Infrared Optical Properties of Ferropericlase (Mg1-xFexO): Experiment and Theory
Tao Sun, Philip B. Allen, David G. Stahnke, Steven D. Jacobsen, Christopher C. Homes
aa r X i v : . [ c ond - m a t . m t r l - s c i ] A p r Infrared Optical Properties of Ferropericlase (Mg − x Fe x O):Experiment and Theory
Tao Sun, ∗ Philip B. Allen, David G. Stahnke, Steven D. Jacobsen, and Christopher C. Homes Department of Physics and Astronomy,State University of New York, Stony Brook, New York 11794 Department of Physics, University of California, San Diego, La Jolla, CA 92093 Department of Earth and Planetary Sciences,Northwestern University, Evanston, IL 60208 Condensed Matter Physics & Materials Science Department,Brookhaven National Laboratory, Upton, New York 11973 (Dated: October 31, 2018)
Abstract
The temperature dependence of the reflectance spectra of magnesium oxide (MgO) and fer-ropericlase (Mg − x Fe x O, for x = 0 .
06 and x = 0 .
27) have been measured over a wide frequencyrange ( ≈
50 to 32 000 cm − ) at 295 and 6 K. The complex dielectric function has been determinedfrom a Kramers-Kronig analysis of the reflectance. The spectra of the doped materials resemblespure MgO in the infrared region, but with much broader resonances. We use a shell model tocalculate the dielectric function of ferropericlase, including both anharmonic phonon-phonon in-teractions and disorder scattering. These data are relevant to understanding the heat conductivityof ferropericlase in the earth’s lower mantle. PACS numbers: 63.20.Kr, 63.20.Mt, 74.25.Kc, 78.30.-j
Typeset by REVTEX 1 . INTRODUCTION
Ferropericlase, (Mg − x Fe x O, with x = 0 . . The name ‘magnesiow¨ustite’is also used, but properly refers to the doping region x close to the w¨ustite (x=1), ratherthan the periclase (x=0) limit. Transport properties of (Mg,Fe)O are therefore importantin modeling the Earth’s thermal state and evolution, where both conduction and convectionare operative.
The two heat carriers in conduction for an insulating mineral like ferroper-iclase are phonons and photons. Phonons (extended or localized) are distributed in the farand mid-infrared frequency range. They can be scattered by various defects (e.g. impuri-ties, grain boundaries, . . . ) and by the intrinsic anharmonic phonon-phonon interactions.Photons are described by Planck’s black-body radiation formula, reaching energies ∼ et al. mea-sured the optical absorption spectra of Mg − x Fe x O (with x=0.06, 0.15, and 0.25) acrossthe high-spin/low-spin transition, which occurs over a pressure range of 40-60 GPa atroom temperature. Their results indicate that low-spin (Mg,Fe)O will exhibit lower (ratherthan higher ) radiative heat conductivity than high-spin phase due to the red-shift of thecharge-transfer edge. The origin of this spin transition and its influence on the radiativeheat conductivity of ferropericlase are further investigated recently. A complete picture ofthe thermal conductivity must include contributions from both phonons and photons.As a solid solution, ferropericlase has a vibrational frequency spectrum similar to that ofpure MgO. However, with strong disorder scattering of propagating vibrational states, theharmonic eigenstates of the disordered crystal do not necessarily have a well defined wavenumber, and may not propagate ballistically. In addition, the anharmonic phonon-phononinteractions causes a shoulder at ∼
640 cm − in the infrared (IR) reflectance spectrum of pureMgO. Thus anharmonicity should also be included in analyzing the infrared reflectance offerropericlase.In this paper we report the temperature-dependent infrared reflectance measurements ofmagnesium oxide and ferropericlase for several Fe concentrations at ambient pressure. We2onstruct a model in which anharmonic phonon-phonon interactions and disorder scatteringare treated separately. Their effects are then combined for comparison with the experimentaldata.
II. EXPERIMENTAL MEASUREMENTS
The samples we examined are homogeneously doped single crystals, in whichFe / P Fe ≈ .
