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- Sprache: Englisch
The International System of Units, the SI, provides the foundation for all measurements in science, engineering, economics, and society. The SI has been fundamentally revised in 2019. The new SI is a universal and highly stable unit system based on invariable constants of nature. Its implementation rests on quantum metrology and quantum standards, which base measurements on the manipulation and counting of single quantum objects, such as electrons, photons, ions, and flux quanta. This book explains and illustrates the new SI, its impact on measurements, and the quantum metrology and quantum technology behind it. The book is based on the book ?Quantum Metrology: Foundation of Units and Measurements? by the same authors. From the contents: -Measurement -The SI (Système International d?Unités) -Realization of the SI Second: Thermal Beam Cs Clock, Laser Cooling, and the Cs Fountain Clock -Flux Quanta, Josephson Effect, and the SI Volt -Quantum Hall Effect, the SI Ohm, and the SI Farad -Single-Charge Transfer Devices and the SI Ampere -The SI Kilogram, the Mole, and the Planck constant -The SI Kelvin and the Boltzmann Constant -Beyond the present SI: Optical Clocks and Quantum Radiometry -Outlook
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Veröffentlichungsjahr: 2019
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Table of Contents
Cover
Foreword
Preface
List of Abbreviations
1 Introduction
References
2 Some Basics
2.1 Measurement
2.2 The SI (Système International d'Unités)
References
3 Realization of the SI Second: Thermal Beam Cs Clock, Laser Cooling, and the Cs Fountain Clock
3.1 The Thermal Beam Cs Clock
3.2 Techniques for Laser Cooling and Trapping of Atoms
3.3 The Cs Fountain Clock
References
4 Flux Quanta, Josephson Effect, and the SI Volt
4.1 Josephson Effect and Quantum Voltage Standards
4.2 Flux Quanta and SQUIDs
4.3 Traceable Magnetic Flux Density Measurements
References
5 Quantum Hall Effect, the SI Ohm, and the SI Farad
5.1 Basic Physics of Three‐ and Two‐Dimensional Semiconductors
5.2 Two‐Dimensional Electron Systems in Real Semiconductors
5.3 The Hall Effect
5.4 Metrology Using the Quantum Hall Effect
5.5 Graphene for Resistance Metrology
References
6 Single‐Charge Transfer Devices and the SI Ampere
6.1 Basic Physics of Single‐Electron Transport
6.2 Quantized Current Sources
6.3 Realization of the SI Ampere
6.4 Consistency Tests: Quantum Metrology Triangle
References
7 The SI Kilogram, the Mole, and the Planck Constant
7.1 From “Monitoring the Stability of the Kilogram” to the Planck Constant
7.2 The Avogadro Experiment
7.3 The Kibble Balance Experiment
7.4 The Mole: Unit of Amount of Substance
7.5 The CODATA Evaluation of the Value of the Defining Planck Constant and the Maintenance and Dissemination of the Kilogram
References
8 The SI Kelvin and the Boltzmann Constant
8.1 Primary Thermometers
8.2 The CODATA Evaluation of the Value of the Defining Boltzmann Constant, Realization and Dissemination of the New Kelvin
References
9 Beyond the Present SI: Optical Clocks and Quantum Radiometry
9.1 Optical Clocks and a New Second
9.2 Single‐Photon Metrology and Quantum Radiometry
References
10 Outlook
References
Index
End User License Agreement
List of Tables
Chapter 2
Table 2.1 List of all units with special name and symbol.
List of Illustrations
Chapter 2
Figure 2.1 Components and quantities considered (left) and the minimum current,...
Figure 2.2 Noise power spectral density,
P
(
T
,
f
)/Δ
f
, versus frequency for differ...
Figure 2.3 Noise power spectral density as measured for a SQUID magnetometer ve...
Figure 2.4 Illustration of the base units of the SI and their defining constant...
Figure 2.5 The Pt/Ir kilogram prototype as stored at the Bureau International d...
Chapter 3
Figure 3.1 Schematic representation of a “thermal” Cs atomic clock. In the lowe...
Figure 3.2 Measured Ramsey fringe pattern for the PTB's CS1 thermal beam clock....
Figure 3.3 Energy‐level scheme of the lower states in
133
Cs.
Figure 3.4 Laser beam arrangement for an optical molasses.
Figure 3.5 Energy levels and laser beam arrangement in a (1D) MOT.
Figure 3.6 Illustration of Sisyphus cooling. (a) The polarization along the
z
d...
Figure 3.7 Schematic setup (simplified) of an atomic fountain clock.
Figure 3.8 Principle of operation of the atomic fountain clock illustrating the...
Chapter 4
Figure 4.1 Schematic drawing of a Josephson junction. Two superconductors are s...
Figure 4.2 Voltage–current characteristic of an ideal Josephson junction illust...
Figure 4.3 Voltage–current characteristic of an ideal Josephson junction for th...
Figure 4.4 RCSJ model to describe a real Josephson junction accounting for disp...
Figure 4.5 Schematic voltage–current characteristic of real Josephson junctions...
Figure 4.6 Schematic layout of an SIS Nb/Al–Al
2
O
3
/Nb Josephson array. Shown are...
Figure 4.7 Photograph of a 10 V Josephson array mounted onto a chip carrier. Th...
Figure 4.8 Schematic layout of a binary divided programmable Josephson array. E...
Figure 4.9 Stepwise approximated 50 Hz sine wave generated by a programmable bi...
Figure 4.10 Scheme of the generation of a quantized waveform with arbitrary sha...
Figure 4.11 Scanning electron microscope image showing part of a Josephson arra...
Figure 4.12 Sine wave with a frequency of 250 Hz, synthesized combining the out...
Figure 4.13 Josephson junction in a magnetic field that points along the
z
‐dire...
Figure 4.14 Critical current of a Josephson junction in a magnetic field (norma...
Figure 4.15 Schematic drawing of a DC SQUID with two Josephson junctions 1 and ...
Figure 4.16 Critical current of a DC SQUID (normalized to the zero‐field critic...
Figure 4.17 Schematic graph of time‐averaged voltage versus bias current of a r...
Figure 4.18 Simplified circuit diagram of a DC SQUID operated in a flux‐locked ...
Figure 4.19 Superconducting flux transformer with primary inductance
L
p
and sec...
