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Partial pair correlation functions for multicomponent systems by EXAFS: A new approach
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 575 (2007) 155–158 www.elsevier.com/locate/nima
Partial pair correlation functions for multicomponent systems by EXAFS: A new approach Yu. A. Babanova,b,, I. Yu. Kamenskya, J.-L. Hazemannc, Y. Calzavarac, D. Raouxd a
Institute of Metal Physics, S. Kovalevskaya Str. 18, Ekaterinburg 620219, Russia b Udmurt State University, Izhevsk, Russia c CNRS, Laboratoire de Crystallographie, Grenoble, France d Synchrotron SOLEIL, Paris, France Available online 13 January 2007
Abstract A method of determining partial pair correlation functions (pPCFs) for multicomponent systems from EXAFS data is presented. The method is based upon the regularization procedure of solving a Fredholm integral equation of the first kind using special iteration procedure. The effectiveness of the method has been tested on the model crystalline solids ZnBr2. We present the result for ZnBr2 aqueous solution obtained from experimental data at room temperature (303 K) and pressure 250 bar. Experiments were performed using an unique equipment at ESRF (Grenoble, France). Peculiar features of the method are high resolution for closely spaced shells and high accuracy in the determination of pPCFs. r 2007 Elsevier B.V. All rights reserved. PACS: 61.10.Ht Keywords: EXAFS; Regularization; Aqueous solution
1. Introduction EXAFS spectroscopy is used to probe the local atomic arrangement around a specific atomic species in amorphous alloys, catalysis, biological molecules, solutions, etc. The conventional procedure of extracting structural information in real space is based on Fourier transforming the normalized oscillating part of the X-ray absorption coefficient. It is well known that the peaks in Fourier transform are shifted to lower r from the positions of the corresponding peaks in partial pair correlation function (pPCF). Besides, the real space contribution of each shell is long range and sharply oscillatory. The contributions made by two closely spaced shells interfere strongly. Corresponding author. Institute of Metal Physics, S. Kovalevskaya Str. 18, GSP-170, Ekaterinburg 620219, Russia. Tel.: +7 343 3783837; fax: +7 343 3745244. E-mail address: [email protected] (Y.A. Babanov).
0168-9002/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2007.01.048
According to theory, the EXAFS can be described as an electron diffraction process where the electron source is the absorbing atom i. The outgoing photoelectron wave and a small part of it, which is backscattered from surrounding atoms j, interfere within the excited atom. This scattering is specified by not only the atomic amplitude fj, as is the case in X-ray diffraction, but also the phase jj. In addition, the scattering on the potential of the central atom i exhibits a phase shift 2di. These scattering characteristics are involved in the integral operator Aij, which, as it acts on the pPCF gij, generates a contribution of a specific atomic pair ij to the normalized oscillating part wi. In this paper, we propose to use the dependence of the integral operator Aij on the atomic scattering characteristics di(k), jj(k,r) for determining pPCFs in multicomponent systems from EXAFS data. Earlier, this idea was applied to extract information about bond lengths for multicomponent systems [1]. Now, we introduce a special iteration procedure for obtaining pPCFs.
ARTICLE IN PRESS Y.A. Babanov et al. / Nuclear Instruments and Methods in Physics Research A 575 (2007) 155–158
156
2. Mathematical
C AB AAB U A ¼ C AB AAB AAA gAA þ gð0Þ AB
EXAFS equation for n-component system has the form Z 1 n X 4pr0 wi ðkÞ ¼ RðkÞ cj f j ðk; rÞeð2r=lðkÞÞ k 0 j¼1
where C AA ¼ ðAAA AAA þ BA Þ1 and C AB ¼ ðAAB AAB þ BB Þ1 are the corresponding inverse operators. The function g(0) is called Tikhonov’s solution. In order to make the structure information more exact, we use the following iteration procedure:
sinð2kr þ cij ðk; rÞÞgij ðrÞ dr
ð1Þ
where k is the wave number of a photoelectron, r0 is the mean atomic density, R(k) is the amplitude reduction factor, cj is the concentration, and cij(k,r) ¼ 2di(k)+jj(k,r) is the total phase shift, which is a peculiar ‘‘mark’’ for a specific pair of atoms. All atomic scattering characteristics are calculated by FEFF 8.1 [2]. The function gij(r) is the pPCF, which is the probability to find atom j from atom i at the distance r by definition. In symbolic form we have Ui ¼
n X
Aij gij
(2)
(5)
ðm1Þ gðmÞ AAA AAB gðm1Þ AA ¼ C AA ðAAA U A þ BA gAA AB Þ ðm1Þ m1Þ gðmÞ AAB AAA gðAA Þ, AB ¼ C AB ðAAB U A þ BB gAB
m ¼ 1; 2; 3 . . .
