26 | NCSLI Measure J. Meas. Sci. www.ncsli.org

SPECIAL FEATURE

Abstract: The fact that the unit of mass might soon be derived from the Planck constant, rather than from an artifact standard, 

can seem daunting and downright baffling when viewed from the vantage point of our day to day perception of mass. After all, 

at measurement levels that register with our human senses, the connection between the quantum mechanics of Planck (atoms) 

and the engineering mechanics of Newton (apples) is less than obvious. However, as the physicist Richard Feynman famously 

observed, “there is plenty of room at the bottom”, and our need to quantify the mass of objects isn’t always limited to the familiar 

quantities we encounter in the produce section of our grocery store. Here, I explore the connection between mass and the Planck 

constant and suggest that a benefit of deriving the unit of mass from a fundamental constant is that it is inherently more scalable 

than the present artifact. For example, scientists and engineers working at the forefront of measurement science are increasingly 

pushing the boundary on what we consider a measurement of mass. In fact, a group now claims to have measured the mass of a 

cesium atom to within well below a yoctogram, which is below the mass of a single proton. This unit of mass is a submultiple of 

our present artifact kilogram so small that it requires 27 zeros after the decimal point before it even registers as a significant digit! 

How are such things possible? Why would you try? Can we even conceive of a traceable yoctogram? To begin grappling with 

these questions, I will attempt to guide you through the physics of Newton and Planck and, I hope, shed some light on how we can 

weigh everything from atoms to apples in a revised SI based on fundamental constants.

1. Introduction

The International System of Units (SI) may 

soon be revised so that all quantitative measurements can be based on fixed values of 

physical constants [1]. This proposed scheme 

for deriving the units of measure is a natural 

progression from ideas that have been with 

the SI since its conception (to appreciate the 

scope, the interested reader should consider 

references [2, 3] for general and entertaining 

accounts of the history of the SI and its coming revision). 

The unit of length, for example, is already 

successfully tied to the speed of light, c (Alan 

Steele, the director of Measurement Science 

and Technology for the National Research 

Council, Canada walked through this as part 

of his presentation at the NCSLI conference 

in 2011 [4], and an interesting and technically 

thorough historical account can be found in 

[5], which points to a number of the seminal 

reviews by the giants of the field). Briefly, the 

process of creating our present unit of length 

began when general consensus emerged over 

the course of the last century that the speed 

of light in vacuum is fixed. Such consensus 

was attained at a time when observations of c

with respect to the prevailing length and time 

standards still showed variability. The generally accepted view was that it was not actually 

c varying in the experiments, but our ability 

to use the standards with which c was being 

measured (talk about a measurement dominated by Type B uncertainty!). Once the notion 

that the speed of light is an absolute constant 

gained traction, its exact numerical value was 

established as a constant in units of meters 

per second, giving metrologists a very useful 

ratio between the units of measure for length 

and time with regards to the physics of light 

(or more generally electromagnetic waves) 

yielding a new definition of the meter in 1983. 

This gradual evolution in thought and practice laid the foundation to measure a length in 

terms of meters by simply (simple being in 

the eye of the beholder here!) using a beam 

of light and a clock. Which begs the question: 

what do we use as a time base for our clock? 

Michelson, who pioneered many elements of 

this scheme, used the frequency of a tuning 

fork calibrated by the frequency of a pendulum whose oscillation period (the inverse of 

its frequency) was fixed in terms of the length 

of time of one mean solar day! Notice that 

a time base is simply the inverse of the frequency of some fixed periodic event. Then, 

the time base was the periodic behavior of 

planet earth. Today, the time base is the periodic behavior of energy associated with the 

electronic state of a Cesium atom, a so-called 

atomic clock. As you might imagine, measurHow to Weigh Everything from Atoms to 

Apples Using the Revised SI

Jon R. Pratt

Author

Jon R. Pratt

jon.pratt@nist.gov

Quantum Measurements Division

National Institute of Standards

 and Technology

100 Bureau Drive

Gaithersburg, MD 20899-8171
Vol. 9 No. 1 • March 2014 NCSLI Measure J. Meas. Sci. | 27

SPECIAL FEATURE

ing length by using the time of flight of light pulses measured in terms 

of a fluctuating atom is not always terribly convenient. In practice the 

fundamental relationship between the length and frequency of light 

waves, as expressed by c = ln, is more commonly exploited, where l is 

the wavelength and n the frequency of the light, respectively. Clearly, 

if we know the frequency of a beam of light, then we also know its 

wavelength, and the light itself becomes an effective length standard 

by virtue of knowing the exact, agreed upon value for c. Importantly, 

the frequency of light can in turn be determined with respect to the 

frequency of our atomic clock, which is just the reciprocal of its period. This frequency has been defined as the primary physical invariant 

in our system of measurement; the ground state hyperfine splitting 

frequency of the cesium 133 atom, Dν(133Cs)hfs (reference [5] can also 

walk you through time, so to speak). This is an example where the 

present SI builds up a unit of measure working from physical constants, rather than physical artifacts. 

A significant hurdle to duplicating this paradigm and basing all of 

our units of measure on physical constants is the historical problem 

of picking a constant for mass that is both believably invariant and 

technically accessible. General consensus is that the most promising 

approach for establishing a unit of mass in terms of a fundamental 

invariant of the universe is to exploit the connection between mass 

and the Planck constant, h. It is worth mentioning that most scientists 

now accept that h is in fact a constant, but that far fewer scientists are 

confident that we have successfully measured its value with sufficient 

accuracy ([6] offers a detailed review of the physics and the state of 

the art, parallel to what will be considered here)! 

For now, let’s agree that the Planck constant is likely a constant, and 

that its value will soon be measured to the desired accuracy within our 

present system of units. The questions that I wish to explore here are:

a. What is mass and how does it relate to the Planck constant, h? 

b. How do we weigh an atom in terms of h?

c. How do we weigh an apple in terms of h?

d. Can an instrument realize and scale the unit of mass?

As it turns out, answers for many of these questions are linked to 

some of the most famous equations and events in science, and the 

impact is no less than the achievement of a system of units capable of 

seamlessly unifying the measurement of all mass and inertial phenomena from the subatomic to the cosmological.

2. What is Mass? And How Does it Relate to the Planck 

Constant, h?

Since Newton first described the physics and mathematics of falling 

apples, our concept of mass has been firmly rooted in his now familiar 

expression

FF = ma, (1)

where F is a force acting on an object, m is the inertia, or mass of the 

object, and a is its acceleration.

Mass, from the vantage point of Newtonian mechanics, is a resistance to motion. It can also be a source of gravitational attraction, 

according to another of Newton’s famous observations about apples

FF = GGmm!mm!

rr! , (2)

where this time F is the gravitational force of attraction between one 

object of gravitational mass m1 and another of gravitational mass m2

separated by a distance r. For this discussion, we will assume the 

equivalence of inertial and gravitational mass, which is still the subject of experimentation, but has been proved to the level of a few 

parts in 1013

.

All we need in order to create a standard of mass from the Newtonian viewpoint of inertia is a known force, and standards of length 

and time with which to measure acceleration. Standards of length and 

time, as we saw in the introduction, have evolved to a point where they 

are based on fixed values of physical invariants, namely the speed of 

light and the atomic frequency of the cesium atom. But what can we 

use as a known force? This turns out to be a very difficult question 

to answer, and one that we will return to throughout this article. For 

now, simply consider that you can’t use the force of gravity, since 

it depends on mass. Instead, the effect of gravity has been used for 

centuries to compare the mass of objects relative to one another by 

comparing the forces gravity exerts on the objects as they balance a 

beam about a fulcrum; the classic equal arm balance.

