X-ray scattering is used extensively for characterizing the structure of the materials. Since the measurement is based on the diffraction of the electromagnetic waves, the length scale must be comparable to the wavelength of the X-rays, 1-10 Angstroms (0.1-1 nm). This is just right for studying the atomic arrangements within crystals. (The medical X-ray machine is a much simpler device. It is not based on diffraction, but instead uses the large penetration depth of X-rays to make photographs. The X-ray tomograph uses similar radiation in a more complex arrangement, and the image processing is done by a computer.)
It is simple to show that the intensity the electromagnetic waves diffracted from a crystal depends on the wavelength, on the relative direction of the incident and scattered beam, and on the orientation of the crystal. One can learn the basic symmetries of the crystal by recording the diffraction pattern of a white radiation (composed of many different wavelengths) on a single crystal. This is called Laue diffraction. Alternatively, one can put a powder sample or a polycrystalline material (with lots of differently oriented micro-crystals) in a monochromatic beam, and the resulting pattern tells us about the principal lattice spacings (Debye-Scherrer method). The full exploration of the crystal structure requires the carefully recording of the diffraction peak angles and intensities for a single crystal in a monochromatic beam.
The X-ray source we use in our spectrometer has a composite spectrum, with narrow lines sitting on the top of the broad background. If we consider the narrow component of the spectrum, we have a nearly monochromatic source. With a polycrystalline sample we could record a Debye - Scherrer spectrum, but the line intensities would be very weak. Therefore we use a single crystal and we rotate it so that we maximize the scattered intensity.
Based on the method of detection, spectrometers can be divided to two groups. The simplest ones have photographic detection; the diffraction pattern is recorded on a Polaroid plate. The more accurate devices have detectors, like the GM tube in our instrument, which can be moved around to record the scattered X-ray intensity as a function of angle. These spectrometers can be further classified according to the number of independent arms or angles one can adjust; a spectrometer with three independently variable angles is called "three axis" spectrometer. In our instrument there is one arm (for the detector) and the sample always rotates by the half of the detector rotation angle. However, there is an independent way to adjust the angle of the sample relative to the X-ray beam. Thus, ours is a "two axis" spectrometer. Traditionally, the angle of the detector relative to the incident beam is called 2q ("two-theta"). The independently adjusted sample angle is denoted by f.
In a top-of-the-line single crystal spectrometer the full control of the sample orientation is done with three independent rotation axis. Together with the detector rotation, the spectrometer have four independently controlled angles. (In addition, these spectrometers may have two, permanently installed crystals with an arrangement similar to our spectrometer, but these serve only to produce monochromatic radiation and to improve detector resolution.) Of course, these instruments are fully computerized.
In summary, we have a spectrometer which is capable to measure the lattice spacing of crystals by using the strong Ka line in the spectrum. Alternatively, if we assume that the crystal structure of our sample is known, we can look at the broad (but less intense) component of the spectrum and we can use the crystal to analyze the intensity of the source as a function of energy.
Figure 1.
IMPORTANT. You are working with potentially dangerous
high voltages and radiation. There is up to 25kV voltage
inside the X-ray machine (this is similar to the voltage in a
TV set, but still very dangerous) and up to 500V to the GM tube.
The X-rays produced here are not very intense, but certainly
should not get to your eyes and it is better to avoid unnecessary
exposure to radiation. Before starting the experiment
Before starting the work, make sure that the cables connecting the various instruments are in place. The bias voltage from the source and the cable from the GM tube goes to a simple resistor-capacitor box (check inside) to separate the high voltage (dc) from the signal (ac). This box is connected to the input of the amplifier. The output drives three instruments: the oscilloscope, the rate meter and the counter. (Fig.1.)
Doing an experiment is like having a conversation with an intelligent person. To each action you do, there should be a response; you should be carefully watching the response, proceeding only if you are satisfied with it.
If everything is all right, you are ready to start the measurements. The timer will switch off the X-rays after the preset time is elapsed. Do not panic, set new time.
After finishing work, switch off in reverse order. Make sure you have recorded every settings, so you can reconstruct your experiment if you want to continue later.
This is not a sequential description of the steps you should take to have an "A" grade for the measurement. You should read this before starting, and seek advice here as you need. The main goals, as summarized above, can be reached in various ways, and significant deviations from the procedures described here are OK, as long as they are sound and safe.
