M¨ ossbauer Spectroscopy Olivia L. Mello (Partner: Jeffrey C. Prouty)∗ MIT Department of Physics (Dated: May 8, 2013) We employ M¨ ossbauer spectroscopy, high resolution resonant absorption spectroscopy of nuclei embedded in a crystal. We measure the Zeeman splitting, ratio of magnetic moments, and internal magnetic field of 57 Fe, Fe2 O3 , and Fe3 O4 . We measure the isomer shift and quadrupole splittings that occur between transitions in these spectra, as well as for FeSO4 and Fe3 (SO4 )2 . In addition, we measure the natural linewidth of the 57 Fe 14.4 keV transition. By measuring the shift in the absorption peak of our spectrum at increased temperature, we demonstrate time dilation in our results due to the second-order Doppler Shift.
I.
INTRODUCTION AND OVERVIEW
In resonance absorption, the incident photon energy should be equal to the energy of a transition to a different state in the atom of choice. Due to recoil of the nuclei emitting a photon, however, the energy of the emitted photon is Doppler shifted. Thus, the resulting energy of the emitted photon is less than what is required for the transition. In addition, this recoil energy for a nuclei is larger than the absorption linewidth. Unless the energy of the emitted photon is greater than that of the transition, the probability for a transition to occur is nearly 0. In 1957, Rudolf M¨ ossbauer devised a way to avoid this effect. By embedding the nuclei in a crystal lattice, the recoil energy includes the mass of the entire lattice rather than a single nuclei. This makes this redshift of the emitted photons much smaller. By moving the source of the photons back and forth on a piston, this generates an additional small doppler shift in the emitted photons in order to scan across a more precise range of energies. [7] The doppler shift is given by E 0 = (1 +
V )E, c
(1)
where V is the velocity of the moving piston and the energy E 0 is in the rest frame of the absorbing species. The resulting resolution is on the order of 1012 , precise enough to measure the hyperfine splitting in ironcontaining species. The energy we are interested in using in this experiment is 14.4 keV γ-ray emission. The energy loss due to recoil of the emitted photon is given as ∆E =
Eγ2 p2 = , 2M 2M
(2)
where Eγ = 14.4keV. For the mass of an iron nuclei, this energy loss is 0.004eV. Embedded in a crystal with a mass of approximately 10−6 g, the recoil energy loss
is only 1.85×10−19 , small enough so it does not affect hyperfine transitions. The absorption and emission spectra of the 14.4keV photons have an intrinsic linewidth, that is usually recoilshifted and Doppler broadened. For an absorber of some thickness, integrating the absorption cross section over energy yields the expression for the count rate as a function of the piston velocity V. ! B C(v) = C0 1 − E0 V (3) ( c − ∆E)2 + Γ2 This is a Lorentzian with a full-width half-maximum with a center energy E0 . II.
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MEASURABLE EFFECTS II.1.
Hyperfine Splitting
A predominant effect in 57 Fe is Zeeman splitting due to interaction between the nucleus’ magnetic dipole moment and internal magnetic field for a total angular momentum j = l + s. l is orbital angular momentum and s is spin angular momentum. This is also known as hyperfine splitting. The hyperfine correction of the energies is given by EHF = gj µN B0 mj ,
(4)
where gj is the dimensionless Land´e g factor, B0 is the internal magnetic field, µN = 3.1526 × 1012 keV/kilogauss is the nuclear magneton, and mj is the z-component of the angular momentum with a range between ±j.[2] The 14.4 keV transition is between states l = 1 and l = 0. For spin- 21 particles, the possible values of mj are ± 21 for the l = 0 ground state, and j = ± 23 , ± 12 for the l = 1 first excited state. This results in six possible transitions, which we will observe in our absorption spectra. II.2.
