Ultraslow calorimetric studies of the martensitic transformation of NiFeGa alloys: detection and analysis of avalanche phenomena

Vol.:(0123456789) Journal of Thermal Analysis and Calorimetry https://doi.org/10.1007/s10973-024-13206-4 Ultraslow calorimetric studies ofthemartensitic transformation ofNiFeGa alloys: detection andanalysis ofavalanche phenomena José‑MaríaMartín‑Olalla1 · AntonioVidal‑Crespo1· FranciscoJavierRomero1· AlejandroF.Manchón‑Gordón2· JhonJ.Ipus1· JavierS.Blázquez1· MaríaCarmenGallardo1· ClaraF.Conde1 Received: 4 October 2023 / Accepted: 8 April 2024 © The Author(s) 2024 Abstract We study the thermal properties of a bulk Ni55Fe19Ga26 Heusler alloy in a conduction calorimeter. At slow heating and cooling rates ( ∼1Kh −1 ), we compare as-cast and annealed samples. We report a smaller thermal hysteresis after the thermal treatment due to the stabilization of the 14M modulated structure in the martensite phase. In ultraslow experiments ( 40 mK h−1 ), we detect and analyze the calorimetric avalanches associated with the direct and reverse martensitic transformation from cubic to 14M phase. This reveals a distribution of events characterized by a power law with exponential cutoff p(u)∝u−𝜀exp(−u∕𝜉) where 𝜀∼2 and damping energies 𝜉=370μJ (direct) and 𝜉=27μJ (reverse) that characterize the asymmetry of the transformation. Keywords Ultraslow calorimetry· Intermittent dynamics· Avalanches· Heusler alloys Introduction TiNi-based alloys are extensively favored in various applications owing to their distinctive shape memory effect and superelasticity[1–3]. Nevertheless, these alloys come with drawbacks such as high cost and a challenging fabrication process[4]. In lieu of these alloys, there has been significant research on Cu-based shape memory alloys due to their costeffectiveness and comparatively straightforward processing[5–7]. In addition to the conventional thermally induced shape memory effect observed in TiNi-based alloys,Ullakko etal.[8] reported a large magnetic field-induced strain in Ni2MnGa single crystals. These possible applications are related to the martensitic transformation (MT) that takes place in this kind of materials, a first-order phase transition which occurs in the solid state from a high temperature (high symmetry) austenite phase to a low temperature (low symmetry) martensite phase. The structure of the martensite phase and the martensitic transformation temperature have been extensively analyzed in NiMn-based Heusler alloys[9, 10], which are very sensitive to both the chemical composition tailoring[11] and the fabrication method[12]. In this sense, the dependence of the valence electron concentration per atom, e/a, is of great significance in the development of Heusler alloys. As an example, the transition temperatures linearly increase with e/a in Ni–Mn–X (X = In, Sn, Ga) systems, i.e., decreasing the X element concentration[13]. On the other hand, it has been shown that the crystal structure of the martensite phase evolves in the sequence 10M → 14M → L10 with the increase of e/a in Ni–Mn–In system[13]. Although the general formula of Heusler alloys includes Ni, Mn and one element of the Ga, In, Sn or Sb quartet, it has been proposed to replace Mn by Fe in order to improve the mechanical properties[14]. The enhanced ductility of these Ni–Fe–Ga alloys is related to the precipitation of the secondary 𝛾− phase. In contrast to the compositional series with Mn, the dependence of the transition temperature is not so obvious. For e∕a>7.8 , transition temperature decreases with the increase of e/a[15, 16]. However, for e∕a<7.8 there is not a clear tendency. Moreover, it has been found that at e∕a=7.8 , the structural transition * José-María Martín-Olalla [email protected] 1 Departamento de Física de la Materia Condensada, Facultad de Física, ICMSE-CSIC Universidad de Sevilla, Avenida de la Reina Mercedes s/n, ES41010Sevilla, Andalucía, Spain 2 Instituto de Ciencia de Materiales de Sevilla, ICMSE-CSIC Universidad de Sevilla, Américo Vespucio 49, ES41092Sevilla, Andalucía, Spain J.