In this section, the basic equations for calculating the elastic constants are presented (for more details see Appendix A). The most easily determined quantity is the bulk modulus B, which provides the behavior of the crystal volume or lattice parameters under hydrostatic pressure. There are several ways to calculate the bulk modulus from the energy-volume E tot ( V ) relation (see References [28] [29] [30] ). In the present work, the bulk modulus B is determined by fitting the total energy calculations to the Birch-Murnaghan equation of state [29] [30] . According to this model, the dependence of the energy on the change in the crystal volume V under hydrostatic pressure p is given by

E = E 0 + 9 16 B V 0 14703.6 [ ( 6 − 4 v 2 ) η 2 + B ′ η 3 ] , (1)

where B ′ = d B / d p is the pressure derivative of the bulk modulus and η = v 2 − 1 , with the ratio v = ( V 0 / V ) 1 / 3 of the actual volume V under pressure to the relaxed volume V 0 at the lowest total energy E 0 (see Reference [31] ). The related pressure is given by

p = 3 2 B ( v 7 − v 5 ) [ 1 + 3 4 ( B ′ − 4 ) η ] . (2)

For tetragonal crystals, the dependence of the bulk modulus B on the elastic stiffness is given by

B = 1 9 ( 2 c 11 + 2 c 12 + 4 c 13 + c 33 ) . (3)

In the case of cubic crystals, c 12 = c 13 , c 11 = c 33 , and c 44 = c 66 (see also Appendix A). Therefore, the equations of the elastic constants for cubic systems are easily obtained from the tetragonal equations.

The remaining elastic properties are determined by applying different types of strain ( e i ) to the tetragonal lattice and by applying proper relations between the total energy and the strain components. The energy E ( e i ) of the strained lattice is calculated using Hooke’s law (see Equation (10)). According to Wallace [31] , if the strains e i are small, the change of the energy is given by

Δ E V 0 = ∑     Δ i e i + 1 2 ∑ ∑     c i , j e i e j . (4)

Here, Δ E = E ( e i ) − E 0 with equilibrium energy E 0 at volume V 0 without strain. The linear terms vanish at equilibrium or if the strain causes no change in the volume of the crystal. The elastic constants c i , j are obtained from the second-order terms and are calculated from the second derivatives of the energy with respect to the strains:

c i , j = 2 ∂ 2 E ∂ e i ∂ e j | V = V 0 (5)

The second derivative relation of elastic constants ( c i j ) with total energy highlights the importance of an accurate calculation of the total energy. Therefore, the choice of the density functional theory solver in the calculation of elastic constants and related properties is significant.

For cubic crystals, three independent elastic constants need at least two different strains to be applied in the calculations, while in tetragonal systems (space group Nos. 89-142), six independent elastic constants need five different strains. The third (for cubic systems) or sixth (for tetragonal systems) condition to solve the linear set of equations are found from the calculated bulk modulus using the equation of state (see Equations (1) and (3)). In fact, there are numerous ways to apply the six different strains and their combinations to the crystal. One condition is that the applied strains (δ) should result in different 2^nd derivatives of the strain energy given by Equation (4) (see Table 1). Another, necessary side condition of Equation (5) is that the volume must be conserved when applying the strain. Therefore, the use of linear strain components ( e i = δ in all possible cases and combinations in the strain matrix of Equation (9) of Appendix A) would lead to large uncertainties because they are not always volume conservative, or they make the use of additional derivatives necessary

(for example, ∂ E ∂ V , ∂ V ∂ e i , and higher orders).

Table 1 and Figure 1 summarize the applied strains that are used to determine the elastic constants in the present work. The applied strains are the same as those reported by Kart et al. [32] . The isotropic strain (0) is not used directly for the calculation of the elastic constants, as it gives the same information as discussed for the bulk modulus B (see discussion above). The five strain types (Equations (1)-(5) in Table 1) are chosen to be volume conservative. The last strain type does not conserve the volume, but it keeps the same symmetry as the crystal and thus can be calculated from the energy versus c/a relation [32] , where c/a is the ratio of the two independent lattice parameters of the tetragonal crystals.

In the present work, six distortions of each type in the range of − 3 % ≤ δ ≤ + 3 % were applied to the relaxed structure with V 0 from the structural optimization using the Birch-Murnaghan equation of state. For tetragonal systems, the energy E ( δ ) versus applied strain curves were fitted to a fourth-order polynomial E ( δ ) = E 0 + a 2 δ 2 + a 3 δ 3 + a 4 δ 4 .

