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

The special least-squares adjustment of the values of the fundamental constants, carried out by the Committee on Data for Science and Technology (CODATA) in the summer of 2017, is described in detail. It is based on all relevant data available by 1 July 2017. The purpose of this adjustment is to determine the numerical values of the Planck constant h , elementary charge e , Boltzmann constant k , and Avogadro constant N A for the revised SI expected to be established by the 26th General Conference on Weights and Measures when it convenes on 13 – 16 November 2018.


Introduction
The 26th General Conference on Weights and Measures (CGPM) is expected to establish at its meeting on 13-16 November 2018 a revised International System of Units (SI) based in part on exact numerical values of the Planck constant h, elementary charge e, Boltzmann constant k, and Avogadro constant N A . To this end, at the invitation of the CGPM (CGPM 2011), the Committee on Data for Science and Technology (CODATA), through its Task Group on Fundamental Constants (the Task Group), carried out a special least-squares adjustment of the values of the fundamental physical constants during the summer of 2017. The required numerical values of h, e, k, and N A resulting from that adjustment are reported by the Task Group (Newell et al 2018). Here we describe the data and analysis underlying the 2017 Special Adjustment.
In keeping with the request of the International Committee for Weights and Measures (CIPM), data to be considered for the adjustment had to be published in a refereed journal or accepted for publication by 1 July 2017 (CIPM 2015). The procedures used for the 2017 Special Adjustment are the same as those used for the 2014 regularly scheduled CODATA adjustment and its 1998, 2002, and 2010predecessors (Mohr and Taylor 2000, 2005, Mohr et al 2008a, 2008b, 2012a, 2012b, 2016a, 2016b.
the input data to the adjusted constants are called observational equations. The standard uncertainties of the input data and the covariances among the data, if any, are also required since they determine the weights of the input data.
An adjustment is characterized by the number of input data N, number of adjusted constants M, degrees of freedom ν = N − M, statistic χ 2 , probability p (χ 2 |ν) of obtaining an observed value of χ 2 by chance that large or larger for the given value of ν, and Birge ratio R B = χ 2 /ν .
The normalized residual of the ith input datum r i = (x i − x i )/u i , where x i is the input datum, x i its adjusted value, and u i its standard uncertainty, is a measure of how well that datum agrees with other data. A value larger than 2 is generally viewed as a cause for concern and may require increasing the uncertainties of some input data by a multiplicative expansion factor.
The self-sensitivity coefficient S c of an input datum is a measure of the influence of that datum on the adjustment. As discussed in CODATA-98, it should be no less than 0.01, or 1%, for the input datum to be included in the final adjustment on which recommended values of constants are based. However, the exclusion rule for a datum is not followed if, for example, a datum with S c < 0.01 is part of a group of data obtained in a given experiment where most of the other data have self-sensitivity coefficients > 0.01. This 1% cutoff implies the input datum's uncertainty should be no more than about a factor of 10 larger than the uncertainty of the adjusted value of the quantity of which the datum is a particular example. That uncertainty follows from the datum's observational equation evaluated with the least-squares adjusted values of the adjusted constants on which the observational equation depends (see CODATA-98, column 1 of p 358 and of p 483 for further discussion).

Review of data
A list of the Symbols and Abbreviations used in this report is given at its end prior to the list of references. This list should be consulted as needed since to keep this report concise, the listed items are in general not fully defined in the text. Further, for the same reason, we sometimes refer the reader to a particular table in CODATA-14 rather than repeat it in this paper.
As in past CODATA adjustments, the uncertainty of a theor etical expression is taken into account by including in the expression an additive correction δ. Each such δ is taken as an adjusted constant and is included as an input datum with initial value zero but with an uncertainty equal to that of its corresponding theoretical expression. If corrections are correlated due to common uncertainty sources, the correlations are taken into account.
The purpose of the CODATA 2017 Special Adjustment is to obtain best numerical values for h, e, k, and N A , not to provide a complete set of recommended values to replace the 2014 set. Nevertheless, to maintain the continuity of CODATA adjustments and to simplify the comparison of the CODATA 2017 Special Adjustment with the CODATA 2014 regular adjustment, the starting point of the 2017 Special Adjustment is the input data included in the final adjustment on which the 2014 CODATA recommended values are based. Consequently, the 22 input data omitted from that final adjustment because of their low weight are not considered for inclusion in the 2017 Special Adjustment. The only other data not considered are four cyclotron frequency ratios that have been superseded by a new atomic mass evaluation, and measurements of the Newtonian constant of gravitation G. The latter data are treated separately in CODATA adjustments because there is no known relationship between G and any other constant, and because the G data are independent of all other input data. Although our data review focuses on the new data, the data used in the 2014 final adjustment are duly noted. For details, including references, CODATA-14 and earlier CODATA reports should be consulted; in general only references to new work are given in this report. Table 1 recalls a number of familiar, exactly-known constants relevant to the 2017 Special Adjustment. It is subject to modification after redefinition of the SI.