Armed with an equal arm balance (or its modern equivalent, the 

compensation balance), we can avoid the problem of finding a known 

force by simply declaring a certain object to be of unit mass and comparing everything else to it using a balance. Of course the object needs 

to be very stable (as must the balance), and there should be a recipe for making copies of it, so that this object, which we will call an 

artifact standard of mass, can be replaced in the event it is damaged 

through use. Most of you will recognize this as the approach to the 

measurement of mass that has been in use for over a century, with all 

measurement of mass defined in terms of one artifact (and its copies), 

the International Prototype Kilogram, or IPK.

Intuitively, mass has also always been thought of as an amount of 

substance, or more colloquially, an amount of “stuff”. From Newton’s 

viewpoint, the way to measure the amount of stuff in an object is to 

apply a force and watch the object move. As we’ve seen, lacking a 

known force, we simply balance objects with respect to a standard 

object using gravitational torques on lever arms. We don’t even need 

to measure the acceleration. We simply observe that if the balance 

doesn’t move, the gravitational force on the unknown object must be 

the same as the gravitational force on our standard, and therefore the 

unknown must have the same amount of stuff as the standard.

From a chemist’s viewpoint, this is highly unsatisfactory, 

and a better way to measure the quantity of stuff in an object is 

to examine its atomic constituents, and then count how many of 

these “indivisible” elements there are in the whole. So many atoms 

of this, plus so many atoms of that, and with a little math you can 

assemble an entire object. From the chemist’s viewpoint, the trick 

is to determine the mass of an atom. Everything else is just ratios. 

Clearly, just as we observed with regard to the Newtonian viewpoint, 

it is possible to declare one object, an atom in this case, as a standard 

(carbon-12 is the IPK of atomic mass units) and ratio ourselves to 

the mass of everything. 

Naturally, at some point chemistry and mechanics must reconcile, 

and this is the path that leads to the Avogadro number, which we will 

take up in more detail later. For now, we simply recognize that we 

can’t declare an atom to have a particular standard value of mass, and 

the IPK another. We will need an experiment that relates the two in 

some fashion.
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Up until fairly recently, these two views, mass as inertia, or mass 

as amount of stuff, captured the totality of our understanding. But in 

the early 1900’s our model of the physical universe took a relative 

quantum leap, so to speak, and a new way of thinking about mass 

in terms of energy was introduced which has gradually called into 

question some of our most basic ideas about matter as substance that 

can be discretized. In the remainder of this section we explore how 

this contemporary viewpoint might answer the question, “what is 

mass?” and introduce the central theme of how mass relates to the 

Planck constant. 

For those of us who did not major in physics, the last encounter 

with the Planck constant may have been a required course somewhere 

in a distant academic past (I find myself rereading chapter 42, “Light 

and Quantum Physics,” of my aging copy of Halliday and Resnick 

fairly regularly these days [7]). So it’s useful to recall that the Planck 

constant (symbol h) is a physical invariant that arose early in the development of quantum mechanics, when Max Planck grappled with 

data from cavity radiation experiments. 

Classical physics of the time could not describe the spectral radiancy of a cavity as a function of wavelength. Something new was 

required to circumvent the so-called ultraviolet catastrophe (models 

predicting infinite radiancy as wavelength decreased), and, in a bold 

bit of empirical insight, Planck began by assuming that the energy E

observed in a cavity was governed by the atoms on the cavity walls. 

He postulated that the atoms behaved as tiny electromagnetic oscillators, each with a specific frequency of oscillation, emitting and absorbing energy to and from the cavity, giving rise to an equilibrium 

radiancy. The radical notion that he introduced was that the energy 

of the oscillators was not continuous, but was quantized in terms of 

the oscillator frequency and a constant that now bears his name. In 

fact, since Planck’s announcement to the Berlin Physical Society in 

December of 1900, it has become increasingly clear that all oscillators 

ranging from atoms to microelectromechanical systems (see [8] for 

just one example of folks probing the quantum behavior of microscale 

objects) cannot have just any value of energy, but are constrained to 

have only values of energy given by

EE = nhνν, (3)

where E is the energy of the oscillator, n is its frequency, n is a number that can take on only positive integer values and here is called the 

quantum number, and h is, of course, the Planck constant. Planck’s 

presentation of the equation above heralded the beginning of quantum mechanics, and provides the first piece of a puzzle connecting h

to mass.

Another key piece of the puzzle connecting h to mass is found in 

Einstein’s 1905 paper on special relativity, which yields the famous 

equation relating the rest mass of a particle to its total energy, or

EE = mmmm!. (4)

As knowledge of quantum mechanics and relativity gained momentum in the early 1900’s, it became evident that in certain limiting 

cases, the particle having mass m in Einstein’s expression of relativity, 

might also be thought of as having a single quantum of energy, or

EE = hνν. (5)

In this limiting case, we see that two famous equations for energy 

both describe the energy of the same object, and it is fairly easy to 

perform some algebra and conclude that the Planck constant and mass 

are linked, at least in terms of scaling, according to

mm = !!

!!, (6)

where n now corresponds to the frequency of a photon whose energy is 

equal to the rest mass equivalent energy of the particle in question. This 

is the so-called Compton frequency, and it is used in expressions from 

quantum mechanics that describe the conversion of mass to energy and 

the interaction of light (or, more specifically photons) with particles.

Summing things up, Nobel prize winner, Frank Wilczek, in his 

book The Lightness of Being: Mass, Ether, and the Unification of 

Forces, somewhat whimsically suggests that Einstein’s Second Law 

(Chapter 3 of the book) ought to be

mm = !

!!, (7)

and asserts that mass is really most deeply understood as energy. So 

according to modern physics, the answer to the question “What is 

mass?” is simply that mass is energy! And as we’ve seen, according 

to the quantum mechanics of Planck, energy is quantized in terms of a 

frequency of oscillation. To quote Jerry Lee Lewis, “There’s a whole 

lotta shakin’ goin on!”

2.1 How do we Weigh an Atom in Terms of h?

Measure its Frequency!

The revolution in our understanding of the complex quantum nature 

of reality witnessed over the last century has lead to the exciting 

possibility that mass can be accurately expressed in terms of frequency, 

at least at the atomic scale. Of course it is far from trivial to measure 

the frequency of a matter wave, and it is only possible because of 

some relatively recent ground breaking work in the fields of optical 

trapping and frequency combs (including some Nobel prize winning 

work at NIST!). 

To appreciate the challenge, consider that the rest mass of a very 

light particle, the proton, is on the order of a yoctogram, or 10-27 kg, 

so that plugging in values for c and h and solving backwards for the 

Compton frequency of the proton we find n = 1.36 × 1023 Hz! As 

a point of reference, typical red laser light has a frequency of only 

4.74 × 1014 Hz. To date, the Compton frequency has been far too high 

to measure directly, but a team recently reported making an indirect 

observation of the Compton frequency of a Cesium atom by using a 

combination of atom interferometry and frequency comb technologies 

[9]. The title of this paper “A Clock Directly Linking Time to a Particle’s Mass,” is a harbinger of the future.