Here, an alternative method of verifying the convergence as well as the accuracy of the results is introduced. In principle, it is sufficient to use either Equations (4) or (5) of Table 1 to calculate all six elastic constants. This is the usually applied method. However, in the present work the elastic constants and all related properties are calculated with both equations to ensure that they provide the same results. This happens, indeed, only if the results are well converged (see also Section 3.1). The system is overdetermined by using both types of strains; however, in this way, the accuracy of the calculated quantities can be estimated. In fact, the values reported here have an error below 0.5%. The combination of different strains as well as different types of equations of state allow determination of the uncertainty of the calculated results, which is expected for a reliable computer experiment.

The ab-initio electronic structure calculations were performed using the Wien2k code [33] . The all-electron full-potential method, FLAPW, with an unbiased basis covers all elements of the periodic table with any spin configuration. This feature is essential for Heusler compounds because they may contain diverse types of atoms, including lanthanide and actinide atoms, together with exotic magnetic ordering. The accuracy of this method makes it suitable for the studied systems. For example, Co₂TiAl fails with spherical potentials or full symmetry potentials together with bare exchange-correlation functionals neglecting gradient corrections [34] [35] [36] . Since elastic constants are calculated from the second derivatives of the total energy, an accurate calculation of total energy is extremely important.

The exchange-correlation functional was taken in the generalized gradient approximation of Perdev, Burke, and Enzerhoff (GGA-PBE) [37] [38] . The number of plane waves was restricted by R MT k max = 9 , and the number of k-points was set to 8000 k-points in the full Brillouin zone. As discussed in Section 3.1, these criteria ensure the convergence of the calculated elastic properties for the investigated systems.

The lattice parameters were optimized before calculating the elastic constants. The results of the structural optimizations are summarized in Table 2 along with some previously reported experimental and theoretical values. Here, the c/a ratio was obtained by a full optimization of the Heusler compounds in tetragonal space groups 119 or 139. In other words, to find the energy minimum, not only the c/a ratio changed (the elongation of c) [32] [39] (see also Figure 2) but also the volume of the structures was relaxed. The assignment of lattice parameters should be performed carefully since the optimization was performed in the tetragonal symmetry. When reducing the cubic fcc cell to a tetragonal fct cell, the cubic lattice parameter a = a c becomes c, and the tetragonal parameter a = a t becomes a c / 2 . To better understand the distortion of the cubic

Heusler structure to the tetragonal Heusler structure, the distortion parameter ε is defined by

ε = c a t 1 2 − 1 , (6)

where a_t is the tetragonal a parameter. At ε = 0 , the structure is cubic; at ε < 0 , the cubic cell is compressed; and at ε > 0 , the cubic cell is elongated along one of the principal axes.

The magnetic state was verified using different settings of the initial magnetization: ferromagnetic (all initial spins parallel) or ferrimagnetic (initial spins partially antiparallel). From the studied systems, all Mn₂YZ compounds exhibit ferrimagnetic order in which one Mn has majority and the other Mn has minority orientation. For all other compounds, the ferromagnetic ground state has the lowest total energy.

Stoichiometric Ni₂MnGa undergoes a structural phase transition from the austenite into the martensite phase [42] . Depending mostly on the composition, the martensite structure is characterized by the tetragonal 5M modulated structure with c / a ≈ 0.94 , the orthorhombic 7M structure with c / a ≈ 0.9 , and the non-modulated tetragonal structure with c / a ≈ 1.2 [39] . Figure 2 shows the appearance of different stable and metastable phases with varying c/a elongation. To focus on the cubic phase, Figure 2(b) shows only the small range of strains with c / a < 1 that cover the cubic phase. The deepest energy minimum is located at a distortion of about ε = 0.27 , corresponding to c / a ≈ 1.26 (non-modulated phase). A shallow minimum (see Figure 2(b)) appears at a distortion of about −0.05 ( c / a ≈ 0.94 , 5M phase). The elastic constants of the structure with c / a > 1 will be discussed in Section 3.2.