Data that determine the Rydberg constant R ∞
The Rydberg constant, which has one of the smallest relative uncertainties u r of any measured constant, plays an important role in the 2017 Special Adjustment. It is obtained by equating measured transition frequencies in hydrogen (H) and Deuterium (D) with their theoretical expressions. Table 2 gives the principal input data that determine the 2017 Special Adjustment value of R ∞ and table 3 gives their correlation coefficients. The adjusted constants in terms of which the R ∞ input data in table 2 are expressed are given in table XXV of CODATA-14 and the observational equation for  each input datum may be found in table XXIII of CODATA-14; thus they need not be repeated here. It should be recalled that the symbol . = in an observational equation indicates that an input datum on the left-hand side is ideally given by the expression on the right-hand side containing adjusted constants. But because the equation is one of an overdetermined set that relates a datum to adjusted constants, the two sides are not necessarily equal.
The discussion of the principal data that determine R ∞ in table 2 follows the order in which they are listed. Items A1-A25 are the additive corrections for the theoretical expressions of the indicated H and D energy levels and items A26.1-A48 are the experimentally measured H and D trans ition frequencies. Items A49 and A50 are the root-mean-square (rms) electric charge radii of the proton r p and deuteron r d required by the theory and obtained from elastic electron scattering data. Differences between table 2 and the corre sponding table XVI  in CODATA-14 (and hence to some extent between table 3 and  its corresponding table XVII in CODATA-14) are as follows.
There are a few changes in the theory of the H and D energy levels as given in CODATA-14 but none of major consequence; none of the uncertainties of the additive corrections to the theory, items A1-A25 of table 2, have changed significantly from their 2014 values. This also applies to their correlation coefficients given in table 3. The changes in the theory are (i) incorporation of the δ 2 B 60 contribution to b L in equation (67) of CODATA-10 based on our reanalysis of the paper by Jentschura (2006); (ii) inclusion of improved relativistic nuclear recoil corrections and the effect of finite nuclear size on them calculated by Yerokhin and Shabaev (2016) and Yerokhin and Shabaev (2015); and (iii) addition of a contribution from light-by-light scattering diagrams to the coefficient B 61 in equation (55) of CODATA-14 calculated by Czarnecki and Szafron (2016) that reduces the ν H (1S 1/2 − 2S 1/2 ) theoretical transition frequency by about 281 Hz.
The only new frequency in table 2 is item A27.2 for the ν H (1S 1/2 − 3S 1/2 ) transition from the group at the MPQ, Garching, as reported by Yost et al (2016). The frequency was determined using two-photon direct frequency comb spectroscopy, or DFCS. This technique employs the high peak intensities of the ultra-short pulse trains of an optical comb to generate the required short-wavelength radiation, in this case 205 nm. However, in contrast to two-photon spectr oscopy using counter propagating continuous-wave (cw) radiation to eliminate first-order Doppler broadening, in DFCS such broadening is not completely eliminated because of varying frequency across the pulses, or 'chirp'. The experimenters thoroughly investigated this systematic effect and developed techniques to mitigate it. The net correction to their initially measured frequency is 80.9 kHz and the three largest uncertainty components contributing to the 17 kHz total uncertainty are, in kHz, 13 for second-order Doppler broadening, 8.4 for the frequency measurement, and 6.3 for pressure shift.   a The values in brackets are relative to the quantities g( 12 C 5+ ), g( 28 Si 13+ ), a e , or ∆ν Mu as appropriate.   The electron-proton and electron-deuteron scattering values of the proton and deuteron rms radii, items B49 and B50, are the same as used in CODATA-14. Because the values of h, e, k, and N A resulting from the CODATA 2017 Special Adjustment are essentially independent of r p and r d , the values chosen are not critical. This is shown in section 5.1, which discusses the result of using values for these radii obtained from measurements of the Lamb shift in muonic hydrogen and muonic deuterium. Table 4 gives the principal input data that determine the 2017 Special Adjustment values of constants other than R ∞ and table 5 gives their correlation coefficients. Many of the data in table 4 are the same as (or fully equivalent to) those in the corresponding table XVIII of CODATA-14. The adjusted constants in terms of which the input data in table 4 are expressed are given in table 6 and their observational equations are given in table 7. As for the Rydberg constant data, the discussion of these data follows the order in which they are listed in table 4.

Atomic masses and binding energies.
The relative atomic masses of the neutron n, 1 H, 2 H (deuterium, D), 3 H (tritium, T), 3 He, 4 He, 28 Si, and 87 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 . 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 r ( 2 H), A r ( 3 H), and A r ( 3 He) from AME2016 are used. The values above are also listed in table 8 together with those for 36 Ar, 38 Ar, and 40 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.
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 calcul ations; 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 r (n) and A r ( 87 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 3 He), alpha particle (nucleus of 4 He), and hydrogenic 28 Si, respectively. Similarly, the denominator on the right-hand-side of observational equation B12 and the term in square brackets on the righthand-side of B14 determine the relative atomic mass of the 12 C nucleus and of hydrogenic 12 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 I (X i+ ). Thus, the relative atomic mass of an atom, its ions, the relative atomic mass of the electron A r (e), and the relativeatomic-mass equivalent of the binding energy of the removed electrons E B (X n+ )/m u c 2 are related according to Table 6. Variables used in the least-squares adjustment of the constants. They are arguments of the functions on the right-hand side of the observational equations in table 7.

Adjusted constant Symbol
Neutron relative atomic mass A r (n) Electron relative atomic mass A r (e) Proton relative atomic mass δ e Muon magnetic moment anomaly a µ Electron-muon mass ratio m e / m µ Electron-proton magnetic moment ratio µ e /µ p Additive correction to ∆ν Mu (th) δ Mu Deuteron-electron magnetic moment ratio µ d /µ e Shielded helion to shielded proton magnetic moment ratio µ h /µ p Neutron to shielded proton magnetic moment ratio µ n /µ p Shielding difference of d and p in HD σ dp Triton-proton magnetic moment ratio µ t /µ p Shielding difference of t and p in HT σ tp Electron to shielded proton magnetic moment ratio µ e /µ p Planck constant h 87  Type of input datum Observational equation

Rb relative atomic mass
where and where m u = m( 12 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 e = A r (e)m u and R ∞ = α 2 m 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 12 C 3+ ; the last digit is now 3 rather than 8 (see table III in CODATA-14). However, the correlation coefficient of the 12 C 6+ and 12 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 12 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, 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 12 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 r (p) obtained from ω c ( 12 C 6+ )/ω c (p) is about 1/3 that of A r (p) obtained from A r ( 1 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.

