Data and analysis for the CODATA 2017 Special Fundamental Constants Adjustment

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1 Metrologia ACCEPTED MANUSCRIPT OPEN ACCESS Data and analysis for the CODATA 1 Special Fundamental Constants Adjustment To cite this article before publication: Peter J Mohr et al 1 Metrologia in press Manuscript version: is the version of the article accepted for publication including all changes made as a result of the peer review process, and which may also include the addition to the article by IOP Publishing of a header, an article ID, a cover sheet and/or an Accepted Manuscript watermark, but excluding any other editing, typesetting or other changes made by IOP Publishing and/or its licensors This is Not subject to copyright in the USA Contribution of NIST As the Version of Record of this article is going to be / has been published on a gold open access basis under a CC BY 0 licence, this Accepted Manuscript is available for reuse under a CC BY 0 licence immediately Everyone is permitted to use all or part of the original content in this article, provided that they adhere to all the terms of the licence Although reasonable endeavours have been taken to obtain all necessary permissions from third parties to include their copyrighted content within this article, their full citation and copyright line may not be present in this version Before using any content from this article, please refer to the Version of Record on IOPscience once published for full citation and copyright details, as permissions may be required All third party content is fully copyright protected and is not published on a gold open access basis under a CC BY licence, unless that is specifically stated in the figure caption in the Version of Record View the article online for updates and enhancements This content was downloaded from IP address on /01/1 at 0:1

2 Page 1 of AUTHOR SUBMITTED MANUSCRIPT - MET-R Data and Analysis for the CODATA 1 Special Fundamental Constants Adjustment Contents Peter J Mohr, David B Newell, Barry N Taylor and Eite Tiesinga National Institute of Standards and Technology, Gaithersburg, Maryland -, USA (Dated: November, 1) 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 1, is described in detail It is based on all relevant data available by 1 July 1 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 th General Conference on Weights and Measures when it convenes on - November 1 I Introduction 1 II Least-Squares Adjustments III Review of Data A Special quantities and units B Data that determine the Rydberg constant R C Data that determine constants other than R 1 Atomic masses and binding energies Hydrogenic carbon and silicon g-factors Electron and muon magnetic moment anomalies Muonium hyperfine splitting Magnetic moment ratios Kibble and joule balance measurements of h Measurement of h/m for Rb 1 Natural silicon measurements Enriched silicon measurements of N A Thermal physical measurements that determine k 1 Tau mass and electroweak quantities 1 This report was prepared by the authors under the auspices of the Committee on Data for Science and Technology CODATA Task Group on Fundamental Constants The members of the task group are F Cabiati, Istituto Nazionale di Ricerca Metrologica, Italy; J Fischer, Physikalisch-Technische Bundesanstalt, Germany; K Fujii, National Metrology Institute of Japan, Japan; S G Karshenboim, Pulkovo Observatory, Russian Federation and Ludwig- Maximilians-Universität, Germany; H S Margolis, National Phyiscal Laboratory, United Kingdom; E de Mirandés, Bureau international des poids et mesures; P J Mohr, National Institute of Standards and Technology, United States of America; D B Newell, National Institute of Standards and Technology, United States of America; F Nez, Laboratoire Kastler-Brossel, France; K Pachucki, University of Warsaw, Poland; T J Quinn, Bureau international des poids et mesures; B N Taylor, National Institute of Standards and Technology, United States of America; M Wang, Chinese Academy of Sciences, People s Republic of China; B M Wood, National Research Council, Canada; Z Zhang, National Institute of Metrology, People s Republic of China; Electronic address: mohr@nistgov Electronic address: dnewell@nistgov Electronic address: barrytaylor@nistgov Electronic address: eitetiesinga@nistgov IV Analysis of Data 1 V Final Results A Proton radius puzzle Acknowledgement List of Symbols and Abbreviations References I INTRODUCTION The th General Conference on Weights and Measures (CGPM) is expected to establish at its meeting on - November 1 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, ), 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 1 The required numerical values of h, e, k, and N A resulting from that adjustment are reported by the Task Group (Newell et al, 1) Here we describe the data and analysis underlying the 1 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 1 (CIPM, ) The procedures used for the 1 Special Adjustment arethe same as those used for the 1regu-

3 larly scheduled CODATA adjustment and its 1, 0, 0, and predecessors (Mohr et al, a,b; Mohr and Taylor, 00, 0; Mohr et al, 0a,b, a,b) II LEAST-SQUARES ADJUSTMENTS Themethodofleast-squaresasusedbytheTaskGroup is discussed in detail in Appendix E of CODATA- A few key points are as follows The numerical input data for an adjustment, obtained from experiment or theory, are expressed in terms of an independent set of quantities called adjusted constants These are the variables of the adjustment and the equations that relate 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, probability p(χ ν) of obtaining an observed value of χ by chance that large or larger for the given value of ν, and Birge ratio R B = χ /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 adjustedvalue,andu i itsstandarduncertainty,isameasure of how well that datum agrees with other data A value larger than 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-, it should be no less than 001, or 1 %, for the input datum to be included in the final adjustment on which recommended values of constants are based However, the exclusion of a datum is not followed if, for example, a datum with S c < 001 is part of a group of data obtained in a given experiment where most of the other data have self-sensitivity coefficients > 001 This 1 % cutoff implies the input datum s uncertainty should be no more than about a factor of 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-, column 1 of p and of p for further discussion) III 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, they are in generalnot fully defined in the AUTHOR SUBMITTED MANUSCRIPT - MET-R text Further, for the same reason, we sometimes refer the reader to a particular table in CODATA-1 rather than repeat it in this paper As in past CODATA adjustments, the uncertainty of a theoretical 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 1 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 1 set Nevertheless, to maintain the continuity of CODATA adjustments and to simplify the comparison of the CODATA 1 Special Adjustment with the CODATA 1 regular adjustment, the starting point of the 1 Special Adjustment is the input data included in the final adjustment on which the 1 CODATA recommended values are based Consequently, the input data omitted from that final adjustment because of their low weight are not considered for inclusion in the 1 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 1 final adjustment are duly noted For details, including references, CODATA-1 and earlier CODATA reports should be consulted; in general only references to new work are given in this report A Special quantities and units Table I recalls a number of familiar, exactly-known constants relevant to the 1 Special Adjustment It is subject to modification after redefinition of the SI B 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 1 Special Adjustment It is obtained by equating measured transition frequencies in hydrogen (H) and Deuterium (D) with their theoretical expressions Table II gives the principal input data that determine the 1 Special Adjustment value of R and Table III gives their correlation coefficients The adjusted constantsintermsofwhichther inputdataintableiiare expressed are given in Table XXV of CODATA-1 and Page of

