Temperature dependence of enthalpies and entropies of formation and migration of mono-vacancy in BCC iron

Accepted Manuscript Temperature Dependence of enthalpies and entropies of formation and migration of mono-vacancy in BCC iron Haohua Wen, C.H. Woo PII: DOI: Reference:

S0022-3115(14)00134-2 http://dx.doi.org/10.1016/j.jnucmat.2014.03.025 NUMA 48039

To appear in:

Journal of Nuclear Materials

Please cite this article as: H. Wen, C.H. Woo, Temperature Dependence of enthalpies and entropies of formation and migration of mono-vacancy in BCC iron, Journal of Nuclear Materials (2014), doi: http://dx.doi.org/10.1016/ j.jnucmat.2014.03.025

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Spin-Dynamics contribution

Haohua Wen and C. H. Woo

Temperature Dependence of enthalpies and entropies of formation and migration of mono-vacancy in BCC iron Haohua Wen and C. H. Woo* Department of Physics and Materials Science, City University of Hong Kong, Hong Kong SAR, China ABSTRACT Entropies and enthalpies of vacancy formation and diffusion in BCC iron are calculated for each temperature directly from free-energies using phase-space trajectories obtained from spin-lattice dynamics simulations. Magnon contributions are found to be particularly substantial in the temperature regime near the α−β (ferro/para-magnetic) transition. Strong temperature dependence and singular behavior can be seen this temperature regime, reflecting magnon softening effects. Temperature dependence of the lattice component in this regime is also much more significant compared to previous estimations based on Arrhenius-type fitting. Similar effects on activation processes involving other irradiation-produced defects in magnetic materials are expected. Keywords: vacancy formation and migration; magnon; enthalpy; entropy.

*

Corresponding author: [email protected]

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Introduction

Irradiation damage begins with the in-cascade direct production of vacancy and interstitial clusters in addition to the freely migrating vacancies and self-interstitials. The re-dissociation of vacancies clusters due to their thermal instability adds to the vacancy production rate and creates a strongly temperature-dependent production bias [1] which accounts for much of the temperature characteristics of the irradiation-damage accumulation [2]. In this regard, the thermodynamics of vacancy formation and migration plays a major role.

Ferritic/martensitic steels and their variants, which are important candidate materials for Generation IV fission reactors (core and pressure vessel applications) as well as fusion reactors (first walls and blankets), are ferromagnetic below the Curie temperature, where the internal degrees of freedom which govern the thermodynamics of atomic processes include the atomic spins. The coupling of the interatomic interaction with the spin states via the exchange and correlation among the shared electrons produces interdependent magnetic and lattice properties, and allows the strong temperature dependence of the former to be transmitted to the latter. Indeed, examples of this kind of coupling are abound, and can be commonly seen in the thermal expansion, elastic anomaly, phonons frequencies and lifetimes, defect configurations and associated properties of ferromagnetic metals.

In a previous paper [3], which will be called paper I from here on, we reported our spin-lattice dynamics (SLD) [4] calculation of the free energies of vacancy migration and formation in ferromagnetic iron. By comparing the results of the magnetic and non-magnetic 2 / 24

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Hamiltonian, we found that the dynamics of the spin degrees of freedom contribute significantly to the properties of the vacancy in terms of its formation and migration. Thus, magnon scattering by the vacancy differs in the equilibrium and saddle-point configurations, and this difference significantly affects both the free energies of formation and migration. In another aspect, the interatomic potential has a component from the exchange of electrons between neighboring atoms, which depends on the strongly temperature-dependent spin-spin correlation. The resulting interatomic force leads to extra temperature dependencies of the interatomic distance and the activation barriers of the vacancy processes, particularly across the magnetic phase boundary. In paper I, the observed non-Arrhenius behaviors of vacancy and self-diffusion were traced to the spin dynamics in this way.

In paper I, the magnetic contribution to the enthalpies and entropies of vacancy migration and formation was estimated by Arrhenius-type fitting. Rigorous justification of such fitting procedure can only be made if the temperature dependency of the enthalpies and entropies is negligible [5], a condition that is clearly not supported by either the calculated results or experimental data, especially near the FM/PM phase boundary. Findings in paper I lack the precision to allow a quantitative conclusion on the effects of ferromagnetism on vacancy migration and self-diffusion processes. A good understand of magnetic effects in vacancy and self-diffusion is a key factor that governs the temperature dependence of the micro-structure and the concomitant materials properties under irradiation damage conditions [6].

