A more accurate measurement of the 28 Si lattice parameter
Enrico Massa, Carlo Paolo Sasso, Giovanni Mana, Carlo Palmisano
aa r X i v : . [ phy s i c s . d a t a - a n ] D ec Accuracy assessment of the measurement of the Si lattice parameter E. Massa, a) C. Sasso, G. Mana, and C. Palmisano INRIM – Istituto Nazionale di Ricerca Metrologica, Str. delle Cacce 91, 10135 Torino,Italy UNITO – Universit`a di Torino, Dipartimento di Fisica, v. P. Giuria, 1 10125 Torino,Italy (Dated: 13 August 2018)
In 2011, a discrepancy between the values of the Planck constant measured by counting Si atoms and bycomparing mechanical and electrical powers prompted a review, among others, of the measurement of thespacing of Si { } lattice planes, either to confirm the measured value and its uncertainty or to identifyerrors. This exercise confirmed the result of the previous measurement and yields the additional value d = 192014711 . I. INTRODUCTION efforts are in progress on accurate determinations ofthe Planck, h , and Avogadro, N A , constants. They areprompted by the proposal of a new kilogram definitionbased on a conventional value of the Planck constant, N A and h being linked by the molar Planck constant, N A h ,which can be accurately measured. The most accurate way to determine N A is by count-ing the atoms in a single-crystal Si ball highly enrichedwith Si.
A measurement was completed in 2011; theuncertainty associated to the determination of h fromthis measurement is 3 . × − h . The measured valuediffered from the result of a watt-balance comparison ofmechanical and electrical powers by 3.7 times the com-bined standard uncertainty. Although subsequent watt-balance determinations gave h values in substantial agreement with that obtainedby atom counting, this prompted a reassessment of the N A uncertainty to identify whether errors were done. Allthe necessary measurements are being scrutinized and re-peated aiming at a smaller uncertainty, thus carrying outstress tests of all the technologies to confirm that the in-tended performances are met. In the present paper wereport about the measurement of the lattice parameter bymeans of combined x-ray and optical interferometry andgive an additional result having a reduced uncertainty. II. N A MEASUREMENT
The N A value is obtained from measurements of themolar volume, V M/m , and lattice parameter, a , of aperfect and chemically pure silicon single-crystal. In aformula, N A = 8 M Va m , (1) a) [email protected] where m and V are the crystal mass and volume, M isthe mean molar mass, a / M and m can beviewed as the molar mass and mass of an ensemble offree atoms. To make the kilogram redefinition possible,the targeted accuracy of the measurement is 2 × − N A .From (1), it follows that the N A determination requiresthe measurement of i) the lattice parameter – by com-bined x-ray and optical interferometry , ii) the amountof substance fraction of the Si isotopes and, then, of themolar mass – by absolute mass-spectrometry , and iii)the mass and volume of nearly perfect crystal-ball havingabout 93 mm diameter. Silicon crystals may contain chemical impurities, in-terstitial atoms, and vacancies, which implies that themeasured mass value does not correspond to that of anideal Si crystal and that the crystal lattice may be dis-torted. This means that crystals must be characterizedboth structurally and chemically, so that the appropri-ate corrections are applied.
The mass, thickness andchemical composition of the oxide layer covering the ballmust be taken into account; they are measured by opticaland x-ray spectroscopy and reflectometry. III. LATTICE PARAMETER MEASUREMENTA. X-ray/optical interferometry
The combined x-ray and optical interferometer usedto measure the lattice parameter is described by Fer-roglio et. al . As shown in Fig. 1, it consists of threeblades, 1.20 mm thick, so cut that the { } planes areorthogonal to the blade surfaces. X rays from a (10 × Mo K α line source are split by the first crystaland recombined, via a transmission crystal, by the third,called analyser. The interference pattern is imaged ontoa multianode photomultiplier tube through a pile of eightNaI(Tl) scintillator crystals. The photomultiplier image FIG. 1. Combined x-ray and optical interferometer. The yel-low line indicates the continuation of the laser beam, at 21mm from the analyser base, where the spacing of the diffract-ing planes was surveyed. The optical interferometer and fixedSi crystal rest on a common silicon plate (not shown). Theanalyser displacement and attitude (pitch and yaw angles)are optically sensed via quadrant detection of the interfer-ence pattern. The transverse, y and z , displacements and rollrotation of the analyser are sensed via a reference 90 ◦ trihe-dron (resting on the same platform as the analyser) and threecapacitive sensors faced to it (not shown). projected on the analyser is (1 × .
2) mm , with a pixelsize of (1 × .