ð6Þ
where m is the iteration number. After obtaining the Tikhonov’s solution (m ¼ 0), it is possible to single out the contributions from specific pairs for iteration process. The algorithm of solving this inverse problem is treated in detail [4].
j¼1
where the function Ui(k) include experimental data wi and asymptote Aij. Using the regularization method [3], the solution of Eq. (1) is obtained as C ij Aij U i ¼ C ij Aij Ai1 gi1 þ . . . þ gij þ . . . þ C ij Aij Ain gin , ðj ¼ 1; 2; . . . ; nÞ.
ð3Þ
The symbol * denotes transposition, C ij ¼ ðAij Aij þ Bj Þ1 is the inverse integral operator, and Bj is the regularization matrix. Mathematical details can be found in Refs. [1,4]. We explain the sense of the procedure by giving an example for binary system AB: C AA AAA U A ¼ gð0Þ AA þ C AA AAA AAB gAB
a
Zn-Br(1) 8
(4)
3. Numerical simulation The effectiveness of the method has been tested on the model crystalline solids ZnBr2 [5]. We calculated pPCFs for Zn–Zn, Zn–Br and Br–Zn, Br–Br pairs with artificial values s2 ¼ 0.022 (Zn–Zn pair), s2 ¼ 0.012 (Zn–Br pair) and s2 ¼ 0.023 (Br–Br pair), s2 ¼ 0.013 (Br–Zn pair). Then Zn and Br EXAFS are calculated with statistical noise. The solutions obtained are shown in Fig. 1 (solid lines). Also we demonstrate Tikhonov’s solutions (dashed lines). Numerical values of the structural parameters of obtaining solutions are given in comparison crystalline ZnBr2 data (Table 1). We can see an excellent agreement.
b
Model Zn-Zn
Model Br-Zn(1)
Zn-Br(2)
8
4
Br-Br
Br-Zn(2)
4
0 0 8
Zn-Zn
4
g (r)
g (r)
8
0
Br-Br
4 0
8
8
Zn-Br
4
4
0
0 2
4
3 r, Å
5
Br-Zn
2
3
4
5
r, Å
Fig. 1. The pPCFs as a result of solution inverse problem for Zn (a) and Br (b) spectra: the model, Tikhonov’s solution (dashed line), the solution obtained after iterations (solid line).
ARTICLE IN PRESS Y.A. Babanov et al. / Nuclear Instruments and Methods in Physics Research A 575 (2007) 155–158
157
Table 1 The parameters of the first coordination shell for crystalline ZnBr2 obtained as a result of solving model inverse problem Zn-edge
Zn–Zn
Zn–Br
R
N
Model Tikhonov’s Iterations
3.86 3.87 3.87
Br-edge
Br–Br
Model Tikhonov’s Iterations
3.94 3.94 3.94
4.0 – 4.0
s2
R
N
s2
0.022 – 0.019
2.41 2.41 2.41
4.0 – 3.8
0.012 – 0.012
2.0 – 2.1
0.013 – 0.013
Br–Zn 11.6 – 11.9
0.023 – 0.022
2.41 2.41 2.41
R—interatomic distance (A˚), N—coordination number (at.), s2—Debye–Waller factor (A˚2).
b
8
Zn-O
4
14
0
0 32
32
Br-O
28
g (r)
g (r)
a
Br-Zn
Zn-Br
16
16
0
0 1
2
3 r, Å
4
5
1
2
3 r, Å
4
5
Fig. 2. The pPCFs as a result of solution inverse problem for Zn (a) and Br (b) spectra of 1 m aqueous ZnBr2 solution at T ¼ 303 K and P ¼ 250 bar.
