## Abstract

We report a complete structural and magneto-thermodynamic characterization of four samples of the Heusler alloy Ni-Co-Mn-Ga-In, characterized by similar compositions, critical temperatures and high inverse magnetocaloric effect across their metamagnetic transformation, but different transition widths. The object of this study is precisely the sharpness of the martensitic transformation, which plays a key role in the effective use of materials and which has its origin in both intrinsic and extrinsic effects. The influence of the transition width on the magnetocaloric properties has been evaluated by exploiting a phenomenological model of the transformation built through geometrical considerations on the entropy versus temperature curves. A clear result is that a large temperature span of the transformation is unfavourable to the magnetocaloric performance of a material, reducing both isothermal entropy change and adiabatic temperature change obtainable in a given magnetic field and increasing the value of the maximum field needed to fully induce the transformation. The model, which is based on standard magnetometric and conventional calorimetric measurements, turns out to be a convenient tool for the determination of the optimum values of transformation temperature span in a trade-off between sheer performance and amplitude of the operating range of a material.

This article is part of the themed issue ‘Taking the temperature of phase transitions in cool materials’.

## 1. Introduction

Over the last two decades, magnetic refrigeration has attracted a great interest as a technological alternative to the conventional gas compression–expansion technique. The finding of suitable magnetocaloric materials, alternative to Gd, with large and reversible magnetocaloric properties, i.e. isothermal entropy change (Δ*s*) and adiabatic temperature change (Δ*T*_{ad}), for cyclic applications in moderate magnetic fields will play a decisive role to bring this technology into the market [1].

The research in this field was boosted by the introduction in 1997 by Pecharsky and Gschneidner of the ‘giant’ magnetocaloric material Gd_{5}(SiGe)_{4}, showing high Δ*s* at room temperature, associated with a first-order magneto-structural phase transition [2,3]. Since then a great effort has been made towards material systems displaying the first-order phase changes, involving a significant latent heat [4–8].

Among them magnetic shape memory Heusler alloys represent a particularly interesting class [9]. They are rare earth-free, easy to prepare and offer large tailoring possibilities. Their interesting phenomenology arises from a martensitic phase transition from a high-temperature cubic phase (austenite) to a low-temperature low-symmetry phase (martensite) that involves a change in both structural and magnetic properties. Remarkably, thanks to the strong discontinuities of the physical properties at the martensitic transformation, caloric effects can be obtained not only by applying magnetic fields but also stress and pressure, enabling multicaloric applications [10–12].

By exploiting suitable compositional changes of Ni_{2+x}Mn_{1+y}X_{1+z} (*X*=Ga, In, Sn, Sb, *x*+*y*+*z*=0) it has been possible to control the main physical properties of this class of materials and consequently tune the magnetocaloric performances: e.g. critical temperatures, field dependence of the transformation temperature, intensity and nature of the magnetocaloric effect (from direct to inverse) [9]. In particular, a crucial goal of the magnetocaloric research has been modelling the magnetic interactions in martensitic and austenitic phases and increasing the magnetization discontinuity (Δ*M*) at the transformation [13,14]. In off-stoichiometric Ga-based compounds by changing the relative amount of the constituent elements, it is possible to merge martensitic and Curie temperatures and obtain a direct first-order transformation from ferromagnetic martensite to paramagnetic austenite, giving rise to a direct magnetocaloric effect [6]. On the other hand, In-, Sn- and Sb-compounds show, in suitable stoichiometric ranges, a martensitic transformation between a paramagnetic-like martensite and a ferromagnetic austenite, a feature which makes them known in the literature as ‘metamagnetic Heuslers’. In this case, an inverse and remarkable magnetocaloric effect has been obtained [9].