4) mm .When the analyser is moved along a direction orthogo-nal to the { } planes, a periodic variation in the trans-mitted and diffracted x-ray intensities is observed, theperiod being the diffracting-plane spacing. The move-ment, up to 5 cm, requires to control the analyser atti-tude to within nanoradians and vibrations and positionto within picometers. The analyser displacement and ro-tation are measured by optical interferometry; the neces-sary picometre and nanoradian resolutions are achievedby phase modulation, polarization encoding, and quad-rant detection of the fringe phase. To eliminate the ad-verse influence of the refractive index of air and to ensuremillikelvin temperature uniformity and stability, the ap-paratus is hosted in a thermo-vacuum chamber.We measured the lattice parameter of a number of nat-ural Si crystals; the link between the results of these mea-surements and the measured value for the enriched crys-tal used to determine N A is given by Massa et. al . All the past measurements relied on the same opticalinterferometer, that served us since 1994. In order toexclude systematic effects, we assembled a new one andintegrated it in the apparatus. The main novelties of theupgraded system are listed hereinbelow.A 532 nm frequency-doubled Nd:YAG laser substi-tuted for the previous 633 nm diode laser – stabilisedby frequency-offset technique against the frequency of anHe-Ne laser which, in turn, was stabilized against com-ponent a of the I transition 11-5 R(127). The laserwas better collimated, thus halving the correction fordiffraction effects. Furthermore, the residual pressure in the vacuum chamber has been reduced by an order ofmagnitude, below 0.01 Pa. This makes any correctionfor the refractive index of the residual gas in the vac-uum chamber inessential and ensures the calibration ofthe optical interferometer with a negligible uncertainty.Contrary to our past measurement, the lattice spac-ing was surveyed along an horizontal line at 21 mm fromthe analyser base (instead of the previous 26 mm) andthe correction for the self-weigh deformation was recal-culated.A new optical bench is clamped to the vacuum cham-ber; it collimates the laser beam, modulates the phaseof the π -polarized component, and delivers it to the in-terferometer by a pointing mirror and a window of thevacuum chamber. The delivery, collimation, modulation,and pointing systems – optical fiber, beam collimator andpolarizer, phase modulator, and injection mirror – havebeen rebuild to conform to the new wavelength.Previously, the orthogonality between the laser beamand the analyser was only occasionally checked. This wasdone by observing simultaneously, via a visual autocol-limator placed – when necessary – outside the vacuumchamber, the analyser and laser beam through the out-put port of the interferometer. To gain the on-line controlof the beam pointing, an home-made telescope picks uppart of the beam delivered to the detector. In order toensure stability, it is clamped on the same base plate asthe x-ray/optical interferometer.A plate beam-splitter was manufactured ad-hoc andsubstitutes for the cube beam-splitter previously usedto ensure that the difference of the transmitted- andreflected-light paths is insensitive to the beam transla-tions and rotations. Therefore, the components of the op-tical interferometer – beam splitter, quarter-wave plates,and fixed mirror – were replaced and assembled anew.In order to make the interfering beams parallel, thecomponents of the optical interferometer are cemented ona glass plate supported by three piezoelectric actuators.As shown in Fig. 2, a noise at a frequency of about1 mHz caused phase instabilities between the x-ray andoptical fringes. A new power supply was realized, havinga sub part-per-million stability over the time scales, from1 s to 1 h, relevant to the lattice parameter measurement.Eventually, the phase noise was reduced to the shot noiselimit of the x-ray photon count.The Physikalische Technische Bundesanstalt found acontamination of the surfaces of the x-ray interferometerby Cu, Fe, Zn, Pb, and Ca caused by the wet etchingused by the INRIM to remove any residual stress due tosurface damage after the crystal machining. The contam-ination was removed by cleaning the crystal in aqueoussolutions of HF and (NH ) S O .The last upgrade concerned the temperature measure-ment. Since, the thermal expansion coefficient of Siis about 2 . × − K − , the measured volumes of theSi balls and unit cell must refer to the same tempera-ture to within a sub-millikelvin accuracy. Absolute tem-perature measurements are not necessary, but the tem- - - - (cid:144) min ph a s e no i s e (cid:144) Π ph a s e no i s e (cid:144) p m FIG. 2. Phase of the x-ray fringes when the analyser is lockedto an integer optical interference-order. Top: noisy supplyvoltage of the optical-interferometer actuators. Bottom: up-graded voltage supply. The integration time of each measure-ment is 30 s. The 192 pm period of the x-ray fringes has beenused to show the analyser displacement (right scale). perature measuring-chains must be linked. We carriedout more accurate and sensitive measurements of the Pt-thermometer resistance and linked our fixed point cellswith those used to calibrate the measurements of the Siball temperature.
B. Measurement procedure
The measurement equation is a = √ d = √ mλ n , (2)where d is the spacing of the { } planes, √ { } and { } planes, and n is the number of x-ray fringes in a step of m optical fringes having period λ/ d is determined by comparing the peri-ods of the x-ray and optical fringes. This is done by mea-suring the x-ray fringe fraction at the ends of increasingsteps mλ/
2, where m = 1, 10, 100, 1000, and 3570. Westart from λ/ (2 d ) = n/m ≈ .
95 and measure thefringe fractions at the step ends with an accuracy suffi-cient for predicting the integer number of fringes in thenext step. Consequently, λ/ (2 d ) is updated at eachstep. Eventually, measurements were carried out over 48subsequent steps of 3570 λ/
2, 0.95 mm each for a totalscanning length of 46 mm.The least-squares method is applied to reconstruct thex-ray fringes and to determine their phases at the endsof each step. Typical input data are 300 photon-countsover 100 ms time windows spaced by 4 pm; a typical sam-ple contains six x-ray fringes, covers 1.2 nm, and lasts 30s. Each d measurement is the average of about ninevalues collected in measurement cycles where the anal-yser is repeatedly moved back and forth along the se-lected step. The visibility of the x-ray fringes approached æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ x (cid:144) mm H d - a m L (cid:144) a m FIG. 3. Lattice spacing values measured on March 14 alongthe line shown in Fig. 1; x rays enter the analyser from theobverse face. All the values are extrapolated to 20 ◦ C, butnot corrected for the systematic errors listed in Table I. Thebars indicate the uncertainty of the linear interpolation of the8 detector-pixel values giving the value at 21 mm from thecrystal base. The linear strain is due to the thermal gradientoriginated by the optical power injected into the crystal (fromleft side of the figure); the best fit line is also shown. Thered dots indicate the outliers that were excluded from thesubsequent analyses.
