Crackling noise: the Barkhausen effect

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The term “crackling noise” refers to the signal that some disordered systems produce as a response to an external driving field smoothly changing in time. Due to the presence of disorder, crackling signals are extremely irregular, despite the steady increase of the external forcing. They are typically characterized by a sequence of pulses of very different sizes and durations, separated by quiescence intervals. Tiny events occur very frequently, while large ones are rare, with power laws probability distributions.

Systems that “crackle” are found in many different situations, and, remarkably, the corresponding signals often share some common characteristic features. Examples of crackling signals include the shear response of a granular media, the acoustic emission during martensitic phase transitions, the bursts of dislocations activity in plastic deformation, the dynamic of superconductors and superfluids, the fluctuations in the stock market, the dielectric polarization of ferroelectric materials, the acoustic emission in fractures, and the seismic activity in earthquakes. Crackling noise signals are expected to encode information on the physical process that generates them. Understanding the statistical properties of these jerky emissions, is therefore a step towards the understanding of the microscopic dynamics taking place in the system that crackles. Moreover, the fact that very diverse systems behave in a remarkably similar manner, suggests that some general basic principle may exist in the underlying physics.

 

 

Barkhausen noise (BN) is probably one of the fiirst crackling signals ever recorded. Indeed the Barkhausen effect has been known for almost a century: its first observation dates back in 1919, when Heinrich Barkhausen noticed that “iron produces a noise when magnetized: as the magnetomotive force is smoothly varied [...] it generates irregular induction pulses in a coil wound around the sample that can be heard as a noise in a telephone”. As the magnetization reverses, the variations of the magnetic flux induce a voltage, that indirectly measures the changes in the magnetization of the sample. The signal recorded in correspondence to jumps in the magnetization, appears to be very irregular, no matter how smooth is the variation of the external field. In soft ferromagnetic materials the magnetization mainly reverses by domain wall displacement. Due to the presence of various types of disorder, like non–magnetic inclusions and dislocations, which act as pinning points, the wall movement is discontinuous. As the external field is smoothly increased, the magnetization changes in steps, in correspondence to jumps of the magnetic interface, or, in microscopic terms, in correspondence to avalanches of spin flips. The hysteresis loops is also discontinuous, and the signal, which is proportional to the derivative of the magnetization, plotted versus time, looks like a disordered series of pulses. The size and duration of magnetization events are power law distributed, and the distributions have been found to be universal in a large class of materials.
Barkhusen experiment

 

Since its first observation, a lot of work has been done in order to “decode” the Barkhausen signal, both on the experimental side and on the theoretical one. For a long time, theoretical studies have been based on a phenomenological approach, which described the signal as a superposition of random elementary jumps. Then, it became clear that BN could represent a powerful tool to investigate the magnetization process and the hysteretic properties on a microscopic scale, and much effort has been devoted in developing physically grounded models that could allow to relate the phenomenology to its microscopic origin. The research on BN has been in part motivated, especially at the beginning, by the applications of Barkhausen effect to material testing. Barkhausen emissions are indeed commonly used to check in a non–destructive way the integrity of magnetic samples: the signal intensity is sensitive to the changes in material microstructure, and to the presence of residual stresses (in magnetostrictive positive materials compressive stresses will decrease the intensity of Barkhausen noise while tensile stresses will increase it). These properties make BN an efficient tool for the detection of micro–imperfections and for the evaluation and mapping of the local distribution of residual stresses. Most of the recent research on the statistical properties of BN is however theoretically oriented, and primarily aimed to gain understanding on the hysteretic properties of ferromagnetic materials, and to investigate the magnetization reversal process on a microscopic scale. Recently, further interest has been raised by the fact that many other driven dissipative systems have been found to respond to a smooth forcing with crackling noise emissions. Clearly, the irregular response is due to the presence of disorder in the system, and the challenge is to get information on the specific system under study by decoding the noise that it generates. As the recent encouraging advances in the understanding of BN compose a rather satisfying picture of the phenomenon, BN emerges as an attractive case study to understand crackling noise in general.

