Structural, Thermophysical, and Magnetic Properties of the γ^′-Fe₄N System: Density Functional Theory and Experimental Study

Abstract

The ${\gamma }^{\prime }$-Fe₄N system has a high technological relevance due to its multiple applications in the field of surface treatment against wear and corrosion of iron in steel parts, as well as in the manufacturing of high-density magnetic recording devices, and so on. In the present work, we present a wide research of the structural, elastic, magnetic, vibrational, and thermophysical properties by means of the phonon analysis. For these purposes, we have compared theoretical and experimental results. The theoretical data were obtained by employing ab initio electronic structure calculations in the framework of density functional theory (DFT), and different experimental measurements, such as X-ray diffraction, magnetization measurements, and calorimetric techniques, were used to characterize the ${\gamma }^{\prime }$-Fe₄N system. The resulting comparison showed an excellent agreement between the theoretical and experimental data reported.

1. Introduction

During the past decades, iron nitride compounds have been widely investigated due to their different physical and chemical properties [1,2,3]. In particular, the ${\gamma }^{\prime }$-Fe₄N compound exhibits excellent magnetic properties, making it suitable for applications in high-density magnetic storage devices [4,5,6,7], perpendicular magnetic recording devices [8,9,10], sensor, actuators, vibration energy harvesting devices [11,12], spintronic-based materials devices [13,14,15], electrocatalysis [16,17,18], ammonia catalysis [19,20], steel treated by plasma nitriding [21,22], wear and corrosion resistance in surface steels [23,24,25,26], and even as potential constituents of the Earth’s core and other terrestrial planets [27,28,29]. In this work, we have studied the structural and elastic properties, thermophysical parameters, and magnetic properties of the ${\gamma }^{\prime }$-Fe₄N material from the theoretical and experimental perspectives, where a good agreement between both points of view has been achieved.

2. Materials and Methods

2.1. Electronic Structure Calculations

The ab initio calculations were conducted within the framework of density functional theory (DFT). The Kohn–Sham self-consistent equations were solved using the pseudopotential plus the plane wave method, as implemented in the QuantumEspresso code [30,31]. Projector augmented waves (PAWs) pseudopotentials from the SSSP pseudopotential library [32] were employed. The kinetic energy cutoff and charge density cutoff were set at 80 Ry and 800 Ry, respectively. The exchange–correlation interaction was treated using the Perdew–Burke–Ernzerhof parametrization of the generalized gradient approximation (GGA-PBE) [33]. In order to investigate the magnetic properties, spin polarization calculations were performed. A dense k-points mesh of $21\times 21\times 21$ was used in the irreducible Brillouin zone (IBZ).

Phonon properties were calculated via linear response theory in the framework of the density functional perturbed theory, using a $4\times 4\times 4$ q-points mesh for a single ${\gamma }^{\prime }$-Fe₄N crystal cell. We have assumed the antiperovskite cubic crystal structure, with space group $Pm\overline{3}m$ (221) (see Figure 1), where two types of Fe atoms can be distinguished, FeI and FeII, located in the corner of the cube and in the center of the face, characterized by 1a and 3c Wyckoff sites, respectively, whilst there is a N atom at the center of the cube, the 1b Wyckoff site.

2.2. Experimental Details

Sample preparation and structural characterization: Synthesis of ${\gamma }^{\prime }$-Fe₄N and X-ray diffraction samples was obtained from Fe_mN powder (Alfa Aesar (Thermo Fisher Scientific, Waltham, MA, USA) with m = 2, 3 and 4) mechanically alloyed in a steel milling (10 cm³) at room temperature (RT) and under Ar atmosphere. The milling process was performed on a Retsch MM2 horizontal mill (LabMakelaar Benelux B.V., Zevenhuizen, The Netherlands) operating at 33 Hz, with one stainless steel ball. The ball-to-powder mass ratio was 10:1, and the milling time was 15 h. To improve the amount of ${\gamma }^{\prime }$-Fe₄N, after the milling process, the resulting powder was annealed at 693 K for 2 h under a dynamic vacuum, with a rate of 20 K/min, and then cooled in the furnace.