50% with a mean brilliance of 500 counts s − mm − .The crystal temperature is simultaneously measured withsub-millikelvin sensitivity and accuracy so that each d value is extrapolated on-line to 20 ◦ C. C. Raw data
Measurements were made over 48 subsequent analysersteps, 0.95 mm long. At each position, the lattice spac-ing was measured in the eight detector pixels and the 8results were processed to obtain, by linear regressions,the lattice spacing values in 48 points of the horizontalline that is the continuation of the laser beam, at 21 mmfrom the analyser base (Fig. 1). A typical result is shownin Fig. 3.The figure shows a gradient of the lattice spacing; wediscovered that it is correlated to the temperature gra-dient caused by the power, about 0.75 mW, injected bythe laser beam in the analyser. The reduced thermalconductivity of the residual gas in the vacuum chamber– because of the otherwise desirable low pressure – con-tributed to worsening the problem. The way we copedwith this problem is described in Sec. IV I.Figure 4 shows that, after the thermal strain is re-moved, the residuals and the outliers are repeatable fromone measurement to the next – also if carried out afterone month. The head-on (obverse) and inverted (reverse)arrangements correspond to the analyser crystal mountedas it was in the boule and in a reversed arrangement;after the reversal, the x rays cross the crystal in the op-posite direction. The residuals and outliers repeatability,the scatter larger than the one expected by statistics, the æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æà à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à - - x (cid:144) mm r e s i du a l s (cid:144) a m æææææææææææææææææææææææææææææææææææææææææææææææææ ààààààààààààààààààààààààààààààààààààààààààààààààà - - x (cid:144) mm r e s i du a l s (cid:144) a m FIG. 4. Comparison of the variations of the measured lattice-spacing values along the line shown in Fig. 1 and after thethermal strain has been removed. Top: obverse face of theanalyser, February 02 (blue) and March 14 (red). Bottom:reverse face of the analyser, May 15 (blue) and June 05 (red). different residual and outlier observed in the head-on andinverted surveys, and additional tests made by shiftingthe x-ray and optical baselines suggest that the outliersand residuals are caused by the analyser surfaces.Apart from the outliers, that were again pinpointed,the profiles shown in Fig. 4 are different from those givenby Massa et al. . Although the support-point variabil-ity could partially explain the difference and in the pastmeasurements the resolution and repeatability were notas good as today (we only spotted a correlation betweenthe profiles taken with the same analyser orientation andno correlation between those taken with opposite orienta-tions), we suspect that the difference is real and this sub-stantiates our surface-effects allegation. A collaborationwith the Leibniz Institutes f¨ur Oberfl¨achenmodifizierungin Liepzig is under way to develop machining technolo-gies based on plasma etching and ion beams to gain abetter control of the geometrical, physical, and chemicalproperties of the crystal surfaces.
IV. ANALYSIS OF THE ERROR BUDGETA. Statistics
Each measurement is the mean, after eliminating theoutliers (red in Fig. 3) and the thermal strain, of the sur-vey results. The uncertainty of the mean is dominatedby the variations of the measured values, that are sup-posed to be caused by local effects of the analyser surface.Therefore, when calculating the uncertainty, we took theresidual correlation into account.
B. Laser beam wavelength
The frequency of the Nd:YAG line is locked to the com-ponent R(56) of the 32-0 transition of the I molecule.It was measured to better than a 10 − relative uncer-tainty; therefore, it does not contribute to the measure-ment uncertainty. To eliminate the influence of the re-fractive index of air, the experiment is carried out invacuo. With respect to our past measurements, the resid-ual pressure has been reduced by a factor of ten, to lessthan 0.01 Pa. Since the air refractivity at the atmo-spheric pressure is 2 . × − , assuming that the pressureis in the interval from zero to 0.01 Pa with a uniformprobability, the relevant correction is 0.015(9) nm/m. C. Laser beam diffraction
The period of the interference fringes is not equal tothe plane-wave wavelength. In the case of the interferenceof two identical paraxial beams – whose angular spectraare strongly concentrated around a wave-vector having ω/c modulus, the difference between the period of theintegrated interference pattern, λ e , and the plane-wavewavelength, λ = 2 πc/ω , is λ e − λλ = (1 − x /w )Tr( Γ )2 + α γ , (3)where the optical-path difference of the interferometerarms is assumed to be much smaller than the Rayleighlength, x is the offset between the beam axes measuredat the beam waists, w is the 1/e spot radius at thebeam waist, Γ is the central second-moment matrix ofthe angular power-spectrum of the beams, 2 α is the mis-alignment between the beam axes, and γ is the beamdeviation from a normal incidence on the analyser.The Tr( Γ ) term originates from diffraction and de-pends on the spread of the transverse impulse of thephotons. It holds for any paraxial beam, no matter itsprofile is Gaussian or not ; actually, it is obtained undera coaxial-beam assumption, i.e., when x /w = α = 0,and subsequently generalized to non-coaxial beams, butunder a Gaussian-beam assumption. It must be notedthat, in the case of Gaussian beams having cylindrical Θ (cid:144) mrad i n t e n s it y (cid:144) a r b it r a r yun it s FIG. 5. Radial profile of the focal plane image of the laserbeam. The solid lines are minimum and maximum valuesof the radial profile of the best bivariate Gaussian functionfitting the data. symmetry, Tr( Γ ) / θ /
