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Recoilless fraction determination by internal standard
Nuclear Instruments and Methods in Physics Research B 268 (2010) 2815–2819
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Recoilless fraction determination by internal standard K. Szyman´ski a,*, L. Dobrzyn´ski a,b, D. Satuła a, W. Olszewski a a b
University of Bialystok, Faculty of Physics, Lipowa 41, 15-424 Bialystok, Poland The Soltan Institute of Nuclear Studies, 05-400 Otwock-Swierk, Poland
a r t i c l e
i n f o
Article history: Received 16 April 2010 Received in revised form 31 May 2010 Available online 11 June 2010 Keywords: Mossbauer spectroscopy Recoilless fraction
a b s t r a c t Novel method of determination recoilless fraction in the transmission geometry is proposed. To avoid difficult problem of extraction of the non-resonant radiation, the a-Fe reference was used as an internal standard for the intensity scale. In order to reduce correlations between parameters fitted to measured spectra, at least three standard measurements have to be carried out. The results of measurements performed on FeSi absorbers with thicknesses varying by more than one order of magnitude demonstrate that proposed procedure correctly reconstructs non-linear dependence of the intensity on the sample thickness. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction The value of recoilless fraction, f, of the Mössbauer effect is intimately connected with site selective lattice dynamics. This quantity is a measure of average square of the amplitude of vibrations of the nuclear probe projected on the direction of the wave vector of the photon. Many methods of the measurement of recoilless fraction have already been reported. Essential difficulty encountered in its measurements consists in taking proper account of the non-resonant radiation detected as a background. Advanced corrections and methods of measurements of recoilless fraction by use of so called ‘‘black absorber” were developed in [1–3]. Recoilless fraction of 57Fe iron impurity in various hosts can be accurately measured if 57Co is introduced into the sample, so it can serve as a source [4,5]. Precise method of determination of absolute recoilless fraction was developed in [6,7]. In the cited papers, the absorber was measured together with the reference sample. The spectra were measured twice: one spectrum when the reference was in resonance with the source and the second one with reference far from the resonance. The difference between the spectra is a measure of the recoilless fraction of the absorber. Although enhanced resolution was observed in this type of measurements, they require the use of two transducers and a special mode of the data acquisition. This complexity of the method makes it not widely used. Recently, new methods of determination of the element selective phonon density of states (DOS), g(x), were developed. These methods involve advanced nuclear resonance scattering and nuclear inelastic resonance scattering of synchrotron radiation [8–12]. * Corresponding author. E-mail address: [email protected] (K. Szyman´ski). 0168-583X/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2010.06.009
The measured densities are directly related to the recoilless fraction and second order Doppler (SOD) shift. An absorber with finite thickness introduces non-Lorentzian absorption line shape, whose explicit form, in general, is not known. However, explicit relations for area and amplitude were derived for single absorption line [13–18]. These expressions were used for determination of the relative recoilless fraction by so called area method [18]. One should note that the shape of absorption cross-sections taken into interpretation of the Mössbauer spectrum influences estimation of effective absorber thicknesses, see Refs. [19,20]. Generalization of the area method was presented in [21]. Some authors uses the area method in simplified form assuming thin absorber approximation and claiming that area under the spectrum is proportional to the recoilless fraction of the absorber [22–24]. This is true in the limit of zero effective thickness. On the other hand in case of thin samples prepared from powders one cannot avoid neither inhomogeneity in thickness nor pinholes, both influencing final results. In another methods temperature dependence of the absorption area is measured and compared with the model of the lattice vibrations [3,25–28]. Then explicit temperature dependence of the average square of the lattice vibration amplitude, hx2i, in the Debye or the Einstein model is used to fit data and find the best value of the recoilless fraction. One can also use more complicated than the single line absorption cross-section to obtain best fitted f value [29] from transmission integral. However strong correlation between thickness and other parameters makes the results ambiguous, as discussed in [30]. It was demonstrated in [31] that simultaneous fitting of spectra measured with different thicknesses substantially improves precision of the results.
