Conference-XAFS Theory-Path Integral Effective Classical Potential Method Applied to Anharmonicity and Quantum Effects in Thermal Expansion of Invar Alloy

Extended X-ray Absorption Fine Structure (EXAFS) spectroscopy is a powerful experimental technique to investigate anharmonic vibrational properties of solids especially when one combines experimental EXAFS with quantum mechanical theoretical evaluations. In this article, the path-integral effective-classical-potential (PIECP) theory is applied to temperature dependence of EXAFS in a real system. The anharmonicity and quantum effects in the Invar alloy Fe64.6Ni35.4 that shows anomalously small thermal expansion are investigated. Experimental Fe and Ni K-edge EXAFS measurements and the computational PIECP simulations have been performed. It is experimentally revealed that the first nearest-neighbor (NN) shells around Fe show almost no thermal expansion, while those around Ni exhibit meaningful but smaller expansion than that of fcc Ni. At low temperature (<100 K), the vibrational quantum effect is found to play an essentially important role, which is confirmed by comparing the quantum mechanical simulations to the classical ones, the latter of which exhibit large (normal) thermal expansion at low temperature. It is also clarified that thermal expansion for the Ni-Ni and Ni-Fe pairs is noticeably suppressed, even though the Ni electronic state may not vary depending on the temperature. On the other hand, the anharmonicity (asymmetric distribution) clearly exist for all the first-NN shells as in the case of the normal thermal expansion system, where thermal expansion originates almost exclusively from the anharmonic interatomic potential. [DOI: 10.1380/ejssnt.2012.486]


I. INTRODUCTION
Extended x-ray-absorption fine-structure (EXAFS) spectroscopy has extensively been applied to local structure determination of a wide variety of systems such as crystalline, noncrystalline and liquid materials, catalysts, solutions, biological materials, surfaces, and so forth.Great attention has also been paid to the temperature dependence of EXAFS, which provides information on thermal vibrations including anharmonicity.Since Eisenberger and Brown [1] argued the importance of deviation from the Gaussian distributions of surrounding atoms for the reliable estimation of interatomic distances by means of EXAFS, several sophisticated analysis method have been proposed.The cumulant-expansion method exploited by Bunker [2] can be regarded as the most general and practical technique to include asymmetric distributions for moderately disordered systems and has extensively been employed so far.
On the other hand, the theory concerning the relationship between the EXAFS cumulants and vibrational potentials of the system has also been developed.The quantum-statistical first-order perturbation theories were proposed by Rabus [3] and Frenkel and Rehr [4] within the framework of the two-body potential (one degree of vibrational freedom).These theories simplify the evaluations of the EXAFS cumulants and have been further improved for practical use.The cumulant calculations based on the two-body simplification, however, requires fitting variables for the force constants since the two-body vibrational potential is just effective in polyatomic systems.Fujikawa and Miyanaga [5,6] and Yokoyama et al. [7][8][9][10] have extended the perturbation theory to account for many degrees of vibrational freedom.The perturbation method, however, includes several intrinsic difficulties for practical use.The formulation and computation become rapidly more cumbersome as the order of the cumulant or the degree of vibrational freedom increases.When one evaluates the nth-order cumulant with a m-dimensional system, the computational load is roughly proportional to m n+1 within the first-order perturbation calculation.
Another quantum-statistical approach to describe thermal properties of molecules and solids is to solve the Feynman's path integral [11,12].This method can handle greater anharmonicity.The quantum-mechanical pathintegral Monte-Carlo (MC) technique has been exploited [13] and should be the most reliable method for the present purpose.It is, however, by far more complicated than the perturbation method and might not be a good candidate for practical applications to EXAFS of complicated systems with many degrees of freedom.The pathintegral effective-classical-potential (PIECP) method has thus been developed to solve the path integral approximately by means of the variational concept [14,15].Fujikawa et al. [16] have applied the PIECP theory to EX-AFS for the first time.Although they have investigated only the two-body quartic double-well potential, some new features have been elucidated in the temperature dependence of the EXAFS cumulants.
Subsequently, the PIECP method has been employed to evaluate the EXAFS cumulants for real systems such as diatomic Br 2 with one degree of freedom and solid fcc Kr and Ni with many degrees of freedom [17].In the case of solid Kr, the potentials are described exclusively with the sum of pairwise interactions.Thus the calculations of the EXAFS cumulants of diatomic Br 2 and solid Kr were able to be performed within the framework of the established PIECP theory.Moreover, in the solid fcc Ni case, although many-body interactions are essentially important for metallic bonds, the embedded atom method (EAM) [18,19] was successfully employed in the PIECP method within the low coupling approximation.This allows one to generally calculate the EXAFS cumulants for metallic systems with many-body interactions.In a meanwhile, Beccara et al. [20,21] have performed more accurate path-integral Monte-Carlo calculations of fcc Cu, though they employed pairwise potential functions.
In the present article, thermal expansion and anharmonicity of an Invar alloy is discussed by means of the experimental EXAFS and theoretical PIECP methods [22].The Invar alloy was discovered by Guillaume in 1897 [23], to whom the Nobel Prize in Physics was awarded in 1920.The material is an iron-nickel alloy with a nickel concentration of around 35% and is widely known to exhibit anomalously small thermal expansion over a wide temperature range.It has been recognized that the Invar effect originates from magnetism.Until now, however, there have been still published many scientific papers concerning the origins of the Invar effect, this implying a lack of full understanding of the effect.A basic concept of the Invar effect is that there exist at least two types of electronic states in Fe, typically high-spin (HS) and low-spin (LS) states [24,25].In this two-state model, the equilibrium potential energy is lower in the HS state than in the LS one, while the equiliburium atomic radius is larger in the former.This results in the compensation of thermal expansion due to increasing density of the LS state at higher temperature.A recent ab initio electronic structure calculation at 0 K, however, reveals a very complicated electronic configuations in a smaller volume region [26].Computational simulations at finite temperatures have also been carried out for the understanding of magnetization and thermal expansion [27][28][29][30][31].There have been, however, no reports concerning quantum-mechanical dynamics calculations like path-integral MC simulations, although in general thermal expansion is inherently resulted from anharmonic vibration, to which the quantum effect is essentially important at low temperature.
The present article is organized in the following manner.Section II summarizes the practical formalism of the PIECP theory with the low coupling approximation.Subsequently, in Sec.III, the experimental results of the EX-AFS analysis are discussed by comparing to the computational results.At low temperature, the vibrational quantum effect is found to play an essentially important role in thermal expansion, which is confirmed by comparing the quantum mechanical simulations to the classical ones.It is also revealed that thermal expansion for the Ni-Ni and Ni-Fe pairs is noticeably suppressed, even though the Ni electronic state may not vary depending on the temperature.On the other hand, the anharmonicity (asymmetric distribution) clearly exist for all the first NN shells as in the case of the normal thermal expansion system, where thermal expansion originates almost exclusively from the anharmonic interatomic potential.