-M.Martín-Olalla et al. temperature and both Curie temperatures of the austenite and the martensite phases are close to room temperature[12], which could make this system candidate for room temperature applications. Therefore, it is of the most interest to perform precise analyses of the martensitic transformation of this system. Low temperature martensitic transformations are characterized by a very fast diffusionless growth that can be detected by magnetic measurements[17, 18], thermal measurements[19–21], XRD-in situ techniques[22], and electrochemical impedance[23]. The diffusionless growth can be described as an autocatalytic kinetic process (e.g., for NiFeGa[24] and NiMnIn Heusler alloys[25]) and yields local symmetry changes that are transmitted to the surrounding at the speed of sound. MT is known to occur intermittently[26–28], due to the existence of kinetic impediments, such as lattice defect, impurities or self-generated heterogeneities that produce a complex energy landscape with different metastable states— multiple minima separated by high energy barriers—in the region of coexistence of the high-symmetry and the lowsymmetry phases. If the energy barriers are large enough, the transition only happens when the system is externally driven (athermal transition), so that the transition extends over a finite interval temperature. The strain energy can be stored in the lattice elastically and blocks subsequent growth of the new phase. The transition happens by a succession of strain relaxations between metastable states and avalanches are linked to these fast relaxation processes. The detection of avalanches has traditionally required the use of experimental techniques sensitive to the strain jumps that occur during relaxation such as acoustic emission[27]; magnetic Barkhausen emission has also provided results in this field[29–31]. Alternatively, calorimetric techniques have proven able to detect and measure the energy associated with the avalanches, which are observed as spikes in DTA traces of high-resolution conduction calorimetry[18, 21, 32, 33] and in differential scanning calorimetry (DSC) traces [34–36]. These techniques provide a straightforward identification of the characteristic energies involved in the jerky events at the prize of more challenging experimental conditions. In this work, an arc-melted Ni55Fe19Ga26 ( e∕ a =7.8 ) Heusler alloy is thermally analyzed before and after a thermal treatment, which leads to the formation of a modulated 14M martensite phase from a previous tetragonal martensite phase L10 . The characteristic jerky behavior observed in single crystals[35] is reproduced in the arc-melted polycrystalline sample after the thermal treatment, with events distributed according to a powerlaw with an exponential cutoff. The characteristic damping energy is smaller (higher damping) in the reverse transformation. Experimental An alloy with nominal, atomic composition Ni55Fe19Ga26 was prepared from a mixture of high-purity constituent elements ( >99.9% ) in argon atmosphere in an arc furnace MAM-1 (Edmund Bühler GmbH). In order to keep the final composition as close as possible to the nominal one, an additional amount of 2% excess of Ga was added to compensate losses connected with its evaporation during fabrication process (melting temperature 303K). To obtain a high chemical homogeneity, the alloy was arcmelted several times inside the furnace. Then, parts of the obtained ingot were cut and annealed at 1073K inside a quartz tube under Ar pressure, adding some Zr wires as getter to prevent oxidation. After an annealing time of 24h, the quartz tube was quenched in water. The chemical composition of the original ingot was analyzed by X-ray fluorescence (XRF) using an EAGLE III instrument with an anticathode of Rh. The average composition of the sample was determined by XRF, obtaining an actual chemical composition (atomic percentage) of Ni54.6 Fe19.4 Ga26 . The averaged chemical composition is very close to the nominal one, with a small depletion of Ni and a small enrichment of Fe. Based on the