Many attempts have been made to ascertain the source of temperature dependence of enthalpies Q and entropies S in vacancy migration and formation in magnetic metals.

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Flynn [7] proposed an empirical equation that can be used to derive the migration enthalpy QM based on the continuum model, in which the effects of ferromagnetism occur through

lattice vibrations via the affected elastic constants. Schober et al. [8] suggested the estimation of QM using the Green function, but did not consider the effects due to the spin dynamics. This problem is also encountered in atomistic simulations using density functional theory (DFT). Although ab initio calculations can be performed to calculate QM (or QF ) of BCC iron at the ground state (0K) [9], there is no appropriate method to include dynamical effects of magnon softening needed to extend the validity to temperatures near the FM/PM transition. In this regard, many authors tend to adopt Girifalco’s model [10], in which the magnetic contributions to QM and QF are taken into account empirically within the framework of the molecular field approximation, i.e.

QM ( F ) (T ) = QM0 ( F ) + α M ( F )

M 2 (T ) , M 2 (0 )

(1)

where QM0 ( F ) is the non-magnetic contributions of the migration (formation) energy. M (T ) is the temperature dependent magnetization that vanishes in the PM state; α M ( F ) is an empirical parameter. Pérez et al. [11] estimated QM and QF , as well as α M ( F ) by fitting to the experimental data. Beside its empirical nature, the influence of short-range magnetic ordering (SRMO) [12,13], which does not vanish in the PM phase, is also absent. Using the Ising model, Fähnle et al. [14,15] and Khatun et al. [16] demonstrated the importance of SRMO in both migration and formation processes of vacancy. In conventional MD simulations such as Mendelev et al. [17], there is no effective scheme to treat contributions from the spin dynamics [4]. 4 / 24

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Based on harmonic approximation, Vineyard [18] proposed a model in which the migration and formation entropies are expressed in terms of the phonon frequencies between the equilibrium and saddle-point configurations,

⎧3 N − 4 S M = kB ln ⎨ ∏ ωivac ⎩ i =1

3 N −4

∏ω

⎧⎪3( N −1) ⎛ k T S F = kB ⎨ ∑ ln ⎜ B vac ⎝ ωi ⎩⎪ i =1

i =1

sad i

⎫ ⎬, ⎭

(2a)

⎞ N − 1 3 N ⎛ kBT ⎟− ∑ ln ⎜ N i =1 ⎝ ωiper ⎠

⎞ ⎫⎪ ⎟⎬ , ⎠ ⎭⎪

(2b)

where ωivac is the ith normal frequency for the crystal containing a single vacancy with N lattice sites in the equilibrium state; ωisad is the ith normal frequency of the same crystal in a saddle-point configuration. ωiper denotes the frequencies in a perfect crystal containing N atoms. Eq. (2b) is widely used. For example, Hatcher et al. [19] calculated S F for vacancy formation in BCC iron and obtained a value ~ 2.1kB . However, there is yet no direct calculation of the migration entropy S M following Eq. (2a) [20]. Instead, it is usually estimated by fitting to the Arrhenius plot of vacancy diffusion, either from experiments or atomic simulations. For example, Tsai et al. [21] performed molecular dynamics simulations to measure the vacancy diffusivity of BCC iron, but obtained a negative S M = −1.4kB , despite that the ground state is generally expected to have a lower entropy than the saddle-point configuration.

Most of the first-principle calculations are performed with static spins in the fully ordered ferromagnetic ground state, from which the spin stiffness for infinitesimal deviations can be estimated. The activation energy and entropy of self-diffusion can be estimated within the

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harmonic approximation. The calculations, without accounting for anharmonicity of the spin dynamics that gives rise to magnon softening, suffer from accuracy issues in the temperature region near the FM/PM phase boundary.

In the following, we will modify the thermodynamic integration method [3], which we have used to obtain energies of vacancy formation and migration in Zr [22], for application within the framework of spin-lattice dynamics (SLD). The methodology will be introduced briefly in the section 2. The results and discussion will be presented in Section 3. The conclusion is given in the end.

2.