´ ski et al. / Nuclear Instruments and Methods in Physics Research B 268 (2010) 2815–2819 K. Szyman
2816
This paper shows that recoilless fraction of the absorber can be measured using rather simple measurements of the absorber jointed with the reference standard. Such sample with known recoilless fraction must be carefully selected. The spectra of the reference and the absorber should additionally be measured and included in simultaneous fit. It is of principal importance to operate with only two common absorption cross-sections during the fitting procedure, one for the reference and one for the absorber. By simultaneous fitting we avoid strong correlation between fitted parameters entering transmission integral. The proposed method uses well characterized reference with known absolute value of the recoilless fraction and relies on simultaneous fitting of spectra, which results in getting absolute value of the recoilless fraction of the absorber. 2. Idea of the proposed method Mössbauer spectrum S(v) in the transmission geometry and with infinitely thin source can be approximated by [32,33]:
Sðv Þ Cs ¼ 1 fs B 2p B
Z
1
1
1 ðE VÞ2 þ C2s =4
ð1 etrðEÞ ÞdE
ð1Þ
constant proportional to the accumulation time of the spectra, activity of the source and geometry of the experiment and electronic absorption in the sample the parameter dependent mainly on the recoilless fraction of the source. This value is strongly influenced by non-resonant radiation and electronic absorption. Thus the measured parameter fs is usually smaller than recoilless fraction of the source width of the Lorentzian distribution of photons emitted by a source, Cs P C0 where C0 is natural width. Broadening of Cs is observed because of self-absorption [32] and distribution of some hyperfine interactions in the source Doppler shift between the source and absorber, V = (v/c)Ec, v is the velocity of the source with respect to the absorber absorption cross-section of the absorber for the energy E the effective thickness of the absorber
fs
Cs
V
r(E) t
The effective thickness of the absorber is given by:
t ¼ r0 fa n
ð2Þ
where
r0
absorption cross-section at resonance [32] the recoilless fraction of the 57Fe in the absorber number of 57Fe atoms per unit area of the absorber
fa n
r(E) is the absorption cross-section proportional to the sum of the Lorentzian lines:
rðEÞ ¼
n X
ai C0 Ca =4
i¼1
ðE Ei Þ2 þ C2a =4
ð3Þ
;
with normalized intensities: n X
ai ¼ 1
ð4Þ
i¼1
or, in case of continuous distribution of the hyperfine interactions:
rðEÞ ¼
Z
1 1
aðxÞC0 Ca =4 2
ðE xÞ þ C
2 a =4
dx;
Z
1
1
aðxÞdx ¼ 1
ð5Þ
Ca is the absorption line width in the absorber, Ca P C0. The broadening of Ca may result from a distribution of some hyperfine interactions in the absorber. Because of the normalization of the amplitudes, the cross-section is normalized: Z
1
1
rðEÞdE ¼
pC0 2
ð6Þ
In widely used so called thin absorber approximation, i.e., small t n X SðVÞ ai ðCa þ Cs Þ=4 ¼ 1 tfs C0 2 2 B ðV E i Þ þ ðCa þ Cs Þ =4 i¼1
ð7Þ
Although transmission integral (1) results in the Lorentzian shape (7) only in the limit of zero thickness, it was shown that reasonably good Lorentzian fits could be obtained for thicknesses as large as t = 10, if the width of the Lorentzian line is free parameter [34]. This is the reason why thin absorber approximation is so successful and widely used. However, for thicknesses larger than one, the fitting with free widths introduces systematic errors in the fitted amplitudes or areas. This is also the reason of observing strong correlations between parameters t, fs Cs and Ca when these parameters are set as free in the fitting procedures [30]. Increase in t and decrease in Cs or Ca results in almost identical shape (1). In the case of thin absorber, Eq. (7), the amplitudes of the absorption lines are proportional to the single factor tfs(Cs + Ca) with unknown values of all four quantities. It is proposed here to measure spectrum of the absorber with unknown recoilless fraction, then measure it together with the reference, and measure finally the reference sample only. Measurements of the absorber prepared with different thicknesses will result in increased final precision. It is important in the last step to fit all the spectra simultaneously, using common absorption cross-section of the reference and common cross-section of the absorber. In the transmission experiment with two absorbers having cross-sections r1 and r2 and thicknesses t1 and t2, respectively, tr(E) in Eq. (1) should be replaced by t1r1(E) + t2r2(E), see [21,33]. The key point of proposed method is to use absorber with well characterized properties and with known thickness. The thickness of the reference sample is kept constant during the fits. This sample serves as a measure of the intensity scale in the spectrum. Thus it is used as an internal standard calibrating non-linear intensity scale. 3. Selection of the standard Natural iron foil is selected as the reference (standard) sample. The advantage of such standard is that it can be easily prepared in any laboratory. The foil or metallic film absorber can be prepared with much more homogeneous thickness than the absorber prepared from powder. When other than Fe foil compounds or alloys are used, one should be aware of the fact that stoichiometry, degree of order, oxidation state, concentration of interstitials and other defects may strongly influence properties of the absorber to be used as a standard. This control is much easier in the case of pure element. Another advantage of the iron foil is large separation between its absorption lines. Thus the standard can be used in experiments with broad as well as with narrow velocity ranges. Still another advantage of choosing iron foil as the standard follows from its chemical stability. Such foil protected simply by adhesive tape layer of organic lacquer against corrosion, is stable within tens of years in ambient laboratory condition. Achievements in the standard Mössbauer spectroscopy, nuclear resonant scattering and inelastic resonant scattering resulted in precise characterization of microscopic properties of iron. They include, e.g., temperature dependences of the recoilless fraction
´ ski et al. / Nuclear Instruments and Methods in Physics Research B 268 (2010) 2815–2819 K. Szyman
[8,9,12,35], and second order Doppler shift [10–12,36]. The cited papers indicate good agreement of these parameters, the average square of the amplitude of vibrations hx2i and average square of the velocity, hv2i, with the ones calculated within harmonic crystal approximation [37,38]:
h x coth gðxÞdx x 2kB T Z 3h hx hv 2 i ¼ 3hv 2x i ¼ x coth gðxÞdx 2m 2kB T hx2 i ¼
h 2m
Z
1
2
d¼
4. The experiment
hx2 i
1 hv 2 i 2 c
The disadvantage of the iron foil standard is its magnetic texture, which is influencing the line intensities nos. 2 and 5. Thus for each foil used as a standard texture must be measured in separate experiment.