∗
2cΓ E0
Isomer Shift and Quadrupole Splitting
If the chemical environment of the Fe nuclei in the source and absorber are different, the resulting electron
2 densities around the nuclei will be different as well. Interactions between the electron density and nuclei cause a small shift in the 14.4 keV transition, known as the isomer or chemical shift δ. This shift only depends on the change in electron density for spherically symmetric states. This shift in the resonant absorption spectrum can be measured as a shift in the maximum of the absorption peak from 0 velocity, or as the shift of two central peaks. [3] Quadrupole splitting arises from the interaction between a non spherically-symmetric charge distribution and surrounding ions with unfilled energy levels. The ground state spectrum is shifted by this interaction, but the actual splitting occurs between first excited state or higher energy levels. For different 57 Fe compounds we observe this effect as the difference in energy ∆Q between two first excited state transitions.
III.
EXPERIMENT AND APPARATUS
The source of 14.4 keV γ photons is a 57 Co source that undergoes β electron capture to become 57 Fe, resulting in the release of the desired photons. This source is connected to a piston that moves back and forth with a velocity determined by the M¨ ossbauer drive circuit. The source moves back and forth with a velocity that is a periodic sawtooth function in time. The result is that the absorption spectrum is swept over a fine range of velocities corresponding to energy values. The absorber is placed between the source and the proportional counter. The proportional counter receives radiation that is not re-absorbed by the absorber, and sends it to the multichannel scaler (MCS) to bin the radiation counts. Figure 1 shows this setup below.
diode. After observing the desired interference pattern, the signal is amplified and send to the computer software to observe the calibration curve.
IV.
DATA PRESENTATION AND ERROR ANALYSIS IV.1.
Velocity Calibration
We obtained three individual velocity calibrations with different discriminator settings and computed a weighted mean of all three calibrations. After performing a linear fit on the left side of the calibration curve and a quadratic fit on the right, we noticed that both fits systematically underestimated many points. A main concern for our raw calibrations were the small wiggles on either side of the curve displayed in the top of figure 2. When we changed the discriminator setting to different values, some of the wiggles disappeared and new ones would form on different areas of the curve. This led us to believe that the irregularities were due to the discriminator outputting to many noise pulses. We resolved this issue by modeling our na¨ıve fits as C(channel) = f (channel) + g(channel), where f (channel) is the number of interference fringes and g(channel) is the number of noise pulses. Thus, the points that our original fits systematically missed were the channels with the least noise. We fit quadratic functions to the minimum points on either side. This decreased our inaccuracy between the two center peaks of the 57 Fe spectrum by 15% to 2%, and yielded much better χ2ν−1 values. The small quadratic terms are to account for the fact that piston does not move with a precisely constant acceleration. 4
2.5
FixedOMirror
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Prop.O Counter
photodiode
Co-57OSource Mirror
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laser MSBOOutput
directOin
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0 −0.5 −500 4 x 10 2
The calibration of the spectrum from bins to velocity values is determined using a Michelson interferometer and a HeNe laser. After passing through a beam splitter, some of the laser light reaches the interferometer mirror that is connected to the piston. The piston moves back and forth, and the laser light eventually reaches a photo-
500
1000 1500 Actual velocity calibration
2000
2500
−24467.6 + 26.5501x −0.00309856x
1.5
χ2ν−1=0.51
1 0.5 0 0
FIG. 1. A block diagram of the Mossbauer setup including the Michelson interferometer.
0
2
Total Counts
Velocity
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Velocity calibration with naive fits
x 10
2
20326.1 − 17.8604x −0.00185189x2 χ2ν−1=1.18
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FIG. 2. The raw velocity calibration overlayed with a naive linear fit on the left side and quadratic fit on the right side. The bottom figure displays the modified fit to points with low noise, with more suitable χ2ν−1 values.
The formula below coverts the count values to an overall velocity calibration. 2Vi C = NT (5) λ
3 N is the number of passes of the piston, T is the dwell time per channel, and λ = 6328˚ Afor the HeNe laser. For our measurements, N=6044 passes and T=0.0001 seconds. Energy values can be obtained by using ∆E = Vc E for E=14.4 keV.
IV.2.
