Determination of the Avogadro constant by counting the atoms in a 28Si crystal

Interferometric sphere surface metrology with cylindrical reference for precision topography

We demonstrate a method for measuring a surface map of a spherical body with interferometric optical point sensors while rotating the test subject. The setup takes advantage of the excellent performance of heterodyne interferometry at nanometer levels and suppression of common-mode errors, as a cylindrical mirror mounted adjacent to the sphere is used as a reference. Future space based missions for gravitational wave research demand an improved inertial reference sensor with reduced acceleration noise levels. Spherical test masses can enable increased performance by suspension-free operation, contrary to cuboid solutions suffering from cross-coupling of attitude control noise into test mass position. However, interferometric readout is affected by surface irregularities and test mass attitude. An accurate surface map for compensation of the center of gravity readout should be established, by characterization either a priori or in-flight, when optical path length changes due to the surface occur in the measurement bandwidth.

Modeling Atom Interferometry Experiments with Bose–Einstein Condensates in Power-Law Potentials

Recent atom interferometry (AI) experiments involving Bose–Einstein condensates (BECs) have been conducted under extreme conditions of volume and interrogation time. Numerical solution of the rotating-frame Gross–Pitaevskii equation (RFGPE), which is the standard mean-field theory applied to these experiments, is impractical due to the excessive computation time and memory required. We present a variational model that provides approximate solutions of the RFGPE for a power-law potential on a practical time scale. This model is well-suited to the design and analysis of AI experiments involving BECs that are split and later recombined to form an interference pattern. We derive the equations of motion of the variational parameters for this model and illustrate how the model can be applied to the sequence of steps in a recent AI experiment where BECs were used to implement a dual-Sagnac atom interferometer rotation sensor. We use this model to investigate the impact of finite-size and interaction effects on the single-Sagnac-interferometer phase shift.

Redefinition of the Mole in the Revised International System of Units and the Ongoing Importance of Metrology for Accurate Chemical Measurements

This Feature highlights the role of metrology, the science of measurement, in maintaining the infrastructure we all rely on for accurate chemical measurements. In particular, the recent change to the definition of the mole, the unit of chemistry, is explained.

High-efficiency ultra-precision comparator for d-spacing mapping measurement of silicon

This article describes a high-efficiency experimental configuration for a self-referenced lattice comparator with a `brush beam' of synchrotron radiation from a bending magnet and two linear position-sensitive photon-counting-type X-ray detectors. The efficiency is more than ten times greater compared with the `pencil-beam' configuration and a pair of zero-dimensional detectors. A solution for correcting the systematic deviation of d-spacing measurements caused by the horizontal non-uniformity of the brush beam is provided. Also, the use of photon-counting-type one-dimensional detectors not only improves the spatial resolution of the measurements remarkably but can also adjust the sample's attitude angles easily.

Empirical model for the temperature dependence of silicon refractive index from O to C band based on waveguide measurements

An accurate model for the silicon refractive index including its temperature and wavelength dependence is critically important for many disciplines of science and technology. Currently, such a model for temperatures above 22°C in the optical communication bands is not available. The temperature dependence in the spectral response of integrated echelle grating filters made in silicon-on-insulator is solely determined by the optical properties of the slab waveguide, making it largely immune to dimensional uncertainties. This feature renders the echelle filters a reliable tool to evaluate the thermo-optic properties of silicon. Here we investigate the temperature dependence of silicon echelle filters for the wavelength range of both O and C bands, measured between 22°C to 80°C. We show that if a constant thermo-optic coefficient of silicon is assumed for each band, as is common in the literature, the predictions show an underestimate of up to 10% in the temperature-induced channel wavelength shift. We propose and assess a model of silicon refractive index that encompasses both the wavelength and temperature dependence of its thermo-optic coefficients. We start from literature data for bulk silicon and further refine the model using the echelle filter measurement results. This model is validated through accurate predictions of device channel wavelengths and their temperature dependence, including the quadratic term, over a wide wavelength and temperature range. This work also demonstrates a new high-precision method for characterizing the optical properties of a variety of materials.