02 for the 6% sample and 0.05 for the 27% sample. However, in our analysisthe influence of Fe and magnesium vacancies is ignored. A detailed description on thesamples’ synthesis, crystallography and elastic properties is in Ref. 11. The samples arerectangular slabs with typical dimensions of 1 mm × ≈ . ≃ ◦ . The reflectance spectra has been measured at a near-normal angleof incidence at 295 and 6 K over a wide frequency range from ≈
50 to about 32 000 cm − on Bruker IFS 66v/S and 113v spectrometers using an in-situ evaporation technique. Themeasured reflectance at 295 and 6 K of pure MgO, and Mg − x Fe x O, for x = 0 .
06 and x = 0 .
27 are shown in Figs. 1(a), (b) and (c), respectively. Although wedging the sampleshas been very effective at reducing interference effects, weak fringes may still be detectedat low temperature below about 150 cm − . The complex dielectric function ǫ = ǫ + iǫ has been determined from a Kramers-Kronig analysis of the reflectance, where extrapola-tions are supplied for ω → , ∞ . At low frequency, an insulating response is assumed and R ( ω → ≃ .
27, 0.28 and 0.31 for MgO, and the 6% and 27% Fe-doped materials, re-spectively. Above the highest measured frequency the reflectance has been assumed to beconstant to approximately 75 000 cm − , above which a free-electron approximation has beenassumed ( R ∝ /ω ). The imaginary part of the resulting dielectric function at 6 and 295 Kof pure MgO, and Mg − x Fe x O, for x = 0 .
06 and x = 0 .
27, are shown in Figs. 1(d), (e) and(f), respectively. The imaginary part of the dielectric function contains most of the physicalinformation, and is the focus of our theoretical analysis. The optical features have been fitto a classical oscillator model using the complex dielectric function ǫ ( ω ) = ǫ ∞ + X j ω p,j ω ,j − ω − i ωγ j , (1)3 R e f l e c t an c e (a)10 -1
0 200 400 600 800 ε (d) (b) 0 200 400 600 800Frequency (cm -1 )(e) (c) 0 200 400 600 800 1000(f) FIG. 1: The measured reflectance R ( ω ); (a) pure MgO, and Mg − x Fe x O for (b) 6% and (c) 27%Fe-doping. The corresponding imaginary part of the dielectric functions ǫ ( ω ); (d) pure MgO, andMg − x Fe x O for (e) 6% and (f) 27% Fe-doping. The solid line corresponds to data measured at6 K, dashed line corresponds to data at 295 K. where ǫ ∞ is a high-frequency contribution, and ω TO ,j , 2 γ j and ω p,j are the frequency, fullwidth and effective plasma frequency of the j th vibration. The results of non-linear least-squares fits to the reflectance and ǫ ( ω ) are shown in Table I. In addition to the strongfeature in ǫ ( ω ) seen at about 400 cm − , other features at ≈
520 and ≈
640 cm − are alsoclearly visible in ǫ ( ω ) shown in Fig. 1; however, these features are very weak and as a resultthe the strengths and widths of these modes are difficult to determine reliably.4 ABLE I: A comparison of the fitted values of the static and high-frequency contributions to thereal part of the dielectric function at room temperature, as well as the fitted frequency, full widthand effective plasma frequency ( ω TO , 2 γ and ω p , respectively) of the feature associated with thestrong TO mode in MgO, and the 6% and 27% Fe-doped materials at 295 and 6 K. The units of ω TO , 2 γ and ω p are in cm − . The strength of the TO mode is also expressed as a dimensionlessoscillator strength S = ω p /ω . 295 K 6 KMg − x Fe x O ǫ a ǫ a ∞ ω b TO γ c ω dp ( S ) ω b TO γ c ω dp ( S )pure 9.2 2.95 396.5 3.44 1010 (6.5) 398.9 1.72 1030 (6.7) x = 0 .
06 10.8 3.10 395.6 30.5 1090 (7.6) 396.7 29.1 1120 (8.0) x = 0 .
