Volume determination of two spheres of the new 28 Si crystal of PTB

In the scope of the redetermination of Avogadro ’ s constant N A , a new isotopically enriched silicon crystal has been produced, from which two spheres were manufactured. After the crystal properties, the lattice parameter and molar mass, as well as the masses of the two spheres have been determined, the volume of the spheres was also measured. For this, the sphere interferometer of PTB was used. The methods of the interferometric measurements have been improved and the major contributions to the uncertainty have been investigated thoroughly. As a result, the total uncertainty could be reduced significantly, yielding a substantial impact on the determination of Avogadro ’ s constant. The mean diameter of each sphere was measured twice with a repeatability of ± 2 × 10 − 10 , and the relative uncertainty of the ‘ apparent ’ volume, which disregards the comparatively small influence of the optical effects of surface layers, was reduced to 7 × 10 − 9 . The final results of the volumes and comments on their uncertainties are given.


Introduction
From 2011 to 2014, the international Avogadro coordination published several measurements for the determination of the Avogadro constant N A . As the Planck constant h and N A can be converted into each other without an increase in uncertainty, N A is not only used to define the mole, but also contributes to the final value for h which will eventually define the new kilogram.
Hitherto, two spheres of the first 28 Si crystal 28Si-10Pr11 were used for the determination of N A . As the first sphere processing in Australia resulted in metal contamination on the surfaces, the spheres were reworked twice and remeasured after each step, always yielding the same Avogadro constant. Due to the complications associated with continually managing the two spheres in agreement with all owners, PTB decided to buy new 28 Si material and to manufacture two new spheres, so that a new measurement campaign with the new spheres could be initiated.
Throughout this time, both the manufacturing of the spheres and the measurement capability of the interferometers could be increased, leading to a significantly reduced uncertainty of the determination of the volume of the silicon spheres. As the uncertainty of the volume determination was the dominant contribution to the overall uncertainty for N A (about 60%), the efforts of the investigations of the sphere interferometers show through in the results of N A .

