Zoltán Klencsár

TRANSMISSION INTEGRAL ANALYSIS OF

MÖSSBAUER SPECTRA DISPLAYING

HYPERFINE PARAMETER DISTRIBUTIONS

WITH ARBITRARY PROFILE

Budapest, Hungary

MSMS 2014 - Hlohovec u Břeclavi – Czech Republic26-30 May, 2014

http://www.mosswinn.hu z.klencsar@mosswinn.hu

Aim: to combine the calculation of arbitrary profile hyperfine

parameter distributions with transmission integral fitting

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

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of an absorption line in the Mossbauer effect due to finite absorber thickness has first been treated by W.M. Visscher (unpublished notes).”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.

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).

The effective thickness, t

0aa fn

an

af

0

Surface number density (in 1/cm2) of Mössbauer nuclei, e.g. 57Fe.

the probability of recoilless nuclear resonance absorption of resonant g radiation by Mössbauer nuclei in the absorber.

Maximum cross section for the resonant absorption per Mössbauer nucleus (e.g. 0 ≈ 25610−20 cm2 for 57Fe).

1

1

12

12π2

g

e20 I

I

- Ie and Ig 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.

Natural Fe surface density, mg / cm2

Eff

ect

ive t

hic

kness

, t 8.0a f

0aa fnFe57

Thin absorber approximation is considered to be valid.

( t < 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).

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 (t = 20)

Typical thickness effectsTr

ansm

issi

on

222exp

2

π)( 10

s0 IIf

I

22exp1

222exp

)(

0

10

0

I

II

W

              

I0 and I1 denote modified Bessel functions of the first kind: deπ

1 π

0

cos0 xxI decos

π

1 π

0

cos1 xxI

(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 57Fe 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

The transmission integral

aan e)de)(1()( )(SSSS

duA u,vuLffvT

4)(

1

π2)( 2

S2

SSS

vu

,vuL

0n 2

π)(

duuA

))(

)()1(()()( b

T

vTbNvN

Mössbauer radiationemitted from the source with recoil.

Mössbauer radiation emitted from the source without recoil, subject to nuclear resonant absorption.

(non-resonant)mass

absorption

aae)( dT 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:

absorptionshape

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. G0 = 0.097 mm/s for 57Fe)

)de)(1()(

)( )(SSSS

n

u,vuLffT

vT uA

To calculate the transmission integral we

need to be able to calculate An(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.

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).

j

iijji vhLhDvN Noise][),()()(~

)( ivN

minimumconstrain)]()(~

[ 2

iii vNvN

NLSLLD~

][ T1T Hesse & Rübartsch:

Smoothing matrix

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 An(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 t An(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: An(u)

A possible general solution to the

problem of thickness effects:

Extract and fit the absorption shape: An(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 An(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 An(u) is not an optimal solution.

Measurement, or a quantity calculated on the basis of the measurement.X

~

Theoretical quantity that can be calculated with arbitrary precision.X

iv Doppler velocity values corresponding to the actual measured counts.

jw 2k equidistantly spaced velocity values.(2048)

)(~

ivN

)( ivN

i i

ii

vN~

vNvN~

)(

)]()([)(

22 p

))()(

)1(()()( bT

vTbNvN i

i

uvuL

f

TvTvU

uAi

s

ii

d)e1(),(

)()/(~

1)(

~

)(~

SSn

P

)de),(1()()( )(

SSSSn

uvuLffT

vT uAi

i

b

bN

vN

T

vTi

i

1)(

)(~

)(

)(~

,,,,),( ssfbN

2048..1),(~

jwU j

range of wj > range of vi

)( fQ

ℱ−1 = 1exp( Ãn(wj))

)(~

n jwA

nT1T ][

~ÃLSLLD

DD~

)(n jwA)(

)(

T

wT jInterpolation Convolution

Hesse & RübartschSubtractcrystalline

)( c

c

e1

e1)( ffa

af

fQ

D.L. Nagy, U. Röhlich, Hyp. Int. 66, 105-126 (1991).

fSπe

Sπa

)(4

4

cs

cs

e1

e1)( ff

f

fQ

rc,s ,,,,,),( ffbN s

An extension of the H-R distribution calculation method to the case of thick absorbers.

To examine the effectiveness of the outlined algorithm, a typical 57Fe 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(vi)

57Fe 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 Bhf) between the isomer shift and hyperfine magnetic field (Bhf), and with zero quadrupole splitting. The single-line Mössbauer source was assumed to have a recoilless fraction of fs = 0.7 and a line width of ΓS = 0.11 mm/s.

