The curve that is really a staircase

In 1919 Heinrich Barkhausen wound a coil round an iron bar, connected it through an amplifier to a loudspeaker, and slowly brought a magnet near. The loudspeaker did not go quiet and then loud. It crackled — a hiss with pops in it, like rain on a roof, while the field outside rose smoothly and continuously.

That was the first direct evidence that magnetisation happens in domains, and it arrived six years before anyone had a theory of a domain wall. What the coil was hearing is what a magnetisation curve is actually made of.

The model behind the figure is a single equation and is worth stating, because everything else follows from it. A domain wall sitting at position $x$ feels three things: the applied field pushing it forward, a pinning field that varies erratically with position because the crystal has defects in it, and a restoring term proportional to its displacement, from the field the displaced wall sets up against itself. Quasi-statically the wall sits wherever those balance. As the applied field rises, the balance point moves — and where the pinning landscape has a local maximum, it does not move smoothly. It disappears, and the wall jumps to the next place it can rest.

Nothing here is random in time #

The most important feature of the model is negative: it has no noise in it. The pinning landscape is fixed once and for all, and the same crystal swept twice gives the same jumps in the same places to every digit the arithmetic carries.

That is not a modelling convenience. It is the physical claim, and it is testable: sweep a real sample twice through the same range and the crackle repeats, jump for jump, in a way that white noise never would. The repeatability was noticed early and is the standard demonstration that the signal is structure rather than interference.

What the jumps are, then, is a deterministic consequence of a wall having to cross a rough landscape. There is a balance point at each field, it moves continuously most of the time, and at particular fields it ceases to exist. This is the same shape of behaviour as the block that sticks and slips against a slowly moving drive, where the drive is smooth and the motion is not; the same as the sudden collapse when a spinning top’s stability boundary is crossed; the same as the jump between two amplitudes when a nonlinear oscillator’s response folds. A smooth parameter crossing a fold produces a discontinuous response, and the fold is where a solution stops existing rather than where anything is disturbed.

What it sounds like #

The induced voltage in a coil is proportional to the rate of change of flux, which is proportional to the wall’s speed. So the coil hears the derivative of the staircase, which is a series of spikes separated by silence. Amplified, it is exactly what Barkhausen described.

The figure also makes visible something the staircase hides: how much of the time nothing is happening. The wall is essentially at rest for eighty-five per cent of the run, and the fifteen per cent when it moves accounts for all of the magnetisation. A measurement that averaged over a second would report a smooth, steady, uneventful process — which is precisely what a slow magnetometer does report, and why the phenomenon needed an amplifier and a loudspeaker to find rather than a better meter.

Reversible and irreversible, and how to tell them apart #

A magnetisation curve has two kinds of motion in it and the crackle separates them cleanly.

Lower the field slightly after any of the jumps above and the wall comes back a little — but not all the way, and not by retracing. The small reversible part is the wall bowing between its pinning points, like a membrane under pressure; it stores energy, returns it, and makes no noise. The irreversible part is the jumps, which do not come back at all: lowering the field returns the wall to a different resting place from the one it left, and the difference is the area of the hysteresis loop.

That is where the dissipation is. The energy lost per cycle — the loop’s area, which is the quantity a transformer designer pays for in heat — is the sum of what the wall gives up at each jump, converted into eddy currents and lattice vibrations as it accelerates and stops. So the crackle is not a curiosity accompanying the loss; it is the loss, resolved into its individual events, and the integral of the noise power over a cycle is the hysteresis loss.

Two consequences follow that are not obvious from the loop alone. A material with finer disorder has smaller, more numerous jumps and the same loop area, so the loss is unchanged while the noise spectrum moves up in frequency. And a material driven very slowly still loses the same energy per cycle, because the jumps happen at the same fields however slowly they are approached — which is why hysteresis loss is quoted per cycle rather than per second and why it is not a viscous effect at all.

