CODATA Recommended Values of the Fundamental Physical Constants: 2014*

REVIEWS OF MODERN PHYSICS, VOLUME 88, JULY–SEPTEMBER 2016 CODATA recommended values of the fundamental physical constants: 2014*

Peter J. Mohr,† David B. Newell,‡ and Barry N. Taylor§ National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8420, USA (published 26 September 2016)

This paper gives the 2014 self-consistent set of values of the constants and conversion factors of physics and chemistry recommended by the Committee on Data for Science and Technology (CODATA). These values are based on a least-squares adjustment that takes into account all data available up to 31 December 2014. Details of the data selection and methodology of the adjustment are described. The recommended values may also be found at physics.nist.gov/ constants.

DOI: 10.1103/RevModPhys.88.035009

CONTENTS A. Relative atomic masses of atoms 5 B. Relative atomic masses of ions and nuclei 6 I. Introduction 2 C. Relative atomic mass of the deuteron, triton, A. Background 2 and helion 7 B. Highlights of the CODATA 2014 adjustment 3 IV. Atomic Transition Frequencies 8 1. Planck constant h, elementary charge e, A. Hydrogen and deuterium transition frequencies, the Boltzmann constant k, Avogadro constant NA, Rydberg constant R∞, and the proton and deuteron and the redefinition of the SI 3 charge radii r , r 8 2. Relative atomic mass of the electron A e 4 p d r 1. Theory of hydrogen and deuterium energy 3. Proton magnetic moment in units of theð nuclearÞ levels 9 magneton μ =μ 4 p N a. Dirac eigenvalue 9 4. Fine-structure constant α 4 b. Relativistic recoil 9 5. Relative atomic masses 4 c. Nuclear polarizability 10 6. Newtonian constant of gravitation G 4 d. Self energy 10 7. Proton radius r and theory of the muon p e. Vacuum polarization 10 magnetic-moment anomaly a 4 μ f. Two-photon corrections 11 C. Outline of the paper 5 g. Three-photon corrections 12 II. Special Quantities and Units 5 h. Finite nuclear size 12 III. Relative Atomic Masses 5 i. Nuclear-size correction to self energy and vacuum polarization 13

* j. Radiative-recoil corrections 13 This review is being published simultaneously by the Journal of k. Nucleus self energy 13 Physical and Chemical Reference Data. l. Total energy and uncertainty 13 This report was prepared by the authors under the auspices of the m. Transition frequencies between levels with CODATA Task Group on Fundamental Constants. The members of n 2 and the fine-structure constant α 13 the task group are F. Cabiati, Istituto Nazionale di Ricerca Metro- 2. Experiments¼ on hydrogen and deuterium 14 logica, Italy; J. Fischer, Physikalisch-Technische Bundesanstalt, 3. Nuclear radii 14 Germany; J. Flowers (deceased), National Physical Laboratory, a. Electron scattering 15 United Kingdom; K. Fujii, National Metrology Institute of Japan, b. Isotope shift and the deuteron-proton radius Japan; S. G. Karshenboim, Pulkovo Observatory, Russian Federation difference 15 and Max-Planck-Institut für Quantenoptik, Germany; E. de Mirandés, c. Muonic hydrogen 15 Bureau international des poids et mesures; P. J. Mohr, National B. Hyperfine structure and fine structure 16 Institute of Standards and Technology, United States of America; V. Magnetic Moments and g-factors 16 D. B. Newell, National Institute of Standards and Technology, United A. Electron magnetic-moment anomaly a and the States of America; F. Nez, Laboratoire Kastler-Brossel, France; e fine-structure constant α 17 K. Pachucki, University of Warsaw, Poland; T. J. Quinn, Bureau 1. Theory of a 17 international des poids et mesures; C. Thomas, Bureau international e 2. Measurements of a 18 des poids et mesures; B. N. Taylor, National Institute of Standards and e B. Muon magnetic-moment anomaly a 18 Technology, United States of America; B. M. Wood, National μ 1. Theory of a 18 Research Council, Canada; and Z. Zhang, National Institute of μ Metrology, People’s Republic of China. 2. Measurement of aμ: Brookhaven 19 †[email protected] 3. Comparison of theory and experiment for aμ 20 ‡[email protected] C. Proton magnetic moment in nuclear magnetons §[email protected] μp=μN 20

0034-6861=2016=88(3)=035009(73) 035009-1 Published by the American Physical Society Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

D. Atomic g-factors in hydrogenic 12C and 28Si A. Comparison of 2014 and 2010 CODATA and Ar e 21 recommended values 64 1. Theoryð Þ of the bound-electron g-factor 22 B. Some implications of the 2014 CODATA 12 5 28 13 2. Measurements of g C þ and g Si þ 24 recommended values and adjustment for VI. Magnetic-moment Ratios andð the Muon-electronÞ ð Þ metrology and physics 65 Mass Ratio 25 1. Conventional electrical units 65 A. Theoretical ratios of atomic bound-particle to 2. Josephson and quantum-Hall effects 66 free-particle g-factors 26 3. The new SI 66 1. Ratio measurements 26 4. Proton radius 66 B. Muonium transition frequencies, the muon-proton 5. Muon magnetic-moment anomaly 66

magnetic-moment ratio μμ=μp, and muon-electron 6. Electron magnetic-moment anomaly, mass ratio mμ=me 27 fine-structure constant, and QED 66 1. Theory of the muonium ground-state C. Suggestions for future work 66 hyperfine splitting 28 List of Symbols and Abbreviations 67 2. Measurements of muonium transition frequencies Acknowledgments 69

and values of μμ=μp and mμ=me 29 References 69 VII. Quotient of Planck Constant and Particle Mass h=m X and α 30 I. INTRODUCTION VIII. Electricalð MeasurementsÞ 31 A. NPL watt balance 31 This report describes work carried out under the auspices of B. METAS watt balance 32 the Task Group on Fundamental Constants, one of several task C. LNE watt balance 32 groups of the Committee on Data for Science and Technology D. NIST watt balance 32 (CODATA) founded in 1966 as an interdisciplinary committee E. NRC watt balance 33 of the International Council for Science (ICSU). It gives a IX. Measurements Involving Silicon Crystals 33 detailed account of the 2014 CODATA multivariate least- A. Measurements with natural silicon 33 squares adjustment of the values of the constants as well as the B. Determination of NA with enriched silicon 34 X. Thermal Physical Quantities 35 resulting 2014 set of over 300 self-consistent recommended A. Molar gas constant R, acoustic gas thermometry 35 values. The cutoff date for new data to be considered for 1. New values 35 possible inclusion in the 2014 adjustment was at the close of a. NIM 2013 35 31 December 2014, and the new set of values first became b. NPL 2013 36 available on 25 June 2015 at physics.nist.gov/constants, part c. LNE 2015 36 of the website of the Fundamental Constants Data Center of 2. Updated values 36 the National Institute of Standards and Technology (NIST), a. Molar mass of argon 36 Gaithersburg, Maryland, USA. b. Molar mass of helium 37 c. Thermal conductivity of argon 37 A. Background B. Quotient k=h, Johnson noise thermometry 37 C. Quotient Aϵ=R, dielectric-constant gas thermometry 37 The compilation of a carefully reviewed set of values of the D. Other data 38 fundamental constants of physics and chemistry arguably E. Stefan-Boltzmann constant σ 38 began over 85 years ago with the paper of Birge (1929). In XI. Newtonian Constant of Gravitation G 38 1969, 40 years after the publication of Birge’s paper, the A. Updated values 38 CODATA Task Group on Fundamental Constants was estab- 1. Huazhong University of Science and Technology 38 lished for the following purpose: to periodically provide B. New values 39 the scientific and technological communities with a self- 1. International Bureau of Weights and Measures 39 consistent set of internationally recommended values of the 2. European Laboratory for Non-Linear basic constants and conversion factors of physics and chem- Spectroscopy, University of Florence 39 istry based on all the data available at a given point in time. 3. University of California, Irvine 40 The Task Group first met this responsibility with its 1973 XII. Electroweak Quantities 41 multivariate least-squares adjustment of the values of the XIII. Analysis of Data 41 A. Comparison of data through inferred values of constants (Cohen and Taylor, 1973), which was followed α, h, and k 41 13 years later by the 1986 adjustment (Cohen and Taylor, B. Multivariate analysis of data 43 1987). Starting with its third adjustment in 1998 the Task 1. Data related to the Newtonian constant of Group has carried them out every 4 years; if the 1998 gravitation G 46 adjustment is counted as the first of the new 4-year cycle, 2. Data related to all other constants 48 the 2014 adjustment described in this report is the 5th of the 3. Test of the Josephson and quantum-Hall-effect cycle. Throughout this article we refer to the detailed relations 53 reports describing the 1998, 2002, 2006, 2010, and 2014 XIV. The 2014 CODATA Recommended Values 55 adjustments, or sometimes the adjustments themselves, as A. Calculational details 55 CODATA-XX, where XX is 98, 02, 06, 10, or 14 (Mohr and B. Tables of values 63 Taylor, 2000, 2005; Mohr, Taylor, and Newell, 2008a, 2008b, XV. Summary and Conclusion 64 2012a, 2012b).

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To help keep this report to a reasonable length, our data B. Highlights of the CODATA 2014 adjustment review focuses on the new results that became available We summarize here the most significant advances made, or between the 31 December 2010 and 31 December 2014 lack thereof, in our knowledge of the values of the funda- closing dates of the 2010 and 2014 adjustments (in this paper the term “past 4 years” means this time interval); our previous mental constants in the past 4 years and, where appropriate, reports should be consulted for discussions of the older data. their impact. The multivariate least-squares methodology Indeed, only new data are given both in the text where they are employed in the four previous adjustments is employed in the 2014 adjustment but in this case with N 141 items of first discussed and in the summary tables in Sec. XIII; data ¼ that have been considered for inclusion in one or more past input data, M 74 variables or unknowns, and ν N − M ¼ ¼ 2 ¼ adjustments are given only in those summary tables. Further, 67 degrees of freedom. The chi square statistic is χ 50.4 with probability p 50.4 67 0.93 and the Birge ratio¼ is extensive descriptions of new experiments and theoretical ð j Þ¼ calculations are generally omitted; comments are made on R 50.4=67 0.87. This adjustment includes data for B ¼ ¼ only their most relevant features. the Newtonian constant of gravitation G, although it is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Readers should also consult the earlier reports for discus- independent of the other constants. sions of the motivation for, and underlying philosophy of, CODATA adjustments, the treatment of numerical calcula- 1. Planck constant h, elementary charge e, Boltzmann tions and uncertainties, etc. With regard to uncertainties, as in constant k, Avogadro constant NA, and the past adjustments they are always given as “standard uncer- redefinition of the SI tainties,” that is, 1 standard deviation estimates, either in the It is planned that at its meeting in the fall of 2018, the 26th unit of the quantity being considered and thus absolute, or as a General Conference on Weights and Measures (CGPM) will relative standard uncertainty, denoted ur. As an aid to the adopt a resolution to revise the International System of Units reader, included near the end of this report is a comprehensive (SI). This new SI, as it is sometimes called, will be defined by list of symbols and abbreviations. assigning exact values to the following seven defining con- Because of its importance, we do once again state that, as a stants: the ground-state hyperfine-splitting frequency of the working principle, the validity of the physical theory under- 133Cs atom ν , the speed of light in vacuum c, the Planck lying the 2014 adjustment is assumed. This includes, as in Δ Cs constant h, the elementary charge e, the Boltzmann constant k, previous adjustments, special and general relativity, quantum the Avogadro constant N , and the luminous efficacy of mechanics, quantum electrodynamics (QED), the standard A monochromatic radiation of frequency 540 THz, K . As a model of particle physics, including CPT invariance, and for cd result of the significant advances made since CODATA-10 in all practical purposes the exactness of the relations K 2e=h J watt-balance measurements of h, x-ray-crystal-density and R h=e2, where K and R are the Josephson¼ and K J K (XRCD) measurements of N using silicon spheres composed von Klitzing¼ constants, respectively, and e is the elementary A charge and h is the Planck constant. of highly enriched silicon, and acoustic-gas-thermometry There continues to be no observed time variation of the (AGT) measurements of the molar gas constant, the relative values of the constants relevant to the data used in adjustments standard uncertainties of the four constants h, e, k, and NA 8 carried out in our current era. Indeed, a recent summary based have been reduced (respectively, in parts in 10 ) from 4.4, 2.2, on frequency ratio measurements of various transitions in 91, and 4.4 in CODATA-10 to 1.2, 0.61, 57, and 1.2, in different atomic systems carried out over a number of years in CODATA-14. (The defining constants ΔνCs, c, and Kcd will several different laboratories gives −0.7 2.1 × 10−17 per year retain their present values.) as the constraint on the time variationð ofÞ the fine-structure This is a truly major development, because these uncer- constant α and −0.2 1.1 × 10−16 per year for the proton-to- tainties are now sufficiently small that the adoption of the new ð Þ SI by the 26th CGPM is expected. It has been made possible to electron mass ratio mp=me (Godun et al., 2014). In general, a result considered for possible inclusion in a a large extent by the resolution of the disagreement between CODATA adjustment is identified by the institution where the different watt-balance measurements of h and the disagree- work was primarily carried out and by the last two digits of the ment of the value of h inferred from the XRCD value of NA year in which it was published in an archival journal. Even if a with one of the watt-balance values. These disagreements led result is labeled with a “15” identifier, it can be safely the Task Group to increase the initial assigned uncertainties of assumed that it was available by the 31 December 2014 the 2010 data that contributed to the determination of h by a closing date for new data. A new result was considered to have factor of 2. Further, the reduction in the relative uncertainty of met this date if published, or if the Task Group received a k from 9.1 × 10−7 to 5.7 × 10−7 is in large part a consequence preprint describing the work by that date and it had already of three new AGT determinations of the molar gas constant been, or was about to be, submitted for publication. However, with relative uncertainties (in parts in 106) of 0.9, 1.0, and 1.4. this closing date does not apply to clarifying information The significant reductions in the uncertainties of h, e, k, and requested from authors. The name of an institution is always NA have also led to the reduction of the uncertainties of many given in full together with its abbreviation when first used, but other constants and conversion factors. for the convenience of the reader the abbreviations and full The values of the constants to be adopted by the CGPM for institutional names are also included in the aforementioned the redefinition will be based on a special least-squares comprehensive list of symbols and abbreviations near the end adjustment carried out by the Task Group during the summer of this report. of 2017. Data for this adjustment must be described in a paper

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-3 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ... that has been published or accepted for publication by values are generally used for the various relative atomic 1 July 2017. masses required for the 2014 adjustment, including that for the neutron. Because AME2012 is a self-consistent evaluation

2. Relative atomic mass of the electron Ar e based on data included in CODATA-10, those data are ð Þ neither discussed nor included in CODATA-14. However, The relative standard uncertainty of the 2014 recommended two new, highly precise pairs of cyclotron frequency ratios value of A e is 2.9 × 10−11, nearly 14 times smaller than that r relevant to the determination of the masses of the of the 2010ð Þ recommended value. It is based on extremely deuteron, triton, and helion (nucleus of the 3He atom) were accurate measurements, using a specially designed triple reported after the completion of AME2012 and are included in Penning trap, of the ratio of the electron spin-precession this adjustment. Yet, because the values of the relative (or spin-flip) frequency in hydrogenic carbon and silicon ions atomic mass of 3He implied by the relevant ratio in each to the cyclotron frequency of the ions, together with the theory pair disagree, the initial uncertainty of each of these ratios is of the electron bound-state g-factor in the ions. The uncer- multiplied by 2.8 to reduce the inconsistency to an accept- tainties of the measurements are so small that the data used to able level. obtain the CODATA-10 value of Ar e are no longer competitive and are excluded from the 2014ð Þ adjustment. Thus, there is no discussion of antiprotonic helium in this report. The 6. Newtonian constant of gravitation G new value of A e will eliminate a potentially significant Three new values of G obtained by different methods have rð Þ source of uncertainty in obtaining the fine-structure constant become available for CODATA-14 with relative standard from anticipated high-accuracy atom-recoil measurements of uncertainties of 1.9 × 10−5, 2.4 × 10−5, and 15 × 10−5, h=m for an atom of mass m. respectively, but have not resolved the considerable disagreements that have existed among the measurements of G for the 3. Proton magnetic moment in units of the nuclear past 20 years. These inconsistencies led the Task Group to magneton μp=μN apply an expansion factor of 14 to the initial uncertainty of The CODATA-10 recommended value of the magnetic each of the 11 values available for the 2010 adjustment and to adopt their weighted mean with its relative uncertainty of moment of the proton in nuclear magnetons μp=μN, where −5 μ eℏ=2m and m is the proton mass, has a relative 12 × 10 as the 2010 recommended value. The expansion N ¼ p p standard uncertainty of 8.2 × 10−9 and is calculated from factor 14 was chosen so that the smallest and largest values other measured constants including the electron to proton would differ from the recommended value by about twice its mass ratio. However, because of the development of a unique uncertainty. For the 2014 adjustment the Task Group has double Penning trap similar to the triple Penning trap decided that its usual practice in such cases, which is to choose mentioned in the previous section, for the first time a value an expansion factor that reduces the normalized residual of of μ =μ from direct measurements of the spin-flip and each datum to less than 2, should be followed instead. Thus an p N expansion factor of 6.3 is chosen and the weighted mean of the cyclotron frequencies of a single proton with an uncertainty 14 values with its relative uncertainty 4.7 × 10−5 is adopted as of 3.3 × 10−9 has become available. As a consequence, the the 2014 recommended value. Because of the three new values uncertainty of the 2014 recommended value is 3.0 × 10−9, of G, the 2014 recommended value is larger than the 2010 which is 2.7 times smaller than that of the 2010 value, and value by 3.6 parts in 105. similar reductions in the uncertainties of other constants that depend on the μp=μN result. 7. Proton radius rp and theory of the muon magnetic-moment anomaly a 4. Fine-structure constant α μ The very precise value of the root-mean-square charge Improved numerical calculations of the 8th- and 10th-order radius of the proton r obtained from spectroscopic measure- mass-independent coefficients of the theoretical expression p ments of a Lamb-shift transition frequency in the muonic for the electron magnetic-moment anomaly a have allowed e hydrogen atom -p was omitted from CODATA-10 because of full advantage to be taken of the 2.4 × 10−10 relative standard μ its significant disagreement with the value from electron- uncertainty of the experimental value of a for the determi- e proton elastic scattering and from spectroscopic measure- nation of the fine-structure constant; the relative uncertainty of ments of hydrogen and deuterium. Although the originally −10 the 2014 recommended value of α is 2.3 × 10 compared measured Lamb-shift frequency has been reevaluated, the with 3.2 × 10−10 for the CODATA-10 value. However, result from a second frequency that gives a value of rp because of the somewhat unexpected large size of the 10th- consistent with the first has been reported, and improvements order coefficient, the 2014 recommended value of α is were made to the theory required to extract r from the Lamb- fractionally smaller than the CODATA-10 value by 4.7 parts p shift frequencies, the disagreement persists. The Task group in 1010. has, therefore, decided to omit the muonic hydrogen result for rp from the 2014 adjustment. 5. Relative atomic masses Similarly, because the value of the muon magnetic-moment

A new atomic mass evaluation, called AME2012, was anomaly aμ th predicted by the theoretical expression for the completed and published by the Atomic Mass Data Center anomaly significantlyð Þ disagreed with the value obtained from (now transferred from France to China), and its recommended a seminal experiment at Brookhaven National Laboratory,

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USA, the theory was omitted from CODATA-10. Even though II. SPECIAL QUANTITIES AND UNITS much effort has been devoted in the past 4 years to improving the theory, the disagreement and concerns about the theory Table I gives the values of a number of exactly known remain. Thus the Task Group has also decided not to employ constants of interest. The speed of light in vacuum c is exact as a consequence of the definition of the meter in the SI and the the theory of aμ in the 2014 adjustment. magnetic constant (vacuum permeability) μ0 is exact because of the SI definition of the ampere (BIPM, 2006). Thus the C. Outline of the paper 2 electric constant (vacuum permittivity) ϵ0 1=μ0c is also exact. The molar mass of carbon 12, M 12¼C , is exact as a Some constants that have exact values in the International ð Þ System of Units (SI) (BIPM, 2006), which is the unit system consequence of the SI definition of the mole, as is the molar 12 used in all CODATA adjustments, are recalled in Sec. II. mass constant Mu M C =12. By definition, the relative atomic mass of the carbon¼ ð 12Þ atom A 12C 12 is exact. The Sections III through XII discuss the input data with an rð Þ¼ emphasis on the new results that have become available quantities KJ−90 and RK−90 are the exact, conventional values during the past 4 years. As discussed in Appendix E of of the Josephson and von Klitzing constants adopted by the CODATA-98, in a least-squares analysis of the values of the International Committee for Weights and Measures (CIPM) in constants, the numerical data, both experimental and theo- 1989 for worldwide use starting 1 January 1990 for mea- retical, also called observational data or input data, are surements of electrical quantities using the Josephson and expressed as functions of a set of independent variables or quantum-Hall effects (BIPM, 2006). Quantities measured in unknowns called adjusted constants. The functions them- terms of these conventional values are labeled with a sub- selves are called observational equations, and the least-squares script 90. methodology yields best estimates of the adjusted constants in the least-squares sense. Basically, the methodology provides III. RELATIVE ATOMIC MASSES the best estimate of each adjusted constant by automatically taking into account all possible ways its value can be The relative atomic masses of some particles and ions are determined from the input data. The best values of other used in the least-squares adjustment. These values are constants are calculated from the best values of the adjusted extracted from measured atom and ion masses by calculating constants. the effect of the bound-electron masses and the binding The analysis of the input data is discussed in Sec. XIII. It is energies, as discussed in the following sections. carried out by directly comparing measured values of the same quantity, by comparing measured values of different quantities A. Relative atomic masses of atoms through inferred values of α, h, and k, and by carrying out least-squares calculations. These investigations are the basis Results from the periodic atomic mass evaluations (AMEs) for the selection of the final input data used to determine the carried out by the Atomic Mass Data Center (AMDC), Centre adjusted constants, and hence the entire 2014 CODATA set of de Spectrométrie Nucléaire et de Spectrométrie de Masse recommended values. (CSNSM), Orsay, France, have long been used as input data in Section XIV provides, in several tables, the set of over 300 CODATA adjustments. Indeed, results from AME2003, the CODATA-14 recommended values of the basic constants and most recent evaluation at the time, were employed in the 2006 conversion factors of physics and chemistry, including the and 2010 CODATA adjustments. In 2008 a memorandum covariance matrix of a selected group of constants. The report between the Institute of Modern Physics, Chinese Academy of concludes with Sec. XV, which includes a comparison of a Sciences (IMP), in Lanzhou, PRC, and CSNSM was signed representative subset of 2014 recommended values with their that initiated the transfer of the AMDC from CSNMS to IMP. 2010 counterparts, comments on some of the implications of The transfer was concluded in 2013 after the completion of CODATA-14 for metrology and physics, and some sugges- AME2012, which supersedes its immediate predecessor, tions for future work, both experimental and theoretical, that AME2003. The results of the 2012 evaluation, which was could advance our knowledge of the values of the fundamental a collaborative effort between IMP and CSNSM, are pub- constants. lished (Audi et al., 2012; Wang et al., 2012) and are also

TABLE I. Some exact quantities relevant to the 2014 adjustment.

Quantity Symbol Value 1 Speed of light in vacuum c, c0 299 792 458 ms− −7 −2 −7 −2 Magnetic constant μ0 4π × 10 NA 12.566 370 614… × 10 NA 2 −1 ¼ −12 −1 Electric constant ϵ0 μ0c 8.854 187 817… × 10 Fm Molar mass of 12C M 12C ð Þ ¼ 12 × 10−3 kg mol−1 ð Þ 12 −3 −1 Molar mass constant Mu M C =12 10 kg mol 12 12 ð Þ ¼ Relative atomic mass of C Ar C 12 ð Þ −1 Conventional value of Josephson constant KJ−90 483 597.9 GHz V Conventional value of von Klitzing constant RK−90 25 812.807 Ω

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-5 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ... available on the AMDC website at amdc.impcas.ac.cn/ with net charge ne, the energy needed to remove n electrons evaluation/data2012/ame.html. from the neutral atom is the sum of the electron ionization i The AME2012 relative atomic mass values of interest for energies EI X þ : the 2014 adjustment are given in Table II; additional digits ð Þ were supplied in 2014 to the Task Group by M. Wang of the n−1 n i AMDC to reduce rounding errors. However, the AME2012 ΔEB X þ EI X þ : 1 ð Þ¼ ð Þ ð Þ values for A 2H and A 3He from which the relative atomic i 0 r r X¼ masses of theð deuteronÞ ð d andÞ helion h (nucleus of the 3He 0 atom) can be obtained are not included. This is because the For a neutral atom we have n 0 and ΔEB X þ 0; for a ¼ ð Þ¼ AME2012 value for A 2H is based to a large extent on bare nucleus n Z. In the 2014 least-squares adjustment, we r ¼ preliminary data from theð Þ group of R. Van Dyck at the use the removal energies expressed in terms of wave numbers University of Washington (UWash), Seattle, Washington, given by USA, that have been superseded by recently reported final 1 7 −1 data (Zafonte and Van Dyck, 2015). Further, the AME2012 ΔEB Hþ =hc 1.096 787 717 4307 10 × 10 m ; 3 ð Þ ¼ ð Þ value for Ar He is partially based on very old UWash data 3 7 −1 ΔEB Hþ =hc 1.097 185 4390 13 × 10 m ; that have beenð supersededÞ by newer and much more accurate ð Þ ¼ ð Þ 4 2 7 −1 data given in the paper that reports the final A 2H -related ΔEB He þ =hc 6.372 195 4487 28 × 10 m ; rð Þ ð Þ ¼ ð Þ data. These new UWash results are discussed below in 12 6 7 −1 ΔEB C þ =hc 83.083 962 72 × 10 m ; Sec. III.C together with new measurements related to the ð Þ ¼ ð Þ 12 5 7 −1 ΔEB C þ =hc 43.563 345 72 × 10 m ; triton and helion from the group of E. Myers at Florida State ð Þ ¼ ð Þ 28 13 7 −1 University (FSU), Tallahassee, Florida, USA. ΔEB Si þ =hc 420.608 19 × 10 m ; The covariances among the AME2012 values in Table II are ð Þ ¼ ð Þ taken from the file covariance.covar available at the AMDC which follow from the data tabulated in Table III. In that table, website indicated above and are used as appropriate in our the value for 1H is from Jentschura et al. (2005), and the rest calculations. They are given in the form of correlation are from the NIST online Atomic Spectra Database (ASD, coefficients in Table XIX, Sec. XIII. 2015), in which the value for 3H is based on a calculation by In the four previous CODATA adjustments, the recom- Kotochigova (2006). In general, because of the relatively mended value of the relative atomic mass of the neutron A n r small size of the uncertainties of the ionization energies given was based on the wavelength of the 2.2 MeV γ ray emittedð inÞ in Table III, any correlations that might exist among them or the reaction n p d γ as measured in the 1990s. In the → with other data used in the CODATA-14 are unimportant. current adjustmentþ the AME2012þ value in Table II is taken as However, there is a significant covariance between the two an input datum and A n as an adjusted constant, because the r carbon binding-energy values, because a large part of the 2012 AME is an internallyð Þ consistent evaluation that uses all available data relevant to the determination of Ar n . ð Þ TABLE III. Ionization energies for 1H, 3H, 3He, 4He, 12C, and 28Si. B. Relative atomic masses of ions and nuclei Atom or ion E =hc 107 m−1 I ð Þ The mass of an atom or ion is the sum of the nuclear mass 1H 1.096 787 717 4307(10) and the masses of the electrons minus the mass equivalent of 3H 1.097 185 4390(13) n 3 the binding energy of the electrons. To produce an ion X þ Heþ 4.388 891 936(3) 4He 1.983 106 6637(20) 4 Heþ 4.389 088 785(2) TABLE II. Relative atomic masses used in the least-squares 12C 0.908 2045(10) 12 adjustment as given in the 2012 atomic mass evaluation and the Cþ 1.966 74(7) 12 12 2 defined value for C. C þ 3.862 410(10) 12 3 C þ 5.201 758(15) Relative atomic Relative standard 12 4 C þ 31.624 233(2) Atom mass Ar X uncertainty ur 12 5 ð Þ C þ 39.520 616 7(5) n 1.008 664 915 85(49) 4.9 × 10−10 28Si 0.657 4776(25) 1 −11 28 H 1.007 825 032 231(93) 9.3 × 10 Siþ 1.318 381(3) 3 −10 28 2 H 3.016 049 2779(24) 7.9 × 10 Si þ 2.701 393(7) 4 −11 28 3 He 4.002 603 254 130(63) 1.6 × 10 Si þ 3.640 931(6) 12 28 4 C 12 (exact) Si þ 13.450 7(2) 28 −11 28 5 Si 27.976 926 534 65(44) 1.6 × 10 Si þ 16.5559(15) 36 −10 28 6 Ar 35.967 545 105(29) 8.1 × 10 Si þ 19.867(8) 38 −9 28 7 Ar 37.962 732 11(21) 5.5 × 10 Si þ 24.492(14) 40 −11 28 8 Ar 39.962 383 1237(24) 6.0 × 10 Si þ 28.318(6) 87 −11 28 9 Rb 86.909 180 5319(65) 7.5 × 10 Si þ 32.374(3) 107 −8 28 10 Ag 106.905 0916(26) 2.4 × 10 Si þ 38.406(6) 109 −8 28 11 Ag 108.904 7553(14) 1.3 × 10 Si þ 42.216 3(6) 133 −11 28 12 Cs 132.905 451 9615(86) 6.5 × 10 Si þ 196.610 389(16)

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-6 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ... uncertainty is due to common uncertainties in the lower ω d cð Þ 0.992 996 654 743 20 2.0 × 10−11 ; 5 ionization stages; this yields the correlation coefficient ω 12C6 ¼ ð Þ½ ð Þ cð þÞ 12 5 12 6 r EB C þ =hc; EB C þ =hc 0.999 976: 2 ωc h −11 ½ ð Þ ð Þ ¼ ð Þ 12ð 6Þ 1.326 365 862 193 19 1.4 × 10 : 6 ωc C þ ¼ ð Þ½ ð Þ The relative atomic mass of an atom, its ions, the relative ð Þ atomic mass of the electron, and the relative atomic mass These ratios are correlated because of the image charge equivalent of the binding energy of the removed electrons are correction applied to each; based on the published uncertainty related according to budgets and additional information provided by Van Dyck (2015), their correlation coefficient is n ΔEB X þ A X A Xn nA e ; 3 r r þ r − ð 2 Þ r ω d =ω 12C6 ; ω h =ω C6 0.306: 7 ð Þ¼ ð Þþ ð Þ muc ð Þ ½ cð Þ cð þÞ cð Þ cð þÞ ¼ ð Þ 12 The relative atomic masses follow from the relations where mu m C =12 is the unified atomic mass constant. Equation (3)¼ isð theÞ form of the observational equation for 12 6 A X used in previous adjustments with A Xn and A e ωc d Ar C þ r r þ r ð Þ ð Þ ; 8 takenð Þ as adjusted constants with the binding-energyð Þ termð Þ ω 12C6 ¼ 6A d ð Þ c þ rð Þ taken to be exact. However, because for 28Si the binding- ð Þ 12 6 energy uncertainty is not negligible compared with the ωc h Ar C þ 28 12ð 6Þ ð Þ ; 9 uncertainty of Ar Si , we adopt the following new approach ωc C þ ¼ 3Ar h ð Þ for treating bindingð energiesÞ in all calculations in which they ð Þ ð Þ are required. Since the binding energies are known most where accurately in terms of their wave number equivalents, and 2 12 6 since R∞ α mec=2h and me Ar e mu, one can write 12 6 ΔEB C þ ¼ ¼ ð Þ Ar C þ 12 − 6Ar e ð 2 Þ ; 10 ð Þ¼ ð Þþ muc ð Þ n 2 n ΔEB X þ α Ar e ΔEB X þ : 4 ð 2 Þ ð Þ ð Þ which takes into account the definition A 12C 12. muc ¼ 2R∞ hc ð Þ r An overview of the University of Washingtonð Þ¼ Penning trap Thus, in the 2014 adjustment we replace the binding-energy mass spectrometer (UW-PTMS), which was developed over term in Eq. (3) by Eq. (4) and take the binding energy several decades, is given by Zafonte and Van Dyck (2015);a n discussion of the various experimental effects that can ΔEB X þ =hc as both an input datum and an adjusted constant,ð Þ thereby obtaining a new form of observational influence UW-PTMS cyclotron frequency measurements is equation for A X expressed solely in terms of adjusted given by Van Dyck, et al. (2006). The later paper also reports a rð Þ 2 constants. Although this requires taking binding energies as preliminary value of Ar H based on the analysis of 12 6 ð Þ input data rather than exactly known quantities, it allows all ωc d =ωc C þ data obtained in three early data runs. ð Þ ð Þ binding-energy uncertainties and covariances to be properly The final result of the UWash deuterium measurements given taken into account. This new form of observational equation is in Eq. (5) is based on 10 data runs, each of which yields one 1 3 4 frequency ratio and lasted more than a month when the time used for the AME2012 values of Ar H , Ar H , Ar He , 28 ð Þ ð Þ ð Þ required to check all experimental effects is included. and Ar Si , and the new way of treating binding energies is used inð the observationalÞ equations for a number of frequency Corrections for six significant experimental effects are applied ratios; see Table XXIV, Sec. XIII. to each of the 10 ratios before their weighted mean is calculated. The largest of these by far is that for image charge; its fractional magnitude is −245 × 10−12 for each C. Relative atomic mass of the deuteron, triton, and helion ratio. Each correction has an uncertainty, but since the −12 We consider here the recent data of the University of 9.9 × 10 relative standard uncertainty ur of the image Washington and Florida State University groups mentioned charge correction is the same for each ratio, it is omitted above relevant to the determination of the relative atomic from the individual ratio uncertainties. Rather, Zafonte and masses of the nuclei of the 2H (deuterium D), 3H (tritium T), Van Dyck (2015) take it into account by combining it with the 12 3 uncertainty u 17.4 × 10− of the weighted mean calcu- and He atoms, or deuteron d, triton t, and helion h, r ¼ respectively. The data are cyclotron frequency ratios obtained lated without the image charge uncertainty, thereby obtaining 12 in a Penning trap and it is these ratios that are used as input the 20 parts in 10 final uncertainty. Although there were seven successful helion runs to data in the adjustment to determine Ar d , Ar t , and Ar h , ð Þ ð Þ ð Þ determine ω h =ω 12C6 , Zafonte and Van Dyck (2015) which are taken as adjusted constants. These new results cð Þ cð þÞ became available shortly before the 31 December 2014 clos- decided to exclude runs three and four from their final analysis 12 6 ing date of the adjustment and were published in 2015. because they were found to contain two C þ ions instead of 12 6 The UWash group reports as the final values of the one. To avoid the problem of isolating a single C þ ion, they 12 6 12 5 cyclotron frequency ratios d and h to C þ (Zafonte and used a single C þ ion in the three other runs and scaled the Van Dyck, 2015) results using the well-known values of A 12C5 and rð þÞ

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-7 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

12 6 Ar C þ without adding any significant uncertainty to what fitting a fourth-order polynomial to the individual cyclotron ð Þ 12 6 3 they would have obtained if a C þ ion had been used. frequencies as a function of time. In total 34 HDþ= Heþ and 3 Zafonte and Van Dyck (2015) treat the five individual HDþ= Hþ runs were carried out over a 5 month period. For 12 6 ωc h =ωc C þ frequency ratios as they did the 10 each frequency ratio the standard uncertainty of the mean of ð Þ ð12 6 Þ ωc d =ωc C þ ratios; the fractional image charge correc- the individual values before correction for two systematic ð Þ ð Þ −12 −12 −12 tion is −515 × 10 with ur 8.9 × 10 , for the weighted effects is 17 × 10 . The correction for cyclotron radius ¼ −12 −12 mean of the five ratios ur 11.2 × 10 , and for the final imbalance for each is 22 45 × 10 and for the polarizability −12 ¼ ð−12Þ value ur 14 × 10 . of the HDþ ion, 94 × 10 with negligible uncertainty. These ¼ 3 The cyclotron frequency ratios of HDþ to Heþ and to t two uncertainty components lead to the final uncertainty for reported by the FSU group are (Myers et al., 2015) each of 48 × 10−12. Since the cyclotron frequencies in Eqs. (11) and (12) are both

ωc HDþ −11 measured with reference to the same molecular ion ð3 Þ 0.998 048 085 153 48 4.8 × 10 ; 11 ωc Heþ ¼ ð Þ½ ð Þ HDþ and there is a sizable correlation coefficient between ð Þ the frequency ratio measurements, Myers et al. (2015) 3 3 ωc HDþ obtain a value for the ratio ω H =ω He with only ð Þ 0.998 054 687 288 48 4.8 × 10−11 : 12 cð þÞ cð þÞ ω t ¼ ð Þ½ ð Þ one-half the 4.8 × 10−11 uncertainty of that for either of cð Þ the ratios determined with HDþ. They are thus able to As for the two UWash ratios, these ratios are correlated, but in deduce for the mass difference between the tritium and this case because of the correction to account for imbalance helium-3 atoms, m 3H − m 3He 1.995 934 7 × 10−5 u between the cyclotron radii of the two ions. Based on the 18 592.01 7 eV=cð2, whichÞ ð has aÞ¼ significantlyð smallerÞ uncer-¼ published uncertainty budgets and additional information tainty thanð anyÞ other value. provided by Myers (2015), their correlation coefficient is 3 The value of Ar He deduced by Myers et al. (2015) from their data, 3.016 029ð 322Þ 43(19), exceeds the value deduced r ω HD =ω 3He ; ω HD =ω t 0.875: 13 ½ cð þÞ cð þÞ cð þÞ cð Þ ¼ ð Þ by Zafonte and Van Dyck (2015) from their data, 3.016 029 321 675(43), by 3.9 times the standard uncertainty of their The relevant equations for these data are difference udiff or 3.9σ. (Throughout the paper, σ as used here is the standard uncertainty u of the difference between 3 diff ωc HDþ Ar Heþ two values.) How this disagreement is treated in the 2014 ð3 Þ ð Þ ; 14 ωc Heþ ¼ Ar HDþ ð Þ adjustment is discussed in Sec. XIII. The THe-Trap experi- ð Þ ð Þ ment currently underway at the Max-Planck-Institut für ω HD A t cð þÞ rð Þ ; 15 Kernphysik (MPIK), Heidelberg, Germany, the aim of ωc t ¼ Ar HDþ ð Þ which is to determine the ratio A 3H =A 3He in order to ð Þ ð Þ r r determine the Q-value of tritium,ð mayÞ clarifyð Þ the cause where of this discrepancy; see Diehl et al. (2011) and Streubel et al. (2014). 3 3 EI Heþ A Heþ A h A e − ð Þ ; 16 rð Þ¼ rð Þþ rð Þ m c2 ð Þ u IV. ATOMIC TRANSITION FREQUENCIES E 3He =hc 43 888 919.36 3 m−1; 17 Ið þÞ ¼ ð Þ ð Þ Comparison of theory and experiment for transition frequencies in hydrogen, deuterium, and muonic hydrogen EI HDþ Ar HDþ Ar p Ar d Ar e − ð 2 Þ ; 18 provides information on the Rydberg constant, and on the ð Þ¼ ð Þþ ð Þþ ð Þ muc ð Þ charge radii of the proton and deuteron. Hyperfine splittings in hydrogen and fine-structure splittings in helium are also E HD =hc 13 122 468.415 6 m−1: 19 Ið þÞ ¼ ð Þ ð Þ briefly considered. The ionization wave number in Eq. (17) is from Table III, and the value in Eq. (19) is from Liu et al. (2010) and Sprecher A. Hydrogen and deuterium transition frequencies, et al. (2010). the Rydberg constant R∞, and the proton and deuteron charge radii r , r In the FSU experiment pairs of individual ions, either HDþ p d and 3H or HD and 3He , are confined at the same time in a þ þ þ The transition frequency between states i and i with energy Penning trap at 4.2 K with an applied magnetic flux density of 0 levels E and E in hydrogen or deuterium is given by 8.5 T. The cyclotron frequency of one ion centered in the trap i i0 in an orbit with a radius of about 45 μm is determined while hν E − E : 20 the other ion is kept in an outer orbit with a radius of about ii0 ¼ i0 i ð Þ 1.1 mm to reduce perturbations on the inner ion due to Coulomb interactions. The two ions are then interchanged. In The energy levels are given by a typical run lasting up to 10 h about 20 cyclotron frequency α2m c2 R hc measurements are made on each ion. The temporal variation e ∞ Ei − 2 1 δi − 2 1 δi ; 21 of the magnet flux density is accounted for by simultaneously ¼ 2ni ð þ Þ¼ ni ð þ Þ ð Þ

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-8 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

where R∞ is the Rydberg constant, ni is the principal quantum b. Relativistic recoil number of state i, and δi, where δi ≪ 1, contains the details The leading relativistic-recoil correction, to lowest order in of the theory of the energy level.j j Zα and all orders in me=mN, is (Erickson, 1977; Sapirstein and Yennie, 1990) 1. Theory of hydrogen and deuterium energy levels 3 5 References to the original works are generally omitted; mr Zα 2 ES 2 ð 3Þ mec these may be found in earlier detailed CODATA reports, in ¼ memN πn Eides, Grotch, and Shelyuto (2001, 2007), and in Sapirstein 1 8 1 7 and Yennie (1990). Uncertainties we assign to the individual × δ ln Zα −2 − ln k n; l − δ − a 3 l0 ð Þ 3 0ð Þ 9 l0 3 n theoretical contributions are categorized as either correlated or & uncorrelated. Correlations we consider arise in two forms. 2 2 me 2 mN − 2 2 δl0 mN ln − me ln ; 27 One case is where the uncertainties are mainly of the form mN − me mr mr ð Þ 3 !' C=ni , where C is the same for all states with the same L and j. Such uncertainties are denoted by u0, while the uncorrelated where uncertainties are denoted by ui. The other correlations we consider are those between corrections for the same state in 2 n 1 1 different isotopes, where the correction only depends on the an −2 ln 1 − δl0 ¼ n þ i 1 i þ 2n mass of the isotope. Calculations of the uncertainties of the X¼ ! energy levels and the corresponding correlation coefficients 1 − δl0 : 28 are described in Sec. IV.A.1.l. þ l l 1 2l 1 ð Þ ð þ Þð þ Þ a. Dirac eigenvalue Values we use for the Bethe logarithms ln k0 n; l in Eqs. (27), (38), and (65) are given in Table IV. ð Þ The Dirac eigenvalue for an electron bound to a stationary Additional contributions to lowest order in the mass ratio point nucleus is and of higher order in Zα are 2 ED f n; j mec ; 22 ¼ ð Þ ð Þ 6 me Zα 2 2 −2 ER ð 3Þ mec D60 D72Zα ln Zα ; 29 where ¼ mN n ½ þ ð Þ þ Á Á Á ð Þ

Zα 2 −1=2 where f n; j 1 ð Þ ; 23 ð Þ¼ þ n − δ 2 ð Þ ð Þ ! 7 n and j are the principal and total angular-momentum D 4 ln 2 − δ 60 ¼ 2 l0 quantum numbers of the bound state, l l 1 2 1 − δl0 3 − ð þ Þ ð Þ ; 30 1 1 2 1=2 þ n2 4l2 − 1 2l 3 ð Þ δ j − j − Zα 2 ; 24 ! ð Þð þ Þ ¼ þ 2 þ 2 ð Þ ð Þ ! 11 and Z is the charge number of the nucleus. D72 − δl0: 31 ¼ 60π ð Þ For a nucleus with a finite mass mN, we have

2 2 The uncertainty in the relativistic recoil correction is taken mr c E H Mc2 f n; j − 1 m c2 − f n; j − 1 2 to be Mð Þ¼ þ½ ð Þ r ½ ð Þ 2M 1 − δ Zα 4m3c2 l0 ð Þ r 25 0.1δl0 0.01 1 − δl0 ER: 32 þ κ 2l 1 2n3m2 þÁÁÁ ð Þ ½ þ ð Þ ð Þ ð þ Þ N Covariances follow from the m =m =n3 scaling of the for hydrogen or e N uncertainty. ð Þ 2 2 2 2 2 mr c EM D Mc f n; j − 1 mrc − f n; j − 1 ð Þ¼ þ½ ð Þ ½ ð Þ 2M TABLE IV. Relevant values of the Bethe logarithms ln k0 n; l . ð Þ 1 Zα 4m3c2 ð Þ r 26 n SP D þ κ 2l 1 2n3m2 þÁÁÁ ð Þ ð þ Þ N 1 2.984 128 556 2 2.811 769 893 −0.030 016 709 for deuterium, where l is the nonrelativistic orbital angular- 3 2.767 663 612 momentum quantum number, δ is the Kronecker delta, κ 4 2.749 811 840 0.041 954 895 0.006 740 939 l0 ¼ − − −1 j−l 1=2 j 1 is the angular-momentum-parity quantum 6 2.735 664 207 −0.008 147 204 ð Þ þ ð þ 2Þ number, M m m , and m m m = m m is the 8 2.730 267 261 −0.008 785 043 e N r e N e N 12 0.009 342 954 reduced mass.¼ þ ¼ ð þ Þ −

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-9 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ... c. Nuclear polarizability Values for G α in Eq. (36) are listed in Table V. See SEð Þ For the nuclear polarizability in hydrogen, we use CODATA-10 for details. The uncertainty of the self-energy contribution to a given level is due to the uncertainty of δ GSE α listed in that table and is taken to be type un. l0 ð Þ EP H −0.070 13 h kHz; 33 Following convention, F Zα is multiplied by the reduced- ð Þ¼ ð Þ n3 ð Þ mass factor m =m 3, exceptð Þ the magnetic-moment term ð r eÞ and for deuterium −1= 2κ 2l 1 in A40 which is instead multiplied by the factor½ mð =mþ 2Þ, and the argument Zα −2 of the logarithms is ð r eÞ ð Þ δl0 replaced by m =m Zα −2. E D −21.37 8 h kHz: 34 e r Pð Þ¼ ð Þ n3 ð Þ ð Þð Þ Presumably the polarizability effect is negligible for states of e. Vacuum polarization higher l in either hydrogen or deuterium. The stationary point nucleus second-order vacuum-polarization level shift is d. Self energy 4 2 α Zα 2 Eð Þ ð Þ H Zα mec ; 42 The one-photon self energy of an electron bound to a VP ¼ π n3 ð Þ ð Þ stationary point nucleus is where H Zα H 1 Zα H R Zα , ð Þ¼ ð Þð Þþ ð Þð Þ α Zα 4 2 2 1 2 −2 Eð Þ ð Þ F Zα mec ; 35 Hð Þ Zα V40 V50 Zα V61 Zα ln Zα SE ¼ π n3 ð Þ ð Þ ð Þ¼ þ ð Þþ ð Þ ð Þ 1 2 Gð Þ Zα Zα ; 43 where þ VPð Þð Þ ð Þ R R 2 Hð Þ Zα Gð Þ Zα Zα ; 44 F Zα A ln Zα −2 A A Zα ð Þ¼ VP ð Þð Þ ð Þ ð Þ¼ 41 ð Þ þ 40 þ 50ð Þ A Zα 2 ln2 Zα −2 A Zα 2 ln Zα −2 with þ 62ð Þ ð Þ þ 61ð Þ ð Þ 2 GSE Zα Zα ; 36 4 þ ð Þð Þ ð Þ V − δ ; 40 ¼ 15 l0 with 5 V50 πδl0; 4 ¼ 48 A41 δl0; 37 2 ¼ 3 ð Þ V − δ : 45 61 ¼ 15 l0 ð Þ 4 10 1 A40 − ln k0 n; l δl0 − 1 − δl0 ; 38 1 ¼ 3 ð Þþ 9 2κ 2l 1 ð Þ ð Þ Values of GVPð Þ Zα are given in Table VI, and ð þ Þ ð Þ 2 139 R 19 π Gð Þ Zα − δl0 A50 − 2 ln 2 πδl0; 39 VP ¼ 32 ð Þ ð Þ¼ 45 27 1 31π2 − π Zα δl0 : 46 A62 −δl0; 40 þ 16 2880 ð Þ þÁÁÁ ð Þ ¼ ð Þ 1 1 28 Higher-order terms are negligible. We multiply Eq. (42) by A61 4 1 ln 2 − 4 ln n 3 ¼ þ 2 þÁÁÁþn þ 3 mr=me and include a factor of me=mr in the argument of ð Þ ð Þ 2 the logarithm in Eq. (43). 601 77 n − 1 2 1 − − − δl0 δj 1 δl1; Vacuum polarization from μþμ pairs is 180 45n2 þ n2 15 þ 3 2 ! 2 4 2 3 96n − 32l l 1 1 − δl0 2 α Zα 4 me mr 2 Eð Þ δ m c ; 47 2 ½ ð þ Þð Þ : 41 μVP ð 3Þ − l0 e þ 3n 2l − 1 2l 2l 1 2l 2 2l 3 ð Þ ¼ π n 15 m me ð Þ ð Þð Þð þ Þð þ Þð þ Þ μ

