Determination of the Avogadro constant by the XRCD method using a ²⁸Si-enriched sphere

The sphere volume was measured by an optical interferometer with a direct optical frequency tuning system [12–15]. The Si sphere was placed in a fused-quartz Fabry–Perot etalon. The sphere and etalon were installed in a vacuum chamber equipped with an active radiation shield to control the sphere temperature. The pressure in the chamber was reduced to 1  ×  10⁻³ Pa. Measurements of the fractional fringe order of interference for the gaps between the sphere and the etalon, d₁ and d₂, and the etalon spacing L were carried out by phase-shifting interferometry. The sphere diameter D was calculated as D  =  L  −  (d₁  +  d₂). The light source of the interferometer was an external cavity diode laser, and the required phase-shift for the diameter measurement was produced by tuning the optical frequency of the laser over the frequency range of 20 GHz [13]. The wavelength reference in the optical frequency tuning system was synthesized by a frequency comb at NMIJ from an atomic clock linked to coordinated universal time (UTC) [12]. The comb is also used as the national standard of length in Japan.

A sphere rotation mechanism installed under the sphere was used to measure the diameter from many different directions. In a set of the diameter measurement, the diameter was measured from 145 directions distributed near-uniformly on the sphere surface [12, 15]. The set of the diameter measurement was repeated 15 times. Between each set, the sphere was oriented to distribute all the measurement directions as uniformly as possible. The temperature of the sphere was measured using small platinum resistance thermometers (PRTs) inserted in copper blocks in contact with the sphere. The PRTs were calibrated using the Ga melting point and the water triple point according to ITS-90. The measured diameters were converted to those at 20.000 °C using the thermal expansion coefficient of the enriched ²⁸Si crystal [16]. The measured diameter and volume were the apparent diameter and volume, which are not corrected for the phase shift due to the surface layers. Table 1 summarizes the apparent diameter D_app and apparent volume V_app. Details of the measurement results and the uncertainty estimation are given in [11]. The values of D_app and V_app in table 1 are slightly different from those in [11] due to a small correction based on the calibration of the temperature sensors after the volume measurement.

The surface layers consisted of an oxide layer (OL), a carbonaceous contamination layer (CL), a physically adsorbed water layer (PWL) and a chemically adsorbed water layer (CWL) as shown in figure 1. To derive the Si core volume V_core from the apparent volume, the thickness of each surface layer was determined by two different methods: x-ray photoelectron spectroscopy and spectroscopic ellipsometry.

3.2.1. X-ray photoelectron spectroscopy (XPS).

3.2.2. Spectroscopic ellipsometry.

To determine the Si core volume, the total phase retardation upon reflection at the sphere surface δ was calculated in accordance with the procedure in [13]. To calculate δ, the thickness and the optical constants of each layer given in table 3 were used. The phase shift upon reflection (δ  −  π), was estimated to be  −0.054(3) rad. The effect of this phase shift on the gap measurement Δd was 2.73(16) nm. This means that the actual diameter including the surface layers is larger than the apparent diameter by 5.45(32) nm. The actual diameter D_actual was therefore obtained using D_actual  =  D_app  +  2 Δd. The phase shift correction to obtain the Si core diameter from the apparent diameter, Δd₀, was determined as Δd₀  =  2 (Δd  −  d_total), where d_total is the sum of the thickness of each layer. The value of Δd₀ was estimated to be  −0.13(8) nm. Table 3 shows the uncertainty budget in the determination of Δd₀. The Si core diameter D_core was determined to be 93.710 811 32(63) mm using D_core  =  D_app  +  Δd₀. The Si core volume V_core was estimated to be 430.891 2905(87)  ×10⁻⁶ cm³ with a relative standard uncertainty of 2.0  ×  10⁻⁸.

A vacuum mass comparator at NMIJ [21] was used to determine the mass of the sphere including the surface layers. The mass of the platinum–iridium kilogram standards of NMIJ was used as the reference in the measurement. Before the mass measurement of the sphere, the mass of the standards was calibrated in the Extraordinary Calibrations conducted by BIPM [22]. The mass of the sphere in vacuum m_sphere was estimated to be 999.698 4595(59) g. Details of the uncertainty evaluation are provided in [21].

