One of the most widespread and most studied groups of solids is the metallic phases. These phases are divided into simple phases (metallic elements), solid solutions and intermetallic phases. Solid solutions include substitution and interstitial solutions. Interstitial solutions are formed when relatively small atoms of a solute element do not occupy atomic positions, but positions in the interstitial voids of the solvent. A characteristic feature of solid solutions is that they retain the crystallographic structure of the solvent. Intermetallic phases assume crystal structures different from those of the individual components of the phase. In general, intermetallic phases are described by the formula A_nB_m and are divided into the following types according to the adopted classification ^[1][2][3][1,2,3]:

• Phases with a dense filling of space (Laves phases), which are classified according to the size of atoms/ions;
• Electronic phases, which are classified according to the electron concentration, i.e., the ratio of the number of valence electrons to the number of atoms in the unit cell;
• Phases with mixed, metallic-ionic and metallic-covalent bonds.

The most important properties of the Laves phases include the following:

• They are metal alloys with the general stoichiometric formula AB₂;
• The main factor influencing their formation is the ratio of the radii of the component atoms, and r_A/r_B is equal, theoretically, to 1.225. In fact, this ratio for the Laves phases is within the range of 1.05–1.68 Å. Here, A refers to larger atoms (e.g., rare earths), and B refers to smaller atoms (e.g., Cu, Zn, Fe, Mn, Cr, etc.);
• They do not create secondary solutions;
• They exhibit an almost purely metallic type of bonding;
• They are phases with the densest space filling;
• They crystallize in one of three homothetic types, C14, C15 and C36, as determined by the value of the electron concentration [4].

The RMn₂ intermetallic compounds crystallize either in the hexagonal C14 type for R = Pr, Nd, Sm, Ho, Er, Tm and Lu or in the cubic C15 type for R = Y, Sm, Gd, Tb, Dy and Ho ^{[5][6][7][8][9][10][11][12][13][14]}[5,6,7,8,9,10,11,12,13,14]. Other elements (A: La, Ce, Eu and Yb) do not form stable intermetallic compounds with Mn [15].

1.2. Structural and Magnetic Properties of RMn

₂

Compounds

The RMn₂ compounds have been widely studied because of their very interesting structural and magnetic properties. Changes of the unit cells for RMn₂ as a function of the temperature are presented in Figure 1a [16]. For the (Tb, Gd, Sm, Nd, Pr, Y)Mn₂ compounds, the first-order phase transition is observed, and temperature dilatometer and X-ray diffraction measurements revealed significant increases in the volume of unit cells of these alloys at temperatures below 105 K. The magnetic ordering was also observed below the temperatures of the structural changes ^[17][18][19][17,18,19]. The largest, an almost 5% change in volume, was recorded for YMn₂. This significant magneto-volume effect is related to the appearance of a well-localized magnetic moment in the Mn sublattice (~2.7μ_B) below the magnetic ordering temperature [20]. The (Er, Ho, Dy)Mn₂ compounds do not show changes in the volume cell; they undergo a second-order phase transition in the temperature range of 15–35 K ^[14][21][14,21]. The type of the phase transition can be related to the occurrence of a magnetic moment of Mn atoms or its absence. The measurements of the hyperfine field on the ⁵⁵Mn nuclei in RMn₂ compounds carried out by Yoshimura (Figure 1b) indicate that below a certain distance of ~2.66 Å between the nearest Mn atoms (d_Mn-Mn), the localized magnetic moment of the Mn atoms is not observed. This distance is called the critical distance and denoted as d_c [22]. Based on the d_Mn-Mn distance, RMn₂ compounds can be divided into three groups [23]:

• d_Mn-Mn < dc, (LuMn₂, ErMn₂, TmMn₂), where the Mn subnet is non-magnetic, and the Er and Tm sublattices are ferromagnetic [24];
• d_Mn-Mn > d_c, (NdMn₂, PrMn₂), where the dominant Mn magnetic sublattice is stable and enforces antiferromagnetic ordering in the sublattice of R-atoms with the R-Mn type interaction [25];
• d_Mn-Mn ~ d_c, (GdMn₂, TbMn₂,31,32,33,34].

