Measurement of the miscut angle in the determination of the Si lattice parameter
MMeasurement of the miscut angle in thedetermination of the Si lattice parameter
C P Sasso, G Mana, and E MassaINRIM – Istituto Nazionale di Ricerca MetrologicaStrada delle Cacce 91, 10135 Torino, [email protected] February 2021
Abstract
The measurement of the angle between the interferometer front mirrorand the diffracting planes is a critical aspect of the Si lattice-parametermeasurement by combined x-ray and optical interferometry. In additionto being measured off-line by x-ray diffraction, it was checked on-line bytransversely moving the analyser crystal and observing the phase shiftof the interference fringe. We describe the measurement procedure andgive the miscut angle of the Si crystal whose lattice parameter was anessential input-datum for, yesterday, the determination of the Avogadroconstant and, today, the kilogram realisation by counting atoms. Thesedata are a kindness to others that might wish to repeat the measurementof the lattice-parameter of this unique crystal.
Keywords : Si lattice parameter, crystal orientation, x-ray interferometry, kilo-gram realisation
The measurement of the surface orientation was a critical aspect of the Silattice-parameter measurement by combined x-ray and optical interferometry[1, 2, 3]. It refers to the parallelism of the diffracting planes to the mirror-polished (front and rear) surfaces of the analyser crystal, whose displacement ismeasured in terms of a calibrated optical wavelength.To achieve a fractional accuracy near to nine significant digits in the meas-urement of the lattice parameter, we polished the (front and rear) analysersurfaces parallel to the { } diffracting planes to within a few microradians.To this end, we realised an apparatus whereby the Si monocrystal was groundand optically polished and the residual misalignment was measured [4]. Sub-sequently, we checked the miscut angle on-line directly by the combined x-rayand optical interferometer. 1 a r X i v : . [ phy s i c s . i n s - d e t ] F e b he paper is organised as follows. After a concise description of combinedx-ray and optical interferometry, in sections 2.1 and 2.2, we model the projectionerrors and detail how the x-ray interferometer is aligned to compensate them.Next, sections 3.1 and 3.2 describe the measurement procedures of the miscutangle. The results are given in section 4. Our x-ray interferometer is like a Mach-Zehnder interferometer of visible op-tics [5]. As shown in Fig. 1, it splits and recombines 17 keV x rays by Lauediffractions in perfect Si crystals. The splitter, mirror, and analyser operate insymmetric Laue geometry, where the { } diffracting planes are perpendicularto their surfaces. When the interfering beams are phase-shifted by moving theanalyser orthogonally to the diffracting planes, interference fringes are observed,their period being the plane spacing, d ≈
192 pm.The analyser displacement is measured by a laser interferometer having pi-cometre sensitivity and accuracy. The lattice spacing is measured as d = mλ/ (2 n ), where n is the number of x-ray fringes observed in m optical ordersof λ/ . × − Since, the x-ray and optical interferometers project the analyser displacement, s , into the unit normals to the diffracting planes and front mirror, h and n , toensure that both projections are equal, the analyser front mirror was polishedas parallel as possible to the diffracting planes. Also, the displacement waselectronically servoed to bisect the misalignment angle, and the laser beam wasaligned as much as possible orthogonally to the front mirror.A model of the projection errors is as follows. In a plane-wave approximation,as shown in Fig. 2, the measurement equation of the laser interferometer, s opt = 2( s · n )( n · o ) , (1)2 d-YAG 512 nm 17 keV Mo K a mirrorsplitter l /4 l /4 90° cornerauxiliary mirroranalyser referenceparallelepiped Figure 1: INRIM’s combined x-ray and optical interferometer. The analyserdisplacement (yellow arrow) is measured vs. an auxiliary mirror mounted onthe same Si base as the fixed splitter-mirror pair of the x-ray interferometer.To achieve picometre resolution, the optical interferometer uses polarisation en-coding and phase modulation. The pitch and yaw angles are measured to withinnanoradian resolution via differential wavefront sensing. The transverse dis-placements (horizontal and vertical) and roll angle are measured via a (coated)glass parallelepiped (having a 90 ◦ corner) and capacitive sensors (red discs),which are integral with the fixed Si base. Feedback loops set the movementorthogonal to the parallelepiped front. 