The Scientific World Journal

Volume 2013 (2013), Article ID 919182, 6 pages

http://dx.doi.org/10.1155/2013/919182

## Phase Stability and Site Preference of Tb-Fe-Co-V Compounds

Institute of Applied Physics, University of Science and Technology, Beijing 100083, China

Received 29 April 2013; Accepted 11 June 2013

Academic Editors: S. V. Dmitriev and Z. Zhang

Copyright © 2013 Jing Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The effect of cobalt on the structural properties of intermetallic with Nd_{3}(Fe,Ti)_{29} structure has been studied by using interatomic pair potentials obtained through the lattice inversion method. Calculated results show that the preferential occupation site of the V atom is found to be the 4i(Fe3) site, and Fe atoms are substituted for Co atoms with a strong preference for the 8j(Fe8) site. The calculated lattice constants coincide quite well with experimental values. The calculated crystal structure can recover after either an overall wide-range macrodeformation or atomic random motion, demonstrating that this system has the stable structure of Nd_{3}(Fe,Ti)_{29}. All these prove the effectiveness of interatomic pair potentials obtained through the lattice inversion method in the description of rare-earth materials.

#### 1. Introduction

In 1993, it had been first found that the structure of the compound suggested to be a Nd_{3}(Fe,Ti)_{29}-type structure with monoclinic symmetry [1]. After lots of investigation and disputation [2–4], it indicates that the Space Group (S.G.) of Nd_{3}(Fe,Ti)_{29}-type structure belongs to A2/m rather than P2_{1}/c.

In the A2/m space group description, the lattice cell with two R_{3}Fe_{29} formulas consists of 64 atoms. The rare-earth ions occupy two nonequivalent crystallographic sites (2a and 4i), and the iron atoms occupy 11 sites (one 2c, one 4g, one 4e, four 4i and four 8j). The crystallographic data for Nd_{3}Fe_{27.5}Ti_{1.5} compound (Space Group no. 12 (A2/m), A12/m1, unique axis b, cell choice 2) is on page 161 of [5]. The 3 : 29 compounds are deemed to consist of tetragonal ThMn_{12}-type (1 : 12) and rhombohedral Th_{2}Ni_{17}-type (2 : 17) segments in a ratio of 1 : 1 [6]. In fact, the binary compounds R_{3}Fe_{29} are metastable, which can be seen as the intrinsic prototype of R_{3}(Fe,T)_{29}, since R_{3}(Fe,T)_{29} can be made stable with a moderate amount of a ternary element T (T = V, Ti, Cr, or Mo, etc.). From that time on, the ternary intermetallic compounds R_{3}(Fe,T)_{29} (R = a rare-earth element or Y, and T = a stabilizing element) have attracted attention, as well as many quaternary compounds [7–14].

As well known, the substitution of cobalt for iron in rare-earth-transition metal intermetallics has a remarkable effect on their magnetic properties. Comparing with iron, cobalt has a different electronic structure and therefore lead to a different local anisotropy. Hence, the structure and the magnetic properties of 3 : 29 compounds are strongly affected by the substitution of iron by cobalt. However, when more than 40% of substitution of iron by cobalt atoms occurs, a large number of stabilizing atoms are demanded. If not, a disordered modification of hexagonal Th_{2}Ni_{17}-type structure will be formed [10, 11, 13, 14].

So far, there are some investigations on the Tb-Fe-Co-V and Tb-Fe-Co-Cr systems [9, 11–14]. For compounds, in the temperature range between 5 K and 200 K, a FOMP occurs [15]. The spin reorientation has been observed in the Tb-Fe-Co-V system, and the Curie temperature and saturation magnetisation decrease with the increase of Co content [13].

In this paper, our investigation focuses on the phase stability and site preference of Co in ( = 0.0, 0.1, 0.2, 0.3 and 0.4) compounds. The interatomic pair potential are obtained by Chen’s lattice inversion method, whose theory is exhibited in the second part. The comparison of the calculated results with the experimental data is shown in Section 3. And the fourth part is the conclusion and discussion.

