Alternative set of defining constants for redefinition of four SI units

Transcription

1 Alternative set of defining constants for redefinition of four SI units V V Khruschov 1,2 1 Centre for Gravitation and Fundamental Metrology, VNIIMS, Moscow, Russia 2 National Research Centre Kurchatov Institute, Moscow, Russia khkon@vniims.ru Abstract We discuss different sets of defining constants, fixed values of which are considered in connection with the transition to new definitions of four SI units (the kilogram, mole, ampere and kelvin). The notion of constant s order in a given system of units is suggested. We propose an alternative set of fixed constants applicable for new definitions of the four SI units. We analyse and discuss in detail the set, which consists of the Planck constant, the Avogadro constant, the Boltzmann constant and the magnetic constant. 1. Introduction The existing systems of units of physical quantities are a part of the necessary toolkit of science, modern technology, industry and international trade, so as any tools, they should adequately correspond to the present-day state of the art. However the International Prototype of the Kilogram (IPK), which is used in the current definition of the kilogram of the International System of Units (SI), is the only artifact made of the platinum-iridium alloy (platinum - 90 % and iridium - 10 %).The temporal instability of the IPK have been revealed on the level of kg per year [1]. It is unacceptable for high precision measurements and long-term storing of obtained results. Redefinition of four SI base units, being now under preparation [2, 3], can be considered as a response to this challenge. The planned revision of SI grounds on the proposal to define the base SI units by fixing the exact values of the corresponding physical constants (PC), following the principle that was already used in the definition of the metre in 1983 [4, 5]. In the SI Brochure 9 th edition Draft [3] the new term is used instead of fixed PC, namely, defining constants (DC). Employment of exact constants values has a great significance in metrology and the proposals to redefine a number of SI units on the basis of DC values have been supported by metrological agencies, workshops and conferences [2, 5]. The main obstacle for adoption of new 1

2 definitions, as soon as it had been realized, was the insufficient precision of experimental DC values. The proposal for redefinition of a number of SI base units was published in 1999 [6] and the complete set of proposals was published in 2006 [4]. Several DC sets were considered. In particular it was suggested to fix without any uncertainty the values of the Planck constant(h), the elementary charge (e), the Boltzmann constant (k)and the Avogadro constant (N A ) for redefinition of the kilogram, the ampere, the kelvin and the mole [3-5]. This DC set is considered the most preferable now. For instance in the SI Brochure 9 th edition Draft the presentation of SI with new definitions of four base units (the New SI) is made on its ground. The current situation with a transition to the New SI is reflected in Resolution 1 of the 24th General Conference on Weights and Measures (CGPM) in 2011 On possible future revision of the International System of Units (SI), where it was proposed to continue work towards improved formulations for the definitions of the SI base units in terms of fundamental constants, having as far as possible a more easily understandable description for users in general, consistent with scientific rigour and clarity. The 25th CGPM in 2014 pointing to the progress from the 24th CGPM nevertheless took into account the insufficient precision of measured DC values and recommended to continue the execution of Resolution 1 of the 24 th CGPM. It is supposed that the New SI adoption will be at the 26th CGPM in 2018 [2]. The purpose of this paper is to discuss critically and in detail proposed new definitions of four base SI units, namely, the kilogram, mole, ampere and kelvin, in the light of the current theoretical and experimental results. Furthermore a new notion is introduced, namely, a constant s order in a considered system of units. We propose also an alternative set of DC constants applicable for new definitions of the four SI units, which is not considered in detail before. The set consists of the Planck constant h, the magnetic constant 0, the Boltzmann constant k and the Avogadro constant N A. Values of this set are discussed and are compared with those of the set with h, e, k and N A. We expect that the consideration of these problems will be useful for the most sensible and relevant choice of the DC set for new definitions of the four SI units. The paper is organized as follows. Section 2 outlines briefly the recent results of measurements of the Planck, Avogadro and Boltzmann constants and some results concerning the creation of a quantum standard of the ampere based on the electron charge. Section 3 suggests a new DC classification and discusses the criteria for a preferable choice of DC set 2