The mass of the surface layers under vacuum m_SL is given by

$$\begin{align} \displaystyle {{m}_{{\rm SL}}}={{m}_{{\rm OL}}}+{{m}_{{\rm CL}}}+{{m}_{{\rm CWL}}},\nonumber \end{align} \tag{ 4 }$$

where m_OL, m_CL and m_CWL are the masses of the OL, CL and CWL, respectively. The masses were calculated from the thickness and density of each layer. The calculated masses are summarized in table 2 along with the density and thickness of each layer. The total mass of the surface layers was estimated to be 120.6(8.9) µg.

To determine the Si core mass, the mass of the surface layers was subtracted from the mass of the sphere. In addition, owing to point defects, there is a difference between the mass of a sphere having Si atoms occupying all the regular sites and the measured mass value. The mass of the Si core m_core to determine the Avogadro constant was therefore calculated using

$$\begin{align} \displaystyle {{m}_{{\rm core}}}={{m}_{{\rm sphere}}}-{{m}_{{\rm SL}}}+{{m}_{{\rm deficit}}},\nonumber \end{align} \tag{ 5 }$$

where m_deficit is the effect of point defects (i.e. impurities and self-point defects in the crystal) on the core mass [8]. The value of m_deficit for AVO28-S5c was estimated to be 3.8(3.8) µg [8]. The values of m_core was determined to be 999.698 343(11) g with a relative standard uncertainty of 1.1  ×  10⁻⁸.

From the values of V_core and m_core determined in this study, the density of the Si core was determined with a relative standard uncertainty of 2.3  ×  10⁻⁸. By combining the density with the lattice constant a and the molar mass of silicon M of AVO28-S5c determined by IAC in 2015 [8], the Avogadro constant was determined to be

$$\begin{align} \displaystyle {{N}_{{\rm A}}}=6.022\,140\,84(15)\times {{10}^{23}}\,{\rm mo}{{{\rm l}}^{-1}},\nonumber \end{align} \tag{ 6 }$$

with a relative standard uncertainty of 2.4  ×  10⁻⁸ using equation (2). Table 4 shows the uncertainty budget of the N_A determination. The largest uncertainty source is the determination of the Si core volume. This determination of N_A from the data measured by IAC in 2015 [8] and from the data measured by NMIJ in 2017 is referred to as 'N_A determination by NMIJ in 2017' in this paper.

Figure 2 shows a comparison of the value of N_A obtained in this work with those in our previous works [7, 8]. The bars on the data show the standard uncertainty. The N_A values converted from the values of the Planck constant h determined by the watt (Kibble) balance experiments at the National Institute of Standards and Technologies of USA (NIST) [23] and the National Research Council of Canada (NRC) [24] are also shown. For the conversion, the molar Planck constant N_Ah  =  3.990 312 7110(18)  ×  10⁻¹⁰ J s mol⁻¹ in the CODATA adjustment in 2014 [5] was used. Because the relative standard uncertainty of N_Ah is as small as 4.5  ×  10⁻¹⁰, the increase of the uncertainty of N_A by the conversion is negligibly small. The value of N_A determined in this work agrees with those in the previous works within their uncertainties.

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In 2011, the Avogadro constant was determined by measuring the lattice constant, molar mass, core volume and core mass of the two ²⁸Si-enriched spheres named AVO28-S5 and AVO28-S8 [7]. The structure of the surface layers was investigated using various surface analysis techniques, and it was found that the spheres were covered with a thin layer contaminated with silicides of the metals Ni and Cu. This unexpected metallic contamination was one of the largest uncertainty sources in the determination of the Avogadro constant in 2011 [7]. To reduce the uncertainty contribution from this source, the metallic contamination was removed by Freckle^™ etching after the 2011 N_A determination [18]. However, the etching process degraded the shape of the spheres [25], and the spheres were therefore repolished to improve their sphericity. The measurements were repeated using the repolished spheres renamed AVO28-S5c and AVO28-S8c, and an N_A value was published in 2015 [8].

In this study, a new value of N_A was determined using AVO28-S5c. To simplify the analysis, we considered the correlations between the N_A values obtained using AVO28-S5 and AVO28-S5c in the followings.

