# Stochastic Approach to Determining the Mass Standard Based on the Fixed Values of Fundamental Physical Constants

## Abstract

**:**

_{x}. When using a fixed value of Planck’s constant h as a proportionality factor, the ratio $h{\tau}_{x}/2{D}_{x}$ has the dimension of a kilogram and can be used as an equivalent of a mass standard. It is proposed to use thermal (i.e., Johnson–Nyquist) noise as a reference Gaussian stationary random process. The theoretical substantiation of the project for the creation of “thermoelectric semiconductor ampere-balances” for balancing the inert mass of a quasi-ideal silicon-28 ball is also given. Combining these two projects can provide the basis for a stable and reproducible mass standard.

## 1. Introduction

^{−34}Js, the speed of light in a vacuum c = 299,792,458 m/s, the frequency of the hyperfine transition of the ground state of the cesium-133 atom v

_{Cs}= 9,192,631,770 Hz, the elementary charge e = 1.602,176,634 × 10

^{−19}C, the Boltzmann constant k

_{B}= 1.380649 × 10

^{−23}J/K and the Avogadro constant N

_{A}= 6.02214076 × 10

^{23}mol

^{−1}.

^{28}Si)” (hereinafter “Counting atoms”) [1,2]. This area of research is also called «The X-ray-crystal-density (XRCD) method».

^{28}Si and the Avogadro constant [1]

_{Si}is the atomic mass of

^{28}Si; v

_{Si}is the number of moles of

^{28}Si in 1 kg.

^{28}Si balls having a mass of 1 kg with a relative error of about 10

^{–8}, it is impossible to compare the readings of “Kibble balances” located in various laboratories around the world. On the other hand, without the “Kibble balance”, it is impossible to verify the equivalence of the inert mass of a quasi-ideal

^{28}Si ball to its gravitational mass [1].

^{28}Si ball must be balanced by Kibble balances with a maximum allowable uncertainty of no more than 10

^{−8}kg. In this case, quasi-ideal

^{28}Si balls, which should be included in the Kibble balance, can be used to compare the readings of similar watt-balances located in different places on the planet.

_{A}.

^{28}Si crystals are never perfect and monoisotopic [1]. Therefore, it is necessary to take into account corrections for the content of impurities, for defects in the

^{28}Si crystal lattice (vacancies and interstices), for the formation of an oxide film, and for adsorbed water molecules by the surface layer of the silicon ball. It is also necessary to take into account the mass defect related to the binding energy of the atoms of the

^{28}Si single crystal [1].

## 2. Relationship between Mass, Planck’s Constant and the Characteristics of a Stationary Random Process

- (a)
- The initial one-dimensional PDF ρ(x) of a stationary random process is represented as a product of two probability amplitudes (PAs):$$\rho \left(x\right)=\psi \left(x\right)\psi \left(x\right)=\psi {\left(x\right)}^{2};$$
- (b)
- two Fourier transforms are performed:$$\varphi \left({x}^{\prime}\right)=\frac{1}{\sqrt{2\pi {\eta}_{x}}}{{\displaystyle \int}}_{-\infty}^{\infty}\psi \left(x\right)exp\{i{x}^{\prime}x/{\eta}_{x}\}dx,$$$${\varphi}^{*}\left({x}^{\prime}\right)=\frac{1}{\sqrt{2\pi {\eta}_{x}}}{{\displaystyle \int}}_{-\infty}^{\infty}\psi \left(x\right)exp\{-i{x}^{\prime}x/{\eta}_{x}\}dx;$$
- (c)
- The desired PDF of the derivative of the studied stationary random process is found:$$\rho \left({x}^{\prime}\right)=\varphi \left({x}^{\prime}\right){\varphi}^{*}\left({x}^{\prime}\right)={\left|\varphi \left({x}^{\prime}\right)\right|}^{2},$$$${\eta}_{x}=\frac{2{\sigma}_{x}^{2}}{{\tau}_{x}}$$

- σ
_{x}is the standard deviation of the original stationary random process x(t); - τ
_{x}is autocorrelation interval of the same random process (see Figure 1).