Mass and volume in analytical chemistry (IUPAC Technical Report)

This technical report reviews measurements of mass and volume, including a review of the SI for mass, length, and amount of substance; principles of mass measurement; calibration of masses and glassware; gravimetry; volumetry; and titrimetry. Measurement uncertainty, metrological traceability and aspects of quality assurance are also treated.

Storing quantum information in spins and high-sensitivity ESR

Quantum information, encoded within the states of quantum systems, represents a novel and rich form of information which has inspired new types of computers and communications systems. Many diverse electron spin systems have been studied with a view to storing quantum information, including molecular radicals, point defects and impurities in inorganic systems, and quantum dots in semiconductor devices. In these systems, spin coherence times can exceed seconds, single spins can be addressed through electrical and optical methods, and new spin systems with advantageous properties continue to be identified. Spin ensembles strongly coupled to microwave resonators can, in principle, be used to store the coherent states of single microwave photons, enabling so-called microwave quantum memories. We discuss key requirements in realising such memories, including considerations for superconducting resonators whose frequency can be tuned onto resonance with the spins. Finally, progress towards microwave quantum memories and other developments in the field of superconducting quantum devices are being used to push the limits of sensitivity of inductively-detected electron spin resonance. The state-of-the-art currently stands at around 65 spins per Hz, with prospects to scale down to even fewer spins.

Fabrication of ultrahigh-precision hemispherical mirrors for quantum-optics applications

High precision, high numerical aperture mirrors are desirable for mediating strong atom-light coupling in quantum optics applications and can also serve as important reference surfaces for optical metrology. In this work we demonstrate the fabrication of highly-precise hemispheric mirrors with numerical aperture NA = 0.996. The mirrors were fabricated from aluminum by single-point diamond turning using a stable ultra-precision lathe calibrated with an in-situ white-light interferometer. Our mirrors have a diameter of 25 mm and were characterized using a combination of wide-angle single-shot and small-angle stitched multi-shot interferometry. The measurements show root-mean-square (RMS) form errors consistently below 25 nm. The smoothest of our mirrors has a RMS error of 14 nm and a peak-to-valley (PV) error of 88 nm, which corresponds to a form accuracy of λ/50 for visible optics.

Interpreting and propagating the uncertainty of the standard atomic weights (IUPAC Technical Report)

In 2009, the Commission on Isotopic Abundances and Atomic Weights (CIAAW) of the International Union of Pure and Applied Chemistry (IUPAC) introduced the interval notation to express the standard atomic weights of elements whose isotopic composition varies significantly in nature. However, it has become apparent that additional guidance would be helpful on how representative values should be derived from these intervals, and on how the associated uncertainty should be characterized and propagated to cognate quantities, such as relative molecular masses. The assignment of suitable probability distributions to the atomic weight intervals is consistent with the CIAAW’s goal of emphasizing the variability of the atomic weight values in nature. These distributions, however, are not intended to reflect the natural variability of the abundances of the different isotopes in the earth’s crust or in any other environment. Rather, they convey states of knowledge about the elemental composition of “normal” materials generally, or about specific classes of such materials. In the absence of detailed knowledge about the isotopic composition of a material, or when such details may safely be ignored, the probability distribution assigned to the standard atomic weight intervals may be taken as rectangular (or, uniform). This modeling choice is a reasonable and convenient default choice when a representative value of the atomic weight, and associated uncertainty, are needed in calculations involving atomic and relative molecular masses. When information about the provenance of the material, or other information about the isotopic composition needs to be taken into account, then this distribution may be non-uniform. We present several examples of how the probability distribution of an atomic weight or relative molecular mass may be characterized, and also how it may be used to evaluate the associated uncertainty.