The magnetism of the RMn₂ compounds is related to the electronic structure of Mn and R atoms. Mn has an unfilled outer 3d electron shell affecting the crystalline field from other ions. However, in the R atoms, the unfilled inner 4f electron shell is shielded by the 5p outer shell, which facilitates the behavior similar to that noted for the free ion. In the RMn₂ compounds, three types of exchange interactions can be distinguished as follows:

• Between 3d–3d magnetic moments (Mn–Mn),
• Between 3d–4f moments (Mn–R),
• , DyMn₂, HoMn₂, YMn₂), where the Mn sublattice is on the verge of stability. Generally, non-collinear magnetic structures are observed in this group of compounds. In some cases (TbMn₂, DyMn₂, HoMn₂), only a part of the Mn atoms carries a magnetic moment ^{[20][26][27][32][33][34]}[20,
• Between localized 4f–4f26^28],27^[29],28^[30],29^[31],30 moments (R^[–R).

The strongest among the above is the 3d–3d exchange interaction, as evidenced by the high temperature of ordering of the magnetic relationship with non-magnetic rare earths (YMn₂, T_N ≈ 105 K). The band model, which is applicable here, assumes that the difference in concentration between electrons with spins ↓ and ↑ is responsible for the formation of the magnetic moment. The weakest is the 4f–4f interaction, as indicated by low temperatures of magnetic ordering of the ErMn₂ and TmMn₂ compounds with the non-magnetic Mn sublattice (Figure 1a,b). The R atoms having an unfilled 4f electron shell are characterized by well-localized magnetic moments; the wave functions that describe them have a short range in comparison to the distance between atoms (magnetic moments). The magnitude of the interaction of 4f–4f depends on the density of the polarized conduction electrons; their oscillating nature as a function of distance is described by the RKKY theory. The magnetic interaction 3d–4f has an intermediate value between 3d–3d and 4f–4f. It occurs mainly through the polarization of electrons of 5d rare earth atoms [35].

1.3. Hydrogen in Metals

Rare earth (R) metals react with hydrogen forming stable di- and trihydrides. The RH₂ dihydrides are very stable, with reported formation enthalpies of about −200 kJ/molH₂; thus, they decompose in a vacuum, releasing hydrogen and forming individual RE metals at very high temperatures only [36]. On the other hand, manganese, which is also the main building block of the discussed RMn₂H_x structures, behaves differently under the influence of hydrogen in comparison to RH_x. At ambient pressure conditions and room temperature, the subhydride Mn₂H is the only stable compound. With increasing pressure, MnH is stabilized, and then hydrogen-rich hydrides appear at higher pressure [37]. A simplified scheme of hydrogen penetration into the crystal structure of the metal (intermetallic compound) can be divided into three stages (Figure 2): (a) Physisorption: gaseous hydrogen molecules interact with electrons on the metal surface using weak attractive van der Waals forces; (b) Chemical adsorption: hydrogen molecules dissociate, overcoming the surface energy barrier, and form a metallic hydrogen bond; (c) Absorption: in this step, the chemisorbed hydrogen atom can jump to the subsurface layer to finally diffuse through the crystalline structure of the metallic host [38]. After absorption, the hydrogen atoms/ions remain in the sites of the crystal structure until they are given a surplus of energy necessary to overcome the diffusion barrier as a result of thermal activation. Under these conditions, the diffusion coefficient satisfies the classical Arrhenius relationship: D = D₀ exp (−E_a/kT), where D₀ is the diffusion constant. Assuming the simple activated diffusion, the typical diffusion constant for H in intermetallic compounds equals 1–10³ mm²/s and E_a = 45 kJ/mol at room temperature ^[39][40][39,40]. Researchers also point to the dependence of the D₀ and the T on the structure on the structure of the metal. For example, for Pd (fcc structure), D₀ = 2.9 × 10⁻³ cm²/s and E_a = 0.230 eV, while D_o = 4.2 × 10⁻⁴ cm²/s and E_a = 0.40 eV for Fe (bcc structure) [41]. One of the first experiments indicating the dissociation of H₂ molecules in the process of its absorption by metals was the study of the p-c-T relationship (hydrogen pressure-concentration-temperature) [39]. In numerous metal hydrides, hydrogen atoms are characterized by high mobility already at room temperature, especially when the number of available positions is greater than the number of hydrogen atoms. Hydrogen atoms can jump from one position to another without creating a crystallographic ordered structure—the positioning is accidental. The frequency of hydrogen jumps is huge (e.g., in vanadium hydrides at room temperature, it is ~10¹⁰–10¹² Hz) [39]. With the lowering of the hydride temperature, the frequency decreases. Therefore, it may also happen that hydrogen atoms occupy selected positions in an orderly manner, and they create a superstructure. Skripov states that in many hydrides of the Laves phase, even two types of hydrogen jumping motions with different characteristic jump rates coexist. While the slower jump process is responsible for long-distance diffusion, the faster process corresponds to the localized movement of hydrogen over small groups of interstitial sites. In some of the Laves-phase compounds, the localized hydrogen motion is not ‘frozen out’ on the frequency scale of 10⁷ to 10⁸ Hz down to 20 K [42]. At low temperatures, there is also the possibility of non-activation, tunnel transitions of the hydrogen atom from one position to another ^[38][43][44][38,43,44]. Figure 3a shows the typical hydrogen absorption isotherms in metal. The isotherms form a phase diagram of the equilibrium pressure as a function of the hydrogen concentration P_eq (x), in which three essential areas can be distinguished: (i) the α region is the “solid” solution of hydrogen in the parent metal, where the hydrogen pressure p increases with the hydrogen content x according to Sievert’s law p = (K_s c)² (K—Sievert’s constant, c—hydrogen concentration), (ii) the two-phase region (α + β) and (iii) the β region—metal hydride. As the temperature increases, the latter narrows and disappears at the critical temperature T_crit, which is characteristic for a given element. The isotherms in the middle region show a hydride characteristic plateau, corresponding to the equilibrium pressure as a function of the hydrogen concentration. The equilibrium pressure zone is ideally suited for the isothermal absorption and desorption of hydrogen from metal. The state of equilibrium can be described by a phase diagram analogous to the Van der Waals diagram of the non-ideal gas. P_eq equilibrium pressure is associated with changes in enthalpy (ΔH) and entropy (ΔS) in the two-phase region via the empirical temperature of the Van’t Hoff relationship (Figure 3b) ^[45][46][45,46]. For a typical equilibrium pressure, e.g., for PdH_x—hydride at 423 K, it is 0.1 MPa, and it is 1.0 MPa at 518 K. The critical point of this hydride is T_crit = 565 K at x ≈ 0.25 H/f.u. [39].