4, where θ is the far-field di-vergence. We measured the angular power-spectrum ofthe beams emerging from the interferometer by using theFourier transforming properties of a lens. Next, (3) iscalculated from the central second-moment matrix of thefocal plane image, which is recorded by a videocamera.Owing to the 8 bit resolution and dark-noise of thecamera that we are presently using, a calculation of Γ based on a discrete approximation of the relevant inte-grals is unreliable. Therefore, it was estimated by fit-ting a bivariate Gaussian function to the focal-plane im-age; an example is shown in Fig. 5. The uncertaintyassociated to the Tr( Γ ) estimate is small, typically, lessthan 1%.To check the correction estimate, we examined the re-sults of a number of d measurements carried out from2010 to 2014 with different beams. The results shown inFig. 6 suggest that we overestimate the correction. Sub-sequent investigations did not shed light on this problem,but a study of the interference of wavefronts differentlyperturbed in the separate arms of the interferometer –where (3) does not hold exactly – seems to support anoverestimation; more details will be given in a separatepaper. Another hypothesis is a wrong estimate of thecenter of mass of the focal-plane image, which implies acorrection always larger than true. In addition, since asingle datum – corresponding to the largest beam diver-gence, see Fig. 6 – dictates the regression line, we may bemislead by a measurement error. Owing to the smallestbeam divergence in the present set-up, the result we arereporting agrees with the value extrapolated to a zerocorrection from the data in Fig. 3. Therefore, we didnot corrected the Tr( Γ ) / . value; but,cautiously, increased its uncertainty to 15%.The interfering beams are kept parallel to within a2 α = 1 µ rad maximum misalignment by levelling thephase in four quadrants of the interference pattern via thepiezoelectric supports (pitch) and inertial drivers (yaw)of the interferometer base-plate. Consequently, the α / æ æææ ŸŸŸŸ H G L (cid:144) mrad H d - a m L (cid:144) a m FIG. 6. Corrected d values vs. the applied correction forthe diffraction of the laser beam. The measurements are madefrom 2010 to 2014 by using laser beams differently collimated.Bullets (red): He-Ne laser source @ 633 nm; squares (green):frequency doubled Nd:YAG laser source @ 532 nm. If thecorrections are correctly estimated, the data are expected tolie on an horizontal line. term in (3) is irrelevant and was omitted. D. Laser beam alignment
When assembling the apparatus, the laser beam devia-tion γ from a normal incidence on the analyser was nulli-fied with the aid of an autocollimator looking at both theanalyser and beam from the interferometer output-port.Next, as shown in Fig. 7, we carried out a number of d measurements while γ was purposely changed along twoorthogonal directions and its variations were recorded byan on-line telescope. The telescope is mounted, insidethe vacuum chamber, on the same base plate of the x-ray/optical interferometer and picks up part of the out-put beam. After two parabola were fitted to the mea-sured d values, the beam direction corresponding tothe maxima – hence, to a supposed normal incidence– was identified to within µ rad uncertainty and main-tained in the telescope optics with an uncertainty of 20 µ rad. Eventually, the laser beam was kept parallel tothat direction and the telescope readings recorded forsubsequent analyses.After we completed the measurements and removedthe interferometer from the apparatus, we realised thatthe operation of the optical interferometer may be liableto a systematic error. This problem will be examined ina separate paper, but, since it relates to the assessmentof the measurement uncertainty, we outline it shortly.In the case of a pointing error γ , an analyser displace-ment s shears the interfering beams by 2 sγ . Hence, themeasure beam goes through different parts of the opticscrossed in its way to the detector. This shear changes theoptical-path length by (2 γs ) β ∆ n , where ∆ n and β arethe refractivity and the relevant component of the verti-cal angle of a wedge that, in a simplified model, substi- - -
100 0 100 200 - - - -
505 beam pitch angle (cid:144) Μ rad ´ D d (cid:144) d FIG. 7. Measured d values vs. the pitch angle of the laserbeam; the angle origin is set in the maximum of the parabola(red line) that best fit the data. tutes for the optics in the way from the analyser to thedetector. In addition, because of the wavefront curva-ture, the beam shear is sensed by the interferometer as arotation equal to Ω = sγR , (4)where 1 /R is the wavefront curvature. Therefore, themeasured d value is d m = d (cid:20) − γ + (cid:18) β ∆ n + b R (cid:19) γ (cid:21) , (5)where b is the Abbe’s offset between the laser and x-raybeam-centroids. According to (5), d m is maximum when γ = β ∆ n + b/ (2 R ), not when γ = 0 as assumed in thealignment procedure. It must be noted that, when d m ismaximum, the sensed rotation is not zero – as expectedif γ = 0 rad – but, Ω = sβ ∆ nR , (6)where, for the sake of simplicity, we assumed b = 0 mm.Since the analyser attitude is servoed so as to nullifysignal of the angle interferometer – the differential phasebetween the quadrants of the interference pattern, theshear is counteracted by an analyser rotation. Eventu-ally, the pitch component of this rotation is disclosed bya d gradient in the different detector pixels. The pitchexplaining the gradient observed with a varying align-ment of the laser beam is shown in Fig. 8. When thebeam is aligned in such a way that the d value is max-imum, the pitch was equal to 0.4 nrad/mm in Februaryand, after the analyser reversal and realignment, to 1.3nrad/mm in May. The yaw rotation might be similar,but, presently, we cannot detect it.Shear strains of the crystal lattice and surface effects(see Sec. III C) mimic the same gradients. Therefore,we cannot unambiguously explain a d gradient by aparasitic pitch rotation. However, a number of additional ƒ ƒ ƒƒ ƒ æ æ ææ æ - -