Principle of the measurement and diameter evaluation
The volume determination of the silicon spheres, 28Sikg01a and 28Sikg01b from the new crystal Si28-23Pr11, was performed at PTB by means of a special spherical interferometer [1]. The measurement of a sphere consists of pairs of alternating measurements, the basic principle of which is explained in the following (details can be found in [2]).
In the scope of the redetermination of Avogadro's constant N A , a new isotopically enriched silicon crystal has been produced, from which two spheres were manufactured. After the crystal properties, the lattice parameter and molar mass, as well as the masses of the two spheres have been determined, the volume of the spheres was also measured. For this, the sphere interferometer of PTB was used. The methods of the interferometric measurements have been improved and the major contributions to the uncertainty have been investigated thoroughly. As a result, the total uncertainty could be reduced significantly, yielding a substantial impact on the determination of Avogadro's constant. The mean diameter of each sphere was measured twice with a repeatability of ±2 × 10 −10 , and the relative uncertainty of the 'apparent' volume, which disregards the comparatively small influence of the optical effects of surface layers, was reduced to 7 × 10 −9 . The final results of the volumes and comments on their uncertainties are given.
Keywords: Avogadro's constant, sphere interferometer, kilogram, volume determination (Some figures may appear in colour only in the online journal) Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Firstly, the space between two opposing spherical reference faces, the etalon, is determined (see figure 1). Then the sphere is inserted and the resulting gaps between the reference surface and the corresponding area of the sphere are measured. If one light ray is assumed for each pixel (x, y) of the camera array, the distance D of the empty etalon and the gaps d 1 and d 2 , respectively, are measured at about 10 000 positions simultaneously. The diameters of the sphere in the current field of view are then obtained from the difference of the measurements: Despite the spherical arrangement, in all three measurement cases the interferences are almost interferences of equal thickness, and the values of D, d 1 or d 2 have only a small spread. Due to the quality of the reference faces and the sphere, the deviations within the 60° field of view are below λ/10-showing a nearly completely dark interference.
With an orientation device, the sphere can be lifted out of the optical beams in order to measure the empty etalon. In this elevated position, it can be rotated around two axes. With about 30 different orientations, the sphere can be covered with measurement segments. Each point of the sphere is therefore measured numerous times and at different areas of the optics [3].
The interferences are evaluated using phase-shifting interferometry with wavelength tuning. For that, an iodine stabilized laser (BIPM recommendation for 633 nm [4]) works as a wavelength reference, whereas a tunable laser is used for the different measurements with changed wavelengths. Each wavelength of the extended cavity laser is controlled by means of a highly stabilized frequency tuner and a phase locked loop. This guarantees that the stability of the tunable laser works at the same 10 −11 uncertainty level as the iodine laser.
With phase-shifting interferometry the interferences are evaluated with high resolution-but only the fractional part inside one interference order. For the integer orders, which amount to about 300 000 for the sphere, a preliminary value of the diameter of the sphere is necessary. As the density of the probes of the crystal has been measured at the 10 −8 level, only a rough mass determination of the sphere at the mg level is necessary to receive a sufficiently good value for the integer interference orders of the sphere.
The laser intensity at the entrance of the interferometer is influenced to a small extent by the thermal stability of the optical fibres. Therefore, the interferometer input was stabilized by means of a noise eater [5].
Each of the 28Sikg01-spheres was measured twice with different starting orientations and therefore with different mapping schemes. This was possible as every sphere was marked by a laser with different marks, two at (1 0 0) and one at (1 1 1) crystal lattice orientations (Miller indices). The temperature stabilization of the interferometer was adjusted in such a way that the measurements could be carried out over several days with only small deviations from 20 °C of maximally ±3 mK. The vacuum pressure was below 0.1 Pa, so that no correction of the refractive index was necessary. Autocollimation of the entrance beam, and of the exit beam was controlled carefully and the adjustment of both the sphere and empty etalon was accomplished to the best possible zero fringe, only limited by the topography of the sphere and the reference face. This scheme is a prerequisite to reach the lowest uncertainty due to optical imperfections [6]. Furthermore, an aperture correction has to be applied to consider the lateral size of the fibre output [7].
To ensure the smallest uncertainty of the temperature measurement of the spheres, the reference Pt-25 thermometer was investigated carefully. The two temperature fixed-points usedthe melting point of gallium and the triple point of water-were prepared several times, and a repeatability of each fixed-point of 50 µK was achieved. An international comparison of several reference thermometers in a highly stabilized 20 °C-referencepoint which overlap generously. yielded standard deviations of better than 30 µK. For a self-heating-free measurement of the sphere, a system of thermocouples was used, detecting the small temper ature differences between the sphere and a copper block containing the Pt-25 reference thermometer. Each pair of thermocouples was checked for zero offset (both sensors together on one isolated copper block) and for slope (the sensing arms of the thermocouples were divided onto two copper blocks, each one measured with a Pt-25 reference thermometer) [8,9]. The main uncertainty contribution of the thermocouples was the noise of the amplifiers, which amounted to 0.3 mK, but was highly averaged through the slow temperature course with time.

Evaluation of the volume
Following the diameter evaluation described, the correct average diameter or better of the volume of each sphere has to be determined. In the present case, two sets of measurements per sphere have been carried out, each of which consists of 31 single measurements and covers the surface of the  respective sphere entirely. The topography of the spheres is then repeatedly represented by approximately 330 000 diameter values, in which influences on each value, as for example the aperture correction [7] or the temperature deviation from 20 °C, are considered by a respective correction. These single diameters are distributed all over the surface and cover the topographies thoroughly, but the densities of the measuring positions are locally different. This is considered in the evaluation by means of a fit of real spherical harmonics, in which the uncertainties of the single values are included as weighting factors. Then the volume of the sphere is the integral over its radius representation ϕ ϑ R , ( ) via real spherical harmonics as described in [2]: The first fit parameter yields the average radius at 20 °C and 0 Pa which gives-multiplied by 2-the mean diameter. This is called the 'apparent' diameter because it is not corrected for the phase shift due to surface layers. To take the influence of the surface layers into account, ellipsometric, XRF and XPS measurements, in combination with a layer model, have to be applied. This is done in [10], and its influence on the optical path length is studied in [2]. As these corrections depend on specific measurements which are based on extrinsic apparatuses in the following, only the 'apparent' diameters of the two spheres will be dealt with. The corrections due to the phase change on reflection are typically on the order of some hundredths of nm and do not alter the final value substantially. Therefore, these corrections are the focus of [10].
Although a 'real' radius topography of each sphere can be calculated [11], here only diameter topographies are considered. Because the form deviations are small compared to the average diameter, the latter is sufficient for the evaluation of the volume. The fits are calculated with a number of 1225 spherical harmonic functions, which correspond to the functions up to the 48th order when only the even orders are used. The selection of the maximum order is based on a threshold in the asymptotic behavior of the figure of merit of the fit. The average diameters (to be understood as the diameter of a mathematical round sphere of volume V) of the measured spheres are listed in table 1 and the corresponding diameter topographies are shown in figure 2.