27 11.8 3.65 384.5 28.6 1100 (8.2) 388.6 25.7 1140 (8.6) a Values at 295 K, the estimated uncertainty is about ± . b The uncertainty in ω TO is ± . − . c The uncertainties for 2 γ are ± . − in the pure material, and ± . − in the Fe-doped materials. d The uncertainty in ω p is ±
20 cm − . III. COMPUTATIONAL METHODSA. General Scheme
Infrared dielectric properties of ionic crystals are contained in the linear response function ǫ αβ ( ω ) = ǫ αβ ( ∞ )+4 πχ αβ ( ω ). Considering only the first-order moment of the electric dipole,the dielectric susceptibility of a crystal can be related to its displacement-displacementretarded Green’s function by: χ αβ ( ω ) = − N V c Z + ∞−∞ θ ( t − t ′ ) i ~ h [ D α ( t ) , D β ( t ′ )] i e iω ( t − t ′ ) d ( t − t ′ )= − N V c X lsγ X l ′ s ′ δ Z αγ ( ls ) Z βδ ( l ′ s ′ ) Z + ∞−∞ θ ( t − t ′ ) i ~ h [ u γ ( ls ; t ) , u δ ( l ′ s ′ ; t ′ )] i e iω ( t − t ′ ) d ( t − t ′ )= − N V c X lsγ X l ′ s ′ δ Z αγ ( ls ) Z βδ ( l ′ s ′ ) G γδ ( ls, l ′ s ′ ; ω ) , (2)where D α ( t ) = X lsβ Z αβ ( ls ) u β ( ls ; t ) is the α component of the first order electric dipole ofthe whole crystal, Z αβ ( ls ) is the Born effective charge tensor of the atom s at site l , and5 β ( ls ; t ) is the atom’s displacement at time t . The volume of a single cell is V c , and N is thenumber of the cells in the whole crystal. The Green’s function G αβ ( ls, l ′ s ′ ; t − t ′ ) is definedas: G αβ ( ls, l ′ s ′ ; t − t ′ ) = θ ( t − t ′ ) i ~ h [ u α ( ls ; t ) , u β ( l ′ s ′ ; t ′ )] i , (3)which can be evaluated from its equation of motion. For a harmonic crystal, the vibrationalHamiltonian is quadratic and can be solved exactly. We denote the eigenvectors of a purecrystal as √ N ˆ e α ( s | q j ) e i q · R ( ls ) , the corresponding eigenvalues as ω q j , those of a disorderedcrystal as e α ( s | j ) and ω j , the Green’s function of the pure as g , the disordered as G . Then g αβ ( ls, l ′ s ′ ; ω ) = X q j ˆ e α ( s | q j )ˆ e ∗ β ( s ′ | q j ) e i q · ( R ( ls ) − R ( l ′ s ′ )) N p M ( s ) M ( s ′ )( ω − ω q j + i ωη ) , (4) G αβ ( ls, l ′ s ′ ; ω ) = X j e α ( ls | j ) e ∗ β ( l ′ s ′ | j ) p M ( ls ) M ( l ′ s ′ )( ω − ω j + i ωη ) , (5)where the mass of the atom s is denoted as M ( s ) in the pure crystal, M ( ls ) in the disorderedcrystal, with the extra label l to specify its site, η is an infinitesimal number ensuringcausality.Anharmonic interaction will couple these modes and make exact solution impossible. Thestandard treatment of this many-body effect uses the Dyson equation to define a self-energyfor each mode. We can either choose e α ( s | j ) as the unperturbed states, then the only inter-action will be anharmonicity, or choose √ N ˆ e α ( s | q j ) e i q · R ( ls ) as the basis and treat disorderas an extra perturbation. The first approach has been used by one of the authors (PBA) tostudy the anharmonic decay of vibrational states in amorphous silicon. In this paper weuse a hybrid approach. We write the dielectric function of a disordered anharmonic crystalin the perfect crystal harmonic basis as ǫ αβ ( ω ) = ǫ αβ ( ∞ ) + 4 πχ αβ ( ω )= ǫ αβ ( ∞ ) + 4 πV c TO X j =1 X sγ Z αγ ( s ) ˆ e γ ( s | j ) p M ( s ) X s ′ δ Z βδ ( s ′ ) ˆ e ∗ δ ( s ′ | j ) p M ( s ′ ) { ω (0 j ) − ω + 2 ω (0 j )(∆(0 j, ω ) − i Γ(0 j, ω ) } , (6)where ω (0 j ) ≡ ω TO ,j is the frequency