# Who discovered the universal constant G.

## Fundamental constants

**4 The problem of SI units**

The precision we ascribe to the constants increased by one decimal place about every 15 years. Such improvements cannot go on indefinitely because there must obviously be a limit to how accurately one can make physical measurements. At the moment we are still far from all natural limits. But there is a practical limitation based on the fact that we cannot implement the definitions of the SI units with arbitrary precision. After measuring the speed of light *c* more accurate than the embodiment of the meter became, went *c* in the definition of the meter. In the past 20 years, the accuracy of electrical constants has increased to the same extent as that of electrical units. The indeterminacy in their embodiment, in connection with our knowledge of the fine structure constants, dominates the current evaluations. It is possible that electrical embodiments of the unit watt achieve such a high level of accuracy that they open up a way of checking the stability of the unit kilogram and creating a new basis for it, which is no longer based on the platinum-iridium original kilogram, but on fundamental constants.

Let's equate a global set of electrical units with the one used by the International Bureau of Weights and Measures (BIPM); Let us also assume that the currently valid units of current, resistance and voltage - i.e. amps, ohms and volts - are the values A._{BI85}, Ω_{BI} or V_{BI} to have. Then the measurements of the fundamental constants also contain these quantities and provide information about the fundamental constants as well as about the embodiment of the SI units. (In the following the indices indicate the year; so was Ω_{BI85} the mean value of a series of standard resistances at the BIPM valid in 1985.) In addition, A applies_{BI85} = V_{BI76} / Ω_{BI85}, and according to the Josephson effect the following applies to the volt: V_{BI76} = [*E.* / (2*e* / *H*)] V. In 1972 the Consultative Committee on Electricity (CCE) presented in the General Conference on Weights and Measures (CGPM) for *E.* exactly the value 483 594 GHz / V.

In 1990 the CCE replaced this value with another quantity, namely the Josephson constant *K*_{J-90} = 483 597,9(1 ± 0,4 · 10^{-6}) GHz / V. The unit of voltage has been adapted accordingly in all countries. The internationally recognized value of Von Klitzing's constants *R.*_{K}which is related to the quantum Hall effect is after *R.*_{K-90} = 25 812,807(1±0,2 · 10^{-6}) Ω. The invariance of magnitude *K*_{J-90} is the basis for V_{BI90}, the embodiment of BIPM for the unit volt, and *R.*_{K-90} is the basis for Ω_{BI90}, the embodiment of the BIPM for the unit ohm. This is not to be confused with the quantized Hall resistance *R.*_{H} = *H* / *e*^{2} or with the constants 2*e* / *H* in Josephson's voltage-frequency relation.

**5 How do we determine the values of the fundamental constants?**

The determination of the values of the physical constants is an important link between theory and experiment. Our knowledge in the individual areas of physics can be checked very effectively and compared with one another, because different combinations of the numerical values of the constants can come from measurements of different physical phenomena. Spectacular new processes, which often brought the scientists concerned the Nobel Prize, replaced older ones and gave physics new impulses. Millikan's measurement of the elementary charge with the help of the oil droplets was replaced by indirect measurements in which combinations of fundamental constants played a role. The speed of light is no longer part of the evaluation and even became the basis for the definition of the length unit, the meter.

Usually one has to include measurements of more quantities than fundamental constants have to be determined. This enables one to recognize discrepancies. When measuring *G* there is currently a particular problem. More recent provisions deviate from one another much more than the error limits correspond to. The result was new considerations and further measurements. So far occurred *G* in no measurable combination with any other fundamental constant that would allow a more accurate determination.

Initially, the values derived from the available data by various researchers were inconsistent. The first systematic evaluation was made in 1929 by R.T. Birge performed. The values he proposed have been recognized internationally. He pointed out some interesting inconsistencies in his work, especially with regard to the measurements of the elementary charge and the Rydberg constant. Long before the era of calculating machines or even computers, Birge had to use comparatively simple calculation methods - often semi-graphic - to evaluate the data. Today we can use computers to help us and we also know a number of very sophisticated methods to obtain a uniform, coherent set of values from an overdetermined amount of data (fundamental constant compensation). But the judgment of the evaluator is always decisive.