4 Page of TABLE I Some exact quantities relevant to the 1 adjustment Quantity Symbol Value speed of light in vacuum c, c 0 m s 1 magnetic constant µ 0 π N A = 01 N A electric constant ǫ 0 (µ 0c ) 1 = 11 F m 1 molar mass of C M( C) kg mol 1 molar mass constant M u M( C)/ = kg mol 1 relative atomic mass of C A r( C) conventional value of Josephson constant K J 0 GHz V 1 conventional value of von Klitzing constant R K 0 0 Ω the observational equation for each input datum may be found in Table XXIII ofcodata-1; thus they need not be repeatedhere It should be recalledthat the symbol = in an observational equation indicates that an input datum of the type 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 II follows the order in which they are listed Items A1-A are the additive corrections for the theoretical expressions of the indicated H and D energy levels and items A1-A are the experimentally measured H and D transition frequencies Items A and A0 are the root-mean-square (rms) bound-state 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 II and the corresponding Table XVI in CODATA-1 (and hence to some extent between Table III and its corresponding Table XVII in CODATA- 1) are as follows There are a few changes in the theory of the H and D energy levels as given in CODATA-1 but none of major consequence; none of the uncertainties of the additive corrections to the theory, items A1-A of Table II, have changed significantly from their 1 values This also applies to their correlation coefficients given in Table III The changes in the theory are (i) incorporation of the δ B 0 contribution to b L in Eq() of CODATA- based on our reanalysis of the paper by Jentschura (0); (ii) inclusion of improved relativistic nuclear recoil corrections and the effect of finite nuclear size on them calculated by Yerokhin and Shabaev () and Yerokhin and Shabaev (); and (iii) addition of a contribution from light-by-light scattering diagrams to the coefficient B 1 in Eq() of CODATA-1 calculated by Czarnecki and Szafron () that reduces the ν H (1S 1/ S 1/ ) theoretical transition frequency by about Hz The only new frequency in Table II is item A for the ν H (1S 1/ S 1/ ) transition from the group at the MPQ, Garching, as reported by Yost et al () The frequency was determined using two-photon direct frequency comb spectroscopy, or DFCS This technique em- AUTHOR SUBMITTED MANUSCRIPT - MET-R ploys the high peak intensities of the ultra-short pulse trains of an optical comb to generate the required shortwavelength radiation, in this case nm However, in contrast to two-photon spectroscopy 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 0kHz and the three largest uncertainty components contributing to the 1 khz total uncertainty are, in khz, for second-order Doppler broadening, for the frequency measurement, and for pressure shift The electron-proton and electron-deuteron scattering values of the proton and deuteron rms radii, items B and B0, are the same as used in CODATA-1 Because the valuesofh, e, k, andn A resultingfromthe CODATA 1 Special Adjustment are essentially independent of r p and r d, the values chosen are not critical This is shown in Sec VA, which discusses the result of using values for these radii obtained from measurements of the Lamb shift in muonic hydrogen and muonic deuterium C Data that determine constants other than R Table IV gives the principal input data that determine the 1 Special Adjustment values of constants other than R and Table V gives their correlation coefficients Many of the data in Table IV are the same as (or fully equivalent to) those in the corresponding Table XVIII of CODATA-1 The adjusted constants in terms of which the input data in Table IV are expressed are given in Table VI and their observational equations are given in Table VII As for the Rydberg constant data, the discussion of these data follows the order in which they are listed in Table IV

