In the last
chapter, we talked about the different types of atomic bonding. Here, we’ll go
one step further and discuss how these atoms are arranged—that is, what types
of crystal structures they can have. This Callister chapter puts emphasis on
the most common metallic crystal structures (FCC, BCC, and HCP), but I will try
to give a broader overview of all Bravais lattices followed by a review of X-ray
diffraction and Bragg’s law.
Bravais lattices
and symmetry
A Bravais lattice is as an infinite set of discrete
points with an arrangement and orientation that appears exactly the same from
whichever of the points the array is viewed. In other words, it is an array of
points with translational symmetry. The basis is the actual atoms that are
positioned on these lattice points. Thus, a crystal = lattice + basis. Crystals
are built out of unit cells, which are the smallest repeating units that show
the full symmetry of the crystal.
Symmetries can be described using space groups, which are
the lattice’s translational symmetry plus other symmetry elements which are
called point groups. This can be summarized by the graphic below:
There are seven crystal systems, each with their own symmetries. We can begin with the cubic system, which has four 3-fold axes and three 4-fold axes. By stretching or compressing the cubic system along one body diagonal, we obtain the trigonal system. Since we lose all the previous symmetries except along the axis we deformed, the trigonal system has only one 3-fold axis. Similarly, by stretching or compressing the cubic system along one axis, we arrive at the tetragonal crystal system, which has one 4-fold axis (along the axis we stretched/compressed). By deforming this system along a second axis, we get an orthorhombic crystal system. This will have three 2-fold axes perpendicular to each of the faces. Next, if we shear one face with respect to the opposite face, we will have a monoclinic system. The monoclinic system will have one 2-fold symmetry, since there is only one face around which you can do a 2-fold rotation. Lastly, by shearing a second face relative to the opposite face, we get a triclinic system, which unsurprisingly has no symmetries. These crystal systems and their parameters are summarized in the table below:
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X-ray diffraction
and Bragg’s law
One of the most powerful tools for understanding the
structure of crystalline solids is X-ray diffraction. Essentially, when a wave
passes through a series of regularly spaced objects that can scatter the wave
and that have spacings on the order of the wavelength of the wave, diffraction
will occur. Since X-rays have high energies and wavelengths that match well
with atomic spacings for solids, X-rays may constructively interfere when they are
directed at a crystalline material, giving us information about the crystal
structure and interplanar distance. The condition under which constructive
interference occurs can be described by Bragg’s law, which is depicted below.
We can imagine that our material of interest is made up
of parallel planes of atoms and that they have some interplanar spacing d. Now
let’s direct a parallel beam of X-rays of wavelength λ on these planes at an
angle of θ. Two of the rays, labeled as 1 and 2 in the figure above, are scattered
by the atoms P and Q. The scattered waves 1’ and 2’ will constructively
interfere if they are still in phase—in other words, if their path difference
is an integer multiple of λ. To put this in mathematical terms, diffraction
will occur when:
which is more commonly written as:
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