Electromagnetic waves form a continuous spectrum covering at least 15 orders of magnitude in wavelength. All travel at 300,000 km/sec in a vacuum, reflect, refract according to Snell's Law, are diffracted by edges, slits and gratings and can be related to their energies by the equation:
Where E is energy, v is frequency, c is velocity of light, lambda is wavelength, and h is Planck's constant. The equation shows the inverse relationship of energy to wavelength; the shorter the wavelength, the more energetic the wave. X-rays are that portion of the spectrum where lambda is between about 100 and 0.02 Angstroms - we are most interested in those X-rays with wavelengths around 1 Angstrom because this is the order of the interplanar spacings in crystals.
Generation of X-rays:
In a vacuum, a tungsten filament (the cathode) is heated and boils off electrons which are then accelerated across a potential and impact onto the anode which can be made of different metals. Above some minimum energy of the incoming electrons, a continuous spectrum of x-rays is generated from the target material. Greater potential gaps lead to both more x-ray emission at all energies and the production of higher energy x-rays (the lower wavelength limit becomes lower). The lower wavelength (higher energy) cutoff corresponds to complete stoppage of the electron in a single interaction; the rest of the "white radiation" or "continuous radiation" is caused by smaller amounts of energy loss per collision or interaction. The x-ray tube is generally water cooled to remove the heat produced by these collisions.
At some critical voltage (dependent on the target element) a line spectrum characteristic of the target material is produced and superimposed on the background radiation. These discrete lines are caused by the incoming electrons having enough energy to dislodge an electron from the K, L, or M shells of the target atoms and then this hole is filled by an outer electron falling back into the hole.
L to K transitions produce 'K alpha' emission while M to K transitions produce 'K beta' emission. Similarly, M to L transitions produce 'L alpha' emissions. Because there are several energy sublevels in the L, M, N levels from which electrons can drop down to fill in the K-shell, there are in fact 'K alpha 1' and 'K alpha 2' peaks which are very close to one another in energy. The value below (e.g. 1.5418 for Cu) is the weighted average of these energies - for careful x-ray work it is important to actually work with the 2 energies separately.
The most common targets and their characteristic 'K alpha average' wavelengths are:
Molybdenum 0.7107 Copper 1.5418 Cobalt 1.7902 Iron 1.9373
For X-ray diffraction we would like to have monochromatic x-rays otherwise we will have multiple peaks for each plane. To attain this goal requires [1] a pure element target and [2] a 'K beta' filter. Metal foils have absorption edges for x-rays which can be used to filter out most of the unwanted white or continuous radiation as well as the 'K beta' peak. In particular, Zr foil has an absorption edge that lies between the 'K beta' and 'K alpha' peaks for Mo. Ni foil works for Cu.
In class we will use a laser and several sieves to demonstrate diffraction. Because atoms are too small to see and x-rays are invisible, we will start with a sieve and a red laser - both of which are visible. As detailed in Brady and Boardman (1995, J. Geol. Education 43, 471-6) it is a simple matter to develop the Fraunhofer Equation which relates the periodic repeat distance to the wavelength of light and the angle of diffraction, i.e.:
By comparing the results from different size sieves and different wavelength lasers, we can gain an intuition into the process of diffraction which can be applied to x-rays and the invisible periodic arrays of atoms that we call crystals.
Because crystals are ordered 3-D periodic structures, the interaction of an x-ray beam with this array of planes of atoms will lead to scattering and can lead to cooperative scattering which we know as diffraction. Read over pages 278-80 in your text for the development of the Laue equations, the simplification made by Bragg allowing us to visualize diffraction using the ideas of reflection, and the origin of the Bragg equation. Remember that the treatment of diffraction as reflection is a simplification - diffraction is really a complex scattering problem. However the reflection approach works and we will use it.
