1. Introduction

In the framework of an international cooperation to determine the Avogadro constant to an accuracy allowing the kilogram definition to be based on the ¹²C mass [1, 2], the relative uncertainty of the (220) Si lattice-plane spacing measurement by combined X-ray and optical interferometry must be reduced to 3×10^-9. With this in view, we developed a guide capable of displacements up to 5 cm with guiding errors commensurate with the requirements of atomicscale positioning and alignment of the interferometer crystals. This breakthrough extends by more than an order of magnitude the operation range and the measurement capabilities of any preceding X-ray interferometer. In order to test our apparatus, assess its performances, and link the measurement values to the previous ones [3, 4, 5, 6], we integrated in it our MO*4 reference crystals and carried out a series of test measurements.

The operation of a combined X-ray and optical interferometer is described in [7, 8, 9]. As is shown in Fig. 1, an X-ray interferometer consists of three flat and parallel crystals cut in such a way that the (220) planes are orthogonal to the crystal surfaces. X rays from a conventional 17 keV Mo Kα source, with a (10×0.1) mm² line focus, are split by the first crystal and are recombined, via two transmission mirrors, by the third, called the analyzer. When the analyzer crystal is moved in a direction orthogonal to the (220) planes, a periodic variation of the transmitted and diffracted X-ray intensities is observed, the period being the diffracting-plane spacing. The analyzer crystal embeds front and rear mirrors, so that its displacement is measured by optical interferometry; the necessary picometer resolution is achieved by polarization encoding and phase modulation. According to the measurement equation

where n is the number of X-ray fringes of d ₂₂₀ period observed in a crystal displacement spanning m optical fringes of λ/2 period, large displacements ensure definite advantages, in terms of both sensitivity and accuracy assessment. The successful operation of a separate-crystal interferometer is a challenge: the fixed and movable crystals must be so faced to allow the atoms to recover their exact position in the initial single crystal and they must be kept aligned notwithstanding the analyzer displacement.

 

Fig. 1. Left: the combined X-ray and optical interferometer. Right: the interferometer slide. On the top of the L shaped carriage is an active tripod, electronically controlled to compensate for sliding errors. The tripod supports the analyzer (XINT) and the reference electrode of a capacitive transducer. The analyzer front-surface is the movable mirror of the optical interferometer.

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2. Experimental apparatus

The progress of X-ray and optical interferometry involved the continual development of more powerful techniques for finer control over experimental conditions, thus making more numerous effects visible and reproducible. With respect to our 2004 apparatus [5, 6], the key development is an interferometer guide (see Fig. 1), 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 as well it shears in the two, x and y, transverse directions, thus allowing compensation of 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 nanometer straightness. After step-by-step improvements - mainly to address stiffness deficiencies, stick-and-slip motion, and creep when the carriage stopped - we achieved measurement capabilities with crystal displacements up to a few centimeters. The details of the measurement and control technologies which have been developed and integrated into the guide will be given in a separate technical paper; descriptions of the precursors can be found in [10, 11, 12].

 

Fig. 2. Top: d ₂₂₀ variation in the MO*4 crystal. The measurements carried out on 2007-12- 18, 2008-02-04, and 2008-02-25 are in blue, green, and red. Bottom: Consistency of the measured d ₂₂₀ values. The average - over the crystal slices indicated by the arrows - of the values in the top part are in red; the results of one-step measurements over the same slices are in green. The graduation is the sequence of possible d ₂₂₀ values compatible with the X-ray fringe fraction in the 20 mm displacement.

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Fig. 3. The MO*4 crystal. The x axis is where the d ₂₂₀ measurements were performed. The red spot is the laser beam, the blue box is the X-ray beam - see also Fig. 5.

