# Diffraction Is A Result Of

Diffraction is a result of A) refraction. C) interference. D) polarization. By definition, diffraction is the process by which a wave is spread out as a result of passing through a narrow aperture or across an edge, typically accompanied by interference between the waveforms produced. The condition to obtain diffraction is that the dimensions of aperture or of the obstacle must be comparable to wavelength.

File:Square diffraction.jpg
The intensity pattern formed on a screen by diffraction from a square aperture
File:Diffraction pattern in spiderweb.JPG
Colors seen in a spider web are partially due to diffraction, according to some analyses.[1]

Diffraction refers to various phenomena associated with the bending of waves when they interact with obstacles in their path. It occurs with any type of wave, including sound waves, water waves, and electromagnetic waves such as visible light, x-rays and radio waves. As physical objects have wave-like properties, diffraction also occurs with matter and can be studied according to the principles of quantum mechanics. While diffraction always occurs when propagating waves encounter obstacles in their paths, its effects are generally most pronounced for waves where the wavelength is on the order of the size of the diffracting objects. The complex patterns resulting from the intensity of a diffracted wave are a result of interference between different parts of a wave that traveled to the observer by different paths.

## Examples of diffraction in everyday life

The effects of diffraction can be readily seen in everyday life. The most colorful examples of diffraction are those involving light; for example, the closely spaced tracks on a CD or DVD act as a diffraction grating to form the familiar rainbow pattern we see when looking at a disk. This principle can be extended to engineer a grating with a structure such that it will produce any diffraction pattern desired; the hologram on a credit card is an example. Diffraction in the atmosphere by small particles can cause a bright ring to be visible around a bright light source like the sun or the moon. A shadow of a solid object, using light from a compact source, shows small fringes near its edges. All these effects are a consequence of the fact that light is a wave.

Diffraction can occur with any kind of wave. Ocean waves diffract around jetties and other obstacles. Sound waves can diffract around objects, this is the reason we can still hear someone calling us even if we are hiding behind a tree. Diffraction can also be a concern in some technical applications; it sets a fundamental limit to the resolution of a camera, telescope, or microscope.

## History

File:Young Diffraction.png
Thomas Young's sketch of two-slit diffraction, which he presented to the Royal Society in 1803

The effects of diffraction of light were first carefully observed and characterized by Francesco Maria Grimaldi, who also coined the term diffraction, from the Latin diffringere, 'to break into pieces', referring to light breaking up into different directions. The results of Grimaldi's observations were published posthumously in 1665.[2][3]Isaac Newton studied these effects and attributed them to inflexion of light rays. James Gregory (1638–1675) observed the diffraction patterns caused by a bird feather, which was effectively the first diffraction grating. In 1803 Thomas Young did his famous experiment observing interference from two closely spaced slits. Explaining his results by interference of the waves emanating from the two different slits, he deduced that light must propagate as waves. Augustin-Jean Fresnel did more definitive studies and calculations of diffraction, published in 1815 and 1818, and thereby gave great support to the wave theory of light that had been advanced by Christiaan Huygens and reinvigorated by Young, against Newton's particle theory.

## The mechanism of diffraction

File:Single-slit-diffraction-ripple-tank.jpg
Photograph of single-slit diffraction in a circular ripple tank

The very heart of the explanation of all diffraction phenomena is interference. When two waves combine, their displacements add, causing either a lesser or greater total displacement depending on the phase difference between the two waves. The effect of diffraction from an opaque object can be seen as interference between different parts of the wave beyond the diffraction object. The pattern formed by this interference is dependent on the wavelength of the wave, which for example gives rise to the rainbow pattern on a CD. Most diffraction phenomena can be understood in terms of a few simple concepts that are illustrated below.

The most conceptually simple example of diffraction is single-slit diffraction in which the slit is narrow, that is, significantly smaller than a wavelength of the wave. After the wave passes through the slit a pattern of semicircular ripples is formed, as if there were a simple wave source at the position of the slit. This semicircular wave is a diffraction pattern.

