Sent: ]4 October 2023; Revised: 15 March 2024; Accepted: 8 April 2024;
Publised: 19 May 2024

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

José-María Martín-Olalla<EMAIL_ADDRESS>https://orcid.org/0000-0002-3750-9113https://ror.org/03yxnpp24 Departamento de
Física de la Materia Condensada, Facultad de Física, ICMSE CSIC, Universidad
de Sevilla, ES41012 Sevilla, Spain  Antonio Vidal-Crespo  Departamento de
Física de la Materia Condensada, Facultad de Física, ICMSE CSIC, Universidad
de Sevilla, ES41012 Sevilla, Spain  Francisco Javier Romero  Departamento de
Física de la Materia Condensada, Facultad de Física, ICMSE CSIC, Universidad
de Sevilla, ES41012 Sevilla, Spain  Alejandro F. Manchón-Gordón  Instituto de
Ciencia de Materiales de Sevilla, ICMSE CSIC, Universidad de Sevilla, Sevilla,
Spain  Jhon J. Ipus  Departamento de Física de la Materia Condensada, Facultad
de Física, ICMSE CSIC, Universidad de Sevilla, ES41012 Sevilla, Spain  Javier
S. Blázquez  Departamento de Física de la Materia Condensada, Facultad de
Física, ICMSE CSIC, Universidad de Sevilla, ES41012 Sevilla, Spain  María
Carmen Gallardo  Departamento de Física de la Materia Condensada, Facultad de
Física, ICMSE CSIC, Universidad de Sevilla, ES41012 Sevilla, Spain  Clara F.
Conde  Departamento de Física de la Materia Condensada, Facultad de Física,
ICMSE CSIC, Universidad de Sevilla, ES41012 Sevilla, Spain

([)

###### Abstract

We study the thermal properties of a bulk Ni_55Fe_19Ga_26 Heusler alloy in a
conduction calorimeter. At slow heating and cooling rates
($\sim$1\text{\,}\mathrm{K}\text{\,}{\mathrm{h}}^{-1}$$), we compare as-cast
and annealed samples. We report a smaller thermal hysteresis after the thermal
treatment due to the stabilization of the 14M modulated structure in the
martensite phase. In ultraslow experiments
($40\text{\,}\mathrm{mK}\text{\,}{\mathrm{h}}^{-1}$), we detect and analyze
the calorimetric avalanches associated with the direct and reverse martensitic
transformation from cubic to 14M phase. This reveals a distribution of events
characterized by a power law with exponential cutoff $p(u)\propto
u^{-\varepsilon}\exp(-u/\xi)$ where $\varepsilon\sim 2$ and damping energies
$\xi=$370\text{\,}\mathrm{\SIUnitSymbolMicro J}$$ (direct) and
$\xi=$27\text{\,}\mathrm{\SIUnitSymbolMicro J}$$ (reverse) that characterize
the asymmetry of the transformation.

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      155+17+0 (2/0/5/0) _top_
      724+1+0 (1/0/12/0) Section: Introduction
      530+1+0 (1/0/8/0) Section: Experimental
      0+1+0 (1/0/0/0) Section: Results
      441+4+157 (1/3/13/0) Subsection: Structural and microstructural characterization
      368+2+177 (1/2/22/0) Subsection: DTA traces
      715+1+235 (1/3/32/0) Subsection: Avalanches
      1034+1+189 (1/2/83/2) Section: Discussion
      300+1+0 (1/0/2/0) Section: Conclusions
      52+1+0 (1/0/0/0) Section: Declarations
      84+4+0 (1/0/0/0) Section: CRediT authorship contribution statement

ultraslow calorimetry; intermittent dynamics; avalanches; Heusler alloys

ultraslow calorimetry; intermittent dynamics; avalanches; Heusler alloys

††preprint: This is the accepted version and does not reflect post-acceptance
improvements, or any corrections. ©The Authors ††preprint: The Version of
Record (OA) is published in the _Journal of Thermal Analysis and Calorimetry_
10.1007/s10973-024-13206-4.

