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## Introduction

In order to teach and learn crystallography at the high school level, one must have a basic understanding in three broad areas.  These important areas are x-ray diffraction, Bragg's law, and crystal structure.  Although each can be incredibly complex, the purpose of this site is to provide a general conceptual overview that can be used as a foundation for understanding the fundamental basis of crystallography in a qualitative way.

# X-Ray Diffraction

• Describing the interaction of x-rays with objects and small openings

X-rays, like other forms of electromagnetic radiation, exhibit unique behavior when exposed to objects and small openings.  When exposed to objects, waves generally bend around them and reform in a constructive and destructive fashion.  Similarly, when waves encounter narrow openings, they widen to produce similar results.  Before the work of Max von Laue, scientists had not witnessed this behavior or confirmed the wave-nature of x-rays.  Von Laue won the Nobel Prize in 1915 for hypothesizing and proving that a crystal lattice could serve as a diffraction grating that would reveal this type of wave-behavior in x-rays.  The image below, posted on Wikipedia by Dick Lyon, illustrates wave diffraction through a slit:

Through the work of Huygens and others, waves are viewed as the summation of new disturbances that are radiating from each point on the wavefront.  This idea is referred to as Huygen's principle.  The University of Cambridge has developed an applet that explains the principle well.  Below is an image of the applet.  Clicking on the image will transfer you to the Cambridge site.

When x-rays come in contact with atoms in a crystal lattice, they diffract and form wavelets much like in the image of the applet above.  Crystallographers are most concerned with the interference that occurs between wavelets that radiate from specific points within the crystal.  Wavelets that are out of phase cancel each other out when they interfere destructively.  Wavelets that are in phase combine constructively and may go on to be detected by the diffractometer.  The image below, posted on Wikipedia by Arne Nordmann, is an illustration of the interference that would be expected when x-rays pass through multiple slits.  This would be similar to x-rays traveling through a plane of crystals.  However, since this image is only one line of "atoms" in two dimensions, it is a gross over-simplification of how x-rays travel through a crystal.

# Bragg's Law

• Using mathematics to make connections between diffraction patterns and crystal structure

William Henry Bragg and William Lawrence Bragg, the father and son team, shared the Nobel Prize in Physics in 1915 for "their services in the analysis of crystal structure by means of x-rays."  Although the elder Bragg, W.H., is credited with inventing the x-ray spectrometer, his son is largely credited with the development of the theoretical and math-based portions of their work.  W.L. developed an equation that allows crystallographers to relate the distance between planes in a crystal lattice to the wavelength of the incoming x-ray radiation if the angle of incidence and reflection are equal and known.  The following diagram is from an Online X-Ray Course developed by Dr. Pat Carroll, director of the University of Pennsylvania's X-Ray Crystallography Facility:

Constructive interference occurs whenever an integer number of wavelengths is present on the right-hand side of the equation.  X-rays that interfere constructively and pass through the crystal produce a detectable diffraction pattern.  Crystallographers use a diffractometer to gradually adjust the angle of incidence, resulting in many strategic reflections that ultimately provide information about the arrangement of atoms in the crystal lattice.  However, the analysis is complicated by the fact that x-rays pass through an excessively large number of planes in the lattice which result in additional constructive and destructive interference.

# Crystal Structure

• The arrangement of atoms within a crystal

Crystals are made up of a repeating lattice of atoms.  The building block of these lattices is called a unit cell.  According to Shriver and Atkins, a unit cell is "an imaginary parallel-sided region from which the entire crystal can be built up by purely translational displacements."  There are 7 crystal systems, which are defined the position of the atoms in space and their relationship to each other.  They diagram above is a common coordinate system used to define unit cells.  Symmetry (point groups and space groups) is also an important consideration.  Auguste Bravais, a French physicist, researched crystal structure extensively.  He classified the 7 crystal systems into 14 lattices which are now called "Bravais lattices."  The following table, from Answers.com, provides an informative summary of the 7 crystal systems and 14 Bravais lattices:

Crystallographers identify imaginary planes within these crystals called Miller indices.  A system of notation exists that allows them to attribute detected spots to specific Miller indices based on their position and intensity.   A symmetric pattern of detected reflections indicates that the atoms in the crystal possess a certain elements of symmetry.  The image below comes from Dr. Patrick Carroll's Online X-Ray Course.

## References

Carroll, P. J., & Carroll, M. H. (n.d.). Structure Determination by X-ray Crystallography . In University of Pennsylvania X-Ray Crystallography Facility [Online Course]. Retrieved January 7, 2009, from http://macxray.chem.upenn.edu/‌course/‌index.html

Dahl, J. (2007). X-ray diffraction pattern of crystallized 3Clpro, a SARS protease [Data file]. Retrieved March 31, 2009, from http://en.wikipedia.org/‌wiki/‌File:X-ray_diffraction_pattern_3clpro.jpg

International Union of Crystallography [Educational web sites and resources of interest]. (n.d.). Retrieved January 7, 2009, from http://www.iucr.org/‌education/‌resources

International Union of Crystallography. (n.d.). Laue’s discovery of x-ray diffraction by crystals (pp. 31-56). Retrieved March 31, 2009, from http://www.iucr.org/‌__data/‌assets/‌pdf_file/‌0010/‌721/‌chap4.pdf

Lyon, D. (2007). Wave diffraction [4 lamba slit]. Retrieved April 8, 2009, from http://en.wikipedia.org/‌wiki/‌File:Wave_Diffraction_4Lambda_Slit.png

Nordmann, A. (2007). Refraction on an aperture [Data file]. Retrieved from http://en.wikipedia.org/‌wiki/‌File:Refraction_on_an_aperture_-_Huygens-Fresnel_principle.svg

Shriver, D., & Atkins, P. (1999). Inorganic chemistry (3rd ed.). New York: W.H. Freeman and Company.

University of Cambridge. (2002). X-ray diffraction. In Cavendish Laboratory educational outreach. Retrieved March 30, 2009, from http://www-outreach.phy.cam.ac.uk/‌camphy/‌xraydiffraction/‌xraydiffraction_index.htm

Wullfman. (2002). Unit cell coordinates [Data file]. Retrieved April 4, 2009, from Center for Theoretical and Computational Materials Science, NIST  Web site: http://www.ctcms.nist.gov/‌wulffman/‌docs_1.2/‌unit_cell.gif