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CODATA recommended values of the fundamental physical constants:

2006*

Peter J. Mohr, Barry N. Taylor, and David B. Newell

National Institute of Standards and Technology, Gaithersburg,

Maryland 20899-8420, USA

Published 6 June 2008

This paper gives the 2006 self-consistent set of values of the basic constants and conversion factors ofphysics and chemistry recommended by the Committee on Data for Science and TechnologyCODATA for international use. Further, it describes in detail the adjustment of the values of theconstants, including the selection of the final set of input data based on the results of least-squaresanalyses. The 2006 adjustment takes into account the data considered in the 2002 adjustment as wellas the data that became available between 31 December 2002, the closing date of that adjustment,and 31 December 2006, the closing date of the new adjustment. The new data have led to a significantreduction in the uncertainties of many recommended values. The 2006 set replaces the previouslyrecommended 2002 CODATA set and may also be found on the World Wide Web atphysics.nist.gov/constants.

DOI: 10.1103/RevModPhys.80.633 PACS numbers: 06.20.Jr, 12.20.m

CONTENTS

I. Introduction 635

A. Background 635

B. Time variation of the constants 635

C. Outline of paper 636

II. Special Quantities and Units 636

Relative Atomic Masses 637

A. Relative atomic masses of atoms 637

B. Relative atomic masses of ions and nuclei 638C. Cyclotron resonance measurement of the electron

relative atomic mass Are 639

V. Atomic Transition Frequencies 639

A. Hydrogen and deuterium transition frequencies, theRydberg constant R, and the proton and deuteron

charge radii Rp, Rd 639

1. Theory relevant to the Rydberg constant 639

a. Dirac eigenvalue 639

b. Relativistic recoil 640c. Nuclear polarization 640

d. Self energy 640

e. Vacuum polarization 641

f. Two-photon corrections 642g. Three-photon corrections 644

h. Finite nuclear size 644

i. Nuclear-size correction to self energy and

vacuum polarization 645 j. Radiative-recoil corrections 645

k. Nucleus self energy 645

l. Total energy and uncertainty 645

m. Transition frequencies between levels with

n=2 6462. Experiments on hydrogen and deuterium 646

3. Nuclear radii 647

B. Antiprotonic helium transition frequencies and Are 647

1. Theory relevant to antiprotonic helium 648

2. Experiments on antiprotonic helium 649

3. Values ofAre inferred from antiprotonic

helium 650C. Hyperfine structure and fine structure 650

1. Hyperfine structure 650

2. Fine structure 651

V. Magnetic Moment Anomalies and g-Factors 651

III.

I

*This report was prepared by the authors under the auspicesof the CODATA Task Group on Fundamental Constants. Themembers of the task group are:

F. Cabiati, Istituto Nazionale di Ricerca Metrologica, ItalyK. Fujii, National Metrology Institute of Japan, JapanS. G. Karshenboim, D. I. Mendeleyev All-Russian ResearchInstitute for Metrology, Russian FederationI. Lindgren, Chalmers University of Technology and GteborgUniversity, SwedenB. A. Mamyrin deceased, A. F. Ioffe Physical-Technical In-stitute, Russian FederationW. Martienssen, Johann Wolfgang Goethe-Universitt, Ger-manyP. J. Mohr, National Institute of Standards and Technology,United States of AmericaD. B. Newell, National Institute of Standards and Technology,United States of AmericaF. Nez, Laboratoire Kastler-Brossel, France

B. W. Petley, National Physical Laboratory, United KingdomT. J. Quinn, Bureau international des poids et mesuresB. N. Taylor, National Institute of Standards and Technology,United States of AmericaW. Wger, Physikalisch-Technische Bundesanstalt, GermanyB. M. Wood, National Research Council, CanadaZ. Zhang, National Institute of Metrology, China Peoples Re-public ofThis review is being published simultaneously by the Journalof Physical and Chemical Reference Data.

[email protected]@[email protected]

REVIEWS OF MODERN PHYSICS, VOLUME 80, APRILJUNE 2008

0034-6861/2008/802/63398 2008 The American Physical Society633
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A. Electron magnetic moment anomaly ae and thefine-structure constant 652

1. Theory ofae 6522. Measurements ofae 654

a. University of Washington 654b. Harvard University 654

3. Values of inferred from ae 654B. Muon magnetic moment anomaly a 655

1. Theory ofa 655

2. Measurement ofa: Brookhaven 656a. Theoretical value ofa and inferred value

of 657C. Bound electron g-factor in 12C5+ and in 16O7+ and

Are 6571. Theory of the bound electron g-factor 6582. Measurements ofge

12C5+ and ge16O7+ 661

a. Experiment on g 12e C5+ and inferred value

ofAre 661b. Experiment on ge

16O7+ and inferred valueofAre 661

c. Relations between ge12C5+ and ge

16O7+ 662VI. Magnetic Moment Ratios and the Muon-Electron Mass

Ratio 662A. Magnetic moment ratios 662

1. Theoretical ratios of atomic bound-particleto free-particle g-factors 662

2. Ratio measurements 663a. Electron to proton magnetic moment ratio

e /p 663b. Deuteron to electron magnetic moment

ratio d / e 664c. Proton to deuteron and triton to proton

magnetic moment ratios p / d and t / p 664d. Electron to shielded proton magnetic

moment ratio e / p 665

e. Shielded helion to shielded protonmagnetic moment ratio h / p 666

f. Neutron to shielded proton magneticmoment ratio n / p 666

B. Muonium transition frequencies, the muon-protonmagnetic moment ratio / p, and muon-electronmass ratio m/ me 666

1. Theory of the muonium ground-statehyperfine splitting 666

2. Measurements of muonium transitionfrequencies and values of /p and m/ me 668

a. LAMPF 1982 668b. LAMPF 1999 668

c. Combined LAMPF results 669Electrical Measurements 669

. Shielded gyromagnetic ratios , the fine-structureconstant , and the Planck constant h 669

1. Low-field measurements 670a. NIST: Low field 670b. NIM: Low field 670c. KRISS/VNIIM: Low field 670

2. High-field measurements 671a. NIM: High field 671b. NPL: High field 671

B. von Klitzing constant RK and 671

VII.A

1. NIST: Calculable capacitor 6712. NMI: Calculable capacitor 6723. NPL: Calculable capacitor 6724. NIM: Calculable capacitor 6725. LNE: Calculable capacitor 672

C. Josephson constant KJ and h 6721. NMI: Hg electrometer 6722. PTB: Capacitor voltage balance 673

D. Product K2JRK and h 673

1. NPL: Watt balance 6732. NIST: Watt balance 674

a. 1998 measurement 674b. 2007 measurement 674

3. Other values 6754. Inferred value ofKJ 675

E. Faraday constant F and h 6751. NIST: Ag coulometer 676

VIII. Measurements Involving Sil icon Crystals 676A. 220 lattice spacing of silicon d220 676

1. X-ray optical interferometer measurementsofd220X 677

a. PTB measurement ofd220W4.2a 678

b. NMIJ measurement ofd220NR3 678c. INRIM measurement ofd220W4.2a and

d220MO* 6782. d220 difference measurements 679

a. NIST difference measurements 679b. PTB difference measurements 680

B. Molar volume of silicon VmSi and the Avogadroconstant NA 680

C. Gamma-ray determination of the neutron relativeatomic mass Arn 681

D. Quotient of Planck constant and particle mass

h / mX and 6821. Quotient h / mn 682

2. Quotient h / m133Cs 6823. Quotient h / m87Rb 683

IX. Thermal Physical Quantities 684A. Molar gas constant R 684

1. NIST: Speed of sound in argon 6852. NPL: Speed of sound in argon 6853. Other values 685

B. Boltzmann constant k 685C. Stefan-Boltzmann constant 686

X. Newtonian Constant of Gravitation G 686A. Updated values 686

1. Huazhong University of Science andTechnology 686

2. University of Zurich 687B. Determination of 2006 recommended value ofG 688C. Prospective values 689

XI. X-ray and Electroweak Quantities 689A. X-ray units 689B. Particle Data Group input 689

XII. Analysis of Data 690A. Comparison of data 690B. Multivariate analysis of data 699

1. Summary of adjustments 6992. Test of the Josephson and quantum Hall

effect relations 704

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XIII. The 2006 CODATA Recommended Values 706A. Calculational details 706B. Tables of values 707

XIV. Summary and Conclusion 716A. Comparison of 2006 and 2002 CODATA

recommended values 716B. Some implications of the 2006 CODATA

recommended values and adjustment for physics andmetrology 719

C. Outlook and suggestions for future work 721Acknowledgments 722Nomenclature 722References 724

I. INTRODUCTION

A. Background

This paper gives the complete 2006 CODATA self-consistent set of recommended values of the fundamen-tal physical constants and describes in detail the 2006

least-squares adjustment, including the selection of thefinal set of input data based on the results of least-squares analyses. Prepared under the auspices of theCODATA Task Group on Fundamental Constants, thisis the fifth such report of the Task Group since its estab-lishment in 19691 and the third in the four-year cycle ofreports begun in 1998. The 2006 set of recommendedvalues replaces its immediate predecessor, the 2002 set.The closing date for the availability of the data consid-ered for inclusion in this adjustment was 31 December2006. As a consequence of the new data that becameavailable in the intervening four years, there has been asignificant reduction of the uncertainty of many con-

stants. The 2006 set of recommended values first becameavailable on 29 March 2007 at http://physics.nist.gov/constants, a web site of the NIST Fundamental Con-stants Data Center FCDC.

The 1998 and 2002 reports describing the 1998 and2002 adjustments Mohr and Taylor, 2000, 2005, re-ferred to as CODATA-98 and CODATA-02 throughoutthis article, describe in detail much of the currentlyavailable data, its analysis, and the techniques used toobtain a set of best values of the constants using thestandard method of least squares for correlated inputdata. This paper focuses mainly on the new informationthat has become available since 31 December 2002 andreferences the discussions in CODATA-98 andCODATA-02 for earlier work in the interest of brevity.More specifically, if a potential input datum is not dis-cussed in detail, the reader can assume that it or aclosely related datum has been reviewed in eitherCODATA-98 or CODATA-02.