##  1 Introduction

TiNi-based alloys are extensively favored in various applications owing to
their distinctive shape memory effect and superelasticity [1, 2, 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 cost-
effectiveness and comparatively straightforward processing [5, 6, 7]. In
addition to the conventional thermally induced shape memory effect observed in
TiNi-based alloys, in 1996, 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 $10\mathrm{M}\to 14\mathrm{M}\to\mathrm{L}1_{0}$ 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 are related to the precipitation of the secondary
$\gamma-$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 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, 20, 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 are known to occur intermittently [26,
27, 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
low-symmetry 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 have also provided
results in this field [29, 30, 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 [21, 32, 33, 18] and in differential scanning
calorimetry (DSC) traces [34, 35, 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 Ni_55Fe_19Ga_26($e/a=7.8$) Heusler alloy is
thermally analyzed before and after a thermal treatment, which leads to the
formation of a modulated $14\mathrm{M}$ martensite phase from a previous
tetragonal martensite phase $\mathrm{L}1_{0}$. 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 power-law with an exponential cutoff. The characteristic
damping energy is smaller (higher damping) in the reverse transformation.

##  2 Experimental

An alloy with nominal, atomic composition Ni_55Fe_19Ga_26 was prepared from a
mixture of high-purity constituent elements ($>$99.9\text{\,}\mathrm{\char
37\relax}$$) 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 $2\text{\,}\mathrm{\char 37\relax}$ excess of Ga
was added to compensate losses connected with its evaporation during
fabrication process (melting temperature $303\text{\,}\mathrm{K}$). To obtain
a high chemical homogeneity, the alloy was arc-melted several times inside the
furnace. Then, parts of the obtained ingot were cut and annealed at
$1073\text{\,}\mathrm{K}$ inside a quartz tube under Ar pressure, adding some
Zr wires as getter to prevent oxidation. After an annealing time of
$24\text{\,}\mathrm{h}$, 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 Ni_54.6Fe_19.4Ga_26. 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
(4s^2 3d^8), Fe (4s^2 3d^6), and Ga (4s^2 4p)— 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
$12.5\text{\,}\mathrm{Hz}$ and is converted into a heat flux $\phi$ by scaling
it with a sensitivity determined in a calibrating experiment [39]. If the rate
of temperature change is $\beta$ then the ratio $\phi/\beta$ 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, 41, 42]. The system operates then as high resolution
differential thermal analyser with a sensitivity around
$1\text{\,}\mathrm{\SIUnitSymbolMicro W}$. 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 Ni_55Fe_19Ga_26 system were previously studied
in Ref. [12].

##  3 Results

###  3.1 Structural and microstructural characterization

Figure 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 14M structure. b) XRD pattern of the TTB (open circles) and its Le
Bail fitting using martensite phase and $\gamma-$phase space groups (red
line). The difference is shown at the bottom (black line). The inset depicts a
horizontally zoomed plot.

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$1_{0}$ tetragonal
structure (space group $I4/mmm$) with traces of a $\gamma-$phase (space group
$Pm\overline{3}m$). 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 14M structure
[43]. Therefore, the thermal stress, excess of vacancies, etc. induced by the
quenching process helps to form the non-thermodynamically stable 14M 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 Figure (see Figure 1b) with GOF=2.03. The
modulated structure can be indexed with the monoclinic space group, obtaining
lattice parameters
$a=$0.445\,53(3)\text{\,}\mathrm{nm}$,b=$3.0329(3)\text{\,}\mathrm{nm}$,c=$0.561\,95(3)\text{\,}\mathrm{nm}$$
and monoclinic angle equal to $None$. The relation $b=7a$ indicates a seven
fold increase in the unit cell length along the $b$ axis, coherently with a
modulation 14M. We obtained a lattice parameter
$a=$0.360\,53(3)\text{\,}\mathrm{nm}$$ for the cubic structure corresponding
to the $\gamma-$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 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.

Figure 2: SEM micrographs obtained using SE —panels a) and d)— and BSE —panels
b) and e)— and corresponding compositional maps obtained by EDS —panels c) and
f)—. Panels a), b) and c) correspond to the ASB sample; and panels d), e) and
f) refers to the TTB sample. Ni, Fe and Ga elements are represented in orange,
green and blue, respectively. The dashed area in panel e) is magnified in
panel f)

The EDS elemental mappings of the produced samples are depicted in Figure 2
—panels 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 $\gamma-$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 $\gamma-$phase, which is slightly enriched in Fe and depleted
in Ga compared to the surrounding martensite matrix.