2. RMn₂H_x Hydrides

2.1. Localization of Hydrogen

The physical properties of rare earth hydrides with manganese RMn₂H_x largely coincide with the properties of metal hydrides (MH_x). Also, alloys of RMn₂ compounds absorb hydrogen very easily. At close to atmospheric hydrogen pressure and temperatures above 100 °C, it is possible to achieve a hydrogen concentration of ~4.5 H/f.u. ^[47][48]. As noted in metal hydrides, hydrogen occupies the interatomic sites of the elements that comprise the RMn₂ unit cell. There are three types of tetrahedral sites in RMn₂ compounds: A2B2, AB3 and B4, where A and B represent atoms of R and Mn, respectively, surrounding the site (Figure 4). The largest and at the same time most willingly filled by hydrogen site (below x ≈ 3.5 H/f.u.) is the A2B2 type. For x ≳ 3.5 H/f.u., there is also a tendency to fill smaller sites, such as AB3 sites ^{[48][49][50][51]}[49,50,51,52]. However, no B4 site filling was observed. The same types of sites occur in both C14 and C15 types of Laves phases. When filling the sites, hydrogen occupies crystallographic positions (Table 1). Although the number of sites is significant, every site cannot be occupied by hydrogen. Two criteria are useful: (i) the Switendick criterion ^[52][53] states that the distance d_H-H must be larger than ~2.1 Å and (ii) the Westlake criterion indicates that the radius of the sphere available for hydrogen cannot be less than ~0.37 Å. ^[53][54]. As an effect, criteria preclude occupancy of A2B2 tetrahedrons that have a common wall, and the B4 site is not available for hydrogen due to size restrictions. The Switendick criterion, confirmed in neutron measurements ^[54][55][55,56], largely determines the maximum content of absorbed hydrogen. The theoretically determined maximum hydrogen concentration in RT₂H_x (R: rare earth, T: transition metal = Y, Mn) compounds is therefore 6.0 H/f.u. ^[56][57]. Only the use of ultra-high hydrogen pressures of the order of 10³ MPa led to the formation of RMn₂H₆ hydrides; the first being YMn₂H₆ ^[57][58][58,59]. Highly hydrogenated (x ≳ 4.5 H/f.u.) compounds show different physical properties compared to low-hydrogen compounds (x ≲ 4.5 H/f.u.) ^[59][60][60,61] and will not be discussed in this study.

