Fundamental Physical Constants 1986 Adjustments E. Richard Cohen, Thousand Oaks CA and Barry N. Taylor, Gaithersburg (Rockwell International Science Center and National Bureau of Standards) Over the past twenty years a surpri­ Table 1. 1986 Adjustment of the fundamental constants. singly strong link has developed bet­ Comparison of the initial and final stages of the multivariate analysis. ween the applied science of metrology and atomic, molecular and solid state Initial Final physics. The techniques of frequency (least-squares, (extended multiplication, and the precise measure­ external error) least-squares) ment of frequency at infrared and visible N 35 22 wavelengths, have reached such a level V 30 17 of development that the metre has been x 2 106.6 17.01 1.89 1.00 redefined in terms of the distance travell­ R b ed by light in a given time 1, 2). The direct α -1 137.035 996(11) 137.035 9895(61) linking of atomic lattice spacings to opti­ Kv 1 -7.24(54) x 10- 6 1 - 7.59(30)x10-6 cal wavelengths has produced a signifi­ KΩ 1 - 1.524(92) x 10-6 1 - 1.563(50)x10-6 cant improvement in the determination d220 192.015553(74) pm 192.015 540(40) pm of the Avogadro constant 3, 4). Impres­ µµ/µP 3.183 345 71(87) 3.183 345 47(47) sive progress has been made in the pre­ cision of measurement of the electron weighted mean of the observational 18). These algorithms use the consisten­ anomalous magnetic moment 5) as well equations that gives the smallest statis­ cy of the data to provide additional infor­ as the numerical evaluation of the asso­ tical variances consistent with the con­ mation with which to improve these a ciated sixth- and eighth-order Feynman straints. priori estimates of the variances. diagrams 6). The most striking metrolo­ The weight ωi associated with each gical advance occurred when K. von experimental datum is the reciprocal of Input Data Klitzing in 1980 observed the quantiza­ the variance, or statistical mean square As in previous adjustments, the data tion of electrical conductance 7) and error, σ2i which is however only available are divided into two categories : achieved not only a direct macroscopic as an a priori estimate s2i and is itself a) the more precise data (auxiliary con­ measurement of the fine-structure con­ uncertain. The usual least-squares pro­ stants) that are not subject to adjust­ stant, but the 1985 Nobel Prize in Phy­ cedure multiplies the calculated uncer­ ment because of their relatively low sics as well 8). tainties of an adjustment by the Birge uncertainties, and The CODATA Task Group on Funda­ ratio, RB = (x2/v)1/2, to rescale the b) the less precise or stochastic data mental Constants has recently comple­ weights and give a value of x2 equal to that are subject to adjustment. ted a new evaluation of the fundamental its expectation value v. This is equiva­ There is no formal basis for separation physical constants 9) taking into ac­ lent to an a posteriori evaluation of the into these two categories except that a count the truly extraordinary amount of 'error associated with unit weight' and is variable with an uncertainty much experimental and theoretical work that valid if the assigned uncertainties have smaller than that of other variables to has become available since the previous only relative significance, or if the syste­ which it is connected will be only slight­ adjustment in 1973 10). matic errors of all input data are roughly ly altered by the adjustment and can Because of past problems associated similar. However, when the data come hence be treated as a constant. The with the statistical treatment of such a from different and unrelated sources uncertainty of an auxiliary constant is diverse set of experimental and theoreti­ with broadly different physical content, typically one twentieth the uncertainty cal data, increased attention was direc­ a uniform expansion of all uncertainties of the stochastic datum with which it ted in the 1986 analysis to questions of can hardly be justified. In such a case, appears. All the auxiliary constants have statistical validity. The least-squares ap­ any rescaling of the assigned weights uncertainties not greater than 0.02 ppm proach to the analysis of such data has should consider all other information while the 38 items of stochastic data been described in detail in previous re­ that may be available concerning the have uncertainties in the range 0.05 views 11, 12, 13, but in brief, each experi­ uncertainty assignment of each indivi­ ppm to 10 ppm mental result with its estimated uncer­ dual datum. In analyzing the 1986 