Hamiltonian and Methodology

As presented in Ref. [3] and [4], the interactive spin and lattice coupling system is described by a Hamiltonian of Heisenberg particles, i.e.,

N

HH = ∑ i =1

pi2 1 N + U ({Ri }) − ∑ J ij ({Ri }) Si ⋅ S j , 2m 2 i≠ j

(3)

Where pi , Ri and Si are respectively the momentum, lattice coordinate and spin of atom i . U is the non-magnetic interatomic potential, and J ij is the exchange interaction

function, which is the function of lattice configuration

{Ri } .

To separate the dynamics

related to the spin and lattice degrees of freedom in the system of Heisenberg particles, the Hamiltonian H H in Eq. (3) is rewritten as H H ≡ H L ( p, R) + H S ( S ) + H Δ ( R, S ) , where H L and H S are the Hamiltonians only containing the lattice and spin degree of freedom, respectively, whereas H Δ represents their interaction, e.g.,

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HL ( R) = ∑ i =1

HS (S ) = −

Haohua Wen and C. H. Woo

pi2 + U ( R) 2mi

(4a)

1 N ∑ J ij ( R ) Si ⋅ S j 2 i≠ j

(4b)

1 N ∑ δ J ij Si ⋅ S j 2 i≠ j

(4c)

H Δ ( R, S ) = −

where δ J ij = J ij ( R ) − J ij ( R ) measures the fluctuation of J ij ( R ) around J ij ( R ) , R being the mean (time-averaged) lattice configuration, which is in general temperature dependent. The corresponding free energies are calculated using the adiabatic switching thermodynamic integration (TI) method [23], with phase-space trajectories of the corresponding activation processes. Details of this method have been given in our previous work [3,5]. We will only give a brief description in the following:

The magnetic free energy of vacancy formation and migration is derived from the spin dynamics (i.e., from H S ) using the adiabatic switching TI method. The corresponding activation entropies and enthalpies are then calculated as a function of temperature using the thermodynamic relations S = − ∂F ∂T and Q = − ∂ ( F / T ) ∂ (1/ T ) . The same procedure,

nevertheless, is not appropriate in the lattice contribution because the volume change during the migration and formation processes cannot be readily taken into account in a dynamical calculation [17]. To solve this problem, we note that the main contributions to the activation enthalpies are configurational. Dynamic contributions from changes in phonon vibrational frequencies during the activation process can be regarded as high-order perturbations [25]. Thus, instead of using dynamical simulations of the MD type, we may consider the enthalpies

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of formation and migration from the configurational energy difference using the modified conjugated gradient (MCG) method. Anharmonic effects are accounted for within the MCG method through the temperature dependence of the interatomic distances due to thermal expansion. Using the free energies calculated using TI method in paper I, the entropies can then be obtained using the alternate thermodynamic relation, i.e., S = (Q − F ) / T . Dynamical contributions arising from the difference between phonon frequencies in the equilibrium and saddle-point configurations are included in the activation entropies. This is consistent with the absolute rate theory [5,18,26], e.g., Vineyard's model (See Eqs. (2a) and (2b)).

In the following calculations, Dudarev-Derlet potential [24] based on embedded atom method for BCC iron is modified to describe non-magnetic interatomic interaction, by subtracting the contributions from the ground-state collinear spin configuration, i.e.,

⎛ 1 ⎞ U ( R ) = U DD ( R ) − ⎜ − ∑ jij ( R ) ⎟ , ⎝ 2 i≠ j ⎠

(5)

where jij ( R ) is the product of the exchange integral J ij ( R ) and the spin magnitudes Si , S j , i.e., jij ( R ) = J ij ( R ) Si S j . The simulation cell contains 16,000 atoms in a box of 20 × 20 × 20 BCC unit cells, with periodic boundary conditions applying to avoid the surface

effect along each dimensions of Cartesian coordinate system. The system is equilibrzted in a canonical ensemble, with the relaxation time set at 1 ns within SLD simulation framework. Temperature is controlled using the Langevin thermostat [4], and atomic volume is temperature-dependent, determined by equilibrating the simulation cell (with a vacancy) under zero pressure. The temperature ranges from 850K to 1300K, which is sufficiently wide

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for a credible analysis around Curie temperature ( TC ~1020K in our calculation,; 1043K experimentally). Based on the equilibrium configurations of spin and lattice, another 2ns is applied to generate the phase-space trajectory for adiabatic switching process, in which both forward and backward processes are performed. Adequacies of both the simulation size and integration time have been confirmed. For example, doubling the simulation size or integration time produces a difference of less than 1%.

3.