A committee of distinguished scientists, the CODATA Task Group on Fundamental Constants, is currently evaluating experimental data. As soon as one agrees here on which measurements are to be included and whether the original assessments of the measurement uncertainties have to be modified, a so to speak impartial set of values and estimates of the uncertainties is determined with statistical methods.

The most recent such evaluation was made in 1986/87, replacing that of 1973. The 1987 values will be adjusted as soon as it is certain that sufficient new data is available. At the moment there are relatively few measurements that have an influence on the values, and since 1987 only a few values have been published that relate to the most important quantities. At the time this article was written (late 1997) it appeared that the 1987 values were essentially correct within the limits of the uncertainties involved. However, some precedents suggest that caution should be exercised in such situations!

The experimental data can be divided into two groups. The so-called auxiliary or supporting constants fall into the first; They include sizes that have been determined a few tens of times more precisely than the others. So deliver sizes like *m*_{p} / *m*_{e} or the Rydberg constant combinations of *e*, *m*_{e}, *m*_{p} and *H*which are known so much more precisely than other measured values that they can be used to determine the relationships between values of the various constants. Therefore, the number of less precisely known constants could be drastically reduced. The supporting constants evaluated in 1986/87 include the relative or reduced atomic masses (given in the form 1 + *m*_{e} / *m*_{a}), the molar mass *M.*_{p} of the proton, furthermore *μ*_{0}, *R.*_{∞}, the muon g-factor *G*_{μ} as *μ*_{e} / *μ*_{p} and *μ*_{p} / *μ*_{B.}. All of these sizes are today with an accuracy of better than 2 to 10^{8} known (for the second group of sizes see Table 4).

One can imagine that every single combination of less precise constants affects the values in a different direction, this trend being greatest for the most precise measurements. Hence 'pull' measurements of the redundant set *e*, *H* / *e*^{2} and 2*e* / *H* the sizes *e* and *H* in different directions.

**6 The less accurate dates that helped establish it**

A preliminary assessment was to determine which of the available measurements were to be excluded because they deviated too much from the others, and which uncertainties needed to be expanded or reduced. Rejecting existing data is always risky, especially since it has already been shown in the past that published values are grouped around an incorrect value. Hence the relative changes in the values of *e*, *H* and *m*_{e} by around 1.5 parts from 10^{5} due to the fact that two measurements of the Faraday constant that were included in 1973 have now been excluded. Most measurements combine a small statistically determined component with a larger uncertainty component, which is determined to a considerable extent by the feeling of the metrologist in question.

The experimental quantities that were used in the evaluation of the constants in 1986 are summarized in Table 4. Each of them is expressed in terms of the supporting constants and the selected unknowns: α, *K*_{Ω}, *K*_{V}, *μ*_{μ} / *μ*_{p} and the lattice parameters *d*_{220} of silicon.

**Fundamental constants 4:** The determinations of the quantities finally included in the evaluation of 1986/87 and the expressions depending on the supporting constants [] and the unknowns α, *K*_{Ω}, *K*_{V}, *μ*_{μ} / *μ*_{p} and *d*_{220}.

| |

1. | Five determinations of K_{Ω}, determined from calculable capacity realizations; Relative accuracy 1.1 to 3.6 to 10^{7}Ω _{BI85} = K_{Ω}Ω |

2. | Six current balance measurements by K_{A.} = A_{BI85} / A = K_{V} / K_{Ω}.Where A is _{BI85} = V_{BI76} / Ω_{BI75}; Relative accuracy 4.1 to 6.1 to 10^{6}A. _{BI85} = K_{V} K_{Ω}^{-1}A. |

3. | Two determinations of K_{V} from voltage balance measurements; relative accuracy 2.4 · 10^{-6} or 6 · 10^{-7}V _{BI76} = K_{V} V |

4. | A measurement of Faraday's constant by coulometry; relative accuracy 1.33 10^{-6}F. _{BI85} = [M_{p}cE / (4R_{∞}m_{p} / m_{e}))] α^{2}K_{V}^{-2}K_{Ω} |

5. | Six measurements of the gyromagnetic ratio (low) of the proton in water with a weak field; relative accuracy between 2.4 · 10 ^{-7} and 3.25 x 10^{-6}(low) _{BI85} = [c (μ ′_{p} / μ_{B.}) E / 4 R_{∞}] α^{-2}K_{Ω}^{-1} |