Within the context of planned revisions to the SI, where h becomes 

a fundamental constant of fixed numerical value, the results of 

Compton clock experiments [9] and atom recoil experiments (see 

Fig. 1 for a schematic explanation of atom recoil) that measure the 

ratio h / m of atomic particles [10, 11] could be used to fix the mass of 

atoms, with the potential for atomic clock like stability and accuracy; 

a traceable unit of mass at the scale of atoms. The preliminary 

experiments already claim the ability to measure the mass of a cesium 

atom in terms of a fixed value of h to four parts per billion (roughly 

2.2 × 10-25 kg ± 10-33 kg)! The upshot is that in the proposed revised 
Vol. 9 No. 1 • March 2014 NCSLI Measure J. Meas. Sci. | 29

SPECIAL FEATURE

SI, atoms can be directly and accurately 

weighed in terms of the SI unit of mass based 

on the fixed value of the Planck constant h, 

the fixed value of the speed of light c, and an 

experimentally observed frequency measured 

in terms of another fixed constant, the 

hyperfine transition frequency of the Cesium 

atom, Dν(133Cs)hfs. 

Why should we care about the mass of 

atoms? Because increasingly the science 

of molecules and atoms informs how 

we approach the development of new 

technologies. In particular, much of today’s 

applied physics research asks the question, 

where do quantum and classical mechanics 

overlap, and how might we exploit this 

overlap to create new technologies? Every 

sensor imaginable is being re-imagined to 

explore how its performance can be coaxed 

to achieve quantum limited behavior (and 

beyond, in the case of quantum oddities 

like entanglement and squeezed states) as 

we shrink its size, or chill it down, or both. 

The manipulation of matter at the atomic 

scale is already an integral part of modern 

manufacturing (think of the semiconductor 

and pharmaceutical industries, e.g., your 

cell phone and your daily meds...), but 

as has been famously observed over the 

years, you cannot make what you cannot 

measure. Today, the width of a “wire” in 

a modern computer chip is measured by 

taking its image using an instrument called 

a Tunneling Electron Microscope and then 

literally counting the number of atoms across 

its width. 

As we begin an era where technologies 

ranging from solar panels to targeted drug 

therapies will exploit the optical, electrical 

and mechanical properties of matter on size 

scales where we have literally counted and 

controlled the number and type of atoms involved, it becomes increasingly important 

to have a system of measurement that can 

bridge between the physics of Planck and the 

physics of Newton. With this in mind, it appears the proposed changes to the SI might 

be well and good for atoms, but what do they 

mean for apples?

3. Bridging Between Atoms and

Apples in the Existing SI

We saw in Section 2 that the connection 

between mass and frequency at the atomic 

scale can be exploited to effectively weigh 

an atom in terms of fundamental constants 

of nature, namely h, c, and the hyperfine 

transition frequency of the Cesium atom, 

Dν(133Cs)hfs. We also noted that modern 

technologies are increasingly exploiting 

the ability to manipulate matter at atomic 

scales, so that knowledge of the optical, 

electrical, and mechanical properties of 

atoms, molecules, and collections of atoms 

is of increasing importance to science and its 

ability to drive next generation technologies. 

But we still live in a physical world where the 

majority of products and measurements occur 

at far more classical or human scales, and the 

question remains as to how revising the unit 

of mass and basing it on the Planck constant 

will impact our measurement of everyday 

objects, like apples. Setting that question 

aside for the moment (we’ll take it up in 

earnest in Section 4), it is helpful to first take 

a look at how we weigh an atom in terms of 

the existing definition of the unit of mass, in 

other words, by considering how we compare 

the mass of an atom to the International 

Prototype Kilogram, IPK.

3.1 Weighing with Artifacts and the 

Accumulation of Uncertainty

The traceability of all weighing operations 

in our present SI begins with an artifact, 

the IPK that realizes the unit of mass at a 

magnitude of a kilogram. This artifact and 

the unit of mass are one and the same, and 

in order to create submultiples of this unit of 

measure, copies of the IPK must be made, 

as well as a series of ever smaller artifacts 

that when stacked back together sum to a 

magnitude of a kilogram (typically beginning 

with a set consisting of 500 g, 200 g, 200 g, 

and 100 g objects). With these artifacts in 

place, a weighing design is executed using 

a mass comparator or compensation balance 

to record the sums and differences of these 

various copies and submultiples of the 

original artifact to compute least squares 

estimates for the relative mass of each object 

in terms of the original IPK. This process 

is repeated at each order of magnitude 

employing ever smaller series of masses. 

Weighing designs, such as the one illustrated 

in Fig. 2, which is reproduced from [12], can 

be performed with amazing precision and 

accuracy. Still, as noted in [12], uncertainty 

inevitably accumulates with each series, so 

that by the time this subdividing process 

reaches the level of milligram sized artifacts, 

the relative uncertainty in terms of the 

original kilogram sized artifact is on the 

order of a few parts in ten thousand, and 

speculatively might be on the order of unity 

by the time we would reach an artifact the 

size of a microgram. How then do we bridge 

nearly another twenty orders of magnitude to 

get to the mass of an atom in this system?

E0

Eex

E0

Eex

E0

Eex

m  2h

c2

 0

2

 1 2 0

... 



 









Figure 1. The above is a cartoon depiction of an atom recoil experiment to determine the mass of an atom. At the upper left of the cartoon, light at frequency n0

(right moving photon wave packet in the upper left hand corner) interacts with an 

isolated atom having energy E0 and sitting at rest. The photon is absorbed, causing the energy of the atom to increase to Eex and to begin moving to the right with 

a velocity indicated by the red arrow. At the lower right, the atom emits a photon 

and returns to initial energy E0, but the velocity is now doubled, owing to the momentum transferred from the ejected photon (the recoil). The motion of the atom 

leads to a Doppler shift in the frequency of the exiting light, Dν, which can be used 

to compute the mass of the atom as shown.
30 | NCSLI Measure J. Meas. Sci. www.ncsli.org

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3.2 The Avogadro Number, a Silicon 

Crystal, and Atoms as Artifacts

It turns out that we can still subdivide the IPK 

using a series of artifacts the sum of which 

is a kilogram. The trick is that the artifacts 

in this case are atoms, and the way that they 

stack up to a kilogram is by forming a perfect, single crystal of silicon. This is the ultimate expression of the chemist’s viewpoint 

introduced in Section 2 and the essence of the 

International Avogadro project, where instead 

of starting at the bottom, and determining the 

mass of an individual atom directly (as we 

might do with a Compton clock), the process 

instead begins at the top, with a copy of the 

IPK that is a perfect sphere of silicon. The 

question then becomes “how many atoms are 

in the silicon sphere?”, and the answer (after 

several decades of work [13]) is related to the 

Avogadro constant NA.

The Avogadro constant relates any quantity 

at the atomic scale to its corresponding macroscopic scale, and it can be written as

NN! = !"

!!!, (8)

where n in this case is again a positive integer, 

this time corresponding to the number of 

atoms per unit cell of the crystal, and r, M, 

and a are the density, molar mass, and lattice 

parameter, respectively. The determination 

of the Avogadro constant using carefully 

manufactured silicon copies of the IPK has 

been a long, and exceedingly challenging 

project involving many workers around the 

globe, but knowledge of this number provides 

a powerful link between the world of atoms 

and molecules and the physical world in 

which we live.

For example, the efforts of the Avogadro 

project have made it possible to state the mass 

of a silicon atom in SI units as

mm!" = !

!!

. (9)

All that is required in order to compute the 

mass of the atom in terms of the SI is knowledge of its molar mass (more complicated 

than it might sound, since it involves determining the isotopic abundance of the given 

sample) and the Avogadro constant. The 

relative uncertainty of the Avogadro constant 

listed in the most recent CODATA recommended values of the physical constants is 45 

parts per billion [14], and this is roughly the 

uncertainty of the mass of a silicon atom in 

the current SI.