Results and Discussion

3.1 Direct magnetic contributions

We consider in this subsection the direct contributions from the spin dynamics, which arises from the difference in magnon scattering by the vacancy in its equilibrium and saddle-point configurations. Fig. 1 shows the free energies of vacancy migration FMS and vacancy formation FFS (from Eq. 4b) calculated from the spin Hamiltonian H S using TI method. Both show similar temperature dependencies, decreasing respectively from 17 to 1meV and from 80 to 5meV with increasing temperature from 850K to 1300K. The inflexions indicate the disappearance of long-range magnetic order near TC . Their values are comparable to the corresponding free-energy differences between magnetic and non-magnetic models in paper I (Figs. 1 and 3), thus establishing the consistency of the two calculations. Activation entropies and enthalpies due to the spin system H S can be deduced from Fig. 1 using thermodynamic relations, i.e., S M ( F ) = −∂FM ( F ) / ∂T and Q = − ∂ ( F / T ) ∂ (1/ T ) , respectively. The results are plotted in Figs. 2 and 3.

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Fig. 2 shows the strongly temperature-dependent migration entropy S MS and migration enthalpy QMS and their singular behavior at the FM/PM phase boundary, caused by the divergence of the correlation length during the spontaneous magnetic phase transition [14,15,16]. S MS increases from ~ 0.6kB at 850K by over 60% to ~ 1.0kB near TC , and the corresponding QMS doubles from ~42meV to ~90meV. Comparison shows that both the entropy and enthalpy in the FM phase obtained in paper I using Arrhenius-type fitting are much overestimated, with S MS ~ 2kB and QMS ~ 180 meV . Peaks near TC for S FS and QFS

for vacancy formation are also seen in Fig. 3, with values doubling from ~ 2.5kB to ~ 5kB and ~250 meV to ~500 meV, respectively. The non-zero values of enthalpies and entropies in the paramagnetic (PM) phase are due to contributions from the SRMO, in good quantitative agreement with other model predictions [14,15,16]. Compared in Fig. 3, values of entropy and enthalpy of the vacancy formation obtained in paper I using Arrhenius fitting are substantially smaller than our current results. Beside the loss of accuracy, the entropies and enthalpies obtained from simple Arrhenius fitting also suffer from the lack of details in the temperature dependence near the FM/PM phase boundary. The activation entropies and enthalpies calculated here are the energy differences of magnon scattering off the vacancy in different configurations (equilibrium and saddle-point). The singular behaviors in Figs. 2 & 3 show the significant role played by the participation of the magnons in the activation process. Magnon softening is mainly responsible for the complicated temperature behavior of self-diffusion in BCC iron near TC (See Fig. 6 in paper I).

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3.2 Direct lattice Contributions

Vacancy migration and formation free energies coming directly from the lattice have been obtained in paper I using TI method. Both are apparently linear functions in an Arrhenius plot (See Figs. 1 and 4 in paper I) from which temperature-independent activation entropies and enthalpies can be derived from the slope and intercept. Yet, the migration and formation enthalpies obtained from the MCG method are both temperature dependent, as shown in Figs. 4(b) and 5(b) in the current paper. Indeed, QML increases from ~0.86eV at 850K to ~0.93eV at 1300K, which is almost 30% larger than the value of 0.64eV found by the Arrhenius fitting in paper I. At the same time, QFL changes from ~1.92eV at 850K to ~1.95eV at 1300K, larger than the estimation of 1.81eV in paper I, via Arrhenius fitting. Here, the temperature dependencies of the enthalpies QML and QFL have a common physical base, namely, the anharmonicity of the lattice vibrations, from which originates thermal expansion. The increase in the interatomic separation raises the energy barriers against both vacancy migration and formation. The difference between our current calculation and those obtained in paper I exposes a possible short-coming of the latter. We will discuss this issue in more detail in the following subsection.

Similarly, from Figs. 4(a) and 5(a), the migration and formation entropies due to lattice dynamics, i.e., S ML and S FL , are also temperature dependent, respectively, from ~3.1 kB and ~4.2 kB at 850K to ~3.2 kB and ~3.9 kB at 1300K, both of which are larger than the constant values obtained in paper I, i.e., 1.13 kB and 3.71 kB , respectively. In this regard, anharmonicity is important in migration and formation processes of the vacancy, producing

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the temperature dependencies of the entropies and enthalpies. Our results in this paper are consistent with experiments and other calculations (See Table 2 in paper I). Note that the direct magnetic contributions to migration entropy S MS is about one-third of that from the lattice S ML . For the formation entropy, S FS has values comparable to S FL in the FM phase. Although the enthalpies are dominated by factors related to the lattice configuration, magnetic contributions near the Curie temperature are still significant.