If we now consider two such narrow apertures, the two radial waves emanating from these apertures can interfere with each other. Consider for example, a water wave incident on a screen with two small openings. The total displacement of the water on the far side of the screen at any point is the sum of the displacements of the individual radial waves at that point. Now there are points in space where the wave emanating from one aperture is always in phase with the other, i.e. they both go up at that point, this is called constructive interference and results in a greater total amplitude. There are also points where one radial wave is out of phase with the other by one half of a wavelength, this would mean that when one is going up, the other is going down, the resulting total amplitude is decreased, this is called destructive interference. The result is that there are regions where there is no wave and other regions where the wave is amplified.

Another conceptually simple example is diffraction of a plane wave on a large (compared to the wavelength) plane mirror. The only direction at which all electrons oscillating in the mirror are seen oscillating in phase with each other is the specular (mirror) direction – thus a typical mirror reflects at the angle which is equal to the angle of incidence of the wave. This result is called the law of reflection. Smaller and smaller mirrors diffract light over a progressively larger and larger range of angles.

Slits significantly wider than a wavelength will also show diffraction which is most noticeable near their edges. The center part of the wave shows limited effects at short distances, but exhibits a stable diffraction pattern at longer distances. This pattern is most easily understood and calculated as the interference pattern of a large number of simple sources spaced closely and evenly across the width of the slit.

This concept is known as the Huygens–Fresnel principle: The propagation of a wave can be visualized by considering every point on a wavefront as a point source for a secondary radial wave. The subsequent propagation and interference of all these radial waves form the new wavefront. This principle mathematically results from interference of waves along all allowed paths between the source and the detection point (that is, all paths except those that are blocked by the diffracting objects).

## Qualitative observations of diffraction

Several qualitative observations can be made of diffraction in general:

• The angular spacing of the features in the diffraction pattern is inversely proportional to the dimensions of the object causing the diffraction, in other words: the smaller the diffracting object the 'wider' the resulting diffraction pattern and vice versa. (More precisely, this is true of the sines of the angles.)
• The diffraction angles are invariant under scaling; that is, they depend only on the ratio of the wavelength to the size of the diffracting object.
• When the diffracting object has a periodic structure, for example in a diffraction grating, the features generally become sharper. The third figure, for example, shows a comparison of a double-slit pattern with a pattern formed by five slits, both sets of slits having the same spacing, between the center of one slit and the next.

## Quantitative description of diffraction

To determine the pattern produced by diffraction we must determine the phase and amplitude of each of the Huygens wavelets at each point in space. That is, at each point in space, we must determine the distance to each of the simple sources on the incoming wavefront. If the distance to each of the simple sources differs by an integer number of wavelengths, all the wavelets will be in phase, resulting in constructive interference. If the distance to each source is an integer plus one half of a wavelength, there will be complete destructive interference. Usually it is sufficient to determine these minimums and maximums to explain the effects we see in nature. The simplest descriptions of diffraction are those in which the situation can be reduced to a 2 dimensional problem. For water waves, this is already the case, water waves propagate only on the surface of the water. For light, we can often neglect one direction if the diffracting object extends in that direction over a distance far greater than the wavelength. In the case of light shining through small circular holes we will have to take into account the full three dimensional nature of the problem.

### Diffraction from an array of narrow slits or a grating

File:TwoSlitInterference.svg
Diagram of two slit diffraction problem, showing the angle to the first minimum, where a path length difference of a half wavelength causes destructive interference.
File:Diffraction of Gaussian beams.svg
Diffraction of a Gaussian beam by a grating. A convergent beam is needed in a real set-up for sharp diffraction maxima for macroscopic gratings demonstrating single slit, double slit, triple slit ...