The reader is also referred to these papers for a dis-cussion of the motivation for and the philosophy behind

the periodic adjustment of the values of the constantsand for descriptions of how units, quantity symbols, nu-merical values, numerical calculations, and uncertaintiesare treated, in addition to how the data are character-ized, selected, and evaluated. Since the calculations arecarried out with more significant figures than are dis-played in the text to avoid rounding errors, data withmore digits are available on the FCDC web site for pos-sible independent analysis.

However, because of their importance, we recall indetail the following two points also discussed in thesereferences. First, although it is generally agreed that thecorrectness and overall consistency of the basic theoriesand experimental methods of physics can be tested bycomparing values of particular fundamental constantsobtained from widely differing experiments, throughoutthis adjustment, as a working principle, we assume thevalidity of the underlying physical theory. This includesspecial relativity, quantum mechanics, quantum electro-dynamics QED, the standard model of particle physics,including combined charge conjugation, parity inversion,

and time reversal CPT invariance, and the theory ofthe Josephson and quantum Hall effects, especially theexactness of the relationships between the Josephsonand von Klitzing constants KJ and RK and the elemen-tary charge e and Planck constant h. In fact, tests ofthese relations K 2J=2e / h and RK =h / e using the inputdata of the 2006 adjustment are discussed in Sec.XII.B.2.

The second point has to do with the 31 December2006 closing date for data to be considered for inclusionin the 2006 adjustment. A datum was considered to havemet this date, even though not yet reported in an archi-val journal, as long as a description of the work was

available that allowed the Task Group to assign a validstandard uncertainty uxi to the datum. Thus, any inputdatum labeled with an 07 identifier because it was pub-lished in 2007 was, in fact, available by the cutoff date.Also, some references to results that became availableafter the deadline are included, even though the resultswere not used in the adjustment.

B. Time variation of the constants

This subject, which was briefly touched upon in

CODATA-02, continues to be an active field of experi-mental and theoretical research, because of its impor-tance to our understanding of physics at the most funda-mental level. Indeed, a large number of papers relevantto the topic have appeared in the last four years; seethe FCDC bibliographic database on the fundamen-tal constants using the keyword time variationat http://physics.nist.gov/constantsbib. For example, seeFortier et al. 2007 and Lea 2007. However, there hasbeen no laboratory observation of the time dependenceof any constant that might be relevant to the recom-mended values.

1The Committee on Data for Science and Technology wasestablished in 1966 as an interdisciplinary committee of theInternational Council for Science.

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C. Outline of paper

Section II touches on special quantities and units, thatis, those that have exact values by definition.

Sections IIIXI review all available experimental andtheoretical data that might be relevant to the 2006 ad-justment of the values of the constants. As discussed inAppendix E of CODATA-98, in a least-squares analysis

of the fundamental constants, the numerical data, bothexperimental and theoretical, also called observationaldata or input data, are expressed as functions of a set ofindependent variables called adjusted constants. Thefunctions that relate the input data to the adjusted con-stants are called observational equations, and the least-squares procedure provides best estimated values, in theleast-squares sense, of the adjusted constants. The focusof the review-of-data sections is thus the identificationand discussion of the input data and observational equa-tions of interest for the 2006 adjustment. Althoughnot all observational equations that we use are expli-citly given in the text, all are summarized in TablesXXXVIII, XL, and XLII of Sec. XII.B.

As part of our discussion of a particular datum, weoften deduce from it an inferred value of a constant,such as the fine-structure constant or Planck constanth. It should be understood, however, that these inferredvalues are for comparison purposes only; the datumfrom which it is obtained, not the inferred value, is theinput datum in the adjustment.

Although just four years separate the 31 Decemberclosing dates of the 2002 and 2006 adjustments, there area number of important new results to consider. Experi-mental advances include the 2003 Atomic Mass Evalua-tion from the Atomic Mass Data Center AMDC,which provides new values for the relative atomicmasses ArX of a number of relevant atoms; a newvalue of the electron magnetic moment anomaly ae frommeasurements on a single electron in a cylindrical Pen-ning trap, which provides a value of the fine-structureconstant ; better measurements of the relative atomicmasses of 2H, 3H, and 4He; new measurements of tran-sition frequencies in antiprotonic helium pAHe+ atomthat provide a competitive value of the relative atomicmass of the electron Are; improved measurements ofthe nuclear magnetic resonance NMR frequencies ofthe proton and deuteron in the HD molecule and of theproton and triton in the HT molecule; a highly accurate

value of the Planck constant obtained from an improvedmeasurement of the product K2JRK using a moving-coilwatt balance; new results using combined x-ray and op-tical interferometers for the 220 lattice spacing ofsingle crystals of naturally occurring silicon; and an ac-curate value of the quotient h / m87Rb obtained bymeasuring the recoil velocity of rubidium-87 atoms uponabsorption or emission of photonsa result that pro-vides an accurate value of that is virtually independentof the electron magnetic moment anomaly.

Theoretical advances include improvements in certainaspects of the theory of the energy levels of hydrogen

and deuterium; improvements in the theory of antipro-tonic helium transition frequencies that, together withthe new transition frequency measurements, have led tothe aforementioned competitive value of Are; a newtheoretical expression for ae that, together with the newexperimental value ofae, has led to the aforementionedvalue of; improvements in the theory for the g-factorof the bound electron in hydrogenic ions with nuclearspin quantum number i=0 relevant to the determinationof Are; and improved theory of the ground-state hy-perfine splitting of muonium Mu the +e atom.

Section XII describes the analysis of the data, with theexception of the Newtonian constant of gravitation,which is analyzed in Sec. X. The consistency of the dataand potential impact on the determination of the 2006recommended values were appraised by comparingmeasured values of the same quantity, comparing mea-sured values of different quantities through inferred val-ues of a third quantity such as or h, and finally byusing the method of least squares. Based on these inves-tigations, the final set of input data used in the 2006adjustment was selected.

Section XIII provides, in several tables, the 2006CODATA recommended values of the basic constantsand conversion factors of physics and chemistry, includ-ing the covariance matrix of a selected group of con-stants.

Section XIV concludes the paper with a comparisonof the 2006 and 2002 recommended values of the con-stants, a survey of implications for physics and metrol-ogy of the 2006 values and adjustment, and suggestionsfor future work that can advance our knowledge of thevalues of the constants.

II. SPECIAL QUANTITIES AND UNITS

Table I lists those quantities whose numerical valuesare exactly defined. In the International System of UnitsSI BIPM, 2006, used throughout this paper, the defi-nition of the meter fixes the speed of light in vacuum c,the definition of the ampere fixes the magnetic constantalso called the permeability of vacuum 0, and thedefinition of the mole fixes the molar mass of the carbon12 atom M12C to have the exact values given in thetable. Since the electric constant also called the permit-tivity of vacuum is related to 0 by 0 =1/0c2, it too isknown exactly.

The relative atomic mass ArX of an entity X is de-fined by ArX =mX / mu, where mX is the mass ofXand mu is the atomic mass constant defined by

mu =1

m12C = 1 u 1.66 2712 10 kg, 1

where m12C is the mass of the carbon 12 atom and u isthe unified atomic mass unit also called the dalton, Da.Clearly, ArX is a dimensionless quantity and Ar

12C=12 exactly. The molar mass MX of entity X, which isthe mass of one mole of X with SI unit kg/mol, is givenby




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MX = NAmX =ArXMu, 2

where NA 6.02 1023/mol is the Avogadro constantand Mu=103 kg/mol is the molar mass constant. Thenumerical value of NA is the number of entities in onemole, and since the definition of the mole states that onemole contains the same number of entities as there are

in 0.012 kg of carbon 12, M12

C =0.012 kg/mol exactly.The Josephson and quantum Hall effects have playedand continue to play important roles in adjustments ofthe values of the constants, because the Josephson andvon Klitzing constants KJ and RK, which underlie thesetwo effects, are related to e and h by

2e h cKJ = , R

0K =

e2= . 3

h 2

Although we assume these relations are exact, and noevidenceeither theoretical or experimentalhas beenput forward that challenges this assumption, the conse-quences of relaxing it are explored in Sec. XII.B.2. Some

references to recent work related to the Josephson andquantum Hall effects may be found in the FCDC biblio-graphic database see Sec. I.B.

The next-to-last two entries in Table I are the conven-tional values of the Josephson and von Klitzing con-stants adopted by the International Committee forWeights and Measures CIPM and introduced on Janu-ary 1, 1990 to establish worldwide uniformity in themeasurement of electrical quantities. In this paper,all electrical quantities are expressed in SI units. How-ever, those measured in terms of the Josephson andquantum Hall effects with the assumption that KJ andRK have these conventional values are labeled with a

subscript 90.For high-accuracy experiments involving the force ofgravity, such as the watt balance, an accurate measure-ment of the local acceleration of free fall at the site ofthe experiment is required. Fortunately, portable andeasy-to-use commercial absolute gravimeters are avail-able that can provide a local value of g with a relativestandard uncertainty of a few parts in 109. That theseinstruments can achieve such a small uncertainty if prop-erly used is demonstrated by a periodic internationalcomparison of absolute gravimeters ICAG carried outat the International Bureau of Weights and Measures

BIPM, Svres, France; the seventh and most recent,denoted ICAG-2005, was completed in September 2005Vitushkin, 2007; the next is scheduled for 2009. In thefuture, atom interferometry or Bloch oscillations usingultracold atoms could provide a competitive or possiblymore accurate method for determining a local value ofgPeters et al., 2001;

2005.

McGuirk et al., 2002; Clad et al.,

III. RELATIVE ATOMIC MASSES

Included in the set of adjusted constants are the rela-tive atomic masses ArX of a number of particles, at-oms, and ions. Tables IIVI and the following sectionssummarize the relevant data.

A. Relative atomic masses of atoms

Most values of the relative atomic masses of neutralatoms used in this adjustment are taken from the 2003atomic mass evaluation AME2003 of the Atomic MassData Center, Centre de Spectromtrie Nuclaire et deSpectromtrie de Masse CSNSM, Orsay, France Audiet al., 2003; Wapstra et al., 2003; AMDC, 2006. The re-sults of AME2003 supersede those of both the 1993atomic mass evaluation and the 1995 update. Table IIlists the values from AME2003 of interest here, whileTable III gives the covariance for hydrogen and deute-rium AMDC, 2003. Other non-negligible covariancesof these values are discussed in the appropriate sections.

