Thursday, 28 January 2010

Bragg's Law of diffraction

Diffraction occurs as waves interact with a regular structure whose repeat distance is about the same as the wavelength. The phenomenon is common in the natural world, and occurs across a broad range of scales. For example, light can be diffracted by a grating having scribed lines spaced on the order of a few thousand angstroms, about the wavelength of light.
It happens that X-rays have wavelengths on the order of a few angstroms, the same as typical interatomic distances in crystalline solids. That means X-rays can be diffracted from minerals which, by definition, are crystalline and have regularly repeating atomic structures.
When certain geometric requirements are met, X-rays scattered from a crystalline solid can constructively interfere, producing a diffracted beam. In 1912, W. L. Bragg recognized a predicatable relationship among several factors.
1. The distance between similar atomic planes in a mineral (the interatomic spacing) which we call the d-spacing and measure in angstroms.
2. The angle of diffraction which we call the theta angle and measure in degrees. For practical reasons the diffractometer measures an angle twice that of the theta angle. Not surprisingly, we call the measured angle '2-theta'.
3. The wavelength of the incident X-radiation, symbolized by the Greek letter lambda and, in our case, equal to 1.54 angstroms.



The Diffractometer

A diffractometer can be used to make a diffraction pattern of any crystalline solid. With a diffraction pattern an investigator can identify an unknown mineral, or characterize the atomic-scale structure of an already identified mineral.
There exists systematic X-ray diffraction data for thousands of mineral species. Much of these data are gathered together and published by the JCPDS-International Centre for Diffraction Data.

The diffractometer in the IPFW Geosciences Department is a Philips APD3520 built in 1986. It consists of several parts.
A. The chiller provides a source of clean water to cool the X-ray tube.
B. The regulator smooths our building current to provide a steady and dependable source of electricity to the diffractometer and its peripherals.
C. The computer sends commands to the diffractometer and records the output from an analysis. We are currently using a 486-100 running DR-DOS7 to run the diffractometer, and provide interfacing with this web page. We process most of the information digitally, although we can make hardcopy analog patterns directly on the;
D. Strip-chart recorder.
E. The tube provides an X-ray source. (An old tube, shown upside down, is on the counter top.) Inside there is a 40,000 volt difference between a tungsten filament and copper target. Electrons from the filament are accelerated by this voltage difference and hit the copper target with enough energy to produce the characteristic X-rays of copper. We use one part of the copper spectrum (with a wavelength of 1.54 angstrom) to make the diffraction pattern. The radiation is monochromatized by a graphite crystal mounted just ahead of the scintillation counter.
F. The theta compensating slit collimates the X-rays before they reach the sample.
G. The sample chamber holds the specimen. We grind our samples to a fine powder before mounting them in the diffractometer, and then close the chamber to allow the collimated X-rays to enter from the left. The X-rays hit and scatter from the sample. The diffracted beams leave the chamber to the right where they can be detected by the;
H. Scintillation counter which measures the X-ray intensity. It is mounted on the;
I. Goniometer which literally means angle-measuring device. The goniometer is motorized and moves through a range of 2-theta angles. Because the scintillation counter is connected to the goniomter we can measure the X-ray intensity at any angle to the specimen. That's how we determine the 2-theta angles for Braggs's Law.

Diffraction Patterns

A diffraction pattern records the X-ray intensity as a function of 2-theta angle. All the diffraction patterns you'll see on this web site were prepared as step-scans. To run a step-scan we mount a specimen, set the tube voltage and current, and enter the following parameters:
--A starting 2-theta angle.
--A step-size (typically 0.005 degrees).
--A count time per step (typically 0.05-1 second).
--An ending 2-theta angle.
Once started, the goniometer moves through its range, stopping at each step for the alotted time. The X-ray counts at each step are saved to a file on the computer. Once finished, the data are smoothed with a weighted moving average and a diffractogram like the one below is printed or displayed.

Consider the following areas on the diffractogram.
A. The diffraction pattern is labelled with the sample name and other information pertinent to the experiment. This happens to be a pattern of ground calcite from the France Stone Quarry in Fort Wayne, Indiana. The sample was randomly mounted using the backpack technique. The diffraction pattern was prepared on March 24, 1993. The diffractometer was running at 40 kv and 30 ma. Steps were in increments of 0.005 degrees, and counts were collected for 0.25 seconds at each step. The data were smoothed with a 15-pt (weighted, moving average) filter.
B. The vertical axis records X-ray intensity. The horizontal axis records angles in degrees 2-theta. Low angles (large d-spacings) lie to the right.
C. This is one of the X-ray peaks. It happens to be the one with the smallest angle which I measured as 23.04 degrees. Solving Bragg's Law (with n=1 and wavelength=1.54 ang) we find that 23.04 degrees 2-theta corresponds to a d-spacing of 3.86 angstrom.
D. This is another peak picked for no special reason. I measured the peak at 39.37 degrees 2-theta. This corresponds to a d-spacing of 2.287 angstrom.
E. This is the largest peak on the pattern. It actually extends several times the height of this image. Many factors affect the intensity of a given peak. Some of these factors are intrinsic to the mineral under study; some of these factors are peculiar to the way a specimen is mounted in the diffractometer. (The random/backpack mounting method limits, but does not eliminate, these peculiarities). You can see a partial list of calcite peaks.
When solving scientific problems it is often useful to ask three questions: What do I Know? What can I measure? What do I want to find out?

1 comments:

Dale Ritter on 29 January 2010 at 07:22 said...

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