TABLE V. Values of the function G α . SEð Þ

n S1=2 P1=2 P3=2 D3=2 D5=2 1 −30.290 240 20 2 −31.185 150ð90Þ −0.973 50 20 −0.486 50 20 3 −31.047 70 90ð Þ ð Þ ð Þ 4 −30.9120 40ð Þ −1.1640 20 −0.6090 20 0.031 63(22) 6 −30.711 47ð Þ ð Þ ð Þ 0.034 17(26) 8 −30.606ð47Þ 0.007 940(90) 0.034 84(22) 12 ð Þ 0.009 130(90) 0.035 12(22)

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-10 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

1 TABLE VI. Values of the function Gð Þ α . VPð Þ

n S1=2 P1=2 P3=2 D3=2 D5=2 1 −0.618 724 2 −0.808 872 −0.064 006 −0.014 132 3 −0.814 530 4 −0.806 579 −0.080 007 −0.017 666 −0.000 000 6 −0.791 450 −0.000 000 8 −0.781 197 −0.000 000 −0.000 000 12 −0.000 000 −0.000 000 and hadronic vacuum polarization is given by 16 71 1 1 B62 − ln 2 γ ψ n − ln n − 2 δl0 2 2 ¼ 9 60 þ þ ð Þ n þ 4n ð Þ ð Þ EhadVP 0.671 15 EμVP: 48 ! ¼ ð Þ ð Þ 4 n2 − 1 2 δl1; 54 Uncertainties are of type u0. The muonic and hadronic þ 27 n ð Þ vacuum-polarization contributions are negligible for higher- l states. 413581 4N nS 2027π2 616ln2 2π2 ln2 B ð Þ − − 61 ¼ 64800 þ 3 þ 864 135 3 & f. Two-photon corrections 40ln22 ζ 3 The two-photon correction is þ 9 þ ð Þ 304 32ln2 3 1 1 2 4 γ ψ n lnn δ 4 α Zα 2 4 − − − 2 l0 Eð Þ ð Þ mec Fð Þ Zα ; 49 þ 135 9 4 þ þ ð Þ n þ 4n ¼ π n3 ð Þ ð Þ !' 4 n2 − 1 31 1 8 N nP δj 1 − ln2 δl1: 55 where þ 3 ð Þþ n2 405 þ 3 2 27 ð Þ ! Values for B used in the adjustment are listed in Table VII. In F 4 Zα B B Zα B Zα 2ln3 Zα −2 61 ð Þð Þ¼ 40 þ 50ð Þþ 63ð Þ ð Þ CODATA-10, the entries for states with l 1 in the corre- 2 2 −2 ¼ B62 Zα ln Zα sponding Table IX are incorrect, which had negligible effect þ ð Þ ð Þ 2 −2 2 on the results. Corrected values are listed here in Table VII. B61 Zα ln Zα B60 Zα þ ð Þ ð Þ þ ð Þ The values of N nL , which appear in Eq. (54), are listed in B Zα 3ln2 Zα −2 ð Þ þ 72ð Þ ð Þ Table VIII. The uncertainties are negligible. 3 −2 3 Values used in the adjustment for B and B are listed in B71 Zα ln Zα B70 Zα 60 60 þ ð Þ ð Þ þ ð Þ Table IX. For the S-state values, the first number in parentheses ; 50 þÁÁÁ ð Þ is the state-dependent uncertainty un B60 , and the second number in parentheses is the state-independentð Þ uncertainty with u B that is common to all S-state values of B . For higher- 0ð 60Þ 60 l states, the notation B60 indicates that the number listed in the 3π2 10π2 2179 9 B ln 2 − − − ζ 3 δ table is the value of the line center shift for the level, in contrast 40 ¼ 2 27 648 4 ð Þ l0 ! to the total real part of the two-photon correction. See 2 2 π ln 2 π 197 3ζ 3 1 − δl0 CODATA-10 for a complete explanation. For S states, the − − − ð Þ ; 51 þ 2 12 144 4 κ 2l 1 ð Þ difference between B60 and B60 is negligible compared to the ! ð þ Þ uncertainty of the value of B60. The uncertainties of B60 for B −21.554 47 13 δ ; 52 higher-l states are taken to be independent. 50 ¼ ð Þ l0 ð Þ For S states, the next term B72 is state independent, but its 8 value is not known. However, the state dependence of the B − δ ; 53 63 ¼ 27 l0 ð Þ following term is

TABLE VII. Values of B61 used in the 2014 adjustment. nBnS B nP B nP B nD B nD 61ð 1=2Þ 61ð 1=2Þ 61ð 3=2Þ 61ð 3=2Þ 61ð 5=2Þ 1 48.958 590 24(1) 2 41.062 164 31(1) 0.157 775 547(1) −0.092 224 453 1 3 38.904 222(1) ð Þ 4 37.909 514(1) 0.191 192 600(1) −0.121 307 400 1 0.0(0) 6 36.963 391(1) ð Þ 0.0(0) 8 36.504 940(1) 0.0(0) 0.0(0) 12 0.0(0) 0.0(0)

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-11 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

TABLE VIII. Values of N used in the 2014 adjustment. As with the one-photon correction, the two-photon correction is multiplied by the reduced-mass factor nNnS N nP 3 ð Þ ð Þ mr=me , except the magnetic-moment term proportional 1 17.855 672 03(1) ð Þ to 1= κ 2l 1 in B40 which is multiplied by the factor 2 12.032 141 58(1) 0.003 300 635(1) m =m½ ð2, andþ Þ the argument Zα −2 of the logarithms is 3 10.449 809(1) r e replacedð Þ by m =m Zα −2. ð Þ 4 9.722 413(1) −0.000 394 332 1 ð e rÞð Þ 6 9.031 832(1) ð Þ 8 8.697 639(1) g. Three-photon corrections 427 16 The three-photon contribution in powers of Zα is ΔB71 nS B71 nS − B71 1S π − ln 2 ð Þ¼ ð Þ ð Þ¼ 36 3 3 1 1 α 3 Zα 4 × − 2 γ ψ n − ln n ; 56 6 2 4 n þ 4n þ þ ð Þ ð Þ Eð Þ ð Þ mec C40 C50 Zα : 57 ! ¼ π n3 ½ þ ð Þ þ Á Á Á ð Þ with a relative uncertainty of 50%. We include this difference, which is listed in Table IX, along with an estimated uncertainty of un ΔB71 ΔB71=2. The leading term C40 is ð Þ¼

2 4 2 2 2 4 2 568a4 85ζ 5 121π ζ 3 84071ζ 3 71ln 2 239π ln 2 4787π ln2 1591π 252251π 679441 C − ð Þ − ð Þ − ð Þ − − − δ 40 ¼ 9 þ 24 72 2304 27 135 þ 108 þ 3240 9720 þ 93312 l0 ! 2 4 2 2 2 4 2 100a4 215ζ 5 83π ζ 3 139ζ 3 25ln 2 25π ln 2 298π ln2 239π 17101π 28259 1 − δl0 − ð Þ − ð Þ − ð Þ − − − ; þ 3 þ 24 72 18 18 þ 18 þ 9 þ 2160 810 5184 κ 2l 1 ! ð þ Þ 58 ð Þ

∞ n 4 where a4 n 1 1= 2 n 0.517 479 061…. Partial re- mr rN mr rN Zα ¼ ¼ ð Þ¼ ENS ENS 1 − Cη Zα − ln sults for C50 have been calculated by Eides and Shelyuto ¼ me ƛC me ƛC n (2004, 2007)P. The uncertainty is taken to be u C & 0 50 5n 9 n − 1 30δ and u C 1, where C would be the coefficientð Þ¼ ψ n γ − ð þ Þð Þ − C Zα 2 ; 59 l0 n 63 63 þ ð Þþ 4n2 θ ð Þ ð Þ of Zα 2 ln3 ðZα −Þ¼2 in the square brackets in Eq. (57). The ! ' ð Þ ð Þ dominant effect of the finite mass of the nucleus is taken where into account by multiplying the term proportional to δl0 by 3 the reduced-mass factor mr=me and the term propor- ð Þ 2 m 3 Zα 2 Zαr 2 tional to 1= κ 2l 1 , the magnetic-moment term, by the E r ð Þ m c2 N ; 60 ½ ð 2 þ Þ NS ¼ 3 m n3 e ƛ ð Þ factor mr=me . e C Theð contributionÞ from four photons is expected to be negligible at the level of uncertainty of current interest. rN is the bound-state root-mean-square (rms) charge radius of the nucleus, ƛC is the Compton wavelength of the electron h. Finite nuclear size divided by 2π, Cη and Cθ are constants that depend on the For S states the leading and next-order correction charge distribution in the nucleus with values Cη 1.7 1 and ¼ ð Þ to the level shift due to the finite size of the nucleus is C 0.47 4 for hydrogen or C 2.0 1 and C 0.38 4 θ ¼ ð Þ η ¼ ð Þ θ ¼ ð Þ given by for deuterium.

TABLE IX. Values of B60, B60, or ΔB71 used in the 2014 adjustment. The uncertainties of B60 are explained in the text.

nB60 nS1=2 B nP B nP B nD B nD ΔB71 nS1=2 ð Þ 60ð 1=2Þ 60ð 3=2Þ 60ð 3=2Þ 60ð 5=2Þ ð Þ 1 −81.3 0.3 19.7 2 −66.2ð0.3Þð19.7Þ −1.6 3 −1.7 3 16(8) 3 −63.0ð0.6Þð19.7Þ ð Þ ð Þ 22(11) 4 −61.3ð0.8Þð19.7Þ −2.1 3 −2.2 3 −0.005 2 25(12) 6 −59.3ð0.8Þð19.7Þ ð Þ ð Þ −0.008ð4Þ 28(14) 8 −58.3ð2.0Þð19.7Þ 0.015(5) −0.009ð5Þ 29(15) 12ð Þð Þ 0.014(7) −0.010ð7Þ ð Þ

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-12 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

For the P states in hydrogen the leading term is 2 2 2 1=2 u δX nLj u0i XnLj uni XnLj ; 66 ½ ð Þ ¼ i ½ ð Þþ ð Þ ð Þ Zα 2 n2 − 1 X E E ð Þ ð Þ : 61 NS ¼ NS 4n2 ð Þ where u XnL and u XnL are the components of 0ið jÞ nið jÞ uncertainty u and u of contribution i, is added to the level. For P states and higher-l states the nuclear-size con- 0 n 3=2 The corrections δ nL , which includes their uncertainties, tribution is negligible. Xð jÞ are taken as input data in the least-squares adjustment. Covariances are taken into account by calculating all the i. Nuclear-size correction to self energy and vacuum polarization covariances and including them in the input data for the For the lowest-order self energy and vacuum polarization adjustment. the correction due to the finite size of the nucleus is Covariances of the δs for a given isotope are

23 ENSE 4 ln 2 − α Zα ENSδl0; 62 u δ n L ; δ n L u Xn L u Xn L : 67 ¼ 4 ð Þ ð Þ X 1 j X 2 j 0i 2 j 0i 1 j ½ ð Þ ð Þ ¼ i ð Þ ð Þ ð Þ X and Covariances between δs for hydrogen and deuterium for states 3 of the same n are E α Zα E δ ; 63 NVP ¼ 4 ð Þ NS l0 ð Þ u δ nL ;δ nL respectively. ½ Hð jÞ Dð jÞ u HnL u DnL u HnL u DnL ; 68 ¼ ½ 0ið jÞ 0ið jÞþ nið jÞ nið jÞ ð Þ j. Radiative-recoil corrections i ic ¼Xf g Corrections for radiative-recoil effects are and for n1 ≠ n2 3 5 mr α Zα 2 ERR 2 ð 2 3Þ mec δl0 ¼ me mN π n u δH n1Lj ; δD n2Lj u0i Hn1Lj u0i Dn2Lj ; 69 ½ ð Þ ð Þ ¼ ð Þ ð Þ ð Þ 2 i ic 2 35π 448 ¼Xf g × 6ζ 3 − 2π ln 2 − ð Þ þ 36 27 where the summation is over the uncertainties common to 2 π Zα ln2 Zα −2 : 64 hydrogen and deuterium. þ 3 ð Þ ð Þ þÁÁÁ ð Þ ! Values of u δX nLj are given in Table XVI of Sec. XIII, and the non-negligible½ ð Þ covariances of the δs are given as The uncertainty is Zα ln Zα −2 relative to the square ð Þ ð Þ correlation coefficients in Table XVII. brackets with a factor of 10 for u0 and 1 for un. Corrections for higher-l states are negligible. m. Transition frequencies between levels with n 2 and the ¼ k. Nucleus self energy fine-structure constant α The nucleus self energy correction for S states is QED predictions for hydrogen transition frequencies between levels with n 2 are obtained from a least-squares 2 4 3 adjustment that does not¼ include an experimental value for the 4Z α Zα mr 2 ESEN ð 3 Þ 2 c transitions being calculated (items A39, A40.1, or A40.2 in ¼ 3πn mN Table XVI), where the constants Ar e , Ar p , Ar d , and α are mN ð Þ ð Þ ð Þ × ln δ − ln k n; l ; 65 assigned the 2014 values. The results are m Zα 2 l0 0ð Þ ð Þ rð Þ ! −6 νH 2P1=2 − 2S1=2 1 057 843.7 2.1 kHz 2.0 × 10 ; with an uncertainty u0 given by Eq. (65) with the factor in the ð Þ¼ ð Þ ½ −7 square brackets replaced by 0.5. For higher-l states, the νH 2S1=2 − 2P3=2 9 911 197.8 2.1 kHz 2.1 × 10 ; correction is negligible. ð Þ¼ ð Þ ½ ν 2P − 2P 10 969 041.530 41 kHz 3.7 × 10−9 ; Hð 1=2 3=2Þ¼ ð Þ ½ l. Total energy and uncertainty 70 ð Þ The energy EX nLj of a level (where L S; P; … and X H, D) is the sumð ofÞ the various contributions¼ listed above. which are consistent with the experimental results given in Uncertainties¼ in the fundamental constants that enter the Table XVI. theoretical expressions are taken into account through the Data for the hydrogen and deuterium transitions yield a formalism of the least-squares adjustment. To take uncertain- value for the fine-structure constant α, which follows from a ties in the theory into account, a correction δ nL that is zero least-squares adjustment that includes all the transition fre- Xð jÞ with an uncertainty that is the rms sum of the uncertainties of quency data in Table XVI, the 2014 adjusted values of Ar e , the individual contributions A p , and A d , but no other input data for α. The resultð isÞ rð Þ rð Þ

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-13 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

α−1 137.035 992 55 4.0 × 10−7 ; 71 achieved using a more stable spectroscopy laser and by reducing ¼ ð Þ½ ð Þ the uncertainties from the principal systematic effects, namely, and is also given in Table XX. the second-order Doppler shift and ac and dc Stark shifts. The result reported by Matveev et al. (2013) has a relative standard uncertainty of 4.4 × 10−15. In contrast to 2. Experiments on hydrogen and deuterium the 2011 measurement, which used an on-site transportable Table X gives the hydrogen and deuterium transition cesium fountain clock as the frequency reference, the 2013 frequencies used to determine the Rydberg constant R∞, measurement used the nonmovable cesium fountain clock items A26 to A48 in Table XVI, Sec. XIII. The only difference at the Phyisikalisch-Technische Budesanstalt (PTB) in between this table and the corresponding Table XI in Braunschweig, Germany, by connecting the MPQ experiment CODATA-10 is that the value for the 1S1=2 2S1=2 hydrogen in Garching to the PTB clock in Braunschweig via a 920 km − transition frequency obtained by the group at the Max-Planck- fiber link. Another difference is the use of an improved Institut für Quantenoptik (MPQ), Garching, Germany, used in detector of the Lyman-α photons emitted when the excited the 2010 adjustment is superseded by two new values obtained hydrogen atom beam is deexcited from the 2S state. The new by the same group but with significantly smaller uncertainties results are consistent with each other and with that used in (first two entries of Table X): 2010. However, the 2011 and 2013 values are correlated, and based on the published uncertainty budgets and information νH 1S1=2 − 2S1=2 2 466 061 413 187.035 10 provided by the researchers (Udem, 2014) the correlation ð Þ¼ ð Þ 4.2 × 10−15 ; 72 coefficient is estimated to be 0.707. This correlation coef- ½ ð Þ ficient is included in Table XVII, Sec. XIII, together with the correlation coefficients of the other correlated frequencies in ν 1S − 2S 2 466 061 413 187.018 11 Hð 1=2 1=2Þ¼ ð Þ Table X as discussed in CODATA-98. 4.4 × 10−15 : 73 ½ ð Þ The result reported by Parthey et al. (2011) has a relative 3. Nuclear radii −15 standard uncertainty ur 4.2 × 10 , about one-third that of It follows from Eqs. (59) and (60) in Sec. IV.A.1.h that the value used in the 2010¼ adjustment. The reduction was transition frequencies in hydrogen and deuterium atoms

TABLE X. Summary of measured transition frequencies ν considered in the present work for the determination of the Rydberg constant R∞ (νH for hydrogen and νD for deuterium). Rel. stand. a Authors Laboratory Frequency interval(s) Reported value ν=kHz uncert. ur Parthey et al. (2011) MPQ ν 1S − 2S 2 466 061 413 187.035(10) 4.2 × 10−15 Hð 1=2 1=2Þ Matveev et al. (2013) MPQ ν 1S − 2S 2 466 061 413 187.018(11) 4.4 × 10−15 Hð 1=2 1=2Þ Weitz et al. (1995) MPQ ν 2S − 4S − 1 ν 1S − 2S 4 797 338(10) 2.1 × 10−6 Hð 1=2 1=2Þ 4 Hð 1=2 1=2Þ ν 2S − 4D − 1 ν 1S − 2S 6 490 144(24) 3.7 × 10−6 Hð 1=2 5=2Þ 4 Hð 1=2 1=2Þ ν 2S − 4S − 1 ν 1S − 2S 4 801 693(20) 4.2 × 10−6 Dð 1=2 1=2Þ 4 Dð 1=2 1=2Þ ν 2S − 4D − 1 ν 1S − 2S 6 494 841(41) 6.3 × 10−6 Dð 1=2 5=2Þ 4 Dð 1=2 1=2Þ Parthey et al. (2010) MPQ ν 1S − 2S − ν 1S − 2S 670 994 334.606(15) 2.2 × 10−11 Dð 1=2 1=2Þ Hð 1=2 1=2Þ de Beauvoir et al. (1997) LKB/SYRTE ν 2S − 8S 770 649 350 012.0(8.6) 1.1 × 10−11 Hð 1=2 1=2Þ ν 2S − 8D 770 649 504 450.0(8.3) 1.1 × 10−11 Hð 1=2 3=2Þ ν 2S − 8D 770 649 561 584.2(6.4) 8.3 × 10−12 Hð 1=2 5=2Þ ν 2S − 8S 770 859 041 245.7(6.9) 8.9 × 10−12 Dð 1=2 1=2Þ ν 2S − 8D 770 859 195 701.8(6.3) 8.2 × 10−12 Dð 1=2 3=2Þ ν 2S − 8D 770 859 252 849.5(5.9) 7.7 × 10−12 Dð 1=2 5=2Þ Schwob et al. (1999) LKB/SYRTE ν 2S − 12D 799 191 710 472.7(9.4) 1.2 × 10−11 Hð 1=2 3=2Þ ν 2S − 12D 799 191 727 403.7(7.0) 8.7 × 10−12 Hð 1=2 5=2Þ ν 2S − 12D 799 409 168 038.0(8.6) 1.1 × 10−11 Dð 1=2 3=2Þ ν 2S − 12D 799 409 184 966.8(6.8) 8.5 × 10−12 Dð 1=2 5=2Þ Arnoult et al. (2010) LKB ν 1S − 3S 2 922 743 278 678(13) 4.4 × 10−12 Hð 1=2 1=2Þ Bourzeix et al. (1996) LKB ν 2S − 6S − 1 ν 1S − 3S 4 197 604(21) 4.9 × 10−6 Hð 1=2 1=2Þ 4 Hð 1=2 1=2Þ ν 2S − 6D − 1 ν 1S − 3S 4 699 099(10) 2.2 × 10−6 Hð 1=2 5=2Þ 4 Hð 1=2 1=2Þ Berkeland, Hinds, and Boshier (1995) Yale ν 2S − 4P − 1 ν 1S − 2S 4 664 269(15) 3.2 × 10−6 Hð 1=2 1=2Þ 4 Hð 1=2 1=2Þ ν 2S − 4P − 1 ν 1S − 2S 6 035 373(10) 1.7 × 10−6 Hð 1=2 3=2Þ 4 Hð 1=2 1=2Þ Hagley and Pipkin (1994) Harvard ν 2S − 2P 9 911 200(12) 1.2 × 10−6 Hð 1=2 3=2Þ Lundeen and Pipkin (1986) Harvard ν 2P − 2S 1 057 845.0(9.0) 8.5 × 10−6 Hð 1=2 1=2Þ Newton, Andrews, and Unsworth (1979) U. Sussex ν 2P − 2S 1 057 862(20) 1.9 × 10−5 Hð 1=2 1=2Þ aMPQ: Max-Planck-Institut für Quantenoptik, Garching, Germany. LKB: Laboratoire Kastler-Brossel, Paris, France. SYRTE: Systèmes de référence Temps Espace, Paris, France. Yale: Yale University, New Haven, Connecticut, USA. Harvard: Harvard University, Cambridge, Massachusetts, USA. U. Sussex: University of Sussex, Brighton, UK.

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-14 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

−2 depend on the bound-state rms charge radius rp and rd of their rp 0.879 11 1.3 × 10 fm; 74 respective nuclei, the proton p and deuteron d. Accurate values ¼ ð Þ½ ð Þ for these radii can be obtained if they are taken as adjusted which is the weighted mean of the two values, but the constants in a least-squares adjustment together with exper- uncertainty is the average of their uncertainties since each imental H and D transition frequency input data and theory. value was based on essentially the same data. The values so determined are often referred to as the H-D The Task Group well recognizes that Eq. (74) is not the spectroscopic values of rp and rd. last word on the subject and that the topic remains one of active interest; see, for example, the papers by Kraus et al. a. Electron scattering (2014), Arrington (2015), Lorenz et al. (2015), and Sick (2015). Particularly noteworthy is the paper by Lee, Values of rp and rd are also available from electron-proton (e-p) and electron-deuteron (e-d) elastic scattering data. If Arrington, and Hill (2015), which did not become available these values are included as input data in an adjustment until well after the 31 December closing date of the 2014 together with the H and D spectroscopic data and theory, a adjustment, that provides a new and improved analysis to obtain r 0.895 20 fm from the MAMI data, r combined least-squares adjusted value for r and for r can be p p p d 0.916 24 fm¼ fromð theÞ remaining available data but not¼ obtained. In the 2010 adjustment the value of rd determined by includingð Þ the MAMI data, and r 0.904 15 fm by com- Sick (2008) from an analysis of the available data on e-d p ¼ ð Þ bining these two values. Three recent papers by Griffioen, elastic scattering was used as an input datum and it is again Carlson and Maddox (2016), Higinbotham et al. (2016), and used as an input datum in the 2014 adjustment. That value Horbatsch and Hessels (2016) obtain results consistent with is r 2.130 10 fm. d ¼ ð Þ the smaller muonic hydrogen radius in Eq. (78). However, The e-p scattering value of rp is more problematic. Two Bernauer and Distler (2016) point out a number of problems such values were used in the 2010 adjustment: r p ¼ with the analyses in these papers. 0.895 18 fm and r 0.8791 79 fm. The first is due to ð Þ p ¼ ð Þ Sick (2008) based on his analysis of the data then available. b. Isotope shift and the deuteron-proton radius difference The second is from a seminal experiment, described in detail From a comparison of experiment and theory for the by Bernauer et al. (2014), carried out at the Johannes hydrogen-deuterium isotope shift, one can extract a value Gutenberg Universität Mainz (or simply the University of for the difference of the squares of the charge radii for the Mainz), Mainz, Germany, with the Mainz microtron electron- deuteron and proton, based on the 2014 recommended values, beam accelerator MAMI. Both values are discussed in given by CODATA-10, as is the significant disagreement of the H-D spectroscopic and e-p scattering values of rp with the value 2 2 2 rd − rp 3.819 48 37 fm : 75 determined from spectroscopic measurements of the Lamb ¼ ð Þ ð Þ shift in muonic hydrogen. The cause of the disagreement The corresponding result based on the 2010 recommended remains unknown, and we review the current situation in the values, following section and in Sec. XIII.B.2. To assist in deciding what scattering value or values of r to 2 2 2 p rd − rp 3.819 89 42 fm ; 76 use in the 2014 adjustment, the Task Group invited a number ¼ ð Þ ð Þ of researchers active in the field to attend its annual meeting in differs from the current value due mainly to the change in the November 2014. It also helped organize the Workshop on the relative atomic mass of the electron used to evaluate the Determination of the Fundamental Constants held in Eltville, theoretical expression for the isotope shift. The electron mass Germany, in February 2015, the proceedings of which are appears as an overall factor as can be seen from the leading published in J. Phys. Chem. Ref. Data 44(3) (2015). A point of term (Jentschura et al., 2011) concern at these meetings was how best to extract rp from the MAMI data and from the combination of the MAMI data with 3 me me Δf1S 2S;d − Δf1S 2S;p ≈− R c − : 77 − − 4 ∞ m m ð Þ the remaining available data. As a result of the discussions at d p these meetings two pairs of knowledgeable researchers, J. Arrington and I. Sick, and J. C. Bernauer and M. O. c. Muonic hydrogen Distler, provided the Task Group with their best estimate of The first reported value of r from spectroscopic measure- r from all the available data. The Arrington-Sick value, p p ments of the Lamb shift in the muonic hydrogen atom -p r 0.879 11 fm, has been published (Arrington and Sick, μ p ¼ ð Þ obtained by the Charge Radius Experiment with Muonic 2015) but was initially transmitted to the Task Group as a Atoms (CREMA) collaboration (Pohl et al., 2010) and the private communication in 2015; the Bernauer-Distler value, problem, now often called the “proton radius puzzle,” that r 0.880 11 fm, was also transmitted to the Task Group in p ¼ ð Þ resulted from its significant disagreement with the scattering 2015 as a private communication but has not been published. and spectroscopic values is discussed in CODATA-10. As a The two values are highly consistent even though different result of this inconsistency the Task Group decided that the approaches and assumptions were used to obtain them. We initial μ-p value of rp should be omitted from the 2010 therefore adopt as the e-p scattering-data input datum for rp in adjustment and the 2010 recommended value should be based the 2014 adjustment on only the two available e-p scattering values (see previous

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-15 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ... section) and the H-D spectroscopic data and theory. α−1 137.035 876 35 [2.6 × 10−7], which differs from the ¼ ð Þ Consequently the disagreement of the μ-p value with the 2014 recommended value by 3.5σ. 2010 recommended value was 7σ and with the H-D spectro- Second, Karshenboim (2014) has proposed a different scopic value, 4.4σ. approach to the calculation of the next-to-leading higher- The μ-p Lamb-shift experiment employs pulsed laser order proton-size contribution to the theory of the muonic spectroscopy at a special muon beam line at the proton hydrogen Lamb shift than was used by Antognini et al. accelerator of the Paul Scherrer Institute (PSI), Villigen, (2013b) to obtain the value of rp in Eq. (78). The approach Switzerland. Muonic atoms in the 2S state (lifetime 1 μs) changes the value in Eq. (78) to r 0.840 22 56 fm p ¼ ð Þ are formed when muons from the beam strike a gaseous H2 (Karshenboim, 2014), for which the disagreement with the target, a 5 ns near-resonance laser pulse (tunable from 50 THz 2014 CODATA recommended value is 5.7σ and with the H-D to 55 THz) induces transitions to the 2P state (lifetime 8.5 ps), spectroscopic value is 4.6σ. [The slightly different value the atoms decay to the 1S ground state by emitting 1.9 keV Kα r 0.840 29 55 fm was subsequently published after the p ¼ ð Þ x rays, and a resonance curve is obtained by counting the closing date of the 2014 adjustment by Karshenboim et al. number of x rays as a function of laser frequency. (Because of (2015).] Because this value of rp is consistent with the value in the large electron vacuum-polarization effect in μ-p, the 2S1=2 Eq. (78), because the disagreement for the two values with the level is far below both the 2P1=2 and 2P3=2 levels.) 2014 CODATA recommended value and H-D spectroscopic The transition initially measured was 2S F 1 − value are very nearly the same, and because the suggested 1=2ð ¼ Þ 2P3=2 F 2 at 50 THz or 6 μm. More recently, the approach has not yet been widely accepted, we have used the ð ¼ Þ CREMA collaboration reported their result for the transition value of rp in Eq. (78) for our discussion here and analysis in 2S1=2 F 0 − 2P3=2 F 1 at 55 THz or 5.5 μm, as well as Sec. XIII.B.2. ð ¼ Þ ð ¼ Þ their reevaluation of the 50 THz data (Antognini et al., A recent review of the proton radius puzzle is given by 2013b). This reevaluation reduced the original frequency Carlson (2015). by 0.53 GHz and its uncertainty from 0.76 GHZ to 0.65 GHz. By comparison, the uncertainty of the 55 THz B. Hyperfine structure and fine structure frequency is 1.05 GHz. The theory that relates the muonic hydrogen Lamb-shift In principle, together with theory, hyperfine-structure measurements other than in muonium, and fine-structure mea- transition frequencies to rp reflects the calculations and critical investigations of many researchers, and a number of issues surements other than in hydrogen and deuterium, could have been resolved since the closing date of the 2010 provide accurate values of some constants, most notably adjustment; a detailed review is given by Antognini et al. the fine-structure constant α. Indeed, it has long been the (2013a). Using the theory in this paper the CREMA collabo- hope that a competitive value of α could be obtained from ration reports, in the same paper in which they give their two experimental measurements and theoretical calculations of measured transition frequencies (Antognini et al., 2013b), the 4He fine-structure transition frequencies. However, as discussed in CODATA-10, no such data were available for the following value as the best estimate of rp from muonic hydrogen: 2010 adjustment and this is also the case for the 2014 adjustment. For completeness, we note that there have been significant rp 0.840 87 39 fm: 78 ¼ ð Þ ð Þ improvements in both the theory and experimental determi- This may be compared with the muonic hydrogen value r nation of the hyperfine splitting in positronium [see Adkins p ¼ 0.841 69 66 fm available for consideration in the 2010 et al. (2015), Eides and Shelyuto (2015), Ishida (2015), and ð Þ Miyazaki et al. (2015) and the references cited therein]. Also, adjustment based on the 2S F 1 − 2P F 2 tran- 1=2ð ¼ Þ 3=2ð ¼ Þ Marsman, Horbatsch, and Hessels (2015a, 2015b) have found sition frequency before reevaluation and the theory as it that the measured values of helium fine-structure transition existed at the time. Because the new result in Eq. (78) is frequencies can be significantly influenced by quantum- smaller and has a smaller uncertainty than this value and that mechanical interference between neighboring resonances neither the H-D data nor theory have changed significantly, it even if separated by thousands of natural linewidths. The is not surprising that the disagreement still persists and that shifts they calculated for reported experimental values of the Task Group has decided to omit the μ-p result for rp 3 3 4 the 2 P1 − 2 P2 fine-structure interval in He improve their in the 2014 adjustment. Indeed, the disagreement of the agreement with theory. value in Eq. (78) with the 2014 CODATA recommended value 0.8751(61) fm is 5.6σ and with the H-D spectroscopic value V. MAGNETIC MOMENTS AND g-FACTORS 0.8759(77) fm is 4.5σ. The negative effect of including the value of rp in Eq. (78) The magnetic-moment vector of a charged lepton is as an input datum in the 2014 adjustment and why the Task Group concluded that it should be excluded is discussed e μl gl s; 79 further in Sec. XIII.B.2. However, we do point out here the ¼ 2ml ð Þ two following facts. First, if the least-squares adjustment that leads to the where l e, μ, or τ, gl is the g-factor, with the convention value of α−1 given in Eq. (71) is carried out with the value that it has¼ the same sign as the charge of the particle, e is the in Eq. (78) included as an input datum, the result is (positive) unit charge, ml is the lepton mass, and s is its spin.

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-16 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

8 Since the spin has projection eigenvalues of sz ℏ=2, the A1ð Þ −1.912 98 84 ; 89 magnetic moment is defined to be ¼Æ ¼ ð Þ ð Þ

gl eℏ 10 μl ; 80 A1ð Þ 7.795 336 : 90 ¼ 2 2ml ð Þ ¼ ð Þ ð Þ and the Bohr magneton is just Higher-order coefficients are assumed to be negligible. Mass-dependent coefficients for the electron, based on the eℏ CODATA-14 values of the mass ratios, are μB : 81 ¼ 2me ð Þ 4 7 Að Þ m =m 5.197 386 76 23 × 10− The g-factor differs from the Dirac value of 2 because of the 2 ð e μÞ¼ ð Þ lepton magnetic-moment anomaly a defined by → 24.182 × 10−10a ; 91 l e ð Þ

gl 2 1 al : 82 j j ¼ ð þ Þ ð Þ 4 −9 A2ð Þ me=mτ 1.837 98 33 × 10 The theoretical value of the anomaly is given by QED, ð Þ¼ ð Þ → 0.086 × 10−10a ; 92 predominately electroweak, and predominantly strong- e ð Þ interaction effects denoted by

6 −6 Að Þ me=m −7.373 941 71 24 × 10 al th al QED al weak al had ; 83 2 ð μÞ¼ ð Þ ð Þ¼ ð Þþ ð Þþ ð Þ ð Þ −10 → −0.797 × 10 ae; 93 respectively. ð Þ

6 8 A. Electron magnetic-moment anomaly a and the Að Þ m =m −6.5830 11 × 10− e 2 ð e τÞ¼ ð Þ fine-structure constant α → −0.007 × 10−10a : 94 e ð Þ A value for the fine-structure constant α is obtained by equating theory and experiment for the electron magnetic- Additional series expansions in the mass ratios yield (Kurz moment anomaly. et al., 2014a)

8 4 1. Theory of ae Að Þ m =m 9.161 970 83 33 × 10− 2 ð e μÞ¼ ð Þ The QED contribution to the electron anomaly is → 0.230 × 10−10a ; 95 e ð Þ a QED A A m =m A m =m eð Þ¼ 1 þ 2ð e μÞþ 2ð e τÞ 8 6 A m =m ;m =m ; 84 Að Þ m =m 7.4292 12 × 10− þ 3ð e μ e τÞ ð Þ 2 ð e τÞ¼ ð Þ → 0.002 × 10−10a . 96 where the mass-dependent terms arise from vacuum- e ð Þ polarization loops. Each term may be expanded in powers of the fine-structure constant as A numerical calculation gives the next term (Aoyama et al., 2012b) ∞ n 2n α Ai Að Þ ; 85 ¼ i ð Þ 10 n 1 π Að Þ m =m −0.003 82 39 X¼ 2 ð e μÞ¼ ð Þ −10 2 2 4 → −0.002 × 10 ae: 97 where A2ð Þ A3ð Þ A3ð Þ 0. ¼ ¼ ¼ 2n ð Þ The mass-independent terms A1ð Þ are known accurately through sixth order: Additional terms have been calculated, but are negligible at the current level of accuracy; see, for example, Aoyama et al. 2 1 (2014) and Kurz et al. (2014a). Að Þ ; 86 1 ¼ 2 ð Þ All contributing terms of each order in α are combined to yield the total coefficients in the series 4 Að Þ −0.328 478 965 579 193…; 87 1 ¼ ð Þ ∞ n 6 2n α a QED Cð Þ ; 98 A1ð Þ 1.181 241 456…: 88 e e ¼ ð Þ ð Þ¼n 1 π ð Þ X¼ Recent numerical evaluations have yielded values for the eighth- and tenth-order coefficients (Aoyama et al., 2014) with

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-17 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

2 where 0.000 37 × 10−12 is the input datum for δ . It is Ceð Þ 0.5; e ¼ noteworthy thatð Þ the uncertainty of a th is significantly 4 eð Þ Ceð Þ −0.328 478 444 00…; smaller than the uncertainty of the most accurate experimental ¼ −10 6 value (see following section), which is 2.4 × 10 ae. Ceð Þ 1.181 234 017…; ¼ 8 Ceð Þ −1.912 06 84 ; 2. Measurements of a ¼ ð Þ e 10 Ceð Þ 7.79 34 ; 99 Two experimental values of ae obtained using a Penning ¼ ð Þ ð Þ trap were initially included in CODATA-10. The first is the where higher-order coefficients are negligible. classic 1987 result from the University of Washington dis- The electroweak contribution, calculated as in CODATA-98 cussed in CODATA-98 with a relative standard uncertainty 2 −9 but with present values of GF and sin θW (see Sec. XII), is ur 3.7 × 10 and which assumes that CPT invariance holds¼ for the electron-positron system (Van Dyck, a weak 0.029 73 23 × 10−12 Schwinberg, and Dehmelt, 1987). The second is the much eð Þ¼ ð Þ −10 more accurate 2008 result from Harvard University discussed 0.2564 20 × 10 ae: 100 −10 ¼ ð Þ ð Þ in CODATA-10 with ur 2.4 × 10 (Hanneke, Fogwell, and Gabrielse, 2008). These¼ results are items B22.1 and B22.2 The hadronic contribution is the sum in Table XVIII, Sec. XIII, with identifications UWash-87 and LO;VP NLO;VP HarvU-08; the values of α that can be inferred from them ae had ae had ae had ð Þ¼ ð Þþ ð Þ based on the theory of ae discussed in the previous section are LL ae had ; 101 given in Table XX, Sec. XIII.A. The University of Washington þ ð ÞþÁÁÁ ð Þ result was not included in the final adjustment on which the LO;VP NLO;VP where ae had and ae had are the leading order and CODATA 2010 recommended values are based because of its next-to-leadingð orderÞ (with anð additionalÞ photon or electron comparatively low weight and is also omitted from the 2014 loop) hadronic vacuum-polarization corrections given by final adjustment. It is only considered to help provide an Nomura and Teubner (2013) overall picture of the data available for the determination of the 2014 recommended value of α. LO;VP −12 ae had 1.866 11 × 10 ; 102 ð Þ¼ ð Þ ð Þ B. Muon magnetic-moment anomaly aμ NLO;VP −12 ae had −0.2234 14 × 10 ; 103 ð Þ¼ ð Þ ð Þ Only the measured value of the muon magnetic-moment NNLO;VP anomaly was included in the 2010 adjustment of the constants and ae had is the next-to-next-to-leading order vacuum-polarizationð Þ correction (Kurz et al., 2014b) due to concerns about some aspects of the theory. The concerns still remain, thus the Task Group decided not to NNLO;VP −12 employ the theoretical expression for a in the 2014 adjust- ae had 0.0280 10 × 10 ; 104 μ ð Þ¼ ð Þ ð Þ ment. The theory and measurement of aμ and the reasons for LL the Task Group decision are presented in the following and where ae had is the hadronic light-by-light scattering term given by ðPrades,Þ de Rafael, and Vainshtein (2010) sections.

LL −12 a had 0.035 10 × 10 : 105 1. Theory of aμ e ð Þ¼ ð Þ ð Þ The relevant mass-dependent terms are The total hadronic contribution is 4 Að Þ m =m 1.094 258 3092 72 −12 2 μ e ae had 1.734 15 × 10 ð Þ¼ ð Þ ð Þ¼ ð Þ → 5.06 × 10−3a ; 110 14.95 13 × 10−10a : 106 μ ð Þ ¼ ð Þ e ð Þ 4 Að Þ m =m 0.000 078 079 14 The theoretical value is the sum of all the terms: 2 ð μ τÞ¼ ð Þ → 3.61 × 10−7a ; 111 μ ð Þ ae th ae QED ae weak ae had ; 107 ð Þ¼ ð Þþ ð Þþ ð Þ ð Þ 6 A2ð Þ m =me 22.868 379 98 17 and has the standard uncertainty ð μ Þ¼ ð Þ → 2.46 × 10−4a ; 112 μ ð Þ u a th 0.037 × 10−12 0.32 × 10−10a ; 108 e e 6 ½ ð Þ ¼ ¼ ð Þ Að Þ m =m 0.000 360 63 11 2 ð μ τÞ¼ ð Þ where the largest contributions to the uncertainty are from the → 3.88 × 10−9a ; 113 8th- and 10th-order QED terms. This uncertainty is taken into μ ð Þ account in the least-squares adjustment by writing 8 Að Þ m =m 132.6852 60 2 ð μ eÞ¼ ð Þ a th a α δ ; 109 → 3.31 × 10−6a ; 114 eð Þ¼ eð Þþ e ð Þ μ ð Þ

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8 LO;VP NLO;VP Að Þ m =m 0.042 4941 53 aμ had aμ had aμ had 2 ð μ τÞ¼ ð Þ ð Þ¼ ð Þþ ð Þ 9 LL → 1.06 × 10− a ; 115 aμ had ; 126 μ ð Þ þ ð ÞþÁÁÁ ð Þ 10 Að Þ m =m 742.18 87 where (Hagiwara et al., 2011) 2 ð μ eÞ¼ ð Þ → 4.30 × 10−8a ; 116 μ ð Þ aLO;VP had 6949 43 × 10−11; 127 μ ð Þ¼ ð Þ ð Þ 10 A2ð Þ mμ=mτ −0.068 5 ð Þ¼ ð Þ aNLO;VP had −98.40 72 × 10−11; 128 → −3.94 × 10−12a ; 117 μ ð Þ¼ ð Þ ð Þ μ ð Þ 6 and (Jegerlehner and Nyffeler, 2009) Að Þ m =m ;m =m 0.000 527 762 94 3 ð μ e μ τÞ¼ ð Þ −9 LL −11 → 5.67 × 10 aμ; 118 a had 116 40 × 10 : 129 ð Þ μ ð Þ¼ ð Þ ð Þ 8 Að Þ m =me;m =m 0.062 72 4 The total hadronic contribution is then 3 ð μ μ τÞ¼ ð Þ −9 → 1.57 × 10 aμ: 119 ð Þ −11 aμ had 6967 59 × 10 : 130 10 ð Þ¼ ð Þ ð Þ A3ð Þ mμ=me;mμ=mτ 2.011 10 ð Þ¼ ð Þ In analogy with the electron we have finally → 1.17 × 10−10a : 120 μ ð Þ a th a QED 7121 59 × 10−11; 131 New mass-dependent contributions reported since CODATA- μð Þ¼ μð Þþ ð Þ ð Þ 10 are Eqs. (114), (116), (117), (119), and (120) from Aoyama et al. (2012a) and Eq. (115) from Kurz et al. (2014a). with the standard uncertainty The total for a QED is μð Þ u a th 59 × 10−11 51 × 10−11a : 132 ∞ n μ μ 2n α ½ ð Þ ¼ ¼ ð Þ aμ QED Ceð Þ ; 121 ð Þ¼n 1 π ð Þ The largest contributions to the uncertainty are from X¼ aLO;VP had and aLL had ; their respective uncertainties of with μ ð Þ μ ð Þ 43 × 10−11 and 40 × 10−11 are nearly the same. By compari- 2 −11 son, the 0.087 × 10 uncertainty of aμ QED is negligible. Cμð Þ 0.5; ð Þ ¼ However, the 63 × 10−11 uncertainty of the experimental 4 Cμð Þ 0.765 857 423 16 ; value of a , which is discussed in the following section, ¼ ð Þ μ 6 and u aμ th are nearly the same. Based on the 2014 Cμð Þ 24.050 509 82 27 ; ½ ð Þ ¼ ð Þ recommended value of α, Eq. (131) yields 8 Cð Þ 130.8774 61 ; μ ¼ ð Þ 10 −3 Cð Þ 751.92 93 : 122 aμ th 1.165 918 39 59 × 10 133 μ ¼ ð Þ ð Þ ð Þ¼ ð Þ ð Þ

Based on the 2014 recommended value of α, this yields for the theoretically predicted value of aμ.

