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The term dimension is difficult, if not impossible, to define in any meaningful way, which raises doubts about the usefulness of the concept, even though it is widely used. This essay examines the significance of a statement of the dimension of a quantity and concludes that it is minimal and possibly misleading.

Values of plane angles are expressed with a choice of several units. Historically the quantity needed a unit because it was, and still is, used as a base quantity. ISO/TC 12 defines it as a derived, dimensionless quantity, and the International System of Units (SI) gives it the 'dimensionless unit' radian, which now means no more than 'one'. This paper discusses the confusion that arises from the dual uses of these terms.

The author argues that solid angle is logically a two-dimensional angle derived from plane angle, and has units naturally derived from units of plane angle. ISO/TC 12, however, defines the quantity independently of plane angle as a dimensionless quantity to which, nevertheless, the SI assigns a 'unit'.

Within an international collaboration of the eight metrological institutes represented by the authors, the dependence of the triple-point temperature of equilibrium hydrogen on the deuterium content at low concentrations has been precisely determined so that the uncertainty in realizing the triple point as a temperature fixed point might be reduced by nearly one order of magnitude. To investigate the thermodynamic properties of the hydrogen–deuterium mixtures and to elucidate the factors that influence the melting temperature, 28 sealed fixed-point cells have been filled and measured, and some of these have been compared with an open-cell system. Hydrogen gas with a deuterium content ranging from 27.2 µmol D/mol H to 154.9 µmol D/mol H was studied using cells containing five different types of spin-conversion catalyst, with different catalyst-to-liquid volume ratios (a few per cent to more than 100%) and of different designs. The latter consideration is especially influential in determining the thermal behaviour of the cells and, thus, the temperature-measurement errors. The cells were measured at the eight participating institutes in accordance with a detailed protocol that facilitates a direct comparison of the results. Through analysis of the measurements, significant inter-institute deviations due to different measurement facilities and methods have been ruled out with respect to the determination of both the melting temperatures and the thermal parameters of the cells. The uncertainty estimates for the determination of the deuterium content have been verified by including isotopic analysis results from four different sources. The slope of the dependence of the triple-point temperature of equilibrium hydrogen isotopic mixtures on the deuterium content has been deduced from the melting temperatures of those sample portions not in direct contact with the catalysts. Evaluation of the data using different mathematical methods has yielded an average value of 5.4₂ µK per µmol D/mol H, with an upper bound of the standard uncertainty of 0.3₁ µK per µmol D/mol H. This is close to the literature value of 5.6 µK per µmol D/mol H that was obtained at higher deuterium concentrations.

The definition of the International Temperature Scale, ITS-90, requires the substances used for the realization of the fixed-points, except ³He, to be of 'natural' isotopic composition. The uncertainty of the best fixed-point realizations for some of these substances is today significantly affected by the natural variability of the isotopic composition of the available samples. The paper discusses the main implications of this problem, which is becoming an obstacle to further improvements for some of the fixed-point realizations. In addition, the two main solutions are compared: (i) to define only a reference composition and (ii) to define a reference composition and allow the use of a correction for the deviations from a reference composition.

A vertical cobalt–carbon (Co–C) eutectic fixed point cell was constructed at PTB to demonstrate its use for improvement of the calibration of noble-metal thermocouples at temperatures above 1100 °C. The melting and freezing temperatures of the Co–C eutectic were measured in different high-temperature furnaces at PTB and INMETRO (Brazil) to show its stability by using a Pt/Pd thermocouple. The reproducibility of all measured electromotive forces at the inflection points of the melting curves amounts to a value of about 0.06 °C. No drift in the melting temperature was observed. Therefore, the Co–C eutectic fixed point cell can be used as an adequate transfer standard for dissemination of the International Temperature Scale of 1990 (ITS-90) in the temperature range above the freezing point of copper.

We measured gas adsorption on native oxides formed on Si(100), Si(111) and Si(110) surfaces and found that the dependence of the gas adsorption on the surface orientation was insignificant. This fact suggests that the atomic structure of SiO₂ near the top surface is almost amorphous. For the physical adsorption of water vapour, we obtained a monolayer capacity of 0.0040 µg cm⁻² and a BET parameter of 4.5, which corresponds to 0.0065 µg cm⁻² water vapour adsorption at 50% humidity with an uncertainty of 0.0005 µg cm⁻² (k = 1). Thus, it can be concluded that the physical adsorption of water on a defect-free surface of a 1 kg silicon sphere amounts to 1.8 µg with an uncertainty of 0.14 µg (k = 1) at 50% humidity. In addition, measurement using ethanol vapour and n-octane vapour showed that gas adsorption decreased after hydroxylation of SiO₂/Si surfaces.

