Effect of vibrations on the pre-edge features of x-ray absorption spectra
Christian Brouder, Delphine Cabaret, Amélie Juhin, Philippe Sainctavit
aa r X i v : . [ c ond - m a t . m t r l - s c i ] D ec Effect of vibrations on the pre-edge features of x-ray absorption spectra
Christian Brouder, Delphine Cabaret, Am´elie Juhin and Philippe Sainctavit Institut de Min´eralogie et de Physique des Milieux Condens´es, CNRS UMR 7590,Universit´es Paris 6 et 7, IPGP, 140 rue de Lourmel, 75015 Paris, France. (Dated: November 2, 2018)The influence of atomic vibrations on x-ray absorption near edge structure (XANES) is calculatedby assuming that vibrational energies are small with respect to the instrumental resolution. Theresulting expression shows that, at the K -edge, vibrations enable electric dipole transitions to 3 s and 3 d final states. The theory is applied to the K -edge of Al in α -Al O and of Ti in TiO rutileand compared with experiment. At the Al K -edge, sizeable transitions towards 3 s final states areobtained, leading to a clear improvement of the agreement with experimental spectra. At the Ti K -edge, electric dipole transitions towards 3 d final states explain the temperature dependence ofthe pre-edge features. PACS numbers: 78.70.Dm, 65.40.-b
Vibronic coupling describes the interaction betweenelectrons and atomic motions. It plays a prominent rolein optical spectroscopy where it is the source of the colorof many pigments and gemstones [1]. For instance, thered color of Black Prince’s ruby is due to “d-d” tran-sitions of chromium impurities in a spinel crystal. Butthese transitions are forbidden because chromium occu-pies an inversion center of the spinel lattice. They be-come allowed when vibrations break inversion symmetry.In the x-ray range, vibrations far from the edgeare taken into account through a Debye-Waller factore − k σ [2]. If the validity of this factor is assumed toextend to the near-edge region, where k ≃
0, then vibra-tions seem negligible in XANES spectra.However, three arguments indicate that vibronic cou-pling can be sizeable in the XANES region: (i) Vibroniccoupling was detected by x-ray resonant scattering exper-iments at the Ge K -edge [3, 4]; (ii) Some XANES peaksseem to be due to forbidden transitions to 3 s states, aprominent example being the Al K -edge in minerals [5];(iii) A temperature dependence of the pre-edge structurewas observed at the Ti K -edge in SrTiO [6] and TiO [7].In the optical range, the effect of vibrations is usuallytaken into account through the Franck-Condon factors.In the x-ray range, Fujikawa and coll. showed in a seriesof papers of increasing sophistication [8, 9, 10] that theeffect of the Franck-Condon factors can be representedby the convolution of the “phonon-less” x-ray absorp-tion spectrum with the phonon spectral function. Sucha convolution leads to a broadening of the peaks with in-creasing temperature but this effect is hardly observablein the pre-edge region.Moreover, in the x-ray range it was shown that thelarge core-electron-phonon coupling of the 1 s core holeof carbon in diamond induces a strong lattice distortionand significant anharmonic contributions [11].The key observation is that all these effects can be eas-ily taken into account if the vibrational energies are smallwith respect to the XANES spectral resolution (core hole lifetime + instrumental resolution). This condition is cer-tainly not satisfied at the C K -edge [11] but it becomesreasonable at the Al and Ti K -edges. In that range theXANES resolution is around one eV whereas the energyof vibrational modes is of the order of a few hundredthsof eV although, of course, several phonons can be simul-taneously present.In this paper, we first use this observation to derivea manageable expression for the vibronically-coupled x-ray absorption spectra. Then, we apply the so-called crude Born-Oppenheimer approximation to further sim-plify this expression, so that only the core-hole motionremains. The resulting equation is compared with ex-periment in two different cases. At the Al K -edge in α -Al O (corundum), vibrations induce transitions to 3 s fi-nal states. These (1 s → s ) monopole transitions explaina pre-edge peak that is completely absent from standardcalculations. At the Ti K -edge in TiO , 1 s → d transi-tions are induced by vibrations. This explains why onlythe first two pre-edge peaks grow with temperature. XANES formula within the Born-Oppenheimer frame-work.
