Condensed Matter Physics

Discovery of X-ray diffraction by Crystals by Van Laue

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Isabela da Costa Tonon

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Created: 2018-10-12 16:49:15.

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After Röntgen’s discovery of x-rays in 1895, scientists speculated that the rays were actually composed of very short electromagnetic waves, but this supposition resisted proof, as it was impossible to construct a diffraction grating with intervals small enough to measure the wavelength. The idea that crystals could be used as a diffraction grating for X-rays arose in 1912 in a conversation between Paul Peter Ewald and Max von Laue in the English Garden in Munich.

Max von Laue.

Ewald had proposed a resonator model of crystals for his thesis, but this model could not be validated using visible light, since the wavelength was much larger than the spacing between the resonators. Von Laue realized that electromagnetic radiation of a shorter wavelength was needed to observe such small spacings, and suggested that X-rays might have a wavelength comparable to the unit-cell spacing in crystals. Von Laue worked with two technicians, Walter Friedrich and his assistant Paul Knipping, to shine a beam of X-rays through a copper sulfate crystal and record its diffraction on a photographic plate. After being developed, the plate showed a large number of well-defined spots arranged in a pattern of intersecting circles around the spot produced by the central beam.

First X-ray diffraction photograph.

The first ever X-ray crystal diffraction photograph made in 1912. The diffraction pattern would then be developed on a photographic medium. Von Laue first thought of using the lattice structure of crystals as a grating. The pattern of spots on the developed photograph proved von Laue's idea.

X-ray diffraction, a phenomenon in which the atoms of a crystal, by virtue of their uniform spacing, cause an interference pattern of the waves present in an incident beam of X rays. The atomic planes of the crystal act on the X rays in exactly the same manner as does a uniformly ruled grating on a beam of light.

Laue diffraction is most often used for mounting single crystals in a precisely known orientation, for example for polishing a surface or for doing measurements. When the incoming beam is parallel to a high-symmetry direction of the crystal, the Laue pattern also has high symmetry. In cubic crystals, an incoming beam parallel to one of the unit cell edges a [001] direction produces Laue patterns with 4-fold symmetry. An incoming parallel to the body diagonal of the unit cell produces a 3-fold symmetrical pattern of Laue spots.

In most applications only the symmetry of the Laue pattern is used. In this lab we will also assign the Laue spots to the crystal planes that give rise to them. We will also determine the wavelength of the Laue beams, and check if this is consistent with the high voltage on the x-ray anode.

Laue diffraction pattern, in X rays, a regular array of spots on a photographic emulsion resulting from X rays scattered by certain groups of parallel atomic planes within a crystal. When a thin, pencil-like beam of X rays is allowed to impinge on a crystal, those of certain wavelengths will be oriented at just the proper angle to a group of atomic planes so that they will combine in phase to produce intense, regularly spaced spots on a film or plate centred around the central image from the beam, which passes through undeviated.

Laue diffraction pattern.

The Laue equations

Let a, b, c be the primitive vectors of the crystal lattice L, whose atoms are located at the points

x = pa + qb + rc

that are integer linear combinations of the primitive vectors and p, q and r are scalar.

Let Kin be the wavevector of the incoming (incident) beam, and let Kout be the wavevector of the outgoing (diffracted) beam. Then the vector 

Kout – Kin = ∆K 

is called the scattering vector (also called transferred wavevector) and measures the change between the two wavevectors.

The three conditions that the scattering vector ∆K must satisfy, called the Laue equations, are the following: the numbers h, k, l determined by the equations

a∆K = 2πh
b∆K = 2πk
c∆K = 2πl

must be integer numbers. Each choice of the integers (h, k, l), called Miller indices, determines a scattering vector ∆K. Hence there are infinitely many scattering vectors that satisfy the Laue equations. They form a lattice L*, called the reciprocal lattice of the crystal lattice. This condition allows a single incident beam to be diffracted in infinitely many directions. However, the beams that correspond to high Miller indices are very weak and can't be observed. Anyway, it is enough to find a basis of the reciprocal lattice, from which the crystal lattice can be determined. This is the principle of x-ray crystallography.

Laue’s discovery was of dual importance: it allowed the subsequent investigation of x-radiation by means of wavelength determination, and it provided the means for the Braggs’ structural analysis of crystals, for which they received the Nobel Prize in 1915. X-ray analysis of crystals, as initially developed by Sir Lawrence Bragg, became the most widely used technique for the investigation of molecular structure, leading to incalculable advances in both inorganic and organic chemistry, as well as molecular biology.

ChemPubSoc Europe. Available in: Access in: 12/10/2018.

Wikipedia. Available in: Access in: 12/10/2018.

Wikipedia. Available in: Access in: 12/10/2018.

New World Encyclopedia. Available in: Access in: 12/10/2018.

Encyclopaedia Britannica. Available in: Access in: 12/10/2018.


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