As shown in Figure 2(c), the metastable cubic phase exists only under an infinitesimal strain and exhibits a very low energy modulation. The optimized lattice constant of cubic Ni₂MnGa with the L2₁ structure is 5.81 Å, in excellent agreement with the experimental value of 5.82 Å [41] . This phase is only stable within ±1 meV energy changes, which confines the lattice distortion to < ±1%. This distortion relates to the tetragonal distortion, providing the c 11 − c 12 combination of elastic constants. To observe such a non-trivial change in energy and to have a smooth dependence on strain, the results need to be precisely converged with very high precision. This does not imply, however, that the results do not need to be converged for a wide energy window with a deep minimum. As an example, the importance of converged results is demonstrated in Figure 3. The c₄₄ shear modulus is stable for a lattice distortion of about ±3% and an energy change of more than ±30 meV. In this case, the rough calculation provides a smooth curve, but the calculated elastic constants significantly deviate from the converged results. The convergence of the results have the same importance for the calculation of the bulk modulus (B) (see Figure 3(c) and Figure 3(d)).

Figure 3 shows the convergence of c₄₄ and B with respect to R MT k max and number of k-points. As shown in (a,c) for constant R MT k max = 7 , increasing the number of k-points increases the c₄₄ and B values. These results converge at 8000 k-points. In contrast, increasing R MT k max decreases c₄₄ and B. Note that a similar result could be obtained at a more relaxed criterion, for example at 2000 k-points and R MT k max = 7 , due to error cancellations. Therefore, R MT k max = 9 and 8000 k-points are the minimum criteria to converge the results for the systems studied in this work.

The calculated elastic constants of Ni₂MnGa with L2₁ structure are given in Table 3. The calculated results show a reasonable agreement with the experimental results and coincide with previously reported theoretical results [32] [39] . Note, however, that good agreement with the experiment results does not guarantee the accuracy of the calculations. First, the experiments were performed at 300 K for Ni₂MnGa in the L2₁ phase, and off-stoichiometry has a significant effect on the measured elastic constants [44] . Moreover, the employed experimental method may result in different measured elastic constants [43] . As an example, the c₄₄ value deviates by about 60 GPa based on the experimental method. In general, the measured elastic constants are inversely related to temperature [45] . Hence, a higher value should be expected for the calculations. Fortunately, c', which is a difference between two constants (c₁₁ and c₁₂), is argued to be less dependent on temperature [45] . In fact, the calculated c ′ = 4.5 GPa exhibits a better agreement with the experiment

(4.1 GPa) than previously reported values [32] [39] .

In addition, c₁₂ and c₁₁ do not have any exclusive physical basis; in other words, no phonon mode directly corresponds to these constants. Mixing with other stiffness ( c i j ), however, results in a meaningful combination. For example, the tetragonal shear modulus c ′ = ( c 11 − c 12 ) / 2 corresponds to cubic-tetragonal distortion. Moreover, it is well established that c'―associated with slow transverse acoustic waves [43] ―plays an important role in the occurrence of structural transformations. Another important quantity is the Cauchy pressure c p = c 12 − c 44 . A negative value of c p ( c 12 < c 14 ) may indicate covalent bonds, where the angular dependence of the inter-atomic forces becomes important.

Furthermore, detailed analysis of elastic constants sheds light on the stability and phase transition in Heusler compounds. Cubic Ni₂MnGa with soft c' is on the border of the phase transition. With the similar interpenetration (see Appendix B), the large elastic anisotropy A e of Ni₂MnGa ( A e = 28 ) hints on its tendency to deviate from the cubic structure. Anisotropy is another indicator for the instability of cubic structures. The elastic anisotropy of crystals is also an important parameter for engineering since it correlates to the possibility of micro-cracks in materials. Unlike the mechanical properties, anisotropy shows the tendency of a system toward phase transitions as it inversely relates to the c’ parameter. In fact, an illustrative way to show the anisotropy is to visualize the rigidity modulus G ( r ^ ) or Young’s modulus E ( r ^ ) . Figure 2(a) shows that the rigidity modulus is largest in the 〈 111 〉 -type direction that is along the tetragonal axes. Such a significant deviation from spherical shape indicates that the moduli of Ni₂MnGa exhibit a large degree of anisotropy. In principle, when c 11 − c 12 → 0 , the rigidity distribution exhibits a stronger directional dependency, as shown for Ni₂MnGa in Figure 2(a).