5 AUTHOR SUBMITTED MANUSCRIPT - MET-R TABLE II Summary of principal input data relevant to the Rydberg constant R for the CODATA 1 Special Adjustment Item Input datum Value Relative standard Identification number uncertainty 1 u r A1 δ H(1S 1/ ) 00() khz [1 ] theory A δ H(S 1/ ) 000() khz [ ] theory A δ H(S 1/ ) 0000() khz [ ] theory A δ H(S 1/ ) 0000() khz [1 ] theory A δ H(S 1/ ) 0000(1) khz [1 ] theory A δ H(S 1/ ) 00000(1) khz [1 ] theory A δ H(P 1/ ) 0000() khz [ 1 ] theory A δ H(P 1/ ) 00000() khz [1 1 ] theory A δ H(P / ) 0000() khz [ 1 ] theory A δ H(P / ) 00000() khz [1 1 ] theory A δ H(D / ) () khz [ ] theory A δ H(D / ) () khz [ ] theory A δ H(D / ) 00000() khz [1 1 ] theory A1 δ H(D / ) 00000() khz [11 1 ] theory A δ H(D / ) () khz [ ] theory A δ H(D / ) () khz [ ] theory A1 δ D(1S 1/ ) 00() khz [ ] theory A1 δ D(S 1/ ) 000() khz [ ] theory A1 δ D(S 1/ ) 0000() khz [1 ] theory A δ D(S 1/ ) 00000() khz [1 ] theory A δ D(D / ) () khz [ ] theory A δ D(D / ) () khz [ ] theory A δ D(D / ) 00000() khz [1 1 ] theory A δ D(D / ) () khz [ ] theory A δ D(D / ) () khz [ ] theory A1 ν H(1S 1/ S 1/ ) 010() khz MPQ- A ν H(1S 1/ S 1/ ) 0101() khz MPQ- A1 ν H(1S 1/ S 1/ ) () khz LKB- A ν H(1S 1/ S 1/ ) (1) khz MPQ- A ν H(S 1/ S 1/ ) 0000() khz 11 LK/SY- A ν H(S 1/ D / ) 0000() khz 11 LK/SY- A0 ν H(S 1/ D / ) 0() khz LK/SY- A1 ν H(S 1/ D / ) 10() khz 1 LK/SY- A ν H(S 1/ D / ) 11(0) khz LK/SY- A ν H(S 1/ S 1/ ) 1 νh(1s 1/ S 1/ ) () khz 1 MPQ- A ν H(S 1/ D / ) 1 νh(1s 1/ S 1/ ) 01() khz MPQ- A ν H(S 1/ S 1/ ) 1 νh(1s 1/ S 1/ ) 0() khz LKB- A ν H(S 1/ D / ) 1 νh(1s 1/ S 1/ ) 0() khz LKB- A ν H(S 1/ P 1/ ) 1 νh(1s 1/ S 1/ ) () khz YaleU- A ν H(S 1/ P / ) 1 νh(1s 1/ S 1/ ) 0() khz 1 YaleU- A ν H(S 1/ P / ) 0() khz 1 HarvU- A01 ν H(P 1/ S 1/ ) 0(0) khz HarvU- A0 ν H(P 1/ S 1/ ) () khz 1 USus- A1 ν D(S 1/ S 1/ ) 00() khz LK/SY- A ν D(S 1/ D / ) 0101() khz LK/SY- A ν D(S 1/ D / ) 0() khz LK/SY- A ν D(S 1/ D / ) 000() khz 11 LK/SY- A ν D(S 1/ D / ) 0() khz LK/SY- A ν D(S 1/ S 1/ ) 1 νd(1s 1/ S 1/ ) 0() khz MPQ- A ν D(S 1/ D / ) 1 νd(1s 1/ S 1/ ) 1(1) khz MPQ- A ν D(1S 1/ S 1/ ) ν H(1S 1/ S 1/ ) 00() khz MPQ- A r p 0() fm 1 rp-1 A0 r d 0() fm rd- 1 The values in brackets are relative to the frequency equivalent of the binding energy of the indicated level Page of

6 Page of TABLE III Correlation coefficients with r(x i,x j) of the input data related to R in Table II For simplicity, the two items of data to which a particular correlation coefficient corresponds are identified by their item numbers in Table II r(a1, A) = 0 r(a, A) = 00 r(a, A) = 0000 r(a1, A) = 00 r(a1, A) = 0 r(a, A1) = 0 r(a, A) = r(a1, A1) = 01 r(a1, A) = 0 r(a, A1) = 0 r(a, A) = r(a1, A) = 00 r(a1, A) = 0 r(a, A1) = 0 r(a, A) = 0000 r(a1, A) = 0001 r(a1, A) = 01 r(a, A) = 01 r(a1, A) = 00 r(a1, A) = 0 r(a1, A1) = 0 r(a, A) = 0000 r(a, A) = 0 r(a, A) = 00 r(a1, A1) = 0 r(a, A) = 0 r(a, A0) = 0 r(a, A) = 00 r(a1, A1) = 01 r(a, A) = 0000 r(a, A1) = 00 r(a, A1) = 0 r(a1, A) = 0 r(a, A) = 0000 r(a, A) = 00 r(a, A) = 00 r(a, A) = 0 r(a, A) = 0 r(a, A) = 00 r(a, A) = 00 r(a, A) = 0 r(a, A1) = 0000 r(a, A) = 00 r(a, A) = 0 r(a, A) = 0 r(a, A) = 0000 r(a, A1) = 0 r(a, A) = 0 r(a, A) = 0 r(a, A) = 0000 r(a, A) = 0 r(a, A) = 0 r(a, A1) = 0 r(a, A) = 0 r(a, A) = 000 r(a, A) = 000 r(a, A1) = 0 r(a, A) = 0000 r(a, A) = 0 r(a, A) = 00 r(a, A1) = 0 r(a, A) = 0000 r(a, A0) = 0 r(a, A) = 00 r(a, A) = 0 r(a1, A) = 0000 r(a, A1) = 00 r(a, A) = 01 r(a, A) = 0 r(a1, A) = 0000 r(a, A) = 0 r(a, A1) = 00 r(a, A) = 0 r(a1, A) = 0000 r(a, A) = 00 r(a, A) = 00 r(a, A) = 0 r(a1, A) = 0000 r(a, A) = 00 r(a, A) = 00 r(a, A1) = 0 r(a1, A) = 0000 r(a, A1) = 0 r(a, A) = 00 r(a, A1) = 0 r(a, A) = 0000 r(a, A) = 0 r(a, A1) = 001 r(a, A1) = 00 r(a, A) = 0000 r(a, A) = 01 r(a, A) = 000 r(a, A) = 0 r(a, A) = 0 r(a, A) = 0 r(a, A) = 00 r(a, A) = 0 r(a, A) = 0000 r(a0, A1) = 00 r(a, A) = 00 r(a, A) = 0 r(a, A) = 0000 r(a0, A) = 0 r(a, A) = 00 r(a, A1) = 01 r(a, A) = 0000 r(a0, A) = 000 r(a1, A) = 0 r(a, A1) = 01 r(a, A) = 0 r(a0, A) = 000 r(a1, A) = 0 r(a, A1) = 0 r(a1, A1) = 0 r(a0, A1) = 0 r(a1, A) = 0 r(a, A) = 01 r(a1, A1) = 0 r(a0, A) = 010 r(a, A) = 01 r(a, A) = 0 r(a1, A) = 01 r(a0, A) = 01 r(a, A) = 01 r(a, A1) = 0 r(a1, A1) = 0 r(a0, A) = 0 r(a, A) = 0 r(a, A1) = 0 r(a1, A) = 0 r(a1, A) = 00 r(a, A) = 000 r(a, A1) = 0 r(a1, A) = 0 r(a1, A) = 00 1 Atomic masses and binding energies The relative atomic masses of the neutron n, 1 H, H (deuterium, D), H (tritium, T), He, He, Si, and Rb, which are data items B1, B, B, B, B, B, B1, and B0 in Table IV, are from AME, the most recent atomic mass evaluation from the AMDC, Lanzhou (Huang et al, 1; Wang et al, 1) AME supersedes AME used in CODATA-1 Since the results from the University of Washington and the Florida State University included in the 1 adjustment as separate input data have been taken into account in AME, they no longer are treated separately and the values of A r ( H), A r ( H), and A r ( He) from AME are used The values above are also listed in Table VIII together with those for Ar, Ar, and 0 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, 1 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 AUTHOR SUBMITTED MANUSCRIPT - MET-R error The covariances among the eight AME values in Table VIII used as input data are taken from the file covariancecovar available at the AMDC Web site and are used as appropriate in calculations; they are given in the form of correlation coefficients in Table V 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 B0 in Table VII for input data A r (n) and A r ( Rb) are quite simple However, observational equations B, B, B, B, B, and B1 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 He), alpha particle (nucleus of He), and hydrogenic Si, respectively Similarly, the denominator on the right-hand-side of observational equation B and the term in square brackets on the right-hand-side of B1 determinetherelativeatomicmassofthe Cnucleusand of hydrogenic C, respectively These equations follow