Table 2 The parameters of the first coordination shell for 1 m aqueous ZnBr2 solution at T ¼ 303 K and P ¼ 250 bar obtained as a result of solving inverse problem Zn-edge
Zn–O
Zn–Br
R
N
s2
R
N
s2
Simonet [7] Present work
2.07 2.09
5.1 5.3
0.009 0.010
2.40 2.34
0.8 0.8
0.010 0.011
Br-edge
Br–O
Simonet [7] Present work
3.33 3.26
0.52 0.42
0.0075 0.015
Br–Zn 4.3 8.0
0.022 0.025
2.39 2.46
R—interatomic distance (A˚), N—coordination number (at.), s2—Debye–Waller factor (A˚2).
4. Experimental conditions and data analysis results We apply a new method for investigation of aqueous ZnBr2 solution. Recently, very peculiar properties of supercritical fluids have attracted much interest both from a fundamental point of view and for application. In normal T, P conditions, water dissolves ionic salts more easily than neutral compounds. This tendency is modified at supercritical conditions since water tends to behave like a nonpolar fluid. Through a change from normal to super-
critical conditions, the interaction strength between ionic electrical charges in aqueous solutions increases and can lead successively to ion pairs, complex formation and eventually to salt precipitation. In this study, we analyse the local atomic structure of one sample only. 1 m solution was prepared by dissolving high-purity anhydrous salt in demineralized water inside a polycrystalline sapphire cell designed for the study of fluids under high P, T conditions. EXAFS measurements at the Zn (around 9659 eV) and Br (around 13474 eV) K edges at
ARTICLE IN PRESS 158
Y.A. Babanov et al. / Nuclear Instruments and Methods in Physics Research A 575 (2007) 155–158
room temperature (303 K) and pressure 250 bar were performed using the spectrometer installed by the French Collaborative Reseach Group on the BM 32 beam line on a bending magnet (0.8 T) at the ESRF (Grenoble, France). Experimental details were given in Refs. [6,7]. If we take into account the small concentration ZnBr2 in solution, it is possible to eliminate the Zn–Zn pair contribution for the Zn EXAFS spectra and similarly with the Br–Br contribution for Br EXAFS spectra. We obtain the solution from Zn EXAFS spectra only Zn–O and Zn–Br pPCFs and, from Br EXAFS spectra only Br–O and Br–Zn pPCFs (Fig. 2). It is necessary to note that for the homogeneous system the pPCF Zn–Br is equal to pPCF Br–Zn. From Fig. 2, it is obvious that Zn–Br and Br–Zn contributions are practically coincident ones. The main structural characteristics are given in Table 2. Thus, the proposed method is characterized by high resolution for closely spaced coordination shells and by high accuracy in determining pPCFs for multicomponent systems. In this context, the method has vast prospects of applications in structural investigations conventional for the EXAFS.
Acknowledgements This work was supported by Grants: Program of Presidium RAS No. 26, RFFI No. 04-02-16464, Scientific School—5869.2006.2.
References [1] Yu.A. Babanov, V.R. Shvetsov, J. Phys. 47 (1986) C8-37. [2] A. Ankudinov, B. Ravel, J.J. Rehr, S. Conradson, Phys. Rev. B 58 (1998) 7565. [3] A.N. Tikhonov, V. Ya. Arsenin, Solution of Ill-Posed Problems, Willey, New York, 1981. [4] Yu.A. Babanov, I.Yu. Kamensky, J.-L. Hazemann, Y. Calzavara, D. Raoux, E. Journal ‘‘Investigated in Russia’’ 32 (2006) 305. /http:// zhurnal.ape.relarn.ru/articles/2006/032.pdfS. [5] C. Chieh, M.A. White, Z. Kristallogr. 166 (1984) 189. [6] Y. Soldo, J.-L. Hazemann, D. Aberdam, M. Inui, K. Tamura, D. Raoux, E. Pernot, J.F. Jal, J. Dupuy-Philon, Phys. Rev. B 57 (1998) 258. [7] V. Simonet, Y. Calzavara, J.-L. Hazemann, R. Argoud, O. Geaymond, D. Raoux, J. Chem. Phys. 117 (2002) 2771.