−10 aμ QED 0.00116584718858 87 7.4 × 10 ; 123 ð Þ¼ ð Þ½ ð Þ 2. Measurement of aμ: Brookhaven −12 where an uncertainty of 0.8 × 10 is included to account for The experimental determination of aμ at Brookhaven uncalculated 12th-order terms (Aoyama et al., 2012a). National Laboratory (BNL), Upton, New York, USA, has As for the electron, there are additional contributions: been discussed in the past four CODATA reports. The quantity measured is the anomaly difference frequency fa fs − fc, a th a QED a weak a had : 124 where f g eℏ=2m B=h is the muon spin-flip¼ (or μ μ μ μ s ¼ j μjð μÞ ð Þ¼ ð Þþ ð Þþ ð Þ ð Þ precession) frequency in the applied magnetic flux density The primarily electroweak contribution is (Czarnecki, B and fc eB=2πm is the corresponding muon cyclotron ¼ μ Marciano, and Vainshtein, 2003; Gnendiger, Stöckinger, frequency. The flux density is eliminated from these expres- and Stöckinger-Kim, 2013) sions by determining its value using proton nuclear magnetic resonance (NMR) measurements. This means that the muon a weak 154 1 × 10−11: 125 μð Þ¼ ð Þ ð Þ anomaly is calculated from Separate contributions (lowest-order vacuum polarization, R next-to-lowest-order, and light-by-light) to the hadronic term a exp ; 134 μð Þ¼ μ =μ − R ð Þ are j μ pj

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-19 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ... where R f =f and f is the free proton NMR frequency others and that lead to a discrepancy of between 4σ and 5σ. ¼ a p p corresponding to the average flux density B seen by the muons However, Davier and Malaescu (2013) give a number of in their orbits in the muon storage ring. reasons why these values may not be reliable. The final value of R obtained in the experiment is, from The value for aLL had used to obtain a th is μ ð Þ μð Þ Table XV of Bennett et al. (2006), 116 40 × 10−11, but there are other credible values in the literature.ð Þ For example, Prades, de Rafael, and Vainshtein R 0.003 707 2063 20 ; 135 (2010) give 105 26 × 10−11 while Melnikov and Vainshtein ¼ ð Þ ð Þ (2004) find 136ð 25Þ × 10−11. A more recent value, due to which is used as an input datum in the 2014 adjustment and is Narukawa et al. (2014)ð Þ , is 118 20 × 10−11, and the largest is the same as used in the 2010 adjustment. It is datum B24 in 188 4 × 10−11 reported by Goecke,ð Þ Fischer, and Williams Table XVIII with identification BNL-06. Based on this value (2013)ð Þ. Others range from 80 40 × 10−11 to 107 17 × 10−11. ð LLÞ ð Þ of R, Eq. (134), and the 2014 recommended value of μμ=μp, Recent brief overviews of a had may be found in μ ð Þ whose uncertainty is negligible in this context, the exper- Dorokhov, Radzhabov, and Zhevlakov (2014a, 2014b), imental value of the muon anomaly is Nyffeler (2014), and Adikaram et al. (2015), which describe the many obstacles to obtaining a reliable estimate of its value. a exp 1.165 920 89 63 × 10−3: 136 The conclusion that emerges from these papers is that μð Þ¼ ð Þ ð Þ aLL had is quite model dependent and the reliability of μ ð Þ Further, with the aid of Eq. (225), the equation for R can be the estimates is questionable. Note that in calculating the value written as 116 40 × 10−11 used to obtain Eq. (133), Nyffeler (2009) addsð theÞ uncertainty components linearly and rounds the result a me μe R − μ ; 137 up from 39 × 10−11 to 40 × 10−11. This result is also thor- ¼ 1 a m μ ð Þ þ e μ p oughly discussed in the detailed review by Jegerlehner and Nyffeler (2009) (see especially Table 13). where use has been made of the relations g −2 1 a , e ¼ ð þ eÞ Although much work has been done over the past 4 years to g −2 1 a , and a is replaced by the theoretical μ ¼ ð þ μÞ e improve the theory of a , the discrepancy between experiment expression given in Eq. (109) for the observational equation. μ and theory remains at about the 3σ level. Further, there is a If the theory of a were not problematic and used in adjust- μ significant spread in the values of aLO;VP had and aLL had , ment calculations, then a in Eq. (137) would be its theoretical μ ð Þ μ ð Þ μ the two most problematic contributions to a th ; it is not expression, which mainly depends on α. If the theory is μð Þ omitted, then a in that equation is simply taken to be an obvious which are the best values. Expanding the uncertainty μ of a th to reflect this spread would reduce its contribution to adjusted constant. The following section discusses why the μð Þ Task Group decided to do the latter. the determination of the 2014 CODATA recommended value of a significantly. Expanding the uncertainties of both a th μ μð Þ and aμ exp to reduce the discrepancy to an acceptable level 3. Comparison of theory and experiment for aμ ð Þ and including both would mean that the recommended value The difference between the experimental value of aμ in would cease to be a useful reference value for future Eq. (136) and the theoretical value in Eq. (133) is comparisons of theory and experiment; it might tend to cover 250 86 × 10−11, which is 2.9 times the standard uncertainty ð Þ up an important physics problem rather than emphasizing it. of the difference or 2.9σ. The terms aLO;VP had and aLL had For all these reasons, the Task Group chose not to include μ ð Þ μ ð Þ are the dominant contributors to the uncertainty of a th , and a th in the 2014 adjustment and to base the 2014 recom- μð Þ μð Þ a smaller value of either will increase the disagreement while a mended value on experiment only as was the case in 2010. larger value will decrease it. The value for aLO;VP had used to obtain a th is μ μ C. Proton magnetic moment in nuclear magnetons μp=μN 6949 43 × 10−11, butð anÞ equally credibleð value,Þ 6923ð42Þ × 10−11, is also available (Davier et al., 2011). If The 2010 recommended value of the magnetic moment of ð Þ it is used instead, the discrepancy is 276 86 × 10−11 or 3.2σ. the proton μp in units of the nuclear magneton μN eℏ=2mp ð Þ ¼−9 Both values are based on very thorough analyses using theory has a relative standard uncertainty ur 8.2 × 10 . It was − ¼ and experimental data from the production of hadrons in eþe not measured directly but calculated from the relation collisions. Davier et al. (2011) also obtain the value μp=μN μp=μB Ar p =Ar e , where Ar e and Ar p are 11 ¼ð Þ½ ð Þ ð Þ ð Þ ð Þ 7015 47 × 10− using both e e− annihilation data and data adjusted constants and μB eℏ=2me is the Bohr magneton. ð Þ þ ¼ from the decay of the τ into hadrons. For this value the The proton magnetic moment in units of μB is calculated discrepancy is reduced to 2.1σ. A result due to Jegerlehner and from μp=μB μe=μB = μe=μp , where μe=μB 1 ae is − ¼ð Þ ð Þ j j ¼ þ Szafron (2011) also obtained using eþe annihilation data and extremely well known and μe=μp is an adjusted constant τ decay data but which is only in marginal agreement with this determined mainly by the experimentally measured value of value is 6910 47 × 10−11. However, it agrees with the first the bound-state magnetic-moment ratio in hydrogen two values, whichð Þ are based solely on e e− data. For this μ H =μ H taken as an input datum. þ eð Þ pð Þ value the discrepancy is 3.3σ. In a lengthy paper Benayoun Now, however, a directly determined value of μp=μN with −9 et al. (2013) used the hidden local symmetry model or HLS to ur 3.3 × 10 obtained from measurements on a single obtain values of aLO;VP had that are significantly smaller than proton¼ in a double Penning trap at the Institut für Physik, μ ð Þ

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Johannes Gutenberg Universität Mainz (or simply the uncertainty assigned by Mooser et al. (2014) for the possible University of Mainz), Mainz, Germany, is available for effect of nonlinear drifts of the magnetic field. inclusion in the 2014 adjustment. The spin-flip transition The relationships given at the beginning of this section frequency in a magnetic flux density B is show how the 2010 recommended value of μp=μN is based on the relation ΔE 2μpB ωs ; 138 ¼ ℏ ¼ ℏ ð Þ μp Ar p μp − 1 a ð Þ ; 142 μ ¼ ð þ eÞ A e μ ð Þ and the cyclotron frequency for the proton is N rð Þ e eB which is the source of the observational equation B29 in ωc : 139 Table XXIV. ¼ mp ð Þ For completeness, we note that a value of μp=μN with ur −6 ¼ The ratio for the same magnetic flux density B is just 8.9 × 10 from the double Penning trap in an earlier stage of development has been published by the Mainz group −6 ω μp (Rodegheri et al., 2012), as has a value with u 2.5×10 s : 140 r ¼ ωc ¼ μN ð Þ obtained by a direct measurement at Harvard University with a different type of Penning trap (Lees et al., 2012). Also The value employed as an input datum in the 2014 adjustment noteworthy is the use of a similar double Penning trap at is CERN to test CPT invariance by comparing the antiproton-to- proton charge-to-mass ratio, demonstrating that they agree μ 12 p 2.792 847 3498 93 3.3 × 10−9 ; 141 within an uncertainty of 69 parts in 10 (DiSciacca μN ¼ ð Þ½ ð Þ et al., 2013). which is the result reported by Mooser et al. (2014) but with D. Atomic g-factors in hydrogenic 12C and 28Si and A e an additional digit for both the value and uncertainty provided rð Þ to the Task Group by coauthor Blaum (2014). This value, The most accurate values of the relative atomic mass of the which we identify as UMZ-14, is the culmination of an electron A e are obtained from measurements of the electron extensive research program carried out over many years. r g-factor inð hydrogenicÞ ions, silicon and carbon in particular, Descriptions of the key advances made in the development and theoretical expressions for the g-factors. In fact, the of the double Penning trap used in the experiment are given in uncertainties of the values so obtained are now so small that a number of publications (Ulmer, Blaum et al., 2011; Ulmer, none of the data previously used to determine A e remain of Rodegheri et al., 2011; Mooser et al., 2013; Mooser, Kracke r interest. ð Þ et al., 2013; Ulmar et al., 2013). For a hydrogenic ion X with a spinless nucleus and atomic The double Penning trap consists of a precision trap of inner number Z, the energy-level shift in an applied magnetic flux diameter 7 mm and an analysis trap of inner diameter 3.6 mm density B in the z direction is given by connected by transport electrodes, all mounted in the horizontal bore of a superconducting magnet with B ≈ 1.89 T. e Storage times for a single proton of no less than 1 year are E −μ · B −g X J B; 143 2m z achieved by enclosing the entire apparatus in a sealed vacuum ¼ ¼ ð Þ e ð Þ chamber cooled to 4 K where pressures below 10−14 Pa are where J is the electron angular-momentum projection in the z reached. The measurement sequence is straightforward but its z direction and g X is the atomic g-factor. In the ground 1S implementation is complex; the cited papers should be ð Þ 1=2 state, J ℏ=2, so the splitting between the two levels is consulted for details. Sputtered atoms created by electrons z ¼Æ from a field emission gun hitting a polyethylene target are eℏ ionized in the center of the precision trap. A single proton is ΔE g X B; 144 obtained from the resulting ion cloud, its cyclotron frequency ¼ j ð Þj 2me ð Þ of approximately 29 MHz is determined, and an electromagnetic signal near the proton’s precession frequency of approx- and the spin-flip transition frequency is imately 81 MHz is applied to it. The proton is then transported to the analysis trap where the spin state of the proton is ΔE eB ωs g X : 145 determined and the proton returned to the precision trap. By ¼ ℏ ¼ j ð Þj 2me ð Þ varying the frequency of the spin-flip signal and repeating the process many times, a resonance curve of the probability In the same flux density, the ion’s cyclotron frequency is P ω =ω as a function of ω is obtained. Its maximum is ð s cÞ s located at ωs=ωc μp=μN. The statistical relative standard qXB ¼ −9 ωc ; 146 uncertainty of the value given in Eq. (141) is 2.6 × 10 , and ¼ mX ð Þ the net fractional correction for systematic effects is −9 −0.64 2.04 × 10 . The largest contributor by far to the where qX Z − 1 e is the net charge of the ion and mX is its uncertaintyð Þ of the net correction is the 2 × 10−9 relative mass. Thus¼ð the frequencyÞ ratio is

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ω g X m g X A X Radiative corrections are given by s j ð Þj X j ð Þj rð Þ ; 147 ωc ¼ 2 Z − 1 me ¼ 2 Z − 1 Ar e ð Þ ð Þ ð Þ ð Þ ∞ n 2n α Δgrad −2 Cðe Þ Zα ; 150 where Ar X is the relative atomic mass of the ion. ¼ n 1 ð Þ π ð Þ ð Þ X¼ where the limits 1. Theory of the bound-electron g-factor The bound-electron g-factor is given by 2n 2n lim Cðe Þ Zα Ceð Þ 151 Zα→0 ð Þ¼ ð Þ

g X gD Δgrad Δgrec Δgns ; 148 ð Þ¼ þ þ þ þÁÁÁ ð Þ are given in Eq. (99). The first term is (Faustov, 1970; Grotch, 1970; Close and where the individual terms on the right-hand side are the Dirac Osborn, 1971; Pachucki, Jentschura, and Yerokhin, 2004; value, radiative corrections, recoil corrections, nuclear-size Pachucki et al., 2005) corrections, and the dots represent possible additional cor- 2 rections not already included. Tables XI and XII give the 2 1 Zα 4 32 2 247 Cð Þ Zα 1 ð Þ Zα ln Zα − numerical values of the various contributions. e;SEð Þ¼2 þ 6 þð Þ 9 ð Þ þ 216 & The Dirac value is (Breit, 1928) 8 8 − ln k − ln k Zα 5R Zα ; 152 9 0 3 3 þð Þ SEð Þ ð Þ ! ' 2 2 gD − 1 2 1 − Zα ¼ 3 ½ þ ð Þ where 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 1 −2 1 − Zα 2 − Zα 4 − Zα 6 ; 149 ¼ 3ð Þ 12ð Þ 24ð Þ þÁÁÁ ð Þ ln k 2.984 128 556; 153 ! 0 ¼ ð Þ where the only uncertainty is due to the uncertainty in α. ln k 3.272 806 545; 154 3 ¼ ð Þ

TABLE XI. Theoretical contributions and total value for the g- RSE 6α 22.160 10 ; 155 factor of hydrogenic carbon 12 based on the 2014 recommended ð Þ¼ ð Þ ð Þ values of the constants. RSE 14α 20.999 2 : 156 Contribution Value Source ð Þ¼ ð Þ ð Þ

Dirac gD −1.998 721 354 392 1 6 Eq. (149) Values for the remainder function RSE Zα are based on 2 −0.002 323 672 435 4ð Þ Eq. (157) extrapolations from numerical calculationsð Þ at higher Z ΔgSEð Þ ð Þ 2 0.000 000 008 511 Eq. (160) (Yerokhin, Indelicato, and Shabaev, 2002, 2004; Pachucki, ΔgVPð Þ 4 Jentschura, and Yerokhin, 2004); see also Yerokhin and Δgð Þ 0.000 003 545 677(25) Eq. (164) 6 Jentschura (2008, 2010). In CODATA-10, the values for Δgð Þ −0.000 000 029 618 Eq. (166) 8 Δgð Þ 0.000 000 000 111 Eq. (167) carbon and oxygen were taken directly from Pachucki, 10 Δgð Þ −0.000 000 000 001 Eq. (168) Jentschura, and Yerokhin (2004). The value for silicon in Δgrec −0.000 000 087 629 Eqs. (169), (170) Eq. (156) is obtained here by extrapolation of the data in Δgns −0.000 000 000 408 1 Eq. (172) Yerokhin, Indelicato, and Shabaev (2004) with a fitting 12 5 ð Þ g C þ −2.001 041 590 183 26 Eq. (173) −2 2 −2 ð Þ ð Þ function of the form a Zα b cln Zα dln Zα . We thus have þð Þ½ þ ð Þ þ ð Þ

2 Cð Þ 6α 0.500 183 606 65 80 ; TABLE XII. Theoretical contributions and total value for the g- e;SEð Þ¼ ð Þ factor of hydrogenic silicon 28 based on the 2014 recommended 2 Cð Þ 14α 0.501 312 630 11 : 157 values of the constants. e;SEð Þ¼ ð Þ ð Þ Contribution Value Source The lowest-order vacuum-polarization correction consists

Dirac gD −1.993 023 571 557 3 Eq. (149) of a wave-function correction and a potential correction, each 2 −0.002 328 917 47 5ð Þ Eq. (157) of which can be separated into a lowest-order Uehling ΔgSEð Þ ð Þ 2 0.000 000 234 81(1) Eq. (160) potential contribution and a Wichmann-Kroll higher-order ΔgVPð Þ 4 contribution. The wave-function correction is (Beier, 2000; Δgð Þ 0.000 003 5521(17) Eq. (164) 6 Δgð Þ −0.000 000 029 66 Eq. (166) Beier et al., 2000; Karshenboim, 2000; Karshenboim, Ivanov, 8 Δgð Þ 0.000 000 000 11 Eq. (167) and Shabaev, 2001a, 2001b) 10 Δgð Þ −0.000 000 000 00 Eq. (168) 2 Δgrec −0.000 000 205 88 Eqs. (169), (170) Ceð;VPwfÞ 6α −0.000 001 840 3431 43 ; Δgns −0.000 000 020 53 3 Eq. (172) ð Þ¼ ð Þ 28 13 ð Þ 2 g Si þ −1.995 348 9581 17 Eq. (173) Cð Þ 14α −0.000 051 091 98 22 : 158 ð Þ ð Þ e;VPwf ð Þ¼ ð Þ ð Þ

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For the potential correction, the Uehling contribution vanishes estimated the uncertainty due to uncalculated higher-order (Beier et al., 2000), and for the Wichmann-Kroll part, we take contributions to be the value of Lee et al. (2005), which has a negligible uncertainty from omitted binding corrections for the present 4 5 4 u Ceð Þ Zα 2 Zα Cðe ÞRSE Zα ; 165 level of accuracy. This gives ½ ð Þ ¼ jð Þ ð Þj ð Þ which we use as the uncertainty. Since the remainder functions 2 Ceð;VPpÞ 6α 0.000 000 008 201 11 ; differ only by about 1% for carbon and silicon, the main Z ð Þ¼ ð Þ 5 2 dependence of the uncertainty is given by the factor Zα .We Cð Þ 14α 0.000 000 5467 11 : 159 ð Þ e;VPpð Þ¼ ð Þ ð Þ shall assume that the uncertainty of the two-photon correction is completely correlated for the two charged ions. As a The total vacuum polarization is the sum of Eqs. (158) consequence, information from the silicon measured value and (159): will effectively be included in the theoretical prediction for the

2 2 2 carbon value through the least-squares adjustment formalism. Cð Þ 6α Cð Þ 6α Cð Þ 6α e;VPð Þ¼ e;VPwf ð Þþ e;VPpð Þ Jentschura (2009) and Yerokhin and Harman (2013) have −0.000 001 832 142 12 ; calculated two-loop vacuum-polarization diagrams of the ¼ ð Þ same order as the uncertainty in Eq. (164). 2 2 2 Cð Þ 14α Cð Þ 14α Cð Þ 14α Equation (163) gives the leading two terms of the higher- e;VPð Þ¼ e;VPwf ð Þþ e;VPpð Þ loop contributions. The corrections are −0.000 050 5452 11 : 160 ¼ ð Þ ð Þ 2 One-photon corrections are the sum of Eqs. (157) 6 6 Zα Ceð Þ Zα Ceð Þ 1 ð Þ and (160): ð Þ¼ þ 6 þÁÁÁ 2 2 2 1.181 611… for Z 6 Ceð Þ 6α Ceð;SEÞ 6α Ceð;VPÞ 6α ¼ ¼ ð Þ¼ ð Þþ ð Þ 1.183 289… for Z 14; 166 0.500 181 774 51 80 ; ¼ ¼ ð Þ ¼ ð Þ 6 2 2 2 where Cðe Þ 1.181 234 017…, Ceð Þ 14α Cð Þ 14α Cð Þ 14α ð Þ¼ e;SEð Þþ e;VPð Þ ¼ 2 0.501 262 085 11 ; 161 8 8 Zα ¼ ð Þ ð Þ Ceð Þ Zα Ceð Þ 1 ð Þ ð Þ¼ þ 6 þÁÁÁ which yields −1.912 67 84 … for Z 6 ¼ ð Þ ¼ 2 2 α … Δgð Þ −2Ceð Þ Zα −1.915 38 84 for Z 14; 167 rad ¼ ð Þ π ¼ ð Þ ¼ ð Þ 8 −0.002 323 663 924 4 for Z 6 where Cðe Þ −1.912 06 84 , and ¼ ð Þ ¼ ¼ ð Þ −0.002 328 682 65 5 for Z 14: 162 2 ¼ ð Þ ¼ ð Þ 10 10 Zα Ceð Þ Zα Ceð Þ 1 ð Þ ð Þ¼ þ 6 þÁÁÁ The leading binding correction is known to all orders in α=π (Eides and Grotch, 1997; Czarnecki, Melnikov, and 7.79 34 … for Z 6 Yelkhovsky, 2001): ¼ ð Þ ¼ 7.80 34 … for Z 14; 168 2 ¼ ð Þ ¼ ð Þ 2n 2n Zα Ceð Þ Zα Ceð Þ 1 ð Þ : 163 10 ð Þ¼ þ 6 þÁÁÁ ð Þ where Ceð Þ 7.79 34 . Recoil of the¼ nucleusð Þ gives a correction proportional to the 4 To order Zα , the two-photon correction for the ground S electron-nucleus mass ratio. It can be written as Δgrec state is (Pachuckið Þ et al., 2005; Jentschura et al., 2006) 0 2 ¼ Δgrecð Þ Δgrecð Þ , where the two terms are zero and first þ þÁÁÁ order in α=π, respectively. The first term is (Eides and Grotch, Zα 2 14 4 4 4 −2 1997; Shabaev and Yerokhin, 2002) Ceð Þ Zα Ceð Þ 1 ð Þ Zα ln Zα ð Þ¼ þ 6 þð Þ 9 ð Þ 4 991 343 2 4 0 2 Zα 5 me ln k ln k grecð Þ Zα ð Þ Zα P Zα − 0 − 3 Δ − 2 2 − þ 155 520 9 3 ¼ ð Þ þ ð Þ ð Þ mN & 3 1 1 − Zα ' 2 2 ½ þ ð Þ 679π 1441π 1441 5 m 2 − ln 2 ζ 3 O Zα 1 Z Zα 2 pe ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 12 960 720 þ 480 ð Þ þ ð Þ þð þ Þð Þ m ! N −0.328 5778 23 for Z 6 −0.000 000 087 70… for Z 6 ¼ ð Þ ¼ ¼ ¼ −0.329 17 15 for Z 14; 164 −0.000 000 206 04… for Z 14; 169 ¼ ð Þ ¼ ð Þ ¼ ¼ ð Þ 4 where Ceð Þ −0.328 478 444 00…, and where ln k0 and ln k3 where mN is the mass of the nucleus. Mass ratios, based are given in¼ Eqs. (153) and (154). Pachucki et al. (2005) have on the current adjustment values of the constants, are

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12 6 28 14 me=m C þ 0.000 045 727 5… and me=m Si þ Si [see Eqs. (155) and (156)]. As a conservative estimate, we 0.000ð 019 613Þ¼ 6… [see Eqs. (10) and (3)]. Forð silicon,Þ¼ we shall assume that 80% of the uncertainty scales as Zα 5 in the use the interpolated value P 14α 7.162 23 1 . case of the two-loop uncertainty. As a result, informationð Þ from 2 ð Þ¼ ð Þ For Δgrecð Þ, we have measurement of the Si g-factor will provide information about the two-loop uncertainty in C and vice versa through the 2 2 α Zα me covariance of the deltas in the least-squares adjustment. The Δgrecð Þ ð Þ covariance of δ and δ is ¼ π 3 mN þÁÁÁ C Si

0.000 000 000 06… for Z 6 −20 ¼ ¼ u δC; δSi 3.4 × 10 ; 177 0.000 000 000 15… for Z 14: 170 ð Þ¼ ð Þ ¼ ¼ ð Þ which corresponds to a correlation coefficient of 2 r δ ; δ 0.79. The uncertainty in Δgrecð Þ is negligible compared to that ð C SiÞ¼ 2 of gð Þ. Δ rad 2. Measurements of g 12C5 and g 28Si13 The nuclear-size correction is given to lowest order in Zα 2 ð þÞ ð þÞ by (Karshenboim, 2000) ð Þ As discussed at the start of Sec. V.D, recent measurements of the electron g-factor in hydrogenic carbon and silicon 2 together with theory provide a value of A e with an 8 4 RN r Δgns − Zα ; 171 uncertainty so small that the data used in theð Þ CODATA ¼ 3 ð Þ ƛC ð Þ 2010 adjustment to determine Ar e are no longer competitive and need not be considered. Asð indicatedÞ at the end of that where RN is the bound-state nuclear rms charge radius and ƛC is the Compton wavelength of the electron divided by 2π. section, the experimental quantities actually determined are Glazov and Shabaev (2002) have calculated additional cor- the ratios of the electron spin precession (or spin-flip) rections within perturbation theory. Scaling their results with frequency in hydrogenic carbon and silicon ions to the cyclotron frequency of the ions, both in the same magnetic the squares of updated values for the nuclear radii RN ¼ flux density. The result used in the 2014 adjustment for 2.4703 22 fm and RN 3.1223 24 fm from the compilation ofðAngeliÞ (2004) for¼ 12C andð 28ÞSi respectively yields hydrogenic silicon is

28 13 12 ωs Si þ 11 Δgns −0.000 000 000 408 1 for C; ð Þ 3912.866 064 84 19 4.8 × 10− : 178 ¼ ð Þ ω 28Si13 ¼ ð Þ½ ð Þ 16 c þ Δgns −0.000 000 020 53 3 for Si: 172 ð Þ ¼ ð Þ ð Þ This value, determined by the group at the Max-Planck- Tables XI and XII list the contributions discussed above and Institut für Kernphysik (MPIK), Heidelberg, Germany, is totals given by based on additional information including a detailed uncertainty budget provided to the Task Group by the experimenters 12 5 g C þ −2.001 041 590 183 26 ; (Sturm, 2015). The information was needed to calculate the ð Þ¼ ð Þ 28 13 covariance between the silicon result and the MPIK carbon g Si þ −1.995 348 9581 17 : 173 ð Þ¼ ð Þ ð Þ result discussed below. The value in Eq. (178) differs slightly For the purpose of the least-squares adjustment, we write from that published by the MPIK group (Sturm et al., 2013) as the theoretical expressions for the g-factors as a consequence of the group’s reassessment of several small corrections for systematic effects (Sturm, 2015). The net 12 5 fractional correction applied to the uncorrected ratio to obtain g C þ gC α δC; ð Þ¼ ð Þþ the final ratio given by Sturm et al. (2013) is −638.9 × 10−12 28 13 g Si þ gSi α δSi; 174 while the net fractional correction applied to obtain Eq. (178) ð Þ¼ ð Þþ ð Þ is −678.5 × 10−12. The largest of these corrections by far, where the first term on the right-hand side of each expression −659 33 × 10−12, is due to image charge and the next largest, ð Þ gives the calculated value along with its functional depend- −20 10 × 10−12, is due to frequency pulling. The statistical ence on α. The second term contains the theoretical uncer- relativeð Þ uncertainty of eight individual measurements is tainty in the calculated value, except for the component due to 33 × 10−12. We identify the result in Eq. (178) as MPIK-15. uncertainty in α, which is taken into account by the least- Both the silicon and carbon ratios were obtained using the squares algorithm through the first term. We thus have MPIK triple cylindrical Penning trap operating at B 3.8 T and in thermal contact with a liquid helium bath. This¼ trap is −11 δC 0.0 2.6 × 10 ; 175 similar in design and operation to the double Penning trap ¼ ð Þ ð Þ used to directly measure μp=μN as discussed in Sec. V.C. The −9 δSi 0.0 1.7 × 10 : 176 first stage of the experiment occurs in the creation trap in ¼ ð Þ ð Þ which ions are created; the second stage is carried out in the In each case, the uncertainty is dominated by uncalculated analysis trap where the spin state of a single ion is determined; two-loop higher-order terms, which are expected to be mainly the third stage occurs in the precision trap where the cyclotron proportional to Zα 5. For the one-loop self energy, approx- frequency is measured and a spin flip is attempted by applying ð Þ imately 85% of the remainder scales as Zα 5 between C and a 105 GHz microwave signal. The ion is transferred back to ð Þ

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-24 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ... the analysis trap where the spin state of the ion is again With the aid of Eq. (4) this becomes the observational 28 13 determined, thereby determining if the spin-flip attempt in the equation for Ar Si þ , which is B19 in Table XXIV. 12 ð Þ precision trap was successful. The process is then repeated. Because Ar C 12 exactly, such an additional observa- The development of the trap and associated measurement tional equationð isÞ¼ unnecessary for carbon; Eq. (181) becomes techniques over a number of years that has allowed uncer- 12 5 12 5 the observational equation for ωs C þ =ωc C þ simply tainties below 5 parts in 1011 to be achieved are discussed in by using Eq. (4) to modify itsð lastÞ termð (see ÞB15 in several papers (Blaum et al., 2009; Sturm et al., 2010; Sturm, Table XXIV). Wagner, Schabinger, and Blaum, 2011; Ulmer, Blaum et al., The silicon frequency ratio yields for the relative atomic 28 13 2011) and results using Si þ that reflect experimental mass of the electron improvements have been reported (Sturm, Wagner, −10 Schabinger, Zatorski et al., 2011; Schabinger et al., 2012; Ar e 0.000 548 579 909 19 46 8.3 × 10 ; 184 Sturm et al., 2013). ð Þ¼ ð Þ½ ð Þ For hydrogenic carbon we use and the carbon frequency ratio gives

12 5 −11 ωs C þ −11 Ar e 0.000 548 579 909 070 17 3.1 × 10 : 185 ð12 5 Þ 4376.210 500 87 12 2.8 × 10 ; 179 ð Þ¼ ð Þ½ ð Þ ωc C þ ¼ ð Þ½ ð Þ ð Þ If both data are used, we obtain which is also based on additional information supplied to the −11 Task Group (Sturm et al., 2015) at the same time as that for Ar e 0.000 548 579 909 069 16 2.9 × 10 : 186 silicon and similar in nature. The high-accuracy carbon result ð Þ¼ ð Þ½ ð Þ was first published in a letter to Nature (Sturm et al., 2014) The slight shift in value and reduction in uncertainty in and differs slightly from Eq. (179) but the value subsequently Eq. (186) as compared to Eq. (185) is due to the information published in the detailed report on the experiment is the same about the higher-order terms in the theory resulting from the (Köhler et al., 2015). The net fractional correction applied to silicon measurement. If the covariance in the theory given by the uncorrected ratio is −283.3 × 10−12, with the largest Eq. (177) is taken to be zero, the result from using both components being for image charge and frequency pulling measurements is the same as the result based only on the at −282.4. × 10−12 and 2.20 × 10−12, respectively. The stat- carbon datum, within the uncertainty displayed in the istical relative uncertainty of the individual measurements is equations. 23 × 10−12. The detailed report gives a comprehensive dis- We note that in the 2010 adjustment, the uncertainty of cussion of the MPIK trap including many possible systematic Ar p was much lower than that of Ar e . As a result, the ratio ð Þ ð Þ effects and their uncertainties. We identify the result in Ar e =Ar p from the analysis of antiprotonic helium tran- ð Þ ð Þ Eq. (179) as MPIK-15. sition frequencies together with the value for Ar p provided a ð Þ The carbon and silicon frequency ratios in Eqs. (178) and competitive value for Ar e . Now that Ar e is more accurately ð Þ ð Þ (179) are correlated. Based on the detailed uncertainty budgets known, the value of Ar e combined with the ratio ð Þ for the two experiments supplied to the Task Group (Sturm, Ar e =Ar p provides a new value for Ar p . However, despite ð Þ ð Þ ð Þ 2015) the correlation coefficient is improvement in the theory of antiprotonic helium (Korobov, Hilico, and Karr, 2014), the uncertainty of the derived value of 12 5 28 13 ω C ω Si Ar p is much too large for it be used in the 2014 adjustment. s þ s þ ð Þ r ð12 5 Þ ; ð28 13 Þ 0.347; 180 ω C ω Si þ ¼ ð Þ cð þÞ cð Þ! VI. MAGNETIC-MOMENT RATIOS AND THE which is mostly due to the image charge correction. MUON-ELECTRON MASS RATIO The frequency ratios are related to the ion and electron masses by Free-particle magnetic-moment ratios can be obtained from experiments that measure moment ratios in bound states by 12 5 12 5 12 5 ωs C þ g C þ ΔEB C þ applying theoretical corrections relating the free moment 12 5A e ; ð12 5 Þ − ð Þ − r ð 2 Þ ratios to the bound moment ratios. ωc C þ ¼ 10Ar e ð Þþ muc ð Þ ð Þ ! The magnetic moment of a nucleus with spin I is 181 ð Þ e and μ g I; 187 ¼ 2mp ð Þ

28 13 28 13 ωs Si þ ge Si þ 28 13 where g is the g-factor of the nucleus, e is the elementary ð Þ − ð Þ Ar Si þ ; 182 ω 28Si13 ¼ 14A e ð Þ ð Þ charge, and m is the proton mass. The magnitude of the cð þÞ rð Þ p magnetic moment is defined to be 28 13 where Ar Si þ is taken to be an adjusted constant and is ð Þ 28 related to the input datum Ar Si by [see Eq. (3)] μ gμ i; 188 ð Þ ¼ N ð Þ E 28Si13 28 13 28 Δ B þ where μN eℏ=2mp is the nuclear magneton, and i is the Ar Si þ Ar Si − 13Ar e ð 2 Þ : 183 ¼ ð Þ¼ ð Þ ð Þþ muc ð Þ maximum spin projection I , defined by I2 i i 1 ℏ2. z ¼ ð þ Þ

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In the Pauli approximation, the Hamiltonian for a hydrogen Muonium (see Sec. VI.B): atom in the ground state in an applied magnetic flux ge Mu 1 1 1 α density B is ð Þ 1 − Zα 2 − Zα 4 Zα 2 g ¼ 3 ð Þ 12 ð Þ þ 4 ð Þ e π 1 m 1 1 α 2 ω μ μ 2 e 4 2 Δ H B N Zα A1ð Þ − Zα H s · I − ge H s · B − gp H I · B; 189 þ 2 ð Þ m þ 2 4 ð Þ ¼ ℏ ð Þ ℏ ð Þ ℏ ð Þ μ π 2 5 α me 1 me − Zα 2 − 1 Z Zα 2 …; 12 ð Þ π m 2 ð þ Þð Þ m þ where ΔωH is the ground-state hyperfine frequency, s is the μ μ electron spin as given in Eq. (79), and μB is given by Eq. (81). 194 The coefficients g H and g H are bound-state g-factors ð Þ eð Þ pð Þ and are related to the corresponding free g-factors g and g by g Mu 1 97 1 m e p μð Þ 1 α Zα α Zα 3 α Zα e the theoretical corrections given below. The analogous cor- − − gμ ¼ 3 ð Þ 108 ð Þ þ 2 ð Þ mμ rections for deuterium, muonium and helium-3 are also 2 1 α me 1 me given. α Zα − 1 Z α Zα …: þ 12 ð Þ m 2 ð þ Þ ð Þ m þ π μ μ 195 A. Theoretical ratios of atomic bound-particle to ð Þ free-particle g-factors Helium-3:

Theoretical binding corrections to g-factors are as follows. 3 μh He References for the calculations are given in previous detailed ð Þ 1 − 59.967 43 10 × 10−6; 196 μ ¼ ð Þ ð Þ CODATA reports. h Hydrogen: which has been calculated by Rudziński, Puchalski, and Pachucki (2009). However, this ratio is not used as an input ge H 1 1 1 α 1 me ð Þ 1 − Zα 2 − Zα 4 Zα 2 Zα 2 datum because it is not coupled to any other data, but allows the ge ¼ 3 ð Þ 12 ð Þ þ 4 ð Þ π þ 2 ð Þ mp Task Group to provide a recommended value for the unshielded 2 helion magnetic moment along with other related quantities. 1 4 1 2 α 5 2 α me Að Þ − Zα − Zα …; Numerical values for the corrections in Eqs. (190) to (195) þ 2 1 4 ð Þ 12 ð Þ m þ π π p are listed in Table XIII; uncertainties are negligible. See Ivanov, 190 Karshenboim, and Lee (2009) for a negligible additional term. ð Þ

1. Ratio measurements The experimental magnetic-moment and bound-state gp H 1 97 3 1 me 3 4ap ð Þ 1 − α Zα − α Zα α Zα þ …; magnetic-moment ratios and magnetic-moment shielding gp ¼ 3 ð Þ 108 ð Þ þ 6 ð Þmp 1 ap þ þ corrections that were used as input data in the 2010 adjustment 191 – ð Þ are used again in the 2014 adjustment; they are B30 B35.1, B37, and B38 in Table XVIII, Sec. XIII. A concise cataloging 4 of these data is given in CODATA-10 and each measurement where Að Þ is given in Eq. (87), and the proton magnetic- 1 has been discussed fully in at least one of the previous detailed moment anomaly is a μ = eℏ=2m − 1 ≈ 1.793. p ¼ p ð pÞ CODATA reports. The observational equations for these data Deuterium: are given in Table XXIV and the adjusted constants in those equations are identified in Table XXVI, both in Sec. XIII. Any ge D 1 1 1 α 1 me ð Þ 1 − Zα 2 − Zα 4 Zα 2 Zα 2 relevant correlation coefficients for these data may be found in ge ¼ 3 ð Þ 12 ð Þ þ 4 ð Þ π þ 2 ð Þ md Table XIX, also in Sec. XIII. The theoretical bound-particle to 2 1 4 1 2 α 5 2 α me free-particle g-factor ratios in the observational equations, A1ð Þ − Zα − Zα …; þ 2 4 ð Þ π 12 ð Þ π md þ which are taken to be exact because their uncertainties are 192 ð Þ TABLE XIII. Theoretical values for various bound-particle to free- particle g-factor ratios relevant to the 2014 adjustment based on the 2014 recommended values of the constants.

Ratio Value gd D 1 97 3 1 me 3 4ad ð Þ 1 − α Zα − α Zα α Zα þ …; −6 gd ¼ 3 ð Þ 108 ð Þ þ 6 ð Þmd 1 ad þ ge H =ge 1 − 17.7054 × 10 ð Þ −6 þ gp H =gp 1 − 17.7354 × 10 193 ð Þ −6 ð Þ ge D =ge 1 − 17.7126 × 10 ð Þ −6 gd D =gd 1 − 17.7461 × 10 ð Þ −6 ge Mu =ge 1 − 17.5926 × 10 where the deuteron magnetic-moment anomaly is ad ð Þ −6 ¼ gμ Mu =gμ 1 − 17.6254 × 10 μ = eℏ=m − 1 ≈−0.143. ð Þ d ð dÞ

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negligible, are given in Table XIII. The symbol μp0 denotes the This result, input datum B36 in Table XVIII and identified as magnetic moment of a proton in a spherical sample of pure StPtrsb-11, replaces the earlier result because of its signifi- H2O at 25 °C surrounded by vacuum; and the symbol μh0 cantly smaller uncertainty. Its observational equation is B36 in denotes the magnetic moment of a helion bound in a 3He Table XXIV. atom. Although the exact shape and temperature of the The reduced uncertainty was achieved by decreasing the gaseous 3He sample is unimportant, we assume that it is magnetic field inhomogeneity across the sample and by measuring the difference ω HD − ω HT using a commercial spherical, at 25 °C, and surrounded by vacuum. pð Þ pð Þ In general, the bound magnetic moment of a particle, for NMR spectrometer operating at 9.4 T and using the result to determine ω HT =ω HT from the measured value of example, that of p, d, t, or h, is related to its free value by tð Þ pð Þ μ bound 1 σ μ free , where σ is the nuclear magnetic ωt HT =ωp HD . The latter frequency ratio was obtained with − ð Þ ð Þ shieldingð Þ¼ð correctionÞ orð parameter.Þ For the hydrogen-deuterium aspeciallydesigned,laboratory-madespectrometeroperatingat molecule HD, σ HD and σ HD are the shielding correc- 2.1 T. Use of HD as a buffer gas in the NMR sample was necessary pð Þ dð Þ tions for the proton and deuteron in HD, respectively. to reduce the diffusion displacement of the HT molecules. The Since σ is small, one may define σ σ HD −σ HD and value given in Eq. (198) is the mean of 10 individual values and its dp ¼ dð Þ pð Þ write μ HD =μ HD 1 σ O σ2 μ =μ . This also assigned uncertainty is the simple standard deviation of these pð Þ dð Þ¼½ þ dp þ ð Þ p d applies to the hydrogen-tritium or HT molecule: σ values rather than that of their mean as initially assigned by tp ¼ Neronov and Aleksandrov (2011).Thisuncertainty,chosenby σ HT −σ HT and μ HT =μ HT 1 σ O σ2 μ =μ . tð Þ pð Þ pð Þ tð Þ¼½ þ tpþ ð Þ p t the Task Group to better reflect possible systematic effects, was Two new relevant data have become available since the discussed with and accepted by Neronov (2015). closing date of the 2010 adjustment. Garbacz et al. (2012) at For completeness, we briefly mention new and potentially the University of Warsaw, Warsaw, Poland determined the relevant data that were not considered for inclusion in the ratio μp HD =μd HD by separately measuring, in the same ð Þ ð Þ 2014 adjustment. More accurate theoretical values for σdp and magnetic flux density B, the nuclear magnet resonance (NMR) σ were reported by Puchalski, Komasa, and Pachucki (2015) frequencies ω HD and ω HD of the proton and deuteron tp pð Þ dð Þ but did not become available until well after the in HD. Their result is 31 December 2014 closing date of the adjustment. They −9 −9 are σdp 20.20 2 × 10 and σtp 24.14 2 × 10 com- μp HD 1 ωp HD ¼ ð Þ −9 ¼ ð Þ −9 ð Þ ð Þ pared with σdp 15 2 × 10 and σtp 20 3 × 10 used μd HD ¼ 2 ωd HD ¼ ð Þ ¼ ð Þ ð Þ ð Þ in the adjustment (input data B37 and B38 in Table XVIII). 3.257 199 514 21 6.6 × 10−9 : 197 ¼ ð Þ½ ð Þ Also, the St. Petersburg NMR researchers reported a value for ω 3He =ω H (Neronov and Seregin, 2012), the NMR The factor 1=2 arises because the spin quantum number for the h p 2 frequencyð ratioÞ ofð theÞ helion in 3He to that of a proton in H , proton is 1=2 while for the deuteron it is 1. We identify this 2 and also for σp H2 σp H2O (Neronov and Seregin, 2014), result, which is taken as an input datum in the 2014 adjustment, ð Þ − ð Þ as UWars-12; it is item B35.2 in Table XVIII and is the second the difference in the magnetic screening constants for the value of this ratio now available with an uncertainty less than 1 proton in H2 and H2O. However, the unavailability of detailed uncertainty budgets for these two results precluded their part in 108. Its observational equation, B35 in Table XXIV, is consideration for possible inclusion in the adjustment. the same as for the other value, identified as StPtrsb-03. The UWars result was obtained using a technique developed at the university and described by Jackowski, Jaszuński, and B. Muonium transition frequencies, the muon-proton magnetic-moment ratio μ =μ , and muon-electron Wilczek (2010). The NMR measurements of the frequencies μ p mass ratio m =m ω HD and ω HD were carried using a variable gaseous μ e pð Þ dð Þ sample that contained HD of a sufficiently low density that the Muonium (Mu) is an atom consisting of a positive muon HD-HD molecular interactions were inconsequential and neon and an electron in a bound state. Measurements of muonium of density between 11 mol=L and 2 mol=L so that the observed ground-state hyperfine transitions in a magnetic field provide frequencies could be extrapolated to zero neon density with an information on the muon-proton magnetic-moment ratio as uncertainty of 0.5 Hz. The neon was used to increase the well as the muon-electron mass ratio. This information is pressure of the sample thereby facilitating the NMR measure- obtained by an analysis of the Zeeman transition resonances in ments. There were no uncertainties of significance from an applied magnetic flux density. systematic effects (Jackowski, 2015). The theoretical expression for the hyperfine splitting may The NMR measurements on the HT molecule carried out in be factorized into a part that exhibits the main dependence on St. Petersburg, Russia that led to a value for μt HT =μp HT various fundamental constants and a function F that depends −9 ð Þ ð Þ with ur 9.4 × 10 and which was used as an input datum in only weakly on them. We write the 2010¼ adjustment have continued (Aleksandrov and Neronov, 2011). The new value reported by Neronov and ΔνMu th ΔνFF α;me=m ; 199 Aleksandrov (2011), which is in agreement with the earlier ð Þ¼ ð μÞ ð Þ value, is where

μ HT 16 m m −3 t −9 3 2 e e ð Þ 1.066 639 8933 21 2.0 × 10 : 198 ΔνF cR Z α 1 200 μ HT ¼ ð Þ½ ð Þ ¼ 3 ∞ m þ m ð Þ pð Þ μ μ

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2 1 is the Fermi formula. In order to identify the source of the where Að Þ , as in Eq. (86). The function G Zα accounts 1 ¼ 2 ð Þ terms, some of the theoretical expressions are for a muon with for all higher-order contributions in powers of Zα; it charge Ze rather than e. can be divided into self-energy and vacuum-polarization contributions, G Zα GSE Zα GVP Zα . Yerokhin and 1. Theory of the muonium ground-state hyperfine splitting Jentschura (2008ð, 2010)Þ¼ haveð calculatedÞþ ð theÞ one-loop self Presented here is a brief summary of the present theory of energy for the muonium hyperfine splitting with the result ΔνMu. Complete results of the relevant calculations are given along with references to new work; references to the original literature included in earlier detailed CODATA reports are not GSE α −13.8308 43 ; 205 repeated. ð Þ¼ ð Þ ð Þ The general expression for the hyperfine splitting is which agrees with the value G α −13.8 3 from an SEð Þ¼ ð Þ ΔνMu th ΔνD Δνrad Δνrec earlier calculation by Yerokhin et al. (2005), as well as with ð Þ¼ þ þ other previous estimates. The vacuum-polarization part is Δνr r Δνweak Δνhad; 201 þ − þ þ ð Þ (Karshenboim, Ivanov, and Shabaev, 1999, 2000) where the terms labeled D, rad, rec, r-r, weak, and had account for the Dirac, radiative, recoil, radiative-recoil, electroweak, and G α 7.227 9 ; 206 hadronic contributions to the hyperfine splitting, respectively. VPð Þ¼ ð ÞþÁÁÁ ð Þ The Dirac equation yields where the dots denote uncalculated Wichmann-Kroll 3 17 2 4 contributions. ΔνD ΔνF 1 aμ 1 Zα Zα ; 202 ¼ ð þ Þ þ 2ð Þ þ 8 ð Þ þÁÁÁ ð Þ For D 4 Zα , we have ! ð Þð Þ where aμ is the muon magnetic-moment anomaly. The radiative corrections are 1 4 4 2 −2 Dð Þ Zα Að Þ 0.770 99 2 πZα − ln Zα ð Þ¼ 1 þ ð Þ þ 3 ð Þ 2 α Δνrad ΔνF 1 aμ Dð Þ Zα ¼ ð þ Þ ð Þ π − 0.6390… × ln Zα −2 10 2.5 Zα 2 ð Þ þ ð Þ ð Þ α 2 α 3 ! D 4 Zα D 6 Zα ; 203 þ ð Þð Þ þ ð Þð Þ þÁÁÁ ð Þ ; 207 π π ! þÁÁÁ ð Þ 2n where the functions Dð Þ Zα are contributions from n virtual ð Þ 4 photons. The leading term is where A1ð Þ is given in Eq. (87), and the coefficient of πZα has been calculated by Mondéjar, Piclum, and Czarnecki (2010). 2 2 5 The next term is Dð Þ Zα Að Þ ln 2 − πZα ð Þ¼ 1 þ 2 2 281 8 2 −2 −2 6 6 − ln Zα − ln 2 ln Zα D Zα Að Þ ; 208 þ 3 ð Þ þ 360 3 ð Þ ð Þ 1 ð Þ¼ þÁÁÁ ð Þ 2 16.9037… Zα 6 þ ð Þ where the leading contribution Að Þ is given in Eq. (88), but ! 1 5 547 only partial results of relative order Zα have been calculated ln 2 − ln Zα −2 π Zα 3 2n þ 2 96 ð Þ ð Þ (Eides and Shelyuto, 2007). Higher-order functions Dð Þ Zα ! with n>3 are expected to be negligible. ð Þ G Zα Zα 3; 204 þ ð Þð Þ ð Þ The recoil contribution is

m me 3 μ Zα 1 −2 65 Δνrec ΔνF − 2 ln 2 ln Zα − 8 ln 2 ¼ m 1 − m =m me π þ 1 m =m ð Þ þ 18 μ ð e μÞ ð þ e μÞ & 9 m 27 m 93 33ζ 3 13 me ln2 μ − 1 ln μ ð Þ − − 12 ln 2 Zα 2 þ 2 2 m þ 2 2 m þ 4 2 þ 2 12 m ð Þ π e π e π π ! μ' 3 m 1 101 Zα 3 − ln μ ln Zα −2 − ln2 Zα −2 − 10 ln 2 ln Zα −2 40 10 ð Þ ; 209 þ 2 m ð Þ 6 ð Þ þ 18 ð Þ þ ð Þ þÁÁÁ ð Þ & e ' π as discussed in CODATA-02.