A. PIECP method
The description of the PIECP method can be found in the literatures [14,15] and is summarized here briefly.According to the Feynman ′ s path-integral theory [11,12], the density matrix ρ(R) (R is the 3N -dimensional Cartesian coordinate and N is the number of atoms) is given by the path integral form using the quantum-mechanical real-space representation as where H and Z are, respectively, the quantum-mechanical Hamiltonian operator and the partition function of the system, β=(k B T ) −1 (k B the Boltzmann constant and T the temperature), and ℏ the Planck constant divided by 2π.A[R(u)] is the Euclidean action such that where M is the diagonal matrix that represents atomic masses, V (R(u)) is the total potential energy of the system, and the left-side superscript t denotes the transposing operator of the matrix ( t Ṙ(u) is the transposed matrix of Ṙ(u)).
The PIECP method is based on the variational path integral theory and assumes a trial Euclidean action A 0 [R(u)] with some variational parameters [14,15].Since the present purpose is a description of anharmonic vibrational properties of solids, the harmonic approximation such that should be a suitable trial function.Here, X=R− R and R is the average path defined by The matrix F is the second-order force constant matrix under the Cartesian coordinate.In the second equation in Eq. 3, the Cartesian coordinate X is transformed into the normal coordinate Q using a linear transformation of Q= t U M 1/2 X, where U the orthogonal matrix that is given as eigenvectors of the matrix . ω 2 is the diagonal eigenfrequency matrix.The scalar quantity w( R) and the eigenfrequency matrix ω 2 are the variational parameters to be optimized.The path-integral calculation of the multidimensional harmonic oscillators can be performed analytically, and the resultant density matrix ρ opt ( R) is given as http://www.sssj.org/ejssnt(J-Stage: http://www.jstage.jst.go.jp/browse/ejssnt/) Volume 10 (2012)

