# What's the difference between a realization, a representation and an implementation in metrology?

In a recent answer, a metrologist casually used the terms 'realization' and 'implementation' of an SI unit as if they were different, which looks very strange to the untrained eye. Some further digging (example) also throws up uses of 'representation' of an SI unit as a technical term with its own distinct meaning.

What is the precise meaning of these terms in metrology, and what are the precise differences between them? What are good examples of currently implemented realizations and representations vs implementations?

The reference document for metrological terms is the International Vocabulary of Metrology (VIM). Definitions there are carefully crafted, but frequently they might seem a bit obscure to non metrologists and further remarks might be needed.

For what concerns realization and reproduction (representation is also found in the literature for reproduction), their meaning is found under the term measurement standard:

Realization of the definition of a given quantity, with stated quantity value and associated measurement uncertainty, used as a reference.

In particular, the related notes 1 and 3 say:

NOTE 1 A “realization of the definition of a given quantity” can be provided by a measuring system, a material measure, or a reference material.

NOTE 3 The term “realization” is used here in the most general meaning. It denotes three procedures of “realization”. The first one consists in the physical realization of the measurement unit from its definition and is realization sensu stricto. The second, termed “reproduction”, consists not in realizing the measurement unit from its definition but in setting up a highly reproducible measurement standard based on a physical phenomenon, as it happens, e.g. in case of use of frequency-stabilized lasers to establish a measurement standard for the metre, of the Josephson effect for the volt or of the quantum Hall effect for the ohm. The third procedure consists in adopting a material measure as a measurement standard. It occurs in the case of the measurement standard of 1 kg.

Therefore, the terms realization and reproduction denote an object or an experiment with specific properties.

To illustrate the difference between a strict realization and a reproduction, let's take the example of a specific quantity, the unit ohm (note that a unit is a quantity, albeit a specially chosen one).

First, we have to define what this quantity is: this can be done in words, possibly with the help of mathematical relationships involving other quantities, and by adding specifications on influence quantities.

The ohm in the SI is defined as follows [CIPM, 1946: Resolution 2]:

The ohm is the electric resistance between two points of a conductor when a constant potential difference of 1 volt, applied to these points, produces in the conductor a current of 1 ampere, the conductor not being the seat of any electromotive force.

So far, so good, or, at least, it seems. Actually we are a bit stuck because we can realize the ampere and the volt respectively through current and voltage balances, but the reproducibility of the ohm realized in this way would be low, roughly at the $10^{-6}$ level. And the procedure would be rather complex. We are saved in 1956 by Thompson and Lampard who discovered a new theorem in electrostatics [1], which is the electrostatic dual of the van der Pauw theorem [2,3]. This theorem essentially says that you can build a standard of capacitance (that is, realize the farad or one of its submultiples), whose capacitance can be accurately calculated (something you cannot do with a parallel plate capacitor, for instance). If we have a standard of capacitance, through the relationships $Y = \mathrm{j}\omega C$ and $Z = 1/Y$, we have the standards of admittance and impedance, that is, we have the siemens and the ohm, however in the AC regime.

Thus, the strict SI realization of the ohm, as standard of resistance, is roughly the following:

1. You build a calculable capacitor (and ten years of your life are gone). Typically a 1 m long calculable capacitor has a capacitance of around 1 pF, which at kHz frequency correspond to a quite high impedance (for a short bibliography on the calculable capacitor, see this page).
2. By means of impedance bridges, you scale the capacitance to higher values (e.g., 1 nF).
3. By means of a quadrature impedance bridge, you compare the impedance value of a standard resistor with calculable AC-DC behaviour to that of the scaled capacitance.
4. You calculate the DC value of the resistance.
5. You scale down the resistance to 1 ohm by means of a resistance bridge.

Once you have all the experiments working (after many years), the realization of the ohm through the above chain of experiments can take more than one month, but the most important issue is that the reproducibility of the ohm realized in this way, though better than that obtainable through the realizations of the volt and the ampere, is just at the $10^{-7}$-$10^{-8}$ level.

Then it arrives the quantum Hall effect (QHE). A QHE element under conditions of low temperature and high magnetic field, realizes a four terminal resistance (or transresistance) with resistance value $R_\mathrm{H} = R_\mathrm{K}/i$, where $R_\mathrm{K}$ is a constant, the von Klitzing constant, and $i$ is an integer, called plateau index (typically we use the plateau corresponding to $i=2$). By the end of the 1980s it was clear that QHE elements could provide resistance standards with much better reproducibility than the other methods described above: at the time, of the order of $10^{-8}$-$10^{-9}$; nowadays, of the order of $10^{-10}$-$10^{-11}$ (two-three order of magnitudes better than that obtainable with a calculable capacitor). It turns out, also, that the von Klitzing constant is linked to two fundamental constants, the Planck constant and the elementary charge, $R_\mathrm{K} = h/e^2$.

