6 Soil Minerals
6.2 X-RAY DIFFRACTION AND GEOTECHNICAL ENGINEERING
6.2.1 Not Your Usual X-Rays
If you ask a medical technician what is meant by X-ray diffraction, you most likely will be met with a querulous look and be asked to repeat the question.
Instead of using penetration of X-rays to observe an inner structure, X-ray diffraction defines relative positions of atoms in minerals or other solid objects.
The strongest diffractions are obtained from crystalline materials that have a regular array of atoms arranged in planes so that the X-rays in effect bounce off as if reflected from a mirror. A critical distinction is made from mirror reflections because X-ray diffraction can occur only at specific angles that depend on distances between the reflecting planes of atoms. By measuring the angles at which reflections occur, one can decipher the distances and identify the minerals. It also is possible to measure exactly how much a clay structure expands when it is wet with water.
6.2.2 Production of X-Rays
X-rays are generated by an electron beam striking a metal plate that is aptly called the ‘‘target.’’ The voltage pulling electrons to the target is high enough to knock electrons out of target atoms, and as orbiting or shell electrons are replaced, energy is given off as X-rays. The target in an X-ray tube therefore glows with a brilliant but invisible light. The voltage is of the order of 50,000 volts. Such high voltages are strictly prohibited in television or computer screens to prevent the production of X-rays.
Replacement of orbiting electrons from outer shells in atoms involves a stepwise energy change, so X-rays have particular wavelengths. Other ‘‘white radiation’’
wavelengths also are produced and are used for medical X-rays, but for diffraction measurements only a single wavelength can be used, so the rest are filtered out.
The strongest monochromatic or single-wavelength X-rays are produced when a position in the innermost or ‘‘K’’ atomic shell is replaced by one falling in from the next outer shell, which generates ‘‘K radiation.’’ The most useful X-rays for mineral identification are from a copper target and are therefore referred to as
‘‘copper K radiation.’’
6.2.3 Geometry of Diffraction
Figure 6.1 shows a diffractometer that is designed to measure diffraction angles from a sample placed in an X-ray beam. A beam is obtained by simply looking at the target through some narrow slits, and is directed toward the sample that is at the center of a circular goniometer. A counter then is moved around the outside of the circle to detect X-rays, and is synchronized with a chart recorder so that when diffraction occurs and the count increases, the angle can be measured.
One of the most famous uses of X-ray diffraction was to determine the structure of DNA.
Unlike reflections in a mirror the diffraction angle is the same on both sides of the crystal and is designated by . As shown in Fig. 6.1, the angle measured around the circumference of a diffractometer is 2.
Figure 6.1 Schematic diagram of an X-ray
diffractometer used for identifying minerals in soils and measuring clay mineral expansion.
6.2.4 Relating Diffraction Angles to Crystal Structure
The interpretation of X-ray diffraction would be relatively simple if each crystalline material had only one set of reflecting planes, and as a matter of fact analyses of clay minerals do emphasize only one set of planes. However, there also are other planes having different interatomic spacings, much as parallel lines drawn at various angles through figures in wallpaper have different spacings (Fig. 6.2). Each mineral species therefore presents a combination of diffraction angles and intensities that serves as a fingerprint to identify that mineral.
Thousands of diffraction patterns have been collected, categorized, and published for purposes of mineral or crystal identification by the American Society for Testing and Materials. Because of its speed and accuracy, X-ray diffraction has many uses including forensic investigations.
6.2.5 Bragg Made It Easy
The principles of X-ray diffraction were described by von Laue in the early 1900s, but a simplified model was needed before diffraction was easily under-stood and utilized. The model was devised by father-and-son English physicists William and Lawrence Bragg, who shared a Nobel Prize for their contribution. The relationship describing diffraction of X-rays from crystals is called Bragg’s Law, but a more accurate description is ‘‘Bragg’s reflection analogy.’’
Figure 6.2
Atoms in a crystal occur in a regular pattern like flowers in wallpaper, so distances
between identical planes depend on the orientation of the planes and the size and arrangement of the flowers and caterpillars.
Bragg, father and son, suggested that even though X-rays actually come from point sources, namely electrons in crystalline atoms, the rays may be thought of as reflecting from atomic planes. As a wave front is reflected from adjacent planes, the interlayer distance imposes an extra path length that is shown in Fig. 6.3.
In order for rays to reinforce, that extra path length must equal an exact number of wavelengths. This enables writing an equation:
nl ¼2d sin ð6:1Þ
where n is a whole number, l is the X-ray wavelength, d is the distance between identical crystal planes, and is the diffraction angle. This is the Bragg Law.
When a reflection occurs, the interlayer distance d is obtained by measuring the angle and knowing the X-ray wavelength.
If a reflection is obtained with n ¼ 1, according to eq. (6.1) another reflection should be obtained with n ¼ 2, or at exactly one-half of the d-spacing. This is called a second-order reflection. In general as the order increases the reflection becomes weaker, but that is not always the case. Reflection intensities are modeled with Fourier analyses for determination of crystal structures.
The unit of measure is the A˚ngstom, or A˚, which equals 108cm or 10 nm (nanometers). As a reference, the diameter of an oxygen ion in crystal structures is 2.64 A˚.
Example 6.1
Mica, the shiny, flakey mineral in mica schist, is closely related to clay minerals and gives an X-ray diffraction angle 2 ¼ 8.88 with an X-ray wavelength of 1.54 A˚. At what angle may one expect a second-order reflection?
Answer: Equation (6.1) solved for the first-order reflection is ð1Þð1:54Þ ¼ 2ðdÞ sin ð4:4Þ
from which d ¼ 10.0 A˚. Substituting this value in eq. (6.1) and solving for n ¼ 2 gives ð2Þð1:54Þ ¼ 2ð10:0Þ sin
2 ¼ 17:7
Figure 6.3 Derivation of the Bragg Law relating diffraction angle to d-spacing.
A similar solution can be made for third-order and higher reflections. Note that diffraction angles always are expressed in terms of 2.
6.2.6 Defining Orientations of Crystal Planes
Planes in a crystal are defined on the basis of their relationships to crystallographic axes. These axes may or may not be orthogonal. In rock salt or diamond, which crystallize in the cubic system, the axes are at right angles. The smallest repeating pattern is called a unit cell, which in this case has the shape of a cube.
Question: Can you identify a unit cell in Fig. 6.1?
The orientation of a crystalline plane is described by a ‘‘Miller index,’’ which is the number shown on the plane orientations in Fig. 6.1. The most useful Miller index in clay mineralogy is 001. The d001 distance is the distance between repeating layers stacked like pages in a book. For mica, d001¼10 A˚, so each layer in a mica crystal is 10 A˚ thick, or about the diameter of three oxygen atoms.