Multiple-slit arrangements can be described as multiple simple wave sources, if the slits are narrow enough. For light, a slit is an opening that is infinitely extended in one dimension, which has the effect of reducing a wave problem in 3-space to a simpler problem in 2-space.

The simplest case is that of two narrow slits, spaced a distance a apart. To determine the maxima and minima in the amplitude we must determine the difference in path length to the first slit and to the second one. In the Fraunhofer approximation, with the observer far away from the slits, the difference in path length to the two slits can be seen from the image to be

${displaystyle Delta S={a}sin theta ,}$

Maxima in the intensity occur if this path length difference is an integer number of wavelengths:

${displaystyle {a}sin theta =nlambda ,}$

where:

n is an integer that labels the order of each maximum,
${displaystyle lambda }$ is the wavelength,
${displaystyle a}$ is the distance between the slits,
and θ is the angle at which constructive interference occurs.

And the corresponding minima are at path differences of an integer number plus one half of the wavelength:

${displaystyle {a}sin theta =lambda (n+1/2),}$

For an array of slits, positions of the minima and maxima are not changed, the fringes visible on a screen however do become sharper as can be seen in the image. The same is true for a surface that is only reflective along a series of parallel lines; such a surface is called a reflection grating.

We see from the formula that the diffraction angle is wavelength dependent. This means that different colors of light will diffract in different directions, which allows us to separate light into its different color components. Gratings are used in spectroscopy to determine the properties of atoms and molecules, as well as stars and interstellar dust clouds by studying the spectrum of the light they emit or absorb.

Another application of diffraction gratings is to produce a monochromatic light source. This can be done by placing a slit at the angle corresponding to the constructive interference condition for the desired wavelength.

### Single-slit diffraction

File:Wave Diffraction 4Lambda Slit.png
Numerical approximation of diffraction pattern from a slit of width four wavelengths with an incident plane wave. The main central beam, nulls, and phase reversals are apparent.

Slits wider than a wavelength will show diffraction at their edges. The pattern is most easily understood and calculated as the interference pattern of a large number of simple sources spaced closely and evenly across the width of the slit. We can determine the minima of the resulting intensity pattern by using the following reasoning. If for a given angle a simple source located at the left edge of the slit interferes destructively with a source located at the middle of the slit, then a simple source just to the right of the left edge will interfere destructively with a simple source located just to the right of the middle. We can continue this reasoning along the entire width of the slit to conclude that the condition for destructive interference for the entire slit is the same as the condition for destructive interference between two narrow slits a distance apart that is half the width of the slit. The result is a formula that looks very similar to the one for diffraction from a grating with the important difference that it now predicts the minima of the intensity pattern.

${displaystyle dsin(theta _{min})=nlambda ,}$

n is now an integer greater than 0; d is the width of the slit.

The same argument does not hold for the maxima. To determine the location of the maxima and the exact intensity profile, a more rigorous treatment is required; a diffraction formalism in terms of integration over all unobstructed paths is required. The intensity profile is then given by

 ${displaystyle I(theta ),}$ ${displaystyle =I_{0}{left[mathrm {sinc} left({frac {pi d}{lambda }}sin theta right)right]}^{2}}$

where the sinc function is given by sinc(x)=sin(x)/x.

### Multiple extended slits

For an array of slits that are wider than the wavelength of the incident wave, we must take into account interference of wave from different slits as well as interference between waves from different locations in the same slit. Minima in the intensity occur if either the single slit condition or the grating condition for complete destructive interference is met. A rigorous mathematical treatment shows that the resulting intensity pattern is the product of the grating intensity function with the single slit intensity pattern.

