X- Ray Sources Diffraction: Bragg's Law Crystal Structure Determination [PDF]

Contents

X- Ray Sources  Diffraction: Bragg’s Law  Crystal Structure Determination

 BRAGG’s  Deviation





EQUATION

= 2

The path difference between ray 1 and ray 2 = 2d Sin For constructive interference:



n = 2d Sin



Braggs' law  = 2dhkl sin hkl

Crystal structure determination methods

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Crystal structure determination

Many s (orientations) Powder specimen

Powder method

Single 

Laue method

 Varied by rotation

Crystal rotating method

X-rays

another set of lattice planes

 Requires random orientation of very fine crystals  Incident beam of a certain X-ray wavelength will diffract from atomic planes oriented at the appropriate θ angles for the characteristic d spacing  Random orientation of crystals will produce more intense diffraction peaks for particular angles that correspond to characteristic atomic planes

Laue Method

Single-Crystal: Laue Method

Laue Diffraction Single crystal placed on a three-axis goniometer in front of narrow beam of x-rays • X-rays are not monochromatic • 2θ and d remain fixed for each set of planes (only λ is varied): • each set diffracts that particular λ which satisfies the Bragg equation for it • each diffracted beam has a different λ • Incident beam passes through photographic film, hits the crystal, and is back-reflected towards the film

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Single crystal mounted with one axis normal to a monochromatic x-ray beam • Cylindrical film placed around the sample • As sample rotates, some sets of planes momentarily satisfy Bragg condition • When film is laid flat, a series of horizontal lines appears • Because crystal rotates about a single axis, possible Bragg angles are limited - not every plane is able to produce a diffracted spot • Sometimes used to determine unknown crystal structures

It is necessary to vary λ or θ to satisfy Bragg’s law

 Summary 

Method

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Laue



λ

of XRD Techniques Use

θ

White radiation (mixed)

Neither sample

Determine the orientation of

nor detector (film) single crystals

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Rotating Crystal Monochromatic (fixed) Some variation (crystal rotates) As above, e.g., Si, Ge, etc moves

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Powder Diffraction Monochromatic (fixed)

with random orientation. Sample and/or detector

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moves. Most common method of analysis 

Sample is fine powder