Hydrogenic carbon and silicon g-factors.
Data item B14 in table 4 is the ratio of the spin-precession (spin-flip) frequency of the 12 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 28 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 r (e) with u r of a few parts in 10 11 .
The observational equations for the 12 C 5+ and 28 Si 13+ frequency ratios are B14 and B17 in table 7. In those equations g C (α) + δ C and g Si (α) + δ Si represent the theoretical expressions of the bound-state g-factors g( 12 C 5+ ) and g( 28 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 SE (6α) = 22.160(10) and R SE (14α) = 20.999(2), which are equations (155) and (156) in CODATA-14, are replaced by the improved values R SE (6α) = 22.166(1) and R SE (14α) = 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 (4) e (Zα) 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, δ C and δ 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(δ C , δ Si ) = 3.4 × 10 −20 and r(δ C , δ Si ) = 0.71. The observational equations for the carbon and silicon additive corrections, B7 and B11 in table 7, are simply δ C . = δ C and δ Si . = δ Si . It is worth noting that the value of A r (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.

Electron and muon magnetic moment anomalies.
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 r = 2.4 × 10 −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 a e (th) = a e (QED) + a e (weak) + a e (had). (3) The electroweak and hadronic contributions are small compared with the quantum electrodynamic contribution a e (QED). The latter is currently written as the sum of five terms of the form C (2n) e (α/π) 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 e (exp). Various terms dependent on the mass ratios m e /m µ and m e /m τ contribute to the coefficients C (2n) e except for n = 1. Following CODATA-14 and using the 2017 adjusted values of these mass ratios, the five coefficients are where two new results have been incorporated in C (8) e and C e . First, in the mass-independent contribution to a e (QED), the numerically calculated eighth-order coefficient used in CODATA-14 due to Aoyama et al (2015), A  (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 0.9σ.
Second, and again in the mass-independent contribution to a e (QED), the numerically calculated tenth-order coefficient employed in CODATA-14, A (10) 1 = 7.795(336), also due to Aoyama et al (2015), is replaced with A (10) 1 = 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 (10) 1 depends on 12 672 Feynman diagrams and its determination is formidable.
The sum of a e (weak) given in CODATA-14 and the values of the various corrections that contribute to a e (had) also given there is 1.735(15) × 10 −12 and is the value used in the 2017 Special Adjustment. It is consistent with the value 1.723(11) × 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 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 R = f a /f 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 f 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 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 .
3.3.4. Muonium hyperfine splitting. 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 magn etic 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 ∆ν Mu and their difference is a frequency denoted ν( f p There has been no change in the theory since the 31 December 2014 closing date of CODATA-14, hence the uncertainty u(δ Mu ) = 85 Hz of the additive correction δ Mu used in the 2014 adjustment to account for the uncertainty of the theory is unchanged. The correction δ Mu is item B27 in 3.3.5. Magnetic moment ratios. 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 µ e /µ 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, µ N = e /2m p is the nuclear magneton, and in items B29-B31 and B34.1 -B35, µ 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 σ dp = σ d (HD) − σ p (HD) and σ tp = σ t (HT) − σ p (HT), the differences between the magnetic shielding correction of the deuteron and of the proto n 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  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 r = 3.6 × 10 −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 r = 3.4 × 10 −8 based on data obtained from mid-December 2015 to early January 2016 using a 1 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 r = 1.3 × 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 10 9 statistical uncertainty and 15.4 parts in 10 9 magnetic field uncertainty of the first result are reduced to 3.0 and 1.8 parts in 10 9 , respectively. The main contributors to the uncertainty of the 2017 result are, in parts in 10 9 , 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 r = 1.8 × 10 −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 10 9 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 r = 3.1 × 10 −7 was obtained in 2014 and an improved result with u r = 1.4 × 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 10 8 uncertainty of the required correction for the refractive index of air and air buoyancy is the dominant contributor to the 5.7 parts in 10 8 total uncertainty. Li et al (2017) have reported a first result for h from the NIM, Beijing, joule balance NIM-2 with u r = 2.4 × 10 −7 and in agreement with other values of h. The joule balance measures h in a manner similar to the Kibble balance but compares electric energy with gravitational potential energy rather than electric power with mechanical power. However, u r of the result is too large for it to be considered for inclusion in the 2017 Special Adjustment, inasmuch as values of h with smaller u r were not sufficiently competitive to be included in the final adjustment on which the CODATA 2014 recommended values are based.
Although the results for the five Kibble-balance results discussed above are given as values of h in table 4, in previous adjustments such results were given as values of K 2 J R K = 4/h to facilitate tests of the exactness of the Josephson and quantum-Hall effect relations K J = 2e/h and R K = h/e 2 . Further, the NIST, NRC, and LNE researchers have found it convenient to analyze their data and also to report their results as deviations from the conventional value of the Planck con- 3.3.7. Measurement of h/m for 87 Rb. Accurate measurement of the quotient of h and the mass of a particle X can provide a competitive value of α through the equation where A r (X) is the relative atomic mass of particle X with mass m(X). This expression is a consequence of the definition of the Rydberg constant R ∞ = α 2 m e c/2h.
Item B39 in table 4 is the value for h/m( 87 Rb) obtained at LKB, Paris, using Bloch oscillations together with atom interferometry. This result is discussed in CODATA-10 and its precursor in CODATA-06. Its observational equation is B39 in table 7 and follows directly from equation (5). The LKB result, with u r = 1.2 × 10 −9 and which is used in both the 2010 and 2014 adjustments, provides a value of α with u r about half as large because of the square root in equation (5) and because u r of A r ( 87 Rb), A r (e), and R ∞ are at the parts in 10 11 to parts in 10 12 level. This value of α is given in table 9 together with that from a e . These data are graphically compared in figure 1.  Dashed values were not included in the final 2017 adjustment. The inner green band is ±20 parts in 10 9 and the outer grey band is ±50 parts in 10 9 . KB: Kibble balance; XRCD: x-ray-crystal-density.
the {220} crystal lattice spacing d 220 (X) of a nearly crystallographically perfect natural silicon crystal X and a similar reference crystal; B49 is the estimated difference between the {220} crystal lattice spacing of an ideal silicon crystal and that of crystal W04, both in vacuum and at a temperature of 22.5 • C; B51-B53.2 are the directly measured d 220 values of the indicated crystals in vacuum and at 22.5 • C; and B59-B62 are the measured ratios of the wavelengths of the indicated x-ray lines to the {220} lattice spacing of the indicated crystal, also under the same conditions. The observational equations for these data are B41-B50, B51-B53, B59, B60, B61, and B62 in table 7. The three conventional x-ray units xu(CuKα 1 ), Å * , and xu(MoKα 1 ) in the last four observational equations are defined by assigning the indicated exact numerical value (the number in the numerator on the righthand-side of each equation) to the CuKα 1 , WKα 1 , and MoKα 1 x-ray lines. Together, the 20 natural silicon x-ray data determine the {220} lattice spacing of each of the eight crystals involved in the measurements, d 220 of an ideal silicon crystal, and the values of the three conventional x-ray units, all in meters.

