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

Authors: Jing Sun; Jiang Shen; Ping Qian
Journal: The Scientific World Journal
 (2013)

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## Abstract

The effect of cobalt on the structural properties of intermetallicTb3Fe27.4−xCoxV1.6 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.


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## Body

## 1. Introduction

In 1993, it had been first found that the structure of the compoundNd2Fe19-xTix 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 Tb3Fe29-xVx 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 inTb3Fe27.4-xCoxV1.6 (x = 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(1)E(x)=12∑ri≠0φ(ri→)=12∑n=1∞r0(n)φ[b0(n)x],
where x is the nearest neighbor distance, ri→ is the position vector of the ith atom in the lattice, b0(n)x is the nth nearest neighbour distance with b0(1)=1, and r0(n) is the nth coordination number. Obviously, {b0} 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 {b} with b(1)=b0(1)=1. Namely, for any b(j), b(k)∈{b}, we have b(j)b(k)∈{b}. Now E(x) can be rewritten as
(2)E(x)=12∑n=1∞r(n)φ[b(n)x],
where the extended coordination number r(n) satisfies the following rule:
(3)r(n)={r0(b0-1[b(n)]),b(n)∈{b0},0,b(n)∉{b0}.It is clear that{r(n)} is uniquely determined by crystal geometrical structure. Now we can extract pair potential φ from (2) as
(4)φ(x)=2∑n=1∞I(n)E[b(n)x],
where I(n), n=1,2,…,∞, and any b(n) satisfying b(n)∣b(t), {I(n)} can be determined by
(5)∑b(n)∣b(t)I(b-1[b(n)])r(b-1[b(t)b(n)])=δt,1(t=1,2,…,∞),
where δ is the Kronecker delta. It is easy to prove that (5) has and only has a single solution set {I(n)}, which, like r{(n)}, 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
(6)φ(x)=D0(e-γ(x/R0-1)-2e-(γ/2)(x/R0-1)),
where x indicates the distance between two atoms and R0 is the equilibrium distance, with D0 and γ being parameters without unit.For the readers’ convenience, several important potential parameters are shown in Figure1.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, (Tb3Fe27.4-xCoxV1.6)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 a=10.525 Å, b=8.433 Å, c=9.696 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
a, b, c (Å)
α, β, γ (deg)
a, b, c (Å)
α, β, γ (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, 90Substitute 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 ofTb3Fe29-xVx 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 x≤2.0. 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 a=10.5433 Å, b=8.4647 Å, c=9.7196 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.
x
=
0
x
=
0.1
x
=
0.2
x
=
0.3
x
=
0.4
Tb3Fe27.4V1.6
Tb3Fe27.3Co0.1V1.6
Tb3Fe27.2Co0.2V1.6
Tb3Fe27.1Co0.3V1.6
Tb3Fe27.0Co0.4V1.6
a (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. (%)
-
0.1298
-
0.0247
0.2168
0.2101
0.2816
b (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. (%)
-
0.3801
-
0.3684
-
0.3027
-
0.1439
0.0295
c (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
β (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
0.3402Table 3
Comparison of related lattice constants before and after atomic random motion of 0.6 Å for Tb3Fe27.4V1.6.
a
(Å)
b  (Å)
c (Å)
β (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)
96.89Figure 2
Calculated cohesive energy ofTb3Fe29-xVx as a function of the V content x.InTb3Fe27.4-xCoxV1.6 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 Tb3Fe27.4-xCoxV1.6 has been investigated. The dependences of the cohesive energy of Tb3Fe27.4-xCoxV1.6 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 a,b (but not c) 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 ofTb3Fe27.4-xCoxV1.6 as a function of the Co content x.The stability of the calculated structure is further checked through molecular dynamics. The cell constants are traced to higher temperatures, as shown in Table4. Using MD (molecular dynamics) NPT ensemble, with P=1 atm, t=0.001 ps, dynamic simulations for Tb3Fe27.4-xCoxV1.6 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 Tb3Fe27.4-xCoxV1.6 under different temperatures.
Compounds
Temperature
a (Å)
b (Å)
c (Å)
β
(deg)
x
=
0.0
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
x
=
0.1
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
x
=
0.2
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
x
=
0.3
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
x
=
0.4
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
96.9180Table 5
The potential and kinetic energies of Tb3Fe27.4-xCoxV1.6 under different temperatures.
Compounds
Temperature
E
p (eV)
E
k (eV)
E
total (eV)
E
p
/
E
total
x
=
0.0
300 K
-
1907.8644
12.4699
-
1895.3944
1.0066
500 K
-
1898.9049
20.1887
-
1878.7162
1.0107
700 K
-
1891.3703
29.0479
-
1862.3224
1.0156
x
=
0.1
300 K
-
1907.4917
12.7817
-
1894.7100
1.0067
500 K
-
1898.5085
20.1824
-
1878.3261
1.0107
700 K
-
1889.7135
27.8954
-
1861.8181
1.0150
x
=
0.2
300 K
-
1907.2399
13.0351
-
1894.2048
1.0069
500 K
-
1898.6019
20.8418
-
1877.7601
1.0111
700 K
-
1890.0060
28.8429
-
1861.1631
1.0155
x
=
0.3
300 K
-
1905.9974
12.4685
-
1893.5289
1.0066
500 K
-
1897.0235
19.9308
-
1877.0927
1.0106
700 K
-
1889.9966
29.3169
-
1860.6797
1.0158
x
=
0.4
300 K
-
1905.6689
12.7901
-
1892.8788
1.0068
500 K
-
1898.0230
21.6214
-
1876.4016
1.0115
700 K
-
1889.3255
29.3861
-
1859.9394
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 ofTb3Fe27.4-xCoxV1.6 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 a,b (but not c) 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 compoundsTb3Fe27.4-xCoxV1.6, which is important for future work on material structure research.



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