7 AUTHOR SUBMITTED MANUSCRIPT - MET-R TABLE IV Summary of principal input data relevant to the fundamental constants other than R for the CODATA 1 Special Adjustment Item Input datum Value Relative standard Identification number uncertainty 1 u r B1 A r(n) 100() AME- B A r( 1 H) 1001() AME- B E B( 1 H + )/hc 100() m 1 1 ASD-1 B A r( H) 001() 0 AME- B E B( H + )/hc 100() m 1 1 ASD-1 B A r( H) 00() AME- B E B( H + )/hc 1010() m 1 1 ASD-1 B A r( He) 00() AME- B E B( He + )/hc 10() m 1 1 ASD-1 B A r( He) 000() 1 AME- B E B( He + )/hc () m 1 ASD-1 B ω c( C + )/ω c(p) 00(1) MPIK-1 B E B( C + )/hc 0() m 1 ASD-1 B1 ω s( C + )/ω c( C + ) 0() MPIK- B E B( C + )/hc 0() m 1 1 ASD-1 B δ C 00() [1 ] theory B1 ω s( Si + )/ω c( Si + ) 0(1) MPIK- B1 A r( Si) () 1 AME- B1 E B( Si + )/hc 0(1) m 1 ASD-1 B δ Si 00(1) [ ] theory B a e 10() HarvU-0 B δ e 0000() [01 ] theory B R 0000() BNL-0 B ν( MHz) (1) khz LAMPF- B ν( MHz) () Hz LAMPF- B1 ν Mu 0() khz LAMPF- B ν Mu 0() Hz 1 LAMPF- B δ Mu 0() Hz [1 ] theory B µ p/µ N () UMZ-1 B µ e(h)/µ p(h) () 10 MIT- B0 µ d (D)/µ e(d) (0) 11 MIT- B1 µ e(h)/µ p () 11 MIT- B µ h/µ p 0() NPL- B µ n/µ p 0() ILL- B1 µ p(hd)/µ d (HD) 1() StPtrsb-0 B µ p(hd)/µ d (HD) 1() WarsU- B µ t(ht)/µ p(ht) 10() 0 StPtrsb- B σ dp () theory B σ tp 1() theory B1 h 01() J s NIST- B h 0() J s NIST- B h 0() J s 1 NIST-1 B h 00(0) J s 1 NRC-1 B h 000() J s LNE-1 B h/m( Rb) () m s 1 1 LKB- B0 A r( Rb) 010() AME- 1 The values in brackets are relative to the quantities g( C + ), g( Si + ), a e, or ν Mu as appropriate Page of