NiCoMnGa-based alloys also belong to the family of metamagnetic Heuslers: stoichiometry controls the critical temperatures, and allows for the realization of materials where the sequence of structural (occurring at the critical temperature *T*_{M}) and magnetic (occurring at and transitions can be swapped: in Ni-Co-Mn-Ga this means that the martensitic transformation can be realized between ferromagnetic phases, between paramagnetic phases and, most interestingly, between a paramagnetic-like martensite and ferromagnetic austenite [15,16]. Additionally, it was found that partial substitution of Ga with In allows one to selectively lower the structural critical temperature while leaving the magnetic critical temperatures almost unaffected [13]. This finding introduces a further degree of freedom in designing the magneto-structural behaviour of these alloys: in particular, it allows one to further separate *T*_{M} and , maximizing the magnetization jump occurring at the transformation. Comparison between quaternary and In-doped quinary compositions allowed to verify that alloys showing higher magnetization jumps displayed also higher structural discontinuities, measured by X-ray diffraction experiments as the relative volume change between the two phases.

Although metamagnetic Heuslers show high values of inverse magnetocaloric effect (adiabatic temperature changes up to Δ*T*_{ad}∼8 K in μ_{0}*H*=1.95 T [17] at the first application of magnetic field) their performances are strongly reduced on subsequent runs of the magnetic field. The hysteretic character of the transition represents a strong drawback for the cyclic use of these materials. It is well assessed that the reversibility of the magnetocaloric effect depends upon two factors: the extent of the hysteresis and the shift in the transition temperature with field [18]. The current research is addressed to systems with low hysteresis and large field dependence of the martensitic transformation temperature, enabling high reversibility rates in moderate magnetic fields (around 1 T). The possible exploitation of minor loops or artificial phase nucleation sites has been proposed; yet, a deeper understanding of thermal and magnetic hysteresis is required to improve materials performances [17,19,20]. Some recent works have been addressed to this [21–23]. Several aspects have to be taken into account both of intrinsic and of extrinsic origin, such as crystalline symmetry and geometric compatibilities of martensite and austenite, local variation of composition, internal stresses, lattice defects, atomic ordering and dynamic properties of the transformation [24–26].

On the other hand, not only hysteresis but also the sharpness of the martensitic transformation plays a crucial role in the effective use of materials, and similarly it is due to both intrinsic and extrinsic effects. The full potential of a material can be exploited only if the applied field is large enough to induce the complete transition, being the magnetocaloric effect proportional to the transformed fraction of phase [8].

In this paper, we report on four samples of the Heusler alloy Ni-Co-Mn-Ga-In; the samples were chosen with similar compositions, critical temperatures and high inverse magnetocaloric effect across their metamagnetic transformation, but different transition widths. We will provide a complete structural and magneto-thermodynamic characterization of the alloys through in-field calorimetry and evaluate the role of the transition width in the magnetocaloric properties by taking advantage of a phenomenological model of the transformation built through geometrical consideration of the entropy versus temperature curves.

## 2. Methods

Ni-Co-Mn-Ga-In samples were prepared by arc melting the stoichiometric amounts of high-purity elements. To prevent oxidation, a protective Ar atmosphere was established through several Ar–vacuum cycles and pure Ti was melted for 3 min prior to every fusion to act as getter of residual oxygen. Melted buttons were turned around and re-melted four times to improve homogeneity; samples where then wrapped in Ta foil and annealed for 72 h at 1173 K in a protective atmosphere, finalized by water quenching. The composition was experimentally determined through energy dispersive spectroscopy (EDS) microanalysis on a Philips 515 scanning electron microscope.

Thermomagnetic analysis (TMA) determined the structural and magnetic critical temperatures of the grown samples by measuring the AC susceptibility in a purpose-built susceptometer working at 0.5 mT and 500 Hz. Temperature-dependent X-ray diffraction patterns were collected with a Thermo ARL X’tra diffractometer equipped with a solid-state Si(Li) Peltier detector and an environmental chamber. Specific heat measurements were performed with a homemade differential scanning calorimeter based on thermoelectric modules [27]. This in-field calorimeter is able to work in 10^{−5} mbar vacuum between 250 and 420 K and in magnetic fields up to 1.8 T. Its temperature control resolution is ±0.01 K and the thermal sweep is controlled by a high-power Peltier cell. The calibration was performed by using a single-crystal sapphire sample. The error of specific heat data is estimated to be about 4%. Such error is due to slightly different vacuum conditions between the calibration and consecutive measurements and due to small oscillations of the temperature sweep rate. The reported measurements were carried out with temperature sweeps in heating and cooling at a rate of 2 K min^{−1}, in zero magnetic field and in a magnetic field μ_{0}*H*=1.8 T.

