What Exactly Are the New Definitions of Kilogram and Other SI Units? Vjera Lopac, and Dario Hrupec

Citation: The Physics Teacher 58, 58 (2020); doi: 10.1119/1.5141976 View online: https://doi.org/10.1119/1.5141976 View Table of Contents: https://aapt.scitation.org/toc/pte/58/1 Published by the American Association of Physics Teachers What Exactly Are the New Definitions of Kilogram and Other SI Units? Vjera Lopac, University of Zagreb, Zagreb, Croatia Dario Hrupec, Josip Juraj Strossmayer University of Osijek, Osijek, Croatia

n November 16, 2018, at the 26th meeting of the each base unit of the SI system. It should be stressed again that General Conference on Weights and Measures the seven defining constants are fixed, i.e., their numerical (CGPM), a resolution was adopted that introduced values are exact. Ohistorical changes into the International System of Units (SI). This decision, effective from May 20, 2019, established that The new definition of kilogram and other base the SI units of measurement will be defined by means of a set units of SI of seven fixed numerical values of natural constants.1-7 The Solving the first equation in Table II gives the definition of kilogram, the basic unit of mass, is no longer defined by a the second: standard prototype, but relies on the Planck constant h. The . definitions of the base SI units ampere, kelvin, and mole will (1) also change. This will accomplish the long-term dream of scientists who have been claiming that measurement systems Combined with the second row in Table II, this gives the solu- should be based on natural units.4,5 In this paper we derive the tion for the meter: natural units as combinations of the seven defining natural constants and determine the numerical relations between the (2) natural units and the units of SI.

The new definitions of the SI units (3) There is plenty of literature dedicated to the history of SI, and the earlier definitions of the base SI units can be found These are the new definitions of the second and the meter, in many publications. Here we focus our attention to the new whereas the expressions definitions1-4 and analyze their meaning. The new definitions of the seven base units are shown in Table I. Each of them is and characterized by one of the seven natural constants. The first constant is the caesium frequency νCs, well known as the measurement standard for precise atomic clocks.8 Next are are the natural units for time and length, respectively. the five universal constants c, h, e, kB, and NA, which are in- By solving the third row in Table II and using the solution volved in the physical laws in classical and quantum physics. for the second and the meter, one obtains the new definition 10 The last constant Kcd is a conveniently chosen fixed numerical of kilogram : value of luminous efficacy.9 The new definitions in Table I are instructive and com- . (4) plete. They are, however, very concise; therefore, the practical guidance is provided on the BIPM website,7 as mentioned The expression is the natural unit for mass. If this in Ref. 6. From them one concludes that the definitions of the second, the meter, and the candela have not changed, equation is multiplied by c2, one obtains although they are now written differently. The change in the . (5) definition of the watt has no practical consequences for the candela. This leads to a physical interpretation of the kilogram, which There remain the four important base units, including the is: kilogram, whose definitions we analyze in the following sec- A body has a mass of 1 kg if its rest energy is equal to tion. We describe the mathematical procedure to obtain for the energy of 1.475521431040 caesium photons. each of them both the natural unit and the factor connecting this natural unit to the corresponding SI unit. This shows that the obtained numerical factors also have a physical meaning and are important parts of the unit defini- Seven natural constants – Seven equations! tions. (It should be said, however, that such an interpretation When the seven constants are written together (Table II), does not suggest a practical way to implement the definition.) one immediately observes that these are actually a system of Continuing with the similar procedure, one solves all seven seven algebraic equations, in which the units s, m, kg, A, K, equations and obtains Table III, where each of the seven base mol, and cd are the unknown variables.10-12 It is obvious that units is expressed as a natural unit (which is a combination of by solving these equations, one obtains expressions contain- the seven natural constants), multiplied by a calculated nu- ing the universal natural units and the numerical factors for merical factor.