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We determined the lattice spacing by comparing the periods of the X-ray and optical fringes; however, counting of all the X-ray fringes in macroscopic analyzer displacements is impractical. In fact, with a maximum flux of only 1000 X-photon/s/mm ² and a beam cross section of (0.5×12) mm², the maximum analyzer velocity, a mere 1 nm/s, makes the fringe counting duration incompatible with the time scale on which the experiment is acceptably stable. We started instead from the first approximation n/m≈1648.28 of the λ/d ₂₂₀ ratio and measured the X-ray fringe fractions at the displacement start and end (zero crossings of the optical signal) with an accuracy sufficient to predict the number of lattice planes in the next displacement in the 1, 10, 100, 1000, 3000, 20000, and 600000 optical-order sequence. To estimate the fringe fractions, we applied the least squares method. We obtained the input data by scanning about five X-ray fringes across both the start and end lines and by sampling about 600 data, with a 100 ms integration time and a sample duration of 60 s. Since it was not possible to keep the drift between the X-ray and optical interferometers as small as desirable, we repeatedly moved the analyzer back and forth along any given displacement and sampled the interferometer signals at each of the ends. In such a way drift - more precisely, its linear component - is eliminated by demodulating the measured fringe fractions.

3. Measurement results

The centimeter displacements opened the way to the mapping of the d ₂₂₀ values in the analyzer crystal. We therefore translated the analyzer step-by-step while maintaining the splitter/mirror crystal and X rays fixed; we carried out measurements over 20 contiguous crystal slices, 1 mm wide. Figure 2 (top) summarizes the measurement results; each data point is the average of about 12 values collected in measurement cycles lasting half an hour during which we translated the analyzer back and forth by about 1 mm (3000 optical orders or 5×10 ⁶ lattice planes). A drawing of the analyzer crystal and the location of the lattice spacing measurement are shown in Fig. 3. According to what reported in [13], the large d ₂₂₀ variations - already, though not so clearly as now, observed - are ascribed to a non-uniform distribution of carbon, which highly contaminates the MO*4 crystal [5]. As we did in past measurements, we obtained the lattice spacing value by averaging the values shown in Fig. 2.

 

Fig. 4. Measurement repetitions of the mean (220) lattice spacing in the MO*4 crystal.

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Subsequently, we fully exploited the centimeter displacement capabilities and carried out measurements by translating the analyzer by about 7 mm (20000 optical orders or 35×10 ⁶ lattice planes) and 20 mm (60000 optical orders or 1×10 ⁸ lattice planes). We measured the mean d ₂₂₀ value over three adjacent crystal slices 7 mm wide and over one single 20 mm slice; results are given in Fig. 2 (bottom). In order to check the consistency of the measured values, the figure compares these measurement results - indicated by green bullets - with the arithmetic means - indicated by red bullets - of the relevant values in the d ₂₂₀ map. Since only the fractional part of the n X-ray fringes in the mλ/2 displacement can be observed, our actual result is a sequence of possible d ₂₂₀ values, spaced by d ₂₂₀/n. The true value was selected on the basis of prior knowledge, obtained via measurements carried on over shorter scans, where the value gap is larger than the prior uncertainty and where the d ₂₂₀ non-uniqueness is not a problem. As shown in Fig. 2 (bottom), when measurements are performed over centimeter displacements, to select the true d ₂₂₀ value requires high resolution and repeatability.

We made six determinations of the mean lattice spacing in the MO*4 crystal, shown in Fig. 4. Three are the averages of 21 values obtained over adjacent 3000 optical-order scans, one is the average of 3 values obtained over adjacent 20000 optical-order scans, and two are the values obtained over 60000 optical-order scans. After we took into account of the corrections listed in Table I, the final lattice-spacing value in a vacuum and at 22.5 °C,

is the average of these values.

Table 1. Crystal MO*4 - corrections and uncertainties of d₂₂₀.

View Table

4. Uncertainty budget

The error analysis was performed in the same way as described in [4, 5, 14]; the error budget is given in Table I. In order to compare our present 6×10^-9 d ₂₂₀ uncertainty with the targeted 3×10^-9 d ₂₂₀ value, the following points deserve to be considered.

4.1. Statistics

The statistical contribution to the error budget is much larger than we expected from the photon counting and optical interferometer noises. Actually, it reflects both the d ₂₂₀ inhomogeneity and measurement repeatability. Inhomogeneities (real or apparent) are related to contamination and local geometrical imperfections of the analyzer, which should be, ideally, a plane parallel crystal slab; repeatability is related to still unexplained variations of the measured value.

