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Calculations of Rare Earth (Y, La and Ce) Diffusivities

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Published: Fri, 15 Sep 2017

First-principles calculations of rare earth (Y, La and Ce) diffusivities in bcc Fe

Xueyun Gaoa,b,[*], Huiping Rena, Chunlong Lia,c, Haiyan Wanga, Yunping Jia, Huijie Tan a

ABSTRACT: The impurity diffusivities of rare earth elements, Y, La and Ce, in bcc Fe have been investigated by the first-principles calculations within nine-frequency model. The microscopic parameters in the pre-factor and activation energies have been calculated. For the three elements, the first nearest-neighbor solute-vacancy interactions are all attractive, in which Y and La solute atoms more favorably bond to the vacancy. The solute-vacancy binding energy can be explained in terms of the combination of the distortion binding energy and the electronic binding energy, and the decomposition results of the total solute-vacancy binding energy suggest that the strain-relief effect accounts for larger portion of the binding energy for Y and La than that for Ce. The diffusion coefficients of Y are one order of magnitude larger than that of La, and predicted to be comparable to that of Fe self-diffusion. Compared with Y and La, Ce shows large migration energy and small solute-vacancy attractive interaction, which accounts for the lowest diffusivity of this element.

Keywords: Diffusion; Rare earth; Bcc Fe; First-principles calculations

1. Introduction

In the past years, the addition of rare earth (RE) elements has been regarded promising in steels. A series of beneficial research for the development of rare earth addition have been focused on the purification and modification of inclusion, since RE elements are characterized by significant negative free energy changes for compound formations [1-3]. RE doping also improve the high-temperature oxidation resistance and the corrosion resistance of steels due to the reactive-element effect (REE) [4,5].In addition, the solidification, phase transformations, recrystallization behavior, of steel can be improved by adding RE [6, 7].

Knowledge of the above mentioned mechanism is essential to understand the influence of RE additions on the physical, chemical and various properties of steels. In spite of the progress so far in RE application, it is apparent that many questions still remain rather controversial. A thorough theoretical study on the diffusivities of RE elements in Fe-based alloy is still lacking, which is essential for understanding the effects of RE on the structure and properties of steel, and is also helpful for designing and preparing RE doped steels [8]. For the three commonly used RE elements, Y, La and Ce, to our knowledge, only the diffusion coefficient of Y has been reported [9].

The diffusion of substitutional-type solute is mainly controlled by vacancy mechanism. In this case, the interaction of solute atom with vacancy plays significant roles in understanding of the diffusion properties of solutes. To understand the microstructure evolution in bcc Fe alloy, DFT method has been applied in calculations of the binding energies of solute atom with vacancy [10]. Based on the nine-frequency model of Le Claire, Ding and Huang et al. [11,12] developed a computational framework to calculate the solute diffusion coefficients in bcc Fe, which presents an effective method in investigation of the diffusion properties in bcc Fe alloy.

The purpose of this work is to investigate the impurity diffusivities of rare earth elements, Y, La and Ce, in bcc Fe by the first-principles calculations within nine-frequency model and the semi-empirical corrections for magnetization[11,13], calculate the associated solute-vacancy binding energies, migration energies, pre-factors and activation energies for these impurity diffusivities, as well as the self-diffusion coefficient of Fe, thus discuss the related factors of the diffusion coefficients.

2. Methodology

The temperature dependence of diffusion coefficient D is expressed in the Arrhenius form D=D0exp(-Q/kBT), where D0 and Q are the pre-factor and activation energy, respectively. Below the Curie temperature, the self-diffusion and solute diffusion coefficients in bcc Fe deviate downward from the Arrhenius type relationship extrapolated from the paramagnetic state [14]. These deviations are attributed to the change of magnetization which affects the diffusion activation energy. To investigate the effect of magnetic disorder on the diffusion activation energy of spin-polarized metals, Ding et al. [15] reported a first-principles approach based on the spin-wave DFT method for studying the self-diffusion of bcc Fe and fcc Co, and the calculated values of α agree well with the experimental data. More recently, by combination of the first-principle calculations and Heisenberg Monte Carlo simulations, Sandberg et al. [16] presented a quasi-empirical model to study the magnetic contribution to the self-diffusion activation energy of bcc Fe. Murali et al. [17] conducted a systematic study of the effects of phonon and electron excitations on the free formation energy of vacancy, the solute-vacancy binding energy, and the vacancy migration energy in bcc Fe. The authors then calculated the Fe self-diffusion coefficient based on the computed free energies, by employing the semi-empirical model presented in Ref. [18]. The diffusion associated data yielded by these methods are in good agreement with experiments. We employ the semi-empirical model to describe the dependence of the diffusion activation energy on the magnetization in the ferromagnetic state [18]:

QF(T)=QP[1+αs(T)2] (1)

where QP is the activation energy in the paramagnetic state; s(T) is the ratio of the magnetization of bcc Fe at a certain temperature T to that at 0K, and has been experimentally measured [19,20]; the constant α quantifies the extent of the influence of magnetic on activation energy, the measured value of α for Fe self-diffusion is 0.156 [11].In the case of the solute species investigated in this context that have no measured α values available, the values can be estimated from a linear correlation with the induced changes in local magnetization on Fe atoms in the first and second neighbor shells of a solute atom [21]. The first-principles calculations give direct access to the magnitude of the activation energies for self- and solute diffusion in the fully ordered ferromagnetic state (T=0K). Given the values of α and s(T), we can compute QP through the relation , and QF(T) from Eq. (1).

We based the diffusion calculations on the assumption that the mechanism of diffusion is primarily monovacancy mediated. The self-diffusion and solute diffusion coefficients on a bcc lattice can be expressed as following [11,22]:

Dself=a2f0Cvw0 (2)

Dsolute=a2f2Cvw2(-ΔGb/kBT) (3)

where a is the bcc lattice constant, f0=0.727 is the self-diffusion correlation factor, f2 is the correlation factor for solute diffusion which depends on the relative vacancy jump frequencies around the solute atom. Cv denotes the equilibrium vacancy concentration, can be written as Cv=exp(ΔSf /kB)exp(-ΔHf /kBT), where ΔSf and ΔHf are the vacancy formation enthalpy and entropy, respectively, the harmonic approximation makes these two become temperature-independent constants. and kB is Boltzmann’s constant. w0 and w2 are the vacancy hopping frequencies for Fe and solute atoms to exchange with a nearest-neighbor vacancy, respectively. Based on transition state theory (TST), the vacancy hopping frequency w is written as , where and are the phonon frequencies in the initial state and transition state, and the product in the denominator ignores the unstable mode; ΔHmig is the migration energy, gives the energy difference for the diffusing atom located at its initial equilibrium lattice position and the saddle-point position. The solute-vacancy binding free energy ΔGb can be expressed as ΔGb =ΔHbTΔSb, where ΔHb and ΔSb are the binding enthalpy and entropy, respectively.

The correlation factor f2 can be calculated using the nine-frequency model developed by Le Claire [13] which involves different jump frequencies of vacancies to their first neighbor position in the presence of the solute atoms, as illustrated in Fig. 1. In this model, the interaction of solute-vacancy is assumed up to second neighbor distance. The nine frequencies shown in Fig. 1 illustrate all of the distinct vacancy jumps in a bcc system with a dilute solute concentration, including the host Fe atom jump w0 without impurity. The detailed calculation procedures could be found in Ref. [11].


Fig. 1. Schematic illustration of the nine-frequency model for the bcc Fe crystalline with a solute atom. The arrows denote the jump directions of the vacancy. The numbers in the circle represent the neighboring site of the solute atom.

For convenience, we can represent the self- and solute diffusion equations (Eqs.(2) and (3)) in Arrhenius form to obtain the pre-factor and activation energy of diffusion. By combining the above Eqs., the diffusion coefficient for Fe self-diffusion and solute diffusion can be expressed as:


For self-diffusion, the pre-factor is, and the activation energy given as . Also, the solute diffusion coefficient can be expressed in an Arrhenius form with the pre-factor is, and.