Enriched silicon measurements of N A .
The x-raycrystal-density (XRCD) method of measuring N A is discussed in CODATA-98 and the progress made over the years is documented in subsequent CODATA reports. The International Avogadro Coordination (IAC) project to determine N A with the smallest possible uncertainty was initiated in 2004. Consisting of a group of researchers working at several different institutions, the project has developed the XRCD method to the point where a relative uncertainty u r approaching 1 part in 10 8 can be achieved. Silicon in the form of highly polished, highly pure, nearly spherical, and nearly crystographically perfect spheres of nominal mass 1 kg composed of very nearly the single iso tope 28 Si are the samples used in IAC work since about 2009. Required measurements include impurity content (point defects), isotopic composition, lattice spacing, sphere surface characterization (mass and thickness of contaminant layers), sphere mean diameter, and sphere mass. The techniques employed include isotope dilution mass spectrometry, combined x-ray and optical interferometry, optical interferometry, weighing with the highest possible accuracy, and for surface characterization x-ray fluorescence (XRF), x-ray reflectometry (XRR), optical spectral ellipsometry (SE), and x-ray photoelectron spectroscopy (XPS).
The four values of N A in table 4, items B54.1 to 54.4, IAC-11, IAC-15, IAC-17, and NMIJ-17, are related. Their treatment in the 2017 Special Adjustment is based on a recommendation to the Task Group by the IAC (Fujii et al 2018) and is motivated by the way the IAC determined IAC-17 (Bartl et al 2017). In the latter experiment the {220} lattice spacing d 220 was not directly measured but was derived from the mean of the d 220 value obtained in the 2011 experiment and that obtained in the 2015 experiment. However, prior to the calcul ation of the mean, the IAC-11 value of d 220 was corrected by the fractional amount 3.6 × 10 −9 and the IAC-15 value by 2.9 × 10 −9 . These corrections to the two d 220 values are due to the discovery of a laser beam diffraction effect and a surface stress effect in the combined x-ray and optical interferometer (XROI) used to determine them. The diffraction effect is considered by Mana et al (2017) and also by Bartl et al (2017); the stress effect is considered in the latter paper and in the references cited therein. As we now discuss, this d 220 'consensus' mean value is used to correct the 2011 and 2015 IAC values of N A as well as the 2017 NMIJ value of N A , recognizing that the fractional change in N A is −3 times the fractional change in d 220 .
As reported by Andreas et al (2011) and reviewed in CODATA-10, item B54.1 in table 4 with identification IAC-11 is the first IAC enriched-silicon result. The two spheres used to obtain IAC-11 are denoted AVO28-S5 and AVO28-S8 and were manufactured from the 28 Si-enriched single crystal boule AVO28. As explained in CODATA-14, IAC-11 has been corrected by Azuma et al (2015) to account for the problem discovered at the BIPM, Sèvres, with the working standards BIPM was employing to disseminate the unit of mass. However, it now also includes a net fractional correction of −6.30 × 10 −9 from two sources: (i) 3 times 0.10 × 10 −9 due to the difference of the IAC-11 d 220 value from the consensus value discussed above; and (ii) −6.61 × 10 −9 due to the difference of the mean of the molar mass values obtained for the two AVO28 spheres used in the 2011 experiment from the more accurate and reliable molar mass value obtained in the 2015 experiment, which is the same for both spheres. Both IAC-15 and NMIJ-17 are based on this molar mass value and the IAC recommended that it be used to correct the 2011 N A value (Fujii et al 2018).
Item B54.2, IAC-15, reported by Azuma et al (2015) and discussed in CODATA-14, was obtained after a number of improvements were incorporated in the initial XRCD experiment including re-etching and re-polishing the AVO28-S5 and AVO28-S8 spheres to remove metallic contaminants in the form of Cu, Ni, and Zn silicide compounds from their surfaces. The spheres were then re-labeled AVO28-S5c and AVO28-S8c. Also, the molar mass was obtained with a significantly reduced uncertainty based on measurements carried out in three different laboratories using improved chemistry. IAC-15 includes a correction of 3 times −3.49 × 10 −9 = −10.47 × 10 −9 due to the difference of the IAC-15 d 220 value from the consensus value discussed above. Item B54.3, IAC-17, is a new result reported by Bartl et al (2017) in a detailed paper with many references. It was obtained using two new spheres called Si28kg01a and Si28kg01b manufactured at PTB, Braunschweig, from a new silicon single crystal boule labeled Si28-23Pr11. The 28 Si enrichment of the boule is 99.9985% compared with the 99.9956% enrichment of the AVO28 boule from which the AVO28-S5 and AVO28-S8 spheres were manufactured. The deviations of the spheres from perfect roundness are less than for the previous spheres and their surfaces show no detectable trace of metallic contaminants.
The crystal properties of the boule and their variations throughout its volume were carefully investigated using 27 samples from strategically selected locations. Measurements of point defects include amount of carbon, oxygen, boron, phosphorus, nitrogen, and vacancies. Measurements of the isotopic composition of the boule to determine the molar mass of the spheres were carried out on different samples bracketing the locations in the boule where the material used to manufacture the spheres was taken. This provided information about the possible variation of molar mass in the boule. The method used to determine the amount of substance fractions of the three silicon isotopes 28 Si, 29 Si, and 30 Si to determine molar mass was basically the same as used to obtain the result IAC-15. The mean diameter of each sphere and hence its volume was determined by optical interferometry and its mass by careful weighing in vacuum.
PTB researchers directly measured all but one of the necessary parameters to determine N A using both spheres Si28kg01a and Si28kg01b, and researchers at NMIJ, Tsukuba, did the same for Si28kg01a. Although both laboratories devoted special effort to determining the properties of the contaminant surface layers on the spheres, PTB by XRR, XRF, and XPS, and NMIJ by XPS and SE, neither directly measured d 220 of Si28-23Pr11 with an XROI. Rather, both obtained their value of d 220 by independently deriving it from the consensus value discussed above. The PTB researchers did this by applying corrections for point defects and the difference in 28 Si enrichment of the AVO28 and Si28-23Pr11 boules. The NIMJ researchers did it by measuring the difference between an AVO28 sample and a Si28-23Pr11 sample using an instrument called a self-referenced lattice comparator and also applying corrections for point defects.
The relative uncertainty of the NMIJ value of N A from sphere Si28kg01a is, in parts in 10 8 , 2.3, and of the PTB values of N A from spheres Si28kg01a and Si28kg01b, 1.4 and 1.4. The final result for N A reported by Bartl et al (2017) is the weighted mean of the three values, which are in agreement, taking correlations into account. The IAC researchers are aware that their new result IAC-17 is not in as good agreement as they would like with their previous results IAC-11 and IAC-15 and with the new NMIJ-17 result discussed next, but to date have been unable to identify any cause.
The value of N A identified as NMIJ-17, item B54.4 in table 4, was measured at NMIJ using sphere AVO28-S5c. Reported by Kuramoto et al (2017), the required parameters determined independently in this experiment are the mass of the sphere, obtained by comparison with a mass standard calibrated by the BIPM as part of the Extraordinary Calibration Campaign; the mean diameter of the sphere, obtained by optical interferometry; and the thickness and mass of its surface layers, obtained by XPS and SE. The other required parameters, molar mass, d 220 , and mass of the point defects, are taken from the IAC-15 experiment (Azuma et al 2015). Thus, the NMIJ-17 value of N A in table 4 includes the same −10.47 × 10 −9 correction for d 220 as does IAC-15. The main contributor to the total relative uncertainty of this result, u r = 2.4 × 10 −8 , is the relative uncertainty of the NMIJ measured volume of the silicon core of the sphere, u r = 2.0 × 10 −8 .
The inclusion of the correction for d 220 and molar mass in the IAC-11 value of N A , and the inclusion of the correction for d 220 in the IAC-15 and NMIJ-17 values of N A , requires a re-evalution of the six covariances among the four N A values based on updated uncertainty budgets. These are given in table 5 in the form of correlation coefficients (Fujii et al 2018).
The observational equation for N A is B54 in table 7 and follows from R ∞ = α 2 m e c/2h. This equation is noteworthy because it can be written as Since c and the molar mass constant M u = 0.001 kg mol −1 are exact and u r of the combination of constants A r (e)α 2 /R ∞ is 4.5 × 10 −10 , a value of N A with, say, u r = 2 × 10 −8 can provide a value of h with essentially the same uncertainty.  4 He, where k is inferred from the Clausius-Mossotti equation and a theoretical evaluation of the static electric polarizability of a helium atom, here listed as B58. In the remainder of this subsection we briefly describe the individual measurements and techniques. The measurement of R by AGT involves determining the speed of sound in a gas, usually Ar or He, at a known temper ature at or near the triple point of water T TPW and at various pressures p, and extrapolating to p = 0. The speed of sound follows from a measurement of the acoustic resonant frequencies of a gas-filled cavity, or resonator, which can be cylindrical, spherical, or often quasispherical in the form of a triaxial ellipsoid. The precise dimensions of the cavity are usually determined by measuring their electromagnetic modes. Finally, the molar mass of the gas is required and involves measuring its isotopic composition and accounting for impurities, sometimes introduced by out-gassing from the walls.