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The radiative-recoil contribution is α Zα 3, and 3 Hz from the uncertainty 2.5 of the number 10 in ð Þ 4 the function Dð Þ Zα . 2 ð Þ α me 2 mμ 13 mμ For Δνrec, the total uncertainty is 64 Hz and is due to three Δνr-r ΔνF −2ln ln ¼ π mμ me þ 12 me components: 53 Hz from 2 times the uncertainty 10 of the & number 40 in Eq. (209) as discussed in CODATA-02; 21 π2 35 4 ζ 3 ln2α−2 34 Hz due to a possible recoil correction of order þ 2 ð Þþ 6 þ 9 þ 3 3 ! ΔνF me=m × Zα ln me=m ; and 6 Hz to reflect a ð μÞ ð Þ ð μÞ 16 341 2 possible recoil term of order ν m =m × Zα 4ln2 Zα −2. ln 2 − ln α− − 40 10 πα Δ F e μ þ 3 180 ð Þ ð Þ ð Þ ð Þ ! The total uncertainty in Δνr−r is 55 Hz, with 53 Hz from 2 times the uncertainty 10 of the number 40 in Eq. (210) as 4 3 mμ 4 2 mμ α − − ln ln above, and 15 Hz as discussed in connection with the newly þ 3 me þ 3 me π ! ' 2 included radiative-recoil contribution, Eq. (211). The uncer- 2 me 13 − ν α 6 ln 2 ; 210 tainty of Δνhad is 1.4 Hz from Eq. (213). The final uncertainty F m þ 6 þÁÁÁ ð Þ μ in ΔνMu th is thus ð Þ where, for simplicity, the explicit dependence on Z is not u ΔνMu th 85 Hz: 214 shown. ½ ð Þ ¼ ð Þ New radiative-recoil contributions arising from all single- logarithmic and nonlogarithmic three-loop corrections are due For the least-squares calculations, we use as the theoretical to Eides and Shelyuto (2014): expression for the hyperfine splitting m α 3 m 2 27 m e e 2 π μ ΔνMu th ΔνMu R∞; α; ;aμ δMu; 215 ΔνF −6π ln 2 ln 68.507 2 ð Þ¼ m þ ð Þ m þ 3 þ 8 m þ ð Þ μ π μ & ! e ' −30.99 Hz: 211 where the input datum for the additive correction ¼ ð Þ δMu 0 85 Hz, which accounts for the uncertainty of the Additional radiative-recoil corrections have been calculated, theoretical¼ ð expression,Þ is data item B28 in Table XVIII. but are negligibly small, less than 0.5 Hz. Uncalculated The above theory yields remaining terms of the same order as those included in Eq. (211) are estimated by Eides and Shelyuto (2014) to −8 ΔνMu 4 463 302 868 271 Hz 6.1 × 10 216 be about 10 Hz to 15 Hz. ¼ ð Þ ½ ðÞ 0 The electroweak contribution due to the exchange of a Z using values of the constants obtained from the 2014 adjust- boson is (Eides, 1996) ment without the two measured values of ΔνMu discussed in the following section. The main source of uncertainty in this Δνweak −65 Hz; 212 ¼ ð Þ value is the mass ratio me=mμ. while for the hadronic vacuum-polarization contribution we have (Nomura and Teubner, 2013) 2. Measurements of muonium transition frequencies and values of μμ=μp and mμ=me

Δνhad 232.7 1.4 Hz: 213 The two most precise determinations of muonium ¼ ð Þ ð Þ Zeeman transition frequencies were carried out at the A negligible contribution (≈0.0065 Hz) from the hadronic Clinton P. Anderson Meson Physics Facility at Los light-by-light correction has been given by Karshenboim, Alamos (LAMPF), New Mexico, USA, and were reviewed Shelyuto, and Vainshtein (2008). in detail in CODATA-98. The results are as follows. The approach used to evaluate the uncertainty of the Data reported in 1982 by Mariam (1981) and Mariam et al. theoretical expression for ΔνMu is described in detail in (1982) are CODATA-02. The only change for CODATA-14 is that the −8 probable error estimates of uncertainties that were sub- ΔνMu 4 463 302.88 16 kHz 3.6 × 10 ; 217 sequently multiplied by 1.48 to convert them to standard ¼ ð Þ ½ ð Þ uncertainties (that is, 1 standard deviation estimates) are now −7 ν fp 627 994.77 14 kHz 2.2 × 10 ; 218 assumed to have been standard uncertainty estimates in the ð Þ¼ ð Þ ½ ð Þ first place and are not multiplied by 1.48. This change was r ΔνMu; ν fp 0.227; 219 motivated by conversations with Eides (2015). ½ ð Þ ¼ ð Þ Four sources of uncertainty in ΔνMu th are Δνrad, Δνrec, ð Þ where f is 57.972 993 MHz, corresponding to the Δνr−r, and Δνhad in Eq. (201), although the uncertainty in the p latter is now so small that it is of only marginal interest. The magnetic flux density of about 1.3616 T used in the total uncertainty in ν is 5 Hz and consists of two experiment, and r ΔνMu; ν fp is the correlation coefficient Δ rad ½ ð Þ components: 4 Hz from an uncertainty of 1 in G α due of ΔνMu and ν fp . The data reported in 1999 by Liu et al. VP ð Þ to the uncalculated Wichmann-Kroll contribution ofð Þ order (1999) are

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−8 ΔνMu 4 463 302 765 53 Hz 1.2 × 10 ; 220 On the other hand, in the full least-squares adjustment the ¼ ð Þ ½ ð Þ value of α is determined by other data which in turn −8 determines the value of m =me with a significantly smaller ν fp 668 223 166 57 Hz 8.6 × 10 ; 221 μ ð Þ¼ ð Þ ½ ð Þ uncertainty than that of Eq. (227).

r ΔνMu; ν fp 0.195; 222 ½ ð Þ ¼ ð Þ VII. QUOTIENT OF PLANCK CONSTANT AND PARTICLE MASS h=m X AND α where fp is 72.320 000 MHz, corresponding to the flux ð Þ density of approximately 1.7 T used in the experiment, and A value of the fine-structure constant α can be obtained r ΔνMu; ν fp is the correlation coefficient of ΔνMu and ½ ð Þ from a measurement of h=m X through the expression ν fp . The data in Eqs. (217), (218), (220), and (221) are ð Þ ð Þ 1=2 data items B27.1, B25, B27.2, and B26, respectively, in 2R∞ Ar X h Table XVIII. α ð Þ ; 229 ¼ c Ar e m X ð Þ The expression for the magnetic-moment ratio is ð Þ ð Þ! which follows from the definition of the Rydberg constant 2 2 −1 2 μ Δν − ν fp 2sefpν fp g Mu R α m c=2h, and where A X is the relative atomic mass μþ Mu ð Þþ ð Þ μð Þ ; 223 ∞ e r 2 of particle¼ X with mass m X .ð TheÞ relative standard uncer- μp ¼ 4sefp − 2fpν fp gμ ð Þ ð Þ tainties of R and A e areð aboutÞ 6 × 10−12 and 3 × 10−11, ∞ rð Þ where ΔνMu and ν fp are the sum and difference of two respectively, and the relative uncertainty of the relative atomic ð Þ −10 measured transition frequencies, fp is the free proton NMR mass of a number of atoms is a few times 10 or less. Hence, −9 reference frequency corresponding to the flux density used in a measurement of h=m X with ur of 1 × 10 can provide a value of α with the highlyð Þ competitive relative uncertainty the experiment, gμ Mu =gμ is the bound-state correction for ð Þ −10 the muon in muonium given in Table XIII, and of 5 × 10 . Two values of h=m X obtained using atom interferometry techniques were initiallyð Þ included as input data in CODATA- μe ge Mu se ð Þ ; 224 10 and the same values are initially included in CODATA-14. ¼ μ g ð Þ p e The first value is the result for h=m 133Cs obtained at Stanford University, Stanford, California,ð USA,Þ reported by where g Mu =g is the bound-state correction for the electron e e −8 in muoniumð Þ given in the same table. Wicht et al. (2002) with ur 1.5 × 10 . This experiment is discussed in CODATA-06 and¼ the result is input datum B46 in The muon to electron mass ratio mμ=me and the muon to proton magnetic-moment ratio μ =μ are related by Table XVIII, Sec. XIII, and is labeled StanfU-02. The value of μ p −9 α inferred from it with ur 7.7 × 10 is given in Table XX. The Stanford result for ¼h=m 133Cs was not included as an m μ μ −1 g μ e μ μ : 225 input datum in the final adjustmentð Þ on which the 2010 m ¼ μ μ g ð Þ e p p e recommended values are based because of its low weight, and is omitted from the 2014 final adjustment for the same A least-squares adjustment using the LAMPF data, the 2014 reason. However, it is discussed in order to provide a complete recommended values of R∞, μe=μp, ge, and gμ, together with picture of the available data relevant to α. Eqs. (199), (200) and Eqs. (223) to (225), yields The second value of h=m X , which is included in the final 2014 adjustment, is the resultð Þ for h=m 87Rb determined at μ μþ −7 −9ð Þ 3.183 345 24 37 1.2 × 10 ; 226 LKB in Paris with ur 1.2 × 10 and reported by μp ¼ ð Þ½ ð Þ Bouchendira et al. (2011).¼ The experiment is discussed in CODATA-10 and the result, labeled LKB-11, is datum B48 in mμ 206.768 276 24 1.2 × 10−7 ; 227 Table XVIII; the value of α inferred from it with ur ¼ me ¼ ð Þ½ ð Þ 6.2 × 10−10 is given in Table XX. Although it has the second smallest uncertainty of the 14 values of α in that table, its α−1 137.036 0013 79 5.8 × 10−8 : 228 uncertainty is still about 2.6 times that of the value with the ¼ ð Þ½ ð Þ smallest uncertainty, that from the Harvard University meas- The muonium value of α in Eq. (228) is compared to other urement of ae. Nevertheless, the comparison of the values of α values in Table XX. from the two experiments provides a useful test of the QED The uncertainty of mμ=me in Eq. (227) is nearly 5 times as theory of ae. Such a comparison is discussed in Sec. XV.B of large as the uncertainty of the 2014 recommended value and the Summary and Conclusions portion of this report. follows from Eqs. (223) to (225). It has the same relative We conclude this section by noting that a value of 133 −9 uncertainty as the moment ratio in Eq. (226). However, taken h=m Cs with ur 4.0 × 10 can in principle be obtained together the experimental value of and theoretical expression fromð the measurementÞ ¼ by Lan et al. (2013) of the Compton 133 for the hyperfine splitting essentially determine the value of frequency of the mass of the cesium-133 atom νC Cs using 2 133 ð2 133Þ the product α me=mμ, as is evident from Eqs. (199) and (200), atom interferometry since h=m Cs c =νC Cs . with an uncertainty dominated by the 1.9 × 10−8 relative However, Müller (2015) informed theð TaskÞ¼ Group thatð smallÞ uncertainty in the theory. corrections have recently been identified that were not

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-30 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

9 included in the reported result and consequently it should not in 10 and the XRCD values of NA had to be increased by a be considered. A new result with a highly competitive similar amount. uncertainty is anticipated. Our third comment has to do with the measurement of g, the local acceleration due to gravity. The basic watt-balance VIII. ELECTRICAL MEASUREMENTS equation is UI msgv, where U is the voltage induced between the terminals¼ of a coil moving in a magnetic flux The principal focus of this portion of the paper is the several density B with velocity v; and I is the current in the coil in the moving-coil watt-balance (or simply watt-balance) measure- same flux density B when the force on the coil due to I and B 2 ments of K R 4=h that have become available in the past just balances the weight msg of a standard of mass ms. J K ¼ 4 years, where KJ 2e=h is the Josephson constant and RK Although commercial absolute free-fall gravimeters that use 2 ¼ ¼ optical interferometry to measure the position of a falling h=e μ0c=2α is the von Klitzing constant. Nevertheless, the 13 legacy¼ electrical input data that were initially included in reflector are capable of determining g with a relative uncer- the 2010 adjustment in order to investigate data robustness and tainty of a few parts in 109, recently there has been concern 2 about the correction due to the finite speed of the propagation the exactness of the relations KJ 2e=h and RK h=e but were omitted from the final adjustment¼ on which¼ the 2010 of light, or so-called “speed of light correction.” This recommended values are based because of their low weight correction for a 20 cm drop is usually about 1 part in 108. are again initially included in the 2014 adjustment for the In particular, Rothleitner, Niebauer, and Francis (2014) claim same purpose. These input data are items B39.1 through that the correction normally applied for this effect is too large B43.5 and B45 in Table XVIII, Sec. XIII. They are five by a third. However, Baumann et al. (2015) have recently measurements of the gyromagnetic ratio of the proton and carried out a very extensive theoretical and experimental helion by the low and high field methods, five measurements investigation of the problem and have convincingly shown of RK using a calculable capacitor, one measurement of KJ that this is not the case. 2 using a mercury electrometer and another using a capacitor The resulting values of KJ RK from the seven watt-balance voltage balance, and one measurement of the Faraday constant experiments we consider are items B44.1–B44.7 in using a silver dissolution electrometer. A brief explanation of Table XVIII, Sec. XIII. This quantity is the actual input these different kinds of measurements and a more detailed datum employed in least-squares calculations using as its 2 cataloging of the 13 data are given in CODATA-10 and each observational equation KJ RK 4=h. The resulting values of measurement has been discussed fully in at least one of the h are compared with each other≐ and with inferred values of h previous four detailed CODATA reports. The observational from other experiments in Table XXI, Sec. XIII.A. equations for these 13 data are B39 − B43 and B45 in Table XXIV and the adjusted constants in those equations A. NPL watt balance are in Table XXVI, both in Sec. XIII. Any relevant correlation coefficients for these data are listed in Table XIX, also in There are two watt-balance results from the National Sec. XIII. Table XX or XXI in Sec. XIII.A compares the data Physical Laboratory (NPL), Teddington, UK, both with −7 among themselves and with other data through the values of relative standard uncertainties ur 2.0 × 10 . NPL-90, item either α or h that they infer. Three comments are in order B44.1 in Table XVIII, is discussed¼ in CODATA-98 and is before beginning the discussion of individual watt-balance from the first truly high-accuracy watt-balance experiment experiments. carried out (Kibble, Robinson, and Belliss, 1990). The design First, some watt-balance researchers find it useful to define of the NPL Mark I apparatus was unique in that the moving the exact, conventional value of the Planck constant h90 coil consisted of two flat rectangular coils above one another 2 −34 ¼ 4=KJ 90RK−90 6.626 068 854… × 10 J s to express the in a vertical plane and hung between the poles of a conven- − ¼ 2 results of their watt-balance determinations of KJ RK and tional electromagnet. hence h (see Table I, Sec. II). Item B44.5, NPL-12, is discussed in CODATA-10 and was Second, it was discovered that the unit of mass dissemi- obtained using the NPL Mark II watt balance. This apparatus nated by the International Bureau of Weights and Measures has cylindrical symmetry about a vertical axis and employs a (BIPM), Sèvres, France, starting about 2000 and extending horizontal circular coil hung in the gap between two concen- into 2014 gradually became offset from the SI unit of mass. tric annular permanent magnets (Robinson, 2012); some The latter is defined by assigning to the mass of the details of the balance are given in CODATA-98. Just prior international prototype of the kilogram (IPK), which is to the transfer of the apparatus to the National Research maintained at the BIPM, the value 1 kg exactly (Stock et al., Council (NRC), Ottawa, Canada, in the summer of 2009, 2015). The discovery was made during the Extraordinary Robinson (2012) identified two possible systematic effects in Calibration Campaign undertaken at the BIPM to ensure that the weighing mode of the experiment. Lack of time neces- watt-balance measurements of h and the x-ray-crystal-density sitated taking them into account by including a comparatively (XRCD) determinations of NA (see Sec. IX.B) are closely tied large additional uncertainty component in the experiment’s to the IPK in preparation for the adoption of the new SI by the uncertainty budget, which in turn led to the final compara- 26th CGPM in 2018. The BIPM working standards used to tively large uncertainty of the final result. Although this value calibrate client standards had lost mass and in 2014 were has not been corrected for the BIPM mass-standard problem offset from the mass of the IPK by 35 μg. As a consequence, discussed above, it is of little consequence because of the some watt-balance values of h had to be decreased by 35 parts small size of the correction compared with the final

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-31 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ... uncertainty. The correlation coefficient of the NPL-90 and D. NIST watt balance NPL-12 results is 0.0025 and thus they are only slightly correlated (Robinson, 2012). There are two watt-balance results from NIST to be −8 considered. NIST-98, item B44.2 with ur 8.7 × 10 , is discussed in CODATA-98 and was obtained¼ using the second B. METAS watt balance generation NIST watt balance called NIST-2 (Williams et al., Reported by Eichenberger et al. (2011), the result from the 1998). In this apparatus an axially symmetric radial magnetic Federal Institute for Metrology (METAS), Bern-Wabern, flux density is generated by a specially designed, 1.5 m high magnet consisting of upper and lower superconducting Switzerland, item B44.3 with identification METAS-11, is solenoids and smaller compensating windings mounted in a discussed in CODATA-10. Its relative uncertainty of Dewar; the moving coil is circular, mounted horizontally in 2.9 × 10−7 was too large for it to be included in the 2010 air, and encircles the Dewar. final adjustment but is initially included in the 2014 adjust- After the publication of this result the NIST researchers not ment for tests of data robustness and the exactness of the only renovated the facility in which the apparatus was used but Josephson and quantum-Hall-effect relations K 2e=h and J completely reconstructed it with little remaining of the earlier R h=e2. No correction for the mass-standard problem¼ has K ¼ NIST-2 watt balance, the major exception being the super- been made to this result; because of the comparatively large conducting magnet. In the new watt balance, called NIST-3 uncertainty of METAS-11, it is of no consequence. and discussed in CODATA-06, the entire balance mechanism and coil are in vacuum. The new apparatus was subsequently 2 C. LNE watt balance used to obtain a result for KJ RK that turned out to be identical −8 to the 1998 result but with ur 3.6 × 10 (Steiner et al., The Laboratoire National de Métrologie et d’Essais (LNE), 2007). This result, with identification¼ NIST-07 and discussed Trappes, France, initiated its watt-balance project in 2001. The in CODATA-06, was included in both the 2006 and 2010 final various elements of the LNE balance were developed with adjustments. However, in the 2006 final adjustment, because continued characterization and improvements of each. The of the inconsistencies among several data that contributed to first result became available in a preprint in December 2014 the determination of h, including NIST-98 and NIST-07 and and is the measurement of the molar volume of natural silicon by the International Avogadro Coordination (IAC), their uncertain- h 6.626 0688 17 × 10−34 Js 2.6 × 10−7 ; 230 ¼ ð Þ ½ ð Þ ties were increased by the expansion factor 1.5 to reduce the relevant residuals to less than 2. Similar inconsistencies were which is equivalent to present in 2010, especially between NIST-07 and the newly reported XRCD result for NA by the IAC denoted IAC-11. As 2 33 −1 −1 −7 KJ RK 6.0367619 15 × 10 J s 2.6 × 10 : 231 a consequence, the expansion factor used in the 2010 final ¼ ð Þ ½ ð Þ adjustment was increased to 2. The result was subsequently published by Thomas et al. NIST researchers were well aware of this problem and of (2015), but although the value of h remained the same as in the the disagreement of NIST-07 with the NRC watt-balance preprint, the relative uncertainty was increased to 3.1 × 10−7 result NRC-12 reported in early 2012 by Steele et al. (2012). with a corresponding increase in the absolute uncertainty. The They were also aware of the fact that NIST-3 had produced preprint uncertainty is used in all calculations, but with no stable values of h from October 2004 to March 2010 when the significant consequence because of its comparatively large value suddenly changed for no apparent reason. To address size. Indeed, because of its low weight, the LNE result as these issues, NIST experimenters carried out six series of new given in Eq. (231), which is item B44.7 with identification measurements lasting between 3 days and 3 weeks starting in LNE-15 in Table XVIII, is omitted from the 2014 final December 2012 and ending in November 2013 with the aim of adjustment. It should also be noted that Thomas et al. producing an independent value. To this end, the measure- (2015) have not corrected the LNE result for the BIPM ments were conducted blindly and NIST-3 was first closely mass-standard shift, which in the LNE case is only −3.7 parts inspected and a number of significant changes made to in 109. it in order to improve its performance, as described by The LNE watt balance uses a cylindrical geometry and a Schlamminger et al. (2014). The result of this effort as permanent magnet as does the NPL Mark II balance. One of reported in the latter paper and denoted NIST-14 is in its unique features is that during the moving-coil mode, the reasonable agreement with NRC-12 and IAC-11, and its −8 8 balance beam and its suspension used in the weighing mode is uncertainty ur 4.5 × 10 is less than 5 parts in 10 . ¼ 2 moved as a single element, which avoids using the balance To answer the question “What is the best value of KJ RK, beam to move the coil. Details of the balance, which was and hence h, that can be deduced from 10 years of NIST-3 operated in air to obtain its first result but has the capability to data?” Schlamminger et al. (2015) thoroughly reviewed all operate in vacuum, are given in the paper by Thomas et al. such data after correcting the new 2012 to 2013 data down- (2015) and the references cited therein. The two largest ward by the fractional amount 35 × 10−9 to account for the uncertainty components, in parts in 107, are 2.4 for the BIPM mass-standard shift. They divide the data into three voltage measurements and 1.2 for the velocity measurement, epochs, 2004 to 2009, 2010 to 2011, and 2012 to 2013, which includes the correction for the refractive index of air calculate the result for each epoch, and take as the best value and the verticality of the laser beams. the average of the three. For the uncertainty they take the

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-32 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ... relative uncertainty 4.5 × 10−8 of NIST-14 and combine it in the four largest in parts in 109 are 9.0 for the mass of the quadrature with an additional component of 3.5 × 10−8, which standard, 6.9 for resistance, 5.9 for alignment, and 5.7 for is one-half the approximate 7 × 10−8 fractional difference gravimetry. The latter two topics are discussed in detail by between NIST-07 and the average of the three individual Liard et al. (2014) and Sanchez and Wood (2014). The total values. This additional component is to account for the lack of relative uncertainty for h=h90 obtained from the 14 individual understanding of the cause of the 7 parts in 108 difference. values determined using this mass standard is 14.4 × 10−9. 9 Thus following Schlamminger et al. (2015), we employ for The 35 parts in 10 reduction of the NRC value of h=h90 the NIST-3 result initially reported by Sanchez et al. (2014) due to the BIPM mass-standard shift is documented by Sanchez et al. (2015) −9 h h90 1 77 57 × 10 ; 232 and it is the value given therein that we take as the final result ¼ ½ þ ð Þ ð Þ of the NRC experiment: which is equivalent to h h 1 189 18 × 10−9 ; 235 ¼ 90½ þ ð Þ ð Þ h 6.626 069 36 38 × 10−34 Js 5.7 × 10−8 ; 233 ¼ ð Þ ½ ð Þ which is equivalent to K2R 6.03676143 34 × 1033 J−1 s−1 5.7 × 10−8 ; 234 J K ¼ ð Þ ½ ð Þ h 6.626 070 11 12 × 10−34 Js 1.8 × 10−8 ; 236 ¼ ð Þ ½ ð Þ where this last value is item B44.4 in Table XVIII with identification NIST-15. The NIST-98 and NIST-15 values of 2 33 −1 −1 −8 K RK 6.03676076 11 × 10 J s 1.8 × 10 ; 237 2 J ¼ ð Þ ½ ð Þ KJ RK are correlated and Schlamminger et al. (2015) estimate their correlation coefficient to be 0.09. This coefficient is where this last value is input datum B44.6 in Table XVIII with included in Table XIX and is used in all relevant calculations. identification NRC-15. Although the value for h=h90 and its uncertainty given by Sanchez et al. (2014) were obtained from the four individual values by a somewhat unusual method, an E. NRC watt balance alternate analysis based on the calculation of a weighted mean The entire NPL Mark II apparatus was dismantled and of correlated values yields essentially the same value and shipped to NRC in the summer of 2009 where in due course it uncertainty (Wood, 2013). As can be seen from Table XXI, was reassembled and recommissioned in a newly constructed the NRC result has the smallest uncertainty of any single −8 determination of h. laboratory. A first result with ur 6.5 × 10 was obtained and reported by Steele et al. (2012)¼ . This result includes For completeness, we note that Xu et al. (2016) at the experimentally measured corrections for the effect of the National Institute of Metrology (NIM), Beijing, PRC, recently published a value for h with u 2.6 × 10−6 in agreement stretching of the beryllium copper flexures that support r ¼ the moving coil under load, and for the effect of the tilting with other values but obtained using the generalized joule of the support base of the balance when the mass lift is loaded balance method. This approach, under development at NIM and unloaded. These are the effects that Robinson (2012) had since 2007, is a variant of the watt-balance approach; the identified but because of a lack of time could only take into reported value is a consequence of the ongoing NIM inves- account through a comparatively large additional uncertainty tigation of the feasibility of using a joule balance to achieve a component. The corrections could not be retroactively applied competitive uncertainty. to the NPL-12 result, because a new set of flexures were installed after the balance arrived at NRC as a result of an IX. MEASUREMENTS INVOLVING SILICON CRYSTALS accident that damaged the original flexures. Subsequently, as described by Sanchez et al. (2013), modifications were made The three naturally occurring isotopes of silicon are 28Si, to the balance that reduced these effects to a negligible level. 29Si, and 30Si, and for natural silicon the amount-of-substance NRC experimenters continued to make important improve- fractions x ASi of these isotopes are approximately 0.92, ments to the NRC watt balance and subsequently carried 0.05, and 0.03,ð Þ respectively. Here we discuss experimental out four measurement campaigns between September and results involving nearly perfect natural silicon single crystals December 2013 using four different mass standards (Sanchez as well as nearly perfect highly enriched silicon single crystals et al., 2014). They are a 1 kg gold-plated copper cylinder, a for which x 28Si ≈ 0.999 96. 500 g diamond turned silicon cylinder, a 500 g gold-plated ð Þ piece of copper, and a 250 g piece of silicon. The data set for A. Measurements with natural silicon each consists of 142, 111, 107, and 148 data points, obtained over 14, 11, 15 and 15, days, respectively. The Type A The natural silicon results employed in the 2010 (statistical) uncertainty for the four values of h=h90 obtained adjustment are used in the 2014 adjustment without change. from each of the four mass standards is taken to be the Natural silicon experimental data have been discussed in standard deviation of the mean of each day’s result from that previous CODATA reports including CODATA-10. The standard. The Type B uncertainty (from systematic effects) for measured quantities of interest are the 220 crystal lattice f g each value of h=h90 is based on an uncertainty budget spacing d220 X of a number of different crystals X deter- containing 51 components distributed over seven major mined in metersð Þ using a combined x ray and optical categories. For the result from the Au-Cu 1 kg mass standard, interferometer or XROI; and the fractional differences

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-33 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

d X − d ref =d ref , where ref is a reference crys- A Si M ½ 220ð Þ 220ð Þ 220ð Þ N rð Þ u : 238 tal, determined using a lattice comparator based on x-ray A 3 ¼ p8d220ρ Si ð Þ double crystal nondispersive diffractometry. The eight natural ð Þ crystals of interest are denoted WASO 4.2a, WASO 04, In the IAC experiment,ffiffiffi the macroscopic silicon mass WASO 17, NRLM3, NRLM4, MO*, ILL, and N, and density ρ Si is obtained from the relation ρ Si m =V , ð Þ ð Þ¼ s s d220 X of each is taken to be an adjusted constant (variable) where ms is the mass of the sphere and is determined by in ourð least-squaresÞ calculations. For simplicity, the simplified 3 weighing, and Vs πds =6 is the volume of the sphere and is forms W4.2a, W04, W17, NR3, and NR4 are used in quantity ¼ð Þ obtained from ds, the sphere’s mean diameter, which is symbols for the first five crystals. determined by optical interferometry. The lattice spacing The CODATA-14 input data for the 220 lattice spacings f g d220 of the silicon boule from which the sphere was fabricated of MO*, WASO 04, and WASO 4.2a are listed in Table XVIII is measured with an XROI using representative silicon and are data items B60, B61, B62.1, and B62.2, respectively; samples from the boule. The mean relative atomic mass of their identifications are either INRIM-08, INRIM-09, or PTB- the silicon atoms Ar Si is determined from representative 81. The input data for the fractional differences of the various samples by measuringð Þ the amount-of-substance ratios crystals of interest are items B50–B59 in the same table and 29 28 30 28 R29=28 n Si =n Si and R30=28 n Si =n Si using are labeled NIST-99, NIST-97, NIST-06, PTB-98, or PTB-03. ¼ ð Þ ð Þ ¼ ð Þ ð Þ isotope dilution mass spectrometry and calculating Ar Si The correlation coefficients of these data are given in A ð Þ from the well-known values of Ar Si . Table XIX and their observational equations may be found Two other aspects of the experimentð Þ are equally important. in Table XXIV. Item B58, the fractional difference between Equation (238) applies only to a pure silicon sphere. In the 220 lattice spacing of an ideal natural silicon crystal d f g 220 practice, the surface of a sphere is contaminated with a and d220 W04 , is discussed in CODATA-06 following physisorbed water layer, a chemisorbed water layer, a carbo- Eq. (312).ð TheÞ laboratories for which INRIM, NIST, and naceous layer, and an SiO2 layer. Thus it is necessary to PTB, as well as for FSUJ and NMIJ in subsequent paragraphs, determine the mass and thickness of these layers so that the are identifiers may be found in the list of symbols and measured value of the mass of the sphere m and the measured abbreviations near the end of this paper. s value of the mean diameter of the sphere ds can be corrected The copper Kα1 x unit, symbol xu CuKα1 , the molybdenum ð Þ for the surface layers and thereby apply to the silicon core. It is Kα1 x unit, symbol xu MoKα1 , and the ångström star, symbol ð Þ also necessary to characterize the material properties of the ÅÃ are historic x-ray units used in the past but still of current silicon, for example, its impurities such as interstial oxygen interest. They are defined by assigning an exact, conventional and substitutional carbon, nonimpurity point defects, dislo- value to thewavelength of the CuKα1, MoKα1, and WKα1 x-ray cations, vacancies, and microscopic voids, and to apply lines when each is expressed in its corresponding unit. corrections to the measured mass of the sphere, 220 lattice These assigned wavelengths for λ CuKα1 , λ MoKα1 , and f g ð Þ ð Þ spacing, and mean diameter as appropriate. λ WKα1 are 1537.400 xu CuKα1 , 707.400 xu MoKα1 , and ð Þ ð Þ ð Þ The IAC researchers continued their work after the pub- 0.209 010 0 ÅÃ, respectively. The four experimental input data lication of their first result and instituted a number of relevant to these units, which are the measured ratios of CuKα1, improvements after carefully examining all significant aspects MoKα1, and WKα1 wavelengths to the 220 lattice spacings of the experiment. This effort is described in detail by Azuma f g of WASO 4.2a and N, are items B68–B71 in Table XVIII; they et al. (2015) and in the references cited therein. It is beyond are labeled either FSUJ/PTB-91, NIST-73, or NIST-79. To the scope of this review to discuss the many advances made, obtain recommended values in meters for the units xu CuKα1 , but two are especially noteworthy. Metallic contaminants in ð Þ xu MoKα1 , and ÅÃ, they are taken as adjusted constants; the the form of Cu, Ni, and Zn silicide compounds were inputð data areÞ then expressed in terms of these constants and the discovered during the course of the work that led to the appropriate 220 lattice spacing of the silicon crystal used to 2011 result, most likely arising from the polishing process, obtain them.f Theg resulting observational equations for these and had to be taken into account. This led to an increased input data are given in Table XXIV. uncertainty for the required surface-layer correction. To overcome this problem the spheres were reetched and carefully repolished. B. Determination of NA with enriched silicon The second improvement involves the determination of the The IAC project to determine NA using the x-ray-crystal- amount-of-substance ratios, and thus the value of Ar Si . In density (XRCD) method was initiated in 2004 and is being the new work the ratios were measured independentlyð at PTB,Þ carried out by a group of researchers from a number of NMIJ, and NIST using a multicollector inductively coupled different institutions, mostly national metrology institutes. plasma mass spectrometer and isotope dilution. The solvent The first enriched silicon result for NA from the IAC and diluent used in the three institutes was tetramethylam- project, formally published in 2011, was included as an input monium hydroxide (TMAH), which significantly reduced the datum in the 2010 final adjustment and is discussed in baseline level of the ion currents to be measured compared CODATA-10. In the IAC work the silicon samples are highly with the levels usually seen with the normally used NaOH. For polished, highly pure, and nearly crystographically perfect this and other reasons discussed in detail by Azuma et al. spheres of nominal mass 1 kg initially designated AVO28-S5 (2015), the ratios obtained by Yang et al. (2012) using NaOH and AVO28-S8. The basic equation for the XRCD determi- were not employed in the calculation of the new IAC value of nation of NA using a perfect silicon crystal is NA. See Kuramoto et al. (2015), Mana et al. (2015), Massa

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-34 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ... et al. (2015), Mizushima et al. (2015), Pramann et al. (2015), of R is recovered from the reported value of k and the Waseda et al. (2015), and Zhang et al. (2015). Avogadro constant NA used by the researchers to calculate k. The two new values of NA reported by Azuma et al. (2015) Finally, some of the input data incorporate changes based and which are used as input data in the 2014 adjustment are upon newly available information, as discussed in Sec. X.A.2. Since there is no serious discrepant input data for the 23 −1 −8 NA 6.022 140 99 18 × 10 mol 3.0 × 10 ; 239 inferred value of the Boltzmann constant for either the 2010 or ¼ ð Þ ½ ð Þ 2014 adjustment, the result for k from refractive index gas 23 −1 −8 NA 6.022 140 76 12 × 10 mol 2.0 × 10 : 240 thermometry (Schmidt et al., 2007) and for k=h from Johnson ¼ ð Þ ½ ð Þ noise thermometry (Benz et al., 2011) that were initially The first result is the 2011 value used in CODATA-10 considered but not included in the final 2010 adjustment due increased by 3 parts in 108 by Azuma et al. (2015) to reflect to their large uncertainties are not considered for the 2014 the recalibration of the mass standards used in its determi- adjustment. nation as a consequence of the extraordinary calibration campaign against the international prototype of the kilogram A. Molar gas constant R, acoustic gas thermometry discussed in Sec. VIII. The second result is the value reported by Azuma et al. (2015) as a consequence of the extensive IAC The measurement of R by the method of acoustic gas thermometry (AGT) is based on the following expressions for efforts of the past 4 years and is the weighted mean of the the square of the speed of sound in a real gas of atoms or results obtained from spheres AVO28-S5c and AVO28-S8c. molecules in thermal equilibrium at thermodynamic temper- (The additional letter c has been added by the researchers to ature T and pressure p and occupying a volume V: distinguish the reetched and repolished spheres used in the new work from the spheres used in the earlier work.) The 2 ca T;p A0 T A1 T p uncertainty assigned to the value for NA obtained from sphere ð Þ¼ ð Þþ ð Þ AVO28-S5c is 21 parts in 109, and for the value obtained from A T p2 A T p3 : 241 þ 2ð Þ þ 3ð Þ þÁÁÁ ð Þ sphere AVO28-S8c is 23 parts in 109. The two largest uncertainty components for AVO28-S5c are 10 parts in 109 Here A1 T , A2 T , etc. are related to the density virial ð Þ ð Þ for surface characterization and 16 parts in 109 for sphere coefficients and their temperature derivatives. In the limit volume as calculated from the mean diameter. It should p → 0, this becomes be noted that the two values of N are correlated; the A γ RT IAC researchers report the correlation coefficient to be 0.17 c2 T;0 A T 0 ; 242 a ð Þ¼ 0ð Þ¼A X M ð Þ (Mana et al., 2015). rð Þ u The two values of N in Eqs. (239) and (240) are items A where γ c =c is the ratio of the specific heat capacity of B63.1 and B63.2 in Table XVIII with identifiers IAC-11 and 0 ¼ p V IAC-15, the values of h that can be inferred from them are the gas at constant pressure to that at constant volume and is given in Table XXI, and their observational equation, which 5=3 for an ideal monotonic gas. The basic experimental approach to determining the speed of sound of a gas, usually also shows how h can be derived from NA, may be found in Table XXIV. How these results compare with other data and argon or helium, is to measure the acoustic resonant frequen- their role in the 2014 adjustment are discussed in Sec. XIII. cies of a cavity at or near the triple point of water, TTPW 273.16 K, at various pressures and extrapolating to p 0¼. The cavities are either cylindrical of fixed or variable X. THERMAL PHYSICAL QUANTITIES length,¼ or spherical, but most commonly quasispherical in the form of a triaxial ellipsoid. This shape removes the degeneracy Table XIV summarizes the 11 results for the thermal of the microwave resonances used to measure the volume of physical quantities R, k=h, and A =R, the molar gas constant, ϵ the resonator in order to calculate c2 T;p from the measured the quotient of the Boltzmann and Planck constants, and the a acoustic frequencies and the correspondingð Þ acoustic resonator quotient of the molar polarizability of a gas and the molar gas eigenvalues known from theory. The cavities are formed by constant, respectively, that are taken as input data in the 2014 carefully joining hemispherical cavities. adjustment. They are data items B64.1 B66 in Table XVIII − In practice, the determination of R by AGT with a relative with correlation coefficients as given in Table XIX and standard uncertainty of order 1 part in 106 is complex; the observational equations as given in Table XXVII. Values of application of numerous corrections is required as well as the the Boltzmann constant k that can be inferred from these data investigation of many possible sources of error. For a review are given in Table XXII and are graphically compared of the advances made in AGT in the past 25 years, see in Fig. 5. Moldover et al. (2014). There are five new input data that contribute to the 2014 determination of the Boltzmann constant, three from acoustic 1. New values gas thermometry (NPL-13, NIM-13, and LNE-15), one from Johnson noise thermometry (NIM/NIST-15), and one from a. NIM 2013 dielectric-constant gas thermometry (PTB-15). Not every Lin et al. (2013) report a result for the Boltzmann constant value in Table XIV appears in the cited references. For some, from an improved version of an earlier experiment additional digits have been provided to the Task Group to (Zhang et al., 2011) using argon in a single 80 mm long reduce rounding errors; for others, the actual measured value fixed cylindrical cavity measured by two-color optical

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-35 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ... interferometry. The shape of the cavity has been made more between the results from LNE-11 and NPL-13 (Yang et al., cylindrical and the thermometry improved. Two different 2015). The LNE-11 value is based on M Ar determinations grades of argon with measured relative isotopic abundances from the Institute for Reference Materialsð andÞ Measurements were used with two different methods of supporting the cavity. (IRMM), Geel, Belgium (Valkiers et al., 2010). The NPL-13 Analysis of the acoustic data was improved by accounting for estimate of M Ar is based on a comparison at the Scottish second-order perturbations to the frequencies from the ther- Universities Environmentalð Þ Research Centre (SUERC), moviscous boundary layer. University of Glasgow, Glasgow, Scotland, of the isotopic The reported calculated value of the Boltzmann constant composition of the experimental gas with the isotopic com- uses the CODATA-10 value of NA, implying the measured position of argon from atmospheric air (de Podesta et al., value 2013), assuming the atmospheric air had the same argon isotopic abundance as the analysis performed by Lee et al. R 8.314 455 31 J mol−1 K−1 3.7 × 10−6 : 243 (2006) at the Korea Research Institute of Standards and ¼ ð Þ ½ ð Þ Science (KRISS), Taedoc Science Town, Republic of The largest uncertainty component, 2.9 parts in 106, is due to Korea. The results of these studies were discussed at the inconsistent values determined from the various acous- meeting of the CIPM Consultative Committee for tic modes. Thermometry Task Group for the SI (CCT TG-SI) in Eltville, Germany, on 6 February 2015 in conjunction with b. NPL 2013 the 2015 Fundamental Constants Meeting co-organized by the de Podesta et al. (2013) used a quasispherical copper Task Group (Karshenboim, Mohr, and Newell, 2015). triaxial ellipsoid cavity with a nominal radius of 62 mm filled In the first study, the isotopic composition of samples of with argon to determine R. The cavity was suspended from argon gas previously measured at the IRMM were remeas- the top of a copper container designed to create a nearly ured. The analysis showed disagreements in values of M Ar 6 ð Þ isothermal environment. The initial value reported is of up to 3.5 parts in 10 . In the second study, the isotopic composition of a sample of argon gas used for isotherm 5 of R 8.314 4787 59 J mol−1 K−1 7.1 × 10−7 : 244 NPL-13 was remeasured. The estimate of M Ar was 2.73 ¼ ð Þ ½ ð Þ parts in 106 lower than the corresponding SUERCð estimate.Þ In The low uncertainty is attributed to the near-perfect shape and the third study, mass spectrometry measurements were made surface condition of the cavity achieved by precise diamond on a series of argon samples from NIM, INRIM, NPL, NMIJ turning techniques during fabrication. The largest uncertainty and LNE on which corresponding speed-of-sound measure- component, 3.5 parts in 107, is from the determination of the ments had been made at LNE. The results showed the molar mass of the argon used in the experiment. expected correlation between the two sets of measurements. From this study an inference of the required correction to the c. LNE 2015 SUERC estimate of the NPL-13 isotherm 5 molar mass of −3.6 parts in 106 was apparent. However, KRISS considered Pitre et al. (2015) used a quasispherical copper triaxial their direct measurement of this sample to have a lower ellipsoid cavity with a nominal radius of 50 mm filled with uncertainty. The inference of the required correction to the helium. The experiment was performed in quasi-adiabatic LNE-11 M Ar was not strong enough to suggest a mean- thermal conditions, instead of standard, constant heat-flux ingful changeð inÞ its value. conditions. The reported calculated value of the Boltzmann Although the estimated relative statistical uncertainty for constant uses the CODATA-10 value of NA, implying the the KRISS M Ar measurements was 0.61 × 10−6, KRISS measured value considered it likelyð Þ that the isotope ratio R 38Ar=36Ar may be in error because it disagrees distinctlyð with measurementsÞ R 8.314 4615 84 J mol−1 K−1 1.0 × 10−6 : 245 ¼ ð Þ ½ ð Þ from Lee et al. (2006). For this reason an additional relative uncertainty of 0.35 × 10−6 was added in quadrature to yield an The dominant source of uncertainty, 6.2 parts in 107, arises overall relative uncertainty for the KRISS M Ar measure- from the acoustic frequency measurements. ments of 0.7 × 10−6. ð Þ Based on the studies at KRISS it was agreed upon by all 2. Updated values participants of the CCT TG-SI meeting that for the 2014 The following updates and corrections for the AGT input CODATA adjustment, the NPL-13 determination of the molar data are summarized in Moldover, Gavioso, and Newell gas constant will rely on the KRISS value and uncertainty of (2015), along with a detailed description of the correlation M Ar , resulting in a fractional correction of −2.73 × 10−6 coefficients given in Table XIX. The AGT values and andð a totalÞ relative uncertainty of 0.9 × 10−6 (de Podesta et al., uncertainties in Table XIV incorporate all the corrections 2015). It was also agreed that while the LNE-11 value of the listed below. molar gas constant will continue to use the determination of M Ar from IRMM, the relative uncertainty will be that of a. Molar mass of argon KRISS,ð Þ namely 0.7 × 10−6. Similarly since the previous NPL- During the period from October to December 2014, three 10 result used the M Ar value from IRMM, the relative important studies on the molar mass of argon M Ar were uncertainty componentð forÞM Ar for NPL-10 was increased performed to investigate the difference of 2.74 partsð Þ in 106 to 0.7 × 10−6. ð Þ

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b. Molar mass of helium where k90 in the SI unit J/K is the result of a Johnson noise Based upon the measured 3He=4He abundance ratios experiment when the voltage and resistance are measured in spanning 0.05 × 10−6 to 0.5 × 10−6 from samples taken from conventional electrical units defined by KJ−90 and RK−90. 12 natural gas wells in the USA (Aldrich and Nier, 1948), the Following the analysis given in CODATA-98 [see Eqs. (29d) and (317)], it can be shown that k =h k=h. Using the expected reduction in M 4He due to naturally occurring 3He 90 90 value of h (see the introduction to Sec. VIII¼ ), the measured is fractionally from 0.012ð × 10Þ −6 to 0.12 × 10−6. The two gas 90 value of k=h is analyses given in the paper by Gavioso et al. (2015) are consistent with this expectation (see Sec. X.D). In contrast, the k=h 2.083 6658 80 × 1010 Hz K−1 3.9 × 10−6 ; 248 ratio of speed-of-sound measurements using two different, ¼ ð Þ ½ ð Þ commercially produced, highly purified helium samples given as given in Table XIV. in the paper reporting LNE-15 differed by the surprisingly In the Qu et al. (2015) experiment, digitally synthesized large value of 0.44 parts in 106. This observation was taken pseudonoise voltages V are generated by means of a pulse- into account by including an additional uncertainty compo- Q 6 biased Josephson junction array. These known voltages are nent of 0.5 parts in 10 . Based upon the possibility that the compared to the unknown thermal-noise voltages V gen- concentration of 3He may be higher than expected, an addi- R erated by a specially designed 200 Ω resistor in a well tional relative uncertainty component of 0.5 × 10−6 was regulated thermal cell at or near TTPW. Since the spectral incorporated in the LNE-09 result. It should finally be noted density of the noise voltage of a 200 resistor at 273.16 K is 3 4 Ω that certain natural gases in Taiwan have He= He abundance only 1.74 nVpHz, it is measured using a low-noise, two- −6 ratios as large as 3.8 × 10 (Sano, Wakita, and Huang, 1986). channel, cross-correlation technique that enables the resistor Clearly, future helium-based low-uncertainty AGT determi- signal to be extractedffiffiffiffiffiffi from uncorrelated amplifier noise of 3 nations of R must measure the He concentration in the gas comparable amplitude and spectral density. The final result is samples used. based on measurements integrated over a bandwidth of 575 kHz and a total integration time of about 33 d. c. Thermal conductivity of argon The dominant uncertainty contributions of 3.2 parts in 106 An improved estimate for the thermal conductivity of argon and 1.8 parts in 106 are from the statistical uncertainty of the became available for the 2014 CODATA adjustment [see V2 =V2 ratio measurement and the ambiguity associated h R Qi supplementary data in Moldover et al. (2014)]. A change in with the spectral mismatch model, respectively. thermal conductivity affects the estimate for the thermal boundary layer thickness close to the wall of the cavity, C. Quotient Aϵ=R, dielectric-constant gas thermometry resulting in a correction for all resonant frequencies at all pressures. For the 2014 CODATA adjustment, these correc- The virial expansion of the equation of state for a real gas of tions are applied to the lowest uncertainty results for AGT amount of substance n in a volume V is using argon. In parts in 106, the corrections to NIST-88 (Moldover, 2015), LNE-11 (Pitre, 2015), and NPL-13 (de p ρRT 1 ρB T ρ2C T ρ3D T ; 249 ¼ ½ þ ð Þþ ð Þþ ð Þ þ Á Á Á ð Þ Podesta et al., 2015) are −0.16, −0.16, and −0.192, respectively, with inconsequential decreases in the uncertainties. where ρ n=V is the amount-of-substance density of the gas at thermodynamic¼ temperature T, and B T , C T , etc. are the ð Þ ð Þ B. Quotient k=h, Johnson noise thermometry virial coefficients. The Clausius-Mossotti equation is

The Nyquist theorem predicts that, with a fractional error of ϵr − 1 6 ρAϵ 1 ρBϵ T less than 1 part in 10 at frequencies less than 10 MHz and ϵr 2 ¼ ½ þ ð Þ temperatures greater than 250 K, þ ρ2C T ρ3D T ; 250 þ ϵð Þþ ϵð Þ þ Á Á Á ð Þ 2 U 4kTRsΔf: 246 h i ¼ ð Þ where ϵr ϵ=ϵ0 is the relative dielectric constant (relative permittivity)¼ of the gas, ϵ is its dielectric constant, ϵ is the Here U2 is the mean-square voltage, or Johnson noise 0 h i exactly known electric constant, Aϵ is the molar polarizability voltage, in a measurement bandwidth of frequency Δf across of the atoms, and Bϵ T , Cϵ T , etc., are the dielectric virial the terminals of a resistor of resistance Rs in thermal coefficients. By appropriatelyð Þ ð Þ combining Eqs. (249) and equilibrium at thermodynamic temperature T. If U2 is h i (250), an expression is obtained from which Aϵ=R can be measured in terms of the Josephson constant KJ 2e=h experimentally determined by measuring ϵ at a known ¼ 2 r and Rs in terms of the von Klitzing constant RK h=e , ¼ constant temperature such as TTPW and at different pressures then the measurement yields a value of k=h. and extrapolating to zero pressure. Continuing the pioneering work of Benz et al. (2011), In practice, dielectric-constant gas thermometry measures Qu et al. (2015) report an improved determination of the the fractional change in capacitance of a specially constructed Boltzmann constant using such a method. The reported capacitor, first without helium gas and then with helium gas at value is a known pressure. The static electric polarizability of a gas atom α0, Aϵ, R, and k are related by Aϵ=R α0=3ϵ0k, which k 1.380 6513 53 × 10−23 JK−1 3.9 × 10−6 ; 247 ¼ 90 ¼ ð Þ ½ ð Þ shows that if α0 is known sufficiently well from theory, then a

