Abstract
This chapter begins with a general description of diffraction and the structure of materials, starting with Bragg’s law. It discusses how diffraction is used to determine crystal structure, but is also sensitive to disorder in the structure. The creation of x-rays by electron accelerations or ionizations of atoms is presented. Some features of x-ray diffractometers are described, especially x-ray detectors, x-ray tubes, and synchrotrons for generating intense x-ray beams. The angular dependence of x-ray diffraction intensities is derived from geometric considerations of a typical laboratory diffractometer. Finally, some aspects of determining the fractions of different crystallographic phases in a material are presented, and the modern method of Rietveld refinement is discussed.
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Notes
- 1.
Precisely and concisely, the diffraction pattern measures the Fourier transform of an autocorrelation function of the scattering factor distribution. The previous sentence is explained with care in Chap. 10. More qualitatively, the crystal can be likened to music, and the diffraction pattern to its frequency spectrum. This analogy illustrates another point. Given only the amplitudes of the different musical frequencies, it is impossible to reconstruct the music because the timing or “phase” information is lost. Likewise, the diffraction pattern alone may be insufficient to reconstruct all details of atom arrangements in a material.
- 2.
Chapter 7 describes how to index diffraction patterns from single crystals.
- 3.
This approximation will be used frequently for high-energy electrons, with their short wavelengths (for 100 keV electrons, λ=0.037 Å), and hence small θ B.
- 4.
A hat over a vector denotes a unit vector: \(\hat{\boldsymbol{x}} \equiv \boldsymbol{x}/x\), where x≡|x|.
- 5.
This is consistent with a photon momentum of \(\boldsymbol{p} = \hat{\boldsymbol{k}} E/c = \hat{\boldsymbol{k}} \hbar \omega /c = \hbar \boldsymbol{k}\), where c is the speed of light and E=ħω is the photon energy.
- 6.
- 7.
The continuum spectrum of Fig. 1.11d is correct qualitatively, but a quantitative analysis requires more details about electron scattering and x-ray absorption.
- 8.
The time-independent Schrödinger equation (1.24) was obtained by the method of separation of variables, specifically the separation of t from r, θ, ϕ. The constant of separation was the energy, E. For the separation of θ and ϕ from r, the constant of separation provides l, and for the separation of θ from ϕ, the constant of separation provides m. The integers l and m involve the angular variables θ and ϕ, and are “angular momentum quantum numbers.” The quantum number l corresponds to the total angular momentum, and m corresponds to its orientation along a given direction. The full set of electron quantum numbers is {n,l,m,s}, where s is spin. Spin cannot be obtained from a constant of separation of the Schrödinger equation, which offers only 3 separations for {r,θ,ϕ,t}. Spin is obtained from the relativistic Dirac equation, however.
- 9.
Additional electron-electron potential energy terms are needed in (1.24), and these alter the energy levels.
- 10.
This result was published in 1914. Henry Moseley died in 1915 at Gallipoli during World War I. The British response to this loss was to assign scientists to noncombatant duties during World War II.
- 11.
Three premier facilities are the European Synchrotron Radiation Facility in Grenoble, France, the Advanced Photon Source at Argonne, Illinois, USA, and the Super Photon Ring 8-GeV, SPring-8 in Harima, Japan [1.4].
- 12.
The alternative arrangement of having the filament at ground and the anode at +40 kV is incompatible with water cooling of the anode. Cooling is required because a typical electron current of 25 mA demands the dissipation of 1 kW of heat from a piece of metal situated in a high vacuum. In a TEM, it is also convenient to keep the specimen and most components at ground potential.
- 13.
The efficiency of x-ray emission, the ratio of emitted x-ray power to electrical power dissipated in the tube, ϵ, is quite low. Empirically it is found that ϵ=1.4×10−9 ZV, where Z is the atomic number and V is accelerating voltage.
- 14.
This “θ–2θ diffractometer” is less versatile than a “θ–θ diffractometer,” but the latter instrument requires precise movement of its x-ray tube.
- 15.
- 16.
The thickness of a filter can be calculated with the method of Sect. 4.2.3.
- 17.
Suppose a detector does not generate noise of its own, but its QE=1/2. For the same x-ray flux as an ideal detector, this detector would have half the signal and half the noise, but SNR actual/SNR ideal is not 1.0. With half the countrate, counting statistics reduce this ratio to \(\sqrt{1/2}\). The DQE of (1.37) would then be 1/2, so DQE=QE for detectors that do not generate false counts.
- 18.
Such as a radioisotope source or a known atomic fluorescence.
- 19.
Throughout this book, an asterisk (*) in a section heading denotes a more specialized topic. For example, the results of the present section, (1.54) and (1.55), are important, but on a first reading the reader may choose to avoid the details of their derivation. Incidentally, a section heading with a double dagger (‡) indicates a higher level of mathematics.
- 20.
- 21.
Even for amorphous solids, diffraction is coherent in the forward direction, for which θ=0.
- 22.
It is a good idea to measure diffraction patterns from at least two samples, each one prepared and mounted independently.
- 23.
A mixture of three or more unknown phases is still amenable to analysis by the ratio method, since another peak ratio equation is provided for each additional unknown.
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Fultz, B., Howe, J. (2013). Diffraction and the X-Ray Powder Diffractometer. In: Transmission Electron Microscopy and Diffractometry of Materials. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29761-8_1
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