Structural parameters were obtained using an X-ray diffraction (XRD) technique using an X-Pert pro PANalytical diffractometer (Malvern Panalytical Ltd., Malvern, UK) equipped with $CuK-\alpha$ radiation ($\lambda$ = 1.5406) at RT. Data were collected in the range $2^{\circ }\le 2\theta \le {100}^{\circ }$ in steps of 0.02°, with a counting time of 0.5 s per step. The present phases were identified through comparison with powder diffraction files. To determine the structural properties of the XRD patterns, such as lattice parameters and internal parameters, Rietveld refinement was carried out.

Calorimetric measurements: The differential scanning calorimetry (DSC) curves were recorded from room temperature to 550 K, under a nitrogen flow of 40 mL min⁻¹ on a DSC-50/Shimadzu instrument (Shimadzu Corporation, Kyoto, Japan). The sample was placed in a sealed aluminum pan and heated at a rate of 10 °C min⁻¹. An empty sealed aluminum pan was used as a reference. The DSC cell was previously calibrated with indium and zinc. Also, for DSC, data were analyzed using TA60WS software (version 2.0, Shimadzu Corporation, Kyoto, Japan) after subtraction of the blank curve (empty sealed aluminum pans) for baseline correction. For low-temperature measurement, the DSC curve was recorded in the temperature range of −109 °C to 200 °C under helium flow of 40 mL min⁻¹ at a rate of 10 °C min⁻¹.

Magnetic measurements: Magnetic characterization was performed using a magnetic property measurement system (MPMS3-SQUID, Quantum Design, San Diego, CA, USA) using the vibration sample magnetization technique (VSM 7404 Lake Shore Cryotronics, Westerville, OH, USA). Magnetization as a function of the applied magnetic field was performed from −70 up to 70 kOe at room temperature. In order to deepen the understanding of the magnetic properties, we fitted the M versus H curves obtained at 5 K using the Brillouin function. Additionally, zero-field cooling (ZFC) and field cooling (FC) measurements were performed to analyze the temperature-dependence of the magnetic behavior. We have used the following unit conversion $1emu/g=1.07828\times {10}^{20}{\mu }_B\times M_{fu}$, where $M_{fu}$ is the mass per formula unit of ${\gamma }^{\prime }$-Fe₄N, $M_{fu}=394.067\times {10}^{-24}g/f.u$.

3. Results

3.1. Structural and Elastic Properties

To determine the equilibrium cell volume $\left(V_0\right)$, a set of total energy calculations for different cell volumes were performed and fitted using the Birch–Murnaghan state equation [34,35]; see Equation (1):

$$E\left(V\right)=E\left(0\right)+{\displaystyle \frac{9V_0{B}_0}{16}}\left\{{\left[{\left({\displaystyle \frac{V_0}{V}}\right)}^{{\displaystyle \frac{2}{3}}}-1\right]}^3{B}_0^{\prime }+{\left[{\left({\displaystyle \frac{V_0}{V}}\right)}^{{\displaystyle \frac{2}{3}}}-1\right]}^2\left[6-4{\left({\displaystyle \frac{V_0}{V}}\right)}^{{\displaystyle \frac{2}{3}}}\right]\right\}$$

(1)

where $V_0$ is the equilibrium lattice parameter, $E\left(0\right)$ the respective total energy, $B_0$ the bulk modulus, and $B_0^{\prime }$ the derivative of $B_0$ with respect to pressure.

In Figure 2, we show a set of calculations of the total energy for different volume values. These data were fitted to Equation 1 in order to obtain the equilibrium volume and the respective lattice parameter of the ${\gamma }^{\prime }$-Fe₄N of $a=3.7910 \mathrm{\AA }$.

