The weight a frequency can find

The third of itself a spring carries found that a spring vibrating with a mass on its end behaves as though a third of its own mass had been added to the load, because the coils near the fixed end hardly move and those near the load move almost as much as it does. The share is set by how much each part of the spring moves in the vibration, squared and added up. It ended by pointing out that this bookkeeping had become an instrument: if adding mass to a vibrating thing lowers its frequency in a known way, then measuring the frequency weighs whatever was added.

That instrument turns out to be the most sensitive balance there is, and its sensitivity has a strange property that no ordinary scale shares. The less the instrument weighs, the better it gets.

Half the fraction added #

Any oscillation near the bottom of a smooth energy well is, to the first approximation that matters, a mass on a spring, with frequency $f = \tfrac{1}{2\pi}\sqrt{k/m}$. Add a small mass $\Delta m$ and the stiffness is unchanged; the frequency falls by

$$\frac{\Delta f}{f} = -\frac{\Delta m}{2m}.$$

A one per cent heavier load lowers the note by half a per cent. The rule is linear for small additions, the exact square root differing from it by a quarter of the square of the fraction, and the frequency is the most precisely measurable quantity in physics: a counter and a stable reference clock read a frequency to parts in a thousand million in a second. A resonator is therefore a converter from mass to the quantity that can be measured best.

Two things in the rule deserve attention. The $m$ in it is not the resonator’s mass but its effective mass — the share of its mass that takes part in the vibration, weighted by how much each part moves, exactly the kind of share the spring essay computed. And $\Delta m$ is not simply the mass added but the mass added weighted in the same way, by where it sits.

Where it lands #

A small beam clamped at one end and free at the other — a cantilever, the diving board of microscopic balances — bends in its lowest mode with the tip moving most and the root not at all. Its mode shape is computed exactly from the equation for a bending beam, and measured against the tip’s motion, the effective mass of every one of its modes comes out at exactly a quarter of the beam’s. A speck landing at the tip adds its whole mass to the vibrating motion, and the first mode’s frequency falls by twice the speck’s mass as a fraction of the beam’s. The same speck landing halfway along adds only the square of the mode’s displacement there, about a tenth, and falls by a tenth as much. Near the clamp it adds nothing that the mode can feel.

So a single frequency shift does not say how much landed. A light speck at the tip and a heavier one further in give the same shift. On a balance large enough to place the load by hand this does not matter, since the load goes where it is put. On a resonator a few micrometres long, onto which molecules drift from a gas or a liquid one at a time, it is the central difficulty — exactly the problem that the mass and where it sits found for the moment of inertia of a rolling object, where the same mass counts for more the further it is from the axis.

Two notes for two unknowns #

The second bending mode of the beam weights positions differently. It has a node about four-fifths of the way out, where it does not move, and a large swing halfway along, where the first mode hardly moves at all.

The ratio of the two shifts depends only on where the speck sits and not on how heavy it is, so measuring both frequencies at once fixes the landing point from their ratio, and the mass from either one. Some ratios occur at two positions, on either side of the node, and a third mode removes the ambiguity. The two pendulums that swap found that a structure’s modes are separate motions each with its own shape; here the separate shapes are separate weightings of the same speck, and the set of them is a coordinate system.

In 2012 Michael Roukes’s group in Pasadena used two modes of a doubly clamped beam a few micrometres long to weigh single protein molecules as they arrived, one at a time, from a spray — a mass spectrometer that reads each particle’s mass directly, rather than its ratio of mass to charge as every conventional instrument does. Each molecule announced itself as a pair of sudden frequency steps, and the ratio of the steps said where it had landed.

A film a fraction of an atom thick #

The oldest everyday use of the principle is on every vacuum coating machine. A thin disc of quartz, cut so that it vibrates in thickness shear — its two faces sliding back and forth past each other — rings at a frequency set by its thickness, a few megahertz for a plate a third of a millimetre thick, and quartz is piezoelectric, so the vibration is driven and read by electrodes on its faces.

A film deposited on the face rides along with the surface and adds to the vibrating mass, and since the plate’s mode is a standing wave across its thickness, a film that is thin compared with the plate simply makes the plate a little thicker. Günter Sauerbrey worked out the relation in 1959: the frequency falls by $2f^2$ times the added mass per area divided by the square root of quartz’s density times its shear stiffness. For a five-megahertz crystal it is 56.6 hertz for every microgram per square centimetre. A nanometre of aluminium, about four atomic layers, shifts it by fifteen hertz, and a frequency read to a tenth of a hertz resolves a few hundredths of an atomic layer.

