The Scientific World Journal

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

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

Research Article

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

Jing Sun, Jiang Shen, and Ping Qian

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 Nd₃(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 Nd₃(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 Nd₃(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 Nd₃(Fe,Ti)₂₉-type structure belongs to A2/m rather than P2₁/c.

In the A2/m space group description, the lattice cell with two R₃Fe₂₉ 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 Nd₃Fe27.5Ti1.5 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 ThMn₁₂-type (1 : 12) and rhombohedral Th₂Ni₁₇-type (2 : 17) segments in a ratio of 1 : 1 [6]. In fact, the binary compounds R₃Fe₂₉ are metastable, which can be seen as the intrinsic prototype of R₃(Fe,T)₂₉, since R₃(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 R₃(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 Th₂Ni₁₇-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.

Figure 1: Potentials in the system of Tb-Fe-Co-V.

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 Tb₃Fe₂₉, which can be seen as the prototype of R₃(Fe,T)₂₉. In the calculation procedure, the initial lattice constants of Tb₃Fe₂₉ 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 Nd₃(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 Tb₃Fe₂₉ has the topological invariability with respect to the existing Nd₃(Fe,Ti)₂₉ structure. Hence, it furnishes convincing evidence that the interatomic pair potentials are reliable for the study of structural material characteristics.

Table 1: Final crystal lattice constants of Tb₃Fe₂₉ corresponding to random initial structures.

Initial state
Final state

, , (Å)
, , (deg)
, , (Å)
, , (deg)

10, 8, 9
90, 95, 90
10.525, 8.433, 9.696
90, 96.91, 90

11, 10, 10
90, 100, 90
10.525, 8.433, 9.696
90, 96.91, 90

10, 10, 10
90, 90, 90
10.525, 8.433, 9.696
90, 96.91, 90

13, 10, 10
90, 95, 90
10.525, 8.433, 9.696
90, 96.91, 90

12, 12, 12
90, 90, 90
10.525, 8.433, 9.696
90, 96.91, 90

10, 10, 10
85, 90, 88
10.525, 8.433, 9.696
90, 96.91, 90

8, 8, 8
90, 90, 90
10.525, 8.433, 9.696
90, 96.91, 90

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 Tb₃Fe27.4V1.6 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 (Tb₃Fe27.4V1.6)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 (Tb₃Fe27.4V1.6)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.

Table 2: Comparison of calculated and experimental unit cell parameters.

Tb₃Fe27.4V1.6
Tb₃Fe27.3Co0.1V1.6
Tb₃Fe27.2Co0.2V1.6
Tb₃Fe27.1Co0.3V1.6
Tb₃Fe27.0Co0.4V1.6

(Å)

    Cal.
10.5433
10.5424
10.5418
10.5411
10.5406

    Exp.
10.557 (1)
10.545 (1)
10.537 (1)
10.519 (1)
10.511 (1)

    Err. (%)

0.2168
0.2101
0.2816

(Å)

    Cal.
8.4647
8.4647
8.4643
8.4638
8.4635

    Exp.
8.497 (1)
8.496 (1)
8.490 (1)
8.476 (1)
8.461 (1)

    Err. (%)

0.0295

(Å)

    Cal.
9.7196
9.7189
9.7185
9.7181
9.7176

    Exp.
9.669 (1)
9.672 (1)
9.669 (1)
9.658 (1)
9.645 (1)

    Err. (%)
0.5233
0.4849
0.5119
0.6223
0.7527

(deg)

    Cal.
97.0681
97.0703
97.0698
97.0690
97.0691

    Exp.
96.89
96.72
96.72
96.71
96.74

    Err. (%)
0.18
0.31
0.3617
0.3712
0.3402

Table 3: Comparison of related lattice constants before and after atomic random motion of 0.6 Å for Tb₃Fe27.4V1.6.

(Å)
(Å)
(Å)
(deg)

Before random motion
10.5433
8.4647
9.7196
97.0681

After random motion of 0.6 Å
10.5433
8.4647
9.7196
97.0681

Experimental [13]
10.557 (1)
8.497 (1)
9.669 (1)
96.89

Figure 2: Calculated cohesive energy of as a function of the V content .

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 Tb₃Fe27.4V1.6 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.

Figure 3: Calculated cohesive energy of as a function of the Co content .

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.

Table 4: The cell constants of Tb₃V1.6 under different temperatures.

Compounds
Temperature
(Å)
(Å)
(Å)
β (deg)

300 K
10.5269
8.4339
9.6954
96.8980

500 K
10.5229
8.4309
9.6962
96.9095

700 K
10.5250
8.4360
9.6958
96.9115

300 K
10.5240
8.4329
9.6952
96.9190

500 K
10.5241
8.4317
9.6952
96.8885

700 K
10.5252
8.4320
9.6950
96.9185

300 K
10.5249
8.4310
9.6954
96.9180

500 K
10.5241
8.4309
9.6954
96.9025

700 K
10.5220
8.4330
9.6962
96.9195

300 K
10.5250
8.4331
9.6959
96.9195

500 K
10.5280
8.4328
9.6962
96.9005

700 K
10.5282
8.4331
9.6950
96.9095

300 K
10.5252
8.4338
9.6962
96.9000

500 K
10.5252
8.4313
9.6960
96.8990

700 K
10.5229
8.4330
9.6956
96.9180

Table 5: The potential and kinetic energies of Tb₃V1.6 under different temperatures.

Compounds
Temperature
(eV)
(eV)
(eV)

300 K

12.4699

1.0066

500 K

20.1887

1.0107

700 K

29.0479

1.0156

300 K

12.7817

1.0067

500 K

20.1824

1.0107

700 K

27.8954

1.0150

300 K

13.0351

1.0069

500 K

20.8418

1.0111

700 K

28.8429

1.0155

300 K

12.4685

1.0066

500 K

19.9308

1.0106

700 K

29.3169

1.0158

300 K

12.7901

1.0068

500 K

21.6214

1.0115

700 K

29.3861

1.0158

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).

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