8 Page of TABLE IV (Continued) AUTHOR SUBMITTED MANUSCRIPT - MET-R Item Input datum Value Relative standard Identification number uncertainty u r B1 1 d 0(W1)/d 0(ILL) () NIST- B 1 d 0(MO )/d 0(ILL) () NIST- B 1 d 0(NR)/d 0(ILL) () NIST- B 1 d 0(N)/d 0(W1) () NIST- B d 0(Wa)/d 0(W0) 1 1() PTB- B1 d 0(W1)/d 0(W0) 1 () PTB- B d 0(W1)/d 0(W0) 1 () NIST-0 B d 0(MO )/d 0(W0) 1 () PTB- B1 d 0(NR)/d 0(W0) 1 () PTB- B d 0(NR)/d 0(W0) 1 () NIST-0 B d 0/d 0(W0) 1 () PTB-0 B0 d 0(NR)/d 0(W0) 1 () NIST-0 B1 d 0(MO ) 10() fm INRIM-0 B d 0(W0) 10() fm 1 INRIM-0 B1 d 0(Wa) 11() fm 1 INRIM-0 B d 0(Wa) 1() fm PTB-1 B1 N A 0(1) mol 1 0 IAC- B N A 00() mol 1 0 IAC- B N A 0(0) mol 1 1 IAC-1 B N A 0() mol 1 NMIJ-1 B1 R () J mol 1 K 1 1 NIST- B R () J mol 1 K 1 LNE-0 B R () J mol 1 K 1 NPL- B R () J mol 1 K 1 1 LNE- B R () J mol 1 K 1 10 LNE- B R () J mol 1 K 1 11 INRIM- B R (1) J mol 1 K 1 0 NIM-1 B R 0() J mol 1 K 1 0 NPL-1 B R 1(0) J mol 1 K 1 0 LNE-1 B R 1() J mol 1 K 1 UVa/CEM-1 B1 k/h 0(0) Hz K 1 NIM/NIST- B k/h 00() Hz K 1 NIM/NIST-1 B k/h 0() Hz K 1 0 NIST-1 B A ǫ( He)/R 0() m K J 1 1 PTB-1 B α 0( He)/πǫ 0a 0 10(1) 10 theory B λ(cukα 1)/d 0(Wa) 00() 0 FSUJ/PTB-1 B0 λ(cukα 1)/d 0(N) 000() NIST- B1 λ(wkα 1)/d 0(N) 0() 0 NIST- B λ(mokα 1)/d 0(N) 000(1) NIST-

9 TABLE V Correlation coefficients r(x i,x j) 0001 of the input data in Table IV For simplicity, the two items of data to which a particular correlation coefficient corresponds are identified by their item numbers in Table IV r(b1, B) = 0 r(b1, B0) = 00 r(b, B1) = 0 r(b, B) = 000 r(b1, B) = 0 r(b, B1) = 0 r(b, B) = 0 r(b, B) = 000 r(b1, B) = 00 r(b, B) = 01 r(b, B1) = 00 r(b, B) = 00 r(b1, B) = 00 r(b1, B) = 000 r(b1, B) = 0 r(b, B) = 0 r(b1, B1) = 00 r(b1, B) = 0 r(b1, B1) = 0 r(b, B) = 000 r(b1, B0) = 000 r(b1, B) = 0 r(b, B) = 00 r(b, B) = 000 r(b, B) = 0 r(b1, B) = 0 r(b, B0) = 00 r(b, B) = 0001 r(b, B) = 0 r(b1, B) = 0 r(b, B1) = 0 r(b, B) = 000 r(b, B) = 0 r(b1, B) = 00 r(b, B0) = 00 r(b, B) = 000 r(b, B1) = 01 r(b1, B0) = 00 r(b1, B) = 0 r(b, B) = 0 r(b, B0) = 00 r(b, B) = 0 r(b1, B) = 01 r(b, B) = 000 r(b, B) = 0 r(b, B) = 00 r(b1, B) = 0 r(b, B) = 000 r(b, B) = 01 r(b, B) = 00 r(b, B) = 00 r(b, B) = 00 r(b, B1) = 00 r(b, B) = 00 r(b, B) = 0 r(b, B) = 00 r(b, B0) = 00 r(b, B0) = 00 r(b, B) = 0 r(b, B) = 00 r(b, B) = 0 r(b, B) = 0 r(b1, B) = 0001 r(b, B) = 00 r(b, B1) = 0 r(b, B) = 00 r(b1, B) = 000 r(b, B) = 000 r(b, B0) = 00 r(b, B) = 0 r(b1, B) = 000 r(b, B) = 001 r(b, B1) = 01 r(b, B0) = 00 r(b1, B) = 000 r(b, B) = 001 r(b, B0) = 000 r(b, B) = 00 r(b, B) = 000 r(b, B) = 0 r(b1, B1) = 0 r(b, B) = 00 r(b, B) = 00 r(b, B) = 00 r(b, B) = 0 r(b, B0) = 00 r(b, B) = 00 fromthe fact that to produceanion X n+ with net charge ne, the energy required to remove n electrons from the neutral atom is the sum of the electron ionization energiese I (X i+ ) Thus, the relativeatomicmass ofan atom, its ions, the relative atomic mass of the electron A r (e), and the relative-atomic-mass equivalent of the binding energy of the removed electrons E B (X n+ )/m u c are related according to where A r (X) = A r (X n+ )+na r (e) E B(X n+ ) m u c, (1) E B (X n+ ) m u c = α A r (e) E B (X n+ ), () R hc and where m u = m( C)/ is the atomic mass constant and α is the fine-structure constant Equation () 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 m e c/h All eight removal energies used in the 1 Special Adjustment, items B, B, B, B, B, B, B, and B1 in Table IV, are obtained from the ionization energies given in the current version of the NIST online Atomic Spectra Database (Kramida et al, 1) The only change since 1 is that for the ionization energy C + ; the last digit is now rather than (see Table III in CODATA-1) However, the correlation coefficient of the C + and C + removal energies is unchanged from its CODATA-1 value 1000 and is included in Table V Item B in Table IV with identification MPIK-1, the ratio of the cyclotron frequency of the C + ion to that AUTHOR SUBMITTED MANUSCRIPT - MET-R of the proton in the same magnetic flux density B, is a new result obtained at MPIK, Heidelberg, and reported by Heiße et al (1) Its observational equation, B in Table VII, shows that, like input datum B with observational equation B, the frequency ratio ω c ( C + )/ω c (p) determines A r (p) The MPIK-1 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 C + ion in the other are alternately shuttled back and forth into the measurement trap where its cyclotron frequency is determined In parts in, the statistical relative standard uncertainty of the measured frequency ratio is 1 and the net correction for various systematic effects is with an uncertainty of The largest contributor to the latter uncertainty is due to an effect termed residual magnetostatic inhomogeneity and the largest correction,, is for image charge As discussed in Sec IV, the uncertainty of A r (p) obtained from ω c ( C + )/ω c (p) is about 1/ that of A r (p) obtained from A r ( 1 H) of AME, 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 B1 in Table IV is the ratio of the spinprecession (spin-flip frequency) of the C + hydrogenic Page of

