IMPROVED DETERMINATION OF BACKSCATTERING AMPLITUDE AND PHASE SHIFT FUNCTIONS

Transcription

1 IMPROVED DETERMINATION OF BACKSCATTERING AMPLITUDE AND PHASE SHIFT FUNCTIONS A. Mckale, S.-K. Chan, B. Veal, A. Paulikas, G. Knapp To cite this version: A. Mckale, S.-K. Chan, B. Veal, A. Paulikas, G. Knapp. IMPROVED DETERMINATION OF BACKSCATTERING AMPLITUDE AND PHASE SHIFT FUNCTIONS. Journal de Physique Colloques, 1986, 47 (C8), pp.c8-55-c8-62. < /physcol: >. <pa > HAL Id: pa Submitted on 1 Jan 1986 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 JOURNAL DE PHYSIQUE Colloque C8, supplgment au no 12, Tome 47, dgcembre 1986 IMPROVED DETERMINATION OF BACKSCATTERING AMPLITUDE AND PHASE SHIFT FUNCTIONS( 1 ) A.G. MCKALE(*), S.-K. CHAN, B.W. VEAL, A.P. PAULIKAS and G. S. KNAPP* Materials Science Division, Argonne National Laboratory, Argonne. XL 60439, U.S.A. 'surface Science Laboratories, 465 National Avenue, Mountain View, CA 94043, U.S.A. Abstract Procedures for analysis of EXAFS data require knowledge of the phase shift and amplitude and B(k). We present procedures to obtain these functions, both experimentally and theoretically, that overcome limitations in presently available methods. A procedure is described that allows for the use of crystallographically complex materials to serve as standards in the experimental determination and B(k). Also we present a convenient algorithm, that uses the full curved wave formalism, to determine the functions theoretically. Improved results are obtained, particularly at low k. We illustrate the use of these procedures with a study of the 3d transition metals. Introduction The FXAFS technique has proven to be an extremely powerful probe of dcroscopic local structure. However, presently available data analysis procedures seriously limit its applicability. In order to obtain the desired structural parameters, knowledge of backscattering amplitude and phase functions, B(k) and b(k), is required. The accuracy with which structural parameters can be obtained is limited by the accuracy of these functions. B(k) and b(k) can be determined both experimentally and theoretically. Use of the plane wa e (PW) approximation limits computational accuracy, especially at small k ((4 A-'). For computation at small k, a full curved-wave (CW) formalism is required. General use of the formalism has been hampered by its complexity and the need for separate calculations for each material studied. We have recast the curved wave formalism into the traditional plane wave form. Using partial-wave phase shifts from atomic calculations, we obtain redefined phase shift and amplitude and B(k,kR). Because these functions have only weak R dependence, they can be used to analyze experimental spectra with existing plane wave type codes. An unknown spectrum can also be analyzed and B(k) extracted from the experimental spectrum of a chemically similar compound of known structure. Experimental determination of these functions was previously limited to those materials ("work supported by the U.S. Department of Energy. BES-Materials Sciences. under contract W Eng-38 (2)~ls0 at the Department of Physics and Astronomy, Northwestern University. Evanston. IL Work performed at* Argonne National Laboratory while a Laboratory-Graduate Thesis Program Participant. Program administered by the Argonne Division of Educational Programs with funding from the U.S. Department of Energy. Article published online by EDP Sciences and available at

3 C8-56 JOURNAL DE PHYSIQUE with simple crystal structures. Materials with closely spaced near-neighbor shells could not serve as references. We have developed a procedure to vercome this limitation thus allowing the use of more complex reference materials.? We illustrate these procedures in a study of the 3d transition metals. Curved Wave Analysis A single scattering theory has been found to be valid in the EXAFS region, i.e., at energies more than 40 ev above the absorption threshold. According to this theory the modu a i n f he absorption coefficient can be described in the standard EXAFS equation:!!-k,g, 1%-15 B.(k) N -2R /X -2k2$ y(k) = (-l)', _ sin [2kR. + 26(k) + b (k)] e e (1) where B. is the backscattering amplitude from each of the N neighboring atoms in the th J shell which are located at an average distance R from the absorbing atom. The first exponential accounts for inelastic processes; \ is the electron mean free path for inelastic scattering. The second exponential, containing o (the mean square radial displacement of the atoms about R), is a Debye-Waller-type term. The parameter 6 is the phase shift due to the central atom and 4 (k) is the phase shift due to the 6ackscattering atom. The parameter 9. is the angular momentum of the outgoing photoelectron with respect to the central atom. Equation (1) was formulated using the plane wave approximation. In this approximation, the backscattering amplitude and phase shift functions are related to the individual partial-wave phase shifts due to scattering by the neighboring atoms by iq (k) i 6%' B(k) e =E l E e sin In this case B(k) and b(k) depend only on k. However if the curvature of the outgoing photoelectron wave function is taken into account, these functions also have an R dependence. Furthermore, the functional relationship between B.(k) and J O.l (k) and the partial phase shifts of the scattering potential depends on the angular momentum of the photoelectron. For K-shell absorption, the final state has p symmetry, and hence9 where h; (kr ) is an outgoing Hankel function.14 For LIII-shell absorption, the final state can have either s or d symmetry. However, it has been shown that the contributfpg to x(k> from the S part is negligible compared to that from the d part. Using only the d contribution, iq (k,r) - i6&1 B.(k,R) J e pkr ~[(~9.*+~) 1 e sin 69.1

