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Diffraction Basics

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The qualitative basics:

Coherent scattering around atomic scattering centers occurs when x-rays interact with material
In materials with a crystalline structure, x-rays scattered in certain directions will be “in-phase” or amplified
Measurement of the geometry of diffracted x-rays can be used to discern the crystal structure and unit cell dimensions of the target material
The intensities of the amplified x-rays can be used to work out the arrangement of atoms in the unit cell

The Bragg Law “bottom line”:

A diffraction direction defined by the intersection of the hth order cone about the a axis, the kth order cone about the b axis and the lth order cone about the c axis is geometrically equivalent to a reflection of the incident beam from the (hkl) plane referred to these axes.

The Reciprocal Lattice

How do we predict when diffraction will occur in a given crystalline material?
How do we orient the X-ray source and detector?
How do we orient the crystal to produce diffraction?
How do we represent diffraction geometrically in a way that is simple and understandable?

The first part of the problem

Consider the diffraction from the (200) planes of a (cubic) LiF crystal that has an identifiable (100) cleavage face.
To use the Bragg equation to determine the orientation required for diffraction, one must determine the value of d200.
Using a reference source (like the ICDD database or other tables of x-ray data) for LiF, a = 4.0270 Å, thus d200 will be ½ of a or 2.0135 Å.
From Bragg’s law, the diffraction angle for Cu K1 ( = 1.54060) will be 44.986 2. Thus the (100) face should be placed to make an angle of 11.03 with the incident x-ray beam and detector.
If we had no more complicated orientation problems, then we would have no need for the reciprocal space concept.
Try doing this for the (246) planes and the complications become immediately evident.

The Powder Diffraction Pattern

Powders (a.k.a. polycrystalline aggregates) are billions of tiny crystallites in all possible orientations
When placed in an x-ray beam, all possible interatomic planes will be seen
By systematically changing the experimental angle, we will produce all possible diffraction peaks from the powder
Diffraction refers to several phenomena that occur when a wave encounters an obstacle or a cleft. It is defined as the bending of light around the corners of an obstacle or aperture in the region of the geometric shadow of the obstacle. In classical physics, the diffraction phenomenon is described as the interference of the waves according to the Huygens-Fresnel principle. These characteristic behaviors are exhibited when a wave encounters an obstacle or a slit that is comparable in size to its wavelength. Similar effects occur when a light wave travels through a medium with a variable refractive index, or when a sound wave travels through a medium with variable acoustic impedance. Diffraction occurs with all waves, including sound waves, water waves, and electromagnetic waves such as visible light, X-rays, and radio waves.

Since physical objects have wave properties (at the atomic level), diffraction also occurs with matter and can be studied according to the principles of quantum mechanics. The Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1660.

While diffraction occurs when propagation waves encounter such changes, their effects are generally more pronounced for waves whose wavelength is approximately comparable to the object's dimensions or diffraction slit. If the obstructing object provides multiple closely spaced openings, a complex pattern of varying intensity may result. This is due to the addition or interference of different parts of a wave traveling to the observer by different paths, where different path lengths result in different phases (see diffraction grid and wave overlap). The diffraction formalism can also describe the way in which finite extension waves propagate in free space. For example, the expansion profile of a laser beam, the beam shape of a radar antenna and the field of view of an ultrasonic transducer can be analyzed using diffraction equations.
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