Table IV gives six values ofArX relevant to the 2006

adjustment reported since the completion and publica-tion of AME2003 in late 2003 that we use in place of thecorresponding values in Table II.

The 3H and 3He values are those reported by theSMILETRAP group at the Manne Siegbahn LaboratoryMSL, Stockholm, Sweden Nagy et al., 2006 using aPenning trap and a time-of-flight technique to detect cy-clotron resonances. This new 3He result is in goodagreement with a more accurate, but still preliminary,result from the University of Washington group in Se-attle, WA, USA Van Dyck, 2006. The AME2003 valuesfor 3H and 3He were influenced by an earlier result for

Value

2 458 m s1

07 N A2 =12.566 370 614. .. 710 N A21 =8.854 187 817.. . 1 10 2 F m1

gmol1

03 kg mol1

conventional value of Josephson constant K 1J90 483 597.9 GHz Vconventional value of von Klitzing constant RK90 25 812.807

TABLE I. Some exact quantities relevant to the 2006 adjustment.

Quantity Symbol

speed of light in vacuum c, c0 299 79magnetic constant 0 41electric constant c20 0 relative atomic mass of 12C Ar

12C 12molar mass constant M 103u kmolar mass of 12C A 12CMu M12r C 12 1




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TABLE II. Values of the relative atomic masses of the neutronand various atoms as given in the 2003 atomic mass evaluationtogether with the defined value for 12C.

Relative atomic Relative standacertainty ur

61010

01010

rdAtom mass ArX un

n 1.008 664 915 7456 5.1H 1.007 825 032 07 10 1. 2

H 2.014 101 777 8536 1.810103H 3.016 049 277725 8.210103He 3.016 029 319126 8.610104He 4.002 603 254 15363 1.6101112C 12 Exact16O 15.994 914 619 5616 1.0101128Si 27.976 926 532519 6.9101129Si 28.976 494 70022 107.61030Si 29.973 770 17132 1.110936Ar 35.967 545 10528 7.8101038Ar 37.962 732 3936 9.510940Ar 39.962 383 122529 7.21011

87Rb 86.909 180 52612 1.41010107Ag 106.905 096846 4.3108109Ag 108.904 752331 82.910133 10

3He from the University of Washington group, which isin disagreement with their new result.

The values for 4He and 16O are those reported by theUniversity of Washington group Van Dyck et al., 2006using their improved mass spectrometer; they are basedon a thorough reanalysis of data that yielded prelimi-

nary results for these atoms that were used inAME2003. They include an experimentally determinedimage-charge correction with a relative standard uncer-tainty ur =7.9 1012 in the case of

4He and ur =4.0 1012 in the case of 16O. The value of Ar

2H is alsofrom this group and is a near-final result based on theanalysis of ten runs carried out over a 4 year periodVan Dyck, 2006. Because the result is not yet final, thetotal uncertainty is conservatively assigned; ur =9.9 1012 for the image-charge correction. This value of

Ar2H is consistent with the preliminary value reported

by Van Dyck et al. 2006 based on the analysis of onlythree runs.

The covariance and correlation coefficient of Ar3H

and Ar3He given in Table V are due to the common

component of uncertainty ur =1.4 1010 of the relativeatomic mass of the H +2 reference ion used in theSMILETRAP measurements; the covariances and cor-

relation coefficients of the University of Washington val-ues of A 2H, A 4He, and A 16r r r O given in Table VIare due to the uncertainties of the image-charge correc-tions, which are based on the same experimentally de-termined relation.

The 29Si value is that implied by the ratioAr

29Si+ /Ar28Si H+=0.999 715 124 181265 obtained

at the Massachusetts Institute of Technology MIT,Cambridge, MA, USA, using a recently developed tech-nique of determining mass ratios by directly comparingthe cyclotron frequencies of two different ions simulta-neously confined in a Penning trap Rainville et al.,2005. The relative atomic mass work of the MIT group

has now been transferred to Florida State University,Tallahassee, FL, USA. This approach eliminates manycomponents of uncertainty arising from systematic ef-fects. The value for Ar

29Si is given in the supplemen-tary information to Rainville et al. 2005 and has a sig-nificantly smaller uncertainty than the correspondingAME2003 value.

B. Relative atomic masses of ions and nuclei

The relative atomic mass ArX of a neutral atom X isgiven in terms of the relative atomic mass of an ion ofthe atom formed by the removal ofn electrons by

TABLE V. The variances, covariance, and correlation coeffi-cient of the values of the SMILETRAP relative atomic massesof tritium and helium 3. The number in bold above the maindiagonal is 1018 times the numerical value of the covariance,the numbers in bold on the main diagonal are 1018 times thenumerical values of the variances, and the number in italicsbelow the main diagonal is the correlation coefficient.

TABLE III. The variances, covariance, and correlation coeffi-cient of the AME2003 values of the relative atomic masses ofhydrogen and deuterium. The number in bold above the maindiagonal is 1018 times the numerical value of the covariance,the numbers in bold on the main diagonal are 1018 times thenumerical values of the variances, and the number in italicsbelow the main diagonal is the correlation coefficient.

A 1 2r H Ar H

Ar1H 0.0107 0.0027

Ar2H 0.0735 0.1272

Cs 132.905 451 93224 1.810

TABLE IV. Values of the relative atomic masses of variousatoms that have become available since the 2003 atomic massevaluation.

AtomRelative atomicmass ArX

Relative standarduncertainty ur

2H 2.014 101 778 04080 4.0 10113H 3.016 049 278725 8.3 10103

He 3.016 029 321726 8.6 10104He 4.002 603 254 13162 1.5 101116O 15.994 914 619 5718 1.1 101129Si 28.976 494 662520 6.9 1011

Ar3H Ar

3He

Ar3H 6.2500 0.1783

Ar3He 0.0274 6.7600




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E X E Xn+A nrX =A X

+r + nAre

b b2 . 4muc

Here EbX / muc2 is the relative-atomic-mass equivalentof the total binding energy of the Z electrons of theatom, where Z is the atomic number proton number,and EbXn+ /muc2 is the relative-atomic-mass equivalentof the binding energy of the Zn electrons of the Xn+

ion. For a fully stripped atom, that is, for n=Z, XZ+ is N,where N represents the nucleus of the atom, andEbX

Z+ /muc2=0, which yields the first few equations ofTable XL in Sec. XII.B.

The binding energies Eb used in this work are thesame as those used in the 2002 adjustment; see Table IVof CODATA-02. For tritium, which is not includedthere, we use the value 1.097 185 439 107 m1 Ko-tochigova, 2006. The uncertainties of the binding ener-gies are negligible for our application.

C. Cyclotron resonance measurement of the electron relative

atomic mass Ar(e)

A value of Are is available from a Penning-trapmeasurement carried out by the University of Washing-ton group Farnham et al., 1995; it is used as an inputdatum in the 2006 adjustment, as it was in the 2002 ad-justment:

Are = 0.000 548 579 911112 2.1 109. 5

IV. ATOMIC TRANSITION FREQUENCIES

Atomic transition frequencies in hydrogen, deute-rium, and antiprotonic helium yield information on theRydberg constant, the proton and deuteron charge radii,and the relative atomic mass of the electron. The hyper-fine splitting in hydrogen and fine-structure splitting inhelium do not yield a competitive value of any constantat the current level of accuracy of the relevant experi-ment and/or theory. All of these topics are discussed inthis section.

A. Hydrogen and deuterium transition frequencies, the

Rydberg constant R

, and the proton and deuteron charge

radii Rp, Rd

The Rydberg constant is related to other constants bythe definition

2 mecR = . 62h

It can be accurately determined by comparing measuredresonant frequencies of transitions in hydrogen H anddeuterium D to the theoretical expressions for the en-ergy level differences in which it is a multiplicative fac-tor.

1. Theory relevant to the Rydberg constant

The theory of the energy levels of hydrogen and deu-terium atoms relevant to the determination of the Ryd-berg constant R, based on measurements of transitionfrequencies, is summarized in this section. Complete in-formation necessary to determine the theoretical valuesof the relevant energy levels is provided, with an empha-sis on results that have become available since the pre-vious adjustment described in CODATA-02. For brevity,references to earlier work, which can be found in Eideset al. 2001b for example, are not included here.

An important consideration is that the theoretical val-ues of the energy levels of different states are highlycorrelated. For example, for S states, the uncalculatedterms are primarily of the form of an unknown commonconstant divided by n3. This fact is taken into account bycalculating covariances between energy levels in addi-tion to the uncertainties of the individual levels as dis-cussed in detail in Sec. IV.A.1.l. In order to take thesecorrelations into account, we distinguish between com-ponents of uncertainty that are proportional to 1/ n3, de-noted by u0, and components of uncertainty that are es-sentially random functions of n, denoted by un.

The energy levels of hydrogenlike atoms are deter-mined mainly by the Dirac eigenvalue, QED effectssuch as self energy and vacuum polarization, and nuclearsize and motion effects, all of which are summarized inthe following sections.

a. Dirac eigenvalue

The binding energy of an electron in a static Coulombfield the external electric field of a point nucleus ofcharge Ze with infinite mass is determined predomi-nantly by the Dirac eigenvalue

E =fn,jm c2D e , 7

where

fn,j = Z2 1/21 +n 2

,

8n and j are the principal quantum number and total an-gular momentum of the state, respectively, and

TABLE VI. The variances, covariances, and correlation coef-ficients of the University of Washington values of the relativeatomic masses of deuterium, helium 4, and oxygen 16. Thenumbers in bold above the main diagonal are 1020 times thenumerical values of the covariances, the numbers in bold onthe main diagonal are 1020 times the numerical values of thevariances, and the numbers in italics below the main diagonalare the correlation coefficients.

Ar

2

H Ar4

He 16

OArAr

2H 0.6400 0.0631 0.1276

Ar4He 0.1271 0.3844 0.2023

16OAr 0.0886 0.1813 3.2400




8/98

1=j+

2 12 1/2

j+ Z2 . 92

Although we are interested only in the case in which thenuclear charge is e, we retain the atomic number Z inorder to indicate the nature of various terms.