Sample | Martensite phase |  $\mathbf{\gamma}-$phase
---|---|---
|  Ni |  Fe |  Ga |  Ni |  Fe |  Ga
ASB | $56(2)$ | $19(3)$ | $25(3)$ | $56(2)$ | $26(3)$ | $18(3)$
TTB | $56(2)$ | $19(3)$ | $25(3)$ | $57(2)$ | $27(3)$ | $16(3)$
Table 1: Chemical composition (atomic percentage) for the bulk samples
obtained from EDS.

###  3.2 DTA traces

The ASB sample with dimensions
$$9.0\text{\,}\mathrm{mm}$\times$9.57\text{\,}\mathrm{mm}$\times$2.35\text{\,}\mathrm{mm}$$
($\text{length}\times\text{width}\times\text{height}$) and
$m=$1.5987(1)\text{\,}\mathrm{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.

While inside the calorimeter the ASB was heated at $335\text{\,}\mathrm{K}$,
in the austenite phase. Then the Julabo controller was set to
$300\text{\,}\mathrm{K}$ and the sample was cooled down against this fixed
setpoint; the martensitic transformation took place. Thereafter, the sample
ASB was further cooled down to $270\text{\,}\mathrm{K}$ and heated back
against a fixed setpoint of $340\text{\,}\mathrm{K}$, 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
$1\text{\,}\mathrm{K}\text{\,}{\mathrm{h}}^{-1}$ (some
$0.015\text{\,}\mathrm{K}\text{\,}{\mathrm{min}}^{-1}$), 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.

Figure 3: The DTA traces (top, a) and the corresponding evolution of the
enthalpy excesses (bottom, b) for the slow runs
($\beta\sim$1\text{\,}\mathrm{K}\text{\,}{\mathrm{h}}^{-1}$$) on a
Ni_55Fe_19Ga_26 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 behaviour 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.

Table 2 lists the start and finish temperatures —determined from the
deviations of the DTA trace— and the peak temperatures. The direct
transformation peaked at $M_{p}=$322.7\text{\,}\mathrm{K}$$, and the reverse
transformation, at $A_{p}=$329.8\text{\,}\mathrm{K}$$, for the ASB sample. On
the other hand, the TTB showed cooler peak temperatures
$M_{p}=$313.1\text{\,}\mathrm{K}$$ and $A_{p}=$319.6\text{\,}\mathrm{K}$$, in
agreement with Ref. [12]. The TTB sample also showed narrower anomalies and
smaller thermal hysteresis $A_{p}-M_{p}=$7.1\text{\,}\mathrm{K}$$ (ASB) vs
$A_{p}-M_{p}=$6.5\text{\,}\mathrm{K}$$ (TTB). Thermal hysteresis was much
smaller than previously reported values [16, 12]
($\sim$25\text{\,}\mathrm{K}$$) for experiments carried out at $1000$ times
faster rates of temperature change.

Both cooling runs showed a distinct rough behaviour in the upper half of the
direct transformation (see the insets I1 and I2 in Figure 3), which was absent
in the reverse transformation. This is evidence of a jerky behaviour in the
direct transformation that will be later analyzed.

| $M_{s}/$\mathrm{K}$$ | $M_{p}/$\mathrm{K}$$ | $M_{f}/$\mathrm{K}$$ | $A_{s}/$\mathrm{K}$$ | $A_{p}/$\mathrm{K}$$ | $A_{f}/$\mathrm{K}$$ | $\Delta h_{d}/$\mathrm{J}\text{\,}{\mathrm{g}}^{-1}$$ | $\Delta h_{r}/$\mathrm{J}\text{\,}{\mathrm{g}}^{-1}$$
---|---|---|---|---|---|---|---|---
ASB | $327$ | $322.7$ | $315$ | $322$ | $329.8$ | $335$ | $-2.7$ | $-2.9$
TTB | $317.3$ | $313.1$ | $308.4$ | $314.2$ | $319.6$ | $323.6$ | $-3.1$ | $-3.3$
Table 2: Start ($s$), peak ($p$), and finish ($f$) temperatures for the
martensite (direct) transformation ($M$) and the austenite (reverse)
transformation ($A$) in the Ni_55Fe_19Ga_26 bulk sample without (ASB) and with
(TTB) thermal treatment. The transition enthalpies $\Delta
h=h_{\text{low}}-h_{\text{high}}$ for the direct (cooling) and reverse
(heating) transformation are also displayed. See Figure 3 for the DTA traces
from which these data originated.

