IMPROVED DETERMINATION OF BACKSCATTERING AMPLITUDE AND PHASE SHIFT FUNCTIONS

HAL Id: jpa-00225992

https://hal.archives-ouvertes.fr/jpa-00225992

Submitted on 1 Jan 1986

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished 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.

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 Col-

loques, 1986, 47 (C8), pp.C8-55-C8-62. �10.1051/jphyscol:1986808�. �jpa-00225992�

JOURNAL DE PHYSIQUE

Colloque C8, supplgment au n o 12, Tome 47, dgcembre 1986

IMPROVED DETERMINATION OF BACKSCATTERING AMPLITUDE AND PHASE SHIFT FUNCTIONS(

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 functions, @(k) and B(k). We present procedures to obtain these func- tions, both experimentally and theoretically, that overcome limitations in presently available methods. A procedure is described that allows for the use of crystallo- graphically complex materials to serve as standards in the experimental determina- tion of @(k) 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 re- quired. 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 experimen- tally and theoretically. Use of the plane wa e (PW) approximation limits computa- tional 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 functions, @(k,kR) 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 using @(k) and B(k) extracted from the experimental spectrum of a chemically similar compound of known structure. Experi- mental 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-31-109-Eng-38

( 2 ) ~ l s 0 at the Department of Physics and Astronomy, Northwestern University. Evanston. IL 60201.

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 http://dx.doi.org/10.1051/jphyscol:1986808

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 limi- tation 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) Nj -2R / X -2k2$

y(k)

(-l)', _ ^{sin [2kR. + 26(k) + b (k)] e} _j ^{j e j (1)}

where B. is the backscattering amplitude from each of the N neighboring atoms in

J j

the jth shell which are located at an average distance R from the absorbing atom.

The first exponential accounts for inelastic processes; j\ is the electron mean free path for inelastic scattering. The second exponential, containing o (the mean square radial displacement of the atoms about Rj), is a Debye-Waller-type term. The j 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 j outgoing photoelectron with respect to the central atom.

Equation (1) was formulated using the plane wave approximation. In this ap- proximation, the backscattering amplitude and phase shift functions are related to the individual partial-wave phase shifts due to scattering by the neighboring atoms by

iQj (k) i 6%'

Bj(k) e = E l E 2 + 1 1 e sin

In this case Bj(k) and bj(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 O .l (k) and the partial phase shifts of the scattering potential depends on the angu- J lar 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 j

For LIII-shell absorption, the final state can have either

or d symmetry.

However, it has been shown that the contributfpg to x(k> from the

part is negligible compared to that from the d part. Using only the d contribution,

i6&1 iQ (k,Rj) -

B.(k,Rj) J e j p k R j ~ [ ( ~ 9 . * + ~ )

e sin 6

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 Rj so that only an approximate interatomic distance need be used for the calculation of Bj(k) and Gj(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) adjustment of E . in the fit to ^j 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 Rj 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 impor- tant 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 func- tions 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 simi- lar to the standard

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 major Limitation on the use of standards for EXAFS analysis. Candidate reference com- pounds 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., Nj , R., and oj, in cases where available standards were not chemically similar to the un4nown system.

We have developed an algorithm for extraction of B(k) and @(k) that is appli- cable to most compounds of known structure. Thus, complex materials of known struc- ture 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

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)

sin [2kr + ^26c(k) + @.(k)l e -2r/ X dI

x(k)

- _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(*)]

j

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 extract- ed, ~ ( 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 ₃ (k)

$(k) - ^{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 j}

j Rj

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 standard). J

The a parameters can be dasured independently j 1820(or determined using the ratio

technique. A precise knowledge of the X value is not needed. The concept of trans-

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 ex- tracting these functions from crystallographically complex standards, we have exam- ined 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 Y

atom. For the elements Ti-CO we used the configurations 3dn4s . For Ni and Cu, the configurations were 3dn4si. The 4s electrons were assumed to be smeared uni- formly 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 evalu- ated 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 ex- pected, 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 pin- holes using the Argonne focusing crystal EXAFS facility." Data were taken at 77 K. After background removal, the data were Fourier filtered to isolate the contri- bution 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 the amplitude and phase functions from both e ~ 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 2.87f0.01 2.91f0.01 2.897

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 2.897 and 2.950 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.

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 imple- mented 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, dramati- cally 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,

(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).

C8-62 JOURNAL DE PHYSIQUE

W. L. Schaich, Phys. Rev. B 3, 6513 (1984). See a l s o S. J. Gurman,

N. Binsted, and I. Ross, J. Phys. C 17, 143 (1984) and J. J. Barton and D. A.

S h i r l e y , Phys. Rev. B 32 1862 (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, f o r i n s t a n c e , A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1966).

E. A. Stern, B. A. Bunker, and S. M. Heald, Phys. R e v B 2 1 , 5521 (1980).

P. Eisenberger and B. Lengeler, Phys. Rev. B 22, 3551 (1980).

B. A. Bunker and E. A. S t e r n , Phys. Rev. B 27, 1017 (1983).

G. B. Bunker, Nucl. I n s t . Methods 207, 437 (1983).

G, Beni and P. M. Platzman, Phys. Rev. B, 2, 1514 (1976). See a l s o , W. Bohmer and P. Rabe, J. Phys. C, 12, 24654 (1979) and R. B. Greegor and F. W. L y t l e , Phys. Rev. B Y E , 4902 (1979).

E. Sevillano, H. Meuth, and J. J. Rehr, Phys. Rev. B%, 4908 (1979).

G. S. Knapp and P. Georgopoulos, i n Laboratory EXAFS-1980 (University of Wash- i n g t o n ) , proceedings of the Workshop i n Laboratory EXAFS F a c i l i t i e s and Their Relationship t o a Synchrotron Source, e d i t e d by E. A. S t e m (AIP, New York, 1980).

C. Tang, e t al.,Phys. Rev. B%, 1000 (1985).