4.2. Wavelength

The frequency of the laser source (an external-cavity single-mode diode laser) is stabilized with respect to the ¹²⁷I₂ (f)=632 991 212.6(1) fm reference wavelength. The measurement is carried out in a vacuum to eliminate the air-refraction effect on the laser beam wavelength, about 3×10-⁹ λPa^-1. The relevant correction and uncertainty takes our present poor vacuum, into account; to reduce the residual pressure, about 1 Pa, by a factor of ten is not a problem.

4.3. Laser beam diffraction

We calculated the effect of diffraction on the basis of an on-line divergence measurement [15], but we set the calculation uncertainty cautiously to a larger value because of the anomalies, which will be related in §5. In order to achieve higher confidence in our capability to cope with diffraction, we are planning detailed investigations into wavefront aberrations and measurement repetitions with a different collimation of the laser beam.

4.4. Laser beam alignment

We directed the laser beam orthogonally to the analyzer front mirror by means of a tilting mirror; subsequently we checked orthogonality by remotely tilting the laser beam - in both the horizontal and vertical planes - and by repeating the d ₂₂₀ measurements. Since the effect of misalignment is quadratic, we identified the orthogonal condition by fitting a parabola to the data and by looking for the maximum measurement value [4]. Accordingly, the laser beam alignment can be further improved by increasing the measurement resolution.

 

Fig. 5. Schemes of the quadrant (left) and multianode (right) detectors. The spacings between the centers of the analyzer areas imaged by the quadrants and anodes are 1.5 mm and 1.4 mm - see also Fig. 3. Hypothetical optical and X-ray interference patterns - with enhanced misalignment fringes - are also rendered by scatter plots.

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4.5. Abbe’s errors

Centimeter displacements and electronic control made the Abbe’s errors, differential displacements sensed by the X-ray and optical interferometers and caused by parasitic rotations and offsets between the centers of the interference patterns, almost harmless. These displacements are given by z ₀Δ_ρ and y ₀Δθ, where Δ_ρ and Δθ are the pitch and yaw rotations and z ₀ and y ₀ are the relevant, vertical and horizontal, offsets.

To cope with the Abbe’s error, the X-ray interference pattern was detected by means of a multianode photomultiplier equipped with a vertical pile of eight NaI(Tl) scintillator crystals (see Fig. 5). In order to eliminate this error, the virtual anode having a zero offset was identified by electronically pitching and yawing - but with a null displacement - the analyzer about the center of the laser-beam spot by Δ_ρ=Δθ=15 nrad. Figure 6 (left) shows the analyzer displacements in the different anodes. The dots and lines can be thought representing a row of atoms and the relevant lattice plane after the pitch (red) and yaw (green) rotations; the same being initially in a vertical position. For the pitch rotation, since any anode has a different vertical location z _i and, therefore, it senses a different displacement, the intersection of the best-fit line with the null displacement axis indicates the location z ₀ of the virtual anode with a zero vertical offset. For the yaw rotation, since all the anodes have the same y ₀ offset, a non-zero Abbe’s error appears as a parallel displacement of the lattice planes. The horizontal offset was nullified by translating the laser beam into the horizontal plane until no displacement was observed. The result is shown in Fig. 6 (left); from these data |y ₀|≤0.25 mm. The slightly different displacements in different anodes - which, when yawing, are not expected - are due to a small coupling, less than 10%, between the yaw and pitch rotations.

 

Fig. 6. Left: Analyzer displacements in the anodes of the photomultiplier tube - at the coordinates z _i≈{-4.2,-2.8,-1.4,0,1.4,2.8,4.2,5.6} mm - when it is pitched (red) and yawed (green) about the center of the laser-beam spot. Right: d ₂₂₀ values measured in the anodes of the photomultiplier tube.

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As Fig. 6 (right) shows, the d ₂₂₀ values measured in each anode were then interpolated to obtain the d ₂₂₀ value in z=z ₀, which is the location of the virtual anode with a zero offset. Significant reduction of the Abbe’s error requires understanding of the inconsistencies related in §5 between the optical and X-ray measurements of the analyzer rotations.

