X-ray Diffraction with a Laue Camera

X-ray diffraction was performed on single crystalline or bi-crystalline specimens either to ensure the unique lattice orientation or to measure the misorientation of large grains. X-rays have a small penetration depth for gold based samples. With a common laboratory source, the X-rays probe a surface layer of severalµm. When an electromagnetic wave strikes a periodic array of atoms, the wave is scattered and reflections of high intensity only appears in special directions where constructive interference occurs. An incoming wave of wave-vector

~k = 2π~n/λ produces a reflection spot in the direction of the outgoing wave of wave-vector k~⁰= 2π~n⁰/λ if the difference of the wave-vectors

K = ~~ k⁰−~k (2.19)

is a vector of the reciprocal lattice [Ashc81c]. The reciprocal lattice is defined as the ensemble of all wave-vectors ~K that fulfil the condition exp¡i ~K · ~R¢ = 1 for all lattice vectors ~R of the direct lattice. The reciprocal lattice of an fcc lattice is a body-centred cubic (bcc) lattice.

In order to make sure that at least some directions produce constructive interference, the Laue method consists in using polychromatic instead of monochromatic X-rays [Ashc81a].

Figure 2.9 a) shows the measurement chamber of a Laue camera. The incoming X-ray beam from the right hits the sample and is back-reflected on the CCD screen surrounding the beam guide on the right. The sample can be moved in space and turned with the help of a goniometer stage.

The CCD camera allows a real time observation of the diffraction pattern produced by the crystal zone hit by the X-ray beam. If the probed part of the sample is single crystalline, the diffraction pattern will be a stereoscopic projection of the reciprocal lattice. An example of a diffraction pattern seen along the [111] crystallographic direction is shown in Figure 2.10.

The crystal orientation is always probed locally since the beam spot usually has a diameter of 0.5 mm. Therefore, an abrupt change in the diffraction pattern when moving the X-ray spot indicates the presence of a GB. If the diffraction spots are broadened or if the pattern turns continuously, the crystal lattice is not unique over large distances and it is a sign for a high dislocation density. A multitude of small diffraction spots is an evidence for a polycrystalline microstructure.

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2.3. X-ray Diffraction with a Laue Camera

Figure 2.10 – a) bcc lattice seen along the [111] crystallographic direction. The cubic unit cell is shown in grey. b) Simulated Laue diffraction pattern for the [111] direction. The spot size reflects the intensity of the reflected beam. c) Diffraction pattern of a gold alloy single crystal aligned along the [111] direction.

In the framework of the present thesis, the Laue camera was used to characterize samples, to show the single or poly-crystalline nature of samples used for mechanical spectroscopy experiments and to characterize grain boundaries for example in bi-crystals. One important parameter for describing bi-crystals is the misorientation angleθ of adjacent grain orientations.

It is defined asθ =q

ψ²₁+ φ²+ ψ²₂, whereψ1,φ and ψ2are the Euler angles needed to rotate the crystal lattice of grain 1 into the crystal lattice of grain 2.

Since the cubic system has a high symmetry, the Euler angles are not uniquely defined. The misorientation angle is then understood to be the smallest angle within the above definition.

Practically, the misorientaiton angle permits to distinguish between low angle grain bound-aries (LAGB) and high angle grain boundbound-aries (HAGB). LAGBs are commonly defined to have a misorientation angle below 10°.

In order to measure the misorientation between two adjacent crystallites, the diffraction pattern of both grains needs to be indexed. A general method to establish sample orientation and to calculate setting angles is described in [Busi67]. The indexing is done automatically by a software, which calculates first angles between reciprocal lattice direction. With the knowledge of the distance between screen and sample, one can attribute an angle to each pair of diffraction spots. These angles are then compared with theoretical angles for an fcc crystal and the software attributes lattice directions to each diffraction spot.

Figure 2.11 shows the diffraction patterns of two grains separated by an LAGB. Due to the indexing of most diffraction spots, the patterns can be compared to each other and the black arrows in Figure 2.11 a) indicate the position of the corresponding spots of b). In order to bring pattern a) in superposition with pattern b), the crystal lattice of a) should be turned around the x-axis and slightly around the y-axis. The program calculated misorientation angles∆x = 5.65°,

∆y = 3.18° and ∆z = 0.44°. The total misorientation of this GB is then p∆x²+ ∆y²+ ∆z²= 6.5°.

Chapter 2. Materials and Experimental Techniques

Figure 2.11 – Indexed Laue patterns of a sample containing a low angle grain boundary. The pattern shown in a) has to be rotated around the y-axis and around the z-axis to obtain the pattern b). The black arrows indicate the new position of the diffraction spots [110] and [210]

in b).

Figure 2.12 – EBSD maps of a) a copper sample with very large grains and b) a gold alloy polycrystal. The grain colouring is the inverse pole figure colouring with respect to the sample axis (corresponds to the horizontal axis of the maps). Twin boundaries are shown in white.

The misorientation measurement for a general boundary was usually done at several positions along the GB by analysing pairs of diffraction patterns at different positions. The scattering along a boundary is typically of the order of 1-3°.

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