Abstract
This chapter explains diffraction contrast in bright- and dark-field conventional transmission electron microscopy, workhorse techniques in materials research. Materials parameters that affect diffraction intensity in kinematical theory are presented, and compared to parameters in dynamical theory. The phase-amplitude diagram is developed for kinematical diffraction theory, and is first used for problems that can be solved analytically, such as intensity fringes in bend and thickness contours. Including the effects of strain fields in phase-amplitude diagrams is shown to be a powerful method for determining qualitatively the diffraction contrast effects from the crystal containing a dislocation. The weak-beam dark-field method for a calculating dislocation strain contrast is described, and the quality of its images is discussed with examples. Phase shifts at interfaces are included in the phase amplitude diagram method. Intensity fringes in the diffraction contrast from moiré effects, stacking faults, and domain boundaries are explained. Some results from dynamical theory are cited without proof, and used to describe diffraction contrast effects from crystals of modest thickness.
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Notes
- 1.
By convention, \(\hat{\boldsymbol{z}}\) points towards the electron gun. This is handy for diffraction patterns and stereographic projections. On the other hand, when integrating wavelet amplitudes from top to bottom of a specimen, we may want \(\hat{\boldsymbol{z}}\) to point down. In such cases it may be necessary to handle with care the sign of the phase 2πsz, as in Sect. 8.9.
- 2.
We can work with unit cells rather than individual atoms because we will consider distortions δr g from internal strains in the crystal that vary over lengths much larger than the unit cell.
- 3.
These two observations can both be understood as how ξ g depends inversely on \(\mathcal{F}_{\boldsymbol{g}}\). Appendix A.5 explains how to convert electron form factors (or \(\mathcal{F}_{\boldsymbol{g}}\)) to values for electrons of different energies. The inverse of the relativistic factor in Appendix A.5, plus (8.17), permits the conversion of ξ g in Table 8.1 to extinction lengths for electrons of other energies.
- 4.
This transforms each phase factor to its complex conjugate exp(i2πsz)→exp(−i2πsz). The diffracted intensities remain correct. When there is another z-dependent contribution to the phase of the diffracted wavelets, however, it is important to ensure that all z-coordinates and s are defined consistently.
- 5.
Don’t laugh. This will happen to your specimens too.
- 6.
Since s≃0 for the (002) and the (020) diffractions at the crossing of the (002) and the (020) bend contours, we expect s≃0 for the (011) too. The (011) bend contour must cross at this same location.
- 7.
The condition s=0 is a poor one for making images of dislocations, because diffraction contrast occurs over a large area around the dislocation core, so the dislocation image is wide and fuzzy.
- 8.
In the case where the Burgers vector of a perfect dislocation lies fully in the plane of the diffraction pattern, the expected phase factor, 2π Δk⋅b, is 2πn, where n is an integer, because b is a lattice translation vector and Δk=g. It is of course possible that this integer is zero when b is perpendicular to g.
- 9.
This is best planned prior to attempting the g⋅b experiments on the microscope.
- 10.
Equation (8.23) pertains to a defect in the center of a specimen of thickness, t.
- 11.
Mathematica code for sy=+0.45 is:
RealA = Integrate[Cos[2*Pi*z - ArcTan[2.2*z]], {z, -2, depth}]
ImagA = Integrate[Sin[2*Pi*z - ArcTan[2.2*z]], {z, -2, depth}]
ParametricPlot[{RealA, ImagA}, {depth, -2, 2}]
- 12.
Mathematica code for s=+0.01g x , \(\boldsymbol{b} = b \hat{\boldsymbol{x}}\), g⋅b=1 and y=100/g x is:
RealA = NIntegrate[Cos[2*Pi*z - ArcTan[z/y]], {z, -2, 2}]
ImagA = NIntegrate[Sin[2*Pi*z - ArcTan[z/y]], {z, -2, 2}]
Plot[RealA*RealA+ImagA*ImagA, {y, -30, 30}]
- 13.
These calculations ignored the thickness of the sample by neglecting the variations of ψ around the asymptotic circles at the left and right sides of the curves in Fig. 8.31.
- 14.
Double images can also occur when two or more diffractions are excited and two different diffraction conditions exist, as illustrated in Fig. 8.33b, but this is poor experimental technique. It is important to have well-defined two-beam kinematical diffraction conditions if the images are to be interpreted by the methods in this section.
- 15.
Dislocations are easiest to see when they are near bend contours. As the specimen is tilted so the bend contour moves away from the dislocation, its contrast weakens.
- 16.
Improvements in sensitivity with imaging plates and CCD cameras help overcome this problem.
- 17.
Note that z<0 since the top of the specimen is at z=0. Since s>0, the slope in Fig. 8.39a is negative.
- 18.
Using a typical extinction length of 500 Å from dynamical theory, the effective deviation parameter of (8.16) is 0.0102 Å−1, which predicts 9.8 wraps, so dynamical theory gives a similar result in what follows.
- 19.
Note again the assumption that δr is the same for all unit cells below the interface, i.e., all unit cells undergo a simple translation by δr. A rotation of the crystal below the interface could cause δr to increase with depth below the interface. This is analyzed as a discontinuity in s in Sect. 8.13.2 on δ boundaries.
- 20.
This is much like placing an identical plane of atoms halfway between two planes with a full wavelength interference, as for example the center atoms in the bcc for (100) diffractions (Fig. 5.9).
- 21.
In constructing Fig. 8.45 we made a few assumptions about the thicknesses of the layers above and below the interface (they were assumed to be the same—N/2 unit cells) and the value of s (which was chosen to produce a half-circle in the phase-amplitude diagram over the thickness of the top or bottom layer).
- 22.
In a similar way, we find the first-nearest-neighbor displacement as: \(1/(\sqrt{2})[110]\cdot a/2 [110]=a/\sqrt {2}\).
- 23.
To perform dynamical calculations of stacking fault contrast, the dynamical “Bloch waves” are transformed into the diffracted beam representation at the depth of the fault. The diffracted beam below the fault is then multiplied by a phase factor such as exp(i4π/3). This multiplication is the equivalent of the kink in the graphical phase-amplitude diagram.
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Fultz, B., Howe, J. (2013). Diffraction Contrast in TEM Images. 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_8
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