Coherent x-ray diffraction imaging of grown-in antiphase boundaries in ${\mathrm{Fe}}_{65}{\mathrm{Al}}_{35}$

Coherent x-ray diffraction has been used to image grown-in antiphase boundaries (APB’s) in a metal alloy, which represent pure phase objects. The fine structure within the (001) superstructure diffraction peak of a $B2$-ordered bulk ${\mathrm{Fe}}_{65}{\mathrm{Al}}_{35}$ sample was inverted by means of iterative algorithms that Fourier transform between reciprocal and real space, applying appropriate constraints in each domain. Since the sample object is noncompact, bigger than the beam footprint, knowing the precise beam profile was essential to define the real-space constraint. Even though a unique long-range structure could not be derived, the algorithm found phase structures that were locally unique. These were identified in all reconstruction runs by means of a cross-correlation analysis. The obtained characteristic APB morphology is confirmed by transmission electron microscopy results from the same sample, revealing nearly planar APB walls on {110} planes terminating at grown-in dislocations.

Figure 1

(Color online) Measured illumination function $f(x,y)$. The size of one pixel was $0.45\times 0.45\mu {\mathrm{m}}^2$.

Figure 2

(Color online) Central part $(200\times 200\text{pixels})$ of the high-resolution image of the ${\mathrm{Fe}}_{65}{\mathrm{Al}}_{35}$ (001) superstructure reflection on a logarithmic pseudocolor intensity scale. The direction of the scattering angle $2\Theta$ is vertical. The full white line is a slice along the $2\Theta$ direction, exhibiting strong intensity modulations. The dashed (pink) line is a Gaussian fit to the slice with a FWHM of $23.9\pm 0.6\text{pixel}$ $(\Delta Q∕\text{pixel}\approx 2.9\times {10}^{-5}{\mathrm{\AA }}^{-1})$.

Figure 3

(Color online) Top: schematic of the scattering at different domains (light and dark gray). Bottom: picture in reciprocal space with ${\mathit{k}}_i$ the incoming and ${\mathit{k}}_f$ the outgoing wave vector. The domains give rise to sheets of diffraction intensity (dark gray). The dashed ellipse indicates the extension of the speckled peak. The width along ${\mathit{Q}}_z$ is related to the information depth by $2\pi ∕t$. The width along ${\mathit{Q}}_{\Vert }$ is determined by the domain size according to $2\pi ∕s$. The typical distance between the sheets, i.e., the mean speckle size along ${\mathit{Q}}_{\Vert }$, is given by $2\pi ∕\left(Ms\right)=2\pi ∕w_{\Vert }$. The CCD detector (transparent blue with its surface normal antiparallel to ${\mathit{k}}_f$) cuts through these sheets under an angle of $\Theta$.

Figure 4

(Color online) Illustration of the applied real-space constraints in the reconstruction procedure. Left-hand side: support (black) for the HIO cycles, defined by Eq. (4). Right-hand side: real-space amplitude used for the GS cycles, defined by Eq. (5).

Figure 5

(Color online) Typical RMS error in real (full red line) and reciprocal space (dashed black line) during 9800 iterations overall. Periodically, 200 GS cycles are followed by 600 HIO steps. The RMS error decreases relatively fast during GS cycles and rises while HIO is applied.

Figure 6

(Color) Typical reconstructed phases after 9800 iterations with periods of 200 GS and 600 HIO steps. Note the nonrandom phase structure within the support region.

Figure 7

Visualization of phase jumps in the reconstructed phases of Fig. 6. If the phase difference between a pixel and any of its four nearest neighbors is $\mid \Delta \varphi \mid ∊[0.35\pi ,\pi ]$, the pixel is shown in black; otherwise white.

Figure 8

(Color) Illustration of the cross-correlation analysis. Top: section of phase-jump pattern shown in Fig. 7 with three chosen masks indicated by the blue and green, respectively, rectangles. Middle: phase-jump pattern from another reconstruction run. The green rectangle indicates, where the corresponding mask is found. Bottom: the corresponding cross-correlation coefficient $\gamma$ with the position of its maximum $({\gamma }_{\max }=0.955)$ surrounded by a green rectangle.

Figure 9

Histogram of ${\gamma }_{\max }$ values from cross correlating the mask indicated by the green rectangle in Fig. 8 and its rotated copy, respectively, with results from 100 reconstruction runs. The histogram is fitted with a Gaussian, full black line.

Figure 10

Top: TEM dark-field image of the ${\mathrm{Fe}}_{65}{\mathrm{Al}}_{35}$ sample, using a weakly reflected beam of the superlattice (010) with a deviation parameter of about $0.04{\mathrm{nm}}^{-1}$. The direction of the imaging vector is indicated by the arrow (direction of incident beam $\approx \left[001\right]$). Antiphase-domain walls occur preferentially on {110} planes. Note that the slowly periodic intensity modulation of the image is caused by the increase of the sample thickness from zero (bottom right-hand side) to $\approx 110\mathrm{nm}$ (top left-hand side). Bottom: zoom of the upper right-hand corner, resolving the characteristic fringe contrast of inclined APB’s imaged using a superlattice reflection. The APB’s are slightly curved and terminate at superpartial dislocations $D_1$ and $D_2$ that show oscillating contrast.

Figure 11

Map of identified APB’s in several adjacent TEM images similar to Fig. 10. The dashed line indicates the area of the support used in the reconstructions for comparison.