# Surface Diffraction

Posted : admin On 1/26/2022**This page is mirrored as supplementary material for BC 530**

- Surface Diffraction Scattering
- Surface Diffraction Gratings
- Surface Wave Diffraction
- Surface X Ray Diffraction

Surface x-ray diflracrion 601 1. Introduction The purpose of this review is to discuss some different ways in which x-ray difiaction is used to study surfaces and interfaces.We begin with an elementary discussion of the theory of diffraction, starting with the scattering hom a single electron. From there the idea of reciprocal space is derived, a concept used in most thinking about. Optical lateral and longitudinal standing waves can be recorded using an optical scanning probe in collection mode. We describe both analytical and numerical methods to determine the image height and the location of a single point scatterer from the recorded surface diffraction image. A surface with periodic variations reflects and transmits light into several different diffraction orders. We can still apply the same domain properties and all of the same boundary conditions. However, if spacing is large enough, then we can have higher-order diffraction.

**What is Bragg's Law and Why is it Important?**

Bragg's Law refers to the simple equation:

(eq 1)n = 2d sin

derived by the English physicists Sir W.H. Bragg and his son Sir W.L. Bragg in 1913 toexplain why the cleavage faces of crystals appear to reflect X-ray beams at certain anglesof incidence (theta, ). The variable *d* is the distancebetween atomic layers in a crystal, and the variable lambda is the **wavelength** of the incident X-ray beam (see applet); n is an integer

This observation is an example of X-ray **wave interference**(Roentgenstrahlinterferenzen), commonly known as X-ray diffraction (XRD), and was directevidence for the periodic atomic structure of crystals postulated for several centuries.The Braggs were awarded the Nobel Prize in physics in 1915 for their work in determiningcrystal structures beginning with NaCl, ZnS and diamond. Although Bragg's law was used toexplain the interference pattern of X-rays scattered by crystals, diffraction has beendeveloped to study the structure of all states of matter with any beam, e.g., ions,electrons, neutrons, and protons, with a wavelength similar to the distance between theatomic or molecular structures of interest.

**How to Use this Applet**

The applet shows two rays incident on two atomic layers of a crystal, e.g., atoms,ions, and molecules, separated by the distance *d*. The layers look like rows becausethe layers are projected onto two dimensions and your view is parallel to the layers. Theapplet begins with the scattered rays in phase and interferring constructively. Bragg'sLaw is satisfied and diffraction is occurring. The meter indicates how well the phases ofthe two rays match. The small light on the meter is green when Bragg's equation issatisfied and red when it is not satisfied.

The meter can be observed while the three variables in Bragg's are changed by clickingon the scroll-bar arrows and by typing the values in the boxes. The *d* and variables can be changed by dragging on the arrows provided on thecrystal layers and scattered beam, respectively.

Sorry. You cannot use this applet because your browser is not Java enabled.

**Deriving Bragg's Law**

Bragg's Law can easily be derived by considering the conditions necessary to make thephases of the beams coincide when the incident angle equals and reflecting angle. The raysof the incident beam are always in phase and parallel up to the point at which the topbeam strikes the top layer at atom z (Fig. 1). The second beam continues to the next layerwhere it is scattered by atom B. The second beam must travel the extra distance AB + BC ifthe two beams are to continue traveling adjacent and parallel. This extra distance must bean integral (n) multiple of the wavelength () for the phasesof the two beams to be the same:

(eq 2)n = AB +BC .

Fig. 1 Deriving Bragg's Law using the reflection geometry and applying trigonometry.The lower beam must travel the extra distance (AB + BC) to continue traveling parallel andadjacent to the top beam.

Recognizing d as the hypotenuse of the right triangle Abz, we can use trigonometry torelate d and to the distance (AB + BC). The distance AB isopposite so,

(eq 3)AB = d sin .

### Surface Diffraction Scattering

Because AB = BC eq. (2) becomes,

(eq 4)n = 2AB

Substituting eq. (3) in eq. (4) we have,

(eq 1)n = 2 d sin

and Bragg's Law has been derived. The location of the surface does not change thederivation of Bragg's Law.

**Experimental Diffraction Patterns **

The following figures show experimental x-ray diffraction patterns of cubic SiC usingsynchrotron radiation.

**Players in the Discovery of X-ray Diffraction **

Friedrich and Knipping first observed Roentgenstrahlinterferenzen in 1912 after a hintfrom their research advisor, Max von Laue, at the University of Munich. Bragg's Lawgreatly simplified von Laue's description of X-ray interference. The Braggs used crystalsin the reflection geometry to analyze the intensity and wavelengths of X-rays (spectra)generated by different materials. Their apparatus for characterizing X-ray spectra was theBragg spectrometer.

### Surface Diffraction Gratings

Laue knew that X-rays had wavelengths on the order of 1 Å. After learning that PaulEwald's optical theories had approximated the distance between atoms in a crystal by thesame length, Laue postulated that X-rays would diffract, by analogy to the diffraction oflight from small periodic scratches drawn on a solid surface (an optical diffractiongrating). In 1918 Ewald constructed a theory, in a form similar to his optical theory,quantitatively explaining the fundamental physical interactions associated with XRD.Elements of Ewald's eloquent theory continue to be useful for many applications inphysics.

**Do We Have Diamonds?**

If we use X-rays with a wavelength () of 1.54Å, and wehave diamonds in the material we are testing, we will find peaks on our X-ray pattern at values that correspond to each of the d-spacings that characterizediamond. These d-spacings are 1.075Å, 1.261Å, and 2.06Å. To discover where to expectpeaks if diamond is present, you can set to 1.54Å in theapplet, and set distance to one of the d-spacings. Then start with at 6 degrees, and vary it until you find a Bragg's condition. Do the same with each of theremaining d-spacings. Remember that in the applet, you are varying ,while on the X-ray pattern printout, the angles are given as 2.Consequently, when the applet indicates a Bragg's condition at a particular angle, youmust multiply that angle by 2 to locate the angle on the X-ray pattern printout where youwould expect a peak.

Text written by Paul J. Schields Center for High Pressure Research Department of Earth & Space Sciences State University of New York at Stony Brook Stony Brook, NY 11794-2100. | Applet created by Konstantin Lukin [email protected] | Project Java WebmasterGlenn A. Richard Center for High Pressure Research SUNY Stony Brook | Applet updated by Jay Painter |

### Surface Wave Diffraction

### Source Code

### Surface X Ray Diffraction

*Mirrored from http://www.journey.sunysb.edu/ProjectJava/Bragg 3 Oct 2001**Last modified October 11, 2004*