**Page updated:**
October 18, 2020 **Author:** Curtis Mobley

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# Units

Almost any human endeavor from subsistence farming to modern science requires an agreed-upon set of standards for measuring things as diverse as the amount of grain harvested or the wavelength of a particular color of light. Over the centuries many diﬀerent measurement standards have been used. These standards were often arbitrary and imprecise. The English inch, for example, was deﬁned in 1324 by King Edward II of England to be “three grains of barley, dry and round, placed end to end, lengthwise.” The metric system was proposed in 1790 with the goal to be “for all times, for all people” because the measurement units would be related to natural physical quantities. Thus the unit of distance, the meter, was deﬁned to be one ten-millionth of the distance from the Earth’s north pole to the equator, measured along a great circle. Such a deﬁnition was adequate for a century or so, but is too imprecise for modern needs.

Modern science uses the International System of Units, or SI units, from the French Système International d’Unités. These units for the commonly needed measures of distance, time, electrical charge, etc. are now all deﬁned in terms of fundamental physical constants such as the speed of light and the charge of the electron.

### The Fundamental Physical Constants

In 2018, after many years of careful measurement and discussion, representatives of 60 nations unanimously agreed on values for seven fundamental physical constants, from which seven SI base units can be deﬁned. Table 1 shows these seven fundamental physical constants. Note that the numerical values shown in the table are by deﬁnition exact.

Most of the quantities in Table 1 should be familiar from introductory physics and chemistry, but a couple warrant comment. The hypeﬁne transition frequency of the cesium-133 atom refers to the frequency of microwave radiation that corresponds to an electron jump between two closely spaced energy levels of a neutral cesium-133 atom (in ”ﬁeld-free space,” that is, in the absence of gravitational, electrical, or magnetic ﬁelds, which can change the atom’s internal energy levels). The value of $\Delta {\nu}_{Cs}$ gives a fundamental standard for specifying frequency, measured in Hertz, which is cycles (or periods) per second. The strangest of these constants is the luminous eﬃcacy ${K}_{cd}$, which is a measure of how well a light source using a given power (in Watts) produces visible light (as seen by a normal human eye), measured in lumens. ${K}_{cd}$ is deﬁned to be the luminous eﬃcacy of monochromatic radiation of frequency $540\times 1{0}^{12}$ Hertz, which is green light. Exact deﬁnitions of these quantities and further discussion can be found in the National Institute of Standards and Technology Special Publication 330.

Physical Constant | Symbol | Exact Numerical Value | Unit |

speed of light in vacuo | $c$ | $299\phantom{\rule{0.3em}{0ex}}792\phantom{\rule{0.3em}{0ex}}458$ | $m\phantom{\rule{0.3em}{0ex}}{s}^{-1}$ |

Planck constant | $h$ | $6.626\phantom{\rule{0.3em}{0ex}}070\phantom{\rule{0.3em}{0ex}}15\times 1{0}^{-34}$ | $J\phantom{\rule{0.3em}{0ex}}H{z}^{-1}$ |

elementary electrical charge | $e$ | $1.602\phantom{\rule{0.3em}{0ex}}176\phantom{\rule{0.3em}{0ex}}634\times 1{0}^{-19}$ | C |

Boltzman constant | $k$ | $1.380\phantom{\rule{0.3em}{0ex}}649\times 1{0}^{-23}$ | $J\phantom{\rule{0.3em}{0ex}}{K}^{-1}$ |