XRF measurements and on the electronic configuration of the outer shells for each element—Ni ( 4s23d8 ), Fe ( 4s 2 3d6 ), and Ga ( 4s24p )—we estimated the electron valence concentration per atom as e∕ a =7.79 . The crystal structure was investigated by X-ray diffraction (XRD, Bruker D8 I diffractometer) using Cu K-alpha radiation. Bruker DIFFRAC.EVA (phase identification) and DIFFRAC.TOPAS (Le Bail refinement) software were employed in order to analyze the XRD patterns. Scanning electron microscopy (SEM) was used to study the microstructural features in a FEI Teneo microscope, using secondary electron (SE), backscattered electron (BSE) modes and energy-dispersive X-ray analysis (EDS) to determine the local composition of the different phases. Before the observation in the microscope, the samples were bonded in an epoxy resin and polished. Thermal properties were studied in a conduction calorimeter described elsewhere[37, 38], which can record high-resolution DTA traces. For this purpose, the measurement device consists of two fluxmeters, with 48 chromel-constantan thermocouples each, which are disposed electrically in series and thermally in parallel. The signal provided by the fluxmeters E is recorded by a Keithley 182 nanovoltmeter at a sampling rate of 12.5 Hz and is converted into a heat flux 𝜙 by scaling it with a Ultraslow calorimetric studies ofthemartensitic transformation ofNiFeGa alloys: detection… sensitivity determined in a calibrating experiment[39]. If the rate of temperature change is 𝛽 then the ratio 𝜙∕ 𝛽 is the heat exchanged by the sample and the calorimetric block per unit change of temperature. After removing a suitable baseline, any excursion of this quantity is associated with the enthalpy change in the event of phase transition[40–42]. The system operates then as highresolution differential thermal analyzer with a sensitivity around 1μW . The device is surrounded by a large thermal bath whose temperature is controlled by a Julabo FP40 or FP45 through a heat exchanger coil. Due to the large thermal inertia of the equipment, the scanning rates range from a few kelvin per hour to few milli-kelvin per hour, much slower than commercial differential scanning calorimeters DSC. The magnetic properties of the Ni55Fe19Ga26 system were previously studied in Ref.[12]. Results Structural andmicrostructural characterization Figure1 depicts XRD patterns at room temperature of the as-prepared bulk sample (ASB) and after annealing treatment and subsequently quenched in water (Thermally Treated Bulk sample, TTB). It has been found that ASB sample reveals the mixture of a non-modulated martensite L 10 tetragonal structure (space group I4/mmm) with traces of a 𝛾− phase (space group Pm3m ). This result confirms the limitation of arc-melting technique to produce monophasic compounds. The tetragonal non-modulated martensite, NM, in the ASB sample is replaced by a modulated structure after thermal treatment and subsequent quenching. Ab initio calculations suggest that the NM structure is thermodynamically more stable than the 14M structure[43]. Therefore, the thermal stress, excess of vacancies, etc., induced by the quenching process help to form the non-thermodynamically stable14M structure at room temperature, which is typically found in Heusler alloys prepared by rapid quenching techniques[44]. In order to evaluate the modulated character of the martensite phase, a Le Bail refinement, which does not require the full crystal structure data but only the space group of the phases, has been performed on the XRD pattern corresponding to the TTB sample (see Fig.1b) with GOF = 2.03. The modulated structure can be indexed with the monoclinic space group, obtaining lattice parameters a=0.44553(3)nm, b=3.0329(3)nm, c=0.56195(3)nm and monoclinic angle equal to 88.009(7)◦ . The relation b=7a indicates a sevenfold increase in the unit cell length along the b axis, coherently with a modulation 14M. We obtained a lattice parameter a=0.36053(3)nm for the cubic structure corresponding to the 𝛾− phase. The microstructure and chemical