The total enthalpy change ($\Delta h=h_{\text{low}}-h_{\text{high}}$) in
either transformation is also listed in Table 2. For the ASB sample we found
$\Delta h_{\mathrm{d}}=$-2.7\text{\,}\mathrm{J}\text{\,}{\mathrm{g}}^{-1}$$
(direct) and $\Delta
h_{\mathrm{r}}=$-2.9\text{\,}\mathrm{J}\text{\,}{\mathrm{g}}^{-1}$$ (reverse).
For the TTB sample, the excesses increased by $20\text{\,}\mathrm{\char
37\relax}$ to $\Delta
h_{\mathrm{d}}=$-3.1\text{\,}\mathrm{J}\text{\,}{\mathrm{g}}^{-1}$$ and
$\Delta h_{\mathrm{r}}=$-3.3\text{\,}\mathrm{J}\text{\,}{\mathrm{g}}^{-1}$$.
These results are inline with those in Ref. [16] and Ref. [12] (see their
Figure 7). The evolution of the partial integration as a function of the
temperature is shown in Figure 3b.

###  3.3 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 Figure 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 Figure 3 show 10 times larger
excursions in the TTB sample. In Discussion we will develop arguments that
link this behavior to the distribution of $\gamma-$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 [21, 32, 33, 18]. 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 $\beta_{2}=$\pm 40\text{\,}\mathrm{mK}\text{\,}{\mathrm{h}}^{-1}$$,
$\sim 25$ slower than the previous run.

Figure 4: A close-up of the heat flux for
$\beta=$-1\text{\,}\mathrm{K}\text{\,}{\mathrm{h}}^{-1}$$ (a,
$0.5\text{\,}\mathrm{h}$),
$\beta=$-40\text{\,}\mathrm{mK}\text{\,}{\mathrm{h}}^{-1}$$ (b,
$12.5\text{\,}\mathrm{h}$); and
$\beta=$+40\text{\,}\mathrm{mK}\text{\,}{\mathrm{h}}^{-1}$$ (c,
$12.5\text{\,}\mathrm{h}$). Notice that the $x-$axis is conformed so that the
time advances from left to right in every panel. Panel (b) and (c) show only
one in every ten data points. The vertical axis in panel (c) is six times
magnified relative to panel (b). The top left label in panels (b) and (c)
display the number of jerks higher than $30\text{\,}\mathrm{nV}$ in the plot.
The jerks are located by vertical lines at the horizontal axis. The
temperature interval shown in panel (b) and (c) are those of the highest
activity in either run, see Figure 6.

Figure 4 shows an interval of $0.5\text{\,}\mathrm{K}$ for the slow and the
ultraslow runs. The panel a) shows the slow cooling run; panel b), the
ultraslow cooling run; 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 $12.5\text{\,}\mathrm{Hz}$
scaled by their sensitivity. In panel c) the standard noise of the signal
($\sim$0.3\text{\,}\mathrm{\SIUnitSymbolMicro W}$$ 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
$1\text{\,}\mathrm{Hz}$. On the other hand, the heating signal was resampled
at $0.1\text{\,}\mathrm{Hz}$ by averaging blocks of $125$ data points.
Thereafter we identified local maxima and local minima: events when the signal
stops 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
($\tau=$70\text{\,}\mathrm{s}$$) [21].

Figure 5: Histogram plot showing the activity —number of events per kelvin—
derived from the number of events larger than
$u_{0}=$13\text{\,}\mathrm{\SIUnitSymbolMicro J}$$ observed in a
$0.2\text{\,}\mathrm{K}$ 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.

We searched for a threshold $E_{0}$ such that events larger or equal than
$E_{0}$ occurred only during the temperature interval of the martensite
transformation. Therefore, they can be attributed to the transformation. The
threshold was identified at $E_{0}=$30\text{\,}\mathrm{nV}$$ in the raw signal
of the voltmeter;
$\phi_{0}=E_{0}/\alpha=$0.2\text{\,}\mathrm{\SIUnitSymbolMicro W}$$ in the
heat flux scale and $u_{0}=\phi_{0}\tau=$13\text{\,}\mathrm{\SIUnitSymbolMicro
J}$$ in the energetic scale. The top, left label in panels B and C of figure 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 $E>E_{0}$ per
kelvin— measured along $0.2\text{\,}\mathrm{K}$-width bins for either
ultraslow run. For the direct (cooling) transformation we collected $N=$1756$$
such events with median value $26\text{\,}\mathrm{\SIUnitSymbolMicro J}$; they
made $U=$74\text{\,}\mathrm{mJ}$$, or $1.5\text{\,}\mathrm{\char 37\relax}$ of
the total enthalpy. For the reverse transformation we identified $N=$830$$
events, with median value $19\text{\,}\mathrm{\SIUnitSymbolMicro J}$ and
making $18\text{\,}\mathrm{mJ}$ or $0.4\text{\,}\mathrm{\char 37\relax}$ of
the total enthalpy.

