Instrumentation & Measurement Magazine 23-6 - 7

Combining (5), (7) and (8) yields:

m a3 c  2
h 
8 V 2 R



Ar  e 
30

(9)
x Ar ( i Si)

i 28 i

So for a praticular Si sample, the values of Ar(Si), a3, V and
m have to be determined with relative uncertainties below 1
× 10−8 to determine the Planck constant with a relative uncertainty of 2 × 10−8.
The measurement of the Si isotopic composition is a
critical part of the XRCD method, and to minimize this uncertainty six crystals of enriched 28Si have been produced at great
expense. Enrichments ranging from 99.9957% to 99.9993%
have been achieved and measured by a number of institutes
[17], [18].
The lattice parameter of a Si crystal is measured by combining X-ray and optical interferometry. The x-ray beams detect
the lattice plane spacings d220 while optical interferometry is
used to establish traceability to the SI meter [19]. Two lamellas,
or etched blades of the single crystal, diffract the x-ray beams
and generate an interferogram with the periodicity of the lattice planes at the position of a third lamella, the "analyzer."
The third lamella is moved through this moiré pattern, the lattice spacings are counted and the movement of the lamella is
measured by an optical interferometer. The lowest uncertainty
has been achieved by INRIM (Istituto Nazionale di Ricerca
Metrologica, Italy), resulting in a relative mass uncertainty
contribution of 5.2 × 10−9 [20].
For the volume measurements, a large part of the single
crystal is shaped as a nearly perfect sphere. Polishing methods
developed at PTB have achieved unroundness values below 30
nm in the radius [21].

For the volume measurement, the sphere is placed between
two reference surfaces and the distances d1 and d2 of the sphere
on both sides to the reference surfaces are measured interferometrically. With a measurement of the distance L between
the reference surfaces, the diameter of the sphere can be calculated: D = L - (d1 + d2). By rotating the sphere, a complete
diameter mapping of the sphere is possible (Fig. 4). Uncertainties as low as 7 × 10−9 V have been achieved.
Equation (9) is valid for perfect silicon single crystals but
corrections must be made for surface layers of contamination,
crystal defects and voids as well as chemical impurities [18],
[20], [22], [23].

Uncertainties Over a Range
The uncertainties over a wide dynamic range of a quantity often display a 'V' shaped profile. This is generally the case when
the apex of traceability is a single artifact, and the lowest uncertainty measurements are made by direct comparison with
that apex artifact and then scaled either up or down. However,
the scaling measurements needed to cover a broader range
inevitably introduce ever more measurements and their accompanying uncertainties. This is particularly true of mass
traceability before 2019 where scaling by just a factor of two requires two additional masses and three measurements.
But the 2019 changes to the SI have already begun to alter
that profile. Kibble balances typically have an optimized operational range of perhaps 20 to 1 over which their statistical
uncertainties are roughly constant. But Kibble balances can
be optimized over other ranges and projects are already underway to extend their operation below 10 g [24]. Other types
of balances, such as the electrostatic force balance, can be implemented for even lower masses while having improved
traceability uncertainties within the new SI [25].

Fig. 4. The new Si-28 ingot Si28-23Pr11 which has an isotopic enrichment of 99.9993% and the cutting scheme for the
spheres Si28kg02a and Si28kg02b. (Reprinted with permission from PTB.)
September 2020

IEEE Instrumentation & Measurement Magazine 7

Instrumentation & Measurement Magazine 23-6

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