The capability of operating a separate crystal x-ray interferometer over centimeter displacements has made it possible to observe minute strain fields of a bent crystal at the atomic scale resolution by means of phase-contrast x-ray topography. Measurement and predictive capabilities of lattice strain are key ingredients of a highly accurate measurement of the Si lattice parameter and of a determination of the number of atoms in a realization of the mass unit based on an atom mass. Here we show that the observed strain can be accurately predicted by a finite-element analysis of the crystal deformation.
© 2009 Optical Society of America
A kilogram definition on the basis of a natural quantity or of a fundamental constant is at the top of the metrological agenda [1, 2]. A way to achieve this goal is to assemble a number of 28Si atoms into a prototype linking atomic and macroscopic masses via the constitutive relation M(28Si)=NAm(28Si), where M and m are the molar and atom masses of 28Si and NA is the Avogadro constant. To this end, it is necessary to count the number of constituent atoms of an almost perfect spherical silicon crystal; this is equivalent to determining the Avogadro constant.
In addition to the determination of the molar volume, this counting requires the measurement of the d 220 spacing of the lattice planes having Miller indices (220) to within an accuracy of 3×10-9 d 220 . This measurement is carried out by combined x-ray and optical interferometry. To achieve the stated uncertainty, we designed and manufactured an x-ray interferometer with an unusually long analyzer crystal, as well as a guide capable of displacements up to many centimeters with guiding errors commensurate with the requirements of atomic-scale positioning and alignment. When the lattice spacing value has to be used to count the atoms in the crystal, via the sphere volume, account must be taken for the different self-weight deformations of the sphere and of the interferometer. Though very small, at the required accuracy level, self weight could influence the measurement result. To account for gravity, we calculated numerically the deformation of the interferometer analyzer-crystal by a finite element modeling package. To check our measurement and predictive capabilities, we intentionally loaded the crystal with two 50 g masses and compared the calculated lattice strain with that observed by phase contrast x-ray topography.
2. Triple Laue x-ray interferometer
A triple Laue x-ray interferometer is similar to a MachZehnder interferometer in classical optics . As shown in Fig. 1, it consists of three crystal slabs so cut that the (220) planes are orthogonal to the crystal surfaces. X rays from a 17 keV Mo Kα source, having a (10×0.1) mm2 line focus, are split by the first crystal and then recombined, via two transmission crystals, by the third, called analyzer. The description of the interference pattern requires the use of the dynamical theory of x-ray scattering by crystals [4, 5], but its basic characteristics can be understood from geometric optics. Accordingly, each ray, when diffracted by any crystal of the interferometer, acquires a phase delay equal to 2πux/d 220, where ux is the displacement of the crystal atoms in a direction normal to the diffracting planes and d220 is the diffracting-plane spacing. No phase delay occurs in the directly transmitted rays.
The interference pattern thus replicates the superposition of the diffracting planes of the crystals forming the interferometer and crystal strains are visible in it . Since four sets of diffracting planes superimpose to form a moirè pattern, an identical deformation in all the crystals is not observable and the phase-contrast image gives no indication about the strain in any single crystal. The capability of operating a separate-crystal interferometer opens the way to surveying the lattice strain of the analyzer crystal separately from that of the splitter/mirror pair. In fact, traveling fringes can be obtained by moving the analyzer crystal in a direction orthogonal to the diffracting planes; actually, by displacing the analyzer atoms. A periodic variation of the transmitted and diffracted x-ray intensities,
is thus observed, the period being the diffracting-plane spacing. Since, as the analyzer is moved, different parts of it are observed, the x displacement, with a reversed sign, is also the horizontal coordinate on the analyzer surface. The analyzer embeds a front mirror, so that the displacement is measured by optical interferometry; the necessary picometer resolution is achieved by polarization encoding and phase modulation. To eliminate the adverse influence of the refractive index of air, the experiment is carried out in a vacuum.
The operation of a separate-crystal interferometer is a challenge: the fixed and movable crystals must be so faced as to allow the atoms to recover their exact position in the initial single crystal and they must be kept aligned notwithstanding the analyzer displacement. The key characteristic of our apparatus is a measurement capability over displacements up to 5 cm. This is obtained by means of a guide where an L shaped carriage slides on a quasi-optical rail. An active tripod with three piezoelectric legs rests on the carriage. Each leg expands vertically and shears in the two transverse directions, thus allowing compensation for the sliding errors and electronic positioning of the x-ray interferometer over six degrees of freedom to atomic-scale accuracy. Crystal displacement, parasitic rotations, and transverse motions are sensed via laser interferometry and by capacitive transducers. Feedback loops provide picometer positioning, nanoradian alignment, and interferometer movement with nanometer straightness.
