The non-smooth, jerky movements of microstructures under external forcing in minerals are explained by avalanche theory in this review. External stress or internal deformations by impurities and electric fields modify microstructures by typical pattern formations. Very common are the collapse of holes, the movement of twin boundaries and the crushing of biominerals. These three cases are used to demonstrate that they follow very similar time dependences, as predicted by avalanche theories. The experimental observation method described in this review is the acoustic emission spectroscopy (AE) although other methods are referenced. The overarching properties in these studies is that the probability to observe an avalanche jerk J is a power law distributed ^{−ε} where

Minerals contain microstructures and much of what minerals can tell us about past geological processes, and about their own intrinsic properties, is related to microstructures. This balances the importance of microstructures with the actual crystallographic structure. Through the enormous progress in nanotechnology over the past decade, our perspective of materials in general and minerals in particular has shifted towards a much better understanding of microstructures. Microstructures cover a huge range of length scales from coarse twinning (mm scale), fine twins (typically on a micrometer scale) and tweed structures with repetition scales between 10 and 100 nm. On an even smaller scale we have structural disruptions, like kinks and domain wall bendings, so-called wobbles, inside these microstructures (Salje et al.

A second development relates to the time scales on which microstructures change. Such changes are either induced by external forcing, like stress, electric or magnetic fields, oxygen fugacities, etc., or during creep experiments without external forcing (Salje et al. ^{3} years but they can also be very fast. Structural changes during radiation damage, for example, take only ca. 5 femto-seconds (10^{–15} s) and the propagation of a twin wall requires times between 10^{–8} s and 10^{–3} s in many cases. The fundamental question is then: what determines the origin of time scales? In this paper we argue that for avalanche processes there is not a ‘typical’ time scale but, instead, a large dynamic range of time scales.

Microstructures often evolve in a non-smooth manner. The shift of a domain boundary is virtually never continuous but occurs in a stop-and-go fashion. Cracks do not progress along straight trajectories but wobble, bifurcate and form complex patterns on an atomistic length scale. The appropriate description of such processes lies in the concept of avalanches. Their discovery, which was sometimes ignored in mineralogy, is probably the most important progress in the design and application of high-tech devices and covers a novel branch of scientific endeavour, referred to a ‘avalanche science’ with several books published in this field (e.g. Salje et al.

Most examples in this review are taken from the field of mineral physics. If the reader wishes to pursue the topic further for other minerals, we recommend consulting Salje and Dahmen (

Crackling noise is encountered when a material is subjected to external forces with jerky responses spanning over a wide range of sizes and energies. The Barkhausen effect of pinned domain walls (Harrison et al.

Crackling noise avalanches, like the well-known snow avalanches, are collective motions, which follow well-defined statistical rules while their exact time-dependent behaviour of any part of the avalanche remains unknown. Collapse avalanches have been thoroughly analysed in porous minerals, like SiO_{2} based glass (Vycor) (Salje et al.

The following fundamental parameters are essential for our further discussions.

The amplitude _{max}. It then decays with a long tail of strain signals until the avalanche terminates. Amplitudes can display very complex evolution patterns, in particular when several avalanches coincide. Sometimes they develop ‘eternal’ avalanches, which never fully end but just diminish and resurge. The obvious analogy to disease spreading mechanisms highlights the close similarity between these two areas of research.

The duration is the time period over which an avalanche survives. Experimental time scales typically extend from a few microseconds to many milliseconds.

The energy is the time integral over the local squared amplitude ^{2}, integrated over the full duration of the avalanche:

This means that for avalanches which represent a short δ-function excitation at the time _{max}, _{max}_{max}) will always display a scaling _{max}^{2}. This is not true for long and smooth ^{X} with 2 < x < 3 are discussed in literature (Casals et al.

The size of the amplitude indicates the number of particles that move during the avalanche. While this parameter appears intuitive in geometrical terms like a ‘patch’ of transformed material, this is not correct. If areas transform, they can do so in compact regions where every atom takes part in the transformation. They can also transform by selecting some of these atoms, forming some ‘sponge-like’ areas. The fractal dimension of these transformed areas becomes then paramount and while ‘size’ is popular in the general description of avalanches, the meaning of such ‘size’ parameter can be surprisingly complex. It is, therefore, recommended to explore the scaling of size with the amplitude or energy as a more fundamental parameter. As an example, if the movement relates to low-dimensional dynamical patterns, the relationship is linear ^{2}. This already highlights that model calculations are often required to determine this

The two names are used interchangeably. They denote the time between avalanches, i.e. the time the system needs to recover after an avalanche has happened. In neural networks, these inter-event times are the ‘sleeping periods’ after high neural activity. Their probability distributions are typically power laws with two different, approximate exponents for short and long times, _{w}) ~ _{w}^{−1} and _{w}) ~ _{w}^{−2}, respectively. Note that in these scaling relationships the negative sign in the exponent is often included in the equation so that the term ‘exponent’ often means the value after the minus sign. These exponents represent the results in the simplest mean field (ML) theory (Salje and Dahmen

In addition, there is a multitude of secondary scaling laws, in particular those describing aftershock activities (Baró et al.