10 Page of TABLE VI 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 VII AUTHOR SUBMITTED MANUSCRIPT - MET-R Adjusted constant Symbol neutron relative atomic mass A r(n) electron relative atomic mass A r(e) proton relative atomic mass A r(p) 1 H + electron removal energy E B( 1 H + ) deuteron relative atomic mass A r(d) H + electron removal energy E B( H + ) triton relative atomic mass A r(t) H + electron removal energy E B( H + ) helion relative atomic mass A r(h) He + electron removal energy E B( He + ) alpha particle relative atomic mass A r(α) He + electron removal energy E B( He + ) C + electron removal energy E B( C + ) C + electron removal energy E B( C + ) additive correction to g C(α) δ C Si + relative atomic mass A r( Si + ) Si + electron removal energy E B( Si + ) additive correction to g Si(α) δ Si fine-structure constant α additive correction to a e(th) 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 Rb relative atomic mass A r( Rb) d 0 of Si crystal WASO 1 d 0(W1) d 0 of Si crystal ILL d 0(ILL) d 0 of Si crystal MO d 0(MO ) d 0 of Si crystal NR d 0(NR) d 0 of Si crystal N d 0(N) d 0 of Si crystal WASO a d 0(Wa) d 0 of Si crystal WASO 0 d 0(W0) d 0 of an ideal Si crystal d 0 d 0 of Si crystal NR d 0(NR) molar gas constant R static electric dipole polarizability of He in atomic units α 0( He) copper Kα 1 x unit xu(cukα 1) ångstrom star Å molybdenum Kα 1 x unit xu(mokα 1) ion to its cyclotron frequency in the same applied magnetic flux density B; and data item B1 is the similar ratio for the hydrogenic ion Si + They are correlated with a correlation coefficient of 0, and the experiments on which they are based are discussed in CODATA-1 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 The observational equations for the C + and Si + frequency ratios are B1 and B1 in Table VII 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( C + ) and g( Si + ) The theory of these g-factors is discussed in CODATA-1 and the only changes are as follows: (i) the remainder terms R SE (α) = 0() and R SE (1α) = (), which are Eqs () and () in CODATA-1, are replaced by the improved values R SE (α) = (1) and R SE (1α) = 000(1) obtained by Yerokhin and Harman (1); and (ii) an additional term from light-by-light scattering diagrams calculated by Czarnecki and Szafron () has been included in Eq () of CODATA-1 for the C e () (Zα) coefficient This contribution is similar to that discussed in Sec IIIB 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 B and B in Table IV, are nearly the same as their CODATA-1 values as is their covariance and hence correlation coefficient; these are now u(δ C,δ Si ) = and r(δ C,δ Si ) = 01 The observational equations for the carbon and silicon additive corrections, B and B in Table VII, are simply δ C = δc and δ Si = δsi It is worth noting that the value of A r (e) recently obtained by Zatorski et al (1) 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 B in Table IV with u r = is the experimental value ofa e with the smallest uncertainty achieved to date Discussed in CODATA- and used in both the and 1 adjustments, it was obtained at Harvard University, Cambridge, using a Penning trap (Hanneke et al, 0); no other result is competitive As discussed in CODATA-1, the theoretical expression for a e can be written as a e (th) = a e (QED)+a e (weak)+a e (had) () The electroweak and hadronic contributions are small

11 TABLE VII Observational equations that express the input data in Table IV as functions of the adjusted constants in Table VI The numbers in the first column correspond to the numbers in the first column of Table IV For simplicity, the lengthier functions are not explicitly given See Sec IIIB for an explanation of the symbol = Type of input datum B1 B B B B B B B B B B B B B1 B B B1 B1 B1 B B B B B,B Observational equation A r(n) = A r(n) A r( 1 H) = A r(p)+a r(e) E B( 1 H + )α A r(e)/r hc E B( 1 H + )/hc = E B( 1 H + )/hc A r( H) = A r(d)+a r(e) E B( H + )α A r(e)/r hc E B( H + )/hc = E B( H + )/hc A r( H) = A r(t)+a r(e) E B( H + )α A r(e)/r hc E B( H + )/hc = E B( H + )/hc A r( He) = A r(h)+a r(e) E B( He + )α A r(e)/r hc E B( He + )/hc = E B( He + )/hc A r( He) = A r(α)+a r(e) E B( He + )α A r(e)/r hc E B( He + )/hc = E B( He + )/hc ω c( C + ) ω c(p) = A r(p) A r(e)+ E B( C + )α A r(e)/r hc E B( C + )/hc = E B( C + )/hc ( ω s C +) ( ω c C +) g = C(α)+δ C [ ( Ar(e)+ E B C +) α A ] r(e)/r hc A r(e) E B( C + )/hc = E B( C + )/hc δ C = δc ( ω s Si +) ( ω c Si +) g = Si(α)+δ Si A r( Si + ) A r(e) A r( Si) = A r( Si + )+A r(e) E B( Si + )α A r(e)/r hc E B( Si + )/hc = E B( Si + )/hc δ Si = δsi a e = ae(α)+δ e δ e = δe R = a µ m e 1+a e(α)+δ e m µ ( ν(f p) = ν m e µ e µ p µ e f p;r,α,,a µ,,δ e,δ Mu m µ µ p ) B ν Mu = νmu (R,α,,a µ m µ B B B B0 AUTHOR SUBMITTED MANUSCRIPT - MET-R δ Mu = δmu µ p µ e(h) µ p(h) µ d (D) µ e(d) m e µ N = (1+ae(α)+δ e) = ge(h) ( gp(h) g e g p = g ( d(d) ge(d) g d g e A r(p) A r(e) ) 1 µe µ p ) 1 µd µ e +δ Mu µ p µ e ) Page of