6. | Four measurements of the gyromagnetic ratio (high) of the proton in water with a strong field; relative accuracy between 1 x 10 ^{-6} and 5.4 x 10^{-6}(high) _{BI85} = [c (μ ′_{p} / μ_{B.}) E / 4 R_{∞}] α^{-2}K_{V}^{-2}K_{Ω} |

7. | Two measurements of the lattice parameter d_{220}(Si) pure silicon by counting the X-ray interference rings; relative accuracies 1 · 10^{-7} and 2.3 x 10^{-6} |

8. | A measurement of the molar mass V_{m}(Si) pure silicon in a vacuum22,5 ^{O}C, determined from the molecular weight and density of the silicon.V _{m}(Si) = [M_{p}μ_{0}c^{2}E.^{2} / {32^{1} / ^{2} (m_{p} / m_{e}) R_{∞} }] · ΑK_{V}^{-2} d^{3}_{220} |

9. | Six measurements of the quantum Hall effect resistance, expressed as a function of Ω_{BI85}; relative accuracy between 1.2 · 10^{-7} and 2.2 x 10^{-7}(R _{H})_{BI85} = [μ_{0}c / 2] · α^{-1}K_{Ω}^{-1} |

10. | Two measurements of the fine structure constant α from a_{e}(expt) and a_{e}(theor); relative accuracy 6.5 x 10^{-8} and 3.25 x 10^{-7} |

11. | Two measurements of the magnetic moment of the muon, expressed as a function of the proton moment, μ_{μ} / μ_{p}; relative accuracy 3.6 · 10^{-7} in the measurements of the muon ground state of the hyperfine splitting ν_{μ (hfs)} and 6.5 · 10^{-8} in a resonance experiment |

12. | A measurement of the hyperfine splitting ν_{μ (hfs)} of the muon, taking into account the theoretical uncertainty in ν_{μ (hfs)}ν _{μ (hfs)} = [16R_{∞}c (μ_{p} / μ_{B.}) / {3 (1 + m_{e} / m_{μ})^{3}}] q · α^{2}(μ_{μ} / μ_{p}); q = 1,000 957 61 (14) |

**7 The values evaluated in 1986/87**

Table 5 contains a selection of the values obtained in the fundamental constant equalization of 1986/87. It is currently believed that there will be enough data before the year 2000 to carry out a new assessment. There is also hope for new realizations of the watt unit *K*_{W.} = *K*_{V} · *K*_{A.} with new relative uncertainty <10^{-7}. This would have a greater impact on the evaluation than measurements (2) to (4) and (6) in Table 4. An initial determination of *K*_{W.} according to the new method, when setting the value of *K*_{J-90} considered. In the next evaluation, the uncertainties of many constants should be around ten times smaller than they are currently.

In the future we can expect that the uncertainties of most fundamental constants will continue to decrease and that some of these will play an increasingly important role in the definition of the SI units. Today's magnitudes characterize 20th century physics, and one should be aware that the very existence of many of them was barely recognized 100 years ago. A far more basic set of constants may well be elaborated as our understanding of physics deepens in the twenty-first century. Findings from quantum chromodynamics (QCD) and regarding the electroweak interaction will be particularly important - possibly as a result of a successful Great Unified Theory. [*annotation*: Some readers probably need the values of the fundamental constants in their quoted accuracies. It should be noted, however, that many of the determined variables are strongly linked to one another. This means: If one of the constants turns out to be incorrect, then others will only have a comparable accuracy. These correlations also mean that, for example, the uncertainty of *m*_{p} / *m*_{e} is much less than the combined uncertainty of *m*_{p} and* m*_{e}. In addition, changes in *m*_{p} and* m*_{e} be much greater than changes in *m*_{p} / *m*_{e}. It is therefore warned against calculating the uncertainty of combinations of constants that are not given in the table without applying the full variance, namely the covariance-uncertainty matrix. For the time being, they must be of the size *G* dependent values should be used with caution.]