#### 2. Methodology

Chen et al. proposed a concise inverse method [16–21] based on the modified Möbius inversion in number theory, which can be applied to obtain the photon density of states [16], to solve the inverse blackbody radiation problem for remote sensing [16], to unify the Debye and Einstein approximations in a general mathematics system [20], and to extract the interatomic pair potentials from ab initio calculated cohesive energy curves in pure metals and intermetallic compounds [17–19, 22, 23], metal/ceramic interface [24], and metal/oxide interface [25], as well as in carbides with complex structures [26–28], with high convergence speed.

Here, we take a single element crystal as an example to explain how to use Chen’s lattice inversion theorem to obtain the interatomic pair potential from the first-principle cohesive energy curve. Suppose that the crystal cohesive energy can be expressed as the sum of interatomic pair potentials where is the nearest neighbor distance, is the position vector of the th atom in the lattice, is the th nearest neighbour distance with , and is the th coordination number. Obviously, is a totally ordered set but, in most cases, not a multiplicative semigroup. In this case, we need to extend it to a totally ordered multiplicative semigroup with . Namely, for any , , we have . Now can be rewritten as where the extended coordination number satisfies the following rule:

It is clear that is uniquely determined by crystal geometrical structure. Now we can extract pair potential from (2) as where , , and any satisfying , can be determined by where is the Kronecker delta. It is easy to prove that (5) has and only has a single solution set , which, like , is uniquely determined by crystal geometrical structure, not related to element category.

This means that we can take advantage of number theory to extract the interatomic pair potentials form cohesive energy curve(s). At this stage, modified Möbius inversion can be adopted to extract the pair potential curve data and then fit these data on the basis of Morse function [29] to determine the potential parameters. Consider where indicates the distance between two atoms and is the equilibrium distance, with and being parameters without unit.

For the readers’ convenience, several important potential parameters are shown in Figure 1.

#### 3. Calculated Results

In this paper, energy minimization is carried out using a conjugate gradient method. The cut-off radius is 14 Å. In order to reduce statistical fluctuation, we take the super-cell containing 1280 atoms, ()_{40}, for simulation.

There are no reports in the literature on the existence of the binary structure Tb_{3}Fe_{29}, which can be seen as the prototype of R_{3}(Fe,T)_{29}. In the calculation procedure, the initial lattice constants of Tb_{3}Fe_{29} are randomly chosen in a certain range. Under the control of the interatomic pair potentials, the energy minimization is carried out. After repeated relaxation, the space group maintains A2/m, the atomic site occupation is similar to that of Nd_{3}(Fe,Ti)_{29}, and the structure will be stabilized with lattice constants Å, Å, Å, and *β* = 96.91° (Table 1). The randomness of the initial structure in a certain range and the stability of the final structure illustrate that Tb_{3}Fe_{29} has the topological invariability with respect to the existing Nd_{3}(Fe,Ti)_{29} structure. Hence, it furnishes convincing evidence that the interatomic pair potentials are reliable for the study of structural material characteristics.

Substitute the atoms of ternary element vanadium for the randomly chosen iron atoms at a certain lattice site, and then make the lattice relaxation. In this calculation procedure, energy minimization is taken as a criterion of the stability. The cohesive energy of on the content of ternary addition is evaluated and illustrated in Figure 2. The values of energy in the figure are statistically averaged over the calculations of 20 samples. Figure 2 shows that the cohesive energy is lower when V atoms are substituted for Fe atoms at the 4i(Fe3) site than the other sites, so V should prefer the 4i(Fe3) site for . The calculated lattice constants of compound Tb_{3}Fe_{27.4}V_{1.6} are compared with the experimental values [13], which are shown in Table 2. Substitute the V atoms for a randomly selected part of Fe atoms at the 4i(Fe3) sites, thus forming the (Tb_{3}Fe_{27.4}V_{1.6})_{40}. Then make use of the conjugate gradient method to minimize the system energy, as an approximation of a practical relaxation process. Furthermore, we take the (Tb_{3}Fe_{27.4}V_{1.6})_{40} structure with each atom random shifted 0.6 Å from their equilibrium position to test the structural stability. Each atom in a disturbed cell can recover its equilibrium position under the interaction of interatomic pair potentials. The results show that the lattice constants are in good agreement with the experimental data [13], still retaining Å, Å, Å, and *β* = 97.0681° (Table 3). Thus, the stability of the lattice and the effectiveness of the interatomic pair potentials are verified.