12 Page of TABLE VII (Continued) Type of input datum B1 B B B B B B B B B0 B1-B0 B1-B µ e(h) Observational equation = ge(h) µ p g e µ h = µ h µ p µ p µ n = µn µ p µ p µ p(hd) = [1+σ dp ] µ d (HD) µ t(ht) = [1 σ tp] µ p(ht) σ dp = σdp h m( Rb) σ tp = σtp h = h = µ e µ p A r(e) A r( Rb) A r( Rb) = A r( Rb) d 0(X) µ p µ e µ e µ d µ t µ p cα R d 1 = d0(x) 0(Y) d 1 0(Y) d 0(X) = d 0(X) B N A = cm ua r(e)α B B B B B, B0 B1 B AUTHOR SUBMITTED MANUSCRIPT - MET-R R = R k h A ǫ( He) R α 0( He) πǫ 0a 0 λ(cukα 1) d 0(X) λ(wkα 1) d 0(N) λ(mokα 1) d 0(N) = R h R R cm ua r(e)α = α0( He) cm ua r(e)α πǫ 0a 0 π RhR = α0( He) πǫ 0a 0 = = = 00 xu(cukα1) d 0(X) 000 Å d 0(N) 01 xu(mokα1) d 0(N)

13 TABLE VIII Relative atomic masses used in the CODATA 1 Special Adjustment as given in the atomic mass evaluation and the defined value for C Atom Relative atomic Relative standard mass A r(x) uncertainty u r n 100() 1 H 1001() H 001() 0 H 00() He 00() He 000() 1 C (exact) Si () 1 Ar () 1 Ar () 0 Ar () 0 Rb 010() compared with the quantum electrodynamic contribution a e (QED) The latter is currently written as the sum of fivetermsoftheformc e (n) (α/π) n with n = 1,,,,and, since terms for n > 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 e (n) except for n = 1 Following CODATA-1 and using the 1 adjusted values of these mass ratios, the five coefficients are C () e = 0, C e () C () = 00, e = 1101, C () C () e = 1 1(), e = 0(), () where two new results have been incorporated in C e () and C e () First, in the mass-independent contribution to a e (QED), the numerically calculated eighth-order coefficient used in CODATA-1 due to Aoyama et al (), A () 1 = 1 (), is replaced with the semianalytical value A () 1 = 1 obtained by Laporta (1) This coefficient depends on 1 -loop Feynman diagrams and its evaluation is difficult; Laporta s result with 00 digit accuracy is the culmination of a -year effort It is noteworthy that the numerically calculated result differs from its essentially exact counterpart by less than 0σ Second, and again in the mass-independent contribution to a e (QED), the numerically calculated tenth-order coefficient employed in CODATA-1, A () 1 = (), alsodue toaoyamaet al(), isreplacedwith A () 1 = () This newly revised result published as an erratum to Aoyama et al () is a consequence of their correctionof an errordiscoveredin a portion of their previous calculation and additional numerical evaluations AUTHOR SUBMITTED MANUSCRIPT - MET-R The coefficient A () 1 depends on Feynman diagrams and its determination is formidable Thesumofa e (weak)givenincodata-1andthevalues of the various corrections that contribute to a e (had) also given there is 1() and is the value used in the 1 Special Adjustment It is consistent with the value 1() obtained by Jegerlehner (1) based on his own independent evaluation of each contribution and employing up-to-date experimental data The uncertainties of the coefficients C e () and C e () together with those of the weak and hadronic contributions yield for the standard uncertainty of the theoretical value u[a e (th)] = 00 The theoretical expression for a e is written as a e (th) = a e (α) + δ e, where the additive correction δ e is equal to 0000() and is input datum B in Table IV The observational equation for a e is B in Table VII and the observational equation for the additive correction is simply δ e = δe, which is equation B in Table VII The value of α that results from equating the Harvard experimental value and a e (th) is given in Table IX, Sec IV 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-1, the experimental value of a µ is derived from the BNL, Upton, measurement of the quantity R = f a / f p, which is item B in Table IV Here f a is the anomaly difference frequency equal to the difference between the muon spin-flip (precession) frequency and cyclotron frequency inthesameappliedmagneticfluxdensityb, and f p isthe free proton nuclear magnetic resonance(nmr) frequency corresponding to the average B seen by the circulating muons in their storage ring The observational equation B in Table VII 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- and 1 because of concerns regarding the theory Inasmuch as these concerns remain, the theory is also omitted from the 1 Special Adjustment However, its omission has no effect on the chosen numerical values of h, e, k, and N A Muonium hyperfine splitting Determination of the ground-state hyperfine-splitting of muonium (µ + e atom) together with the theory of the splittingdeterminesm e /m µ, whichinturnentersthetheory 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 1 and 1 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 Page of

