Calculations of Rare Earth (Y, La and Ce) Diffusivities
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Firstprinciples calculations of rare earth (Y, La and Ce) diffusivities in bcc Fe
Xueyun Gao^{a,b,}[*], Huiping Ren^{a}, Chunlong Li^{a,c}, Haiyan Wang^{a}, Yunping Ji^{a}, Huijie Tan^{ a}ABSTRACT: The impurity diffusivities of rare earth elements, Y, La and Ce, in bcc Fe have been investigated by the firstprinciples calculations within ninefrequency model. The microscopic parameters in the prefactor and activation energies have been calculated. For the three elements, the first nearestneighbor solutevacancy interactions are all attractive, in which Y and La solute atoms more favorably bond to the vacancy. The solutevacancy 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 solutevacancy binding energy suggest that the strainrelief 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 selfdiffusion. Compared with Y and La, Ce shows large migration energy and small solutevacancy attractive interaction, which accounts for the lowest diffusivity of this element.
Keywords: Diffusion; Rare earth; Bcc Fe; Firstprinciples 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 [13]. RE doping also improve the hightemperature oxidation resistance and the corrosion resistance of steels due to the reactiveelement 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 Febased 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 substitutionaltype 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 ninefrequency 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 firstprinciples calculations within ninefrequency model and the semiempirical corrections for magnetization[11,13], calculate the associated solutevacancy binding energies, migration energies, prefactors and activation energies for these impurity diffusivities, as well as the selfdiffusion 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=D_{0}exp(Q/k_{B}T), where D_{0} and Q are the prefactor and activation energy, respectively. Below the Curie temperature, the selfdiffusion 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 spinpolarized metals, Ding et al. [15] reported a firstprinciples approach based on the spinwave DFT method for studying the selfdiffusion of bcc Fe and fcc Co, and the calculated values of Î± agree well with the experimental data. More recently, by combination of the firstprinciple calculations and Heisenberg Monte Carlo simulations, Sandberg et al. [16] presented a quasiempirical model to study the magnetic contribution to the selfdiffusion 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 solutevacancy binding energy, and the vacancy migration energy in bcc Fe. The authors then calculated the Fe selfdiffusion coefficient based on the computed free energies, by employing the semiempirical model presented in Ref. [18]. The diffusion associated data yielded by these methods are in good agreement with experiments. We employ the semiempirical model to describe the dependence of the diffusion activation energy on the magnetization in the ferromagnetic state [18]:
where Q^{P} 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 Î± quantifies the extent of the influence of magnetic on activation energy, the measured value of Î± for Fe selfdiffusion is 0.156 [11].In the case of the solute species investigated in this context that have no measured Î± 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 firstprinciples 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 Î± and s(T), we can compute Q^{P} through the relation , and Q^{F}(T) from Eq. (1).
We based the diffusion calculations on the assumption that the mechanism of diffusion is primarily monovacancy mediated. The selfdiffusion and solute diffusion coefficients on a bcc lattice can be expressed as following [11,22]:
D_{self}=a^{2}f_{0}C_{v}w_{0} (2)
D_{solute}=a^{2}f_{2}C_{v}w_{2}(Î”G_{b}/k_{B}T) (3)
where a is the bcc lattice constant, f_{0}=0.727 is the selfdiffusion correlation factor, f_{2} is the correlation factor for solute diffusion which depends on the relative vacancy jump frequencies around the solute atom. C_{v} denotes the equilibrium vacancy concentration, can be written as C_{v}=exp(Î”S_{f} /k_{B})exp(Î”H_{f} /k_{B}T), where Î”S_{f} and Î”H_{f} are the vacancy formation enthalpy and entropy, respectively, the harmonic approximation makes these two become temperatureindependent constants. and k_{B} is Boltzmann's constant. w_{0} and w_{2} are the vacancy hopping frequencies for Fe and solute atoms to exchange with a nearestneighbor 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; Î”H_{mig} is the migration energy, gives the energy difference for the diffusing atom located at its initial equilibrium lattice position and the saddlepoint position. The solutevacancy binding free energy Î”G_{b} can be expressed as Î”G_{b} =Î”H_{b}TÎ”S_{b}, where Î”H_{b} and Î”S_{b} are the binding enthalpy and entropy, respectively.
The correlation factor f_{2} can be calculated using the ninefrequency 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 solutevacancy 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 w_{0} without impurity. The detailed calculation procedures could be found in Ref. [11].
Fig. 1. Schematic illustration of the ninefrequency 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 prefactor and activation energy of diffusion. By combining the above Eqs., the diffusion coefficient for Fe selfdiffusion and solute diffusion can be expressed as:
(4)
For selfdiffusion, the prefactor is, and the activation energy given as . Also, the solute diffusion coefficient can be expressed in an Arrhenius form with the prefactor is, and.
The firstprinciples 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 PerdewBurkeErnzerhof functional(GGAPBE) [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 planewave cutoff and 12ï‚´12ï‚´12 kpoint 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 forceconstant approach as implemented in the Alloy Theoretic Automated Toolkit (ATAT) [25] package. Similar cutoff energy, kpoint 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 Î± for solute species, the associated paramagnetic activation energies and fully ordered ferromagnetic activation energies for both self and solutediffusion. For pure bcc Fe, the vacancy formation energy and migration energy obtained here are consistent with the reported range of values, Î”H_{f}=2.162.23 eV and Î”H_{mig}=0.550.64 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 Î”H_{b}=0.73 eV. It can be seen that Y and La have smaller activation energy than that for Fe selfdiffusion, while Ce is predicted to have a lager value of activation energy than that for Fe selfdiffusion, in both the ordered ferromagnetic and paramagnetic state.
Vacancy formation energy Î”H_{f} , solutevacancy binding energy Î”H_{b}, migration energy Î”H_{mig}, the ferromagnetic activation energy and the paramagnetic activation energy Q^{P}; the variable dependence parameter of activation energy on magnetization Î±.
Fe 
Y 
La 
Ce 