3 for new definitions of the SI units. Section 4 analyses different DC sets proposed for a redefinition of the four SI units. In Section 5, new definitions of the kilogram, mole, ampere and kelvin are considered in detail on the basis of fixation of the h, 0, k and N A values. The conclusion summarizes our main proposals, which are related to the future optimum choice of the New SI variant, i.e. the New SI with the definite DC set. 2. Resent results concerning the Planck, Avogadro and Boltzmann constants and a quantum standard of the ampere For adoption of new definitions for the units of mass and amount of substance the most important conditions are achieving of the level of for the relative standard uncertainty u r of the Planck and Avogadro constants values and a consistency between their values obtained by different methods on the 95 % C.L. (Recommendation G1 of the Consultative Committee on Mass and Related Quantities (CCM) adopted at the session in 2010 [2]). Until 2014 these conditions had not been achieved so at the 25 CGPM the New SI adoption was postponed, it is now planned to occur at the 26 CGPM in 2018 [2]. At the end of 2014 and at the beginning of 2015 new results of measurements of the Planck and Avogadro constants were obtained with u r = [8, 9] (see also [10]) that is decisive for adoption and realization of new definitions of the kilogram and mole. The conditions for passing over to a new definition of the kelvin were formulated by Consultative Committee for Thermometry in 2014 (Recommendation T 1) [11]: 1. the relative standard uncertainty of the adjusted value of k is less than ; 2. the determination of k is based on at least two fundamentally different methods, of which at least one result for each shall have a relative standard uncertainty less than A value of k is found by measuring = kt at the temperature of the triple point of water ( tpw ), the thermodynamic temperature of which T tpw is fixed exactly as T tpw = K. So the uncertainty of k is determined by the uncertainty of tpw. In other thermo physical experiments values are measured, then T values are determined with the help of k. For the time being the adjusted value of k is k 2014 = JK -1 with u r = [12]. This value is obtained by the special procedure of the data adjustment of the precise k values obtained in the national metrological institutes experiments. The most precise one was obtained in the NPL experiment in 2013 by the method of the acoustic gas thermometry [13], namely k NPL-2013 = JK -1, u r = During the latest two years the method for determining of the molar argon mass in the NPL experiment 3

4 had been improved and in 2015 u r was increased: k NPL-2015 = JK -1, u r = [14]. This improvement will influence on the CODATA value of k [15]. For the fixation of the precise electron charge value e for the ampere definition the problem of the so-called quantum triangle is of the decisive importance [15, 16, 17]. Solving of this problem must result in the creation of a quantum standard of the ampere based on e [18-23]. Year to date this line of investigation do not achieve the assigned mark for the New SI, namely the u r 10-8 level for the current strength. The up-to-date results tell us about some new proposed technical devices but the achieved now level of uncertainty is about 10-7 [22, 23]. 3. DC classification and criteria for a preferable choice of DC In the SI Brochure 9 th edition Draft [3] the presentation of the New SI is made on the basis of DC. This notion is used instead PC or constants of nature, the IPK, the material parameter such as the triple point of water and so on. The totality of constants can be divided in four classes such as fundamental constants of nature, special atomic parameters, conversion factors, and technical constants. However this classification has some shortcomings. For instance the speed of light in vacuum c is the fundamental constant of nature, but it becomes the conversion factor after fixing its value. The Avogadro constant N A corresponds to a conversion factor between the unit for amount of substance and the unit for counting entities (unit 1) [3] (see also[24]), but it can be considered as the constant of nature (Avogadro s law). It is known that the number of constants increases or decreases along with the number of the base units [25]. Besides that the number of constants of the definite system of units depends on the assumed physical theory or model. For example, there are not the c and h constants in the frame of the nonrelativistic classical mechanics, but they are in the relativistic quantum mechanics. Different classifications of PC are suggested taking into account their role in description of physical phenomena (see, e.g. [26]). In the present paper we propose to use a new classification of constants that is connected with the base units of the given system of units. The constant, which does not depend on the base units of the given system, is named as the constant of the order zero. The constant, which depends on a base unit of the given system, is named as the constant of the order one and so on. At first glance the PC value cannot be fixed by virtue of the fact that it contains both the estimated value and the uncertainty associated with that value. However the PC value can be fixed for the redefined unit if the PC relative uncertainty is less or equal to the min- 4