at q = 0 of the j th TO branch. The terms ∆(0 j, ω )and Γ(0 j, ω ) correspond to the real and imaginary part of the mode’s self-energy Σ(0 j, ω ).Then we split this self-energy into two parts: Σ = Σ anharm + Σ disorder . Each piece is cal-culated independently. This is equivalent to omitting all the diagrams where the disorder6cattering vertex appears inside an anharmonic interaction loop. The anharmonic interac-tion of ferropericlase is assumed to be the same as that of pure MgO, i.e. the influence ofdisorder on anharmonic coefficients is totally ignored. Disorder is treated by exact diago-nalization without anharmonicity, it is then converted to a self-energy of the TO mode inthe perfect crystal harmonic basis. These approximations are tested by comparing with theexperimental results. B. Shell Model
The scheme described above is general. It does not depend on which microscopic modelis chosen to get harmonic phonons, disorder scattering strength, and higher-order force con-stants. Here we use an anharmonic shell model, with shell parameters fitted to experiments.The harmonic phonon properties in this paper are calculated with the general utility latticeprogram (GULP) code. Two sets of shell parameters are used for MgO: S-I and B, and one for FeO: S-II.S-I and S-II are rigid shell models in which O − has the same set of parameters, thus theycan be conveniently used to simulate ferropericlase. B is an isotropic breathing shell modelwhich gives better fit to the experimental data. However, it can not be directly used forferropericlase. For FeO the elastic constants C > C , while the isotropic breathing shellmodel is only suitable for cases where C < C . We treat B as a reference to checkour anharmonic calculations based on S-I. All the model parameters are listed in Table II.Table III contains the calculated physical properties and corresponding experimental values.Phonon dispersion curves for the pure crystals of MgO and FeO are shown in Fig. 2.
C. Anharmonicity
A complete calculation of anharmonicity is tedious, even for a pure crystal. Thus weignore the less important terms and focus on the dominant one. From Eq. (6) it is clearthat since | Σ | = | ∆ − i Γ | is small compared to ω TO , the real part of the self-energy ∆ hasnegligible influence on ǫ ( ω ), except to shift its resonant frequency. The shell models we useare fitted to the experimental data measured at room temperature. The anharmonic shift issmall, compared with the shift caused by impurity scattering. Thus, we ignore it completely7 ABLE II: Shell model parameters used in the calculation.
The short-range repulsive potentialis assumed to be a two-body Buckingham type: for S-I and S-II, V ( r ) = A exp( − r/ρ ) − C/r ; forB, V ( r ) = A exp( − ( r − r ) /ρ ) − C/r . The parameter k represents the spring constant betweencore and shell. Rows in which atomic symbols have a star (*) are for the B model. The label‘shell’ denotes a potential that acts on the central position of the shell, while ‘bshell’ denotes aninteraction that acts on the radius of the shell which was fixed at 1.2 ˚A. An extra parameter in Bmodel is k BSM = 351 .
439 eV˚A − . The equilibrium shell radius r is 1.1315 ˚A after optimization.Z core (e) Z shell (e) k (eV)O 0 . − . . − − Fe − . . . ∗ . − . . ∗ − − A (eV) ρ (˚A) C (eV · ˚A )O shell-O shell 22764 . .