100 0 100 200 - - (cid:144) Μ rad p it c h r o t a ti on (cid:144) n r a d mm - FIG. 8. Parasitic pitch of the analyser motion vs. the pitchangle of the laser beam. The analyser motion is servoed soas to nullify the differential signals detected by the optical in-terferometer. Measurements were done in February (squares,obverse face) and May (bullets, reverse face). The lines (solidand dashed, respectively) that best fit the data are also shown.The angle origin is set in the maximum of the parabola thatbest fit the measured d values (see Fig. 7). tests excluded large strains. For instance, Fig. 9 showsthe parasitic pitch of the analyser that explains the d gradients observed in the March 14 and May 15 surveys,where the laser beam was aligned in such a way thatthe d measure is maximum. Since the only surveydifference is the reversed analyser alignment, if we hadobserved a lattice strain, the two plots should be similar.Contrary, they are not; the mean pitch was 1.0 nrad/mmin March and − . R = 25(1) m, can be estimated fromthe mean slope, s/R = 40(2) × − mm − , of the linesthat best fit the data in Fig. 8. By explaining the d gradients in terms of parasitic pitch rotations and (6),we estimate that the pitch component of β is 50 µ radfrom the March data and − µ rad from the May one.Since no change was made in the optical interferometer,these contradictory estimates exclude wedge angles muchgreater than, say, 50 µ rad.In order to estimate the needed correction and the as-sociated uncertainty, we used ∆ n ≈ / c γ = ( γ − β ) γd / β = 0(50) µ rad and γ = 0 . β ± µ radare independently normally distributed. The final resultis c γ = − . . E. Laser beam walk
The beam walk refers to the transverse motion of theinterfering beams through the optical components. Itoriginates from different effects causing the beams tomove across imperfect surfaces or wedged optics. Theeffect of walks caused by a tilted incidence on the anal-yser was investigated in Sec. IV D. In our previous set-up, the beam-splitter imperfection, combined with tilts of äääääääääääääääääääääääääääääääääääääääääääääääää ••••••••••••••••••••••••••••••••••••••••••••••••••0 10 20 30 40 50 - - x (cid:144) mm p it c h r o t a ti on (cid:144) n r a d mm - FIG. 9. Parasitic pitch of the analyser motion measured over0.95 steps in 48 subsequent analyser positions. Surveys werecarried out on February 02 (bottom, obverse face) and May15 (top, reverse face). The laser beam was pointed so as toensure that the measured d value was maximum and theanalyser was servoed so as to nullify the differential signalsdetected by the optical interferometer. the apparatus baseplate with respect to the laser beam,caused systematic differential variations of the opticalpaths through the interferometer that required ad hoc corrections. In the new apparatus, we made this prob-lem harmless by using a plate beam-splitter, having aparallelism error less than 10 µ rad, and by controllingelectronically the baseplate level and tilt to within 25nm and 70 nm/m, over any analyser (short or long) dis-placement. The differential beam walk due to analyserparasitic rotations is irrelevant because rotations are lessthan 1 nrad/mm (see Sec. IV D) and the detector dis-tance is less than 0.5 m.The mechanical load driving the analyser carriage doesthe apparatus to sag and to yaw with respect to the laserbeam. The relevant beam walks are quite large, up to 1 µ m and 5 µ rad, but, no correlation with the carriage dis-placement was observed. In addition, after any displace-ment, the link between the apparatus and the drivingsystem is removed, thus allowing the same equilibriumposition to be restored. It is difficult to estimate if thereis a residual systematic differential-walk of the interfer-ing beams. If, over 1 mm analyser step, the systematicwalk is in the [ − . , .
1] mm interval, a 10% of whatobserved, and the differential wedge-angle between theend surfaces of the separate paths through the interfer-ometer is in the [ − , µ rad interval, the beam walkcontribution to the uncertainty budget is 0.577 nm/m. F. Abbe’s error
The Abbe error refers to the difference, ˆ s · ( b × Ω ) = b · ( Ω × ˆ s ) = Ω · (ˆ s × b ), of the displacements sensedby the laser and x-ray interferometers, where b is theinterferometer offset and Ω and ˆ s are the rotation andmovement-direction of the analyser.As regards as Ω × ˆ s , it was zeroed to within 1 nrad æ æ æ æ æ æ æ æ ì ì ì ì ì ì ì ì pitch rotationyaw rotation - - - ph a s e o f x - r a y fr i ng e s (cid:144) H Π L FIG. 10. Vertical variations of the x-ray fringe phase when theanalyser is rotated – while keeping the displacement measuredby the optical interferometer null – about the y (solid line)and z (dashed line) axes. æ æ æ æ æ æ æ æà à à à à à à àì ì ì ì ì ì ì ì x=6 mmx=11 mmx=24 mm H d - a m L (cid:144) a m FIG. 11. Vertical variations of the d value measured overdifferent 0.95 mm steps. The coordinate of the step center isalso given. The solid lines are the linear regressions used tointerpolate the measured values. æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ - r e s i du a l v a r i a n ce (cid:144) a m c o rr e l a ti on FIG. 12. Solid line: variance of the residuals from the linebest fitting the d values interpolated at different detectorpixels. Dashed line: correlation between the same residualsand the residuals from the line best fitting the analyser pitch.The solid lines are expected variance and correlation. (see Sec. IV D) by servoing the motion so as the signals ofthe angle interferometer – the phase differences betweenthe vertical and horizontal quadrants of the interferencepattern – are null.As regards as ˆ s × b , the vertical offset was nullified bycarrying out off-line measurements of the variations ofthe x-ray fringe phase in different detector pixels whilethe