## 3. Results and discussion

Figure 1 shows the TMA of four Co- and In-doped NiMnGa Heusler alloys of general formula Ni_{50−x}Co_{x}Mn_{50−y}(Ga,In)_{y}: their measured compositions are reported in table 1, as well as their critical temperatures, measured as the inflection points on the susceptibility curves recorded by TMA. All the samples show a similar transformation between a paramagnetic-like martensite, characterized by a null signal of the susceptibility, and a ferromagnetic austenite, evidenced by the high susceptibility region in the TMA curves. The martensitic Curie temperatures occur well below room temperature, between 170 and 223 K; the transformation temperatures are all above room temperature and in a narrow interval, ranging between 350 K for sample S2 and 388 K for sample S1. The austenitic Curie temperatures, which are mostly influenced by the Co content, occur between 430 K (sample S1) and 476 K (sample S3). Besides these similarities, TMA highlights quite different transformation widths and hysteresis.

In order to evaluate the structural properties, powder X-ray diffraction patterns were collected for each sample in a wide temperature range across the transformation. The powders used for the experiments were heat treated to reduce crystal defects and stresses introduced by grinding.

All samples transform from cubic austenite to tetragonal martensite; the diffraction patterns have been fitted through the LeBail algorithm [28] to extrapolate the lattice parameters of the two phases at various temperatures. In the following, we discuss the results for sample S3, which is representative of the whole series. Figure 2 shows a notable region of the diffraction patterns and their evolution with temperature: the lowering of the austenitic reflections and the onset of the tetragonal martensitic phase are clearly visible. The patterns were collected on cooling from austenite to martensite in the range 453–283 K: the martensitic diffraction peaks become clearly visible below 370 K, and traces of the austenitic phase are detectable at the lowest temperature of the series. Figure 3 shows the temperature evolution of the martensitic lattice parameters (figure 3*a*) and the tetragonal distortion *c*_{M}/*a*_{M}√2 (figure 3*b*): the tetragonal plane (the *a*_{M} lattice parameter) shows almost negligible thermal expansion in the measurement range. On the other hand, the tetragonal *c*_{M} axis shows a quite strong contraction coming from high temperature down to 343 K, which is the closest value in the measured series to the cooling martensitic temperature measured by TMA (*T*_{A−M}=350 K) (figure 1). On further cooling, the *c*_{M} axis reverts the trend and increases again to higher values. The anomalous behaviour of the tetragonal axis is evidenced in the evolution of the tetragonal distortion and of the martensitic volume, which show a minimum at the transformation temperature *T*_{A−M}.

The analysis of the temperature dependence of the lattice parameters allows for the determination of several features useful for characterizing the transformation [13]: besides the tetragonal distortion of the martensitic lattice, described above, the relative volume change Δ*V*/*V* provides a measure of the structural discontinuity between martensite and austenite, while the middle eigenvalue of the transformation matrix λ_{2}, defined for the cubic–tetragonal transformation as λ_{2}=*a*_{M}√2/*a*_{A}, is a good parameter [24] for describing the lattice mismatch at the transformation invariant plane. The calculated unit cell volumes and relative changes are reported in figure 3*c*,*d*. From figure 3*c*, it appears that the two phases have different thermal expansion factors: thus, the relative volume change is not constant over temperature. Additional variability, visible as scattering of the computed quantities in the graphs of figure 3, is generated by the thermal drift of the experimental set-up during the pattern acquisitions and by the error propagation induced by calculations; it is therefore sensible to estimate the mean transformation Δ*V*/*V* by considering the linear fit of the computed values at different temperatures and extrapolating at *T*_{A−M}. For sample S3, we estimate Δ*V*/*V* ≈0.85±0.05%. The same mathematical approach has been followed for the estimation of the mean tetragonal distortion and the λ_{2} values.