149 20 . . . . . . . ∗ shell-O ∗ shell 0.0 0.3 54.038Mg ∗ core-O ∗ bshell 28.7374 0.3092 0.0TABLE III: Physical properties of pure MgO and FeO, compared with shell model results.a (˚A) C (GPa) C (GPa) C (GPa) ǫ ǫ ∞ TO (cm − )MgO (exp ) 4.212 297.0 95.2 155.7 9.86 2.96 401S-I 4.225 370.9 163.0 163.0 9.88 2.94 399B 4.212 297.0 95.0 155.7 9.89 2.94 392FeO (exp ) 4.310 359 156 56 14.2 5.4 320S-II 4.324 327 149 149 14.18 5.34 327 F r equen cy ( c m - ) Γ Χ [00 ξ ] ∆ Γ (a) [ ξξ Σ
0 0.5 L [ ξξξ ] Λ F r equen cy ( c m - ) Γ Χ [00 ξ ] ∆ Γ (b) [ ξξ Σ
0 0.5 L [ ξξξ ] Λ FIG. 2: Phonon dispersions of the pure crystals. (a) MgO, solid line corresponds to the rigid-shellmodel S-I, dashed line to the isotropic breathing-shell model B, dots are the experimental datataken from Ref. 19; (b) FeO, solid line corresponds to the rigid-shell model S-II, dots are theexperimental data taken from Ref. 20. and only consider the imaginary part of the self energy Γ anharm (0 j, ω ). To the lowest orderΓ anharm (0 j, ω ) can be written as Γ anharm (0 j, ω ) = 18 π ~ X q j j (cid:12)(cid:12)(cid:12)(cid:12) V (cid:18) q − q j j j (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) { ( n + n + 1)[ δ ( ω + ω − ω ) − δ ( ω + ω + ω )]+( n − n )[ δ ( ω − ω + ω ) − δ ( ω − ω + ω )] } , (7)where n = n ( q j ) is the Bose-Einsein population factor of the mode, and ω = ω ( q j ) isthe corresponding frequency. The anharmonic coefficient V (cid:0) q − q j j j (cid:1) is V (cid:18) q − q j j j (cid:19) = 13! X αβγ X s s s X l l Φ αβγ (cid:18) l l s s s (cid:19) ˆ e α ( s | j )ˆ e β ( s | q j )ˆ e γ ( s | − q j ) (cid:18) ~ ω (0 j ) ω ( q j ) ω ( − q j ) M ( s ) M ( s ) M ( s ) (cid:19) exp { i q · [ R ( l s ) − R ( l s )] } . (8)The third-order force constants Φ αβγ (cid:0) l l s s s (cid:1) are large only for the nearest neighbors. Sym-metry will restrict most of them to be zero, and among those nonzero terms only two areindependent. The general formula for third order force constants is Φ αβγ (cid:18) l ′ s s s ′ (cid:19) = BR α R β R γ + C ( R α δ βγ + R β δ αγ + R γ δ αβ ) ,B = φ ′′′ R − φ ′′ R + 3 φ ′ R ,C = φ ′′ R − φ ′ R , (9)9here R is the lattice distance between the ion (cid:0) s (cid:1) and (cid:0) l ′ s ′ (cid:1) , and R α is its projection along α direction. The term φ ( r ) is the two-body pair potential, and φ ′ , φ ′′ . . . are derivatives withrespect to r . Following E. R. Cowley, we compute Φ αβγ (cid:0) l ′ s s s ′ (cid:1) by direct differentiation overthe nearest-neighbour short-range potentials and Coulomb potentials. For the rigid-shellmodel S-I, φ ( r ) = A exp( − r/ρ ) − e r . For the breathing-shell model B, φ ( r ) = A exp( − ( r − r ) /ρ ) − e r . To be more specific, if we take a Mg as the origin and denote it as 1, itsnearest neighbor O − along the [100] direction as 2, then from symmetry we can determineΦ xxx (112) = Φ xxx (121) = − Φ xxx (122) = Φ yyy (112) = · · · , Φ xyy (112) = Φ xzz (112) = · · · .Putting in numbers from Table II we obtain Φ xxx (112) = − .
34 eV/˚A , Φ xyy (112) = − .
79 eV/˚A for the S-I model, and Φ xxx (112) = − . , Φ xyy (112) = − .
78 eV/˚A forthe B model. If we do not include the Coulomb interaction, these values will be Φ xxx (112) = − .
70 eV/˚A Φ xyy (112) = 6 .
88 eV/˚A for the S-I model, and Φ xxx (112) = − .