ð8Þ
where g(x) is the phonon DOS. Indeed, if we take experimentally determined phonon DOS, perform integration (8) and calculate
f ¼ ek
2817
ð9Þ
then the temperature dependencies of f and d turn out to be very slightly dependent on whether g(x) was determined by means of the neutron inelastic scattering [39] or by means of the inelastic nuclear resonant scattering [11], see Fig. 1. We note that no free parameters are involved in (8) and (9). It is a common practice to use the Debye approximation for calculations of the integrals in Eq. (8). Results of such calculations are shown in Fig. 1 by black lines. Because all the discussed lines overlap, the differences between them are shown in the insets. It is interesting to note that the Debye temperature which fits well to the data is 435 K in Fig. 1a and 419 K in Fig. 1b. The difference in the Debye temperatures must result from approximation of the real shape of the DOS, g(x), by quadratic dependence, see discussion in [40]. Doubtless, a-Fe could be considered as a standard for which predictions (8) and (9) in relatively large temperature range agree well with the experimental data. From the solid lines representing (9) we get at T = 295 K f = 0.7985(5), df/dT = 5.43(5) 104 K1, dd/dT = 6.61(5) 104 mm s1 K1.
Standard spectrometer working in constant acceleration mode equipped in proportional counter was used for measurement the Mössbauer spectra. Proportional counter with resolution not better than 10% was used as a detector. Thus non-resonant and resonant radiation was registered together and – in contrast to many reported methods in [1–3] – their fraction could be quite arbitrary. The thickness (about 8 lm) of natural a-Fe foil was measured by two independent methods. Thickness was measured by micrometer at many points forming a grid (1 mm 1 mm). In the second method we have measured the area and the weight. Results of both methods agrees within the experimental error and the effective thickness at T = 295 K is t = 3.025(35). To demonstrate that the effective thickness can be unambiguously determined a set of FeSi absorbers with different mass per unit area was prepared. The thin FeSi absorber was measured without iron foil and this spectrum served as information source for the absorption cross-section of FeSi. The differences of spectra of all absorbers measured jointly with iron foil followed uniquely from different thicknesses of absorbers. Spectra were measured at low (Fig. 2) as well as at large (Fig. 3) velocity range to demonstrate that Fe reference can be used at any requested velocity ranges. The reference was measured within broad velocity range in order to determine relative line intensities. The simultaneous fitting methods are well developed in our laboratory [41–43] software is used in cases when the standard Mössbauer packages [44] does not have enough flexibility for arranging the constraints between common parameters describing
Fig. 1. (a) recoilless fraction of a-Fe, 1 – [30]; 2 – [36]; squares – [8] corrected as in [12]; triangles and diamonds – [9]. (b) Second order Doppler shift 1 – [11]; 2 – W. Sturhahn, Argonne National Laboratory, unpublished, see [12]; 3 – [10]; triangles – [37] corrected for zero point vibrations; squares – [12]. Three strongly overlapping lines shows results of (8) and (9) with DOS taken from [40] (dotted line), taken from [11] (dashed line). Solid lines represent the Debye approximation with the Debye temperature 435 K in (a) and 419 K in (b). The insets show the differences between the values obtained from given line and the arithmetic average of all three values obtained from the three overlapping lines.
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´ ski et al. / Nuclear Instruments and Methods in Physics Research B 268 (2010) 2815–2819 K. Szyman
Fig. 4. Effective thicknesses obtained from simultaneous fit plotted as function of FeSi mass per unit area in the absorbers.
with mass per unit area. Proportionality of both parameters show consistency of the method in quite wide thickness range. Recoilless fraction of FeSi could thus be determined from the slope of straight line in Fig. 4, see Eq. (2). It equals to f = 0.70(2) at room temperature. 5. Discussion
Fig. 2. Examples of Mössbauer transmission spectra of (a) FeSi absorber (4.89 mg/ cm2) (b) FeSi absorber (9.97 mg/cm2) measured together with the reference (c) the reference measured at small velocity range. Solid lines show results of the best simultaneous fit for all measured spectra of FeSi absorbers and the reference. Thin lines represent Lorentzian components of the absorption cross-section r(E) (1).