Yokoyama
where α k,µ denotes the pure quantum fluctuation of the normal vibrational phonon mode (k,µ) (k the wavenumber and µ the polarization) defined by Thus the thermal average of a certain physical quantity O within the present approximation is evaluated as where Z opt is the optimzed partition function of the harmonic oscillator system and ≪ ≫ denotes the 3Ndimensional integral average concerning the pure quantum fluctuations (see Eq. 6).V eff ( R) is the effective classical potential defined as For the optimization of w and ω k,µ , one can employ the Jensen-Feynman inequality [11,12] as where F and F 0 are, respectively, the true and trial free energies.The eventual variational conditions can be given by [14,15] and (11) Although considerable approximations have already been employed and the resultant density ρ opt ( R) has been reduced to rather a simple expression, Eqs. 5 and 7 still look too complicated to employ in numerical calculations, because they contain the 3N -dimensional integration originating from the quantum fluctuations.For the one-dimensional system, no further approximation is required, while in the multidimensional system like solids (N ∼10 23 ) the quantum fluctuation integrals should be separated out.Here we should come closer to the harmonic approximation, which is the so-called low coupling approximation [14,15].This approximation consequently assumes that the variational parameters of w and ω k,µ are independent of R.
In the monatomic Bravais lattice with pairwise potentials (atomic mass of m), the lattice dynamical theory consequently yields the scalar potential w( R) as where u eff (R ij ) is the effective classical pair potential given by Here, σ (2)L ij and σ (2)L ij are respectively the longitudinal and transverse pure quantum mean square relative displacements as the former (Eq.14) corresponding to a pure quantum EXAFS Debye-Waller factor [38].In the right side of Eq. 13, the first term corresponds to the original adiabatic classical potential, while the latter two to the quantum mechanical correction.

B. EAM
For the description of the thermal properties of metals, the pair-potential approximation is not sufficient since the metallic bond should inherently be of many body.The EAM idea was firstly developed by Daw and Baskes [18], these being based on the density-functional theory.They argue that the total adiabatic potential energy of the system E can be written as where ρ h,i is the total electron density at atom i due to the host (the rest of the atoms in the system), F i is the embedding energy for placing atom i into the electron density, and ϕ ij is the short-range pairwise core-core repulsion that is defined as a function of the interatomic distance R ij .Although the embedding energy F i should be described as a complicated density functional, a socalled local-density approximation can be introduced and F i was simply given as a normal function of the host density ρ h,i at the position of atom i.In the case of isotropic metals, ρ h,i can be given as a sum of the spherically averaged atomic densities.In the previous EAM simulations, the numerical values of F i and ϕ ij were determined empirically instead of the density functional calculations.The EAM does not require the three-dimensional periodicity and has thus been applied to alloys, defects, and surfaces.When one tries to apply the EAM scheme to the PIECP method, the harmonic force constant matrix should at first be calculated.For this purpose, the EAM energy E is simply Taylor expanded around the equilibrium host electron density ρh,i .For monatomic systems, the resultant formula is given as where u is the pairwise potential between atoms i and j, which is given as Note that the subscripts i and j are neglected for simplicity because of the monatomic system.Here we have reached a very important conclusion.Although the EAM scheme inherently treats many-body forces, the harmonic approximation is found to be described within the pairpotential scheme.This comes from the assumption that the embedding energy F i is described only as a function of the sum of the spherically averaged atomic density of electrons, which depends only on the interatomic distance.
No angular dependence such as a bending force needs to be taken into account.This enables us to apply the EAM to the PIECP theory in a straightforward manner.Namely, the pairwise effective classical potential in Eq. 13 can be replaced with Eq. 18.

