Diffraction Of X Rays By Crystals

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1. Infrared
2. Definition Of X-rays
3. X Ray Diffraction Explained

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

Bragg's Law refers to the simple equation:

(eq 1)n = 2d sin

• Review of x-ray diffraction Crystals act as three-dimensional gratings; they scatter the wave and produce observable interference effects. In fact, x-rays scatter from the lattice planes (or Bragg planes) of the crystal as shown in Figure 2. Electron Waves Reflected from Atomic Planes.
• The closely related techniques of Grazing Incidence Diffraction (GID), also called Grazing Incidence X-ray Scattering (GIXS) and X-ray Reflectivity (XR) utilize the fact that, when the beam of X-rays impinges on a surface at very low incident angle (α i in the picture to the right), the reflectivity is greatly enhanced and the beam penetrates only a short distance into the surface.
• Practically everything we know about the structure of crystals has come from the use of x-ray diffraction. X-ray diffraction is the scattering of x-rays by atoms in the crystal lattice.

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 .

Because AB = BC eq. Columbia valley ava map. (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.

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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.

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.

Infrared

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 .

Because AB = BC eq. Columbia valley ava map. (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.

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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.

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.

Infrared

Definition Of X-rays

Source Code

X Ray Diffraction Explained

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