5 imal relative uncertainty of direct measurements of the corresponded quantity. Such a fixed PC will be the DC for the considered unit. It is desirable that a DC set be consistent and minimal, i.e. any DC value do not lead to a contradiction with the usage of other constants with fixed values within the used theory. It is also clear that the status of any DC is not absolute and can change due to generalization of a relevant theory or when the measurement accuracy increases. Anyway, the key requirement to the New SI is that it should not aggravate the situation in any respect for any users as compared with the current SI. This means, above all, a necessary succession with respect to the current SI, which implies establishing the same set of base units and the same values of all units as they exist by the day of revision in order that the whole enormous set of the existing measurement data could be preserved without correction. Evidently, the revision of SI should not also worsen the stability properties of the standards of any units against the old ones. For instance, new BIPM prototypes of the mass unit should have a confirmed temporal instability not worse than per year, and this requirement should hold for any new prototype of the kilogram at any time and place. In this respect, it is necessary to recall that the detected temporal instability of the IPK copies for 100 years [1] is mainly accounted for the planned transition to the New SI. So it is clear that similar requirements concern the remaining three base SI units. For concordance between the system of measurement units with fixed values and physical theories, it is desirable that the number of fixed constants should be minimum possible, and the relation between the DC and the corresponding unit should be as simple as possible. The same conditions are desirable for successful teaching of the fundamentals of metrology at universities and colleges. This goal is achieved if the base unit and the corresponding DC have the same physical dimension or their dimensions relate each other as simple as possible. For example, the dimensions of the velocity of light and the metre, the electron charge and the ampere differ only by a certain power of time. The dimension of the kilogram is the same as the dimension of the unified atomic mass unit m u (= (mass 12 C)/12). In terms of constants of different orders having introduced above the criterion needed for this purpose can be formulated as follows: the DC order should be as low as possible. The maximum possible simplicity of the new SI and its succession with respect to the present SI also assumes that, if possible, the base measurement units should be mutually independent, and that no new factors or constants should be introduced for the transition to 5

6 new definitions. Thus we suggest the following criteria for choosing the optimal set of DC for using in the new definitions of SI units (see also [15, 17]): a) succession between the old and new definitions, b) the stability requirement for transfers of the unit values, c) a minimum of DC, d) use of DC with the minimal possible orders. 4. Analysis of DC sets proposed for redefinition of four base SI units At present time discussions about strengths and shortcomings of different new definitions of base SI units still continue (see, e.g. [7, 15, 17, 27, 28]). The preferred version is the DC set consisted of h, e, k, and N A [4]. There are some other versions. For instance, the version with m u, 0, k, and N A is considered in [7] taking into account the latest CODATA adjustment of PC by the least squares method [12]. In the references [6, 17, 29] the version with the m u, e, k, and N A fixation is considered in detail. It is pointed out in Ref. [7] that the kilogram redefinition based on m u is more preferable now than it was ten years ago at forming the four base SI units new definitions. Each version of a DC set has its advantages and disadvantages, which is desirable to consider and discuss at a preparation of a 26 CGPM decision. In the present paper an alternative set of fixed constants applicable for new definitions of four SI units is proposed for the first time. This set consists of h, 0, k, and N A (see below and section 5). The final decision about the four base SI units redefinition would be well to accomplish by taking into account the topical experimental and theoretical results. Let us consider, for example, the results of the work [7] which take into account recent experimental achievements in DC measurements. The previous analysis of possible new definitions of SI units [30] was based on the CODATA-2006, and the 2006 proposals [4] were prepared using the CODATA The level of precision of the CODATA-2014 data set is significantly improved [12]. These improvements provide a different context for the choice of a optimal DC set than was possible in 2007 at the 23 CGPM. Five possible versions of the SI with different DC sets (including the present SI) were considered in the work [7]. For instance, in the present SI the definitions of the kilogram, the ampere, the kelvin and the mole are based on the fixed values of the IPK mass (m(k)), 0, the triple point of water (T tpw ) and the molar mass of 12 C (M( 12 C)). In the work [7] the present SI is named as variant A or system A. Remaining four variants of the New SI are based on the fixed values of different sets containing h, e, k, N A, m u, 0. Since in the frameworks of all these variants k and N A are always fixed then we can distinguish these variants picking two 6

