Structural analysis and magnetic properties of lattice distortions from hexagonal to tetragonal systems in non-equilibrium Y–Fe alloys

Intermetallics 119 (2020) 106713

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Structural analysis and magnetic properties of lattice distortions from hexagonal to tetragonal systems in non-equilibrium Y–Fe alloys Hiroyuki Suzuki (

)∗

Magnetic Materials Research Laboratory, Hitachi Metals, Ltd., 5200 Mikajiri, Kumagaya, Saitama 360-8577, Japan

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Keywords: Lattice distortions Structural analysis Rietveld Mössbauer spectroscopic Modulated-CaCu5 structure Iron dumbbell Permanent magnets

ABSTRACT A structural analysis model, which was specialized by space group 𝐼𝑚𝑚𝑚 and denoted by substituting a pair of iron atoms (iron dumbbell) for rare-earth (𝑅) elements continuously, was devised and used to treat continuous lattice distortions from hexagonal TbCu7 to tetragonal ThMn12 structures observed in non-equilibrium Y–Fe alloys. Trajectories of lattice distortions due to annealing, which were drawn on a map arranged in terms of axial ratios 𝑏∕𝑐 and 𝑏∕𝑎, could be classified into four types according to composition ratio (𝑇 ∕𝑅) of the Y–Fe alloys. Especially, type-I distortion (with composition range of 11.5 ≤ 𝑇 ∕𝑅 ≤ 13.5) transformed from a hexagonal structure with 𝑏∕𝑎 ∼ 1.002 to a tetragonal one with 𝑏∕𝑎 = 1.000 via a structure with a large distortion, i.e., 𝑏∕𝑎 = 1.008. Meanwhile, type-II distortion (with composition range of 10.5 < 𝑇 ∕𝑅 < 11.5) transformed from a hexagonal structure with 𝑏∕𝑎 ∼ 1.001 to an orthorhombic one as close as possible to a tetragonal one. Moreover, the continuous-lattice-distortion model could bring geometric insights on structural stability and magnetic properties of not only the alloys but also previously recognized compounds with modulated-CaCu5 structures, which are configured by substituting iron dumbbells for 𝑅 elements.

1. Introduction Metastable phases generated through quenching processes often include disordered atomic arrangements. Post annealing changes such phases into ordered ones stabilized thermally via certain states. Unlike the transition temperature of a first-order structural transition like crystallization, which releases latent heat rapidly, the transition temperature is unclear when the transition is continuous and diffuse. It was reported that such a structural change from disordered state into ordered state occurs in metastable YFe12 fabricated by a rapid quenching [1]. It was also reported that an as-spun TbCu7 structure of a non-equilibrium phase approaches a ThMn12 one as annealing temperature is increased, and latent heat cannot be detected during the structural distortion [1]. A converged ThMn12 structure decomposed into equilibrium phases after annealing at 1000 ◦ C because it does not include stabilizing elements such as silicon (Si), titanium (Ti), vanadium (V), and tungsten (W) etc [1]. Recently, it was reported that Sm(Fe0.8 Co0.2 )12 fabricated by a thin-film method has a large magnetization (𝐼s , 1.78 T) and a magnetic anisotropy field (𝜇0 𝐻a , 12 T) at room temperature (RT) [2], so it can be said that Sm(Fe0.8 Co0.2 )12 is superior to Nd2 Fe14 B. Under these circumstances, structure analysis of such iron-rich rare-earth (𝑅-𝑇 ) compounds is of great significance. While the TbCu7 structure is configured by substituting a pair of iron atoms (iron dumbbell) for rare-earth (𝑅) elements at random in a

CaCu5 structure, and it has a wide range of compositions, the ThMn12 structure is configured by substituting iron dumbbells for half of the 𝑅 elements in the CaCu5 structure. Since the ThMn12 structure is regarded as one of the ordered TbCu7 structures, its diffraction patterns consist of the fundamental reflections of the TbCu7 structure and super-lattice ones from its ordered arrangement. Such substitutions in accordance with certain rules allow descriptions on continuous lattice distortion between modulated-CaCu5 structures, which are defined as related structures configured by substituting iron dumbbells for 𝑅 elements on a CaCu5 structure, such as the other Th2 Zn17 , Th2 Ni17 , and Nd3 (Fe, Ti)29 structures. Structural models that can treat continuous changes from CaCu5 to TbCu7 [3,4] or Th2 Ni17 [5,6] structures by substituting iron dumbbells for 𝑅 elements have been developed. While these structural models can only be applied to lattice distortions with the same rotation symmetry and rotation axis, they cannot treat a tetragonal system with a four-fold rotation axis perpendicular to three-fold or six-fold rotation axes of hexagonal systems. Although a relationship between ThMn12 and Th2 Ni17 structures was revealed by using the transformation matrix, a structural model that could analyze continuous lattice distortions was not established [7]. Analyzing a ThMn12 structure by using the above-mentioned models resulted in unreliable fittings of Rietveld analysis [8]. Thus, a structural model that can treat hexagonal and tetragonal systems simultaneously has been pursued.

∗ Correspondence to: Research & Development Group, Hitachi, Ltd., 2520 Akanuma, Hatoyama, Saitama 350-0395, Japan. E-mail address: [email protected].

https://doi.org/10.1016/j.intermet.2020.106713 Received 4 December 2019; Received in revised form 16 January 2020; Accepted 16 January 2020 Available online 30 January 2020 0966-9795/© 2020 Elsevier Ltd. All rights reserved.