The theoretical studies on BN have followed two kind of approaches: one based on a microscopic description in terms of spins in a random magnetic field; the other one describes the system in terms of a fluctuating magnetic interface that moves under the action of the external field. A central role in our current uderstanding of BN is played by a model of avalanche dynamics known as ABBM model (named after the authors of the original papers, B. Alessandro, C. Beatrice, G. Bertotti and A. Montorsi). The ABBM model, corresponds to a mean field description of an elastic magnetic wall moving in a disordered ferromagnet under the effect of an external driving field. The system is ultimately described by a simple Langevin equation for the velocity of the center of mass of the wall. Despite its simplicity, the ABBM model is able to reproduce with striking accuracy most of the phenomenology observed in BN experiments on a large class of magnetic materials. Moreover, thanks to a mapping onto a simple stochastic process, all the results concerning the noise statistics can be derived analytically, and have a clear and direct interpretation. In particular, this simple mean field description allows to explain the origin of the power law distributions of size and duration of pulses, both in the quasi–static limit, where the applied field variations are extremely slow, and at finite driving field rates. In this second case, the model is also able to predict how these distributions depend on the driving field rate.


A more refined analysis of the experimental data against theoretical predictions includes the comparison of the shape of the avalanches. This is where the ABBM model unexpectedly fails, being unable to predict the characteristic leftward asymmetric form of BN pulses. The origin of this asymmetry lies in the non–instantaneous response of the eddy fields to the domain wall displacement. To understand and evaluate the effect of such delay, one as to take into account the dynamical effect of eddy currents in the Maxwell equations for the eddy field. Eddy currents retardation gives rise to an “anti–inertial” effect, that can be accounted for, to the first order, by associating a negative effective mass to the wall.

It is interesting to underline the role of universality versus non– universality in the theory of Barkhausen effect. The standard statistical mechanics approach usually focuses on universal quantities. Definitely, universality is a key and extremely powerful concept at the basis of this approach, as it allows to predict the essential behavior of self–similar systems by means of very simple models. This consideration indeed applies to the case of BN, where, in analogy to critical phenomena, most of the statistical properties of the signals only depend on general properties of the system, while they are independent of the microscopic details: different magnetic samples respond to the forcing by an external field by producing events of magnetization reversal characterized by the same power law distributions, regardless of the specific microscopic structure of the material. However, some interesting features of BN turn out to be of microscopic origin. Although, consistently with universality, microscopic details would become negligible on extremely large scales, they have an unusually large effect in Barkhausen signals, an effect that is still significant at the experimental scale. On one hand this can be seen as an inconvenient and a limitation of the methods of statistical mechanics, since it means that not all the relevant aspects of the phenomenon can be captured by simple models. On the other hand, the identification of some macroscopic effect of the microscopic details, provides a mean to extract information on microscopic quantities by Barkhausen measurements. The asymmetry of the pulses gives indeed a measure of this type. As an example, the skewness of the pulses measured as a function of the pulse duration shows a peak, that allows to identify a characteristic timescale for relaxation, which corresponds to the ratio between mass and damping constant.


The seminal paper on crackling noise is Crackling noise, Nature 410 (2001), pp. 242–250, by J.P. Sethna, K.A. Dahmen, and C.R. Myers.

For an excellent review see The Barkhausen effect in "The Science of Hysteresis", vol. II, G. Bertotti and I. Mayergoyz eds, Elsevier, Amsterdam, pp. 181-267 (2006) by Gianfranco Durin and Stefano Zapperi.

For a review article focusing on ABBM model and beyond see Exactly solvable model of avalanches dynamics for Barkhausen crackling noise in Advances in physics (in press) by Francesca Colaiori (will be avaliable shortly).