Corrections to the Dirac eigenvalue that approxi-mately take into account the finite mass of the nucleusm

Nare included in the more general expression for

atomic energy levels, which replaces Eq. 7 Barker andGlover, 1955; Sapirstein and Yennie, 1990,

m2c2EM= Mc2 + fn,j 1m 2 fn,j 12

rrc 2M

1 2l0 Z4m3

+ rc

2 + , 102l+ 1 2n3mN

where l is the nonrelativistic orbital angular momentumquantum number, is the angular-momentum-parityquantum number 1= 1jl+1/2j+ 2, M=me+ mN, andmr =memN /me+ mN is the reduced mass.

b. Relativistic recoil

Relativistic corrections to Eq. 10 associated withmotion of the nucleus are considered relativistic-recoilcorrections. The leading term, to lowest order in Z andall orders in me / mN, is Erickson, 1977; Sapirstein andYennie, 1990

m3 Z5 1 8ES =

rm 22 l0 ln 3 ec Z 2 ln kmemN n 3 3 0n,l

1 7 2 l0 a

9 3n

m2

m2 l0

N e

m2 me 2 mNN ln r me ln

m

mr , 11

where

an = 2ln 2n

n

+ 1 1

+ 1 i=1 i 2n

1 l0 + l0 .ll+ 12l+ 1

12

To lowest order in the mass ratio, higher-order correc-tions in Z have been extensively investigated; the con-tribution of the next two orders in Z is

m 6E = e

ZR m

mN n3 ec

2

D60 + D2 2

72Z ln Z + , 13

where for nS1/2 states Pachucki and Grotch, 1995; Eidesand Grotch, 1997c

D = 4 ln 2 760 2 14

and Melnikov and Yelkhovsky, 1999; Pachucki andKarshenboim, 1999

11D72 = , 1560

and for states with l 1 Golosov et al., 1995; Elkhovski,1996; Jentschura

D60 = and Pachucki, 1996

ll+ 13

n2 2

4l2. 16

12l+ 3

In Eq. 16 and subsequent discussion, the first subscripton the coefficient of a term refers to the power of Zand the second subscript to the power of lnZ2. Therelativistic recoil correction used in the 2006 adjustmentis based on Eqs. 1116. The estimated uncertainty forS states is taken to be 10% of Eq. 13, and for stateswith l 1 it is taken to be 1% of that equation.

Numerical values for the complete contribution of Eq.13 to all orders in Z have been obtained by Shabaevet al. 1998. Although the difference between the all-orders calculation and the truncated power series for Sstates is about three times their quoted uncertainty, thetwo results are consistent within the uncertainty as-

signed here. The covariances of the theoretical valuesare calculated by assuming that the uncertainties arepredominately due to uncalculated terms proportionalto me /mN / n3.

c. Nuclear polarization

Interactions between the atomic electron and thenucleus which involve excited states of the nucleus giverise to nuclear polarization corrections. For hydrogen,we use the result Khriplovich and Senkov, 2000

EPH = 0.07013h

l0

n3kHz. 17

For deuterium, the sum of the proton polarizability, theneutron polarizability Khriplovich and Senkov, 1998,and the dominant nuclear structure polarizability Friarand Payne, 1997a gives

EPD = 21.378h

l0

n3kHz. 18

We assume that this effect is negligible in states ofhigher l.

d. Self energy

The one-photon electron self energy is given by

E2 Z

4

SE = 3 FZ2m

nec , 19

where

FZ =A 241 lnZ +A40 +A50Z

+A Z 2 ln2Z 2 +A Z 2 262 61 lnZ

+ GSEZZ2. 20

From Erickson and Yennie 1965 and earlier paperscited therein,




9/98

TABLE VII. Bethe logarithms ln k0n , l relevant to the deter-mination ofR.

n S P

1 2.984 128 5562 2.811 769 893 0.030 016 7 093 2.767 663 6124 2.749 811 840 0.041 954 895 0.006 740 939

6 2.735 664 207 0.008 147 2048 2.730 267 261 0.008 785 04312 0.009 342 954

TABLE VIII. Values of the function GSE.

4A41 = 3 l0

,

4 10 1A40 = ln k0n,l + 3 9 l0

1 22l+ 1 l0

,

139A50 = 2 ln 232

l0 ,

A62 =

l0,

A61 = 4 11 + +2 1 28 601

+ + ln 2 4 ln n

n

3 180

77 1

45n2 l0+ 1

n2 2 1+

15 3 j1/2l1

96n2 32ll+ 1+

3n21

2l 12l2l+ 12l+ 22l+ 3 l0.

21The Bethe logarithms ln k0n , l in Eq. 21 are given inTable VII Drake and Swainson, 1990.

The function GSEZ in Eq. 20 is the higher-ordercontribution in Z to the self energy, and the valuesfor GSE that we use here are listed in Table VIII. ForS and P states with n 4, the values in the table arebased on direct numerical evaluations by Jentschura andMohr 2004, 2005 and Jentschura et al. 1999, 2001. Thevalues of GSE for the 6S and 8S states are based onthe low-Z limit of this function GSE0=A60 Jentschura,

Czarnecki, and Pachucki, 2005 together with extrapola-tions of complete numerical calculation results of FZsee Eq. 20 at higher ZKotochigova and Mohr, 2006.The values ofGSE for D states are from Jentschura,Kotochigova, Le Bigot, et al. 2005.

The dominant effect of the finite mass of the nucleuson the self energy correction is taken into account bymultiplying each term ofFZ by the reduced-mass fac-tor m

r/m

e3, except that the magnetic moment term

1/ 22l+1 in A40 is multiplied instead by the factormr /me2. In addition, the argument Z2 of the loga-rithms is replaced by me / mrZ2 Sapirstein andYennie, 1990.

The uncertainty of the self energy contribution to agiven level arises entirely from the uncertainty ofGSElisted in Table VIII and is taken to be entirely of type un.

e. Vacuum polarization

The second-order vacuum-polarization level shift is

E2 Z

4

VP = m2

3 HZn

ec , 22

where the function HZ is divided into the part corre-sponding to the Uehling potential, denoted here byH1Z, and the higher-order remainder HRZ,where

H1Z = V40 + V50Z + V61Z2 lnZ2

+ G1VPZZ2, 23

HRZ = GRVPZZ2 , 24

with

V40

= 4 15 l0

,

V50 =5

48l0 ,

V61 = 215 l0. 25

The part G1VPZ arises from the Uehling potentialwith values given in Table IX Mohr, 1982; Kotochigovaet al., 2002. The higher-order remainder GRVPZ hasbeen considered by Wichmann and Kroll, and the lead-ing terms in powers of Z are Wichmann and Kroll,1956; Mohr, 1975, 1983

D

n S1/2 P1/2 P3/2 D3/2 D5/2

1 30.290 240202 31.185 15090 0.973 5020 0.486 50203 31.047 70904 30.912040 1.164020 0.609020 0.031 63226 30.71147 0.034 17268 30.60647 0.007 94090 0.034 842212 0.008020 0.035030




10/98

TABLE IX. Values of the function G1VP.

n S1/2 P1/2 P3/2 D3/2 D5/2

1 0.618 7242 0.808 872 0.064 006 0.014 1323 0.814 5304 0.806 579 0.080 007 0.017 666 0.000 0006 0.791 450 0.000 0008 0.781 197 0.000 000 0.000 00012 0.000 000 0.000 000

2

G 31R 19 2 1 VPZ = l0 + 45 27 16 2880Zl0+ . 26

Higher-order terms omitted from Eq. 26 are negligible.In a manner similar to that for the self energy, the

leading effect of the finite mass of the nucleus is taken

into account by multiplying Eq. 22 by the factormr /me3 and including a multiplicative factor ofme / mr in the argument of the logarithm in Eq. 23.

There is also a second-order vacuum polarizationlevel shift due to the creation of virtual particle pairsother than the ee+ pair. The predominant contributionfor nS states arises from +, with the leading termbeing Eides and Shelyuto, 1995; Karshenboim, 1995

Z2 4 4 mEVP = n3 15 e m

2m 3r m c2. 27 m

ee

The next-order term in the contribution of muon

vacuum polarization to nS states is of relative orderZme / m and is therefore negligible. The analogouscontribution E2VP from

+ 18 Hz for the 1S state isalso negligible at the level of uncertainty of current in-terest.

For the hadronic vacuum polarization contribution,we use the result given by Friar et al. 1999 that utilizesall available e+e scattering data,

E2 VP = 0.67115E2

had VP, 28

where the uncertainty is of type u0.The muonic and hadronic vacuum polarization contri-

butions are negligible for P and D states.

f. Two-photon corrections

Corrections from two virtual photons have been par-tially calculated as a power series in Z,

2

E4Z 4

=

n3mec

2F4Z, 29

where

F4Z = B40 + B50Z + B63Z2 ln3Z2

+ B62Z2 ln2Z2 + B61Z

2 lnZ2

+ B60Z2 + . 30

The leading term B40 is well known,

32 102 2179 9B40 =

ln 2 3

2 27 648 4

l0

2 ln 2 2 197 33 1 + l0 . 31

2 12 144 4 2l+ 1

The second term is Pachucki, 1993a, 1994; Eides andShelyuto, 1995; Eides et al., 1997

B50 = 21.556131l0, 32

and the next coefficient is Karshenboim, 1993; Manoharand Stewart, 2000; Yerokhin, 2000; Pachucki, 2001

B63 = 827 l0. 33

For S states,

the coefficient B

62is given by

16 71 1 1B62 = ln 2 + + n ln n + ,9 60 n 4n2

34

where =0.577... is Eulers constant and is the psifunction Abramowitz and Stegun, 1965. The differenceB621 B62n was calculated by Karshenboim 1996and confirmed by Pachucki 2001, who also calculatedthe n-independent additive constant. For P states, thecalculated value is Karshenboim, 1996

4 n2 1

B62 = . 3527 n2

This result has been confirmed by Jentschura and Nn-dori 2002, who also showed that for D and higher an-gular momentum states B62=0.