During 100 years of research many experimental methods were developed to quantify avalanches. They range from magnetic measurements to electrical depolarization currents in ferroelectrics and optical observations of crack patterns and the determination of fractal dimensions (Lung and Zhang

Figure _{AE}(

Schematic representation of the composition of an avalanche signal in acoustic emission experiments

While AE spectroscopy is probably the best way to detect avalanches under in-vivo conditions, it has a serious drawback. The measured AE spectrum, i.e. the macroscopic jerk spectrum of a sample, is not exactly the initial avalanche distribution

Minerals are often lighter than their chemical composition would suggest. Defect chemistry traditionally points to vacancies as a reason for the weight loss. This is not always the case. In fact, vacancies are simply the smallest version of holes and cavities in mineral structures. Holes can also be envisaged as empty inclusions, so that much what is known about holes can be extended to other inclusions. Holes are also structural elements in porous materials, which are widely used as filters, fillers, low thermal conduction materials and so on. Porous materials are particularly important due to their relevance in the collapse forecast of both natural and artificial structures such as mines (Jiang et al.

A prototype of porous materials is the tuff-like Vycor, which is a porous material based on SiO_{2} with holes covering a very wide range of diameters. The smallest hole diameter is ca. 5 nm. Understanding holes requires to understand their strain fields. Vacancies (and other inclusions) exert large strain fields, which were summarized in Markenscoff et al. (^{n} in space with

Sample height (_{2} (Gelsile 2.6). The vertical scales are logarithmic in (

Cracks and hole–hole interactions have in common that they do not form simple microstructures. Like cracks in scattered window glass, they form complex patterns where the crack propagation does not follow linear trajectories but progress by junctions, bifurcations, spirals and specific patterns, like Turing patterns (Scott

We now explore what happens when stress is applied to a porous sample and refer to the extended literature for samples with crack propagation (here the crack propagation in granite is a particularly nice example how AE and avalanche physics helped to determine the thermal stability of minerals (Xie et al.

The AE during the uniaxial compression experiments of SiO_{2}-Vycor with 40% porosity is shown as an example in Fig. _{i} when the preamplified signal _{i} associated with each event ^{2}(

_{2}-Vycor at a rate

Figure ^{3} s. The distributions show a power-law behaviour

Distribution of avalanche energies of SiO_{2}-Vycor during the full experiment with _{min} for the three experiments. [after Baró et al. (

The next step in a typical AE analysis is the computation of the number of aftershocks (AS) in order to compare with Omori’s law for avalanches (also in Earthquakes). We define as mainshocks (MS) all the events with energies in a certain predefined energy interval. After each MS we study the sequence of subsequent events until an event with an energy larger than the energy of the MS is found. This terminates the AS sequence. Then we divide the timeline from the MS towards the future in intervals, for which we count the number of AS in each interval. Averages of the different sequences corresponding to all MS in the same energy range are performed, normalizing each interval by the number of sequences that reached such a time distance. The results compiled in Fig.

_{min} and the compression rate

One of the most intricate examples of collapsing holes, which mix with sliding dislocations, is described in Ho-doped Mg metal. The microstructure is shown in Fig.

TEM figure shows Ho granule with surrounding dislocations (

The AE spectrum is dominated by a separation of signals which correspond to the hole collapse and sliding dislocations. Their respective fingerprints are very different and relate to their individual signal strengths. Porous collapse generates very strong AE signals while dislocation movements create more but weaker signals. This allows a separation of the two processes even though they almost always coincide temporarily. The porous collapse follows approximately the predictions of mean-field behaviour of short, independent avalanches (

The observation that many minerals are twinned is as old as mineralogy itself. So why became the investigation of twins, or more precisely of boundaries between twins, so popular during the past decade? There are two aspects to clarify. First, we know almost nothing about the detailed structure of boundaries between growth twins and research in this field has hardly started. Boundaries between ferroelastic twins, on the other hand, are much better understood (Janovec et al.

where _{0} is its bulk value and the tanh-functions describe the profile of the boundary. When the space coordinate

Much progress is expected from research on mineral structures over the coming years, not only in the discovery of novel twin boundaries based on growth phenomena, but also in the exploration of boundary mobilities. Mineral physics follows metallurgy where such studies are more advanced. Martensites and many alloys were investigated using AE spectroscopy to identify how twin boundaries move (Salje et al. _{3} and some other ferroelectric materials were measured (Salje et al.