Fig. 3. Example of Mössbauer transmission spectra of FeSi absorber (17.30 mg/cm2) measured together with the reference within broad velocity range. Solid lines are results of the best simultaneous fit for all measured spectra of FeSi absorbers and the reference.
the different spectra. One of advantage of the Mathematica package is that the convolution (1) can be performed easily and precisely since built-in +infinity and infinity symbols can be used as limits of integration. Thus the use of transmission integral is almost as simple as use of thin absorber approximation. All measured spectra (the reference, the absorber, the absorbers of different thicknesses plus the reference) were fitted simultaneously by transmission integral. During spectra analysis the formulas contained only two common absorption cross-sections, Zeeman sextet for the reference and doublet for the absorber. The thickness parameter of the reference was fixed while the thicknesses of the absorbers were free. Examples of the best fits are shown in Figs. 2 and 3 while the best values of the fitted thicknesses in Fig. 4 together
Novel and easy method of measurement of recoilless fraction is described. The proportionality of the thicknesses obtained from the simultaneous fit to the mass per unit area shows that the non-linear intensity scale is correctly reproduced irrespectively of the uncontrolled fraction of non-resonant radiation, due to the use of proportional counter with relatively low, about 10%, resolution. This is an advantage since measurements are simple and experiments can be performed fast. The foil of a-Fe was selected as the reference. The analysis of collection of earlier published papers show that using experimentally determined phonon density of states both, the recoilless fraction and the second order Doppler shift, can be predicted with no free parameters and with accuracy sufficient for the presented method or typical Mössbauer measurements. One concludes that observation of anharmonic effect in aFe larger than shown in Fig. 1 will require very precise experiment. As was mentioned, magnetic texture of the selected standard is important in precise experiment. This has to be determined in experiment performed within large velocity range. In addition, one requires detailed knowledge of the constraints introduced during the simultaneous fitting of transmission integral. So far, no commercial software which permits the simultaneous fits with constraints as in this paper is available. However, because software is developing quickly, one may expect that in the future proposed options for fitting will be included in commercial packages. Acknowledgements The work was supported as a research project NN202172335. References [1] R.M. Housley, N.E. Erickson, J.G. Dash, Measurement of recoil-free fractions in studies of the Mössbauer effect, Nuclear Instruments and Methods 27 (1964) 29–36. [2] R.M. Housley, Discussion of factors affecting the absolute accuracy of Mössbauer f measurements, Nuclear Instruments and Methods 35 (1965) 77–82. [3] C. Hohenemser, Measurements of the Mössbauer recoilless fraction in beta-Sn for 1.3 to 370 K, Physical Review 139 (1965) A185–A196. [4] R.H. Nussbaum, D.G. Howard, W.L. Nees, C.S. Steen, Lattice-dynamical properties of 57Fe impurity atoms in Pt, Pd and Cu from precision measurements of Mössbauer fractions, Physical Review 173 (1968) 653–663. [5] W.A. Steyert, R.D. Taylor, Lattice dynamical studies using absolute measurements of the Lamb Mössbauer recoil-free fraction, Physical Review 134 (1964) A716–A722.
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[25] L.D. Lafleur, C. Goodman, Characteristic temperatures of the Mössbauer fraction and thermal shift measurements in iron and iron salts, Physical Review B 4 (1971) 2915–2920. [26] D.J. Erikson, L.D. Roberts, J.W. Burton, J.O. Thomson, Precision determinations of the Mössbauer Recoilless fraction for metallic gold in the temperature range 4.2 < T < 100 K, Physical Review B 3 (1971) 2180–2186. [27] W. Keune, Lattice-dynamical study of SnTe by Mössbauer spectroscopy, Physical Review B 10 (1974) 5057–5062. [28] I. Nowik, I. Jacob, R. Moreh, Mössbauer study of crystallographic and magnetic phase transitions, phonon softening and hyperfine interactions in Zr(AlxFe1x)2, Physical Review B 47 (1993) 723–726. [29] H. Vogel, H. Spiering, W. Irler, U. Volland, G. Ritter, The lamb-mössbauer factor of metal iron foils at 4. 2K, Journal de Physique, Colloque 3 (Suppl.) (1979) C2–676. [30] G.K. Shenoy, J.M. Friedt, H. Maletta, S.L. Ruby, Curve fitting and the transmission integral: warnings and suggestions, Mössbauer Effect Methodology 9 (1974) 277–305. [31] D.G. Agresti, M. Belton, J. Webb, S. Long, Determination of transmission integral parameters by simultaneous fits to thick-absorber mössbauer spectra: application to 151Eu2O3, Mössbauer Effect Methodology 9 (1974) 225–243. [32] S. Margulies, J.R. Ehrman, Transmission and line broadening of resonance radiation incident on a resonance absorber, Nuclear Instruments and Methods in Physics Research 12 (1961) 131–137. [33] S. Margulies, P. Debrunner, H. Frauenfelder, Transmission and line broadening in the Mössbauer effect, II, Nuclear Instruments and Methods in Physics Research 21 (1963) 217–231. [34] J.W. Robinson (Ed.), Handbook of Spectroscopy, CRC Press, Boca Raton, FL, 1981. [35] T.A. Kovats, J.C. Walker, Mössbauer absorption in 57Fe in metallic iron from the Curie point to the g–d transition, Physical Review 181 (1969) 610–618. [36] R.S. Preston, S.S. Hanna, J. Heberle, Mössbauer effect in metalic iron, Physical Review 128 (1962) 2207–2218. [37] R.M. Housley, F. Hess, Analysis of Debye–Waller-factor and Mössbauerthermal-shift measurements. I General theory, Physical Review 146 (1966) 517–526. [38] R.D. Taylor, P.P. Craig, Correlation