Intermetallics 119 (2020) 106713

H. Suzuki

We constructed a structural model that can describe continuous lattice distortions from the hexagonal to the tetragonal systems and applied it to powder X-ray-diffraction (XRD) data of the non-equilibrium Y–Fe alloys. Using that model, we clarified that there are peculiar trajectories of the lattice distortions that depend on compositions of the Y–Fe alloys. Moreover, we revealed that TbCu7 and ThMn12 structures with various compositions can be regulated on a map arranged in terms of axial ratio.

their site occupancies are the same, space group 𝐼𝑚𝑚𝑚 can also be described ones with a higher rotation symmetry. The lattice constants described by each space group have a relationship given as 𝑐hex = 𝑏ortho ∕2 = 𝑎tetra and 𝑎hex = 𝑐ortho = 𝑐tetra . Unless otherwise noted in the following, the structures are described by using space group 𝐼𝑚𝑚𝑚, where the prohibitions 𝑎ortho ≠ 𝑏ortho ≠ 𝑐ortho are removed. Incidentally, cobalt favors the 8𝑓 and 8𝑗 sites in the ThMn12 structure [9,10], whose sites correspond to the 4𝑓 , 4ℎ, and 8𝑘 sites described in space group 𝐼𝑚𝑚𝑚. We imposed the following assumptions and constraints on structure analyses. Cobalt and iron they have almost the same atomic-scattering factors (due to their proximity on the periodic table), it is difficult to distinguish cobalt from iron by XRD. We imposed the simplification that cobalt locates at the 8𝑘 site only although it, in principle, has no influence on results. Two geometric relationships in regard to site occupancies, namely, 0 ≤ 𝑔2𝑎 + 𝑔4𝑔2 ≤ 1 and 0 ≤ 𝑔2𝑑 + 𝑔4𝑔1 ≤ 1, were established. Moreover, the condition 𝑔2𝑎 > 𝑔2𝑑 was imposed without loss of generality due to a body-centered translational symmetry. The 4𝑔 site, divided into the 4𝑔1 and 4𝑔2 sites by self-interstitial splitting, is located on the 𝑏 axis. There is no relation between internal coordinates of the 4𝑔1 and 4𝑔2 sites. Since 4𝑔 site often moved around on the 𝑏 axis during the Rietveld analyses, it was assumed that distances of 4𝑔 site from the 𝑅 site are equal (In other words, the distance between two iron atoms in an iron dumbbell is equal). Although occupancies of the other iron sites could be taken freely, the occupancies were fixed to 1 when they varied within 𝜎𝑖 , which is the standard deviation of the 𝑔𝑖 . Incidentally, it is noted that the occupancies 𝑔𝑖 of 4𝑓 , 4ℎ, and 8𝑘 sites are constant not variable (the values equal to one). Since the structural model are expressed by substituting iron dumbbells for 𝑅 elements, only the occupancies related with the 𝑅 elements (2𝑎 and 2𝑑 sites) and the iron dumbbells (4𝑔1 , 4𝑔2 , and 4𝑒 sites) can be variable.

2. Experiments 2.1. Lattice-distortion model While TbCu7 structure belonging to the space group 𝑃 6∕𝑚𝑚𝑚 has a six-fold rotation symmetry, ThMn12 structure belonging to that of 𝐼4∕𝑚𝑚𝑚 has a four-fold rotation symmetry as characteristic symmetry element. Both structures have mirror planes perpendicular to each axis [001], [100], and [110]. To build a structural model that can describe continuous lattice distortions from TbCu7 to ThMn12 structures by substituting iron dumbbells for 𝑅 elements, the symmetry of the structural model should be lower than those of both structures due to orthogonal rotation axes. However, if the symmetry of the structure model is set too low, the number of fitting parameters is too large to analyze XRD data. Continuous distortions can be described by removing the rotation symmetries with keeping the mirror symmetries. Then, 𝑃 𝑚𝑚𝑚 and 𝐼𝑚𝑚𝑚 are candidate space groups for treating such distortions. While Bravais lattice symbol 𝑃 denotes a simple translational symmetry in a unit cell, symbol 𝐼 denotes a body-centered translational symmetry. Symbol 𝐼, which can connect 𝑃 6∕𝑚𝑚𝑚 and 𝐼4∕𝑚𝑚𝑚 with the shortest path, is suitable for analyzing XRD data in order not to excessively increase the number of fitting parameters. The relationship between the TbCu7 and ThMn12 structures when space group 𝐼𝑚𝑚𝑚 is used is shown in Fig. 1. The TbCu7 structure has four crystallographic sites, and the 1𝑎 site (for 𝑅 elements) and the 2𝑒 site (for iron dumbbells) are substituted at random mutually. When the ThMn12 structure is described from the viewpoint of the TbCu7 one, it becomes clear that the 𝑅 element lines and iron-dumbbell lines ̄ direction and pile up along the [001] locate alternately along the [210] direction in antiphase. The intermediate structure can be described by eight crystallographic sites, and iron dumbbells are partly substituted for 𝑅 elements between the 2𝑎 and 4𝑔2 sites and between the 2𝑑 and 4𝑔1 sites. The former sites are rich in the amount of the 2𝑎 site and the later sites are rich in the amount of the 4𝑔1 site. As the lattice distortions progress from the TbCu7 to ThMn12 structures, the sites are divided and joined as follows: 1𝑎 site → 2𝑎&2𝑑 sites → 2𝑎 site & none; 2𝑒 site → 4𝑔2 &4𝑔1 sites → none & 8𝑖 site; 2𝑐 site (forming Honeycomb lattices) → 4𝑒&4𝑓 sites → 8𝑖&8𝑗 sites; and 3𝑔 (forming Kagome lattices) → 4ℎ&8𝑘 sites → 8𝑗&8𝑓 sites. This distortion can be described by changing substitution rate (𝑠) of the iron dumbbells for the 𝑅 elements continuously, and lattice constants and internal coordinates of the distortion also change continuously in accordance with rate 𝑠. The larger the occupancy (𝑔𝑖 ) of the 4𝑔2 site (the 2𝑑 site) becomes, the more similar the structure described by space group 𝐼𝑚𝑚𝑚 becomes TbCu7 structure. Conversely, the larger the occupancy of the 2𝑎 site (the 4𝑔1 site) becomes, the more similar the structure described by space group 𝐼𝑚𝑚𝑚 becomes ThMn12 structure. In case of the space group 𝐼𝑚𝑚𝑚, when a four-fold symmetry condition such as 𝑎 = 𝑏, 𝑔4𝑔2 = 0, 𝑔2𝑑 = 0, 4𝑔1 (𝑥) = 4𝑒(𝑥), and 4ℎ(𝑦) = 4𝑓 (𝑥) is satisfied, the structure acquires the four-fold symmetry around the 𝑐 axis and forms the ThMn12 structure specialized by space group 𝐼4∕𝑚𝑚𝑚.√On the other hand, when a six-fold symmetry condition such as 𝑎 = 3𝑐, 𝑔2𝑎 = 𝑔2𝑑 , 𝑔4𝑔1 = 𝑔4𝑔2 , 4𝑒(𝑥) = 4𝑓 (𝑥), and 4ℎ(𝑦) = 8𝑘(𝑥) = 1∕4 is satisfied, the structure acquires the six-fold symmetry around the 𝑏 axis and forms the TbCu7 structure specialized by space group 𝑃 6∕𝑚𝑚𝑚. Namely, when certain correlations between the lattice constants occur, atoms located in general coordinates translate into special ones, and