composition of prepared samples have been analyzed by SEM. Figure2 shows —panels a and d—the representative SEM secondary electron images of the polished samples. A small black dot in the panel d corresponds to pores created during the fabrication process. The existence of gamma precipitates dispersed inside the martensite grains can be inferred from the SE images, which are clearly observed in the Intensity/a.u.Intensity/a.u. (125) (125) (0014) Y(111) (127) (129) 42 44 46 (127) 42 44 46 as-prepared quenched Y(200) (2 0 12) (2 0 14) (2 0 14) (0 4 0) (0 0 16) Y(220) (1 2 21) (2 4 0) 30 40 50 60 70 80 30 40 50 60 70 80 L10(101) Y(111) LI0(110) Y(200) LI0(111) LI0(200) Y(220) LI0(201) 2/ θ ° 2/ θ ° 2/ θ ° 2/ θ ° Fig. 1 a XRD patterns of Ni55Fe19Ga26 as-prepared (ASB) and quenched bulk (TTB) samples. The Bragg peaks of the modulated phase have been labeled according to the notation of the monoclinic system that usually describes the modulated 14M structure. b XRD pattern of the TTB (open circles) and its Le Bail fitting using martensite phase and 𝛾− phase space groups (red line). The difference is shown at the bottom (black line). The inset depicts a horizontally zoomed plot J.-M.Martín-Olalla et al. case of the backscattered images. The observation of the precipitates, which exhibit a larger size in the case of the TTB sample, is consistent with the XRD analysis reported above. The EDS elemental mappings of the produced samples are depicted in Fig.2—c and f, in which the signals of the Ni, Fe and Ga elements are recorded. Table1 lists the composition of the martensite and 𝛾− phases obtained by EDS point analysis of representative micro-areas of the samples. Despite the thermal treatment of the sample, which leads to the transformation from the non-modulated to the modulated martensitic structure, martensite phase maintains a stable composition. Accordingly, the same results have been obtained for the 𝛾− phase, which is slightly enriched in Fe and depleted in Ga compared to the surrounding martensite matrix. DTA traces The ASB sample with dimensions 9.0 mm ×9.57 mm ×2.35 mm (length ×width ×height) and m=1.5987(1)g in mass was placed in the conduction calorimeter to characterize its thermal properties. Thereafter, the sample was removed from the calorimeter, quenched and transformed into the TTB sample and was placed back in the calorimeter. In the calorimeter the ASB sample was heated up to 335K, in the austenite phase. Then, the Julabo controller was set to 300K and the sample was cooled down against this fixed setpoint; the martensitic transformation took place. Thereafter, the sample ASB was further cooled down to 270K and heated back against a fixed setpoint of 340K, while the reverse transformation took place. A similar experiment was carried out for the TTB sample. The rate of temperature change in these runs was around 1 Kh −1 (some 0.015 Kh −1 ), much slower than the rates in conventional DSC. We will refer this as a slow rate. Figure3a shows the excesses of the DTA traces in a cooling (blueish) and a heating (reddish) runs. The exothermic and endothermic excursions correspond to the direct (cooling) and reverse (heating) martensitic transformation. Table2 lists the start and finish temperatures—determined form the deviations of the DTA trace— and the peak temperatures. The direct transformation attained the maximum DTA tracte at Mp=322.7 K , and the reverse transformation, at Ap=329.8 K , for the ASB sample. On the other hand, the TTB showed cooler peak temperatures Mp=313.1 K and Ap=319.6 K , in agreement with Ref.