7 DC of h, e, m u, 0. So variant B contains h, e, variant C contains e, m u, variant D contains h, 0, variant E contains m u, 0. System B is the variant of the New SI proposed in the work [4] and noted in Resolution 1 of the 24 CGPM. System B provides zero uncertainty for the Josephson constant K J and the von Klitzing constant R K, which are used as practical electromagnetic standards, i.e. this variant settles the current problem of electrical metrology, namely, the existence of the fixed K J and R K values accepted in In 1990 the K J and R K values conform to the experimental ones within their uncertainties, but now it is not so. Thus the 1990 values should be abrogated; the K J and R K values should be measured anew and then be fixed in system B. Notice that the constant 0 must change depending on experimental values and uncertainties. This dependence can be parameterized as 4 (1+ ) 10-7 N/A 2, where the value of = ( / ), 2018 being the best available experimental value of at the time of redefinition [7]. u r of is the same as u r of, taking the 2018 value to be exact. However, the factor (1+ ) is difficult to explain when 0 needs to be introduced in textbooks in electromagnetism. This is similar to the problem with the varied molar mass constant M u in the framework of system B. So if deviations of 0 and M u from the values N/A 2 and 1 g mol -1 will be small enough then we shall have no problems in practice. However, it is difficult to explain the deviations of 0 and M u from the exact values in educational process and it is not compatible with Resolution 1 of the 24 CGPM that the New SI should be easily understandable for users in general, consistent with scientific rigour and clarity [2]. Moreover, the experimental variation of 0 will result in difference from systems of units used in theoretical physics. System E maps these problems onto the problems connected with the K J and R K values, which are important only for high level users [7]. New definitions of the kilogram and the mole in systems E and C can be realized with the help of both the watt balance and the silicon sphere devices. Equivalence of these methods follows from the known relation between the Planck constant and the Avogadro constant: N A h= A r (e)m u cα 2 /(2R ), where the molar mass constant M u = M( 12 C)/12, M u = N A m u coincides with M u0 = 1 g mol -1 in the present SI, A r (e) is the relative atomic mass of the electron (=m e /m u ), R is the Rydberg constant. For instance, in the framework of the CODATA 2014 the value of the molar Planck constant is known with u r = 4, [12]: (N A h) CODATA-2014 = 3, (18) J s mol -1. So there is not any loss of precision at determination of the N A value with the help of h value and vise versa. u r (N A h) gives a limitation for the new correction factor к in system B: к = M u /M u0 1 = N A m( 12 С)/(12M u0 7