The analyzer is mounted on a silicon plate and it is kept in position by a thin film of high viscosity silicon oil. This prevents accidental movements without stressing the crystal; the only crystal deformation is due to the self-weight. The analyzer design was based on the finite element analysis of self-weight bending and of the response to surface stress and thermal loads; the compromise between the requirements of a beam-like crystal and minimum bending ended into the shape shown in Fig. 2. The analyzer is supported on the Airy’s points, whose location was optimized to minimize bending or droop. As shown in Fig. 3, the residual strain is minimum at the crystal top, but it is nevertheless of the same magnitude as the wanted d 220 uncertainty and, therefore, it is worth while correcting the measurement result. This raises a question about the accuracy of the numerical calculation. Although our d220 measurements should be sensitive sensitive enough to detect the self-weight deformation, other effect are still prevailing and prevented a direct check of the numerical results . Therefore, in order to make the crystal strain easily observable, we decided to magnify the crystal bending.
3. Phase contrast topography
To assess the predictions of the finite element analysis, we exploited the interferometer size and the centimeter scan capability to observe the lattice deformation of the analyzer when it is loaded with a Si bridge carrying two identical 50 g masses at its ends (Fig. 2). The strain field εxx=∂xux=(d eff-d 220)/d 220 is easily obtained from a survey of the effective lattice spacing d eff=d 220(1+∂xux) at different crystal points. According to the measurement equation
where n is the number of x-ray fringes of d eff(x, z) period observed in a millimeter crystal displacement centered in (x, z) and spanning m optical fringes of λ/2 period, the effective lattice spacing is determined by comparing the periods of the x-ray and optical fringes, a comparison made by measuring the x-ray fringe phases at the displacement ends. The analyzer was then shifted step-by-step while the splitter/mirror crystal and x rays were maintained fixed and the measurement of the effective lattice spacing repeated; d eff measurements were thus carried out over 52 contiguous crystal slices, about 1 mm wide. Since a temperature difference of 1 mK implies a lattice spacing variation of about 2.5×10-9 d 220, all measurement results have been reduced to the same 22.5 °C temperature. The traveling x-ray fringes were recorded by means of a multianode photomultiplier having a vertical pile of eight NaI(Tl) scintillator crystals. Eventually, the x-ray fringes were processed to obtain, first, the effective lattice spacing value in each of the 52×8 image pixels – of size (1×1.75) mm2 – and, then, the strain field. The reason for this procedure resides in the impossibility of keeping the coupling between the x-ray and optical interferometers as stable as desirable; a differential measurement technique, at each step of which the analyzer is repeatably moved back and forth to measure d eff, allows the effects of these instabilities to be removed.
In the measurement, we can not trust linear vertical-strains. The reason is a deficiency of the optical interferometer. A parasitic analyzer-tilt associated to the displacement (the ρ angle in Fig. 1) imitates an intrinsic tilt of the lattice planes. To avoid this, the analyzer tilt is sensed by the optical interferometer, via the differential displacement across the beam spot, and it is electronically nullified. Since the necessary nanoradian sensitivity of the tilt measurement corresponds to a picometer sensitivity of the differential measurement, misalignments and aberrations easily compromise the interferometer operation and make it to detect non-existent tilts which, at the lowest approximation order, are proportional to the analyzer displaceme This makes it impossible to distinguish between a uniform rotation of the analyzer during the measurement and a uniform tilt of the diffracting planes, which planes will appear fanlike and uniformly strained in the vertical direction .
Consequently, we removed a meaningless linear vertical-strain from the (d eff-d 220)/d 220 data. In order to retain the information about the strain magnitude, the removed strain was set to zero at the vertical coordinate of the virtual pixel having a zero offset with respect to the center of the laser-beam spot. In fact, in this pixel, both the x-ray and optical interferometers sense the same displacement and the measured strain is unaffected by faults in the detection and compensation of the analyzer tilt. Finally, in order to preserve only the strain due to the load, we subtracted the result of a previous survey of the unloaded analyzer . The result is shown in Fig. 4 (middle), where the data have been smoothed by the
fitting polynomial. The polynomial degree is the minimum one which satisfactorily approximates both the experimentally observed and numerically calculated strains. For what concerns the vertical direction, also because only eight data points are available, no more than a cubic polynomial is meaningful. Fit residuals can be assessed from Fig. 5.