3.3 Contribution from spin-lattice coupling

Effects of spin-lattice interactions show up as the difference between the sum of spin and lattice contributions from H L + S ( ≡ H L + H S ) in Eq. (4) and H H in Eq. (3). This difference comes

from

the

dependence

of

the

interatomic

exchange

interaction

on

the

temperature-dependent correlation between neighboring spins, which is present in H H while absent in H L + S . In calculating activation enthalpies of H H , the same scheme as H L is adopted.

Thermal expansion due to anharmonicity of both phonons and magnons are

included, except that which is associated with H H is different from that with H L because of the involvement of magnons.

We plot the temperature dependence of the migration entropy and enthalpy respectively in Figs. 4(a) & 4(b) for Hamiltonians H L , H L + S and H H , in comparison with those obtained in paper I. Here, the cusps present in S MH and QMH , involving the phonon-magnon interactions, demonstrate the effects of magnon softening as the spontaneous FM/PM phase transition is approached, which cannot be reproduced by Arrhenius-type fitting as shown in Fig. 4. Furthermore,

S ML + S

and QML + S

are, by definition, contributions from the 12 / 24

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non-interactive phonon and magnon excitations, from which the contributions due to the phonon-magnon interaction can be estimated by comparing with S MH and QMH . Note that, through the exchange interaction, the phonon-magnon interaction leads to additional anharmonicity of the phonon subsystem, e.g., the anomalous thermal expansivity near TC , thus giving rise to extra contributions to migration entropy and enthalpy. Shown in Fig. 4, such interactive effect increases the migration entropy by ~ 0.3kB and decreases the migration enthalpy by ~40meV, which are far smaller than the values of direct contributions from the magnon and phonon systems.

In Figs. 5(a) & 5(b) we compare the temperature dependencies of migration and formation entropies calculated from H L , H L + S and H H . Similar to the migration process, both S FH and QFH for vacancy formation also show singular behavior at TC , with values larger than those in Paper I. Furthermore, the phonon-magnon interaction provides a positive contribution to formation entropy, but a negative contribution to formation enthalpy, thus a negative contribution to formation free energy. Although the values of this contribution is small compared to those of pure phonons and magnons, the phonon-magnon interaction plays important role on vacancy formation, by reducing the energy barrier and increasing the heat dissipation. Our results from SLD simulations are in good agreement with results from experiments and other calculations, as listed in Table 2 of paper I.

3.4 Issues with Arrhenius fitting

Spin contributions to entropies and enthalpies in paper I deviates significantly from those in the present paper. This confirms that Arrhenius-type fitting in strongly temperature 13 / 24

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dependent non-Arrhenius diffusion phenomena may not be appropriate. Thus, apparent from the Arrhenius plot of the vacancy-diffusivity for the non-magnetic Hamiltonian in Fig. 3 of paper I, both the migration entropy and enthalpy, i.e., S ML and QML , are constant. However, considering the anharmonicity of phonons, both S ML and QML obtained in the present paper are temperature dependent from 850K to 1300K, as shown in Figs. 4. In addition, the singular behaviors in Figs. 2 and 3 near TC are lost in the Arrhenius fitting. Indeed, Arrhenius fitting fails to distinguish between the temperature dependencies of the entropy and enthalpy, which is clear from the following example. Thus, if one may expand the migration entropy and enthalpy to linear terms in temperature, namely,

Q (T ) = Q0 + Q1T ,

(6a)

S (T ) = S0 + S1T ,

(6b)

where Q0 , Q1 , S0 and S1 are all temperature independent parameters, the free energy is thus written as F (T ) = Q (T ) − TS (T ) = Q0 + (Q1 − S0 ) T + S1T 2

,

(7)

so that migration enthalpy and entropy estimated by Arrhenius fitting are respectively Q = Q0 and S = (Q1 − S0 ) , and the term of S1T 2 is usually neglected. The enthalpy and entropy can thus be overestimated or underestimated, depending on the sign of Q1 . In this regard, Arrhenius fitting is only a convenient approach to obtain a preliminary estimation of entropies and enthalpies of activation processes. There is no guarantee that the final results have real

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physical meaning. They should be checked carefully using other treatment, especially when one is dealing with cases near phase transitions.