Figure 4.20 First‐order SQUID gradiometer.
Figure 4.21 Cross section of a superconducting tube (gray). A current
I
is pass...
Figure 4.22 Schematic circuit diagram of a resistance bridge based on a cryogen...
Figure 4.23 Eighty‐three‐channel SQUID system aligned above a patient for MCG m...
Figure 4.24 Free‐precession (FP) signal of protons in a water sample at a magne...
Chapter 5
Figure 5.1 Schematic illustration of the density of states of three‐, two‐, one...
Figure 5.2 TEM image of a GaAs/AlAs/GaAs heterostructure. The individual dots r...
Figure 5.3 Schematic representation of an MBE chamber for the epitaxial growth ...
Figure 5.4 Spatial variation of the band gap energy
E
g
of an Al
x
Ga
1−
x
As/G...
Figure 5.5 Modulation‐doped Al
0.3
Ga
0.7
As/GaAs heterostructure. (a) Layer sequen...
Figure 5.6 Experimental arrangement for the observation of the Hall effect in a...
Figure 5.7 Experimentally determined Hall resistance
R
xy
(left scale) and longi...
Figure 5.8 Schematic representation of the density of states of inhomogeneously...
Figure 5.9 Realization of the SI ohm starting with a calculable capacitor witho...
Figure 5.10 Schematic representation of a typical Hall bar with two current con...
Figure 5.11 Photograph of a GaAs/AlGaAs quantum Hall resistance standard showin...
Figure 5.12 Realization of the SI farad starting from a DC quantum Hall resista...
Figure 5.13 Schematic top view of a doubly shielded AC quantum Hall resistance ...
Figure 5.14 Energy splitting between the two lowest Landau levels normalized to...
Figure 5.15 Atomic force microscope (left) and Raman (right) images of monolaye...
Figure 5.16 Experimentally determined Hall resistance
R
xy
(black, left scale) a...
Chapter 6
Figure 6.1 Comparison of a closed metallic circuit without charge quantization ...
Figure 6.2 Equivalent circuit of an SET transistor. The charge island is shown ...
Figure 6.3 Chemical potential across a metallic SET transistor for a fixed gate...
Figure 6.4 Schematic representation of the source–drain current
I
SD
(top) and t...
Figure 6.5 Stability diagram of an SET transistor: number of excess electrons
n
Figure 6.6 SET pump consisting of two charge islands and three tunnel junctions...
Figure 6.7 Schematic representation of the stability diagram of an SET pump tha...
Figure 6.8 SEM image of an Al/AlO
x
/Al/AlO
x
/Al SET transistor. Each structure is...
Figure 6.9 Source–drain current versus source–drain voltage of an Al/AlO
x
/Al/Al...
Figure 6.10 Schematic layout of a GaAs/AlGaAs SET pump. Typical parameters: wid...
Figure 6.11 Schematic representation of the pumping cycle of a GaAs/AlGaAs SET ...
Figure 6.12 Current in units of
ef
as a function of the exit gate voltage. Cloc...
Figure 6.13 Operating principle of an SINIS turnstile. Shown is the density of ...
Figure 6.14 SEM image of a self‐referenced quantized current source. Three GaAs...
Figure 6.15 The quantum metrology triangle in the current version applying Ohm'...
Chapter 7
Figure 7.1 Mass difference of different National Kilogram Prototypes (black) an...
Figure 7.2 Photo of a single crystal Si sphere used in the Avogadro experiment....
Figure 7.3 Schematic drawing of the spherical Fizeau interferometer constructed...
Figure 7.4 Diameter variations of a single crystal Si sphere.
Figure 7.5 Schematic layout of the combined optical and X‐ray interferometer fo...
Figure 7.7 Photo of the central part of the INRIM X‐ray interferometer showing ...
Figure 7.6 Operation principle of a Bonse–Hart X‐ray interferometer. The three ...
Figure 7.8 Principle of the Watt balance experiment.
Figure 7.9 (a) Schematic drawing of the NIST‐4 Kibble balance [72]. The magneti...
Figure 7.10 Values of the Planck constant inferred from
Kibble balance
(
KB
) and...
Figure 7.11 Previous and present traceability chain for the dissemination of th...
Chapter 8
Figure 8.1 Schematic sketch of the DCGT setup used at PTB [13].
Figure 8.2 Photo of the assembled
National Physics Laboratory
(
NPL
) 1 l copper ...
Figure 8.3 Schematic of the laser spectrometer setup for DBT.
Figure 8.4 Schematic block diagram for the switched‐input noise correlator: (a)...
Figure 8.5 Schematic illustration of the differential conductance (b) of an SET...
Figure 8.6 Values of the Boltzmann constant contributing to the final value of ...
Chapter 9
Figure 9.1 Principle of operation of an optical clock.
Figure 9.2 Scheme of a Bordé–Ramsey atom interferometer. The laser beams are sh...
Figure 9.3 Time trace of a mode‐locked laser pulse train (a) and the correspond...
Figure 9.4 Setup of a mode‐locked Ti‐sapphire laser with a linear resonator.
Figure 9.5 Mode‐locked Er:fiber laser oscillator (a) and amplifier (b) for opti...
Figure 9.6 Two‐dimensional sectional view of the electrode configuration and el...
Figure 9.7 Configuration of linear Paul traps with additional ring electrodes (...
Figure 9.8 Yb
+
ions (seen by their fluorescence) in a linear Paul trap. The...
Figure 9.9 Potential landscape of a two‐dimensional optical lattice.
Figure 9.10 Simplified diagram of the relevant energy levels of
40
Ca.
Figure 9.11 Simplified term scheme of neutral
88
Sr, indicating the cooling tran...
Figure 9.12 Partial energy scheme of
171
Yb
+
indicating the cooling transiti...
Figure 9.13 Excitation spectrum of the
2
S
1/2
(
F
= 0) →
2
F
7/2
(
F
= 3) transition...
Figure 9.14 Time variation of measured transition frequencies in different atom...
Figure 9.15 Set‐up of the Hanbury Brown and Twiss interferometer.
Figure 9.16 Second‐order correlation function
g
2
(
τ
)
versus delay time
τ
...
Figure 9.17 Energy‐level structure (not to scale) of the (NV)
−
center.
