Metre

The metre (symbol: m), spelled meter in the United States, is the SI unit of length and one of the seven base units in the International System of Units. The name is derived from the Greek μέτρον (metron) meaning "measure", and gives rise to the term "metric system", a system of units of which the metre is one.

The metre was conceived in the aftermath of the French Revolution (1789) as a replacement for the old units of length that were associated with the ancien régime. Although there was initially considerable resistance to the adoption of the new units in France (including an official reversion to the mesures usuelles ["normal units"] for a period), the metre gained following in continental Europe during the mid nineteenth century, particularly in scientific usage, and was consacrated as an international measurement unit by the Metre Convention of 1875.

Definition

The current definition of the metre was agreed at the 17th General Conference on Weights and Measures (CGPM) in 1983:

The metre is the length of the path travelled by light in vacuum during a time interval of ¹⁄_{299 792 458} of a second.

This definition has the effect that the speed of light in vacuum has an exact, defined value in SI units: c₀ = 299 792 458 m s⁻¹.

Relativity

The metre is considered to be a unit of proper length. As such, the definition is only adequate in itself for measurements in an area where the local gravitational field can be considered to be uniform, a condition which is met by almost all measurements on the surface of the Earth (the Earth's gravitational field varies by about one part in 10¹⁶ per vertical metre, which is far less than the measurement uncertainty in Earth-based length measurements). For astronomical measurements, the effects of general relativity cannot be ignored and the definition of the metre must be completed with a description of the metric tensor which defines the gravitational field.

Realization

The metre can be realized either directly, by measuring the time taken for light to travel a certain distance ("time-of-flight"), or by interferometry. Time-of-flight measurements are usually used for long distances (more than a few tens of metres), while interferometry is the normal technique for the measurement of lengths less than a few metres. In either case, corrections may be necessary for gravitational effects, diffraction and, especially, the refractive index of the medium: for measurements in air, there are conventionally accepted formulae to calculate the refractive index based on local temperature, pressure and carbon dioxide content.

There are, of course, many other methods for the routine measurement of length, such as rulers, tape measures, calipers, micrometers, etc. These are properly considered as representations of the metre and, for accurate work, their calibration should be traceable to one of the realizations of the metre (usually through interferometry).

Time-of-flight measurements

The best-known example of time-of-flight realizations of the metre is radar, where a pulse of electromagnetic radiation (in this case, usually radio waves) is sent out towards a distant object and the time interval t between the outgoing pulse and the reflected radiation is measured. This gives the distance d between the observer and the object as:

d = t/2c

Similar measurements have been made of the distance to the Moon and to the inner planets of the Solar System. It is also possible, with adequate clock synchronization, to measure the distance from a distant emittor to a receiver; this is the principle behind the Global Positioning System (GPS) and similar systems such as GLONASS and Galileo.

Time-of-flight measurements are the only practical solution to measure long distances: it is possible with modern electronics (e.g. a good oscilloscope) to measure lengths of the order of one metre by time-of-flight methods, but the accuracy and precision are much lower than for interferometric measurements. In the context of general relativity, raw time-of-flight measurements (unlike interferometry) do not necessarily correspond to proper length, and so may need correction for gravitational effects to correspond to the definition of the metre.

Interferometry

Interferometry relies on the calculation of the difference in the number of wavelengths of monochromatic light between two unequal optical paths: by multiplying by the wavelength of the light, the difference in path length is obtained in metres. The wavelength may be calculated by measuring the frequency, but it is much more usual to use one of the "recommended radiations" on a list that is published periodically by the International Committee for Weights and Measures (CIPM). The vacuum wavelength is given by

λ = c₀/f

Two radiations in particular are relevant for high-precision interferometry:

Source	Stabilizing line (127I2)	Frequency THz	Wavelength nm	ur
Helium–neon laser	a16 component, R(127) 11-5 transition	473.612 353 604(10)	632.991 212 579(13)	2.1 × 10−11
Nd:YAG laser	a10 component, R(56) 32-0 transition	563.260 223 513(5)	532.245 036 104(5)	8.9 × 10−12

The uncertainty in the frequencies (and hence the wavelengths) of the recommended radiations is far lower than is required for most length measurements. For routine interferometry, it is usual to use the neon 3s₂→2p₄ transition of an unstabilized helium–neon laser, and the CIPM has also published a recommended frequency for this line to avoid an excessive need for calibration of individual sources:

f = 473.6127(7) THz

λ = 632.9908(9) nm

u_r = 1.5 × 10⁻⁶

History

Universal measure

The standard measures of length in Europe diverged from one another following the fall of Charlemagne's Empire: while measures could be standardized within a given jurisdiction (which was often little more than a single market town), they were numerous varieties of measure between regions. Indeed, as the measures were often used as the basis for taxation (of cloth, for example), the use of a certain measure was associated with the sovereignty of a given ruler and often dictated by law.