An array of N slits, each of width d and spaced a distance a apart results in the following intensity pattern:

${displaystyle Ileft(theta right)=I_{0}left[operatorname {sinc} left({frac {pi d}{lambda }}sin theta right)right]^{2}cdot left[{frac {sin left({frac {Npi a}{lambda }}sin theta right)}{sin left({frac {pi a}{lambda }}sin theta right)}}right]^{2}}$

When doing experiments with gratings that have a slit width being an integer fraction of the grating spacing, this can lead to 'missing' orders. If for example the width of a single slit is half the separation between slits (i.e. the duty cycle of the grating is 50%), the first minimum of the single slit diffraction pattern will line up with the second maximum of the grating diffraction pattern. This expected diffraction peak will then not be visible. The same is true in this case for any even-numbered grating-diffraction peak.

## Particle diffraction

Quantum theory tells us that every particle exhibits wave properties. In particular, massive particles can interfere and therefore diffract. Diffraction of electrons and neutrons stood as one of the powerful arguments in favor of quantum mechanics. The wavelength associated with a particle is the de Broglie wavelength

${displaystyle lambda ={frac {h}{p}},}$

where h is Planck's constant and p is the momentum of the particle (mass × velocity for slow-moving particles) . For most macroscopic objects, this wavelength is so short that it is not meaningful to assign a wavelength to them. A Sodium atom traveling at about 3000 m/s would have a De Broglie wavelength of about 5 pico meters.

Because the wavelength for even the smallest of macroscopic objects is extremely small, diffraction of matter waves is only visible for small particles, like electrons, neutrons, atoms and small molecules. The short wavelength of these matter waves makes them ideally suited to study the atomic crystal structure of solids and large molecules like proteins.

Relatively recently, larger molecules like buckyballs,[4] have been shown to diffract. Currently, research is underway into the diffraction of viruses, which, being huge relative to electrons and other more commonly diffracted particles, have tiny wavelengths so must be made to travel very slowly through an extremely narrow slit in order to diffract.

## Bragg diffraction

Following Bragg's law, each dot (or reflection), in this diffraction pattern forms from the constructive interference of X-rays passing through a crystal. The data can be used to determine the crystal's atomic structure.

Diffraction from a three dimensional periodic structure such as atoms in a crystal is called Bragg diffraction.It is similar to what occurs when waves are scattered from a diffraction grating. Bragg diffraction is a consequence of interference between waves reflecting from different crystal planes. The condition of constructive interference is given by Bragg's law:

${displaystyle mlambda =2dsin theta ,}$

where

λ is the wavelength,
d is the distance between crystal planes,
θ is the angle of the diffracted wave.
and m is an integer known as the order of the diffracted beam.

Bragg diffraction may be carried out using either light of very short wavelength like x-rays or matter waves like neutrons (and electrons) whose wavelength is on the order of (or much smaller than) the atomic spacing[5]. The pattern produced gives information of the separations of crystallographic planes d, allowing one to deduce the crystal structure. Diffraction contrast, in electron microscopes and x-topography devices in particular, is also a powerful tool for examining individual defects and local strain fields in crystals.

## Coherence

The description of diffraction relies on the interference of waves emanating from the same source taking different paths to the same point on a screen. In this description, the difference in phase between waves that took different paths is only dependent on the effective path length. This does not take into account the fact that waves that arrive at the screen at the same time were emitted by the source at different times. The initial phase with which the source emits waves can change over time in an unpredictable way. This means that waves emitted by the source at times that are too far apart can no longer form a constant interference pattern since the relation between their phases is no longer time independent.

The length over which the phase in a beam of light is correlated, is called the coherence length. In order for interference to occur, the path length difference must be smaller than the coherence length. This is sometimes referred to as spectral coherence as it is related to the presence of different frequency components in the wave. In the case light emitted by an atomic transition, the coherence length is related to the lifetime of the excited state from which the atom made its transition.

If waves are emitted from an extended source this can lead to incoherence in the transversal direction. When looking at a cross section of a beam of light, the length over which the phase is correlated is called the transverse coherence length. In the case of Young's double slit experiment this would mean that if the transverse coherence length is smaller than the spacing between the two slits the resulting pattern on a screen would look like two single slit diffraction patterns.