Zeeman Splitting in
57
Fe and Fe2 O3
After acquiring an overnight run, we obtained the following spectrum for 57 F e below. In order to determine the peak channels and FWHM of each peak, we fit the spectra too six Lorentzians. We calibrated the channel number corresponding to each fitting parameter to obtain the energy difference in the ground state ∆E0 as the difference between the outer second and third peak on either side. We calculated a weighted mean of these two peaks to yield the resulting value in table II, with uncertainties due to the standard error between the two values and propagated errors from fitting and the calibration added in quadrature. We did a similar procedure for ∆E1 , except with the difference in energy being between the two outermost peaks on either side. We calculated the magnitude of the ratio of g factors gg01 = µµ01 as the ratio of ∆E0 over ∆E1 . With the value g0 = 0.0903, we calculated the internal magnetic field using eq. II.1. The difference between the energies for ∆E1 was small enough that the quadrupole splitting occurred in digits beyond our level of accuracy. We applied the same procedure to the overnight spectrum we acquired for Fe2 O3 . In this case, the +3 oxidation state of the Fe ions and the presence of oxygen anions introduces an expected isomer shift and quadrupole splitting.
second order Doppler shift effect. δE 3kB T = E 2M c2
(6)
Where T is the temperature of the sample. To measure this, we decreased the M¨ossbauer Drive from 80 to 35 and acquired a 57 Fe spectrum both at room temperature and 120o C, only showing four of the six peaks for both as in figure 4. These serve as a calibration. At increased dispersion, we acquired spectra for stainless steels 14.4 keV peak at room temperature and 120o C to measure the energy shift. We determined the time dilation shift, d δE −1 . One caveat to this dT E , to be (−2.46 ± 1.32) K calculation is that the uncertainty was too large using our velocity calibration, so we needed to rely on the known splittings of 57 Fe. Using these known splittings, however, we are 1.8σ confident we determined the second order Doppler shift, as we are 0.28σ away from the accepted value of -2.09K−1 in [4].
Mossbauer 57Fe Spectra
1400
χ2ν−1=1.29
1300
Counts
1200
FIG. 4. An observation of the shift in the linewidth shift with increasing temperature.
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2000
IV.4.
Superposition of Zeeman Effects in Fe3 O4
Uncalibrated Velocity Bins
FIG. 3. The absorption spectrum of fit of six Lorenztians.
IV.3.
57
Fe, overlayed with a
Time Dilation of Absorption Peak
Increasing the temperature of a sample increases the velocity of particles relative to a stationary frame. This introduces time dilation that can be measured with the
Magnetite, or Fe3 O4 , has a unique structure with tetrahedral sites where there are eight Fe3+ ions, and octahedral lattice sites where there are eight Fe2+ and and Fe3+ ions each. This leads to a superposition of of two Zeeman patterns, one for each lattice. Collecting a weekend run at room temperature, we were only able to resolve the superposition for three known splittings. However, we were able to determine the relevant parameters of the Zeeman splitting at these sites by fitting nine Lorentzians in the appropriate regions. The measurements of gg10 ,∆E0 , ∆E1 , and B0 for both the octahedral and tetrahedral sites in
4 table II. In addition we observe small isomer shifts and quadrupole splittings. Magnetite Spectra
4
3.65
x 10
TABLE I. Values of the isomer shift in Fe2 (SO4 )3 ) in the first row, and the isomer shift and quadrupole splitting of FeSO4 in the second row, compared against their accepted values. Value δ [cm/s] ∆Q[cm/s] δ [cm/s]
3.6
Counts
3.55 3.5 3.45
Measured 0.043 ± 0.0035 0.260±0.015 0.138 ± 0.050
Accepted 0.055 ± 0.005 0.280 ± 0.05 .140 ± 0.05
3.4 3.35 3.3 0
500
1000
1500
2000
2500
Velocity bins (uncalibrated)
FIG. 5. Lorentzian fits on the two superimposed spectra of lattice sites in magnetite. Blue curves represent the absorption peaks of the octahedral sites and green curves represent the absorption peaks of the tetrahedral sites. χ2ν−1 varied between 1.17 and 1.6 for all fits.