During the measurement you should immediately plot the results, so that you can better judge the quality of the data and reach firm conclusions about the next steps.
The counter counts the pulses for the time interval preset on the right hand side. The discriminator on the left side is used to eliminate small amplitude noise. You have to set the discriminator levels so that when the X-ray tube is off, the counter rarely ticks (remember, background radiation from cosmic rays is always around.) The time should be selected by finding a compromise between the speed of the measurement and accuracy: if the intensity is high, 10sec may be enough, but sometimes it may be necessary to increase the time to more the a minute. Typically you do not change this time within a single measurement run, except if widely different intensities are measured, like in the absorption studies. Never forget the "golden rule": the relative error is decreasing as N-1/2.
Remove sample. Place the detector at a small angle to the direct beam. Set detector voltage and amplifier gain to reasonable numbers and record them. Set X-ray tube current to a value below 80mA, record. Observe the pulses on the oscilloscope (compare to Melissinos), count them with the counter. If two pulses come too close, the GM tube will lose the second one. This event is more likely if the intensity is high. Therefore, if you wish to measure the intensity accurately, it is not wise to use the instrument above a certain count rate. Estimate this rate (rmax) from the "dead time" t seen on the oscilloscope as rmax =1/t. Watch the pulses on the oscilloscope as you vary the detector position and the count rate exceeds rmax. What do you see? Plot the count rate vs. tube current at a detector position where the maximum rate does not exceed r max. Move the detector closer to the beam, so that r max corresponds to about 40mA current. Repeat the measurement, and scale the plot so that the low intensity parts coincide. Why are the high intensity parts different? Note rmax, and never exceed it when you need accurate intensity readout. The rate meter provides an audible signal of the counts, mostly for entertainment purposes. This signal may also warn you if you are overloading the detector.
Position the detector so that you get a reasonable count rate. Record the count rate vs. detector voltage. Do you see the plateau described in Melissinos? For the rest of the experiment operate the detector around 400V.
Once the X-ray peak of the sample is found, it may be possible to change the width of the slit #3 to obtain the optimum angular resolution (Fig. 2). For the study of the continuous component of the X-ray radiation a lower resolution / higher intensity setting is appropriate.
Since the relative accuracy is proportional to N-1/2, if the intensity goes down by a factor of 2, you need 4 times more time to obtain the same error. This may "make or break" a measurement (spending three hours taking data is OK; spending 12 hours is not a good idea). In fact it is practically impossible to complete all of the measurements discussed here if you are not prepared to change the collimation by switching slit #3.
The position of the slits and the sample determines the "zero" for the angle of the detector. A convenient way to make sure that the zero readout corresponds to zero scattering angle is to align the slits and the sample by eye (remove the detector for this; see Fig. 2). Keep in mind that each time the slits are changed or moved, the zero adjustment may also change! His work provided insight to the interaction between the charge carriers and the light in an interesting system of reduced dimensionality.
According to the rules of diffraction, the angle between the lattice planes and the incident beam (q) should be equal to the angle between the scattered beam and the lattice planes. The spectrometer tries to ensure this by rotating the crystal as the detector angle is changed. There is a pair of thin marker lines on the base plate of the sample holder, and a corresponding scale of angles on the diffractometer table. Watch the angle readout of the sample and the detector as you move the detector!
There is no guarantee that the sample is cut and positioned in the sample holder in such a way that the crystal planes are in ideal position. Therefore an independent adjustment is provided to set the angle between the sample and the incident X-ray beam. This adjustment is best done when the detector is in the 2q=0 position. Open the cover of the spectrometer, release the thumbscrew at the base of the sample holder. The sample can be rotated without moving the detector; this angle is traditionally called f. After resetting f, the thumbscrew should be tightened.
When starting with a new sample, make sure that zero detector angle (2q=0) corresponds to zero sample angle (f=0), but the optimum setting may be different for each sample. If the sample angle is not selected properly, the diffraction peak intensities will be very low, or the peaks may totally disappear. Of course, the better is the collimation the more important is to have the right alignment for the sample. Work out a systematic way to align the sample, and do it for each new crystal you want to study!