3.3 Atoms, Apples, and Things in 

Between

As we’ve seen so far, all appears well and 

good with our knowledge of the atomic mass 

of silicon within the current SI, and we might 

rightly wonder what all the fuss is about with 

regards to revising the system of measurement, since it already covers everything from 

atoms to apples, at least as far as mass goes. 

This is a topic of considerable debate, but to 

motivate a view towards change, it helps to 

focus on some of the limitations of the present system. 

First, it is worth noting that it took a team 

of national metrology institutes (NMIs) working around the globe three decades to achieve 

a system capable of bringing the uncertainty 

in the expression of the SI traceable mass of 

a silicon atom to the present level of 45 parts 

per billion. The groups involved in this work 

hope to push this to the level of perhaps 20 

parts per billion over the next three years, 

but at that point the project will have pushed 

the work as far as is thought humanly possible (at least within the budgets the world’s 

governments are willing to expend on such 

tasks; they do have countries to run). And yet, 

Compton clock and atom recoil experiments 

have already yielded an estimate of atomic 

mass ten times more precise, if only we accept the entirely reasonable assumption that 

the Planck constant is a constant having a 

fixed value.

A second, perhaps more practical, reason 

to consider rethinking the unit of mass is 

that subdivision of the unit of mass typically 

leads to the accumulation of uncertainty we 

discussed in Section 3.1. Through the heroic 

NIST reference

standard (restraint)

NIST check

standard Unknown weights

1 kg 1 kg 1 kg 1 kg

500g

200g 200g 100g

Restraint

NIST check

standard

Unknown weights

500g 200g 200g 100g 100g

50g 20g 20g 10g

100 g

50g 20g 20g 10g

Restraint

10g

NIST check

standard 5g 2g 2g 1g

10 g

Unknown weights

Restraint

etc...

Series 1

Series 2

Series 3

Series 4

etc...

Figure 2. A schematic showing a typical weighing series reproduced from [12]. 

The total mass grows smaller with each series of weighings, with the unknown 

weights in any series being determined in terms of the known values from the 

previous series. At the top of the design, the reference standard can be accurate 

to within a few parts per billion, but as the series reaches levels of a milligram the 

resulting calibration is accurate to only a few parts in ten thousand.
Vol. 9 No. 1 • March 2014 NCSLI Measure J. Meas. Sci. | 31

SPECIAL FEATURE

efforts of the Avogadro project it is possible to say something about 

the mass of atoms using a silicon sphere roughly 90 mm in diameter, 

but this is a scaling trick based on the ability of a silicon crystal to 

act as a “low-noise amplifier” [13]. Crystalline uniformity allows the 

Avogadro project to traverse some 27 orders of magnitude using what 

can creatively be interpreted as a weighing series where the summation is of an Avogadro sized number of single atom masses. Unfortunately, this atom amplification trick has yet to be exploited at other 

size scales, owing to the extreme technical difficulties associated with 

obtaining absolutely pure samples, processing the samples so as to 

know the fraction of the various isotopes in the sample (those neutrons 

add up!), controlling the purity of the surface of the crystal (everything 

tends to form an oxide, or as the musician Neil Young put it, “Rust 

Never Sleeps”), and then accurately measuring the volume of the bulk 

sample in terms of the meter. For now, suffice to say that the technical hurdles associated with producing Avogadro style silicon masses, 

other than the handful of copies of the IPK already in existence, are 

unlikely to be surmounted by the private sector, and that the NMIs capable of such a feat are still squarely focused on understanding things 

at the size scale of a kilogram.

What we have is an existing system of units that can get from apples 

to atoms, but that for a variety of technological reasons struggles to 

provide accurate, precise determinations of mass at scales of micrograms and below, at least until the submultiple of the IPK is back to 

being an individual silicon atom. On the other hand, the proposed new 

system has the potential to connect the mass of atoms to the most accurately measured quantity in recorded history, atomic frequency, via 

Compton clocks and atom recoil experiments. Questions remain however, as to what good this might be for measuring apples, and whether 

or not a revised SI offers us any hope of improving the certainty with 

which masses on the order of micrograms can be determined. In other 

words, it’s time to consider how the forces of Newton can be accurately expressed in terms of the energies of Planck.

4. Balancing Newton in Terms of Planck

In Section 2 we learned that mass can be linked to frequency at the 

atomic scale as a consequence of the fundamental particle wave duality 

described by quantum mechanics. This presents the exciting possibility 

of fixing the Planck constant and realizing the mass of individual 

atoms with the resolution, stability, and accuracy of atomic clocks 

using the technologies developed for atom recoil and Compton clock 

experiments. In Section 3 we learned how the Avogadro project has 

scaled the existing system of mass to the level of atoms through an 

amazing international effort that capitalized on the natural perfection 

of silicon crystals to effectively count the atoms in a kilogram. At the 

end of Section 3, we grappled with the technological difficulties of 

duplicating this feat to realize the unit of mass at other nontraditional 

scales, such as at the level of micrograms and below. This sets the 

stage to consider other methods for scaling mass, or how the forces of 

Newton can be accurately expressed in terms of the energies of Planck.

4.1 Balances for Comparing Mass

When we think of mass in the present system of units, the concept 

is firmly rooted in a Newtonian viewpoint, where force equals mass 

times acceleration. In principle, if we can apply a known force to an 

object, mass can be determined simply by recording the resulting acceleration. However, known forces are hard to come by, so instead 

of realizing mass from a known force and a measured acceleration, 

the unit of mass is disseminated through a relative measurement that 

compares the gravitational mass of an object to the gravitational mass 

of the IPK, taking advantage of the weak equivalence principle which 

allows us to treat gravity as acceleration.

Gravimetric mass measurement is carried out using comparison 

weighing (see, Jabbour et al. [12]) where a balance compares the effect of gravity on an unknown artifact X to the effect of gravity on 

the IPK, or a suitable reference copy, R. In mathematical terms, the 

comparison (accounting for the buoyancy of air) can be described as

CC! = mm!gg! − ρρ!VV!gg!, (10a)

CC! = mm!gg! − ρρ!VV!gg!, (10b)

where CR and CX, mR and mX, VR and VX, and gR and gX are the balance 

reading, the mass, and the volume of the reference R and the unknown 

X, respectively, gR and gX are the local accelerations of gravity at the 

centers of gravity for the reference R and unknown X, respectively, and 

ra refers to the air density during the measurement. As written here, 

the comparison is assumed to be aligned along the axis of gravity, and 

this alignment is assumed to remain the same during the comparison. 

Although the acceleration of gravity can be measured accurately in 

terms of c and Dν(133Cs)hfs [15], the goal for now is simply to compare 

the mass of one artifact relative to the other, not to actually quantify 

the forces. In most cases, gR and gX are sufficiently equal to be factored 

out of the comparison. Then the relative magnitude of the forces is 

obtained by considering the difference between CR and CX, or

mm! = mm! − ρρ! VV! − VV! − CC, (11)

where C = CR - CX, and the air density is assumed to be constant during 

the weighing. This is an example of the most basic weighing process 

in the present system of units.

Gravimetric mass metrology is simple and robust but utterly and 

completely wed to one artifact in Paris. As such, it involves the rather 

inconvenient and complicated shipping of artifacts around the globe in 

order to facilitate the necessary chain of comparisons back to the only 

actual realization of mass, the IPK. The miraculous achievements of 

the Avogadro Project aside, the existing artifact mass definition also 

scales poorly, as we saw in Section 3.1. This presents a far more genuine problem that looms for nascent nano and molecular technologies. 

