Powder Method

The powder method is the standard technique for identifying minerals by X-ray diffraction. The fundamental problem it solves is that a single crystal grain must be oriented precisely to diffract X-rays from any given set of atomic planes - for an unknown mineral, the chance of a single grain being in exactly the right orientation is negligible. By using a finely ground powder of the mineral instead, grains in every possible crystallographic orientation are present simultaneously, ensuring that every set of atomic planes capable of diffracting X-rays will produce its peak as the detector sweeps through its angular range. The result is a diffraction pattern that is a fingerprint unique to each mineral’s crystal structure and unit cell dimensions. For routine identification work, the method is both fast and powerful; it is not normally used for detailed crystal structure determination, for which single-crystal methods are better - except in the case of minerals like clays and zeolites that are inherently too fine-grained to yield adequate single crystals, where the Rietveld refinement method can extract structural details from powder patterns. ^[1]

Sample Preparation

A few tenths of a gram or more of the unknown mineral is crushed in a mortar to produce a fine powder. Grains passing through a 200-mesh (<0.074 mm) or 400-mesh (<0.037 mm) sieve are used; grain sizes below 0.010 mm are best for intensity accuracy. The powder is either adhered to a glass microscope slide or packed into a well in a plastic or metal sample holder. Grains should be as randomly oriented as possible for most purposes, so that no set of atomic planes is over-represented in the diffraction pattern. The one exception is analysis of clay minerals, for which the prominent basal cleavage is deliberately aligned parallel to the holder surface to enhance diffraction from basal planes. ^[1]

The sample holder is mounted on a pivot in the diffractometer, allowing the angle of incidence θ of the X-ray beam to be varied from near zero up to nearly 90°. The X-ray detector is on a concentric goniometer that moves at exactly twice the angular speed of the sample, always maintaining the 2θ angle required to satisfy the Bragg equation for the current θ. ^[1]

Instrumental Output

For routine identification with CuKα radiation, the detector is swept through a 2θ range of 5° to 70°. This corresponds to a detectable d-spacing range of about 17.7 to 1.34 Å. Angles below 5° are avoided because the intense direct beam from the X-ray tube could damage some detectors; angles above 70° are rarely needed for identification as they pick up only higher-order or weak minor reflections. As the detector sweeps through its range, the X-ray intensity is recorded continuously by computer. A peak in intensity at a given 2θ signals that the mineral has a set of atomic planes whose d-spacing satisfies the Bragg equation at that angle. ^[1]

Because the powder sample contains grains in all orientations, each distinct set of atomic planes capable of diffracting X-rays produces its own peak at the appropriate angle as the detector moves. The intensity of each peak is controlled by mineral structure - some atomic planes are more effective reflectors than others. An important practical complication arises from the Kα1/Kα2 split in the X-ray source: at small and intermediate 2θ, the Kα2 peak shows only as a slight bulge on the shoulder of the larger Kα1 peak and the two cannot be distinguished. At large 2θ, they separate into two distinct peaks, with Kα2 at slightly higher 2θ and about half the intensity of Kα1. In quartz, for example, the &#123;100&#125; reflection shows a Kα1-Kα2 separation of only 0.06° at 2θ ≈ 26.6°, while the &#123;212&#125; reflection at 2θ ≈ 67.8° shows a separation of 0.21° that a well-aligned instrument can readily resolve. When the peaks substantially overlap they are treated as one entity and the d-spacing calculated using the weighted average Kᾱ wavelength. ^[1]

Data Reduction

From the raw diffractometer trace, two values are recorded for each peak: the 2θ angle and the peak intensity. The 2θ angle is taken as the arithmetic center of the peak at a point one-half to two-thirds of the way from the base to the top. The intensity may be reported as peak height - the height above background - or as integrated intensity, which is the total area under the peak (equivalent to the total number of counts). For manual work, the area can be approximated by treating the peak as an isosceles triangle; computer systems account for the actual peak shape and compute areas precisely. ^[1]

From these raw values, two derived quantities are calculated. The d-spacing for each peak is calculated from the Bragg equation assuming n = 1 (see below). The relative intensity is I/I₁ × 100, where I is the intensity of a given peak and I₁ is the intensity of the strongest peak. Because each mineral has its own unique crystal structure and unit cell dimensions, its set of d-spacings and relative intensities is unique. Identification is accomplished by matching the experimental set with reference data for known minerals.

Bragg Reflection Indices

The hkl values listed on reference cards and in the PDF are Bragg reflection indices, not Miller indices. They are related by: Bragg reflection index = nh, nk, nl - where n is the order of the reflection in the Bragg equation. When calculating d-spacings from an experimental pattern, all reflections are assumed to be first-order (n = 1). This is both practical and necessary: first-order reflections are typically the strongest, and it is impossible to know which peaks are higher-order until after the mineral is identified. Accepting this assumption means the calculated d-spacings may not all correspond to planes that physically exist in the crystal - some will be “hypothetical” higher-order harmonics - and this must be kept in mind when interpreting results. ^[1]

For example, in quartz the &#123;100&#125; planes have a d-spacing of 4.26 Å. This plane produces a first-order CuKα reflection at 2θ = 26.85°, a second-order at 42.48°, and a third-order at 65.87°. If all three are naively assumed to be first-order, the Bragg equation yields hypothetical d-spacings of 4.26, 2.128, and 1.418 Å - exactly 1, 1/2, and 1/3 of the true &#123;100&#125; spacing, equivalent to hypothetical &#123;100&#125;, &#123;200&#125;, and &#123;300&#125; planes. The Bragg reflection index notation (written without braces) encodes this: d₂₀₀ = second-order reflection from &#123;100&#125;; d₂₀₂ = second-order reflection from &#123;101&#125;. A useful diagnostic: if two or more experimental d-values are related by integer factors of 2, 3, or 4, they are likely different orders from the same set of planes. A complication is that some minerals possess real &#123;200&#125; and &#123;300&#125; planes at 1/2 and 1/3 of their &#123;100&#125; spacing, which produce first-order reflections at exactly the same 2θ as the second- and third-order &#123;100&#125; reflections, making them indistinguishable.