III. RESULTS AND DISCUSSION
The Fe and Ni K-edge EXAFS oscillation functions k 3 χ(k) (k the photoelectron wave number) and their Fourier transforms are depicted in Fig. 1.To perform quantitative analysis, the first-and third-NN shells were Fourier filtered, inverse Fourier transformed, and curve fitted in k space.The theoretical EXAFS function used in the curve-fitting analysis is given as where R is the bond distance, N the coordination number, C 2 =<(r − R) 2 > the mean square relative displacement, C 3 =<(r − R) 3 > the mean cubic relative displacement, S 2 0 the intrinsic loss factor, F (k) the backscattering amplitude, and ϕ(k) the total phase shift for the X-rayabsorbing and photoelectron-scattering atoms.F (k) and ϕ(k) are considered to include inelastic scattering events.C 2 and C 3 correspond to the thermal and static variance of the bond distance and the asymmetry of the pair distribution function, respectively [2].To evaluate the relative quantities like thermal expansion, the empirical analysis using the lowest temperature data as a reference was conducted, while the determination of the absolute values was carried out by employing the FEFF8 [32] standards that were obtained by the calculations of clusters assuming some randomly distributed alloy structures.Note that all the quantities are the average of the Fe and Ni scattering atoms.
Figure 2 shows the thermal expansion of the first-NN bond distances determined by the experimental EXAFS spectra.The dashed line corresponds to the experimental equilibrium distance R lattice =a 0 / √ 2 (a 0 is the lattice constant determined by the x-ray diffraction [33]).The solid line is the bond distance R bond estimated within the harmonic approximation as R bond =R lattice +<u 2 ⊥ >/R lattice , where <u 2 ⊥ > is the average squared displacement of each atom along the perpendicular direction to the corresponding bond axis.The difference of the distance between R lattice and R bond originates from the vibration perpendicular to the equilibrium bond direction [34][35][36][37].R bond is always slightly larger than R lattice .The <u 2 ⊥ > can be evaluated using the experimental C 2 in Fig. 3 with the correlated Debye model [38].We obtained Θ D =331 and http://www.sssj.org/ejssnt(J-Stage: http://www.jstage.jst.go.jp/browse/ejssnt/) 409 K for Cu and Ni, respectively.Both the first-and third-NN shells yielded essentially the same Debye temperature.The Debye temperature of the Invar alloy was estimated to be Θ D =430 K.The fcc Cu and Ni metals show normal thermal expansion in good agreement with estimated R bond , indicating high accuracy of the present analysis as in the previous reports [10,17,[35][36][37]39].The estimated C 2 value for the first-NN shells of both Fe and Ni K-edge EXAFS was C 2 =0.256×10 −2 Å2 at T =12.5 K. On the other hand, the above FEFF analysis gave C 2 =0.415×10 −2 Å2 and 0.357×10 −2 Å2 at T =12.5 K for the first-NN shells of Fe and Ni K-edge EXAFS, respectively.The differences between the FEFF and empirical analyses may originate from the static disorder, and in the present system one could recognize at least partially that the static disorder of around 0.10-0.15×10−2 Å2 is caused by the difference in the bond lengths among the Fe-Fe, Fe-Ni, and Ni-Ni atomic pairs.In the Invar alloy, thermal expansion is hardly seen in the local structure around Fe, while that around Ni is observed clearly.The magnitude of thermal expansion around Ni is, however, significantly smaller than that of fcc Ni, indicating the suppression of thermal expansion around Ni as well as Fe. A we expected from the two- given by the PIECP and classical MC simulations, together with the experimental literature data (red circle and dotted line) [33].
state model given by Weiss [24], almost no thermal expansion around Fe can be ascribed to the direct effect of increasing population of the LS state in Fe with increasing temperature.On the other hand, the behavior around Ni may be attributed to the indirect effect in the two-state model.Although Ni is likely to expand normally with the temperature rise, this is also suppressed by the almost fixed lattice constant.This will be further discussed in detail below.
PIECP MC simulations within the low coupling approximation [14,15,17] were performed under constant number of particles, pressure, and temperature (N P T ) condition.The constant-pressure condition is essentially important to reveal thermal expansion [10,40].Although the PIECP theory can treat only periodic lattices [14,15,17] and is strictly not applicable to the present alloy system with random distribution, the Invar alloy exhibits clear fcc structure and the atomic weights of Fe and Ni are not very different, allowing us to adopt the theory to the present simulation.The total number of atoms was 500 (5  provide consequent physical quantities.The potentials of Fe and Ni were based on the empirical embedded-atom method (EAM) [18,41].The potential parameters of Ni were optimized by using known thermodynamic quantities like elastic constants [41].Those of Fe were assumed by referring to the literatures [42][43][44] and also the present experimental results.For comparison, the MC simulations based on the classical thermodynamics and also the PIECP simulations using only the HS potential were carried out.
Figure 4 shows the simulated bond distances and the lattice constants by the PIECP and classical MC methods.Figure 4(a) gives the binding energies of Invar Fe 65 Ni 35 and fcc Fe at a temperature of 0 K as a function of the first-NN distance.In the present atomic potentials, the fcc Fe system shows that the LS state is more stable by 8.0 meV than the HS state with the bond distances of R HS =2.530 Å and R LS =2.492 Å, while the Invar case exhibits a more stable HS state by 25.0 meV with the bond distances of of R HS =2.530 Å and R LS =2.490 Å.In Figs.2(b-d), the agreement between the PIECP and experiments is good: almost no thermal expansion around Fe and meaningful thermal expansion around Ni. On the contrary, the classical method is found to give fatal discrepancies at low temperature below ∼100 K; the bond and lattice distances significantly increase with the temperature rise.These findings implies the essential importance of the vibrational quantum effect, which is recognized as a zero-point vibration.
To get further insights into local thermal expansion, the bond distance of each component (Fe-Fe, Ni-Ni and Ni-Fe) pair is shown in Fig. 5.In this plot, the PIECP MC results by using only the HS Fe state are also depicted to recognize hypothetical normal thermal expansion in this system.As expected, the Fe-Fe pair shows the largest discrepancies between the two-state (HS+LS) and the HS-only models.This is caused by the increasing population of the Fe LS state with the temperature rise, yielding compensation of the thermal expansion with the one originating from anharmonic vibration.The most im- Finally, let us discuss the third-order anharmonicity in the Invar alloy.Figure 6 shows the mean cubic relative displacements C 3 for the average first-and third-NN coordinations around Fe and Ni.The agreement between the experiments and the PIECP simulations is not perfect, but C 3 increases gradually with the increase in temperature for both the Fe and Ni data.Clear anharmonicity in the Invar alloy is thus confirmed by comparing the PIECP simulations between the two-state (HS+LS) and HS-only models.Both the simulated data exhibit essentially the same C 3 values, implying no suppression of C 3 due to Volume 10 (2012) Yokoyama the contribution of the LS state, as observed in the thermal expansion.Since the asymmetric radial distribution for the first-NN shell almost exclusively originates from the anharmonic interatomic potential, the present result implies that the third-order anharmonicity clearly exists even in the case of no thermal expansion.Note that in the present PIECP simulations within the low coupling approximation, the quantum effect in C 3 is not properly taken into consideration because of the neglect of the phonon-phonon coupling.The important qualitative finding of the presence of C 3 without thermal expansion is however clearly exemplified.Although in a simple twobody model the anharmonicity of the interatomic potential directly corresponds to thermal expansion [3,4,45], the Invar alloy is not the case.On the other hand, the third-NN shells do not show significant asymmetry either for Fe or Ni, although the experiments gives large errors.This effect is already known [10,40] since there exist almost no direct interaction with the third-NN shell and the pair distribution function between non-correlated atoms should be given as a Gaussian function according to the central limit theorem.