In the case of particles like electrons, neutrons and atoms, the coherence length is related to the spacial extent of the wave function that describes the particle.

## Diffraction limit of telescopes

File:Zboo lucky image 1pc.png
The Airy disc around each of the stars from the 2.56m telescope aperture can be seen in this lucky image of the binary starzeta Boötis.

For diffraction through a circular aperture, there is a series of concentric rings surrounding a central Airy disc. The mathematical result is similar to a radially symmetric version of the equation given above in the case of single-slit diffraction.

A wave does not have to pass through an aperture to diffract; for example, a beam of light of a finite size also undergoes diffraction and spreads in diameter. This effect limits the minimum diameter d of spot of light formed at the focus of a lens, known as the diffraction limit:

${displaystyle d=1.22lambda {frac {f}{a}},}$

where λ is the wavelength of the light, f is the focal length of the lens, and a is the diameter of the beam of light, or (if the beam is filling the lens) the diameter of the lens. The diameter given is enough to contain about 70% of the light energy; it is the radius to the first null of the Airy disk, in approximate agreement with the Rayleigh criterion. Twice that diameter, the diameter to the first null of the Airy disk, within which 83.8% of the light energy is contained, is also sometimes given as the diffraction spot diameter.

By use of Huygens' principle, it is possible to compute the far-field diffraction pattern of a wave from any arbitrarily shaped aperture. If the pattern is observed at a sufficient distance from the aperture, or projected onto a screen with a collimating lens, it will appear as the two-dimensional Fourier transform of the function representing the aperture.

## References

1. Dietrich Zawischa. 'Optical effects on spider webs'. Retrieved 2007-09-21.
2. Jean Louis Aubert (1760). Memoires pour l'histoire des sciences et des beaux arts. Paris: Impr. de S. A. S.; Chez E. Ganeau. p. 149.
3. Sir David Brewster (1831). A Treatise on Optics. London: Longman, Rees, Orme, Brown & Green and John Taylor. p. 95.
4. Brezger, B. (2002). 'Matter-Wave Interferometer for Large Molecules'(reprint). Physical Review Letters. 88 (10): 100404. doi:10.1103/PhysRevLett.88.100404. Retrieved 2007-04-30.Unknown parameter ` coauthors=` ignored (help); Unknown parameter ` month=` ignored (help)
5. John M. Cowley (1975) Diffraction physics (North-Holland, Amsterdam) ISBN 0 444 10791 6

## External links

 Wikimedia Commons has media related to Diffraction.
• On Diffraction at MathPages.
• Wave Optics - A chapter of an online textbook.
• 2-D wave Java applet - Displays diffraction patterns of various slit configurations.
• Diffraction Java applet - Displays diffraction patterns of various 2-D apertures.
• Diffraction approximations illustrated - MIT site that illustrates the various approximations in diffraction and intuitively explains the Fraunhofer regime from the perspective of linear system theory.
• GapObstacleCorner - Java simulation of diffraction of water wave.
• Google Maps - Satellite image of Panama Canal entry ocean wave diffraction.
Retrieved from 'https://www.wikidoc.org/index.php?title=Diffraction&oldid=713743'
,
Christine M. Clark, Eastern Michigan University

## What is X-ray Powder Diffraction (XRD)

### Diffraction Is A Result Of Interaction Of Light Coming From

X-ray powder diffraction (XRD) is a rapid analytical technique primarily used for phase identification of a crystalline material and can provide information on unit cell dimensions. The analyzed material is finely ground, homogenized, and average bulk composition is determined.

## Fundamental Principles of X-ray Powder Diffraction (XRD)

Max von Laue, in 1912, discovered that crystalline substances act as three-dimensional diffraction gratings for X-ray wavelengths similar to the spacing of planes in a crystal lattice. X-ray diffraction is now a common technique for the study of crystal structures and atomic spacing.