Recent work has led to new results for B61 and higher-order coefficients. In the paper ofJentschura, Czarnecki,and Pachucki 2005, an additional state-independentcontribution to the coefficient B61 for S states is given,which differs slightly 2% from the earlier result of Pa-chucki 2001 quoted in CODATA 2002. The revised co-efficient for S states is




11/98

TABLE X. Values of N used in the 2006 adjustment.

n NnS NnP

1 17.855 672 0312 12.032 141 581 0.003 300 63513 10.449 80914 9.722 4131 0.000 394 33216 9.031 83218 8.697 6391

413 581 4NnS 20272 616 ln 2B61 = + + 64 800 3 864 135

22 ln 2 40 ln2 2 + + 3 +

3 9304 32 ln 2

135 9

3 1 1 + + n ln n + ,4 n 4n2 36where is the Riemann zeta function Abramowitz andStegun, 1965. The coefficients NnS are listed in TableX. The state-dependent part B61nS B611S was con-firmed by Jentschura, Czarnecki, and Pachucki 2005 intheir Eqs. 4.26 and 6.3. For higher-l states, B61 hasbeen calculated by Jentschura, Czarnecki, and Pachucki2005; for P states,

4 n2 1 166 8B61nP1/2 = NnP +3 n2

ln 2405 27

, 374 n2 1 31 8

B61nP3/2 = NnP +3 n2 ln 2 38

405;

27

and for D states,

B61nD = 0. 39

The coefficient B61 also vanishes for states with l 2.The necessary values of NnP are given in Eq. 17 ofJentschura 2003 and are listed in Table X.

The next term is B60, and recent work has also beendone for this contribution. For S states, the state depen-dence is considered first, and is given by Czarnecki et al.2005 and Jentschura, Czarnecki, and Pachucki 2005,

B60nS B601S = bLnS bL1S +An, 40

where

An = 38 4 337 043 ln 2NnS N1S 45 3 129 600

94 261 902 609 4 16 4 + + + ln2 2

21 600n 129 600n2

3 9

n 9n2

76 304 76 53 35

+ + 45 135n 135n2

ln 2 + +

15 2n

419 28 003 11

30n22ln 2 +

10 800 2n

31 397 53 419+ + 2 2 + 35 2 310 800n 60 8n 120n+ 37 793 16 304+ ln2 2 ln 2 + 82ln 2

10 800 9 135

13 2 23

3+ n ln n . 41

The term An makes a small contribution in the range0.3 to 0.4 for the states under consideration.

The two-loop Bethe logarithms bL in Eq. 40 arelisted in Table XI. The values for n=1 to 6 are fromJentschura 2004 and Pachucki and Jentschura 2003,and the value at n=8 is obtained by extrapolationof the calculated values from n=4 to 6 bL5S=60.68 with a function of the form

b cbLnS = a + + , 42

n nn + 1

which yields

24bLnS = 55.8 . 43

n

It happens that the fit gives c =0. An estimate for B60given by

B60nS = bLnS +109 NnS + 44

was derived by Pachucki 2001. The dots represent un-calculated contributions at the relative level of 15% Pa-chucki and Jentschura, 2003. Equation 44 givesB601S =61.69.2. However, more recently Yerokhin etal. 2003, 2005a, 2005b, 2007 have calculated the

1S-state two-loop self energy correction for Z 10. Thisis expected to give the main contribution to the higher-order two-loop correction. Their results extrapolated toZ=1 yield a value for the contribution of all terms oforder B60 or higher of 127 10.3, which corre-sponds to a value of roughly B60=12939, assuming alinear extrapolation from Z=1 to 0. This differs byabout a factor of 2 from the result given by Eq. 44. Inview of this difference between the two calculations, forthe 2006 adjustment, we use the average of the two val-ues with an uncertainty that is half the difference, whichgives

TABLE XI. Values of bL, B60, and B71 used in the 2006adjustment. See the text for an explanation of the uncertainty33.7.

n bLnS B60nS B71nS

1 81.40.3 95.30.333.72 66.60.3 80.20.333.7 1683 63.50.6 77.00.633.7 2211

4 61.80.8 75.30.833.7 25126 59.80.8 73.30.833.7 28148 58.82.0 72.32.033.7 2915




12/98

B601S =95.30.333.7. 45

In Eq. 45, the first number in parentheses is the state-dependent uncertainty unB60 associated with the two-loop Bethe logarithm, and the second number in paren-theses is the state-independent uncertainty u0B60 thatis common to all S-state values of B60. Values ofB60 forall relevant S states are given in Table XI. For higher-lstates, B60 has not been calculated, so we take it

to be zero, with uncertainties unB60nP =5.0 andunB60nD =1.0. We assume that these uncertainties ac-count for higher-order P- and D-state uncertainties aswell. For S states, higher-order terms have been esti-mated by Jentschura, Czarnecki, and Pachucki 2005with an effective potential model. They find that thenext term has a coefficient of B72 and is state indepen-dent. We thus assume that the uncertainty u0B60nS issufficient to account for the uncertainty due to omittingsuch a term and higher-order state-independent terms.In addition, they find an estimate for the state depen-dence of the next term, given by

B71nS = B

71nS B711S

427 16= ln 2

36 3

3 1 1

+ + 2 + n ln n4 n 4n 46

with a relative uncertainty of 50%. We include this ad-ditional term, which is listed in Table XI, along with theestimated uncertainty unB71 =B71/2.

The disagreement of the analytic and numerical calcu-lations results in an uncertainty of the two-photon con-tribution that is larger than the estimated uncertainty

used in the 2002 adjustment. As a result, the uncertain-ties of the recommended values of the Rydberg constantand proton and deuteron radii are slightly larger in the2006 adjustment, although the 2002 and 2006 recom-mended values are consistent with each other. On theother hand, the uncertainty of the 2P state fine structureis reduced as a result of the new analytic calculations.

As in the case of the order self energy and vacuum-polarization contributions, the dominant effect of the fi-nite mass of the nucleus is taken into account by multi-plying each term of the two-photon contribution by thereduced-mass factor mr / me3, except that the magneticmoment term, the second line of Eq. 31, is instead mul-tiplied by the factor m

r/m

e2. In addition, the argument

Z2 of the logarithms is replaced by me / mrZ2.

g. Three-photon corrections

The leading contribution from three virtual photons isexpected to

have

E6 =

the form3Z4

3 mec2C40 + C50Z + , 47

n

in analogy with Eq. 29 for two photons. The leadingterm C40 is Baikov and Broadhurst, 1995; Eides and

Grotch, 1995a; Laporta and Remiddi, 1996; Melnikovand van Ritbergen,

2000

568a 5 24 85 121 3 84 0713C40 = + 9 24 72 2304

71 ln4 2 2392 ln2 2 47872 ln 2 +

27 135 108

15914 252 2512

679 441+ +3240 9720 93 312

l0

100a 24 2155 83 3 1393+ + 3 24 72 18

25 ln4 2 252 ln2 2 2982 ln 2 2394 + + +

18 18

9 2160

17 1012 28 259 1 l0 , 48

810 5184 2l+ 1

where a = n4 n=11/ 2 n4=0.517 479 061.. . . Higher-order

terms have not been calculated, although partial results

have been obtained Eides and Shelyuto, 2007. An un-certainty is assigned by taking u0C50=30l0 andunC63=1, where C63 is defined by the usual convention.The dominant effect of the finite mass of the nucleus istaken into account by multiplying the term proportionalto l0 by the reduced-mass factor mr /me3 and the termproportional to 1/ 2l+1, the magnetic moment term,by the factor mr / me2.

The contribution from four photons is expected to beof order

44 Z n3

mec2, 49

which is about 10 Hz for the 1S state and is negligible atthe level of uncertainty of current interest.

h. Finite nuclear size

At low Z, the leading contribution due to the finitesize of the nucleus is

E0NS = ENSl0, 50

with

2 3 2E = m 2r ZNS m c2 ZRN , 513 m 3e n e C

where RN is the bound-state root-mean-square rmscharge radius of the nucleus and C is the Comptonwavelength of the electron divided by 2. The leadinghigher-order contributions have been examined by Friar1979b, Friar and Payne 1997b, and Karshenboim1997 see also Borisoglebsky and Trofimenko 1979and Mohr 1983. The expressions that we employ toevaluate the nuclear size correction are the same asthose discussed in more detail in CODATA-98.

For S states, the leading and next-order correctionsare given by




13/98

R ZE mNS = ENS 1 C r N m R Z r

Cln

N

me me C n

5n + 9n 1+ n + 2 CZ24n ,

52

where C and C are constants that depend on the de-tails of the assumed charge distribution in the nucleus.The values used here are C=1.71 and C=0.474 forhydrogen or C=2.01 and C=0.384 for deuterium.

For the P1/2 states in hydrogen, the leading term is

Z2n2 1ENS = ENS .4n2

53

For P3/2 states and D states, the nuclear-size contribu-tion is negligible.

i. Nuclear-size correction to self energy and vacuum polarization

For the self energy, the additional contribution due tothe finite size of the nucleus is Pachucki, 1993b; Eides

and Grotch, 1997b; Milstein et al., 2002, 2003a

E23

NSE = 4 ln 2 4 ZENSl0, 54and for the vacuum polarization it is Friar, 1979a, 1981;Hylton, 1985; Eides and Grotch, 1997b

ENVP =3

ZENS4 l0. 55

For the self energy term, higher-order size correctionsfor S states Milstein et al., 2002 and size corrections forP states have been calculated Jentschura, 2003; Milsteinet al., 2003b, but these corrections are negligible for thecurrent work, and are not included. The D-state correc-tions are assumed to be negligible.

j. Radiative-recoil corrections

The dominant effect of nuclear motion on the self en-ergy and vacuum polarization has been taken into ac-count by including appropriate reduced-mass factors.The additional contributions beyond this prescriptionare termed radiative-recoil effects with leading termsgiven by

m3r Z5 352ERR =

m2emN 2 m c

2 6 3 22 ln 2 +n3

e l0 36

448 2 + Zln

2Z

2

+27 3 . 56The constant term in Eq. 56 is the sum of the analyticresult for the electron-line contribution Czarnecki andMelnikov, 2001; Eides et al., 2001a and the vacuum-polarization contribution Eides and Grotch, 1995b; Pa-chucki, 1995. This term agrees with the numerical valuePachucki, 1995 used in CODATA-98. The log-squaredterm has been calculated by Pachucki and Karshenboim1999 and by Melnikov and Yelkhovsky 1999.