Finally, we computed the empirical cumulative complementary distribution
function (empirical $\mathrm{CCDF}$): the shares of avalanches larger or equal
than a given size. The empirical $\mathrm{CCDF}$ was determined by ranking the
avalanches from the largest (rank one) to the smallest (rank $N$, assigned to
$u_{0}$). Then, the empirical $\mathrm{CCDF}$ is the ratio of the rank to the
sample size. Figure 6 shows the empirical $\mathrm{CCDF}$ against the
avalanche size $u$ in the energetic scale for the direct (dark blue) and
reverse (dark red) transformation. The empirical $\mathrm{CCDF}$ shows a quasi
linear behaviour in the $\log-\log$ plot, indicating a power law distribution
of events $p(u)\propto u^{-\varepsilon}$, where the larger avalanches are
increasingly scarce.

Notably, as observed in Figure 4 and in Figure 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 behaviour 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 [47, 45, 33]. In section 4, we will
further discuss this issue.

Figure 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 behaviour in the $\log-\log$ scale suggest a power law
distribution with exponent $\varepsilon$ in the range $(2,3)$. The lack of
very large energetic sizes suggest an exponential cutoff with characteristic
energy $\hat{\xi}$.

##  4 Discussion

Aside from the development of the 14M martensite phase, the main effect of
thermal treatment is the coarsening of the $\gamma-$phase and the austenite
crystals, which alter the differences between the direct and reverse
transformations [35]. In Figure 2, this is evidenced by the increase in the
distances between $\gamma-$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 Figure 4, Figure 5, Figure 6, see also Ref.
[33], which put forward the intrinsic nature of the phenomenon. Analytically,
the slope of the $\mathrm{CCDF}$ at the top of the distribution characterizes
the asymmetry. Our results in Figure 6 show $\varepsilon\sim 2.3$ (direct) and
$\varepsilon\sim 3$ (reverse). Either case the exponents are larger than
$\varepsilon=2$, expected for monoclinic to cubic transformation [48] and
larger that the exponents reported previously on a NiFeGaCo sample from DSC
cooling measurements ($\varepsilon\sim$1.9$$), see Ref. [49, 35].

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 distribution 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 $\varepsilon,\xi$ for several choices of
$E_{\min}$.

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 $u_{0}$
(see also Ref. [50]). The model is characterized by an exponent $\varepsilon$
and a damping energy $\xi$ which further hinders the probability of observing
large events: $p(u;\varepsilon,\xi)\propto
u^{-\varepsilon}\times\exp(-u/\xi)$. For our analysis we derived the empirical
complementary cumulative distribution function $\mathrm{ECCDF}$ for the $N$
observed events larger than $u_{0}$ (see Section 3.3). On the other hand, we
set an array $200\times 200$ of $\varepsilon_{i},\xi_{j}$ pairs and computed:

$\mathrm{CCDF}(u;\varepsilon_{i},\xi_{j})=\dfrac{\displaystyle\int\limits_{u}^{\infty}t^{-\varepsilon_{i}}\times\exp(-t/\xi_{j})dt}{\displaystyle\int\limits_{u_{0}}^{\infty}t^{-\varepsilon_{i}}\times\exp(-t/\xi_{j})dt},$
(1)

by numerical integration. The $\varepsilon$ values were arranged linearly from
$1$ to $2.5$ (direct) and $1$ to $3.5$ (reverse); while the $\xi$ values were
logarithmically arranged from $10^{0.5}u_{0}$ to $10^{3}u_{0}$.

Finally we computed the Kolmogorov-Smirnov distance given by the maximum
absolute distance between the two, empirical and model, $\mathrm{CCDF}$:

$\begin{split}&D(\varepsilon_{i},\xi_{j})=D_{ij}=\\\
&\sqrt{N}\times\max(|\mathrm{ECCDF}(u)-\mathrm{CCDF}(u;\varepsilon_{i},\xi_{j})|).\end{split}$
(2)

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 $\alpha=$0.05$$ the null hypothesis sustains when $D$ is
below $D_{0}=$1.358$$ [51].