A critical review of the proposed definitions of fundamental chemical quantities and their impact on chemical communities (IUPAC Technical Report)

In the proposed new SI, the kilogram will be redefined in terms of the Planck constant and the mole will be redefined in terms of the Avogadro constant. These redefinitions will have some consequences for measurements in chemistry. The goal of the Mole Project (IUPAC Project Number 2013-048-1-100) was to compile published work related to the definition of the quantity ‘amount of substance’, its unit the ‘mole’, and the consequence of these definitions on the unit of the quantity mass, the kilogram. The published work has been reviewed critically with the aim of assembling all possible aspects in order to enable IUPAC to judge the adequateness of the existing definitions or new proposals. Compilation and critical review relies on the broadest spectrum of interested IUPAC members.

Quadrupole shift of nuclear magnetic resonance of donors in silicon at low magnetic field

Shifts from the expected nuclear magnetic resonance frequencies of antimony and bismuth donors in silicon of greater than a megahertz are observed in electrically detected magnetic resonance spectra. Defects created by ion implantation of the donors are discussed as the source of effective electric field gradients generating these shifts via quadrupole interaction with the nuclear spins. The experimental results are modeled quantitatively by molecular orbital theory for a coupled pair consisting of a donor and a spin-dependent recombination readout center.

Interferometric surface mapping of a spherical proof mass for ultra precise inertial reference sensors

In the context of our investigations on novel inertial reference sensors for space applications, we have explored a design utilizing an optical readout of a spherical proof mass. This concept enables full drag-free operations, hence reducing proof mass residual acceleration noise to a minimum. The main limitations of this sensor are errors in position determination of the center of mass of the proof mass due to the surface topography and the involved path length changes upon rotation. One solution is to apply a surface map for correction of the measurement data, thus improving the precision of position determination. This article presents the results of our one-dimensional interferometric surface topography measurements of a sphere, achieving uncertainties of ≈10  nm, as a first step to realize a complete surface map. The measurement setup consists of two heterodyne interferometers positioned in an opposing configuration, which measure the surface topography while the sphere is continuously rotated by a rotation stage.

Mass Spectrometric Investigation of Silicon Extremely Enriched in (28)Si: From (28)SiF4 (Gas Phase IRMS) to (28)Si Crystals (MC-ICP-MS)

A new generation of silicon crystals even further enriched in (28)Si (x((28)Si) > 0.999 98 mol/mol), recently produced by companies and institutes in Russia within the framework of a project initiated by PTB, were investigated with respect to their isotopic composition and molar mass M(Si). A modified isotope dilution mass spectrometric (IDMS) method treating the silicon as the matrix containing a so-called virtual element (VE) existing of the isotopes (29)Si and (30)Si solely and high resolution multicollector inductively coupled plasma mass spectrometry (MC-ICP-MS) were applied in combination. This method succeeds also when examining the new materials holding merely trace amounts of (29)Si (x((29)Si) ≈ 5 × 10(-6) mol/mol) and (30)Si (x((30)Si) ≈ 7 × 10(-7) mol/mol) extremely difficult to detect with lowest uncertainty. However, there is a need for validating the enrichment in (28)Si already in the precursor material of the final crystals, silicon tetrafluoride (SiF4) gas prior to crystal production. For that purpose, the isotopic composition of selected SiF4 samples was determined using a multicollector magnetic sector field gas-phase isotope ratio mass spectrometer. Contaminations of SiF4 by natural silicon due to storing and during the isotope ratio mass spectrometry (IRMS) measurements were observed and quantified. The respective MC-ICP-MS measurements of the corresponding crystal samples show-in contrast-several advantages compared to gas phase IRMS. M(Si) of the new crystals were determined to some extent with uncertainties urel(M) < 1 × 10(-9). This study presents a clear dependence of the uncertainty urel(M(Si)) on the degree of enrichment in (28)Si. This leads to a reduction of urel(M(Si)) during the past decade by almost 3 orders of magnitude and thus further reduces the uncertainty of the Avogadro constant NA which is one of the preconditions for the redefinition of the SI unit kilogram.

