Data and analysis for the CODATA 2017 special fundamental constants adjustment^*

The relative atomic masses of the neutron $\rm n$, ${\hspace{0pt}}^1{\rm H}$, ${\hspace{0pt}}^2{\rm H}$ (deuterium, D), ${\hspace{0pt}}^3{\rm H}$ (tritium, T), ${\hspace{0pt}}^3{\rm He}$, ${\hspace{0pt}}^4{\rm He}$, ${\hspace{0pt}}^{28}{\rm Si}$, and ${\hspace{0pt}}^{87}{\rm Rb}$, which are data items B1, B2, B4, B6, B8, B10, B18, and B40 in table 4, are from AME2016, the most recent atomic mass evaluation from the AMDC, Lanzhou (Huang et al 2017, Wang et al 2017). AME2016 supersedes AME2012 used in CODATA-14. Since the results from the University of Washington and the Florida State University included in the 2014 adjustment as separate input data have been taken into account in AME2016, they no longer are treated separately and the values of $A_{\rm r}({\hspace{0pt}}^2{\rm H})$, $A_{\rm r}({\hspace{0pt}}^3{\rm H})$, and $A_{\rm r}({\hspace{0pt}}^3{\rm He})$ from AME2016 are used. The values above are also listed in table 8 together with those for ${\hspace{0pt}}^{36}{\rm Ar}$, ${\hspace{0pt}}^{38}{\rm Ar}$, and ${\hspace{0pt}}^{40}{\rm Ar}$. However, the argon values are not actual input data but are employed in the calculation of the molar mass of a particular argon sample used in the NIST, Gaithersburg, 1988 acoustic gas thermometry measurement of the molar gas constant R.

Table 8. Relative atomic masses used in the CODATA 2017 Special Adjustment as given in the 2016 atomic mass evaluation and the defined value for ${\hspace{0pt}}^{12}$C.

Atom	Relative atomic mass $A_{\rm r}({\rm X})$	Relative standard uncertainty ur
n	1.008 664 915 82(49)	$4.9\times 10^{-10}$
${\hspace{0pt}}^{1}$H	1.007 825 032 241(94)	$9.3\times 10^{-11}$
${\hspace{0pt}}^{2}$H	2.014 101 778 11(12)	$6.0\times 10^{-11}$
${\hspace{0pt}}^{3}$H	3.016 049 281 98(23)	$7.7\times 10^{-11}$
${\hspace{0pt}}^{3}$He	3.016 029 322 65(22)	$7.3\times 10^{-11}$
${\hspace{0pt}}^{4}$He	4.002 603 254 130(63)	$1.6\times 10^{-11}$
${\hspace{0pt}}^{12}$C	12	(exact)
${\hspace{0pt}}^{28}$Si	27.976 926 534 99(52)	$1.9\times 10^{-11}$
${\hspace{0pt}}^{36}$Ar	35.967 545 105(29)	$8.1\times 10^{-10}$
${\hspace{0pt}}^{38}$Ar	37.962 732 10(21)	$5.5\times 10^{-9}$
${\hspace{0pt}}^{40}$Ar	39.962 383 1238(24)	$6.0\times 10^{-11}$
${\hspace{0pt}}^{87}$Rb	86.909 180 5312(65)	$7.4\times 10^{-11}$

Extra digits were supplied to the Task Group by Task Group and AMDC member M. Wang to reduce rounding error. The covariances among the eight AME2016 values in table 8 used as input data are taken from the supplementary files at the AMDC website http://amdc.impcas.ac.cn/web/masseval.html and are used as appropriate in calculations; they are given in the form of correlation coefficients in table 5. However, the correlation coefficients for the argon relative atomic masses are negligible in the context in which the masses are used and are not included in the table.