Mount a LiF crystal to the sample holder. The flat face of the crystal is approximately in (100) orientation. Keep in mind that the X-rays penetrate only a fraction of a millimeter to the surface, so most of the crystal is inactive during the measurement. After choosing appropriate (not too narrow!) collimation and setting 80mA filament current, take a full scan of the scattering (detector) angle in steps of 1 . Try to find ALL peaks, do not miss the ones at high angles! Explain the results. You may want to repeat scans around the peaks with better collimations to improve accuracy. Use the known wavelength of the copper Ka and Kb rays to determine the spacing of the diffraction planes. What is the relationship between this results and the lattice spacing of the LiF? Take similar data on other alkali-halide crystals. Determine the lattice constant and compare the results to the hard sphere model of ions.
The X-ray counts observed between the sharp peaks arise from the broad band, continuous spectrum of X-rays, known as bremsstahlung. The highest photon energy in this spectrum is equal to eV, where V is the accelerating voltage of the X-ray tube. You can set this voltage to two values by the red switch inside the machine (notice that the voltage depends on the filament current, as indicated on the label next to the switch). Measure the low angle (high energy) cut-off of the X-ray intensity. Using the lattice constant of the LiF, evaluate the cut-off wavelength. Evaluate the Planck's constant from the wavelength and the accelerating voltage. Take precautions to block X-rays reaching the detector without scattering from the crystal!
Measure the absorption of the copper Ka radiation by aluminum. Use the LiF crystal to select the Ka line, insert layers of kitchen Al foil to the beam and record the intensity (thickness is about 0.001"). What is the role of the background radiation in this measurement, and how can one minimize and take into account the background effects? Is the observed decay exponential? If you observe deviation from exponential decay, it may be due to the fact the the radiation is not monochromatic (the absorption coefficient depends on the wavelength, and for non-monochromatic radiation the intensity decays in a non-exponential fashion). How is it possible to have different wavelengths in your beam, if you use a beam diffracted from a crystal? Compare your results to the accepted values of the absorption coefficient.
The electronic processes leading to the generation of the Ka radiation if the X-ray tube also lead to characteristic features in X-ray absorption. Investigate the critical absorption edges of Cu and Ni foils! Use the crystal as an energy analyzer, and compare the intensity with and without a foil inserted to the beam.
These measurements are best done with a reduced collimation (wide slit at position #3). Use the LiF crystal as an energy analyzer, i.e take a 2q scan with no absorber. Repeat the scan with the Ni foil in the beam, and take the point by point ratio of the two spectra! This ratio will be independent of the source spectrum, and it shows the absorption of the Ni foil. Plot the curve against the energy of the X-rays. Try to find a step in the curve, characteristic of critical absorption. Do the same for the Cu foil. Show that an appropriately selected material can be used to suppress the Ka line!
Crystal: An infinite array of objects (like atoms or molecules). The crystal is constructed by repeating the basis at every Bravais lattice points.
Basis: The building block of crystals. May be a simple, spherical atom - or as complex as a DNA molecule. Sometimes we have to use a basis made up of two (or more) atoms, even if there is only one type of atom in the crystal (see example at bottom of this page).
Bravais lattice: An abstract mathematical concept. Equivalent definitions:
There are infinite number of different choices for the primitive vectors of a given lattice. For example, a'1 = a1 + a2; a'2 = a1 -a2 ; a'3 = a3 describes the same lattice.
Two dimensional examples:
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| square | rectangular | centered rect. | hexagonal | oblique |
Symmetry: An operation that maps the crystal (or the lattice) onto itself. Examples: reflection, inversion, rotations around an axis with a lattice point, rotation around an arbitrary axis. A Bravais lattice always has inversion symmetry.
Symmetry group: Another mathematical concept. The elements of the symmetry group are the symmetry operations. The important thing is the relationship between the elements - i.e. what happens if two symmetry operations are applied subsequently. In the language of the group theory, this relationship is described by the multiplication table. The symmetry group can be represented in many ways (collection of matrices, symmetry operations of a simple geometrical object etc.); as long as the multiplication table is the same, we are dealing with the same group. The name of the symmetry group is used to identify the Bravais lattices or crystals; sometimes people also use funny shorthand notations.