[12]. The TTB sample also showed narrower anomalies Fig. 2 SEM micrographs obtained using SE—a and d—and BSE—b and e—and corresponding compositional maps obtained by EDS—c and f—. a–c correspond to the ASB sample; and d–f refers to the TTB sample. Ni, Fe and Ga elements are represented in orange, green and blue, respectively. The dashed area in e is magnified in f. (Color figure online) Table 1 Chemical composition (atomic %) for the bulk samples obtained from EDS Sample Martensite phase 𝛄− phase Ni Fe Ga Ni Fe Ga ASB 56±2 19±3 25±3 56±2 26±3 18±3 TTB 56±2 19±3 25±3 57±2 27±3 16±3 Ultraslow calorimetric studies ofthemartensitic transformation ofNiFeGa alloys: detection… and smaller thermal hysteresis Ap−Mp=7.1 K (ASB) vs Ap−Mp=6.5 K (TTB). Thermal hysteresis was much smaller than previously reported values[12, 16] ( ∼25 K ) for experiments carried out at 1000 times faster rates of temperature change. Both cooling runs showed a distinct rough behavior in the upper half of the direct transformation (see the insets I1 and I2 in Fig.3), which was absent in the reverse transformation. This is evidence of a jerky behavior in the direct transformation that will be later analyzed. The total enthalpy change ( Δh=hlow −hhigh ) in either transformation is also listed in Table2. For the ASB sample we found Δ h d=−2.7 Jg−1 (direct) and Δhr=−2.9 Jg−1 (reverse). For the TTB sample, the excesses increased Fig. 3 The DTA traces (top, a) and the corresponding evolution of the enthalpy excesses (bottom, b) for the slow runs ( 𝛽∼1Kh −1 ) on a Ni55Fe19Ga26 bulk sample. The vertical axis displays excess heat per unit rate of temperature change and per unit mass (a) and excess enthalpy per unit mass (b). The ASB runs are shown in lighter shades; the TTB runs in darker shades pale. Blue is used for the direct, cooling, exothermic transformations; red for the reverse, heating, endothermic transformations. Start, finish and peak temperatures are listed in Table2. The insets zoom in the peak of the DTA anomaly to highlight the rough behavior of the direct transformation in the ASB (c) and TTB (d) samples. The peak temperatures and the total enthalpy excesses are listed in Table2. (Color figure online) trace TTB (d) c) d) c) ASB (d) TTB (r) ASB (r) exothermic endothermic d) Temperature/ Temperature DTA Enthalpy excess TTBASB Table 2 Start (s), peak (p), and finish (f) temperatures for the martensite (direct) transformation (M) and the austenite (reverse) transformation (A) in the Ni55Fe19Ga26 bulk sample without (ASB) and with (TTB) thermal treatment The transition enthalpies Δh=hlow −hhigh for the direct (cooling) and reverse (heating) transformation are also displayed. See Fig.3 for the DTA traces from which these data originated Ms∕K Mp∕K Mf∕K As∕K Ap∕K Af∕K Δ h d∕Jg −1 Δ h r∕Jg −1 ASB 327 322.7 315 322 329.8 335 − 2.7 − 2.9 TTB 317.3 313.1 308.4 314.2 319.6 323.6 − 3.1 − 3.3 J.-M.Martín-Olalla et al. by 20% to Δ h d =−3.1 Jg −1 and Δ h r =−3.3 Jg −1 . These results are inline with those in Ref.[12] (see Fig.7). The evolution of the partial integration as a function of the temperature is shown in Fig.3b. Avalanches The scanning rates of the previous experiments were much smaller than conventional DSC scans. The DTA cooling traces showed a distinct rough behavior which is magnified in the insets of Fig.3. This is the result of the coalescence of a myriad of fast, microscopic exchanges, called jerks or avalanches that characterize the martensitic transformation. In our experiment, they occurred in a short scale of time compared to the scale of time of the experiment. Moreover, the insets of Fig.3 show 10 times larger excursions in the TTB sample. In Discussion, we will develop arguments that link this behavior to the distribution of 𝛾− phase in either sample. The jerks or avalanches characterize, at the macroscopic scale, the energies associated with the intermittent dynamic of the phase transition as shown by acoustic emission measurements[26], calorimetry[34, 45] and by high precision calorimetry[18, 21, 32, 33]. They are best analyzed in the adiabatic limit, when the thermal driving is slow enough to prevent overlapping. In that circumstances events are individualized, and their distribution is physically meaningful. Therefore, ultraslow experiments were conducted on the TTB sample following the strategy of Romero etal. [46]. Now the rate of temperature change was controlled by the Julabo FP45 and was set to 𝛽2=±40 mK h−1 , ∼25 slower than the previous run. Figure4 shows an interval of 0.5K for the slow and the ultraslow runs. The