But perhaps the most glaring flaw is that the underlying physics is 

showing its age. Gravimetric mass metrology enshrines notions leftover from a previous century that are now known to be patently false: 

that the mass of an object depends only on the number of atoms present (this ignores binding energy and relativistic effects, an inconsequential thing to Newton, a big thing to Einstein and Planck) and, 

more practically, that the mass of a macroscopic hunk of metal washed 

and handled by human hands is somehow a constant of the universe. 

So what is the alternative?

4.2 Balancing the Electrical and Mechanical Units:

the Watt Experiment

To ease ourselves into the complex world of trying to redefine the unit 

of mass, it helps to dig a little deeper into some of the messy finicky 

details of balances used to execute a gravimetric weighing, and to appreciate that such balances are at the heart of establishing a consistent 
32 | NCSLI Measure J. Meas. Sci. www.ncsli.org

SPECIAL FEATURE

set of units of measure. For example, consider the compensation balance shown schematically in Fig. 3. 

Compensation balances are so named 

because the net downward force exerted on the 

weighing pan is compensated by an upward 

electromagnetic force (or electrostatic, or 

optical, as we’ll see later in Section 5) that 

cancels the downward force to maintain a null 

position. Early on in the making of precision 

balances it was recognized that one of the 

chief limitations of mechanical balances was 

the fact that motion of the balance mechanics 

inevitably leads to hysteretic restoring 

forces, producing weighing errors. As shown 

in Fig. 3, a solution was to introduce a 

compensating force using a feedback control 

system to monitor the balance deflection. The 

compensation control acts to maintain null, 

minimizing deflections and the erroneous 

forces they contribute to the weighing, by 

controlling the current i that flows in a coil 

suspended in a magnetic flux f. Conveniently, 

this compensation current also provides a 

signal proportional to the mass that can be 

used as the readout Cx.

For the gravimetric weighing experiments 

described in the last section, absolute measurements of the gravity, the current in the 

coil, and the magnetic flux are unnecessary, 

since the current in the coil merely serves as a 

readout signal for comparing masses, the other 

variables being constant on the time scale of a 

typical comparison. But what if we did want to 

know the absolute value of the compensating 

force in terms of the electromagnetic quantities? Could we use this as a known force to 

directly realize the unit of mass? Can a balance 

be constructed so that the compensation force 

can be measured in terms of a set of fundamental constants? The answer is yes, of course, 

but it takes a good bit of explanation, and the 

devil is in details that I must mostly brush over 

here (detailed technical accounts abound, and 

recent examples are available by some of the 

pioneers of the field [16, 17]). 

Imagine that the balance of Fig. 3 has been 

constructed so that the coil and the balance 

mechanics result in the magnetic and gravitational forces aligning perfectly to cancel 

one another along the same vertical axis in 

a vacuum chamber. With the buoyant forces 

conveniently removed by the vacuum pump, 

the resulting balance of forces can be written 

explicitly as

ii !"

!" = mm!gg, (12)

where the expression on the left hand side of 

the equation is recognized as a scalar component of the more general Lorentz force law, 

realized experimentally by painstakingly designing and fabricating a magnet system and 

balance mechanics that make the effects of 

the other force components effectively zero. 

Now, if the current and magnetic flux 

gradient responsible for the compensating 

force can be measured, then the compensating 

force could in principle be known independent 

of the mass. Clearly, knowing this force 

and the local gravity, we could realize the 

mass of objects placed on the weighing 

pan directly from the current and gradient 

measurements. No need for IPK. Everyone 

who had such a balance could realize the 

unit of mass themselves, with traceability to 

the SI through electrical quantities, length, 

and time (ignoring the magnetic flux, which 

will conveniently disappear a little later). In 

principle, the scaling of mass in this scheme 

would be limited only by the accuracy 

with which we could realize and scale the 

ampere, perhaps offering a way to avoid the 

accumulation of uncertainty associated with 

subdividing the IPK. As we might guess, 

this turns out to be quite a bit trickier than it 

would seem at first glance.

Intuitively, a force should be a force, 

whether electromagnetic, or gravitational, 

and the “electrically” determined value of a 

mass should equal that determined through 

gravimetric comparison to the IPK. In setting 

up international measurement systems (well 

before we had electrical units it turns out, see 

Maxwell’s paper presented to the Royal Society [18]) it has always been understood that 

the electrical and mechanical units of measure must be defined such that like quantities 

balance, regardless of their phenomenological origin. This is a big part of what it means 

to have a consistent set of units. However, 

the price for this consistency is that the base 

electrical unit, the ampere, is actually defined 

in terms of mechanical force, which derives 

from the IPK. So an electrically determined 



mR

R Voltage readout, Vw

Coil position Current source measurement

system Feedback

i

z, u

M

g

Figure 3. Above are the basic elements of a compensation balance. In this cartoon, the balance is an equal arm type where the additional weight of the moving 

coil system that applies the electromagnetic compensation force is offset by a 

tare mass M. When a mass, mR is placed on the balance, the beam starts to tip 

about the fulcrum, causing the coil to move. A highly sensitive position sensor 

detects the motion immediately, signaling the feedback control to adjust the current and restore the coil position to null, effectively canceling any motion of the 

balance beam. The compensation current is proportional to the mass, and is read 

by measuring the voltage drop across the resistor, R.
Vol. 9 No. 1 • March 2014 NCSLI Measure J. Meas. Sci. | 33

SPECIAL FEATURE

mass is ultimately linked to a mechanically determined mass, at least 

in the present system of units, and this would seem to have made for 

a short and circular trip. The proposed electrical mass is simply a mechanical mass in disguise, at least as far as the present system of units. 

Nevertheless, the idea of an electrically realized mass is still attractive. 

Particularly since it turns out there are ways to define an ampere that 

tie its value to fundamental constants, rather than the IPK. As we will 

see, when all is said and done, a great deal is possible if we embrace 

a system of units based on a set of fixed fundamental constants where 

the Planck constant is a constant (a familiar theme!).

The development of balances to realize the electrical units and 

establish the consistency of the measurement system has a long 

history (see [19] for a history of the development of electrical 

standards) beginning with Maxwell in the 1860s (e.g., [18] again) 

and culminating in the first absolute value of the ampere in 1934 

[20]. Over the years, this so-called ampere balance experiment has 

evolved, and in 1975 a two-part experiment to relate the electrical 

and mechanical units through a virtual power measurement was 

conceived [21]. This experiment, known as a watt balance, began as 

an attempt to overcome the limitations in measuring the geometrical 

quantities necessary to establish the magnetic flux gradient in 

ampere experiments. 

In addition to the weighing mode, a watt balance uses a second 

velocity mode, where the mass is removed and the coil is translated 

in the same magnetic flux at a velocity u. The motion of the coil 

induces a voltage across the now open terminals of the coil (feedback 

control is disconnected) such that

VV = uu !"

!" , (13)

which is simply Faraday’s law of induction. 

If the magnetic field present during the experiment is the same for 

both the weighing and velocity modes, the results can be combined to 

yield a virtual balance between mechanical and electrical power, so 

that the basic measurement equation for a watt balance becomes

mm!gggg = iiii. (14)

Watt balance experiments make it possible to establish the 

consistency of the electrical units (i.e., the volt and ampere) and 

mechanical units (i.e., the kilogram, meter, and second) without 

requiring the determination of the magnetic flux gradient, which 

improves the realization of the ampere in terms of the IPK.