Bragg recognized that the diffracted x-rays act as though they were "reflected" from planes of atoms in the structure. Unlike the continuous reflection of light however, the x-ray "reflection" took place only at certain angles that were controlled by the spacings between atomic planes and the wavelength of the radiation. He showed that this "reflection" took place only when the equation:
n(lambda) = 2d sin (theta)
was satisfied where n is an integer, lambda is the wavelength of the radiation, d is the interplanar spacing, and theta is the incident angle relative to the plane of atoms. For "reflection" to occur from a set of parallel planes of atoms in a structure, these "reflections" must be in phase so that they constructively reinforce each other and generate a measurable signal. The geometry required for constructive interference can be seen in the following figure:
If the path difference (ABC) for the diffracted rays 1 and 2 is exactly one wavelength then they will reinforce each other by constructive interference. Under these same conditions the path difference (A'B'C') for rays 1 and 3 will be two wavelengths and they also will constructively interfere.
The powder x-ray diffractometer uses Bragg's Law and the fact that we know the wavelength (lambda) and by varying the detector location (theta) can determine the interplanar spacings 'd'. This allows us to identify the mineral from information about its structure.
The electron microprobe uses Bragg's Law in a different way. In this instrument, electrons are accelerated and impact upon an unknown mineral causing x-rays to be emitted by the various elements constituting the unknown. These x-rays may be collected and analyzed using a crystal with a known d-spacing and a detector for counting the x-rays. Effectively we set up a geometry such that we know 'd' and theta and thus can solve for the wavelength lambda. The measured characteristic x-rays identify the elements present - the use of standards of known composition allows the amounts of the elements to be determined as well. Thus we can identify an unknown based on its chemistry.
These two techniques are complementary to one another and both rely on Bragg's Law and the diffraction of x-rays.
X-ray fluorescence analysis is similar to the electron microprobe in that both instruments collect and count x-rays emitted from a sample using Bragg's Law and a crystal of known d-spacing to separate out specific wavelengths of radiation. The difference between the techniques is that in the electron probe individual grains of a largely intact sample are excited using an accelerated electron beam while in x-ray fluorescence, a ground up sample is excited using the same kind of x-rays used in powder diffraction studies. The electron probe is used for individual mineral analyses, x-ray fluorescence is used for whole rock analyses.
See page 146 in your text for the example of NaCl.
What do we need to know?
What the atoms are and where they are = the 'solution' of the crystal structure.
What is the composition?
Density of halite is 2.165 g/cm3
What is the structure?
Return to NaCl
Consider the view below of a face-centered cubic array of atoms and the interaction of x-rays with the planes of atoms in the structure. At some angle theta, constructive interference will occur from the 'reflections' from the top and the bottom (100) planes of atoms related by d(100), the unit cell translation. At this angle, AB + BC = lambda. However there is another equally dense plane of atoms half way between each pair of (100) planes and for these (200) planes, A'B + BC' = lambda/2. Thus the (200) reflection will be 180 degrees out of phase with the (100) reflection and cancel it out - we say that the (100) reflection is extinct. This is a general truth for all face-centered lattices irrespective of the crystal system and it can be generalized by saying that all observed (non-extinct) reflections have (hkl) values where all the digits are either odd or even (where 'zero' is even). (The (200) reflection will be present at the appropriate angle.) The proof of this is beyond this course but rules like this prove the power of understanding crystallography. There are other rules governing the extinctions in c-centered lattices; no systematic extinctions occur in primitive lattices.
Orienting a single crystal of a mineral such that even a single plane satisfies Bragg's Law is tedious work and thus 2 possibilities arise:
See your book for a detailed description of the powder camera and the strips of film it produces. In a lab on the 3rd floor we have an automated powder diffractometer which does away with the film and makes the determination of intensity and location of the various diffracted beams much easier. The intensity of a particular diffracted ray is proportional to the density of atoms defining the plane in the structure. This means that in general the intense reflections are caused by planes with simple indices and relatively large d-spacing. Each of the construction lines in the figure below is the same length - count the number of atoms per line and note the relationship to d-spacing.