From the experimental part, in Figure 3, we present the X-ray pattern for samples as supplied and after a thermal treatment of 420 °C; in both cases, the lattice parameter obtained is 3.7909 Å, and there is an excellent agreement between our theoretical and experimental values and other experimental data results reported in different samples: 3.7970 Å in crystals [36], 3.7900 Å in crystals [37], 3.7910 Å in nanocomposites [38], 3.7950 Å in crystals [39], 3.7800 Å in nanocrystals [40], 3.7990 Å in nanoparticles [41], 3.7900 Å in powders [42,43], and 3.7950 Å in thin films [44,45].

To calculate the elastic constants for ${\gamma }^{\prime }$-Fe₄N, we have used a numerical calculation of the stress tensor $\left(\sigma \right)$ for small strains $\left(\epsilon \right)$ from $V_0$. For cubic crystals, only three independent elastic constants are defined: $C_{11}$, $C_{12}$, and $C_{44}$. So, a set of three equations are needed to determinate all elastic constants. The first equation involves the bulk modulus $\left(B\right)$ obtained from (2):

$$B={\displaystyle \frac{1}{3}}(C_{11}+2C_{12})$$

(2)

The others equations involve the volume-conservative tetragonal $\left({\epsilon }_{tetra}\right)$ and rhombohedral $\left({\epsilon }_{rhom}\right)$ strain tensor:

$${\epsilon }_{tetra}=\left(\begin{array}{ccc}\epsilon & 0& 0\\ 0& \epsilon & 0\\ 0& 0& {\displaystyle \frac{1}{{(1-\epsilon )}^2}}-1\end{array}\right)$$

(3)

$${\epsilon }_{rhom}={\displaystyle \frac{\epsilon }{3}}\left(\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right)$$

(4)

The application of these strains has an effect on the total energy from its unstrained value as follows:

$$E_{tetra}\left(\epsilon \right)=E_0+3(C_{11}-C_{12})V_0{\epsilon }^2+\mathcal{O}\left({\epsilon }^3\right)$$

(5)

$$E_{rhom}\left(\epsilon \right)=E_0+{\displaystyle \frac{1}{6}}(C_{11}+2C_{12}+4C_{44})V_0{\epsilon }^2+\mathcal{O}\left({\epsilon }^3\right)$$

(6)

In Figure 4, we show the variation of the total energy with $\epsilon$ for (5) and (6), and we have obtained the values of the elastic constants $C_{11}$, $C_{12}$, and $C_{44}$. Using the Voigt-Reuss-Hill (VRH) formalism [46], the shear modulus $\left(G\right)$ is expressed as:

$$G={\displaystyle \frac{G_V+G_R}{2}};G_V={\displaystyle \frac{C_{11}-C_{12}+3C_{44}}{5}};G_R={\displaystyle \frac{5(C_{11}-C_{12})C_{44}}{4C_{44}+3(C_{11}-C_{12})}}$$

(7)

where $G_V$ and $G_R$ are the shear modulus in the Voigt and Reuss approximation, respectively. From B and G, it is possible to determine other important elastic parameters, such as the Young’s modulus $\left(E\right)$, Poisson’s radio $\left(v\right)$, Pugh’s parameter $\left(P\right)$, Lame’s parameter $(\lambda ,\mu )$, and microhardness parameter $\left(H\right)$:

$$E={\displaystyle \frac{9BG}{G+3B}}$$

(8)

$$v={\displaystyle \frac{3B-2G}{2(3B+G)}}$$

(9)

$$P={\displaystyle \frac{G}{B}}$$

(10)

$$\lambda ={\displaystyle \frac{vE}{(1+v)(1-2v)}};\mu ={\displaystyle \frac{E}{2(1+v)}}$$

(11)

$$H={\displaystyle \frac{(1-2v)E}{6(1+v)}}$$

(12)

In

Table 1. Elastic properties calculated for ${\gamma }^{\prime }$-Fe₄N. Compared with other DFT-based theoretical values and experimental data reported in the literature.