Figure 7: The Kolmogorov-Smirnov distance (color depth) from the empirical
$\mathrm{CCDF}$ to the $\mathrm{CCDF}$ 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
$200\times 200$ grid. The inset b) shows $\min(D_{ij})$ as a function of
$\varepsilon$. The inset c) shows $D_{ij}$ as a function of $\xi$ for the
condition $\varepsilon=2$, 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 $D_{ij}$ at $\varepsilon=2$ (b),
whereas the damping parameter differs in one order of magnitude (c), see also
Figure 6.

Figure 7a shows the bi-dimensional grid $\varepsilon,\xi$ and, by the color
depth, the Kolmogorov-Smirnov distance $D_{ij}$. The heatmap highlights the
region where $D_{ij}<2$: 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 $u_{0}/\xi\to 0$—
the exponent goes to $\sim$2.3$$ (direct) and $\sim$3$$ (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
$\min(D_{ij})$ and $\varepsilon$ for the direct (solid blue circles) and the
reverse (open red circles) transformations. The dashed line signals $D_{0}$
and, remarkably, both transformations sustains values below $D_{0}$ for
$\varepsilon\sim 2$. The inset c) shows the distribution of $D_{ij}$ as a
function of $\xi$ at the condition $\varepsilon=2$. The local minimum
$\hat{\xi}$ for either transformation can easily be spotted; they differ
roughly in one order of magnitude.

In summary a power-law with exponent $\varepsilon=2$ and an appropriate
damping energy suffices to describe the distribution of events both in the
direct and in the reverse transformation; the ratio
$\Xi=\hat{\xi}_{d}/\hat{\xi}_{r}\sim$14$$ of the damping energies
characterizes the asymmetry in the transformation. Figure 6 also shows the
$\mathrm{CCDF}$ for these choices of $\varepsilon=2,\hat{\xi}$ in either
transformation (dashed lines), compared with the empirical distribution and
the pure power-law behaviour.

| $\log_{10}(u_{0}/\xi)$
---|---
System | AE | Calorimetry
| Direct | Reverse | Direct | Reverse
Cu-Zn-Al [47, 21] | $-2.0$ | $-2.5$ | $-1.3$ | $-1.3$
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) |  |  | $-1.5$ | $-0.3$
Table 3: Deduced upper bounduaries (black) and reported values (blue) for
$\log_{10}(u_{0}/\xi)$ in acoustic emission (AE) jerks and calorimetric jerks
for system that undergo cubic to monoclinic martensitic transitions. The
boundaries are given by $\log_{10}(u_{0}/u_{\max})$ when pure power-law is
reported. It provides a lower bound since $\xi$ must be larger than
$u_{\max}$. 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.

Table 3 lists reported values from a set of previous AE and calorimetry
studies. Pure power-law reports correspond to $\xi\to\infty$, but the study is
always limited by the largest event in the catalog $u_{\max}$. Eventually the
largest event is an experimental lower bound for $\xi$. The table reproduces
this deduced lower bounds in black, and reported values of $\xi$ in blue ink.
Values are shown as $\log_{10}(u_{0}/\xi)$.

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 $\hat{\xi}$. Our estimate is some ten times smaller than
results extracted from Ref. [35].