Nuclear Instruments and Methods in Physics Research A 575 (2007) 155–158 www.elsevier.com/locate/nima
Partial pair correlation functions for multicomponent systems by EXAFS: A new approach Yu. A. Babanova,b,, I. Yu. Kamenskya, J.-L. Hazemannc, Y. Calzavarac, D. Raouxd a
Institute of Metal Physics, S. Kovalevskaya Str. 18, Ekaterinburg 620219, Russia b Udmurt State University, Izhevsk, Russia c CNRS, Laboratoire de Crystallographie, Grenoble, France d Synchrotron SOLEIL, Paris, France Available online 13 January 2007
Abstract A method of determining partial pair correlation functions (pPCFs) for multicomponent systems from EXAFS data is presented. The method is based upon the regularization procedure of solving a Fredholm integral equation of the first kind using special iteration procedure. The effectiveness of the method has been tested on the model crystalline solids ZnBr2. We present the result for ZnBr2 aqueous solution obtained from experimental data at room temperature (303 K) and pressure 250 bar. Experiments were performed using an unique equipment at ESRF (Grenoble, France). Peculiar features of the method are high resolution for closely spaced shells and high accuracy in the determination of pPCFs. r 2007 Elsevier B.V. All rights reserved. PACS: 61.10.Ht Keywords: EXAFS; Regularization; Aqueous solution
1. Introduction EXAFS spectroscopy is used to probe the local atomic arrangement around a specific atomic species in amorphous alloys, catalysis, biological molecules, solutions, etc. The conventional procedure of extracting structural information in real space is based on Fourier transforming the normalized oscillating part of the X-ray absorption coefficient. It is well known that the peaks in Fourier transform are shifted to lower r from the positions of the corresponding peaks in partial pair correlation function (pPCF). Besides, the real space contribution of each shell is long range and sharply oscillatory. The contributions made by two closely spaced shells interfere strongly. Corresponding author. Institute of Metal Physics, S. Kovalevskaya Str. 18, GSP-170, Ekaterinburg 620219, Russia. Tel.: +7 343 3783837; fax: +7 343 3745244. E-mail address: [email protected] (Y.A. Babanov).
0168-9002/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2007.01.048
According to theory, the EXAFS can be described as an electron diffraction process where the electron source is the absorbing atom i. The outgoing photoelectron wave and a small part of it, which is backscattered from surrounding atoms j, interfere within the excited atom. This scattering is specified by not only the atomic amplitude fj, as is the case in X-ray diffraction, but also the phase jj. In addition, the scattering on the potential of the central atom i exhibits a phase shift 2di. These scattering characteristics are involved in the integral operator Aij, which, as it acts on the pPCF gij, generates a contribution of a specific atomic pair ij to the normalized oscillating part wi. In this paper, we propose to use the dependence of the integral operator Aij on the atomic scattering characteristics di(k), jj(k,r) for determining pPCFs in multicomponent systems from EXAFS data. Earlier, this idea was applied to extract information about bond lengths for multicomponent systems [1]. Now, we introduce a special iteration procedure for obtaining pPCFs.
ARTICLE IN PRESS Y.A. Babanov et al. / Nuclear Instruments and Methods in Physics Research A 575 (2007) 155–158
156
2. Mathematical
C AB AAB U A ¼ C AB AAB AAA gAA þ gð0Þ AB
EXAFS equation for n-component system has the form Z 1 n X 4pr0 wi ðkÞ ¼ RðkÞ cj f j ðk; rÞeð2r=lðkÞÞ k 0 j¼1
where C AA ¼ ðAAA AAA þ BA Þ1 and C AB ¼ ðAAB AAB þ BB Þ1 are the corresponding inverse operators. The function g(0) is called Tikhonov’s solution. In order to make the structure information more exact, we use the following iteration procedure:
sinð2kr þ cij ðk; rÞÞgij ðrÞ dr
ð1Þ
where k is the wave number of a photoelectron, r0 is the mean atomic density, R(k) is the amplitude reduction factor, cj is the concentration, and cij(k,r) ¼ 2di(k)+jj(k,r) is the total phase shift, which is a peculiar ‘‘mark’’ for a specific pair of atoms. All atomic scattering characteristics are calculated by FEFF 8.1 [2]. The function gij(r) is the pPCF, which is the probability to find atom j from atom i at the distance r by definition. In symbolic form we have Ui ¼
n X
Aij gij
(2)
(5)
ðm1Þ gðmÞ AAA AAB gðm1Þ AA ¼ C AA ðAAA U A þ BA gAA AB Þ ðm1Þ m1Þ gðmÞ AAB AAA gðAA Þ, AB ¼ C AB ðAAB U A þ BB gAB
m ¼ 1; 2; 3 . . .