pitch component of Ω is purposely changed whilekeeping the analyser displacement null. As shown in Fig.10, we identified the virtual pixel having a zero offset towithin a 0.1 mm uncertainty. The horizontal offset wasset to zero to within the same uncertainty by rotating theanalyser about the vertical and by shifting horizontallythe laser beam up to no phase variation is detected, asshown in Fig. 10.The angle interferometer and attitude control displayimperfections; we took advantage of the resulting pitchnoise – which is shown in Fig. 9 – to check the verticaloffset by data analysis. The eight detector-pixels havea linearly increasing offset and, as shown in Fig. 11, thelinear regressions of the d values intersect, ideally, inthe pixel having a null offset. Since the d value in thispixel is insensitive to the pitch noise, the best way tofind it is to look at the minimum variance of the resid-uals from the best-fit lines of the values interpolated ineach detector pixel or, which is the same, at the zero-correlation between the same residuals and the residualsfrom the best-fit line of the pitch noise. Examples ofthe residual variance and correlation are shown in Fig.12. In half of the surveys the null offset locates in apixel that differs from where the phase variation of thex-ray fringe was found – in previous off-line experiments– insensitive to the analyser pitch rotation. After theprojection on the analyser, the maximum differences are0.5 mm. Since the x-ray/optical interferometer rests onan double anti-vibration system (a table supported byair springs, in turn, mounted on a giant pendulum), weexplained these shifts by instabilities of the relative lev-elling between the x-ray source and interferometer whichaffect the pixel looking at the analyser point having zerovertical offset.We trusted the zero-offset pixel identified by the dataanalysis. Since we carried out this analysis after the in-terferometer was removed from the apparatus and, con-sequently, we did not investigated experimentally theproblem, we assumed a null-pixel error uniform in the[ − . , +0 .
5] mm interval with an uncertainty of 0.29 mm.By combining this uncertainty with the uncertainty ofthe zeroing of the horizontal offset component, 0.1 mm,and parasitic rotation, 2 nrad, we estimated that theAbbe error was nullified to within a total uncertaintyof 6 . × − d . G. Movement direction
The analyser moves orthogonally to the front mirrorof a trihedron; straightness errors are nullified to within - - - -
20 20 40 - - - - Μ r a d Μ rad nh s - -
12 8 - - - -
14 14 - - FIG. 13. Relative directions of the reciprocal vector h ofthe diffracting planes, unit normal to the analyser n (obversemirror), and displacement direction s . The dates indicatethe s checks; after 2014/4/14 the analyser was removed andrealigned in reverse. The circles indicate the measurementuncertainties. The dashed line is the locus of the s directionwhere the projection error is null. nanometers by servoing the motion with the signals ofcapacitive transducers that sense the transverse displace-ments of the trihedron top- and side-face.The misalignment between the optical and x-ray inter-ferometers causes them to measure different componentsof the displacement. With s indicating the displacement,the x-ray interferometer senses s · ˆ h , where ˆ h is the unitnormal to the diffracting planes, whereas the optical in-terferometer senses s · ˆ n , where ˆ n is the unit normal tothe analyser, see Sec. IV D. The difference, s · (ˆ n − ˆ h ), isnull when the displacement is orthogonal to (ˆ n − ˆ h ), thatis, when it bisects the angle formed by ˆ n and ˆ h . Thelattice constant is linked to the measured mλ/ (2 n ) ratioby d = mλ n h s · (ˆ n − ˆ h ) i . (7)The two (obverse and reverse) analyser-mirrors are pol-ished parallel to the { } planes. The residual misalign-ments, | ˆ n − ˆ h | = 13 . . µ rad and | ˆ n − ˆ h | = 10 . . µ rad for the obverse and reverse mirrors, respectively,were estimated by a least-squares adjustment of the mis-alignment between x-ray and light reflections on the mir-rors and lattice planes, the phase shift of the x-rayfringes when the analyser motion lies in the mirror planes,and the measured angle between the two mirrors. In or-der to calculate the relevant correction and the associateduncertainty, the (ˆ s − ˆ n ) angle – that is, the angle betweenthe trihedron and analyser – was periodically measuredto within a 10 µ rad uncertainty; examples of the mea-surement results are shown in Fig. 13. H. Analyser temperature
The analyser temperature is measured by a capsulestandard Pt resistance thermometer inserted into a wellin a copper block in thermal contact with the crystal.Resistance measurement were carried out by a FLUKE1595A multimeter; to minimize the self-heating, the mea-surement current was 0.3 mA. Each temperature datumis the average of 15 measurements pairs, carried out withboth positive and negative currents and integrated over30 s. The 1595A linearity was checked by a resistor net-work made in such a way that the voltages across anynumber of resistors in a resistor series are read to getfour-terminal values interrelated by the formula for theseries connection. The test showed that linearity is bet-ter than 10 µ Ω – corresponding to 25 µ K – for resistancemeasurements from 90 Ω to 120 Ω.The lattice constant measurements were carried outfrom February to June 2014; on April 08, the thermome-ter was calibrated in situ – that is, by moving the ther-mometer from the apparatus to the fixed-point cells with-out changing the measuring chain and cables. We ex-trapolated the resistance readings to a zero current andcorrected the cell temperatures for the immersion depthand hydrostatic pressure.The temperature measurements require sub-mK accu-racies and any difference between the temperature scalesused to extrapolate the molar volume and lattice con-stant to 20 ◦ C