8 ) 1. Appearance of к in this variant of the New SI is under serious criticism (see [31-33] and references therein). In the work [7] it was suggested to change to the New SI on the base of system E. In the framework of this system the new definition of the mass unit using m u is evident. The uncertainties of DC were determined in system E when the current value of 0 is fixed [7]. The system E has advantages for definitions of the kilogram and the mole [34] which did not early take into account due to large uncertainties for R K and K J leading to degradation of electromagnetic measurements accuracy. System C, that also uses the new definition of the mass unit with the help of m u, is considered in detail in the works [17, 28, 35, 36]. This system is intermediate between system B [4] and system E [7], and is optimal with respect to the criteria presented in Sec. 3. Note that it is very preferential to fix m u, mainly due to the coincidence of the physical dimensions of DC and the mass unit. While choosing h as DC for the new definition of the mass unit we obtain a more complicated relation between these dimensions and it needs to use quantum physics for explanation of using of h. There are no difficulties with the choice between h or m u as DC for the dissemination of the macroscopic mass unit. This is because the ratio h/m u is now known with u r, which is negligible compared to u r achievable by either the watt balance or silicon sphere methods [12]. System C was proposed, in particular, to keep the exact value of the molar mass constant M u, but in the frameworks of systems B and D M u turns into an experimental quantity, that leads to some difficulties in educational process. Choosing m u as DC we detour these difficulties and come to the fixed values of M u and N A and the relations: M u = 1 g mol -1, N A = M u /m u. The analysis in detail of all problems concerning the redefinition of the kilogram based on the fixed e value was published in the works [30, 37, 17]. The choice between h or m u as DC has no significance of principle, however the situation was different in 2002, when the redefinition of the four SI units was proposed. Two arguments in favour of h were offered [6], but they do not work now [7]. Firstly, the diversity and accuracy of the silicon sphere method has significantly improved and the availability of such spheres wil increase in the near future [9, 38]. Secondly, the improved uncertainty of h/m u [39, 40] means that both methods can realize the kilogram equally well regardless of the choise made for the definition of the kilogram. Moreover, with the silicon sphere method the kilogram is realized as the inertial mass unit while with the watt balance method as the gravitational mass unit which depends on the local value of the acceleration varying with place and time of 8

9 measurement [15]. So the procedure of comparison of different spheres is well known whereas it is more complicated for different watt balances. 5. New definitions of the kilogram, ampere, kelvin and mole in system D In system D the new definitions of the mole and the kelvin are based on fixed values of N A and k as well as in systems B, C and E. By reason of the non-fixed value of the electron charge e in system D we have experimental uncertainties for R K and K J, however they are very small ( ) and negligible. Notice, it is only needed for precise electromagnetic measurements that the R K and K J values be stable and known with uncertainties smaller than some limits. The fine structure constant plays a central role in determining the uncertainties of R K and K J. Between CODATA 2006 and CODATA 2010, the reported value for changed by 6.5 times the 2006 uncertainty (a relative shift of ).Fortunately, the reliability of experimental determinations of has now improved. The reason for the 2006 error consists in the theoretical value of the electron magnetic moment anomaly, which was reliant on a single calculation [7]. More recently the value of has been confirmed by the completely independent, non QED, route of atomic recoil experiment (e.g. measurements of h/m( 87 Rb) [40]). Now we confident in the CODATA 2014 uncertainty for the measured value of. Improvements in determinations will continue. It will give us the high accuracy of determination of the R K and K J constants. Moreover, let us draw attention to the advantage of keeping the fixed 0 value in system D, 0 = N/A 2. In so far as the dielectric and magnetic permittivity of free space 0 and 0 obey the relation: 0 0 c 2 = 1, then 0 must be fixed too and be equal to the SI value: 0 = F m -1. So, taking into account = e 2 /(2 0 h c) we have u r (e 2 ) = u r ( ). Afterwards for estimations of different constants uncertainties in system D one can use that the results of spectroscopic measurements depends on two units only (the metre and the second), which do not change with changing of considered DC. So, for instance, it is sufficient to represent m u and e through the fixed constants h and 0 and the measured constants:, A r (e) and R. 6. Conclusions We have analyzed the suggested versions of new definitions of the kilogram, ampere, kelvin and mole based on fixing the exact values of certain DC, and formulated the 9