The calculation of the correlation coefficient between the two N_A determinations requires the correlation of each pair of input data: molar mass, lattice constant, core volume and core mass. To explain the estimation procedure of the correlation, let us consider the correlation between two output quantities obtained in 2015 and 2017, y₂₀₁₅ and y₂₀₁₇, as an example [9, 26]. The two output quantities are represented by

$$\begin{align} \displaystyle {{y}_{2015}}={{f}_{2015}}\left({{x}_{1-2015}},{{x}_{2-2015}} \right),\nonumber \end{align} \tag{ 7 }$$

$$\begin{align} \displaystyle {{y}_{2017}}={{f}_{2017}}\left({{x}_{1-2017}},{{x}_{2-2017}} \right),\nonumber \end{align} \tag{ 8 }$$

where f₂₀₁₅ and f₂₀₁₇ are the measurement functions in 2015 and in 2017, respectively, x_1-2015 and x_2-2015 are the input parameters for y₂₀₁₅, and x_1-2017 and x_2-2017 are the input parameters for y_2017. The variance and covariance matrix associated with y, U_y,y, is represented by

$$\begin{align} \displaystyle {{\boldsymbol{U}}_{y,y}}=\left(\begin{matrix} V\left({{y}_{2015}} \right) & {\rm Cov}\left({{y}_{2015}},{{y}_{2017}} \right)\nonumber \\ {\rm Cov}\left({{y}_{2015}},{{y}_{2017}} \right) & V\left({{y}_{2017}} \right) \nonumber \\ \end{matrix} \right),\nonumber \end{align} \tag{ 9 }$$

where V(y₂₀₁₅) and V(y₂₀₁₇) are the variances of y₂₀₁₅ and y₂₀₁₇, respectively, and Cov(y₂₀₁₅,y₂₀₁₇) is the covariance between y₂₀₁₅ and y₂₀₁₇.

U_y,y, is given by

$$\begin{align} \displaystyle {{\boldsymbol{U}}_{y,y}}={{\boldsymbol{C}}_{{x}}}{{\boldsymbol{U}}_{{x,x}}}\boldsymbol{C}_{{x}}^{\rm T},\nonumber \end{align} \tag{ 10 }$$

where C_x is the sensitivity matrix obtained by evaluating

$$\begin{align} \displaystyle {{\boldsymbol{C}}_{{x}}}=\left(\begin{matrix} \frac{\partial {{f}_{2015}}}{\partial {{x}_{1-2015}}} & \frac{\partial {{f}_{2015}}}{\partial {{x}_{2-2015}}} & 0 & 0 \nonumber \\ 0 & 0 &\frac{\partial {{f}_{2017}}}{\partial {{x}_{1-2017}}} & \frac{\partial {{f}_{2017}}}{\partial {{x}_{2-2017}}} \end{matrix} \right)\nonumber \end{align} \tag{ 11 }$$

and U_x,x is the variance and covariance matrix between the input parameters given by

$$\begin{align} \displaystyle {{\boldsymbol{U}}_{{x,x}}}=\left(\begin{matrix} V\left({{x}_{1-2015}} \right) & {\rm Cov}\left({{x}_{1-2015}},{{x}_{2-2015}} \right) & {\rm Cov}\left({{x}_{1-2015}},{{x}_{1-2017}} \right) & {\rm Cov}\left({{x}_{1-2015}},{{x}_{2-2017}} \right) \nonumber \\ {\rm Cov}\left({{x}_{2-2015}},\,{{x}_{1-2015}} \right) & V\left({{x}_{2-2015}} \right) & {\rm Cov}\left({{x}_{2-2015}},{{x}_{1-2017}} \right) & {\rm Cov}\left({{x}_{2-2015}},{{x}_{2-2017}} \right) \nonumber \\ {\rm Cov}\left({{x}_{1-2017}},\,{{x}_{1-2015}} \right) & {\rm Cov}\left({{x}_{1-2017}},\,{{x}_{2-2015}} \right) & V\left({{x}_{1-2017}} \right) & {\rm Cov}\left({{x}_{1-2017}},{{x}_{2-2017}} \right) \nonumber \\ {\rm Cov}\left({{x}_{2-2017}},\,{{x}_{1-2015}} \right) & {\rm Cov}\left({{x}_{2-2017}},\,{{x}_{2-2015}} \right) & {\rm Cov}\left({{x}_{2-2017}},\,{{x}_{1-2017}} \right) & V\left({{x}_{2-2017}} \right) \end{matrix} \right).\nonumber \end{align} \tag{ 12 }$$

