2.1 THE INTERNATIONAL SYSTEM OF UNITS (SI)

The International System of Units (SI) was adopted in 1960 [3]. The SI is set up on the basis of seven base units, which are given in Section 2.2. The SI derived units are expressed as products of the base units. As of May 20, 2019, the SI is defined by fixing the numerical values of seven physical constants [3].

The SI is the system of units in which: - the unperturbed ground state hyperfine transition frequency of the caesium-133 atom Cs is 9 192 631 770 Hz, - the speed of light in vacuum c is 299 792 458 m/s, - the Planck constant h is 6.626 070 15×10−34 J s, - the elementary charge e is 1.602 176 634×10−19 C, - the Boltzmann constant k is 1.380 649×10−23 J/K, - the Avogadro constant NA is 6.022 140 76×1023 mol−1, - the luminous efficacy of monochromatic radiation of frequency 540×1012 Hz, Kcd, is 683 lm/W.

Some of these physical constants are also given in the abbreviated list of fundamental constants in the cover page of this book.

In the International System of Units there is only one coherent SI unit for each physical quantity. This is either the appropriate SI base unit itself or the appropriate SI derived unit. However, any of the approved decimal prefixes, called SI prefixes, may be used to construct decimal multiples or submultiples of SI units (see Section 2.5 below). It is recommended that units of the SI be used in science and technology (for more details, see [3]). When other units are used, they should be clearly defined in relation to the SI units.

2.2 NAMES AND SYMBOLS FOR THE SI BASE UNITS

The symbols listed here are internationally agreed and shall not be changed in other languages or scripts. See sections 1.3.2 and 1.6, p. 3 and p. 5 on the printing of symbols for units.

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2.3 COHERENT UNITS AND CHECKING DIMENSIONS

If equations between numerical values have the same form as equations between physical quantities, then the system of units defined in terms of base units avoids numerical factors between units, and is said to be a coherent system. For example, the kinetic energy T of a particle of mass m moving with a speed v is defined by the equation

T = (1/2) mv2

but the SI unit of kinetic energy is the joule, defined by the equation

J = kg (m/s)2 = kg m2 s−2

where it is to be noted that the factor (1/2) is omitted. In fact the joule is simply a special name and its symbol J stands for the product of units kg m2 s−2.

The International System (SI) is a coherent system of units. The advantage of a coherent system of units is that if the value of each quantity is substituted for the quantity symbol in any quantity equation, then the units may be canceled, leaving an equation between numerical values which is exactly similar (including all numerical factors) to the original equation between the quantities. Checking that the units cancel in this way is sometimes described as checking the dimensions of the equation.

The use of a coherent system of units is not essential. In particular the use of multiple or submultiple prefixes destroys the coherence of the SI, but is nonetheless often convenient.

2.4 PHYSICAL CONSTANTS USED AS ATOMIC UNITS

Sometimes fundamental physical constants, or other well defined physical quantities, are used as though they were units in certain specialized fields of science. For example, in astronomy it may be more convenient to express the mass of a star in terms of the mass of the sun. In atomic and molecular physics it is similarly more convenient to express masses in terms of the electron mass, me, or in terms of the unified atomic mass unit, 1 u, and to express charges in terms of the elementary charge e, and energies in terms of the electronvolt, eV.

The electronvolt is the kinetic energy acquired by an electron in passing through a potential difference of 1 V in vacuum, 1 eV = 1.602 176 634×10−19 J. The numerical value of a quantity expressed in this unit may be converted into its value when expressed in the SI by multiplication with the value of the physical constant in the SI.

The dalton and the unified atomic mass unit are alternative names for the same unit, therefore 1 u = 1 Da 1.6605×10−27 kg. The dalton may be combined with the SI prefixes to express the masses of large molecules in kilodalton (kDa) or megadalton (MDa).

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(1) = h/2 ; a0 = 4 0 2/mee2; Eh = 2/mea02. (2) The numerical value of the speed of light, when expressed in atomic units, is equal to the reciprocal of the fine-structure constant ; c/(a0Eh/ ) = c /a0Eh = −1 = 137.035 999 084(21). (3) The atomic unit of magnetic dipole moment is twice the Bohr magneton, μB.

One particular group of physical constants that are used as though they were units deserve special mention. These are the so-called atomic units and arise in calculations of electronic wavefunctions for atoms and molecules, i.e. in quantum chemistry. Only the first five atomic units in the table above have special names and symbols.

The relation of atomic units to the corresponding SI units involves the values of the fundamental physical constants, and is therefore not exact. The numerical values in the table are rounded from the CODATA compilation [17, 18]. The numerical results of calculations in theoretical chemistry are frequently quoted in atomic units, or as numerical values in the form physical quantity divided by atomic unit, so that the reader may make the conversion using the current best estimates of the physical constants.

Many authors make no use of the symbols for the atomic units listed in the tables above, but instead use the symbol “a.u.” or “au” for all atomic units. This custom should not be followed. It leads to confusion, just as it would if we were to write “SI” as a symbol for every SI unit, or “cgs” as a symbol for every cgs unit (the ‘centimetre, gram, second’ system of units, see Section 3.2, p. 16).

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2.5 SI PREFIXES AND PREFIXES FOR BINARY MULTIPLES

The following prefixes [3] are used to denote decimal multiples and submultiples of SI units.

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Prefix symbols shall be printed in roman (upright) type with no space between the prefix and the unit symbol.

Example kilometre, km

When a prefix is used with a unit symbol, the combination is taken as a new symbol that can be raised to any power without the use of parentheses.

Examples 1 cm3 = (10−2 m)3 = 10−6 m3

A prefix shall never be used on its own, and prefixes are not to be combined into compound prefixes.

Example pm, not m

The names and symbols of decimal multiples and submultiples of the SI base unit of mass, the kilogram, symbol kg, which already contains a prefix, are constructed by adding the appropriate prefix to the name gram and symbol g.

Examples mg, not kg; Mg, not kkg

The International Electrotechnical Commission (IEC) has standardized the following prefixes for binary multiples, mainly used in information technology, to be distinguished from the SI prefixes for decimal multiples [7].

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