and absorber nuclei, irregularities in the drive circuit, and the finite thickness of our absorbers. To approximate the natural linewidth of the 14.4 keV transition, we can measure the linewidth of Na3 Fe(CN)6 absorbers with varying thickness and extrapolate to 0. We measured the linewidth by doing overnight runs of 5 absorbers and performing a Lorentzian fit to each spectrum to measure the linewidth. Figure 7 shows the linear fit to this data. The linewidth at 0 thickness is the Zero Thickness Linewidth Extrapolation Linewidth HeVL
IV.5.
Isomer Shifts and Quadrupole Splitting in FeSO4 and Fe2 (SO4 )3
1.4 ´ 10-8 1.2 ´ 10-8 1. ´ 10-8
As both of these compounds have different chemical environments than the source, we can expect them to have isomer shifts. Least squares Lorentzian fitting yields these parameters for both FeSO4 and Fe2 (SO4 )3 , shown in table I compared against the known values. In addi-
8. ´ 10-9 6. ´ 10-9 4. ´ 10-9 2. ´ 10-9 0
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Thickness HmgL
FeSO Absorption Spectrum 4
2200
FIG. 7. A linear fit to determine the natural linewidth of the 14.4 keV transition at 0 thickness. χ2ν−1 = 0.51.
2000 1800 1600 Counts
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Fe2(SO4)3 Absorption Spectrum
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y-intercept, Γn = (7.70 ± 2.34) × 10−9 eV. This value is 1.8σ away from the accepted value in [7]. Additionally, we can measure the lifetime of the first excited state using the Heisenberg uncertainty principle for time-energy measurements. Γn (τ 12 ) = ~log2. Using this formula, τ = (5.92 ± 1.80) × 10−8 s.
Uncalibrated Velocity Bin
IV.7. FIG. 6. The absorption spectra of FeSO4 and Fe3 (SO4 )2 with Lorentzian fits superimposed to find the quadrupole splitting and isomer shifts. The fits were decent, with χ2ν−1 values of 0.77 and 0.78 for the top and bottom fits, respectively.
tion, FeSO4 exhibits quadrupole splitting within 1.3σ of the known value. IV.6.
Natural Linewidth Measurement
The natural linewidth of the 14.4 keV line experiences broadening due to inhomogeneities in the source
Error Analysis
From the uncertainties on the fitting parameters of our velocity calibration, we propagated uncertainties through each of our calibrations from channel number to velocity. This served as our systematic error. The uncertainty from the Lorentzian fitting parameters comes from the Poisson errors incorporated as weights in our nonlinear least squares fitting routines. These errors serve as our systematic uncertainty, as this uncertainty will decrease with repeated measurements and longer integration times. We propagated the uncertainty in the channel number of the Lorentzian fitting parameters through the ve-
5 TABLE II. The values of the relevant M¨ ossbauer parameters for 57 F e, Fe2 O3 , the octahedral sites in Fe3 O4 , and the tetrahedral sites in Fe3 O4 , respectively, compared against their accepted values. Value g0 g1
∆E0 [eV] ∆E1 [eV] B0 [kG] ∆Q [eV] δ [eV] g0 g1
∆E0 [eV] ∆E1 [eV] B0 [kG] ∆Q [eV] g0 g1
∆E0 [eV] ∆E1 [eV] B0 [kG] ∆Q [eV] δ [eV] g0 g1
∆E0 [eV] ∆E1 [eV] B0 [kG] ∆Q [eV] δ [eV]
Measured 1.78 ±0.086
Accepted 1.75
(2.00 ±0.038) × 10−7 ( 1.11 ±0.049) × 10−7 351.28 ±6.67 0.00 ±0.02 (0.0060 ±0.0043) × 10−7 1.68±0.23
1.90×10−7 -1.075×10−7 330 0.00 0.0 1.77
(2.94±0.23) × 10−7 (1.74±0.24) × 10−7 517±41 (0.5 ± 2.61) × 10−8 1.83±0.0043