For the uncertainty, we take a term of orderZlnZ2 relative to the square brackets in Eq. 56

with numerical coefficients 10 for u0 and 1 for un. Thesecoefficients are roughly what one would expect for thehigher-order uncalculated terms. For higher-l states inthe present evaluation, we assume that the uncertaintiesof the two- and three-photon corrections are muchlarger than the uncertainty of the radiative-recoil correc-tion. Thus, we assign no uncertainty for the radiative-recoil correction for P and D states.

k. Nucleus self energy

An additional contribution due to the self energy ofthe nucleus has been given by Pachucki 1995,

4Z2Z4 m3ESEN =

r

3n3 m2c2

Nln mN

mrZ2l0

ln k0n,l . 57This correction has also been examined by Eides et al.2001b, who consider how it is modified by the effect ofstructure of the proton. The structure effect would lead

to an additional model-dependent constant in the squarebrackets in Eq. 57.

To evaluate the nucleus self energy correction, we useEq. 57 and assign an uncertainty u0 that corresponds toan additive constant of 0.5 in the square brackets for Sstates. For P and D states, the correction is small and itsuncertainty, compared to other uncertainties, is negli-gible.

l. Total energy and uncertainty

The total energy EXnLj of a particular level where L=S,P ,. . . and X=H,D is the sum of the various contri-

butions listed above plus an additive correction XnLj thataccounts for the uncertainty in the theoretical expres-sion for EXnLj. Our theoretical estimate for the value of

XnLj for a particular level is zero with a standard uncer-tainty of u XnLj equal to the square root of the sum ofthe squares of the individual uncertainties of the contri-butions; as they are defined above, the contributions tothe energy of a given level are independent. Compo-nents of uncertainty associated with the fundamentalconstants are not included here, because they are deter-mined by the least-squares adjustment itself. Thus, wehave for the square of the uncertainty, or variance, of aparticular level

u2 2

u2 X = 0iXLj + uniXLj

nLj , 58i n

6

where the individual values u0iXLj /n3 and uniXLj / n3

are the components of uncertainty from each of the con-tributions, labeled by i, discussed above. The factors of1 /n3 are isolated so that u0iXLj is explicitly indepen-dent ofn.

The covariance of any two s follows from Eq. F7 ofAppendix F of CODATA-98. For a given isotope X, wehave




14/98

2X jun Lj,

Xn Lj = u0iXL

3 , 591 2i n1n2

which follows from the fact that uu0i ,uni =0 anduun i ,un i =0 for n1 n2. We also set1 2

u X Xn L j ,n L j = 0 601 1 1 2 2 2

if L1 L2 or j1 j2.

For covariances between s for hydrogen and deute-rium, we have for states of the same n

H D u0iHLju DLj + u HLju DLj

unLj,nLj =0i ni ni

i=i n6 ,

c

61

and for n1 n2

HLu H

j n Lj,

D u0i u0iDLjn Lj =1 2i=ic

n1n23 , 62

where the summation is over the uncertainties common

to hydrogen and deuterium. In most cases, the uncer-tainties can in fact be viewed as common except for aknown multiplicative factor that contains all of the massdependence. We assume

u Hn L j ,Dn L j = 0 631 1 1 2 2 2

if L1 L2 or j1 j2.The values of u XnLj of interest for the 2006 adjust-

ment are given in Table XXVIII of Sec. XII, and thenon-negligible covariances of the s are given in theform of correlation coefficients in Table XXIX of thatsection. These coefficients are as large as 0.9999.

Since the transitions between levels are measured infrequency units Hz, in order to apply the above equa-tions for the energy level contributions we divide thetheoretical expression for the energy difference E ofthe transition by the Planck constant h to convert it to afrequency. Further, since we take the Rydberg constantR =2mec /2h expressed in m1 rather than the elec-tron mass me to be an adjusted constant, we replace thegroup of constants 2m 2ec /2h in E /h by cR.

m. Transition frequencies between levels with n = 2

As an indication of the consistency of the theory sum-

marized above and the experimental data, we list belowvalues of the transition frequencies between levels withn=2 in hydrogen. These results are based on values ofthe constants obtained in a variation of the 2006 least-squares adjustment in which the measurements of thedirectly related transitions items A38, A39.1, and A39.2in Table XXVIII are not included, and the weakly de-pendent constants Are, Arp, Ard, and are as-signed their 2006 adjusted values. The results are

H2P1/2 2S1/2 = 1 057 843.92.5 kHz

2.3 106,

H2S1/2 2P3/2 = 9 911 197.62.5 kHz

2.5 107,

H2P1/2 2P3/2 = 10 969 041.47599 kHz

9.0 109 , 64

which agree well with the relevant experimental resultsof Table XXVIII. Although the first two values in Eq.

64 have changed only slightly from the results of the2002 adjustment, the third value, the fine-structure split-ting, has an uncertainty that is almost an order of mag-nitude smaller than the 2002 value, due mainly to im-provements in the theory of the two-photon correction.

A value of the fine-structure constant can be ob-tained from data on the hydrogen and deuterium transi-tions. This is done by running a variation of the 2006least-squares adjustment that includes all the transitionfrequency data in Table XXVIII and the 2006 adjustedvalues ofAre, Arp, and Ard. The resulting value is

1 = 137.036 00248 3.5 107, 65

which is consistent with the 2006 recommended value,although substantially less accurate. This result is in-cluded in Table XXXIV.

2. Experiments on hydrogen and deuterium

Table XII summarizes the transition frequency datarelevant to the determination ofR. With the exceptionof the first entry, which is the most recent result for the1S1/22S1/2 transition frequency in hydrogen from thegroup at the Max-Planck-Institute fr QuantenoptikMPQ, Garching, Germany, all these data are the sameas those used in the 2002 adjustment. Since these data

are reviewed in CODATA-98 or CODATA-02, they arenot discussed here. For a brief discussion of data notincluded in Table XII, see Sec. II.B.3 of CODATA-02.

The new MPQ result,

H1S1/2-2S1/2 = 2 466 061 413 187.07434 kHz

1.4 1014, 66

was obtained in the course of an experiment tosearch for a temporal variation of the fine-structureconstant Fischer et al., 2004a, 2004b; Hnschet al., 2005; Udem, 2006. It is consistent with, but has asomewhat smaller uncertainty than, the previousresult from the MPQ group, H1S1/2 2S1/2

=2 466 061 413 187.10346 kHz 1.9 1014 Niering etal., 2000, which was the value used in the 2002 adjust-ment. The improvements that led to the reduction inuncertainty include a more stable external referencecavity for locking the 486 nm cw dye laser, thereby re-ducing its linewidth; an upgraded vacuum system thatlowered the background gas pressure in the interactionregion, thereby reducing the background gas pressureshift and its associated uncertainty; and a significantlyreduced within-day Type A i.e., statistical uncertaintydue to the narrower laser linewidth and better signal-to-noise ratio.




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TABLE XII. Summary of measured transition frequencies considered in the present work for the determination of the Rydbergconstant R H is hydrogen and D is deuterium.

Reported value Rel. stand.Authors Laboratory Frequency intervals /kHz uncert. ur

Fischer et al., 2004a, 2004b MPQ H1S1/22S1/2 2 466 061 413 187.07434 1.4 1014

Weitz et al., 1995 MPQ H2S1/24S1/214 H1S1/22S1/2 4 797 33810 2.1 10

6

H2S1/24D5/2 14 H1S1/22S1/2 6 490 14424 3.7 10

6

D2S1/24S1/2 14 D1S1/22S1/2 4 801 69320 4.2 106

D2S1/24D5/2 14 D1S1/22S1/2 6 494 84141 6.3 10

6

Huber et al., 1998 MPQ D1S1/22S1/2 H1S1/22S1/2 670 994 334.6415 2.2 1010

de Beauvoir et al., 1997 LKB/SYRTE H2S1/28S1/2 770 649 350 012.08.6 1.1 1011

H2S1/28D3/2 770 649 504 450.08.3 1.1 1011

H2S1/28D5/2 770 649 561 584.26.4 8.3 1012

D2S1/28S1/2 770 859 041 245.76.9 8.9 1012

D2S1/28D3/2 770 859 195 701.86.3 8.2 1012

D2S1/28D5/2 770 859 252 849.55.9 7.7 1012

Schwob et al., 1999, 2001 LKB/SYRTE H2S1/212D3/2 799 191 710 472.79.4 1.2 1011

H2S1/212D5/2 799 191 727 403.77.0 8.7 1012

D

2S1/2

12D3/2

799 409 168 038.08.6 1.1 1011

D2S1/212D5/2 799 409 184 966.86.8 8.5 1012

Bourzeix et al., 1996 LKB H2S1/26S1/214 H1S1/23S1/2 4 197 60421 4.9 10

6

H2S1/26D5/2 14 H1S1/23S1/2 4 699 09910 2.2 10

6

Berkeland et al., 1995 Yale H2S1/24P1/214 H1S1/22S1/2 4 664 26915 3.2 10

6

H2S1/24P3/214 H1S1/22S1/2 6 035 37310 1.7 10

6

Hagley and Pipkin, 1994 Harvard H2S1/22P3/2 9 911 20012 1.2 106

Lundeen and Pipkin, 1986 Harvard H2P1/22S1/2 1 057 845.09.0 8.5 106

Newton et al., 1979 U. Sussex H2P1/22S1/2 1 057 86220 1.9 105

The MPQ result in Eq. 66 and Table XII for

H1S1/22S1/2 was provided by Udem 2006 ofthe MPQ group. It follows from the measuredvalue H1S1/22S1/2 =2 466 061 102 474.85134 kHz1.4 1014 obtained for the 1S ,F=1, mF= 1 2S , F =1, mF = 1 transition frequency Fischer etal., 2004a, 2004b; Hnsch et al., 2005 by using the wellknown 1S and 2S hyperfine splittings Ramsey, 1990;Kolachevsky et al., 2004 to convert it to the frequencycorresponding to the hyperfine centroid.