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competitive value of k can be obtained if the quotient Aϵ=R other adjusted constants, they contribute only to the determi- can be measured with a sufficiently small uncertainty. nation of the 2014 recommended value of G. The calculation Piszczatowski et al. (2015) have calculated the static of this value is discussed in Sec. XIII.B.1. electric polarizability of 4He in atomic units to be While three new values for G have become available for the 2014 CODATA adjustment, the data remain discrepant. The 4 −7 α0Ã He 1.383 760 77 14 a:u: 1.0 × 10 ; 251 first is a competitive result from the International Bureau of ð Þ¼ ð Þ ½ ð Þ Weights and Measures (BIPM) (Quinn et al., 2013, 2014) from which the static electric polarizability of 4He in SI units obtained using a similar but completely rebuilt apparatus as is was used to obtain the BIPM 2001 result. The second, based on a unique technique involving atom interferometry, is from 4 3 4 α0 He 4πϵ0a0α0Ã He ; 252 the European Laboratory for Non-linear Spectroscopy ð Þ¼ ð Þ ð Þ (LENS) (Prevedelli et al., 2014; Rosi et al., 2014). where a0 and ϵ0 are the Bohr radius and electric constant, Although not competitive, the conceptually different approach respectively. could help identify errors that have proved elusive in other Superseding previous preliminary results (Fellmuth et al., experiments. The third is from the University of California, 2011; Gaiser and Fellmuth, 2012), Gaiser et al. (2013) report a Irvine (Newman et al., 2014) and is a highly competitive result final value of Aϵ=R from dielectric-constant gas thermometry from data collected over a 7-year span using a cryogenic with an updated uncertainty given by Gaiser, Zandt, and torsion balance. Fellmuth (2015): The previously reported measurements of G as discussed in past Task Group reports remain unchanged with one −8 3 −1 −6 Aϵ=R 6.221128 25 × 10 m KJ 4.0 × 10 : 253 exception. It was discovered that the reported correlation ¼ ð Þ ½ ð Þ coefficient between the 2005 and 2009 result from the The dominant uncertainty components are the fitted coeffi- Huazhong University of Science and Technology, HUST-05 cient from the 10 isotherms (statistical) and the effective and HUST-09, was unphysical. As described below, a compressibility of the capacitor assembly at 2.6 parts in 106 reexamination of the uncertainty analysis has led to slight and 2.4 parts in 106, respectively. reductions in the HUST-05 value and in the correlation coefficient of the two results. D. Other data For simplicity, in the following text, we write G as a numerical factor multiplying G0, where For completeness we note the following result that became −11 −1 3 −2 available only after the 31 December 2014 closing date of G0 10 kg m s : 255 the 2014 adjustment. Gavioso et al. (2015) obtained the ¼ ð Þ very competitive value R 8.314 4743 88 J mol−1 K−1 1.06 × 10−6 using acoustic gas¼ thermometryð Þ with a mis- A. Updated values ½ aligned spherical cavity with a nominal radius of 90 mm filled 1. Huazhong University of Science and Technology with helium. This value is 1.47 parts in 106 larger than the 2014 CODATA value. The initially assigned covariance of the HUST-05 and HUST-09 values of G exceeded the variance of the HUST- E. Stefan-Boltzmann constant σ 09 value which had the smaller uncertainty of the two. As a result the weighted mean of the two values was outside the The Stefan-Boltzmann constant is related to c, h, and k by interval between them, which is unphysical (Cox et al., 2006; σ 2π5k4=15h3c2, which, with the aid of the relations Bich, 2013). ¼ 2 In collaboration with the HUST researchers, the uncertainty k R=NA and NAh cAr e Muα =2R∞, can be expressed in¼ terms of the molar¼ gasð Þ constant and other adjusted budgets and corrections of both the HUST-05 and HUST-09 constants as measurements were reviewed. Upon further examination, the 2010 correlation coefficient between HUST-05 and HUST-09 5 4 of 0.234 contained a misassigned contribution due to fiber 32π h R∞R σ 6 2 : 254 anelasticity of 0.098. While the suspension fibers in both ¼ 15c Ar e Muα ð Þ ð Þ experiments were 25 μm diameter tungsten wire, the individ- Since no competitive directly measured value of σ is available ual wires used were different. Moreover the HUST-05 for the 2014 adjustment, the 2014 recommended value is anelasticity correction was estimated from the pendulum obtained from this equation. quality factor Q as predicted by Kuroda (1995), whereas for HUST-09 it was directly measured (Luo et al., 2009). After XI. NEWTONIAN CONSTANT OF GRAVITATION G careful reevaluation of all correlations and removing the anelasticity correction, the correlation coefficient between Table XV summarizes the 14 measured values of the HUST-05 and HUST-09 was reduced to 0.134. Newtonian constant of gravitation G of interest in the 2014 Since the Q for the HUST-05 torsion pendulum was adjustment. Because the values are independent of the other approximately 3.6 × 104, the positive bias due to fiber data relevant to the current adjustment, and because there is no anelasticity was originally neglected. For the 2014 adjustment, known quantitative theoretical relationship between G and the HUST-05 value has been reduced by 8.8 parts in 106 based

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-38 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ... on the Kuroda correction with the result that the HUST-05 source masses were reduced in height and remeasured, and the input datum is now experiment was also rebuilt in a different BIPM laboratory. Quinn et al. (2014) conclude that the 2014 and 2001 BIPM G 6.672 22 87 G 1.3 × 10−4 : 256 values of G are not correlated. ¼ ð Þ 0 ½ ð Þ

B. New values 2. European Laboratory for Non-Linear Spectroscopy, University of Florence 1. International Bureau of Weights and Measures A novel measurement technique to measure G using atom A new result from the BIPM, labeled BIPM-14, has been interferometry instead of a precision mechanical balance has reported by Quinn et al. (2013, 2014) using the same recently been completed by the European Laboratory for Non- principles of a flexure strip torsion balance operating in either Linear Spectroscopy at the University of Florence (Prevedelli of two different modes as in the previous BIPM experiment: et al., 2014; Rosi et al., 2014). Labeled LENS-14, the compensation mode (cm) and deflection mode (dm) (Quinn experiment combines two vertically separated atomic clouds et al., 2001). However, almost all of the apparatus was rebuilt forming a double atom-interferometer-gravity gradiometer or replaced. Extensive tests were performed and improve- that measures the change in the gravity gradient when a ments made on key parameters, including test and source mass well-characterized source mass is displaced. coordinates, calibration of angle measurements, calibration of The experimental design uses a double differential con- ac voltage and capacitance electrical instruments, timing figuration that greatly reduces the sensitivity to common- measurements for period of oscillation, and precision of mode spurious signals. Two atomic rubidium clouds are torque measurements. With all identified errors taken into launched in the vertical direction with a vertical separation account, the results for the two modes are of approximately 30 cm in a juggling sequence. The two clouds are simultaneously interrogated by the same Raman −5 Gcm 6.675 15 41 G0 6.1 × 10 ; 257 three-pulse interferometric sequence. The difference in the ¼ ð Þ ½ ð Þ phase shifts between the upper and lower interferometers −5 measures the gravity gradient. The gravity gradient is then Gdm 6.675 86 36 G0 5.4 × 10 : 258 ¼ ð Þ ½ ð Þ modulated by the symmetric placement of the 516 kg tungsten The largest uncertainty component for each mode is 47 parts source mass in two different vertical positions around the in 106 from angle measurements, but the angle measurements double atom interferometer. To further cancel common-mode are anticorrelated between the two modes. Taking into spurious effects the two-photon recoil used to split and consideration all correlations, the reported weighted mean recombine the wave packets in the interferometers is reversed. and uncertainty from the two modes is The value of G is extracted by calculating the source mass gravitational potential and the phase shift for single-atom −5 trajectories, carrying out Monte Carlo simulations of the G 6.675 54 16 G0 2.4 × 10 ; 259 ¼ ð Þ ½ ð Þ atomic cloud, and estimating other corrections not taken into which agrees well with the 2001 BIPM result. This agreement account by the Monte Carlo simulation. The final result is is noteworthy, because only the source masses and their G 6.671 91 99 G 1.5 × 10−4 : 260 carousel in the original apparatus were not replaced; the ¼ ð Þ 0 ½ ð Þ

TABLE XIV. Summary of thermal physical measurements relevant to the 2014 adjustment (see text for details). AGT: acoustic gas thermometry; JNT: Johnson noise thermometry; cylindrical, spherical, quasispherical: shape of resonator used; JE and QHE: Josephson effect voltage and quantum-Hall-effect resistance standards; DCGT: dielectric-constant gas thermometry

Rel. stand. a Source Ident. Quant. Method Value uncert ur Colclough, Quinn, and Chandler (1979) NPL-79 R AGT, cylindrical, argon 8.314 504 70 J mol−1 K−1 8.4 × 10−6 Moldover et al. (1988) NIST-88 R AGT, spherical, argon 8.314 470ð15Þ J mol−1 K−1 1.8 × 10−6 Pitre et al. (2009) LNE-09 R AGT, quasispherical, helium 8.314 467ð23Þ J mol−1 K−1 2.7 × 10−6 Sutton et al. (2010) NPL-10 R AGT, quasispherical, argon 8.314 468ð26Þ J mol−1 K−1 3.2 × 10−6 Gavioso et al. (2010) INRIM-10 R AGT, spherical, helium 8.314 412ð63Þ J mol−1 K−1 7.5 × 10−6 Pitre et al. (2011) LNE-11 R AGT, quasispherical, argon 8.314 455ð12Þ J mol−1 K−1 1.4 × 10−6 Lin et al. (2013) NIM-13 R AGT, cylindrical, argon 8.314 455ð31Þ J mol−1 K−1 3.7 × 10−6 de Podesta et al. (2013) NPL-13 R AGT, quasispherical, argon 8.314 4544ð 75Þ J mol−1 K−1 9.0 × 10−7 Pitre et al. (2015) LNE-15 R AGT, quasispherical, helium 8.314 4615ð84Þ J mol−1 K−1 1.0 × 10−6 Qu et al. (2015) NIM/NIST-15 k=h JNT, JE and QHE 2.083 6658ð80Þ × 1010 Hz K−1 3.9 × 10−6 ð Þ −8 3 −1 −6 Gaiser, Zandt, and Fellmuth (2015) PTB-15 Aϵ=R DCGT, helium 6.221 128 25 × 10 m KJ 4.0 × 10 ð Þ aNPL: National Physical Laboratory, Teddington, UK; NIST: National Institute of Standards and Technology, Gaithersburg, Maryland, and Boulder, Colorado, USA; LNE: Laboritoire Commun de Métrologie (LCM), Saint-Denis, France, of the Laboratoire National de Métrologie et d’Essais (LNE); INRIM: Instituto Nazionale di Ricerca Metrologica, Torino, Italy; NIM: National Institute of Metrology, Beijing, PRC; PTB: Physikalisch-Technische Bundesanstalt, Braunschweig and Berlin, Germany.

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TABLE XV. Summary of the results of measurements of the Newtonian constant of gravitation relevant to the 2014 adjustment.

a 11 3 −1 −2 Source Identification Method 10 G m kg s Rel. stand. uncert. ur ð Þ Luther and NIST-82 Fiber torsion balance, 6.672 48(43) 6.4 × 10−5 Towler (1982) dynamic mode Karagioz and TR&D-96 Fiber torsion balance, 6.672 9(5) 7.5 × 10−5 Izmailov (1996) dynamic mode Bagley and LANL-97 Fiber torsion balance, 6.673 98(70) 1.0 × 10−4 Luther (1997) dynamic mode Gundlach and UWash-00 Fiber torsion balance, dynamic 6.674 255(92) 1.4 × 10−5 Merkowitz (2000, 2002) compensation Quinn et al. (2001) BIPM-01 Strip torsion balance, 6.675 59(27) 4.0 × 10−5 compensation mode, static deflection Kleinevoß (2002) UWup-02 Suspended body, displacement 6.674 22(98) 1.5 × 10−4 and Kleinvoß et al. (2002) Armstrong and MSL-03 Strip torsion balance, 6.673 87(27) 4.0 × 10−5 Fitzgerald (2003) compensation mode Hu, Guo, and Luo (2005) HUST-05 Fiber torsion balance, 6.672 22(87) 1.3 × 10−4 dynamic mode Schlamminger et al. (2006) UZur-06 Stationary body, weight change 6.674 25(12) 1.9 × 10−5 Luo et al. (2009) and HUST-09 Fiber torsion balance, 6.673 49(18) 2.7 × 10−5 Tu et al. (2010) dynamic mode Parks and Faller (2010) JILA-10 Suspended body, displacement 6.672 34(14) 2.1 × 10−5 Quinn et al. (2013, 2014) BIPM-14 Strip torsion balance, 6.675 54(16) 2.4 × 10−5 compensation mode, static deflection Prevedelli et al. (2014) LENS-14 Double atom interferometer 6.671 91(99) 1.5 × 10−4 and Rosi et al. (2014) gravity gradiometer Newman et al. (2014) UCI-14 Cryogenic torsion balance, 6.674 35(13) 1.9 × 10−5 dynamic mode aNIST: National Institute of Standards and Technology, Gaithersburg, Maryland, and Boulder, Colorado, USA; TR&D: Tribotech Research and Development Company, Moscow, Russian Federation; LANL: Los Alamos National Laboratory, Los Alamos, New Mexico, USA; UWash: University of Washington, Seattle, Washington, USA; BIPM: International Bureau of Weights and Measures, Sèvres, France; UWup: University of Wuppertal, Wuppertal, Germany; MSL: Measurement Standards Laboratory, Lower Hutt, New Zealand; HUST: Huazhong University of Science and Technology, Wuhan, PRC; UZur: University of Zurich, Zurich, Switzerland; JILA: JILA, University of Colorado and National Institute of Standards and Technology, Boulder, Colorado, USA; LENS: European Laboratory for Non-Linear Spectroscopy, University of Florence, Florence, Italy; UCI: University of California, Irvine, Irvine, California, USA.

The leading uncertainty components arise from the determi- vacuum dewar), thus greatly reducing the period-change nation of the atomic cloud size, center, and launch direction, signal of the torsion balance. The torsion balance test mass and the tungsten source mass position, and in parts in 106 are is a thin fused silica plate as pioneered by Gundlach and 61, 38, 36, and 38, respectively. Although the final uncertainty Merkowitz (2000) that, when combined with the ring source is not presently competitive, determinations of G using atom masses, minimizes the sensitivity to test mass shape, mass interferometry could be more competitive in the future. distribution, and placement. Over the 7 year span three fibers were used. Fiber 1 was an 3. University of California, Irvine as-drawn CuBe fiber, fiber 2 a heat-treated CuBe fiber, and A highly competitive result from data collected over a fiber 3 an as-drawn 5056 aluminium-alloy fiber. It was 7 year span using a cryogenic torsion balance operating below observed that the Birge ratio for the data within each run 4 K in a dynamic mode with two orientations for the source was much larger than expected. A total of 27 variants of data mass has recently been reported by researchers from the analysis methods were used for each fiber to test the robust- University of California, Irvine (Newman et al., 2014), labeled ness of the data, with the resulting 27 values of G varying over 6 UCI-14. The advantages of cryogenic operation are a much a range of 14, 24, and 20 parts in 10 for fibers 1, 2, and 3, higher torsion pendulum Q, which greatly reduces the respectively. The final analysis uses an unweighted average systematic bias predicted by Kuroda (1995), much lower for each run, a weighted average over runs for each fiber with thermal noise acting on the balance, greatly reduced fiber- a Birge ratio uncertainty expansion, and an outlier identifi- property dependence on temperature variation, excellent cation protocol. An additional uncertainty component equal to temperature control, easy to maintain high vacuum, and ease half of the range of G values determined during a robustness of including effective magnetic shielding with superconduct- test is included in the final values of G from the three fibers, ing material. The source mass is a pair of copper rings that which are produces an extremely uniform gravity gradient over a large region centered on the torsion balance test mass. However, by G 6.674 350 97 G 1.5 × 10−5 ; 261 necessity it is located 40 cm from the test mass (i.e. outside the 1 ¼ ð Þ 0 ½ ð Þ

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-40 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

G 6.674 08 15 G 2.2 × 10−5 ; 262 used in CODATA-10. The input data are given in Tables XVI, 2 ¼ ð Þ 0 ½ ð Þ XVII, XVIII, and XIX. For ease of presentation the relevant G 6.674 55 13 G 2.0 × 10−5 : 263 covariances among the data are given in the form of corre- 3 ¼ ð Þ 0 ½ ð Þ lation coefficients, but the actual covariances are used in all Instead of an averaged value of G from the three fibers as calculations. There are 15 types of input data with two or more published by Newman et al. (2014), the Task Group decided values and the data of the same type generally agree among to use a weighted mean of the three values with a correlated themselves; that is, there are no differences between like data relative uncertainty of 8.6 parts in 106 between each pair of that exceed 2udiff , the standard uncertainty of the difference. fibers due to uncertainties associated with the source and test The major exception is the values of the Newtonian masses. The final UCI-14 input datum is constant of gravitation G. These are listed in Table XXVII and because the G data are independent of all other data, they G 6.674 35 13 G 1.9 × 10−5 ; 264 are treated separately in Sec. XIII.B.1. A minor exception is ¼ ð Þ 0 ½ ð Þ the difference between items B44.2 and B44.6, the 2 where the uncertainty is taken to be the average of the three NIST-98 and NRC-15 watt-balance values of KJ RK; for these u 2.04. uncertainties as assigned by the researchers rather than that of diff ¼ the weighted mean since it better reflects the researchers view of the reliability of their measurements. A. Comparison of data through inferred values of α, h, and k

XII. ELECTROWEAK QUANTITIES The extent to which the data agree is shown in this section by directly comparing values of α, h, and k that can be inferred There are a few cases in the 2014 adjustment, as in previous from different experiments. However, the inferred value is for adjustments, where an inexact constant that is used in the comparison purposes only; the datum from which it is analysis of input data is not treated as an adjusted quantity, obtained, not the inferred value, is used in the least-squares because the adjustment has a negligible effect on its value. calculations. Three such constants, used in the calculation of the theoretical Table XX and Figs. 1 and 2 compare values of α calculated expression for the electron magnetic-moment anomaly ae, are from the indicated input data. They are obtained using the appropriate observational equation for the corresponding the mass of the tau lepton mτ, the Fermi coupling constant GF, 2 and sine squared of the weak mixing angle sin θW; they are input datum as given in Table XXIV and the 2014 recom- obtained from the most recent report of the Particle Data mended values of the constants other than α that enter that Group (Olive et al., 2014): equation. The table and figures show that a large majority of the values of α agree, and thus the data from which they are m c2 1776.82 16 MeV 9.0 × 10−5 ; 265 obtained agree; of the 91 differences between the 14 values of τ ¼ ð Þ ½ ð Þ α, there are only eight that exceed 2udiff and these are in the G range 2.02 to 2.60. Six are between α from item B39.1, the F 1.1663787 6 × 10−5 GeV−2 5.1 × 10−7 ; 266 NIST-89 result for Γ lo , and α from B22.1, B22.2, B43.1, ℏc 3 ¼ ð Þ ½ ð Þ p0 −90ð Þ ð Þ B43.3, B46, and B48. The other two are between α from item 2 −3 B40, the KR/VN-98 result for Γ0 lo , and α from items sin θW 0.2223 21 9.5 × 10 : 267 h−90 ¼ ð Þ½ ð Þ B43.1 and B43.3. ð Þ 2 2 The inconsistency of these two gyromagnetic ratios has We use the definition sin θW 1 − mW=mZ , where mW ¼ ð Þ 0 been discussed in previous CODATA reports and is not a and mZ are, respectively, the masses of the WÆ and Z bosons, because it is employed in the calculation of the electroweak serious issue because their self-sensitivity coefficients Sc (a measure of their weights in an adjustment, see Sec. XIII.B) are contributions to ae (Czarnecki, Krause, and Marciano, 1996). The Particle Data Group’s recommended value for the mass less than 0.01. Therefore the two ratios are omitted from the final adjustment on which the 2014 CODATA recommended ratio of these bosons is mW=mZ 0.8819 12 , which leads to 2 ¼ ð Þ values are based. This was also the case in the 2006 and 2010 the value of sin θW given above. The values of these constants are the same as used in adjustments. They are initially considered again, as are other CODATA-10, which were taken from the 2010 Particle Data data, to test data robustness and the exactness of the relations K 2e=h and R h=e2 (see Sec. XIII.B.3). Group report (Nakamura et al., 2010), except that for J ¼ K ¼ 3 Because of the large uncertainties of most of the values of α GF= ℏc . The 2014 value exceeds the 2010 value by the fractionalð Þ amount 1.3 × 10−5 and its uncertainty is about one compared with the uncertainties of those from B22.2, the eighth that of the 2010 value. The value for G = c 3 is taken HarvU-08 result for ae, and B48, the LKB-11 result for F ℏ 87 from p. 139 of Olive et al. (2014). ð Þ h=m Rb , the 2014 recommended value of α is essentially determinedð Þ by these two input data. Figure 2 compares them XIII. ANALYSIS OF DATA through their inferred values of α and shows how their consistency has changed since 2010, but not because the 87 The input data discussed in the previous sections are measured values of ae and h=m Rb have changed. Rather, ð Þ analyzed in this section, and based on that analysis the data it is because the ae QED theoretical expression and the value 87 used to determine the 2014 CODATA recommended values of of Ar e required to determine α from h=m Rb have the constants are selected. We closely follow the approach changed.ð Þ The UWash-87 result for a , the h=m 133ð Cs Þresult, e ð Þ

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-41 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

TABLE XVI. Summary of principal input data for the determination of the 2014 recommended value of the Rydberg constant R∞. Relative standard a Item number Input datum Value uncertainty ur Identification Sec. −13 A1 δH 1S1=2 0.0(2.5) kHz 7.5 × 10 Theory IV.A.1.l ð Þ ½ −13 A2 δH 2S1=2 0.00(31) kHz 3.8 × 10 Theory IV.A.1.l ð Þ ½ −13 A3 δH 3S1=2 0.000(91) kHz 2.5 × 10 Theory IV.A.1.l ð Þ ½ −13 A4 δH 4S1=2 0.000(39) kHz 1.9 × 10 Theory IV.A.1.l ð Þ ½ −13 A5 δH 6S1=2 0.000(15) kHz 1.6 × 10 Theory IV.A.1.l ð Þ ½ −13 A6 δH 8S1=2 0.0000(63) kHz 1.2 × 10 Theory IV.A.1.l ð Þ ½ −14 A7 δH 2P1=2 0.000(28) kHz 3.5 × 10 Theory IV.A.1.l ð Þ ½ −14 A8 δH 4P1=2 0.0000(38) kHz 1.9 × 10 Theory IV.A.1.l ð Þ ½ −14 A9 δH 2P3=2 0.000(28) kHz 3.5 × 10 Theory IV.A.1.l ð Þ ½ −14 A10 δH 4P3=2 0.0000(38) kHz 1.9 × 10 Theory IV.A.1.l ð Þ ½ −15 A11 δH 8D3=2 0.000 00(44) kHz 8.5 × 10 Theory IV.A.1.l ð Þ ½ −15 A12 δH 12D3=2 0.000 00(13) kHz 5.7 × 10 Theory IV.A.1.l ð Þ ½ −14 A13 δH 4D5=2 0.0000(35) kHz 1.7 × 10 Theory IV.A.1.l ð Þ ½ −14 A14 δH 6D5=2 0.0000(10) kHz 1.1 × 10 Theory IV.A.1.l ð Þ ½ −15 A15 δH 8D5=2 0.000 00(44) kHz 8.5 × 10 Theory IV.A.1.l ð Þ ½ −15 A16 δH 12D5=2 0.000 00(13) kHz 5.7 × 10 Theory IV.A.1.l ð Þ ½ −13 A17 δD 1S1=2 0.0(2.3) kHz 6.9 × 10 Theory IV.A.1.l ð Þ ½ −13 A18 δD 2S1=2 0.00(29) kHz 3.5 × 10 Theory IV.A.1.l ð Þ ½ −13 A19 δD 4S1=2 0.000(36) kHz 1.7 × 10 Theory IV.A.1.l ð Þ ½ −13 A20 δD 8S1=2 0.0000(60) kHz 1.2 × 10 Theory IV.A.1.l ð Þ ½ −15 A21 δD 8D3=2 0.000 00(44) kHz 8.5 × 10 Theory IV.A.1.l ð Þ ½ −15 A22 δD 12D3=2 0.000 00(13) kHz 5.6 × 10 Theory IV.A.1.l ð Þ ½ −14 A23 δD 4D5=2 0.0000(35) kHz 1.7 × 10 Theory IV.A.1.l ð Þ ½ −15 A24 δD 8D5=2 0.000 00(44) kHz 8.5 × 10 Theory IV.A.1.l ð Þ ½ −15 A25 δD 12D5=2 0.000 00(13) kHz 5.7 × 10 Theory IV.A.1.l ð Þ ½ −15 A26.1 νH 1S1=2 − 2S1=2 2 466 061 413 187.035(10) kHz 4.2 × 10 MPQ-11 IV.A.2 ð Þ −15 A26.2 νH 1S1=2 − 2S1=2 2 466 061 413 187.018(11) kHz 4.4 × 10 MPQ-13 IV.A.2 ð Þ −12 A27 νH 1S1=2 − 3S1=2 2 922 743 278 678(13) kHz 4.4 × 10 LKB-10 IV.A.2 ð Þ −11 A28 νH 2S1=2 − 8S1=2 770 649 350 012.0(8.6) kHz 1.1 × 10 LK/SY-97 IV.A.2 ð Þ −11 A29 νH 2S1=2 − 8D3=2 770 649 504 450.0(8.3) kHz 1.1 × 10 LK/SY-97 IV.A.2 ð Þ −12 A30 νH 2S1=2 − 8D5=2 770 649 561 584.2(6.4) kHz 8.3 × 10 LK/SY-97 IV.A.2 ð Þ −11 A31 νH 2S1=2 − 12D3=2 799 191 710 472.7(9.4) kHz 1.2 × 10 LK/SY-98 IV.A.2 ð Þ −12 A32 νH 2S1=2 − 12D5=2 799 191 727 403.7(7.0) kHz 8.7 × 10 LK/SY-98 IV.A.2 A33 ð Þ 1 4 797 338(10) kHz −6 MPQ-95 IV.A.2 νH 2S1=2 − 4S1=2 − 4 νH 1S1=2 − 2S1=2 2.1 × 10 A34 ð Þ 1 ð Þ 6 490 144(24) kHz −6 MPQ-95 IV.A.2 νH 2S1=2 − 4D5=2 − 4 νH 1S1=2 − 2S1=2 3.7 × 10 A35 ð Þ 1 ð Þ 4 197 604(21) kHz −6 LKB-96 IV.A.2 νH 2S1=2 − 6S1=2 − 4 νH 1S1=2 − 3S1=2 4.9 × 10 A36 ð Þ 1 ð Þ 4 699 099(10) kHz −6 LKB-96 IV.A.2 νH 2S1=2 − 6D5=2 − 4 νH 1S1=2 − 3S1=2 2.2 × 10 A37 ð Þ 1 ð Þ 4 664 269(15) kHz −6 YaleU-95 IV.A.2 νH 2S1=2 − 4P1=2 − 4 νH 1S1=2 − 2S1=2 3.2 × 10 A38 ð Þ 1 ð Þ 6 035 373(10) kHz −6 YaleU-95 IV.A.2 νH 2S1=2 − 4P3=2 − 4 νH 1S1=2 − 2S1=2 1.7 × 10 ð Þ ð Þ −6 A39 νH 2S1=2 − 2P3=2 9 911 200(12) kHz 1.2 × 10 HarvU-94 IV.A.2 ð Þ −6 A40.1 νH 2P1=2 − 2S1=2 1 057 845.0(9.0) kHz 8.5 × 10 HarvU-86 IV.A.2 ð Þ −5 A40.2 νH 2P1=2 − 2S1=2 1 057 862(20) kHz 1.9 × 10 USus-79 IV.A.2 ð Þ −12 A41 νD 2S1=2 − 8S1=2 770 859 041 245.7(6.9) kHz 8.9 × 10 LK/SY-97 IV.A.2 ð Þ −12 A42 νD 2S1=2 − 8D3=2 770 859 195 701.8(6.3) kHz 8.2 × 10 LK/SY-97 IV.A.2 ð Þ −12 A43 νD 2S1=2 − 8D5=2 770 859 252 849.5(5.9) kHz 7.7 × 10 LK/SY-97 IV.A.2 ð Þ −11 A44 νD 2S1=2 − 12D3=2 799 409 168 038.0(8.6) kHz 1.1 × 10 LK/SY-98 IV.A.2 ð Þ −12 A45 νD 2S1=2 − 12D5=2 799 409 184 966.8(6.8) kHz 8.5 × 10 LK/SY-98 IV.A.2 A46 ð Þ 1 4 801 693(20) kHz −6 MPQ-95 IV.A.2 νD 2S1=2 − 4S1=2 − 4 νD 1S1=2 − 2S1=2 4.2 × 10 A47 ð Þ 1 ð Þ 6 494 841(41) kHz −6 MPQ-95 IV.A.2 νD 2S1=2 − 4D5=2 − 4 νD 1S1=2 − 2S1=2 6.3 × 10 ð Þ ð Þ −11 A48 νD 1S1=2 − 2S1=2 − νH 1S1=2 − 2S1=2 670 994 334.606(15) kHz 2.2 × 10 MPQ-10 IV.A.2 ð Þ ð Þ −2 A49 rp 0.879(11) fm 1.3 × 10 rp-14 IV.A.3 −3 A50 rd 2.130(10) fm 4.7 × 10 rd-98 IV.A.3 aThe values in brackets are relative to the frequency equivalent of the binding energy of the indicated level.

the three Γx0 −90 lo results, and the five RK results are omitted Table XXI and Figs. 3 and 4 compare values of h obtained from the 2010ð finalÞ adjustment because of their low weights from the indicated input data. They show that the vast majority and are omitted from the 2014 final adjustment for the same of the values of h agree, and thus the data from which they are reason. obtained agree; of the 91 differences between h values, only

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-42 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

TABLE XVII. Correlation coefficients r xi;xj ≥ 0.0001 of the input data related to R∞ in Table XVI. For simplicity, the two items of data to which a particular correlation coefficientð correspondsÞ are identified by their item numbers in Table XVI.

r A1;A2 0.9905 r A6;A19 0.7404 r A26.1;A26.2 0.7069 r A31;A44 0.0901 rðA1;A3Þ¼0.9900 rðA6;A20Þ¼0.9851 ð r A28;A29Þ¼0.3478 rðA31;A45Þ¼0.1136 rðA1;A4Þ¼0.9873 ðr A7;A8Þ¼0.0237 rðA28;A30Þ¼0.4532 rðA32;A35Þ¼0.0278 rðA1;A5Þ¼0.7640 r Að 9;A10Þ¼0.0237 rðA28;A31Þ¼0.0899 rðA32;A36Þ¼0.0553 rðA1;A6Þ¼0.7627 r Að 11;A12Þ¼0.0006 rðA28;A32Þ¼0.1206 rðA32;A41Þ¼0.1512 r Að 1;A17Þ¼0.9754 rðA11;A21Þ¼0.9999 rðA28;A35Þ¼0.0225 rðA32;A42Þ¼0.1647 rðA1;A18Þ¼0.9656 rðA11;A22Þ¼0.0003 rðA28;A36Þ¼0.0448 rðA32;A43Þ¼0.1750 rðA1;A19Þ¼0.9619 rðA12;A21Þ¼0.0003 rðA28;A41Þ¼0.1225 rðA32;A44Þ¼0.1209 rðA1;A20Þ¼0.7189 rðA12;A22Þ¼0.9999 rðA28;A42Þ¼0.1335 rðA32;A45Þ¼0.1524 ðr A2;A3Þ¼0.9897 rðA13;A14Þ¼0.0006 rðA28;A43Þ¼0.1419 rðA33;A34Þ¼0.1049 rðA2;A4Þ¼0.9870 rðA13;A15Þ¼0.0006 rðA28;A44Þ¼0.0980 rðA33;A46Þ¼0.2095 rðA2;A5Þ¼0.7638 rðA13;A16Þ¼0.0006 rðA28;A45Þ¼0.1235 rðA33;A47Þ¼0.0404 rðA2;A6Þ¼0.7625 rðA13;A23Þ¼0.9999 rðA29;A30Þ¼0.4696 rðA34;A46Þ¼0.0271 r Að 2;A17Þ¼0.9656 rðA13;A24Þ¼0.0003 rðA29;A31Þ¼0.0934 rðA34;A47Þ¼0.0467 rðA2;A18Þ¼0.9754 rðA13;A25Þ¼0.0003 rðA29;A32Þ¼0.1253 rðA35;A36Þ¼0.1412 rðA2;A19Þ¼0.9616 rðA14;A15Þ¼0.0006 rðA29;A35Þ¼0.0234 rðA35;A41Þ¼0.0282 rðA2;A20Þ¼0.7187 rðA14;A16Þ¼0.0006 rðA29;A36Þ¼0.0466 rðA35;A42Þ¼0.0307 ðr A3;A4Þ¼0.9864 rðA14;A23Þ¼0.0003 rðA29;A41Þ¼0.1273 rðA35;A43Þ¼0.0327 rðA3;A5Þ¼0.7633 rðA14;A24Þ¼0.0003 rðA29;A42Þ¼0.1387 rðA35;A44Þ¼0.0226 rðA3;A6Þ¼0.7620 rðA14;A25Þ¼0.0003 rðA29;A43Þ¼0.1475 rðA35;A45Þ¼0.0284 r Að 3;A17Þ¼0.9651 rðA15;A16Þ¼0.0006 rðA29;A44Þ¼0.1019 rðA36;A41Þ¼0.0561 rðA3;A18Þ¼0.9648 rðA15;A23Þ¼0.0003 rðA29;A45Þ¼0.1284 rðA36;A42Þ¼0.0612 rðA3;A19Þ¼0.9611 rðA15;A24Þ¼0.9999 rðA30;A31Þ¼0.1209 rðA36;A43Þ¼0.0650 rðA3;A20Þ¼0.7183 rðA15;A25Þ¼0.0003 rðA30;A32Þ¼0.1622 rðA36;A44Þ¼0.0449 ðr A4;A5Þ¼0.7613 rðA16;A23Þ¼0.0003 rðA30;A35Þ¼0.0303 rðA36;A45Þ¼0.0566 rðA4;A6Þ¼0.7600 rðA16;A24Þ¼0.0003 rðA30;A36Þ¼0.0602 rðA37;A38Þ¼0.0834 r Að 4;A17Þ¼0.9625 rðA16;A25Þ¼0.9999 rðA30;A41Þ¼0.1648 rðA41;A42Þ¼0.5699 rðA4;A18Þ¼0.9622 rðA17;A18Þ¼0.9897 rðA30;A42Þ¼0.1795 rðA41;A43Þ¼0.6117 rðA4;A19Þ¼0.9755 rðA17;A19Þ¼0.9859 rðA30;A43Þ¼0.1908 rðA41;A44Þ¼0.1229 rðA4;A20Þ¼0.7163 rðA17;A20Þ¼0.7368 rðA30;A44Þ¼0.1319 rðA41;A45Þ¼0.1548 ðr A5;A6Þ¼0.5881 rðA18;A19Þ¼0.9856 rðA30;A45Þ¼0.1662 rðA42;A43Þ¼0.6667 r Að 5;A17Þ¼0.7448 rðA18;A20Þ¼0.7366 rðA31;A32Þ¼0.4750 rðA42;A44Þ¼0.1339 rðA5;A18Þ¼0.7445 rðA19;A20Þ¼0.7338 rðA31;A35Þ¼0.0207 rðA42;A45Þ¼0.1687 rðA5;A19Þ¼0.7417 rðA21;A22Þ¼0.0002 rðA31;A36Þ¼0.0412 rðA43;A44Þ¼0.1423 rðA5;A20Þ¼0.5543 rðA23;A24Þ¼0.0001 rðA31;A41Þ¼0.1127 rðA43;A45Þ¼0.1793 rðA6;A17Þ¼0.7435 rðA23;A25Þ¼0.0001 rðA31;A42Þ¼0.1228 rðA44;A45Þ¼0.5224 rðA6;A18Þ¼0.7433 rðA24;A25Þ¼0.0002 rðA31;A43Þ¼0.1305 rðA46;A47Þ¼0.0110 ð Þ¼ ð Þ¼ ð Þ¼ ð Þ¼

one exceeds 2udiff . The two values are from input datum B. Multivariate analysis of data B44.2 and B44.6, the NIST-98 and NRC-15 watt-balance 2 Our multivariate analysis of the data employs a well-known results for KJ RK, but the difference is only 2.04udiff. The first five values of h in the table, each with relative standard least-squares method that allows correlations among the input −7 data to be properly taken into account. Used in the four uncertainty ur < 10 , are compared in Fig. 4. The five input data from which these values are obtained are included in the previous adjustments, it is described in Appendix E of final adjustment and determine the 2014 recommended value CODATA-98 and the references cited therein. It is recalled of h. The input data from which the next nine values of h with from that appendix that a least-squares adjustment is characterized by the number of input data N, number of variables or u from 2.0 × 10−7 to 1.6 × 10−6 are omitted from the final r adjusted constants M, degrees of freedom ν N M, statistic adjustment because of their low weight. − χ2, probability p χ2 ν of obtaining an observed¼ value of χ2 Table XXII and Fig. 5 compare values of k obtained from ð j Þ the indicated input data. Although most of the source data are that large or larger for the given value of ν, Birge ratio 2 2 2 acoustic gas thermometry measurements of R, values of RB χ =ν (χ =ν is often called the reduced χ ), and the ¼ k R=N are compared, because k is one of the defining normalized residual of the ith input datum ri xi − xi =ui, ¼ A pffiffiffiffiffiffiffiffiffi ¼ð h iÞ constants of the new SI (see Sec. I.B.1) and u N u R . where xi is the input datum, xi its adjusted value, and ui its r A ≪ r h i The table and figure show that all the valuesð ofÞ k areð inÞ standard uncertainty. excellent agreement, and thus so are the data from which they The observational equations for the input data are given in are obtained; of the 55 differences between k values, none Tables XXIII and XXIV. These equations are written in terms exceed 2udiff and the largest is only 0.96. Moreover, it turns of a particular independent subset of constants (broadly out that only the NPL-79 and INRIM-10 results for R have interpreted) called adjusted constants. These are the variables insufficient weight to be included in the 2014 final adjustment, (or unknowns) of the adjustment. The least-squares calcu- although they had sufficient weight to be included in the 2010 lation yields values of the adjusted constants that predict final adjustment. values of the input data through their observational equations

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-43 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

TABLE XVIII. Summary of principal input data for the determination of the 2014 recommended values of the fundamental constants (R∞ and G excepted).

Relative standard a Item number Input datum Value uncertainty ur Identification Sec. and Eq. −10 B1 Ar n 1.008 664 915 85(49) 4.9 × 10 AME-12 III.A ð1 Þ −11 B2 Ar H 1.007 825 032 231(93) 9.3 × 10 AME-12 III.A ð 1Þ 7 −1 −13 B3 ΔEB Hþ =hc 1.096 787 717 4307 10 × 10 m 9.1 × 10 ASD-14 III.B b 3 ð Þ ð Þ −10 B4 Ar H 3.016 049 2779(24) 7.9 × 10 AME-12 III.A b ð 3Þ 7 −1 −9 B5 ΔEB Hþ =hc 1.097 185 4390 13 × 10 m 1.2 × 10 ASD-14 III.B 4 ð Þ ð Þ −11 B6 Ar He 4.002 603 254 130(63) 1.6 × 10 AME-12 III.A ð 4 Þ 2 7 −1 −10 B7 ΔEB He þ =hc 6.372 195 4487 28 × 10 m 4.4 × 10 ASD-14 III.B ð 12Þ 6 ð Þ −11 B8 ωc d =ωc C þ 0.992 996 654 743(20) 2.0 × 10 UWash-15 III.C (5) c ð Þ ð12 6 Þ −11 B9 ωc h =ωc C þ 1.326 365 862 193(19) 1.4 × 10 UWash-15 III.C (6) ð Þ12 6ð Þ 7 −1 −7 B10 ΔEB C þ =hc 83.083 962 72 × 10 m 8.7 × 10 ASD-14 III.B c ð Þ 3 ð Þ −11 B11 ωc HDþ =ωc Heþ 0.998 048 085 153(48) 4.8 × 10 FSU-15 III.C (11) ð Þ ð Þ −11 B12 ωc HDþ =ωc t 0.998 054 687 288(48) 4.8 × 10 FSU-15 III.C (12) ð 3 Þ ð Þ −1 −10 B13 ΔEI Heþ =hc 43 888 919.36 3 m 6.8 × 10 ASD-14 III.C (17) ð Þ ð Þ −1 −10 B14 ΔEI HDþ =hc 13 122 468.415 6 m 4.6 × 10 Literature III.C (19) 12ð 5 Þ 12 5 ð Þ −11 B15 ωs C þ =ωc C þ 4376.210 500 87(12) 2.8 × 10 MPIK-15 V.D.2 (179) ð 12 5Þ ð Þ 7 −1 −6 B16 ΔEB C þ =hc 43.563 345 72 × 10 m 1.7 × 10 ASD-14 III.B ð Þ ð −Þ11 −11 B17 δC 0.0 2.6 × 10 [1.3 × 10 ] Theory V.D.1 (175) 28 13 28 13 ð Þ −11 B18 ωs Si þ =ωc Si þ 3912.866 064 84(19) 4.8 × 10 MPIK-15 V.D.2 (178) ð28 Þ ð Þ −11 B19 Ar Si 27.976 926 534 65(44) 1.6 × 10 AME-12 III.A ð 28Þ 13 7 −1 −5 B20 ΔEB Si þ =hc 420.608 19 × 10 m 4.4 × 10 ASD-14 III.B ð Þ ð Þ −9 −10 B21 δSi 0.0 1.7 × 10 [8.3 × 10 ] Theory V.D.1 (176) b ð Þ −3 −9 B22.1 ae 1.159 652 1883 42 × 10 3.7 × 10 UWash-87 V.A.2 ð Þ −3 −10 B22.2 ae 1.159 652 180 73 28 × 10 2.4 × 10 HarvU-08 V.A.2 −12ð Þ −10 B23 δe 0.000 37 × 10 [0.32 × 10 ] Theory V.A.1 (109) B24 R 0.003ð 707Þ 2063(20) 5.4 × 10−7 BNL-06 V.B.2 (135) B25 ν 58 MHz 627 994.77(14) kHz 2.2 × 10−7 LAMPF-82 VI.B.2 (218) B26 νð72 MHzÞ 668 223 166(57) Hz 8.6 × 10−8 LAMPF-99 VI.B.2 (221) ð Þ −8 B27.1 ΔνMu 4 463 302.88(16) kHz 3.6 × 10 LAMPF-82 VI.B.2 (217) −8 B27.2 ΔνMu 4 463 302 765(53) Hz 1.2 × 10 LAMPF-99 VI.B.2 (220) −8 B28 δMu 0(85) Hz [1.9 × 10 ] Theory VI.B.1 (215) −9 B29 μp=μN 2.792 847 3498(93) 3.3 × 10 UMZ-14 V.C (141) −8 B30 μe H =μp H −658.210 7058 66 1.0 × 10 MIT-72 VI.A.1 ð Þ ð Þ ð Þ −4 −8 B31 μd D =μe D −4.664 345 392 50 × 10 1.1 × 10 MIT-84 VI.A.1 ð Þ ð Þ ð Þ −8 B32 μe H =μp0 −658.215 9430 72 1.1 × 10 MIT-77 VI.A.1 ð Þ ð Þ −9 B33 μh0 =μp0 −0.761 786 1313 33 4.3 × 10 NPL-93 VI.A.1 ð Þ −7 B34 μn=μp0 −0.684 996 94 16 2.4 × 10 ILL-79 VI.A.1 ð Þ −9 B35.1 μp HD =μd HD 3.257 199 531(29) 8.9 × 10 StPtrsb-03 VI.A.1 ð Þ ð Þ −9 B35.2 μp HD =μd HD 3.257 199 514(21) 6.6 × 10 WarsU-12 VI.A.1 (197) ð Þ ð Þ −9 B36 μt HT =μp HT 1.066 639 8933(21) 2.0 × 10 StPtrsb-11 VI.A.1 (198) ð Þ ð Þ −9 B37 σdp 15 2 × 10 StPtrsb-03 VI.A.1 ð Þ −9 B38 σtp 20 3 × 10 StPtrsb-03 VI.A.1 b ð Þ 8 −1 −1 −7 B39.1 Γp0 −90 lo 2.675 154 05 30 × 10 s T 1.1 × 10 NIST-89 VIII b ð Þ ð Þ 8 −1 −1 −7 B39.2 Γp0 −90 lo 2.675 1530 18 × 10 s T 6.6 × 10 NIM-95 VIII b ð Þ ð Þ 8 −1 −1 −7 B40 Γh0 −90 lo 2.037 895 37 37 × 10 s T 1.8 × 10 KR/VN-98 VIII b ð Þ ð Þ 8 −1 −1 −6 B41.1 Γp0 −90 hi 2.675 1525 43 × 10 s T 1.6 × 10 NIM-95 VIII b ð Þ ð Þ 8 −1 −1 −6 B41.2 Γp0 −90 hi 2.675 1518 27 × 10 s T 1.0 × 10 NPL-79 VIII b ð Þ ð Þ −1 −7 B42.1 KJ 483 597.91 13 GHz V 2.7 × 10 NMI-89 VIII b ð Þ −1 −7 B42.2 KJ 483 597.96 15 GHz V 3.1 × 10 PTB-91 VIII b ð Þ −8 B43.1 RK 25 812.808 31 62 Ω 2.4 × 10 NIST-97 VIII b ð Þ −8 B43.2 RK 25 812.8071 11 Ω 4.4 × 10 NMI-97 VIII b ð Þ −8 B43.3 RK 25 812.8092 14 Ω 5.4 × 10 NPL-88 VIII b ð Þ −7 B43.4 RK 25 812.8084 34 Ω 1.3 × 10 NIM-95 VIII b ð Þ −8 B43.5 RK 25 812.8081 14 Ω 5.3 × 10 LNE-01 VIII b 2 ð Þ 33 −1 −1 −7 B44.1 KJ RK 6.036 7625 12 × 10 J s 2.0 × 10 NPL-90 VIII.A 2 ð Þ 33 −1 −1 −8 B44.2 KJ RK 6.036 761 85 53 × 10 J s 8.7 × 10 NIST-98 VIII.D b 2 ð Þ 33 −1 −1 −7 B44.3 KJ RK 6.036 7617 18 × 10 J s 2.9 × 10 METAS-11 VIII.B B44.4 K2R 6.036 761 43ð 34Þ × 1033 J−1 s−1 5.7 × 10−8 NIST-15 VIII.D (234) J K ð Þ (Table continued)

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-44 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

TABLE XVIII. (Continued)

Relative standard a Item number Input datum Value uncertainty ur Identification Sec. and Eq. b 2 33 −1 −1 −7 B44.5 KJ RK 6.036 7597 12 × 10 J s 2.0 × 10 NPL-12 VIII.A 2 ð Þ 33 −1 −1 −8 B44.6 KJ RK 6.036 760 76 11 × 10 J s 1.8 × 10 NRC-15 VIII.E (237) b 2 ð Þ 33 −1 −1 −7 B44.7 KJ RK 6.036 7619 15 × 10 J s 2.6 × 10 LNE-15 VIII.C (231) b ð Þ −1 −6 B45 F 90 96 485.39 13 C mol 1.3 × 10 NIST-80 VIII B46b h=m 133Cs 3.002 369ð 432Þ 46 × 10−9 m2 s−1 1.5 × 10−8 StanfU-02 VII b 133ð Þ ð Þ −11 B47 Ar Cs 132.905 451 9615(86) 6.5 × 10 AME-12 III.A B48 h=mð 87RbÞ 4.591 359 2729 57 × 10−9 m2 s−1 1.2 × 10−9 LKB-11 VII 87ð Þ ð Þ −11 B49 Ar Rb 86.909 180 5319(65) 7.5 × 10 AME-12 III.A ð Þ −9 B50 1 − d220 W17 =d220 ILL −8 22 × 10 NIST-99 IX.A ð Þ ð Þ ð Þ −9 B51 1 − d220 MOÃ =d220 ILL 86 27 × 10 NIST-99 IX.A ð Þ ð Þ ð Þ −9 B52 1 − d220 NR3 =d220 ILL 33 22 × 10 NIST-99 IX.A ð Þ ð Þ ð Þ −9 B53 1 − d220 N =d220 W17 7 22 × 10 NIST-97 IX.A ð Þ ð Þ ð Þ −9 B54 d220 W4.2a =d220 W04 − 1 −1 21 × 10 PTB-98 IX.A ð Þ ð Þ ð Þ −9 B55.1 d220 W17 =d220 W04 − 1 22 22 × 10 PTB-98 IX.A ð Þ ð Þ ð Þ −9 B55.2 d220 W17 =d220 W04 − 1 11 21 × 10 NIST-06 IX.A ð Þ ð Þ ð Þ −9 B56 d220 MOÃ =d220 W04 − 1 −103 28 × 10 PTB-98 IX.A ð Þ ð Þ ð Þ −9 B57.1 d220 NR3 =d220 W04 − 1 −23 21 × 10 PTB-98 IX.A ð Þ ð Þ ð Þ −9 B57.2 d220 NR3 =d220 W04 − 1 −11 21 × 10 NIST-06 IX.A ð Þ ð Þ ð Þ −9 B58 d220=d220 W04 − 1 10 11 × 10 PTB-03 IX.A ð Þ ð Þ −9 B59 d220 NR4 =d220 W04 − 1 25 21 × 10 NIST-06 IX.A ð Þ ð Þ ð Þ −8 B60 d220 MOÃ 192 015.5508(42) fm 2.2 × 10 INRIM-08 IX.A ð Þ −8 B61 d220 W04 192 015.5702(29) fm 1.5 × 10 INRIM-09 IX.A ð Þ −8 B62.1 d220 W4.2a 192 015.5691(29) fm 1.5 × 10 INRIM-09 IX.A ð Þ −8 B62.2 d220 W4.2a 192 015.563(12) fm 6.2 × 10 PTB-81 IX.A ð Þ 23 3 −1 −8 B63.1 NA 6.022 140 99 18 × 10 m mol 3.0 × 10 IAC-11 IX.B (239) ð Þ 23 3 −1 −8 B63.2 NA 6.022 140 76 12 × 10 m mol 2.0 × 10 IAC-15 IX.B (240) B64.1b R 8.314 504 70ð JÞ mol−1 K−1 8.4 × 10−6 NPL-79 X.A B64.2 R 8.314 470ð15Þ J mol−1 K−1 1.8 × 10−6 NIST-88 X.A B64.3 R 8.314 467ð23Þ J mol−1 K−1 2.7 × 10−6 LNE-09 X.A B64.4 R 8.314 468ð26Þ J mol−1 K−1 3.2 × 10−6 NPL-10 X.A B64.5b R 8.314 412ð63Þ J mol−1 K−1 7.5 × 10−6 INRIM-10 X.A B64.6 R 8.314 455ð12Þ J mol−1 K−1 1.4 × 10−6 LNE-11 X.A B64.7 R 8.314 455ð31Þ J mol−1 K−1 3.7 × 10−6 NIM-13 X.A (243) B64.8 R 8.314 4544ð 75Þ J mol−1 K−1 9.0 × 10−7 NPL-13 X.A B64.9 R 8.314 4615ð84Þ J mol−1 K−1 1.0 × 10−6 LNE-15 X.A (245) B65 k=h 2.083 6658ð80Þ × 1010 Hz K−1 3.9 × 10−6 NIM/NIST-15 X.B (248) ð Þ −8 3 −1 −6 B66 Aϵ=R 6.221 128 25 × 10 m KJ 4.0 × 10 PTB-15 X.C (253) 4 3 ð Þ −7 B67 α0 He =4πϵ0a0 1.383 760 77(14) 1.0 × 10 Theory X.C (251) ð Þ −7 B68 λ CuKα1 =d220 W4.2a 0.802 327 11(24) 3.0 × 10 FSUJ/PTB-91 IX.A ð Þ ð Þ −7 B69 λ CuKα1 =d220 N 0.802 328 04(77) 9.6 × 10 NIST-73 IX.A ð Þ ð Þ −7 B70 λ WKα1 =d220 N 0.108 852 175(98) 9.0 × 10 NIST-79 IX.A ð Þ ð Þ −7 B71 λ MoKα1 =d220 N 0.369 406 04(19) 5.3 × 10 NIST-73 IX.A ð Þ ð Þ a 12 5 28 13 The values in brackets are relative to the quantities g C þ , g Si þ , ae, or ΔνMu as appropriate. bDatum not included in the final least-squares adjustmentð thatÞ providesð theÞ recommended values of the constants. cDatum included in the final least-squares adjustment with an expanded uncertainty.

that best agree with the data themselves in the least-squares such as δe and R, the observational equation is simply δe δe sense. The adjusted constants used in the 2014 calculations are and R R. ≐ given in Tables XXV and XXVI. The≐ bound-state g-factor ratios in the observational equa- The symbol in an observational equation indicates that an tions of Table XXIV are treated as fixed quantities with input datum of≐ the type on the left-hand side is ideally given negligible uncertainties (see Table XIII, Sec. VI.A). The by the expression on the right-hand side containing adjusted frequency fp is not an adjusted constant but is included in constants. But because the equation is one of an overdeter- the equation for data items B25 and B26 to indicate that they mined set that relates a datum to adjusted constants, the two are functions of fp. Finally, the observational equations for sides are not necessarily equal. The best estimate of the value items and B25 and B26, which are based on Eqs. (223)–(225) of an input datum is its observational equation evaluated with of Sec. VI.B.2, include the theoretical expressions for ae and the least-squares adjusted values of the adjusted constants on ΔνMu which are given as observational equations B22 and which its observational equation depends. For some input data B27 in Table XXIV.