_{i}of this process, the random variable x is distributed according to the Gaussian law:

_{x}

^{2}= D

_{x}and a

_{x}are the variance and expected value, resp., of the random process x(t) under study.

_{x}(i.e., the derivative of its coordinate x with respect to time);

^{2}/s].

## 3. Statistical Approach to the Determination of the Mass Standard

_{k}

_{1}(into section t

_{1}) and x

_{k}

_{2}(into section t

_{2}) (see Figure 1), at a distance between sections ${\tau}_{xcor}={t}_{2}-{t}_{1}$ (where ${\tau}_{xcor}$ is an estimate of the autocorrelation interval${\tau}_{x}$), at which Pearson’s autocorrelation coefficient $r\left({x}_{k1},{x}_{k2}\right)$ is less than some critical parameter ${\epsilon}_{cr}$ steadily tending to zero (${\epsilon}_{cr}\to 0$) (see Figure 2):

_{1};

_{2}.

- 1.
- Choose a stable stationary random process with PDF close to a Gaussian distribution, which can be carried out in any metrological laboratory;
- 2.
- Estimate its variance ${D}_{x}$ (with a confidence level corresponding to nσ
_{x}= n$\sqrt{{D}_{x}}$, where n is any natural number providing a given level of accuracy); - 3.
- Estimate the correlation interval ${\tau}_{xcor}$ at a fixed value of the critical parameter ${\epsilon}_{cr}$.

## 4. Reference Thermal Noise

_{B}and c, nothing prevents us from taking as a mass standard the value corresponding to the temperature, for example, the triple point of water T

_{i}= 273.16 K, or any other referent thermal scale points, for example, the crystallization point of aluminum T

_{i}= 933.473 K or the crystallization point of copper T

_{i}= 1357.77 K. Then expression (42) for j = 1 takes the form

## 5. Thermoelectric Semiconductor Ampere Balance

_{k}= Lρ is the resistance of the looped conductor (L is the length of the conductor, i.e., the circumference; ρ is the linear resistance of the conductor).

_{k}will be determined by an expression like (41), while the accuracy of determining the thermal current (48) will increase significantly:

_{1}and F

_{2}, taking into account expression (49), leads to the relation

^{28}Si ball with a mass of 1 $g$ can be used, with the help of which a unit of mass can be transferred to other measuring instruments, according to an appropriate verification scheme, and comparison of readings with thermoelectric semiconductor ampere-scales located in various laboratories of the world.

_{i}= 692.677 K:

^{2}and n = 1. Then

^{−10}m, then a stack of $n=7\times {10}^{9}$such layers will turn out to be a height of the order of $H\approx 0.14\times {10}^{-9}\times 7\times {10}^{9}\approx 1\mathrm{m}.$

_{i}, and a decrease in R

_{k}. For example, at R

_{k}= 100 Ω, B = 1T and T

_{i}= 692.677 K, we obtain the value

^{−8}m.

## 6. Conclusions

_{B}, c and N

_{A}on the basis of the resolutions of the 26th GCWM opens up wide possibilities for applying various physical principles to determine the standards of physical quantities.

_{i}= 273.16 K (or any other reference point on the temperature scale, for example, aluminum crystallization point T

_{i}= 933.473 K, or zinc crystallization point T

_{i}= 692.677 K):

_{i}= 692.677 K, B = 1T, l = 1 m, $g$ = 9.81 m/s

^{2}, n = 1 and j = 1, according to (53), $m\approx 1.3\times {10}^{-10}\mathrm{g}$. However, it all depends on what value to take as the mass standard. For example, if we take the Planck mass M

_{p}= 2.176434 × 10

^{−5}g as a standard, then by selecting the parameters T

_{i}, B, l, n and j, you can achieve m ≈ M

_{p}with a given accuracy at a quite acceptable temperature, magnetic field and thickness of the looped semiconductor layer of the stack (Figure 7).

