X-Ray Diffraction

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X-ray diffraction is coherent elastic scattering of x-rays by atoms or ions in a crystal. Because the wavelength of photons with energy of order 10 KeV is a little smaller than the spacing of atoms in solids, a crystal will act as a sort of diffraction grating for x-ray. As a crystal is three dimensional, the diffraction conditions are more stringent than for a two-dimensional grating. For certain alignments of the crystal and detector relative to the x-ray beam, all atoms in the crystal scatter in phase*. This is called Bragg diffraction (see references).
[* for a simple lattice with just one atom per primitive cell. The primitive cell is the smallest structure which can be translated periodically to generate the crystal.]

Reading:

*Kittel, Introduction to Solid State Physics, 5th - 7th ed.

chap. 1 (crystal structure) and chap. 2 (reciprocal lattice). Copy in lab.

Ashcroft and Mermin, Solid State Physics

chaps. 4 (crystal lattices), 5 (reciprocal lattice), and 6 (x-ray diffraction)

Woodson, An Introduction to X-ray Crystallography [for reference, see Figs. in sec. 3.2 and 3.3]
Cullity and Stock, Elements of X-ray Diffraction, 3rd. ed. Copy in lab.

Things you should know about from your reading:

Bravais Lattice

basis

Primitive cell
Conventional unit cell of cubic lattices: fcc, bcc

structure of other cubic structures: diamond, NaCl

Bragg diffraction condition [real space]
Miller indices (for cubic crystals) in terms of lattice planes.

same, in terms of reciprocal lattice vectors

X-ray Spectroscopic Notation

Characteristic x-rays can be generated by bombarding a metal target with electrons sufficiently energetic to knock a core electron out of a metal ion. (In our x-ray tube, electrons of typically 35-40 KeV are directed at a copper target). A characteristic x-ray is emitted when an electron from a higher energy level makes a transition to fill the vacancy.

The principal quantum number of the vacancy (n = 1, 2, 3,...) is designated by K, L, M, ... respectively. The principal quantum number of the upper level of the transition, relative to the vacancy, is indicated by a subscript a, b ,d ,... Fine structure levels are indicated by a further numerical subscript. Thus K_a1 and K_a2 designate x-rays due to vacancies in the lowest core level (n=1) being filled by a transition from the next level (n=2), specifically from the P_3/2 and P_1/2 angular momentum states of n=2, respectively.

The wavelengths for copper are:

K_a1	1.540
K_a2	1.544
K_b	1.392

Here a designates the first level above the vacancy, etc. The K-absorption-edge (K-to-continuum threshold) for nickel is at 1.488 Angstrom. As a result, nickel absorbs Cu K_b radiation much more strongly than it absorbs Cu K_a radiation. A Ni foil filter is often used to attenuate Cu K_b radiation, and is included in our tube housing. You will still see double peaks due to K_a1 and K_a2 at least at large scattering angles.

Experiments:

Study Bragg scattering from various powder samples, including gold (f.c.c.), silicon (diamond structure), NaCl (rock salt sructure), graphite (hexagonal with a basis), etc. Identify the peaks and determine the lattice parameter. Determine or confirm the structure, on the basis of which sets of Miller indicies do and which do not give diffraction peaks.

An "unknown" sample "X" is provided (something from the cubic class). Obtain evidence for its crystal structure and determine the lattice constant.

Look at Pb-Sn solder. Compare to diffraction expected for pure Pb and pure Sn. What does this tell you about the alloy?

Other things to check: Separation of CuK_a1 and CuK_a2 peaks. Effect of Ni filter: If you remove it, do you see CuK_b peaks? Effects of scan rate; peak widths and precision with which centers of peaks can be located. Calculate the resulting uncertainty in the lattice parameter, as calculated from a peak at small and at large angle, and from all the peaks combined. [i.e., do the error analysis!]

Safety: Do not bypass interlocks or other safety features.

Instructions for operating the X-ray Diffractometer

A. Starting Procedure

1. With the high voltage off, open the front door of the enclosure. Install your sample. Check that the limit switches are securely in position. Turn on power switches of the two interlock modules inside the enclosure. (The nickel filter, collimating slit, and two detector slits should be already in position. Do not attempt to change the x-ray tube alignment). Close door of enclosure.

2. Turn on the water at the ball valve on the wall.

3. Turn on power to the stepper motor controller, detector electronics, and strip chart recorder. Position the goniometer at the desired starting angle (You could do this manually when the enclosure is open).

Turn on the high voltage generator "control power" (bottom button).

4. Set the strip chart recorder speed and sensitivity (push button). Set the pen on a cm line by moving the paper manually. Record necessary information on the chart.

5. Set the desired stepper motor rate. Select "Scan".

6. Turn on high voltage (top button: "x-ray on"). The generator should automatically ramp up to 40 kV, 20 ma (or as preset ) within 1 min.

7. Open the shutter: Turn on the power switch of the external black box and then push the

"x-ray on" button.

8. Push the "start" button on stepper motor controller to start scan and strip chart recorder.

9. Enter starting time & other information in the log book.

10. Calculate the completion time of scan and be present to stop the scan. Do not depend on the limit switches.

B. Running an additional sample

1. Close the shutter by pushing the "x-ray OFF" button on the external box.

2. Turn off the high voltage by pushing the middle button on the HV generator: "X-ray OFF".

3. Open & latch the enclosure door. [If you fail to do steps 1 or 2, they will be done by the interlock system, but only after the door is partially open].

4. Install new sample.

5. Close the enclosure door.

6. Proceed from step A.4.

C. Shutting down 1. Close the shutter.

2. Turn off the high voltage by pushing the middle button on the HV generator: "X-ray OFF".

3. Turn off the high voltage generator "control power" (bottom button).

4. Enter time of stopping in the log book.

5. Open the enclosure. Remove your sample. Turn off the two interlock power switches inside enclosure. Return the goniometer to a typical starting position.

6. Turn off all other power switches.

7. Close the water valve on the wall.

Note: Some alignment procedures require bypassing the enclosure interlock.

This is not part of the normal operating procedure and should be done only by persons authorized by Larry Suddarth.

Likewise, do not bypass the water flow interlock without Larry's knowledge.

Suggestions for data acquisition:

Note that the strip chart recorder is driven by a stepper motor which is driven by the same source as the goniometer stepper motor. Therefore you can change the scan rate and the scale on the chart remains the same (This is the "external" mode). The standard settings marked on the recorder are good for most situations.

Maybe do a fast scan ( 5^o /min) before standard scan (2 or 1^o /min). It may be interesting to scan an individual peak even more slowly (say 1/2^o /min) to see how much the quality of data improves.
Calculate time of completion of your scans. It is risky to depend on limit switch to terminate scan.
The x-rays are actually scattered by the electrons in the atoms, so, to the extent that all the electrons in an atom scatter in phase (the path differences for electrons in the same atom are small compared to the wavelength), the scattered intensity is proportional to Z², where Z is the number of electrons in the atom or ion. This means that low-Z atoms (or ions) are relatively hard to see.

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