X-ray diffraction is based on constructive interference of monochromatic X-rays and a crystalline sample. These X-rays are generated by a cathode ray tube, filtered to produce monochromatic radiation, collimated to concentrate, and directed toward the sample. The interaction of the incident rays with the sample produces constructive interference (and a diffracted ray) when conditions satisfy Bragg's Law (nλ=2d sin θ). This law relates the wavelength of electromagnetic radiation to the diffraction angle and the lattice spacing in a crystalline sample. These diffracted X-rays are then detected, processed and counted. By scanning the sample through a range of 2θangles, all possible diffraction directions of the lattice should be attained due to the random orientation of the powdered material. Conversion of the diffraction peaks to d-spacings allows identification of the mineral because each mineral has a set of unique d-spacings. Typically, this is achieved by comparison of d-spacings with standard reference patterns.

All diffraction methods are based on generation of X-rays in an X-ray tube. These X-rays are directed at the sample, and the diffracted rays are collected. A key component of all diffraction is the angle between the incident and diffracted rays. Powder and single crystal diffraction vary in instrumentation beyond this.

## X-ray Powder Diffraction (XRD) Instrumentation - How Does It Work?

X-ray diffractometers consist of three basic elements: an X-ray tube, a sample holder, and an X-ray detector.
Bruker's X-ray Diffraction D8-Discover instrument. Details
X-rays are generated in a cathode ray tube by heating a filament to produce electrons, accelerating the electrons toward a target by applying a voltage, and bombarding the target material with electrons. When electrons have sufficient energy to dislodge inner shell electrons of the target material, characteristic X-ray spectra are produced. These spectra consist of several components, the most common being K

### What Is The Cause Of Diffraction

α and Kβ. Kα consists, in part, of Kα1 and Kα2. Kα1

### Interference Is A Property Of

has a slightly shorter wavelength and twice the intensity as Kα2. The specific wavelengths are characteristic of the target material (Cu, Fe, Mo, Cr). Filtering, by foils or crystal monochrometers, is required to produce monochromatic X-rays needed for diffraction. Kα1and Kα2 are sufficiently close in wavelength such that a weighted average of the two is used. Copper is the most common target material for single-crystal diffraction, with CuKα radiation = 1.5418Å. These X-rays are collimated and directed onto the sample. As the sample and detector are rotated, the intensity of the reflected X-rays is recorded. When the geometry of the incident X-rays impinging the sample satisfies the Bragg Equation, constructive interference occurs and a peak in intensity occurs. A detector records and processes this X-ray signal and converts the signal to a count rate which is then output to a device such as a printer or computer monitor.
X-ray powder diffractogram. Peak positions occur where the X-ray beam has been diffracted by the crystal lattice. The unique set of d-spacings derived from this patter can be used to 'fingerprint' the mineral. Details

The geometry of an X-ray diffractometer is such that the sample rotates in the path of the collimated X-ray beam at an angle θ while the X-ray detector is mounted on an arm to collect the diffracted X-rays and rotates at an angle of 2θ. The instrument used to maintain the angle and rotate the sample is termed a goniometer. For typical powder patterns, data is collected at 2θ from ~5° to 70°, angles that are preset in the X-ray scan.

## Applications

X-ray powder diffraction is most widely used for the identification of unknown crystalline materials (e.g. minerals, inorganic compounds). Determination of unknown solids is critical to studies in geology, environmental science, material science, engineering and biology.

