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Zoltán Klencsár TRANSMISSION INTEGRAL ANALYSIS OF MÖSSBAUER SPECTRA DISPLAYING HYPERFINE PARAMETER DISTRIBUTIONS WITH ARBITRARY PROFILE Budapest, Hungary MSMS 2014 - Hlohovec u Břeclavi – Czech Republic 26-30 May, 2014 http://www.mosswinn.huz.klencsar@mosswinn.hu

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Aim: to combine the calculation of arbitrary profile hyperfine parameter distributions with transmission integral fitting

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Outline Introduction effective thickness, thickness effects, transmission integral The problem of combining arbitrary profile distribution calculation methods with transmission integral fitting Solution of the problem, algorithm workflow Test cases, demonstration of the capabilities of the algorithm Related issues in the MossWinn program

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Introduction „The broadening of an absorption line in the Mossbauer effect due to finite absorber thickness has first been treated by W.M. Visscher (unpublished notes).” H. Frauenfelder: The Mössbauer Effect, New York: W.A. Benjamin, Inc., 1962 S. Margulies, J.R. Ehrman, Nucl. Instr. Meth. 12, 131-137 (1961). S. Margulies, P. Debrunner, H. Frauenfelder, Nucl. Instr. Meth. 21, 217-231 (1963). The shape of the Mössbauer spectrum to be expected in a transmission- type experiment was treated by Margulies et al. for various experimental conditions including thick and thin as well as split and unsplit source and absorber: The thickness dependence of the apparent Mössbauer line / spectrum parameters were widely investigated and approximate formulas were derived among others for line intensity and line width parameters. See, e.g., J.M. Williams, J.S. Brooks, Nucl. Instr. Meth. 128, 363-372 (1975). and references therein.

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1. H. Frauenfelder, The Mössbauer Effect, New York: W.A. Benjamin, Inc., 1962, pp. 45- 2. S. Margulies, J.R. Ehrman, Nucl. Instr. Meth. 12, 131-137 (1961). 3. R.E. Meads, B.M. Place, F.W.D. Woodhams, R.C. Clark, Nucl. Instr. Meth. 98, 29-35 (1972). 4. S.A. Wender, N. Hershkowitz, Nucl. Instr. Meth. 98, 105-107 (1972). 5. N. Hershkowitz, R.D. Ruth, S.A. Wender, A.B. Carpenter, Nucl. Instr. Meth. 102, 205-217 (1972). 6. G. Hembree, D.C. Price, Nucl. Instr. Meth. 108, 99-106 (1973). 7. G.K. Shenoy, J.M. Friedt, Phys. Rev. Lett. 31, 419-422 (1973). 8. G.K. Shenoy, J.M. Friedt, Nucl. Instr. Meth. 116, 573-578 (1974). 9. S. Mørup, E. Both, Nucl. Instr. Meth. 124, 445-448 (1975). 10. R. Zimmermann, R. Doerfler, Hyperfine Interactions 12, 79-93 (1982) 11. D. Gryffroy, R.E. Vandenberghe, Nucl. Instr. Meth. 207, 455-458 (1983). 12. O. Ballet, Hyperﬁne Interactions 23 (1985) 133-177. 13. D.G. Rancourt, Nucl. Instr. Meth. Phys. Res. B 44, 199-210 (1989). 14. S. Margulies, P. Debrunner, H. Frauenfelder, Nucl. Instr. Meth. 21, 217-231 (1963). 15. J.D. Bowman, E. Kankeleit, E.N. Kaufmann, B. Persson, Nucl. Instr. Meth. 50, 13-21 (1967). 16. J. Heberle, Nucl. Instr. Meth. 58, 90-92 (1968). 17. B.T. Cleveland, J. Heberle, Physics Letters 36A, 33-34 (1971). 18. J.M. Williams, J.S. Brooks, Nucl. Instr. Meth. 128, 363-372 (1975). 19. P. Jernberg, Nucl. Instr. Meth. B4, 412-420 (1984). 20. J.G. Mullen, A. Djedid, G. Schupp, D. Cowan, Y. Cao, M.L. Crow, W.B. Yelon, Phys. Rev. B 37, 3226-3245 (1988). 21. J.G. Mullen, A. Djedid, D. Cowan, G. Schupp, M.L. Crow, Y. Cao, W.B. Yelon, Physics Letters A 127, 242-246 (1988). 22. M.C. Dibar-Ure, P.A. Flinn, A Technique for the Removal of the “Blackness” Distortion of Mössbauer Spectra, in Mössbauer Effect Methodology 7, edited by I.J. Gruverman, New York–London: Plenum Press, 1971, pp. 245-262. 23. T-M. Lin, R.S. Preston, Comparison of Techniques for Folding and Unfolding Mössbauer Spectra for Data Analysis, in Mössbauer Effect Methodology 9, edited by I.J. Gruverman, C.W. Seidel, D.K. Dieterly, New York–London: Plenum Press, 1974, pp. 205-224. 24. G.K. Shenoy, J.M. Friedt, H. Maletta, S.L. Ruby, Curve Fitting and the Transmission Integral: Warnings and Suggestions, in Mössbauer Effect Methodology 9, edited by I.J. Gruverman, C.W. Seidel, D.K. Dieterly, New York– London: Plenum Press, 1974, pp. 277-305. 25. D.L. Nagy, Physical and Technical Bases of Mössbauer Spectroscopy, in Mössbauer Spectroscopy of Frozen Solutions, edited by A. Vértes & D.L. Nagy, Budapest: Akadémiai Kiadó, 1990, pp. 34-39. 27. S.S. Hanna, R.S. Preston: Phys. Rev. 139, A722-725 (1965). 28. R.M. Housley, R.W. Grant, U. Gonser: Phys. Rev. 178, 514-522 (1969). 29. J.M. Williams, J.S. Brooks: Nucl. Instr. Meth. 128, 363-372 (1975). 30. U. Gonser, H. Fischer: Hyp. Int. 72, 31-44 (1992).