Application of the double paddle oscillator for quantifying environmental, surface mass variation

Sub-monolayer sensitivity to controlled gas adsorption and desorption is demonstrated using a double paddle oscillator (DPO) installed within an UHV (ultra-high vacuum) environmental chamber equipped with in situ film deposition, (multi)gas admission and temperature control. This effort is intended to establish a robust framework for quantitatively comparing mass changes due to gas loading and unloading on different materials systems selected or considered for use as mass artifacts. Our apparatus is composed of a UHV chamber with gas introduction and temperature control and in-situ materials deposition for future materials testing enabling in situ preparation of virgin surfaces that can be monitored during initial exposure to gasses of interest. These tools are designed to allow us to comparatively evaluate how different materials gain or lose mass due to precisely controlled environmental excursions, with a long term goal of measuring changes in absolute mass. Herein, we provide a detailed experimental description of the apparatus, an evaluation of the initial performance, and demonstration measurements using nitrogen adsorption and desorption directly on the DPO.

(30)Si mole fraction of a silicon material highly enriched in (28)Si determined by instrumental neutron activation analysis

The latest determination of the Avogadro constant, carried out by counting the atoms in a pure silicon crystal highly enriched in (28)Si, reached the target 2 × 10(-8) relative uncertainty required for the redefinition of the kilogram based on the Planck constant. The knowledge of the isotopic composition of the enriched silicon material is central; it is measured by isotope dilution mass spectrometry. In this work, an independent estimate of the (30)Si mole fraction was obtained by applying a relative measurement protocol based on Instrumental Neutron Activation Analysis. The amount of (30)Si isotope was determined by counting the 1266.1 keV γ-photons emitted during the radioactive decay of the radioisotope (31)Si produced via the neutron capture reaction (30)Si(n,γ)(31)Si. The x((30)Si) = 1.043(19) × 10(-6) mol mol(-1) is consistent with the value currently adopted by the International Avogadro Coordination.

A Clock Directly Linking Time to a Particle's Mass

Historically, time measurements have been based on oscillation frequencies in systems of particles, from the motion of celestial bodies to atomic transitions. Relativity and quantum mechanics show that even a single particle of mass m determines a Compton frequency ω0 = mc2/ℏ, where c is the speed of light and ℏ is Planck's constant h divided by 2π. A clock referenced to ω0 would enable high-precision mass measurements and a fundamental definition of the second. We demonstrate such a clock using an optical frequency comb to self-reference a Ramsey-Bordé atom interferometer and synchronize an oscillator at a subharmonic of ω0. This directly demonstrates the connection between time and mass. It allows measurement of microscopic masses with 4 × 10−9 accuracy in the proposed revision to SI units. Together with the Avogadro project, it yields calibrated kilograms.

History and progress on accurate measurements of the Planck constant

The measurement of the Planck constant, h, is entering a new phase. The CODATA 2010 recommended value is 6.626 069 57 × 10(-34) J s, but it has been a long road, and the trip is not over yet. Since its discovery as a fundamental physical constant to explain various effects in quantum theory, h has become especially important in defining standards for electrical measurements and soon, for mass determination. Measuring h in the International System of Units (SI) started as experimental attempts merely to prove its existence. Many decades passed while newer experiments measured physical effects that were the influence of h combined with other physical constants: elementary charge, e, and the Avogadro constant, N(A). As experimental techniques improved, the precision of the value of h expanded. When the Josephson and quantum Hall theories led to new electronic devices, and a hundred year old experiment, the absolute ampere, was altered into a watt balance, h not only became vital in definitions for the volt and ohm units, but suddenly it could be measured directly and even more accurately. Finally, as measurement uncertainties now approach a few parts in 10(8) from the watt balance experiments and Avogadro determinations, its importance has been linked to a proposed redefinition of a kilogram unit of mass. The path to higher accuracy in measuring the value of h was not always an example of continuous progress. Since new measurements periodically led to changes in its accepted value and the corresponding SI units, it is helpful to see why there were bumps in the road and where the different branch lines of research joined in the effort. Recalling the bumps along this road will hopefully avoid their repetition in the upcoming SI redefinition debates. This paper begins with a brief history of the methods to measure a combination of fundamental constants, thus indirectly obtaining the Planck constant. The historical path is followed in the section describing how the improved techniques and discoveries in quantum mechanics steadily reduced the uncertainty of h. The central part of this review describes the technical details of the watt balance technique, which is a combination of the mechanical and electronic measurements that now determine h as a direct result, i.e. not requiring measured values of additional fundamental constants. The first technical section describes the basics and some of the common details of many watt balance designs. Next is a review of the ongoing advances at the (currently) seven national metrology institutions where these experiments are pursued. A final summary of the recent h determinations of the last two decades shows how history keeps repeating itself; there is again a question of whether there is a shift in the newest results, albeit at uncertainties that are many orders of magnitude less than the original experiments. The conclusion is that there is room for further development to resolve these differences and find new ideas for a watt balance system with a more universal application. Since the next generation of watt balance experiments are expected to become kilogram realization standards, the historical record suggests that there is yet a need for proof that Planck constant results are finally reproducible at an acceptable uncertainty.