57 eV/˚A ,Φ xyy (112) = 7 eV/˚A for the B model. Although Φ xyy (112) is small compared with Φ xxx (112),it can still have non-negligible influence on the amplitude of the Γ(0 j, ω ) near 640 cm − .Other parameters (Born effective charge tensors, harmonic eigenvectors) are obtained fromGULP. The integration over q-space is done with the tetrahedron method, using 1/48 of theBrillouin zone, and averaging over x, y, and z polarizations. We use 3345 q-points, equivalentto 160560 q-points in the whole Brillouin zone. D. Disorder Scattering
The self-energy of a vibrational mode caused by disorder scattering is definedstatistically, hh G ii = g + gΣ hh G ii , where hh G ii denotes the Green’s function averaged over different impurity distributions.We slightly modify this definition by including the Born effective charge. From Eqs. (4) and(5), we define the following equation hh X γ X δ Z αγ ( ls ) G γδ ( ls, l ′ s ′ ; ω ) Z δβ ( l ′ s ′ ) ii = X γ X δ Z αγ ( s )˜ g γδ ( ls, l ′ s ′ ; ω ) Z δβ ( s ′ ) , (10)where ˜ g γδ ( ls, l ′ s ′ ; ω ) = X q j ˆ e γ ( s | q j )ˆ e ∗ δ ( s ′ | q j ) e i q · ( R ( ls ) − R ( l ′ s ′ )) N p M ( s ) M ( s ′ )( ω − ω q j − i ω q j Σ disorder ( q j, ω )) . (11)10he self-energy defined in this way guarantees that the dielectric susceptibilities calcu-lated from G and ˜g are the same. Summing over all sites of the crystal leaves only TOmodes on the right hand side of Eq. (10). Thus, once we get the averaged dielectric sus-ceptibility hh χ αβ ii from the exact eigenvectors of the disordered crystal, we can extract theself-energy of its TO phonon.We expand an orthogonal 8-atom MgO unit cell in each direction by 5 times, which givesa 5 × × by Fe . The shell parameters of Mg are from S-I model, those of Fe are from S-II model, those of O − are the same in both models. From Eqs. (2) and (5), foreach configuration we have a harmonic susceptibility χ αβ ( ω ) = 1 N V c modes X j =1 X lsγ Z αγ ( ls ) e γ ( ls | j ) p M ( ls ) X l ′ s ′ δ Z βδ ( l ′ s ′ ) e ∗ δ ( l ′ s ′ | j ) p M ( l ′ s ′ ) ω j − ω − i ωη . (12)We can choose a small value for η and evaluate Eq. (12) directly (Lorentzian broadening).However, insofar as η is finite, it is equivalent to have each mode j in Eq. (12) an imaginaryself-energy (life time) linear in frequency ω . The self-energy of the TO phonon Σ disorder extracted from this approach will depend on frequency linearly. Replacing the factor 2 ω by2 ω j won’t help either, as each mode j now has a life time independent of frequency, andΣ disorder will be a constant depending on η when ω →
0. To avoid such artifacts we use1 ω j − ω − i ωη = 1 ω j − ω + iπ ω [ δ ( ω − ω j ) + δ ( ω + ω j )]to separate the real ( χ ) and imaginary part ( χ ) of the dielectric susceptibility. Thenwe divide the vibrational spectrum into equally sized bins (1 cm − ) and compute χ as ahistogram. The real part χ is obtained from χ from the Kramers-Kronig relation. Manysuch super-cells are built and their ǫ ∞ and χ calculated. We find that 10 configurationsare sufficient to give a well converged average. The final ǫ ∞ and χ are assumed to be theaveraged values of all configurations. To remove the unphysical spikes caused by the finitesize of our super-cells, while keeping the main features unchanged, we further smooth thedielectric susceptibility by averaging over adjacent bins iteratively, χ n +12 ( j ) = 16 [ χ n ( j −
1) + 4 χ n ( j ) + χ n ( j + 1)] . (13)11 -2 -1
200 300 400 500 600 700 800 ε Frequency (cm -1 )(a) S-IBExp Γ ( j , ω )( c m - ) Frequency (cm -1 )(b) S-IBExp
FIG. 3: Computed anharmonic properties compared with experimental data for pure MgO. (a) Theimaginary part of dielectric function at 295 K; the experimental data are the same as those inFig. 1(d). (b) The imaginary part of self energy at 295 K; the experimental data are digitized fromRef. 25, which are fit to infared spectra based on a semi-quantum dielectric function model.