input On the basis of a preliminary screen­ tainty represents a constraint on the data we have considered not only the ing, three of the 38 items were identified values of a set of physical quantities, ex­ usual least-squares algorithm, but also as inappropriate for further considera­ pressed as an algebraic relationship in­ the algorithm proposed by Tuninskii and tion in the adjustment: one because it volving the auxiliary constants, the Kholin in 1975 14) of the Mendeleyev In­ represented an uncompleted measure­ unknown stochastic data and their un­ stitute of Metrology in Leningrad, as well ment; a second because it had recently known experimental and theoretical as a modification of it suggested by been shown to be in error; and the third errors. The least-squares solution de­ Taylor in 1982 15), and the extended because it required for its interpretation termines the "best" values of these least-squares algorithms described by an inadequately developed theoretical unknown quantities by finding the Cohen in 1976 16), 1978 17) and 1980 expression. The remaining 35 items for- 65 med the basis for an analysis with five Table 2. Summary of the 1986 recommended values of the fundamental physical unknowns. These unknowns were the constants. inverse fine-structure constant a-1, the A list of some fundamental constants of physics and chemistry based on a least-squares ratio of the standard volt (defined and adjustment with 17 degrees of freedom. The digits in parentheses are the one-standard- maintained by the Josephson effect) to deviation uncertainty in the last digits of the given value. Since the uncertainties of many of these entries are correlated, the full covariance matrix must be used in evaluating the the SI volt Kv, the ratio of the standard uncertainties of quantities computed from them. ohm (maintained in terms of a set of standard resistance coils, corrected for drift to the value on 1 Jan, 1985) to the Relative SI ohm KΩ, the lattice spacing of the Quantity Symbol Value Units uncertainty silicon lattice (at 22.5°C in vacuum) (ppm) d220, and the ratio of the muon magne­ tic moment to the proton magnetic mo­ speed of light in vacuum c 299 792 458 m s-1 (exact) ment, µµ/µP. The multivariate analysis permeability of vacuum µ0 4δπx10-7 N A- 2 was carried out using the several algo­ =12.566 370 614... 10-7 N A- 2 (exact) rithms mentioned above ; the details are permittivity of vacuum ϵ0 1/µ0c² (exact) given in CODATA Bulletin No. 63 9). =8.854187 817... 10-12 F m- 1 Newtonian constant of gravitation G 6.672 59(85) 10-11 m3 kg-1 s- 2 128 From the results of applying the Planck constant h 6.626 0755(40) 10-34 J s 0.60 various algorithms to the data and from h/2π h 1.054 572 66(63) 10- 34J s 0.60 a consideration of the effects on con­ elementary charge e 1.602 177 33(49) 10- 19C 0.30 sistency produced by eliminating data, magnetic flux quantum, h/2e Φ0 2.067 834 61(61) 10-15 Wb 0.30 we deleted 11 observations of very low Josephson frequency-voltage ratio 2e/h 4.835 9767(14) 1014 Hz V- 1 0.30 weight and two other older observations quantized Hall resistance, RH 25 812.8056(12) 0.045 which were of low weight and some­ h/e2 = µ0c/2a electron mass me 9.109 3897(54) 10-31 kg 0.59 what discrepant relative to their claimed 5.485 799 03(13) 10-4 u 0.023 precision. As a result, 22 items of sto­ in electron volts, mec2/{e} 0.510 999 06(15) MeV 0.30 chastic data remained to define the data electron specific charge — e/m e -1.758 819 62(53) 1011 C kg-1 0.30 set for the 1986 recommended values of muon mass mµ 1.883 5327(11) 10-28 kg 0.61 the fundamental physical constants. 0.113428 913(17) u 0.15 MeV 0.32 This final solution is compared with the in electron volts, mµc2/{e} 105.658 389(34) muon-electron mass ratio mµ/me 206.768 262(30) 0.15 initial adjustment with 35 observations proton mass mP 1.672 6231(10) 10- 27kg 0.59 in Table 1. The uncertainties given for the 1.007 276 470(12) u 0.012 initial adjustment (with X2 = 106.6 and in electron volts, mpc2/{e} 938.272 31(28) MeV 0.30 RB = 1.89) are computed from external proton-electron mass ratio mp/me 1836.152 701(37) 0.020 consistency. These uncertainties, which neutron mass mn 1.674 9286(10) 10- 27kg 0.59 are expanded by the factor RB from the 1.008 664 904(14) u 0.014 939.565 63(28) MeV 0.30 internally computed values, are still in electron volts, mnc2/{e} neutron-electron mass ratio mn/me 1838.683 662(40) 0.022 significantly smaller than the correspon­ neutron-proton mass ratio mn/m P 1.001 378 404(9) 0.009 ding uncertainties of the 1973 adjust­ α 10- 3 0.045 ment.