In the next step, a compound that is not stable in the cubic structure was examined for comparison. In the case of Rh₂FeSn shown in Figure 4, the cubic structure exhibits a maximum of the total energy, and any tetragonal distortion will lead to a different stable structure. Based on Figure 4, the appearance of energy minima are expected for two different tetragonally distorted systems with c / a > 1 and c / a < 1 . Larger distortions show that c / a > 1 is the stable phase, while c / a < 1 is a metastable phase. Previous works only reported the structure

with ε > 0 [27] [46] . However, the calculated elastic constants also supported the instability of the cubic phase from negative values of tetragonal shear modulus c' and anisotropy A_e. The metastable structure with ε < 0 appears when expanding the in-plane lattice parameter a while keeping c at the cubic lattice parameter. Such a situation may be artificially initialized by epitaxial or pseudomorphic thin film growth on a substrate with an appropriate lattice parameter. Similar metastable situations may exist in many other tetragonal Heusler compounds and will open the field of lattice parameter engineering to enlarge the number of properties on demand.

The tetragonal Heusler compounds studied in this work along with their elastic constants are summarized in Table 4 and Table 5. The Heusler intermetallics Mn₂YGa (Y = Fe, Ni) and X₂MnGa (X = Fe, Ni) undergo tetragonal magneto-structural transitions that result in half-metallicity, magnetic shape memory, or magneto-electric effects. In this section, the off-stoichiometric compositions are briefly discussed, and then, the elastic constants and related properties of the Heusler compounds are analyzed. Calculating the elastic properties of the tetragonal phases illuminates the structural transformations, chemical bonding, and mechanical stability of these intermetallic compounds for applications. Likewise, the elastic properties of Rh-based Heusler compounds synthesized by Suits [27] are calculated. Although Ni₂MnGa has been widely studied experimentally at different phases, there is no experimental measurement

of the elastic modulus of the non-modulated tetragonal phase, and only theoretical works on this phase have already been reported [32] [39] .

As shown in Table 4, in the case of Ni₂MnGa, the results show a qualitative agreement with the previous theoretical report (values in brackets). However, a quantitative comparison of the results reveals some significant deviations. These differences can be traced back to the calculation method and the method of performing structural optimization. To address this problem, the cubic phase is briefly considered. In the cubic phase, the present as well as other calculations are performed for the same lattice parameters using different calculation

schemes. Here, the results of FPLAPW calculations are slightly larger than those of projected augmented wave (PAW) calculations, and the deviations range from 1% for c₁₁ up to 7% for c₄₄. These small differences are expected because of the selected methods and convergence criteria. In contrast, in the case of the tetragonal system (see Table 4), the deviations between calculations range up to 30%, such as the case of c₄₄ and c₆₆. Indeed, the different calculation methods should not lead to such a large discrepancy (if all factors are set carefully), and these observed differences mainly arise from the underlying structural optimization. Here, all initial tetragonal structures are fully optimized at their relevant symmetries (I4/mmm or I 4 ¯ m 2 ), and thus, they differ from the simply elongated cubic structures (see also Section 2.2 for more details about the calculations).

The elastic constants of all studied tetragonal Heusler compounds follow the inequality B > c 44 > G > c ′ > 0 , so that the tetragonal shear modulus c' is the main constraint on the stability and properties. Pugh’s and Poisson’s ratio (see Appendix A) supply valuable information about the malleability and the type of bonding in crystals. Small values of Pugh’s ratio indicate low malleability of crystals [47] , meaning they are brittle. Pugh’s ratio indicates the type of bonding, namely covalent or metallic bonding, because changes in the angle of a covalent bond require more energy than stretching-stressing the bond, meaning G becomes larger compared to B; this leads to smaller values of k. On the other hand, Poisson’s ratio for covalent bonding is about ν = 0.1 and increases for ionic crystals up to ν = 0.25 . For instance, Pugh’s and Poisson’s ratios of strong covalent compounds such as diamond are assumed to be k = 0.83 and ν = 0.069 , respectively, while in the strongly ionic KCl, Pugh’s and Poisson’s ratios are k = 1.19 and ν = 0.27 , respectively [48] [49] . Despite having different types of bonding, both KCl and diamond are brittle. On the other hand, strongly malleable gold exhibits ratios of k = 6.14 and ν = 0.42 [47] [50] , and gold is the most ductile metallic element. According to Christensen [51] , the critical value for the ductile-brittle transition appears at a Pugh’s ratio of

k B / D = 2 3 1 1 − 1 / 2 ≈ 2.3 , (7)

which corresponds to a critical Poisson’s ratio of ν B / D = ( 3 2 − 1 ) − 1 ≈ 0.31 (compare equations for k and ν in the Appendix). Based on the calculated elastic constants, the studied tetragonal compounds exhibit Pugh’s ratios between 1.38 and 2.2 and Poisson’s ratios between 0.21 and 0.3, indicating that they have an intermediate behavior between ductile and brittle type behavior. The values of k and n indicate the covalent or metallic character of these systems. More details about the bonding type may be found from a Bader analysis of the charge densities [52] [53] [54] . It should be kept in mind that ductility and brittleness are rather subjective criteria and usually related to the elastic constants by empirical or semi-empirical rules.