Î”H_{f}(eV) 
2.31 
â€’ 
â€’ 
â€’ 
Î”H_{b} (eV) 
â€’ 
0.69 
0.66 
0.43 
Î”H_{mig} (eV) 
0.54 
0.09 
0.17 
1.09 
Î± 
0.156 
0.088 
0.038 
0.125 
(eV) 
2.85 
1.71 
1.82 
2.97 
Q^{P} (eV) 
2.47 
1.57 
1.75 
2.64 
Solutevacancy 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 nearestneighbor solutevacancy pairs, the binding energies are all negative, which implies the solutevacancy pairs are favorable. Specifically, Y and La impurities are computed to have higher values of solutevacancy 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 solutevacancy pair at the 2nn shells tend to be smaller in magnitude than that of 1nn, and that of Cevacancy predicted to be repulsive. The interactions of the 3nn solutevacancy are relatively weak, indicating that the interactions of the solutevacancy are local. According to Le Clair model [13], in the situation that the interactions of the first and second nearest solutevacancy 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 solutevacancy pair, and can be expressed as:
(5)
where and can be calculated as follows: after the supercell containing a solutevacancy pair (or a substitutional atom) has been fully relaxed, the solutevacancy 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 solutevacancy binding energies of 1nn, 2nn and 3nn are shown in Table 2, along with the energy decomposition for 1nn solutevacancy 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 solutevacancy 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 solutevacancy, and the lattice relaxation around the vacancy is local. For the case of Cevacancy , 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 solutevacancy binding energy into distortion binding energy and electronic binding energy. Units are eV.
Solute element 
Y 
La 
Ce 