Parameter	This Work	Theoretical Data	Experimental Data
$C_{11}$ (GPa)	325.4	307.2 [47], 313.0 [48],
		306 [49], 337 [50]
$C_{12}$ (GPa)	131.3	134.1 [47], 137.0 [48],
		99 [49], 131 [50]
$C_{44}$ (GPa)	50.1	46.0 [47], 46.0 [48],
		47 [49], 58.5 [50]
B (GPa)	196.0	191.8 [47], 196.0 [48], 200 [51]	196.0 [48], 198.0 [52]
		168 [49], 194.1 [53], 199.4 [50]	155.0 [54] 155.8 [55], 144.4 [29]
		195.8 [56], 198.7 [57], 190.6 [58]	141 [59], 169 [27], 162 [28]
E (GPa)	184.9	161.5 [47], 162.0 [48], 168.9 [53]	159.0 [48], 165.0 [48]
		195.6 [60], 197.0 [50], 156.4 [56]	172.0 [61], 200.0 [62]
		184.0 [49], 168.5 [47], 173.8 [57]	211.0 [63], 204 [64], 195 [65]
		178.0 [58]
G (GPa)	65.5	63.6 [60], 59.4 [47], 59.0 [48]	59.0 [48]
		70.0 [49], 73.6 [50], 57.2 [56]
		62.2 [47], 70 [49], 62.3 [53]
		64.2 [57]
v	0.318	0.325 [47], 0.360 [48], 0.321 [53]	0.360 [48]
		0.325 [60], 0.37 [56], 0.34 [58]
		0.350 [57], 0.320 [49], 0.336 [50]
H (GPa)	8.5	10.15 [53], 6.01 [57]	6.6 [66], 6.76 [58], 8.0 [48]

, we report our theoretical values of the elastic properties for the ${\gamma }^{\prime }$-Fe₄N compound, compared with other theoretical and experimental values reported in the literature.

An effective isotropic value of v can be explained as a measurement of the material to resist shear and could be associated with the volume change during uniaxial deformation. A value of 0.318 is an average value between 0.25, the lower limit for central-force solids, and 0.5, the upper limit, which corresponds to infinite elastic anisotropy. The high v value for ${\gamma }^{\prime }$-Fe₄N indicates that the interatomic forces in the compound are central. The Pugh’s parameter $\left(P\right)$ is related to the brittleness of the material, with lower values suggesting that material is more brittle, suggesting that when $P<0.57$, the material has ductile features; when $P>0.57$, the material shows brittle character.

Young’s modulus $\left(E\right)$ is one of the most important elastic parameters; in Figure 5, we represent the spacial variation of the Young’s modulus, showing that this material is isotropic in the directions [100], [010], and [001], where the Young’s modulus is at the maximum, and the minumum value is in the [111] direction.

3.2. Thermophysical and Magnetic Properties

The thermophysical properties of materials are intrinsically linked to their atomic vibrational characteristics. It is well established that heat capacity increases with temperature as a consequence of enhanced phonon thermal vibrations. At high temperatures, anharmonic effects tend to be negligible, and the constant-volume heat capacity $C_v$ asymptotically approaches the classical Dulong–Petit limit, given by $C_v=3N{k}_B$ for monoatomic solids, where N is the number of atoms per cell and $k_B$ is Boltzmann’s constant. Conversely, at low temperatures, $C_v$ follows a cubic dependence with temperature, $C_v\sim T^3$. In the intermediate temperature regime, the specific temperature dependence of $C_v$ is governed by detailed lattice vibrations, which are typically characterized by experimental techniques. We have performed two different theoretical frameworks to determine the thermophysical properties, especially the heat capacity $\left(C_v\right)$, which was compared with experimental measurements.