Nevertheless, with the increasing scientific activity of the seventeenth century came calls for the institution of a "universal measure" (as Englishman John Wilkins called it) or "metro cattolico" (Italian Tito Livio Burattini), which would be based on a natural phenomenon rather than royal decree, and would also be decimal rather than the various systems of multipliers, often duodecimal, that coexisted at the time.

Wilkins' idea was to choose the length of a "seconds pendulum" (a pendulum with a half-period of one second) as the unit length: such pendulums had recently been demonstrated by Christiaan Huygens, and their length is very close to one modern metre (as well as to length units which were then in use, such as the yard). However, it was soon discovered that the length of a seconds pendulum varies from place to place: French astronomer Jean Richer had measured the 0.3% difference in length between Cayenne (in French Guiana) and Paris.

Little practical progress was made towards the establishment of the "universal measure" until the French Revolution of 1789. France was particularly affected by the proliferation of length measures, and the need for reform was widely accepted across all political viewpoints, even if needed the push of revolution to bring it about. Talleyrand resurrected the idea of the seconds pendulum before the Constituant Assembly in 1790, suggesting that the new measure be defined at 45°N (a latitude that, in France, runs just north of Bordeaux and just south of Grenoble): dispite the support of the Assembly, and of Great Britain and the newly independent United States, nothing came of Talleyrand's proposal.

Meridional definition

The question of measurement reform was placed in the hands of the Academy of Sciences who appointed a commission chaired by Jean-Charles de Borda. Borda could be said to have been a fanatic for decimalization: he had invented the "repeating circle", a surveying instrument which allowed a much-improved precision in the measurement of angles between landmarks, but insisted that it be calibrated in "grades" (¹⁄₁₀₀ of a quarter-circle) rather than degrees, with 100 minutes to a grade and 100 seconds to a minute. For Borda, the seconds pendulum was a poor choice for a standard because the second (as a unit of time) was insufficiently decimal: he preferred a system of 10 hours to the day, 100 minutes to the hour and 100 seconds to the minute…

Instead, the commission – whose members included Lagrange, Laplace, Monge and Condorcet – decided that the new measure should be equal to one ten-millionth of the distance from the North Pole to the Equator (the quadrant of the Earth's circumference), measured along the meridian passing through Paris. Apart from the obvious nationalistic considerations, the Paris meridian was also a sound choice for practical scientific reasons: a portion of the quadrant from Dunkerque to Barcelona (about 1000 km, or one-tenth of the total) could be surveyed with start- and end-points at sea level, and that portion was roughly in the middle of the quadrant, where the effects of the Earth's oblateness were expected to be the largest.

The task of surveying the meridian fell to Pierre Méchain and Jean-Baptiste Delambre, and took more than six years (1792–98). In the meantime, the commission calculated a provisional value from older surveys of 443.44 lignes. Méchain and Delambre's result came out at 0.144 lignes shorter than this, a difference of about 0.03%.

Mètre des Archives

While Méchain and Delambre were completing their survey, the comission had ordered a series of platinum bars to be made based on the provisional metre. When the final result was known, the bar whose length was closest to the meridional definition of the metre was selected and placed in the National Archives on 22 June 1799 (4 messidor An VII in the Republican calendar) as a permanent record of the result: this standard metre bar became known as the mètre des Archives.

The metric system, that is the system of units based on the metre, was officially adopted in France on 10 December 1799 (19 frimaire An VIII) and became the sole legal system of weights and measures from 1801. After the instauration of the Empire, in 1812, the old names for units of length were revived but the units redefined in terms of the metre: this system was known as mesures usuelles, and lasted until 1840 when the decimal metric system was again made the sole legal measure. In the meantime, the Netherlands had adopted the metric system from 1816, the first of several countries to follow the French lead.

It soon became apparent that Méchain and Delambre's result (443.296 lignes) was slightly too short for the meridional definition of the metre. Arago and Biot extended the survey to the island of Formentera in the western Mediterranean Sea in 1806–9, and found that one ten-millionth of the Earth's quadrant should be 443.31 lignes: later work increased the value to 443.39 lignes. The modern value, for the WGS 84 reference spheroid, is 1.000 196 57 m or 443.383 08 lignes.

Nevertheless, the mètre des Archives remained the legal and practical standard for the metre in France, even once it was known that it did not exactly correspond to the meridional definition. When it was decided (in 1867) to create a new international standard metre, the length was taken to be that of the mètre des Archives "in the state in which it shall be found".

