Lecture 19: Slicing a Crystal: the Miller Planes

Lecture 19: Slicing a Crystal: the Miller Planes Summary A crystal lattice is a map that indicates how identical points are arranged in space. In the last lecture, we placed a single atom at each lattice site, and from there we could calculate atomic packing fraction. However, anything could be placed at the lattice sites, as long as the same thing is put at every lattice site. The lattice is “how” to repeat; “what” to repeat is called the basis. The unit cell serves as a way to think about where we are in the crystal. For points in the lattice, Cartesian coordinates (h,k,l) are used: conventionally, they’re scaled by the lattice constant. A direction within the crystal is described by a vector: these vectors always start at the origin, so directions are completely described by the end point of the vector, which lies on a face of the unit cell. The notation is slightly different for crystallographic vectors: the end points are scaled to all be integers, and they’re enclosed by brackets with no spaces, [hkl]. For example, the vector from (0,0,0) to (1/2,1,0) in the diagram is the crystallographic direction [120]. This notation is called Miller Indices. A negative direction is indicated with an overbar: the vector from (0,0,0) to ¯ (-1,0,0) is written as [100]. In a cubic system, all directions with the same set of Miller Indices are equivalent, regardless of order: these can be grouped into families of directions: these are indicated with angle brackets, hhkli. Miller indices can also be used to describe crystallographic planes, which can be determined by following 4 steps: 1. Read off the points at which the plane intercepts the axes in terms of fraction of the unit cell length 2. Take reciprocals of the intercepts 3. Divide by the greatest common factor to yield integer values 4. Enclose in parentheses, with no commas (hkl) For example, this plane intercepts the x-axis at x=1, the y-axis at y=1, and it never crosses the z-axis, so the z intercept is z=∞. The reciprocals of these values are 1, 1, and 0, so thus is the (110) plane. A family of planes in the cubic system contains all of the planes with the same set of Miller Indices but in any order. Families of planes are indicated by curly brackets, {hkl}. The packing density in a given plane is calculated as Planar packing density = ( atoms in the plane)/(area of the plane) [atoms/area] Note that the planar packing density is not the same as the planar packing fraction, which is unitless! Finally, the distance between planes described by the same set of Miller indices, d, can be determined 53 Diagram of Miller Indices and (110) plane © sources unknown. All rights reserved. This content is excluded from our Creative Commons license. For information, see https://ocw.mit.edu/fairuse. in terms of the lattice constant, a: a d= p (h2 + k 2 + l2 ) Why this matters One of the key properties that depends on the density in a plane is the energy to cleave a crystal. This turns out to be extremely important since cleaving is just what it sounds like: cutting, or in other words, breaking. If a crystal can be cleaved in a certain crystallographic direction more easily than another, then that is the direction in which the crystal will break most easily. In fact, this is why crystals are so often “faceted,” even when they’re just dug right out of the ground. The Why This Matters for today isn’t about how a crystal looks, but rather how it breaks. That planar packing density can tell us which plane gives in to a strain put on a crystal, and therefore how the crystal winds up deforming under strain. Take a look at the crystal being pulled here in this picture, before a tensile strain is applied and afterwards. Notice that when it’s stretched, it has those tilted disclike shapes. That’s because it breaks along the Miller planes that have the lowest packing density since those are most weakly bonded. Not only does this inform us as to when a crystalline material will break, but it gives us the atomic-scale picture of how it breaks and what shape it takes as it breaks. Now, a lot of solids may not have large differences in the energy required to break different Miller planes, so this type of phenomenon may not occur when it’s under tensile strain. For many metals the bonds are not so directional and the material can simply deform easily because of that sea of electrons accommodating the stretch (remember that wire drawing example from Lecture 16?). Then there’s the other end of the spectrum, where a material is so anisotropic that in one direction the planar density and bonding type are completely different than in the other directions. Graphite (pictured here) is a beautiful example of this. Thin long wires can be made out of graphite, too. In fact, carbon fibers, as they’re called, make up a $1.7B market today, and it’s a market that’s growing strong. If we could just get the cost down by about a factor of 10 or so, then carbon fibers would be able to compete with materials like steel on cost while being stronger, lighter, and more resilient. Just for comparison, today’s carbon fibers are 10x stronger than steel while being 5x lighter! If the cost of carbon fibers could be lowered, then we’d replace everything we could with them because they’re just that good. Carbon fibers are not graphite but they are like graphite, in the sense that they have large sp2 -bonded planes of carbon that are aligned together. But in order to give it added strength either the planes are kind of crumpled, or they’re bonded to one other through covalent links. Either way, it’s the ability to engineer the chemistry of the Miller planes that makes all the difference. 54 Gems and micrograph of wire deformation © source unknown. All rights reserved. This content is excluded from our Creative Commons license. For information, see https://ocw.mit.edu/fairuse. Why this employs Let’s keep going with the carbon fibers concept from the Why This Matter section above. As I mentioned, carbon fibers could completely revolutionize structural materials and make so many aspects of our lives lighter, stronger, and more efficient. But a lot of work is yet to be done on the fibers themselves to lower their cost, as well as to integrate them into other products, from fabrics to plastics to concrete to metal. A lot of companies are doing extremely interesting work related to these problems. And yes, it all starts from a solid understanding of those Miller planes and crystallographic directions. Hexcel has annual revenue over $2B and their flagship product is carbon fiber and composites. Note that both their name as well as logo are all about those hexagons in graphite! Hexcel is also cool because on their website they directly list student internship opportunities. Another company in this space is Solvay which makes thousands of products, many of which involve crystal planes of graphite. There’s Toray, with its origins in race cars, which lists carbon fibers as one of their 5 core businesses, and there’s Nippon Steel which has a whole subsidiary devoted to Granoc, a light-weight fabric or yarn made from carbon fibers. Its proprietary carbon fiber chemistry, they promise, is, “spinning the future.” Example Problems 1. Give the Miller indices of the shaded plane: 2. Draw the following in a cubic unit cell: a) (210) b) [121] 55 Diagram © source unknown. All rights reserved. This content is excluded from our Creative Commons license. For information, see https://ocw.mit.edu/fairuse. Solutions 1. 2. a) b) 56 Diagrams © sources unknown. All rights reserved. This content is excluded from our Creative Commons license. For information, see https://ocw.mit.edu/fairuse.