1. X-Ray Spectra
X-Rays are only a small part of the electromagnetic spectum with wavelengths (l) ranging from 0.02 A to 100 A (A = Angstroms = 10^-8 m). X-Rays used to study crystals have l on the order of 1 to 2 A (i.e. Copper Ka = 1.5418 A).  Visible light has much larger l's (4000-7200 A) and thus, x-rays are much more energetic (i.e. can penetrate deeper into a material).  This can easily be seen by inspection of the Einstein equation (E = hn = hc/l; E is Energy, n frequency, c speed of light which is constant for electromagnetic radiation, l wavelength, h Plank's constant).
 

2. Diffraction and the Bragg Equation

• Diffraction of an x-ray beam striking a crystal occurs because the l of the x-ray beam is similar to the spacing of atoms in minerals (1-10 A).  When an x-ray beam encounters the regular, 3-D arrangement of atoms in a crystal most of the x-rays will destructively interfere with each other and cancel each other out, but in some specific directions they constructively interfere and reinforce one another.  It is these reinforced (diffracted) x-rays that produce the characteristic x-ray diffraction patterns that used for mineral ID.
• W.L. Bragg (early 1900's) showed that diffracted x-rays act as if they were "reflected" from a family of planes within crystals.  Bragg's planes are the rows of atoms that make up the crystal structure.  These "reflections" were shown to only occur under certain conditions which satisfy the equation:

where n is an interger (1, 2, 3, ......, n), l the wavelength, d the distance between atomic planes, and q the angle of incidence of the x-ray beam and the atomic planes.  2dsinq is the path length difference between two incident x-ray beams where one x-ray beam takes a longer (but parallel) path because it "reflects" off an adjacent atomic plane.  This path length difference must equal an integer value of the l of the incident x-ray beams for constructive interference to occur such that a reinforced diffracted beam is produced.
 

For a given l of incident x-rays and interplanar spacing (d) in a mineral, only specific q angles will satisfy the Bragg equation.  Example:  focus a monochromatic x-ray beam (x-rays with a single l) on a cleavage fragment of calcite and slowly rotate crystal.  No "reflections" will occur until the incident beam makes an angle q that satisfies the Bragg equation with n = 1.  Continued rotation leads to other "reflections" at higher values of q and correspond to when n = 2, 3, ... etc.; these known as 1st, 2nd, 3rd order, etc., "reflections".

3. X-Ray Diffraction Techniques

• Photographic plates were traditionally used to record the intensity and position of diffracted x-rays.  Modern systems use diffractometers which are electronic x-ray counters (detectors) that can measure intensities much more accurately.  Computers are used to process data and make necessary complex calculations.
• There are two main techniques.
• Single-Crystal Methods:  (x-ray beam is focused on a single crystal).
• Primary application is to determine atomic structure (symmetry, unit cell dimensions, space group, etc.,).
• Older methods (Laue method) used a stationary crystal with "white x-ray" beam (x-rays of variable l) such that Bragg's equation would be satisfied by numerous atomic planes.  The diffracted x-rays exiting the crystal all have different q and thus produce "spots" on a photographic plate.  The diffraction spots show the symmetry of the crystal.
• Modern methods (rotation, Weissenberg, precession, 4-circle) utilize various combination of rotating-crystal and camera setup to overcome limitations of the stationary methods (mainly the # of diffractions observed).  These methods use monochromatic x-rays, but vary q by moving the crystal mounted on a rotating stage.  Usually employ diffractometers and computers for data collection and processing.
• Powder Methods:  (x-ray beam focused on a powder pellet or powder smeared on a glass slide). Essential for minerals that do not form large crystals (i.e. clays) and eliminates the problem of precise orientation necessary in single-crystal methods.
• Primary application is for mineral identification.  Also can be used to determine mineral compositions (if d-spacing is a function of mineral chemistry) and to determine relative proportions of minerals in a mixture.
• Monochromatic x-rays are focused on pellet or slide mounted on rotating stage.  Since sample is powder, all possible diffractions are recorded simultaneously from hypothetical randomly oriented grains.  Mount is then rotated to ensure all diffractions are obtained.
• Older methods used photographic techniques. Most modern applications employ X-Ray Powder Diffractometers.

4.  X-Ray Powder Diffractometry

• Uses monochromatic x-rays on powder mounted on glass slide that is attached to a stage which systematically rotates into the path of the x-ray beam through q = 0 to 90°.
• The diffracted x-rays are detected electronically and recorded on a inked strip chart.  The detector rotates simultaneously with the stage, but rotates through angles = 2q.  The strip chart also moves simultaneously with the stage and detector at a constant speed.
• The strip chart records the intensity of x-rays as the detector rotates through 2q.  Thus, the angle 2q at which diffractions occur and the relative intensities can be read directly from the position and heights of the peaks on the strip chart.
• Then use the Bragg equation to solve for the interplanar spacings (d) for all the major peaks and look up a match with JCPDS cards.  JCPDS = Joint Committee on Powder Diffraction Standards.