Abstract
The essence of image-formation theory is presented in order to illustrate the relation of the diffraction pattern and the image. Thereby the background is given for the treatment of imaging/microscopic and diffraction methods. After discussing light microscopy, transmission and scanning electron microscopy, including important variants thereof, is discussed. Composition analysis on highly local, also (almost) atomic, scale in transmission and scanning electron microscope is dealt. Special attention is paid to so-called supermicroscopy as a tool to bypass the diffraction limit for image resolution. X-ray diffraction, as the important method to analyze the crystal imperfection, is emphasized by presenting basic approaches to determine the size of and microstrain within diffracting crystallites, as well as the (macro) internal, residual stress in specimens/workpieces.
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Notes
- 1.
Light is considered here as an electromagnetic wave propagation, which can be characterized by its amplitude and phase. With reference to the discussion on the dualistic nature of light in Sect. 2.4, and noting that there is no such medium, as “ether”, through which the “wave” would propagate, the only observable quantity of the light is the (time averaged) intensity which is proportional to the squared amplitude.
- 2.
Note that in case in object space n1≠1: sinϕ = mλo/(n1p).
- 3.
Fermat’s principle says that the light follows always the course of minimal optical path length. The optical path length is defined by the product of path length and index of refraction. Constant optical path length then implies that the number of wavelengths corresponding to the length of the path followed is the same, thereby accounting for possible variations of the value of the index of refraction along the path followed.
- 4.
This is the same resolving power as achieved from self-luminous objects.
- 5.
In principle the entire field of light microscopy can obviously be described as interference microscopy, because image formation implies the interaction (interference) of transmitted/reflected and diffracted beams of light (cf. Sect. 6.2)
- 6.
Propagation of a planar wavefront in an anisotropic, possibly absorbing, crystalline medium leads to splitting into two parallel planar wavefronts of different index of refraction and different polarization (always mutually perpendicularly, linearly or elliptically polarized). This is the phenomenon of “double refraction”, also called “birefringe”. There are, however, in general two directions for the incident planar wavefront, with respect to the crystal axes, for which such splitting does not occur. These directions are called “optical axes”. Both optical axes may coincide. Thus, one distinguishes biaxial and uniaxial materials.
- 7.
The currently commercially available transmission electron microscopes have accelerating voltages in this range.
- 8.
In crystallography, the “Ångstrom” is still often used as a distance/length unit and has even been formally sanctioned as such by the International Union of Crystallography (IUCr); cf. Footnote 8 in Sect. 4.1.1.
- 9.
Spherical aberration occurs when, on imaging an object point on the lens axis, the rays refracted by different lens zones do not converge in a single image point on the lens axis: the edge zone of the lens refracts too strongly: an (already) curved wavefront becomes (even) more curved. Abbe was the first to demonstrate in 1872 that in light optical microscopy spherical aberration in principle can be eliminated by use of a compensating, composite lens system: a doublet consisting of a convergent lens and a divergent lens. For transmission electron microscopy, with magnetic lenses, it took until 1998 before such first, spherical-aberration-corrected lens systems were used for the objective lens system. However, this important step forward still does not lead to direct (atomic structure) image formation of the structure analysed, in a way as holds for image formation in the bright field mode in a light optical microscope: phase information in the diffracted electron waves has to be converted into amplitude information (see first footnote in Sect. 6.1 and the discussion on phase-contrast microscopy in Sect. 6.6.3). A corresponding discussion is beyond the scope of this book (see an overview by Urban 2007).
- 10.
The “viewing direction” in the electron microscope is perpendicularly through the specimen/foil. Atoms on top of each other, arranged in a row in the “viewing direction”, are projected on top of each other in the image.
- 11.
For example, for a wavelength of 0.003 nm and a lattice spacing of 0.3 nm, the diffraction angle, 2θ, is less than 0.6°.
- 12.
An image of the source (also called “cross-over”) is produced at the front focal plane of a final condenser lens which then produces a truly parallel electron beam hitting the specimen (cf. Sect. 6.4)
- 13.