The only significant international use of the meridional definition of the metre, apart from Méchain and Delambre's original survey, was the initial work conducted by the British Association for the Advancement of Science (B.A.) on electrical units which was to lead to the International System of Electrical and Magnetic Units. It was often claimed that the international electrical units formed a coherent set of absolute units in the "QES system", where the unit length was the quadrant of the Earth's polar circumference, the unit mass was the "eleventh-gram" or 10⁻¹¹ grams and the unit time was the second. Nevertheless, the precision of absolute electrical measurements in the late nineteenth century was not such that the 0.02% difference in the definitions of the metre had any practical significance.

International prototype metre

With increasing international adoption of the metre, the short-comings of the mètre des Archives as a standard became ever more apparent. Countries which adopted the metre as a legal measure purchased standard metre bars that were intended to be equal in length to the mètre des Archives, but there was no systematic way of ensuring that the countries were actually working to the same standard. The meridional definition, which had been intended to ensure international reproduceability, quickly proved so impractical that is was all but abandoned in favour of the artefact standards, but the mètre des Archives (and most of its copies) were "end standards": such standards (bars which are exactly one metre in length) are prone to wear with use, and different standard bars could be expected to wear at different rates.

The International Conference on Geodesy in 1867 called for the creation of a new, international prototype metre and to arrange a system where national standards could be compared with it. The international prototype would also be a "line standard", that is the metre was defined as the distance between two lines marked on the bar, so avoiding the wear problems of end standards. The French government gave practical support to the creation of an International Metre Commission, which met in Paris in 1870 and again in 1872 with the participation of about thirty countries.

The international nature of the standards was ensured by a treaty – the Metre Convention – signed in Paris on 20 May 1875. This set up an international organization, the Bureau international des poids et mesures (BIPM), to conserve the prototypes (which were to be the joint property of the signatory nations) and to carry out regular comparisons with national standards. In recognition of the role of France in designing the metric system, the BIPM is based in Sèvres, just outside Paris, but it enjoys the usual privileges of an international organization and is under the ultimate control of a diplomatic conference (the Conférence générale des poids et mesures, CGPM) and not of the French government.

The construction of the international prototype metre and the copies which would be national standards was at the limits of the technology of its time. The bars were to be made of a special alloy, 90% platinum and 10% iridium, which is significantly harder than pure platinum, and have a special X-shaped cross section (a "Tresca section", named after French engineer Henri Tresca) to minimise the effects of torsional strain during length comparisons. The first castings proved unsatisfactory, and the job was given to the London firm of Johnson Matthey who succeded in producing thirty bars to the required specification: one of these (No. 6) was determined to be identical in length to the mètre des Archives, and was consacrated as the international prototype metre at the first meeting of the CGPM in 1889. The other bars, duely calibrated against the international prototype, were distributed to the signatory nations of the Metre Convention for use as national standards: for example, the United States received No. 27 with a calibrated length of 0.999 9984 m ± 0.2 µm (1.6 µm short of the international prototype).

The first (and only) follow-up comparison of the national standards with the international prototype was carried out between 1921 and 1936, and indicated that the definition of the metre was preserved to with 0.2 µm. At this time, it was decided that a more formal definition of the metre was required (the 1889 decision had said merely that the "prototype, at the temperature of melting ice, shall henceforth represent the metric unit of length"), and this was agreed at the 7th CGPM in 1927:

The unit of length is the metre, defined by the distance, at 0°, between the axes of the two central lines marked on the bar of platinum–iridium kept at the Bureau International des Poids et Mesures and declared Prototype of the metre by the 1st Conférence Générale des Poids et Mesures, this bar being subject to standard atmospheric pressure and supported on two cylinders of at least one centimetre diameter, symmetrically placed in the same horizontal plane at a distance of 571 mm from each other.

The support requirements represent the Airy points of the prototype, that is the points, separated by ⁵⁄₉ of the total length of the bar, at which the bending or droop of the bar is minimized.

Krypton standard

The first interferometric measurements carried out using the international prototype metre were those of Michelson and Benoît (1892–93) and of Benoît, Fabry and Perot (1906), both using the red line of cadmium. These results, which gave the wavelength of the cadmium line (λ ≈ 644 nm), led to the definition of the angstrom as a secondary unit of length for spectroscopic measurements, first by the International Union for Solar Research (1907) and later by the CIPM (1927). Michelson's work in "measuring" the prototype metre to within ¹⁄₁₀ of a wavelength (< 0.1 µm) was one of the reasons for which he was awarded the Nobel Prize in Physics in 1907.