The acoustic-gas-thermometry values of R numbered B55.1-B55.5 were discussed and used in the CODATA-14 adjustment. Measurement B55.6 at INRIM, Torino  was the first competitive data point to appear after our closing deadline of the 2014 adjustment. They used a slightly-misaligned spherical copper resonator with an internal radius of 90 mm filled with helium. The quoted uncertainty for R is dominated (≈90%) by their measurement of the speed of sound. Smaller (nearly-equal) contributions to the uncertainty are due to the uncertainties in the molar mass of the helium gas and its temperature. The six evaluations B55.1-B55.6 are not independent. We have employed the approach of Moldover et al (2015) to evaluate the covariances among these values. The covariances in the form of correlation coefficients are listed in table 5. Input datum B55.7, labeled NIM-17, is based on combining five AGT measurements with argon carried out over the last five years at NIM, Beijing, with help from researchers at NIST, Gaithersburg. The individual data can be found in Lin et al (2013), Feng et al (2015), and Feng et al (2017). In all cases steel cylindrical resonators of different lengths rather than the more common spherical or quasispherical resonator were used. It is worth noting that in Feng et al (2017) resonator lengths were obtained from measurements of resonant microwave frequencies while in the earlier experiments lengths were obtained by two-color laser interferometry. Feng et al (2017) also carefully reviewed and updated as appropriate the values and uncertainty budgets of their previous values of R. In addition, they evaluated the covariances among the five values and calculated a weighted mean following the CODATA procedure given in appendix E of CODATA-98. This mean with u r = 2.0 × 10 −6 is used in our special 2017 adjustment.
Recently reported by NPL, Teddington (Podesta et al 2017), item B55.8 corresponds to a revision of argon-based AGT measurements in a quasi-spherical cavity published in 2013 (de Podesta et al 2013(de Podesta et al 2015. It is updated for argon isotopic composition by using a novel sampling procedure from its storage cylinder, which eliminates any possibility of contamination. The sampling was repeated three times and the samples were examined with two different mass spectrometers. The spectrometer measurements were traceable to gravimetrically prepared argon samples. The end result is a fractional increase of 5.0 parts in 10 7 in the molar mass of the argon compared to that found by de Podesta et al (2015) leading to a fractional increase of 7.1 parts in 10 7 for R with uncertainty u r = 7.0 × 10 −7 . This value is labeled NPL-17 in table 4 and supersedes the AGT datum NPL-13 in CODATA-14.
Item B55.9 is an AGT measurement at LNE, La Plaine-St. Denis (Pitre et al 2017). Three older AGT values of R by LNE are given in table 4. Datum B55.4 used argon, while all others used 4 He. The 2017 helium experiment employed a copper quasispherical triaxial ellipsoid resonator with a volume of about 3 L and a radius of 90 mm and was, in nearly equal parts, limited by uncertainties in the determination of temperature and acoustic resonances. This result has the smallest uncertainty u r = 6.0 × 10 −7 of any of the 14 data that contribute to the determination of k.
We have included the four LNE values for R as separate input data rather than using their weighted mean also given in Pitre et al (2017) in order to account for correlations noted by Moldover et al (2015). Nevertheless, the weighted mean of the four LNE values including their correlation coefficients as given in tables 4 and 5 agrees with the weighted mean given by Pitre et al (2017).
Item B55.10, UVa/CEM-17, was determined in a joint project of the University of Valladolid, Valladolid, Spain, and the National Metrology Institute of Spain, Centro Español de Metrologia (CEM), Madrid. Reported by Segovia et al (2017), this AGT experiment used purified Ar gas and a steel ellipsoidal cavity of nominal 40 mm radius formed from two hemispheres bolted together and operated at T TPW . Their inside walls were coated with a 1.5 µm layer of gold. Acoustic transducers in the lower hemisphere and microwave transducers in the upper hemisphere enabled simultaneous measurements of the acoustic and microwave frequencies.
A determination of the isotopic composition of the argon gas was not possible, so the UVa/CEM researchers used the results from Lee et al (2006) for the isotopic composition for Ar in atmospheric air and included additional comp onents of uncertainty to account for possible impurities in the argon employed in the experiment and deviations of its isotopic composition from that of argon in air. The two largest contributors by far to their total relative uncertainty are the 4.7 × 10 −6 uncertainty of the mean cavity radius and the 4.4 × 10 −6 uncertainty of the measured frequencies of the acoustic modes.
The Johnson noise thermometry (JNT) method of determining k is discussed in CODATA-14 and is based on the Nyquist theorem U 2 = 4kTR J ∆f , where U 2 is the Johnson noise voltage in a frequency bandwidth ∆f across a resistance R J at temperature T. For frequencies f less than 10 MHz and temperatures greater than 250 K this theorem holds with a fractional uncertainty significantly less than 1 × 10 −6 , well below current experimental uncertainties.
The constant determined in JNT is the kelvin-hertz conversion factor k/h, when the noise voltage is measured in terms of the conventional value of the Josephson constant K J−90 and the resistance in terms of the conventional value of the von Klitzing constant R K−90 . In practice, the measured quantity is the ratio S R /S Q , where S R is the thermal noise power spectral density of the resistor and S Q is the pseudo noise power spectral density of approximately equal power synthesized by a pulsed-bias Josephson junction array also known as a quantum voltage noise source. The uncertainty of current JNT experiments is limited by statistics.
Item B56.1, NIM/NIST-15, in table 4 is a JNT measurement reported by Qu et al (2015), and was used in the 2014 final adjustment. This joint NIM-NIST thermometry project has continued and an improved value for k/h, item B56.2, NIM/NIST-17, was recently reported by Qu et al (2017). Some important changes were made to the apparatus, for example, the 200 Ω resistor was replaced with a 100 Ω resistor to increase the measurement bandwidth as were the leads used to connect the resistor and Josephson junction array to the measurement electronics. Nevertheless, the reduction in relative uncertainty is mainly the result of decreasing the statistical uncertainty by increasing the integration time from 33 d to 100 d. Based on the uncertainty budgets for NIM/NIST-15 and NIM/NIST-17 given by Qu et al (2017), the correlation coefficient between the two experiments is 0.021. Item B56.3, NIST-17, is the JNT value of k/h determined at NIST, Boulder, reported by Flowers-Jacobs et al (2017). This is the latest result of continued work at NIST over the last decade. The most significant improvement is the input circuitry from the two noise sources to the measurement system amplifiers and the method employed to match the frequency responses of the two sources. The reported result with u r = 5.0 × 10 −6 is based on 49 d of data taking.
Item B57, PTB-17, is a multi-year effort of PTB, Berlin, to accurately measure the Boltzmann constant using di electric constant gas thermometry (DCGT) of a 4 He gas. The method relies on the virial expansion of the equation of state and the Clausius-Mossotti equation to find an expression for A ( 4 He)/R, the ratio of the molar polarizability of a 4 He gas and the molar gas constant R, in terms of the measurable relative dielectric constant r = ( 4 He)/ 0 at a known temperature T and various pressures p, and extrapolating to p = 0 (see . The Boltzmann constant is then obtained from the equation A ( 4 He)/R = α 0 ( 4 He)/(3 0 k), where α 0 ( 4 He) is the static electric polarizability of a helium atom in SI units. It is equal to 4π 0 a 3 0 α * 0 ( 4 He), where a 0 is the Bohr radius and α * 0 ( 4 He) is the static electric polarizability of 4 He in atomic units. The latter is item B58 in table 4 calculated by Piszczatowski et al (2015) and also used in CODATA-14.
In the PTB experiments, carried out at T TPW , the fractional change in capacitance of a specially constructed capacitor, first filled with helium gas at a known pressure and then without, is determined. Their first high-accuracy value for A ( 4 He)/R  was used in CODATA-14. More recently, Gaiser et al (2017) reported results for two new capacitors, and a final value from the PTB DCGT project with u r = 1.9 × 10 −6 obtained by calculating the weighted mean of the values from the three capacitors and their covariances. The value for PTB-17, item B57 in table 4, was inadvertently omitted from Gaiser et al (2017) but was provided to the Task Group by Gaiser (2017).
For completeness we note that two other results relevant to the determination of k have recently been published. By measuring the optical refractive index of helium at a known temperature and pressure, Egan et al (2017) determined k with u r = 12.5 × 10 −6 . Also, Urano et al (2017) determined k with u r = 10.2 × 10 −6 from JNT measurements using a superconducting circuit. However, their uncertainties are too large to be considered for inclusion in the 2017 Special Adjustment. The two results are consistent with the other Boltzmann constant data.
3.3.11. Tau mass and electroweak quantities. The mass of the tau lepton m τ , the Fermi coupling constant G F , and sine squared of the weak mixing angle sin 2 θ W , enter the calculation of the theoretical expression for the electron magnetic moment anomaly a e . The values employed in the CODATA 2017 Special Adjustment are taken from the 2016 biennial report of the Particle Data Group (Patrignani et al 2016): The definition sin 2 θ W = 1 − (m W /m Z ) 2 , where m W and m Z are, respectively, the masses of the W ± and Z 0 bosons, is adopted because it is used in the calculation of the electroweak contributions to a e (Czarnecki et al 1996). The value for the mass ratio given by the Particle Data Group (Patrignani et al 2016), m W /m Z = 0.881 53(17), leads to the above value of sin 2 θ W .