panel a shows the slow cooling run; panel b, the ultraslow cooling run; and panel c), the ultraslow heating run. Notice that in panels a and b the temperature increases from right to left so that in every data set the time goes from left to right. Figure4 shows the raw signal provided by the fluxmeters at a sampling rate of 12.5Hz scaled by their sensitivity. In panel c the standard noise of the signal ( ∼0.3 μW full height) is evident due to the high magnification of the vertical axis. In order to characterize the spikes, we smoothed the cooling signal with a fifth-order all-pole Butterworth filter with a cut-off frequency of 1 Hz . On the other hand, the heating signal was resampled at 0.1Hz by averaging blocks of 125 data points. Thereafter, we identified local maxima and local minima: events when the signal stops Fig. 4 A close-up of the heat flux for 𝛽=−1Kh −1 (a, 0.5 h), 𝛽=−40 mK h−1 (b, 12.5h); and 𝛽 =+40 mK h− 1 (c, 12.5h). Notice that the x− axis is conformed so that the time advances from left to right in every panel. b, c show only one in every ten data points. The vertical axis in c is six times magnified relative to b. The top left label in b, c display the number of jerks higher than 30nV in the plot. The jerks are located by vertical lines at the horizontal axis. The temperature interval shown in b, c are those of the highest activity in either run, see Fig.6 Ultraslow calorimetric studies ofthemartensitic transformation ofNiFeGa alloys: detection… increasing (maxima) or stops decreasing (minima). The size of the event (spike, jerk or avalanche) is associated with the size of a monotonous up rise from one minimum to the following maximum. The energy of the event is determined by scaling the heat flux with the time constant of the calorimeter ( 𝜏=70 s )[21]. We searched for a threshold E0 such that events larger or equal than E0 occurred only during the temperature interval of the martensite transformation. Therefore, they can be attributed to the transformation. The threshold was identified at E0=30 nV in the raw signal of the voltmeter; 𝜙0=E0∕ 𝛼 =0.2 μW in the heat flux scale and u0=𝜙0𝜏=13 μJ in the energetic scale. The top, left label in panels B and C of Fig.4 shows the number of events larger than the threshold found in the interval. Figure5 shows histogram plots of the activity—number of events E>E0 per kelvin—measured along 0.2K-width bins for either ultraslow run. For the direct (cooling) transformation we collected N=1756 such events with median value 26 μJ ; they made U=74 μJ , or 1.5% of the total enthalpy. For the reverse transformation we identified N=830 events, with median value 19μJ and making 18 μJ or 0.4% of the total enthalpy. Finally, we computed the empirical cumulative complementary distribution function (empirical CCDF ): the shares of avalanches larger or equal than a given size. The empirical CCDF was determined by ranking the avalanches from the largest (rank one) to the smallest (rank N, assigned to u0 ). Then, the empirical CCDF is the ratio of the rank to the sample size. Figure6 shows the empirical CCDF against the avalanche size u in the energetic scale for the direct (dark blue) and reverse (dark red) transformation. The empirical CCDF shows a quasilinear behavior in the log −log plot, indicating a power Fig. 5 Histogram plot showing the activity—number of events per kelvin—derived from the number of events larger than u0=13 μJ observed in a 0.2K interval for the ultraslow experiments. The bluish histogram shows the direct (cooling) run; the reddish histogram shows the reverse (heating) run. The x− axis shows the austenite and martensite start and finish temperatures, see Table2 direct reverse Temperature/ Activity/ Fig. 6 The energetic distributions of avalanche sizes for the ultraslow runs in the direct (dark blue) and reverse (dark red) transformations. The vertical axis displays the empirical complementary distribution function (ECCDF). The horizontal axis displays the energetic size of the avalanche. The quasilinear behavior in the log −log scale suggests a power law distribution with exponent 𝜀 in the range (2,3). The lack of very large energetic sizes suggests an exponential cutoff with characteristic energy  𝜉 . (Color figure online) Empirical CCDF (direct) Empirical CCDF (reverse) (pure power law) CCDF with cutoff Avalanche energy J.