2.2. Sample preparation Samples with compositions of Y(Fe0.83 Co0.17 )z (10.5 ≤ 𝑧 ≤ 17.0; 𝑧 = 𝑇 ∕𝑅) were fabricated using 99.9 + % grade materials by a rapid quenching.1 Specifically, the composition of 𝑧 = 10.5, 11.0, 11.25, 11.5, 12.0, 12.5, 13.0, 13.5, 14.0, 15.0, 16.0, and 17.0 was selected. A copper-roll velocity of 40 m/s was used for preparing TbCu7 structure, and the thickness of the formed samples was 30-40 μm. To investigate trajectories of structure distortions, these ‘‘ribbon’’ samples were annealed at temperatures from 700 ◦ C to 1000 ◦ C for 0.5 h in an argon (Ar) atmosphere. After being introduced in a furnace keeping a target temperature, a silica tube (including the ribbon samples and replaced by an Ar gas) was cooled by launching into water. The ‘‘ribbon’’ samples were pulverized to particle size under 75 μm with a hand grinding by an alumina mortar. ‘‘Powder’’ samples were used in the following measurements. Incidentally, it was confirmed that lattice strains from such a mild pulverization have no effect on the following results. 2.3. Measurement methods XRD patterns of pulverized powders were measured at RT with a wide-angle X-ray diffractometer (Bruker, D8 ADVANCED/TXS). A copper-rotor was induced with a voltage of 45 kV and a current of 360 mA. A nickel-filter was used for absorbing 𝐾𝛽 . Both incident and receive solar slits with 2.5◦ , a divergence slit with 1.5◦ , a receive slit with 0.1 mm, and no scattering slit were installed. XRD data covering a diffraction–angle 2𝜃 range from 20◦ to 100◦ were collected by scanning interlocking 2𝜃∕𝜃 continuously. Rietveld analysis was performed by using the source code DIFFRACplus Professional TOPAS 4 (Bruker). Solution convergence was judged on the basis of fitting-accuracy condition

1

2

Cobalt was added to improve magnetic properties.

Intermetallics 119 (2020) 106713

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Fig. 1. Relationship between TbCu7 and ThMn12 structures when space group 𝐼𝑚𝑚𝑚 is used. 𝐾𝑎𝑔𝑜𝑚𝑒 lattices in the structural diagrams were omitted for visibility.

value for a ThMn12 structure [11], was supposed. Quadrupole splitting (𝛥) can be described with the following formula by using elementary charge (𝑒), electric quadrupole (𝑄), and electric scalar potential (𝑉 ) when the 𝑥 and 𝑦 directions perpendicular to quantum axis 𝑧 can be defined as isotropic.

𝑅wp ∕𝑅e < 1.5, where 𝑅wp and 𝑅e represent the sum of statistically weighted squares of residuals and the lowest statistically expected 𝑅wp given as the follows: √∑ √ 2 𝑁 −𝑃 i 𝑤i (𝑦i − 𝑓i (𝒙)) 𝑅wp = , 𝑅 = , ∑ ∑ e 2 2 𝑤 𝑦 i i i i 𝑤i 𝑦 i (1) √ ∑ 2 𝑅wp i 𝑤i (𝑦i − 𝑓i (𝒙)) 𝑆= = 𝑅e 𝑁 −𝑃

𝛥=

57 𝑒𝑄𝑉zz 1∑∑ 𝜕2 𝑉 𝑄𝑖𝑗 = 6 1 𝑥,𝑦,𝑧 𝜕𝑥𝑖 𝜕𝑥𝑗 2

(2)

where electric gradient 𝑉zz is defined as 𝑉 with the steepest slope of any directions. In the notation of the space group 𝐼𝑚𝑚𝑚, 𝑧 directions in structures similar to the TbCu7 and ThMn12 structures correspond to the 𝑏 and 𝑐 directions, respectively. Then, it is worth noting that the 𝑧 direction changes from the 𝑏 to 𝑐 directions continuously during the lattice distortions. Magnetic anisotropy energy (𝐾u ) of an iron sublattice is generated by survival of a part of orbital angular momentum due to a spin–orbit coupling. 𝐾u can be described by the following approximate formula [12].