Observational equations B1 and B40 in table 8 for input data $A_{\rm r}(\rm n)$ and $A_{\rm r}({\hspace{0pt}}^{87}{\rm Rb})$ are quite simple. However, observational equations B2, B4, B6, B8, B10, and B18 for the other relative atomic masses are not because their purpose is to provide values of the relative atomic masses of the proton p, deuteron d, triton t, helion h (nucleus of ${\hspace{0pt}}^3{\rm He})$, alpha particle (nucleus of ${\hspace{0pt}}^4{\rm He})$, and hydrogenic ${\hspace{0pt}}^{28}\rm Si$, respectively. Similarly, the denominator on the right-hand-side of observational equation B12 and the term in square brackets on the right-hand-side of B14 determine the relative atomic mass of the ${\hspace{0pt}}^{12}{\rm C}$ nucleus and of hydrogenic ${\hspace{0pt}}^{12}{\rm C}$, respectively. These equations follow from the fact that to produce an ion $X^{n+}$ with net charge ne, the energy required to remove n electrons from the neutral atom is the sum of the electron ionization energies $E_{\rm I}(X^{i+})$. Thus, the relative atomic mass of an atom, its ions, the relative atomic mass of the electron $A_{\rm r}(\rm e)$, and the relative-atomic-mass equivalent of the binding energy of the removed electrons $E_{\rm B}(X^{n+})/m_{\rm u}c^2$ are related according to

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle A_{\rm r} (X) = A_{\rm r}(X^{n+}) + n A_{\rm r} ({\rm e}) -\frac{\Delta E_{\rm B}(X^{n+})}{{m_{\rm u} c^2}}, \label{eq:araxn} \nonumber \end{align} \tag{ 1 }$$

where

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{\Delta E_{\rm B}(X^{n+})}{{m_{\rm u} c^2}} = \frac{\alpha^2A_{\rm r}({\rm e})}{2R_\infty}\, \frac{\Delta E_{\rm B}(X^{n+})}{hc}, \label{eq:ebxn} \nonumber \end{align} \tag{ 2 }$$

and where $m_{\rm u} = m({^{12}{\rm C}})/12$ is the atomic mass constant and α is the fine-structure constant. Equation (2) accounts for the fact that the binding energies are known most accurately in terms of their wavenumber equivalents and follows from the relations $m_{\rm e} = A_{\rm r}({\rm e})m_{\rm u}$ and $R_\infty = \alpha^2 m_{\rm e}c/2h$.

All eight removal energies used in the 2017 Special Adjustment, items B3, B5, B7, B9, B11, B13, B15, and B19 in table 4, are obtained from the ionization energies given in the current version of the NIST online Atomic Spectra Database (Kramida et al 2017). The only change since 2014 is that for the ionization energy ${\hspace{0pt}}^{12}{\rm C}^{3+}$; the last digit is now 3 rather than 8 (see table III in CODATA-14). However, the correlation coefficient of the ${\hspace{0pt}}^{12}{\rm C^{6+}}$ and ${\hspace{0pt}}^{12}{\rm C^{5+}}$ removal energies is unchanged from its CODATA-14 value 1.000 and is included in table 5.

Item B12 in table 4 with identification MPIK-17, the ratio of the cyclotron frequency of the ${\hspace{0pt}}^{12}{\rm C^{6+}}$ ion to that of the proton in the same magnetic flux density B, is a new result obtained at MPIK, Heidelberg, and reported by Heiße et al (2017). Its observational equation, B12 in table 7, shows that, like input datum B2 with observational equation B2, the frequency ratio $\omega_{\rm c}({\hspace{0pt}}^{12}{\rm C^{6+}})/\omega_{\rm c}({\rm p})$ determines $A_{\rm r}(\rm p)$.

The MPIK-17 datum was obtained using a cryogenic Penning trap optimized for mass measurements of light ions. It has a measurement trap on either side of which is a storage trap. A proton stored in one and a ${\hspace{0pt}}^{12}{\rm C^{6+}}$ ion in the other are alternately shuttled back and forth into the measurement trap where its cyclotron frequency is determined. In parts in $10^{11}$, the statistical relative standard uncertainty of the measured frequency ratio is 1.52 and the net correction for various systematic effects is 3.82 with an uncertainty of 2.89. The largest contributor to the latter uncertainty is 2.75 due to an effect termed residual magnetostatic inhomogeneity and the largest correction, 9.10, is for image charge.