In the 1980’s the question of unit consistency in watt balance experiments revolved around assuring that the representations of the 

electrical units (historically banks of batteries and resistors) were 

maintained and adjusted so that the balance of these virtual powers 

was as accurate as possible, with the understanding that the electrical 

units would always adjust to insure consistency, since length and time 

standards used in characterizing the mechanical power were already 

based on fixed fundamental constants, and the unit of mass, the IPK, 

is by definition a fixed quantity. It is worth reiterating that in the existing system of units, measuring the “electrical” force independent of 

IPK simply isn’t possible, because the units used to quantify electrical 

measurements are themselves based on the IPK as determined in this 

experiment! This Gordian knot of the electrical and mechanical units 

loosened a bit in 1990 (or got more complicated, depending on your 

viewpoint), when it became practical to measure electrical quantities 

in terms of quantum effects.

In 1990 it became common practice to measure both voltage and 

resistance in terms of entities thought to be fundamental constants 

themselves, namely the charge on the electron, e, and the Planck constant h. Voltage and resistance standards based on the Josephson effect 

(inversely proportional to the Josephson constant, KJ = 2e / h) and the 

quantum Hall resistance (proportional to the von Klitzing constant, 

RK = h / e2

) became available, and suddenly it was possible to measure 

voltages and resistances using very precise quantum phenomena that 

were codified into a system of conventional units that exist in parallel with the SI (that’s right, the electrical units are effectively outside 

the SI!). The traditional basis of measurement, where the mechanical 

quantities define the electrical quantities, began to be called into question [22] and the stage was set to finally pursue a dramatic redefinition 

of the kilogram by fixing the value of the Planck constant.

4.3 Measuring the Planck Constant

As it stands now, in order to balance the watt balance equation, we 

must effectively test the equivalence, or consistency, of the SI and 

the quantum electrical effects, leaving the Planck constant as a free 

variable, since by definition, IPK is invariant (the Avogadro constant 

offers another approach, but that’s another story). Here’s how it works 

in terms of the physics of the electromagnetic compensation balance 

we have been discussing. 

The weighing current i is observed using the voltage drop across a 

resistor, with the voltage measured by comparison to a quantum invariant, the Josephson effect, so

VV! = !!

!!

, (15)

where fw is a microwave frequency used to excite the Josephson 

voltage junctions. The resistance is likewise measured in terms of 

a quantum invariant. In fact, we can use a so-called quantum Hall 

resistance standard, where the resistance of the device is expressed in 

terms of the quantum Hall effect, so

RR = !!

! , (16)

where n again is an integer, this time having to do with the quantization 

of the quantum Hall resistance standard (which deep in the theory 

links up with quantization of energy, of course). 

In electrical metrology, we routinely use Ohm’s law V = iR to 

realize, or measure, a current from the quantum standards. In this case 

the current during weighing is

ii = !!

! = !"!!!!!

= !!!"

! . (17)

While the voltage induced (again measured in terms of the Josephson effect) during the velocity mode is

VV = !!

!!

= !!!

!! . (18)

Substituting for V and i, and solving the watt balance equation for 

the Planck constant, we find

h = !!!!"

!!!!! , (19)
34 | NCSLI Measure J. Meas. Sci. www.ncsli.org

SPECIAL FEATURE

where the result, and the complex chain of measurements necessary to 

achieve it, provides a rationale for understanding the forces of Newton 

(mRg) in terms of the energy of Planck. Stepping back for the moment, 

let us consider the relative uncertainties believed to be associated with 

the standards used to measure the Planck constant in this fashion, since 

they will determine the ultimate resolution of such an experiment.

The quantum electrical effects used as our standards for measuring 

the virtual electrical power are known to be exceedingly precise and 

stable. For instance, a review of Josephson comparison results [23] 

found comparisons achieving a precision of a few parts in 1011, as 

well as no evidence contradicting that the relation KJ= 2e / h is exact 

and universal. The situation with the quantum Hall effect is similar, 

with a recent comparison between a conventionally fabricated GaAs/

AlGaAs heterostructure device and a new graphene device in agreement to within 8.6 parts in 1011 [24]. A recent theory paper [25] finds 

the relation RK = h / e2

 is not exact, but suggests a whopping correction 

of perhaps 2 parts in 1025 may be required at the magnetic field levels 

employed for metrology!

On the mechanical power side of things, our ability to measure the 

local gravity and to likewise measure the velocity of a moving coil 

relies on standards of length and time already exceedingly well expressed in terms of fundamental constants, c and Dν(133Cs)hfs. Optical 

frequencies (and hence wavelengths) derived from the cesium standard can be stable to a few parts in 1014. The IPK has no uncertainty, 

but the IPK cannot be used directly, so one must consider the uncertainty of available artifacts, which is typically on the order of 4 to 7 

parts in 109

. Furthermore, although there is no systematic uncertainty 

stated for the IPK, this strains credulity, since the IPK is well known 

to have drifted by as much as 50 parts in 109

 with respect to its copies.

If the Planck constant is a universal constant, then its value must 

correspond to some exact fixed number, and by performing enough 

watt balance experiments of ever increasing precision, you eventually 

converge on its exact value. Ultimately, such an endeavor can go only 

as far as the precision of the least precise quantity in the experiment, 

limited in the end by the stability of the standards used to make the 

measurements. As the procedures surrounding watt balance experiments become increasingly refined, it is more and more evident that 

our knowledge of the Planck constant will ultimately be limited by the 

mass standards used in the evaluation! If this sounds like the situation 

I described for the speed of light circa 1983 at the beginning of the 

article, it is because it is entirely analogous.

5. A New System of Mass Where Scaling can be 

Accomplished with “Scales”

At this point in the discussion, I will take the liberty to declare the 

Planck constant a constant, and begin exploring how we realize a mass 

from an instrument, in anticipation of a redefined system of units. In 

other words, from here on out, I am speaking as if we are working in 

the new SI. 

For starters, it is a simple matter to turn the watt balance equation 

for the Planck constant around and write the value of any unknown 

mass as

mm! = !!!!!!

!!" . (20)

Here then is a method to weigh apples in terms of fundamental 

constants. But it is not limited to apples. We are at liberty to build 

other similar balances, whatever size we want, and to use them to 

derive a realization of the unit of mass from the Planck constant as well 

(think about the atom recoil experiments way back at the start of this 

article). And the balances need not be limited to electromagnetic force. 

The force could be electrostatic, or even optical. All that will change 

is the nature of the compensating force and the methods required 

to characterize its strength and direction in terms of the quantum 

electrical standards. The only limiting factor is our imagination. The 

watt balance is simply a special case of a balance between forces 

due to gravity, and those arising from electromagnetic fields where 

the characterization of the field gradient has been made accurate by a 

clever use of the reciprocal nature of Maxwell’s equations. So far, the 

most successful balances have been demonstrated for measuring at the 

scale of apples, but there is already evidence that balances could be 

constructed to realize mass at the scale of say, apple seeds.

5.1 Electrostatic Force Balances and an Electrically 

Derived Milligram

The challenge for creating new balances to derive a realization of 

mass is that they must be constructed so that all the fields and forces 

align and cancel along the weighing axis with sufficient accuracy to 

maintain the precision we have come to expect from mass measurements. At the level of apples, the challenge is to do everything at the 

level of a few parts per billion, in order to be competitive with the 

existing SI. At the level of apple seeds, however, the present mass system struggles to achieve accuracy better than a few parts in 104

. Here 

is a flaw in the existing system that could be addressed more directly 

in the new system.