Table 2 reports the computed values of relative volume discontinuity, Δ*V*/*V* , tetragonal distortion, *c*_{M}/*a*_{M}√2, and invariant plane mismatch, λ_{2}=*a*_{M}√2/*a*_{A}, for the presented samples. The relative volume change is the quantity with the highest variability within the series: the maximum volume discontinuity is ≈1.2±0.1% for sample S2, while the smallest one is ≈0.6±0.1% for sample S1. The values of tetragonal distortion and transformation matrix eigenvalue show a much smaller variance. A similar trend as the one observed for the volume discontinuity is however established: sample S1 shows the least tetragonal distortion and the highest compatibility factor (λ_{2} is the closest to unity of the series), while sample S2 shows the highest tetragonal distortion and the lowest value of λ_{2}.

The thermodynamic and thermomagnetic properties of the four presented samples are explored by measuring their specific heat under magnetic field across the martensitic transformation. In-field differential scanning calorimetry (DSC) offers a large amount of information concerning the thermodynamic and magnetocaloric features of materials exploiting first-order transitions [29]. Figure 4 shows the specific heat of the samples measured with temperature sweeps on heating and cooling in zero and in a 1.8 T applied magnetic field. The specific heat of martensitic and austenitic phases, before and after the transition, is nearly the same for the four samples ( at 300 K and at 400 K), and it shows a slow variation with temperature. At the transition temperature, the presence of peaks in the specific heat confirms that this magnetic transition is of first order; the heating and cooling peaks are separated by the transformation hysteresis, while the magnetic field, as expected for inverse magnetocaloric effect alloys, promotes the high-temperature magnetic phase thus shifting the transformations to lower temperatures. From the shift of the heat flow peaks, we can deduce the values of d*T*/μ_{0}d*H*. The values, calculated for the transition on heating and on cooling, are reported in table 3. In all the samples, the d*T*/μ_{0}d*H* across the cooling transformation is higher than that in the heating branch. This confirms the already reported behaviour [14,30] according to which the magnetic field shifts more the transformation temperature on cooling due to the larger magnetization jump. Observing the temperature of peaks in the cooling and heating measurements, we can deduce also the values of the thermal hysteresis (reported in table 3), which characterize the first-order transition. The different field sensitivity of the cooling and heating critical temperatures affects also the thermal hysteresis: the in-field hysteresis is larger than the zero-field one.

The peaks’ shape, width and height are different for every sample and do not seem to be correlated with the stoichiometric composition. The area under the peaks corresponds to the latent heat (*L*) of the fully transformed phase, which can be calculated by integrating the specific heat data after subtraction of the baseline (*c*_{p,baseline}) between the start and finish temperature of the transformation (*T*_{s} and *T*_{f}):
3.1The calculated latent heat values are comparable with those measured in samples with similar composition [14]. Both the application of magnetic field and the shift to lower temperatures of the transformation observed in the cooling curves result in a sizeable reduction of *L*. The strong action of the magnetic field on the latent heat was already observed in Ni-Mn-Co-Ga-In [14]. The absence of In in the quaternary alloys reduces this effect, which disappears in the parent Ni_{2}MnGa alloy, showing a ferro–ferro martensitic transformation [14].