4 The functions in Eqs. (3) and (4) can be calculated relatively easily even on a microcomputer given the individual partial-wave phase shifts and the interatomic distances. The functions are weakly dependent on R so that only an approximate interatomic distance need be used for the calculation of B(k) and G(k) for a given element. The type of error that is made by the use of an approximate value of R can be largely compensated for by a small (<lev) adustment of E. in the fit to Equation (1). Structural parameters obtained by fitting Eq. (1) to measured spectra (with E, as usual, being a fit parameter) are quite independent of reasonable variations in R. from the crystallographic values. J The use of the full curved wave (CW) formalism, Eqs. (3) and (4), with suitably chosen R provides a capability for analyzing EXAFS data using a single scattering theory much closer to the absorption edge than was previously possible using the plane wave (PW) functions. The ability to analyze low k data is critically important for the study of disordered systems and materials that contain backscattering atoms of low atomic number. These EXAFS spectra decay rapidly with increasing k with the consequence that the usable spectral window is small and furthermore, it is confined to low k. Complex Standards It is common practice to extract the backscattering amplitude and phase functions B(k) and 8(k) from measured EXAFS spectra of an element or compound with known crystallographic structure. These functions may then be used with Eq. 1 to analyze an unknown material arly if the unknown is believed to be chemically similar to the standard 39PS:fb:YJ. Unfortunately, present algorithms require that the known compounds be relatively simple, containing only one shell of atoms of a single atomic species. This shell must be adequately separated from neighboring shells to be resolved in the Fourier spectrum of ~(k). These requirements have posed a maor Limitation on the use of standards for EXAFS analysis. Candidate reference compounds frequently have complex structures with multiple shells too closely spaced to be resolvable in the Fourier spectrum. A lack of suitable standards has restricted the range of materials to which the EXAFS technique can be satisfactorily applied and has limited accuracy of the derived parameters, i.e., N, R., and o, in cases where available standards were not chemically similar to the un4nown system. We have developed an algorithm for extraction of B(k) that is applicable to most compounds of known structure. Thus, complex materials of known structure can now serve as standards for analysis of EXAFS data. The most common method of data analysis using a standard involves Fourier filtering of the reference sample ~(k) to isolate the contribution to ~(k) coming from a single shell of atoms. This is generally possible if the separation between two neighboring shells of the absorber is greater than 3.7 A. Comparison can then be made with the spectra of the unknown compound. The backscattering amplitude and phase need not be explicitly calculated if the unkno spectrum can also be Fourier filtered to isolate a single shell; the ratio methodplcan be applied to obtain the structural parameters. If a single shell cannot be isolated from the unknown spec- trum, however, it is necessary to extract B(k) and G(k) from the reference compound. These functions are then used with Eq. (1) to analyze the unknown sample. If a single shell can be isolated from the reference spectrum the summation of Eq. (1) reduces to a single term. Then the backscattering amplitude and phase functions are related to the experimental amplitude and phase, A(k) and +(k), where ~(k) = A(k) sin by