As discussed in section 4, the uncertainty of $A_{\rm r}(\rm p)$ obtained from $\omega_{\rm c}({\hspace{0pt}}^{12}{\rm C^{6+}})/\omega_{\rm c}({\rm p})$ is about $1/3$ that of $A_{\rm r}(\rm p)$ obtained from $A_{\rm r}({\hspace{0pt}}^1{\rm H})$ of AME2016, but the two values disagree. The MPIK researchers were aware of this and performed measurements on other ions that confirmed literature values; they were unable to identify any systematic effects in their method that would explain the inconsistency.

Data item B14 in table 4 is the ratio of the spin-precession (spin-flip) frequency of the ${\hspace{0pt}}^{12}{\rm C}^{5+}$ hydrogenic ion to its cyclotron frequency in the same applied magnetic flux density B; and data item B17 is the similar ratio for the hydrogenic ion ${\hspace{0pt}}^{28}{\rm Si}^{13+}$. They are correlated with a correlation coefficient of 0.347, and the experiments on which they are based are discussed in CODATA-14. These ratios, which were obtained at MPIK, Heidelberg, together with the theory of the electron bound-state g-factor in the C and Si hydrogenic ions, provide a value for the electron relative atomic mass $A_{\rm r}({\rm e})$ with u_r of a few parts in $10^{11}$.

The observational equations for the ${\hspace{0pt}}^{12}{\rm C}^{5+}$ and ${\hspace{0pt}}^{28}{\rm Si}^{13+}$ frequency ratios are B14 and B17 in table 7. In those equations $g_{\rm C}({\alpha}) + \delta_{\rm C}$ and $g_{\rm Si}({\alpha}) + \delta_{\rm Si}$ represent the theoretical expressions of the bound-state g-factors $g({\hspace{0pt}}^{12}{\rm C}^{5+})$ and $g({\hspace{0pt}}^{28}{\rm Si}^{13+})$. The theory of these g-factors is discussed in CODATA-14 and the only changes are as follows: (i) the remainder terms $R_{\rm SE}(6\alpha) = 22.160(10)$ and $R_{\rm SE}(14\alpha) = 20.999(2)$, which are equations (155) and (156) in CODATA-14, are replaced by the improved values $R_{\rm SE}(6\alpha) = 22.166(1)$ and $R_{\rm SE}(14\alpha) = 21.0005(1)$ obtained by Yerokhin and Harman (2017); and (ii) an additional term from light-by-light scattering diagrams calculated by Czarnecki and Szafron (2016) has been included in equation (164) of CODATA-14 for the $C_{\rm e}^{(4)}(Z\alpha)$ coefficient. This contribution is similar to that discussed in section 3.2 in connection with the changes in the H and D energy level theory

The uncertainties of the additive corrections to the carbon and silicon theory, $\delta_{\rm C}$ and $\delta_{\rm Si}$, items B7 and B11 in table 4, are nearly the same as their CODATA-14 values as is their covariance and hence correlation coefficient; these are now $u(\delta_{\rm C}, \delta_{\rm Si}) = 3.4\times10^{-20}$ and $r(\delta_{\rm C}, \delta_{\rm Si}) = 0.71$. The observational equations for the carbon and silicon additive corrections, B7 and B11 in table 7, are simply $\delta_{\rm C} \doteq \delta_{\rm C}$ and $\delta_{\rm Si} \doteq \delta_{\rm Si}$.

It is worth noting that the value of $A_{\rm r}({\rm e})$ recently obtained by Zatorski et al (2017) from their treatment of the MPIK frequency ratios and review of the g-factor theory and the value resulting from our treatment are essentially identical.

The most accurate value of α is obtained by equating the experimental value of the electron magnetic moment anomaly a_e and its theoretical expression. Item B21 in table 4 with $u_{\rm r} = 2.4\times10^{-10}$ is the experimental value of a_e with the smallest uncertainty achieved to date. Discussed in CODATA-10 and used in both the 2010 and 2014 adjustments, it was obtained at Harvard University, Cambridge, using a Penning trap (Hanneke et al 2008); no other result is competitive.