10 notion of the constant s order in the given system of units. Following the results of the works [7, 15] we have considered the five versions of the New SI (systems A, B, C, D and E, system A corresponds to the present SI) taking into account progress in experimental determinations of DC values. New experimental results for the Planck and Avogadro constants [8, 9, 10], as well as the CODATA 2014 adjustment of PC, certainly, will have an influence on a choice of four possible formulations of the base SI units. Thus the arguments for and against different formulations should be changed since they were first proposed in In the present paper the new classification of DC based on the notion of the order of DC in the given system of units is introduced and some criteria are formulated for choosing the set of DC suitable for redefinition of the base SI units. These include (A) succession between the new and old definitions, (B) stability in transferring the value of a unit, (C) a minimum number of DC, and (D) use of DC with the minimal possible orders. The New SI version with fixed h, 0, k, N A (system D) and its advantages are considered in detail for the first time. We can say that fixing of h and 0 leads to some fixation of space-time properties associated with quantum phenomena, while fixing of m u and e leads to some fixation of properties of atomic particles. When system D is compared with system B, it is apparent that system B has advantage for electromagnetic measurements because the practical units 90 and V 90 (or, more precisely, 2018 and V 2018 ) will turn to the New SI units. However, all this cannot be realized until the quantum metrological triangle is closed, and it can happen in principle that it will not be closed quite exactly. Therefore there are serious reasons to attribute the electron charge constant e to the class of being measured electromagnetic quantum constants, together with the K J and R K constants. Then a new definition of the ampere will be provided by a new quantum standard on the basis of single-electron tunneling and a fixed value of h only. In this case, the values of K J and R K will be found by reconciling the results of various experiments with the same or better accuracies than the current ones. The practice of using fixed values of K J and R K can be preserved at a certain accuracy level, leaving open the opportunity of measuring e or more and more accurately for testing the existing theories and a search for new ones. It is believed that the Boltzmann constant k is a conversion factor between two temperature scales, namely, the thermodynamic scale and the energy scale. So it is convenient to fix the value of k for the new definition of the kelvin. Moreover, this can be done regardless of the fixation of other three DC. The current precision of the k determination 10

11 guaranties the succession between the new and old definitions as well as keeping a total of thermo-physical data for various materials within different temperature ranges [15]. The adopted about ten years ago aim, the determination of the most precise and concordant with each other values of the Planck and Avogadro constants, had been achieved by 2015, when these constants had been measured with u r = by the silicon sphere method [9] and the watt balance method [8, 10]. Thus, the results of the recent NMIJ, BIPM, PTB, INRIM, NIST and NRC experiments [8, 9, 10] allow to achieve the 10-8 level demanded by the 24 and 25 CGPM decisions [2] and, most likely, will bring to the adoption of new definitions of the kilogram and mole in There is so far enough time to consider the advantages and shortcomings of the suggested new definitions and their different versions taking into account the fulfillment of all necessary criteria and further progress in reducing the relative uncertainties of the relevant DC measurements results. References 1. Girard G 1994 The third periodic verification of nationals prototypes of the kilogram Metrologia Bureau International des Poids et Mesures SI Brochure: The International System of Units (SI) (Draft 9 th edition) Mills I M et al 2006 Redefinition of the kilogram, ampere, kelvin and mole: a proposed approach to implementing CIPM recommendation 1 (CI-2005) Metrologia Milton M J T, Davis R and Fletcher N 2014 Towards a new SI: a review of progress made since 2011 Metrologia 51 R21-R30 6. Taylor B N and Mohr P J 1999 On the redefinition of the kilogram Metrologia Fletcher N, Davis R S, Stock M and Milton M J T 2015 Modernizing the SI- implications of recent progress with the fundamental constants arxiv: Sanchez C A et al 2014 A determination of Planck's constant using the NRC watt balance Metrologia 51 S5-S14 9. Azuma Y et al 2015 Improved measurement results for the Avogadro constant using a 28 Si-enriched crystal Metrologia Sanchez C A et al 2015 Corrigendum to the 2014 NRC determination of Planck's constant Metrologia 52 L23 11

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14 Abbreviations CGPM Coference Generale des Poids et Mesures. BIPM Bureau International des Poids et Mesures. INRIM Istituto Nazionale di Ricerca Metrologica, Italy. NIST National Institute of Standards and Technology, USA. NMIJ National Metrology Institute of Japan. NPL National Physical Laboratory, United Kingdom. NRC National Research Council, Canada. PTB Physikalisch-Technische Bundesanstalt, Germany. 14