In this work, we have applied two different theoretical approaches to investigate thermophysical properties, with particular emphasis on the heat capacity $C_v$, and compared the theoretical predictions with available experimental data. One of the methodologies employed is the Debye quasi-harmonic approximation (QHA), which provides a framework to estimate the Debye temperature $\left({\Theta }_D\right)$. This parameter is a fundamental parameter, as it correlates with a variety of physical properties, including elastic moduli, melting temperature, heat capacity, and other thermophysical properties. At low temperature, the vibrational excitations arise solely from acoustic phonon modes. Consequently, ${\Theta }_D$ derived from elastic constants is equivalent to that obtained from calorimetric experiments.

A widely used approach for estimating ${\Theta }_D$ involves the use of elastic constant data, wherein the Debye temperature can be calculated from the average sound velocity $\left(v_m\right)$ using the following relation:

$${\Theta }_D={\displaystyle \frac{2\pi \hslash }{k_B}}{\left[{\displaystyle \frac{3n}{4\pi }}{\displaystyle \frac{N_A\rho }{M}}\right]}^{1/3}{v}_m$$

(13)

where ℏ is the reduced Planck’s constant, n the number of atoms per formula unit, $N_A$ the Avogadro’s number, M the molecular weight, and $\rho$ the density of the material. Here, we assume that three acoustic branches contribute to the low-temperature $C_v$. The average wave velocity is:

$$v_m={\left[{\displaystyle \frac{2}{3}}\left({\displaystyle \frac{1}{v_s^3}}+{\displaystyle \frac{1}{v_{\ell }^3}}\right)\right]}^{-1/3}$$

(14)

where $v_s$ and $v_{\ell }$ are the shear and compressional wave velocity, respectively, which are obtained from Navier’s equation:

$$v_s={\left({\displaystyle \frac{G}{\rho }}\right)}^{1/2};v_{\ell }={\left({\displaystyle \frac{3B+4G}{3\rho }}\right)}^{1/2}$$

(15)

To study the thermophysical properties of ${\gamma }^{\prime }$-Fe₄N in the context of the QHA-Debye model implemented in the Gibbs2 code [67,68], the non-equilibrium Gibbs function can be written as:

$$G(V,P,T)=E\left(V\right)+PV+A_{vib}({\Theta }_D,T)$$

(16)

where $E\left(V\right)$ is the total energy dependence on the volume of the unit cell of ${\gamma }^{\prime }$-Fe₄N, $PV$ represents the constant hydrostatic pressure condition, and $A_{vib}({\Theta }_D,T)$ represents the vibrational Helmholtz free energy, expressed as [69,70,71]:

$$A_{vib}({\Theta }_D,T)=n{k}_{B}T\left[{\displaystyle \frac{9}{8}}{\displaystyle \frac{{\Theta }_D}{T}}+3ln\left(1-exp(-{\Theta }_D/T)\right)-\vartheta ({\Theta }_D/T)\right]$$

(17)

where $\vartheta ({\Theta }_D/T)$ is the Debye integral, defined as:

$$\vartheta ({\Theta }_D/T)=3{\left({\displaystyle \frac{{\Theta }_D}{T}}\right)}^3{\int }_0^{{\Theta }_D/T}{\displaystyle \frac{x^3}{e^x-1}}dx$$

(18)

For isotropic solids, ${\Theta }_D$ can be expressed as:

$${\Theta }_D={\displaystyle \frac{\hslash }{k_B}}{\left[6n{\pi }^2{V}^{1/2}\right]}^{1/3}\sqrt{{\displaystyle \frac{B}{M}}}f\left(v\right)$$

(19)

where M is the molecular mass per formula unit; B the adiabatic bulk modulus, which can be approximated by the static compressibility; and $f\left(v\right)$ is a function of the Poisson’s ratio [69,70].