Uncertainty assessment
An overview of the relevant contributions for the determination of the apparent diameters or volumes is listed in table 2.
As has been noted in [2], the statistical uncertainty contribution of the evaluation of the interferometric data was negligibly small in relation to the other contributions. But since compared to the uncertainty budget in [2], some advances in the assessment of the influences lead to a reduction of the major uncertainty contributions, the statistical uncertainty has now became significant.
The noted contribution relating to the data evaluation sums up the outcome of the light intensity measurement, the frequency control, the effect of parasitic interferences-as the major contributions-and includes the statistical share resulting from the fit. The stated number is the maximum value occurring in the fitted topography.
It was possible to reduce the uncertainty of the temperature measurement to 0.6 mK due to the results of the examination which is described above in section 2 and will be discussed in more detail in a seperate paper.
The formerly dominating contribution regarding the influence of wavefront aberrations on the measured lengths has been investigated in detail via optical simulations [6]. The wavefront effect is divided into two parts: one covers unavoidable misalignments of the optical system, and the other handles the retrace errors due to the form deviations of the spheres. As the former term is typical for the optical design of the interferometer and can therefore be determined principally, the latter has to be calculated individually for  each sphere. For this, the results of the measurement of the specific sphere, more precisely its representation in the form of spherical harmonics, were used as an input for an optical simulation. The simulation program applies the physical laws of propagation, refraction and reflection for every ray in the field of view and calculates the complete beam paths, interferometry and projection onto the camera sensor. Following the measurement with subapertures and subsequent reconstruction, one obtains a completely calculated topography of the whole sphere. This fully simulated sphere can-in the case of deviations-be used for a correction, or-in this case-to check the goodness of the actual measurement. The data of both spheres have been evaluated in this way. The results in the form of difference topographies between the measurement result and the simulated sphere are shown in figure 3. These differences amount to maximally 0.07 nm and 0.13 nm for the peak values, but are zero for the mean diameters. Therefore, no corrections for the final volumes have been applied.
The uncertainties for the wavefront aberrations due to the specific form of the sphere amount to 0.01 nm.
The total standard uncertainty is underpinned by the fact that the reproducibility of the diameter measurements is on the order of 0.1 nm, as has been observed for more than ten years. This involves the short-term comparison of measured diameters (i.e. over a few weeks or months) as well as a long-term comparison, as was reported in [12]. The latter case includes several fundamental realignments of the whole setup and also the replacement of key components of the optical system. An example of a short-term repetition is given by the measurement sets A and B in table 1.

Summary
The volumes of two spheres of a new 28 Si crystal have been measured interferometrically as a contribution to the redetermination of Avogadro's constant N A . The interferometer used spherical wavefronts which allowed high-resolution diameter topographies to be measured. Compared to former approaches in the past, the measurement uncertainty could be reduced significantly. For the volume, a relative standard uncertainty of 7 × 10 −9 could be achieved. The stated uncertainties are underpinned by the long-term reproducibility of the diameter measurements. In the overview article [10] in which all individual quantities are put together for the evaluation of N A , the influence of the surface layers on the volume determination is also addressed. The largest uncertainty contribution is given by the temperature measurement. Its direct influence on the dimensional quantities can be reduced by at least one order of magnitude in Avogadro's constant by means of a joint consideration, where the quotient of the macroscopic volume (i.e. of the sphere) and of the microscopic volume (i.e. of the crystal's unit cell) appears.