The mild movements still constitute avalanches in the description of (Salje and Dahmen

An important step forward was the idea that domain wall transport includes chemical changes during electronic conduction. This impacts on the origin of memristor properties of ferroelastic domain walls (Bibes and Barthelemy

As an example, we now consider a perovskite structure, BaTiO_{3}, which is not only ferroelastic but also ferroelectric. The relevant domain boundaries are twin walls. They can be shifted by electric fields and by external stress. In this example we use the electric field as driving field. The experiment takes typically one night to increase and decrease the field six times (blue line in Fig.

Jerk spectrum of acoustic noise of BaTiO3 during domain switching (red signals). The noise is measured as time evolution of the energy of AE signals. The electric field (blue line) is ramped between − 1000 and 1000 V with a rate of 0.5 V/s (right axis). In total six loops (from 1000 to − 1000 V back to 1000 V) were measured. The scale for the jerk energy (in attojoule) is logarithmic and stretches over five decades (Salje et al.

The energy of the jerks that constitute the avalanche is power law distributed with an overall exponent

The probability distribution function (binned data, top panel) shows a power-law distribution. The maximum likelihood graph (bottom panel) is used to determine the energy exponent. The energy exponent is 1.65 as average over all loops (Salje et al.

To discuss avalanches in bio-minerals we choose microbially induced calcite precipitation (MICP), which is a common process in bio-geotechnical engineering. Laboratory tests have demonstrated that MICP treatment of granular soils improves their strength by three orders of magnitude (DeJong et al.

In order to test the MICP material, uniaxial compression was applied to calcareous sand grains, sands without cementation and bio-cemented sand samples treated by MICP. The sand composition was mainly aragonite and Mg-bearing calcite, the carbonate component was above 97%. The microstructure is characterized by accumulated insoluble MICP calcite bridging sand grains. The SEM image of a bio-cemented sand sample shows that sand particles are bridged by microbial induced calcite (Xiao et al.

Stress–strain relationships and AE spectra for calcareous sand grains and a bio-cemented sand sample are shown in Fig.

The probability distribution function (PDF) of avalanche energies is shown in Fig. ^{−ε}. Figure _{min} indicating that the AE signals are damped by absorption or scattering of the acoustic signals (Salje et al.

_{min} for the three experiments during the full experiment.

Grains show excellent plateaus with

Changes of microstructures often progress in a wild, non-smooth manner. Experimental evidence rules out simple catastrophic events, like one big step when a twin wall moves. Instead, we find universal behaviour with multitudes of small ‘jerks’ which can cut down the big step into millions of small steps. This phenomenon appears in many systems, and only three of them were briefly reviewed here. The overall behaviour of the totality of the jerks follows very strict rules. These rules are the same as what is theoretically expected for avalanches, which establishes a close link between avalanches and microstructural evolution. As the probability to find a ‘jerk’ with an energy ^{−ε}. Consider an energy interval between E and 2E; then the probabilities are between ^{−ε} and (2E)^{−ε}. We now scale the energy by a factor x. The interval is now from xE to 2xE and the probabilities change to (x E)^{−ε} and 2^{−ε} (x E)^{−ε}. The common numerical prefactor ^{−ε} is irrelevant for the functional form, which remains exactly the same as before. This proves that the power-law distributions are scale invariant. Note that this is a special property of the power law and that other functions are not scale invariant. Furthermore, combinations of power laws are not power laws and hence not scale invariant.

The scale invariance in avalanches is not restricted to energies but holds equally for the amplitudes, durations and, with some modification, for waiting times. In practical terms, anything we see in a space (or time) interval is exactly the same as in any other. The limits are given by cut-offs, such as the atomic diameter or the sample size, but the region between these cut-offs can reach many orders of magnitude. AE allows us to estimate this range and we find that 6–8 orders of magnitude of energy are not uncommon for microstructural changes. The question on which length scale do structural changes happen is hence ill-posed: there is a large interval of length scales and wherever we situate our experiment we will see the same change.

This powerful approach is important for minerals where defects and lattice imperfections favour avalanches. The induced behaviour is then independent of these obstacles and significant similarities are found in a multitude of different minerals. For reasons alluded to in the introduction, the full power of this method has been used to solve several problems in solid-state physics and metallurgy, but much less in mineral physics. There is a wide range of mineralogical research waiting to be done in future.

E.K.H. Salje is grateful to EPSRC (No. EP/P024904/1) for support. The project has received funding from the EU’s Horizon 2020 programme under the Marie Skłodowska-Curie grant agreement No 861153. X. Jiang thanks the financial support from the Natural Science Foundation of China (Nos. 51908088).

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