Other applications include:

• characterization of crystalline materials
• identification of fine-grained minerals such as clays and mixed layer clays that are difficult to determine optically
• determination of unit cell dimensions
• measurement of sample purity
With specialized techniques, XRD can be used to:
• determine crystal structures using Rietveld refinement
• determine of modal amounts of minerals (quantitative analysis)
• characterize thin films samples by:
• determining lattice mismatch between film and substrate and to inferring stress and strain
• determining dislocation density and quality of the film by rocking curve measurements
• measuring superlattices in multilayered epitaxial structures
• determining the thickness, roughness and density of the film using glancing incidence X-ray reflectivity measurements
• make textural measurements, such as the orientation of grains, in a polycrystalline sample

## Strengths and Limitations of X-ray Powder Diffraction (XRD)?

### Strengths

• Powerful and rapid (< 20 min) technique for identification of an unknown mineral
• In most cases, it provides an unambiguous mineral determination
• Minimal sample preparation is required
• XRD units are widely available
• Data interpretation is relatively straight forward

### Limitations

• Homogeneous and single phase material is best for identification of an unknown
• Must have access to a standard reference file of inorganic compounds (d-spacings, hkls)
• Requires tenths of a gram of material which must be ground into a powder
• For mixed materials, detection limit is ~ 2% of sample
• For unit cell determinations, indexing of patterns for non-isometric crystal systems is complicated
• Peak overlay may occur and worsens for high angle 'reflections'

## User's Guide - Sample Collection and Preparation

Determination of an unknown requires: the material, an instrument for grinding, and a sample holder.

• Obtain a few tenths of a gram (or more) of the material, as pure as possible
• Grind the sample to a fine powder, typically in a fluid to minimize inducing extra strain (surface energy) that can offset peak positions, and to randomize orientation. Powder less than ~10 μm(or 200-mesh) in size is preferred
• Place into a sample holder or onto the sample surface:
Packing of fine powder into a sample holder. Details
• smear uniformly onto a glass slide, assuring a flat upper surface
• pack into a sample container
• sprinkle on double sticky tape
Typically the substrate is amorphous to avoid interference
• Care must be taken to create a flat upper surface and to achieve a random distribution of lattice orientations unless creating an oriented smear.
• For analysis of clays which require a single orientation, specialized techniques for preparation of clay samples are given by USGS.
• For unit cell determinations, a small amount of a standard with known peak positions (that do not interfere with the sample) can be added and used to correct peak positions.
• ## Data Collection, Results and Presentation

Data Collection The intensity of diffracted X-rays is continuously recorded as the sample and detector rotate through their respective angles. A peak in intensity occurs when the mineral contains lattice planes with d-spacings appropriate to diffract X-rays at that value of θ. Although each peak consists of two separate reflections (Kα1 and Kα2), at small values of 2θ the peak locations overlap with Kα2 appearing as a hump on the side of Kα1. Greater separation occurs at higher values of θ. Typically these combined peaks are treated as one. The 2λ position of the diffraction peak is typically measured as the center of the peak at 80% peak height.

Data Reduction

Results are commonly presented as peak positions at 2θ and X-ray counts (intensity) in the form of a table or an x-y plot (shown above). Intensity (I) is either reported as peak height intensity, that intensity above background, or as integrated intensity, the area under the peak. The relative intensity is recorded as the ratio of the peak intensity to that of the most intense peak (relative intensity = I/I1 x 100 ).

Determination of an Unknown

The d-spacing of each peak is then obtained by solution of the Bragg equation for the appropriate value of λ. Once all d-spacings have been determined, automated search/match routines compare the ds of the unknown to those of known materials. Because each mineral has a unique set of d-spacings, matching these d-spacings provides an identification of the unknown sample. A systematic procedure is used by ordering the d-spacings in terms of their intensity beginning with the most intense peak. Files of d-spacings for hundreds of thousands of inorganic compounds are available from the International Centre for Diffraction Data as the Powder Diffraction File (PDF). Many other sites contain d-spacings of minerals such as the American Mineralogist Crystal Structure Database. Commonly this information is an integral portion of the software that comes with the instrumentation.

Determination of Unit Cell Dimensions

For determination of unit cell parameters, each reflection must be indexed to a specific hkl.