On the other hand, if the transition mechanisms were associated with an
elastic, surface energy [45] then $\sqrt{\Xi}\sim 4$ would be a rough estimate
of the ratio in the longitudinal size of the surface driving the avalanches.
Alternatively, if they were associated with martensite phase developed in an
austenite crystal [47], $\sqrt[3]{\Xi}\sim 2.5$ would be a rough estimate of
the longitudinal size of the characteristic volume driving the avalanches.
Furthermore, in this case a rough estimation of an upper bound for the volume
of sample affected in a single event can be obtained from the following
argument. We first take the total specific energy of the transformation
($|\Delta h|\sim$3\text{\,}\mathrm{J}\text{\,}{\mathrm{g}}^{-1}$$, see Table
2), the sample mass $m$ and the sample volume $V$ (see Section 2). Then, the
volume size of an event $e$ is proportional count $v=Ve/|\Delta h|m$ and the
diameter of a sphere of equal volume is $d=6(Ve/|\Delta h|m)^{1/3}/\pi$. For
the median value of the event distribution —$\langle
e_{d}\rangle=$26\text{\,}\mathrm{\SIUnitSymbolMicro J}$$— we get
$d\sim$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ which is roughly in
agreement with the size of the austenite crystals that can be deduced from
Figure 2 as this grain boundaries should stop the progression of martensite
fronts during the transformation and would give rise to
$\hat{\xi}_{d}=$372.2\text{\,}\mathrm{\SIUnitSymbolMicro J}$$, equivalent to
the volume of a sphere $300\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in
diameter. We note that less than $1\text{\,}\mathrm{\char 37\relax}$ of the
recorded events were larger than $\hat{\xi}_{d}$. In contrast, the reverse
transition which occurs by fast shrinkage of the martensitic domains [47]
showed a smaller damping energy
$\hat{\xi}_{r}=$25\text{\,}\mathrm{\SIUnitSymbolMicro J}$$, similar to the
median value of the event distribution ($\langle
e_{r}\rangle=$19\text{\,}\mathrm{\SIUnitSymbolMicro J}$$) and resulted in
roughly $20\text{\,}\mathrm{\char 37\relax}$ of the recorded reverse events
larger than $\xi_{r}$.

Accordingly, the lack of intermittent dynamics in the ASB samples suggests
$\xi$ smaller than the observed values in the TTB sample. Understandably, the
larger distribution of $\gamma-$phase should have further hindered the size of
the events and resulted in a smoother distribution, see Figure 3.

##  5 Conclusions

We have thermally characterized a Ni_55Fe_19Ga_26 alloy produced by the arc
melting technique with and without a posterior thermal treatment that promotes
the metastable, adaptative 14M phase at room temperature.

The thermal treatment narrows the thermal hysteresis and the temperature
distance for the start to finish condition. Interestingly, the thermal
treatment enhances the intermittent dynamics (avalanches) of the martensite
transition both in the direct and in the reverse transformations. The jerky
behaviour was more prominent in the direct (cubic to 14M) transformation.

The distribution of avalanches follows a power law with an exponential cutoff
both in the direct and reverse transformation. The exponent $\varepsilon=2$
associated with the cubic to monoclinic transformation suffices to explain the
power law decay both in the direct and reverse transformations. This result is
in line with previous observations. We identified the damping energies
associated with the transformations and found that the reverse transformation
is $10$ times more damped than the direct transformation. This is a
quantification of the asymmetry of the processes involved in the
transformation.

###### Acknowledgements.

A. Vidal-Crespo acknowledges a VI-PPITU fellowship from Universidad de Sevilla
(Spain). The authors wish to thank the staff at Departamento de Física de la
Materia Condensada and Facultad de Física for their continued support. The
X-ray fluorescence, the X-ray diffraction and SEM measurements were taken at
the Centre for Research, Technology and Development (CITIUS) of the
Universidad de Sevilla, with the financial support by Universidad de Sevilla.
The color squeme in Figure 7 is bilbao from Fabio Crameri’s _Scientific colour
maps_ https://www.fabiocrameri.ch/colourmaps/
https://doi.org/10.5281/zenodo.1243862. Red and blue inks that represent the
reverse and direct transformation come from Crameri’s roma color squeme.

## Declarations

This work was supported by PAI of the Regional Government of Andalusia, VI
PPITU and VII PPITU of Universidad de Sevilla, and by Junta de Andalucía-
Consejería de Conocimiento, Investigación y Universidad project
ProyExcel_00360.

The authors declare no competing interests.

Data are available upon reasoned request to the corresponding author.

## CRediT authorship contribution statement

Conceptualization: Francisco Javier Romero; Javier S. Blázquez.

Methodology: José-María Martín-Olalla; Francisco Javier Romero.

Data Curation: José-María Martín-Olalla; Antonio Vidal-Crespo.

Formal Analysis and Investigation: José-María Martín-Olalla; Antonio Vidal-
Crespo; Francisco Javier Romero; Alejandro F. Manchón-Gordón; Jhon J. Ipus;
María Carmen Gallardo.

Original draft preparation: José María Martín-Olalla; Alejandro F. Manchón-
Gordón; Javier S. Blázquez.

Review and editing: Antonio Vidal-Crespo; Francisco J Romero; Jhon J. Ipus;
María Carmen Gallardo; Clara F Conde.

Supervision: Maria Carmen Gallardo; Clara F Conde.

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