Embracing the quantum limit in silicon computing

Quantum computers hold the promise of massive performance enhancements across a range of applications, from cryptography and databases to revolutionary scientific simulation tools. Such computers would make use of the same quantum mechanical phenomena that pose limitations on the continued shrinking of conventional information processing devices. Many of the key requirements for quantum computing differ markedly from those of conventional computers. However, silicon, which plays a central part in conventional information processing, has many properties that make it a superb platform around which to build a quantum computer.

Is the proton radius a player in the redefinition of the International System of Units?

It is now recognized that the International System of Units (SI units) will be redefined in terms of fundamental constants, even if the date when this will occur is still under debate. Actually, the best estimate of fundamental constant values is given by a least-squares adjustment, carried out under the auspices of the Committee on Data for Science and Technology (CODATA) Task Group on Fundamental Constants. This adjustment provides a significant measure of the correctness and overall consistency of the basic theories and experimental methods of physics using the values of the constants obtained from widely differing experiments. The physical theories that underlie this adjustment are assumed to be valid, such as quantum electrodynamics (QED). Testing QED, one of the most precise theories is the aim of many accurate experiments. The calculations and the corresponding experiments can be carried out either on a boundless system, such as the electron magnetic moment anomaly, or on a bound system, such as atomic hydrogen. The value of fundamental constants can be deduced from the comparison of theory and experiment. For example, using QED calculations, the value of the fine structure constant given by the CODATA is mainly inferred from the measurement of the electron magnetic moment anomaly carried out by Gabrielse's group. (Hanneke et al. 2008 Phys. Rev. Lett. 100, 120801) The value of the Rydberg constant is known from two-photon spectroscopy of hydrogen combined with accurate theoretical quantities. The Rydberg constant, determined by the comparison of theory and experiment using atomic hydrogen, is known with a relative uncertainty of 6.6×10(-12). It is one of the most accurate fundamental constants to date. A careful analysis shows that knowledge of the electrical size of the proton is nowadays a limitation in this comparison. The aim of muonic hydrogen spectroscopy was to obtain an accurate value of the proton charge radius. However, the value deduced from this experiment contradicts other less accurate determinations. This problem is known as the proton radius puzzle. This new determination of the proton radius may affect the value of the Rydberg constant . This constant is related to many fundamental constants; in particular, links the two possible ways proposed for the redefinition of the kilogram, the Avogadro constant N(A) and the Planck constant h. However, the current relative uncertainty on the experimental determinations of N(A) or h is three orders of magnitude larger than the 'possible' shift of the Rydberg constant, which may be shown by the new value of the size of the proton radius determined from muonic hydrogen. The proton radius puzzle will not interfere in the redefinition of the kilogram. After a short introduction to the properties of the proton, we will describe the muonic hydrogen experiment. There is intense theoretical activity as a result of our observation. A brief summary of possible theoretical explanations at the date of writing of the paper will be given. The contribution of the proton radius puzzle to the redefinition of SI-based units will then be examined.