The Powder Diffraction File

Reference d-spacing data for minerals and other compounds are compiled in the Powder Diffraction File (PDF), maintained by the International Centre for Diffraction Data (12 Campus Boulevard, Newtown Square, PA 19073-3273, USA). The PDF is available in most university and industrial X-ray laboratories. Its historical format was 3×5-inch paper cards or microfiche; modern editions are digital and distributed on CD-ROM and other media, updated regularly, and are usually bundled with commercial diffractometer software. Each entry in the PDF lists the three most intense d-values and the largest d-spacing at the top, followed on the left by crystallographic data, optical and physical properties, and references, and on the right by columns of d-spacing (in Å), relative intensity (I/I₀ × 100), and Bragg reflection index, listed in order of decreasing d-spacing. ^[1]

Mineral Identification

Computer-based search routines in modern diffractometer software handle the comparison of experimental data with the PDF automatically: they identify peaks, calculate d-spacings and relative intensities, and return a ranked list of matches or near-matches. However, considerable caution is required. Contrary to expectations from popular media, computers rarely return an unambiguous single identity - they return a list of possibilities, and which possibilities appear near the top is strongly influenced by the search parameters the operator specifies. Prior knowledge of the likely identity based on physical properties, optical data, rock type, and associated minerals is essential to choosing correctly from that list. It is also necessary to manually verify any proposed match by comparing the full experimental dataset with the PDF entry and accounting for any discrepancies, since computers cannot apply geologic judgment - they are only as good as the data and instructions they receive. ^[1]

Sources of Error

The powder method is straightforward but is subject to several categories of error. Instrumental errors include misalignment of the instrument and problems with sample positioning; computer-controlled systems can introduce additional errors through improper calibration. Instrumental errors can be partially detected by including a well-characterized standard - quartz, fluorite, or another mineral with accurately known d-spacings - alongside the unknown sample. Preferred orientation is an important source of intensity error: minerals with prominent cleavage tend to settle into preferred orientations when mounted, causing intensities to differ systematically from a random-orientation sample. Optimal grain size helps: sizes below 0.010 mm minimize preferred-orientation effects. Recording errors arise simply from measurement variability - multiple runs on the same sample yield slightly different peak heights and 2θ positions. ^[1]

The geometric consequences of errors in 2θ are strongly angle-dependent: a 0.4° reading error at 2θ = 5° produces an error of 1.31 Å in d, while the same angular error at 2θ = 50° produces only 0.014 Å error. This is a strong argument for using larger-2θ peaks for precise d-spacing determinations. Additional complications arise because not all PDF peaks for a mineral necessarily appear in every experimental run - missing peaks are typically the weakest ones, but their absence introduces ambiguity. Compositional variation in solid-solution minerals shifts d-spacings away from those of the PDF reference sample, potentially preventing easy matching. Prior knowledge of the likely identity helps manage all these issues, and d-spacing variation can sometimes be turned around to estimate mineral composition. ^[1]

Mixed Samples

X-ray powder diffraction can identify individual minerals in a mixed rock or mineral sample. The diffraction pattern of a mixture contains peaks from all constituent minerals, with intensities that are roughly proportional to the relative amounts present. The identification procedure is sequential: identify the most prominent mineral, remove its d-spacings from the list, recalculate relative intensities for the remaining peaks, and proceed to identify the next mineral. This process is complicated when peaks from different minerals overlap, or when weak peaks from a minor mineral are masked by adjacent strong peaks from a dominant one. Prior knowledge of the probable mineralogy greatly simplifies the task. ^[1]

Estimation of Relative Mineral Abundance

For minerals in mixtures that cannot be physically separated - clay minerals being the most common example - XRD peak intensities offer the only means of estimating the relative amounts of the different species present. Calibration is the key difficulty: X-ray intensity depends on sample preparation details and the absorption properties of the constituent minerals. A standard approach is to prepare a calibration series of mixtures of known composition and measure peak intensities for one or more characteristic peaks of each mineral. If the calibration and unknown samples are prepared under identical conditions, reasonable accuracy can be obtained. ^[1]

Estimation of Mineral Composition

Many minerals show significant solid solution. Because substituting ions typically differ in size, solid solution changes the unit cell dimensions, and therefore shifts d-spacings for all atomic planes. Diagrams relating d-spacing variation to composition are compiled in reference literature alongside descriptions of specific mineral groups and can be used to estimate a mineral’s composition once its d-spacings are accurately measured. ^[1]

Determining Unit Cell Parameters

The mathematical relationships between unit cell parameters and d-spacings depend on the crystal system. For each crystal system, d-spacings for specific sets of planes are functions of some combination of unit cell dimensions a, b, c and/or angles α, β, γ. The number of independent d-spacing measurements needed to determine the unit cell equals the number of unit cell parameters - one for isometric, two for tetragonal, three for orthorhombic, four for monoclinic, and six for triclinic. For an orthorhombic mineral, for example, three different d-spacing measurements such as d₁₀₀, d₁₁₁, and d₀₁₀ are simultaneously solved to yield a, b, and c. For a triclinic mineral, all six parameters (a, b, c, α, β, γ) must be determined from six different d-spacings. Finding the set of unit cell parameters that best fits the full experimental diffraction pattern with minimum error is a task that modern computers handle routinely. ^[1]

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