According to the Born-Oppenheimer approxima-tion [12, 13], the wavefunction of the system of electronsand nuclei can be written as the product χ jn ( Q ) ψ n ( r ; Q ),where r is the electronic variable and Q = ( Q , . . . , Q N )collectively denotes the position vectors of the N nuclei ofthe system. The electronic wavefunction ψ n ( r ; Q ), withenergy ǫ n ( Q ), describes the state of the electrons in apotential where the nuclei are fixed at position Q . Theground state corresponds to n = 0. The origin of the nu-clear variables is chosen so that Q = is the equilibriumposition, i.e. ǫ ( ) is the minimum of ǫ ( Q ).For each n , the vibrational wavefunctions χ jn ( Q ) arethe orthonormal solutions of the Schr¨odinger equation (cid:0) H kin ( Q ) + ǫ n ( Q ) (cid:1) χ jn ( Q ) = E jn χ jn ( Q ) . The total energy of the electrons + nuclei system is E jn .Within the Born-Oppenheimer approximation, the x-rayabsorption cross-section is σ ( ω ) = 4 π α ~ ω X fj (cid:12)(cid:12)(cid:12) Z d Q d r χ jf ( Q ) ∗ ψ f ( r , Q ) ∗ ε · r × χ ( Q ) ψ ( r , Q ) (cid:12)(cid:12)(cid:12) δ ( E jf − E − ~ ω ) , where α is the fine structure constant, ~ ω the energy ofthe incident x-rays and ε their polarization vector. Thecore-hole lifetime and the instrumental resolution can berepresented by the convolution of the absorption cross-section with a Lorentzian function (Γ /π ) / ( ω +Γ ). Thisgives us σ γ ( ω ) = 4 πα X fj (cid:12)(cid:12)(cid:12) Z d Q d r χ jf ( Q ) ∗ ψ f ( r , Q ) ∗ ε · r × χ ( Q ) ψ ( r , Q ) (cid:12)(cid:12)(cid:12) ( E jf − E ) γ ( E jf − E − ~ ω ) + γ , where γ = ~ Γ.The energy E jf can be written as the sum of the elec-tronic energy at equilibrium position ǫ f and a vibrationalenergy E jf = ǫ f + E fj vib . When γ is much larger thanthe vibrational energy we can neglect the contributionof E fj vib and sum over the vibrational states χ jf ( Q ). Thecompleteness relation gives us X j χ jf ( Q ) ∗ χ jf ( Q ′ ) = δ ( Q − Q ′ ) . Therefore, σ γ ( ω ) = 4 πα Z d Q X f (cid:12)(cid:12)(cid:12) Z d r ψ f ( r , Q ) ∗ ε · r × χ ( Q ) ψ ( r , Q ) (cid:12)(cid:12)(cid:12) ( ǫ f − ǫ ) γ ( ǫ f − ǫ − ~ ω ) + γ . (1)Note that we derived this result without making the har-monic approximation. Therefore, the possible anhar-monic behavior due to the core hole [11] is taken intoaccount. Note also that the final and initial energies ǫ f and ǫ do not depend on Q . Therefore, eq. (1) is notthe average of standard XANES spectra over various nu-clear positions Q . In other words, we have here a way todistinguish thermal disorder from static disorder due toimpurities and structural defects.Now we make a different approximation for the initialand final electronic states. For a K -edge, the 1 s core levelwavefunction is highly localized around the nucleus andit weakly depends on the surrounding atoms. Therefore,we can approximate ψ ( r , Q ) by φ ( r − Q a ), where Q a isthe position vector of the absorbing atom and where φ is the 1 s wavefunction of the absorbing atom at equilib-rium position. For the final electronic states ψ f ( r , Q ) inthe presence of a core hole, the variation of the nuclear coordinates Q in eq. (1) is ruled by the vibrational wave-function χ of the initial state, which is expected to berather smooth. Therefore, we make the standard crude Born-Oppenheimer approximation, according to whichthe electronic wavefunction does not significantly varywith Q for small vibrational motions. In other words, ψ f ( r , Q ) ≃ φ f ( r ), where φ f ( r ) = ψ f ( r , ). This gives us σ γ ( ω ) = 4 πα ~ ω Z d Q | χ ( Q ) | X f (cid:12)(cid:12)(cid:12) Z d r ψ f ( r ) ∗ ε · r × φ ( r − Q a ) (cid:12)(cid:12)(cid:12) ~ γ ( ǫ f − ǫ − ~ ω ) + γ . When the crude Born-Oppenheimer approximation is notvalid, it is possible to Taylor-expand ψ f ( r , Q ) as a func-tion of Q [14]. The integral over electronic variables de-pends only on the position of the absorbing atom, fromnow on denoted by R . Therefore, we can integrate overthe other nuclear variables and the expression becomes σ γ ( ω ) = 4 πα ~ ω Z d R ρ ( R ) X f (cid:12)(cid:12)(cid:12) Z d r ψ f ( r ) ∗ ε · r × φ ( r − R ) (cid:12)(cid:12)(cid:12) ( ǫ f − ǫ ) γ ( ǫ f − ǫ − ~ ω ) + γ , (2)where ρ ( R ) = R d Q | χ ( Q ) | δ ( Q a − R ). Within the har-monic approximation, the core displacement distributionhas the form [15]. ρ ( R ) = exp (cid:16) − R · U − · R (cid:17) , where U is the thermal parameter matrix [16] that ismeasured in x-ray or neutron scattering experiments. Calculation of the matrix element . For a hydrogenoidatom, the 1 s core-hole radial wavefunction is propor-tional to e − ar , where a = Z/a , Z is the atomic numberand a the Bohr radius. The 1 s wavefunction φ ( r ) of atrue atom is close to that of a hydrogenoid one and canbe written as a fast converging linear combination of ex-ponentials. Thus, φ ( r − R ) becomes a sum of shifted ex-ponentials that can be described by the Barnett-Coulsonexpansion [17]e − a | r − R | = X n (2 n + 1) P n (ˆ r · ˆ R ) c n ( r, R ) , with P n a Legendre polynomial and c n ( r, R ) = − √ rR (cid:16) r < I ′ n +1 / ( ar < ) K n +1 / ( ar > )+ r > I n +1 / ( ar < ) K ′ n +1 / ( ar > ) (cid:17) , where r < ( r > , resp.) is the smaller (larger, resp.) of r and R , I ν ( z ) and K ν ( z ) are the modified Bessel functionsand I ′ ν ( z ) and K ′ ν ( z ) their derivatives with respect to z . For notational convenience, we consider that the corewavefunction can be represented by a single exponential φ ( r ) = C e − ar .To calculate the matrix element, we expand the fi-nal state wavefunction over spherical harmonics ψ f ( r ) = P ℓm f ℓm ( r ) Y mℓ (ˆ r ). The matrix element over the elec-tronic variable is Z d r ψ f ( r ) ∗ ε · r φ ( r , R ) = X ℓm X mℓ ( R ) , with X mℓ ( R ) = C ∞ X n =0 Z d r f ∗ ℓm ( r ) Y mℓ (ˆ r ) ∗ ε · r (2 n + 1) c n ( r, R ) P n (ˆ r · ˆ R ) . Standard angular momentum recoupling leads to X mℓ ( R ) = (4 π ) C X n = | ℓ ± | Z r d rf ∗ ℓm ( r ) c n ( r, R ) X λ ( − m Y − λ ( ε ) Y λ − mn ( ˆ R ) C ℓm λnm − λ , (3)where C kκ λnp are Gaunt coefficients. Equation (3) showsthat all values of the final state angular momentum ℓ arenow allowed. The core-hole wavefunction is still spher-ical, but with respect to a shifted centrum. Thus, withrespect to the original spectrum, it is a sum over all an-gular momenta given by the Barnett-Coulson expansion.Thus, all final states angular momenta are available inspite of the fact that only electric dipole transitions areallowed. In particular, vibrations allow for dipole transi-tions to the 3 s and 3 d final states at the K -edge. General features of vibrational transitions . The fore-going approach enables us to draw some general conclu-sions concerning the effect of vibrations on XAS pre-edgestructure. This effect is measurable if the density of non- p states of the system in the final state (i.e. in the presenceof a core hole) is large and well localized near the Fermienergy (vibrational transitions towards p -states would bemasked by the allowed vibrationless transitions). For ex-ample, just above the Fermi level, many aluminum orsilicon compounds have a strong density of 3 s states andmany transition metal compounds have a large density of3 d states. In the first case, vibrational transitions appearas monopole 1 s → s transitions, which are completelyexcluded with electromagnetic transitions. In the secondcase, vibrational transitions superimpose upon electricquadrupole 1 s → d transitions. Thus, vibrations inducetransitions at specific energies in the pre-edge but hardlymodify the rest of the XANES spectrum.A temperature dependence of vibrational transitionsis expected if U varies with temperature. This occursbetween 0 K and room temperature if the sample has softmodes (i.e. low energy phonons). Otherwise, vibrational Energy (eV) A b s o r p t i on ( a r b i t. un i t s ) exp.calc. no vib.calc. vib. U exp calc. vib. U fit AA FIG. 1: Experimental [23] and calculated Al K -edge isotropicspectrum of corundum at 300 K. U exp and U fit are the exper-imental and fitted thermal parameters (see text). transitions are only due to the zero-point motion of thenuclei. In other words, vibrational transitions are then aconsequence of the fact that, even at 0 K, nuclei are notlocalized at a single point.We test these conclusions with two examples: the Al K -edge in corundum (where