As discussed in CODATA-14, the theoretical expression for a_e can be written as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle a_{\rm e}({\rm th}) = a_{\rm e}({\rm QED}) + a_{\rm e}({\rm weak}) + a_{\rm e}({\rm had}). \label{eq:aeth} \nonumber \end{align} \tag{ 3 }$$

The electroweak and hadronic contributions are small compared with the quantum electrodynamic contribution $a_{\rm e}({\rm QED})$. The latter is currently written as the sum of five terms of the form $C_{\rm e}^{(2n)}(\alpha/\pi){\hspace{0pt}}^n$ with $n = 1, 2, 3, 4,$ and 5, since terms for $n > 5$ are assumed to be negligible at the level of uncertainty of $a_{\rm e}({\rm exp})$. Various terms dependent on the mass ratios m_e/m_µ and m_e/m_τ contribute to the coefficients $C_{\rm e}^{(2n)}$ except for $n = 1$. Following CODATA-14 and using the 2017 adjusted values of these mass ratios, the five coefficients are

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle C_{\rm e}^{(2)} &= 0.5, \nonumber \\ C_{\rm e}^{(4)} &= -0.328\,478\,444\,00\ldots, \nonumber \\ C_{\rm e}^{(6)} &= 1.181\,234\,017\ldots, \nonumber \\ C_{\rm e}^{(8)} &= -1.911\,322\,138\,91(88), \nonumber \\ C_{\rm e}^{(10)} &= 6.60(22), \label{eq:appaecs} \nonumber \end{align} \tag{ 4 }$$

where two new results have been incorporated in $C_{\rm e}^{(8)}$ and $C_{\rm e}^{(10)}$. First, in the mass-independent contribution to $a_{\rm e}({\rm QED}$), the numerically calculated eighth-order coefficient used in CODATA-14 due to Aoyama et al (2015), $A_1^{(8)} = -1.912\, 98(84)$, is replaced with the semi-analytical value $A_1^{(8)} = -1.912\, 245\, 764\ldots$ obtained by Laporta (2017). This coefficient depends on 891 4-loop Feynman diagrams and its evaluation is difficult; Laporta's result with 1100 digit accuracy is the culmination of a 20-year effort. It is noteworthy that the numerically calculated result differs from its essentially exact counterpart by less than $0.9\sigma$.

Second, and again in the mass-independent contribution to $a_{\rm e}({\rm QED}$), the numerically calculated tenth-order coefficient employed in CODATA-14, $A_1^{(10)} = 7.795(336)$, also due to Aoyama et al (2015), is replaced with $A_1^{(10)} = 6.599(223)$. This newly revised result published as an erratum (Aoyama et al 2017) is a consequence of their correction of an error discovered in a portion of their previous calculation and additional numerical evaluations. The coefficient $A_1^{(10)}$ depends on 12 672 Feynman diagrams and its determination is formidable.

The sum of $a_{\rm e}({\rm weak})$ given in CODATA-14 and the values of the various corrections that contribute to $a_{\rm e}({\rm had})$ also given there is $1.735(15)\times 10^{-12}$ and is the value used in the 2017 Special Adjustment. It is consistent with the value $1.723(11)\times 10^{-12}$ obtained by Jegerlehner (2017) based on his own independent evaluation of each contribution and employing up-to-date experimental data.

The uncertainties of the coefficients $C_{\rm e}^{(8)}$ and $C_{\rm e}^{(10)}$ together with those of the weak and hadronic contributions yield for the standard uncertainty of the theoretical value $u[a_{\rm e}({\rm th})] = 0.021 \times 10^{-12}$. The theoretical expression for a_e is written as $a_{\rm e}({\rm th}) = a_{\rm e}(\alpha) + \delta_{\rm e}$, where the additive correction $\delta_{\rm e}$ is equal to $0.000(21)\times 10^{-12}$ and is input datum B22 in table 4. The observational equation for a_e is B21 in table 7 and the observational equation for the additive correction is simply $\delta_{\rm e} \doteq \delta_{\rm e}$, which is equation B22 in table 7. The value of α that results from equating the Harvard experimental value and $a_{\rm e}({\rm th})$ is given in table 9.