Analysis of data
There are 10 types of input data with two or more values and in general data of the same type agree among themselves; that is, the difference between any two data of the same type does not exceed the standard uncertainty of their difference u diff by more than 2u diff , or 2σ. However, there are two exceptions among the h data and one among the N A data. As can be seen from their respective observational equations in table 7, items B2 and B12, which are A r ( 1 H) and ω c ( 12 C 6+ )/ω c (p), determine A r (p). However, the two inferred values of A r (p) differ by 3.5σ.
The inferred values of α are compared in table 9 and figure 1, of h in table 10 and figure 2, and of k in table 11 and figure 3. The two values of α in table 9, the first from a e (HarvU-08) and the second from h/m( 87 Rb) (LKB-11), differ by less than 2σ; and none of the 14 values of k in table 11 differs from another by as much as 2σ. However there are some differences greater than 2σ between XRCD inferred values of h and directly measured Kibble-balance values of h. The most significant of these are the differences between h from IAC-17 and the NIST-17, NRC-17, and NIST-15 values of h, which are 4.0σ, 2.8σ, and 2.7σ, respectively.
In summary, these comparisons have identified a few cases where data differ by significantly more than 2σ, implying that there may be input data with normalized residuals r i with absolute values greater than two in least-squares calculations.
Our initial multivariate least-squares adjustment includes all of the input data in tables 2 and 4 and their correlation coefficients in tables 3 and 5. It is characterized by N = 138 input data, M = 74 adjusted constants, and ν = N − M = 64 degrees of freedom; χ 2 = 72.9, p (χ 2 |ν) = 0.21, and Birge ratio R B = χ 2 /ν = 1.07. Five input data have |r i | greater than two. They are B2 in To address these large residuals, we follow the usual CODATA Task-Group practice of reducing the absolute values of residuals to two or less by multiplying the uncertainties of the input data related to the problem by an expansion factor. If the data have covariances with other data, the covariances are multiplied by the same factor squared if the uncertainties of the other data are multiplied by the same expansion factor, or by only the expansion factor if the uncertainties of the other data with which they are correlated are not multiplied by the expansion factor. This ensures that the correlation coefficients of the data remain the same and the expansion factor does not significantly alter the values of the adjusted constants (variables) determined by the adjustment, only their uncertainties.
For the data that play the dominant role in determining the adjusted constant h, namely the Kibble-balance values of h, items B38.1-B38.5 , and the XRCD values of N A , items B54.1-B54.4 , an expansion factor of 1.7 brings all the Kibble balance and XRCD data into agreement. The relative uncertainties of the first five values of h in table 10 are, in parts in 10 9 , only 15, 20, 23, 34, and 42, respectively.
The data that play the dominant role in determining the adjusted constant A r (p), namely B2 (AME-16) and B12 (MPIK-17), also require an expansion factor. Coincidentially it is 1.7 and reduces |r i | of B2 to 1.9.
We also follow the usual CODATA Task-Group practice of omitting data with self sensitivity coefficients S c less than 1% in the final adjustment on which recommended values of constants are based (see section 2). Thus we delete input data