The comparison between zero and in-field specific heat data gives the possibility to obtain a complete magnetocaloric effect characterization of the samples. The integration of the calorimetry data provides the entropy–temperature curves across the transition at different magnetic fields:
3.2The adiabatic temperature change Δ*T*_{ad}(*T*) and the isothermal entropy change Δ*s*(*T*) can be deduced from the obtained *s*–*T* curves. The errors correlated with this numerical manipulation of specific heat data can be estimated following the discussion of Porcari *et al*. [27] and Pecharsky & Gschneidner [31]. The temperature behaviours of Δ*s*(*T*) and Δ*T*_{ad}(*T*) for a μ_{0}Δ*H* of 1.8 T are reported in figure 5. The results of samples S2 and S4 have been compared with the Δ*s*(*T*) obtained by magnetic measurements using the Maxwell relation and with the Δ*T*_{ad}(*T*) directly measured with a probe based on a Cernox temperature sensor [32]. The results obtained from the different techniques, provided that they have been used with the proper measurement protocol and on strictly the same sample, turn out to be consistent, as already demonstrated in [27]. The peak values Δ*s*_{peak} and Δ*T*_{adpeak} for a magnetic field span μ_{0}Δ*H*=1.8 T across the transformation on heating are shown in table 4. The Δ*T*_{adpeak} values reported in this paper are the highest of all the Ga-based Heuslers [14,27,33,34], reaching almost 2 K T^{−1} for sample S2. For all the samples, the measured Δ*s*_{peak} results lower than the maximum entropy change of the fully induced phase, estimated from latent heat Δ*s*_{full}≈*L*/*T*_{p}. This means that a magnetic field of 1.8 T does not fully induce the transformation in those samples. We can observe in table 4 that there is no close relationship between the Δ*s*_{peak} and the Δ*s*_{full} values: for instance, S4 has the biggest Δ*s*_{full}=14 J kg^{−1} K^{−1} among the four samples but it shows a Δ*s*_{peak} lower than that of S2. This fact underlines that further quantities characterizing the transitions play a role in determining the Δ*s*(*T*) and Δ*T*_{ad}(*T*) of real materials. A large Δ*s*_{full}, which on the basis of the Clausius–Clapeyron relation is proportional to the magnetization difference between the two phases, is not enough to ensure a large Δ*s*_{peak} exploitable in thermomagnetic cycles. There is instead a correlation between Δ*T*_{adpeak} and (d*T*/μ_{0}d*H*); however also in this case none of the samples reaches the maximum expected value of Δ*T*_{ad}, calculated as Δ*H*⋅(d*T*/d*H*) (table 4).

The key to understand the behaviour of these materials is the transformation width (*W*): we can observe in figure 4 that for all the samples the transition occurs over a large temperature range rather than at a well-defined temperature, as expected in principle for a first-order transition. We estimated *W* from the calorimetric measurements as the FWHM of the transformation peaks (table 3), because it is difficult to exactly determine the initial and final temperature of the transition. The quantity *W* assumes a relevant role in determining both Δ*s* and Δ*T*_{ad}. We can observe that S1, which has the narrowest *W*, is the sample in which the 1.8 T magnetic field manages to transform almost all the phase (Δ*s*_{peak}/Δ*s*_{full}=91%). At the same time, this sample has the highest ratio between the measured Δ*T*_{peak} and the maximum exploitable Δ*T*_{ad}, as deduced from the relation Δ*T*_{max}=Δ*H*(d*T*/d*H*).

As for thermal and magnetic hysteresis, several features contribute to the smearing of the transformation over temperature, both extrinsic and intrinsic to the material. Lattice mismatch (e.g. λ_{2}) and volume differences Δ*V*/*V* between the two phases contribute to the total free energy with an elastic strain energy term that plays a major role in broadening the transition and in determining the two-phase stability regions. Coherently, among the measured samples, sample S1, which shows a less pronounced discontinuity of the structural parameters at the transformation (table 2), displays a smaller transformation width (i.e. *W*=3.1±0.3 K). However, a direct correlation between intrinsic structural features and transition width cannot be drawn for all the samples, highlighting the crucial role of extrinsic properties in giving rise to variations of the effective local transition temperature.

