PSI - Issue 40

A. Barannikov et al. / Procedia Structural Integrity 40 (2022) 40–45 A. Barannikov at al. / Structural Integrity Procedia 00 (2022) 000 – 000

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A. Barannikov t l. / Struc ural Integrity Proc dia 00 (2022) 0 0 – 000

2. Material and methods Under monochromatic, collimated X-ray beam illumination of a slightly deformed thin single crystal plate placed in Bragg condition with diffraction angle 2 θ , only a limited region will diffract. This is due to arising misorientation of the lattice planes deviating some parts of the crystal from the perfect Bragg condition. Thus, only a small region corresponding to the same misorientation angle Δ θ is accepted for diffraction. As the result, a so-called equi misorientation contour (isoline showing the region of equal Δ θ ) can be observed on a 2D high-resolution X-ray camera. A small rotation of the sample on the different angles ±Δ θ close to the diffraction angle 2 θ make it possible to image appropriate isolines forming by the rest of the single-crystal plate. The summation of the obtained images allows forming a contour map representing the deformation field of the sample. In the first approximation, a deformation profile of the thin plate can be reconstructed using the following expression: where z ( x , y ) is the deformation amplitude at a specific point on the sample surface ( x , y ), also corresponding to the spatial coordinates of the contour map, ū = ( u x , u y ) is a unit vector in Cartesian coordinates defining the direction of the deformation profile reconstruction, and Δ θ ū is the tangential angle of the deformation profile at the observation point. It should be noted that the sensitivity of this approach to the thin single crystal plate deformation is determined by the smallest rotation angle of the sample provided by the goniometer, and the camera's spatial resolution, which typical value can reach several micrometers. Thus, the deformation amplitude z of the thin single crystal plate can be measured with the accuracy of some nanometers. The experimental demonstration of the considered technique was implemented using a circular X-ray FZP as the sample (Fig. 1). This optical element is manufactured by MEMS technology including the following main processes: electron beam- and photolithography, deep plasma etching, and anisotropic wet etching of silicon. The FZP consists of a silicon pl ate with an etched 1080 μm size square window and T m = 12 μm thick silicon membrane. The [110] crystal direction of the membrane is parallel to its surface. The membrane has 242 circular-shaped channels (zones) with 9 μm depth h . The diameter of the area with etched channels A is equal to 387 μm. The focal distance of the FZP at 4 keV radiation energy is 0.5 m. The parameters of the FZP are presented in Table 1. 2. Material and metho s Under mo ochromati , c llimated X-r y beam illuminat on of slightly deforme thin single crystal plate placed in Bragg condition with diffraction angle 2 θ , on y a limited region will diffract. This is due to arising misorientati n of the lattice planes deviating some parts of the crystal from he perfect Bragg ondition. Thus, only a sma l region corresponding t the same mi ori ntati n a gle Δ θ is accepted for diffraction. As the resul , a so-called equi misorientati n contour (isoline showi g t e region of equal Δ θ ) c n be observ d on a 2D high-resolution X-ray camera. A s all rotation of he sample on the different angles ±Δ θ close to the diffraction angle 2 θ mak it possible to image appropri te is lines forming by the rest of the ingle-crysta plate. The summation of the obtained im ges allows forming a co tour map representing th deformation field f th sample. In the first approximation, a deformation profile of the thin plate can be re o structed using the following expression: ( , ) tan( ) u x y u z z x y u x y         (1) where z ( x , y ) is the deformation a pl tude at a sp cific point on the sample surface ( x , y ), also corresp nding t the spatial coord nates of the contour map, ū = ( u x , u y ) is a unit vector n Cartes an coordi ates defining the d rec ion of the deformation profile rec nstructi , and Δ θ ū is the tangential angle of the deformation profile at the observati n point. It should be noted that the sensitivity of th s approach to the thin single crysta plate deformation is determined by th smallest rot tion angle of the sample provid d by th goniometer, and he camera's spati l resolution, which typical value can reach sev ral micrometers. Thus, the deformation a pl tude z of th thin single crystal plate c n be me sur d with the accuracy of some nan ters. The experimental demonstration of the consid red technique was implemented using a circular X- ay FZP as the sample (Fig. 1). This optical element is a ufactured by MEMS technology i c uding the following main processes: el ctron b am- and photolithography, dee plasma etching, and anisotropic wet etching of silicon. The FZP consists of a silic n pl ate with n etched 1080 μm size square window and T m = 12 μm thick silicon membrane. The [110] crystal di ec ion of the membrane is p rallel to its surface. The membrane has 242 circular-shaped channels (zo es) with 9 μm depth h . The diamet r of th a ea with etched chann ls A is equal to 387 μm. The focal distance of he FZP at 4 keV radiation energy is 0.5 m. The parameters of the FZP ar present d in Table 1. ( , )          tan( ) u x y u z x z y z x y u u 

(1)

Fig.1. The principal scheme of the FZP (left) and SEM image of the FZP (rig t).

Fig.1. The principal scheme of the FZP (left) and SEM image of the FZP (right).

Table 1. The parameters of the FZP. Aperture diam ter ( μm ) 387

Table 1. The parameters of the FZP. Aperture diameter ( μm ) 387 M mbrane thickness ( μm ) 12

Membrane thickness ( μm ) 12

Out rmost zon width ( μm ) 0.4

Outermost zone width ( μm ) 0.4

Zone height ℎ ( μm ) 9

Zone height ℎ ( μm ) 9

Number of zones

Number of zones

F cal distance t the 4 keV (m)

Focal distance at the 4 keV (m)

242

242

0.5

0.5

3. Experiment The experiment was performed at the ID06 beamline of the ESRF. The beam was produced by an in-vacuum undulator with 40 μm x 900 μm (FWHM) source size in the vertical and horizontal directions, respectively. The 3. Experiment The experiment was performed at the ID06 b amline of the ESRF. The beam was produced by an in-vacuum undulator with 40 μm x 900 μm (FWHM) source size in the vertical and horizontal d rections, respectively. The