X-RAY ABSORPTION IN IONIC MATERIALS

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X-RAY ABSORPTION IN IONIC MATERIALS

L. Wille, P. Durham, P. Sterne

To cite this version:

L. Wille, P. Durham, P. Sterne. X-RAY ABSORPTION IN IONIC MATERIALS. Journal de Physique

Colloques, 1986, 47 (C8), pp.C8-43-C8-47. �10.1051/jphyscol:1986806�. �jpa-00225990�

JOURNAL DE PHYSIQUE

Colloque C8, supplbment au n o 12, Tome

47,

dkcembre 1986

X-RAY ABSORPTION IN IONIC MATERIALS

L.T. WILLE, P.J. DURHAM and P.A.

STERNE

SERC Daresbury Laboratory, Daresbury, GB-Warrington WA4

4 A D ,

Great-Britain

,

,

Resume. Nous pr&sentons des calculs des spectres XANES de l'oxyde de calcium, qui constitue un exemple typique de solide ionique. Nous discutons les prescriptions pour une mod6lisation du transfert de charge sans proc&dure d'itgration par comparaison avec le calcul complet self- consistant de la structure de bande. Outre son application,directe aux systsmes ioniques, cette Gtude est importante pour la comprehension des limites thgoriques des calculs r&alise/s dans le cas de syst&mes complexes pour lesquels des potentiels self-consistants ne sont pas disponibles (tels les silicates et les biomoldcules)..

Abstract. Calculations of the XANES spectra for calcium oxide, as a typical ionic solid, are presented. Non-self-consistent prescriptions for modelling charge transfer are judged against a fully self-consistent band structure calculation. Apart from its immediate relevance to ionic systems, this study is important for an understanding of the theoretical limitations in calculations for complex systems (such as silicates or biomolecules) where no self-consistent potentials are available.

From the theoretical point of view, X-ray absorption spectra can conveniently be thought of as depending on three phenomena. First there is the crystal structure, or more generally, the spatial arrangement of the atoms. Then there is the electronic structure which determines the potential and hence the phase shifts. Finally there are several many body effects, such as the photoelectron self-energy, core hole relaxation effects, screening of the photon field, etc. One-electron calculations have been very succesful, especially for metallic systems, and the experience is that the spectra are most sensitive to the crystal structure. But in order to improve the usefulness of X-ray absorption as a structural tool one needs to investigate quantitatively the approximation schemes used to handle the electronic and many body part of the problem. The present work focuses on the accuracy of the potentials and phase shifts used in onerelectron calculations, choosing as a test case CaO. This ionic system, which has been investigated before ( 1 1 , is structurally sufficiently simple so that a self- consistent band structure calculation can be performed and compared to

results obtained by the non-self-consistent Matthels prescription.

Three sets of non-self-consistent potentials were generated, corresponding to CaO,

c ~ + o -

and

c~++o--.

The atomic potentials were obtained from a relativisric Hartree-Fock-Slater calculation using the Perdew-Zunger form of the local exchange-correlation potential 121- The Mattheis prescription [ 3 , 4 ] consists in superposing and spherically

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986806

JOURNAL

DE

PHYSIQUE

averaging atomic charge densities. By solving Poisson's equation and adding a local exchange-correlation contribution one then obtains a spherically symmetric potential in the muffin-tin form. In systems with more than one atom per unit cell one has a certain freedom in the choice of the muffin-tin radii. These have to be chosen so as to minimise the discontinuity in the potential across the (touching) muffin-tin boundaries. In the present case it was found that a ratio of 2/3 between the volumes of an 0 and a Ca sphere gave good results. In our calculations the charge densities of a finite number (e.g. 8 ) of shells of atoms or ions were superposed and spherically averaged. In the neutral atom case this procedure is sufficient to yield the muffin-tin potential of the central atom in the cluster. When charged ions are superposed, however, the Coulomb potential from ions outside the cluster, however distant, cannot be neglected, and in fact gives rise to the Madelung potential. The calculation of this term is non-trivial, since it involves a slowly converging series, but it is well-known how to accelerate the convergence by means of the Ewald summation method.

The Madelung term has been tabulated for a number of crystal structures, such as the NaC1-structure in which CaO crystallises. The Coulomb potential for the cluster can thus be straightforwardly augmented, so as to contain the correct Madelung contribution, and when this is done the muffin-tin potential of the central ion is invariant if more then approximately eight shells are included in the cluster. (More precisely, this holds if the cluster is large enough for the Coulomb potentialF(r) from the external ions, for r within the central ion, to be accurately approximated by a point charge model.)

For comparison a self-consistent charge density was obtained using the LMTO method ( 5 1 with the von Barth-Hedin local density exchange- correlation potential 1 6 ) . To facilitate comparison with the non-self- consistent potentials, the Ca and 0 atomic sphere radii were chosen to be the same as the muffin-tin radii above. The remaining interstitial volume was incorporated in two equivalent "empty" spheres at the cubic sites in the rocksalt structure. At self-consistency, there was a deficit of 1.76 electrons in the Ca sphere and a net negative charge of 0.45 on the 0 sphere. The remaining 1.31 electrons were divided between the two empty spheres. Because of the large spatial extent of the Ca core, it was found necessary to treat both the 3s and 3p core levels as bound states. Thls was accomplished in the usual way [ 5 , 7 ] by using separate panels for the core and valence states.