Figure 9.18 (a) Confocal microscopy raster scan of a part of a diamond sample w...
Figure 9.19 Photoluminescence spectra (
T
= 2.3K) of a single InGaAs/GaAs quantu...
Figure 9.20 cw‐Photoluminescence of an InAs/GaAs quantum dot and second‐order c...
Figure 9.21 Operation principle of a TES depicting the resistance,
R
, versus te...
Figure 9.22 ETF‐TES bias circuit with a dc SQUID read‐out.
Figure 9.23 Schematic of two setups used to calibrate single‐photon avalanche d...
Guide
Cover
Table of Contents
Begin Reading
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E1
The New International Systemof Units (SI)
Quantum Metrology and Quantum Standards
Ernst O. Göbel and Uwe Siegner
Copyright
Authors
Prof. Dr. Ernst O. Göbel
Physikalisch‐Technische Bundesanstalt
Emeritus
Oscar‐Fehr‐Weg 16
38116 Braunschweig
Germany
Prof. Dr. Uwe Siegner
Physikalisch‐Technische Bundesanstalt
Bundesallee 100
38116 Braunschweig
Germany
All books published by Wiley‐VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.
Library of Congress Card No.:
applied for
British Library Cataloguing‐in‐Publication Data
A catalogue record for this book is available from the British Library.
Bibliographic information published by
the Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at <http://dnb.d‐nb.de>.
© 2019 Wiley‐VCH Verlag GmbH & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany
All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.
Print ISBN: 978‐3‐527‐34459‐8
ePDF ISBN: 978‐3‐527‐81451‐0
ePub ISBN: 978‐3‐527‐81449‐7
oBook ISBN: 978‐3‐527‐81448‐0
Cover Design SCHULZ Grafik‐Design, Fußgönheim, Germany
Foreword
The International System of Units (Système International d'Unités, SI) provides the basis for internationally harmonized measurements that are indispensable for scientific, economic, and social progress. The SI was established in the Metre Convention, which was signed in 1875 and presently has 60 Signatory States as well as 42 Associate States and Economies, who together represent more than 97% of the world economy. It is thus the cornerstone of global trade and quality infrastructure. Since 1875, the SI has been continuously advanced by the organs of the Metre Convention: the General Conference on Weight and Measures (Conférence Générale des Poids et Mesures, CGPM) and the International Committee for Weights and Measures (Comité International des Poids et Mesures, CIPM), including its Consultative Committee for Units (CCU) and the International Bureau of Weights and Measures (Bureau International des Poids et Mesures, BIPM), a scientific institute in Sèvres near Paris.
In 2018, the evolution of the SI took a quantum leap forward: in a landmark decision in November 2018, the 26th CGPM voted to fundamentally revise the SI by abandoning all physical artifacts, material properties, and measurement descriptions used to date to define the kilogram/mole, the kelvin, and the ampere, respectively. On 20 May 2019, the revised SI, which is defined by fixing the numerical values of seven “defining constants,” will come into force. Among these are fundamental constants such as the Planck constant, the speed of light in vacuum, and the elementary charge, which together form the fine‐structure constant α. The units will thus be independent of space and time with a relative accuracy below 10−17 per year, according to the state‐of‐the‐art experiments on the constancy of α. The revised SI guarantees long‐term stability and realization of the units anywhere in the known universe with ever‐increasing accuracy as technology develops, thus opening the door to innovation in science, industry, and technology.
This book provides a complete review of the revised SI. The definition of units based on the defining constants is examined alongside the realization of the units, which often incorporates the most recent progress in quantum technologies. The book explains and illustrates the physics and technology behind the definitions and their impact on measurements, emphasizing the decisive role quantum metrology has played in the revision. It also reviews what progress based on quantum metrology is anticipated. The book is thus indispensable and highly topical – indeed, it is urgently needed in order to communicate the background and consequences of the revised SI to the broad scientific community and to other interested readers, including lecturers and teachers.
The authors are well qualified for this undertaking. Both have extensive experience and an excellent track record in metrology: Ernst Göbel was president of PTB, the national metrology institute of Germany, for more than 16 years. He was also a member of the CIPM for more than 15 years and served as its president from 2004 to 2010. Uwe Siegner joined the PTB in 1999, working on metrological applications of femtosecond laser technology and on electrical quantum metrology. He has been the head of the electricity division of PTB since 2009. Both authors are experienced university lecturers; in fact, this book is based on lectures they have given at the Technische Universität Braunschweig.
I have studied the book with great interest and pleasure, and I wish the same to a broad readership.
Braunschweig
November 2018
Prof. Dr. Joachim Ullrich
President of PTB, Vice President of CIPM,
President of the Consultative Committee for Units (CCU)
Preface
The General Conference on Weights and Measures (Conférence Générale des Poids et Mesures, CGPM) is the governing body of the Metre Convention. The CGPM rules the International System of Units (Système International d'Unités), the SI, which provides the basis for all measurements worldwide. At its 26th meeting in November 2018, the CGPM decided that all SI units would be based on seven “defining constants,” among them fundamental constants of nature, such as the Planck constant, the speed of light in vacuum, and the elementary charge. To a significant extent, quantum metrology has provided the scientific foundation for this revolutionary change of the system of measurement units. The essence of quantum metrology is to base measurements on counting of discrete quanta.
The concept of some indivisible discrete single particles that are the basic building blocks of all matter goes back to philosophers many centuries BCE. In particular, the Greek philosopher Demokrit and his students specified the idea of atoms (from the Greek àtomos) as the base elements of all matter.
These concepts found support in natural science beginning in the eighteenth century. This was particularly driven by chemistry (e.g. A. Lavoisier, J. Dalton, and D. Mendeleev), kinetic gas theory (e.g. J. Loschmidt and A. Avogadro), and statistical physics (e.g. J. Stefan, L. Boltzmann, and A. Einstein).
The discovery of the electron by J.J. Thomson (1897) and the results of the scattering experiments by J. Rutherford and his coworkers (1909) opened a new era in physics, based on their conclusions that atoms are not indivisible but instead composite species. In the atomic model developed by N. Bohr in 1913, the atom consists of electrons carrying a negative elementary charge (−e) and a tiny nucleus which carries almost all the mass of an atom composed of positively charged (+e) protons and electrically neutral neutrons. In Bohr's model, the electrons in an atom can only occupy discrete energy levels, consistent with the experimental findings of atomic spectroscopy.