4.6. Trajectory

Since the lattice spacing is measured by comparing the projections, nd ₂₂₀ and mλ/2, of the crystal displacement over the normals to the diffracting planes (X-ray interferometer) and the front mirror (optical interferometer), when the projection angles are different an error occurs. This error is given by αβ d ₂₂₀, where α is the angle between the measuring directions of the X-ray and optical interferometers and β is the angle between the movement direction and the bisector of α. Therefore, not only the parasitic rotations, but also the trajectory must be carefully controlled. In the MO*4 crystal, α=0.17 mrad. This relatively large value makes the interferometer quite sensitive to movement straightness and direction; we took advantage of this sensitivity to test our motion control capabilities.

 

Fig. 7. Consistency of the measured lattice spacing values. The movement direction was intentionally varied and measurement results are plotted vs. the projection error. The solid line is the expected d ₂₂₀ variation.

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In the first place, we measured the average lattice spacing in the same crystal slice, but with different locations of the interferometer guide. Measurements were made possible by translating the X-ray source and detectors to different positions; then we moved the analyzer to place the same crystal slice again in front of X rays. No significant difference between the measured values was observed, to within the present ±5×10^-9 d ₂₂₀ sensitivity. In the second place, we varied the movement direction and changed the bisection error from β=6 µrad to β=0.26 mrad. The figure 7 compares the measured d ₂₂₀ values with the αβ d ₂₂₀ error; the nearly perfect agreement between the d ₂₂₀ variations and the expected error confirms our excellent motion control.

In the recently machined prototype of the ²⁸Si interferometer, we demonstrated a parallelism between the crystal front mirror and diffracting planes better than 10 µrad, to be compared with the 170 µrad error in the MO*4 crystal; this good parallelism will further reduce the error by a factor of higher than ten.

4.7. Temperature

The volumetric expansion of Si is about 8×10^-9 mK^-1. Consequently, in order to determine the Avogadro constant to within the targeted 2×10^-8 relative uncertainty, the cell and molar volumes - which are separately measured by the INRIM and the Physikalisch-Technische Bundesanstalt (PTB) - must refer to the same temperature to within 0.5 mK. Our temperature measurement capability was tested by circulation of a 20 °C electronic reference between the PTB and INRIM. Results demonstrated that the temperature measurements are presently consistent to within 1 mK.

4.8. Aberrations

Variations in the analyzer thickness, ΔT _A, and analyzer to mirror distance (defocusing), ΔF, cause additional phase shifts of the X-ray fringes, which are detected as variations of (220) plane spacing. With a 0.8 mm thickness of the analyzer and the use of the 17 keV radiation, sensitivities to defocusing and analyzer thickness are

$\frac{\Delta d_{220}}{d_{220}}=2\times {10}^{-6}\frac{\Delta F}{m\lambda /2}$

and

$\frac{\Delta d_{220}}{d_{220}}={10}^{-6}\frac{\Delta T_A}{m\lambda /2},$

respectively [16]. For instance, with 1 mm measurement baseline, 1 µm error causes an apparent lattice spacing variation slightly larger than 10^-9 d ₂₂₀. Centimeter interferometry, together with accurate manufacturing and dimensional characterization - e.g., by a coordinate measuring machine - can relieve this problem. Additionally, the ²⁸Si interferometer is so designed as to allow it to operate also in reverse; in this way, we can compare obverse and reverse measurement results and increase our confidence in the assessment of these errors.

4.9. C and O contamination

Since the purpose of the present investigation was to assess the performances of our apparatus and check the consistency of the measured d ₂₂₀ value with the previous ones, we are not correcting the result for carbon and oxygen contamination of the MO*4 crystal and, consequently, the relevant contribution to the error budget is not taken into account.

5. Measurement anomalies

The centimeter scan capability allowed us to investigate anomalies already suggested by the use of our previous apparatus.