5 CS-58 JOURNAL DE PHYSIQUE Using standard nonlinear least square methods, the backscattering amplitude and phase functions can then be used with equation (1) to fit the unknown spectrum. However, if the spectrum of,' the known compound cannot be Fourier filtered to isolate single shell contributions to ~(k), the above method does not work. To obtain a more generally applica le procedure, we consider a variant of the original BXAFS equation of Stern et al. and follow the approach of L?unker18 to derive an alternate algorithm for extracting 1(k) and B(k) from the reference sample. We can write equation (1) in the form: B(k) x(k) = - sin [2kr + 26c(k) e -2r/ X di J where pl(r) is the radial probability distribution of atoms in the shell, normalized so that /pl(r) dr = N, the total number of atoms in the shell. If we then define an effective distribution -2r/ X P(r) z -T e (7) r and its Fourier transform?(k) = 1 P(r) ei2krdr we can rewrite equation (6) as x(k) = I?(k) I sin [26,(k) + 1 (k) + arg F(*)] From the measured reference spectrum, with no restriction imposed on the number of backscattering shells, an amplitude and phase, A(k) and $(k), can still be extracted, ~(k) = A(k) sin #(k), but now the backscattering amplitude and phase shift functions are related to the experimental amplitude and phase by: 26c(k) + 1 (k) = $(k) 3 - arg?(k) The radial probability distribution can be written as a sum over shells; each with a Gaussian distribution of positions about a single distance: and hence.. N -2k2% 2ikR -2R / X ~(k) = I 4 e J e 'e R If we assume small disorder and a single distance, Eq. (6) reduces to the standard EXAFS equation [Eq. (l)] for a single shell. In order to calculate p(k) we need to know N, X, and o. for each shell in the unresolved set. The N and R values are kn " '~rystallographic J standard). The a parameters can be dasured independently 1820(or determined using the ratio technique. A precise knowledge of the X value is not needed. The concept of trans-

6 ferability requires that the same X be used for the known and the unknown. This extraction procedure is applicable to materials that have a set of backscattering shells provided that all atoms of the set are the same, i.e., all the atoms have the same backscattering amplitude and phase shift functions. Application to Experimental Data To test our calculations of B(k) and B(k) and to test our procedure from extracting these functions from crystallographically complex standards, we have examined the 3d transition metals. To obtain the partial-wave phase shifts, tit, for the transition metals, the radiai 3chrcSdinger equation for an outgoing electron was solved at energy E = 6 k /2m in the Hartree-Fock approximation. Herman-Skillman wave functions were used for the core electrons. A muffin tin sphere with radius R, taken to be 40% of the interatomic distance in the elemental solid, was placed abo t the scattering atom. For the elements Ti-CO we used the configurations 3dn4s Y. For Ni and Cu, the configurations were 3dn4si. The 4s electrons were assumed to be smeared uniformly within the muffin tin sphere. Charge associated with electron wave functions that extended beyond the muffin tin radius was also added uniformly back into the muffin tin. Thus the potential V(r) is constant outside the muffin tin radius. The radial SchrcSdinger equation was then integrated outwards from the origin and matched at the muffin tin boundary R to the outgoing free-space solution to obtain No inelastic processes are included in this calculation. The partial-wave phase shifts were then used to generate the plane wave and curved wave backscattering amplitude and phase shift functions (Eqs. 2 and 3) for each element. Results are shown in Fig. 1. The curved-wave functions were evaluated at 2 = 2.75 A. This distance was held fixed so that systematic variations in the funct ons with Z could be reliably monitored. When fitting the spectra of an unknown sample it is not necessary that these functions be calculated at exactly the correct distance; using the functions at an approximate distance results in a fit with the same structural parameters and a slight variation in the fit Eo. As expected, large deviations occur between the FW and CW results for the low-k region. The PW and CW functions are in near agreement at high k. These calculated functions were then used with Eq. (1) to fit the measured spectra. Spectra were acquired from thin high purity fo s with no apparent pinholes using the Argonne focusing crystal EXAFS facility." Data were taken at 77 K. After background removal, the data were Fourier filtered to isolate the contribution rom the first shell (or unresolved group of shells) of backscattering atoms." The filtered data were then fit to Eq. 1, using the central atom phase shifts as determined by Teo and ~ e and e the ~ amplitude and phase functions from both PW and CW calculations. The radial distances obtained are listed in Table 1. In every case the value obtained using the curved wave formalism is closer to the known crystallographic value. Furthermore the quality of the fit of the theoretical function to experimental data was noticeably improved in the low k region. Table 1. Radial Distances in A Derived from EXAFS Spectra and Known Crystallographic Values PW CW Diffraction Ti 2.87f f

7 Figure 1. The calculated backscattering amplitude and phase shifts for the 3d transition metals. Solid lines-plane wave; Dashed lines-curved wave. We also demonstrate our mlti-shell extraction procedure with a study of Fe and Ti metal. Fe is a bcc material whose first two neighboring shells from an absorber site are se arated by almost 0.4 A. Unequal numbers of atoms occupy the shells. Different 3 parameters have been predicted.20 These shells are generally not resolvable k the Fourier transform of ~(k) unless data are taken for an extremely large range of k. Ti is hcp with the first two shells located at and A from an absorber. With Fourier filtering, we were able to isolate for both Fe and Ti, the contribution to ~(k) from the first two shells. Using the algorithm described above with the known structural parameters, central atom phase shifts of Teo and Lee theoretical Debye-Waller type factors:" and an electron mean free path of 8 A,15'we obtain extracted backscattering amplitude and phase functions for Ti and Fe. Using the theoretical functions calculated using the full curved wave formalism we generated a theoretical ~(k) which was filtered through the same windows as were used for the experimental data and the backscattering amplitude and phase shift functions were exracted from the resulting filtered ~(k). The results of this procedure are shown in Figure 2. Reasonable agreement is obtained.

