Metrology and Precision Measurement and Units

Introduction

As scientists we take the normal human desire to understand the world to quantitative extremes. We demand agreement of theory and experiment to the greatest accuracy possible. We measure quantities way beyond the current level of theoretical understanding in the hope that this measurement will be valuable as a reference point or that the difference between our value and some other nearly equal or simply related quantity will be important. The science of measurement, called metrology, is indispensable to this endeavor because the accuracy of measurement limits the accuracy of experiments and their intercomparison. In fact, the construction, intercomparison, and maintenance of a system of units is really an art (with some, a passion), often dependent on the latest advances in the art of physics (e.g.. quantized Hall effect, cold atoms, trapped particle frequency standards).

As a result of this passion, metrological precision typically marches forward a good fraction of an order of magnitude per decade. Importantly, measurements of the same quantity (e.g. $\alpha$ or $e/h$) in different fields of physics (e.g. atomic structure, QED, and solid state) provide one of the few cross-disciplinary checks available in a world of increasing specialization. Precise null experiments frequently rule out alternative theories, or set limits on present ones. Examples include tests of local Lorentz invariance, and the equivalence principle, searches for atomic lines forbidden by the exclusion principle, searches for electric dipole moments (which violate time reversal invariance), and the recent searches for a "fifth [gravitational] force".

A big payoff, often involving new physics, sometimes comes from attempts to achieve routine progress. In the past, activities like further splitting of the line, increased precision and trying to understand residual noise have lead to the fine and hyperfine structure of $H$, anomalous Zeeman effect, Lamb shift, and the discovery of the 3K background radiation. One hopes that the future will bring similar surprises. Thus, we see that precision experiments, especially involving fundamental constants or metrology not only solidify the foundations of physical measurement and theories, but occasionally open new frontiers.

Dimensions and Dimensional Analysis

Oldtimers were brought up on the mks system - meter, kilogram, and second. This simple designation emphasized an important fact: three dimensionally independent units are sufficient to span the space of all physical quantities. The dimensions are respectively $\ell$ - length, $m$-mass, and $t$-time. These three dimensions suffice because when a new physical quantity is discovered (e.g.. charge, force) it always obeys an equation which permits its definition in terms of m,k, and s. Some might argue that fewer dimensions are necessary (e.g.. that time and distance are the same physical quantity since they transform into each other in moving reference frames); we'll keep them both, noting that the definition of length is now based on the speed of light. Practical systems of units have additional units beyond those for the three dimensions, and often additional "as defined" units for the same dimensional quantity in special regimes (e.g. x-ray wavelengths or atomic masses). Dimensional analysis consists simply in determining for each quantity its dimension along the three dimensions (seven if you use the SI system rigorously) of the form

${\text{Dimension}}(G)\equiv [G]=m^{-1}l^{3}t^{-2}$

Dimensional analysis yields an estimate for a given unknown quantity by combining the known quantities so that the dimension of the combination equals the dimension of the desired unknown. The art of dimensional analysis consists in knowing whether the wavelength or height of the water wave (both with dimension $l$) is the length to be combined with the density of water and the local gravitational acceleration to predict the speed of the wave.

SI units

A single measurement of a physical quantity, by itself, never provides information about the physical world, but only about the size of the apparatus or the units used. In order for a single measurement to be significant, some other experiment or experiments must have been done to measure these "calibration" quantities. Often these have been done at an accuracy far exceeding our single measurement so we don't have to think twice about them. For example, if we measure the frequency of a rotational transition in a molecule to six digits, we have hardly to worry about the calibration of the frequency generator if it is a high accuracy model that is good to nine digits. And if we are concerned we can calibrate it with an accuracy of several more orders of magnitude against station WWV or GPS satellites which give time valid to 20 ns or so.

Time/frequency is currently the most accurately measurable physical quantity and it is relatively easy to measure to $10^{-12}$. In the SI (System International, the agreed-upon systems of weights and measures) the second is defined as 9,192,631,770 periods of the Cs hyperfine oscillation in zero magnetic field. Superb Cs beam machines at places like NIST-Boulder provide a realization of this definition at about $10^{-14}$. Frequency standards based on laser-cooled atoms and ions have the potential to do several orders of magnitude better owing to the longer possible measurement times and the reduction of Doppler frequency shifts. To give you an idea of the challenges inherent in reaching this level of precision, if you connect a 10 meter coaxial cable to a frequency source good at the $10^{-16}$ level, the frequency coming out the far end in a typical lab will be an order of magnitude less stable - can you figure out why?

The meter was defined at the first General Conference on Weights and Measures in 1889 as the distance between two scratches on a platinum-iridium bar when it was at a particular temperature (and pressure). Later it was defined more democratically, reliably, and reproducibly in terms of the wavelength of a certain orange krypton line. Most recently it has been defined as the distance light travels in 1/299,792,458 of a second. This effectively defines the speed of light, but highlights the distinction between defining and realizing a particular unit. Must you set up a speed of light experiment any time you want to measure length? No: just measure it in terms of the wavelength of a He-Ne laser stabilized on a particular hfs component of a particular methane line within its tuning range; the frequency of this line has been measured to about a part in $10^{-11}$ and it may seem that your problem is solved. Unfortunately the reproducibility of the locked frequency and problems with diffraction in your measurement both limit length measurements at about $10^{-11}$.