3. Nuclear radii

The theoretical expressions for the finite nuclear size

correction to the energy levels of hydrogen H and deu-terium D see Sec. IV.A.1.h are functions of the bound-state nuclear rms charge radius of the proton Rp and ofthe deuteron Rd. These values are treated as variables inthe adjustment, so the transition frequency data, to-gether with theory, determine values for the radii. Theradii are also determined by elastic electron-proton scat-tering data in the case of Rp and from elastic electron-deuteron scattering data in the case of Rd. These inde-pendently determined values are used as additionalinformation on the radii. There have been no new re-sults during the last four years and thus we take as input

data for these two radii the values used in the 2002 ad-

justment,Rp = 0.89518 fm, 67

Rd = 2.13010 fm. 68

The result for Rp is due to Sick 2003 see also Sick2007. The result for Rd is that given in Sec. III.B.7 ofCODATA-98 based on the analysis of Sick and Traut-mann 1998.

An experiment currently underway to measure theLamb shift in muonic hydrogen may eventually providea significantly improved value of Rp and hence an im-proved value ofR Nebel et al., 2007.

B. Antiprotonic helium transition frequencies and Ar(e)

The antiprotonic helium atom is a three-body systemconsisting of a 4He or 3He nucleus, an antiproton, andan electron, denoted by pHe+. Even though the Bohrradius for the antiproton in the field of the nucleus isabout 1836 times smaller than the electron Bohr radius,in the highly excited states studied experimentally, theaverage orbital radius of the antiproton is comparable tothe electron Bohr radius, giving rise to relatively long-




16/98

TABLE XIII. Summary of data related to the determination of Are from measurements on anti-protonic helium.

Transition Experimental Calculated a bn , ln , l value MHz value MHz 2cR 2cR

p 4He+ : 32,3131,30 1 132 609 20915 1 132 609 223.5082 0.2179 0.0437

p 4He+ : 35,3334,32 804 633 059.08.2 804 633 058.01.0 0.1792 0.0360

p 4He+ : 36,3435,33 717 474 00410 717 474 001.11.2 0.1691 0.0340

p 4He+ : 37,3436,33 636 878 139.47.7 636 878 151.71.1 0.1581 0.0317p 4He+ : 39,3538,34 501 948 751.64.4 501 948 755.41.2 0.1376 0.0276

p 4He+ : 40,3539,34 445 608 557.66.3 445 608 569.31.3 0.1261 0.0253

p 4He+ : 37,3538,34 412 885 132.23.9 412 885 132.81.8 0.1640 0.0329

p 3He+ : 32,3131,30 1 043 128 60813 1 043 128 579.7091 0.2098 0.0524

p 3He+ : 34,3233,31 822 809 19012 822 809 170.91.1 0.1841 0.0460

p 3He+ : 36,3335,32 646 180 43412 646 180 408.21.2 0.1618 0.0405

p 3He+ : 38,3437,33 505 222 295.78.2 505 222 280.91.1 0.1398 0.0350

p 3He+ : 36,34 37,33 414 147 507.84.0 414 147 509.31.8 0.1664 0.0416

lived states. Also, for the high-l states studied, becauseof the vanishingly small overlap of the antiproton wavefunction with the helium nucleus, strong interactions be-tween the antiproton and the nucleus are negligible.

One of the goals of antiprotonic helium experimentsis to measure the antiproton-electron mass ratio. How-ever, since we assume that CPT is a valid symmetry, forthe purpose of the least-squares adjustment we take themasses of the antiproton and proton to be equal and usethe data to determine the proton-electron mass ratio.Since the proton relative atomic mass is known moreaccurately than the electron relative atomic mass fromother experiments, the mass ratio yields information pri-marily on the electron relative atomic mass. Other ex-

periments have demonstrated the equality of the charge-to-mass ratio of p and p to within 9 parts in 1011; seeGabrielse 2006.

1. Theory relevant to antiprotonic helium

Calculations of transition frequencies of antiprotonichelium have been done by Kino et al. 2003 and byKorobov 2003, 2005. The uncertainties of calculationsby Korobov 2005 are of the order of 1 MHz to 2 MHz,while the uncertainties and scatter relative to the experi-mental values of the results of Kino et al. 2003 aresubstantially larger, so we use the results of Korobov

2005 in the 2006 adjustment. See also the remarks inHayano 2007 concerning the theory.The dominant contribution to the energy levels is the

nonrelativistic solution of the Schrdinger equation forthe three-body system together with relativistic and ra-diative corrections treated as perturbations. The nonrel-ativistic levels are resonances, because the states can de-cay by the Auger effect in which the electron is ejected.Korobov 2005 calculated the nonrelativistic energy byusing one of two formalisms, depending on whether theAuger rate is small or large. In the case in which the rateis small, the Feshbach formalism is used with an optical

potential. The optical potential is omitted in the calcula-tion of higher-order relativistic and radiative corrections.For broad resonances with a higher Auger rate, the non-relativistic energies are calculated with the complex co-ordinate rotation method. In checking the convergenceof the nonrelativistic levels, attention was paid to theconvergence of the expectation value of the the deltafunction operators used in the evaluation of the relativ-istic and radiative corrections.

Korobov 2005 evaluated the relativistic and radia-tive corrections as perturbations to the nonrelativisticlevels, including relativistic corrections of order 2R,anomalous magnetic moment corrections of order

3

R

and higher, one-loop self energy and vacuum-polarization corrections of order 3R, and higher-orderone-loop and leading two-loop corrections of order4R. Higher-order relativistic corrections of order 4Rand radiative corrections of order 5R were estimatedwith effective operators. The uncertainty estimates ac-count for uncalculated terms of order 5 ln R.

Transition frequencies obtained by Korobov 2005,2006 using the CODATA-02 values of the relevant con-stants are listed in Table XIII under the column headercalculated value. We denote these values of the fre-quencies by 0pHen , l:n

3 , l, where He is either He+ or

4He+. Also calculated are the leading-order changes in

the theoretical values of the transition frequencies as afunction of the relative changes in the mass ratiosArp /Are and ArN /Arp, where Nis either

3He2+ or4He2+. If we denote the transition frequencies as func-tions of these mass ratios by pHen , l: n , l, then thechanges can be written as

A p 0pHen,l:n,la n, rpHen,l: l = , 69

Are

ArpAre




17/98

A He0r pHen,l:n,l

bpHen,l:n,l = . 70Arp ArN

Arp

Values of these derivatives, in units of 2cR, are listed inTable XIII in the columns with the headers a and b,respectively. The zero-order frequencies and the deriva-tives are used in the expression

pHen,l:n,l = 0p

He

n,l:n,l + apHen, l:n,l

A re0 Arp

1Arp

Are

A p 0

+ bpHen,l:n r,lArN

ArN 1 + , 71

Arp

which provides a first-order approximation to the tran-sition frequencies as a function of changes to the massratios. This expression is used to incorporate the experi-mental data and calculations for the antiprotonic systemas a function of the mass ratios into the least-squaresadjustment. It should be noted that even though themass ratios are the independent variables in Eq. 71 andthe relative atomic masses Are, Arp, and ArN arethe adjusted constants in the 2006 least-squares adjust-ment, the primary effect of including these data in theadjustment is on the electron relative atomic mass, be-cause independent data in the adjustment provide valuesof the proton and helium nuclei relative atomic masseswith significantly smaller uncertainties.

The uncertainties in the theoretical expressions forthe transition frequencies are included in the adjustment

as additive constants pHen , l:n , l. Values for the the-oretical uncertainties and covariances used in the adjust-ment are given in Sec. XII, Tables XXXII and XXXIII,respectively Korobov, 2006.

2. Experiments on antiprotonic helium

Experimental work on antiprotonic helium began inthe early 1990s and it continues to be an active field ofresearch; a comprehensive review through 2000 hasbeen given by Yamazaki et al. 2002 and a concise re-view through 2006 by Hayano 2007. The first measure-ments of pHe+ transition frequencies at CERN with ur

106

were reported in 2001 Hori et al., 2001, im-proved results were reported in 2003 Hori et al., 2003,and transition frequencies with uncertainties sufficientlysmall that they can, together with the theory of the tran-sitions, provide a competitive value of Are, were re-ported in 2006 Hori et al., 2006.

The 12 transition frequenciesseven for 4He and fivefor 3He given by Hori et al. 2006which we take asinput data in the 2006 adjustment, are listed in column 2of Table XIII with the corresponding transitions indi-cated in column 1. To reduce rounding errors, an addi-tional digit for both the frequencies and their uncertain-

ties as provided by Hori 2006 have been included. All12 frequencies are correlated; their correlation coeffi-cients, based on detailed uncertainty budgets for each,also provided by Hori 2006, are given in Table XXXIIIin Sec XII.

In the current version of the experiment, 5.3 MeV an-tiprotons from the CERN Antiproton Decelerator ADare decelerated using a radio-frequency quadrupole de-celerator RFQD to energies in the range 10 keV to 120keV controlled by a dc potential bias on the RFQDselectrodes. The decelerated antiprotons, about 30% ofthe antiprotons entering the RFQD, are then diverted toa low-pressure cryogenic helium gas target at 10 K by anachromatic momentum analyzer, the purpose of which isto eliminate the large background that the remaining70% of undecelerated antiprotons would have pro-duced.

About 3% of the p stopped in the target form pHe+,in which a p with large principle quantum number n38 and angular momentum quantum number l ncirculates in a localized, nearly circular orbit around theHe2+ nucleus while the electron occupies the distributed1S state. These p energy levels are metastable with life-times of several microseconds and de-excite radiatively.There are also short-lived p states with similar values ofn and l but with lifetimes on the order of 10 ns andwhich de-excite by Auger transitions to form pHe2+ hy-drogenlike ions. These undergo Stark collisions, whichcause the rapid annihilation of the p in the heliumnucleus. The annihilation rate versus time elapsed sincepHe+ formation, or delayed annihilation time spectrumDATS, is measured using Cherenkov counters.

With the exception of the 36,34 35,33 transitionfrequency, all frequencies given in Table XIII were ob-

tained by stimulating transitions from the pHe+

meta-stable states with values of n and l indicated in columnone on the left-hand side of the arrow to the short-lived,Auger-decaying states with values of n and l indicatedon the right-hand side of the arrow.