The watt balance: determination of the Planck constant and redefinition of the kilogram

Since 1889, the international prototype of the kilogram has served as the definition of the unit of mass in the International System of Units (SI). It is the last material artefact to define a base unit of the SI, and it influences several other base units. This situation is no longer acceptable in a time of ever-increasing measurement precision. It is therefore planned to redefine the unit of mass by fixing the numerical value of the Planck constant. At the same time three other base units, the ampere, the kelvin and the mole, will be redefined. As a first step, the kilogram redefinition requires a highly accurate determination of the Planck constant in the present SI system, with a relative uncertainty of the order of 1 part in 10(8). The most promising experiment for this purpose, and for the future realization of the kilogram, is the watt balance. It compares mechanical and electrical power and makes use of two macroscopic quantum effects, thus creating a relationship between a macroscopic mass and the Planck constant. In this paper, the operating principle of watt balance experiments is explained and the existing experiments are reviewed. An overview is given of all available experimental determinations of the Planck constant, and it is shown that further investigation is needed before the redefinition of the kilogram can take place. Independent of this requirement, a consensus has been reached on the form that future definitions of the SI base units will take.

Learning from the time and length redefinitions, and the metre demotion

After discussing several issues in a future redefinition of the kilogram, this paper considers the lessons that one might have learned from the analogous redefinitions of the metre and the second. The progress of length metrology was slow and steady, from seven digits reproducibility with the 1889 X-shaped metre prototype, to nine digits with Kr lamps, to 11 digits with the 1983 redefinition of the metre using the speed of light. With laser cooling, the Cs clock improved to 15, now 16, digits (and so also astronomical distance measurements could improve). Laser-cooled ions, and now atoms captured and cooled in an optical lattice, enable accuracy capability of three different optical frequency references to exceed 17 digits, i.e. better than time itself. The optical comb and related techniques vastly simplify frequency comparisons. Such progress stimulates a new satellite experiment, the STAR Mission (Space-Time Asymmetry Research). The goal is to test at the 1E-18 level frequency shifts owing to spatial anisotropy, position, gravitational potential and boost. The onboard optical clock will use stabilization to a molecular transition in I(2) or HCCH or CO(2). The length etalons will be multiply redundant, with stability at the thermo-mechanical mirror motion limit. For a ULE glass etalon spacer (1987), I measure length creep approximately -1.5E-12/d, i.e. below 1E-14 over the 500 s satellite spin period.

The role of the international prototype of the kilogram after redefinition of the International System of Units

Since 1889, the international prototype of the kilogram has served to define the unit of mass in what is now known as the International System of Units (SI). This definition, which continues to serve mass metrology well, is an anachronism for twenty-first century physics. Indeed, the kilogram will no doubt be redefined in terms of a physical constant, such as the Planck constant. As a practical matter, linking the quantum world to the macroscopic world of mass metrology has, and remains, challenging although great progress has been made. The international prototype or, more likely, a modern ensemble of reference standards, may yet have a role to play for some time after redefinition, as described in this paper.

A new definition for the mole based on the Avogadro constant: a journey from physics to chemistry

The mole is the most recent addition to the set of base units that form the International System of Units, although its pre-cursor the 'gram-molecule', had been in use by both physicists and chemists for more than 120 years. A proposal has been published recently to establish a new definition for the mole based on a fixed value for the Avogadro constant. This would introduce consistent relative uncertainties for the molar and the atomic masses while making no change to the system of relative atomic masses ('atomic weights'). Although the proposal would have little impact on the measurement uncertainty of practical work, it has stimulated considerable debate about the mole and the nature of the quantity amount of substance. In this paper, the rationale for the new definition is explained against the background of changes in the way the quantity amount of substance has been used, from its first use during the early development of thermodynamics through to the use of the 'number of gram-molecules' at the end of the nineteenth century.