In compounds, substitute the atoms of quaternary element cobalt for the randomly chosen iron atoms at a certain lattice site, and then make the lattice relaxation. Based on the ternary compounds Tb_{3}Fe_{27.4}V_{1.6} and the site occupation of cobalt in quaternary system has been investigated. The dependences of the cohesive energy of on the content of quaternary addition are evaluated and illustrated in Figure 3. It is shown that the calculated cohesive energy increases least significantly while cobalt atoms occupy 8j(Fe8) site. Therefore, Fe atoms are substituted for Co atoms with a strong preference for the 8j(Fe8) site. The calculated lattice constants are compared with the experimental values [13], which are shown in Table 2. It states clearly that the lattice parameters (but not ) of the compounds are decreased with increasing cobalt concentration due to the fact that the iron atoms are replaced by the smaller cobalt atoms.

The stability of the calculated structure is further checked through molecular dynamics. The cell constants are traced to higher temperatures, as shown in Table 4. Using MD (molecular dynamics) NPT ensemble, with atm, ps, dynamic simulations for are carried out at temperatures of 300, 500, and 700 K. The lattice constants change very little with respect to temperature variation, thus the structural stability is again verified. The comparisons of potential energy and kinetic energy at different temperatures are shown in Table 5. It can be seen from the table that both the values of potential energy and kinetic energy increase with the increasing temperature, but the contribution of interatomic potentials to the internal energy of the system is much larger than that from thermal motion energy. Therefore, the crystal structure at different temperature is basically determined by the interatomic pair potentials.

#### 4. Conclusions

In this paper, we have made an investigation of the phase stability and effect of cobalt atoms element on the structural properties of crystals. The energies of the Tb-Fe-Co-V systems are calculated from these effective interatomic pair potentials. The calculation results show that vanadium atoms preferentially occupy 4i(Fe3) site, and cobalt prefer the 8j(Fe8) site. Besides, due to the fact that iron atoms are replaced by the smaller cobalt atoms, the lattice parameters (but not ) of the compounds decreased with increased concentration.

Above all, with the interatomic potentials from lattice inversion, the structural properties have been well reproduced and are in agreement with the experimental data. This suggests that the potentials are successfully used to calculate the structural properties of rare-earth compounds , which is important for future work on material structure research.

#### Acknowledgments

This work was supported by the 973 Project in China (no. 2011CB606401) and the National Natural Science Foundation of China (Grant no. 50971024).