-M.Martín-Olalla et al. law distribution of events p(u)∝u−𝜀 , where the larger avalanches are increasingly scarce. Notably, as observed in Fig.4 and in Fig.3, the size of the largest avalanches during the reverse transformation is one order of magnitude smaller than those observed during the direct transformation. These results show different behavior in the direct and reverse transformations with greater activity, larger sizes and smaller exponent in the direct experiment. These differences have already been reported and studied in martensitic transformations[33, 45, 47]. In Sect.4, we will further discuss this issue. Discussion Aside from the development of the 14M martensite phase, the main effect of thermal treatment is the coarsening of the 𝛾− phase and the austenite crystals, which alter the differences between the direct and reverse transformations[35]. In Fig.2, this is evidenced by the increase in the distances between 𝛾− phase crystals. The intermittent dynamics of the direct and the reverse transitions in martensitic transformations are characteristically asymmetric: exponents and accumulated energy differ in either transformations. This has been previously associated with the fact that nucleation is required in the direct transformation, whereas the reverse transformation occurs by fast shrinkage of martensitic domains[47]. Alternatively, the different mechanism of the relaxation of elastic strain energy in either transition was proposed as a source of the asymmetry[45]. The asymmetry manifests itself even in the adiabatic conditions, see here Figs.4–6, see also Ref.[33], which put forward the intrinsic nature of the phenomenon. Analytically, the slope of the CCDF at the top of the distribution characterizes the asymmetry. Our results in Fig.6 show 𝜀∼2.3 (direct) and 𝜀∼3 (reverse). Either case the exponents are larger than 𝜀=2 , expected for monoclinic to cubic transformation[48] and larger that the exponents reported previously on a NiFeGaCo sample from DSC cooling measurements ( 𝜀∼1.9 ), see Ref.[35, 49]. Planes and Vives [47] using acoustic emission characterized the asymmetry in, among others, Cu–Zn–Al shape-memory alloy single crystals, which undergoes a cubic to monoclinic phase transition. They reported different exponents for the direct and reverse transformation and different distributions of energy sizes—power law (direct) and power law with exponential cutoff (reverse)—from a maximum likelihood analysis of the empirical observations. The analysis included a bidimensional chart of 𝜀,𝜉 for several choices of Emin . Inspired by this analysis, and considering that the grain boundaries must stop the growing of avalanches, we tested the distribution of calorimetric events against a power law with exponential cutoff for events larger than u0 (see also Ref.[50]). The model is characterized by an exponent 𝜀 and a damping energy 𝜉 which further hinders the probability of observing large events: p(u;𝜀,𝜉)∝u−𝜀×exp(−u∕𝜉) . For our analysis, we derived the empirical complementary cumulative distribution function ECCDF for the N observed events larger than u0 (see Sect.3.3). On the other hand, we set an array 200 ×200 of 𝜀i,𝜉j pairs and computed: by numerical integration. The 𝜀 values were arranged linearly from 1 to 2.5 (direct) and 1 to 3.5 (reverse); while the 𝜉 values were logarithmically arranged from 100.5 u 0 to 103u0 . Finally, we computed the Kolmogorov–Smirnov distance given by the maximum absolute distance between the two, empirical and model, CCDF : The Kolmogorov–Smirnov distance provides a p value for the null hypothesis “the empirical distribution originates from the tested model”: at the standard level of confidence 𝛼=0.05 the null hypothesis sustains when D is below D0=1.358 [51]. Figure7a shows the bi-dimensional grid 𝜀,𝜉 