The situation in the late 1980s is thus the following:

1. A QHE experiment is much easier to implement than that of a calculable capacitor (and much less expensive).
2. The resistance realized by a QHE experiment has a much better reproducibility than that realizable by a calculable capacitor experiment.
3. The accuracy of the von Klitzing constant, however, is only at the level of the SI ohm realization, that is, of about $10^{-7}$, and the relationship $R_\mathrm{H} = R_\mathrm{K}/i = h/(e^2 i)$ has not yet enough sound theoretical foundation to be exploited.

The first two points suggest the adoption of a conventional unit of resistance, by defining a conventional value of the von Klitzing constant [CIPM, 1988: Recommendation 2]. This conventional value of the von Klitzing constant is denoted by $R_{\mathrm{K}-90}$ (because it was adopted in 1990) and has value

$$R_\mathrm{K-90} = 25\,812.807\,\Omega\quad \text{(exact)}.$$

The conventional unit of resistance is the $\mathit{\Omega}_{90}$, defined as 1

$$\mathit{\Omega}_{90} = \frac{R_\mathrm{K}}{\{R_\mathrm{K-90}\}} = \frac{R_\mathrm{K}}{25\,812.807}.$$

At present, virtually all national resistance scales are traceable to this conventional unit.

It is now worth pointing out that the quantity $\mathit{\Omega}_{90}$ has no links to the SI ohm: it's close to (the relative discrepancy is of the order of $10^{-8}$), but quite not the same thing. Thus, the $\mathit{\Omega}_{90}$ is called a reproduction (or a representation) of the ohm, because it realizes somehow the ohm, but not according to its definition.

At present, this is not the only reproduced unit: the volt is currently reproduced by means of the Josephson effect through a conventional value of the Josephson constant, and the thermodynamic temperature scale is reproduced through two conventional temperature scales, the International Temperature Scale of 1990 (ITS-90) and the Provisional Low Temperature Scale of 2000 (PLTS-2000).

Instead, with the forthcoming revision of the International System of Units, the so called "new SI", the quantum Hall effect and the Josephson effect will really provide SI realizations of the ohm and the volt (see this draft of the mise en pratique of the electrical units).

Finally, for what concerns the term implementation, as far as I know, it has no specific technical meaning within the community of metrologists, and it is used in the common English meaning (whereas realization has a somehow different connotation). Thus, for instance, we can speak of two different implementations of a quantum Hall resistance experiment (because some details might be different).

## Notes

1 A note on notation: the quantity $\mathit{\Omega}_{90}$ is typeset in italics because it's not an SI unit; braces denote the numerical value of a quantity, according to the notation $Q = \{Q\}[Q]$ [4,5, and this question].

## References

[1] A. M. Thompson and D. G. Lampard (1956), "A New Theorem in Electrostatics and its Application to Calculable Standards of Capacitance", Nature, 177, 888.

[2] L.J. van der Pauw (1958), "A method of measuring specific resistivity and Hall effect of discs of arbitrary shape", Philips Research Reports, 13, 1–9.

[3] L.J. van der Pauw (1958), "A method of measuring the resistivity and Hall coefficient on lamellae of arbitrary shape", Philips Technical Review, 20, 220–224.

[4] E. R. Cohen et al. (2008), Quantities, Units and Symbols in Physical Chemistry, IUPAC Green Book, 3rd Edition, 2nd Printing, IUPAC & RSC Publishing, Cambridge [Online]

[5] E R Cohen and P. Giacomo (1987), Symbols, Units, Nomenclature and Fundamental Constants in Physics, IUPAP SUNAMCO Red Book, 1987 revision, IUPAP & SUNAMCO, Netherlands [Online]

• You know, I was just about to start a bounty on this one, but I realized that it's likely that you're the only one here who can answer this, so I was wondering whether to start it right away or to tell flippiefanus to bug you, to see whether you were back ;-). So, here goes, but I still want to know about that QHE stuff. – Emilio Pisanty Aug 25 '16 at 13:06
• I also want to quiz you on the quantum metrological triangle, but it'll take a bit of thinking to phrase a good question. – Emilio Pisanty Aug 25 '16 at 13:08
• @EmilioPisanty I went back just yesterday ;-) QHE will arrive (historically it's not the first reproduction of the ohm, because of course we somehow had the ohm before the calculable capacitor, but I wanted to start from the strict realization). However, I'm a slow writer, I also want to complete the other answer: keep your other questions for later ;-) – Massimo Ortolano Aug 25 '16 at 13:11
• @EmilioPisanty Added the QHE part: let me know if there is something unclear or some part that you would like to see expanded (in reference to the original question). – Massimo Ortolano Aug 26 '16 at 12:52