The first-principles calculations presented here are carried out using the Vienna Ab initio Simulation Package(VASP) with the projector augmented wave(PAW) method and the generalized gradient approximation of Perdew-Burke-Ernzerhof functional(GGA-PBE) [23]. All calculations were performed in spin polarized. The computations performed within a 4ï‚´4ï‚´4 supercell including 128 atoms. The binding, vacancy formation and migration energies were calculated with 300eV plane-wave cutoff and 12ï‚´12ï‚´12 k-point meshes. The residual atomic forces in the relaxed configurations were lower than 0.01eV/Å. The transition states with the saddle point along the minimum energy diffusion path for vacancy migration were determined using nudged elastic band (NEB) method [24] as implemented in VASP. We adopt the harmonic approximation (HA) to consider the contribution of normal phonon frequencies to free energy. The normal phonon frequencies were calculated using the direct force-constant approach as implemented in the Alloy Theoretic Automated Toolkit (ATAT) [25] package. Similar cutoff energy, k-point mesh size and supercell size used for the total energies were used for the vibrational calculations.

3. Result and discussion

Table 1 illustrates our calculated energies for vacancy formation, migration and binding, as well as the constant α for solute species, the associated paramagnetic activation energies and fully ordered ferromagnetic activation energies for both self- and solute-diffusion. For pure bcc Fe, the vacancy formation energy and migration energy obtained here are consistent with the reported range of values, ΔHf=2.16-2.23 eV and ΔHmig=0.55-0.64 eV [11,26,27]. For Y impurity in bcc Fe, the calculated vacancy binding energy in full ordered ferromagnetic state also compare well with the previous first principles work [28], in which ΔHb=-0.73 eV. It can be seen that Y and La have smaller activation energy than that for Fe self-diffusion, while Ce is predicted to have a lager value of activation energy than that for Fe self-diffusion, in both the ordered ferromagnetic and paramagnetic state.

Table 1

Vacancy formation energy ΔHf , solute-vacancy binding energy ΔHb, migration energy ΔHmig, the ferromagnetic activation energy and the paramagnetic activation energy QP; the variable dependence parameter of activation energy on magnetization α.










ΔHb (eV)





ΔHmig (eV)















QP (eV)





Solute-vacancy binding energy plays a crucial role in understanding solute diffusion kinetics. Table 2 presents the binding energies of Y, La and Ce atoms with vacancy in their 1nn, 2nn and 3nn coordinate shells. From Table 2 it can be seen that referring to the first nearest-neighbor solute-vacancy pairs, the binding energies are all negative, which implies the solute-vacancy pairs are favorable. Specifically, Y and La impurities are computed to have higher values of solute-vacancy binding energies -0.69 eV and -0.66 eV in 1nn configuration, respectively, while that for Ce is -0.43 eV. Correspondingly, we found that Y, La and Ce atoms relax towards the 1nn vacancy by 22.3%, 19.6% and 12.2% of the initial 1nn distance (2.488 Å) after the structure optimization. The interactions of the solute-vacancy pair at the 2nn shells tend to be smaller in magnitude than that of 1nn, and that of Ce-vacancy predicted to be repulsive. The interactions of the 3nn solute-vacancy are relatively weak, indicating that the interactions of the solute-vacancy are local. According to Le Clair model [13], in the situation that the interactions of the first and second nearest solute-vacancy neighbors are appreciable, the nine different jump frequencies should be considered.

To obtain information on the origin of these attractive behaviors, we decompose the total binding energy into the distortion binding energyand the electronic binding energy as [29] . The distortion binding energy can be obtained by the distortion reducing of the bcc Fe matrix when a solute atom and a vacancy combine to form a solute-vacancy pair, and can be expressed as:


where and can be calculated as follows: after the supercell containing a solute-vacancy pair (or a substitutional atom) has been fully relaxed, the solute-vacancy pair (or the substitutional atom) is removed from the system, then the total energy can be calculated. denotes the total energy of the pure bcc Fe supercell, and denotes the total energy of the supercell containing a vacancy. Then can be calculated from . The calculated solute-vacancy binding energies of 1nn, 2nn and 3nn are shown in Table 2, along with the energy decomposition for 1nn solute-vacancy binding. The distortion energies (-0.31 to -0.65 eV) for all solute elements(Y, La and Ce) are negative, and much bigger than their corresponding electronic binding energies (-0.04 to -0.12 eV). This implies that the distortion energy accounts for a major part of the total solute-vacancy binding energy, i.e. the strain relief effect contributes significantly to the interaction between the impurity atom and the vacancy, especially for the solute Y and La, which accounted for 94.2% and 97.0% of the total binding energy, respectively. Furthermore, there is a strong correlation between the binding energy and the distance of the solute-vacancy, and the lattice relaxation around the vacancy is local. For the case of Ce-vacancy , specifically, we found that Ce atom relax away from the 2nn vacancy by 4.3% of the initial 2nn distance, which leads to the positive binding energy.