and, by the color depth, the Kolmogorov–Smirnov distance Dij . The heatmap highlights the region where Dij <2 : darker shades show smaller Kolmogorov–Smirnov distance and therefore greater similarity between the empirical distribution and the tested model. The heatmap shows that for pure power law analysis— that is u0∕ 𝜉→ 0 —the exponent goes to ∼2.3 (direct) and ∼3 (reverse), as noted earlier. Understandably, a finite damping energy yields a smaller exponent in the power law component. The inset b shows the relationship between min(Dij) and 𝜀 for the direct (solid blue circles) and the reverse (open red circles) transformations. The dashed line signals D0 and, remarkably, both transformations sustain values below D0 for 𝜀∼2 . The inset c shows the distribution of Dij as a function of 𝜉 at the condition 𝜀=2 . The local minimum  𝜉 for either transformation can easily be spotted; they differ roughly in one order of magnitude. (1) CCDF (u;𝜀i,𝜉j)= ∞ ∫ u t−𝜀i×exp(−t∕𝜉j)dt ∞ ∫ u 0 t−𝜀i×exp(−t∕𝜉j)dt , (2) D (𝜀 i ,𝜉 j )=D ij = √ N×max( � ECCDF(u)−CCDF(u;𝜀 i ,𝜉 j ) � ) . Ultraslow calorimetric studies ofthemartensitic transformation ofNiFeGa alloys: detection… In summary a power-law with exponent 𝜀=2 and an appropriate damping energy suffices to describe the distribution of events both in the direct and in the reverse transformation; the ratio Ξ=  𝜉 d∕ 𝜉 r∼14 of the damping energies characterizes the asymmetry in the transformation. Figure6 also shows the CCDF for these choices of 𝜀=2,  𝜉 in either transformation (dashed lines), compared with the empirical distribution and the pure power-law behavior. Table3 lists reported values from a set of previous AE and calorimetry studies. Pure power-law reports correspond to 𝜉 → ∞ , but the study is always limited by the largest event in the catalog umax . Eventually the largest event is an experimental lower bound for 𝜉 . The table reproduces this deduced lower bounds inlight, and reported values of 𝜉 in bold. Values are shown as log10(u0∕𝜉) . In our case, we hypothesize that the arc melting technique followed by the thermal treatment has resulted in larger presence of local free energy barriers that lead to intermittent dynamics and asymmetry, as well as eventually prevented the growing of larger avalanches thus giving rise to a smaller, finite  𝜉 . Our estimate is some ten times smaller than results extracted from Ref.[35]. (b) (a) (c) Direct Direct Direct Reverse Reverse Reverse Fig. 7 The Kolmogorov–Smirnov distance (color depth) from the empirical CCDF to the CCDF of a biparametric power-law distribution with exponential cutoff (a). The parameters of the model are represented in the x− axis (exponent) and y− axis (damping energy) in a 200 ×200 grid. The inset b shows min(Dij) as a function of 𝜀 . The inset c shows Dij as a function of 𝜉 for the condition 𝜀=2 , representative of the cubic to monoclinic transformation. Either inset shows the direct transformation by solid blue circles and the reverse transformation by open red circles. Notice that either transformation attains similar, low values of Dij at 𝜀=2 (b), whereas the damping parameter differs in one order of magnitude (c), see also Fig.6 Table 3 Deduced upper boundaries (black) and reported values (bold) for log10( u 0∕𝜉) in acoustic emission (AE) jerks and calorimetric jerks for system that undergo cubic to monoclinic martensitic transitions The boundaries are given by log10(u0∕umax) when pure power-law is reported. It provides a lower bound since 𝜉 must be larger than umax . The rate of temperature change in AE experiments was usually in the range of several kelvin per minute. Calorimetric experiments were carried out in DSC calorimetry [35] at similar rates, and in conduction calorimetry (otherwise) at ultraslow rates. Light numbers reproduce deduced lower bounds; bold numbers, reported values log10(u0∕𝜉) System AE Calorimetry Direct Reverse Direct Reverse Cu–Zn–Al[21, 47] −2.0 −2.5−1.3 −1.3 Cu–Al–Be[33] −5.0 −2.0−2.5 −1.0 Ni–Fe–Ga–Co[35] −4.5 −3.0 −2.0 n.a. Ni–Mn–In[18] −3.0 −3.0 −3.0 −3.0 Ni–Fe–Ga (this work) −1.5−0.3