In the standard model of modern particle physics, electrons are in fact elementary particles belonging to the group of leptons. Protons and neutrons are composite particles composed of fractionally charged elementary particles, named quarks, which are bound together by the strong force.
In the last 50 years or so, scientists have learned to handle single quantum objects, for example, atoms, ions, electrons, and Cooper pairs, not least due to the tremendous progress in laser physics and nanotechnology. This progress has also laid the base for “quantum metrology.” The paradigm of quantum metrology is to base measurements on the counting of discrete quanta (e.g. charge or magnetic flux quanta). In contrast, in classical metrology, the values of continuous variables are determined. Proceeding from classical to quantum metrology, the measurement of real numbers is replaced by counting of integers.
The progress in quantum metrology stimulated the discussion about a revision of the SI more than 10 years ago. In particular, it was recognized early on that quantum metrology would allow a new definition of the base units of the SI in terms of constants of nature. This concept was implemented by the decision of the CGPM in November 2018 to revise the SI and to base it on seven defining constants. This book describes this new SI, which will be used from 20 May 2019, its definitions and the underlying physics and technology.
The discrete nature of a physical system is sometimes obvious, for example, by counting cycles when microwave or optical transitions between discrete energy states in atoms or ions are considered. The discrete quantum character of solid‐state systems is less obvious because their single‐particle energy spectra are quasi‐continuous energy bands. Discrete quantum entities can then result from collective effects called macroscopic quantum effects.
The paradigm of quantum metrology becomes particularly obvious when the new definition of the electrical units (ampere, volt, and ohm) is considered. We, therefore, give a more comprehensive description of the underlying solid‐state physics and the relevant macroscopic quantum effects. For example, we partly summarize the textbook knowledge and deduce results starting from general principles in Chapter 4 where we introduce superconductivity, the Josephson effect, and quantum interference phenomena in superconductors.
This book addresses advanced students, research workers, scientists, practitioners, and professionals in the field of modern metrology as well as a general readership interested in the foundations of the new SI definition. However, we consider this book as an overview that shall not cover all subjects in the same detail as it covers the electrical units. For further reading, we refer to the respective literature.
This book is based on the previous book by the same authors “Quantum Metrology: Foundation of Units and Measurements,” however, reorganized and revised by including the final wording of the new SI definitions and the final values of the defining constants as decided by the 26th CGPM. The differences between the previous and the present SI are highlighted. Further, the individual chapters are updated by including latest results and progress.
This book would not have been possible without the support of many colleagues and friends. We would like to especially mention Stephen Cundiff (JILA, now University of Michigan), Wolfgang Elsäßer (University of Darmstadt), Peter Michler (University Stuttgart), and Alfred Leitenstorfer (University Konstanz) as well as our PTB colleagues Franz Ahlers, Peter Becker, Ralf Behr, Bernd Fellmuth, Joachim Fischer, Christian Hahn, Frank Hohls, Oliver Kieler, Johannes Kohlmann, Stefan Kück, Andre Müller, Ekkehard Peik, Klaus Pierz, Hansjörg Scherer, Piet Schmidt, Sibylle Sievers, Lutz Trahms, Stephan Weyers, and Robert Wynands. We are also grateful for the technical support provided by Alberto Parra del Riego and Jens Simon. We further acknowledge the support of the Wiley‐VCH staff members.
Braunschweig
December 2018
Ernst O. Göbel and Uwe Siegner
List of Abbreviations
2DEG
two‐dimensional electron gas
AGT
acoustic gas thermometer/thermometry
AIST
National Institute of Advanced Industrial Science and Technology (National Metrology Institute of Japan)
APD
avalanche photo diode
BIPM
International Office for Weights and Measures (
Bureau International des Poids et Mesures
)
CBT
Coulomb blockade thermometer/thermometry
CCC
cryogenic current comparator
CCEM
Consultative Committee for Electricity and Magnetism
CCL
Consultative Committee for Length
CCM
Consultative Committee for Mass
CCT
Consultative Committee for Temperature
CCU
Consultative Committee for Units
CERN
European Organization for Nuclear Science
CGPM
General Conference on Weights and Measures (
Conférence Générale des Poids et Mesures
)
CIPM
International Committee for Weights and Measures (
Comité International des Poids et Mesures
)
CIPM MRA
CIPM mutual recognition arrangement
CODATA
International Council for Science: Committee on Data for Science and Technology
CVD
chemical vapor deposition
CVGT
constant volume gas thermometer/thermometry
DBT
Doppler broadening thermometer/thermometry
DCGT
dielectric constant gas thermometer/thermometry
ECG
electrocardiography
EEG
electroencephalography
EEP
Einstein's equivalence principle
FQHE
fractional quantum Hall effect
GUM
guide to the expression of uncertainties in measurements
HEMT
high electron mobility transistor
IAC
international Avogadro coordination
IDMS
isotope dilution mass spectroscopy
IERS
International Earth Rotation and Reference Systems Service
INRIM
National Institute of Metrology of Italy (
Istituto Nazionale di Ricerca Metrologia)
ISO
International Organization for Standards
ITS
international temperature scale
JNT
Johnson noise thermometer/thermometry
KRISS
Korea Research Institute of Standards and Science
LED
light emitting diode
LNE
French Metrology Institute (
Laboratoire National de Métrologie et d'Essais
)
MBE
molecular beam epitaxy
MCG
magnetocardiography
MEG
magnetoencephalography
METAS
Federal Institute of Metrology, Switzerland
MOCVD
metalorganic chemical vapor deposition
MODFET
modulation‐doped field‐effect transistor
MOS
metal‐oxide‐semiconductor
MOSFET
metal‐oxide‐semiconductor field‐effect transistor
MOT
magneto‐optical trap
MOVPE
metalorganic vapor‐phase epitaxy
MSL
Measurement Standards Laboratory of New Zealand
NBS
National Bureau of Standards
NEXAFS
near‐edge absorption fine structure
NIM
National Institute of Metrology (National Metrology Institute of China)
NININ
normal metal/insulator/normal metal/insulator/normal metal
NIST
National Institute of Standards and Technology (National Metrology Institute of the United States)
NMIJ
National Metrology Institute of Japan
NMR
nuclear magnetic resonance
NPL
National Physics Laboratory (National Metrology Institute of the United Kingdom)
NRC
National Research Council, Canada
NV