between Mössbauer resonance strength and second order Doppler shift: estimate of zero-point velocity, Physical Review 175 (1968) 782–787. [39] V.J. Minkiewicz, G. Shirane, R. Nathans, Phonon dispersion relation for iron, Physical Review 162 (1967) 528–531. [40] B. Kolk, in: Studies of Dynamical Properties of Solids with the Mössbauer Effect, North-Holland, Amsterdam, 1984. [41] K. Szymanski, D. Satula, L. Dobrzynski, E.V. Voronina, E.P. Yelsukov, T. Miyanaga, Arrangements of magnetic moments in nanocrystalline Fe48Al52, Physical Review B 72 (2005) 104409-1–104409-12. [42] K. Szyman´ski, W. Olszewski, D. Satuła, L. Dobrzyn´ski, K. Brzózka, J. JankowskaKisielin´ska, Microscopic properties of c-FeMn studied by Mössbauer spectroscopy, Journal of Physics: Condensed Matter 78 (2009) 124708. [43] Wolfram Research Inc., Mathematica, Version 7.0, Champaign, IL, 2008. [44] R.A. Brand, NORMOS-98 Mössbauer Fitting Program Package, 1998.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Recoilless fraction determination by internal standard K. Szyman´ski a,*, L. Dobrzyn´ski a,b, D. Satuła a, W. Olszewski a a b
University of Bialystok, Faculty of Physics, Lipowa 41, 15-424 Bialystok, Poland The Soltan Institute of Nuclear Studies, 05-400 Otwock-Swierk, Poland
a r t i c l e
i n f o
Article history: Received 16 April 2010 Received in revised form 31 May 2010 Available online 11 June 2010 Keywords: Mossbauer spectroscopy Recoilless fraction
a b s t r a c t Novel method of determination recoilless fraction in the transmission geometry is proposed. To avoid difficult problem of extraction of the non-resonant radiation, the a-Fe reference was used as an internal standard for the intensity scale. In order to reduce correlations between parameters fitted to measured spectra, at least three standard measurements have to be carried out. The results of measurements performed on FeSi absorbers with thicknesses varying by more than one order of magnitude demonstrate that proposed procedure correctly reconstructs non-linear dependence of the intensity on the sample thickness. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction The value of recoilless fraction, f, of the Mössbauer effect is intimately connected with site selective lattice dynamics. This quantity is a measure of average square of the amplitude of vibrations of the nuclear probe projected on the direction of the wave vector of the photon. Many methods of the measurement of recoilless fraction have already been reported. Essential difficulty encountered in its measurements consists in taking proper account of the non-resonant radiation detected as a background. Advanced corrections and methods of measurements of recoilless fraction by use of so called ‘‘black absorber” were developed in [1–3]. Recoilless fraction of 57Fe iron impurity in various hosts can be accurately measured if 57Co is introduced into the sample, so it can serve as a source [4,5]. Precise method of determination of absolute recoilless fraction was developed in [6,7]. In the cited papers, the absorber was measured together with the reference sample. The spectra were measured twice: one spectrum when the reference was in resonance with the source and the second one with reference far from the resonance. The difference between the spectra is a measure of the recoilless fraction of the absorber. Although enhanced resolution was observed in this type of measurements, they require the use of two transducers and a special mode of the data acquisition. This complexity of the method makes it not widely used. Recently, new methods of determination of the element selective phonon density of states (DOS), g(x), were developed. These methods involve advanced nuclear resonance scattering and nuclear inelastic resonance scattering of synchrotron radiation [8–12]. * Corresponding author. E-mail address: [email protected] (K. Szyman´ski). 0168-583X/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2010.06.009
The measured densities are directly related to the recoilless fraction and second order Doppler (SOD) shift. An absorber with finite thickness introduces non-Lorentzian absorption line shape, whose explicit form, in general, is not known. However, explicit relations for area and amplitude were derived for single absorption line [13–18]. These expressions were used for determination of the relative recoilless fraction by so called area method [18]. One should note that the shape of absorption cross-sections taken into interpretation of the Mössbauer spectrum influences estimation of effective absorber thicknesses, see Refs. [19,20]. Generalization of the area method was presented in [21]. Some authors uses the area method in simplified form assuming thin absorber approximation and claiming that area under the spectrum is proportional to the recoilless fraction of the absorber [22–24]. This is true in the limit of zero effective thickness. On the other hand in case of thin samples prepared from powders one cannot avoid neither inhomogeneity in thickness nor pinholes, both influencing final results. In another methods temperature dependence of the absorption area is measured and compared with the model of the lattice vibrations [3,25–28]. Then explicit temperature dependence of the average square of the lattice vibration amplitude, hx2i, in the Debye or the Einstein model is used to fit data and find the best value of the recoilless fraction. One can also use more complicated than the single line absorption cross-section to obtain best fitted f value [29] from transmission integral. However strong correlation between thickness and other parameters makes the results ambiguous, as discussed in [30]. It was demonstrated in [31] that simultaneous fitting of spectra measured with different thicknesses substantially improves precision of the results.