The LMTO-calculation, being based on the density functional theory, should give a very good approximation to the ground state density; in Figs. l and 2 we compare the results of our three non-self-consistent models with it. In the case of oxygen one notices that the extra electrons,are put in the 2p level and that there is a corresponding depletion of the interstitial region (Fig. la). The self-consistent result is quite close to the result for neutral atoms around the 2p shell, but the charge density extends more towards the muffin-tin radius. For calcium (Fig. lb) all cases are very similar, except near the muffin-tin radius where the LMTO-result is very close to the singly ionised case. Similar conclusions can be drawn from the integrated charge densities (Figs. 2a and 2b). The corresponding phase shifts are shown in Fig. 3. These are referred to the potential energy zero, rather than to the muffin-tin zero. For oxygen the most prominent feature is the binding and unbinding of a p-state as extra electrons are put in the 2p level, reflected in the shift of the p-resonance.

Certainly both the charge densities and the phase shifts are similar for all cases, but there are sufficient differences to make the corresponding XANES calculations, shown in Fig. 4, interesting.

--- -- -. 0

----

_0-

0-

- -

LMTO

-. - - -. - Ca

----

Ca+

Ca+ ⁺ --- LMTO

0 '

40 0:4 0:8 1:2

18

2b f 2.4 r ( a 4 ^RMT

. .

Charge density for oxygen (a) and calcium (b) obtained from non-self-consistent prescriptions and an W O calculation. The muffin- tin radius (RMT) has been indicated.

i / ,.; ^.---

!,,',.,. ⁰

// ^,:' 0-

i; ^{4: - -}

/>

1'; - LMTO

014 0:8 1.'2 116 210 2.'4 r (a.u.)

Fig. 2a

---. Ca ---- Ca +

-. - Ca+ +

-

LMTO

r (a.u.) - Fig. 2b FLg.

2 .

Integrated charge density for oxygen (a) and calcium (b).

C8-46 JOURNAL DE PHYSIQUE

2.5- : l..

..

- LMTO

. . . . . . - . . Ca

----.

.

^{,, Ca+}

Ca++

LMTO

E (H) Fig. 3 b Fig. 3a

F_Lp.

2-

Phase shifts (referred to potential en&rgy zero) for oxygen (a) and calcium (b).

- - - - -. . --- 0

0-

o--

- LMTO

O ~ O 0:4 ^' 018 ^' 1:2 ' 1:6 ^' 2.0 ' E (H)

Fig. 4 a

E (H) Fig. 4b

F_Ag. 4. XANES in CaO : (a) 0 K-edge, (b) Ca K-edge. The energy zero is arbitrary.

Our computational scheme, fully described elsewhere [8], solves the full multiple scattering equations for a finite number of shells of atoms surrounding the absorber. Many body lifetime effects are included by giving the potential a small imaginary part, so that spectral features are appropriately broadened. In fact, the broadening we have used is rather small, since we are interested in this paper in bringing out the sensitivity of spectral features to the underlying model charge density, rather than in realistic comparison with experiment. Much of the structure shown in Fig. 4 would in practice be smeared out by lifetime and resolution effects. It is obvious that the differences are quantitative, rather than qualitative, and fairly small, particularly for the Ca spectra (note the good agreement with experrment for the Ca- edge [ l ] ) . It would certainly be difficult to deduce on the basis of experimental information which is the "best" potential. Clearly the Mattheis prescription, startin8 from neutral atomic charge densities, is a reasonable approximat~on to a fully self-consistent calculation, as far as XANES is concerned. There appears to be little advantage in constructing non-self-consistent model potentials from ionic charge densities, and taking the nominal valence of the ions certainly overestimates the effects of charge transfer. All this reinforces the view that XANES is mainly a structural probe. It also means that XANES analysis will be useful for structurally complex systems where self- consistent calculations are not feasible at present, especially if used to trace the effects of structural changes on spectra. On the other hand, it is also clear that if highly accurate theoretical fLgg to experimental data are required, then self-consistency in the construction of the scattering potentials cannot be ignored (and the same must apply to the many-body effects mentioned at the outset). Such considerations clearly limit the precision with which quantitative structural information can be derived from EXAFS or XANES.

In conclusion, we have compared and contrasted the XANES spectra for three sets of non-self-consistent potentials and a self-consistent one, within the one-electron approximation. The effects of self-consistency (allowing for charge transfer) on the shape of the spectra are fairly small which shows once again that XANES carries mainly structural information. A fuller description of these calculations, together with a discussion of the EXAFS spectra and threshold shifts, will be given elsewhere.

[l] D. Norman, K. B. Garg and P. J. Durham, Solid State Commun. 55 ,895 (1985).

[2] J. P. Perdew and A. Zunger, Phys. Rev. B

22

,5048 (1981).

[3] L. Mattheis, Phys. Rev.

134

.A970 (1964).

[4] T. L. Loucks, "Augmented Plane Wave Method" (Benjamin, New York, 1967).

(51 H. L. Skriver, "The LMTO-method", (Berlin, Springer, 1984).

[6] U. von Barth and L. Hedin, J. Phys. C: Solid. St. Phys.

5 ,

¹⁶²⁹

(1972).

[7] G. B. Bachelet and N. E. Christensen, Phys. Rev. B

21 ,

879 (1985).

(81 P. J. Durham, J. B. Pendry and C. H. Hodges Comput. Phys. Commun.

25 ,193 (1982).

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