8 Figure 2. Theoretical (dashed lines) and experimentally determined (solid lines) backscattering amplitude and phase functions for Ti and Fe at 77K. Conclusions We have presented algorithms for obtaining backscattering amplitude and phase shift functions for analysis of EXAFS data. Using curved wave theory, a formulation for B(k) and 8(k) is presented that preserves the form of the conventional EXAFS single scattering equation. The curved wave modification can be thus easily implemented into existing data analysis codes. With use of the curved wave formalism, accuracy of the backscattering functions is improved, particularly at low k. This capability is important for analysis of low Z elements. The calculated functions were tested by analyzing EXAFS data for 3d transition elements. For all elements studied, interatomic distances obtained using curved wave theory were more accurate than the distances obtained using plane wave theory. An approach is also described for extracting B(k) and b(k) from experimental data acquired from samples of known crystallography, even when the samples have closely spaced multiple near-coordination shells. The procedure is illustrated with application to elemental Fe and Ti which have bcc and hcp structures, respectively. This procedure, valid when the multiple near-shells have identical atoms, dramatically increases the number of standard materials which can be used by the experimentalist. The increased flexibility should increase the value of EXAFS for chemical and materials studies since the most accurate results can be obtained when standards are chemically similar to the unknown. References l. Apparently the method has been independently discovered by Boyce (J. B. Boyce et al., Phys. Rev. B 33, 7314 (1986)) and by Ma, Stern, and Bouldin (Y. Ma, E. A. Stern, C. E. p ox din, Phys. Rev. Lett. 57, xxx (1986)) who applied it to a study of quasicrystals. 2. D. E. Sayers, E. A. Stern, and F. W. Lytle, Phys. Rev. Lett. 27, 1204 (1971). 3. E. A. Stern, D. E. Sayers, and F. W. Lytle, Phys. Rev. B 2, 4836 (1975). 4. P. A. Lee and J. B. Pendry, Phys. Rev. B 11, 2795 (1975). 5. B. K. Teo and P. A. Lee, J. Am. Chem. SOC~E, 2815 (1979). 6. E. A. Stem, Phys. Rev. B g, 3027 (1974). 7. J. E. Miiller and W. L. Schaich, Phys. Rev. B 27, 6489 ( 1983) 8. G. B. Bunker and E. A. Stern, Phys. Rev. Lett. 52, 1990 (1984).

9 C8-62 JOURNAL DE PHYSIQUE W. L. Schaich, Phys. Rev. B 3, 6513 (1984). See also S. J. Gurman, N. Binsted, and I. Ross, J. Phys. C 17, 143 (1984) and J. J. Barton and D. A. Shirley, Phys. Rev. B (1985). A. G. McKale, G. S. Knapp, S.-K. Chan, Phys. Rev. B 33, 841 (1986). W. L. Schaich, Phys. Rev. B 8, 4028 (1973). C. A. Ashley and S. Doniach, Phys. Rev. B 11, 1279 (1975). J. J. Boland, S. E. Crane, and J. D. Baldeschwieler, J. Chem. Phys. 77, 142 (1982). See, for instance, A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1966). E. A. Stern, B. A. Bunker, and S. M. Heald, Phys. RevB21, 5521 (1980). P. Eisenberger and B. Lengeler, Phys. Rev. B 22, 3551 (1980). B. A. Bunker and E. A. Stern, Phys. Rev. B 27, 1017 (1983). G. B. Bunker, Nucl. Inst. Methods 207, 437 (1983). G, Beni and P. M. Platzman, Phys. Rev. B, 2, 1514 (1976). See also, W. Bohmer and P. Rabe, J. Phys. C, 12, (1979) and R. B. Greegor and F. W. Lytle, Phys. Rev. BYE, 4902 (1979). E. Sevillano, H. Meuth, and J. J. Rehr, Phys. Rev. B%, 4908 (1979). G. S. Knapp and P. Georgopoulos, in Laboratory EXAFS-1980 (University of Wash- ington), proceedings of the Workshop in Laboratory EXAFS Facilities and Their Relationship to a Synchrotron Source, edited by E. A. Stem (AIP, New York, 1980). C. Tang, et al.,phys. Rev. B%, 1000 (1985).