In this way we successfully simulate the dielectric function of a ‘real’ crystal (real in the sensethat except for finite size, disorder scattering is treated without any further approximations).These results, together with anharmonicity, are summarized in the next section.
IV. COMPARISONS AND DISCUSSION
The anharmonic effects in pure MgO are shown in Fig. 3. The computational results andexperimental values are quite close, especially near 640 cm − which corresponds to TO+TAcombination mode. It is not surprising, since the procedure we followed was originallydeveloped and worked well for alkali-halide salts, which are similar to MgO. Below 800 cm − ,the rigid shell model S-I and breathing shell model B give almost identical self-energies. Thediscrepancy in the high-frequency range indicates that the dispersion relations from empiricalmodels are less accurate for high-frequency optical branches. The width of TO mode at thereststrahlen frequency ω TO is less accurate, as Γ anharm (0 j, ω TO ) is intrinsically small andhigher-order anharmonic effects become important. Figures 4 and 5 show how anharmonicity and disorder scattering influence the dielectricfunction. For the 6% sample it is clear that the shoulder near 640 cm − is caused byanharmonicity, while the shoulder at about 520 cm − is due to disorder scattering. Disorderscattering becomes stronger for the 27% sample and seems contributes to all the shoulders.12 -2 -1
100 300 500 700 900 ε Frequency (cm -1 )(a) Γ ( j , ω )( c m - ) Frequency (cm -1 ) (b) FIG. 4: The anharmonic and disordering scattering effects in Mg − x Fe x O for the 6% Fe-dopedsample. (a) Imaginary part of the dielectric function. The labels ‘6 K Calc’ and ‘295 K Calc’denote the calculated curves, including both disorder scattering and anharmonic interactions atthe corresponding temperature. Experimental data are the same as those in Fig. 1(e). The label‘No anharm’ denotes the dielectric function calculated from disorder scattering only. (b) Imaginarypart of self energy. The labels ‘6 K’ and ‘295 K’ denote the self-energies caused by anharmonicinteraction at the corresponding temperature; ‘disorder’ denotes the self-energy due to disorderscattering, which is computed by histogram method where the bin size equals 1 cm − , then itera-tively averaged 30 times. The total self-energies are the sum of these two pieces, and are used incalculating the ‘6 K Calc’ and ‘295 K Calc’ dielectric functions shown in (a). The shoulder caused by anharmonicity corresponds to a peak in the two-phonon DOS, whileshoulders caused by disorder scattering are related to peaks in the one-phonon DOS.Figure 6 contains the reflectance computed from the dielectric functions at 295 and 6 Kshown in Figs. 4 and 5. As in the case of pure MgO, the agreement between theory andexperiment is better in the region where the self-energy caused by lowest-order pertubationis large. Near the reststrahlen frequency ω TO , the self-energy is smaller, and R ( ω ) is moresensitive to details. Our model underestimates the broadening of the resonance, but correctlyidentifies the sources of broadening.It is of interest to determine whether the disorder scattering is mainly due to the differ-ences in mass or in the inter-atomic potential. Thus we repeat the above procedure witha model which only contains mass disorder, i.e. Fe is treated as an isotope of Mg, its shellparameters are the same as Mg in S-I model. It turns out the most significant factor13 -2 -1
100 300 500 700 900 ε Frequency (cm -1 )(a) Γ ( j , ω )( c m - ) Frequency (cm -1 ) (b) FIG. 5: The anharmonic and disordering scattering effects in Mg − x Fe x O for the 27% Fe-dopedsample. (a) Imaginary part of the dielectric function; (b) Imaginary part of self energy. Thecomputation procedure is the same as for the 6% Fe doping. R e f l e c t an c e Frequency (cm -1 ) (a)
6K Calc295K Calc6K Exp295K Exp R e f l e c t an c e Frequency (cm -1 ) (b)