#### References

- S. J. Collotcott, R. K. Day, J. B. Dunlop, and R. L. Davis, “Preparation and Properties of Iron-Rich Nd–Fe–Ti Intermetallic Compounds and Their Nitrides,” in
*Proceedings of the 7th International Symposium on Magnetic Anisotropy and Coercivity in R-T alloys*, p. 437, Canberra, Australia, 1992. - H. Pan, Y. Chen, X. Han, C. Chen, and F. Yang, “Structural characterization of the ternary phase R
_{3}(Fe,Mo)_{29}(R = rare earth),”*Journal of Magnetism and Magnetic Materials*, vol. 185, no. 1, pp. 77–84, 1998. View at Publisher · View at Google Scholar - V. Psycharis, O. Kalogirou, E. Devlin, M. Gjoka, A. Simopoulos, and D. Niarchos, “Structural and intrinsic magnetic material parameters of Pr
_{3}(Fe,Ti)_{29}and Pr_{3}(Fe,Ti)_{29}N_{x},”*Journal of Magnetism and Magnetic Materials*, vol. 153, p. 75, 1996. View at Google Scholar - S.-Q. Hao, N.-X. Chen, and J. Shen, “The Space Group of Nd
_{3}Fe_{29-x}Ti_{x}: A2/m or P2_{1}/c,”*Physica Status Solidi B*, vol. 234, no. 2, pp. 487–495, 2002. View at Google Scholar - International Tables for Crystallography,
*Vol. A, Space-Group Symmetry*, D. Reidel, Dordrecht, The Netherlands, 1983. - X. F. Han, H. G. Pan, H. Liu, F. Yang, and Y. Zheng, “Syntheses and magnetic properties of Tb
_{3}Fe_{29-x}Cr_{x}compounds,”*Physical Review B*, vol. 56, no. 14, pp. 8867–8875, 1997. View at Google Scholar · View at Scopus - H. Z. Luo, Y. X. Li, L. Jia et al., “Structure and magnetic properties of Er
_{3}Fe_{29-x}Cr_{x}compounds,”*Journal of Alloys and Compounds*, vol. 397, no. 1-2, pp. 31–36, 2005. View at Publisher · View at Google Scholar · View at Scopus - L. Jia, H. Z. Luo, J. Shen et al., “Structure and magnetic properties of Er
_{3}Fe_{29-x}Nb_{x},”*Journal of Physics D*, vol. 38, no. 5, p. 667, 2005. View at Publisher · View at Google Scholar - A. Gholizadeh, N. Tajabor, M. Behdani et al., “Anisotropy and FOMP in Tb
_{3}(Fe_{28-x}Co_{x}) V1.0 (*x*= 0, 3 and 6) compounds,”*Physica B*, vol. 406, no. 18, pp. 3465–3469, 2011. View at Publisher · View at Google Scholar · View at Scopus - N. Sheloudko, C. Sarafidis, M. Gjoka, K. G. Efthimiadis, and O. Kalogirou, “Magnetic anisotropy of Ho-Fe-Co-Cr intermetallic compounds,”
*Journal of Alloys and Compounds*, vol. 482, no. 1-2, pp. 19–22, 2009. View at Publisher · View at Google Scholar · View at Scopus - K. G. Efthimiadis, C. Sarafidis, M. Gjoka, and O. Kalogirou, “Existence and properties of Co-rich 3:29-type of compounds synthesized with heavy rare earths,”
*Journal of Magnetism and Magnetic Materials*, vol. 316, no. 2, pp. e458–e461, 2007. View at Publisher · View at Google Scholar · View at Scopus - O. Kalogirou, C. Sarafidis, K. G. Efthimiadis, and M. Gjoka, “Influences of Co on structural and magnetic properties of R
_{3}(Fe_{1-x}Co_{x})_{29-y}M_{y}(R = rare earth metal, M = transition metal) intermetallic compounds,”*Journal of Alloys and Compounds*, vol. 423, no. 1-2, pp. 4–9, 2006. View at Publisher · View at Google Scholar · View at Scopus - O. Kalogirou, C. Sarafidis, M. Gjoka, and G. Litsardakis, “Effects of Co substitution on structural and magnetic properties of R3(Fe1-xCox)29-y Vy (R = Tb, Dy),”
*Journal of Magnetism and Magnetic Materials*, vol. 247, no. 1, pp. 34–41, 2002. View at Publisher · View at Google Scholar · View at Scopus - D. Hadjiapostolidou, M. Gjoka, C. Sarafidis, E. Pavlidou, T. Bakas, and O. Kalogirou, “Structural and magnetic properties of rare earth-iron-cobalt-vanadium intermetallic compounds (R: Tb, Dy),”