We review the development and status of GPS geodetic methods for high-precision global time and frequency comparisons. A comprehensive view is taken, including hardware effects in the transmitting satellites and tracking receiver stations, data analysis and interpretation, and comparisons with independent results. Other GPS techniques rely on single-frequency data and/or assume cancellation of most systematic errors using differences between simultaneous observations. By applying the full observation modelling of modern geodesy to dual-frequency observations of GPS carrier phase and pseudorange, the precision of timing comparisons can be improved from the level of several nanoseconds to near 100 ps. For an averaging interval of one day, we infer a limiting Allan deviation of about 1.4 × 10⁻¹⁵ for the GPS geodetic technique. The accuracy of time comparisons is set by the ability to calibrate the absolute instrumental delays through the GPS receiver and antenna chain, currently about 3 ns. Geodetic clock measurements are available for most of the major timing laboratories, as well as for many other tracking stations and the satellites, via the routine products of the International GPS Service.

As part of the research activity carried on at the Institute of Metrology 'G Colonnetti' IMGC-CNR in Turin on the development of the IMGC02 absolute gravimeter, a method to estimate the time occurrences of interference fringes was developed and tested. The capabilities of the IMGC02 have been enhanced by: increased flexibility in timing measurement, the possibility of real-time analysis and the removal of frequency-dependent time delays introduced by the timing electronics. A Monte Carlo simulation has shown, in the case of an ideal frequency and length reference, that the resolution of the waveform digitizer that samples the interference signal can limit the accuracy of the timing measurement. Nevertheless, the expected systematic bias in the absolute measurement of the acceleration due to gravity, g, introduced by this data processing technique is negligible when compared to the current instrument uncertainty.

For an accurate assessment of the f'₁ value of a photometer head it is important to determine the maximum step size for sampling and an appropriate bandwidth of the monochromator for an acceptable uncertainty contribution. A simulation is carried out to evaluate these uncertainty contributions for fine photometer heads, with a mismatch characteristic f' ⩽ 3% and uncertainty contributions f' ⩽ 0.1 % taken as the limits for acceptance. Assuming a stable measurement set-up, the results show that a 5 nm step size and a 5 nm bandwidth or less are needed for this assessment. The uncertainty of the f'₁ value caused by the wavelength uncertainty of the monochromator is analysed additionally. For extended bandwidths a method is proposed to correct the spectral responsivity data of the photometer head and reduce the bandpass uncertainty contribution effectively.

The use of special units for logarithmic ratio quantities is reviewed. The neper is used with a natural logarithm (logarithm to the base e) to express the logarithm of the amplitude ratio of two pure sinusoidal signals, particularly in the context of linear systems where it is desired to represent the gain or loss in amplitude of a single-frequency signal between the input and output. The bel, and its more commonly used submultiple, the decibel, are used with a decadic logarithm (logarithm to the base 10) to measure the ratio of two power-like quantities, such as a mean square signal or a mean square sound pressure in acoustics. Thus two distinctly different quantities are involved. In this review we define the quantities first, without reference to the units, as is standard practice in any system of quantities and units. We show that two different definitions of the quantity power level, or logarithmic power ratio, are possible. We show that this leads to two different interpretations for the meaning and numerical values of the units bel and decibel. We review the question of which of these alternative definitions is actually used, or is used by implication, by workers in the field. Finally, we discuss the relative advantages of the alternative definitions.

We examine chi-squared statistics that are appropriate for analysing the adequacy of different key comparison reference value (KCRV) candidates in accounting for the observed dispersion of results of a key comparison, about the candidate estimator and within the stated uncertainty claims. We extend the analysis to cover cases where the uncertainty budgets incorporate low degrees of freedom or have significant correlations. In this context, we discuss when it is important to view the KCRV as a method and not merely as a number. To use these statistics for the usual chi-squared tests of consistency, the required distributions (that can depart from the exact chi-squared distribution) can readily be evaluated by Monte Carlo simulation, for any KCRV algorithm that uses only the peer results of the comparison. Similarly, the effects of non-Gaussian distributions (such as the Student distribution implied when a participant has reported finite degrees of freedom) can be evaluated with the requisite precision.

Since the signing of the Mutual Recognition Arrangement, national metrology institutes (NMIs) have carried out many key comparisons in a wide range of metrological areas to establish the equivalence of their measurement standards. The determination of a key comparison reference value (KCRV) and its associated uncertainty is the central task in the evaluation of key comparison data. One of the most popular ways to estimate the KCRV is to use a weighted mean of the NMI's reported values, with weights inversely proportional to the variances of the reported value. One potential problem with the use of the weighted mean is its reliance on weights, which may vary greatly across NMIs. Consequently, some of the NMIs can be influential in the determination of the weighted-mean KCRV. Thus it is of interest to identify the influential NMIs based on some simple and well-defined criteria. In this paper, we present several easy-to-use criteria for detecting influential data in the calculation of the weighted-mean KCRV.

