(Braggs law)假设入射波从晶体中的平行原子平面作镜面反射，每个平面反射很少一部分辐射，就像一个轻微镀银的镜子一样。在这种类似镜子的镜面反射中，其反射角等于入射角。当来自平行原子平面的反射发生相长干涉时，就得出衍射束。 　　考虑间距为d的平行晶面，入射辐射线位于纸面平面内。相邻平行晶面反射的射线行程差是2dsinx，式中从镜面开始量度。当行程差是波长的整数倍时，来自相继平面的辐射就发生了相长干涉。 这就是布拉格定律。 　　布拉格定律用公式表达为：2dsinx=n*波长(d为平行原子平面的间距，f为入射波频率，x为入射光与晶面之夹角) 　　布拉格定律的成立条件是波长小于2d

 

 

衍射条件与布拉格定律
             ../images/img2/formula2-17.gif   （n为衍射级数）    （2-17）    [k空间描述]
     从以上结论，可得图2-13：
     对倒格矢空间中任一矢量 ../images/img2/formula97.gif ，正格子在必有一垂直平分面等分 ../images/img2/formula98.gif ，其晶面密勒指数为( ../images/img2/formula99.gif )，由倒格子性质3)，这簇晶面的面间距为：

       ../images/img2/formula99a.gif

即：../images/img2/formula99b.gif （2-1）， 为布拉格定律 。   [实空间描述]

 

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

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Bragg's Law refers to the simple equation:

nλ = 2d sinΘ

derived by the English physicists Sir W.H. Bragg and his son Sir W.L. Bragg in 1913 to explain why the cleavage faces of crystals appear to reflect X-ray beams at certain angles of incidence (Θ, λ). The variable d is the distance between 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 direct evidence 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 determining crystal structures beginning with NaCl, ZnS and diamond. Although Bragg's law was used to explain the interference pattern of X-rays scattered by crystals, diffraction has been developed 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 the atomic 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 because the layers are projected onto two dimensions and your view is parallel to the layers. The applet begins with the scattered rays in phase and interferring constructively. Bragg's Law is satisfied and diffraction is occurring. The meter indicates how well the phases of the two rays match. The small light on the meter is green when Bragg's equation is satisfied and red when it is not satisfied.

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

Deriving Bragg's Law

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

nλ = AB +BC (2).

Recognizing d as the hypotenuse of the right triangle Abz, we can use trigonometry to relate d and q to the distance (AB + BC). The distance AB is opposite Θ so,

AB = d sinΘ(3).

Because AB = BC eq. (2) becomes,

nλ = 2AB (4)

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

nλ = 2 d sinΘ, (1)

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

Experimental Diffraction Patterns

The following figures show experimental x-ray diffraction patterns of cubic SiC using synchrotron radiation. http://www.eserc.stonybrook.edu/ProjectJava/Bragg/SiC1.gif

http://www.eserc.stonybrook.edu/ProjectJava/Bragg/SiC2.gif

Players in the Discovery of X-ray Diffraction

Friedrich and Knipping first observed Roentgenstrahlinterferenzen in 1912 after a hint from their research advisor, Max von Laue, at the University of Munich. Bragg's Law greatly simplified von Laue's description of X-ray interference. The Braggs used crystals in 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 the Bragg spectrometer.

Laue knew that X-rays had wavelengths on the order of 1 ? After learning that Paul Ewald's optical theories had approximated the distance between atoms in a crystal by the same length, Laue postulated that X-rays would diffract, by analogy to the diffraction of light from small periodic scratches drawn on a solid surface (an optical diffraction grating). 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 in physics.

Do We Have Diamonds?

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

See Also

• Wikipedia: Bragg's law
• Wikipedia: Diffraction
• Wikipedia: Bragg diffraction
• Wikipedia: Diffraction grating
• Wikipedia: X-ray crystallography
• SERC: X-ray reflection in accordance with Bragg's Law
• HyperPhysics: Bragg's Law

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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.

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Source Code

• Arrow.java
• Box.java
• Bragg.java
• Details.java
• GraphCanvas.java
• Handle.java
• Message.java
• MyMath.java
• Point2D.java
• Segment.java
• SineWave.java
• UserInterface.java

• Zip archive with all Java source code

Last modified January 29, 2010

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