The correlation coefficient between y₂₀₁₅ and y₂₀₁₇, Corr (y₂₀₁₅, y₂₀₁₇), is given by

$$\begin{align} \displaystyle {\rm Corr}\left({{y}_{2015}},{{y}_{2017}} \right)=\frac{{\rm Cov}\left({{y}_{2015}},{{y}_{2017}} \right)}{\sqrt{V\left({{y}_{2015}} \right)V\left({{y}_{2017}} \right)}}.\nonumber \end{align} \tag{ 13 }$$

In our previous paper [9], the relative variance and covariance matrix associated with y, U_{r y,y}, was used instead of U_y,y,

$$\begin{align} \displaystyle {{\boldsymbol{U}}_{{\rm r}~y,y}}=\left(\begin{matrix} {{V}_{{\rm r}}}\left({{y}_{2015}} \right) & {\rm Co}{{{\rm v}}_{{\rm r}}}\left({{y}_{2015}},{{y}_{2017}} \right) \nonumber \\ {\rm Co}{{{\rm v}}_{{\rm r}}}\left({{y}_{2015}},{{y}_{2017}} \right) & {{V}_{{\rm r}}}\left({{y}_{2017}} \right) \end{matrix} \right),\nonumber \end{align} \tag{ 14 }$$

where V_r (y₂₀₁₅) and V_r (y₂₀₁₇) are the relative variances of y₂₀₁₅ and y₂₀₁₇, respectively, and Cov_r (y₂₀₁₅,y₂₀₁₇) is the relative covariance between y₂₀₁₅ and y₂₀₁₇. Equation (10) is therefore modified to

$$\begin{align} \displaystyle {{\boldsymbol{U}}_{{\rm r}\,y,y}}={{\boldsymbol{C}}_{{\rm r}\,x}}{{\boldsymbol{U}}_{{\rm r}\,x,x}}\boldsymbol{C}_{{\rm r}\,x}^{\rm T},\nonumber \end{align} \tag{ 15 }$$

where

$$\begin{align} \displaystyle {{\boldsymbol{C}}_{{\rm r}\,{x}}}=\left(\begin{matrix} 1 & 1 & 0 & 0 \nonumber \\ 0 & 0 & 1 & 1 \end{matrix} \right)\nonumber \end{align} \tag{ 16 }$$

and

$$\begin{align} \displaystyle {{\boldsymbol{U}}_{{\rm r}\,{x,x}}}=\left(\begin{matrix} {{V}_{{\rm r}}}\left({{x}_{1-2015}} \right) & {\rm Co}{{{\rm v}}_{{\rm r}}}\left({{x}_{1-2015}},{{x}_{2-2015}} \right) & {\rm Co}{{{\rm v}}_{{\rm r}}}\left({{x}_{1-2015}},{{x}_{1-2017}} \right) & {\rm Co}{{{\rm v}}_{{\rm r}}}\left({{x}_{1-2015}},{{x}_{2-2017}} \right) \nonumber \\ {\rm Co}{{{\rm v}}_{{\rm r}}}\left({{x}_{2-2015}},{{x}_{1-2015}} \right) & {{V}_{{\rm r}}}\left({{x}_{2-2015}} \right) & {\rm Co}{{{\rm v}}_{{\rm r}}}\left({{x}_{2-2015}},{{x}_{1-2017}} \right) & {\rm Co}{{{\rm v}}_{{\rm r}}}\left({{x}_{2-2015}},{{x}_{2-2017}} \right) \nonumber \\ {\rm Co}{{{\rm v}}_{{\rm r}}}\left({{x}_{1-2017}},{{x}_{1-2015}} \right) & {\rm Co}{{{\rm v}}_{{\rm r}}}\left({{x}_{1-2017}},{{x}_{2-2015}} \right) & {{V}_{{\rm r}}}\left({{x}_{1-2017}} \right) & {\rm Co}{{{\rm v}}_{{\rm r}}}\left({{x}_{1-2017}},{{x}_{2-2017}} \right) \nonumber \\ {\rm Co}{{{\rm v}}_{{\rm r}}}\left({{x}_{2-2017}},{{x}_{1-2015}} \right) & {\rm Co}{{{\rm v}}_{{\rm r}}}\left({{x}_{2-2017}},{{x}_{2-2015}} \right) & {\rm Co}{{{\rm v}}_{{\rm r}}}\left({{x}_{2-2017}},{{x}_{1-2017}} \right) & {{V}_{{\rm r}}}\left({{x}_{2-2017}} \right) \end{matrix} \right).\nonumber \end{align} \tag{ 17 }$$