Avagadro constant | ${N}_{A}$ | $6.022\phantom{\rule{0.3em}{0ex}}140\phantom{\rule{0.3em}{0ex}}76\times 1{0}^{23}$ | $mo{l}^{-1}$ |

hyperﬁne transition frequency of ${}^{133}Cs$ | $\Delta {\nu}_{Cs}$ | $9\phantom{\rule{0.3em}{0ex}}192\phantom{\rule{0.3em}{0ex}}631\phantom{\rule{0.3em}{0ex}}770$ | Hz |

luminous eﬃcacy | ${K}_{cd}$ | 683 | $lm\phantom{\rule{0.3em}{0ex}}{W}^{-1}$ |

### The SI Base Units

The fundamental physical constants seen in Table 1 can be used to deﬁne seven SI base units, which are more convenient for practical applications. Thus one second is deﬁned at the duration of $9\phantom{\rule{0.3em}{0ex}}192\phantom{\rule{0.3em}{0ex}}631\phantom{\rule{0.3em}{0ex}}770$ periods of the radiation corresponding to the transition between the two hyperﬁne levels of the ground state of the cesium-133 atom, or

$$1\phantom{\rule{1em}{0ex}}second\equiv \frac{9\phantom{\rule{0.3em}{0ex}}192\phantom{\rule{0.3em}{0ex}}631\phantom{\rule{0.3em}{0ex}}770}{\Delta {\nu}_{Cs}}\phantom{\rule{0.3em}{0ex}}.$$ |

Similarly, the meter is deﬁned by

$$1\phantom{\rule{1em}{0ex}}meter\equiv \frac{9\phantom{\rule{0.3em}{0ex}}192\phantom{\rule{0.3em}{0ex}}631\phantom{\rule{0.3em}{0ex}}770}{299\phantom{\rule{0.3em}{0ex}}792\phantom{\rule{0.3em}{0ex}}458}\frac{c}{\Delta {\nu}_{Cs}}\phantom{\rule{0.3em}{0ex}},$$ |

and so on. The deﬁnition of the remaining base units are given in NIST Special Publication 330, cited above.

Table 2 shows the seven SI base units, plus two supplementary units that are convenient for measurement of plane and solid angle. All other quantities are derivable from these units. With the exception of the candela, which is needed only for the discussion of photometry, we presume that the reader is familiar with these SI units from basic physics and chemistry.

Physical quantity | Base Unit | Symbol |

length | meter | m |

mass | kilogram | kg |

time | second | s |

electric current | ampere | A |

temperature | kelvin | K |

amount of substance | mole | mol |

luminous intensity | candela | cd |

Supplementary units | ||

plane angle | radian | rad |

solid angle | steradian | sr |

The nomenclature and symbols most widely used today in optical oceanography follow the recommendations of the Committee on Radiant Energy in the Sea of the International Association of Physical Sciences of the Ocean (IAPSO; see Morel and Smith (1982)). However, neither the SI units nor the recommended IAPSO notation are entirely satisfactory. In particular, they are sometimes inconvenient for measurements and mathematical manipulations; consequently we occasionally shall make minor deviations from the IAPSO recommendations. Several derived units that we shall need are shown in Table 3.

Physical quantity | Derived Unit | Symbol | Deﬁnition |

wavelength of light | nanometer | nm | $1{0}^{-9}$ m |

energy | joule | J | 1 kg m${}^{2}$ s${}^{-2}$ |

power | watt | W | 1 kg m${}^{2}$ s${}^{-3}$ |

number of photons | einstein | einst | 1 mol of photons |

There are other non-SI units that are commonly used and acceptable. These include minutes, hours, and days, which are multiples of the second; the liter, which is one-thousandth of a cubic meter; degrees, minutes, and seconds of angles, which are fractions of a radian; and so on. Again, it is assumed that the reader is familiar with these units.

#### Historical Note

The United States is one of only three countries that do not use the SI system (the other two are Myanmar and Liberia). If you live in the USA, you have to learn, for example, that there are 5280 feet in a mile. Where does such a number come from? In England, a (statute) mile was originally deﬁned as 8 furlongs. A furlong (a furrow long) was deﬁned as the distance a team of oxen could plow without resting. A furlong was divided into 40 rods, and a rod was 16.5 feet, where a foot was deﬁned as the average length of the left feet of 16 men chosen at random as they left church on Sunday. Seriously, you can’t make this stuﬀ up! So a mile is 8 x 40 x 16.5 = 5280 feet. No wonder the rest of the world makes fun of Americans for not converting to metric units. Fortunately, American scientists have enough sense to use SI units, even if the average American does not.