nitrogen vacancy
OM
optical molasses
PLTS
provisional low temperature scale
PMT
photomultiplier tube
PTB
Physikalisch‐Technische Bundesanstalt (National Metrology Institute of Germany)
PTR
Physikalisch‐Technische Reichsanstalt (former National Metrology Institute of Germany)
QED
quantum electrodynamics
QHE
quantum Hall effect
QMT
quantum metrology triangle
QVNS
quantized voltage noise source
RCSJ
resistively and capacitively shunted junction
RHEED
reflection high‐energy electron diffraction
RIGT
refractive index gas thermometer/thermometry
rms
root‐mean‐square
RT
radiation thermometry
SEM
scanning electron microscopy
SET
single‐electron transport
SI
International System of Units (
Système International d'Unités
)
SINIS
superconductor/insulator/normal metal/insulator/superconductor
SIS
superconductor/insulator/superconductor
SNS
superconductor/normal metal/superconductor
SNT
shot noise thermometer
SOI
silicon‐on‐insulator
SPAD
single‐photon avalanche diode
SQUID
superconducting quantum interference device
TAI
international atomic time (
temps atomique international
)
TES
transition‐edge sensor
TEM
transmission electron microscope
TPW
triple point of water
ULCA
ultrastable low‐noise current amplifier
UME
TÜBITAK Ulusal Metroloji Enstitüsü
UV
ultraviolet
UTC
coordinated universal time
XRCD
X‐ray crystal density
XRF
X‐ray fluorescence
XPS
X‐ray photoelectron spectroscopy
XXR
X‐ray reflectometry
YBCO
yttrium barium copper oxide
1Introduction
Metrology is the science of measurement including all theoretical and experimental aspects, in particular, the experimental and theoretical investigations of uncertainties in measurement results. According to Nobel Prize Winner J. Hall, “metrology truly is the mother of science” [1].
Metrology is almost as old as humankind. When people began to exchange goods, they had to agree on commonly accepted standards as a base for their trade. Indeed, many of the ancient cultures such as China, India, Egypt, Greece, and the Roman Empire had a highly developed measurement infrastructure. Examples are the Nippur cubit from the third millennium BCE found in the ruins of a temple in Mesopotamia and now exhibited in the archeology museum in Istanbul and the famous Egyptian royal cubit as the base length unit for the construction of pyramids. However, the culture of metrology faded during the Middle Ages when many different standards were in use. In Germany, for instance, at the end of the eighteenth century, 50 different standards for mass and more than 30 standards for length were used in different parts of the country. This, of course, had been a barrier to trade and led to abuse and fraud. It was then during the French Revolution that the French Académie des Sciences took the initiative to define standards independent of the measures taken from the limbs of royal representatives. Instead, their intent was to base the standards on stable quantities of nature available for everyone at all times. Consequently, in 1799, the standard for length was defined as the ten millionth part of the quadrant of the earth, and a platinum bar was fabricated to represent this standard (Mètre des Archives). Subsequently, the kilogram, the standard of mass, was defined as the mass of one cubic decimeter of pure water at the temperature of its highest density at 3.98 °C. This can be seen as the birth of the metric system, which, however, at that time was not generally accepted through Europe or even in France. It was only with the signature of the Metre Convention in 1875 by 17 signatory countries that the metric system based on the meter and the kilogram received wider acceptance [2]. At the time of this writing, the Metre Convention was signed by 60 states with another 42 states being associated with the General Conference on Weights and Measures (Conférence Générale des Poids et Mesures, CGPM) (as of November 2018). At the General Conferences, following the first one in 1889, the system of units was continuously extended. Finally, at the 11th CGPM in 1960, the previous SI (Système International d'Unités) (see Section 2.2) with the kilogram, second, meter, ampere, kelvin, and candela as base units was defined. The mole, unit of amount of substance, was added at the 14th CGPM in 1971. Within the SI, the definition of some units has been adopted according to progress in science and technology; for example, the meter was defined in 1960 based on the wavelength of a specific emission line of the noble gas krypton. But then, in 1983, it was replaced by the distance light travels in a given time and by assigning a fixed value to the speed of light in vacuum. Similarly, the second, originally defined as the ephemeris second, was changed by the 13th CGPM and defined via an electronic transition in the Cs isotope 133. Thus, in the previous SI, the meter and the second were defined by constants of nature. In the present revised SI, as accepted by the 26th CGPM in 2018, all units are based on constants of nature [3–7]. In fact, in this context, single quanta physics has a decisive role as will be outlined in this book.
We shall begin with introducing some basic principles of metrology in Chapter 2. We start in Section 2.1 by repeating some basic facts related to measurement and discuss the limitations for measurement uncertainty. The present SI is then presented in Section 2.2 . The previous definitions of the respective units are also given for comparison.
Chapter 3 treats the realization of the present definition of the second employing atomic clocks based on the hyperfine transition in the ground state of 133Cs applying thermal beams and laser‐cooled atoms, respectively.
Chapter 4 is devoted to superconductivity and its utilization in metrology. Because of its prominent role for electrical metrology, we introduce superconductivity, the Josephson effect, magnetic flux quantization, and quantum interference. By means of the Josephson effect, the volt (the unit for the electrical potential difference) is traced back to the Planck constant and the elementary charge as realized in today's most precise voltage standards. We further discuss magnetic flux quantization and quantum interference allowing the realization of quantum magnetometers (superconducting quantum interference devices) with unprecedented resolution and precision.
The underlying solid‐state physics and the metrological application of the quantum Hall effect are discussed in Chapter 5. In the present SI, the unit of electric resistance, ohm, is traced back to the Planck constant and the elementary charge by the quantum Hall effect.
In Chapter 6, we describe the physics of single‐electron transport devices, which allow the realization of the unit of electric current, the ampere, according to its present definition based on the elementary charge and frequency. We further discuss the so‐called metrological triangle experiment aimed to prove the consistency of the present realizations of the volt, ampere, and ohm.