The megawatt-scale light intensities needed to inducethe pHe+ transitions, which cover the wavelength range265 nm to 726 nm, can only be provided by a pulsedlaser. Frequency and linewidth fluctuations and fre-quency calibration problems associated with such laserswere overcome by starting with a cw seed laser beamof frequency , known with u 4 1010cw r through itsstabilization by an optical frequency comb, and then am-

plifying the intensity of the laser beam by a factor of 106in a cw pulse amplifier consisting of three dye cellspumped by a pulsed Nd:YAG laser. The 1 W seed laserbeam with wavelength in the range 574 nm to 673 nmwas obtained from a pumped cw dye laser, and the 1 Wseed laser beam with wavelength in the range 723 nm to941 nm was obtained from a pumped cw Ti:sapphire la-ser. The shorter wavelengths 265 nm to 471 nm forinducing transitions were obtained by frequency dou-bling the amplifier output at 575 nm and 729 nm to 941nm or by frequency tripling its 794 nm output. The fre-quency of the seed laser beam cw, and thus the fre-




18/98

quency pl of the pulse amplified beam, was scannedover a range of 4 GHz around the pHe+ transition fre-quency by changing the repetition frequency frep of thefrequency comb.

The resonance curve for a transition was obtained byplotting the area under the resulting DATS peak versuspl. Because of the approximate 400 MHz Dopplerbroadening of the resonance due to the 10 K thermal

motion of the pHe

+

atoms, a rather sophisticated theo-retical line shape that takes into account many factorsmust be used to obtain the desired transition frequency.

Two other effects of major importance are the so-called chirp effect and linear shifts in the transition fre-quencies due to collisions between the pHe+ and back-ground helium atoms. The frequency pl can deviatefrom cw due to sudden changes in the index of refrac-tion of the dye in the cells of the amplifier. This chirp,which can be expressed as ct =pltcw, can shiftthe measured pHe+ frequencies from their actual values.Hori et al. 2006 eliminated this effect by measuringct in real time and applying a frequency shift to the

seed laser, thereby canceling the dye-cell chirp. This ef-fect is the predominant contributor to the correlationsamong the 12 transitions Hori, 2006. The collisionalshift was eliminated by measuring the frequencies of tentransitions in helium gas targets with helium atom den-sities in the range 2 1018/ cm3 to 3 1021/cm3 to de-termine d/d. The in vacuo =0 values were ob-tained by applying a suitable correction in the range 14MHz to 1 MHz to the initially measured frequenciesobtained at 2 1018 /cm3.

In contrast to the other 11 transition frequencies inTable XIII, which were obtained by inducing a transitionfrom a long-lived, metastable state to a short-lived,

Auger-decaying state, the 36,34

35,33 transitionfrequency was obtained by inducing a transition fromthe 36,34 metastable state to the 35,33 metastablestate using three different lasers. This was done by firstdepopulating at time t1 the 35, 33 metastable state byinducing the 35,33 34,32 metastable state to short-lived-state transition, then at time t2 inducing the36,34 35,33 transition using the cw pulse-amplifiedlaser, and then at time t3 again inducing the 35,33 34,32 transition. The resonance curve for the36,34 35,33 transition was obtained from theDATS peak resulting from this last induced transition.

The 4 MHz to 15 MHz standard uncertainties of the

transition frequencies in Table XIII arise from the reso-nance line-shape fit 3 MHz to 13 MHz, statistical orType A, not completely eliminating the chirp effect 2MHz to 4 MHz, nonstatistical or Type B, collisionalshifts 0.1 MHz to 2 MHz, Type B, and frequency dou-bling or tripling 1 MHz to 2 MHz, Type B.

3. Values ofAr(e) inferred from antiprotonic helium

From the theory of the 12 antiprotonic transition fre-quencies discussed in Sec IV.B.1, the 2006 recommendedvalues of the relative atomic masses of the proton, alpha

particle , nucleus of the 4He atom, and the helion h,nucleus of the 3He atom, Arp, Ar, and Arh, re-spectively, together with the 12 experimental values forthese frequencies given in Table XIII, we find the fol-lowing three values for Are from the seven p

4He+ fre-quencies alone, from the five p3He+ frequencies alone,and from the 12 frequencies together:

Are = 0.000 548 579 910312 2.19

10 , 72

Are = 0.000 548 579 905315 2.7 109, 73

Are = 0.000 548 579 908 8191 1.7 109. 74

The separate inferred values from the p4He+ and p3He+

frequencies differ somewhat, but the value from all 12frequencies not only agrees with the three other avail-able results for Are see Table XXXVI, Sec. XII.A,but has a competitive level of uncertainty as well.

C. Hyperfine structure and fine structure

1. Hyperfine structure

Because the ground-state hyperfine transition fre-quencies H, Mu, and Ps of the comparativelysimple atoms hydrogen, muonium, and positronium, re-spectively, are proportional to 2Rc, in principle avalue of can be obtained by equating an experimentalvalue of one of these transition frequencies to its pre-sumed readily calculable theoretical expression. How-ever, currently only measurements of Mu and thetheory of the muonium hyperfine structure have suffi-ciently small uncertainties to provide a useful result forthe 2006 adjustment, and even in this case the result is

not a competitive value of , but rather the most accu-rate value of the electron-muon mass ratio me /m. In-deed, we discuss the relevant experiments and theory inSec. VI.B.

Although the ground-state hyperfine transition fre-quency of hydrogen has long been of interest as a poten-tial source of an accurate value of because it is experi-mentally known with ur 1012 Ramsey, 1990, therelative uncertainty of the theory is still of the order of106. Thus, H cannot yet provide a competitive valueof the fine-structure constant. At present, the mainsources of uncertainty in the theory arise from the inter-nal structure of the proton, namely i the electric charge

and magnetization densities of the proton, which aretaken into account by calculating the protons so-calledZemach radius; and ii the polarizability of the protonthat is, protonic excited states. For details of theprogress made over the last four years in reducing theuncertainties from both sources, see Carlson 2007, Pa-chucki 2007, Sick 2007, and references therein. Be-cause the muon is a structureless pointlike particle, thetheory ofMu is free from such uncertainties.

It is also not yet possible to obtain a useful value of from Ps since the most accurate experimental resulthas ur =3.6 106 Ritter et al., 1984. The theoretical un-




19/98

certainty of Ps is not significantly smaller and may infact be larger Adkins et al., 2002; Penin, 2004.

2. Fine structure

As in the case of hyperfine splittings, fine-structuretransition frequencies are proportional to 2Rc andcould be used to deduce a value of. Some data relatedto the fine structure of hydrogen and deuterium are dis-cussed in Sec. IV.A.2 in connection with the Rydbergconstant. They are included in the adjustment becauseof their influence on the adjusted value ofR. However,the value of that can be derived from these data is notcompetitive; see Eq. 65. See also Sec. III.B.3 ofCODATA-02 for a discussion of why earlier finestructure-related results in H and D are not considered.

Because the transition frequencies corresponding tothe differences in energy of the three 2 3P levels of4He can be both measured and calculated with reason-able accuracy, the fine structure of 4He has long beenviewed as a potential source of a reliable value of .

The three frequencies of interest are 0129.6 GHz,122.29 GHz, and 0231.9 GHz, which correspond tothe intervals 2 3P12

3P 3 3 3 30, 2 P22 P1, and 2 P22 P0, re-spectively. The value with the smallest uncertainty forany of these frequencies was obtained at HarvardZelevinsky et al., 2005,

01 = 29 616 951.6670 kHz 2.4 108. 75

It is consistent with the value of 01 reported by Georgeet al. 2001 with ur=3.0 108, and that reported byGiusfredi et al. 2005 with ur =3.4 108. If the theoret-ical expression for 01 were exactly known, the weightedmean of the three results would yield a value of with

ur 8 109

.However, as discussed in CODATA-02, the theory ofthe 2 3PJ transition frequencies is far from satisfactory.First, different calculations disagree, and because of theconsiderable complexity of the calculations and the his-tory of their evolution, there is general agreement thatresults that have not been confirmed by independentevaluation should be taken as tentative. Second, thereare significant disagreements between theory and ex-periment. Recently, Pachucki 2006 has advanced thetheory by calculating the complete contribution to the2 3PJ fine-structure levels of order m

7 or 5 Ry, withthe final theoretical result for 01 being

01 = 29 616 943.0117 kHz 5.7 109. 76

This value disagrees with the experimental value givenin Eq. 75 as well as with the theoretical value 01=29 616 946.4218 kHz 6.1 109 given by Drake2002, which also disagrees with the experimental value.These disagreements suggest that there is a problemwith theory and/or experiment that must be resolved be-fore a meaningful value of can be obtained from thehelium fine structure Pachucki, 2006. Therefore, as inthe 2002 adjustment, we do not include 4He fine-structure data in the 2006 adjustment.

V. MAGNETIC MOMENT ANOMALIES AND g-FACTORS

In this section, theory and experiment for the mag-netic moment anomalies of the free electron and muonand the bound-state g-factor of the electron in hydro-genic carbon 12C5+ and in hydrogenic oxygen 16O7+are reviewed.

The magnetic moment of any of the three chargedleptons =e, , is written as

e =g s , 772m

where g is the g-factor of the particle, m is its mass,and s is its spin. In Eq. 77, e is the elementary chargeand is positive. For the negatively charged leptons , gis negative, and for the corresponding antiparticles +, gis positive. CPT invariance implies that the masses andabsolute values of the g-factors are the same for eachparticle-antiparticle pair. These leptons have eigenvaluesof spin projection sz = /2, and it is conventional towrite, based on Eq. 77,

g e = , 782 2m

where in the case of the electron, B =e /2me is theBohr magneton.

The free lepton magnetic moment anomaly a is de-fined as

g = 21 + a, 79

where gD =2 is the value predicted by the free-electronDirac equation. The theoretical expression for a may bewritten as

ath = aQED + aweak + ahad , 80

where the terms denoted by QED, weak, and had ac-count for the purely quantum electrodynamic, predomi-nantly electroweak, and predominantly hadronic that is,strong interaction contributions to a, respectively.

The QED contribution may be written as Kinoshitaet al., 1990

aQED =A1 +A2m/m +A2m/m

+A3m/m ,m/m , 81

where for the electron, , ,= e, , and for themuon, , ,= , e ,. The anomaly for the , whichis poorly known experimentally Yao et al., 2006, is notconsidered here. For recent work on the theory of a,see Eidelman and Passera 2007. In Eq. 81, the term

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