´ ski et al. / Nuclear Instruments and Methods in Physics Research B 268 (2010) 2815–2819 K. Szyman
2816
This paper shows that recoilless fraction of the absorber can be measured using rather simple measurements of the absorber jointed with the reference standard. Such sample with known recoilless fraction must be carefully selected. The spectra of the reference and the absorber should additionally be measured and included in simultaneous fit. It is of principal importance to operate with only two common absorption cross-sections during the fitting procedure, one for the reference and one for the absorber. By simultaneous fitting we avoid strong correlation between fitted parameters entering transmission integral. The proposed method uses well characterized reference with known absolute value of the recoilless fraction and relies on simultaneous fitting of spectra, which results in getting absolute value of the recoilless fraction of the absorber. 2. Idea of the proposed method Mössbauer spectrum S(v) in the transmission geometry and with infinitely thin source can be approximated by [32,33]:
Sðv Þ Cs ¼ 1 fs B 2p B
Z
1
1
1 ðE VÞ2 þ C2s =4
ð1 etrðEÞ ÞdE
ð1Þ
constant proportional to the accumulation time of the spectra, activity of the source and geometry of the experiment and electronic absorption in the sample the parameter dependent mainly on the recoilless fraction of the source. This value is strongly influenced by non-resonant radiation and electronic absorption. Thus the measured parameter fs is usually smaller than recoilless fraction of the source width of the Lorentzian distribution of photons emitted by a source, Cs P C0 where C0 is natural width. Broadening of Cs is observed because of self-absorption [32] and distribution of some hyperfine interactions in the source Doppler shift between the source and absorber, V = (v/c)Ec, v is the velocity of the source with respect to the absorber absorption cross-section of the absorber for the energy E the effective thickness of the absorber
fs
Cs
V
r(E) t
The effective thickness of the absorber is given by:
t ¼ r0 fa n
ð2Þ
where
r0
absorption cross-section at resonance [32] the recoilless fraction of the 57Fe in the absorber number of 57Fe atoms per unit area of the absorber
fa n
r(E) is the absorption cross-section proportional to the sum of the Lorentzian lines:
rðEÞ ¼
n X
ai C0 Ca =4
i¼1
ðE Ei Þ2 þ C2a =4
ð3Þ
;
with normalized intensities: n X
ai ¼ 1
ð4Þ
i¼1
or, in case of continuous distribution of the hyperfine interactions:
rðEÞ ¼
Z
1 1
aðxÞC0 Ca =4 2
ðE xÞ þ C
2 a =4
dx;
Z
1
1
aðxÞdx ¼ 1
ð5Þ
Ca is the absorption line width in the absorber, Ca P C0. The broadening of Ca may result from a distribution of some hyperfine interactions in the absorber. Because of the normalization of the amplitudes, the cross-section is normalized: Z
1
1
rðEÞdE ¼
pC0 2
ð6Þ
In widely used so called thin absorber approximation, i.e., small t n X SðVÞ ai ðCa þ Cs Þ=4 ¼ 1 tfs C0 2 2 B ðV E i Þ þ ðCa þ Cs Þ =4 i¼1
ð7Þ
Although transmission integral (1) results in the Lorentzian shape (7) only in the limit of zero thickness, it was shown that reasonably good Lorentzian fits could be obtained for thicknesses as large as t = 10, if the width of the Lorentzian line is free parameter [34]. This is the reason why thin absorber approximation is so successful and widely used. However, for thicknesses larger than one, the fitting with free widths introduces systematic errors in the fitted amplitudes or areas. This is also the reason of observing strong correlations between parameters t, fs Cs and Ca when these parameters are set as free in the fitting procedures [30]. Increase in t and decrease in Cs or Ca results in almost identical shape (1). In the case of thin absorber, Eq. (7), the amplitudes of the absorption lines are proportional to the single factor tfs(Cs + Ca) with unknown values of all four quantities. It is proposed here to measure spectrum of the absorber with unknown recoilless fraction, then measure it together with the reference, and measure finally the reference sample only. Measurements of the absorber prepared with different thicknesses will result in increased final precision. It is important in the last step to fit all the spectra simultaneously, using common absorption cross-section of the reference and common cross-section of the absorber. In the transmission experiment with two absorbers having cross-sections r1 and r2 and thicknesses t1 and t2, respectively, tr(E) in Eq. (1) should be replaced by t1r1(E) + t2r2(E), see [21,33]. The key point of proposed method is to use absorber with well characterized properties and with known thickness. The thickness of the reference sample is kept constant during the fits. This sample serves as a measure of the intensity scale in the spectrum. Thus it is used as an internal standard calibrating non-linear intensity scale. 