When a current-to-voltage (I/V) converter is used with a photodetector, the ratio of the photodetector impedance (shunt impedance) to the input impedance of the detection circuit defines the ratio of the photogenerated current that will produce the output voltage. Ideally, the input impedance should be negligible in comparison with the shunt impedance as only in this case will the whole photogenerated current produce the output signal. With high quality, high band gap detectors in most of the cases this requirement is automatically met, but if lower band gap detectors are used this requirement may not be satisfied.

The input impedance increases linearly with increasing feedback impedance of the I/V converter; i.e. at a sufficiently high feedback impedance value the output signal will deviate from the ideal case—it will be lower. Since the measurement of low intensity radiation requires a high feedback impedance, the lower the irradiance value on the detector the higher this deviation will be. If all the parameters of the detection system are kept constant during a measurement series, only calibration is required to achieve low uncertainty, but if any of the impedances or the gain of the operational amplifier changes during a measurement series this will result in increased measurement error. It should be noted that the values of these impedances can change rapidly with temperature. By applying a bootstrap circuit to the detector the impedance of the bootstrapped detector—depending on the frequency of the operation—will be 3–6 decades higher, compared with the detector alone; i.e. bootstrapping virtually increases the impedance of the detector. Consequently, this error will appear only at a much higher sensitivity range, greatly reducing this problem. Here we analyse the bootstrap solution.

The parametric down conversion of photons generated in a non-linear crystal gives rise to two correlated photons. When associated with a system of coincidence counting, this phenomenon allows one to measure the quantum efficiencies of detectors working at a normal rate of photon counting, without the need for reference sources or detectors. At the Institut National de Métrologie (INM) (France), this method was implemented with the aim of developing a new standard detector for the absolute measurement of very weak radiation. The validation of this method is presented from an international comparison between the laboratories of the INM (France) and the Istituto Elettrotecnico Nazionale Galileo Ferraris (IENGF) (Italy) and from a comparison of methods at INM using the French national standard detector, the cryogenic radiometer.

The triple point of xenon is shown to be a suitable fixed point for incorporation into the next revision of the International Temperature Scale (ITS) as a means of reducing the non-uniqueness in the important 84 K to 273 K range. Isotopic effects, once thought to be a limiting factor, are shown to be negligible. Purity remains the overriding concern, but high-purity xenon gas is obtainable, giving a very flat plateau with a melting range within ±10 µK from 50% to 90% of the melted sample fraction. The triple point temperature of xenon is shown to be 161.405 96 K ± 0.32 mK (k = 1) on the ITS-90. The propagated calibration uncertainty of the platinum resistance thermometers and the component attributed to the non-uniqueness of the ITS-90, as evidenced by the differences among the seven calibrated thermometers, dominate the overall uncertainty of the estimated triple point temperature. The xenon triple point itself is highly reproducible, with a standard deviation of 48 µK for the eight melts of this study and a total realization uncertainty of just 76 µK.

The magnitude of subrange inconsistency among calibrated capsule-style platinum resistance thermometers is explored using recent published data.

To reduce the uncertainties in measurements obtained by means of capillary viscometers, we developed an absolute falling-ball viscometer. The metrological characterization of this experimental bench is presented for a mineral oil with a viscosity of 30 Pa s by means of two distinct methods described in the GUM (Guide to the Expression of Uncertainty in Measurement). First, the calculation is performed using the law of propagation of uncertainties. Then, because of the increasing use of the techniques of propagation of distributions in metrology, we calculated the measurement uncertainties by a numerical Monte Carlo simulation. Both results were then compared for a confidence interval of 95%. Finally, the viscosity measurements and uncertainties obtained by the falling-ball viscometer were compared against those obtained by a capillary viscometer.

We demonstrate a simplified set-up for control of the carrier-envelope offset frequency (f_ceo) in a Ti : sapphire femtosecond laser frequency comb. A periodically poled KTiOPO₄ crystal is used for second harmonic generation with zero walk-off angle, which enables collinear propagation of beating beams in the self-referencing set-up. A beat signal with a signal-to-noise ratio of more than 40 dB can be obtained within a 300 kHz bandwidth. Using this signal, f_ceo could be tracked to a frequency synthesizer within a few millihertz for an averaging time of 1 s, which contributed a relative stability of 8 × 10⁻¹⁸ in the visible region around 532 nm.

We present an ultra-precise flotation method (±0.1 ppm at 99% level-of-confidence) for density-of-water comparisons close to its temperature of maximum density. The method consists of observing two equilibrium temperatures of the given silica float: below and above the maximum density point. Discussion of measurement sensitivity and results for various water samples are given.

In 2003, the International Committee for Weights and Measures (CIPM) recommended updated values of the frequency for certain optical frequency standards recommended for the practical realization of the definition of the metre. The text of this CIPM Recommendation and details of the updated radiations are given here. The complete updated set of recommended radiations, including frequencies, wavelengths, uncertainties and operating conditions where appropriate, is available on the BIPM website.