The correlation coefficient Corr (y₂₀₁₅, y₂₀₁₇) is calculated from the elements of U_{r y,y} as

$$\begin{align} \displaystyle {\rm Corr}\left({{y}_{2015}},{{y}_{2017}} \right)\boldsymbol{=}\frac{{\rm Co}{{{\rm v}}_{{\rm r}}}\left({{y}_{2015}},{{y}_{2017}} \right)}{\sqrt{{{V}_{{\rm r}}}\left({{y}_{2015}} \right){{V}_{{\rm r}}}\left({{y}_{2017}} \right)}}.\nonumber \end{align} \tag{ 18 }$$

For the N_A determination by IAC in 2015, the lattice constant a of AVO28-S5c was determined at the Istituto Nazionale di Ricerca Metrologica of Italy (INRIM) using a combined x-ray and optical interferometer with a relative uncertainty of 1.8  ×  10⁻⁹ [27], and the relative uncertainty contribution of a to the N_A determination was 5.5  ×  10⁻⁹. As described in section 5.1, the same value of a was used in the N_A determination by NMIJ in 2017. The correlation coefficient between the uncertainty contribution of a to the N_A determination in 2015 and that in 2017 was therefore estimated to be 1.

For the N_A determination by IAC in 2015, the molar mass M of AVO28-S5c was measured independently by the Physikalisch–Technische Bundesanstalt of Germany (PTB), NIST and NMIJ using isotope dilution and multicollector inductively coupled plasma mass spectrometers [28–31]. The measured amount of substance of ³⁰Si was confirmed by the measurement results with instrumental neutron activation analysis (INAA) by INRIM [32]. By combining these measurement data, the value of M for AVO28-S5c was determined with a relative uncertainty of 5.4  ×  10⁻⁹ [8]. The relative uncertainty contribution of M to the N_A determination was also 5.4  ×  10⁻⁹. As described in section 5.1, the same value of M was used in the N_A determination by NMIJ in 2017. The correlation coefficient between the uncertainty contribution of M to the N_A determination in 2015 and that in 2017 was therefore estimated to be 1.

For the N_A determination by IAC in 2015, the V_core value of AVO28-S5c was determined independently by PTB and NMIJ. For the estimation of the correlation between the V_core value obtained by IAC in 2015 V_IAC-2015 and that by NMIJ in 2017 V_NMIJ-2017, the correlations for the following three combinations were examined:

• 1.  
  Corr (V_NMIJ-2015, V_NMIJ-2017),
• 2.  
  Corr (V_PTB-2015, V_NMIJ-2017),
• 3.  
  Corr (V_NMIJ-2015, V_{PTB 2015}),

where V_NMIJ-2015 and V_PTB-2015 are the V_core values determined by NMIJ and PTB in 2015, respectively.

7.3.1. Correlation between V_NMIJ-2015 and V_NMIJ-2017.

7.3.2. Correlation between V_PTB-2015 and V_NMIJ-2017 and correlation between V_PTB-2015 and V_NMIJ-2015.

7.3.3. Correlation between V_IAC-2015 and V_NMIJ-2017.

Table 6 summarizes the relative uncertainties and correlations of the Si core volume determinations by PTB and NMIJ in 2015 and 2017.

Table 6. Correlations between the Si core volume measurements by PTB and NMIJ in 2015 and 2017.

Institute, year	Relative uncertainty ur	Correlation coefficient
		VNMIJ-2015	VPTB-2015	VNMIJ-2017
VNMIJ-2015	20.1  ×  10−9	1	0	0.17
VPTB-2015	26.7  ×  10−9	0	1	0
VNMIJ-2017	20.1  ×  10−9	0.17	0	1