## Literature

The following literature can be used to further explore X-ray Powder Diffraction (XRD)

• Bish, DL and Post, JE, editors. 1989. Modern Powder Diffraction. Reviews in Mienralogy, v. 20. Mineralogical Society of America.
• Cullity, B. D. 1978. Elements of X-ray diffraction. 2nd ed. Addison-Wesley, Reading, Mass.
• Klug, H. P., and L. E. Alexander. 1974. X-ray diffraction procedures for polycrystalline and amorphous materials. 2nd ed. Wiley, New York.
• Moore, D. M. and R. C. Reynolds, Jr. 1997. X-Ray diffraction and the identification and analysis of clay minerals. 2nd Ed. Oxford University Press, New York.

## Related Links

For more information about X-ray Powder Diffraction (XRD) follow the links below.

• For additional information on X-ray basics; Materials Research Lab, University of California- Santa Barbara
• Rigaku Journal; an on-line journal that describes and demonstrates a wide range of applications using Xray diffraction.
• Introduction to X-ray Diffraction--University of California, Santa Barbara
• Introduction to Crystallography--from LLNL
• X-ray Crystallography Lecture Notes--from Steve Nelson, Tulane University
• Reciprocal Net--part of the National Science Digital Library. Use the 'Learn About' link to find animations of the structures of common molecules (including minerals), crystallography learning resources (tutorials, databases and software), resources on crystallization, and tutorials on symmetry and point groups.

## Teaching Activities and Resources

### Diffraction Is A Result Of

Teaching activities, labs, and resources pertaining to X-ray Powder Diffraction (XRD).

• Weathering of Igneous, Metamorphic, and Sedimentary Rocks in a Semi-Arid Climate - An Engineering Application of Petrology - This problem develops skills in X-ray diffraction analysis as applied to clay mineralogy, reinforces lecture material on the geochemistry of weathering, and demonstrates the role of petrologic characterization in site engineering.
• A Powerpoint presentation on use of XRD in Soil Science(PowerPoint 1.6MB Sep7 07) by Melody Bergeron, Image and Chemical Analysis Laboratory at Montana State University.
• Brady, John B., and Boardman, Shelby J., 1995, Introducing Mineralogy Students to X-ray Diffraction Through Optical Diffraction Experiments Using Lasers. Jour. Geol. Education, v. 43 #5, 471-476.
• Brady, John B., Newton, Robert M., and Boardman, Shelby J., 1995, New Uses for Powder X-ray Diffraction Experiments in the Undergraduate Curriculum. Jour. Geol. Education, v. 43 #5, 466-470.
• Dutrow, Barb, 1997, Better Living Through Minerals X-ray Diffraction of Household Products, in: Brady, J., Mogk, D., and Perkins D. (eds.) Teaching Mineralogy, Mineralogical Society of America, p. 349-359.
• Hovis, Guy, L., 1997, Determination of Chemical Composition, State of Order, Molar Volume, and Density of a Monoclinic Alkali Feldspar Using X-ray Diffraction, in: Brady, J., Mogk, D., and Perkins D. (eds.) Teaching Mineralogy, Mineralogical Society of America, p. 107-118.
• Brady, John B., 1997, Making Solid Solutions with Alkali Halides (and Breaking Them) , in: Brady, J., Mogk, D., and Perkins D. (eds.) Teaching Mineralogy, Mineralogical Society of America, p. 91-95.
• Perkins, Dexter, III, and Sorensen, Paul, Mineral Synthesis and X-ray Diffraction Experiments, in: Brady, J., Mogk, D., and Perkins D. (eds.) Teaching Mineralogy, Mineralogical Society of America, p. 81-90.
• Hollecher, Kurt, A Long-Term Mineralogy Practical Exam, in: Brady, J., Mogk, D., and Perkins D. (eds.) Teaching Mineralogy, Mineralogical Society of America, p. 43-46.
• Hluchy, M.M., 1999, The Value of Teaching X-ray Techniques and Clay Mineralogy to Undergraduates, Jour. Geoscience Education, v. 47, p. 236-240.