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-45 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

TABLE XIX. Correlation coefficients r xi;xj ≥ 0.001 of the input data in Table XVIII. For simplicity, the two items of data to which a particular correlation coefficient correspondsð areÞ identified by their item numbers in Table XVIII. r B1;B2 −0.133 r B42.1;B64.2 0.068 r B52;B57.2 −0.367 r B64.2;B64.7 0.001 rðB1;B19Þ¼ −0.015 rðB44.1;B44.5Þ¼0.003 rðB52;B59 Þ¼0.065 rðB64.2;B64.8Þ¼0.003 rðB1;B47Þ¼−0.007 rðB44.2;B44.4Þ¼0.090 rðB53;B55.Þ¼2 0.504 rðB64.3;B64.4Þ¼0.002 rðB1;B49Þ¼−0.007 rðB47;B49 Þ¼0.102 rðB53;B57.2Þ¼0.066 rðB64.3;B64.5Þ¼0.001 rðB2;B19Þ¼0.165 rðB50;B51Þ¼0.421 rðB53;B59 Þ¼0.066 rðB64.3;B64.6Þ¼0.011 rðB2;B47Þ¼0.058 rðB50;B52Þ¼0.516 rðB54;B55Þ¼.1 0.469 rðB64.3;B64.8Þ¼0.006 rðB2;B49Þ¼0.063 rðB50;B53Þ¼−0.288 rðB54;B56 Þ¼0.372 rðB64.3;B64.9Þ¼0.018 rðB8;B9 Þ¼0.306 rðB50;B55Þ¼.2 −0.367 rðB54;B57Þ¼.1 0.502 rðB64.4;B64.6Þ¼0.113 rðB10;BÞ¼16 1.000 rðB50;B57.2Þ¼0.065 rðB55.1;B56Þ¼0.347 rðB64.4;B64.8Þ¼0.007 rðB11;B12Þ¼0.875 rðB50;B59 Þ¼0.065 rðB55.1;B57Þ¼.1 0.469 rðB64.4;B64.9Þ¼0.005 rðB15;B18Þ¼0.347 rðB51;B52Þ¼0.421 rðB55.2;B57.2Þ¼0.509 rðB64.5;B64.9Þ¼0.002 rðB17;B21Þ¼0.791 rðB51;B53Þ¼0.096 rðB55.2;B59 Þ¼0.509 rðB64.6;B64.7Þ¼0.001 rðB19;B47Þ¼0.033 rðB51;B55Þ¼.2 0.053 rðB56;B57.1Þ¼0.372 rðB64.6;B64.8Þ¼0.016 rðB19;B49Þ¼0.041 rðB51;B57.2Þ¼0.053 rðB57.2;B59Þ¼0.509 rðB64.6;B64.9Þ¼0.254 rðB25;B27Þ¼.1 0.227 rðB51;B59 Þ¼0.053 rðB63.1;B63.Þ¼2 0.170 rðB64.7;B64.8Þ¼0.084 rðB26;B27.2Þ¼0.195 rðB52;B53Þ¼0.117 rðB64.2;B64.4Þ¼0.001 rðB64.8;B64.9Þ¼0.016 rðB39.2;B41Þ¼.1 −0.014 rðB52;B55Þ¼.2 0.065 rðB64.2;B64.6Þ¼0.002 ð Þ¼ ð Þ¼ ð Þ¼ ð Þ¼

Also recalled from Appendix E of CODATA-98 is the self- multivariate analysis becomes simply the calculation of their sensitivity coefficient Sc for an input datum, which is a weighted mean. measure of the influence of that datum on the adjusted value of the quantity of which the datum is an example. As in 1. Data related to the Newtonian constant of gravitation G previous adjustments, in general, for an input datum to be included in the final adjustment on which the 2014 recom- The 14 values of G to be considered are summarized in mended values are based, its value of Sc must be greater than Table XVof Sec. XI and are discussed in the text accompany- 0.01, or 1%, which means that its uncertainty must be no more ing it. For easy reference they are listed in Table XXVII of this than about a factor of 10 larger than the uncertainty of the section and are graphically compared in Fig. 6. The last adjusted value of that quantity; see Sec. I.D of CODATA-98 three results in the table and figure, BIPM-14, LENS-14, and for the justification of this 1% cutoff. However, the exclusion UCI-14, have become available since 2010. Although the of a datum is not followed if, for example, a datum with BIPM-14 and UCI-14 results have the comparatively small S < 0.01 is part of a group of data obtained in a given relative standard uncertainties u 24 × 10−6 and u c r ¼ r ¼ experiment where most of the other data have self-sensitivity 19 × 10−6, respectively, they have not reduced the quite large coefficients >0.01. It is also not followed for G, but in this inconsistencies among the G data that have plagued them case it is because of the significant disagreement of the for some 20 years and which are evident in the table and available data and hence lack of motivation for anything figure. beyond a simple weighted mean. Indeed, because the G data Indeed, of the 91 differences among the 14 results, 45 are are independent of all other data and can be treated separately, larger than 2udiff and 22 are larger than 4udiff . The five largest, and because they determine only one variable, in this case the 15.1udiff , 11.4udiff , 10.7udiff , 10.5udiff , and 10.4udiff , are

TABLE XX. Inferred values of the fine-structure constant α in order of increasing standard uncertainty obtained from the indicated experimental data in Table XVIII.

Relative standard −1 Primary source Item number Identification Sec. and Eq. α uncertainty ur −10 ae B22.2 HarvU-08 V.A.2 137.035 999 160(33) 2.4 × 10 h=m 87Rb B48 LKB-11 VII 137.035 998 996(85) 6.2 × 10−10 ð Þ −9 ae B22.1 UWash-87 V.A.2 137.035 998 27(50) 3.7 × 10 h=m 133Cs B46 StanfU-02 VII 137.036 0000(11) 7.7 × 10−9 ð Þ −8 RK B43.1 NIST-97 VIII 137.036 0037(33) 2.4 × 10 −8 Γp0 −90 lo B39.1 NIST-89 VIII 137.035 9879(51) 3.7 × 10 ð Þ −8 RK B43.2 NMI-97 VIII 137.035 9973(61) 4.4 × 10 −8 RK B43.5 LNE-01 VIII 137.036 0023(73) 5.3 × 10 −8 RK B43.3 NPL-88 VIII 137.036 0083(73) 5.4 × 10 −8 ΔνMu B27.1, B27.2 LAMPF VI.B.2 (228) 137.036 0013(79) 5.8 × 10 −8 Γh0 −90 lo B40 KR/VN-98 VIII 137.035 9852(82) 6.0 × 10 ð Þ −7 RK B43.4 NIM-95 VIII 137.036 004(18) 1.3 × 10 −7 Γp0 −90 lo B39.2 NIM-95 VIII 137.036 006(30) 2.2 × 10 ð Þ −7 νH, νD IV.A.1.m (71) 137.035 992(55) 4.0 × 10

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-46 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

FIG. 1. Values of the fine-structure constant α with u < 10−7 r FIG. 3. Values of the Planck constant h with u < 10−6 inferred inferred from the input data in Table XVIII in order of decreasing r from the input data in Table XVIII and the 2014 CODATA uncertainty from top to bottom (see Table XX). recommended value in chronological order from top to bottom (see Table XXI). between JILA-10 and BIPM-14, UWash-00, BIPM-01, −6 UCI-14, and UZur-06, respectively. The weighted mean of results with ur < 30 × 10 . These are, in order of increasing −6 the 14 results is 6.674 083 50 G0 [7.5 × 10 ], where uncertainty, UWash-00, UZur-06, UCI-14, JILA-10, BIPM- −11 −1 3 −2 ð Þ 2 G0 10 kg m s . For this calculation χ 319.3, 14, and HUST-09. Their weighted mean is 6.674 077 52 G0 ¼ ¼ −6 2 ð Þ p 319.3 13 ≈ 0, and RB 4.96. Nine data have normalized [7.8×10 ], with χ 258.6, p 258.6 13 ≈0, and RB 7.19; residualsð j rÞ > 2: JILA-10,¼ BIPM-14, BIPM-01, NIST-82, their respective normalized¼ residualsð j rÞ are 1.9, 1.4,¼ 2.2, j ij i HUST-09, TR&D-96, LENS-14, HUST-05, and UCI-14; their −12.4, 9.1, and −3.3. The significant disagreement of the respective values are −12.5, 9.1, 5.6, −3.7, 3.3, 2.4, 2.2, 2.1, JILA-10 and BIPM-14 results with the four other low- and 2.1. uncertainty results is apparent. Additional calculations have Because of their comparatively small uncertainties, there is been carried out, for example, one in which the JILA-10, little impact if this calculation is repeated with just the six G BIPM-01, and BIPM-14 results are omitted. The weighted mean of the remaining 11 data is 6.674 121 57 G0 [8.6 × 10−6], with χ2 49.8, p 49.8 13 2.9 × 10−ð6,Þ and ¼ ð j Þ¼ RB 2.2. The value of G is not significantly different from¼ the two other weighted-mean values and deleting the three data increases the χ2 probability by 10 orders of magnitude. Nevertheless, for all practical purposes it is still very small. In 2010 the Task Group decided to take as the recommended value of G the weighted mean and its uncertainty of the 11 values then available (essentially the first 11 values in Tables XV and XXVII), but after multiplying the initially assigned uncertainty of each value by the factor 14, called the expansion factor. The number 14 was chosen so that the smallest and largest of the 11 values differed from the recommended value by about twice its uncertainty. This reduced each ri to less than 1. To achieve this level of consistency forj thej 14 values now available would require an expansion factor of about 16. After due consideration the Task Group decided that it would be more appropriate to follow its FIG. 2. Comparison of input data B22.2 (HarvU-08) and B48 usual approach of treating inconsistent data, namely, to choose (LKB-11) through their inferred values of α. QED-10 and QED- an expansion factor that reduces each ri to less than 2. It 14 mean the QED theoretical expression for ae at the time of the j j 2010 and 2014 CODATA constants adjustments, and A e -10 concluded that the resulting uncertainty would better reflect rð Þ the current situation in light of the new low-uncertainty and Ar e -14 have the same meaning for Ar e . Both B22.2 and ð Þ ð Þ UCI-14 result with u 19 × 10−6, which agrees well with B48 have the same value in the 2010 and 2014 adjustments and r ¼ are essentially the sole determinants of the recommended value of the low-uncertainty UWash-00 and UZur-06 results with u r ¼ α in each. 14 × 10−6 and u 19 × 10−6, respectively. Thus based on an r ¼

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-47 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

TABLE XXI. Inferred values of the Planck constant h in order of increasing standard uncertainty obtained from the indicated experimental data in Table XVIII.

Relative standard Primary source Item number Identification Sec. and Eq. h= Js uncertainty u ð Þ r 2 −34 −8 KJ RK B44.6 NRC-15 VIII.E (237) 6.626 070 11 12 × 10 1.8 × 10 28 ð Þ −34 −8 NA Si B63.2 IAC-15 IX.B (240) 6.626 070 15 13 × 10 2.0 × 10 ð28 Þ ð Þ −34 −8 NA Si B63.1 IAC-11 IX.B (239) 6.626 069 89 20 × 10 3.0 × 10 2 ð Þ ð Þ −34 −8 KJ RK B44.4 NIST-15 VIII.D (234) 6.626 069 36 38 × 10 5.7 × 10 2 ð Þ −34 −8 KJ RK B44.2 NIST-98 VIII.D 6.626 068 91 58 × 10 8.7 × 10 2 ð Þ −34 −7 KJ RK B44.5 NPL-12 VIII.A 6.626 0712 13 × 10 2.0 × 10 2 ð Þ −34 −7 KJ RK B44.1 NPL-90 VIII.A 6.626 0682 13 × 10 2.0 × 10 2 ð Þ −34 −7 KJ RK B44.7 LNE-15 VIII.C (231) 6.626 0688 17 × 10 2.6 × 10 2 ð Þ −34 −7 KJ RK B44.3 METAS-11 VIII.B 6.626 0691 20 × 10 2.9 × 10 ð Þ −34 −7 KJ B42.1 NMI-89 VIII 6.626 0684 36 × 10 5.4 × 10 ð Þ −34 −7 KJ B42.2 PTB-91 VIII 6.626 0670 42 × 10 6.3 × 10 ð Þ −34 −6 Γp0 −90 hi B41.2 NPL-79 VIII 6.626 0730 67 × 10 1.0 × 10 ð Þ ð Þ −34 −6 F 90 B45 NIST-80 VIII 6.626 0658 88 × 10 1.3 × 10 ð Þ −34 −6 Γ0 hi B41.1 NIM-95 VIII 6.626 071 11 × 10 1.6 × 10 p−90ð Þ ð Þ

TABLE XXII. Inferred values of the Boltzmann constant k in order of increasing standard uncertainty obtained from the indicated experimental data in Table XVIII.

Relative standard Primary source Item number Identification Section k= JK−1 uncertainty u ð Þ r RB64.8 NPL-13 X.A 1.380 6476 12 × 10−23 9.0 × 10−7 RB64.9 LNE-15 X.A 1.380 6487ð14Þ × 10−23 1.0 × 10−6 RB64.6 LNE-11 X.A 1.380 6477ð19Þ × 10−23 1.4 × 10−6 RB64.2 NIST-88 X.A 1.380 6501ð25Þ × 10−23 1.8 × 10−6 RB64.3 LNE-09 X.A 1.380 6497ð38Þ × 10−23 2.7 × 10−6 RB64.4 NPL-10 X.A 1.380 6498ð44Þ × 10−23 3.2 × 10−6 RB64.7 NIM-13 X.A 1.380 6477ð51Þ × 10−23 3.7 × 10−6 k=h B65 NIM/NIST-15 X.B 1.380 6513ð53Þ × 10−23 3.9 × 10−6 ð Þ −23 −6 Aϵ=R B66 PTB-15 X.C 1.380 6509 55 × 10 4.0 × 10 RB64.5 INRIM-10 X.A 1.380 641 10ð Þ× 10−23 7.5 × 10−6 RB64.1 NPL-79 X.A 1.380 656ð12Þ × 10−23 8.4 × 10−6 ð Þ expansion factor of 6.3, the 2014 CODATA recommended 2. Data related to all other constants value is Tables XXVIII and XXIX summarize 12 least-squares analyses of the input data in Tables XVI and XVIII, including G 6.674 08 31 × 10−11 kg−1 m3 s−2 47 × 10−6 : 268 their correlation coefficients in Tables XVII and XIX; they are ¼ ð Þ ½ ð Þ discussed in the following paragraphs. Because the adjusted (Note that when the same expansion factor is applied to both value of R∞ is essentially the same for all five adjustments members of a correlated pair, its square is also applied to their summarized in Table XXVIII and equal to that of adjustment 3 covariance so their correlation coefficient is unchanged. When of Table XXIX, the values are not listed in Table XXVIII. the expansion factor is applied to one member of a correlated (Note that adjustment 3 in Tables XXVIII and XXIX is the pair, just the expansion factor is applied to the covariance.) For same adjustment.) 2 Adjustment 1. The initial adjustment includes all of the input this calculation χ 8.1, p 8.1 13 0.84, and RB 0.79. As might be expected,¼ JILA-10ð j andÞ¼ BIPM-14 still have¼ the data, four of which have values of ri that are problematically larger than 2. They are B2, the AME-12j j result for A 1H , B11 largest values of ri: 1.98 and 1.45, respectively. The other 12 r and B12, the FSU-15 results for ω HD =ω 3Heð Þ and values of ri are less than 0.01. c þ c þ j j ω HD =ω t , and B39.1, the NIST-89ð resultÞ forð Γ Þ lo . In this calculation Sc is less than 0.01 for five of the 14 input cð þÞ cð Þ p0 −90ð Þ data. If they are omitted, the value for G in Eq. (268) increases Their respective values of ri are 2.61, 4.06, 3.57, and 2.20. The by 2 in the last place, but its two-digit uncertainty is unchanged. NIST-89 datum was discussed above in connection with Although excluding such data is the Task Group’s usual inferred values of α and because it is of no real concern it is practice, it is not implemented in this case, because of the not discussed further. The other three residuals are due to the 12 6 significant disagreements among the data and the desirability inconsistency of B9, the UWash-15 result for ωc h =ωc C þ , −11 ð −Þ11 ð Þ of having the recommended value reflect all the data. and B11. Although ur is 1.4 × 10 and 4.8 × 10 for B9 and

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-48 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

TABLE XXIII. Observational equations that express the input data related to R∞ in Table XVI as functions of the adjusted constants in Table XXV. The numbers in the first column correspond to the numbers in the first column of Table XVI. Energy levels of hydrogenic atoms are discussed in Sec. IV.A. Note that EX nLj =h is proportional to cR∞ and independent of h, hence h is not an adjusted constant in these equations. See Sec. XIII.B for an explanationð of theÞ symbol . ≐ Type of input datum Observational equation

A1–A16 δH nLj δH nLj ð Þ ≐ ð Þ A17–A25 δD nLj δD nLj ð Þ ≐ ð Þ A26–A32, A39;A40 νH n1L1j − n2L2j EH n2L2j ; R ; α;Ar e ;Ar p ;rp; δH n2L2j ð 1 2 Þ ≐ ½ ð 2 ∞ ð Þ ð Þ ð 2 ÞÞ − EH n1L1j1 ; R∞; α;Ar e ;Ar p ;rp; δH n1L1j1 =h 1 ð ð Þ ð Þ ð ÞÞ A33–A38 νH n1L1j − n2L2j − νH n3L3 − n4L4j EH n2L2j ; R ; α;Ar e ;Ar p ;rp; δH n2L2j ð 1 2 Þ 4 ð j3 4 Þ ≐ f ð 2 ∞ ð Þ ð Þ ð 2 ÞÞ − E n L ; R ; α;A e ;A p ;r ; δ n L Hð 1 1j1 ∞ rð Þ rð Þ p Hð 1 1j1 ÞÞ − 1 E n L ; R ; α;A e ;A p ;r ; δ n L 4 ½ Hð 4 4j4 ∞ rð Þ rð Þ p Hð 4 4j4 ÞÞ − E n L ; R ; α;A e ;A p ;r ; δ n L =h Hð 3 3j3 ∞ rð Þ rð Þ p Hð 3 3j3 ÞÞg A41–A45 νD n1L1j − n2L2j ED n2L2j ; R ; α;Ar e ;Ar d ;rd; δD n2L2j ð 1 2 Þ ≐ ½ ð 2 ∞ ð Þ ð Þ ð 2 ÞÞ − ED n1L1j1 ; R∞; α;Ar e ;Ar d ;rd; δD n1L1j1 =h 1 ð ð Þ ð Þ ð ÞÞ A46–A47 νD n1L1j −n2L2j − νD n3L3 −n4L4j ED n2L2j ;R ;α;Ar e ;Ar d ;rd;δD n2L2j ð 1 2 Þ 4 ð j3 4 Þ ≐ f ð 2 ∞ ð Þ ð Þ ð 2 ÞÞ −E n L ;R ;α;A e ;A d ;r ;δ n L Dð 1 1j1 ∞ rð Þ rð Þ d Dð 1 1j1 ÞÞ − 1 E n L ;R ;α;A e ;A d ;r ;δ n L 4½ Dð 4 4j4 ∞ rð Þ rð Þ d Dð 4 4j4 ÞÞ −E n L ;R ;α;A e ;A d ;r ;δ n L =h Dð 3 3j3 ∞ rð Þ rð Þ d Dð 3 3j3 ÞÞg A48 νD 1S1=2 − 2S1=2 − νH 1S1=2 − 2S1=2 ED 2S1=2; R ; α;Ar e ;Ar d ;rd; δD 2S1=2 ð Þ ð Þ ≐ f ð ∞ ð Þ ð Þ ð ÞÞ − E 1S ; R ; α;A e ;A d ;r ; δ 1S Dð 1=2 ∞ rð Þ rð Þ d Dð 1=2ÞÞ − E 2S ; R ; α;A e ;A p ;r ; δ 2S ½ Hð 1=2 ∞ rð Þ rð Þ p Hð 1=2ÞÞ − E 1S ; R ; α;A e ;A p ;r ; δ 1S =h Hð 1=2 ∞ rð Þ rð Þ p Hð 1=2ÞÞg A49 rp rp ≐ A50 rd rd ≐

3 B11, respectively, which are quite small, their inconsistency weight, the value of ΔEB Hþ =hc, item B5, which is not 3 ð Þ becomes apparent by comparing the values of Ar He that they relevant to any other input datum, is also deleted and omitted as infer; the result is that the value from B11 exceedsð thatÞ from B9 an adjusted constant. The situation is exactly the same for 133 133 by 3.9udiff (Myers et al., 2015; Zafonte and Van Dyck, 2015). h=m Cs , item B46, and Ar Cs , item B47. This brings Because the reason for this discrepancy is unknown and both the totalð numberÞ of omitted datað to 24.Þ Table XXVIII shows that frequency ratios are credible, the Task Group decided to include deleting them has inconsequential impact on the values of α and both in the final adjustment with a sufficiently large expansion h. The data for the final adjustment are quite consistent, as 2 factor so that ri < 2 for both. It was also decided to include the demonstrated by the value of χ : p 42.4 54 0.87. 12 6 ð j Þ¼ companion ratios ωc d =ωc C þ (B8) and ωc HDþ =ωc t Adjustments 4 and 5. The purpose of these adjustments is to ð Þ −ð11 Þ −11 ð Þ ð Þ (B12) with ur 2.0 × 10 and 4.8 × 10 , respectively, test the robustness of the 2014 recommended values of α and h without an expansion¼ factor. by omitting the most accurate data that determine these Adjustment 2. This adjustment uses all the data with an constants. Adjustment 4 differs from adjustment 2 in that expansion factor of 2.8 applied to the uncertainties of data B9 the four data that provide values of α with the smallest and B11, resulting in ri 1.95 for B11 and ri 1.73 for B12. uncertainties are deleted, namely, items B22.2, B48, B22.1, This also results in the reduction¼ of r of B2 from¼ 2.61 to 0.49. and B46, which are the two values of a and the h=m X values i e ð Þ The complex relationships, apparent from their observational for 133Cs and 87Rb; see the first four entries of Table XX. [For equations, between input data B2, B8 and B9, and B11 and B12, the same reason as in adjustment 3, in adjustment 4 the value of 133 133 and the adjusted constants Ar p , Ar d , and Ar h , are respon- Ar Cs is also deleted as an input datum and Ar Cs as an sible for the somewhat surprisingð Þ effectð Þ of the expansionð Þ factor adjustedð Þ constant; the same applies to A 87Rb .] Adjustmentð Þ 5 rð Þ on ri (see Table XXIV). The expansion factor has no effect on differs from adjustment 2 in that the five data that provide the values of α and h, as can be seen from Table XXVIII. values of h with the smallest uncertainties are deleted, namely, Adjustment 3. Adjustment 3 is the adjustment on which the items B44.6, B63.2, B63.1, B44.4, and B44.2, which are three 2 2014 CODATA recommended values are based and as such is watt-balance values of KJ RK and two XRCD-enriched silicon called the “final adjustment.” It differs from adjustment 2 in values of NA; see the first five entries of Table XXI. The results that, following the prescription described above, 22 of the of these two adjustments are reasonable: Table XXVIII shows initial input data, all from Table XVIII, with values of Sc < that the value of α from the less accurate α-related data used in 0.01 in adjustment 2 are omitted. These are B4, B22.1, B39.1 to adjustment 4, and the value of h from the less accurate B44.1, B44.3, B44.5, B44.7 to B46, B64.1, and B64.5. (The h-related data used in adjustment 5, agree with the corre- range in values of Sc for the deleted data is 0.0000 to 0.0096, sponding recommended values from adjustment 3. and no datum with a value of Sc > 1 was “converted” to a value Adjustments 6 to 12. The purpose of the seven adjustments with Sc < 1 due to the expansion factor.) Further, because summarized in Table XXIX is to investigate the data that A 3H , item B4, is deleted as an input datum due to its low determine the recommended values of R , r , and r . Results rð Þ ∞ p d

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-49 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

TABLE XXIV. Observational equations that express the input data in Table XVIII as functions of the adjusted constants in Table XXVI. The numbers in the first column correspond to the numbers in the first column of Table XVIII. For simplicity, the lengthier functions are not explicitly given. See Sec. XIII.B for an explanation of the symbol . ≐ Type of input datum Observational equation Sec.

B1 Ar n Ar n III.A ð1 Þ ≐ ð Þ 1 2 B2 Ar H Ar p Ar e − ΔEB Hþ α Ar e =2R∞hc III.B ð 1Þ ≐ ð Þþ ð Þ1 ð Þ ð Þ B3 ΔEB Hþ =hc ΔEB Hþ =hc III.B 3 ð Þ ≐ ð Þ 3 2 B4 Ar H Ar t Ar e − ΔEB Hþ α Ar e =2R∞hc III.B ð 3Þ ≐ ð Þþ ð Þ3 ð Þ ð Þ B5 ΔEB Hþ =hc ΔEB Hþ =hc III.B 4 ð Þ ≐ ð Þ 4 2 2 B6 Ar He Ar α 2Ar e − ΔEB He þ α Ar e =2R∞hc III.B ð 4 Þ ≐2 ð Þþ ð Þ4 2 ð Þ ð Þ B7 ΔEB He þ =hc ΔEB He þ =hc III.B ð Þ ≐ ð Þ 12 6 2 ωc d 12 − 6Ar e ΔEB C þ α Ar e =2R∞hc B8 12ð 6Þ ð Þþ ð Þ ð Þ III.C ωc C þ ≐ 6Ar d ð Þ 12ð 6Þ 2 ωc h 12 − 6Ar e ΔEB C þ α Ar e =2R∞hc B9 12ð 6Þ ð Þþ ð Þ ð Þ III.C ωc C þ ≐ 3Ar h ð 12 6Þ 12 6 ð Þ B10 ΔEB C þ =hc ΔEB C þ =hc III.B ð Þ ≐ ð Þ 3 2 ωc HDþ Ar h Ar e − EI Heþ α Ar e =2R∞hc B11 ð3 Þ ð Þþ ð Þ ð Þ ð2 Þ III.C ω Heþ ≐ A p A d A e − E HDþ α A e =2R hc cð Þ rð Þþ rð Þþ rð Þ Ið Þ rð Þ ∞ ωc HDþ Ar t B12 ð Þ ð Þ 2 III.C ωc t ≐ Ar p Ar d Ar e − EI HDþ α Ar e =2R∞hc 3 ð Þ ð Þþ3 ð Þþ ð Þ ð Þ ð Þ B13 EI Heþ =hc EI Heþ =hc III.B ð Þ ≐ ð Þ B14 EI HDþ =hc EI HDþ =hc III.B ð12 5 Þ ≐ ð Þ ωs C þ gC α δC 12 5 2 B15 ð12 5 Þ − ð Þþ 12 − 5Ar e ΔEB C þ α Ar e =2R∞hc V.D.2 ωc C þ ≐ 10Ar e ½ ð Þþ ð Þ ð Þ ð 12 5Þ ð Þ12 5 B16 ΔEB C þ =hc ΔEB C þ =hc III.B ð Þ ≐ ð Þ B17 δC δC V.D.1 ≐28 13 ωs Si þ gSi α δSi 28 13 B18 ð28 13 Þ − ð Þþ Ar Si þ V.D.2 ωc Si þ ≐ 26Ar e ð Þ ð28 Þ 28 13 ð Þ 28 13 2 B19 Ar Si Ar Si þ 13Ar e − ΔEB Si þ α Ar e =2R∞hc III.B ð 28Þ ≐13 ð Þþ 28 13ð Þ ð Þ ð Þ B20 ΔEB Si þ =hc ΔEB Si þ =hc III.B ð Þ ≐ ð Þ B21 δSi δSi V.D.1 ≐ B22 ae ae α δe V.A.1 ≐ ð Þþ B23 δe δe V.A.1 ≐ aμ me μe B24 R − V.B.2 ≐ 1 a α δ m μ þ eð Þþ e μ p me μe B25;B26 ν fp ν fp; R ; α; ;a ; ; δe; δMu VI.B.2 ð Þ ≐ ∞ m μ μ μ p me B27 ΔνMu ΔνMu R ; α; ;a δMu VI.B.1 ≐ ∞ m μ þ μ B28 δMu δMu VI.B.1 ≐ μp Ar p μp B29 − 1 ae α δe ð Þ V.C μN ≐ ð þ ð Þþ Þ Ar e μe −1ð Þ μ H g H gp H μ B30 eð Þ eð Þ ð Þ e VI.A.1 μ H ≐ g g μ pð Þ e p p μ D g D g D −1 μ B31 dð Þ dð Þ eð Þ d VI.A.1 μ D ≐ g g μ eð Þ d e e μ H g H μ B32 eð Þ eð Þ e VI.A.1 μp0 ≐ ge μp0 μ μ B33 h0 h0 VI.A.1 μp0 ≐ μp0 μn μn B34 VI.A.1 μp0 ≐ μp0

μp HD μp μe B35 ð Þ 1 σdp VI.A.1 μ HD ≐ ½ þ μ μ dð Þ e d (Table continued)

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-50 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

TABLE XXIV. (Continued)

Type of input datum Observational equation Sec.

μt HT μt B36 ð Þ 1 − σtp VI.A.1 μ HT ≐ ½ μ pð Þ p B37 σdp σdp VI.A.1 ≐ B38 σtp σtp VI.A.1 ≐ 3 −1 KJ−90RK−90 1 ae α δe α μe B39 Γ0 lo − ½ þ ð Þþ VIII p−90ð Þ ≐ 2μ R μ 0 ∞ p0 3 −1 KJ−90RK−90 1 ae α δe α μe μh0 B40 Γ0 lo ½ þ ð Þþ VIII h−90ð Þ ≐ 2μ R μ μ 0 ∞ p0 p0 2 −1 c 1 ae α δe α μe B41 Γ0 hi − ½ þ ð Þþ VIII p−90ð Þ ≐ K R R h μ J−90 K−90 ∞ p0 8α 1=2 B42 KJ VIII ≐ μ ch 0 μ0c B43 RK VIII ≐ 2α 2 4 B44 K RK VIII J ≐ h 2 cMuAr e α B45 F 90 ð Þ VIII ≐ KJ−90RK−90R∞h h A e cα2 B46;B48 rð Þ VII m X ≐ A X 2R ð Þ rð Þ ∞ B47;B49 Ar X Ar X III.A ð Þ ≐ ð Þ d220 X d220 X B50–B59 ð Þ − 1 ð Þ − 1 IX.A d Y ≐ d Y 220ð Þ 220ð Þ B60–B62 d220 X d220 X IX.A ð Þ ≐ ð 2Þ cMuAr e α B63 NA ð Þ IX.B ≐ 2R∞h B64 R R X.A ≐ k 2R∞R B65 2 X.B h ≐ cMuAr e α 4 ð Þ 5 Aϵ α0 He cMuAr e α B66 ð 3Þ 2 ð Þ4 X.C R ≐ 4πϵ0a0 96π RhR∞ 4 4 α0 He α0 He B67 ð 3Þ ð 3Þ X.C 4πϵ0a0 ≐ 4πϵ0a0 λ CuKα1 1537.400xu CuKα1 B68;B69 ð Þ ð Þ IX.A d X ≐ d X 220ð Þ 220ð Þ λ WKα1 0.209 010 0 ÅÃ B70 ð Þ IX.A d N ≐ d N 220ð Þ 220ð Þ λ MoKα1 707.831xu MoKα1 B71 ð Þ ð Þ IX.A d N ≐ d N 220ð Þ 220ð Þ

from adjustment 3, the final adjustment, are included in the rp 0.8759 77 fm; 269 table for reference purposes. We begin with a discussion of ¼ ð Þ ð Þ adjustments 6 to 10, which are derived from adjustment 3 by r 2.1416 31 fm: 270 d ¼ ð Þ ð Þ deleting selected input data. We then discuss adjustments 11 and 12, which examine the impact of the value of the proton It is evident from a comparison of the results of this adjust- rms charge radius derived from the measurement of the Lamb ment and adjustment 3 that the scattering values of the radii shift in muonic hydrogen discussed in Sec. IV.A.3.c and given play a comparatively small role in determining the 2014 recommended values of R , rp and rd. in Eq. (78). Note that the value of R∞ depends only weakly on ∞ the data in Table XVIII. Adjustment 7 is based on only hydrogen data, including the scattering values of r but not the difference between the In adjustment 6, the electron scattering values of rp and rd, p data items A49 and A50 in Table XVI, are deleted from 1S1=2 2S1=2 transition frequencies in H and D, item A48 in − adjustment 3. Thus, the values of these two quantities from Table XVI, known as the isotope shift. Adjustment 8 differs adjustment 6 are based solely on H and D spectroscopic data from adjustment 7 in that the scattering value of rp is deleted. and are called the spectroscopic values of rp and rd: Adjustments 9 and 10 are similar to 7 and 8 but are based on

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-51 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

TABLE XXV. The 28 adjusted constants (variables) used in the least-squares multivariate analysis of the Rydberg-constant data given in Table XVI. These adjusted constants appear as arguments of the functions on the right-hand side of the observational equations of Table XXIII.

Adjusted constant Symbol

Rydberg constant R∞ Bound-state proton rms charge radius rp Bound-state deuteron rms charge radius rd Additive correction to EH 1S1=2 =h δH 1S1=2 Additive correction to E ð2S Þ=h δ ð2S Þ Hð 1=2Þ Hð 1=2Þ Additive correction to EH 3S1=2 =h δH 3S1=2 Additive correction to E ð4S Þ=h δ ð4S Þ Hð 1=2Þ Hð 1=2Þ Additive correction to EH 6S1=2 =h δH 6S1=2 Additive correction to E ð8S Þ=h δ ð8S Þ Hð 1=2Þ Hð 1=2Þ Additive correction to EH 2P1=2 =h δH 2P1=2 Additive correction to E ð4P Þ=h δ ð4P Þ Hð 1=2Þ Hð 1=2Þ Additive correction to EH 2P3=2 =h δH 2P3=2 Additive correction to E ð4P Þ=h δ ð4P Þ H 3=2 H 3=2 −7 ð Þ ð Þ FIG. 4. Values of the Planck constant h with ur < 10 inferred Additive correction to EH 8D3=2 =h δH 8D3=2 Additive correction to E ð12D Þ =h δ ð12D Þ from the input data in Table XVIII and the 2014 CODATA Hð 3=2Þ Hð 3=2Þ recommended value in chronological order from top to bottom Additive correction to EH 4D5=2 =h δH 4D5=2 Additive correction to E ð6D Þ=h δ ð6D Þ (see Table XXI). The input data from which these values are Hð 5=2Þ Hð 5=2Þ inferred are included in the final adjustment on which the 2014 Additive correction to EH 8D5=2 =h δH 8D5=2 Additive correction to E ð12D Þ =h δ ð12D Þ recommended values are based. Hð 5=2Þ Hð 5=2Þ Additive correction to ED 1S1=2 =h δD 1S1=2 Additive correction to E ð2S Þ=h δ ð2S Þ Dð 1=2Þ Dð 1=2Þ scattering results is further demonstrated by the comparatively Additive correction to E 4S =h δ 4S 2 Dð 1=2Þ Dð 1=2Þ low probability of χ for adjustment 11: p 72.8 55 0.0054. Additive correction to ED 8S1=2 =h δD 8S1=2 ð j Þ¼ ð Þ ð Þ The 2014 recommended value of rp and the purely spectro- Additive correction to ED 8D3=2 =h δD 8D3=2 Additive correction to E ð12D Þ =h δ ð12D Þ scopic value, which is that from adjustment 6, exceed the Dð 3=2Þ Dð 3=2Þ Additive correction to E 4D =h δ 4D muonic hydrogen value by 5.6udiff and 4.5udiff , respectively. Dð 5=2Þ Dð 5=2Þ Additive correction to E 8D =h δ 8D The impact of the muonic hydrogen value of rp can also be Dð 5=2Þ Dð 5=2Þ Additive correction to ED 12D5=2 =h δD 12D5=2 seen by examining for adjustments 3, 11, and 12 the ð Þ ð Þ normalized residuals and self-sensitivity coefficients of the principal experimental data that determine R∞, namely, items only deuterium data; that is, adjustment 9 includes the A26.1 to A50 in Table XVI. In brief, ri for these data in the final adjustment range from near 0 to 1.13j j for item A50, the r scattering value of rd but not the isotope shift, while for d adjustment 10 the scattering value is deleted. The results of scattering result, with the vast majority being less than 1. For these four adjustments show the dominant role of the hydro- the three greater than 1, ri is 1.03, 1.02, and 1.02. The value j j gen data and the importance of the isotope shift in determining of Sc is 1.00 for items A26.1 and A26.2 together, which are the two hydrogen 1S 2S transition frequencies; it is also the recommended value of rd. Further, the four values of R∞ 1=2 − 1=2 from these adjustments agree with the 2014 recommended 1.00 for A48, the H-D isotope shift. For item A49, the value, and the two values of rp and of rd also agree with their scattering value of rp, it is 0.31. Most others are a few respective recommended values: the largest difference from percent, although some values of Sc are near 0. the recommended value for the eight results is 1.4udiff . The situation is markedly different for adjustment 12. First, Adjustment 11 differs from adjustment 3 in that it includes r for item A30, the hydrogen transition frequency involving j ij the muonic hydrogen value rp 0.840 87 39 fm, and adjust- the 8D5=2 state, is 3.12 compared to 1.03 in adjustment 3; and ment 12 differs from adjustment¼ 11 in thatð theÞ two scattering items A41, A42, and A43, deuterium transitions involving the values of the nuclear radii are deleted. Because the muonic 8S1=2, 8D3=2, and 8D5=2 states, are now 2.54, 2.47, and 3.12, hydrogen value is significantly smaller and has a significantly respectively, compared to 0.56, 0.34, and 0.86. Further, ten smaller uncertainty than the purely spectroscopic value of other transitions have residuals in the range 1.04 to 1.80. As a adjustment 6 as well as the scattering value, it has a major result, with this proton radius, the predictions of the theory for impact on the results of adjustments 11 and 12, as can be seen hydrogen and deuterium transition frequencies are not gen- from Table XXIX: for both adjustments the value of R∞ shifts erally consistent with the experiments. Equally noteworthy is down by over 5 standard deviations and its uncertainty is the fact that, although Sc for items A26.1 and A26.2 together reduced by a factor grater than 6. Moreover, and not surpris- and A48 remain equal to 1.00, for all other transition frequen- ingly, the values of rp and of rd from both adjustments are cies Sc is less than 0.01, which means that they play an significantly smaller than the recommended values and have inconsequential role in determining R∞. The results for adjust- significantly smaller uncertainties. The inconsistency between ment 11, which includes the scattering values of the nuclear the muonic hydrogen result for rp and the spectroscopic and radii as well as the muonic hydrogen value, are similar.

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TABLE XXVI. 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 XXIV.