58 THE PHYSICS TEACHER ◆ Vol. 58, January 2020 DOI: 10.1119/1.5141976 Table I. New definitions of the seven base units of SI. and the derived units. In principle, all SI SECOND The second, symbol s, is the SI unit of time. It is defined by taking the fixed units can be derived from other sets of numerical value of the caesium frequency ΔvCs, the unperturbed ground-state seven basic units. (For example, in the hyperfine transition frequency of the caesium-133 atom, to be 9192631770 revised SI the joule could have replaced –1 when expressed in the unit Hz, which is equal to s . the kilogram and the ohm could have METER The meter, symbol m, is the SI unit of length. It is defined by taking the fixed replaced the ampere as basic units, with numerical value of the speed of light in vacuum c to be 299792458 when no change needed to the seven defining −1 expressed in the unit m.s , where the second is defined in terms of the caesium constants.) This is stressed by the fact frequency ΔvCs. that some of the natural units are ex- KILOGRAM The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the tremely simple, containing only one or −34 fixed numerical value of the Planck constant h to be 6.62607015×10 when two constants. These are the units for expressed in the unit J.s, which is equal to kg.m2.s−1, where the meter and the the frequency ( νCs), velocity (c), action second are defined in terms of c and ΔvCs. (h), or electric charge (e), as well as those AMPERE The ampere, symbol A, is the SI unit of electric current. It is defined by taking for the magnetic flux (h/e) and the elec- −19 the fixed numerical value of the elementary charge e to be 1.602176634×10 tric conductance (e2/h). Furthermore, . when expressed in the unit C, which is equal to A s, where the second is defined in the natural units we recognize the el- in terms of Δ . vCs ements of some fundamental equations KELVIN The kelvin, symbol K, is the SI unit of thermodynamic temperature. It is 2 for the energy (E = hν, E = mc , E = kBT, defined by taking the fixed numerical value of the Boltzmann constant kB to be and E = eU, where v, m, T, and U denote −23 . −1 . 2. −2. 1.380649×10 when expressed in the unit J K , which is equal to kg m s the frequency, mass, temperature and K−1, where the kilogram, meter, and second are defined in terms of h, c, and voltage, respectively), appearing in clas- Δv . Cs sical, quantum, relativistic, statistical, MOLE The mole, symbol mol, is the SI unit of amount of substance. One mole contains and atomic physics. exactly 6.02214076×1023 elementary entities. This number is the fixed numeri- −1 cal value of the Avogadro constant, NA, when expressed in the unit mol and is Educational benefits from the called the Avogadro number. The amount of substance, symbol n, of a system is a measure of the number of specified elementary entities. An elementary entity new definitions may be an atom, a molecule, an ion, an electron, any other particle or specified Teachers of introductory physics group of particles. in high schools and colleges have now CANDELA The candela, symbol cd, is the SI unit of luminous intensity in a given direction. It a great responsibility, since they will is defined by taking the fixed numerical value of the luminous efficacy of mono- be the first who will have to explain to 12 chromatic radiation of frequency 540×10 Hz, Kcd, to be 683 when expressed students and to the general public the in the unit lm.W−1, which is equal to cd.sr.W−1, or cd.sr.kg−1.m−2.s3, where the importance of the new definitions of kilogram, meter, and second are defined in terms of h, c, and ΔvCs. SI units. As stressed in Ref. 5, the new definitions offer many topics for further Relations between the natural units and the study: among others, the history of SI derived units of SI and experimental methods developed for obtaining accurate The method used for the base units and the obtained re- values of the natural constants13 or the experimental setups to sults shown in Table III are applied to calculations of the cor- be used in the classroom,14 for which, however, the absolute responding relations for units of the derived physical quanti- precision can not be expected. The literature recommended to ties. The results for a number of derived units are presented in teachers and students should be completed with the 9th edi- Table IV. tion of the SI brochure,6 which describes the revised SI and is Several important conclusions follow from the analysis of published online on the BIPM website.7 Tables III and IV. The natural units for the base and derived To these subjects we add the investigations of the natural quantities are obtained by solving equations of Table II. They units by solving appropriate equations and calculations of are combinations Table II. Natural constants involved in the new their numerical relationships. Their numerical values can be of the natural definitions of the SI base units. obtained by using the pocket calculator, but when high preci- constants, with The value of the natural constants expressed sion values are needed, the best solution is a computer code. the correct di- in the SI base units (Even then the obtained numbers are only approximations mension, and no –1 but, if needed, the additional digits can always be calculated additional dimen- vCs = 9192631770 s using the described method.) The program code is simple and –1 sional analysis c = 299792458 m∙s can be realized with any of the computer languages accessible is necessary. It h = 6.62607015310–34 kg∙m2 ∙ s–1 to students. The exponents of the seven constants contained should be pointed e = 1.602176634 3 10–19A∙s in a natural unit are read as input. We did our calculations 15 out that now there –23 2 –2 –1 with codes in Python and FORTRAN. In the appendix our kB = 1.380649 310 kg∙m ∙ s ∙ K is no essential Python code is listed, with the necessary explanations. 23 –1 distinction be- NA = 6.02214076 3 10 mol At the end, we conclude that the specific choice of a set of tween the base –1 –2 3 universal constants and properties of the resulting natural Kcd = 683 cd ∙ sr ∙ kg ∙ m ∙ s