According to convention (see Sect. 4.5), the reflection observed from the (hkl) family of crystallographic planes is designated with Laue indices as HKL (with H = nh, K = nk and L = nl; n = order of reflection), without brackets or braces, in the (X-ray, or electron, or …) diffraction pattern.
- 14.
Actually, this statement only holds for the so-called Laue zone of order zero (further, see Williams and Carter 1996).
- 15.
Note that electrons scattered in same directions, but originating from different locations in the specimen foil, converge in the back focal plane of the objective lens in a single point: parallel diffracted rays intersect the back focal plane of the objective lens at a location that is determined by the point of intersection of the line drawn through the lens centre parallel to the diffracted rays considered and the back focal plane. Hence, although the probe in STEM mode moves (it scans the specimen foil surface), the generated CBED pattern in STEM mode is stationary. Normally, for CBED application, e.g. to analyse the diffraction pattern of a very small specimen volume, the electron beam is convergent but it is not moved, i.e. it is not scanning the foil surface!
- 16.
Scanning (modes of) microscopes in general do not produce images in the sense as discussed in Sect. 6.2; such (modes of) microscopes produce “response maps” of the specimen with respect to one specific type of response to the action of a scanning probe (here a focussed electron beam, or, as holds for STED (Sect. 6.7), a laser beam), as, for example, the amount of (specifically) diffracted or of not diffracted electrons (as in STEM), the amount of secondary or of back-scattered electrons (as in SEM; cf. Sect. 6.9), or the amount of generated X-rays originating from a specific element, as in composition analysis (EPMA and EDS; cf. Sects. 6.8.7.1 and 6.9.3) or the amount of fluorescence (STED; cf. Sect. 6.7).
- 17.
In this sense, the future of high resolution in microscopy in general needs not in the first place lie in hardware instrumental advancements leading to nicer, i.e. sharper and contrast richer, images but rather in the development of computational models for the image-formation process in non-ideal microscopes, leading to algorithms for the processing of the enormous quantities of data contained in recorded non-ideal images.
- 18.
Actually, the fringe spacing is given by the reciprocal of the distance between the two diffraction spots in the diffraction pattern as enclosed by the objective aperture. Only if one of the two diffraction spots is the non-diffracted one and the other diffraction spot is a first order reflection, the fringe spacing is equal to the spacing of the crystallographic planes giving rise to the diffraction spot.
- 19.
Regarding the usage of “lattice planes” versus “crystallographic planes”, see (again) the begin of Sect. 4.1.4.
- 20.
Conventional TEM images also reveal the presence of such distortions, albeit at a spatially less resolved scale. See Fig. 6.19 case (b), the bright field image shown in the top part of the figure: the nitride (VN) platelets in the ferrite (b.c.c. iron) matrix are surrounded by dark contrast along their faces. This contrast is caused by precipitate/matrix misfit strains inducing local bending of lattice planes in the matrix leading to the contrast observed.
- 21.
Elastic scattering involves the interaction of the incident electrons with the nuclei of atoms in the specimen, which is associated with no loss of energy but a change of direction (momentum). Diffraction is an elastic scattering process. Inelastic scattering involves the interaction of the incident electrons with the electrons of atoms in the specimen, which is associated with both energy loss and change of direction (momentum). An incident electron can eject an electron out of a near core orbital of an atom, leaving a “hole” in the near core orbital. Next, an electron in a higher orbital of the atom concerned can jump into the near core orbital with the “hole”. This process is associated with the emittance of the energy difference between the higher orbital and the lower orbital in the form of either X-rays of characteristic energy (in this way the X-rays from X-ray tubes used in X-ray diffraction analysis are produced; cf. Sect. 4.5 and Sect. 6.10) or, less frequently, in the form of a so-called Auger electron of characteristic energy which is ejected from a relatively high atomic orbital.
- 22.
In fact, the methods based on using a scanning electron probe and measuring the energies of, e.g., the emitted Auger electrons or the inelastically scattered electrons (as in EELS) could have also been called EPMA, but because of the historical development this is not usual and the designation EPMA is reserved for the analysis of the X-ray radiation induced by a scanning electron probe.
