $V_{\text{u.c.}}$:

Calculating Distances in a Unit Cell

Lastly, before moving on to some real crystal structures, we want to be sure we can compute distances $d$ between different locations in unit cells for many different reasons. The simplest and quickest way to do this is to use the distance formula in three-dimensions:

where $qrs$ are the crystallographic point coordinates and $a$, $b$, $c$ are the unit cell lengths.

Simple Cubic Structure

Figure 5.10.2 shows a static image of a simple cubic structure. (A perhaps more helpful rotatable graphic is embedded in Section 5.10.9.) This structure has a cubic-P Bravais lattice with a single atom decorating each lattice site. Importantly, this structure is close-packed (atoms touch in a space-filling configuration) along the $\langle 100 \rangle$ family of directions.

Simple cubic structures are rare in nature - the only element known to crystalize in this form at standard temperature and pressure is polonium (and the reason for this isn't simple).

In the embedded webpage (Section 5.10.14, at the end of the page) we show an interactive model for viewing the various crystal structures. Select "SC" and read the directions on the right (or press "Help" to look at different views, activate different features, or manipulate controls. Note, this model allows for viewing of lattice positions, directions, and planes and extending the unit cell ("mirroring" - which should say "translate"). (Warning - some of the notation in the right-hand window is a bit different than I have in my text. I'll change that ASAP.)

Let's show some example calculates for this simple cubic structure.

• Atomic Coordination: The hard-sphere model (or space-filling) model shows how atoms touch each other - or are coordinated - in this structure. The red atom (there is one at each lattice point) touches each of it's nearest neighbors along the $\langle 100 \rangle$ family of directions, making its coordination number 6.
• Number of Atoms in Unit Cell: The "default view" is the best for showing this. There are not 8 atoms in this unit cell. At each lattice point there is an atom that is partially positioned within the unit cell, so we can calculate the number of atoms in the unit cell using Eq. 5.10.1: $$N_{\text{SC}} = N_{\text{i}} + \frac{N_{\text{f}}}{2}+\frac{N_{\text{c}}}{8} = 0 + 0 + \frac{8}{8} = 1$$ One atom per unit cell in a simple cubic structure.
• We'll want an expression for the lattice parameter $a$ in terms of the radius of the atom $r_{\text{SC}}$, where I've added the "SC" subscript to show this is always true for a simple cubic structure. You can just look at the so-called close-packed family of directions and clearly recognize that it takes $2r_{\text{SC}} = a$. However, I always recommend using the distance formula to find the relationship between $r$ and $a$ generally for any structure because sometimes they won't be so simple. So, I'll find the close-packed direction and compute the distance between two close packed atoms. Here, I'll use Eq. 5.10.4 I'll select the atoms at $q_1r_1s_1 = 100$ and $q_2r_2s_2 = 110$, noting that for this cubic unit cell $a = b = c$: $$ \begin{align} d = 2r_{\text{SC}} &= \sqrt{(q_2 a-q_1 a)^2 + (r_2 a-r_1 a)^2 + (s_2 a-s_1 a)^2}\\ 2r_{\text{SC}} &= \sqrt{(1 a-1 a)^2 + (1 a-0a)^2 + (0 a-0 a)^2}\\ 2r_{\text{SC}} &= \sqrt{0^2 +a^2 + 0^2}\\ 2r_{\text{SC}} &= \sqrt{a^2}\\ 2r_{\text{SC}} &= a\\ \end{align} Indeed, $2r_{\text{SC}} = a$, as we could see by inspection of Figure 5.10.2.
• The Atomic Packing Factor is calculated using Eq. 5.10.2: $$\text{APF} = \frac{N_{\text{SC}} V_{\text{sphere}}}{V_{\text{u.c.}}}$$ and we know $a = b = c$ and $\alpha = \beta = \gamma = 90^{\circ}$, so $V_{\text{u.c.}} = a^3 = 8r_{\text{SC}}^3$, $V_{\text{sphere}} = \frac{4}{3}\pi r^3$, and $N_{\text{SC}} = 1$:

\begin{align} \text{APF} &= \frac{1 \times \frac{4}{3}\pi r_{\text{SC}}^3}{8r_{\text{SC}}^3}\\ \text{APF} &= \frac{1}{6}\pi =0.52\\ \end{align} - For the calculation of density - (and our first clear structure-property relationship) we need to actually add specific atoms to get the values of $r_{\text{SC}}$ and $A_{\text{r}}^{\circ}$. In this case, we'll work with Po because, well, there aren't really many other simple cubic structures. The atomic radius for polonium is $r_{\ce{Po}} = 0.166\,\text{nm}$, and its standard atomic weight is $A_{\text{r}}^{\circ}(Po) = 209 \frac{\text{g}}{\text{mol}}$. Avogadro's constant is $6.022 \times 10^{23}\, \frac{\text{atoms}}{\text{mol}}$ . Using Eq. 5.10.3.

$$ \begin{align} \rho_\text{Po} &= \frac{N_{\text{SC}} A_r^{\circ}(\text{Po})}{a^3 N_{\mathrm{A}}}\\ \rho_\text{Po} &= \frac{1 \,\text{atom} \times 209 \frac{\text{g}}{\text{mol}}}{(2 \times 0.166\,\text{nm})^3 \times 6.022 \times 10^{23}\, \frac{\text{atoms}}{\text{mol}}}\\ \rho_\text{Po} &= \frac{209\,\text{g}}{0.0365 \times 10^{-21}\,\text{cm}^3 \times 6.022 \times 10^{23}}\\ \rho_{\ce{Po}} &= 9.48\,\text{g/cm} \end{align} $$

Other Basic Crystal Structures

The definitions above and the ability to navigate a unit cell is indeed the only information you need to compute relevant values for any perfect (more on what this means later) elemental crystal. Indeed, you could use this approach for non-cubic crystals and compounds as well, but we'll only do elemental cubic ones right now... I'll encourage you to try Question 5.10.1.1 for either BCC or FCC (and an element of your choice: I suggest Li for BCC and Al for FCC). Compute these values on your own. You can compute element-independent values like the coordination, the relationship between $a$ and $r$, $\text{APF}$, and $N_{\text{Tot}}$ without elemental information. To calculate density, you need to know the element.

Let's just briefly look at the BCC and FCC structures in Figure 5.10.3 before continuing, but we've essentially seen these before (apart from their packing configurations) when going over the Bravais lattices. You can also view these structures interactively in Section 5.10.14.

The FCC Crystal Structure

The face-centered cubic (FCC) crystal structure has a cubic-F lattice with one atom decorating each lattice position. View the rotatable graphic below.

In the embedded webpage we show an interactive model for viewing the simple cubic crystal structure. Look at the directions on the right (or press "Help" to look at different views, activate different features, or manipulate controls. Note, this model allows for viewing of lattice positions, directions, and planes and extending the unit cell ("mirroring" - which should say "translate"). (Warning - some of the notation in the right-hand window is a bit different than I have in my text. I'll change that ASAP.)

The BCC Crystal Structure

The body-centered cubic crystal (BCC) structure has a cubic-I lattice with one atom decorating each lattice position. View the rotatable graphic below.

In the embedded webpage we show an interactive model for viewing the simple cubic crystal structure. Look at the directions on the right (or press "Help" to look at different views, activate different features, or manipulate controls. Note, this model allows for viewing of lattice positions, directions, and planes and extending the unit cell ("mirroring" - which should say "translate"). (Warning - some of the notation in the right-hand window is a bit different than I have in my text. I'll change that ASAP.)

Interative Graphics