The first anomaly is related to aberrations of the laser beam wavefront. The electronic control of the analyzer parasitic-rotations relies on the displacement of four points, spaced by about 1.5 mm, of the front mirror surface; these displacements are sensed with the aid of a quadrant detector (see Fig. 5). The displacements in the top-bottom and left-right quadrant pairs are made identical to within picometers by the electronic control of the analyzer attitude, but nothing imposes that these two - vertical and horizontal - displacement pairs must be identical. Therefore, the comparison between the d ₂₂₀ values in the vertical and horizontal quadrants delivers information about aberrations in the optical interferometer. If the interfering wavefronts are perfectly plane or spherical, no difference should be observed. On the contrary, results in Fig. 8 show a modulated difference between the effective wavelength in different parts of the laser beam, the origin of which is, at present, unknown.

The second anomaly is related to a discrepancy between X-ray and optical measurements of the analyzer pitch. In Fig. 6 (right), the slope of the best-fit line to the observed d ₂₂₀ values, which is expected zero, delivers information about the congruity of the analyzer pitch and (220) plane tilts. In fact, the measured d ₂₂₀ value in the i-th analyzer slice - mλ/2 wide and centered at {x _i, z _k}, where x _i is the slide position and z _k is the anode offset - is

where d ₀ is the actual d ₂₂₀ value, z _kΔ_ρ(x _i) is the Abbe’s differential displacement between the centers of the {x _i, z _k} slice and the laser beam spot, and Δ_ρ(x _i) is the analyzer (or lattice-plane) pitching. Therefore, Δ_ρ(x _i) can be obtained from the slope of the best-fit line to the d(x _i, z _k) values. The sum ρ(x _n)=∑ⁿ _iΔρ(x _i) of each partial rotation, which is shown in Fig. 9 (top), displays the intrinsic tilt of (220) planes - if we trust the optical measurement of the analyzer pitch. On the contrary, if we assume that the analyzer lattice is perfect, the same sum discloses errors in the optical measurement. As yet, we have no hints about an unambiguous choice between these two hypotheses. Clearly, the linear trend of the data in Fig. 9 (top) is not due to lattice deformations; it is caused by non-orthogonal incidence of the laser beam on the analyzer front mirror [17]. If the laser beam is imperfectly aligned, when the analyzer is moved over the mλ/2 distance, the interfering beams drift by γmλ, where γ is the deviation from orthogonality. Owing to the wavefront curvature, this drift gives rise to an interference pattern imitating the pattern originated by the analyzer pitch. In this case, we detect and compensate for a non-existent pitch ρ=γmλ/(2R), where R≈50 m is the wavefront curvature-radius [17]. In Fig. 9 (top), the worst slope is 15 nrad/cm, to which a possible alignment error of 50 µrad will correspond. The plane tilt after the trend has been removed, shown in the bottom part of Fig. 9, is a better candidate to display lattice deformation.

 

Fig. 8. Difference between the d ₂₂₀ values in the vertical and horizontal pairs of the quadrant detector; measurements carried out on 2007-12-18 and 2008-02-04 are in blue and green.

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Fig. 9. Top: Analyzer or (220) plane pitching - as sensed by the X-ray interferometer-when the analyzer pitch - as sensed by the optical interferometer - is kept to zero. Measurements carried out on 2008-02-04 and 2008-02-25 are in green and red. Bottom: Residuals from the best-fit line to the data.

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6. Conclusions

The test measurements of the lattice parameter of INRIM’s MO*4 reference crystal demonstrate that the new combined X-ray and optical interferometer, capable of centimeter displacements, was successfully put into operation. The results of measurement carried out over displacements from less than 1 mm to more than 1 cm proved excellent consistency, better than 10^-8 d ₂₂₀. Figure 10 illustrates the history of our progress towards the 3×10^-9 d ₂₂₀ accuracy necessary to obtain a kilogram prototype made by a 1 kg ²⁸Si crystal-sphere.

 

Fig. 10. History of the measurements of the mean (220) lattice spacing in the MO*4 crystal. The labels record the increasing scan-length and control-capability; numbers in brackets are the relevant references.

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The new apparatus allows us to investigate finer points in the interferometer operation, for which we have no satisfactory explanations. The determination of the Avogadro constant will be carried out with the aid of a highly enriched ²⁸Si crystal; the design and manufacturing of the relevant X-ray interferometer are under way. In the near future we will integrate and test into the experimental apparatus a prototype of the ²⁸Si interferometer having an unusually long analyzer, to exploit the large displacement capabilities of our apparatus.