Thinner crystals ring higher and respond more, as the square of the frequency, because a thinner plate is lighter and the same film is a larger fraction of it. Every machine that coats lenses, mirrors or the metal layers of a chip holds such a crystal beside the work and watches its note fall as the film grows, stopping the deposition at the frequency that means the right thickness. The same crystals immersed in liquid weigh the films of protein or polymer that settle on them, with the complication — that the shear that only reaches so far into a viscous liquid also drags on the face — that the liquid’s own viscosity must be untangled from the film’s mass.

A scale with no weight #

None of this needs gravity. The frequency measures the resistance of the mass to being accelerated — its inertial mass — and a resonator in orbit weighs exactly as well as one on the ground.

On a space station a bathroom scale reads nothing, and the loss of mass in the body’s muscles and bones over months of weightlessness is exactly what the doctors need to track. The crews of Skylab in 1973 were weighed on a chair mounted on springs: the astronaut strapped in, set it swinging, and the period was timed. The figure’s chair, with springs of 900 newtons per metre, swings with a period of about two seconds and changes it by eleven milliseconds for each kilogram of occupant. The calibration has to remove the chair and the part of the springs that moves with it — the third of the spring, again. What it cannot remove is the person’s own interior: blood and organs move a little relative to the skeleton, the body is not a rigid mass, and the method is good to a few tenths of a kilogram however precisely the swing is timed.

That such a device weighs at all is a statement about the fall that does not depend on what is falling. What the chair measures is inertia, and what a scale on the ground measures is the pull of gravity; that the two give the same number to parts in $10^{15}$ is the equivalence principle, and it is why a frequency in orbit can be read as a weight on the ground.

The one oscillator that cannot weigh #

There is a resonator for which the whole scheme fails, and it is the oldest one. A pendulum’s frequency is $\tfrac{1}{2\pi}\sqrt{g/L}$, with no mass in it. Load a pendulum bob with extra lead and its period does not change, because the restoring force is the bob’s own weight, which grows in proportion to the inertia it must move. The “stiffness” of a pendulum is made of the same mass that resists being moved, and the two cancel.

That cancellation is not a coincidence of the formula; it is the equivalence of gravitational and inertial mass, and a pendulum is an instrument for testing it rather than for weighing. Newton timed pendulums with bobs of gold, silver, lead, glass, sand, salt, wood, water and wheat, all hollow boxes of the same size so that air resistance was the same, and found their periods equal to about a part in a thousand. Friedrich Bessel repeated the test in 1832 to a part in sixty thousand. Every spring-based balance in this essay works because its stiffness comes from something other than the load’s own weight — the elasticity of steel, quartz or a carbon lattice — and so does not cancel against it. The pendulum and its small lie found the pendulum’s period nearly independent of its swing; it is exactly independent of its mass, and that exactness is why no pendulum has ever weighed anything.

A quarter, by counting motion #

Where does the quarter for a cantilever come from? The same place the third came from for a spring. The vibrating beam has kinetic energy spread along its length, each slice contributing its mass times the square of its speed, and the speed of each slice is the tip’s speed times the mode’s displacement at that slice. Adding the slices gives the tip’s speed squared times the beam’s mass times the average of the mode’s squared displacement — and for a clamped beam’s bending modes that average, with the tip set to one, is exactly a quarter.

The quarter applies to every bending mode of the cantilever, not only the first, because the higher modes, though they wiggle more, have their largest swing at the tip in exactly the proportion that makes the average come out the same. A speck at the tip therefore weighs equally on every mode, and that is why the ratio of shifts at the tip in the two-mode figure is exactly one. The effective mass is not a property of the beam alone: it is a property of the beam and of where the motion is measured. Measured at the midpoint, where the first mode moves only a third as far as at the tip, its effective mass would be nearly nine times larger, and a speck there would weigh correspondingly less. A balance’s calibration is a statement about where it is read. In the protein-weighing beams, clamped at both ends rather than one, the reference point is the middle, where the first mode swings most, and the same counting gives an effective mass of about two-fifths of the beam; the arithmetic changes with the shape, and the principle does not.

Smaller is better #

The limit on what a resonator can weigh is the smallest fractional frequency change it can resolve. Set the fraction to that limit and the rule gives the smallest detectable mass: twice the effective mass times the resolution.