vibrational transitions to 3 s states are expected) and the Ti K -edge in rutile (wherevibrational transitions to 3 d states and a temperaturedependence are expected). The Al K -edge in corundum . The X-ray absorptioncross section within the crude Born-Oppenheimer ap-proximation has been implemented in the XSpectra pack-age [18] of the Quantum-espresso suite of codes [19]. Tocalculate the integral over R in eq. (2), it is found suf-ficient to compute the integral over a cube of size 2Λ,where Λ is the largest eigenvalue of the matrix U . Thiscube is cut into 27 smaller cubes where the integral iscarried out using eq. (25.4.68) of Ref. 20. The technicaldetails of the self-consistent calculation are the same asin Refs. [21, 22].Figure 1 shows the result for experimental mean dis-placements σ = σ =0.048 ˚A, σ =0.049 ˚A [24]. Themean displacements σ i are the square root of the eigen-values of U and they have a more direct physical meaningthan U . The vibration transitions are observed exactlyat the position of the pre-edge peak which is absent fromthe calculation without vibrations. However, the vibra-tional transitions are overestimated and a better agree-ment is obtained when setting the mean displacementsto 0.026 ˚A. Therefore, computed vibrational transitionsshow up at the right position but with a too large inten- Energy (eV) A b s o r p t i on ( a r b . un i t s ) exp. 250Kexp. 8Kcalc. vib. U2calc. vib. U1calc. no vib. E1 + E2 ε // [001] E2A1 A2 A3
FIG. 2: Experimental [7] and calculated Ti K pre-edge spec-trum of rutile. E E sity. A part of this discrepancy might be due to the factthat the experimental thermal parameters include someamount of static disorder, but most of it probably comesfrom the crude Born-Oppenheimer approximation. Themain vibrational transitions come from the relative dis-placement of the aluminum atom with respect to the sixoxygen first neighbors. The vibrational modes involvingan overall motion of the aluminum atom with its oxy-gen octahedron contribute to the thermal parameter U although they do not contribute to the vibrational transi-tions. Because of the crude Born-Oppenheimer approx-imation, the vibrations are summarized in the thermalparameter, which overestimates the real effect of vibra-tions. A similar phenomenon occurs with the EXAFSDebye-Waller factor which is different from the one de-termined by x-ray diffraction because only relative dis-placements must be taken into account. This point willnow be confirmed with the Ti K -edge absorption in ru-tile. The Ti K -edge of rutile . With the Ti K -edge of rutile,we test the limit of the crude Born-Oppenheimer approx-imation. Indeed, this approximation assumes that the fi-nal state wavefunction does not change when the crystalvibrates. This might be reasonable for the Al 3 s statesbecause they are poorly localized and overlap the oxygen2 p orbitals, but this is not true for the 3 d states of Ti inrutile, which are localized near the Ti nucleus.Figure 2 shows the experimental and theoretical Ti K -edge spectrum of rutile with the polarization parallelto the c axis and at two temperatures. A similar re-sult is obtained when ε is perpendicular to the c axis.The mean displacements σ , σ , σ used in the calcula-tions are, in ˚A, 0.0028, 0.0027, 0.0023 for U U [1] A. J. Bridgeman and M. Gerloch, Coord. Chem. Rev. , 315 (1997).[2] F. D. Vila, et al., Phys. Rev. B , 014301 (2007).[3] A. Kirfel, et al., Phys. Rev. B , 165202 (2002).[4] V. E. Dmitrienko, et al., Acta Cryst. A , 481 (2005).[5] D. Li, et al., Amer. Mineral. , 432 (1995).[6] S. Nozawa, et al., Phys. Rev. B , 121101 (2005).[7] O. Durmeyer, et al., unpublished.[8] T. Fujikawa, J. Phys. Soc. Japan , 87 (1996).[9] T. Fujikawa, J. Phys. Soc. Japan , 2444 (1999).[10] H. Arai, et al., in AIP Conference Proceedings , edited byB. Hedman and P. Painetta (AIP, 2007), vol. 882, pp.108–10.[11] K. A. M¨ader and S. Baroni, Phys. Rev. B , 9649(1997).[12] M. Born and R. Oppenheimer, Ann. Phys. , 457(1927).[13] B. Henderson and G. Imbusch, Optical Spectroscopy ofInorganic Solids (Clarendon Press, Oxford, 1989).[14] M. Born and K. Huang,
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