where 𝑤i is statistical weight, 𝑦i and 𝑓i (𝒙) represent observed and theoretical diffraction intensities, respectively, and 𝑁 and 𝑃 are the number of total data and refined parameters, respectively. In Rietveld analysis, the main phase and other phases of disordered-Y2 (Fe, Co)17 (𝑑-Y2 (Fe, Co)17 ) [6] and 𝛼-(Fe, Co) were included. Curie temperatures (𝑇C ) were measured with a thermal magnetic balance introducing a magnetic field of about 0.01 T in an Ar gas flow. 𝑇C was defined as an inflection point of a temperature dependence of magnetic force (TM) in temperature-decreasing processes from 650 ◦ C . TM measurement can detect each similar phase as a difference in 𝑇C , even in the case of indistinguishable XRD patterns. 57 Fe Mössbauer spectra were measured at RT and fitted to Lorentzian sextets by least-squares minimization under an intensity constraint of 3:2:1:1:2:3 due to magnetic and texture isotropy. To estimate magnetic moments, a linear relation 15.7 T/𝜇B between the inner magnetic field and the magnetic moment, which is a common

𝜉2 (3) (𝐷 − 𝐷Feasy ) 𝛺 Fhard where 𝛺 is system volume, and 𝐷F𝑥 is electron density of states (DOS) at Fermi level when magnetic moments are aligned in the 𝑥 direction. 2 𝜉 is a coefficient of spin–orbit coupling described as 𝜉 = 2𝑚ℏ 2 𝑐 2 ( 1𝑧 𝑑𝑉 )= 𝑑𝑧 𝐾u ∝

ℏ2 𝑉 . 2𝑚e 2 𝑐 2 zz

e

Therefore, 𝐾u is proportional to 𝑉zz 2 . This indicates that when

𝐷F hardly changes, or magnetic moments do not change, 𝛥2 normalized 3

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still has randomly substituted parts with iron dumbbells for yttrium elements. The convergent tetragonal structure of type-I distortion can thus be called a ‘‘disordered-ThMn12 (𝑑-ThMn12 )’’ structure. The 𝑑ThMn12 structure was generated by annealing at temperatures higher than 850 ◦ C. Type-II: A lattice in alloys with composition range of 10.5 < 𝑧 < 11.5 is distorted from a hexagonal structure with 𝑏∕𝑎 ∼ 1.001 to an orthorhombic one as close as possible to a tetragonal one. Lattice distortion during the lattice distortion is much smaller than that of type-I distortion. The orthorhombic structure as close as possible to a tetragonal one also has randomly substituted parts with iron dumbbells for yttrium elements. The orthorhombic structure as close as possible to the tetragonal one of type-II distortion can thus be called a ‘‘pseudodisordered-ThMn12 (𝑝𝑑-ThMn12 )’’ structure. The word ‘‘pseudo’’ is used to indicate that the structure is not a tetragonal one but one as close as possible to it. It was defined as an orthorhombic structure with a difference between the 𝑎 and 𝑏 constants of less than 0.1%. It is worth noting that the 𝑝𝑑-ThMn12 structure with 𝑏∕𝑐 > 1.761 cannot be formed. Type-III: A lattice in alloys with composition range of 𝑧 ≤ 10.5 is distorted from a hexagonal structure to an orthorhombic one with a shorter 𝑏 axis than the 𝑎 one. Judging from a composition ratio 𝑧, it can be assumed that the orthorhombic structure is similar to the Nd3 (Fe, Ti)29 structure. However, since the structural model cannot treat the Nd3 (Fe, Ti)29 structure, it would have further discussions on distortion of the Type-III structure. Type-IV: A lattice in alloys with composition range of 𝑧 ≥ 14.0 is also distorted from a hexagonal structure to an orthorhombic one with a shorter 𝑏 axis than the 𝑎 one. It is noted that lattice constants of type-IV distortion were estimated with a low accuracy due to low production of the main phase and diffuse full-widths at half maximum (FWHM) of its XRD peaks. It is concluded from the above results that the as-spun structures are distorted into ones with a large 𝑏∕𝑐 only at a certain 𝑇 ∕Y ratio by substituting iron dumbbells for yttrium elements. Type-I and II distortions are treated precisely in the following subsections.

Fig. 2. Trajectories of lattice distortions arranged in terms of axial ratios for Y(Fe0.83 Co0.17 )𝑧 (10.5 ≤ 𝑧 ≤ 17.0). Data plots are interpolated to display lattice distortion by annealing.

by 𝛺 behaves in the same way as 𝐾u . Since the number of sites described in space group 𝐼𝑚𝑚𝑚 was too large to analyze the Mössbauer spectra, they were analyzed by the three Mössbauer components, which was the same as the number of sites in both end structures (TbCu7 and ThMn12 structures) of the lattice distortions. This number was still enough to acquire quantitative results. However, since it was too difficult to assign sites due to low estimations of 𝑔𝑖 , a simple average of three components was estimated. In Mössbauer spectrum analyses, the Mössbauer components of the equilibrium phase 𝑑-Y2 (Fe, Co)17 and 𝛼-(Fe, Co) were first estimated by using the data of samples annealed at over 1000 ◦ C, which are composed of only 𝑑-Y2 (Fe, Co)17 and 𝛼-(Fe, Co). Next, with the Mössbauer components of 𝑑-Y2 (Fe, Co)17 and 𝛼-(Fe, Co) fixed, the data of samples including the main phase were analyzed.