Table 9. Inferred values of the fine-structure constant α in order of increasing standard uncertainty obtained from the indicated experimental data in table 4.

Primary source	Item number	Identification	$\alpha^{-1}$	Relative standard uncertainty ur
ae	B21	HarvU-08	$137.035\, 999\, 150(33)$	$2.4\times 10^{-10}$
$h/m({\hspace{0pt}}^{87}{\rm Rb})$	B39	LKB-11	$137.035\, 998\, 995(85)$	$6.2\times 10^{-10}$

The muon magnetic moment anomaly a_µ is required to obtain m_e/m_µ from measurements involving muonium (see next section). As discussed in CODATA-14, the experimental value of a_µ is derived from the BNL, Upton, measurement of the quantity $\bar R = f_{\rm a}/\bar f_{\rm p}$, which is item B23 in table 4. Here f_a is the anomaly difference frequency equal to the difference between the muon spin-flip (precession) frequency and cyclotron frequency in the same applied magnetic flux density B, and $\bar f_{\rm p}$ is the free proton nuclear magnetic resonance (NMR) frequency corresponding to the average B seen by the circulating muons in their storage ring. The observational equation B23 in table 7 for the input datum $\bar R$ shows its relationship with the adjusted constants a_µ and m_e/m_µ as well as others.

The theoretical expression for a_µ is omitted from both CODATA-10 and 14 because of concerns regarding the theory. Inasmuch as these concerns remain, the theory is also omitted from the 2017 Special Adjustment. However, its omission has no effect on the chosen numerical values of h, e, k, and N_A.

Determination of the ground-state hyperfine-splitting of muonium (µ⁺ e atom) together with the theory of the splitting determines m_e/m_µ, which in turn enters the theory of the electron and muon magnetic moment anomalies. The available experimental data are from two experiments carried out at LAMPF, Los Alamos, and reported in 1982 and 1999.

In these experiments two Zeeman transition frequencies are measured with the muonium atoms in an applied magnetic flux density B known in terms of a free proton NMR frequency f_p. The sum of the two frequencies is the hyperfine splitting frequency $\Delta\nu_{\rm Mu}$ and their difference is a frequency denoted $\nu(\,f_{\rm p})$. The experiments are discussed in CODATA-98 and there have been no changes in the data since then. The two 1982 and two 1999 frequencies are items B24 and $B26.1$, and B25 and $B26.2$, in table 4. Their observational equations are $B24, B25$ and B26 in table 7. For each experiment the two frequencies are correlated; the correlation coefficients are $r[\Delta\nu_{\rm Mu}, \, \nu(\,f_{\rm p})] = 0.227$ for the 1982 data and $r[\Delta\nu_{\rm Mu}, \, \nu(\,f_{\rm p})] = 0.195$ for the 1999 data and are taken into account in calculations. For the 1982 data $f_{\rm p} = 57.972\, 993\, {\rm MHz}$ corresponding to B of about $1.4\, {\rm T}$ and for the 1999 data $f_{\rm p} = 72.320\, 000\, {\rm MHz}$ corresponding to B of about  =  $1.7\, {\rm T}$.

There has been no change in the theory since the 31 December 2014 closing date of CODATA-14, hence the uncertainty $u(\delta_{\rm Mu}) =85\, {\rm Hz}$ of the additive correction $\delta_{\rm Mu}$ used in the 2014 adjustment to account for the uncertainty of the theory is unchanged. The correction $\delta_{\rm Mu}$ is item B27 in table 4 and its observational equation is simply $\delta_{\rm Mu} \doteq \delta_{\rm Mu}$, which is equation B27 in table 7.