The sizes, and the absence of a typical one #

A power law means there is no typical size, in the same sense and with the same consequences as a distribution of forces in a granular pile. That is a stronger statement than “the sizes vary a lot”: it means that asking for the average jump gives a number that describes nothing, because the distribution has no scale in it over the range where the law holds. Doubling the size range examined roughly triples the number found, and the largest one found keeps growing with the length of the observation.

The exponent is close to minus three halves, and it is not fitted. It comes out of the computation, and its origin is one of the more satisfying arguments in the subject: a wall in a landscape whose pinning field is a random walk in position is being asked, at each field, how far ahead the running maximum lies — and the distribution of return times of a random walk is a power law with exponent minus three halves. The same exponent turns up wherever a first-passage problem of that kind is underneath, which is a large family.

Nearly all of the magnetisation is delivered by a handful of the biggest events. That is the practical consequence of the exponent being between one and two: the sum is dominated by the largest terms, so the count of events says almost nothing and the size of the largest says almost everything. Anyone reading a crackling signal for a purpose — and Barkhausen noise is used for exactly this, as a non-destructive probe of residual stress and grain structure — has to know that the useful information is in the tail.

What sets the largest one #

The cutoff is the interesting end, because it is the end that is not universal. Below it, the distribution is the same shape for every sample of every material; above it, there is nothing. And what sets it is not the disorder at all. It is the restoring term — the demagnetising field, which grows as the wall displaces and therefore chases the wall down, ending the avalanche.

That has a directly checkable consequence: the same material in a different shape crackles differently, because the demagnetising factor is a property of the shape. A long thin sample has a small restoring term and large avalanches; a squat one has a large restoring term and small ones. Nothing about the metal has changed. The connection to the load line a magnet’s own shape imposes on it is exact — it is the same demagnetising factor, doing the same job, at a different point in the same story.

The separation is worth stating as a principle, since it recurs. The exponent belongs to the mechanism and the cutoff belongs to the sample. Two quantities extracted from the same measurement, one of which is a fact about a large class of systems and the other a fact about the object in hand. Reporting only the first is reporting nothing about the sample; reporting only the second is throwing away the evidence that the mechanism is understood at all.

The loop the staircase lives inside #

Placing the staircase back inside the loop is worth doing, because the two descriptions answer different questions and neither replaces the other.

The loop is a statement about memory. It says that magnetisation is not a function of field, that there is a set of accessible states at each field, and that which one the material is in depends on its history — which is what makes iron useful for recording anything and what a magnet’s working point on its own load line is a statement about. Nothing in that argument needs the jumps: a population of independent units with a spread of switching thresholds reproduces the loop, the minor loops and the coercivity perfectly well, and does so with each unit switching smoothly.

The staircase is a statement about mechanism. It says the states are metastable — separated by barriers, with the wall genuinely stuck between them — and that the transitions are collective, involving many units at once with a size distribution that has no scale. A picture of independent smoothly-switching units cannot produce a power law, and the measured crackle is how the two descriptions are told apart.

That the smoother description gets the engineering quantities right and the microscopic story wrong is a common arrangement and a reason to be careful about what a successful model has established. Fitting a hysteresis loop is not evidence about what a domain wall does. Listening to it is.

The same distribution elsewhere #

Crackling is not a peculiarity of iron, and the reason for saying so here is that the family it belongs to is a good illustration of what a universality claim is and is not.

Earthquakes obey the Gutenberg–Richter law — the number of events above a given size falls as a power of the size, with an exponent that is remarkably stable across regions and across four orders of magnitude of energy. Fracture in a heterogeneous solid emits acoustic bursts with the same kind of distribution. So do vortex avalanches in a superconductor, plastic deformation of a small crystal, and the slow compaction of a granular pile — which is where this collection’s jamming of a granular heap meets the same statistics from a different direction.

What is shared is not a mechanism in the ordinary sense. It is a structure: a slowly driven system, with many metastable configurations, and a threshold at each of them. Drive it slowly and it responds in bursts whose sizes are distributed as a power law up to a cutoff set by whatever restores it. The details of what is stuck, what unsticks it and what carries it are different in every case; the exponent survives, in the same way and for the same reason as the constant that describes a period-doubling cascade survives a change of map.