ð6Þ
where m is the iteration number. After obtaining the Tikhonov’s solution (m ¼ 0), it is possible to single out the contributions from specific pairs for iteration process. The algorithm of solving this inverse problem is treated in detail [4].
j¼1
where the function Ui(k) include experimental data wi and asymptote Aij. Using the regularization method [3], the solution of Eq. (1) is obtained as C ij Aij U i ¼ C ij Aij Ai1 gi1 þ . . . þ gij þ . . . þ C ij Aij Ain gin , ðj ¼ 1; 2; . . . ; nÞ.
ð3Þ
The symbol * denotes transposition, C ij ¼ ðAij Aij þ Bj Þ1 is the inverse integral operator, and Bj is the regularization matrix. Mathematical details can be found in Refs. [1,4]. We explain the sense of the procedure by giving an example for binary system AB: C AA AAA U A ¼ gð0Þ AA þ C AA AAA AAB gAB
a
Zn-Br(1) 8
(4)
3. Numerical simulation The effectiveness of the method has been tested on the model crystalline solids ZnBr2 [5]. We calculated pPCFs for Zn–Zn, Zn–Br and Br–Zn, Br–Br pairs with artificial values s2 ¼ 0.022 (Zn–Zn pair), s2 ¼ 0.012 (Zn–Br pair) and s2 ¼ 0.023 (Br–Br pair), s2 ¼ 0.013 (Br–Zn pair). Then Zn and Br EXAFS are calculated with statistical noise. The solutions obtained are shown in Fig. 1 (solid lines). Also we demonstrate Tikhonov’s solutions (dashed lines). Numerical values of the structural parameters of obtaining solutions are given in comparison crystalline ZnBr2 data (Table 1). We can see an excellent agreement.
b
Model Zn-Zn
Model Br-Zn(1)
Zn-Br(2)
8
4
Br-Br
Br-Zn(2)
4
0 0 8
Zn-Zn
4
g (r)
g (r)
8
0
Br-Br
4 0
8
8
Zn-Br
4
4
0
0 2
4
3 r, Å
5
Br-Zn
2
3
4
5
r, Å
Fig. 1. The pPCFs as a result of solution inverse problem for Zn (a) and Br (b) spectra: the model, Tikhonov’s solution (dashed line), the solution obtained after iterations (solid line).
ARTICLE IN PRESS Y.A. Babanov et al. / Nuclear Instruments and Methods in Physics Research A 575 (2007) 155–158
157
Table 1 The parameters of the first coordination shell for crystalline ZnBr2 obtained as a result of solving model inverse problem Zn-edge
Zn–Zn
Zn–Br
R
N
Model Tikhonov’s Iterations
3.86 3.87 3.87
Br-edge
Br–Br
Model Tikhonov’s Iterations
3.94 3.94 3.94
4.0 – 4.0
s2
R
N
s2
0.022 – 0.019
2.41 2.41 2.41
4.0 – 3.8
0.012 – 0.012
2.0 – 2.1
0.013 – 0.013
Br–Zn 11.6 – 11.9
0.023 – 0.022
2.41 2.41 2.41
R—interatomic distance (A˚), N—coordination number (at.), s2—Debye–Waller factor (A˚2).
b
8
Zn-O
4
14
0
0 32
32
Br-O
28
g (r)
g (r)
a
Br-Zn
Zn-Br
16
16
0
0 1
2
3 r, Å
4
5
1
2
3 r, Å
4
5
Fig. 2. The pPCFs as a result of solution inverse problem for Zn (a) and Br (b) spectra of 1 m aqueous ZnBr2 solution at T ¼ 303 K and P ¼ 250 bar.