Adjusted constant Symbol

Neutron relative atomic mass Ar n Electron relative atomic mass A ðeÞ rð Þ Proton relative atomic mass Ar p 1 ð Þ1 Hþ electron removal energy ΔEB Hþ ð Þ Triton relative atomic mass Ar t 3 ð Þ 3 Hþ electron removal energy ΔEB Hþ ð Þ Alpha particle relative atomic mass Ar α 4 2 ð Þ4 2 He þ electron removal energy ΔEB He þ Deuteron relative atomic mass A dð Þ rð Þ Helion relative atomic mass Ar h 12 6 ð Þ12 6 C þ electron removal energy ΔEB C þ 3 ð3 Þ Heþ electron ionization energy ΔEI Heþ ð Þ HDþ electron ionization energy ΔEI HDþ 12 5 ð 12 5Þ C þ electron removal energy ΔEB C þ ð Þ Additive correction to gC α δC 28 13 ð Þ 28 13 Si þ relative atomic mass Ar Si þ FIG. 5. Values of the Boltzmann constant k inferred from the input 28 13 ð 28 13Þ Si þ electron removal energy ΔEB Si þ data in Table XVIII and the 2010 and 2014 CODATA recom- Additive correction to g α δ ð Þ mended values in chronological order from top to bottom (see Sið Þ Si Fine-structure constant α Table XXII). AGT: acoustic gas thermometry; DCGT: dielectric- Additive correction to a th δ eð Þ e constant gas thermometry; JNT: Johnson noise thermometry. Muon magnetic-moment anomaly aμ Electron-muon mass ratio m =m e μ 3. Test of the Josephson and quantum-Hall-effect relations Electron-proton magnetic-moment ratio μe=μp Additive correction to ΔνMu th δMu As in the three previous CODATA adjustments, the exact- ð Þ 2 Deuteron-electron magnetic-moment ratio μd=μe ness of the relations K 2e=h and R h=e is investigated J ¼ K ¼ Shielded helion to shielded proton μh0 =μp0 by writing magnetic-moment ratio Neutron to shielded proton magnetic-moment ratio μ =μ n p0 2e 8α 1=2 Shielding difference of d and p in HD σdp K 1 ϵ 1 ϵ ; 271 J ¼ h ð þ JÞ¼ μ ch ð þ JÞ ð Þ Triton-proton magnetic-moment ratio μt=μp 0 Shielding difference of t and p in HT σtp h μ c Electron to shielded proton magnetic-moment ratio μe=μp0 R 1 ϵ 0 1 ϵ ; 272 Planck constant h K ¼ e2 ð þ KÞ¼ 2α ð þ KÞ ð Þ 133 133 Cs relative atomic mass Ar Cs 87Rb relative atomic mass A ð87Rb Þ where ϵ and ϵ are unknown correction factors taken rð Þ J K d220 of Si crystal WASO 17 d220 W17 to be additional adjusted constants. Replacing the relations ð Þ d220 of Si crystal ILL d220 ILL 2 ð Þ KJ 2e=h and RK h=e in the analysis leading to the d220 of Si crystal MOÃ d220 MOÃ ¼ ¼ ð Þ d220 of Si crystal NR3 d220 NR3 d of Si crystal N d ðN Þ TABLE XXVII. Summary of values of G used to determine the 220 220ð Þ d220 of Si crystal WASO 4.2a d220 W4.2a 2014 recommended value (see also Table XV, Sec. XI). d of Si crystal WASO 04 d ðW04 Þ 220 220ð Þ Relative d220 of an ideal Si crystal d220 a d of Si crystal NR4 d NR4 Item Value standard 220 220 −11 3 −1 −2 Molar gas constant R ð Þ number (10 m kg s ) uncertainty ur Identification 4 5 Static electric dipole polarizability α0Ã He G1 6.672 48(43) 6.4 × 10− NIST-82 4 ð Þ of He in atomic units G2 6.672 9(5) 7.5 × 10−5 TR&D-96 Copper Kα1 x unit xu CuKα1 G3 6.673 98(70) 1.0 × 10−4 LANL-97 Ångstrom star ð Þ ÅÃ G4 6.674 255(92) 1.4 × 10−5 UWash-00 Molybdenum Kα1 x unit xu MoKα1 −5 ð Þ G5 6.675 59(27) 4.0 × 10 BIPM-01 G6 6.674 22(98) 1.5 × 10−4 UWup-02 G7 6.673 87(27) 4.0 × 10−5 MSL-03 Because of the impact of the latter value on the internal G8 6.672 22(87) 1.3 × 10−4 HUST-05 consistency of the R∞ data and its continued disagreement G9 6.674 25(12) 1.9 × 10−5 UZur-06 with the spectroscopic and scattering values, the Task Group G10 6.673 49(18) 2.7 × 10−5 HUST-09 decided, as it did for the 2010 adjustment, that it was G11 6.672 34(14) 2.1 × 10−5 JILA-10 premature to include it as an input datum in the 2014 final G12 6.675 54(16) 2.4 × 10−5 BIPM-14 adjustment; it was deemed more prudent to continue to wait G13 6.671 91(99) 1.5 × 10−4 LENS-14 and see if further research can resolve what has come to be G14 6.674 35(13) 1.9 × 10−5 UCI-14 called the “proton radius puzzle”; see Sec. IV.A.3.c for aCorrelation coefficients: r G1;G3 0.351; r G8;G10 additional discussion. 0.134. ð Þ¼ ð Þ¼

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not included in the tests of the relations KJ 2e=h and 2 ¼ RK h=e , because of its comparatively large uncertainty. The¼ results of six different adjustments are summarized in Table XXXI. An entry of 0 in the ϵK column means that it is assumed R h=e2 in the corresponding adjustment; sim- K ¼ ilarly, an entry of 0 in the ϵJ column means that it is assumed KJ 2e=h in the corresponding adjustment. The following remarks¼ apply to the six adjustments. Adjustment (i) differs from adjustment 2 summarized in Table XXVIII only in that the assumption KJ 2e=h and R h=e2 is relaxed. For this adjustment, N 153¼ , M 77, K ¼ ¼ ¼ ν N − M 76, χ2 64.1, p 64.1 76 0.83, and R 0.92. ¼ ¼ ¼ ð j Þ¼ B ¼ Examination of the table shows that ϵK is consistent with −8 0 within 1.2 times its uncertainty of 1.8 × 10 , while ϵJ is consistent with 0 within its uncertainty of 1.5 × 10−8. Adjustments (ii) and (iii) focus on ϵK; ϵJ is set equal to 0 and values of ϵK are calculated from data whose observational equations are independent of h. Adjustment (ii) uses the five FIG. 6. Values of the Newtonian constant of gravitation G in results for R , items B43.1 to B43.5, and (iii) uses the three low- Table XXVII and the 2010 and 2014 CODATA recommended K values in chronological order from top to bottom. field gyromagnetic ratio results, items B39.1;B39.2, and B40 [the three together are denoted by Γ lo ]. We see from p0 ;h−90ð Þ Table XXXI that the values of ϵK resulting from the two observational equations in Table XXIV with the generaliza- adjustments not only have opposite signs but disagree signifi- tions in Eqs. (271) and (272) leads to the modified observa- cantly: their difference is 3.0udiff. Their disagreement reflects tional equations given in Table XXX. the fact that while the five inferred values of α from RK are Although the NIM/NIST-15 result for k=h, item B65, was consistent among themselves and with the highly accurate value obtained using the Josephson and quantum-Hall effects, it is from a , the inferred value from the NIST-89 result for Γ lo e p0 −90ð Þ

TABLE XXVIII. Summary of the results of some of the least-squares adjustments used to analyze the input data given in Tables XVI, XVII, XVIII, and XIX. The values of α and h are those obtained in the adjustment, N is the number of input data, M is the number of adjusted 2 constants, ν N − M is the degrees of freedom, and RB χ =ν is the Birge ratio. See the text for an explanation and discussion of each adjustment, but¼ in brief, adjustment 1 is all the data; 2 is the¼ same as 1 except with the uncertainties of the two cyclotron frequency ratios related to the helion multiplied by 2.8; 3 is 2 with the low-weight inputpffiffiffiffiffiffiffiffiffi data deleted and is the adjustment on which the 2014 recommended values are based; 4 is 2 with the input data that provide the most accurate values of α deleted; and 5 is 1 with the input data that provide the most accurate values of h deleted as well as the low-weight data for α.

2 −1 −1 Adj. NM ν χ RB α u α h= Js ur h rð Þ ð Þ ð Þ 1 151 75 76 85.6 1.06 137.035 999 136(31) 2.3 × 10−10 6.626 070 031 81 × 10−34 1.2 × 10−8 2 151 75 76 65.5 0.93 137.035 999 136(31) 2.3 × 10−10 6.626 070 031ð81Þ × 10−34 1.2 × 10−8 3 127 73 54 42.4 0.89 137.035 999 139(31) 2.3 × 10−10 6.626 070 040ð81Þ × 10−34 1.2 × 10−8 4 145 73 71 58.4 0.90 137.035 9997(13) 9.5 × 10−9 6.626 070 00 10ð Þ× 10−34 1.5 × 10−8 5 135 74 61 41.0 0.82 137.035 999 138(31) 2.3 × 10−10 6.626 069 39ð72Þ × 10−34 1.1 × 10−7 ð Þ

TABLE XXIX. Summary of the results of some of the least-squares adjustments used to analyze the input data related to R∞. The values of R∞, rp, and rd are those obtained in the indicated adjustment, N is the number of input data, M is the number of adjusted constants, ν N − M 2 ¼ is the degrees of freedom, and RB χ =ν is the Birge ratio. See the text for an explanation and discussion of each adjustment, but in brief, adjustment 6 is 3, but the scattering¼ data for the nuclear radii are omitted; 7 is 3, but with only the hydrogen data included (but not the isotope pffiffiffiffiffiffiffiffiffi shift); 8 is 7 with the rp datum deleted; 9 and 10 are similar to 7 and 8, but for the deuterium data; 11 is 3 with the muonic Lamb-shift value of rp included; and 12 is 11, but without the scattering values of rp and rd.

Adj. NM ν χ2 R R =m−1 u R r =fm r =fm B ∞ rð ∞Þ p d 3 127 73 54 42.4 0.89 10 973 731.568 508(65) 5.9 × 10−12 0.8751(61) 2.1413(25) 6 125 73 52 41.0 0.89 10 973 731.568 517(82) 7.4 × 10−12 0.8759(77) 2.1416(31) 7 109 63 46 38.0 0.91 10 973 731.568 533(74) 6.7 × 10−12 0.8774(69) 8 108 63 45 38.0 0.92 10 973 731.568 523(94) 8.6 × 10−12 0.8764(89) 9 92 56 36 28.0 0.88 10 973 731.568 36(13) 1.1 × 10−11 2.1287(93) 10 91 56 35 28.0 0.89 10 973 731.568 27(30) 2.7 × 10−11 2.121(25) 11 128 73 55 72.8 1.15 10 973 731.568 157(10) 9.4 × 10−13 0.841 00(39) 2.127 65(18) 12 126 73 53 60.8 1.07 10 973 731.568 157(10) 9.4 × 10−13 0.840 95(39) 2.127 63(18)

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TABLE XXX. Generalized observational equations that express because of the disagreement between the NIST-07 watt- input data B32 − B38 in Table XVIII as functions of the adjusted balance measurement of h and the IAC-11 XRCD enriched constants in Tables XXV and XXVI with the additional adjusted silicon measurement of NA. With the resolution of the constants ϵJ and ϵK as given in Eqs. (271) and (272). The numbers in disagreement by the replacement of the earlier NIST result the first column correspond to the numbers in the first column of Table XVIII. For simplicity, the lengthier functions are not explicitly with NIST-15 and the agreement of its inferred value of h with given. See Sec. XIII.B for an explanation of the symbol . other values (see Table XXI, Sec. XIII.A), the only remaining ≐ issue is the values of ϵ and ϵ from Γ lo . However, K J p0 ;h−90ð Þ Type of input these three data are dominated by the NIST-89 Γ lo result datum Generalized observational equation p0 −90ð Þ and as discussed in Sec. XIII.A, the inconsistency of this 3 −1 B39Ã KJ−90RK−90 1 ae α;δe α μe− Γ0 lo − ½ þ ð Þ datum has been of concern in past adjustments and because of p−90ð Þ≐ 2μ R 1 ϵ 1 ϵ μ 0 ∞ð þ JÞð þ KÞ p0 its low weight is not included in the 2006 and 2010 final 3 −1 B40Ã KJ−90RK−90 1 ae α;δe α μe− μh0 adjustments, or in the 2014 final adjustment. It can thus be Γh0 −90 lo ½ þ ð Þ ð Þ≐ 2μ0R 1 ϵJ 1 ϵK μp0 μp0 concluded that the current data show the Josephson and ∞ð þ Þð þ Þ 2 −1 8 B41Ã c 1 ae α;δe α μe− quantum-Hall-effect relations to be exact within 2 parts in 10 . Γp0 −90 hi − ½ þ ð Þ 1 ϵJ 1 ϵK ð Þ≐ KJ−90RK−90R∞h ð þ Þð þ Þ μp0 One way to test the universality of the Josephson and μ c B43Ã 0 quantum-Hall-effect relations is to investigate their material RK 1 ϵK ≐ 2α ð þ Þ dependence. Recently, Ribeiro-Palau et al. (2015) obtained 1=2 B42Ã 8α agreement between the quantized Hall resistance in a graphene KJ 1 ϵJ ≐ μ ch ð þ Þ 0 (two-dimensional graphite) device and a GaAs=AlGaAs hetero- 11 B44Ã 2 4 2 structure device well within the 8.2 parts in 10 uncertainty of K RK 1 ϵJ 1 ϵK J ≐ h ð þ Þ ð þ Þ their measurement. This is slightly better than the previous best 2 B45Ã cMuAr e α graphene-GaAs=AlGaAs comparison, which obtained agree- F 90 ð Þ 1 ϵJ 1 ϵK ≐ KJ−90RK−90R∞h ð þ Þð þ Þ ment within the 8.7 parts in 1011 uncertainty of the experiment (Janssen et al.,2012). Another way is to “close the metrology triangle” by using a single electron tunneling (SET) device that (item B39.1), which has the smallest uncertainty of the three generates a quantized current I ef when an alternating voltage low-field gyromagnetic ratios, is not (see Table XX, Sec. XIII.A) of frequency f is applied to it. The¼ current I is then compared to a Adjustments (iv) to (vi) focus on ϵJ; ϵK is set equal to current obtained from Josephson and quantum-Hall-effect 0 and values of ϵJ are calculated from data whose observational devices. The status of such efforts was briefly discussed in equations, with the exception of adjustment (iv), are dependent the same section of CODATA-10 and little has changed since. on h. Because ϵJ and ϵK enter the observational equations Two relevant papers not referenced in CODATA-10 are by for the gyromagnetic ratios in the symmetric form Devoille et al. (2012) and Scherer and Camarota (2012). 1 ϵ 1 ϵ , the numerical result from adjustments (iii) ð þ JÞð þ KÞ and (iv) are identical. Although ϵJ from adjustment (iv) has the XIV. THE 2014 CODATA RECOMMENDED VALUES opposite sign of ϵJ from adjustments (v) and (vi), it agrees with ϵ from adjustment (v) because of that result’s extremely large J A. Calculational details uncertainty. However, it does disagree with the adjustment (vi) result: their difference is 2.8udiff . The 151 input data and their many correlation coefficients In summary, we recall that rather limited conclusions could initially considered for inclusion in the 2014 CODATA adjust- be drawn from the similar analysis presented in CODATA-10, ment of the values of the constants are given in Tables XVI, XVII, XVIII, and XIX. The 2014 recommended values are TABLE XXXI. Summary of the results of several least-squares based on adjustment 3, the final adjustment, summarized in adjustments to investigate the relations KJ 2e=h 1 ϵJ and Table XXVIII and discussed in the associated text. Adjustment R h=e2 1 ϵ . See the text for an explanation¼ð andÞð discussionþ Þ 3 omits 22 of the 151 initially considered input data, namely, K ¼ð Þð þ KÞ of each adjustment, but in brief, adjustment (i) uses all the data, items B4, B22.1, B39.1 to B44.1, B44.3, B44.5, B44.7 to B46, (ii) assumes K 2e=h (that is, ϵ 0) and obtains ϵ from the five J ¼ J ¼ K B64.1, and B64.5, because of their low weight (self-sensitivity measured values of RK, (iii) is based on the same assumption and obtains ϵ from the two values of the proton gyromagnetic ratio and coefficient Sc less than 0.01). However, because the observa- K 3 3 one value of the helion gyromagnetic ratio, (iv) is (iii) but assumes tional equation for Ar H , item B4, depends on ΔEB Hþ =hc 2 ð Þ ð Þ RK h=e (that is, ϵK 0) and obtains ϵJ in place of ϵK, (v) and (vi) and item B4 is deleted due to its low weight, the value of ¼ ¼ 3 are based on the same assumption and obtain ϵJ from all the measured ΔEB Hþ =hc, item B5, is also deleted as an adjusted constant. values given in Table XVIII for the quantities indicated. Theð sameÞ statement applies to h=m 133Cs , item B46, and 133 ð Þ 8 8 A Cs , item B47. Further, the initial uncertainties of two Adj. Data included 10 ϵK 10 ϵJ r inputð data,Þ items B11 and B12, are multiplied by the expansion (i) All 2.2(1.8) −0.9 1.5 (ii) R 2.8(1.8) 0ð Þ factor 2.8. As a consequence, the normalized residual ri of each K as well as that of item B2 is reduced to below 2. (iii) Γ0 lo −25.5 9.3 0 p;h−90ð Þ ð Þ Each input datum in this final adjustment has a self- (iv) Γ0 lo 0 −25.5 9.3 p;h−90ð Þ ð Þ (v) Γ hi , K , K2R , F 0 8.2(71.9) sensitivity coefficient Sc greater than 0.01, or is a subset of p0 −90ð Þ J J K 90 (vi) 2 0 0.7(1.2) the data of an experiment or series of experiments that provide Γp0 −90 hi , KJ, KJ RK, F 90, NA ð Þ an input datum or input data with Sc > 0.01. Not counting

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TABLE XXXII. An abbreviated list of the CODATA recommended values of the fundamental constants of physics and chemistry based on the 2014 adjustment.

Relative std. Quantity Symbol Numerical value Unit uncert. ur −1 Speed of light in vacuum c, c0 299 792 458 ms Exact −7 −2 Magnetic constant μ0 4π × 10 NA 12.566 370 614… × 10−7 NA−2 Exact 2 ¼ −12 −1 Electric constant 1=μ0c ϵ0 8.854 187 817… × 10 Fm Exact Newtonian constant of gravitation G 6.674 08 31 × 10−11 m3 kg−1 s−2 4.7 × 10−5 Planck constant h 6.626 070ð 040Þ 81 × 10−34 Js 1.2 × 10−8 ð Þ h=2π ℏ 1.054 571 800 13 × 10−34 Js 1.2 × 10−8 Elementary charge e 1.602 176 6208ð 98Þ × 10−19 C 6.1 × 10−9 ð Þ −15 −9 Magnetic flux quantum h=2e Φ0 2.067 833 831 13 × 10 Wb 6.1 × 10 2 ð Þ −5 −10 Conductance quantum 2e =h G0 7.748 091 7310 18 × 10 S 2.3 × 10 ð Þ −31 −8 Electron mass me 9.109 383 56 11 × 10 kg 1.2 × 10 ð Þ −27 −8 Proton mass mp 1.672 621 898 21 × 10 kg 1.2 × 10 ð Þ −11 Proton-electron mass ratio mp=me 1836.152 673 89(17) 9.5 × 10 2 −3 −10 Fine-structure constant e =4πϵ0ℏc α 7.297 352 5664 17 × 10 2.3 × 10 inverse fine-structure constant α−1 137.035 999 139(31)ð Þ 2.3 × 10−10 2 −1 −12 Rydberg constant α mec=2hR∞ 10 973 731.568 508(65) m 5.9 × 10 23 −1 −8 Avogadro constant NA, L 6.022 140 857 74 × 10 mol 1.2 × 10 ð Þ −1 −9 Faraday constant NAeF96 485.332 89(59) C mol 6.2 × 10 Molar gas constant R 8.314 4598(48) J mol−1 K−1 5.7 × 10−7 −23 −1 −7 Boltzmann constant R=NA k 1.380 648 52 79 × 10 JK 5.7 × 10 2 4 3 2 ð Þ Stefan-Boltzmann constant π =60 k =ℏ c σ 5.670 367 13 × 10−8 Wm−2 K−4 2.3 × 10−6 ð Þ ð Þ Non-SI units accepted for use with the SI Electron volt e=C J eV 1.602 176 6208 98 × 10−19 J 6.1 × 10−9 (Unified) atomicð massÞ unit 1 m 12C u 1.660 539 040 20ð Þ× 10−27 kg 1.2 × 10−8 12 ð Þ ð Þ

TABLE XXXIII. The CODATA recommended values of the fundamental constants of physics and chemistry based on the 2014 adjustment.

Relative std. Quantity Symbol Numerical value Unit uncert. ur UNIVERSAL −1 Speed of light in vacuum c; c0 299 792 458 ms Exact −7 −2 Magnetic constant μ0 4π × 10 NA 12.566 370 614… × 10−7 NA−2 Exact 2 ¼ −12 −1 Electric constant 1=μ0c ϵ0 8.854 187 817… × 10 Fm Exact Characteristic impedance of vacuum μ0cZ0 376.730 313 461… Ω Exact −11 3 −1 2 −5 Newtonian constant of gravitation G 6.674 08 31 × 10 m kg s− 4.7 × 10 G=ℏc 6.708 61ð31Þ × 10−39 GeV=c2 −2 4.7 × 10−5 Planck constant h 6.626 070ð 040Þ 81 × 10−34 Jsð Þ 1.2 × 10−8 4.135 667 662ð25Þ × 10−15 eV s 6.1 × 10−9 ð Þ h=2π ℏ 1.054 571 800 13 × 10−34 Js 1.2 × 10−8 6.582 119 514ð40Þ × 10−16 eV s 6.1 × 10−9 ℏc 197.326 9788(12)ð Þ MeV fm 6.1 × 10−9 1=2 −8 −5 Planck mass ℏc=G mP 2.176 470 51 × 10 kg 2.3 × 10 ð Þ 2 ð Þ 19 −5 energy equivalent mPc 1.220 910 29 × 10 GeV 2.3 × 10 5 1=2 ð Þ 32 −5 Planck temperature ℏc =G =k TP 1.416 808 33 × 10 K 2.3 × 10 ð Þ 3 1=2 ð Þ −35 −5 Planck length ℏ=mPc ℏG=c lP 1.616 229 38 × 10 m 2.3 × 10 ¼ð 5 1=2 Þ ð Þ −44 −5 Planck time lP=c ℏG=c tP 5.391 16 13 × 10 s 2.3 × 10 ¼ð Þ ð Þ ELECTROMAGNETIC Elementary charge e 1.602 176 6208 98 × 10−19 C 6.1 × 10−9 e=h 2.417 989 262 15ð Þ× 1014 AJ−1 6.1 × 10−9 ð Þ −15 −9 Magnetic flux quantum h=2e Φ0 2.067 833 831 13 × 10 Wb 6.1 × 10 2 ð Þ −5 −10 Conductance quantum 2e =h G0 7.748 091 7310 18 × 10 S 2.3 × 10 −1 ð Þ −10 inverse of conductance quantum G0 12 906.403 7278(29) Ω 2.3 × 10 a 9 −1 −9 Josephson constant 2e=h KJ 483 597.8525 30 × 10 Hz V 6.1 × 10 b 2 ð Þ −10 von Klitzing constant h=e μ0c=2α RK 25 812.807 4555(59) Ω 2.3 × 10 ¼ (Table continued)

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-56 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

TABLE XXXIII. (Continued)

Relative std. Quantity Symbol Numerical value Unit uncert. ur −26 −1 −9 Bohr magneton eℏ=2me μB 927.400 9994 57 × 10 JT 6.2 × 10 5.788 381 8012ð 26Þ × 10−5 eV T−1 4.5 × 10−10 ð Þ 9 −1 −9 μB=h 13.996 245 042 86 × 10 Hz T 6.2 × 10 ð Þ −1 −1 −9 μB=hc 46.686 448 14(29) m T 6.2 × 10 −1 −7 μB=k 0.671 714 05(39) KT 5.7 × 10 −27 −1 −9 Nuclear magneton eℏ=2mp μN 5.050 783 699 31 × 10 JT 6.2 × 10 3.152 451 2550ð 15Þ × 10−8 eV T−1 4.6 × 10−10 ð Þ −1 −9 μN=h 7.622 593 285(47) MHz T 6.2 × 10 −2 −1 −1 −9 μN=hc 2.542 623 432 16 × 10 m T 6.2 × 10 ð Þ −4 −1 −7 μN=k 3.658 2690 21 × 10 KT 5.7 × 10 ð Þ ATOMIC AND NUCLEAR General 2 −3 −10 Fine-structure constant e =4πϵ0ℏc α 7.297 352 5664 17 × 10 2.3 × 10 inverse fine-structure constant α−1 137.035 999 139(31)ð Þ 2.3 × 10−10 2 −1 −12 Rydberg constant α mec=2hR∞ 10 973 731.568 508(65) m 5.9 × 10 15 −12 R∞c 3.289 841 960 355 19 × 10 Hz 5.9 × 10 ð Þ −18 −8 R∞hc 2.179 872 325 27 × 10 J 1.2 × 10 13.605 693 009(84)ð Þ eV 6.1 × 10−9 2 2 −10 −10 Bohr radius α=4πR∞ 4πϵ0ℏ =mee a0 0.529 177 210 67 12 × 10 m 2.3 × 10 2 ¼ 2 2 ð Þ −18 −8 Hartree energy e =4πϵ0a0 2R∞hc α mec Eh 4.359 744 650 54 × 10 J 1.2 × 10 ¼ ¼ 27.211 386 02(17)ð Þ eV 6.1 × 10−9 −4 2 −1 −10 Quantum of circulation h=2me 3.636 947 5486 17 × 10 m s 4.5 × 10 ð Þ −4 2 −1 −10 h=me 7.273 895 0972 33 × 10 m s 4.5 × 10 ð Þ Electroweak c 3 −5 −2 −7 Fermi coupling constant GF= ℏc 1.166 3787 6 × 10 GeV 5.1 × 10 d ð Þ ð Þ Weak mixing angle θW (on-shell scheme) 2 2 2 2 −3 sin θW s 1 − mW=mZ sin θ 0.2223(21) 9.5 × 10 ¼ W ≡ ð Þ W Electron, e− −31 −8 Electron mass me 9.109 383 56 11 × 10 kg 1.2 × 10 5.485 799 090ð 70Þ 16 × 10−4 u 2.9 × 10−11 2 ð Þ −14 −8 energy equivalent mec 8.187 105 65 10 × 10 J 1.2 × 10 0.510 998 9461(31)ð Þ MeV 6.2 × 10−9 −3 −8 Electron-muon mass ratio me=mμ 4.836 331 70 11 × 10 2.2 × 10 ð Þ −4 −5 Electron-tau mass ratio me=mτ 2.875 92 26 × 10 9.0 × 10 ð Þ −4 −11 Electron-proton mass ratio me=mp 5.446 170 213 52 52 × 10 9.5 × 10 ð Þ −4 −10 Electron-neutron mass ratio me=mn 5.438 673 4428 27 × 10 4.9 × 10 ð Þ −4 −11 Electron-deuteron mass ratio me=md 2.724 437 107 484 96 × 10 3.5 × 10 ð Þ −4 −11 Electron-triton mass ratio me=mt 1.819 200 062 203 84 × 10 4.6 × 10 ð Þ −4 −11 Electron-helion mass ratio me=mh 1.819 543 074 854 88 × 10 4.9 × 10 ð Þ −4 −11 Electron to alpha particle mass ratio me=mα 1.370 933 554 798 45 × 10 3.3 × 10 ð Þ 11 −1 −9 Electron charge to mass quotient −e=me −1.758 820 024 11 × 10 C kg 6.2 × 10 ð Þ −7 −1 −11 Electron molar mass NAme M e ;Me 5.485 799 090 70 16 × 10 kg mol 2.9 × 10 ð Þ ð Þ −12 −10 Compton wavelength h=mec λC 2.426 310 2367 11 × 10 m 4.5 × 10 2 ð Þ −15 −10 λC=2π αa0 α =4πR∞ ƛC 386.159 267 64 18 × 10 m 4.5 × 10 ¼ ¼ 2 ð Þ −15 −10 Classical electron radius α a0 re 2.817 940 3227 19 × 10 m 6.8 × 10 2 ð Þ −28 2 −9 Thomson cross section 8π=3 re σe 0.665 245 871 58 91 × 10 m 1.4 × 10 ð Þ ð Þ −26 −1 −9 Electron magnetic moment μe −928.476 4620 57 × 10 JT 6.2 × 10 ð Þ −13 to Bohr magneton ratio μe=μB −1.001 159 652 180 91 26 2.6 × 10 ð Þ −11 to nuclear magneton ratio μe=μN −1838.281 972 34 17 9.5 × 10 Electron magnetic-moment ð Þ −3 −10 anomaly μe =μB − 1 ae 1.159 652 180 91 26 × 10 2.3 × 10 j j ð Þ −13 Electron g-factor −2 1 ae ge −2.002 319 304 361 82 52 2.6 × 10 ð þ Þ ð Þ −8 Electron-muon magnetic-moment ratio μe=μμ 206.766 9880(46) 2.2 × 10 −9 Electron-proton magnetic-moment ratio μe=μp −658.210 6866 20 3.0 × 10 Electron to shielded proton magnetic-moment ð Þ −8 ratio (H2O, sphere, 25 °C) μe=μp0 −658.227 5971 72 1.1 × 10 ð Þ −7 Electron-neutron magnetic-moment ratio μe=μn 960.920 50(23) 2.4 × 10 Electron-deuteron magnetic-moment ratio μ =μ −2143.923 499 12 5.5 × 10−9 e d ð Þ (Table continued)

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-57 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

TABLE XXXIII. (Continued)

Relative std. Quantity Symbol Numerical value Unit uncert. ur Electron to shielded helion magnetic-moment −8 ratio (gas, sphere, 25 °C) μe=μh0 864.058 257(10) 1.2 × 10 11 −1 −1 −9 Electron gyromagnetic ratio 2 μe =ℏ γe 1.760 859 644 11 × 10 s T 6.2 × 10 j j ð Þ −1 −9 γe=2π 28 024.951 64(17) MHz T 6.2 × 10 Muon, μ− −28 −8 Muon mass mμ 1.883 531 594 48 × 10 kg 2.5 × 10 0.113 428 9257(25)ð Þ u 2.2 × 10−8 energy equivalent m c2 1.692 833 774 43 × 10−11 J 2.5 × 10−8 μ ð Þ 105.658 3745(24) MeV 2.3 × 10−8 −8 Muon-electron mass ratio mμ=me 206.768 2826(46) 2.2 × 10 −2 −5 Muon-tau mass ratio mμ=mτ 5.946 49 54 × 10 9.0 × 10 ð Þ −8 Muon-proton mass ratio mμ=mp 0.112 609 5262(25) 2.2 × 10 −8 Muon-neutron mass ratio mμ=mn 0.112 454 5167(25) 2.2 × 10 −3 −1 −8 Muon molar mass NAmμ M μ ;Mμ 0.113 428 9257 25 × 10 kg mol 2.2 × 10 ð Þ ð Þ −15 −8 Muon Compton wavelength h=mμc λC;μ 11.734 441 11 26 × 10 m 2.2 × 10 ð Þ −15 −8 λC;μ=2π ƛC;μ 1.867 594 308 42 × 10 m 2.2 × 10 ð Þ −26 −1 −8 Muon magnetic moment μμ −4.490 448 26 10 × 10 JT 2.3 × 10 ð Þ −3 −8 to Bohr magneton ratio μμ=μB −4.841 970 48 11 × 10 2.2 × 10 ð Þ −8 to nuclear magneton ratio μμ=μN −8.890 597 05 20 2.2 × 10 Muon magnetic-moment anomaly ð Þ μ = eℏ=2m − 1 a 1.165 920 89 63 × 10−3 5.4 × 10−7 j μj ð μÞ μ ð Þ Muon g-factor −2 1 a g −2.002 331 8418 13 6.3 × 10−10 ð þ μÞ μ ð Þ Muon-proton magnetic-moment ratio μ =μ −3.183 345 142 71 2.2 × 10−8 μ p ð Þ Tau, τ− e −27 −5 Tau mass mτ 3.167 47 29 × 10 kg 9.0 × 10 1.907 49(17)ð Þ u 9.0 × 10−5 2 −10 −5 energy equivalent mτc 2.846 78 26 × 10 J 9.0 × 10 1776.82(16)ð Þ MeV 9.0 × 10−5 −5 Tau-electron mass ratio mτ=me 3477.15(31) 9.0 × 10 −5 Tau-muon mass ratio mτ=mμ 16.8167(15) 9.0 × 10 −5 Tau-proton mass ratio mτ=mp 1.893 72(17) 9.0 × 10 −5 Tau-neutron mass ratio mτ=mn 1.891 11(17) 9.0 × 10 −3 −1 −5 Tau molar mass NAmτ M τ ;Mτ 1.907 49 17 × 10 kg mol 9.0 × 10 ð Þ ð Þ −15 −5 Tau Compton wavelength h=mτc λC;τ 0.697 787 63 × 10 m 9.0 × 10 ð Þ −15 −5 λC; =2π ƛC; 0.111 056 10 × 10 m 9.0 × 10 τ τ ð Þ Proton, p −27 −8 Proton mass mp 1.672 621 898 21 × 10 kg 1.2 × 10 1.007 276 466ð 879(91)Þ u 9.0 × 10−11 energy equivalent m c2 1.503 277 593 18 × 10−10 J 1.2 × 10−8 p ð Þ 938.272 0813(58) MeV 6.2 × 10−9 −11 Proton-electron mass ratio mp=me 1836.152 673 89(17) 9.5 × 10 −8 Proton-muon mass ratio mp=mμ 8.880 243 38(20) 2.2 × 10 −5 Proton-tau mass ratio mp=mτ 0.528 063(48) 9.0 × 10 −10 Proton-neutron mass ratio mp=mn 0.998 623 478 44(51) 5.1 × 10 7 −1 −9 Proton charge-to-mass quotient e=mp 9.578 833 226 59 × 10 C kg 6.2 × 10 ð Þ −3 −1 −11 Proton molar mass NAmp M p , Mp 1.007 276 466 879 91 × 10 kg mol 9.0 × 10 ð Þ ð Þ −15 −10 Proton Compton wavelength h=mpc λC;p 1.321 409 853 96 61 × 10 m 4.6 × 10 ð Þ −15 −10 λC;p=2π ƛC;p 0.210 308 910 109 97 × 10 m 4.6 × 10 −15ð Þ −3 Proton rms charge radius rp 0.8751 61 × 10 m 7.0 × 10 ð Þ −26 −1 −9 Proton magnetic moment μp 1.410 606 7873 97 × 10 JT 6.9 × 10 ð Þ −3 −9 to Bohr magneton ratio μp=μB 1.521 032 2053 46 × 10 3.0 × 10 ð Þ −9 to nuclear magneton ratio μp=μN 2.792 847 3508(85) 3.0 × 10 −9 Proton g-factor 2μp=μN gp 5.585 694 702(17) 3.0 × 10 −7 Proton-neutron magnetic-moment ratio μp=μn −1.459 898 05 34 2.4 × 10 Shielded proton magnetic moment ð Þ −26 −1 −8 (H2O, sphere, 25 °C) μp0 1.410 570 547 18 × 10 JT 1.3 × 10 ð Þ −3 −8 to Bohr magneton ratio μp0 =μB 1.520 993 128 17 × 10 1.1 × 10 ð Þ (Table continued)

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-58 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

TABLE XXXIII. (Continued)

Relative std. Quantity Symbol Numerical value Unit uncert. ur −8 to nuclear magneton ratio μp0 =μN 2.792 775 600(30) 1.1 × 10 Proton magnetic shielding correction −6 −4 1 − μp0 =μp (H2O, sphere, 25 °C) σp0 25.691 11 × 10 4.4 × 10 ð Þ 8 −1 −1 −9 Proton gyromagnetic ratio 2μp=ℏ γp 2.675 221 900 18 × 10 s T 6.9 × 10 ð Þ −1 −9 γp=2π 42.577 478 92(29) MHz T 6.9 × 10 Shielded proton gyromagnetic ratio 8 −1 −1 −8 2μp0 =ℏ (H2O, sphere, 25 °C) γp0 2.675 153 171 33 × 10 s T 1.3 × 10 ð Þ −1 −8 γp0 =2π 42.576 385 07(53) MHz T 1.3 × 10 Neutron, n −27 −8 Neutron mass mn 1.674 927 471 21 × 10 kg 1.2 × 10 1.008 664 915ð 88(49)Þ u 4.9 × 10−10 2 −10 −8 energy equivalent mnc 1.505 349 739 19 × 10 J 1.2 × 10 939.565 4133(58)ð Þ MeV 6.2 × 10−9 −10 Neutron-electron mass ratio mn=me 1838.683 661 58(90) 4.9 × 10 −8 Neutron-muon mass ratio mn=mμ 8.892 484 08(20) 2.2 × 10 −5 Neutron-tau mass ratio mn=mτ 0.528 790(48) 9.0 × 10 −10 Neutron-proton mass ratio mn=mp 1.001 378 418 98(51) 5.1 × 10 −30 −7 Neutron-proton mass difference mn − mp 2.305 573 77 85 × 10 kg 3.7 × 10 0.001 388 449ð 00(51)Þ u 3.7 × 10−7 energy equivalent m − m c2 2.072 146 37 76 × 10−13 J 3.7 × 10−7 ð n pÞ ð Þ 1.293 332 05(48) MeV 3.7 × 10−7 −3 −1 −10 Neutron molar mass NAmn M n ;Mn 1.008 664 915 88 49 × 10 kg mol 4.9 × 10 ð Þ ð Þ −15 −10 Neutron Compton wavelength h=mnc λC;n 1.319 590 904 81 88 × 10 m 6.7 × 10 ð Þ −15 −10 λC;n=2π ƛC;n 0.210 019 415 36 14 × 10 m 6.7 × 10 ð Þ −26 −1 −7 Neutron magnetic moment μn −0.966 236 50 23 × 10 JT 2.4 × 10 ð Þ −3 −7 to Bohr magneton ratio μn=μB −1.041 875 63 25 × 10 2.4 × 10 ð Þ −7 to nuclear magneton ratio μn=μN −1.913 042 73 45 2.4 × 10 ð Þ −7 Neutron g-factor 2μn=μN gn −3.826 085 45 90 2.4 × 10 ð Þ −3 −7 Neutron-electron magnetic-moment ratio μn=μe 1.040 668 82 25 × 10 2.4 × 10 ð Þ −7 Neutron-proton magnetic-moment ratio μn=μp −0.684 979 34 16 2.4 × 10 Neutron to shielded proton magnetic-moment ð Þ −7 ratio (H2O, sphere, 25 °C) μn=μp0 −0.684 996 94 16 2.4 × 10 ð Þ 8 −1 −1 −7 Neutron gyromagnetic ratio 2 μn =ℏ γn 1.832 471 72 43 × 10 s T 2.4 × 10 j j ð Þ −1 −7 γn=2π 29.164 6933(69) MHz T 2.4 × 10 Deuteron, d −27 −8 Deuteron mass md 3.343 583 719 41 × 10 kg 1.2 × 10 2.013 553 212ð 745(40)Þ u 2.0 × 10−11 2 −10 −8 energy equivalent mdc 3.005 063 183 37 × 10 J 1.2 × 10 1875.612 928(12)ð Þ MeV 6.2 × 10−9 −11 Deuteron-electron mass ratio md=me 3670.482 967 85(13) 3.5 × 10 −11 Deuteron-proton mass ratio md=mp 1.999 007 500 87(19) 9.3 × 10 −3 −1 −11 Deuteron molar mass NAmd M d ;Md 2.013 553 212 745 40 × 10 kg mol 2.0 × 10 ð Þ −15ð Þ −3 Deuteron rms charge radius rd 2.1413 25 × 10 m 1.2 × 10 ð Þ −26 −1 −9 Deuteron magnetic moment μd 0.433 073 5040 36 × 10 JT 8.3 × 10 ð Þ −3 −9 to Bohr magneton ratio μd=μB 0.466 975 4554 26 × 10 5.5 × 10 ð Þ −9 to nuclear magneton ratio μd=μN 0.857 438 2311(48) 5.5 × 10 −9 Deuteron g-factor μd=μN gd 0.857 438 2311(48) 5.5 × 10 −4 −9 Deuteron-electron magnetic-moment ratio μd=μe −4.664 345 535 26 × 10 5.5 × 10 ð Þ −9 Deuteron-proton magnetic-moment ratio μd=μp 0.307 012 2077(15) 5.0 × 10 Deuteron-neutron magnetic-moment ratio μ =μ −0.448 206 52 11 2.4 × 10−7 d n ð Þ Triton, t −27 −8 Triton mass mt 5.007 356 665 62 × 10 kg 1.2 × 10 3.015 500 716ð 32(11)Þ u 3.6 × 10−11 2 −10 −8 energy equivalent mtc 4.500 387 735 55 × 10 J 1.2 × 10 2808.921 112(17)ð Þ MeV 6.2 × 10−9 −11 Triton-electron mass ratio mt=me 5496.921 535 88(26) 4.6 × 10 −11 Triton-proton mass ratio mt=mp 2.993 717 033 48(22) 7.5 × 10 −3 −1 −11 Triton molar mass NAmt M t ;Mt 3.015 500 716 32 11 × 10 kg mol 3.6 × 10 ð Þ ð Þ (Table continued)

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-59 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

TABLE XXXIII. (Continued)

Relative std. Quantity Symbol Numerical value Unit uncert. ur −26 −1 −9 Triton magnetic moment μt 1.504 609 503 12 × 10 JT 7.8 × 10 ð Þ −3 −9 to Bohr magneton ratio μt=μB 1.622 393 6616 76 × 10 4.7 × 10 ð Þ −9 to nuclear magneton ratio μt=μN 2.978 962 460(14) 4.7 × 10 −9 Triton g-factor 2μt=μN gt 5.957 924 920(28) 4.7 × 10 Helion, h −27 −8 Helion mass mh 5.006 412 700 62 × 10 kg 1.2 × 10 3.014 932 246ð 73(12)Þ u 3.9 × 10−11 2 −10 −8 energy equivalent mhc 4.499 539 341 55 × 10 J 1.2 × 10 2808.391 586(17)ð Þ MeV 6.2 × 10−9 −11 Helion-electron mass ratio mh=me 5495.885 279 22(27) 4.9 × 10 −11 Helion-proton mass ratio mh=mp 2.993 152 670 46(29) 9.6 × 10 −3 −1 −11 Helion molar mass NAmh M h ;Mh 3.014 932 246 73 12 × 10 kg mol 3.9 × 10 ð Þ ð Þ −26 −1 −8 Helion magnetic moment μh −1.074 617 522 14 × 10 JT 1.3 × 10 ð Þ −3 −8 to Bohr magneton ratio μh=μB −1.158 740 958 14 × 10 1.2 × 10 ð Þ −8 to nuclear magneton ratio μh=μN −2.127 625 308 25 1.2 × 10 ð Þ −8 Helion g-factor 2μh=μN gh −4.255 250 616 50 1.2 × 10 Shielded helion magnetic moment ð Þ −26 −1 −8 (gas, sphere, 25 °C) μh0 −1.074 553 080 14 × 10 JT 1.3 × 10 ð Þ −3 −8 to Bohr magneton ratio μh0 =μB −1.158 671 471 14 × 10 1.2 × 10 ð Þ −8 to nuclear magneton ratio μh0 =μN −2.127 497 720 25 1.2 × 10 Shielded helion to proton magnetic- ð Þ −8 moment ratio (gas, sphere, 25 °C) μh0 =μp −0.761 766 5603 92 1.2 × 10 Shielded helion to shielded proton ð Þ magnetic-moment ratio −9 (gas=H2O, spheres, 25 °C) μh0 =μp0 −0.761 786 1313 33 4.3 × 10 Shielded helion gyromagnetic ratio ð Þ 8 −1 −1 −8 2 μh0 =ℏ (gas, sphere, 25 °C) γh0 2.037 894 585 27 × 10 s T 1.3 × 10 j j ð Þ −1 −8 γh0 =2π 32.434 099 66(43) MHz T 1.3 × 10 Alpha particle, α −27 −8 Alpha particle mass mα 6.644 657 230 82 × 10 kg 1.2 × 10 4.001 506 179ð 127(63)Þ u 1.6 × 10−11 2 −10 −8 energy equivalent mαc 5.971 920 097 73 × 10 J 1.2 × 10 3727.379 378(23)ð Þ MeV 6.2 × 10−9 −11 Alpha particle to electron mass ratio mα=me 7294.299 541 36(24) 3.3 × 10 −11 Alpha particle to proton mass ratio mα=mp 3.972 599 689 07(36) 9.2 × 10 −3 −1 −11 Alpha particle molar mass NAm M α ;M 4.001 506 179 127 63 × 10 kg mol 1.6 × 10 α ð Þ α ð Þ PHYSICOCHEMICAL 23 −1 −8 Avogadro constant NA;L 6.022 140 857 74 × 10 mol 1.2 × 10 Atomic mass constant ð Þ 1 12 −27 −8 mu 12 m C 1 u mu 1.660 539 040 20 × 10 kg 1.2 × 10 ¼ ð Þ¼ 2 ð Þ −10 −8 energy equivalent muc 1.492 418 062 18 × 10 J 1.2 × 10 931.494 0954(57)ð Þ MeV 6.2 × 10−9 f −1 −9 Faraday constant NAeF96 485.332 89(59) C mol 6.2 × 10 −10 −1 −10 Molar Planck constant NAh 3.990 312 7110 18 × 10 J s mol 4.5 × 10 ð Þ −1 −10 NAhc 0.119 626 565 582(54) J m mol 4.5 × 10 Molar gas constant R 8.314 4598(48) J mol−1 K−1 5.7 × 10−7 −23 −1 −7 Boltzmann constant R=NA k 1.380 648 52 79 × 10 JK 5.7 × 10 8.617 3303 50ð ×Þ 10−5 eV K−1 5.7 × 10−7 k=h 2.083 6612ð12Þ × 1010 Hz K−1 5.7 × 10−7 k=hc 69.503 457(40)ð Þ m−1 K−1 5.7 × 10−7 Molar volume of ideal gas RT=p −3 3 −1 −7 T 273.15 K, p 100 kPa Vm 22.710 947 13 × 10 m mol 5.7 × 10 ¼ ¼ ð Þ 25 −3 −7 Loschmidt constant NA=Vm n0 2.651 6467 15 × 10 m 5.7 × 10 Molar volume of ideal gas RT=p ð Þ −3 3 −1 −7 T 273.15 K, p 101.325 kPa Vm 22.413 962 13 × 10 m mol 5.7 × 10 ¼ ¼ ð Þ 25 −3 −7 Loschmidt constant NA=Vm n0 2.686 7811 15 × 10 m 5.7 × 10 ð Þ (Table continued)

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-60 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

TABLE XXXIII. (Continued)

Relative std. Quantity Symbol Numerical value Unit uncert. ur Sackur-Tetrode (absolute entropy) constantg 5 2 3=2 2 ln 2πmukT1=h kT1=p0 Tþ ½ð1 K, p 100Þ kPa S =R −1.151 7084 14 1.2 × 10−6 1 ¼ 0 ¼ 0 ð Þ T 1 K, p 101.325 kPa −1.164 8714 14 1.2 × 10−6 1 ¼ 0 ¼ ð Þ Stefan-Boltzmann constant π2=60 k4=ℏ3c2 σ 5.670 367 13 × 10−8 Wm−2 K−4 2.3 × 10−6 ð 2 Þ ð Þ −16 2 −8 First radiation constant 2πhc c1 3.741 771 790 46 × 10 Wm 1.2 × 10 2 ð Þ −16 2 −1 −8 First radiation constant for spectral radiance 2hc c1L 1.191 042 953 15 × 10 Wm sr 1.2 × 10 ð Þ −2 −7 Second radiation constant hc=k c2 1.438 777 36 83 × 10 mK 5.7 × 10 Wien displacement law constants ð Þ −3 −7 b λmaxT c2=4.965 114 231… b 2.897 7729 17 × 10 mK 5.7 × 10 ¼ ¼ ð Þ 10 −1 −7 b0 νmax=T 2.821 439 372…c=c2 b0 5.878 9238 34 × 10 Hz K 5.7 × 10 ¼ ¼ ð Þ aSee Table XXXV for the conventional value adopted internationally for realizing representations of the volt using the Josephson effect. bSee Table XXXV for the conventional value adopted internationally for realizing representations of the ohm using the quantum-Hall effect. cValue recommended by the Particle Data Group (Olive et al., 2014). d Based on the ratio of the masses of the W and Z bosons mW=mZ recommended by the Particle Data Group 2 (Olive et al., 2014). The value for sin θW they recommend, which is based on a particular variant of the modified minimal 2 ˆ subtraction MS scheme, is sin θW MZ 0.231 26 5 . e ð Þ ð Þ¼ ð Þ 2 This and all other values involving mτ are based on the value of mτc in MeV recommended by the Particle Data Group (Olive et al., 2014). fThe numerical value of F to be used in coulometric chemical measurements is 96 485.3251(12) [1.2 × 10−8] when the relevant current is measured in terms of representations of the volt and ohm based on the Josephson and quantum-Hall effects and the internationally adopted conventional values of the Josephson and von Klitzing constants KJ−90 and RK−90 given in Table XXXV. gThe entropy of an ideal monoatomic gas of relative atomic mass A is given by S S 3 R ln A − R ln p=p 5 R ln T=K . r ¼ 0 þ 2 r ð 0Þþ2 ð Þ

TABLE XXXIV. The variances, covariances, and correlation coefficients of the values of a selected group of constants based on the 2014 CODATA adjustment. The numbers in bold above the main diagonal are 1016 times the numerical values of the relative covariances; the numbers in bold on the main diagonal are 1016 times the numerical values of the relative variances; and the numbers in italics below the main diagonal are the correlation coefficients.a

α heme NA me=mμ F α 0.0005 0.0005 0.0005 −0.0005 0.0005 −0.0010 0.0010 h0.0176 1.5096 0.7550 1.5086 −1.5086 −0.0010 −0.7536 e0.0361 0.9998 0.3778 0.7540 −0.7540 −0.0010 −0.3763 me 0.0193 0.9993 0.9985 1.5097 −1.5097 0.0011 −0.7556 − NA 0.0193 0.9993 0.9985 1.0000 1.5097 −0.0011 0.7557 − − − me=mμ 0.0202 0.0004 0.0007 0.0004 0.0004 4.9471 −0.0021 F0− .0745 −0.9957 −0.9939 0.9985− 0.9985 0.0015 0.3794 − − − − a The relative covariance is ur xi;xj u xi;xj = xixj , where u xi;xj is the covariance of xi and xj; the relative variance is u2 x u x ;x : and the correlationð Þ¼ coefficientð Þ isðr x ;xÞ u x ð;x = Þu x u x . r ð iÞ¼ rð i iÞ ð i jÞ¼ ð i jÞ ½ ð iÞ ð jÞ

TABLE XXXV. Internationally adopted values of various quantities.