0.69 
0.66 
0.43 

(1nn) 
0.65 
0.64 
0.31 

(1nn) 
0.04 
0.02 
0.12 

Î”H_{b} (2nn) 
0.16 
0.21 
0.10 

Î”H_{b} (3nn) 
0.06 
0.09 
0.05 
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 w_{2} jump for Y and La are lower than that of w_{0} jump for host Fe atom (0.54 eV), while the migration barrier of Ce in bcc Fe is higher than that of Fe selfdiffusion. The migration barrier of w_{2} 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 solutevacancy and the migration energy, i.e. the strong attraction of solutevacancy 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 solutevacancy, the migration barriers of which the 1nn vacancy jump away from the solute atom, i.e. w_{3}, w'_{3}, and w"_{3}, are higher than that of the opposite jumps, i.e. w_{4}, w'_{4}, and w"_{4}, as well as that of Fe selfdiffusion in pure bcc Fe. And the same tendency can be found in the results of jump w_{5} and w_{6}.
Table 3
Migration energies for different jumps in the presence of Y, La and Ce in bcc Fe matrix. Units are eV.
Jump 
Y 
La 
Ce 
w_{2} 
0.09 
0.17 
1.08 
w_{3} 
1.81 
1.84 
1.55 
w_{4} 
0.91 
0.99 
0.92 
w'_{3} 
0.93 
1.23 
1.07 
w'_{4} 
0.04 
0.03 
0.08 
w"_{3} 
0.86 
0.92 
0.87 
w"_{4} 
0.12 
0.05 
0.11 
w_{5} 
0.94 
0.98 
0.89 
0.69 
0.67 
0.82 
The correlation factor f_{2} 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 3.3Ã10^{5} at 1000K, close to the value of 1Ã10^{}^{6} 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 (f_{2}) for Y, La and Ce solutediffusion at representative temperatures of 850, 1000 and 1150K.
T(K) 
Y 
La 
Ce 

f_{2}/ f_{0} 
f_{2} 
f_{2}/ f_{0} 
f_{2} 
f_{2}/ f_{0} 

850 
6.4Ã10^{}^{6} 
1.111 
2.9Ã10^{}^{7} 
1.264 
0.379 
1.373 
1000 
3.3Ã10^{5} 
1.070 
2.4Ã10^{}^{6} 
1.223 
0.381 
1.370 
1150 
1.2Ã10^{}^{4} 
1.034 
1.4Ã10^{5} 
1.188 
0.383 
1.367 
Table 5 lists the calculated diffusion activation energies and prefactors for Fe selfdiffusion 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 prefactor 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), firstprinciples 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 FeLa and FeCe, is lacking.
Table 5
Activation energies in the fully ordered ferromagnetic state () and paramagnetic state (Q^{P}), along with diffusion prefactors for Fe selfdiffusion and impurity diffusion of Y, La and Ce in bcc Fe.
Reference 
(kJ mol^{1}) 
Q^{P}(kJ mol^{1}) 
D_{0}(m^{2}/s) 

Fe 
Present work 
275.3 
238.1 

Huang et al. [11] 
277 
239 

Nitta et al. [32] 
289.7Â±5.1 
250.6Â±3.8 
2.76Ã10^{4} 

Seeger[33] 
280.7 
242.8 
6.0Ã10^{4} 

Y 
Present work 
165.9 
159.9 
1.09Ã10^{}^{9} 
Murali et al. [9] 
218.1 
â€’ 
8.0Ã10^{}^{7} 