Discussed in this section are the six input data B28-B35 in table 4 with observational equations B28-B35 in table 7. Some of these data contribute to the determination of the adjusted constant $\mu_{\rm e}/\mu_{\rm p}$, which in turn contributes to the determination of the adjusted constant m_e/m_µ. The latter is required for the theoretical expression of a_e and a_µ. In item B28, $\mu_{\rm N}= e\hbar/2m_{\rm p}$ is the nuclear magneton, and in items B29-B31 and $B34.1$ - B35, $\mu_X(Y)$ is the magnetic moment of particle X in atom or molecule Y. These data have been discussed in earlier CODATA reports and all are used in CODATA-14. The exception is input data B36 and B37 for adjusted constants $\sigma_{\rm dp} = \sigma_{\rm d}({\rm HD}) - \sigma_{\rm p}({\rm HD})$ and $\sigma_{\rm tp} = \sigma_{\rm t}({\rm HT}) - \sigma_{\rm p}({\rm HT})$, the differences between the magnetic shielding correction of the deuteron and of the proton in the HD molecule and of the triton and of the proton in the HT molecule. They are the improved theoretical values reported by Puchalski et al (2015) and although discussed in CODATA-14, they were not included in the 2014 adjustment because they became available only after its 31 December 2014 closing date.

The observational equations in table 7 for items B29 to B31 contain four bound-particle to free particle ratios of the form $g_X(Y)/g_X$, where X is either e, p, or d, and Y is either H or D. These ratios are calculated theoretically and have sufficiently small uncertainties in the context in which they are used to be taken as exact. Their values have not changed since CODATA-02 and are given in table XIII of CODATA-14.

The watt balance was conceived by Bryan Kibble of NPL, Teddington, in 1975. With the discovery of the quantum-Hall effect in 1980 it became a means of measuring h. Upon the passing of Kibble in April 2016 the Consultative Committee for Units (CCU) of the CIPM decided to honor him by renaming the 'watt balance' the 'Kibble balance' and this is the name used herein (CCU 2016).

How the Kibble balance determines h is discussed in CODATA-98, and values of h so obtained have been included in all CODATA adjustments ever since. The five values considered for inclusion in the CODATA 2017 Special Adjustment are items $B38.1$-$B38.5$ in table 4.

The NIST-98 value of h, item $B38.1$, was obtained by Williams et al (1998) using a Kibble balance called NIST-2 that operated in air and used a superconducting magnet to generate the required magnetic flux density B; it is discussed in CODATA-98, but see also CODATA-06. NIST-2 was replaced by a new Kibble balance called NIST-3 that operated in vacuum but used the same superconducting magnet. The value of h obtained from this apparatus with $u_{\rm r} = 3.6\times10^{-8}$, identified as NIST-07, was taken as an input datum in both the 2006 and 2010 adjustments. However, as discussed in CODATA-14, issues with the NIST-07 result were uncovered that led to a number of improvements in the NIST-3 apparatus, additional data, and a thorough review of all previous NIST-3 data. This led to the value identified as NIST-15, item $B38.2$ in table 4. As reported by Schlamminger et al (2015), it is viewed as the final result from NIST-3 and is used as an input datum in CODATA-14. The NIST-98 and NIST-15 values are correlated with a correlation coefficient of 0.09.

NIST-3 was replaced by a completely new Kibble balance called NIST-4. It uses a permanent magnet rather than a superconducting magnet to generate B but as in NIST-2 and NIST-3, a wheel is used as the balance beam and to displace the moving coil. Haddad et al (2016) discuss the new apparatus, measurement procedures, and sources of uncertainty and report a first result with $u_{\rm r} = 3.4 \times 10^{-8}$ based on data obtained from mid-December 2015 to early January 2016 using a $1\, {\rm kg}$ 90% Pt-10% Ir mass standard. After obtaining this first result the NIST researchers continued improving NIST-4 and acquiring additional data. Based on all 1174 individual determinations of h obtained from mid-December 2015 through April 2017, Haddad et al (2017) report the result identified as NIST-17, item $B38.3$ in table 4, with $u_{\rm r} = 1.3 \times 10^{-8}$. Each measurement was carried out with one of five different 0.5 kg stainless steel mass standards, a 1 kg stainless steel standard, or one of two 1 kg Pt-Ir standards, and various combinations to provide standards in the range 0.5 kg to 2 kg.