must be excluded or identified. Conse-quently, on April, our fixed-point cells were comparedwith those of the Physikalish Technische Bundesanstalt.After the corrections for the immersion depth and hydro-static pressure were taken into account, the differences ofthe resistance readings were R TPW (PTB) − R TPW (INRIM) = 5(3) µ Ω R Ga (PTB) − R Ga (INRIM) = − µ Ω . Unfortunately, it was not possible to investigate the non-uniqueness associated with the readings of the two ther-mometers at 20 ◦ C; it was cautiously set to 0.1 mK. Taken note of these differences, the uncertainties of thetriple point of water and melting point of Ga realisationsare irrelevant; the repeatability of the cells is 50 µ K. Ow-ing to the huge data averaging, the noise of the resistancemeasurements is irrelevant; the stability of the reference100 Ω resistor over the two months before and after thecalibration is 33 µ Ω, the linearity of the measurement ofthe 107 Ω thermometer-resistance is better than 10 µ Ω,the measurement non-uniqueness is 0.1 mK. All together,the uncertainty of the temperature measurements, esti-mated by Monte Carlo simulation, is 0.17 mK.Each d measurement was extrapolated to 20 ◦ C ac-cording to d ( T ) = d ( T ) (cid:2) α ( T − T ) + α ( T − T ) (cid:3) , (8)where T = 20 ◦ C, α = 2 . × − K − , and α = 4 . × − K − . All measurements were car-ried out in the temperature range from 19.9 ◦ C to 20.3 - - x (cid:144) mm s t r a i n FIG. 14. Comparison of the finite-element calculation of theminimum (the thermal flux is grounded through the ther-mometer copper block) and maximum (the thermal flux isgrounded through the crystal support points) thermal strains(with respect to the the measured crystal temperature) dueto the optical power, 0.75 mW, injected into the analyser bythe laser beam (solid lines, red) and the least-squares adjust-ment (filled area) of the d gradients and variations at thecrystal ends (dots). ◦ C; therefore, the average extrapolation uncertainty is0 . × − d .The thermometer self-heating was identified by repeat-ing d measurements with varying currents; the rel-evant correction for the 0.3 mA current is 0 . × − d , the measured value being smaller than the trueone.The calibration history, dating back to December 2007,shows a linear drift of 14(5) µ Ω/month or 0.035(13)mK/month that was taken into account to extrapolatethe calibration to the actual measurement date.Eventually, the total uncertainty of the lattice constantextrapolation to 20 ◦ C is 0 . × − d . I. Thermal strain
A linear approximation of thermal strain due to the op-tical power injected into the analyser by the laser beam,∆ d d = a − b ( x − x ) , (9)where a = 0 . × − , b = 0 . × − mm − ,and x = 22 . d gradients andthe results of repeated d measurements carried outwith varying optical powers. The comparison of (9) withthe numerical calculations of the thermal strain is shownin Fig. 14. To correct for the thermal strain, we trustedthe value given by (9), but increased its uncertainty tothe one half of the 1 . × − gap between the minimumand maximum strain predicted by the numerical calcula-tion. Therefore, the interpolated value at 22.8 mm wasreduced by 0 . × − d . More details about the0 ææææææææ
25 30 35 40 45 - - - point distance (cid:144) mm s t r a i n (cid:144) a m (cid:144) m FIG. 15. Residual mean self-weigh strain of the analyser cal-culated at 21 mm from the base as a function of the distancebetween the support points. The filled curve is the distanceprobability-distribution, given three contact points uniformlydistributed in the (5 ×
5) mm support areas. numerical and experimental investigations of the analyserresponse to the thermal load will be given in a separatepaper. J. Self-weight deformation
To average the d measurements over the largestcrystal part, a long analyser has been used. The sim-ulation of the gravitational bending allowed the analyserto be optimally designed, the residual lattice strain tobe predicted, and the contribution of the self-weight de-formation to the uncertainty budget estimated. Thesimulation purposes were to find the maximum height ofthe analyser lamella consistent with a non-strained lat-tice and the support points minimizing bending or sag-ging. In order to estimate the necessary correction andits uncertainty, the residual strain at the 20.5 mm heightwas recalculated for points randomly located inside the(5 ×
5) mm support areas. The results show that themean strain, which is shown in Fig. 15, depends onlyon the distance between the support points. Since thesimulation indicates that the everted-inverted transitionoccurs with a slightly expanded lattice, we reduced themeasured values by 0 . × − d – which relateswith the distance distribution shown in Fig. 15. K. Aberrations of the x-ray interferometer
Geometric aberrations contribute to the phase of thex-ray fringes by less than 0 . d per 1 µ m changes ofthe analyser thickness or focusing. The root-mean-square roughness of the analyser surfaces is less than 1 µ m, with its main components in the neighbour of the0.1 mm wavelength. The effect of the surface roughness– included the large local variations of the angle with æ æ æ ææ æ æ