FIG. 2 :
FIG.2: Experimental thermal expansion of the first-and third-NN bond distances of fcc Cu metal (green triangle), fcc Ni metal (purple diamond), and the Invar alloy around Ni (red circle) and around Fe (blue square).Error bars are also indicated.The dashed line is the equilibrium distance given by the lattice constant.The solid lines correspond to the bond distances that are corrected from the equilibrium lattice distance by adding the contribution of the vibrations perpendicular to the bond direction.For the equation, see the text.

FIG. 3 :
FIG.3: Experimental mean square relative displacements C2 of the first-and third-NN shells of fcc Cu metal (green triangle), fcc Ni metal (purple diamond), and the Invar alloy around Ni (red circle) and around Fe (blue square).Error bars are indicated.The solid lines correspond to the fitted ones assuming the correlated Debye model[38] with the Debye temperature as a fitting parameter.The absolute values at the lowest temperature were estimated by the Debye model and possible structural disorder expected by the FEFF analysis is not added.The plotted C2 can thus be regarded as a thermal mean square relative displacement.

FIG. 4 :
FIG. 4: (a) Binding energies of Invar Fe64.6Ni35.4(top), fcc Fe (bottom), and fcc Ni (bottom, green dotted line) as a function of the first-NN distance at a temperature of 0 K.For Fe, two types of the potentials for the HS (red solid line) and LS (blue dashed line) states are depicted.(b,c) Simulated first-NN bond distance around Fe (b) and Ni (c) given by the PIECP (blue circle and solid line, quantum) and the classical MC (green diamond and dashed line, classic) methods, together with the experimental EXAFS data (red open circle with an error bar).(d) Equilibrium first-NN distance (a0/ √ 2)given by the PIECP and classical MC simulations, together with the experimental literature data (red circle and dotted line)[33].

FIG. 5 :
FIG. 5: Simulated bond distances of Fe-Fe (blue square and slid line), Ni-Ni (red square and solid line), and Ni-Fe (green square and solid line) pairs, together with the average ones around Fe (pink circle ans solid line) and Ni (orange circle and solid line).The experimental data for the average one around Fe and Ni are again shown.The dashed lines are the PIECP results by using only the HS state in Fe.

FIG. 6 :
FIG. 6: Mean cubic relative displacement C3 given by the experimental EXAFS (red open circle with an error bar) and by the PIECP simulations using the two-state (blue circle and dashed line) and the HS-only (green solid line) models for the average first-NN bonds around Fe (a) and Ni (b).
3fcc cubic unit cells), and the distributions of Fe and Ni were chosen randomly.11 types of the superlattices were simulated and the results were averaged to http://www.sssj.org/ejssnt(J-Stage: http://www.jstage.jst.go.jp/browse/ejssnt/) e-Journal of Surface Science and Nanotechnology