The line is straight and has slope one. A resonator a million times lighter notices a million times smaller masses at the same fractional resolution. A tenth-of-a-gram quartz crystal, read to a part in a hundred million, notices two nanograms. A microcantilever of a nanogram notices twenty attograms. A carbon nanotube a few hundred nanometres long, weighing about $10^{-22}$ kilograms, is so small that it is fragile and noisy and can be read only to about a part in a hundred thousand, but that is enough: in 2012 Adrian Bachtold’s group in Barcelona reported a nanotube resonator that resolved 1.7 yoctograms, about the mass of a single proton.

No ordinary balance works this way. A beam balance weighing a speck must be lighter to be more sensitive too, but it must also be stiff, rigid and protected from draughts, and the improvement runs out. A resonator turns lightness directly into sensitivity because what it compares is not two forces but a mass against its own inertia, and the comparison is read through a frequency.

What sets the floor #

The fractional resolution is not a fixed number. It is set by noise, and the noise is set by temperature, by the resonator’s losses, and by how long the measurement is allowed to take.

A resonator at temperature $T$ jiggles with an energy of $kT$ in its mode, and that random motion makes its phase wander, so its frequency can only be read to a precision that improves with the square root of the measurement time, with the quality factor $Q$ of the resonance, and with the ratio of the energy deliberately driven into the vibration to $kT$. Smaller resonators are more affected: their thermal jiggle is a larger fraction of any motion they can be driven to without bending nonlinearly, and their large surfaces adsorb and release stray molecules, whose arrival and departure shift the frequency in exactly the way the measurement is looking for. Nanotube resonators are therefore cooled to a few kelvin and operated in ultra-high vacuum, and they are read for seconds to reach their best resolution.

There is also a floor no engineering can lower. Measuring a frequency more precisely than its intrinsic width requires time, and the time needed to weigh a mass grows in inverse proportion to the precision wanted. A balance that weighs one proton per second cannot weigh a molecule per microsecond. For the protein-weighing beams, which must catch molecules as they fly past, the trade between how fast and how finely they weigh decides what they can see.

What the beam leaves out #

The figures assume a perfectly clamped, uniform beam vibrating in a vacuum at small amplitude, a speck that is a point mass and rides along rigidly, and a quartz film that is thin, rigid and uniformly spread. Real specks have size and stiffness and change the beam’s stiffness as well as its mass; a large one, or one landing near the clamp where the beam bends most, can raise the frequency rather than lower it. Real beams are driven to amplitudes where their stiffness starts to depend on how far they swing, and the frequency then depends on the amplitude as well as the mass. Thick films on quartz, soft films, and films in liquid break Sauerbrey’s assumption that the film simply moves with the surface. And every resonator’s frequency depends on temperature, so that a millikelvin’s drift can look like the arrival of a molecule; precise instruments are thermostatted, or use pairs of modes whose temperature responses cancel.

The domain of the argument is a small added mass that moves rigidly with a resonator whose stiffness it does not change, measured for long enough to read the frequency to the resolution claimed. Within it, the shift is mass weighted by the mode’s shape, divided by twice the effective mass, and nothing else.

Still open: how close to a single atom, and how fast #

Weighing at the proton scale has been reached in a few laboratory resonators under ideal conditions, and the open questions now are about doing it routinely and quickly: weighing single large molecules — viruses, protein complexes, particles of soot or plastic in air — fast enough and at room temperature, inside fluids as well as in vacuum. One approach puts the fluid inside the resonator, flowing through a channel carved along a hollow cantilever, so that cells and particles passing through are weighed one at a time while the resonator itself stays in vacuum; such resonators have weighed single bacteria to a few femtograms. How far mass resolution, speed and robustness can be pushed together, and whether a resonator can weigh a molecule while also telling what it is from its mass alone, are being worked out device by device.

The rule beneath all of it is the spring’s. Adding a mass Δm at a point where a mode’s displacement is φ lowers its frequency by $\Delta m\,\phi^2/2m_{\text{eff}}$, so a frequency weighs inertia — astronauts in orbit, films a fraction of an atom thick, single molecules — and a resonator notices masses twice its own effective mass times the fraction it can resolve, which is why the lightest resonators are the best balances; two modes, weighting the landing point differently, say where the mass sat. The scale that improves as it shrinks is the oscillator’s single most useful property, and it was hiding in a correction for the mass of a spring.