The compositional inhomogeneity, caused either by improper melting or by phase splitting due to solubility limits of the various elements in the alloy, can be one of the most important contributions. Compositional mapping performed through EDS analysis shows that compositional fluctuations are present in all samples, mainly involving Mn: the composition errors reported in table 1 are the propagation of the standard deviations calculated for all elements with the experimental error of the EDS technique, which is in our case ±0.1 at.%. Uncertainties on Mn top 0.4 at.%, while the other elements show much lower deviations, in some cases comparable to the experimental error of EDS. These values, although numerically limited, can be significant in these alloys, where variations of 1 at.% on Mn can in some cases shift the martensitic critical temperature for tens of degrees [15].

Besides compositional inhomogeneities, microstructural features such as defects and grain boundaries strongly influence the martensitic transformation process that proceeds through nucleation and growth of one phase into the other following an avalanche criticality type of path [17]. Further analyses specifically targeted to these aspects are needed to improve the comprehension of the phenomenon, aiming at a better exploitation of magnetocaloric materials.

In order to understand the role that each thermodynamic and thermomagnetic parameter, characterizing the transition, plays in determining the magnetocaloric effect features, a simple geometrical model is constructed in the *s*–*T* plane. A similar model, based on magnetization data, has been introduced in [33] in order to correlate the isothermal (Δ*s*) and adiabatic (Δ*T*_{ad}) features of the magnetocaloric effect; in this paper, we generalize its construction to take into account also the transformation width.

The model is built drawing the tangent lines at the inflection points of the two entropy curves across the transition, both in zero field and under applied magnetic field, and the tangent lines at the entropy curves below and above the transition region (figure 6*a*). In this way, the area in the *s*–*T* plane where the magnetocaloric effect is significant looks like a parallelogram. This model physically means that we are considering a linear variation of the phase fraction, the order parameter of the process, over a temperature range *W* centred at the peak temperature *T*_{p} of the *c*_{p}(*T*) curve. Figure 6*b* shows the comparison between the measured specific heat data of sample S2 and the model constructed on such data. The specific heat peaks are approximated by a rectangular shape, which after integration gives rise to the linear trend of the entropy curves at the transition. The rectangle width, *W*, is determined by the latent heat, which must remain the same obtained from the specific heat data, and by the height of the specific heat peak, calculated as the peak value of a Gaussian best fit of the experimental curve. The temperature dependence of the specific heat below and above the transition is considered to be linear and the entropy curves of the austenitic phase with and without magnetic field are assumed to be overlapping: the effect of applied magnetic field on the austenitic phase, where a direct magnetocaloric effect contribution is expected, turns out to be negligible. Both positive Δ*T*_{ad} and negative Δ*s* contributions on the high-temperature peak-tails of the transformation, reported in figure 5 for all the samples, are experimentally observed but below the experimental error. Therefore, the latent heat, represented as the segment *DB*’ in figure 6*c*, is assumed to be the same in zero and applied magnetic field. Although this approximation might seem far from reality, a possible field dependence of latent heat (leading to the *s*(*T*) curves not overlapping in the austenitic range) would not affect the determination of peak values of both Δ*T*_{ad} and Δ*s*: by model construction peak values are realized before the in-field austenitic line starts. The only effect of the magnetic field is to shift the transition temperature *T*_{p} to lower temperature. For clarity reasons, we apply this construction on the entropy curves in heating, only. Curves in cooling and the effects due to thermal hysteresis can be introduced too, as was done by Gottschall *et al.* [26], but this is outside the purpose of this paper.