Primitive unit cell (or primitive cell): A volume which fills the space completely and without overlap, if shifted by each lattice vectors. The primitive unit cell contains exactly one lattice point. A possible choice is the volume made up by the primitive vectors, but there are many other possibilities. Typically does not reflect the symmetry of the lattice.
Unit cell: A volume which fills up the space with an integer multiplicity, if shifted by each lattice vectors. It contains an integer number of lattice points. Sometimes it is more convenient then the primitive unit cell:
Wave vector: Conveniently characterizes the wavelength and the direction of a plane wave. The wave vector k points to the direction of the propagation of the wave, and the length of the vector is k = 2p/ l, where l is the wavelength. The condition for constructive interference can be expressed in terms of wave vectors as (k1 - k2)d = 2pn, where n is an integer, k1, k2 are the incident and scattered wave vectors and d is the vector pointing from one scattering center to the other. (Fig. 4)
Reciprocal lattice:
The volume of the primitive unit cell in the reciprocal lattice is (2p)3 / V.
Miller indices: The Miller indices h,k,l are obtained from the "coordinates" of a reciprocal lattice vector G = hg1 + kg2 + lg3 . By definition the Miller indices are integers. For a simple cubic lattice these numbers are real coordinates in a Descartes coordinate system.
There is also an interesting relationship between Miller indices and lattice planes (although this issue has more historical importance than real use). Looking at a Bravais lattice, one can easily recognize lattice planes, i.e. planes containing infinite number of lattice points. For any plane there is an infinite number of other, parallel lattice planes, separated by a distance d. It is easy to see that the ratio x : y : z is the same for all parallel planes, where x, y, z are the intercepts of a given plane with the coordinate axis defined by the primitive vectors a1, a2, a3.
Sometimes the need arises to classify these planes. There is a convenient mapping between a given class of lattice planes and a lattice vector in the reciprocal space: for any family of lattice planes separated by a distance d there is a reciprocal vector with length G = 2p/d, and this vector is perpendicular to the lattice planes. One can show (non-trivial) that h : k : l = (1/x) : (1/y) : (1/z).
Bragg condition: nl = 2d sin q, where d is the spacing between subsequent lattice planes and q is the angle between the incident beam and the lattice planes. The scattered beam will have the same angle to the planes, so the total angle of scattering is 2q.
Laue condition, Ewald construction: If the difference between the incident (k1) and scattered (k2) wave vectors is equal to a reciprocal lattice vector, we may have non-zero diffracted intensity. With K = k2 - k1 , this simply leads to K = G, or k1G = ½ |G| . The Ewald construction is a geometrical represent-ation of these equations (see textbooks on crystallography).
Structure factor, form factor: Tells the diffraction intensities for a real crystal. The input for the Laue condition is the Bravais lattice, so this condition is independent of the basis. The structure factor and the form factor are calculated as a sum (or integral) within the unit cell; therefore they may be totally different for two different crystals, even if the crystals have the same Bravais Lattice. The scattering intensity is proportional to the absolute value square of
and 
where r(r) is the atomic charge distribution, the sum is over the atoms in the unit cell and the integration is over the volume of an atom. S is the structure factor, f is the atomic form factor. The charge density appears due to the a simple assumption concerning the electromagnetic interaction between the radiation and the charge.
Similar formula works for electron and neutron scattering, except the f integral is different, depending on the microscopic interaction. Even for X-rays, the calculation of the form factor as an integral over the charge density works only for the most simple cases. It becomes more complicated, for example, if there is a matching between the energy of atomic transitions and the X-ray quanta. However, it remains a calculation where only one atom is involved.
The crystals investigated in this measurement has two atoms in the primitive unit cell and the Bravais lattice is cubic. To illustrate how are the basis and the Bravais lattice related to the original crystal, let us look at a simple, two dimensional example:
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Crystal
Basis
Bravais lattice (the basis is added at a few points
to illustrate the concept) |
Notice, that the crystal can be reconstructed by placing the basis at every Bravais lattice point. The lattice spacing is the distance between the Bravais lattice points; this is NOT equal to the distance between the atoms in the original crystal!
In three dimensions similar ideas apply. The situation is even more interesting, since there are three different cubic Bravais lattices. To obtain more insight, study the corresponding chapters of the recommended textbooks.