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The effective thickness, Surface number density (in 1/cm 2 ) of Mössbauer nuclei, e.g. 57 Fe. the probability of recoilless nuclear resonance absorption of resonant radiation by Mössbauer nuclei in the absorber. Maximum cross section for the resonant absorption per Mössbauer nucleus (e.g. 0 ≈ 25610 −20 cm 2 for 57 Fe). -I e and I g are the nuclear spin quantum numbers of the excited and ground state of the Mössbauer transition, - = / 2 where is the wavelength of the resonant radiation, and - is the internal conversion coefficient associated with the nuclear transition.

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Natural Fe surface density, mg / cm 2 Effective thickness, Thin absorber approximation is considered to be valid. ( < 1 ) “Clearly, the thin absorber limit will not normally be valid in Fe-57 work.” D.G. Rancourt: Nucl. Instr. Meth. Phys. Res. B 44, 199-210 (1989).

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Compared to the thin absorber case: broadening of lines change of relative line widths change of relative line intensities change of relative line amplitudes change of relative subspectrum areas (in case of multiple components) change of line shape profile Compared to the thin absorber case: broadening of lines change of relative line widths change of relative line intensities change of relative line amplitudes change of relative subspectrum areas (in case of multiple components) change of line shape profile thin thick ( = 20) Typical thickness effects Transmission

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I 0 and I 1 denote modified Bessel functions of the first kind: (see, e.g., D.L. Nagy, Physical and Technical Bases of Mössbauer Spectroscopy, in Mössbauer Spectroscopy of Frozen Solutions, edited by A. Vértes & D.L. Nagy, Budapest: Akadémiai Kiadó, 1990, pp. 34-39.) Considering the 57 Fe Mössbauer spectrum of a random powder sample displaying a full-blown sextet, the six individual absorption peaks “share” the total effective thickness in proportion to their “ideal” thin-absorber intensity ratios, and in the absence of polarization effects they display intensity and peak width values that to a good approximation correspond to the effective thicknesses 1,6 = 3 /12, 2,5 = 2 /12 and 3,4 = /12 for the 1-6, 2-5 and 3-4 peaks of the sextet, respectively. For s = a = 0 : = 8

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The transmission integral Mössbauer radiation emitted from the source with recoil. Mössbauer radiation emitted from the source without recoil, subject to nuclear resonant absorption. (non-resonant) mass absorption The counts of the experimental spectrum will scatter around the curve: Baseline Background fraction of detected counts due to radiation other than the Mössbauer radiation emitted from the source (e.g. X-rays, -rays admitted through the SCA window). G.K. Shenoy, J.M. Friedt, H. Maletta, S.L. Ruby, Curve Fitting and the Transmission Integral: Warnings and Suggestions, in Mössbauer Effect Methodology 9, edited by I.J. Gruverman, C.W. Seidel, D.K. Dieterly, New York–London: Plenum Press, 1974, pp. 277-305. For Lorentzian as well as for split and/or non-Lorentzian absorption shapes: absorption shape Scope: - Thin, single-line, unpolarized source. -Unpolarized, uniform absorber (or polarization and granularity effects may be neglected) R.M. Housley, R.W. Grant, U. Gonser: Phys. Rev. 178, 514-522 (1969). J.D. Bowman, E. Kankeleit, E.N. Kaufmann, B. Persson: Nucl. Instr. Meth. 50, 13-21 (1967). Natural line width of the Mössbauer transition. (e.g. 0 = 0.097 mm/s for 57 Fe)