3. Selection of the standard Natural iron foil is selected as the reference (standard) sample. The advantage of such standard is that it can be easily prepared in any laboratory. The foil or metallic film absorber can be prepared with much more homogeneous thickness than the absorber prepared from powder. When other than Fe foil compounds or alloys are used, one should be aware of the fact that stoichiometry, degree of order, oxidation state, concentration of interstitials and other defects may strongly influence properties of the absorber to be used as a standard. This control is much easier in the case of pure element. Another advantage of the iron foil is large separation between its absorption lines. Thus the standard can be used in experiments with broad as well as with narrow velocity ranges. Still another advantage of choosing iron foil as the standard follows from its chemical stability. Such foil protected simply by adhesive tape layer of organic lacquer against corrosion, is stable within tens of years in ambient laboratory condition. Achievements in the standard Mössbauer spectroscopy, nuclear resonant scattering and inelastic resonant scattering resulted in precise characterization of microscopic properties of iron. They include, e.g., temperature dependences of the recoilless fraction
´ ski et al. / Nuclear Instruments and Methods in Physics Research B 268 (2010) 2815–2819 K. Szyman
[8,9,12,35], and second order Doppler shift [10–12,36]. The cited papers indicate good agreement of these parameters, the average square of the amplitude of vibrations hx2i and average square of the velocity, hv2i, with the ones calculated within harmonic crystal approximation [37,38]:
h x coth gðxÞdx x 2kB T Z 3h hx hv 2 i ¼ 3hv 2x i ¼ x coth gðxÞdx 2m 2kB T hx2 i ¼
h 2m
Z
1
2
d¼
4. The experiment
hx2 i
1 hv 2 i 2 c
The disadvantage of the iron foil standard is its magnetic texture, which is influencing the line intensities nos. 2 and 5. Thus for each foil used as a standard texture must be measured in separate experiment.
ð8Þ
where g(x) is the phonon DOS. Indeed, if we take experimentally determined phonon DOS, perform integration (8) and calculate
f ¼ ek
2817
ð9Þ
then the temperature dependencies of f and d turn out to be very slightly dependent on whether g(x) was determined by means of the neutron inelastic scattering [39] or by means of the inelastic nuclear resonant scattering [11], see Fig. 1. We note that no free parameters are involved in (8) and (9). It is a common practice to use the Debye approximation for calculations of the integrals in Eq. (8). Results of such calculations are shown in Fig. 1 by black lines. Because all the discussed lines overlap, the differences between them are shown in the insets. It is interesting to note that the Debye temperature which fits well to the data is 435 K in Fig. 1a and 419 K in Fig. 1b. The difference in the Debye temperatures must result from approximation of the real shape of the DOS, g(x), by quadratic dependence, see discussion in [40]. Doubtless, a-Fe could be considered as a standard for which predictions (8) and (9) in relatively large temperature range agree well with the experimental data. From the solid lines representing (9) we get at T = 295 K f = 0.7985(5), df/dT = 5.43(5) 104 K1, dd/dT = 6.61(5) 104 mm s1 K1.
Standard spectrometer working in constant acceleration mode equipped in proportional counter was used for measurement the Mössbauer spectra. Proportional counter with resolution not better than 10% was used as a detector. Thus non-resonant and resonant radiation was registered together and – in contrast to many reported methods in [1–3] – their fraction could be quite arbitrary. The thickness (about 8 lm) of natural a-Fe foil was measured by two independent methods. Thickness was measured by micrometer at many points forming a grid (1 mm 1 mm). In the second method we have measured the area and the weight. Results of both methods agrees within the experimental error and the effective thickness at T = 295 K is t = 3.025(35). To demonstrate that the effective thickness can be unambiguously determined a set of FeSi absorbers with different mass per unit area was prepared. The thin FeSi absorber was measured without iron foil and this spectrum served as information source for the absorption cross-section of FeSi. The differences of spectra of all absorbers measured jointly with iron foil followed uniquely from different thicknesses of absorbers. Spectra were measured at low (Fig. 2) as well as at large (Fig. 3) velocity range to demonstrate that Fe reference can be used at any requested velocity ranges. The reference was measured within broad velocity range in order to determine relative line intensities. The simultaneous fitting methods are well developed in our laboratory [41–43] software is used in cases when the standard Mössbauer packages [44] does not have enough flexibility for arranging the constraints between common parameters describing
Fig. 1. (a) recoilless fraction of a-Fe, 1 – [30]; 2 – [36]; squares – [8] corrected as in [12]; triangles and diamonds – [9]. (b) Second order Doppler shift 1 – [11]; 2 – W. Sturhahn, Argonne National Laboratory, unpublished, see [12]; 3 – [10]; triangles – [37] corrected for zero point vibrations; squares – [12]. Three strongly overlapping lines shows results of (8) and (9) with DOS taken from [40] (dotted line), taken from [11] (dashed line). Solid lines represent the Debye approximation with the Debye temperature 435 K in (a) and 419 K in (b). The insets show the differences between the values obtained from given line and the arithmetic average of all three values obtained from the three overlapping lines.