What must not be claimed from that is a shared cause. Iron does not crackle because of anything an earthquake does. The claim is that both are instances of a class defined by a small number of features, and the value of that claim is predictive: measuring the exponent in one system tests the classification, and finding the wrong exponent says a feature is missing.

The instrument it became #

Barkhausen noise is measured commercially, and what it is used for is a good illustration of a measurement whose value comes from the parts of the signal a physicist would call uninteresting.

The signal is sensitive to anything that pins a domain wall: dislocations, grain boundaries, precipitates, and — most usefully — internal stress, because a stressed region has a different anisotropy and therefore a different pinning strength. So the amplitude and spectrum of the crackle from a steel component report on its residual stress and its microstructure, without cutting it open and without any contact beyond a coil held near.

The applications are exactly where those matter: grinding burn on a bearing race, which changes the surface microstructure without changing the shape; case-hardening depth; residual stress in a weld; fatigue damage before a crack exists. In each the alternative measurement is destructive.

What the technique cannot do is tell those apart from each other. A change in the signal says something about the material has changed and not what, so the method is a comparator against a known-good sample rather than an absolute measurement. That is a limitation worth stating plainly, and it comes from the same fact the physics is about: the signal is a convolution of a universal mechanism with everything local, and the universal part carries no information about the sample at all.

Where the model stops #

The wall is a rigid plane and it is not. A real domain wall is a surface that bows, and part of the magnetisation is that bowing rather than any translation — a reversible part, which returns when the field is lowered and makes no noise at all. The model here has only the irreversible part, so it describes the crackle correctly and the initial susceptibility not at all.

There is one wall, and a real sample has an enormous number. Domains nucleate, walls meet and annihilate, and the whole domain structure rearranges. Below the coercive field the single-wall picture is fair; near saturation, the magnetisation proceeds by rotation of whole domains and there is no wall to pin.

The pinning landscape is a random walk in position, which is a choice. It is the standard one and it is what produces the minus-three-halves exponent; a landscape with long-range correlations gives a different one, and real materials with strong pinning centres of a definite size are not well described by either. The exponent is therefore a claim about the class of disorder as well as about the mechanism.

And the drive here is quasi-static. At a finite sweep rate the avalanches begin to overlap, the individual bursts stop being separable, and the measured distribution is distorted at the large end — which is exactly the end that matters. Every measurement of this kind has to demonstrate that the answer stops changing as the sweep is slowed.

What the pictures cannot show #

The staircase draws the wall’s position and not the wall. What a jump physically is — a sheet of atoms a few tens of nanometres thick sweeping through some volume of the crystal, reversing the moments in it — is not in any of these figures, and the model has replaced it by a single coordinate. That replacement is what makes the problem tractable and it is also what makes the model unable to say anything about why walls have the thickness they do.

The noise trace draws a speed against time, and it draws it after choosing a viscosity — how fast the wall moves when the pinning is overcome. That constant sets the width of each burst and therefore the whole shape of the audible signal, and it comes from eddy-current damping in the metal rather than from anything in the pinning. So the figure’s horizontal scale is a property of the conductivity, and the vertical structure is a property of the disorder; the picture mixes two quite separate pieces of physics and cannot label which is which.

Where the ladder goes next #

The magnetisation ladder began with a magnet fighting its own field and the magnetism classical physics forbids. This rung asks what the magnetisation curve is made of. The rungs after it: the domain wall’s own width, which is set by the competition between exchange and anisotropy and is the first length in the subject that is not a sample dimension; the Preisach description, which builds a hysteresis loop out of a population of independent switching units and gets the shape right while getting the noise wrong; and thermal after-effect, where a wall pinned at a barrier a few kT high eventually gets over it on its own and the magnetisation creeps for hours.

The habit worth carrying away is about smooth curves. A measured curve is smooth at the resolution it was measured at, and nowhere else. The question worth asking of any monotonic response is what it looks like magnified fifty times — and if the answer is a staircase, the interesting physics is in the steps rather than in the average through them.