*Journal of Alloys and Compounds*, vol. 367, no. 1-2, pp. 255–261, 2004. View at Publisher · View at Google Scholar · View at Scopus - D. Courtois, H. S. Li, J. M. Cadogan, D. Givord, and E. Bourgeat-Lami, “Magnetization curves of a Tb
_{3}(Fe,V)_{29}single crystal,”*IEEE Transactions on Magnetics*, vol. 33, no. 5, pp. 3844–3846, 1997. View at Publisher · View at Google Scholar · View at Scopus - N. X. Chen, “Modified Möbius inverse formula and its applications in physics,”
*Physical Review Letters*, vol. 64, no. 11, pp. 1193–1195, 1990. View at Publisher · View at Google Scholar - N. X. Chen and G. B. Ren, “Carlsson-Gelatt-Ehrenreich technique and the Möbius inversion theorem,”
*Physical Review B*, vol. 45, no. 14, pp. 8177–8180, 1992. View at Publisher · View at Google Scholar · View at Scopus - N. X. Chen, Z. D. Chen, Y. N. Shen, S. Liu, and M. Li, “3D inverse lattice problems and Möbius inversion,”
*Physics Letters A*, vol. 184, no. 4-5, pp. 347–351, 1994. View at Google Scholar · View at Scopus - N. X. Chen, Z. D. Chen, and Y. C. Wei, “Multidimensional inverse lattice problem and a uniformly sampled arithmetic Fourier transform,”
*Physical Review E*, vol. 55, no. 1, pp. R5–R8, 1997. View at Google Scholar - N. X. Chen and E. Q. Rong, “Unified solution of the inverse capacity problem,”
*Physical Review E*, vol. 57, no. 2, pp. 1302–1308, 1998. View at Publisher · View at Google Scholar - Y. Liu, N. X. Chen, and Y. M. Kang, “Virtual lattice technique and the interatomic potentials of
zinc-blend-type binary compounds,”
*Modern Physics Letters B*, vol. 16, pp. 187–194, 2002. View at Publisher · View at Google Scholar - H. Y. Wang, F. A. Zhao, and N. X. Chen, “Theoretical investigation on the phase stability of Nd
_{2}Fe_{14}B and site preference of V, Cr, Mn, Zr and Nb,”*Journal of Magnetism and Magnetic Materials*, vol. 295, no. 3, pp. 219–229, 2005. View at Publisher · View at Google Scholar - Y. Y. Li, J. Shen, and Y. Chen, “Atomistic simulation for disordered TbCu
_{7}-type compounds SmCo_{7}and Sm(Co,T)_{7}(T = Ti, Ga, Si, Cu, Hf, Zr),”*Solid State Sciences*, vol. 12, no. 1, pp. 33–38, 2010. View at Publisher · View at Google Scholar - Y. Long, N. X. Chen, and W. Q. Zhang, “Pair potentials for a metal-ceramic interface by inversion of adhesive energy,”
*Journal of Physics*, vol. 17, no. 12, pp. 2045–2058, 2005. View at Google Scholar - Y. Long and N. X. Chen, “Traces of semigroups associated with interacting particle systems,”
*Journal of Physics*, vol. 19, no. 2, pp. 196–216, 2007. View at Google Scholar - J. Y. Xie, N. X. Chen, J. Shen, L. D. Teng, and S. Seetharaman, “Atomistic study on the structure and thermodynamic properties of Cr
_{7}C_{3}, Mn_{7}C_{3}, Fe_{7}C_{3},”*Acta Materialia*, vol. 53, no. 9, pp. 2727–2732, 2005. View at Publisher · View at Google Scholar · View at Scopus - J. Y. Xie, N. X. Chen, L. D. Teng, and S. Seetharaman, “Atomistic study on the site preference and thermodynamic properties for Cr
_{23-x}Fe_{x}C_{6},”*Acta Materialia*, vol. 53, no. 20, pp. 5305–5312, 2005. View at Publisher · View at Google Scholar · View at Scopus - J. Y. Xie, J. Shen, N. X. Chen, and S. Seetharaman, “Site preference and mechanical properties of Cr
_{23-x}T_{x}C_{6}and Fe_{21}T_{2}C_{6}(T = Mo, W),”*Acta Materialia*, vol. 54, no. 18, pp. 4653–4658, 2006. View at Publisher · View at Google Scholar · View at Scopus - V. L. Moruzzi, J. F. Janak, and A. R. Williams,
*Calculated Electronic Properties of Metals*, Pergamon, Oxford, UK, 1978.