- 23.
It is even possible to make maps of the state of bonding and type of chemical environment of atoms by selective energy filtering, i.e. (again) selecting those electron energy-loss ranges which are particularly sensitive to the effects of interest.
- 24.
So-called low-voltage SEM (LVSEM) allows high-resolution imaging of delicate biological structures sensitive to electron radiation induced damage.
- 25.
Often the word “crystallite” instead of “crystal” is used to allow the finite size of the coherently diffracting crystalline domain, giving rise to the observed “size broadening”, to be smaller than the size of a grain in a polycrystalline specimen. This can be relevant if, for example, specific defect (e.g. dislocation) arrangements occur in an otherwise perfect crystal which induces incoherency of diffraction at the location of such a defect arrangement. Then the “size” leading to the size broadening in the measured diffraction-line profile is smaller than the grain size.
- 26.
Equation (6.33) to a large extent parallels Eq. (6.10). However, there occurs a significant difference. The path difference of two rays diffracted by two neighbouring slits of the grating in Fig. 6.4 equals psinφ = \( d_{hkl}^{\rm ref}\)sin(2θ), recognizing the similar roles of p and \( d_{hkl}^{\rm ref}\) and of φ and 2θ. However, the path difference between two (X-) rays diffracted from two neighbouring lattice planes equals 2\( d_{hkl}^{\rm ref}\)sinθ = 2p sin(φ/2). This difference is a consequence of the grating in Fig. 6.4 being oriented not symmetrical with respect to the incoming and diffracted rays, whereas the lattice planes in the X-ray diffraction experiment are oriented symmetrical with respect to the incident and diffracted X-rays.
- 27.
More precisely: the microstrain distribution must be Gaussian for all correlation distances. The correlation distance is the distance between two points in the specimen, in a direction perpendicular to the diffracting lattice planes, for which the strain is considered.
- 28.
The local strain is the strain for which the correlation distance is nil (see Footnote 27).
- 29.
Here it is assumed that no hkl dependence of e, ε and D occurs, so the subscripts “hkl” have been omitted.
- 30.
The total structural line broadening cannot be equated with the measured diffraction-line broadening because instrumental broadening occurs as well and is included in the measured line profile. Various more or less exact approaches exist to correct for the instrumental broadening. Within the context of the discussion in this section, the following procedure is indicated. The instrumental line broadening is measured using a standard specimen that does not show structural line broadening. Then the integral breadth of the total structural line broadening may approximatively be obtained according to: \(\beta_{\rm{total}} = \beta_{\rm{measured}} - \beta_{\rm{instrumental}}\).
- 31.
- 32.
The orientation of the measured strain (lattice spacing), in a certain coordinate system called here the “specimen frame of reference”, is fully described by two angles: φ representing the rotation angle of the specimen about the specimen surface normal and ψ representing the tilt (inclination) angle of the specimen surface normal.
- 33.
Intrinsic elastic anisotropy means that if a constant uniaxial (state of) stress is applied to a single crystal of the material considered, then the resulting strain in the direction of the stress depends on the orientation of the crystal with respect to the direction of the stress. Elastic isotropy involves that the strain in the direction of the stress is the same independent of the orientation of the crystal, for the same value of applied stress.
- 34.
The diffraction experiment as described above relies on the determination of the lattice spacing of the diffracting lattice planes and thus the “measurement direction” is perpendicular to the diffracting lattice planes (cf. the above discussion of the sin2ψ method and Fig. 6.32).
- 35.
The compressive nature of a stress component is, by convention, expressed by deliberately positioning a minus sign before a corresponding stress value (cf. Sect. 12.2).
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Mittemeijer, E.J. (2021). Analysis of the Microstructure; Analysis of Structural Imperfection: Light and Electron Microscopical and (X-Ray) Diffraction Methods. In: Fundamentals of Materials Science. Springer, Cham. https://doi.org/10.1007/978-3-030-60056-3_6
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