In this simplified model, five fundamental parameters are enough to describe the transition: the temperature of the transition peak in zero applied magnetic field (*T*_{p}), the total latent heat of the transition (λ), the shift of the transition temperature due to the applied magnetic field Δ*H*⋅(d*T*/d*H*), the transition width (*W*) and the specific heat value of the martensitic phase before the transition (*c*_{pmart}). Thanks to some geometrical proportions (figure 6*c*), considering the triangles *ABD*, *ACE* and *FC’E*, we can link the magnetocaloric features at the transition, Δ*T*_{ad} and Δ*s*, to these five parameters:
3.3and
3.4The segments *CC*^{′} and *BB*^{′} depend on the entropy rate before the transition and can be approximated as
3.5and
3.6The validity of equation (3.3) for real materials was demonstrated in Porcari *et al*. [27] by comparing the Δ*T*_{ad} values obtained from (3.3) and those directly measured and derived from in-field specific heat data. The proportions (3.3) and (3.4) are strictly valid only for purely first-order systems and when the field-induced transition shift is smaller than the transformation width [34]: all the samples presented in this paper comply with these restrictions.

By inserting (3.5) and (3.6) in (3.3) and (3.4), we obtain
3.7and
3.8By combining equations (3.7) and (3.8), it is possible to correlate Δ*s*_{peak} and Δ*T*_{adpeak}:
3.9One can appreciate how equation (3.9) deviates from previous derivations on the same matter [35]. The reason for such difference originates from the substantially different approximations employed to describe the first-order transformation: we must remark that the derivation appearing in Pecharsky *et al.* is obtained outside the range of validity of the present model, i.e. by assuming idealized sharp transitions (*W*=0) where the application of the magnetic field is sufficient to complete the transformation, while in this paper we are dealing with partial phase transformations and finite transition ranges. The denominator of equation (3.9) represents an *effective* specific heat inside the transition region, as is shown in figure 6*b*, with a factor (*L*/*W*) taking into account a contribution of the latent heat that is spread over the whole temperature range of the transition.

Equations (3.7) and (3.8) can be used to estimate the magnetocaloric features of materials from standard magnetometric and conventional zero-field DSC measurements. The only required parameters to perform the calculation are the specific heat before the transition, the latent heat of the transformation in zero applied field, the peak temperature of the transition, its span width and its change with the applied magnetic field. In table 4, we compare the Δ*T*_{ad} calculated using equation (3.8) and those obtained from in-field specific heat data. The calculated values show the same trend as the measured ones but they turn out to be overestimated by about 20%. This overestimation is due to the difference between the smoother trend of the experimental entropy curves when compared with the series of line segments of our geometrical model.

Equations (3.7) and (3.8) also give some simple indication of how the various parameters affect the Δ*s*_{peak} and Δ*T*_{adpeak} values. The first aspect we notice is that higher values of (d*T*/d*H*) and *L* increase both Δ*s*_{peak} and Δ*T*_{adpeak}. The former has a primary role in determining the Δ*T*_{adpeak} and it represents its upper limit when the product *W*⋅*c*_{pmart} tends to zero. Regarding instead Δ*s*_{peak}, it reaches its maximum value, limited by the entropy variation of the fully induced phase *L*/*T*_{p}, when the ratio (d*T*/d*H*)/*W* tends to 1. For a lower width of the transition, this model is no longer valid: anyhow, the value of Δ*s*_{peak} cannot grow more and it is expected to remain constant over a finite temperature range. On the contrary, we can observe that an enlargement of the transition width always decreases both the Δ*s*_{peak} and Δ*T*_{adpeak} values.

These general considerations can be visualized in figure 7. In figure 7*a*, a series of possible *s*–*T* diagrams is represented differing in the transition width *W*. All the other parameters (d*T*/μ_{0}d*H*,*T*_{p},*c*_{p},*L*) are kept constant and equal to those of sample S2. Figure 7*b*,*c* reports the variations of Δ*s*(*T*) and Δ*T*_{ad}(*T*) curves on changing the *W* value. As discussed above, we can observe that Δ*s* reaches its upper limit when *W*=Δ*H*⋅(d*T*/d*H*) (=8.3 K, in this case), while Δ*T*_{ad} continues to grow for *W* tending to zero. It is evident indeed that the transition width *W* plays a key role in changing the Δ*s*_{peak} and Δ*T*_{adpeak} values as compared to their upper limit: *L*/*T*_{p} and , respectively.