2.2. Sample Preparation and Dependence of ΔV/V versus Hydrogen Content

In the discussed hydrides, the sample preparation was similar and usually involved the following scenario. The host materials (RMn₂) were prepared from high purity elements using the standard induction melting technique under argon atmosphere. Next, materials were usually annealed to obtain a single-phase compound. The RMn₂ samples were saturated with hydrogen using a standard volumetric method (Sievert) to obtain RMn₂H_x hydrides in the typical range of x: 0 < x < 4.3–4.5 ^{[7][8][9][54][61][62][63][64][65][66]}[7,8,9,55,62,63,64,65,66,67]. The first test after hydrogen saturation was XRD measurements at 300 K. As a representative example, the HoMn₂H_x_C15 diffractogram is presented in Figure 5a. The most visible effect of hydrogen absorption by the sample is the shift of the diffraction lines towards smaller angles (in 2 theta), which means an increase in the distance between atoms in the cell, and consequently an increase in the unit volume of the cell with x. Even at room temperature, the RMn₂H_x hydrides do not persist in the crystallographic structure of the host compound. For the HoMn₂H_x_C15 hydride, a combination of the cubic (α) and rhombohedral distortion (β) is observed (Figure 5a). The obtained results are in agreement with ^[67][68]. To compare lattice parameters of hydrides with different crystallographic structures, their parameters were converted into parameters of the pseudo-cubic cell considering abundance phases, if the system is not a single-phase system. For example, the lattice parameters of the rhombohedral phase were converted into lattice parameters 𝑎*𝛿  of the pseudo-cubic phase according to the relationship reported below: where a_δ and c_δ are lattice parameters of the rhombohedral phase (space group: 𝑅3, nr166, setting: 1) ^[65][66]. The relative volume change in unit cell versus hydrogen content in HoMn₂H_x_C15 is depicted in Figure 5b. The first ‘model’ that attempted to describe the relation between ΔV/V₀ and x was just a linear relationship. The next one, the Hirata model, only correctly reproduces the lowest range of x (0 < x < 3.0) (Figure 5b, dashed line) ^[68][69], and its development was proposed by Figiel et al. ^[50][51]. In the last one (farther: Hirata–Figiel model), the authors assume a two-stage approach to occupying interstitial sites (Figure 5b red line). Initially, with a characteristic parameter X_c < 3.0–3.5 H/f.u. (see below), the A2B2 type sites are filled (this part is identical with Hirata model). Next, above X_c > 3.0–3.5 H/f.u., the AB3 type sites are also occupied. It is represented by a characteristic break in the red curve (x~3.2 H/f.u., Figure 5). The Hirata-Figiel model is described according to the relationship described below ^[50][51]. where ΔV/V₀ is the relative increase in unit cell volume; B₀ is a parameter of the lattice contraction, a quantity analogous to the bulk modulus at starting concentration related to the intrinsic pressure; b is the first derivative with respect to concentration x: (B_x = B₀ + b_x); and P is the probability of filling the AB3 sites by hydrogen atoms for concentrations greater than X_c (P = 0 for x ≤ X_c) ^[34][50][34,51]. Fitted parameters for the RMn₂H_x (R: Y, Dy, Gd, Tb, Ho) hydrides are presented in Table 2. For all compounds, the characteristic X_c belongs to the region of ~3.0–3.5 and probabilities of entering the AB3 interstitial sites P~0.6–0.9. This type of filling sites was also postulated in ^[49][50][50,51]. The relative volume increase (ΔV/V₀) of unit cells of hydrides determined at room temperature and depending on the hydrogen content in the sample is non-linear and can be described by the same function, indicating that this effect is determined only by the presence of hydrogen. The fits of the Hirata–Figiel model to the change in (ΔV/V₀) depending on the hydrogen concentration for the various studied hydrides RMn₂H_x are shown in Figure 6. For x < 3.0 H/f.u., the curves, except for those corresponding to (Gd and Nd)Mn₂H_x, are almost identical. At values greater than x ≈ 3.0 H/f.u., distinct behavior becomes visible with increasing x, especially for concentrations close to the maximum, where the relative increase in volume increases with the atomic number of the rare earth element.

2.3. Examples of Structural and Magnetic Transformations in RMn

₂

H

_x

With a temperature change, the RMn₂H_x hydrides undergo complex phase transformations in many cases, which, combined with the often occurring multi-phase states, makes the interpretation of their X-ray diffraction magnetic transformation results extremely difficult. Some authors have even argued that it will never be possible [39]. One of the first hydride families to be studied over a wide range of x and temperature changes was YMn₂H_x, showing a very wide range of structural transformations as a function of temperature. In contrast, TbMn₂H_x hydride families show a very ‘modest’ spectrum of structural transformations. Very interesting are the structural and magnetic properties of HoMn₂H_x hydrides, in which the base compound can crystallize in a C14 or C15 Laves phase structure, depending on the sample preparation [10].