Table 2

Decomposition of the total solute-vacancy binding energy into distortion binding energy and electronic binding energy. Units are eV.

Solute element




ΔHb (1nn)












ΔHb (2nn)




ΔHb (3nn)




The calculated migration energies of the different vacancy jumps corresponding to the paths in Fig. 1 are listed in Table 3. The migration energies of w2 jump for Y and La are lower than that of w0 jump for host Fe atom (0.54 eV), while the migration barrier of Ce in bcc Fe is higher than that of Fe self-diffusion. The migration barrier of w2 jump for Y is 0.09 eV, comparable to the reported value of 0.03 eV and 0.02 eV [9, 30]. The results indicate that there is a correlation between the binding energy of solute-vacancy and the migration energy, i.e. the strong attraction of solute-vacancy in 1nn configuration gives rise the low migration energy of the corresponding vacancy jump. For the three solute atoms, because of the strong attraction of 1nn solute-vacancy, the migration barriers of which the 1nn vacancy jump away from the solute atom, i.e. w3, w’3, and w”3, are higher than that of the opposite jumps, i.e. w4, w’4, and w”4, as well as that of Fe self-diffusion in pure bcc Fe. And the same tendency can be found in the results of jump w5 and w6.

Table 3

Migration energies for different jumps in the presence of Y, La and Ce in bcc Fe matrix. Units are eV.









































The correlation factor f2 is related to the probability of the reverse jump of a solute atom to its previous position [31]. Table 4 lists the calculated values of correlation factors for Y, La and Ce at representative temperatures of 850, 1000 and 1150K. The correlation factor of Y is 3.3Ã-10-5 at 1000K, close to the value of 1Ã-106 obtained by Murali [9]. For the three elements, the correlation factors of Ce have the highest values, and the correlation factors of La are one order of magnitude lower than that of Y. Therefore, Ce atom is the most difficult to return back to its original position in the temperature range of our investigation. Including the smallest binding energy, highest migration energy and correlation factor, provides an explanation for the low diffusivity of Ce atom.

Table 4

Correlation factors (f2) for Y, La and Ce solute-diffusion at representative temperatures of 850, 1000 and 1150K.






f2/ f0


f2/ f0


f2/ f0






















Table 5 lists the calculated diffusion activation energies and pre-factors for Fe self-diffusion and Y, La and Ce impurity diffusion. For pure bcc Fe, we find our calculated results are in good agreement with the published values. For Y impurity in bcc Fe, the calculated activation energy in full ordered ferromagnetic state is lower than the previous first principles work, and the pre-factor is as much as two orders of magnitude lager than the reported value. The experimental or calculated diffusion coefficients of La and Ce are not available to the best of our knowledge. For the case of experimental investigation, due to the very small solubilities of La and Ce in iron, the measured data may be affected by segregation of solutes, grain boundary, other impurities and the method of detection. Besides, the theory calculations, e.g. molecular dynamics (MD), first-principles etc. have not been applied widely in the study of RE contained steel yet, so the fundamental data of RE elements in iron, such as the potential functions of Fe-La and Fe-Ce, is lacking.

Table 5

Activation energies in the fully ordered ferromagnetic state () and paramagnetic state (QP), along with diffusion pre-factors for Fe self-diffusion and impurity diffusion of Y, La and Ce in bcc Fe.


(kJ mol-1)

QP(kJ mol-1)



Present work




Huang et al. [11]




Nitta et al. [32]









Present work




Murali et al. [9]





Present work





Present work




Fig. 2 presents a direct comparison between the calculated and published temperature dependent diffusion coefficients for Fe self-diffusion and Y solute diffusion. For Fe self-diffusion, the calculated values are in good agreement with Huang et al. [11] and Nitta et al. [29], but smaller than the measured d

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