3.1.1. Type-I distortion (11.5 ≤ 𝑧 ≤ 13.5) Type-I distortion was classified into three areas according to 𝑏∕𝑐, which consist of ranges 𝑏∕𝑐 < 1.746 (area I-A) for a TbCu7 structure, 1.746 < 𝑏∕𝑐 < 1.761 (area I-B) for an immediate body-centered orthorhombic one, and 𝑏∕𝑐 > 1.761 (area I-C) for a 𝑑-ThMn12 one. Dependences of (a) site occupancies and (b) internal coordinates on 𝑏∕𝑐 ratio for type-I distortion are shown in Fig. 3. Triangles and circles indicate as-spun and annealed samples, respectively. It is noted that the structure within area I-A is not TbCu7 one due to a non-six-fold rotation√symmetry around the 𝑏 axis (since it only meets the conditions of 𝑎 ≃ 3𝑐 and 𝑔2𝑎 ≃ 𝑔2𝑑 ). Moreover, the structure within area I-C is not the ThMn12 one due to a non-four-fold rotation symmetry around the 𝑐 axis (since it only meets the conditions of 𝑎 = 𝑏, 𝑔4𝑔2 = 0, and 4𝑔1 (𝑥) ≃ 4𝑒(𝑥)). However, these may be analytical errors caused by the increase of fitting parameters, since the number of degree of freedom in the space group 𝐼𝑚𝑚𝑚 is larger than that in the original space groups 𝑃 6∕𝑚𝑚𝑚 and 𝐼4∕𝑚𝑚𝑚. The former can be eliminated by imposing a strong condition 𝑔2𝑎 + 𝑔4𝑔2 = 1, and the latter can be eliminated by applying the structural model under the same coordinates in the 𝑎 and 𝑏 directions. Only a weak constraint, 0 ≤ 𝑔2𝑎 + 𝑔4𝑔2 ≤ 1, was imposed in Rietveld analysis to eliminate allegory. Imposition of the strong constraint condition and a slight revision of the structural model would lead to detailed knowledge on the lattice distortions. Incidentally, if the analytical errors are ignored, the situation of 𝑔2𝑎 + 𝑔4𝑔2 < 1 implies atom deficiencies. It was revealed that the equivalent Y sites (or iron dumbbell site) divide into Y-rich (or iron dumbbell-poor) and Y-poor (or iron dumbbell-rich) sites when 𝑏∕𝑐 ratio increase (i.e., annealing temperature is increased). Dependence of lattice constants 𝑎, 𝑏, and 𝑐 on 𝑏∕𝑐 ratio for type-I distortion are shown in Fig. 4. In the case that 𝑏∕𝑐 ratio was increased, the following results for each area were confirmed. Area I-A: the lattice constant 𝑎 was shrunk, the lattice constant 𝑏 was enlarged, and the

3. Results 3.1. Lattice distortions by annealing Regulated by axial ratios with dimensionless quantity, the lattice distortions could be recognized visually. Among combinations of axial ratios, the axial ratios of 𝑏∕𝑎 and 𝑏∕𝑐 were adopted to √ describe structures as a monovalent function and visualize lines 𝑐 = 3𝑎 and 𝑎 = 𝑏. Especially, 𝑏∕𝑐 represents a tetragonality when condition 𝑎 = 𝑏 is satisfied. The lattice distortions were described by axial ratio 𝑏∕𝑐 as a variable as follows. Analyses of Y(Fe0.83 Co0.17 )𝑧 (10.5 ≤ 𝑧 ≤ 17.0) alloys illuminated trajectories of lattice distortion by annealing (obtained by applying the continuous-lattice-distortion model), as shown in Fig. 2. It was revealed that the rise of annealing temperature makes structures with low 𝑏∕𝑐 ratio transform into ones with high 𝑏∕𝑐 ratio regardless of their compositions. The lattice distortions caused by annealing depend on the alloys’ compositions and consist of the following four types: (It should be noted that composition boundaries between each type are narrow enough to appear discontinuous.) Type-I: A lattice in alloys with composition range of 11.5 ≤ 𝑧 ≤ 13.5 is distorted from a hexagonal structure with 𝑏∕𝑎 ∼ 1.002 to a tetragonal one with 𝑏∕𝑎 = 1.000 via one with large distortion of 𝑏∕𝑎 = 1.008. The convergent tetragonal structure 4

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Fig. 5. Dependences of (a) site occupancies and (b) internal coordinates on 𝑏∕𝑐 ratio for type-II distortion. Triangle and circle symbols correspond to as-spun and subsequent annealing samples, respectively.

Fig. 3. Dependences of (a) site occupancies and (b) internal coordinates on 𝑏∕𝑐 ratio for type-I distortion. Triangle and circle symbols correspond to as-spun and subsequent annealing samples, respectively.