**Bibliography**

E. R. Cohen, K. M. Crowe, J. W. M. DuMond: *Fundamental Constants of Physics*(New York: Interscience, 1957);

K. D. Froome, L. Essen: *Velocity of light and radio waves*. (London: Academic Press, 1969);

E. R. Cohen, B. N. Taylor: *J. Phys. Chem. Ref. Data ***17**, 1795-1803 (1988);

B. W. Petley: *Fundamental constants and the frontier of measurement*. Update 1988 (Bristol: Adam Hilger, 1988) ISBN 0-85274-388-2;

J. Bordtfeldt, B. Kramer: *Units and Fundamental Constants in Physics and Chemistry*. Sub-volume b: *Fundamental Constants in Physics and Chemistry*. (Berlin: Springer-Verlag, 1992).

**Fundamental constants 5:** Some values of fundamental physical constants recommended by CODATA in 1986.

| ||||

Speed of light in a vacuum | c | 299 792 458 (exact) | m s^{-1} | |

magnetic field constant | μ_{0} | 4π (exact) | · 10^{-7} N / A^{-2} | |

electric field constant | ε_{0} | 8.854 187 817 ... (exact) | · 10^{-12} F m^{-1} | |

Planck's quantum of action | H | 6,626 075 5(40) | · 10^{-34} J Hz^{-1} | |

Elementary charge (of the proton) | e | 1,602 177 33(49) | · 10^{-19} C. | |

Avogadro's constant | N_{A.} | 6,022 136 7(36) | · 10^{23} mol^{-1} | |

Newtonian constant of gravity | G | 6,672 59(85) | · 10^{-11} m^{3} kg^{-1} s^{-2} | |

Boltzmann constant | k_{B.} | 1,380 658(12) | · 10^{-23} J K^{-1} | |

Molar gas constant | R. | 8,314 510(70) | J mol^{-1} K^{-1} | |

Rest mass of the electron | m_{e} | 9,109 389 7(54) | · 10^{-31} kg | |

Rest mass of the proton as a multiple of the electron rest mass | m_{p}m _{p} / m_{e} | 1,672623(10) 1836,152 701(37) | · 10^{-27} kg | |

Fine structure constant inverse fine structure constant | α α ^{-}^{1} | 7,29735308(33) 137,035 9895(6) | · 10^{-3} | |

Rydberg constant | R._{∞} | 10 973 731,534(13) | m^{-1} | |

Bohr radius | a_{0} | 0,529 177 249(24) | · 10^{-10} m | |

magnetic flux quantum | Φ_{0} | 2,067 834 61(61) | · 10^{-15} Wb | |

Stefan-Boltzmann constant | σ | 5,670 51(19) | · 10^{-8} W m^{-2} K^{-4} | |

magnetic moment of the electron in Bohr's magnetons | μ_{e}μ _{e} / μ_{B.} | 928,477 01(31) 1,001 159 652 193(10) | · 10^{-26} J T^{-1} | |

magnetic moment of the proton in nuclear magnetons | μ_{p}μ _{p} / μ_{N} | 1,410 607 61(47) 2,792 847 386(63) | · 10^{-26} J T^{-1} | |

gyromagnetic ratio of the proton (spherical water sample at 25 ^{O}C) | 42,576 375(13) | · 10^{6} Hz T^{-1} | ||

Some quantities that serve as "units": | ||||

atomic mass unit | u | 1,660 540 · 2(10) | · 10^{-27} kg | |

Electron volts | eV | 1,602 177 · 33(49) | · 10^{-19} J | |

Planck mass | m_{P.} | 2,176 71(14) | · 10^{-8} kg | |

Planck's elementary length | l_{P.} | 1,616 05(10) | · 10^{-35} m | |

Planck time | t_{P.} | 5,390 56(34) | · 10^{-44} s | |

Josephson's frequency-voltage relationship | 2e / h | 4,835 976 7(14) | · 10^{14} Hz / V | |

Josephson's constant | K_{J-90} | 483 597,9(1±0,4 · 10^{-6}) | GHz / V | |

quantized Hall resistance | R._{H} | 25 812,805 6(12) | Ω | |

Von Klitzing's constant | R._{K-90} | 25812,807(1±0,2 · 10^{-6}) | Ω |

^{1)} Numbers in brackets are the estimated standard deviations of the last digit indicated.

For example, 6.626 075 5 (4) reads as 6.626 075 5 ± 0.000 000 4

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