Model spectra were then created by adding normally distributed random spectral noise to the spectra. Whereas the N() = 106 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 fc,r (“Filter cutoff”) and l (“Smoothing factor”). These were fitted to their optimal values that provided the lowest 2.

Preparation of test spectra

t = 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:t = 5, 10, 20, 30, 40, 50b = 0.8

t = 20, b = 0.8

)(~

ivN

)( ivN

i i

ii

vN~

vNvN~

)(

)]()([)(

22 p

))()(

)1(()()( bT

vTbNvN i

i

uvuL

f

TvTvU

uAi

s

ii

d)e1(),(

)()/(~

1)(

~

)(~

SSn

)de),(1()()( )(

SSSSn

uvuLffT

vT uAi

i

b

bN

vN

T

vTi

i

1)(

)(~

)(

)(~ 2048..1),(

~jwU j

range of wj > range of vi

)( fQ

ℱ−1 = 1exp( Ãn(wj))

)(~

n jwA

nT1T ][

~ÃLSLLD

DD~

)(n jwA)(

)(

T

wT jInterpolation Convolution

Hesse & RübartschSubtractcrystalline

Algorithm Workflow Stages

10

Ãn(wj)An(wj)

)(4

4

cs

cs

e1

e1)( ff

f

fQ

max,

rc,c

c

f

ff 10

rc, f

min)(rc,

2f un

der-

smoo

thed

over-

smoothed

Optimal value of the cutoff frequency

)(4

4

cs

cs

e1

e1)( ff

f

fQ

= 10, b = 0.3

Ãn(wj)An(wj)

GIF animations were created with http://gifmaker.me/

Automatic adaptation to the S/N ratio

10

10

Ãn(wj)An(wj)

Optimal S/N Ratio of Ãn(w)

u,vuLf

TvTvU uA

s

d)e1()()()/(1

)( )(SS

n

= 5, b = 0.1

))(

)()1(()()( b

T

vTbNvN

= 5, b = 0.9

Broadening due to filtering

fc,r = fc / (250 s/mm)

GG

(F

WH

M)

The effect of the relative cutoff frequency of the applied filter function (with Gs = 0.11 mm/s) on the width of a Gaussian with original FWHM of GG = 0.1 mm/s.

)(4

4

cs

cs

e1

e1)( ff

f

fQ

before filtering

after filtering

Q(f) filter

I. Vincze, Nucl. Instr. Meth. 199, 247-262 (1982).

S/Nln

2

2min

(S/N)log

2

2

sG

For S/N = 16 … 64 :

GG ≳ Gs

fc,r(optimal) 0.27

Effect of the cutoff frequency

t = 10b = 0.8

n, Number of distribution data points

n, Number of distribution data points

The effect of the number of

distribution data points(t = 10, b=0.8)

n, Number of distribution data points

n, Number of distribution data points

t = 40b = 0.9

The effect of the number of

distribution data points(t = 40, b=0.9)

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

t = 5, 10, 20, 30, 40, 50

b = 0.8

singlet

t = 5 t = 10

t = 20 t = 30

t = 40 t = 50

b = 0.8Thin absorber approximation

t = 5 t = 10

t = 20 t = 30

t = 40 t = 50

b = 0.8Transmission integral

Thin absorber approximation Transmission integral

Fit result

Theory

b = 0.8

Thin absorber approximationTransmission integral

Handling unfolded spectra

Ũ- +(wj)

Ũ+ -(wj)

Ũ- +(wj) + Ũ+ -(wj)

2

How to take into account both parts of

an unfolded spectrum

Fitting unfolded spectra

Mössbauer line sharpening

)de),(1()(

)( )(SSSS

n

uvuLffT

vT uA))(

)()1(()()( b

T

vTbNvN

)()()( 0n ,vvLvPvA jij

ji

mm/s097.00 where for 57Fe

Distribution of singlets.

)( jvPincludes neither the source nor the absorber intrinsic line width.

n

~ÃLSLLPD T1T ][

),( 0 jiij vvLL

4)(

4),(

20

2

20

0 /vv

/vvL

jiji

1801...j

t = 5, b = 0.2 t = 20, b = 0.5

t = 40, b = 0.6

)()()( 0n ,vvLvPvA jj

j t = 5, b = 0.2

t = 20, b = 0.5

t = 40, b = 0.6

)( jvP

t = 20, b = 0.5

t = 20, b = 0.8

t = 20, b = 0.8t = 20, b = 0.8

singlet)()()( 0n ,vvLvPvA jj

j

The algorithm can be applied…

• for 57Fe 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.

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.