lattice constant 𝑐 was shrunk slightly. Area I-B: the lattice constant 𝑎 was enlarged, the lattice constant 𝑏 was almost constant, and the lattice constant 𝑐 was shrunk with a linear slope. Area I-C: the lattice constants 𝑎 and 𝑏 were same and enlarged slightly, and the lattice constant 𝑐 was shrunk with the same slope as that observed in area I-B. It is worth noting that changes of the 𝑎 and 𝑐 axes, which is perpendicular to an axial direction of substitutable iron dumbbells, were larger than that of the 𝑏 axis, which are parallel to the axial direction of the iron dumbbells. 3.1.2. Type-II distortion (10.5 < 𝑧 < 11.5) Type-II distortion was also classified into three areas according to 𝑏∕𝑐, which consist of ranges 𝑏∕𝑐 < 1.740 (area II-A) for TbCu7 structure, 1.740 < 𝑏∕𝑐 < 1.761 (area II-B) for 𝑝𝑑-ThMn12 one, and 𝑏∕𝑐 > 1.761 (area II-C) for nothing. Dependences of (a) site occupancies and (b) internal coordinates on 𝑏∕𝑐 ratio for type-II distortion are shown in Fig. 5. Although the √ structure within area II-A was satisfied with conditions 𝑎 ≃ 3𝑐 and 𝑔2𝑎 ≃ 𝑔2𝑑 , the structure was not TbCu7 one due to a non-six-fold rotation symmetry around the 𝑏 axis by judging from condition 𝑔4𝑔1 ≠ 𝑔4𝑔2 . This may be due to analytical errors caused by the increase of fitting parameters. Shifts of site occupancies and internal coordinates for type-II distortion clearly were differed from those for type-I one . Dependences of lattice constants 𝑎, 𝑏, and 𝑐 on 𝑏∕𝑐 ratio for type-II distortion are shown in Fig. 6. This plot demonstrated that when 𝑏∕𝑐

Fig. 4. Dependences of lattice constants 𝑎, 𝑏, and 𝑐 on 𝑏∕𝑐 ratio for type-I distortion. Triangle and circle symbols correspond to as-spun and subsequent annealing samples, respectively.

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Fig. 6. Dependences of lattice constants 𝑎, 𝑏, and 𝑐 on 𝑏∕𝑐 ratio for type-II distortion. Triangle and circle symbols correspond to as-spun and subsequent annealing samples, respectively.

ratio was increased, 𝑎 and 𝑏 (with almost the same length) increased, and 𝑐 decreased linearly. The lattice constants for type-II distortion behaved differently from those for type-I distortion. Type-I (11.5 ≤ 𝑧 ≤ 13.5) and type-II (10.5 < 𝑧 < 11.5) distortions could be distinguished in terms of composition ratio 𝑇 ∕Y and were also entirely different. As annealing temperatures was increased, both distortions tended to enlarge in the axial direction of substitutable iron dumbbells and shrink along in the orthogonal direction (𝑐 axis) of the iron dumbbells; therefore, it is concluded that an order parameter is enhanced by substituting the iron dumbbells for yttrium elements. This conclusion might also be supported by a result that the 𝑑-ThMn12 and 𝑝𝑑-ThMn12 structures form in smaller volumes than the TbCu7 structure including the disorder parts (which are configured by substituting iron dumbbells for 𝑅 elements randomly). Meanwhile, although it was observed that 𝑇 ∕𝑌 tended to decrease as the TbCu7 structures approach the ThMn12 structures, this is discussed within analytical errors. Incidentally, estimated accuracies of the order parameter, which could be defined by site occupancies, were not enough to organize the above figures.

Fig. 7. Dependences of 𝑇C on 𝑏∕𝑐 ratio for (a) type-I and (b) type-II distortions. Triangle and circle symbols correspond to as-spun and subsequent annealing samples, respectively.

3.2. Magnetic-property shifts by lattice distortions Shifts of 𝑇C , magnetic moments, and 𝐾u during type-I and typeII distortions are still described in terms of 𝑏∕𝑐 ratio as a variable. The magnetic moments and 𝐾u were estimated with Mössbauer spectroscopy by using the samples with representative composition of 𝑧 = 12.0 (type-I distortion) and 𝑧 = 11.0 (type-II distortion). Dependences of 𝑇C on 𝑏∕𝑐 ratio for the (a) type-I and (b) typeII distortions are shown in Fig. 7. 𝑇C for type-I distortion increased linearly slope and saturated after reaching area I-C, which indicates that convergence of a structural distortion also leads to that of the 𝑇C shift. Meanwhile, 𝑇C for type-II distortion spiked at an boundary between areas II-A and II-B (𝑏∕𝑐 ≃ 1.740) and saturated within area II-B (𝑏∕𝑐 ≳ 1.755). It should be noted that the structure is distorted continuously despite the spike in 𝑇C . The saturation value of 𝑇C for type-I distortion is about 10 ◦ C higher than that for type-II distortion, and this difference was independent of substitution amount of cobalt. This result seems to indicate that amount of iron dumbbells involved during type-I distortion is larger than that involved during type-II distortion.

Fig. 8. Dependences of average inner magnetic field (left axis) and magnetic moment (right axis) on 𝑏∕𝑐 ratio for type-I and type-II distortions.

Dependences of average inner magnetic field (left axis) and magnetic moment (right axis) on 𝑏∕𝑐 ratio for type-I and type-II distortions are shown in Fig. 8. When 𝑏∕𝑐 ratio increases (or annealing temperatures increased), the inner magnetic fields (magnetic moments) for both type-I and type-II distortions increase and reach 33.0 T (2.10 𝜇B ). 6

