Scherrer's equation
The Scherrer equation (after the Swiss physicist Paul Scherrer ) offers the possibility of experimentally determining the crystal size in X-ray diffraction .
In general, the diffraction pattern of X-ray diffraction can be described by the Bragg equation . The prerequisite for this, however, is that the crystals examined have a certain thickness and thus a sufficient number of parallel lattice planes with a distance d hkl are present. With powder diffractometry and other powder methods such as the Debye-Scherrer method , the crystals should therefore have a grain size of at least 0.1 μm. When analyzing the crystal structure of single crystals , the crystals are usually 50–500 μm in size.
If the crystals are very small (crystal size ), this results in a broadening of the X-ray reflections, which is described by the Scherrer equation:
It is
- the full width at half maximum of the reflex, measured in radians
- the Scherrer form factor with a value of approximately 1
- the wavelength of the X-rays
- the expansion of the crystal perpendicular to the lattice planes of the reflex
- the Bragg angle (sometimes referred to as ).
literature
- P. Scherrer: Determination of the size and the internal structure of colloidal particles by means of X-rays . In: News from the Society of Science in Göttingen, Mathematical-Physical Class . Weidmannsche Buchhandlung, Berlin 1918, p. 98-100 ( uni-goettingen.de ).
- A. Guinier: X-Ray Diffraction in Crystals, Imperfect Crystals, and Amorphous Bodies. Dover Publications, New York 1994, ISBN 0-486-68011-8 , Chapter 5.
- U. Holzwarth and N. Gibson: The Scherrer Equation versus the 'Debye-Scherrer Equation'. In: Nature Nanotechnology . Volume 6, No. 534, 2011, p. 534, doi: 10.1038 / nnano.2011.145 .