Chapter 7 is then devoted to the present definition of the kilogram and the mole based on, respectively, the Planck constant and the Avogadro constant. We present the Kibble balance and the silicon single‐crystal experiment, which have been seminal for the precise determination of the Planck constant and are now primary realizations of the kilogram replacing the International Kilogram Prototype (IKP).
Various experiments that have contributed to the precise determination of the value of the Boltzmann constant and that are potential realizations of the unit of thermodynamic temperature, kelvin, are described inChapter 8.
In Chapter 9, we take an even further look into the future of the SI when we discuss optical clocks, which may in due time cause a change of the defining constant for the unit of time, the second, resulting in an improved realization. Further, we discuss the prospect of single‐photon emitters for a possible new definition of radiometric and photometric quantities, for example, for (spectral) irradiance and luminous intensity.
In an outlook in Chapter 10, we finally discuss a few examples how the present definitions of the SI pave the way to bring quantum metrology and quantum technology to the “workbench,” thereby considerably improving the quality of measurements for industry, science, and society.
References
1 Hall, J. (2011). Learning from the time and length redefinition, and the metre demotion.
Philos. Trans. R. Soc. London, Ser. A
369: 4090–4108.
2 For a review on the development of modern metrology see e.g.: Quinn, T. and Kovalevsky, J. (2005). The development of modern metrology and its role today.
Philos. Trans. R. Soc. London, Ser. A
363: 2307–2327.
3 Discussion Meeting issue “The new SI based on fundamental constants”, organized by Quinn, T. (2011).
Philos. Trans. R. Soc. London, Ser. A
369: 3903–4142.
4 Mills, I.M., Mohr, P.J., Quinn, T.J. et al. (2006). Redefinition of the kilogram, ampere, kelvin and mole: a proposed approach to implementing CIPM recommendation 1 (CI‐2005).
Metrologia
43: 227–246.
5 BIPM (2019). Draft of the ninth SI‐Brochure.
https://www.bipm.org/utils/en/pdf/si‐revised‐brochure/Draft‐SI‐Brochure‐2018.pdf
(accessed November 2018).
6 Fischer, J. and Ullrich, J. (2016). The new system of units.
Nat. Phys.
12: 4–7.
7 Stenger, J. and Ullrich, J.H. (2016). Units based on constants: the redefinition of the international system of units.
Annu. Rev. Condens. Matter Phys.
7: 35–59.
2Some Basics
2.1 Measurement
Measurement is a physical process to determine the value or magnitude of a quantity. The quantity value can be calculated as follows:
where {q} is the numerical value and [Q] the unit (see Section 2.2). The unit is thus simply a particular example of a quantity value. Equation (2.1) also applies for Q being a constant. If the numerical value of a constant is fixed, it defines the unit because their product must be equal to the quantity value, Q. This is the underlying concept of the present SI.
Repeated measurements of the same quantity, however, will generally result in slightly different results. In addition, systematic effects that impact the measurement result must be considered. Thus, any measurement result must be completed by an uncertainty statement. This measurement uncertainty quantifies the dispersion of quantity values being attributed to a measurand, based on the information used. Measurement uncertainty comprises, in general, many components. Some of the components may be evaluated by type A evaluation of measurement uncertainty from the statistical distribution of quantity values from a series of measurements and can be characterized by standard deviations. The other components, which may be evaluated by type B evaluation of measurement uncertainty, can also be characterized by standard deviations, evaluated from probability density functions based on experience or other information. For the evaluation of uncertainties in measurements, an international agreed guide has been published jointly by ISO and the Bureau International des Poids et Mesures (BIPM), the Guide to the Expression of Uncertainty in Measurement (GUM) [1–3]. Generally, precision measurements are those with smallest measurement uncertainty.
2.1.1 Limitations of Measurement Uncertainty
One might tend to believe that measurement uncertainty can be continuously decreased as more efforts are put in the respective experiment. However, this is not the case since there are fundamental as well as practical limitations for measurement precision. The fundamental limit is a consequence of the Heisenberg uncertainty principle of quantum mechanics, and the major practical limit is due to noise.
2.1.1.1 The Fundamental Quantum Limit
Note that throughout this book, we use the letter f to denote technical frequencies and the Greek letter ν to denote optical frequencies.
The Heisenberg uncertainty principle is a fundamental consequence of quantum mechanics stating that there is a minimum value for the physical quantity action, H:
where h is the Planck constant. Action has the dimensions of energy multiplied by time and its unit is joule seconds. From the Heisenberg uncertainty principle, it follows that conjugated variables, such as position and momentum or time and energy, cannot be measured with ultimate precision at a time. For example, if Δx and Δp are the standard deviation for position, x, and momentum, p, respectively, the inequality relation holds (ℏ = h/2π):
Applied to measurement, the argument is as follows: during a measurement, information is exchanged between the measurement system and the system under consideration. Related to this is an energy exchange. For a given measurement time, τ, or bandwidth of the measurement system, Δf = 1/τ, the energy extracted from the system is limited according to Eq. (2.2) [4]:
Let us now consider, for example, the relation between inductance, L, and, respectively, magnetic flux, Φ, and current, I (see Figure 2.1). The energy is given by E = (1/2)LI2 = (1/2)(Φ2/L), and consequently,
Figure 2.1 Components and quantities considered (left) and the minimum current, Imin, and the minimum magnetic flux, Φmin, versus inductance, L, for an ideal coil.
Source: Kose and Melchert 1991 [4] . Reproduced with permission of John Wiley and Sons.
These relations are also depicted in Figure 2.1 . The gray area corresponds to the regime that is accessible by measurement. Note that this is a heuristic approach that does not consider a specific experiment. Nevertheless, it may provide useful conclusions on how to optimize an experiment. For instance, if an ideal coil (without losses) is applied to measure a small current, inductance should be large (e.g. L = 1 H, τ = 1 s, and then Imin = 3.5 × 10−17 A). If instead the coil is applied to measure magnetic flux, L should be small (e.g. L = 10−10 H, τ = 1 s, and then Φmin = 4 × 10−22 V s = 2 × 10−7 × Φ0, where Φ0 = h/2e is the flux quantum = 2.067 × 10−15 V s).