3 {220} h o s n s·n ) s·n ) ·o s s · h Figure 2: Two-dimensional sketch of a hypothetical arrangement of the vectorsdetermining the projection errors. The dashed frame shows the analyser beforethe displacement s . Green: laser beam, brown: normal to the analyser frontmirror, blue: displacement (not in scale), red: normal to the lattice planes.where o is a unit vector parallel to the beam axis, is two times the componentof the analyser displacement along the front-mirror normal, s · n , projected onthe beam axis.The measurement equation of the x-ray interferometer is s x = s · h . Therefore,considering separately the projections on the laser beam, n · o , and on the front-mirror normal, s · n , the projection errors compensate if s · n = s · h . If n = h ,the displacement direction is irrelevant. Otherwise, it is required that s = s (cid:107) (cid:100) ( n + h ) + s ⊥ (cid:100) ( n × h ) , (2)where the hats indicate normalised vectors and the s = s (cid:107) + s ⊥ decompositionis arbitrary. Otherwise said, s must lie in the plane bisecting the normals to thediffracting planes and front mirror. The fractional measurement-error, ˆ s · ( n − h ),is proportional to the miscut angle, | n − h | , and bisection error, ˆ s · (cid:100) ( n − h ). To achieve a nine-digit accuracy, the miscut angle | n − h | must be as small aspossible. At the same time, the normals to the front mirror and laser beam, n and o , and the movement direction, ˆ s , must be aligned each other so thatˆ s · (cid:100) ( n − h ) is minimum.Figure 3 shows the way we aligned the x-ray interferometer. A glass paral-lelepiped rests on the same platform as the analyser (see Fig. 1). It has the topand right-side optically polished to within λ/
20 flatness and metallic coated.Capacitive sensors (rendered as red discs in Fig. 1) monitor its horizontal andvertical positions and drive piezoelectric actuators to keep them fixed to withinsub-nanometer accuracy. These feedback loops ensure that the parallelepipedmoves orthogonally to its front, which is orthogonal to the top and side to within12 µ rad and 8 µ rad, respectively. 4 h n s six degrees of freedom platform reference parallelepiped capacitive transducer 90° Figure 3: Two-dimensional sketch of a hypothetical alignment of the analysercrystal. A feedback loop locks the reading of the capacitive transducer; there-fore, the analyser displacement occurs orthogonally to the front (sky-blue) face.Brown: normal to the analyser front mirror, blue: displacement, red: normal tothe lattice planes.As shown in Fig. 3, the analyser is mounted in such a way the { } planesand front mirror are symmetrically placed to the parallelepiped front. Thisalignment is made with the aid of an autocollimator looking at the parallelepipedfront and the analyser rear. Obviously, the analyser miscut and parallelepipedorthogonality are taken into account. The raw crystal – a Si block (55 × ×
28) mm – was oriented by x-raydiffraction. Two opposite faces – which will be the interferometer front andrear mirrors – were ground and optically polished parallel to the { } latticeplanes.Figure 4 shows how the angles between the lattice planes and the crystalsurfaces were measured. The main components of the apparatus are a crystal-holder made by a two-axis tilter integrated into a precision rotary table andan autocollimator measuring the wobble of the crystal surface. Figure 5 showsthe silicon disc embedding the Si block (having the same crystallographicorientation) for the orientation and optical polishing.By tilting the crystal to keep aligned the Bragg’s reflection of the x rays whenit is rotated, the lattice planes are set orthogonal to the rotation axis. Next,the autocollimator measures the wobble of the optically-polished surface. Moredetails are given in [4]. The accuracy achieved is near to 1 µ rad, depending onthe surface flatness: a 2 cm wide part of a λ/
20 surface has a sagitta of 5 nmand an end-to-end tilt of 5 µ rad. 5igure 4: Diagram of the apparatus for crystal-orientation and measurement ofthe miscut angle (adapted from [4]). We repeated the measurement of the miscut angle on-line directly by the com-bined x-ray and optical interferometer. This measurement repetition strengthenedour confidence in the measured angle and checked the miscut of the surface areaactually sensed by the laser beam.As shown in Fig. 6, the measurement was done by translating the analysertransversally, in the vertical and horizontal directions. Because of the presentlimited operation range of the supporting platform, the translations were limitedto a few micrometres. To be safe, we verified not to lose integer orders ofthe x-ray interference by carrying out preliminary displacements of increasingamplitude. To ensure that the translations occur in the plane of the front mirror,a feedback loop locks to zero (to within 1 pm and 1 nrad) the axial displacementand the pitch and yaw rotations.Figure 7 shows the x-ray fringes observed before and after a vertical displace-ment of the analyser equal to 3.2 µ m. As shown in Fig. 6, the phase differencebetween the x-ray fringes detected at the start and end positions are propor-tional to