4.

Summary and Conclusion

Spin-lattice dynamics simulations have been performed to clarify the roles of phonon and magnon excitations on vacancy migration and formation processes in BCC iron. In a previous paper, entropies and enthalpies were estimated by Arrhenius-type fitting, which may not be appropriate. In this paper, entropies and enthalpies are recalculated for each temperature directly from the free-energies using alternate thermodynamic relations. Values of the magnetic entropy and enthalpy presently obtained are quite different from those due to Arrhenius-type fitting. We found that the differences in magnon dynamics with the vacancy in its equilibrium and saddle-point configurations contribute significantly to the entropies and enthalpies in these activation processes. In particular, the softening of magnons near FM/PM phase boundary gives rise to a strong temperature dependence with a singular behavior. This is a typical consequence of magnetic phase transition, which has been demonstrated by theoretical prediction of Ising model. Similar effects are expected of activation processes involving other irradiation-generated defects in magnetic materials. In this temperature regime, magnon contributions are very substantial. Effects due to phonon-magnon interaction reduce the energy barrier and increase entropy of both vacancy formation and migration. They are weak except near the Curie temperature. Comparison between current results and those in Paper I show that the slope and intercept from Arrhenius-type fitting on non-Arrhenius diffusion phenomena does not give good estimates of the activation enthalpies and entropies.

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Temperature dependence of the lattice component, are also found to be much more significant, confirming our doubt of the accuracy of the Arrhenius-type fitting. We also found that the temperature dependence due to the anharmonicity of lattice vibrations is relatively weak in this investigation, even up to the FCC/BCC transition temperatures where phonon softening is expected. This is perhaps a sign of inadequacy of the interatomic part of the potential.

Acknowledgement

This project was funded by the Grant 124013 from the Hong Kong Research Grant Commission, to which the authors are grateful.

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Reference

[1]

C.H. Woo, B.N. Singh, Phil. Mag. A65 (1992) 889-912.

[2]

C. H. Woo, B.N. Singh, A.A. Semenov, J. Nucl. Mater.239 (1996) 7-23.

[3]

Haohua Wen, Pui-Wai Ma, C. H. Woo, J. Nucl. Mater. 440 (2013) 428.

[4]

P.-W. Ma, C. H. Woo, S. L. Dudarev, Phys. Rev. B 78 (2008) 024434.

[5]

C. Zener, in: Imperfections in Nearly Perfect Crystals, edited by W. Shockley, J. H.

Hollomon, R. Maurer and F. Seitz, (John-Wiley & Sons, New York, 1952), pp. 289. [6]

T. G. Langdon, Materals Transactions, 46 (2005) 1951.

[7]

C. P. Flynn, Phys. Rev. 171 (1968) 682.

[8]

H. R. Schober, W. Petry, J. Trampenau, J. Phys.: Condens. Matter 4 (1992) 9321.

[9]

C.-C. Fu, J. D. Torre, F. Willaime, J.-L. Bocquet, A. Barbu, Nature Mater. 4 (2005) 68.

[10] L. Ruch, D. R. Sain, H. L. Yeh, L. A. Girifalco, J. Phys. Chem. Solids 37 (1976) 649. [11] R. A. Pérez, M. Weissmann, J. Phys.: Condens. Matter 16 (2004) 7033. [12] G. Hettich, H. Mehrer, K. Maier, Scr. Metall. 11 (1977) 795. [13] J. Kučera, Czech. J. Phys. B 29 (1979) 797. [14] M. Fähnle, A. Seeger, Phys. Stat. Sol. (a) 91 (1985) 609. [15] M. Fähnle, M. Khatun, Phys. Stat. Sol. (a) 126 (1991) 109. [16] M. Khatun, J. W. Emert, Phys. Stat. Sol. (b) 231 (2002) 341. [17] M. I. Mendelev, Y. Mishin, Phys. Rev. B 80 (2009) 144111. [18] G. H. Vineyard, J. Phys. Chem. Sol. 3 (1957) 121. [19] R. D. Hatcher, R. Zeller, P. H. Dederichs, Phys. Rev. B 19 (1979) 5083. [20] S. Huang, D. L. Worthington, M. Asta, V. Ozolins, G. Ghosh, P. K. Liaw, Acta Mat. 58 (2010) 1982. [21] D. H. Tsai, R. Bullough, R. C. Perrin, J. Phys. C: Solid St. Phys. 3 (1970) 2022. [22] Haohua Wen, C. H. Woo, J. Nucl. Mater. 420 (2012) 362. [23] M. de Koning, A. Antonelli, Phys. Rev. E 53 (1996) 465. [24] S. L. Dudarev, P. M. Derlet, J. Phs. Condens. Mater. 17 (2005) 7097. [25] H. R. Schober, W. Petry, J. Trampenau, J. Phys.: Condens. Matter 4 (1992) 9321. [26] C. A. Wert, Phys. Rev. 79 (1950) 601.

