Three-Dimensional X-Ray Fourier Transform Holography: The Bragg Case

A novel approach to determine the structure of nanoscale crystals in three dimensions is proposed by the use of coherent x-ray Fourier transform holography in Bragg geometry. The full internal description is directly obtained by a single Fourier transform of the 3D intensity hologram. Together with the morphology, Bragg geometry gives access to the 3D displacement field within the crystal. This result opens great possibilities for the investigation of strain fields inside nanocrystals in a simple way.

Figure 1

(a) AFM characterization of the holographic sample (the arrow indicates the beam direction, the height is given in nm). (b) Sketch of the 3D acquisition in Bragg geometry. (c)–(e) 2D intensity slices taken for different values of ${\mathbf{q}}_3$. (f) Normalized intensity line cuts extracted along the black lines from (c) (blue stars) and (e) (pink circles).

Figure 2

2D slice taken in the center of the 3D direct space matrix obtained by inverse Fourier transforming the 3D intensity matrix. One of the two object images is emphasized by a pink line (inset: the same image in its 3D representation).

Figure 3

(a)–(c) 3D isosurface views of the object image density. (d) 2D convolution of OC with RC obtained from the AFM measurements (Fig. 1); the black dotted line is the enlarged contour plot obtained from the 3D view shown in (a).

Figure 4

Stack of 2D slices extracted from the 3D direct space matrix. (a) Internal view of the object image density [the color scale is linear (arbitrary unit)] and (b) phase (the angular color scale is given in radians). (c) Mean values of the object image phases (black circles) as a function of $r_3$ (in nm), calculated from (b); expected phase behavior (solid line) calculated for a RC density function with linear phase shift. This phase behavior results from a tilt ($\alpha$) of the RC lattice planes with respect to OC. The inset at the bottom shows two values of ${\mathbf{u}}_{004}$ (pink and blue arrows), which increases linearly with ${\mathbf{r}}_3$ when the RC is tilted.