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Mana, G.; Schlamminger, S. The kilogram: Inertial or gravitational mass? arXiv
**2022**, arXiv:2201.12136. [Google Scholar] [CrossRef] - Bronnikov, K.A.; Ivashchuk, V.D.; Kalinin, M.I.; Khrushchev, V.V. Evolution of the System of Measurement Units. Toward a Future Revision of the International System of Units (SI)/Legislative and Applied Metrology, No. 2. 2018. Available online: https://www.vniims.ru/activities/redakcionno-izdatelskaya-deyatelnost/zakon_and_prikl/archive/main_2_2018.php (accessed on 6 July 2022).
- Kibble, B.P.; Robinson, I.A.; Belliss, J.H. A Realization of the SI Watt by the NPL Moving-coil Balance. Metrologia
**1990**, 27, 173. [Google Scholar] [CrossRef] - Schlamminger, S.; Haddad, D. The Kibble balance and the kilogram. Comptes Rendus Physique
**2019**, 20, 55–63. [Google Scholar] [CrossRef] - Li, S.S.; Zhang, Z.H.; Zhao, W.; Li, Z.K.; Huang, S.L. Progress on accurate measurement of the Planck constant: Watt balance and counting atoms. Chin. Phys. B
**2014**, 24, 010601. [Google Scholar] [CrossRef] [Green Version] - Batanov-Gaukhman, M. Derivation of The Generalized Time-Independent Schrödinger Equation. The New Stochastic Quantum Mechanics: “Think and Calculate”. Av. Cienc. Ing.
**2020**, 11, Articulo 6. Available online: https://www.executivebs.org/publishing.cl/avances-en-ciencias-e-ingenieria-vol-11-nro-4-ano-2020-articulo-6/ (accessed on 2 July 2022). - Yakimov, A.V. Physics of Noise and Fluctuations of Parameters. Nizhny Novgorod State University. N.I. Lobachevsky-th, 85C. 2013. Available online: http://www.unn.ru/books/met_files/Yakimov_Noise.pdf (accessed on 22 June 2022).
- Verin, O.G. Quantum Hall Effect and Superconductivity. 2019. Available online: http://www.sciteclibrary.ru/yabb26.pdf (accessed on 13 June 2022).
- Gavrilenko, V.I.; Ikonnikov, A.V. Quantum Hall Effect. Nizhny Novgorod State University. N.I. Lobachevsky. 2010. Available online: http://www.unn.ru/books/met_files/QHE.pdf (accessed on 4 June 2022).

**Figure 1.**N realizations of a stationary random process with an autocorrelation interval τ

_{x ≈}τ

_{xcor}. These implementations can be interpreted, for example, as changes over time t of the projection onto the X axis of the location of a particle randomly wandering in a closed region of 3-dimensional space.

**Figure 2.**The autocorrelation interval. ${\tau}_{x}\approx {\tau}_{xcor}$ is the time interval between the values of the correlation function R(τ = 0) and R(${\tau}_{xcor})<{\epsilon}_{cr}$, where ${\epsilon}_{cr}$ is a small critical parameter steadily tending to zero (${\epsilon}_{cr}$ → 0).

**Figure 7.**The scheme of thermoelectric semiconductor ampere-balances for transferring a measure of mass to a reference or exemplary measuring instrument (in particular, a quasi-ideal silicon ball).

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Batanov-Gaukhman, M.
Stochastic Approach to Determining the Mass Standard Based on the Fixed Values of Fundamental Physical Constants. *Thermo* **2022**, *2*, 289-301.
https://doi.org/10.3390/thermo2030020

**AMA Style**

Batanov-Gaukhman M.
Stochastic Approach to Determining the Mass Standard Based on the Fixed Values of Fundamental Physical Constants. *Thermo*. 2022; 2(3):289-301.
https://doi.org/10.3390/thermo2030020

**Chicago/Turabian Style**

Batanov-Gaukhman, Mikhail.
2022. "Stochastic Approach to Determining the Mass Standard Based on the Fixed Values of Fundamental Physical Constants" *Thermo* 2, no. 3: 289-301.
https://doi.org/10.3390/thermo2030020