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To calculate the transmission integral we need to be able to calculate A n (u): This is straightforward for models for crystalline phases („crystalline subspectra”) that can be fully parameterized, i.e. their shape is fully determined by the corresponding elements of the parameter vector p under evaluation. But how to handle hyperfine parameter distributions? Assume a certain shape for the distribution, e.g. Gaussian, binomial etc., that can be fully parameterized just like a crystalline subspectrum. But what if the distribution shape is not known in advance? Do successive approximation by adding up more and more of the fully parameterized distributions, and hope that their shape is suitable for a fast convergence, or … Use one of the “model-independent” distribution evaluation methods, that make use of the measured spectrum data to derive arbitrary profile distributions. B. Window, J. Phys. E: Sci. Instrum. 4, 401-402 (1971). J. Hesse, A. Rübartsch, J. Phys. E: Sci. Instrum. 7, 526-532 (1974). G. Le Caër, J.M. Dubois, J. Phys. E: Sci. Instrum. 12, 1083-1090 (1979). C. Wivel, S. Mørup, J. Phys. E: Sci. Instrum. 14, 605-610 (1981). L. Dou, R.J.W. Hodgson, D.G. Rancourt, Nucl. Instr. Meth. Phys. Res. B 100, 511-518 (1995). There is a problem though: all of these methods assume that the measured spectrum can be modeled with a fit envelope calculated as the weighted sum of elementary subspectra, and make use of the measured spectrum data accordingly in order to derive the elementary subspectrum weights in question. Such use of the measured data is not justified when the spectrum is subject to thickness effects.

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Use one of the “model-independent” distribution evaluation methods, that make use of the measured spectrum data to derive arbitrary profile distributions. B. Window, J. Phys. E: Sci. Instrum. 4, 401-402 (1971). J. Hesse, A. Rübartsch, J. Phys. E: Sci. Instrum. 7, 526-532 (1974). G. Le Caër, J.M. Dubois, J. Phys. E: Sci. Instrum. 12, 1083-1090 (1979). C. Wivel, S. Mørup, J. Phys. E: Sci. Instrum. 14, 605-610 (1981). L. Dou, R.J.W. Hodgson, D.G. Rancourt, Nucl. Instr. Meth. Phys. Res. B 100, 511-518 (1995). Hesse & Rübartsch: Smoothing matrix

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M.C. Dibar-Ure, P.A. Flinn, A Technique for the Removal of the “Blackness” Distortion of Mössbauer Spectra, in Mössbauer Effect Methodology 7, edited by I.J. Gruverman, New York–London: Plenum Press, 1971, pp. 245-262. - used discrete Fourier transformation (FFT) - in conjunction with Gaussian noise filtering - to extract the A n (u) absorption shape from spectra. D.G. Rancourt, Accurate Site Populations From Mössbauer Spectroscopy, Nucl. Instr. Meth. Phys. Res. B 44, 199-210 (1989). - suggested a clever (fitting-based) noise filtering and deconvolution procedure to extract A n (u) from spectra without numerical calculation of the Fourier transform of measured data. A possible general solution to the problem of thickness effects: Extract and fit the absorption shape: A n (u)

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A possible general solution to the problem of thickness effects: Extract and fit the absorption shape: A n (u) Problems The selection of the noise filter function has a degree of arbitrariness. No clue concerning the “best” filter function for a particular spectrum & fit problem. The effect of the unwanted spectral deformation/broadening (appearing in A n (u) due to the combined effect of the deconvolution and noise filtering) on the final fit parameters is dubious. The deconvolution and noise filtering will change the spectrum and statistics of the spectral noise, whose effect on the final fit parameters is dubious. Fitting of A n (u) is not an optimal solution.

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Measurement, or a quantity calculated on the basis of the measurement. Theoretical quantity that can be calculated with arbitrary precision. Doppler velocity values corresponding to the actual measured counts. 2 k equidistantly spaced velocity values. (2048)

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P range of w j > range of v i ℱ −1 = 1 exp( Ã n (w j )) Interpolation Convolution Hesse & Rübartsch Subtract crystalline D.L. Nagy, U. Röhlich, Hyp. Int. 66, 105-126 (1991). An extension of the H-R distribution calculation method to the case of thick absorbers.