Table 2 The parameters of the first coordination shell for 1 m aqueous ZnBr2 solution at T ¼ 303 K and P ¼ 250 bar obtained as a result of solving inverse problem Zn-edge
Zn–O
Zn–Br
R
N
s2
R
N
s2
Simonet [7] Present work
2.07 2.09
5.1 5.3
0.009 0.010
2.40 2.34
0.8 0.8
0.010 0.011
Br-edge
Br–O
Simonet [7] Present work
3.33 3.26
0.52 0.42
0.0075 0.015
Br–Zn 4.3 8.0
0.022 0.025
2.39 2.46
R—interatomic distance (A˚), N—coordination number (at.), s2—Debye–Waller factor (A˚2).
4. Experimental conditions and data analysis results We apply a new method for investigation of aqueous ZnBr2 solution. Recently, very peculiar properties of supercritical fluids have attracted much interest both from a fundamental point of view and for application. In normal T, P conditions, water dissolves ionic salts more easily than neutral compounds. This tendency is modified at supercritical conditions since water tends to behave like a nonpolar fluid. Through a change from normal to super-
critical conditions, the interaction strength between ionic electrical charges in aqueous solutions increases and can lead successively to ion pairs, complex formation and eventually to salt precipitation. In this study, we analyse the local atomic structure of one sample only. 1 m solution was prepared by dissolving high-purity anhydrous salt in demineralized water inside a polycrystalline sapphire cell designed for the study of fluids under high P, T conditions. EXAFS measurements at the Zn (around 9659 eV) and Br (around 13474 eV) K edges at
ARTICLE IN PRESS 158
Y.A. Babanov et al. / Nuclear Instruments and Methods in Physics Research A 575 (2007) 155–158
room temperature (303 K) and pressure 250 bar were performed using the spectrometer installed by the French Collaborative Reseach Group on the BM 32 beam line on a bending magnet (0.8 T) at the ESRF (Grenoble, France). Experimental details were given in Refs. [6,7]. If we take into account the small concentration ZnBr2 in solution, it is possible to eliminate the Zn–Zn pair contribution for the Zn EXAFS spectra and similarly with the Br–Br contribution for Br EXAFS spectra. We obtain the solution from Zn EXAFS spectra only Zn–O and Zn–Br pPCFs and, from Br EXAFS spectra only Br–O and Br–Zn pPCFs (Fig. 2). It is necessary to note that for the homogeneous system the pPCF Zn–Br is equal to pPCF Br–Zn. From Fig. 2, it is obvious that Zn–Br and Br–Zn contributions are practically coincident ones. The main structural characteristics are given in Table 2. Thus, the proposed method is characterized by high resolution for closely spaced coordination shells and by high accuracy in determining pPCFs for multicomponent systems. In this context, the method has vast prospects of applications in structural investigations conventional for the EXAFS.
Acknowledgements This work was supported by Grants: Program of Presidium RAS No. 26, RFFI No. 04-02-16464, Scientific School—5869.2006.2.
References [1] Yu.A. Babanov, V.R. Shvetsov, J. Phys. 47 (1986) C8-37. [2] A. Ankudinov, B. Ravel, J.J. Rehr, S. Conradson, Phys. Rev. B 58 (1998) 7565. [3] A.N. Tikhonov, V. Ya. Arsenin, Solution of Ill-Posed Problems, Willey, New York, 1981. [4] Yu.A. Babanov, I.Yu. Kamensky, J.-L. Hazemann, Y. Calzavara, D. Raoux, E. Journal ‘‘Investigated in Russia’’ 32 (2006) 305. /http:// zhurnal.ape.relarn.ru/articles/2006/032.pdfS. [5] C. Chieh, M.A. White, Z. Kristallogr. 166 (1984) 189. [6] Y. Soldo, J.-L. Hazemann, D. Aberdam, M. Inui, K. Tamura, D. Raoux, E. Pernot, J.F. Jal, J. Dupuy-Philon, Phys. Rev. B 57 (1998) 258. [7] V. Simonet, Y. Calzavara, J.-L. Hazemann, R. Argoud, O. Geaymond, D. Raoux, J. Chem. Phys. 117 (2002) 2771.