A list of spectral lines whose frequency is known to better than $10^{-9}$ is given in Appendix II of the NIST special publication number 330 (http://physics.nist.gov/Pubs/SP330/cover.html). The latest revision of the fundamental constants CODATA is available from NIST at http://physics.nist.gov/constants , and a previous version is published in Reference \cite{codata2002}. The third basic unit of the SI system is kilogram, the only fundamental SI base unit still defined in terms of an artifact - in this case a platinum-iridium cylinder kept in clean air at the Bureau de Poids et Measures in Severes, France. The dangers of mass change due to cleaning, contamination, handling, or accident are so perilous that this cylinder has been compared with the dozen secondary standards that reside in the various national measurement laboratories only two times in the last century. Clearly one of the major challenges for metrology is replacement of the artifact kilogram with an atomic definition. This could be done analogously with the definition of length by defining Avogadro's number. While atomic mass can be measured to $10^{-11}$, there is currently no sufficiently accurate method of realizing this definition, however. (The unit of atomic mass, designated by $\mu$ is 1/12 the mass of a $^{12}$C atom.)

There are four more base units in the SI system - the ampere, kelvin, mole, and the candela - for a total of seven. While three are sufficient (or more than sufficient) to do physics, the other four reflect the current situation that electrical quantities, atomic mass, temperature, and luminous intensity can be and are regularly measured with respect to auxiliary standards at levels of accuracy greater than that with which can be expressed in terms of the above three base units. Thus measurements of Avogadro's constant, the Boltzmann constant, or the mechanical equivalents of electrical units play a role of interrelating the base units of mole (number of atoms of $^{12}$C in 0.012 kg of $^{12}$C), kelvin, or the new de facto volt and ohm (defined in terms of Josephson and quantized Hall effects respectively). In fact independent measurement systems exist for other quantities such as x-ray wavelength (using diffraction from calcite or other standard crystals), but these other de facto measurement scales are not formally sanctioned by the SI system.

Metrology

The preceding discussion gives a rough idea of the definitions and realizations of SI units, and some of the problems that arise in trying to define a unit for some physics quantity (e.g.. mass) that will work across many orders of magnitudes. However, it sidesteps questions of the border between metrology and precision measurements. (Here we have used the phrase "precision measurement" colloquially to indicate an accurate absolute measurement; if we were verbally precise, precision would imply only excellent relative accuracy.) It is clear that if we perform an experiment to measure Boltzmann's constant we are not learning any fundamental physics; we are just measuring the ratio of energy scales defined by our arbitrary definitions of the first three base units on the one hand and the thermal energy of the triple point of water on the other. This is clearly a metrological experiment. Similarly, measuring the hfs frequency of Cs would be a metrological experiment in that it would only determine the length of the second.

If we measure the hfs frequency of $H$ with high accuracy, this might seem like a physics experiment since this frequency can be predicted theoretically. Unfortunately theory runs out of gas at about $10^{-6}$ due to lack of accurate knowledge about the structure of the proton (which causes a 42 ppm shift). Any digits past this are just data collection until one gets to the 14th, at which point one becomes able to use a $H$ maser as a secondary time standard. This has stability advantages over Cs beams for time periods ranging from seconds to days and so might be metrologically useful -- in fact, it is widely used in very long baseline radio astronomy.

One might ask "Why use arbitrarily defined base units when Nature has given us quantized quantities already?" Angular momentum and charge are quantized in simple multiples of $\hbar$ and $e$, and mass is quantized also although not so simply. We then might define these as the three main base units - who says we have to use mass, length and time? Unfortunately the measurements of $e,\hbar$, and $N_{A}^{-1}$ (which may be thought of as the mass quantum in grams) are only accurate at about the $10^{-8}$ level, well below the accuracy of the realization of the current base units of SI.

The preceding discussion seems to imply that measurements of fundamental constants like $e,\hbar$, and $N_{A}^{-1}$ are merely determinations of the size of SI units in terms of the quanta of Nature. In reality, the actual state of affairs in the field of fundamental constants is much more complicated. The complication arises because there are not single accurate experiments that determine these quantities - real experiments generally determine some combination of these, perhaps with some other calibration type variables thrown in (e.g.. the lattice spacing of Si crystals). To illustrate this, consider measurement of the Rydberg constant,

$Ry_{\infty }={\frac {m_{e}e^{4}}{(4\pi )^{3}\epsilon _{0}^{2}\hbar _{c}^{3}}}$

a quantity which is an inverse length (an energy divided by $\hbar c$), closely related to the number of wavelengths per cm of light emitted by a $H$ atom (the units are often given in spectroscopists' units, cm$^{-1}$, to the dismay of SI purists). Clearly such a measurement determines a linear combination of the desired fundamental quantities. Too bad, because it has recently been measured by several labs with results that agree to $10^{-11}$. (In fact, the quantity that is measured is the Rydberg frequency for hydrogen, $cRy_{H}$, since there is no way to measure wavelength to such precision.)

This example illustrates a fact of life of precision experiments: with care you can trust the latest adjustment of the fundamental constants and the metrological realizations of physical quantities at accuracies to $10^{-7}$; beyond that the limit on your measurement may well be partly metrological. In that case, what you measure is not in general clearly related to one single fundamental constant or metrological quantity. The results of your experiment will then be incorporated into the least squares adjustment of the fundamental constants, and the importance of your experiment is determined by the size of its error bar relative to the uncertainty of all other knowledge about the particular linear combination of fundamental and metrological variables that you have measured.