the lattice displacements at the end, u x ( y, z = const . ) = s y · h and u x ( y = const ., z ) = s z · h , relative to the start.Since it was not possible to eliminate the drift between the optical and x-raysignals, the analyzer was moved repeatedly forwards and backwards, and the twosignals were repeatedly sampled at each end. Next, the drift was identified and6igure 5: Photograph of the silicon disc embedding the Si block for the opticalpolishing of the front and rear surfaces. From this Si block, both the INRIM’sand PTB’s interferometers were manufactured.7 h s s·h s Figure 6: Schematic diagram of the miscut-angle measurement (horizontal com-ponent) by combined x-ray and optical interferometry. The dashed frame showsthe analyser after a transverse (horizontal) displacement s . A feedback loop usesthe optical-interferometer signal to lock to zero the axial displacement and en-sures that s lies in the mirror surface. The phase shift of the x-ray interferencebefore and after the displacement is s · h . - -
200 0 200 400 axial displacement / pm c oun t s c oun t s u x ( y = const,z ) Figure 7: Scans of the x-ray fringes before (orange) and after (blue) a verticalanalyser displacement s y = 3 . µ m of the analyser. The dots are the x-photons counted in 100 ms. The solid lines are the best-fit sinusoids approximat-ing the data. The observed phase difference is u x ( y = const ., z ) = 0 . d (see the first two measured phases in Fig. 7).8 - - / min pha s e / ( π ) u x ( y = const,z ) Figure 8: Phases of the x-ray fringes measured before (orange) and after (blue)a vertical analyser displacement of 3.2(3) µ m. The dots are the measured phases(see Fig. 7). The bars are the associated standard uncertainties. The same best-fit polynomial has been subtracted from the measured phases. The observeddifference is u x ( y = const ., z ) = 0 . d , to which a miscut angle (cid:15) xy =8 . µ rad will correspond. 9ubtracted by fitting the phases of the x-ray fringes with polynomials differingonly by the sought phase difference. Figure 8 shows the results.After calculating the ratios to the displacements, we obtain the shear strains (cid:15) xy = u x ( y, z = const . ) / s y and (cid:15) xz = u x ( y = const ., z ) /s z , which are nothingelse that the horizontal and vertical components of the sought miscut angle. The measurement results are given in Fig. 9. The horizontal and vertical com-ponents of the angle between the front and rear mirrors were measured with theaid of a precision rotary table having a sub-arcsecond resolution, a calibratedpolygon, and an autocollimator.The measured values of the front-to-rear angles are systematically smallerthan the values inferred by the diffractometric and interferometric measure-ments, which are in good agreement. We did not investigate the origin of thisdiscrepancy; besides, it was irrelevant in aligning the x-ray and optical interfer-ometers and correcting the lattice parameter measurement. However, togetherwith the potentially higher resolution achievable, it might be a clue of the su-perior accuracy of Si-monocrystal polygons having the angles calibrated to thelattice planes [6]. 10 µ rad 2.6(9) µ rad µ rad 7.8(6) µ rad µ rad 10.6(3) µ rad µ rad 11.6(3) µ rad µ rad 6.2(1.0) µ rad {220} {220} Figure 9: Measured values of the miscut angles of the INRIM’s Si interfer-ometer; the analyser sketch shows the angle signs. Blue: x-ray diffraction.Black: combined x-ray and optical interferometry. The lattice planes are as-sumed everywhere parallel better than 0.1 µ rad. The measured components ofthe angle between the front and rear mirrors are also given.11 Conclusions
The measurement of the Si lattice parameter in terms of optical wavelengthsopened the way to determine the Avogadro and Planck constants and to realisethe kilogram by counting Si atoms.We reported about two independent measurements of the miscut angle ofthe front a rear surface of the Si block from which the x-ray interferometerused for this measurement was obtained. A second x-ray interferometer has beenrecently manufactured from the same block, and an additional lattice parametermeasurement is underway [7].The value of this angle is critical to quantify the projection errors of thex-ray and optical interferometers and to align the crystal under measurementin such a way to make these projections identical. Therefore, it lets these x-rayinterferometers be future proof, in that who may wish to remeasure the latticespacing of these unique crystals does not need to redetermine the miscut angle.
References [1] Basile G, Bergamin A, Cavagnero G, Mana G, Vittone E and Zosi G 1991
IEEE Transactions on Instrumentation and Measurement Metrologia S37–S43[3] Massa E, Sasso C P, Mana G and Palmisano C 2015
Journal of Physical andChemical Reference Data MeasurementScience and Technology Zeitschrift fuer Physik B
Metrologia Measurement Science and Technology31