When most users come to the facility, the first thing they want to do is identify the phase
of their material. Whether you synthesize the material yourself or received it from
someone else, it is vitally important to make sure that the material is what you believe it
to be. For example, here's a tungsten disulfide powder and here is a silicon oxide and manganese
oxide mixture that I made for a later portion of this video. Often, we can't tell what a material
is simply by looking at it. Fortunately, X-ray diffraction can show the differences that our
eyes alone can't detect. A useful aspect of diffraction is that each material has its own
diffraction pattern. The example shown here is for silicon, and each peak corresponds to a particular
atomic plane, as indicated. After collecting a diffraction pattern for your material,
you can compare it to the PDF-4+ database, which contains over 380,000 other patterns,
to find a match. Access to the PDF-4+ database is available in the Diffraction Facility. This
diffraction pattern can also be used to determine the size of the unit cell. If we take Bragg's law
and recognize that the wavelength remains constant throughout the experiment and "n" can be ignored
for the purposes of calculation, our remaining variables are the d-spacing and the angle. We can
get the angles at which Bragg's law is obeyed by determining the peak locations in the diffraction
pattern. We can then solve Bragg's law for the d-spacing of each plane for which we have a
diffraction peak, which allows for the calculation of the lattice parameters. In the simple cubic
example from earlier, the d-spacing of the (100) plane gives all three lattice parameters, a, b,
and c. For the example shown on the left of the screen, it took less than 2 minutes to
load the data into the analysis software and obtain the lattice parameters with incredible
accuracy. Granted, this is an ideal reference material, so your experience may not be as fast,
but the process is quite straightforward. If your material is a solid solution,
utilizing Vegard's law along with the lattice parameters of the pure constituents and your
sample could help you determine the composition of your sample. The previous slide covered the
case in which the material was a single phase. Many times, however, users come to the lab with
samples composed of multiple phases. Shown here is a diffraction pattern for the sample
containing both silicon oxide and manganese oxide that I showed you earlier. If you look at the unit
cells on the right of the screen, you can see that the shapes are very different. Because of this,
each phase has a distinct diffraction pattern. The blue lines in the diffraction pattern designate
the silicon oxide peaks while the green lines designate the manganese oxide peaks.
The software available in the Diffraction Facility greatly helps in the determination
of which peaks belong to which phase. If the sample is prepared and tested appropriately,
you may also be able to determine how much of each phase is in your material. When making the sample,
I had weighed 50.4% silicon oxide and 49.6% manganese oxide. As you can see,
the analysis is very close. Phase quantification utilizes peak intensities. We are able to obtain
information from the breath of the reflection too. For instance, if we have crystallite sizes of a
hundred nanometers, the full width half maximum or FWHM of the reflection should be around 0.1
degrees. If the crystallite size is reduced to five nanometers, the full width half maximum
should be around 1.75 degrees. Typically, we like samples with a crystallite size of a few microns;
however, we can certainly accommodate crystallite sizes on the nanometer scale as long as you don't
mind the peaks looking very broad. In order to calculate the crystallite size,
a popular method is to utilize Sherrer's equation, shown here. It is a pretty straightforward
equation where the crystallite size equals the shape factor, typically taken to be around 0.9,
times the wavelength of the X-rays divided by the product of the full width half maximum in
radians of a particular peak and the cosine of the Bragg angle of the same peak. This equation
appears so simple but you can't let the simplicity tempt you to use it with less consideration than
you should. For instance, based off of what I have told you so far, you would think that the
sample that produced the black curve should have a much smaller crystallite size than a
sample that produced the blue curve. Surprisingly, both patterns were made with the same sample. The
difference is that a different diffractometer and optics were used. This illustrates the fact that
there is some inherent instrumental broadening of the data you collect related to the incident and
diffracted beam optics that must be accounted for before further analysis. Another issue is that,
in addition to instrumental broadening and broadening due to crystallite size,
microstrain also affects the breadth of your peaks. For instance, imagine that you have a
crystal lattice with an extra half plane of atoms. The d-spacings for the atomic planes
around this extra half plane of atoms now varies instead of being the same. If we were to take
these d-spacings and apply them to the diffraction pattern, we can see why microstrain broadens the
peak. Recall that every d-spacing corresponds to a particular value of two-theta that brings about a
peak. If the d-spacing of a particular plane now has a variety of values larger and smaller than
the average, this means that the angles larger and smaller than the Bragg angle will also bring
about some sort of intensity, thereby broadening the peak. Fortunately, there is an equation that
also governs the effects of microstrain. The broadening is equal to 4 times the strain times
the sine of theta divided by the cosine of theta. If we combine this with the broadening due to the
crystallize size, we can make a Williamson Hall plot. If we plot it so that the y-axis is the
full width half maximum after having corrected for the instrumental broadening times the cosine of
theta and put the sine of theta on the x axis and then fit the data with a straight line,
the y-intercept gives the crystallite size and the slope gives the microstrain. For this sample,
it was determined that the crystallite size is approximately 100 angstroms and
the microstrain 0.12%. Another function of XRD is the determination of the degree of crystallinity.
Crystallinity refers to the amount of order in your material and can have a large influence on
material properties. Gases have atoms that are completely random, which means no crystallinity.
Amorphous materials have limited short-range order on the scale of a few atomic or molecular
distances. Crystalline materials have a long-range order of at least tens or hundreds of atomic
distances. There are some materials, such as polymers, organics, and ion-irradiated materials,
that have a mixture of amorphous and crystalline regions. To this point, we have looked only at
crystalline materials. The images shown here, however, correspond to semi-crystalline materials.
The diffraction patterns of semi-crystalline materials have sharp crystalline peaks along with
broad amorphous peaks. Determining the degree of crystallinity of such a semi-crystalline material
requires first deconvoluting the diffraction pattern. For instance, an example of the raw
data is shown in the figure. This pattern is a sum of all peaks. Utilizing software such as HighScore
Plus or Origin, we are able to determine the contribution from each peak. Such analysis
should also give the area beneath each peak, which can then be used to calculate the degree
of crystallinity by dividing the total area of the crystalline peaks by the total area beneath
all peaks. If you would like more information about the JIAM Diffraction Facility or have any
questions, here's the address for our website and my email address. Thank you for watching!