Similarly, for a capacitor with capacitance, C, the energy is given by
and thus,
Finally, for a resistor with resistance, R, the energy is given by
and thus, for the minimum current and voltage, respectively, we obtain
2.1.1.2 Noise
In this chapter, we briefly summarize some aspects of noise theory. For a more detailed treatment of this important and fundamental topic, the reader is referred to, for example, [5].
Noise limits the measurement precision in most practical cases. The noise power spectral density, P(T, f)/Δf, can be approximated by (Planck formula)
where f is the frequency, k the Boltzmann constant, and T the temperature. Two limiting cases can be considered as follows.
(i)
Thermal noise
(Johnson noise) (kT ≫ hf)
:
According to this “Nyquist relation,” the thermal noise power spectral density is independent of frequency (white noise) and increases linearly with temperature. Thermal noise was first studied by Johnson [6]. It reflects the thermal agitation of, for example, carriers (electrons) in a resistor.
(ii)
Quantum noise
(hf ≫ kT)
:
The quantum noise power spectral density in this limit is determined by the zero point energy, hf, and is independent of temperature and increases linearly with frequency.
Thermal noise dominates at high temperatures and low frequencies (see Figure 2.2). The transition frequency, fc(T), where both contributions are equal depends on temperature and is given by
This transition frequency amounts to 4.3 THz at T = 300 K and 60.6 GHz at the temperature of liquid He at T = 4.2 K.
The thermal noise in an electrical resistor at temperature T generates under open circuit or short circuit, respectively, a voltage or current with effective values:
To reduce thermal noise, the detector equipment should be cooled to low temperatures. Decreasing the temperature from room temperature (300 K) to liquid He temperature (4.2 K) reduces the thermal noise power by a factor of about 70. In addition, both thermal and quantum noise can be reduced by reducing the bandwidth, that is, by integrating over longer times, τ. This, however, requires stable conditions during the measurement time, τ. Unfortunately, however, other noises may be observed such as shot noise and at low frequencies the so‐called 1/f noise.
(iii)
Shot noise
: Shot noise originates from the discrete nature of the species carrying energy (e.g., electrons, photons). It was first discovered by Schottky
[7]
when studying the fluctuations of current in vacuum tubes. Shot noise is observed when the number of particles is small such that the statistical nature describing the occurrence of independent random events is described by the Poisson distribution. The Poisson distribution transforms into a normal (Gaussian) distribution as the number of particles increases. At low frequencies, shot noise is white; that is, the noise spectral density is independent of frequency and, in contrast to thermal noise, also independent of temperature. The shot noise spectral density of an electric current,
S
el
, at sufficiently low frequencies is given by
where I is the average current. Similarly, for a monochromatic photon flux, we have the shot noise spectral density of photon flux, Sopt,
where hν is the photon energy and P the average power.
(iv)
Low‐frequency noise
(1/f noise)
: 1/
f
noise (sometimes also called pink noise or flicker noise) occurs widely in nature but might have quite different origins. More precisely, the relation between noise power spectral density and frequency is often given by
with β usually close to 1. In contrast to thermal or quantum noise, the noise power of 1/f noise decreases with increasing frequency (by 3 dB per octave of frequency). Figure 2.3 shows, for example, the noise power spectral density measured for a superconducting quantum interference device (SQUID) magnetometer versus frequency.
Figure 2.2 Noise power spectral density, P(T,f)/Δf, versus frequency for different temperatures.
Source: Kose and Melchert 1991 [4] . Reproduced with permission of John Wiley and Sons.
Figure 2.3 Noise power spectral density as measured for a SQUID magnetometer versus frequency.
Source: Kose and Melchert 1991 [4] . Reproduced with permission of John Wiley and Sons.
2.2 The SI (Système International d'Unités)
According to the decision of the 26th meeting of the Conférence Générale des Poids et Mesures (CGPM), the present system of units is set up by seven defining constants, namely, the frequency of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom (ΔνCs), the Planck constant (h), the velocity of light in vacuum (c), the elementary charge (e), the Boltzmann constant (k), the Avogadro constant (NA), and the luminous efficacy of monochromatic radiation of frequency 540 × 1012 Hz (Kcd) [8].
The respective wording together with the numerical values of the defining constants based on the 2017 CODATA evaluation [9,10] is presented in the recent edition of the SI brochure [11] and it reads as follows:
The International System of Units, the SI, is the system of units in which
the unperturbed ground state hyperfine transition frequency of the caesium 133 atom
Δν
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
N
A
is 6.022 140 76 × 10
23
mol
−1
,
the luminous efficacy of monochromatic radiation of frequency 540 × 10
12
hertz
K
cd
is 683 lm/W.
These seven constants set the scale of the entire SI (Figure 2.4), and they are all needed to fully define it. These numerical values do not exhibit uncertainty and have been chosen to ensure continuity between the previous and present SI. These definitions of the present SI disconnect the definition of a unit from its realization, thus leaving room for improved realizations as science and technology advance. The nature of the constants is quite different, ranging from fundamental constants such as the Planck constant, h, and the speed of light in vacuum, c, to a technical constant such as the luminous efficacy, Kcd. The set of the seven defining constants has been chosen such that they provide a most fundamental, stable, and universal reference and simultaneously allow for practical realizations with smallest uncertainties [11] . All physical artifacts are abandoned in the present SI.
Figure 2.4 Illustration of the base units of the SI and their defining constants.
Source: Courtesy of PTB.
The previous SI consisted of 7 base units and 22 derived units with specific names. A formal distinction between the base units and derived units does not exist in the present SI, however, is maintained since it is historically established and proven to be useful and in particular to maintain consistency with international written standards such as the ISO/IEC 80000 series. The system is called coherent, which means that the derived units are given as a product of powers of the base units with only “1” as the numerical factor (e.g., for the derived unit for energy, joule, we have 1 J = 1 m2 kg s−2). Consequently, numerical equations do have the same format as quantity equations.
Now we briefly describe the present (and the previous) seven base units: second, meter, kilogram, kelvin, ampere, mole, and candela. In the present definition, one of the seven defining constants is explicitly assigned to each base unit as described in the following sections. For further reading, we refer to the SI brochure of the BIPM [11] .