The Si core volume obtained by IAC in 2015 V_IAC-2015 was the weighted mean of V_NMIJ-2015 and V_PTB-2015. The weights of V_NMIJ-2015 and V_PTB-2015 in V_IAC-2015 are (1/[u_r(V_NMIJ-2015)]²)/(1/[u_r(V_NMIJ-2015)]²  +  1/[u_r(V_PTB-2015)]²)  =  0.64 and (1/[u_r (V_PTB-2015)]²)/(1/[u_r(V_NMIJ-2015)]²  +  1/[u_r(V_NMIJ-2015)]²)  =  0.36, respectively. Equation (15) is therefore modified to

$$\begin{align} \displaystyle {{\boldsymbol{U}}_{{\rm r}\,{{\boldsymbol{V}}_{{\rm IAC-2015}}},{{\boldsymbol{V}}_{{\rm NMIJ-2017}}}}}=\boldsymbol{C}\,{{\boldsymbol{U}}_{{\rm r}\,{{\boldsymbol{V}}_{{\rm NMIJ-2015}}},{{\boldsymbol{V}}_{{\rm PTB-2015}}},{{\boldsymbol{V}}_{{\rm NMIJ-2017}}}}}{{\boldsymbol{C}}^{{\rm T}}},\nonumber \end{align} \tag{ 19 }$$

where

$$\begin{align} \displaystyle {{\boldsymbol{U}}_{{\rm r}\,{{V}_{{\rm IAC-2015}}}{\rm,}{{V}_{{\rm PTB-2015}}}{\rm,}{{V}_{{\rm NMIJ-2017}}}}}=\left(\begin{matrix} {{\left[{{u}_{{\rm r}}}\left({{V}_{{\rm NMIJ-2015}}} \right) \right]}^{2}} & {\rm Co}{{{\rm v}}_{{\rm r}}}\left({{V}_{{\rm NMIJ-2015}}},{{V}_{{\rm PTB-2015}}} \right) & {\rm Co}{{{\rm v}}_{{\rm r}}}\left({{V}_{{\rm NMIJ-2015}}},{{V}_{{\rm NMIJ-2017}}} \right) \nonumber \\ {\rm Co}{{{\rm v}}_{{\rm r}}}\left({{V}_{{\rm NMIJ-2015}}},{{V}_{{\rm PTB-2015}}} \right) & {{\left[{{u}_{{\rm r}}}\left({{V}_{{\rm PTB-2015}}} \right) \right]}^{2}} & {\rm Co}{{{\rm v}}_{{\rm r}}}\left({{V}_{{\rm PTB-2015}}},{{V}_{{\rm NMIJ-2017}}} \right) \nonumber \\ {\rm Co}{{{\rm v}}_{{\rm r}}}\left({{V}_{{\rm NMIJ-2015}}},{{V}_{{\rm NMIJ-2017}}} \right) & {\rm Co}{{{\rm v}}_{{\rm r}}}\left({{V}_{{\rm PTB-2015}}},{{V}_{{\rm NMIJ-2017}}} \right) & {{\left[{{u}_{{\rm r}}}\left({{V}_{{\rm NMIJ-2017}}} \right) \right]}^{2}} \end{matrix} \right),\nonumber \end{align} \tag{ 20 }$$

$$\begin{align} \displaystyle \boldsymbol{C}=\left(\begin{matrix} 0.64 & 0.36 & 0 \nonumber \\ 0 & 0& 1 \end{matrix} \right).\nonumber \end{align} \tag{ 21 }$$

The correlation coefficient between the volume measurements by IAC in 2015 and by NMIJ in 2017 was estimated to be 0.14.

Table 7 shows the relative contributions and correlations of uncertainty sources to the Si core mass determinations of AVO28-S5c by IAC in 2015 and by NMIJ in 2017. The Si mass obtained by IAC in 2015 was the weighted mean of the values obtained by three laboratories, BIPM, PTB and NMIJ, in which the weight of NMIJ's result was 0.19 [8]. On the other hand, the Si mass obtained by NMIJ in 2017 was determined only by NMIJ.

The uncertainty sources of the mass measurements were composed of the BIPM mass scale, the mass difference measurement, the surface layers mass and the point defect mass. The mass measurement results obtained by IAC in 2015 and by NMIJ in 2017 were both traceable to the BIPM mass scale established at the Extraordinary Calibrations from 2013 to 2014 [36, 37], which provided us the newest linkage to the IPK. The uncertainty due to this BIPM mass scale was estimated to be 3.0 µg [9] and 3.3 µg for IAC's result in 2015 and NMIJ's result in 2017, respectively, and the common systematic uncertainty between them was estimated to be 3.0 µg. Consequently, the correlation coefficient was given by (3.0 µg)²/(3.0 µg  ×  3.3 µg)  =  0.91.