14 Page of is the hyperfine splitting frequency ν Mu and their difference is a frequency denoted ν(f p ) The experiments are discussed in CODATA- and there have been no changes in the data since then The two 1 and two 1 frequencies are items B and B1, and B and B, in Table IV Their observational equations are B,B and B in Table VII For each experiment the two frequencies are correlated; the correlation coefficients are r[ ν Mu, ν(f p )] = 0 for the 1 data and r[ ν Mu, ν(f p )] = 01 for the 1 data and are taken into account in calculations For the 1 data f p = MHz corresponding to B of about 1T and for the 1 data f p = 000MHz corresponding to B of about = 1T There has been no change in the theory since the 1 December 1 closing date of CODATA-1, hence the uncertainty u(δ Mu ) = Hz of the additive correction δ Mu used in the 1 adjustment to account for the uncertainty of the theory is unchanged The correction δ Mu is item B in Table IV and its observational equation is simply δ Mu = δmu, which is equation B in Table VII Magnetic moment ratios Discussed in this section are the six input data B- B in Table IV with observational equations B-B in Table VII 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 B, µ N = e h/m p is the nuclear magneton, and in items B-B1 and B1 - B, 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-1 The exception is input data B and B 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 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 () and although discussed in CODATA-1, they were not included in the 1 adjustment because they became available only after its 1 December 1 closing date The observational equations in Table VII for items B to B1 contain four bound-particle to free particle ratios of the form g X (Y)/g X, where X is ether 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-0 and are given in Table XIII of CODATA-1 AUTHOR SUBMITTED MANUSCRIPT - MET-R Kibble and joule balance measurements of h The watt balance was conceived by Bryan Kibble of NPL, Teddington, in 1 With the discovery of the quantum-hall effect in 10 it became a means of measuring h Upon the passing of Kibble in April 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, ) How the Kibble balance determines h is discussed in CODATA-, and values of h so obtained have been included in all CODATA adjustments ever since The five values considered for inclusion in the CODATA 1 Special Adjustment are items B1-B in Table IV The NIST- value of h, item B1, was obtained by Williams et al (1) using a Kibble balance called NIST- that operated in air and used a superconducting magnet to generate the required magnetic flux density B; it is discussed in CODATA-, but see also CODATA- 0 NIST- was replaced by a new Kibble balance called NIST- that operated in vacuum but used the same superconducting magnet The value of h obtained from this apparatus with u r =, identified as NIST-0, was taken as an input datum in both the 0 and adjustments However, as discussed in CODATA-1, issueswith thenist-0resultwereuncoveredthat ledto a number of improvements in the NIST- apparatus, additional data, and a thorough review of all previous NIST- data This led to the value identified as NIST-, item B in Table IV As reported by Schlamminger et al (), it is viewed as the final result from NIST- and is used as an input datum in CODATA-1 The NIST- and NIST- values are correlated with a correlation coefficient of 00 NIST- was replaced by a completely new Kibble balance called NIST- It uses a permanent magnet rather than a superconducting magnet to generate B but as in NIST- and NIST-, a wheel is used as the balance beam and to displace the moving coil Haddad et al () discuss the new apparatus, measurement procedures, and sources of uncertainty and report a first result with u r = based on data obtained from mid-december to early January using a 1kg 0 % Pt-%Ir mass standard After obtaining this first result the NIST researchers continued improving NIST- and acquiring additional data Based on all individual determinations of h obtained from mid-december through April 1, Haddad et al (1) report the result identified as NIST-1, item B in Table IV, with u r = 1 Each measurement was carried out with one of five different 0kg stainless steel mass standards, a 1kg stainless steel standard, or one of two 1 kg Pt-Ir standards, and various combinations to provide standards in the range 0kg to kg The uncertainty of the initial NIST- result was reduced by more than a factor of by (i) making significantly more measurements of h, thus allowing a more

15 realistic statistical analysis of the data; (ii) using mass standards as large as 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 parts in statistical uncertainty and parts in magnetic field uncertainty of the first result are reduced to 0 and 1 parts in, respectively The main contributors to the uncertainty of the 1 result are, in parts in, for electrical, 1 for mass metrology, 0 for magnetic field profile fitting, 0 for balance mechanics, for alignment, and for the local gravitational acceleration Item B in Table IV from NRC, Ottawa, identified as NRC-1 and reported by Wood et al (1), 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 0, is discussed in CODATA-1 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-1 with u r = 1 used as an input datum in CODATA-1 This value was based on four measurement campaigns carried out between September and December using four different mass standards The new NRC-1 result is based on a reanalysis of the measurements and three new determinations carried out in February, November, and December with the same 00g silicon, 1kg AuCu, and 00g AuCu mass standards used in However, additional improvements were made to the apparatus between the and 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 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- 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 parts in was applied to each of the seven values The total uncertainty is not dominated by any one component Item B, reported by Thomas et al (1) and identified as LNE-1, is the most recent and accurate result from the LNE, Trappes, Kibble-balance project initiated in 01 A first result reported by Thomas et al () with u r = 1 was obtained in 1 and an improved result with u r = 1 in in the course of the BIPM s Extraordinary Calibration Cam- AUTHOR SUBMITTED MANUSCRIPT - MET-R 1 paign Improvements in the apparatus and measurement procedures continued, including better calibration of the 00g 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-1 result was obtained over a day period during the spring of 1 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 parts in uncertainty of the required correctionfor the refractive index of air and air buoyancy is the dominant contributor to the parts in total uncertainty Li et al (1) have reported a first result for h from the NIM, Beijing, joule balance NIM- with u r = 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 1 Special Adjustment, inasmuchasvaluesofhwith smalleru r were not sufficiently competitive to be included in the final adjustment on which the CODATA 1 recommended values are based Although the results for the five Kibble-balance results discussed above are given as values of h in Table IV, in previous adjustments such results were given as values of K J R K = /h to facilitate tests of the exactness of the Josephson and quantum-hall effect relations K J = e/h and R K = h/e 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 constant h 0 = /K J 0 R K 0 = 0 Js, where K J 0 and R K 0 are the exact, conventional values of the Josephson and von Klitzing constants given in Table I The reported values of (h/h 0 ) 1 for the five results are, in parts in, (), (), (), 10(1), and () The five Kibble-balance values of h together with the four values of h that can be inferred from the four values of N A discussed in Sec IIIC are compared in Table X and Fig in Sec IV The observational equation for Kibble-balance measurements of h is B in Table VII and is simply h = h Measurement of h/m for Rb Accurate measurement of the quotient of h and the mass of a particle X can provide a competitive value of Page 1 of