2.2.1 The Second: Unit of Time
The unit second was originally defined as the 86 400th part of the duration of a mean solar day. However, at the 11th CGPM in 1960, after it had been shown that the rotation of the earth was not stable, the second was referred to the duration of the tropical year in 1900 (ephemeris second). In 1967 [12], however, the definition of the second was changed and no longer based on an astronomic timescale but refers to the frequency of electromagnetic radiation of a magnetic dipole transition in the hyperfine split |F = 3, mF = 0〉 ↔ |F = 4, mF = 0〉 ground state 62S1/2 of the isotope 133Cs (for the energy level scheme of 133Cs; see Figure 3.3). The definition of the second is unchanged in the present SI and is still defined via the hyperfine transition frequency in 133Cs, ΔνCs, which is now one of the defining constants of the SI. According to the SI‐Brochure [11] , the wording for the definition of the second is as follows:
The second, symbol s, is the SI unit of time. It is defined by taking the fixed numerical value of the caesium frequency ΔνCs, the unperturbed ground‐state hyperfine transition frequency of the caesium 133 atom, to be 9 192 631 770 when expressed in the unit Hz, which is equal to s−1.
This definition implies the exact relation ΔνCs = 9 192 631 770 Hz. Inverting this relation gives an expression for the unit second in terms of the value of the defining constant ΔνCs:
The second is equal to the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the unperturbed ground state of the 133Cs atom [11] .
The wording in the present SI is intended to make it clear that the definition refers to an isolated cesium atom unperturbed by any external field such as electric and magnetic fields and black body radiation. This wording thus defines the idealized value of the Cs hyperfine transition frequency and defines the “proper time” valid for any gravitational potential [13]. For the provision of a coordinated timescale, the signals of different primary clocks in different locations are combined, which must be corrected for relativistic frequency shifts.
In the previous SI, the definition of the second read as follows:
The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.
Despite the different wording, the definitions in the previous and the present SI are identical.
By this definition of the second, it had been ensured that at the time of the definition the new “atomic clock second” did agree with the ephemeris second. To keep the astronomic timescale and the atomic timescale (coordinated universal time (UTC)) identical, leap seconds are added (or subtracted) to the international atomic time (TAI) occasionally whenever their difference becomes larger than 0.9 s. Up to today, 27 leap seconds have been added to UTC since 1972 (see also [14]). The responsibility for adding or subtracting leap seconds lies with the International Earth Rotation and Reference Systems Service (IERS), and changing the procedure by taking much longer time intervals for coordinating these two timescales is under discussion.
The definition of the second is put into praxis, that is, realized, as metrologists use to say, by atomic clocks. The basic concept of atomic clocks is to lock the frequency of a local oscillator to the frequency of an electronic resonance of the respective atom, which in the classical Cs atomic clocks lies in the microwave regime. The two different versions of primary Cs atomic clocks currently in use operate with a thermally generated beam of Cs atoms and laser‐cooled Cs atoms, respectively, which are described in detail in Chapter 3.
2.2.2 The Meter: Unit of Length
The definition of the meter is also unchanged in the present SI and based on the defining constant speed of light in vacuum [11] .
The metre, symbol m, is the SI unit of length. It is defined by taking the fixed numerical value of the speed of light in vacuum c to be 299 792 458 when expressed in the unit m s−1, where the second is defined in terms of the caesium frequency ΔνCs.
One meter is defined as the length of the path traveled by light in vacuum during a time interval of 1/299 792 458 of a second [11] .
Thus
Since 1983, the following definition had been used for the meter in the previous SI:
The metre is the length of the path traveled by light in vacuum during a time interval of 1/299 792 458 of a second.
It follows that the speed of light in vacuum is exactly 299 792 458 m/s, c0 = 299 792 458 m/s.
Though customary in astronomy to measure distances in the path length light travels in a given time (e.g. light‐year), it is not very convenient for daily life purposes. Therefore, the Consultative Committee for Length (CCL) of the International Committee for Weights and Measures (CIPM) recommended three different procedures to realize the meter:
(i) According to its definition by measuring the distance light travels within a certain time interval.
(ii) Via radiation sources (in particular lasers) with known wavelength (or frequency). A list of respective radiation sources (Mise en Pratique) is published by the CCL and frequently updated [
15
–
17
].
(iii) Via the vacuum wavelength,
λ
, of a plane electromagnetic wave with frequency
f
. The wavelength is obtained according to
λ
=
c
/
f
.
According to procedures (ii) and (iii), interferometry can then be applied to calibrate the length of a gauge block [18]. Gauge blocks made of metals or ceramics exhibit two opposing precisely flat, parallel surfaces. For calibration of its length, the gauge block is wrung on an auxiliary platen forming one of the end mirrors of a modified Michelson interferometer (Twyman–Green interferometer, Kösters comparator). Since interference can be obtained from both end surfaces of the gauge block, its length can be measured in terms of the wavelength of the radiation used by counting the interference orders. Iodide‐stabilized He–Ne lasers are often applied for this purpose. For the highest precision, the interferometer is placed in vacuum to avoid uncertainties due to the refractive index of air. In addition, the temperature must be precisely known and stable. In any case, the frequency (and hence the wavelength) of the respective laser must be known in terms of the frequency of the Cs hyperfine transition frequency, ΔνCs, which defines the second. Today, these many orders of frequency are bridged by optical frequency combs. This technique for which T. Hänsch and J. Hall were awarded the 2005 Nobel Prize in physics can be considered as a gear, which transfers the microwave frequency of the Cs atomic clock into the visible and adjacent spectral regimes. The name “optical frequency comb” refers to the emission spectrum of mode‐locked lasers generating ultrafast (fs) laser pulses. Femtosecond frequency combs are discussed in more detail in Section 9.1.1.
At present, the meter can be realized according to recommendations (ii) and (iii) of the CCL with a relative uncertainty of the order of 10−11, and gauge calibrations can reach fractional uncertainties as low as 10−8 [ 18 ,19].
2.2.3 The Kilogram: Unit of Mass
The definition of the kilogram reads as follows [11] :
The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed numerical value of the Planck constant h to be 6.626 070 15 × 10−34 when expressed in the unit J s, which is equal to kg m2 s−1, where the metre and the second are defined in terms of c and ΔνCs.
This definition implies the exact relation h = 6.626 070 15 × 10−34 kg m2 s−1. Inverting this relation gives an exact expression for the kilogram in terms of the three defining constants h, ΔνCs, and c [11] :