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To examine the effectiveness of the outlined algorithm, a typical 57 Fe hyperfine magnetic field distribution, sampled in 70 equidistantly distributed points spanning the range of 0…35 T, was created by the addition of three Gaussians, and corresponding N(v i ) 57 Fe Mössbauer spectra were created for various values of b and by calculating the transmission integral via direct numerical integration with a relative precision of 10 −8. Sextets with equal ( a = 0.097 mm/s) line widths and relative line areas of 3:2:1:1:2:3 were used as the elementary pattern of the distribution, with a correlation ( = 0.4 mm/s − 0.02 mm s −1 T −1 B hf ) between the isomer shift and hyperfine magnetic field (B hf ), and with zero quadrupole splitting. The single-line Mössbauer source was assumed to have a recoilless fraction of f s = 0.7 and a line width of Γ S = 0.11 mm/s. Model spectra were then created by adding normally distributed random spectral noise to the spectra. Whereas the N() = 10 6 baseline and therefore the variance of the noise distribution are the same in all of the spectra, the S/N ratio of the Mössbauer signal depends on b as well as on . The algorithm, as realized in the MossWinn program, was used to fit the spectra by fixing the (above given) theoretical value for all the parameters in p with the exception of f c,r (“Filter cutoff”) and l (“Smoothing factor”). These were fitted to their optimal values that provided the lowest 2. Preparation of test spectra = 5, 10, 20, 30, 40, 50 b = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95 Spectra with singlet: = 5, 10, 20, 30, 40, 50 b = 0.8 = 20, b = 0.8

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range of w j > range of v i ℱ −1 = 1 exp( Ã n (w j )) Interpolation Convolution Hesse & Rübartsch Subtract crystalline

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Algorithm Workflow Stages Ãn(wj)Ãn(wj) An(wj)An(wj)An(wj)An(wj) under- smoothed over- smoothed

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Optimal value of the cutoff frequency = 10, b = 0.3 Ãn(wj)Ãn(wj) An(wj)An(wj)An(wj)An(wj) GIF animations were created with http://gifmaker.me/

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Automatic adaptation to the S/N ratio

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Ãn(wj)Ãn(wj) An(wj)An(wj)An(wj)An(wj)

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Optimal S/N Ratio of Ã n (w) = 5, b = 0.1 = 5, b = 0.9

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Broadening due to filtering f c,r = f c / (250 s/mm) G (FWHM) The effect of the relative cutoff frequency of the applied filter function (with s = 0.11 mm/s) on the width of a Gaussian with original FWHM of G = 0.1 mm/s. before filtering after filtering Q(f) filter I. Vincze, Nucl. Instr. Meth. 199, 247-262 (1982). For S/N = 16 … 64 : G ≳ s

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f c,r (optimal) 0.27 Effect of the cutoff frequency

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= 10 b = 0.8 n, Number of distribution data points The effect of the number of distribution data points ( = 10, b=0.8)

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n, Number of distribution data points = 40 b = 0.9 The effect of the number of distribution data points ( = 40, b=0.9)

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Comparison of results obtained with thin absorber approximation and with transmission integral fitting Distribution shape Relative area fraction of the crystalline component Goodness of the fit = 5, 10, 20, 30, 40, 50 b = 0.8 singlet

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= 5 = 10 = 20 = 30 = 40 = 50 b = 0.8 Thin absorber approximation

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= 5 = 10 = 20 = 30 = 40 = 50 b = 0.8 Transmission integral

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Thin absorber approximationTransmission integral Fit result Theory b = 0.8

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Thin absorber approximation Transmission integral

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Handling unfolded spectra

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Ũ (w j ) Ũ (w j ) Ũ (w j ) + Ũ (w j ) 2 How to take into account both parts of an unfolded spectrum

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Fitting unfolded spectra

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Mössbauer line sharpening where for 57 Fe Distribution of singlets. includes neither the source nor the absorber intrinsic line width.

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= 5, b = 0.2 = 20, b = 0.5 = 40, b = 0.6 = 5, b = 0.2 = 20, b = 0.5 = 40, b = 0.6

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= 20, b = 0.5 = 20, b = 0.8 singlet

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The algorithm can be applied… for 57 Fe as well as other Mössbauer nuclides available in MossWinn. to fit multiple distributions to the same spectrum. with various elementary patterns selected for the distribution, including patterns calculated via the numerical diagonalization of the static Hamiltonian for powder samples, as well as patterns displaying dynamic (relaxation) effects. in conjunction with the simultaneous fitting of several spectra allowing the application of constraints among the distribution and transmission integral parameters associated with the spectra fitted together.

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Conclusions An algorithm has been developed that successfully combines the model-independent distribution evaluation method of Hesse and Rübartsch with the calculation of the transmission integral for unpolarized absorbers, and enables the extraction of arbitrary- profile hyperfine parameter distributions from Mössbauer spectra of thin as well as of thick samples. An automatic treatment of noise filtering was successfully realized on one hand by binding the cutoff steepness of the applied filter function to the FWHM width of the source Lorentzian, on the other hand by treating the filter’s relative cutoff frequency as a fit parameter. As the algorithm handles the required noise filtering quasi automatically, the fit of arbitrary-profile hyperfine parameter distributions to Mössbauer spectra of unpolarized thick absorbers becomes straightforward in practice.

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Mossbauer spectroscopy

Mossbauer spectroscopy

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