As a general consideration aimed at future materials’ design, the sole reduction of *W* should not be the main goal of research: in fact, *W* acts also on the width of the Δ*s*(*T*) and Δ*T*_{ad}(*T*), enlarging the area where the magnetocaloric effect is sizable and can thus be exploited in thermodynamic cycles. A guideline for this analysis comes from a careful consideration of the denominator in equation (3.8). Following a straightforward mathematical manipulation, one can also rewrite equation (3.8) as
3.10where it is evidenced the combined effect of quantities *W* and *L* to determine the Δ*T*_{adpeak} value. Considering that *c*_{pmart} is almost constant for all the samples of this series, we deduce from equation (3.10) that a large latent heat allows a relevant Δ*T*_{adpeak} value to be kept even in the case of a non-negligible transition width. This statement is valid if d*T*/μ_{0}d*H* is kept constant. Instead, it was observed that for some materials, like for the Fe_{2}P-based compounds, a larger latent heat decreases the Δ*T*_{adpeak}, due to its significant effect on d*T*/μ_{0}d*H* [36]. As an example, for sample S1, *L*=2600 J kg^{−1} and the relatively small *W*=3.1 K corresponds to Δ*T*_{adpeak}=2.7 K, while for sample S4 it is *L*=5150 J kg^{−1} and, in spite of a larger width *W*=7.7 K, from equation (3.10) it results in Δ*T*_{adpeak}=3.7 K.

In practical terms, higher values of transformation width, and thus higher working ranges, can be tolerated without excessive decrease of Δ*T*_{adpeak} and Δ*s*_{peak} as long as high transformation latent heat is obtained.

## 4. Conclusion

In this contribution, we have presented a thorough calorimetric and structural characterization of four samples of the Heusler alloy Ni-Co-Mn-Ga-In. The studied materials show high values of inverse magnetocaloric effect, triggered by magnetic field in the surroundings of their martensitic transformation temperatures: compared to the other members of the Co- and In-doped Ni-Mn-Ga alloys up to now reported in the literature, we have measured the highest values of adiabatic temperature change (up to almost 2 K T^{−1}).

The presented samples display similar compositions and magneto-structural phenomenology, yet their martensitic transformations realize in different temperature spans. The different transformation width has to be ascribed to both intrinsic (e.g. structural discontinuity between martensite and austenite) and extrinsic reasons (e.g. sample inhomogeneities, defects, grain boundaries).

The role of the transformation broadening in the magnetocaloric properties has been investigated by developing a geometrical model, which traces the transformation coordinates on the entropy–temperature plane. The model is readily applicable, as it relies on standard magnetometric and conventional DSC measurements.

It is found that the transition width is always detrimental to the magnetocaloric performances of a material, reducing the amount of both isothermal entropy change and adiabatic temperature change obtainable in a given magnetic field and increasing the value of the maximum field needed to fully induce the transformation; yet, the presented model is a convenient tool for estimating the effects of the transition width on the magnetocaloric properties, allowing for the determination of the optimum values of transformation width in a trade-off between sheer performance and amplitude of the operating range of a material.

## Authors' contributions

All the authors contributed to the data analysis and helped draft and revise the manuscript. F.C. and G.P. performed the calorimetric characterization and developed the geometrical model. S.F. synthesized the samples and performed the TMA measurements and the temperature-dependent X-ray diffraction experiments. F.A. and M.S. motivated the study and supervised the research activity. All authors gave final approval for publication.

## Competing interests

The authors declare no competing interests.

## Funding

F.C. thanks Fondazione Cariparma for financial support.

## Acknowledgements

The authors acknowledge Dr Davide Calestani (IMEM-CNR) and Dr Tiziano Rimoldi (University of Parma) for their contributions to the experimental characterization (composition measurements).

## Footnotes

One contribution of 16 to a discussion meeting issue ‘Taking the temperature of phase transitions in cool materials’.

- Accepted May 9, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.