Next, the displacement field ux(x, z) was calculated by integrating (3) along the x axis; the result is shown in Fig. 4 (top). Since the fixed and movable crystal-lattices superimpose to form the interference pattern, we have no indication about the absolute displacements; only displacement differences are meaningful. This is expressed by the ambiguity of the origin x 0 in
where we have ux(x 0, z)=0. Our choice was x 0=35 mm, because the finite-element analysis indicates an almost perfect symmetry axis at this point.
The shear strain should be calculated as εxz=(∂zux+∂zux)/2, where ux and uz are the horizontal and vertical displacements of the crystal atoms. However, the z displacement is invisible to the x-ray interferometer. Therefore, we calculated εxz as the rotation of the lattice planes about the y-axis by differentiating ux(x, z) with respect to the z variable. Hence,
The result is shown in Fig. 4 (bottom). Obviously, the same εxz definition was used in the finite-element analysis.
4. Finite element analysis
The predictions of the finite-element model are shown in Fig. 4. Silicon is an anisotropic material; consequently, the Young modulus and Poisson ratio depend on the crystal direction along which it is stretched. The values of the non-zero elements of the stiffness tensor, in the  crystal-axes, are cii=165.6 GPa, cij=63.9 GPa (i, j=1,2, 3), and ckk=79.6 GPa (k=4,5, 6) . In order to compare theory with practice, the results of the finite-element analysis have been post-processed to simulate the results of phase-contrast topography. Consequently, in Fig. 4 (top), the unobservable rigid-body displacement and rotation have been eliminated. Similarly, in Fig. 4 (middle), the unobservable vertical strain gradient has been eliminated and so have been eliminated the rigid-body rotation and uniform tilt of lattice planes (indistinguishable from an analyzer rotation proportional to the displacement) in Fig. 4 (bottom). Since the geometry of the actual analyzer was not as symmetric as designed – besides, the use of three support points will broke any symmetry, the simulated displacement, strain, and shear are not perfectly symmetric.
In Fig. 4, the comparison of prediction and observations points out the agreement between the calculated and observed deformation, strain, and shear. In order to carry out a more severe test, Fig. 5 compares some horizontal and vertical section of the simulated and observed strain fields. The figure evidences that in the vertical sections the agreement between theory and experiment is less satisfactory. The reason is the correlation between the errors of the eight strain-values measured in the same vertical strip. In fact, they are all obtained by using the same optical signal as a reference. Consequently, any failure in the optical-interferometer adds to all data the same offset and an apparent strain-gradient; these errors are smoothed only by the global polynomial fit (3). For this reason, and because only eight data per vertical strip are available, a polynomial degree in z higher that three is meaningless and some detail of the shear field is unobservable. This explains the less satisfactory agreement between the observed and calculated tilts displayed in Fig. 4.
This investigation can be improved if remedies are applied for the following weak points. First, surface effects limited the sensitivity of the interferometer to lattice strains greater than 10 nm/m and prevented the observation of the self-weight deformation of the analyzer crystal . In the second place, when assembling the interferometer we were not fully aware of its capability to detect the minute differences between the lattice deformations when the location of the support points or of the load forces is changed by only one millimeter. Consequently, the contact areas of the analyzer with the base plate and the load, (4×4) mm2 and (5×15) mm2, respectively, were too large. For this reason, in the numerical simulation, the x coordinates of the lines where the crystal had been constrained and loaded were adjusted, to within the contact areas, to best fitting of the experimental data. This is equivalent to adjust the elasticity coefficients of silicon and prevented the Young modulus in the  crystal direction to be measured from atomic-scale lattice deformations.
Recent progress in technology allowed an x-ray interferometer to be moved and positioned to atomic-scale resolution over centimeter displacements. This opened the way to measuring the Si lattice parameter to previously unattained accuracy and, in turn, to count the number of atoms in a crystal of pure silicon to within the accuracy necessary to relate the kilogram to the mass of a certain number of atoms. Since they ultimately limit the measurement accuracy and the atom count, the capabilities to predict and detect minute lattice strains have been assessed by measuring the deformation, strain, and rotation of the diffracting planes of a loaded x-ray interferometer by phase-contrast topography. The results are in good agreement with the predictions of a finite-element analysis of the crystal bending, which were thus proved accurate enough to support a correction of the measurement value of d 220 for the lattice strain due to the self-weight.
This work was supported by the European Community’s Seventh Framework Programme ERA-NET Plus (grant 217257), the Regione Piemonte (grant D64), and by the Compagnia di San Paolo.
References and links
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