2.93×10−7 1.66×10−7 513 5.7×10−9 1.76 ± 0.10
(2.94±0.036) × 10−7 (1.59±0.092) × 10−7 443.54±54.34 (8.95 ± 5.78) × 10−9 (3.11±1.26) × 10−8 1.75±0.0071
2.83×10−7 -1.61 ± 0.07 × 10−7 500 ± 20 (0.00 ±0.01) × 10−8 0.0 ±0.5 × 10−9 1.7 ± 0.1
(2.61±0.017) × 107 (1.49 ±0.071)107 409.86±26.57 (7.45 ± 6.83) × 10−9 1.99±0.72 × 10−8
(2.41 ±0.06) × 107 (-1.49 ±0.05) × 107 450 ± 20 (0.00 ±0.01) × 10−8 (0.00 ±0.01) × 10−8
locity calibration as well, and added it in quadrature
[1] Preston, R.S. et al. ”Mossbauer Effect in Metallic Iron”, Phys. Rev., 128, Number 5, pp. 2207-2218, (1962). [2] Griffiths, D. J. Introduction to Quantum Mechanics. Prentice Hall, 2005. [3] De Benedetti,S. et al. ”Electric Quadrupole Splitting and the Nuclear Volume Effect in the Ion of FE57”, Phys. Rev., V6N4, January, pp. 60-62, (1961) [4] Frauenfelder.H. The Moessbauer Effect: Collection of Reprints. WW. A. Benjamin, Inc., 1962. [5] Bauminger, R. et al. Study of the low-temperature transition in magnetite using moessbauer absorption. Phys.Rev., 122(5), 1961. [6] Kistner, O.C. and Sunyar, A. W. ”Evidence for Quadrupole Interaction of Fe57 and Influence of Chemical Binding on Nuclear Gamma-Ray Energy”, Phys. Rev., 4, pp. 412-415, (1960). note:Better copy than the one in
with the uncertainties we propagated with the calibration parameters. This was done consistently with all of our measurements apart from the second order Doppler shift, which included only statistical uncertainties due to the constraints we had. For a velocity calibration V (x) = ax2 + bx + c for a channel number x corresponding to a Lorentzian fitting parameter, our errors are as follows. 2 σtot
=
∂v ∂a
2
σa2
+
∂v ∂b
2
σb2
+
∂v ∂c
2
σc2
+
∂v ∂x
2
σx2 (7)
V.
CONCLUSIONS
Table II shows the compounded results for every Ironbearing compound that exhibited hyperfine splitting. We qualitatively observe and measure hyperfine splitting in 57 F e, Fe2 O3 , and Fe3 O4 . We determine the linewidth of the 14.4 keV line and observe the second order Doppler shift by increasing temperature. Additionally, we determine the differences of Zeeman parameters in the different structural regions of Fe3 O4 , qualitatively verifying the high precision of this spectroscopic technique. Generally, our results are in good quantitative agreement with the accepted results published in [1], [3], [5],[6], and [8]. Some of our uncertainties, especially for the quadrupole splitting of Fe2 O3 , are large and may lead to somewhat inconclusive results despite the fact that the actual value is close to the accepted value. Deviations of 3σ or more may be attributed to the fact that the samples are not perfectly homogeneous, particularly in the case of magnetite and its observed isomer shift and quadrupole splitting.
Frauenfelder [7] Junior Lab Staff, “M¨ ossbauer Spectroscopy”, 2012. [8] Hargrove, R.S. and K¨ undig,W. ”M¨ ossbauer Measurements of Magnetite below the Verwey Transition”, Solid State Communications, Vol. 8, pp. 303-308, 1970. Pergamon Press.
ACKNOWLEDGMENTS
The author acknowledges her partner, Jeffrey Prouty, in his equal participation in this experiment and its analysis. In addition, the author is grateful for the advice and input of 8.14 instructor Professor Nahn and T.A. Matthew Chan. All curve fitting was performed using least squares methods in MATLAB and Mathematica.