The mass of the Si sphere including the surface layers was determined independently by the mass difference measurement by each laboratory by comparison with a reference weight traceable to the BIPM mass scale. The uncertainty due to this mass difference measurement was estimated to be 1.8 µg [8, 9] and 4.9 µg for IAC's result in 2015 and NMIJ's result in 2017, respectively, and the common systematic uncertainty between them was estimated to be 0.19  ×  3.1 µg  =  0.6 µg, where the value of 0.19 is the weight of NMIJ's result in IAC's result in 2015 and the value of 3.1 µg is the estimated common systematic uncertainty between NMIJ's result in 2015 and 2017. Consequently, the correlation coefficient for the mass difference measurement was estimated to be (0.6 µg)²/(1.8 µg  ×  4.9 µg)  =  0.04.

The uncertainty due to the surface layers mass was estimated to be 10.0 µg [8, 9] and 8.9 µg for IAC's result in 2015 and NMIJ's result in 2017, respectively. The same values of the density of the OL and the mass of the CWL were used for IAC's result in 2015 and NMIJ's result in 2017, and the corresponding common systematic uncertainties were estimated to be 3.4 µg and 2.2 µg [9], respectively. Consequently, the correlation coefficient for the surface layers mass was estimated to be [(3.4 µg)²  +  (2.2 µg)²]/[(10.0 µg  ×  8.9 µg]  =  0.18.

The uncertainty due to the point defect mass was estimated to be 3.8 µg [8, 9] both for IAC's result in 2015 and NMIJ's result in 2017. Since the same value of the point defect mass was used for IAC's result in 2015 and NMIJ's result in 2017, the correlation coefficient between them was estimated to be 1.

In conclusion, the correlation coefficient between the Si mass determinations by IAC in 2015 and by NMIJ in 2017 was estimated to be 0.32.

Table 8 summarizes the relative contributions and correlations of the uncertainty sources to the determination of N_A by IAC in 2015 and that by NMIJ in 2017. The correlation coefficient between the N_A determinations was estimated to be 0.28, indicating that the correlation is not particularly strong even though the same ²⁸Si sphere was used for the N_A determinations. This result is explained by the following considerations. Though the same values of M and a of AVO28-S5c were used for the N_A determinations as described in sections 7.1 and 7.2, the uncertainty contributions from a and M are small compared with those of the Si core volume and mass as shown in table 8. In addition, the covariances between many of the large uncertainty sources in the determinations of the Si core volume and mass are relatively small as shown in tables 5 and 7. The correlations between the Si core volume and mass determinations in 2015 and 2017 are therefore not particularly strong. Consequently, the correlation coefficient between the N_A determinations in 2015 and 2017 is relatively small.

The correlation between the uncertainty contributions of the lattice constant to the 2011 and 2015 N_A determinations by IAC was examined thoroughly, and the correlation coefficient was estimated to be 0.15 in [9]. As described in section 5.1, the same lattice constant was used for the N_A determinations by IAC in 2015 and by NMIJ in 2017. The correlation coefficient between the uncertainty contribution of the lattice constant to the N_A determination in 2011 and that in 2017 was therefore also estimated to be 0.15.

The correlation between the uncertainty contributions of the molar mass to the 2011 and 2015 N_A determinations was examined thoroughly, and the correlation coefficient was estimated to be 0 in [9]. As described in section 5.1, the same molar mass was used for the N_A determinations by IAC in 2015 and by NMIJ in 2017. The correlation between the uncertainty contributions of the molar mass to the N_A determinations in 2011 and 2017 was therefore also estimated to be 0.

In the 2011 N_A determination, the Si core volume of AVO28-S5 was determined independently at NMIJ and PTB. To estimate the correlation between the uncertainty contributions of the Si core volume to the N_A determinations in 2011 and 2017, the correlation coefficients of the following three combinations were estimated:

• 1.  
  Corr (V_NMIJ-2011, V_NMIJ-2017),
• 2.  
  Corr (V_PTB-2011, V_NMIJ-2017),
• 3.  
  Corr (V_NMIJ-2011, V_PTB-2011),

where V_NMIJ-2011 and V_PTB-2011 are the V_core values determined by NMIJ and PTB in 2011, respectively.