Final results
The values for the constants h, e, k and N A from the CODATA 2017 Special Adjustment are h = 6.626 070 150(69) × 10 −34 J s where it is understood that the units here are those of the revised SI. These values have been approved at the 106 th meeting of the CIPM, with concurrence from its relevant consultative committees. The exact numerical values in equations (14)- (17) agree within the uncertainties with the adjusted numerical values in equations (10)-(13). The Consultative Committee for Thermometry recommended that eight digits be chosen for k; however, since the eighth digit is a zero, only seven digits are required (CCT 2017). The CIPM notes that at the time of the adoption of the revision: the mass of the international prototype of the kilogram (IPK), m(K), will be 1 kg with the same relative standard uncertainty as h in equation (10); the vacuum magnetic permeability µ 0 will be 4π × 10 −7 H m −1 with the same relative uncertainty as α in the Special Adjustment (2.3 × 10 −10 ); the thermodynamic temperature of the triple point of water will be 273.16 K with the same relative uncertainty as k in (12); and the molar mass of carbon 12 M( 12 C) will be 0.012 kg mol −1 with the same relative uncertainty as N A h in the Special Adjustment (4.5 × 10 −10 ).
These expectations are indeed confirmed by the following considerations. In the case of the Planck constant and the IPK, Kibble balance measurements determine the constant C, in the relationship h = Cm d (K), in terms of quantities with units that are the same before and after the redefinition, such as the local acceleration of gravity. The subscript d on the symbol for the IPK indicates that it is a defining quantity in the cur rent SI with the value 1 kg. Similarly, x-ray-crystal-density measurements determine the constant D, in the relationship h = Dm d (K), in terms of quantities with units that do not change in the redefinition and relatively accurately known constants that are unchanged by the redefinition. The adjusted value of h in (10) is a weighted mean of the various Kibble balances (labeled by i) and x-ray-crystal-density measurements (labeled by j) that yield values and uncertainties of C i and D j possibly with covariances. Just after the redefinition, the mass of the IPK will be an adjusted value based on the set of relations m(K) = C −1 i h d and m(K) = D −1 j h d . In this case, h d is the defining constant given in (14) and m(K) is the adjusted quantity. Hence, the relative uncertainty of m(K) will be the that of the set of quantities C −1 i and D −1 j , including their correlations, which is essentially the same as the uncertainty of h in equation (10).
For the quantity µ 0 , we use the relationship h/e 2 = (c/2α)µ 0d , where µ 0d is the current defining constant for electrical quantities with its exact value. Just after the redefinition, the equation defining µ 0 is µ 0 = (2α/c)h d /e 2 d , where e d is the defining constant in (15). Thus its relative uncertainty is that of the adjusted value of α, which is the same as that of h/e 2 .
In the case of the Boltzmann constant, the various measurements determine the constant F in the relationship k = F/T TPWd where T TPWd is the triple point of water and F is a quantity that depends on constants that do not change in the redefinition or whose changes are small compared to the uncertainties of the measurements. As in the case of mass, this relation can be inverted to yield T TPW = F/k d , where k d is the defining constant in (16). Evidently the relative uncertainty of the derived value of T TPW has the same relative uncertainty as k in (12).
For the molar mass of carbon, the relation N A h = cA r (e)α 2 /24R ∞ M d ( 12 C), where M d ( 12 C) is the exact defining constant 0.012 kg mol −1 in the current SI, gives M( 12 C) = cA r (e)α 2 /24R ∞ −1 N Ad h d , where N Ad is the defining constant in (17) just after the redefinition. As in the cases above, the uncertainty in the determination of M( 12 C) is that of N A h before the redefinition.