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[27] J. E. Sinclair, R. Fletcher, J. Phys. C 7 (1974) 864. [28] C. H. Woo, X. L. Liu, Philos. Mag. 87 (2007) 2355.

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Figure Caption:

Fig. 1: The temperature dependence of free energies of vacancy migration (red open circles) and formation (blue open squares) arising from contributions of magnon scattering by vacancy.

Fig. 2: The temperature dependence of (a) entropy and (b) enthalpy of vacancy migration arising from the changes of magnon states directly, derived from the free energies shown in Fig. 1. The dashed lines are the estimation of Arrhenius fitting in Ref. [3].

Fig. 3: The temperature dependence of (a) entropy and (b) enthalpy of vacancy formation arising from the changes of magnon states directly, derived from the free energies shown in Fig. 1. The dashed lines are the estimation of Arrhenius fitting in Ref. [3].

Fig. 4: The temperature dependence of vacancy migration (a) entropy and (b) enthalpy, in comparison with Arrhenius fitting estimations in Ref. [3] (dashed green lines). Here, the solid lines are guides for eye.

Fig.5:

The temperature dependence of vacancy formation (a) entropy and (b) enthalpy, in comparison with Arrhenius fitting estimations in Ref. [3] (dashed green lines). Here, the solid lines are guides for eye.

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20 S

FM

80

S

16

S

FF 60

12 40 8 20 4 0 0 800

Magnon Formation Free Energy, FF (meV)

Haohua Wen and C. H. Woo

S

Magnon Migration Free Energy, FM (meV)

Spin-Dynamics contribution

900

1000

1100

1200

Temperature (Kelvin)

FIG. 1

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1300

Haohua Wen and C. H. Woo

(a)

2.0

Ref. [3] Current

S

Magnon Migration Entropy, SM (kB)

Spin-Dynamics contribution

1.5

1.0

0.5

0.0 800

900

1000

1100

1200

1300

200

(b) Ref. [3] Current

S

Magnon Migration Enthalpy, QM (meV)

Temperature (Kelvin)

150

100

50

0 800

900

1000

1100

1200

Temperature (Kelvin)

FIG. 2

21 / 24

1300

Spin-Dynamics contribution

Haohua Wen and C. H. Woo

S

Magnon Formation Entropy, SF (kB)

6

(a) 5

Ref. [3] Current

4 3 2 1 0 800

900

1000

1100

1200

1300

S

Magnon Formation Enthalpy, QF (meV)

Temperature (Kelvin)

600

(b) 500

Ref. [3] Current

400 300 200 100 0 800

900

1000

1100

1200

Temperature (Kelvin)

FIG. 3

22 / 24

1300

Spin-Dynamics contribution

Haohua Wen and C. H. Woo

Migration Entropy, SM (kB)

5

(a) 4 3 Ref. [3] L SM

2

S

SM

L+S

SM

1200

1300

SM

H

1 0 800

900

1000

1100

Temperature (Kelvin)

Migration Enthalpy, QM (eV)

1.1

(b) 1.0 0.9 0.8

Ref. [3] L QM

L+S

QM

H

QM

0.7 0.6 800

900

1000

1100

1200

Temperature (Kelvin)

FIG. 4

23 / 24

1300

Spin-Dynamics contribution

Haohua Wen and C. H. Woo

12

Formation Entropy, SF (kB)

(a)

Ref. [3] L SF

10

L+S

S

SF H

SF

SF

1200

1300

8 6 4 2 0 800

900

1000

1100

Temperature (Kelvin)

Formation Enthalpy, QF (eV)

2.5 2.4

Ref. [3] L QF

(b)

L+S

QF

H

QF

2.3 2.2 2.1 2.0 1.9 1.8 800

900

1000

1100

1200

Temperature (Kelvin)

FIG. 5

24 / 24

1300