Quantity Symbol Numerical value Unit Relative std. uncert. ur a 12 12 Relative atomic mass of C Ar C 12 Exact ð Þ −3 −1 Molar mass constant Mu 1 × 10 kg mol Exact Molar mass of 12C M 12C 12 × 10−3 kg mol−1 Exact b ð Þ −1 Conventional value of Josephson constant KJ−90 483 597.9 GHz V Exact c Conventional value of von Klitzing constant RK−90 25 812.807 Ω Exact Standard-state pressure 100 kPa Exact Standard atmosphere 101.325 kPa Exact a 12 The relative atomic mass Ar X of particle X with mass m X is defined by Ar X m X =mu, where mu m C =12 M =N 1 u is the atomic massð constant,Þ M is the molar mass constant,ð Þ N is the Avogadroð Þ¼ constant,ð Þ and u is the unified¼ atomicð Þ mass¼ u A ¼ u A unit. Thus the mass of particle X is m X Ar X u and the molar mass of X is M X Ar X Mu. bThis is the value adopted internationallyð Þ¼ forð realizingÞ representations of the volt usingð Þ¼ theð JosephsonÞ effect. cThis is the value adopted internationally for realizing representations of the ohm using the quantum-Hall effect.

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-61 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

TABLE XXXVI. Values of some x-ray-related quantities based on the 2014 CODATA adjustment of the values of the constants.

Relative std. Quantity Symbol Numerical value Unit uncert. ur −13 −7 Cu x unit: λ CuKα1 =1 537.400 xu CuKα1 1.002 076 97 28 × 10 m 2.8 × 10 ð Þ ð Þ ð Þ −13 −7 Mo x unit: λ MoKα1 =707.831 xu MoKα1 1.002 099 52 53 × 10 m 5.3 × 10 ð Þ ð Þ ð Þ −10 −7 Ångström star: λ WKα1 =0.209 010 0 ÅÃ 1.000 014 95 90 × 10 m 9.0 × 10 ða Þ ð Þ Lattice parameter of Si (in vacuum, 22.5 °C) a 543.102 0504 89 × 10−12 m 1.6 × 10−8 ð Þ −12 −8 220 lattice spacing of Si a=p8 (in vacuum, 22.5 °C) d220 192.015 5714 32 × 10 m 1.6 × 10 f g 3 ð Þ −6 3 −1 −8 Molar volume of Si M Si =ρ Si NAa =8 (in vacuum, 22.5 °C) Vm Si 12.058 832 14 61 × 10 m mol 5.1 × 10 ð Þ ð Þ¼ffiffiffi ð Þ ð Þ aThis is the lattice parameter (unit cell edge length) of an ideal single crystal of naturally occurring Si free of impurities and imperfections and is deduced from measurements on extremely pure and nearly perfect single crystals of Si by correcting for the effects of impurities.

TABLE XXXVII. The values in SI units of some non-SI units based on the 2014 CODATA adjustment of the values of the constants.

Relative std. Quantity Symbol Numerical value Unit uncert. ur Non-SI units accepted for use with the SI Electron volt: (e=C) J eV 1.602 176 6208 98 × 10−19 J 6.1 × 10−9 (Unified) atomic mass unit: 1 m 12C u 1.660 539 040 20ð Þ× 10−27 kg 1.2 × 10−8 12 ð Þ ð Þ Natural units (n.u.) −1 n.u. of velocity c, c0 299 792 458 ms Exact n.u. of action: h=2π ℏ 1.054 571 800 13 × 10−34 Js 1.2 × 10−8 6.582 119 514ð40Þ × 10−16 eV s 6.1 × 10−9 ℏc 197.326 9788(12)ð Þ MeV fm 6.1 × 10−9 −31 −8 n.u. of mass me 9.109 383 56 11 × 10 kg 1.2 × 10 2 ð Þ −14 −8 n.u. of energy mec 8.187 105 65 10 × 10 J 1.2 × 10 0.510 998 9461(31)ð Þ MeV 6.2 × 10−9 −22 −1 −8 n.u. of momentum mec 2.730 924 488 34 × 10 kg m s 1.2 × 10 0.510 998 9461(31)ð Þ MeV=c 6.2 × 10−9 −15 −10 n.u. of length: ℏ=mec ƛC 386.159 267 64 18 × 10 m 4.5 × 10 n.u. of time ℏ=m c2 1.288 088 667 12ð 58Þ × 10−21 s 4.5 × 10−10 e ð Þ Atomic units (a.u.) a.u. of charge e 1.602 176 6208 98 × 10−19 C 6.1 × 10−9 ð Þ −31 −8 a.u. of mass me 9.109 383 56 11 × 10 kg 1.2 × 10 ð Þ a.u. of action: h=2π ℏ 1.054 571 800 13 × 10−34 Js 1.2 × 10−8 ð Þ −10 −10 a.u. of length: Bohr radius (bohr) α=4πR∞ a0 0.529 177 210 67 12 × 10 m 2.3 × 10 a.u. of energy: Hartree energy (hartree) ð Þ 2 2 2 −18 −8 e =4πϵ0a0 2R∞hc α mec Eh 4.359 744 650 54 × 10 J 1.2 × 10 ¼ ¼ ð Þ −17 −12 a.u. of time ℏ=Eh 2.418 884 326 509 14 × 10 s 5.9 × 10 ð Þ−8 −8 a.u. of force Eh=a0 8.238 723 36 10 × 10 N 1.2 × 10 ð Þ 6 −1 −10 a.u. of velocity: αca0Eh=ℏ 2.187 691 262 77 50 × 10 ms 2.3 × 10 ð Þ −24 −1 −8 a.u. of momentum ℏ=a0 1.992 851 882 24 × 10 kg m s 1.2 × 10 ð Þ −3 −9 a.u. of current eEh=ℏ 6.623 618 183 41 × 10 A 6.1 × 10 3 ð Þ 12 −3 −9 a.u. of charge density e=a0 1.081 202 3770 67 × 10 Cm 6.2 × 10 ð Þ −9 a.u. of electric potential Eh=e 27.211 386 02(17) V 6.1 × 10 11 −1 −9 a.u. of electric field Eh=ea0 5.142 206 707 32 × 10 Vm 6.1 × 10 2 ð Þ 21 −2 −9 a.u. of electric field gradient Eh=ea0 9.717 362 356 60 × 10 Vm 6.2 × 10 ð Þ −30 −9 a.u. of electric dipole moment ea0 8.478 353 552 52 × 10 Cm 6.2 × 10 2 ð Þ −40 2 −9 a.u. of electric quadrupole moment ea0 4.486 551 484 28 × 10 Cm 6.2 × 10 2 2 ð Þ −41 2 2 −1 −10 a.u. of electric polarizability e a0=Eh 1.648 777 2731 11 × 10 C m J 6.8 × 10 3 3 2 ð Þ −53 3 3 −2 −9 a.u. of 1st hyperpolarizability e a0=Eh 3.206 361 329 20 × 10 C m J 6.2 × 10 4 4 3 ð Þ −65 4 4 −3 −8 a.u. of 2nd hyperpolarizability e a0=Eh 6.235 380 085 77 × 10 C m J 1.2 × 10 2 ð Þ 5 −9 a.u. of magnetic flux density ℏ=ea0 2.350 517 550 14 × 10 T 6.2 × 10 ð Þ −23 −1 −9 a.u. of magnetic dipole moment: 2μB ℏe=me 1.854 801 999 11 × 10 JT 6.2 × 10 2 2 ð Þ −29 −2 −9 a.u. of magnetizability e a0=me 7.891 036 5886 90 × 10 JT 1.1 × 10 7 2 2 ð Þ −10 −1 a.u. of permittivity: 10 =c e =a0Eh 1.112 650 056… × 10 Fm Exact

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-62 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

TABLE XXXVIII. The values of some energy equivalents derived from the relations E mc2 hc=λ hν kT, and based on the 2014 1 ¼12 ¼ −3 ¼ ¼−1 CODATA adjustment of the values of the constants; 1 eV e=C J, 1 u mu 12 m C 10 kg mol =NA, and Eh 2R∞hc 2 2 ¼ð Þ ¼ ¼ ð Þ¼ ¼ ¼ α mec is the Hartree energy (hartree). Relevant unit J kg m−1 Hz 1J 1 J 1 J 1 J =c2 1 J =hc 1 J =h ð Þ¼ 1ð.112Þ 650¼ 056… × 10−17 kg ð5.034Þ 116¼ 651 62 × 1024 m−1 1ð.509Þ 190¼ 205 19 × 1033 Hz ð Þ ð Þ 1 kg 1 kg c2 1 kg 1 kg 1 kg c=h 1 kg c2=h ð8.987Þ 551¼ 787… × 1016 J ð Þ¼ ð4.524Þ 438 411¼ 56 × 1041 m−1 1ð.356Þ 392 512¼ 17 × 1050 Hz ð Þ ð Þ 1 m−1 1 m−1 hc 1 m−1 h=c 1 m−1 1 m−1 1 m−1 c ð1.986 445Þ 824¼ 24 × 10−25 J ð2.210 219Þ 057¼ 27 × 10−42 kg ð Þ¼ ð299 792Þ 458¼ Hz ð Þ ð Þ 1 Hz 1 Hz h 1 Hz h=c2 1 Hz c 1 Hz 1 Hz 6ð.626Þ 070¼ 040 81 × 10−34 J 7ð.372Þ 497 201¼ 91 × 10−51 kg ð3.335Þ 640¼ 951… × 10−9 m−1 ð Þ¼ ð Þ ð Þ 1K 1 K k 1 K k=c2 1 K k=hc 1 K k=h ð1.380Þ 648¼ 52 79 × 10−23 J ð1.536Þ 178 65¼ 88 × 10−40 kg ð69.503Þ 457 ¼40 m−1 ð2.083Þ 6612¼12 × 1010 Hz ð Þ ð Þ ð Þ ð Þ 1 eV 1 eV 1 eV =c2 1 eV =hc 1 eV =h ð1.602Þ¼ 176 6208 98 × 10−19 J 1ð.782Þ 661 907¼ 11 × 10−36 kg ð8.065Þ 544 005¼ 50 × 105 m−1 ð2.417Þ 989¼ 262 15 × 1014 Hz ð Þ ð Þ ð Þ ð Þ 1u 1 u c2 1 u 1 u c=h 1 u c2=h 1ð.492Þ 418¼ 062 18 × 10−10 J 1ð.660Þ¼ 539 040 20 × 10−27 kg ð7.513Þ 006¼ 6166 34 × 1014 m−1 2ð.252Þ 342 7206¼ 10 × 1023 Hz ð Þ ð Þ ð Þ ð Þ 2 1 Eh 1 Eh 1 Eh =c 1 Eh =hc 1 Eh =h ð4.359Þ¼ 744 650 54 × 10−18 J 4ð.850Þ 870 129¼ 60 × 10−35 kg ð2.194Þ 746 313¼ 702 13 × 107 m−1 ð6.579Þ 683¼ 920 711 39 × 1015 Hz ð Þ ð Þ ð Þ ð Þ such input data with S < 0.01, the five data with r > 1.2 B. Tables of values c j ij are B11, B12, B44.2, B44.4, and B48; their values of ri are 1.95, 1.71, 1.96, 1.84, and 1.68, respectively. Tables XXXII, XXXIII, XXXIV, XXXV, XXXVI, The 2014 recommended values are calculated from the set XXXVII, XXXVIII, and XXXIX give the 2014 CODATA of best estimated values, in the least-squares sense, of 75 recommended values of the basic constants and conversion adjusted constants, including G, and their variances and factors of physics and chemistry and related quantities. They covariances, together with (i) those constants that have exact are identical in form and content to their 2010 counterparts in that no constants are added or deleted. values such as μ0 and c; and (ii) the values of mτ, GF, and 2 It should be noted that the values of the four helion-related sin θW given in Sec. XII of this report. See Sec. V.B of CODATA-98 for details. constants are calculated from the adjusted constant μh0 =μp0

TABLE XXXIX. The values of some energy equivalents derived from the relations E mc2 hc=λ hν kT and based on the 2014 1 ¼12 ¼ −3 ¼ ¼−1 CODATA adjustment of the values of the constants; 1 eV e=C J, 1 u mu 12 m C 10 kg mol =NA, and Eh 2R∞hc 2 2 ¼ð Þ ¼ ¼ ð Þ¼ ¼ ¼ α mec is the Hartree energy (hartree). Relevant unit

K eV u Eh 1J 1 J =k 1 J 1 J =c2 1 J ð Þ ¼ 22 ð Þ¼ 18 ð Þ ¼ 9 ð Þ¼ 17 7.242 9731 42 × 10 K 6.241 509 126 38 × 10 eV 6.700 535 363 82 × 10 u 2.293 712 317 28 × 10 Eh ð Þ ð Þ ð Þ ð Þ 1 kg 1 kg c2=k 1 kg c2 1 kg 1 kg c2 6ð.509Þ 6595 ¼37 × 1039 K 5ð.609Þ 588¼ 650 34 × 1035 eV ð6.022Þ¼ 140 857 74 × 1026 u ð2.061Þ 485¼ 823 25 × 1034 E ð Þ ð Þ ð Þ ð Þ h 1 m−1 1 m−1 hc=k 1 m−1 hc 1 m−1 h=c 1 m−1 hc ð1.438 777Þ 36 83¼ × 10−2 K ð1.239 841Þ 9739¼ 76 × 10−6 eV ð1.331 025Þ 049¼ 00 61 × 10−15 u ð4.556 335Þ 252¼ 767 27 × 10−8 E ð Þ ð Þ ð Þ ð Þ h 1 Hz 1 Hz h=k 1 Hz h 1 Hz h=c2 1 Hz h ð Þ ¼ −11 ð Þ ¼ −15 ð Þ ¼ −24 ð Þ ¼ −16 4.799 2447 28 × 10 K 4.135 667 662 25 × 10 eV 4.439 821 6616 20 × 10 u 1.519 829 846 0088 90 × 10 Eh ð Þ ð Þ ð Þ ð Þ 1K 1 K 1 K 1 K k 1 K k=c2 1 K k ð Þ¼ ð Þ ¼ −5 ð Þ ¼ −14 ð Þ ¼ −6 8.617 3303 50 × 10 eV 9.251 0842 53 × 10 u 3.166 8105 18 × 10 Eh ð Þ ð Þ ð Þ 1 eV 1 eV =k 1 eV 1 eV 1 eV =c2 1 eV ð Þ ¼ 4 ð Þ¼ ð Þ ¼ −9 ð Þ¼ −2 1.160 452 21 67 × 10 K 1.073 544 1105 66 × 10 u 3.674 932 248 23 × 10 Eh ð Þ ð Þ ð Þ 1u 1 u c2=k 1 u c2 1 u 1 u 1 u c2 1ð.080Þ 954 38¼ 62 × 1013 K 931ð .Þ494¼ 0954 57 × 106 eV ð Þ¼ 3ð.423Þ 177¼ 6902 16 × 107 E ð Þ ð Þ ð Þ h 2 1 Eh 1 Eh =k 1 Eh 1 Eh =c 1 Eh 1 Eh 3ð.157Þ 7513¼18 × 105 K 27ð .211Þ¼ 386 02 17 eV 2ð.921Þ 262 3197¼ 13 × 10−8 u ð Þ¼ ð Þ ð Þ ð Þ

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-63 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

and the theoretically predicted shielding correction σh A. Comparison of 2014 and 2010 CODATA 59.967 43 10 × 10−6 due to Rudziński, Puchalski, and¼ recommended values Pachuckið (2009)Þ using the relation μ μ 1 σ ; see h0 h − h Table XL compares the 2014 and 2010 recommended Sec. VI.A. ¼ ð Þ values of a representative group of constants. The regularities Table XXXII is a highly abbreviated list of the values of the observed in the numbers in columns 2 to 4 arise because the constants and conversion factors most commonly used. values of many constants are obtained from expressions Table XXXIII is a much more extensive list of values catego- proportional to the fine-structure constant α, Planck constant rized as follows: UNIVERSAL; ELECTROMAGNETIC; ATOMIC AND NUCLEAR; and PHYSICOCHEMICAL. The ATOMIC AND NUCLEAR category is subdivided into TABLE XL. Comparison of the 2014 and 2010 CODATA recom- 11 subcategories: General; Electroweak; Electron, e; Muon, μ; mended values of a representative group of constants. Here Dr is the Tau, τ; Proton, p; Neutron, n; Deuteron, d; Triton, t; Helion, h; 2014 value minus the 2010 value divided by the standard uncertainty and Alpha particle, α. Table XXXIV gives the variances, u of the 2010 value. covariances, and correlation coefficients of a selected group of constants. (Use of the covariance matrix is discussed in 2014 rel. std. Ratio 2010 ur Appendix E of CODATA-98.) Table XXXV gives the interna- Quantity uncert. ur to 2014 ur Dr tionally adopted values of various quantities; Table XXXVI α 2.3 × 10−10 1.4 −1.4 −10 lists the values of a number of x-ray-related quantities; RK 2.3 × 10 1.4 1.4 −10 Table XXXVII lists the values of various non-SI units; and a0 2.3 × 10 1.4 −1.4 −10 Tables XXXVIII and XXXIX give the values of various energy λC 4.5 × 10 1.4 −1.4 −10 re 6.8 × 10 1.4 −1.4 equivalents. −9 σe 1.4 × 10 1.4 −1.4 All of the values given in these tables are available on the h 1.2 × 10−8 3.6 1.6 website of the Fundamental Constants Data Center of the −8 me 1.2 × 10 3.6 1.6 −8 NIST Physical Measurement Laboratory at http://physics.nist mh 1.2 × 10 3.6 1.6 −8 .gov/constants. In fact, this electronic version of the 2014 mα 1.2 × 10 3.6 1.6 −8 CODATA recommended values of the constants enables NA 1.2 × 10 3.6 −1.6 −8 users to obtain the correlation coefficient of any two constants Eh 1.2 × 10 3.6 1.6 −8 listed in the tables. It also allows users to automatically c1 1.2 × 10 3.6 1.6 9 convert the value of an energy-related quantity expressed in e 6.1 × 10− 3.6 1.6 K −9 3.6 1.6 one unit to the corresponding value expressed in another unit J 6.1 × 10 − F 6.2 × 10−9 3.6 −1.7 (in essence, an automated version of Tables XXXVIII −8 γp0 1.3 × 10 2.0 −1.5 and XXXIX). −9 μB 6.2 × 10 3.6 1.5 −9 μN 6.2 × 10 3.6 1.5 −9 XV. SUMMARY AND CONCLUSION μe 6.2 × 10 3.6 −1.5 −9 μp 6.9 × 10 3.4 1.3 7 Here we (i) compare the 2014 to the 2010 recommended R 5.7 × 10− 1.6 −0.3 k 5.7 × 10−7 1.6 −0.2 values of the constants and identify those new results that have −7 Vm 5.7 × 10 1.6 −0.3 contributed most to the changes in the 2010 values; (ii) present −7 c2 5.7 × 10 1.6 0.3 several conclusions that can be drawn from the 2014 recom- σ 2.3 × 10−6 1.6 −0.3 mended values and the input data from which they are G 4.7 × 10−5 2.6 0.3 −12 obtained; and (iii) identify new experimental and theoretical R∞ 5.9 × 10 0.8 −0.6 −11 work that can advance our knowledge of the values of the me=mp 9.5 × 10 4.3 −1.9 −8 constants. me=mμ 2.2 × 10 1.1 0.3 −11 Topic (iii) is relevant to the plan of the 26th General Ar e 2.9 × 10 13.8 −1.8 ð Þ −11 Conference on Weights and Measures (CGPM) to adopt at its Ar p 9.0 × 10 1.0 0.7 ð Þ −10 meeting in Paris in the fall of 2018 a resolution that will revise Ar n 4.9 × 10 0.9 −0.3 ð Þ −11 “ ” Ar d 2.0 × 10 1.9 0.4 the SI. In the new SI, as it has come to be called to ð Þ −11 Ar t 3.6 × 10 22.8 1.2 distinguish it from the present SI, the definitions of the ð Þ −11 Ar h 3.9 × 10 21.3 −0.0 kilogram, ampere, kelvin, and mole are linked to exact values ð Þ −11 Ar α 1.6 × 10 1.0 0.0 ð Þ −8 of the Planck constant h, elementary charge e, Boltzmann d220 1.6 × 10 1.0 0.0 −13 constant k, and Avogadro constant NA, in much the same way ge 2.6 × 10 1.0 −0.5 −10 as the present definition of the meter is linked to an exact value gμ 6.3 × 10 1.0 0.0 −9 of the speed of light in vacuum c. CODATA, through its Task μp=μB 3.0 × 10 2.7 −0.3 −9 Group on Fundamental Constants, is to provide the values of μp=μN 3.0 × 10 2.7 −0.2 μ =μ 2.4 × 10−7 1.0 0.0 h, e, k, and NA for the new definitions by carrying out a n N − −9 special least-squares adjustment during the summer of 2017. μd=μN 5.5 × 10 1.5 0.0 −9 Details of the proposed new-SI may be found on the BIPM μe=μp 3.0 × 10 2.7 −0.3 μ =μ 2.4 × 10−7 1.0 0.0 website at bipm.org/en/measurement‑units/new‑si/ [see also n p − μ =μ 5.0 × 10−9 1.5 0.3 Milton, Davis, and Fletcher (2014)]. d p

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-64 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ... h, or molar gas constant R raised to various powers. For the normalized residual of each datum to less than 2. Thus in example, the first six quantities are obtained from expressions 2014 the expansion factor is 6.3 and the weighted mean of the proportional to αa, where a 1, 2, 3, or 6. The next 15 14 available values, which is the 2014 recommended value, j j ¼ −5 quantities, from h to the magnetic moment of the proton μp, has a relative uncertainty of 4.7 × 10 . The shift of the 2014 are calculated from expressions containing the factor ha, value from the 2010 value is due to the three new results that where a 1 or 1=2. And the five quantities from R to became available for the 2014 adjustment. j j ¼ a the Stefan-Boltzmann constant σ are proportional to R , where The reduction in uncertainty of me=mp is a consequence of a 1 or 4. the large reduction in uncertainty of A e , which resulted rð Þ j jAdditional¼ comments on some of the entries in Table XL from measurement of the ratio of the electron spin-flip 12 5 are as follows. frequency of the C þ ion to the cyclotron frequency of (i) The shift and uncertainty reduction of the 2014 recom- the ion in the same magnetic flux density, and the same ratio 28 13 mended value of α is mainly due to the availability for the first for Si þ, with the extraordinarily small relative uncertain- time of a numerically calculated result for the tenth-order ties of 2.8 × 10−11 and 4.8 × 10−11, respectively. The signifi- 10 cant reduction in uncertainty of A t and A h is the result of coefficient A1ð Þ (12 672 Feynman diagrams) in the theoretical r r the measurement of two pairs ofð cyclotronÞ ð frequencyÞ ratios expression for ae; see Sec. V.A.1. The value used in 2010, based on a procedure described in CODATA-98, is 0.0(4.6) carried out in two different laboratories, the first being that of 12 6 −11 compared with the newly available value 7.79(34) used d and h to C þ with relative uncertainties of 2.0 × 10 and −11 3 in 2014. 1.4 × 10 ; and the second is that of HDþ to Heþ and to t (ii) In the 2010 adjustment inconsistencies among watt- with relative uncertainties for each of 4.8 × 10−11. However, balance measurements of h and the value inferred from an because of the significant inconsistency of the values of x-ray-crystal-density (XRCD) measurement of N using A 3He implied by the second ratio of the first pair and A rð Þ highly enriched silicon led the Task Group to expand the the first ratio of the second pair, the Task Group applied a uncertainties assigned to these data by a multiplicative factor, multiplicative factor of 2.8 to the uncertainties of these two or expansion factor, of 2. These inconsistencies have since ratios in order to reduce the residual of each to less than 2 in been resolved and further, an improved watt-balance result for the final adjustment on which the 2014 recommended values h with a relative uncertainty of 1.8 × 10−8 and an improved are based. Finally, the reduction in the uncertainties of μp=μB, XRCD-Avogadro constant inferred value of h with a relative μp=μN, and μe=μp are a consequence of a new, directly 8 uncertainty of 2.0 × 10− have become available for the 2014 measured value of μp=μN with a relative uncertainty of adjustment. Because the data are now sufficiently consistent 3.3 × 10−9. that it is no longer necessary to increase uncertainties as in 2010, the relative uncertainty of the 2014 recommended value B. Some implications of the 2014 CODATA recommended of h is 1.2 × 10−8 compared to 4.4 × 10−8 for the 2010 value. values and adjustment for metrology and physics (iii) The 2010 recommended value of R is based on six acoustic-gas-thermometry (AGT) results with relative uncer- 1. Conventional electrical units 6 6 tainties in the range 1.2 × 10− to 8.4 × 10− while the 2014 The conventional values KJ 90 483 597.9 GHz=V and − ¼ recommended value is based mainly on seven AGT results RK 90 25 812.807 Ω adopted in 1990 for the Josephson and − ¼ with relative uncertainties in the range 0.90 × 10−6 to von Klitzing constants established conventional units of −6 voltage and resistance, V and Ω , given by V 3.7 × 10 . Of these seven, three are new results, two of 90 90 90 ¼ −6 −6 KJ 90=KJ V and Ω90 RK=RK 90 Ω. Other conventional which have uncertainties of 0.90 × 10 and 1.0 × 10 , ð − Þ ¼ð − Þ respectively. Also contributing to the determination of the electric units follow from V90 and Ω90, for example, 2014 recommended value of R is a Johnson noise thermom- A90 V90=Ω90, C90 A90 s, W90 A90V90, F90 C90=V90, and¼H Ω s, which¼ are the conventional¼ units¼ of current, etry measurement of k=h and a dielectric-constant gas 90 ¼ 90 thermometry measurement of A =R with relative uncertainties charge, power, capacitance, and inductance, respectively, ϵ (Taylor and Mohr, 2001). The 2014 adjustment gives for of 3.9 × 10−6 and 4.0 × 10−6, respectively. It is the significant the relations between KJ and KJ 90, and for RK and RK−90, advances made in AGT in the past four years that have led to − the reduction of the uncertainty of the recommended value of K K 1 − 9.83 61 × 10−8 ; 273 R by nearly 40%. The consistency of the new and previous J ¼ J−90½ ð Þ ð Þ AGT results have led to the comparatively small shift of the R R 1 1.765 23 × 10−8 ; 274 2014 value from that of 2010. K ¼ K−90½ þ ð Þ ð Þ (iv) Other constants in Table XL whose changes are worth which lead to noting are G, m =m , A e , A t , A h , μ =μ , μ =μ , and e p rð Þ rð Þ rð Þ p B p N μe=μp. The 2010 recommended value of G with relative −8 V90 1 9.83 61 × 10 V; 275 uncertainty 12 × 10−5 is the weighted mean of 11 results ¼½ þ ð Þ ð Þ −8 whose a priori uncertainties were increased by an expansion Ω90 1 1.765 23 × 10 Ω; 276 factor of 14 so that the smallest and largest result differed from ¼½ þ ð Þ ð Þ −8 the recommended value by about twice the latter’s uncertainty. A90 1 8.06 61 × 10 A; 277 For the 2014 adjustment the Task Group decided to follow its ¼½ þ ð Þ ð Þ usual practice and to choose an expansion factor that reduces C 1 8.06 61 × 10−8 C; 278 90 ¼½ þ ð Þ ð Þ

Rev. Mod. Phys., Vol. 88, No. 3, July–September 2016 035009-65 Peter J. Mohr, David B. Newell, and Barry N. Taylor: CODATA recommended values of the fundamental ...

−8 W90 1 17.9 1.2 × 10 W; 279 Task Group to be still so severe that the only recourse was to ¼½ þ ð Þ ð Þ once again exclude it from the final adjustment. −8 F90 1 − 1.765 23 × 10 F; 280 ¼½ ð Þ ð Þ 5. Muon magnetic-moment anomaly H 1 1.765 23 × 10−8 H: 281 90 ¼½ þ ð Þ ð Þ The long-standing significant difference between the theoretically predicted, standard-model value of aμ and the Equations (275) and (276) show that V90 exceeds V and Ω90 experimentally determined value that led to the exclusion exceeds Ω, which means that measured voltages and resis- of the theoretical expression for aμ from the 2010 adjustment tances traceable to the Josephson effect and KJ 90 and the remains; the difference is still at about the 3σ level depending quantum-Hall effect and R , respectively, are− too small K−90 on the way the all-important lowest-order hadronic vacuum relative to the SI. However, the differences are well within the polarization and hadronic light-by-light contributions are −8 −8 40 × 10 uncertainty assigned to V90= V and the 10 × 10 evaluated. Because of the continuing difficulty of reliably uncertainty assigned to Ω90=Ω by the Consultative Committee calculating these terms, the Task Group decided to omit the for Electricity and Magnetism (CCEM) of the International theory from the 2014 adjustment as in 2010. The 2014 Committee for Weights and Measures (CIPM) (Quinn, 1989; recommended values of a and those of other constants that Quinn, 2001). μ depend on it are, therefore, based on experiment.

2. Josephson and quantum-Hall effects 6. Electron magnetic-moment anomaly, fine-structure Watt-balance and XRCD-NA advances have led to tests of constant, and QED the exactness of the quantum-Hall and Josephson effect A useful test of the QED theory of a is to compare two relations R h=e2 and K 2e=h that are less clouded e K J values of α: The first, with relative uncertainty 2.4 × 10−10, is by inconsistencies¼ of the data.¼ Indeed, based on the that obtained by equating the experimental value of a with its 2014 data as used in adjustment 2 summarized in e QED theoretical expression; the second, which is only weakly Table XXVIII, Sec. XIII.B.2, the possible corrections ϵ K dependent on QED theory and with relative uncertainty and ϵ to the quantum-Hall and Josephson effect relations are J 6.2 × 10−10, is that obtained from the measurement of 2.2 1.8 × 10−8 and −0.9 1.5 × 10−8, respectively; see ð Þ ð Þ h=m 87Rb using atom interferometry. These two values Table XXXI, Sec. XIII.B.3. Thus the exactness of these ð Þ relations is experimentally confirmed within about 2 parts (see the first and second rows in Table XX, Sec. XIII.A) differ by 1.8 times the uncertainty of their difference, or 1.8σ. in 108. However, as Table XXXI indicates, some of the initial Although this is acceptable agreement and supports the QED 2014 input data are not as supportive of the exactness of the theory of a , the result of the same comparison in CODATA- relations, most notably the NIST-89 result for Γ ; on the e p0 −90 10 based on the same experimental values of a and other hand, its normalized residual in the adjustment that e h=m 87Rb is 0.4σ. The two main reasons for this rather produced the above values of ϵ and ϵ is 2.3 and because of K J significantð Þ change are the new and somewhat surprisingly its low-weight it is omitted from the 2006 and 2010 final 10 adjustments as well as the 2014 final adjustment. large value of the A1ð Þ coefficient in the theoretical expression for ae which decreased the ae value of α; and the decreased but 3. The new SI highly accurate new value of Ar e which increased the h=m 87Rb value of α. ð Þ The impact of the new data that have become available for ð Þ the 2014 adjustment on the establishment of the new SI by the C. Suggestions for future work CGPM is discussed in detail in the Introduction section of this report (see Sec. I.B.1) and need not be repeated here. Suffice it As discussed, to deal with data inconsistencies the Task to say that the uncertainties of the 2014 recommended values Group decided to (i) omit from the 2014 adjustment the value of the four new defining constants h, e, k, and N , which in A of rp obtained from measurements of the Lamb shift in parts in 108 are 1.2, 0.61, 57, and 1.2, are already sufficiently muonic hydrogen; (ii) omit the theory of the muon mag- small to meet the requirements deemed necessary by the netic-moment anomaly aμ; (iii) increase the initial uncertain- CIPM and CGPM for the adoption of the new SI. 12 6 ties of the cyclotron frequency ratios ωc h =ωc C þ and 3 ð Þ ð Þ ωc HDþ =ωc Heþ relevant to the determination of Ar h by 4. Proton radius anð expansionÞ ð factorÞ of 2.8; and (iv) increase theð initialÞ The severe disagreement of the proton rms charge radius rp uncertainties of the 14 values of G by an expansion factor determined from the Lamb shift in muonic hydrogen with of 6.3. Issues (i), (ii), and (iv) have been with us for some time values determined from H and D transition frequencies and and suggestions for their resolution were given in CODATA- electron-proton scattering experiments present in the 2010 10. Updated versions follow together with a suggestion adjustment remain present in the 2014 adjustment. Although regarding (iii). the uncertainty of the muonic hydrogen value is significantly As also discussed, the data now available provide values of smaller than the uncertainties of these other values, its the defining constants h, e, k, and NA of the new SI with negative impact on the internal consistency of the theoretically uncertainties sufficiently small for its adoption by the 26th predicted and experimentally measured frequencies, as well as CGPM in the fall of 2018 as planned. The final values are to on the value of the Rydberg constant, was considered by the be based on a special adjustment carried out by the Task

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Group during the summer of 2017. Because of its importance, LIST OF SYMBOLS AND ABBREVIATIONS the CIPM has decided that all the data to be used in that adjustment must have been published in an archival journal or ASD NIST Atomic Spectra Database (online) be available in a preprint accepted for publication by 1 July AMDC Atomic Mass Data Center, transferred in 2013 2017. Nevertheless, we include suggestions for work that will to Institute of Modern Physics (IMP), Chinese improve the robustness of the currently available data that Academy of Sciences, Lanzhou, PRC, from determine these important constants. Centre de Spectrométrie Nucléaire et de (i) Work currently underway could solve the “proton radius Spectrométrie de Masse (CSNSM), Orsay, puzzle” and should continue to be pursued as vigorously as France possible. This includes the measurement of hydrogen tran- AME Atomic mass evaluation from the AMDC sition frequencies, the analysis of μ-p and μ-d data, and (completed in year specified) possible new Lamb-shift measurements in μ-h and μ-α. New Ar X relative atomic mass of X: Ar X m X =mu scattering data from experiments such as MUSE (Downie, A ð Þ conventional unit of electricð current:Þ¼ ð AÞ 90 90 ¼ 2014) and PRad (Gasparian, 2014) and improved methods to V90=Ω90 extract rp from such data as well as verification of the ÅÃ ångström star: λ WKα1 0.209 010 0 ÅÃ theory of H, D, and muonic hydrogenlike energy levels could a electron magnetic-momentð Þ¼ anomaly: a e e ¼ also help. ge − 2 =2 (ii) Because the disagreement between a theory and a ðmuonj j magnetic-momentÞ anomaly: a μ μ μ ¼ experiment remains even after many years of effort devoted g − 2 =2 ðj μj Þ to improving the theory and the experimental results on which BIPM International Bureau of Weights and Mea- the theory relies, the solution to the problem may have to wait sures, Sèvres, France until the completion over the next 5 to 10 years of the two BNL Brookhaven National Laboratory, Upton, new experiments underway to remeasure aμ (Mibe, 2011; New York, USA Logashenko et al., 2015). Improved measurements of cross CERN European Organization for Nuclear Research, − sections for the scattering of eþe into hadrons and better data Geneva, Switzerland on the decay of the τ into hadrons could also be useful. CGPM General Conference on Weights and Measures (iii) The two cyclotron frequency ratios in question were CIPM International Committee for Weights and obtained from experiments at the University of Washington Measures and at Florida State University (FSU). A careful study of the CODATA Committee on Data for Science and Technol- University of Washington apparatus that might uncover an ogy of the International Council for Science overlooked systematic effect is not possible because it is no CPT combined charge conjugation, parity inver- longer available. However, the research program at FSU sion, and time reversal continues and the researchers are encouraged to search for c speed of light in vacuum a possible explanation of the disagreement. An experiment d deuteron (nucleus of deuterium D, or 2H) under way to measure the Q value of tritium at the Max- d220 220 lattice spacing of an ideal crystal of f g Planck Institut für Kernphysik, Heidelberg, Germany, could naturally occurring silicon resolve the problem (Streubel et al., 2014). d220 X 220 lattice spacing of crystal X of naturally ð Þ f g (iv) One or more measurements of G with an uncertainty of occurring silicon 1 part in 105 using new and innovative approaches might e symbol for either member of the electron- − finally resolve some of the problems that have plagued the positron pair; when necessary, e or eþ is used reliable determination of G over the past three decades. The to indicate the electron or positron possibliity of transferring the apparatus used by Quinn et al. e elementary charge: absolute value of the (2014) and by Parks and Faller (2014) to other laboratories to charge of the electron F Faraday constant: F N e be used there by new researchers in the hope of uncovering ¼ A overlooked systematic effects could be helpful as well. FSU Florida State University, Tallahassee, Florida, (v) Watt-balance and watt-balance-like measurements of h USA FSUJ Friedrich-Schiller University, Jena, Germany and the XRCD measurement of NA currently under way should continue to be vigorously pursued with the goal of F 90 F 90 F=A90 A G Newtonian¼ð constantÞ of gravitation achieving uncertainties of no more than a few parts in 108. g local acceleration due to gravity This also applies to experiments to determine R, k=h, and g deuteron g-factor: g μ =μ A =R with an uncertainty goal of no more than a few parts in d d d N ϵ g electron g-factor: g ¼2μ =μ 6 8 10 e e ¼ e B 10 . An independent calculation of the A1ð Þ and Að1 Þ g proton g-factor: g 2μ =μ p p ¼ p N coefficients in the theoretical expression for ae would increase g shielded proton g-factor: g 2μ =μ p0 p0 ¼ p0 N confidence in the value of α from ae. For this same reason, g triton g-factor: g 2μ =μ results from experiments currently under way to determine t t ¼ t N gX Y g-factor of particle X in the ground (1S) state h=m X with an uncertainty small enough to provide a value ð Þ of hydrogenic atom Y ð Þ 10 of α with an uncertainty of 5 parts in 10 or less would be g muon g-factor: g 2μ = eℏ=2m valuable. μ μ ¼ μ ð μÞ

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GSI Gesellschaft für Schweironenforschung, NIM National Institute of Metrology, Beijing, PRC Darmstadt, Germany NIST National Institute of Standards and Technol- HD HD molecule (bound state of hydrogen and ogy, Gaithersburg, Maryland and Boulder, deuterium atoms) Colorado, USA HT HT molecule (bound state of hydrogen and NMI National Metrology Institute, Lindfield, tritium atoms) Australia HUST Huazhong University of Science and Tech- NMIJ National Metrology Institute of Japan, nology, Wuhan, PRC Tsukuba, Japan h helion (nucleus of 3He) NPL National Physical Laboratory, Teddington, h Planck constant UK ℏ reduced Planck constant; h=2π NRC National Research Council of Canada, HarvU Harvard University, Cambridge, Massachusetts, Measurement Science and Standards, Ottawa, USA Ontario, Canada IAC International Avogadro Coordination n neutron ILL Institut Max von Laue-Paul Langevin, PTB Physikalisch-Technische Bundesanstalt, Grenoble, France Braunschweig and Berlin, Germany INRIM Istituto Nazionale di Ricerca Metrologica, p proton Torino, Italy QED quantum electrodynamics IRMM Institute for Reference Materials and Mea- p χ2 ν probability that an observed value of surements, Geel, Belgium ð j Þ chi-square for ν degrees of freedom would JILA JILA, University of Colorado and NIST, exceed χ2 Boulder, Colorado, USA R molar gas constant KRISS Korea Research Institute of Standards and R ratio of muon anomaly difference frequency to Science, Taedok Science Town, Republic of free proton NMR frequency Korea 2 1 R Birge ratio: R χ =ν 2 KR/VN KRISS-VNIIM collaboration B B ¼ð Þ rd bound-state rms charge radius of the deuteron KJ Josephson constant: KJ 2e=h 2 ¼ RK von Klitzing constant: RK h=e KJ−90 conventional value of the Josephson constant R conventional value of the¼ von Klitzing con- −1 K−90 KJ: KJ−90 483 597.9 GHz V ¼ stant RK: RK−90 25 812.807 Ω k Boltzmann constant: k R=NA r bound-state rms charge¼ radius of the proton LAMPF Clinton P. Anderson Meson¼ Physics Facility at p R Rydberg constant: R m cα2=2h Los Alamos National Laboratory, Los ∞ ∞ e r x ;x correlation coefficient of¼ estimated values x Alamos, New Mexico, USA ð i jÞ i and x : r x ;x u x ;x = u x u x LANL Los Alamos National Laboratory, Los j ð i jÞ¼ ð i jÞ ½ ð iÞ ð jÞ Alamos, New Mexico, USA Sc self-sensitivity coefficient LENS European Laboratory for Non-Linear Spec- SI Système international d’unités (International troscopy, University of Florence, Italy System of Units) LKB Laboratoire Kastler-Brossel, Paris, France StPtrsb various institutions in St. Petersburg, Russian LK/SY LKB and SYRTE collaboration Federation LNE Laboratoire national de métrologie et d’essais, StanfU Stanford University, Stanford, California, Trappes, France USA MIT Massachusetts Institute of Technology, SUREC Scottish Universities Environmental Research Cambridge, Massachusetts, USA Centre, University of Glasgow, Glasgow, METAS Federal Institute for Metrology, Bern-Wabern, Scotland Switzerland SYRTE Systèmes de référence Temps Espace, Paris, MPIK Max-Planck-Institut für Kernphysik, France Heidelberg, Germany T thermodynamic temperature MPQ Max-Planck-Institut für Quantenoptik, TR&D Tribotech Research and Development Com- Garching, Germany pany, Moscow, Russian Federation MSL Measurement Standards Laboratory, Lower Type A uncertainty evaluation by the statistical analy- Hutt, New Zeland sis of series of observations Type B uncertainty evaluation by means other than M X molar mass of X: M X Ar X Mu ð Þ − ð Þ¼ ð Þ the statistical analysis of series of observations Mu muonium (μþe atom) 3 M molar mass constant: M 10−3 kg mol−1 t triton (nucleus of tritium T, or H) u u UCI University of California, Irvine, Irvine, m unified atomic mass constant:¼ m m 12C =12 u u California, USA m , m X mass of X (for the electron e, proton¼ ð p,Þ and X UMZ Institut für Physik, Johannes Gutenberg ð Þ other elementary particles, the first symbol is Universität Mainz (or simply the University used, i.e., m , m , etc.) e p of Mainz), Mainz, Germany N Avogadro constant A USussex University of Sussex, Brighton, UK

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UWash University of Washington, Seattle, δX nLj additive correction to the theoretical expres- Washington, USA ð Þ sion for an energy level of either hydrogen H UWup University of Wuppertal, Wuppertal, Germany or deuterium D with quantum numbers n, L, UZur University of Zurich, Zurich, Switzerland and j u unified atomic mass unit (also called the ϵJ hypothetical correction to the Josephson effect 21 dalton, Da): 1 u mu m C =12 relation: K 2e=h 1 ϵ ¼ ¼ ð Þ J ¼ð Þð þ JÞ udiff standard uncertainty of the difference between ϵK hypothetical correction to the quantum-Hall- two values (σ is sometimes used in place effect relation: R h=e2 1 ϵ K ¼ð Þð þ KÞ of udiff ) ϵ0 electric constant (vacuum electric permittiv- 2 u xi standard uncertainty (i.e., estimated standard ity): ϵ 1=μ c ð Þ 0 ¼ 0 deviation) of an estimated value xi of a symbol used to relate an input datum to its ≐ quantity Xi (also simply u) observational equation ur xi relative standard uncertainty of an estimated λ XKα1 wavelength of Kα1 x-ray line of element X ð Þ ð Þ value xi of a quantity Xi: ur xi μ symbol for either member of the muon- ð Þ¼ − u xi = xi ;xi ≠ 0 (also simply ur) antimuon pair; when necessary, μ or μþ is u x ;x covarianceð Þ j j of estimated values x and x used to indicate the negative muon or positive ð i jÞ i j u x ;x relative covariance of estimated values x and muon rð i jÞ i x : u x ;x u x ;x = x x μB Bohr magneton: μB eℏ=2me j rð i jÞ¼ ð i jÞ ð i jÞ ¼ V Si molar volume of naturally occurring silicon μN nuclear magneton: μN eℏ=2mp mð Þ ¼ VNIIM D. I. Mendeleyev All-Russian Research Insti- μX Y magnetic moment of particle X in atom or tute for Metrology, St. Petersburg, Russian ð Þ molecule Y Federation μ0 magnetic constant (vacuum magnetic per- −7 2 V conventional unit of voltage based on meability): μ0 4π × 10 N=A 90 ¼ the Josephson effect and K : V μX, μ0 magnetic moment, or shielded magnetic mo- J−90 90 ¼ X KJ−90=KJ V ment, of particle X WarsUð UniversityÞ of Warsaw, Warsaw, Poland ν degrees of freedom of a particular adjustment 2 W conventional unit of power: W V =Ω ν fp difference between muonium hyperfine split- 90 90 ¼ 90 90 ð Þ XROI combined x-ray and optical interferometer ting Zeeman transition frequencies ν34 and ν12 xu CuKα1 Cu x unit: λ CuKα1 1 537.400 xu CuKα1 at a magnetic flux density B corresponding to ð Þ ð Þ¼ ð Þ xu MoKα1 Mo x unit: λ MoKα1 707.831 xu MoKα1 the free proton NMR frequency fp ð Þ ð Þ¼ ð Þ x X amount-of-substance fraction of X σ Stefan-Boltzmann constant: σ 2 5k4= ð Þ π YaleU Yale University, New Haven, Connecticut, 15h3c2 ¼ ð Þ USA τ symbol for either member of the tau-antitau 2 − α fine-structure constant: α e =4πϵ0ℏc ≈ pair; when necessary, τ or τþ is used to ¼ 1=137 indicate the negative tau or positive tau α alpha particle (nucleus of 4He) χ2 the statistic “chi square” Γ lo Γ lo γ A A−1, X p or h Ω conventional unit of resistance based on X0 −90ð Þ X0 −90ð Þ¼ð X0 90Þ ¼ 90 Γ hi Γ hi γ =A A the quantum-Hall effect and R ∶ Ω p0 −90ð Þ p0 −90ð Þ¼ð p0 90Þ K−90 90 ¼ γ proton gyromagnetic ratio: γ 2μ =ℏ RK=RK−90 Ω p p ¼ p ð Þ γp0 shielded proton gyromagnetic ratio: γp0 ¼ ACKNOWLEDGMENTS 2μp0 =ℏ γ shielded helion gyromagnetic ratio: γ h0 h0 As always, we gratefully acknowledge the help of our 2 μ = ¼ h0 ℏ many colleagues throughout the world who provided the A n j j ΔEB X þ energy required to remove n electrons from a ð Þ CODATA Task Group on Fundamental Constants with results neutral atom prior to formal publication and for promptly and patiently A i ΔEI X þ electron ionization energies, i 0 to n − 1 ð Þ ¼ answering our many questions about their work. We wish to ΔνMu muonium ground-state hyperfine splitting thank our fellow Task Group members for their invaluable δC additive correction to the theoretical expression guidance and suggestions during the course of the 2014 12 5 for the electron ground-state g-factor in C þ adjustment effort. δe additive correction to the theoretical expression for the electron magnetic-moment anomaly ae REFERENCES δ additive correction to the theoretical expres- Mu Adikaram, D., et al., 2015, Phys. Rev. Lett. 114, 062003. sion for the ground-state hyperfine splitting of Adkins, G. S., M. Kim, C. Parsons, and R. N. Fell, 2015, Phys. Rev. muonium ν Δ Mu Lett. 115, 233401. δSi additive correction to the theoretical expres- Aldrich, L. T., and A. O. Nier, 1948, Phys. Rev. 74, 1590. sion for the electron ground-state g-factor in Aleksandrov, V. S., and Y. I. Neronov, 2011, Zh. Eksp. Teor. Fiz. 93, 28 13 Si þ 337 [JETP 93, 305 (2011)].

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