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to TbCu7 , Th2 Ni17 , Th2 Zn17 or CaCu5 structures. In terms of the axial ratios, 𝑏∕𝑐 ratios of the TbCu7 structure within the range of 1.70 ≲ 𝑏∕𝑐 ≲ 1.75 are larger than those of Th2 Ni17 and Th2 Zn17 structures within the range of 𝑏∕𝑐 ∼ 1.68 and CaCu5 within the range of 𝑏∕𝑐 ∼ 1.60, which implies that the TbCu7 structure can take ironricher compositions than its stoichiometric composition due to random substitution of iron dumbbells for 𝑅 elements. It is concluded that a compound including large amounts of iron dumbbells takes a high axial ratio, 𝑏∕𝑐. From the viewpoint of lattice geometry, it was understood that a TbCu7 structure moving away from the line 𝑏∕𝑎 = 1.000 such as SmFe10.1 Nx [13] and Sm0.7 Zr0.3 (Fe0.8 Co0.2 )11 B0.5 [14] tends to have large magnetic anisotropy in comparison √ with SmFe10.1 [13] and Sm0.75 Zr0.25 (Fe0.7 Co0.3 )9.3 [15] near point ( 3, 1) even if effects from composition differences were removed. Although magnetic anisotropy changes have been understood mainly √ in terms of effects of element insertions, an effect far from point ( 3, 1) was also expected √ on the basis of the following mechanism. When both conditions 𝑐 = 3𝑎 and 𝑎 = 𝑏 are satisfied, a regular hexagonal prism and a regular square pillar coexist in the structure, which results in conflict of easy-magnetization directions (i.e., the structure becomes more isotropic) and tends to cause the smallest 𝐾u of those of neighboring structures with disordered parts. This conflict affects 𝐾u from not only a spin–orbit coupling but also a crystal field. However, it should be noted that the perspective of a lattice geometry on 𝐾u from a spin–orbit coupling should be limited on distortions from disordered to ordered systems without composition changes since 𝐾u is highly sensitive to the number of 𝑑 electrons [16]. From the structural viewpoint, the probability that iron dumbbells with the axial direction parallel to the 𝑎 axis occur should increase due to the equivalent length of the 𝑎 and 𝑏 axes. Whereas, the ThMn12 structure can take 𝑏∕𝑐 in the range of 1.756 ≲ 𝑏∕𝑐 ≲ 1.808, although most structures within a range of 𝑏∕𝑐 ≳ 1.790 are inserted by elements such as nitrogen (N) [11]. Representative equilibrium phases such as ZrFe10 Si2 [17], YFe10 Si2 [11], YFe11 Ti [11], SmFe11 Ti [11], YFe10.5 W1.5 [11], and Sm0.8 Zr0.2 (Fe0.8 Co0.2 )11.5 Ti0.5 [18] are just located within a stable range of 1.756 ≲ 𝑏∕𝑐 ≲ 1.790. Non-equilibrium phases such as SmFe12 * [2], Sm(Fe0.9 Co0.1 )12 * [2], and SmFe12 † [19] fabricated by thin-layer methods are located within a range of 𝑏∕𝑐 ≲ 1.755, which is considered to come from substrates restraint. This consideration allows SmFe12 * [2], SmFe12 † [19], SmFe12 ‡ [20], and SmFe12 § [21] fabricated by different conditions to take various 𝑏∕𝑐 ratios. If the non-equilibrium phases are analyzed by utilizing the proposed continuous-lattice-distortion model, trajectories free from the line of 𝑏∕𝑎 = 1.000 would √be illuminated. Substrate restraint or leaving from the special point ( 3, 1) is necessary to explain the finding that SmFe12 * [2] near the special point has a large magnetic anisotropy. Incidentally, it was also understood that increasing the tetragonality 𝑏∕𝑐 should tend to increase 𝐾u if condition 𝑎 ∼ 𝑏 is satisfied. From the structural viewpoint, it is significant that 𝑏∕𝑐 ratio of Sm(Fe, Co)12 * [2] approach the stable range (1.756 ≲ 𝑏∕𝑐 ≲ 1.790) of a ThMn12 structure as an amount of cobalt substitution increases, which is consistent with the result of a first-principles calculation that cobalt stabilizes a ThMn12 structure [22]. As mentioned above, the previously recognized compounds with high magnetic properties could be understood on the map arranged in terms of axial ratios comprehensively. Meanwhile, it is hard to say that structural differences between type-I and type-II distortions are completely clear. The converged structure in the case of type-II distortion is similar to SmFe10 Ti [23] or 𝑅Fe9 Ti [24] with iron-poor structures. It was pointed out that although these compounds have the same space group 𝐼4∕𝑚𝑚𝑚 as a ThMn12 structure, they have 10–15 ◦ C lower 𝑇C and smaller magnetization due to their iron-poorer compositions than those of the ThMn12 structure [24], which are consistent with our results. Since the lengths of 𝑎 and 𝑏 in the case of type-II distortion are almost the same, type-II distortion might be analyzed by utilizing another continuous-lattice-distortion model that also includes partial substitutions of 𝑅 elements for a pair of iron atoms in 4𝑒 sites. On the

Fig. 9. Dependences of quadrupole splitting 𝛥 on 𝑏∕𝑐 ratio for type-I and type-II distortions. 𝐻a estimated by SPD is inset.