Anecdotal evidence suggests there is need for improvement in the 

measurement of ever smaller masses, particularly to serve the pharmaceutical industries. At the 2013 NCSLI meeting in Nashville, Tennessee, a panel of mass experts from industry and governmental organizations suggested that manufacturers are increasingly interested in the 

accurate weighing of samples as small as a few tens of micrograms, 

with one paper presented on the issues of assessing balance performance at the low end of readability [26] and a second presentation that 

focused on proposed changes to the ASTM E617 standard, which includes the addition of sub-milligram weights [27]. Unless something 

changes, we appear to be at the beginning of a new era of ever shrinking mass artifacts. This perceived pressure from the bottom of the 

mass scale may be causing ripples at the top, as the revisions to ASTM 

E617 also include a new ultra-high precision weight class, perhaps 

aimed at the accumulation of uncertainty discussed in Section 3. 

Extrapolating into the future, if we factor in the usual accumulation 

of uncertainty from all the weighing series between you and the IPK 

at the level of a microgram, it is quite plausible that we will need to 

improve the relative uncertainty of a kilogram working standard by 

perhaps two orders of magnitude if we want to quantify the mass of 

a microgram sample with even modest uncertainty. Further, we must 

somehow create a series of weighing designs and artifacts that can be 

reliably employed to calibrate instruments without risk of bias due to 

contamination and environmental factors, such as charged particles 

(think of the lint accumulated in your dryer). A microgram is roughly 

the size of a speck of dust, so the potential for contamination is high. 

For that reason, technical approaches that minimize or eliminate the 

need to handle such tiny mass artifacts are attractive. A balance that 

can directly read the traceable mass of sub-milligram samples on the 
Vol. 9 No. 1 • March 2014 NCSLI Measure J. Meas. Sci. | 35

SPECIAL FEATURE

factory floor might ameliorate some of these 

problems, particularly if it can be calibrated 

using only reference to robust intrinsic electrical, length, and time standards.

By the end of the 1990’s, NIST was well 

aware that a watt balance experiment effectively 

enables one to compute a compensation force 

of 10 N with reference to intrinsic standards to 

within less than 50 parts per billion. Perhaps, 

we reasoned, a smaller balance could be 

constructed to realize a micronewton to within 

a few parts per million, since in principle all 

of the underlying metrology scales with little 

loss in relative accuracy (e.g., [28]). In other 

words, can a balance weigh milligrams with 

more accuracy by tracing to the intrinsic 

electrical, length and time standards than 

through conventional direct sub-division of 

an artifact kilogram? We were not seeking a 

mass standard at the time. What we sought was 

a known, calculable force that could be used 

as a reference for a variety of nanotechnology 

instruments, such as atomic force microscopes. 

We were trying to avoid using a mass, because 

we believed that at the levels of force we 

needed to eventually reach, nanonewtons and 

below, mass artifacts would never be accurate, 

or stable enough to serve as a deadweight 

reference. We were also cognizant that many 

new sensors were based on so-called Micro 

Electro Mechanical devices, or MEMs, where 

the readout and actuation were capacitive, 

and electrostatic, respectively. It seemed to 

us at the time that demonstrating methods for 

the accurate realization of electrostatic forces 

might prove useful to this community. The 

experiment which we eventually constructed 

is known as the NIST Electrostatic Force 

Balance, or EFB for short [29]. 

The EFB, shown schematically in Fig. 4, is 

used to compare mechanical and electrostatic 

forces that act on a cylindrical, variable capacitor in a compensation balance scheme very 

similar to the one described for a watt balance. Those familiar with electrical metrology in the days before the Josephson volt will 

recognize this as a variant on the volt balance. 

One electrode of this capacitor is attached to 

the weighing pan, and this moving electrode 

nests inside of another electrode fixed to the 

stationary balance frame. To a first approximation, the balance of forces acting on the 

suspended electrode can be expressed mathematically as

( ) gmV dz

dC Fe = = R

2

2

1 , (21)

where Fe is the electrostatic force, dC / dz is 

the gradient of capacitance between the two 

cylinders, with dz being the differential displacement of the moving electrode along the 

balance axis and dC the corresponding differential change in capacitance, V is a voltage 

applied between the two electrodes, and mRg

is the mechanical force exerted on the weighing pan due to an unknown mass mR. Motion 

of the electrodes with respect to one another 

is measured along the axis z, which, as in the 

watt balance experiment, must be painstakingly aligned to the motion axis and to gravity, so that all the forces are acting along the 

same direction (one trick we use to minimize 

the off-axis forces is to make the capacitor as 

perfectly cylindrical and symmetric as possible, so that electrical forces in the x and y

directions are effectively zero). The experiment is performed in vacuum to improve the 

accuracy of the electrical and length measurements, and has the added benefit that it 

eliminates the buoyant forces, which would 

otherwise complicate the determination of an 

absolute mass.

As in the discussion of compensation balances in section 4.2, we observe that the electrical quantities used to characterize the compensating force can be measured in terms of 

intrinsic standards. The voltage is traceable to 

the Josephson constant, and the capacitance is 

traceable to the von Klitzing constant. So the 

compensating force, expressed in the revised 

system of units, can be known without any 

reference to IPK. As a result, we can realize 

the mass of objects placed on the weighing 

pan. Here’s how. 

In the first step, or calibration mode, the 

switch (SW in Fig. 4) is set so that the capacitance meter records the capacitance between 

the suspended and fixed electrode. The capacitance gradient of the instrument is measured in order to calibrate the instrument’s 

ability to convert, or transduce, electrical 

voltage into mechanical force. This is done 

by simply displacing the suspended electrode 

a measured distance with respect to the fixed 

electrode (SW connects the position controller to another pair of electrodes to swing the 

balance arm, as shown in Fig. 4) and then recording the capacitance as a function of the 

displacement. Of course, both the displacement and the capacitance must be measured 

using instruments that have been calibrated 

in terms of appropriate intrinsic standards. 

We can employ off the shelf capacitance 

meters and laser interferometers (such as the 

Displacement Measuring Interferometer, or 

DMI in Fig. 4) and still improve traceability 

compared to an artifact mass approach. This 

is because commercial instruments for electrical and length metrology now routinely 

transfer their respective dimensional quantities with accuracy and stability of a few parts 

mT

mR

Feedback

null/position

control

Voltage source

Capacitance meter

Lo High

Voltage source SW

SW

Laser

z

DMI

g

Volt meter

Figure 4. A schematic illustration of an Electrostatic Force Balance for the 

calibration of masses in terms of intrinsic electrical, length, and time standards. 

Operation is described in the text.
36 | NCSLI Measure J. Meas. Sci. www.ncsli.org

SPECIAL FEATURE

per million, reliably traced to intrinsic standards across a very large 

dynamic range.

For the second step, the instrument is operated as a compensation 

balance and used to weigh objects of unknown mass. Feedback control is used to adjust the voltage on the electrodes so that the signal 

from the interferometer stays constant as the unknown mass is placed 

on and off the balance. The voltage necessary to maintain the balance 

at this null position serves as the readout. Since the voltage is ultimately used to compute the compensating force, it must be measured using 

an instrument that has been calibrated or is traceable to the appropriate 

intrinsic standard, a Josephson volt. Again, high-quality multimeters 

are often calibrated in this fashion, and are available off the shelf with 

an uncertainty of a few parts per million. 