The uncertainty of the initial NIST-4 result was reduced by more than a factor of 2.5 by (i) making significantly more measurements of h, thus allowing a more realistic statistical analysis of the data; (ii) using mass standards as large as 2 kg to more thoroughly investigate the effect of the current in the moving coil on the magnet generating the magnetic flux density B; and (iii) carefully investigating the effect of coil velocity on the measurements, thereby enabling reduction of the uncertainty due to time-dependent leakage of current in the coil. As a result of these advances, the 24.9 parts in 10⁹ statistical uncertainty and 15.4 parts in 10⁹ magnetic field uncertainty of the first result are reduced to 3.0 and 1.8 parts in 10⁹, respectively. The main contributors to the uncertainty of the 2017 result are, in parts in 10⁹, 6.2 for electrical, 6.1 for mass metrology, 5.0 for magnetic field profile fitting, 5.0 for balance mechanics, 4.7 for alignment, and 4.3 for the local gravitational acceleration.

Item $B38.4$ in table 4 from NRC, Ottawa, identified as NRC-17 and reported by Wood et al (2017), was obtained using the NRC Kibble balance. The history of this balance, which was originally the NPL, Teddington, Mark II balance but was transferred to NRC in the summer of 2009, is discussed in CODATA-14. That discussion covers the measurements made with it prior to its transfer, the many significant improvements made to it at NRC, and the result identified as NRC-14 with $u_{\rm r} = 1.8\times10^{-8}$ used as an input datum in CODATA-14. This value was based on four measurement campaigns carried out between September and December 2013 using four different mass standards.

The new NRC-17 result is based on a reanalysis of the 2013 measurements and three new determinations carried out in February, November, and December 2016 with the same 500 g silicon, 1 kg AuCu, and 500 g AuCu mass standards used in 2013. However, additional improvements were made to the apparatus between the 2013 and 2016 campaigns. The reported final result, which has the smallest uncertainty of any value of h available to date, was obtained by combining all seven values from the seven measurement campaigns taking into account the covariances among them from the 58 uncertainty components that contribute to the uncertainty of each value. The procedure used is the same as used by the Task Group to carry out least-squares adjustments as discussed in appendix E of CODATA-98. A previously unidentified systematic effect, the hysteresis of the magnetization of the permanent magnet used to generate B due to the current in the required moving coil of a Kibble balance, was quantified and a correction that averaged $3.8$ parts in 10⁹ was applied to each of the seven values. The total uncertainty is not dominated by any one component.

Item $B38.5$, reported by Thomas et al (2017) and identified as LNE-17, is the most recent and accurate result from the LNE, Trappes, Kibble-balance project initiated in 2001. A first result reported by Thomas et al (2015) with $u_{\rm r} = 3.1 \times 10^{-7}$ was obtained in 2014 and an improved result with $u_{\rm r} = 1.4 \times 10^{-7}$ in 2016 in the course of the BIPM's Extraordinary Calibration Campaign. Improvements in the apparatus and measurement procedures continued, including better calibration of the 500 g pure iridium mass standard used in the experiment's weighing phase, reduction of the Type A (statistical) uncertainty, better alignment of the balance's various laser beams, and use of a programmable Josephson effect voltage standard. The LNE-17 result was obtained over a 35 d period during the spring of 2017.

An important feature of the LNE Kibble balance is that instead of having the beam of the balance used in the weighing phase also move the coil in the dynamic phase, the entire balance and suspended coil are moved by a translation stage. In contrast to the NIST and NRC Kibble balances that are operated in vacuum, to date the LNE balance has only been operated in air. The 4.7 parts in 10⁸ uncertainty of the required correction for the refractive index of air and air buoyancy is the dominant contributor to the $5.7$ parts in 10⁸ total uncertainty.