The shear anisotropy factor A_e provides a measure of the degree of anisotropy of the bonds between atoms in different planes. Tetragonal systems are described by two different shear anisotropic factors A₀₀₁ and A₁₀₀ (or equivalent A₀₁₀). Young’s moduli of some of the studied Heusler compounds are shown in Figure 5. The shear anisotropy for {100} planes is considerably higher compared to {001} planes for Mn₂NiGa and Rh₂FeSn. This behavior is similar for Mn₂FeGa and other Rh₂-based compounds, as shown in Table 4. In contrast, in the case of Fe₂MnGa and Mn₂FeGa, the anisotropy for {001} is higher than that for {100}. Moreover, Rh₂FeSn has an interesting distribution of Young’s modulus: it is isotropic in the square x-y planes of the tetragonal structure, arising from the value of A 001 = 0.95 , which is close to unity.

One interesting feature of Heusler compounds, including Ni₂MnGa, is their sensitivity to stoichiometric compositions. An infinitesimal deviation may lead to a phase transition or to changes in the electronic structure properties. Here, VCA was applied to explore off-stoichiometric systems. This approximation is valid for small changes of components with nearly the same radius, which holds for the considered systems. Moreover, VCA is only valid for neighboring elements and for small differences in the number of valence electrons ( Δ e − < 0.1 ).

In VCA, the atom ^ZA with charge Z is replaced by atom A Z ′ with the virtual charge Z ′ = Z ± ϵ to reflect that the average charge deviates from the original value at a certain position. In Ni₂MnGa, if the site where ²⁸Ni resides is partially occupied by ²⁵Mn, the charge at that site will be lower; accordingly, the charge will be higher when ³⁰Ga occupies the same site. In parallel to the change in the nuclear charge, the number of electrons in the primitive cell changes to remain neutral. Thus, for N 28 ± ϵ i 2 M 25 n G 31 a , the number of valence electrons will be n v = 30 ± 2 ϵ in the primitive cell, whereas the number of core plus semi-core

electrons stays fixed at 72. The following cases are used in the present work:

・ ^30.9Ga Þ Ni or Mn at Ga site,

・ ^25.1Mn Þ Ni or Ga at Mn site,

・ ^27.9Ni Þ Mn at Ni site, and

・ ^28.1Ni Þ Ga at Ni site.

All elongated structures of off-stoichiometric Ni₂MnGa have been fully optimized within the VCA approximation. As shown in Figure 6, a small change

in the number of electrons at the Ni site has the most drastic effect on the energy landscape. The largest difference appears between Ga-rich (^28.1Ni) and Ni-poor (^27.9Ni) compounds. Increasing Ni at Ga and Mn sites lowers the energy minimum at c / a > 1 compared to the stoichiometric compound. Conversely, increasing Ga at the Ni site increases the energy of c / a > 1 with respect to the stoichiometric compound. These results are in agreement with previously reported calculations [55] . Differences in the E ( δ ) dependence are more pronounced when the distorted structure is far from the initial structure. In the next step, the elastic constants of the off-stoichiometric Ni₂MnGa were calculated using VCA for the tetragonal distorted structures with c / a > 1 at the lowest total energy (see Figure 6).