The third and final step converts the voltage readout to mass using 

the measured value of gravity (which can be estimated using global 

positioning coordinates to within a few parts per million using the 

US Geodetic Survey website [30]), the previously measured capacitance gradient, and the equation for the balance of forces. All of the 

references used to support the measurement are readily available and 

accurate to within a few parts per million (even in the present SI, it 

turns out), so that the combined uncertainty of the references is potentially quite low compared to the uncertainty accumulated in trying 

to subdivide from the IPK. Interestingly, there is no reason you could 

not employ this approach in the present system of units to leap frog 

the accumulated uncertainties. But an electrically derived mass has 

generated little interest among the metrology community; apparently 

due to the conceptual break with a century’s worth of experience using artifacts. 

The scaling of mass using an EFB is limited by the accuracy with 

which we can measure the local gravity and how well we can scale 

the volt, meter, and the farad. The balance mechanism and electrodes 

are of course also limiting factors, but their performance can, in principle, always be designed to achieve a desired sensitivity. Confounding forces, due to surface potentials, stray charge, magnetic interactions, mechanical hysteresis and the like must also be accounted for, 

and mitigated, as necessary. Nevertheless, in 2006, NIST succeeded 

in weighing a 20 mg artifact in terms of the second, volt, meter, and 

farad (ohm) with a combined relative uncertainty of a few parts in 

105

 [31]. So at the scale of a milligram, traceability of mass to intrinsic standards has already demonstrated precision that competes with 

the existing mass scale, and points the way for instruments that could 

weigh samples into the microgram regime with traceable accuracy, 

without the need to manipulate ever shrinking mass artifacts.

5.2 Back to Where We Started: Mass and Frequency

Compton clocks, watt balances and the NIST EFB experiment make 

a strong case that a new class of instruments can be developed to 

scale mass using traceability to electrical, length, and time standards 

that have their basis in fixed values of fundamental constants, scaled 

by frequency and integer multiples of the Planck constant. Broadly 

speaking, this situation should not surprise us. The view of modern 

physics is that the amount of stuff in the universe is not discretized in 

terms of ever smaller pieces of stuff, but in terms of energy, which is 

related to frequency through integer multiples of the Planck constant. 

Viewed in this light, it is perhaps inevitable that our system of measurement evolve to one where all dimensional quantities are linked to 

fixed constants through the observation of various frequencies. 

Regarding mass, sometimes the link to frequency will be direct, as 

in the Compton clock experiments described at the beginning of the 

article. In other instances, the link will be less obvious, as in the balance experiments, where a variety of frequencies come into play as we 

quantify electrical measures in terms of the Josephson and quantum 

Hall effects. In the future, and between the scale extremes represented 

by Compton clocks and EFB’s, from say yoctograms to micrograms, 

there will be opportunities to measure mass using shifts in the natural 

frequency of resonant oscillators ranging from trapped ions, to micromechanical cantilevers. Again, even in these systems, the ability to 

conjure up a known force will prove useful. Here’s how it might work.

Recently, accurate forces on the order of piconewtons have been 

demonstrated in experiments using both electrostatics [32], and photon 

momentum [33], where the forces were ultimately traceable to intrinsic standards. The forces in these experiments were used to calibrate 

the absolute force sensitivity of mechanical springs using Hooke’s law, 

or F = kx, where k is the unknown spring constant or sensitivity, x is 

the spring displacement as measured using an optical reference to the 

speed of light, and F is a known electrostatic force or photon pressure 

measured in terms of appropriate electrical and length standards. The 

sensors of these studies were intended to measure atomic forces, but 

could be used to sense the mass of very small objects. For example, 

such sensors can be employed to directly weigh tiny particles, using 

the sensor as a spring balance or scale. Of course, it is also possible to 

measure the sensor natural frequency by observing its thermomechanical noise spectrum. Knowing the natural frequency, we can back out 

the effective inertial mass of the sensor using the relation

mm = !

!! , (22)

where w is the natural frequency of the oscillator in radians per second. 

Having identified the oscillator mass, stiffness, and natural frequency, 

it is possible to measure the mass of objects placed on the oscillator 

simply by noting the resulting shift in natural frequency. This is the 

approach taken by quartz crystal microbalances, and by new sensors 

being developed to “weigh” cells, viruses, and even single molecules. 

At present, the traceability of such measurements is, at best, ambiguous. Accurate force, traced to intrinsic standards, offers a path towards 

clarity as new technologies develop to explore the measurement of 

mass at ever smaller scales.

6. Conclusions: Redefinition and the Outlook for the 

Near Future

The redefinition of mass in terms of the Planck constant is still in the 

future, and it remains to be seen just when plans for revision of the 

SI will come to pass. For the moment, all eyes are focused on results 

emerging from the various watt balance experiments and the corresponding efforts of the International Avogadro Collaboration (IAC). 

The question on most people’s minds is whether or not the measured 

Planck values will agree to within 50 parts per billion, and whether 

or not one (or more?) of the experiments can succeed at measuring 

the constant with relative uncertainty below 20 parts per billion. 

These conditions have been suggested by the International Bureau of 

Weights and Measure’s Consultative Committee on Mass as a benchmark to achieve before redefinition should proceed, and a variety of 

groups around the globe are working towards satisfying these requirements. We will know better the success of these ongoing endeavors 
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within the next year, as many of the groups have plans in place to 

announce new numbers in 2014. 

Assuming the CCM benchmarks are met before 2015, and redefinition moves forward, the SI could be revised as early as 2017 or 2018. 

Clearly, watt balances to realize the unit of mass at the level of a kilogram will be few and far between, and other instruments to realize the 

unit of mass at other scale points are unlikely to exist. How then will 

mass be disseminated?

It is helpful to step back and remind ourselves that in the existing system of mass we have but one realization of the unit of mass in the entire 

world (one mass to rule them all!). If redefinition is successful, we will 

immediately have at least three groups capable of realizing the unit and 

calibrating artifacts for use as primary standards. Plans are in place to 

accumulate a pool of these calibrated artifacts at the BIPM. The pool 

will help average variations that will inevitably occur among groups 

capable of deriving the unit of mass from the Planck constant, while 

providing the opportunity for new groups to develop and verify their 

own methods through comparison to the pool value. In other words, the 

BIPM will continue to be a source of mass calibration for those working to realize the unit, and those who are unable to derive the unit themselves. In fact, other than at the highest level of realization, the system 

can persist as it always has, by using artifacts, weighing designs, and 

compensation balances to disseminate and subdivide the unit of mass.

The day after redefinition, if all goes according to plan, nothing has 

to change as far as the world of mass dissemination. And yet everything can change with regards to scaling of the unit, since, as we’ve 

seen in this article, it will be possible to derive a unit of mass using whatever means is convenient and at whatever scale point desired 

by using intrinsic standards of length, time, and electrical quantities. 

More broadly, the entire system of measurement will finally be tied 

to a set of physical invariants, creating a basis for quantifying the 

physical universe across all dimensional scales. The scientific appeal 

is obvious, and perhaps you will agree, the ability to use instruments 

to realize the unit of mass and thereby weigh everything from atoms 

to apples has the potential to grow a new sector of metrology, where 

companies vie to produce economical solutions that accurately quantify ever smaller objects in terms of a vital, new SI.

7. Acknowledgements

I would like to thank NCSLI and Measure (in particular Carol Hockert 

and Michael Lombardi) for encouraging me to share my thoughts about 

this topic. I also thank my numerous colleagues in the watt balance 

and Avogadro communities for their conversations and the insights 

they have shared with me over the last decade. I wish to express a 

special thanks to Edwin Williams and Clayton Teague, mentors who, 

perhaps unwittingly, pointed me down this path so many years ago. 

Helpful reviews of this text by Carl Williams, David Newell, Stephan 

Schlamminger, and Peter Mohr were deeply appreciated.

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