Proton radius 'puzzle'
The bound state proton and deuteron rms charge radii r p and r d obtained from Lamb shift measurements in muonic hydrogen and deuterium atoms (p µ − , d µ − ) are not included in the 2017 Special Adjustment. The muonic hydrogen value of r p is smaller than the H-D spectroscopic value and the e-p scattering value by 4.5σ and 3.5σ, respectively; see CODATA-14 for further discussion of this so called proton radius puzzle. The muonic deuterium result for r d became available only last year and it is smaller than the H-D spectroscopic value by 5.0σ. The muonic hydrogen and deuterium values are r p = 0.840 87(39) fm (Antognini et al 2013) and r d = 2.125 62(78) fm . If these are used in the final 2017 Special Adjustment in place of input data items A51 and A52 of table 2, then R ∞ decreases a fractional amount of about 5.3 times its u r of 5.9 × 10 −12 , or about 3.1 parts in 10 11 . However, there are no changes in the exact numerical values for h, e, k, or N A .

Acknowledgment
The Task Group gratefully acknowledges the help provided to it by the many colleagues throughout the world who provided results by the 1 July 2017 due-date and who answered questions both promptly and completely. The Task Group also thanks the General Conference on Weights and Measures (CGPM) for inviting it to play a significant role in the international effort to establish a revised SI for the 21st century. Unified atomic mass unit (also called the dalton, Da): 1 u = m u = m( 12 C)/12 u diff Standard uncertainty of the difference between two values (σ is sometimes used in place of u diff ) u(x i ) Standard uncertainty (i.e., estimated standard deviation) of an estimated value x i of a quantity X i (also simply u) u r (x i ) Relative standard uncertainty of an estimated value x i of a quantity X i : u r (x i ) = u(x i )/|x i |, x i = 0 (also simply u r ) u(x i , x j ) Covariance of estimated values x i and x j u r (x i , x j ) Relative covariance of estimated values x i and x j :

WarsU
University of Warsaw, Warsaw, Poland XROI Combined x-ray and optical interferometer YaleU Yale University, New Haven, Connecticut, USA α Fine-structure constant: α = e 2 /4π 0 c ≈ 1/137 ∆E B ( A X n+ ) Energy required to remove n electrons from a neutral atom ∆E I ( A X i+ ) Electron ionization energies, i = 0 to n − 1 ∆ν Mu Muonium ground-state hyperfine splitting δ C Additive correction to the theoretical expression for the electron ground-state g-factor in 12 C 5+ δ e Additive correction to the theoretical expression for the electron magnetic moment anomaly a e δ Mu Additive correction to the theoretical expression for the ground-state hyperfine splitting of muonium ∆ν Mu δ Si Additive correction to the theoretical expression for the electron ground-state g-factor in 28 Si 13+ δ X (nL j ) Additive correction to the theoretical expression for an energy level of either hydrogen H or deuterium D with quantum numbers n, L, and j 0 Electric constant (vacuum electric permittivity): Symbol used to relate an input datum to its observational equation µ Symbol for either member of the muon-antimuon pair; when necessary, µ − or µ + is used to indicate the negative muon or positive muon µ B Bohr magneton: µ B = e /2m e µ N Nuclear magneton: µ N = e /2m p µ X (Y) Magnetic moment of particle X in atom or molecule Y µ 0 Magnetic constant (vacuum magnetic permeability): µ 0 = 4π × 10 −7 N/A 2 µ X Magnetic moment of particle X ν Degrees of freedom of a particular adjustment ν( f p ) Difference between muonium hyperfine splitting Zeeman transition frequencies ν 34 and ν 12 at a magnetic flux density B corresponding to the free proton NMR frequency f p τ Symbol for either member of the tau-antitau pair; when necessary, τ − or τ + is used to indicate the negative tau or positive tau The statistic 'chi square'