8.3.1. Correlation between V_NMIJ-2011 and V_NMIJ-2017.

8.3.2. Correlation between V_PTB-2011 and V_NMIJ-2017 and correlation between V_PTB-2011 and V_NMIJ-2011.

8.3.3. Correlation between V_IAC-2011 and V_NMIJ-2017.

Table 11 shows the relative contributions and correlations of the uncertainty sources to the Si core mass determination of AVO28-S5 by IAC in 2011 and that of AVO28-S5c by NMIJ in 2017. The Si mass obtained by IAC in 2011 was the weighted mean of the values obtained by three laboratories, BIPM, PTB and NMIJ in 2011, in which the weight of NMIJ's result was 0.25 [7]. The mass measurement result by each laboratory was revised in 2015 on the basis of the Extraordinary Calibrations from 2013 to 2014 [36, 37]. On the other hand, the Si mass for the N_A determination by NMIJ in 2017 was determined only by NMIJ.

The mass measurement results obtained by IAC in 2011 and by NMIJ in 2017 were both traceable to the BIPM mass scale established at the Extraordinary Calibrations from 2013 to 2014 [36, 37]. The uncertainty due to this BIPM mass scale was estimated to be 3.0 µg [9] and 3.3 µg for IAC's result in 2011 and NMIJ's result in 2017, respectively, and the common systematic uncertainty between them was estimated to be 3.0 µg. Consequently, the correlation coefficient was given by (3.0 µg)²/(3.0 µg  ×  3.3 µg)  =  0.91.

The mass of the Si sphere including the surface layers was independently determined by the mass difference measurement by each laboratory by comparison with a reference weight traceable to the BIPM mass scale. The uncertainty due to this mass difference measurement was estimated to be 1.8 µg [9] and 4.9 µg for IAC's result in 2011 and NMIJ's result in 2017, respectively, and the common systematic uncertainty between them was estimated to be 0.25  ×  3.1 µg  =  0.8 µg, where the value of 0.25 is the weight of NMIJ's result in IAC's result in 2011 and the value of 3.1 µg is the estimated common systematic uncertainty between NMIJ's results in 2011 and 2017. Consequently, the correlation coefficient for the mass difference measurement was estimated to be (0.8 µg)²/(1.8 µg  ×  4.9 µg)  =  0.07.

The uncertainty due to the surface layers mass was estimated to be 14.5 µg [7, 9] and 8.9 µg for IAC's result in 2011 and NMIJ's result in 2017, respectively. The same values of the density of the OL and the mass of the CWL were used for IAC's result in 2011 and NMIJ's result in 2017, and the corresponding common systematic uncertainties were estimated to be 3.4 µg and 2.2 µg [9], respectively. Consequently, the correlation coefficient for the surface layers mass was estimated to be [(3.4 µg)²  +  (2.2 µg)²]/[(14.5 µg  ×  8.9 µg]  =  0.13.

The uncertainty due to the point defect mass was estimated to be 2.4 µg [7, 9] and 3.8 µg for IAC's result in 2011 and NMIJ's result in 2017, respectively, and the common systematic uncertainty between them was estimated to be 2.4 µg. Consequently, the correlation coefficient was given by (2.4 µg)²/(2.4 µg  ×  3.3 µg)  =  0.63.

In conclusion, the correlation coefficient between the Si mass determinations by IAC in 2011 and by NMIJ in 2017 was estimated to be 0.19.

Table 12 summarizes the relative contributions and correlations of the uncertainty sources to the N_A determinations by IAC in 2011 and by NMIJ in 2017. The correlation coefficient between the two N_A determinations was estimated to be 0.07, which is smaller than that between the N_A determinations in 2015 and 2017 given in table 8. This result is explained by the following considerations. One of the largest uncertainty sources in the N_A determination in 2011 was the mass determination of the surface layers. Because of the metallic contamination layer, it was not possible to determine the mass of the surface layers accurately [7]. After the measurements in 2011, this contamination layer was removed [18]. Consequently, the correlation between the Si core mass measurements in 2011 and 2017 is smaller than that in 2015 and 2017 as shown in tables 8 and 12. In addition, the correlations between the values of a and M in 2011 and 2017 are also significantly small compared to that in 2015 and 2017 as described in sections 8.1 and 8.2. As to the determination of the Si core volume determinations, the correlation coefficient between the Si volume measurements in 2011 and 2017 is smaller than that in 2015 and 2017 as shown in tables 8 and 12 because the weight of V_NMIJ-2011 in V_IAC-2011 is smaller than that of V_NMIJ-2015 in V_IAC-2015. Consequently, the correlation coefficient between the N_A determinations in 2011 and 2017 is smaller than that between the N_A determinations in 2015 and 2017.