$$f\left(v\right)={\left\{3{\left[2{\left({\displaystyle \frac{2}{3}}{\displaystyle \frac{1+v}{1-2v}}\right)}^{3/2}+{\left({\displaystyle \frac{1}{3}}{\displaystyle \frac{1+v}{1-v}}\right)}^{3/2}\right]}^{-1}\right\}}^{1/3}$$

(20)

Then, minimize the Gibbs function with respect to the volume V:

$${\left({\displaystyle \frac{\partial G(V,P,T)}{\partial V}}\right)}_{P,T}=0$$

(21)

It is possible to express the isothermal bulk modulus as:

$$B_T(P,T)=V{\left({\displaystyle \frac{{\partial }^{2}G(V,P,T)}{\partial V^2}}\right)}_{P,T}=0$$

(22)

and other thermophysical properties such as the heat capacity at constant volume $\left(C_v\right)$, thermal expansion $\left(\alpha \right)$, and Grüneisen’s parameter $\left(\gamma \right)$ as:

$$C_v=3n{k}_B\left[4\vartheta \left({\displaystyle \frac{{\Theta }_D}{T}}\right)-{\displaystyle \frac{{\Theta }_D}{T}}{\displaystyle \frac{3}{exp({\Theta }_D/T)-1}}\right]$$

(23)

Phonon calculations based on density functional perturbation theory (DFPT) enable the computation of thermodynamic potentials such as enthalpy $\left(H\right)$, free energy $\left(F\right)$, and temperature–entropy term $\left(TS\right)$, where S is derived from internal energy $\left(U\right)$ and free energy. Furthermore, these calculations allow one to evaluate the lattice heat capacity $\left(C_v\right)$ as a function of temperature. Density functional theory (DFT)-based codes provide the total electronic energy in the ground state $\left(0K\right)$, and the vibrational contributions derived from these energies are used to determine thermodynamic properties at finite temperatures, including H, F, $TS$, and $C_v$.

The temperature function of the thermophysical potentials can be found in [70,71]. Defining the variable ${\displaystyle \beta =\frac{1}{k_{B}T}}$, the enthalpy H is:

$$H\left(T\right)=E_{tot}+E_{zp}+\int {\displaystyle \frac{\hslash \omega }{exp\left(\beta \hslash \omega \right)-1}}\varrho \left(\omega \right)d\omega$$

(24)

where $E_{zp}$ is the zero-point vibrational energy, and $\varrho \left(\omega \right)$ is the phonon density of states.

The vibrational contribution to the free energy is:

$$F\left(T\right)=E_{tot}+E_{zp}+{\displaystyle \frac{1}{\beta }}\int ln\left[1-exp(-\beta \hslash \omega )\right]\varrho \left(\omega \right)d\omega$$

(25)

and the heat capacity at constant volume as a function of T and at zero pressure is:

$$C_v\left(T\right)=k_B\int {\displaystyle \frac{{(\beta \hslash \omega )}^{2}exp(\beta \hslash \omega )}{{\left[exp(\beta \hslash \omega )-1\right]}^2}}\varrho \left(\omega \right)d\omega$$

(26)

In Figure 6, we show the theoretical curve of $C_v$ obtained from Debye quasi-harmonic approximation (Equation (23)) and from phonon calculations using DFPT (Equation (26)), compared with experimental measurements. At low temperatures, low-frequency acoustic modes predominate. This is the region of the harmonic approximation, where DFPT can accurately calculate the phonon spectra. Meanwhile, the QHA-Debye model assumes isotropy in the material, resulting in $T^3$ behavior of $C_v$ rather than that obtained from the phonon spectra, and it depends on the Debye temperature [72,73,74]. However, both methods can be used to describe $C_v$ at temperatures greater than 300 K. Experimental measurements were performed in the range of 300–550 K, where both theoretical values agree very well with the experimental data, and in the limit of high temperatures, both theoretical models are equivalent, converged in a classical limit of the Dulong–Petit value.