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´ ski et al. / Nuclear Instruments and Methods in Physics Research B 268 (2010) 2815–2819 K. Szyman
Fig. 4. Effective thicknesses obtained from simultaneous fit plotted as function of FeSi mass per unit area in the absorbers.
with mass per unit area. Proportionality of both parameters show consistency of the method in quite wide thickness range. Recoilless fraction of FeSi could thus be determined from the slope of straight line in Fig. 4, see Eq. (2). It equals to f = 0.70(2) at room temperature. 5. Discussion
Fig. 2. Examples of Mössbauer transmission spectra of (a) FeSi absorber (4.89 mg/ cm2) (b) FeSi absorber (9.97 mg/cm2) measured together with the reference (c) the reference measured at small velocity range. Solid lines show results of the best simultaneous fit for all measured spectra of FeSi absorbers and the reference. Thin lines represent Lorentzian components of the absorption cross-section r(E) (1).
Fig. 3. Example of Mössbauer transmission spectra of FeSi absorber (17.30 mg/cm2) measured together with the reference within broad velocity range. Solid lines are results of the best simultaneous fit for all measured spectra of FeSi absorbers and the reference.
the different spectra. One of advantage of the Mathematica package is that the convolution (1) can be performed easily and precisely since built-in +infinity and infinity symbols can be used as limits of integration. Thus the use of transmission integral is almost as simple as use of thin absorber approximation. All measured spectra (the reference, the absorber, the absorbers of different thicknesses plus the reference) were fitted simultaneously by transmission integral. During spectra analysis the formulas contained only two common absorption cross-sections, Zeeman sextet for the reference and doublet for the absorber. The thickness parameter of the reference was fixed while the thicknesses of the absorbers were free. Examples of the best fits are shown in Figs. 2 and 3 while the best values of the fitted thicknesses in Fig. 4 together
Novel and easy method of measurement of recoilless fraction is described. The proportionality of the thicknesses obtained from the simultaneous fit to the mass per unit area shows that the non-linear intensity scale is correctly reproduced irrespectively of the uncontrolled fraction of non-resonant radiation, due to the use of proportional counter with relatively low, about 10%, resolution. This is an advantage since measurements are simple and experiments can be performed fast. The foil of a-Fe was selected as the reference. The analysis of collection of earlier published papers show that using experimentally determined phonon density of states both, the recoilless fraction and the second order Doppler shift, can be predicted with no free parameters and with accuracy sufficient for the presented method or typical Mössbauer measurements. One concludes that observation of anharmonic effect in aFe larger than shown in Fig. 1 will require very precise experiment. As was mentioned, magnetic texture of the selected standard is important in precise experiment. This has to be determined in experiment performed within large velocity range. In addition, one requires detailed knowledge of the constraints introduced during the simultaneous fitting of transmission integral. So far, no commercial software which permits the simultaneous fits with constraints as in this paper is available. However, because software is developing quickly, one may expect that in the future proposed options for fitting will be included in commercial packages. Acknowledgements The work was supported as a research project NN202172335. References [1] R.M. Housley, N.E. Erickson, J.G. Dash, Measurement of recoil-free fractions in studies of the Mössbauer effect, Nuclear Instruments and Methods 27 (1964) 29–36. [2] R.M. Housley, Discussion of factors affecting the absolute accuracy of Mössbauer f measurements, Nuclear Instruments and Methods 35 (1965) 77–82. [3] C. Hohenemser, Measurements of the Mössbauer recoilless fraction in beta-Sn for 1.3 to 370 K, Physical Review 139 (1965) A185–A196. [4] R.H. Nussbaum, D.G. Howard, W.L. Nees, C.S. Steen, Lattice-dynamical properties of 57Fe impurity atoms in Pt, Pd and Cu from precision measurements of Mössbauer fractions, Physical Review 173 (1968) 653–663. [5] W.A. Steyert, R.D. Taylor, Lattice dynamical studies using absolute measurements of the Lamb Mössbauer recoil-free fraction, Physical Review 134 (1964) A716–A722.
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