State Bragg’s diffraction condition in reciprocal lattice.
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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 to explain why the cleavage faces of crystals appear to reflect X-ray beams at certain angles of incidence (theta, ). 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  variables can be changed by dragging on the arrows provided on the crystal layers and scattered beam, respectively.
Sorry. You cannot use this applet because your browser in not Java enabled.
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 () for the phases of 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 and adjacent to the top beam.
Recognizing d as the hypotenuse of the right triangle Abz, we can use trigonometry to relate d and  to the distance (AB + BC). The distance AB is opposite  so,
(eq 3) AB = d sin .
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 the derivation of Bragg's Law.
(eq 1) n = 2d sin
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 (theta, ). 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  variables can be changed by dragging on the arrows provided on the crystal layers and scattered beam, respectively.
Sorry. You cannot use this applet because your browser in not Java enabled.
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 () for the phases of 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 and adjacent to the top beam.
Recognizing d as the hypotenuse of the right triangle Abz, we can use trigonometry to relate d and  to the distance (AB + BC). The distance AB is opposite  so,
(eq 3) AB = d sin .
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 the derivation of Bragg's Law.
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