Concerning the magnetic properties, in order to establish the theoretical DFT-based magnetic ground state, we have performed different magnetic configurations between Fe atoms, FeI and FeII, the ferromagnetic case (FM), and two antiferromagnetic configurations AF01 and AF02, respectively; see Figure 7. In

Table 2. Difference in energy $(\Delta E)$, with respect to FM configuration, total magnetic moment $M_T$, and magnetic moment per atom $(m_{FeI},m_{FeII})$, for the magnetic configurations considered for ${\gamma }^{\prime }$-Fe₄N, represented in Figure 7.

	FM	AF01	AF02
$\Delta E$ (meV)	0.0	1022.1	695.6
$M_T({\mu }_B/fu)$	9.80	4.33	−1.65
$m_{FeI}\left({\mu }_B\right)$	3.00	−2.12	−2.60
$m_{FeII}\left({\mu }_B\right)$	2.00	1.99	−1.91/1.54

, we report the total energy variation ($\Delta E$, respect to FM case), total magnetic moment $M_T$, and magnetic moment per atom, $m_{FeI}$ and $m_{FeII}$, for FM, AF01, and AF02 magnetic cases.

From the DFT results, in Table 2, the minimal total energy configuration belongs to the FM case, where the magnetic coupling between the FeI and FeII atoms is parallel, with a net total magnetization of $9.8 {\mu }_B/f.u.$. From experimental measurements, the total magnetization is 207.45 emu/g (see Figure 8), equivalent to $8.83 {\mu }_B/f.u.$, with coercivity $H_c=40Oe$ (see inset Figure 8); a similar value has been reported in nanoparticles sample [75]. In order to analyze temperature-dependent magnetic properties, ZFC and FC measurements were performed up to 300 K and using an external magnetic field of 0.1 T, see Figure 9. Low-temperature ZFC diverges and FC remains constant, which could be related to collective spin behavior, similar to spin-glass or superparamagnetism. The ZFC magnetization increases with the temperature up to 50 K, then remains almost constant parallel to the FC curve. Above 150 K, both curves collapse and decrease as the temperature increases; these behaviors could be attributed to the volume distribution of nanoparticles. Up to 300 K, there are no magnetic transitions; the Curie temperature reported is around 760 K [36,76,77]. From DFT-based calculations, the value of the total magnetization are in total agreement with our experimental measurements and with other values published for different samples and at different measurement temperatures: $8.9 {\mu }_B/f.u.$ at 4.2 K for a powder sample [78], $9.22 {\mu }_B/f.u.$ at 5 K for a $\mathsf{\mu }$m film sample [79], $9.1 {\mu }_B/f.u.$ at 0 K for powder [80], $10.3 {\mu }_B/f.u.$ at RT for nm film [81], $8.85 {\mu }_B/f.u.$ at RT for nanocrystalline [82], $9.20 {\mu }_B/f.u.$ at 55 K for nanocomposites [38], and $8.83 {\mu }_B/f.u.$ at 5 K for powder [43]. The magnetic moment per atom is $3.0 {\mu }_B$ and $2.0 {\mu }_B$, and these values are also in great agreement with neutron diffraction measurement data from [36,83,84].

4. Conclusions

The structural, elastic, thermophysical, and magnetic properties of ${\gamma }^{\prime }$-Fe₄N have been studied using DFT-based calculation and experimental techniques, with excellent agreement, where the radio between experimental and theoretical values for the lattice parameter is $a_{exp}/a_{DFT}=1.00$ and for the total magnetization is $M_{exp}/M_{DFT}=0.90$, respectively. The heat capacity was described using two different theoretical approaches, both yielding the same accuracy. One of the advantages of Debye’s QHA model is the non-expensive computational requirement compared with DFPT phonon calculation. With regard to the structural and magnetic ground state of ${\gamma }^{\prime }$-Fe₄N, our theoretical and experimental results are consistent.