By the 1950s, interferometry had become the method of choice for precise measurements of length but there remained a practical problem imposed by the system of units used. The natural unit for expressing a length measured by interferometry was the angstrom, but this result then had to be converted into metres using an experimental conversion factor – the wavelength of light used, but measured in metres rather than in angstroms. This added an additional measurement uncertainty to any length result in metres, over and above the uncertainty of the actual interferometric measurement. The solution was to define the metre in the same manner as the angstrom had been defined in 1907, that is in terms of the best interferometric wavelength available.

Advances in both experimental technique and theory showed that the cadmium line was actually a cluster of closely separated lines, and that this was due to the presence of different isotopes in natural cadmium (eight in total). To get the most precisely defined line, it was necessary to use a monoisotopic source and this source should contain an isotope with even numbers of protons and neutrons (so as to have zero nuclear spin). Several isotopes of cadmium, krypton and mercury both fulfill the condition of zero nuclear spin and have bright lines in the visible region of the spectrum. Krypton is a gas at room temperature, allowing for easier isotopic enrichment and lower operating temperatures for the lamp (which reduces broadening of the line due to the Doppler effect), and so it was decided to select the orange line of krypton-86 (λ ≈ 606 nm) as the new wavelength standard. Accordingly, the 11th CGPM in 1960 agreed a new definition of the metre:

The metre is the length equal to 1 650 763.73 wavelengths in vacuum of the radiation corresponding to the transition between the levels 2p₁₀ and 5d₅ of the krypton 86 atom.

The measurement of the wavelength of the krypton line was not made directly against the international prototype metre: instead the ratio of the wavelength of the krypton line to that of the cadmium line was determined in vacuum, and this was compared to the 1906 Fabry–Perot determination of the wavelength of the cadmium line in air (with a correction for the refractive index of air). In this way, the new definition of the metre was traceable to both the old prototype metre and the old definition of the angstrom.

Speed of light standard

The krypton-86 discharge lamp operating at the triple point of nitrogen (63.14 K, −210.01 °C) was the state of the art light source for interferometry in 1960, but it was soon to be supeceded by a new invention – the laser, of which the first working version was constructed in the same year as the redefinition of the metre. Laser light is usually highly monochromatic, and is also coherent (all the light has the same phase, unlike the light from a discharge lamp), both of which are advantageous for inteferometry.

The shortcomings of the krypton standard were demonstrated by the measurement of the wavelength of the light from a methane-stabilized helium–neon laser (λ ≈ 3.39 µm). The krypton line was found to be asymmetrical, so two different wavelengths could be found for the laser light depending on which point on the kyrpton line was taken for reference. The asymmetry also affected the precision to which the wavelengths could be measured.

Developments in electronics also made it possible for the first time to measure the frequency of light in or near the visible region of the spectrum, instead of inferring the frequency from the wavelength and the speed of light. Although visible and infrared frequencies were still too high to be directly measured, it was possible to construct a "chain" of laser frequencies that, by suitable multiplication, differ from each other by only a directly measureable frequency in the microwave region. The frequency of the light from the methane-stabilized laser was found to be 88.376 181 627(50) THz.

Independent measurements of frequency and wavelength are, in effect, a measurement of the speed of light (c = fλ), and the results from the methane-stabilized laser gave the speed of light with an uncertainty almost 100-times lower than previous measurements in the microwave region. Or, somewhat inconveniently, the results gave two values for the speed of light, depending on which point on the krypton line was chosen to define the metre. The ambiguity was resolved in 1975, when the 15th CGPM approved a conventional value of the speed of light as exactly 299 792 458 m s⁻¹.

Nevertheless, the infrared light from a methane-stabilized laser was inconvenient for use in practical interferometry. It was not until 1983 that the chain of frequency measurements reached the 633 nm line of the helium–neon laser, which can be stabilized (if necessary) against absorptions of molecular iodine. That same year, the 17th CGPM adopted the current definition of the metre, in terms of the 1975 conventional value for the speed of light:

The metre is the length of the path travelled by light in vacuum during a time interval of ¹⁄_{299 792 458} of a second.

The concept of define a unit of length in terms of a time received some comment, although it is no different from Wilkins' original proposal in 1668 to define the universal unit of length in terms of the seconds pendulum. In both cases, the practical issue is that time can be measured more accurately than length (one part in 10¹³ as opposed to four parts in 10⁹ in 1983). The definition in terms of the speed of light also means that the metre can be realized using any light source of known frequency, rather than defining a "preferred" source in advance: given that there are more than 22,000 lines in the visible spectrum of iodine, any of which could be potentially used to stabilize a laser source, the advantages of flexibity are obvious.

External links

• meter (metre) at Sizes.com