THE PHYSICS TEACHER ◆ Vol. 58, January 2020 59 Table III. Numerical relations between the natural units and the base units of SI. methods with the laws of physics, confirm that Base quantity Base unit Relations between the base units of SI and the redefinition of the SI units has a huge edu- of SI the natural units cational potential. 9 time second 1 s = (1 / vCs) ∙ 9.192631770310 References length meter 1 m = (c / v ) ∙ 3.066331899 3101 Cs 1. “On the revision of the International System of 2 40 mass kilogram 1 kg = (h vCs /c ) ∙ 1.475521400 310 Units (SI),” (“Sur la révision du Système international 8 d’unités [SI]”), https://www.bipm.org/utils/common/ electric current ampere 1 A = (e vCs ) ∙ 6.789686817310 pdf/CGPM-2018/26th-CGPM-Resolutions.pdf. thermodynamic kelvin 0 1 K = (h vCs/kB) ∙ 2.266665265 3 10 2. “2019 redefinition of SI base units,” https:// temperature en.wikipedia.org/wiki/2019_redefinition_of_SI_ 23 amount of sub- mole 1 mol = (1/ NA) ∙ 6.022140760 3 10 base_units. stance 3. “Redefining the SI units,” https://www.npl.co.uk/ luminous intensity candela 2 10 si-units/the-redefinition-of-the-si-units. 1 cd = [Kcd h( vCs ) ] ∙ 2.614830482 3 10 /sr 4. Gordon J. Aubrecht II, “Changes coming to the Table IV. Numerical relations between the natural units and the derived SI units. International System of Units,” Phys. Teach. 50, 338–342 (Sept. 2012). Derived quantity Derived unit of SI Relations between the derived units of SI and the natural units 5. Sandra Knotts, Peter J. Mohr, and William D. Phillips, “An intro- -1 –9 velocity, speed meter per second 1 m∙s = (c) ∙ 3.335640952310 duction to the new SI,” Phys. Teach. –2 –19 55, 16–21 (Jan. 2017). acceleration meter per second 1 m∙ s = (c vCs) ∙ 3.628602815 310 squared 6. BIPM Brochure, 9th ed., “Le –1 31 Système international d’unités, momentum kilogram × meter per kg∙m ∙s = (h vCs/c) ∙ 4.921809606 310 second The International System of Units,” https://www.bipm.org/utils/ angular momentum kilogram × meter 2 –1 33 1 kg∙ m ∙ s = (h) ∙ 1.509190180 310 common/pdf/si-brochure/SI-Bro- squared per second chure-9.pdf. –10 frequency hertz 1 Hz = ( vCs) ∙ 1.0878277573 10 7. BIPM website, https://www. 23 bipm.org/. energy, work joule 1 J = (h vCs ) ∙ 1.641738968310 8. John M. Richardson and James power watt 2 13 1 W = h ( vCs ) ∙ 1.785929219 3 10 F. Brockman, “Atomic standards of 2 21 force newton 1 N = [h ( vCs) /c] ∙ 5.354081105 310 frequency and time,” Phys. Teach. 4, 247–256 (Sept. 1966). pressure pascal 1 Pa = [h ( v )4/c3 ] ∙ 5.694382340 3 1018 Cs 9. Nathaniel R. Greene, “Shedding 33 action joule × second 1 J ∙s = (h) ∙ 1.509190180 3 10 light on the candela,” Phys. Teach. electric charge coulomb 1 C = (e) ∙ 6.24150907431018 41, 409–414 (Oct. 2003). 10. Richard Davis, “How to define electric field newton per coulomb 1 N∙ C-1 = [h( v )2/ (e c)]∙ 8.5781836433102 Cs the units of the revised SI starting 4 voltage volt 1 V = (h vCs/e) ∙ 2.630355814 310 from seven constants with fixed electric resistance ohm 1  = (h /e2) ∙ 3.874045865 310–5 numerical values,” J. Res. Nat. Inst. Stand. Technol. 123021 (2018). electric capacitance farad 2 3 14 1 F = [e / (h vCs)] ∙ 2.372876339 10 11. Barry N. Taylor, “Quantity cal- magnetic flux weber 1 Wb = (h/ e) ∙ 2.41798924231014 culus, fundamental constants, and SI units,” J. Res. Nat. Inst. Stand. Tech- magnetic induction tesla 1 T = [(h ( v )2/(e c2)] ∙ 2.571674759 31011 Cs nol. 123008 (2018). electric conductance siemens 2 4 1 S = (e /h) ∙ 2.581280746310 12. Peter J. Mohr, “Defining units 2 5 inductivity henry 1 H = [h /( vCs e )] ∙ 3.561267710 3 10 in the quantum based SI,” Metrologia 45 (2), 129–133 (2008). luminous flux lumen 1 lm = [h( v )2 K ] ∙ 2.614830482 3 1010 Cs cd 13. Peter J. Mohr and David B. illuminance lux 4 2 7 1 lx = [h ( vCs) Kcd/c ] ∙ 2.781027076 3 10 Newell, “Resource Letter FC-1: The –10 physics of fundamental constants,” activity becquerel 1 Bq = ( vCs) ∙ 1.087827757 310 Am. J. Phys. 78, 338–358 (April absorbed dose gray 2 3 –17 1 Gy = (c ) ∙ 1.112650056 10 2010). 2 –17 dose equivalent sievert 1 Sv = (c ) ∙ 1.112650056 3 10 14. L. S. Chao, S. Schlamminger, D. B. Newell, J. R. Pratt, F. Seifert, X. units can lead to inspiring arguments in the discussions with Zhang, G. Sineriz, M. Liu, and D. Haddad, “A LEGO watt bal- students. The scaling factors obtained for various quantities ance: An apparatus to determine a mass based on the new SI,” may differ by 58 orders of magnitude, and this can be another Am. J. Phys. 83, 913–922 (Nov. 2015). creative idea for discussion. A wide choice of possible topics 15. Readers can view the appendix at TPT Online, http://dx.doi. for discussion, connecting the measurement principles and org/10.1119/1.5141976 under the Supplemental tab. [email protected]; [email protected]

60 THE PHYSICS TEACHER ◆ Vol. 58, January 2020