50 100 150711.0711.5712.0712.5713.0 days since 2014 January 01 H d - a m L (cid:144) a m FIG. 16. Measured d values. Each measurement is themean, after eliminating the outliers and correcting for thethermal strain, of a survey of a 48 mm long crystal part. respect to the diffracting planes – and of local surfacestrain, if any, were washed out by the survey averaging.The linear gradient of the mean analyser thickness overthe 48 mm measurement distance is less than 10 µ m andcontribute by less than 0 . × − to the d measure-ment. This error is nullified by repeating the measure-ment after a 180 ◦ rotation of the analyser and by aver-aging the results. If the crystal displacement does not liein the mean surface of the analyser, the interferometerdefocuses. The out-of-plane angle is less than 2 µ m/cm,to which a zero-mean uniform error having 0.23 nm/mstandard deviation will correspond.A stress exists in the crystal surfaces even if the bulkmaterial is stress-free. This problem was investigated byQuagliotti et. at by using an elastic-film model to pro-vide a surface load in a finite element analysis. The studyshowed that, if the tensile stress is 1 N/m, the measuredlattice spacing is 6 × − d smaller than the valuein an unstrained crystal. Literature values of the (001)surface-stress obtained from ab initio and molecular dy-namics calculations are given by Quagliotti et. at ; thestress of the (110) surface is expected to be 60% smaller.Owing to the value and sign scatters of the literaturedata, we do not propose a correction and associate to anull stress an uncertainty of 0.1 N/m. Therefore, the rel-evant contribution to the lattice constant uncertainty is0 . × − d . Further experimental investigations andatomistic calculations are under way to confirm that sur-face stress effects are irrelevant or to quantify and correctfor them. V. MEASUREMENT RESULTS
Three d surveys were made on February 12 andMarch 10 and 14 with the analyser in the head-on ar-rangement; four were made on May 14 and 15 and June05 and 12 with the analyser in the back orientation. Ex-amples are given in Figs. 3 and 4. Next, after eliminatingthe outliers and correcting for the thermal strain, each1 ææ H d - L (cid:144) amref. @ D this work FIG. 17. Comparison of the d values (10) and (11). d profile was averaged to obtain the mean lattice spac-ing.The results are shown in Fig. 16; an example of theerror budget is given in Table I. With respect to our pre-vious measurement, in addition to the reduction of thetotal uncertainty, the Table I shows a significant redis-tribution of the uncertainty contributions. The valuesbelonging to each obverse/reverse faces set are signifi-cantly correlated; this explains the repeatability, whichis much better than the uncertainty.The analyser reversal required a full realignment of thetwo interferometers. Therefore, only the wavelength andtemperature uncertainties, laser-beam diffraction, self-weigh deformation, and aberrations of the x-ray interfer-ometer combine in the same way; the remaining contri-butions to the total uncertainty are largely independent.Figure 16 shows a difference between the values mea-sured with the analyser mounted in the head-on and in-verted arrangements. We are not yet able to say if thisdifference is real, e.g., due to a different physical and/orchemical structure of the analyser surfaces, or if it in-dicates that the control of the systematic errors is lessgood than what we estimated. The first option is sup-ported by the repeatable observation of different obverse-and reverse-profile, which was statistically anticipated byMassa et al. . The second is supported by the fact that adifference between the obverse and reverse mean-valuesof d was not reported.The final measured value, d (2014) = 192014711 . , (10)at 20 ◦ C and 0 Pa, is the mean of the data in Fig. 16.The relative uncertainty is 1.75 nm/m. In the average,we did not take the data uncertainty and correlation intoaccount, but, to avoid that the different number of ob-verse/reverse surveys biases the result, firstly, we aver-aged the obverse and reverse data and, subsequently, av-eraged the two results. To be conservative, we associatedto the mean the worst uncertainty of the input data.As shown in Fig. 17, (10) is slightly smaller than d (2011) = 192014712 . TABLE I. Relative correction and uncertainty, in parts per10 , of the 2014/02/12 d value.Contribution Correction Uncertaintydata averaging 0.000 0.722wavelength − .
015 0.009laser beam diffraction 3.978 0.597laser beam alignment − .
110 0.480beam walks 0.000 0.577Abbe’s errors 0.000 0.611movement direction 0.699 0.214temperature − .
500 0.497thermal strain − .
948 0.641self-weigh − .
543 0.377aberrations 0.000 0.642total 2.56 1.75 given by Massa et al. . A reasons might be a positive biasof the correction for diffraction applied to (11), as shownin Fig. 6. In addition, in (11), the pointing stability of thelaser beam was not monitored on-line and, in retrospect,we might have overestimated the pointing error. Since apointing error causes a χ measurement-underestimate,we consistently – but, perhaps incorrectly – applied arelatively large positive correction. Eventually, the re-analysis of the effect of a non-orthogonal incidence of thelaser beam on the analyser mirror in Sec. IV D shows thatwedged optics in the beam path combine with a wrongpointing to originate a positive error. Therefore, contraryto what we did in the past, the correction for the laserbeam alignment in Table I is negative.To evaluate the correlation between (10) and (11), thefollowing considerations are in hands. The optical in-terferometer used for the two measurements were com-pletely different as regards the wavelength, laser sourcebeam geometry and power, and optical components; thecombined x-ray and optical system was also completelyknocked down and realigned. Furthermore, the temper-ature measurements relied on a different, of higher qual-ity, apparatus and were calibrated anew. Eventually, allthe contributions to the error budget were re-examined;some – the thermal strain, self-weigh deformation, sur-face stress, and walks of the laser beam – were reconsid-ered or calculated anew. VI. CONCLUSIONS
We re-checked the measurement uncertainty of thelattice parameter of the Si crystal used to determinethe Avogadro constant. Additional measurements andstress-tests were carried out by using of an upgraded mea-surement apparatus. No error was identified; this workconfirms the value given by Massa et al. and yields anadditional results having a reduced uncertainty.2 ACKNOWLEDGMENTS
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