La 
Present work 
175.6 
169.2 

Ce 
Present work 
286.3 
275.8 
7.66Ã10^{}^{6} 
Fig. 2 presents a direct comparison between the calculated and published temperature dependent diffusion coefficients for Fe selfdiffusion and Y solute diffusion. For Fe selfdiffusion, the calculated values are in good agreement with Huang et al. [11] and Nitta et al. [29], but smaller than the measured data of Seeger [31], this discrepancy may be due to the impurity, defects and grain sizes of the testing materials, and instrumental accuracy. For Y solute diffusion, the diffusion coefficient values above 1043K are in good agreement with Murali et al. [9], while the calculated values are lower than the reported values below the Curie temperature. This discrepancy can be attributed to the harmonic vibration and the influence of the bulk magnetization on the activation energy.
Fig.2. Calculated temperature dependence of diffusion coefficients for Fe and Y in bcc Fe vs. published experimental data.
The temperature dependence of the solute diffusion coefficients for Y, La and Ce in bcc Fe matrix are plotted in Fig. 3, where the Fe selfdiffusion coefficient is also included. It can be clearly seen that over the entire investigated temperature range which below the Î±Î³ phase transition, the diffusion coefficients for La and Ce are lower than the selfdiffusion in pure bcc Fe, and the Ce is the slowest diffuser. For the case of Ce solute diffusivity, the lowest value of diffusion coefficient originates from the largest migration energy and the smallest solutevacancy binding energy. For the fastest diffuserY, we can observe that the diffusion rate of Y is lower than that of Fe selfdiffusion at temperatures above 970 K. This trend is comparable to the previous experimental data [34] which indicate that the diffusivity of Y in bcc Fe is about two orders of magnitude lower than that of Fe at 1125 K. But unlike Ref. [34], the relation of diffusivity of Y and Feself reverses with temperature dropping below 970 K in our investigation. This is attributed to the effects of the diffusion prefactors of these two elements. Therefore, the investigation for diffusion should also be based on detailed considerations of the diffusion prefactor, including the contribution of the correlation factor, rather than the vacancy formation energy and migration energy.
Fig. 3. Calculated temperature dependence of diffusion coefficients for Y, La and Ce impurity in comparison with Fe selfdiffusion in bcc Fe.
The results of this paper are important for the better understanding of nanostructure evolution in ferritic Fe alloys. It is well known that Oxide dispersion strengthened (ODS) alloys have attracted significant interest due to their high resistance to irradiationinduced damage and enhanced high temperature mechanical properties, which mainly originate from the formation of ultra high density of YTiOenriched nanoclusters [34]. The formation enthalpy of La_{2}Ti_{2}O_{7}(3855.5 kJ/mol) is close to that of Y_{2}Ti_{2}O_{7} (3874.2 kJ/mol) [35],with considering the similarity of diffusivity for Y and La in bcc Fe, one can assume that La is a potential candidate to form nanoparticles in ODS alloys. In this regard, Pasebani et al. [36] developed La contained ODS alloy, and the data analysis revealed that La could be an alternative rare earth element for ODS alloys. Besides, in the case of trap helium and thus slow down the helium bubble growth in ODS alloy, in addition to the oxide nanoclusters, Ono and Hao et al. [3739] reported that Cr, Au and Ce can segregate to the He bubble surface and prevent He bubble growth. Based on our calculated result, the slow diffuser Ce can serve as a extra nucleation sites for small helium, thereby suppressing the heliuminduced degradation.
4. Summary
The diffusion properties of solute elements (Y, La and Ce) in bcc Fe matrix have been studied using DFT calculations in conjunction with the Le Claire ninefrequency model. Of the three impurities that we investigated, the diffusion coefficients of Y solutediffusion are lower than that of Fe selfdiffusion above 970 K and higher than the later below 970 K. This direction deviation between the trends of the diffusion coefficients of Y and the host atom Fe mainly comes from the contribution of the diffusion correlation factor. The diffusivity of La solute is predicted to be about one order of magnitude less than that of Y. The calculated results indicate that Ce diffuse slower than La and Y solute, as well as Fe selfdiffusion. The slowest diffusion rate of Ce originates from that the solutevacancy binding energy of this element is lower than that of Y and La, and the former has higher migration energy as well.
Acknowledgments
The authors are grateful for the financial support of the National Natural Science Foundation of China (No. 51101083, 51261018).
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Figure captions
Fig. 1. Schematic illustration of the ninefrequency 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.
Fig.2. Calculated temperature dependence of diffusion coefficients for Fe and Y in bcc Fe vs. published experimental data.
Fig. 3. Calculated temperature dependence of diffusion coefficients for Y, La and Ce impurity in comparison with Fe selfdiffusion in bcc Fe.
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