Differences between both types of distortions could not be clarified. Increase in 𝑇C caused by the lattice distortions suppresses thermal fluctuations of the magnetic moments measured at RT. However, a temperature of 450 ◦ C for 𝑇C is much higher than RT; thus, the suppression of thermal fluctuations can be ignored, and the magnetic moments themselves would increase. It was revealed that the structures similar to ThMn12 ones tend to carry larger magnetic moments than the TbCu7 ones. Within a range of 𝑏∕𝑐 ≳ 1.746, average magnetic moment did not change, even though the lattice distortion progressed. Taking into consideration that the ThMn12 structures occupy a smaller volume than TbCu7 ones, it is considered that although the magnetic moments are decreased after volume reduction by a magneto-volume effect, they are increased after the change of electron DOS by progressing order. Dependences of quadrupole splitting 𝛥 on 𝑏∕𝑐 ratio for the type-I and type-II distortions are shown in Fig. 9. Magnetic anisotropy fields 𝜇0 𝐻a estimated by other measurements by singular point detection (SPD) are inset. It is demonstrated by these plots that while values of 𝛥 are small within the range of 𝑏∕𝑐 < 1.740, in which TbCu7 structures are formed, they are maximized around ranges of 𝑏∕𝑐 ∼ 1.754 and 1.746 < 𝑏∕𝑐 < 1.752 for both type-I and type-II distortions. Since the magnetic moments change little, as shown in Fig. 8, the behavior of 𝐾u should be inferred partly from one of 𝛥 according to the formulas (2) and (3), which are supported by the change of 𝜇0 𝐻a and small 𝛥 for the TbCu7 structure. It is noteworthy that quantum axis 𝑧 of TbCu7 and ThMn12 are in the 𝑏 and 𝑐 directions, respectively, and orthogonalized to each other. It is thus confirmed that 𝑑-ThMn12 for type-I distortion and 𝑝𝑑-ThMn12 for type-II distortion tend to have the largest 𝐾u compared to that of neighboring structures. 4. Discussion By utilizing the proposed continuous-lattice-distortion model, it was shown that non-equilibrium Y–Fe alloys are distorted in four lattice patterns, which depend on the alloy compositions. Since the model can treat continuous distortions from hexagonal to tetragonal via orthorhombic structures, the modulated-CaCu5 structure, except for a monoclinic Nd3 (Fe, Ti)29 structure, can be analyzed and regulated on a map arranged in terms of axial ratios. Especially, plotting the TbCu7 and ThMn12 structures with various compositions on the map awake new perspectives, as shown in Fig. 10. It is noted that lattice constants were defined by the continuous-lattice-distortion model specialized by space group 𝐼𝑚𝑚𝑚, and approximate annealing temperatures of the Y–Fe√ alloys were also inset. Structures on a line with a gradient of 1∕ 3, whose basal plane forms hexagons, belong 7

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𝑏∕𝑐 ∼ 1.754 and 1.746 < 𝑏∕𝑐 < 1.752 in the case of both type-I and typeII distortions, respectively. The structures on a line with a gradient of √ 1∕ 3, the ThMn12 structure, and their immediate structures could be arranged in terms of axial ratios by utilizing the proposed continuouslattice-distortion model. The continuous-lattice-distortion-model could bring geometric insights on structural stability and magnetic properties of not only the alloys but also previously recognized compounds with modulated-CaCu5 structures, which are configured by substituting iron dumbbells for 𝑅 elements. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement Hiroyuki Suzuki: Conceptualization, Methodology, Validation, Formal analysis, Investigation, Resources, Data curation, Writing - original draft, Writing - review & editing, Visualization, Supervision.

Fig. 10. A map arranged in terms of axial ratios on TbCu7 and ThMn12 structures with various compositions.

Acknowledgments other hand, lattice distortions from hexagonal to tetragonal systems are considered to occur because the small atomic radius of yttrium and lattice size related to lanthanide contraction allow iron dumbbells to be substituted easily. It also seems to contribute essentially that a ThMn12 structure is stabilized at a high temperature and that chemical characteristics of yttrium are located between those of dysprosium (Dy) and terbium (Tb) in the periodic table. These considerations are supported by our another experimental result indicating that when samarium and gadolinium, especially gadolinium, with similar atomic radius to yttrium were used, similar lattice distortions also were observed. It was also affirmed by results of a first-principles calculation that the enthalpy of 𝑅Fe12 correlates to an atomic radius of 𝑅 strongly, and a part substitution of yttrium, dysprosium, holmium (Ho), erbium (Er), or thulium (Tm) helps 𝑅Fe12 to form [25].

We would like to express our deepest appreciation to Dr. Takeshi Nishiuchi, Dr. Masakuni Okamoto, and Dr. Akira Sugawara for their support and helpful discussions and Prof. Hiroyuki Nakamura for the Mössbauer spectra measurement. References [1] [2] [3] [4] [5] [6] [7] [8]

5. Summary A structural analysis model, which was specialized by space group 𝐼𝑚𝑚𝑚 and denoted by substituting iron-dumbbells for 𝑅 elements, was devised to treat lattice distortions from hexagonal TbCu7 to tetragonal ThMn12 structures observed in non-equilibrium Y–Fe alloys. Trajectories of lattice distortions due to annealing could be classified into four types according to composition ratio (𝑇 ∕𝑅) of the Y–Fe alloys. Especially, type-I distortion with a composition range of 11.5 ≤ 𝑇 ∕𝑅 ≤ 13.5 transformed from a hexagonal structure with 𝑏∕𝑎 ∼ 1.002 to a tetragonal one with 𝑏∕𝑎 = 1.000 via one with large distortion, i.e., 𝑏∕𝑎 = 1.008. The convergent tetragonal structure still had randomly substituted parts with iron dumbbells for yttrium elements. The convergent tetragonal structure could thus be called a disorder-ThMn12 (𝑑-ThMn12 ) structure. Meanwhile, type-II distortion with a composition range of 10.5 < 𝑇 ∕𝑅 < 11.5 transformed from a hexagonal structure with 𝑏∕𝑎 ∼ 1.001 to an orthorhombic one as close as possible to a tetragonal one. The orthorhombic structure also still had an arrangement of the random substitution of iron dumbbells for yttrium elements. It could thus be called a pseudo-disorder-ThMn12 (𝑝𝑑-ThMn12 ) structure. The saturation value of Curie temperature (𝑇C ) for type-I distortion was about 10 ◦ C higher than that for type-II distortion. It was revealed from inner magnetic fields of the Mössbauer measurement that the structures similar to ThMn12 structure tend to have larger magnetic moments than TbCu7 structure. Moreover, while average quadrupole splitting (𝛥), which infers the strength of magnetic anisotropy energy (𝐾u ) partly, was small within the range 𝑏∕𝑐 < 1.740, it maximized within ranges

[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

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