Table 4 summarizes the results of the elastic constant calculations for stoichiometric and off-stoichiometric Ni₂MnGa. An extreme effect of the off-stoichiometry is reflected in the anisotropy ratio (A_B). As shown in Table 4, the ^27.9Ni and ^30.9Ga compounds exhibit negative anisotropies of −0.02 and −0.4, respectively. In fact, the negative anisotropy highlights the extreme instability of these systems. In particular, ^30.9Ga has a large negative value. The results explain why ^30.9Ga (Ga-poor) and ^27.9Ni (Mn-rich) Ni₂MnGa synthesis is difficult. Moreover, the ^28.1Ni (Ga-rich) and ^25.1Mn (Mn-poor) compounds have large anisotropies, which are twice the value of the stoichiometric anisotropies. The large value also explains the tendency of phase transitions in these materials. Therefore, off-stoichiometric Ni₂MnGa―nearly all synthesized samples are slightly off-stoichiometric―is expected to have pronounced phase transitions depending on the composition [56] . Thus, deficiency of valence electrons at the Mn or Ga sites in these systems leads to a negative anisotropy ratio and thus structural instability. Among the elastic constants, c₁₁ and c₂₂ are more strongly influenced when the composition changes compared to c₆₆ and c₄₄, which remain nearly constant. Therefore, a small excess of each of the elements (small change of the valence electron concentration in the vicinity of that site) will not change the shear in the {100} direction. However, the asymmetries A_B and A₁₀₀ significantly change compared to A₀₀₁, reflecting a change of the in-plane chemical bonding. Here, Ni₂MnGa is used as an example of the sensitivity of Heusler compounds on their stoichiometry. Disorder-induced phase transitions have been reported for other Heusler compounds such as iron-based compounds [57] .

Finally, some physical properties and material parameters of the compounds are derived from the calculated elastic properties. The velocity of sound is an important quantity. Its averaged values can be directly determined from the calculated elastic constants. In experiments, on the other hand, the sound velocities can be used to measure elastic constants. Therefore, the sound velocities v are nearly synonymous with the elastic stiffness constants c. Further, sound velocities have been used to study various solid-state properties and processes [45] . Therefore, having the sound velocities predicted by calculations in advance could be quite important for experimental measurements. Usually, the directionally dependent acoustic properties are analyzed in terms of the slowness that is the inverse of the phase velocity. The group velocities are found from the derivatives of the slowness.

Figure 7 compares the slowness surfaces of Fe₂MnGa and Ni₂MnGa. The slowness surfaces reflect the elastic anisotropy in comparison to Figure 5, which shows the distribution of Young’s modulus. Three slowness surfaces appear in both cases, representing different polarizations of the sound wave. The pressure (p) wave is longitudinal polarized. Moreover, p has the highest phase velocity and thus the smallest slowness (Figure 7(a) and Figure 7(d)). The remaining two surfaces belong to the fast (s₁) and slow (s₂) shear waves that are transversely polarized. The slowness surfaces of the p waves have a similar shape for both materials, and their maxima are found along the {001}-type principle axes. The

shapes of the slowness surfaces of the shear waves differ between the two compounds. The observed differences reflect the differences in the anisotropy of both materials, and it is clearly seen that Ni₂MnGa has a much lower anisotropy in the x-y plane.

Further material parameters are derived from the average sound velocities as described in Appendix C. At low temperatures, where only acoustic vibrational modes contribute, the Debye temperature θ D can be estimated from the average sound velocity [58] . The values estimated in this way are generally larger than Debye temperatures determined from phonon calculations or in experiments [59] because the optical phonon branches are neglected when the elastic constants are used for calculating the “acoustical” Debye temperature θ D acc . In a similar way, the average sound velocities can be used to estimate the “acoustical” Grüneisen parameter ζ ac [60] .

The calculated average sound velocities together with the estimated Debye temperatures and Grüneisen parameters are listed in Table 5. The average sound velocities are rather similar for all compounds, ranging from about 3100 to 3800 m/s. The acoustical Debye temperatures are all above room temperature, ranging from 376 to 490 K. As expected, compounds consisting of heavier elements tend to have lower values. With the exception of Mn₂NiGa, all ζ ac values are about 2. This demonstrates that the anharmonicity of the lattice vibrations is nearly the same for all compounds.

As further shown in Table 5, the theory and experimental values have about a 20% discrepancy for the case of Ni₂MnGa. As reported in Reference [61] , however, the stoichiometry of the compound has a large effect on the measured Debye temperature. As shown in Table 4, the Debye temperature decreases by about 20% in ^27.9Ni. However, in the calculations, the changes in stoichiometry are extremely small. The changes in the calculated values may be more evident with larger variations in the stoichiometry. For example, in experiments, the results change from 261 K in the case of Ni_49.6Mn_21.9Ga_28.5 to 345 K in the case of Ni_53.1Mn_26.6Ga_20.3. Therefore, having an ideal 2:1:1 system, like that assumed in most theories, is not easily possible or may even be impossible from an experimental standpoint. However, the estimation of measurable properties should provide information about the studied system and its potential for applications.