6K Calc295K Calc6K Exp295K Exp
FIG. 6: The calculated infrared reflectance, compared with the experimental data (same as inFig. 1(b) and (c)) for Mg − x Fe x O. (a) 6% Fe doping; (b) 27% Fe doping. is ǫ ∞ : For the isotope model (S-I) ǫ ∞ is the same as pure MgO (2.94), for S-II model ǫ ∞ increases to 3.05 for 6% Fe and 3.47 for 27%, in reasonable agreement with the results shownin Table I. Thus the LO frequency predicted from the isotope model is larger than the ex-perimental value. The differences in the inter-atomic potentials change the relative strengthof the self-energy, but in both cases the self-energy spectra carry features of the one phononDOS of pure MgO.In addition to phonons, electronic transitions may also influence the infrared dielectricproperties of ferropericlase. Wong measured the far-infrared absorption spectra of iron-doped MgO. A line at 105 cm − was observed with a peak absorption coefficient of 1.5 cm − ≃ − at 20 K in a sample with 0.2% Fe. This feature is attributed to thetransition Γ → Γ , Γ of MgO: Fe at cubic sites. If we assume the absorption coefficientis proportional to the impurity concentration, then we can estimate the corresponding ǫ at 105 cm − by ǫ ( ω ) = nα ( ω )2 πω , where n is the refractive index (for pure MgO, n ≃ . − ), α ( ω ) is the absorption coefficient at frequency ω (in units of cm − ). The value of ǫ is about 0.22 for 6% Fe concentration, 0.98 for 27%. As the iron concentration increases,the electronic transitions of Fe should show greater influence on the far-infrared spectraof ferropericlase. In our measurement the spectra below 200 cm − are complicated due tothe presence of fringes, consequently we can not confirm this tendency. Henning et al. measured the infrared reflectance of Fe x Mg − x O for x = 1 .
0, 0.9, 0.8, 0.7, 0.5 and 0.4 atroom temperature. The ǫ curves reported in their paper do not show a monotonic rise inthe far-infrared region as the iron concentration x increases from 0.4 to 1.0, while they areall in the range of 6-10 near 100 cm − . It is difficult to explain such large ǫ with latticevibrations alone, and the accuracy of these data has been questioned. Further experimentsare needed to clarify this issue.
V. CONCLUSIONS
The infrared reflectance spectra of magnesium oxide and ferropericlase has been measuredat 295 and 6 K. It is found that ǫ ∞ increases as Fe concentration increases, while the widthof the TO modes remains the same in the doped materials. We construct a theoretical modelwhich includes both disorder scattering and anharmonic phonon-phonon interactions. Themodel shows fairly good agreement with the experiment in the regions where the lowest-order perturbation is relatively large. Near the resonance, theory and experiment both havesmaller self-energies, which makes the reflectance quite sensitive to the details. We do notknow whether the disagreements with experiment in the region are caused by neglect ofhigher order corrections, or by inaccuracy of the underlying model. However, the modelidentifies the global features reasonably well, and may provide a good basis for the study ofphonon decay needed for a theory of heat conductivity.15 cknowledgments This work was supported by the Office of Science, U.S. Department of Energy, underContract No. DE-AC02-98CH10886. SDJ is supported by NSF EAR-0721449. ∗ Electronic address: [email protected] T. Yagi and N. Funamori, Phil. Trans. , 1711 (1996). A. M. Hofmeister, Science , 1699 (1999). A. M. Hofmeister, J. Geodyn. , 51 (2005). A. F. Goncharov, V. V. Struzhkin, and S. D. Jacobsen, Science , 1205 (2006). D. M. Sherman, J. Geophys. Res. , 14299 (1991). J. Badro, G. Fiquet, F. Guyot, J.-P. Rueff, V. V. Struzhkin, G. Vank´o, and G. Monaco, Science , 789 (2003). J.-F. Lin, V. V. Struzhkin, S. D. Jacobsen, M. Y. Hu, P. Chow, J. Kung, H. Liu, H.-K. Mao,and R. J. Hemley, Nature , 377 (2005). R. G. Burns,
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