To determine the Avogadro constant N_A to an accuracy allowing the kilogram definition to be based on the atomic mass of the ²⁸Si atom, several metrology institutes are participating in a research project (International Avogadro Coordination, IAC) for the determination of N_A using a highly enriched ²⁸Si crystal. In this framework, the relative uncertainty of the Si lattice parameter measured by combined X-ray and optical interferometry (fig. 1) must be reduced to 3·10^-9.

Fig.1 Schematic drawing of the combined X-ray and optical interferometer.

In addition, an accurate value of the Si lattice parameter is relevant in determining the relative atomic mass of the neutron and the fine structure constant. The measurement capabilities were extended and an X-ray interferometer prototype was manufactured from a high-purity natural Si crystal, named WASO04, which was grown on purpose for N_A determination.
An X-ray interferometer consists of three thin crystals so cut that the diffracting planes are orthogonal to the crystal surfaces. X rays from a conventional 17 keV Mo Kα source are split by the first crystal and then recombined, via two transmission crystals, by the third, called the analyzer. The analyzer crystal embeds front and rear mirrors, so that its displacement is measured by optical interferometry. The laser source realizes the metre by definition; it operates in single-mode configuration and its frequency is stabilized against that of a recommended transition of the ¹²⁷I₂ molecule. This ensures the calibration of the optical interferometer with negligible uncertainty.
When the analyzer is moved along a direction orthogonal to the diffracting planes, a periodic variation in the transmitted and diffracted X-ray intensities is observed, the period being the diffracting-plane spacing (lattice parameter). According to the measurement equation

d₂₂₀ = (m/n) λ/2

where n is the number of X-ray fringes of d₂₂₀ 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. This magnification makes more numerous effects visible and reproducible. In addition, it allows wider crystal parts to be surveyed, thus increasing confidence in the crystal perfection and in the mean value of the lattice parameter.
With this in view, an X-ray interferometer with an unusually long analyzer crystal was designed and manufactured (fig. 2), 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. To operate a separate-crystal interferometer of such a size was a formidable task: 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 vibrations and displacements. The result is a system that accurately measures and controls changes in the position and alignment of the interferometer crystals at sub-atomic level and over distances as large as 5 cm.

Fig.2 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.

This measurement capability is obtained by means of a guide where an L shaped carriage slides on a quasi-optical rail. The carriage displacement is measured with picometer sensitivity by optical interferometry; the necessary resolution is achieved by polarization encoding and phase modulation. 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.
We determined (fig. 3) the lattice spacing by comparing the unknown period of the X-ray fringes against the known period of the optical fringes. This is done by measuring the X-ray fringe phases at the ends of increasing analyzer displacements. To measure the X-ray fringe fraction, the least-squares method is applied; the input data are about 300 samples of six fringes, with a 100 ms integration time and a sample duration of 30 s. Since it is not possible to keep the drift between the X-ray and the optical interferometers as small as desirable, the analyzer is repeatedly moved back and forth along any given displacement, sampling 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.

Fig.3 Experimental set-up.

L. Ferroglio, G. Mana, E. Massa: "Si lattice parameter measurement by centimeter X-ray interferometry", Optics Express, Vol. 16, pp. 16877-16888, 2008.
E. Massa, G. Mana, U. Kuetgens: "Comparison of the INRIM and PTB lattice-spacing standards",, Metrologia, Vol. 46, pp. 249-253, 2009.
E. Massa, G. Mana, U. Kuetgens, L. Ferroglio: "Measurement of the Lattice Parameter of a Silicon Crystal", New J. Phys, Vol. 11, Article number 053013, 2009.