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Mossbauer Spectroscopy of 57Fe
Margot Fitz Axen and Jake Novotny
Abstract:
Mossbauer spectroscopy was used to probe the hyperfine energy levels in 57Fe compounds. Due the to recoil free gamma ray absorptions and emissions of solids, the 10-9 eV hyperfine structure of these compounds' energy levels could be studied. By Doppler shifting gamma rays from a 57Co source, a spectrum of gamma rays was produced with 1 part in 1013 resolution, allowing the nuclear Zeeman effect, the isomer shift, and quadrupole splitting of 57Fe compounds to be measured. Using on a known calibration factor, we were able to achieve 10-9 eV precision, and our results ranged from 0.00-2.50 standard deviations from literature values, with the exception of a single 5.50 sigma event. Extensions of this work would include an attempt to measure the known calibration factor directly and to investigate the temperature dependence of these energy levels.
Introduction:
An atom's nucleus undergoes energy transitions through photon emission and absorption. Nuclear spectroscopic experiments are designed to observe the hyperfine energy levels of the nucleus, where it is known nuclear photon emissions/absorptions have a Lorentzian lineshape [1-6]. They involve bombarding a sample nucleus with a continuous spectrum of photon energies and observing the number of photons which pass through. Since an excited nucleus will generally re-emit in a random direction, we have a recipe for determining what the energy of the excitation was, but only if the emitted photon has the same energy as the photon which excited the nucleus in the first place. In fact, when this is the case it is most common to bombard a sample with a source of the same element in an excited state.
In order for this to work, resonant absorption must be satisfied. Resonant absorption simply means the energy emission spectrum of the source must overlap with the absorption spectrum of the sample. However, in a free atom, resonant absorption is cannot be achieved because in both emission and absorption, some of the energy of the transition goes into recoil kinetic energy of the nucleus, which occurs to satisfy momentum conservation. The recoil Kinetic energy of the nucleus is given by
where Eγ is the energy of the photon and m is the mass of the nucleus in the sample [5]. As long as this recoil kinetic energy is smaller than the natural line width of the transition resonant absorption will not be achieved. For example, for a free 57Fe nucleus the recoil kinetic energy is about 0.004 eV while the natural linewidths are on the order of 10−8 eV [6]: certainly this says an attempt at nuclear spectroscopy will be fruitless. Mossbauer spectroscopy solves this issue by using nuclei embedded in a solid lattice. Using this technique, nuclear recoil is negligible (on the order of 10-19 eV). This experiment used the technique of Mossbauer to observe the hyperfine nuclear energy levels of 57Fe, Fe2O3, and Fe3O4.
Theory:
57Co is a radioactive source which decays into an excited state of 57Fe through electron capture, shown in the above figure. As 57Fe decays to the ground state, it makes several transitions between nuclear energy levels characterized by their nuclear angular momentum j, each time emitting a photon of a certain energy [2]. For example, when it transitions from the j=3/2 to j=1/2 state, it emits a photon of 14.41 keV [3]. This was the transition of interest in this experiment. For spectroscopy to work at all, we require a spectrum of photon emissions which is approximately continuous with respect to the linewidth of the absorbers ( 57Fe, Fe2O3, and Fe3O4). The typical method for creating this spectrum is the Doppler effect. By giving the sample a linear velocity towards and away from the sample (so a velocity-time curve would be a triangle wave) with a max velocity on the order of cm/sec, an energy shift of 1 part in 1013 eV is possible. To first order in the limit of small velocities, we have ∆E = (v/c) E, where E is the energy of photons emitted by the source at rest, and v is the linear velocity of the source [3].
The line width Γ can be estimated by the Heisenberg uncertainty principle and is related to the lifetime of the excited state, τ through the relationship Γ = h/τ, although this point is subtle and a much more detailed discussion is required [harry]. Since Γ is known to increase linearly with sample thickness, the natural line width is often quoted to be the extrapolated linewidth at zero thickness, by observing samples of varying thickness. This was not of principal interest in our investigations, but a linewidth on the order of 10-8 eV was expected [6].
The hyperfine energy levels in the nucleus are primarily determined by three effects: the Zeeman effect, the isomer shift, and quadrupole splitting. The Zeeman effect ∆mj is caused by the movement of unpaired electrons around the nucleus, which generates an internal magnetic field [5]. The quadruple splitting δ arises because interactions with surrounding atoms create an antisymmetric electric field gradient, effectively creating a preferred direction [5]. Finally, the isomer Shift ε refers to the net effect of the nucleus interacting with the surrounding electrons, increases the magnitude of the energy and depends on the electron radius [5]. Since the radius of the first excited state, in general, is larger than the ground state, the net effect will be a scaling upwards of the energy levels [5]. The combined effect of these interactions is shown in the figure below.
It is experimentally known that the Zeeman effect is present in each of 57Fe, Fe2O3, and Fe3O4, but of these, quadrupole splitting only occurs in Fe2O3 while the isomer shift occurs in both Fe2O3 and Fe3O4 [5]. Additionally, Fe3O4 is known to split into 2 overlapping Zeeman spectra ("a" and "b") at room temperature. For more detail, see [5].
Therefore, on a histogram of photon counts vs energy shift (setting 14.41 keV = 0 on an absolute energy scale), the 6 arrows will appear as 6 peaks (dips) in photon counts, and the four parameters ∆3/2, ∆1/2, δ, and ε can be calculated.
Experimental Setup:
The figure below depicts the experimental setup used. The function generator powered the Mossbauer driver, which produced a linear velocity about zero and drove the motor. The motor held a 57Co source which decayed to an excited state of 57Fe, emitting photons at transition energies as it decayed to the ground state. In addition, the motor provided velocity amplitude data to the DAQ as a proportional Voltage. The sample was a solid bit of with an Fe nucleus, causing the source photons to either pass through in a straight line or be absorbed and reemitted in a random direction. Photons were observed with the detector and recorded only when a pre-calibrated single channel analyzer output a trigger pulse. This was done to ensure the DAQ was not flooded with counts from the 122 keV and 136 keV photons. The observed photons, with the velocity signal at the time of detection, went to the DAQ and were put into a Labview program. The Labview program made a histogram of channel number vs. number of photon counts.
Notice the mj values are flipped in the j = 1/2 state. This is because the alignment of the nuclear magnetic moment flips when transitioning to an excited state [6].
In the above figure, each line, numbered (1-6) represents a photon emission of energy 14.41 keV + ∆E# , where # takes one of the vales in {1,2,3,4,5,6}. The energy at each line can be calculated by summing the contributions from ∆, δ, ε. For example, the expected transition energy value for the first arrow is given by
Results and Analysis:
The Labview program provided histograms of number of photon counts vs. channel number for each sample tested. The first step taken for analysis was to recalibrate these plots in terms of energy. For calibration, we used the histogram provided by the pure 57Fe sample, first fitting Lorentzians to the peaks to determine the channel number of the peak locations. It is known that for this sample, the energy difference between the two outermost peaks is 10.657 mm/s. Using this information along with the observed energy difference between the peaks, we were able to convert every channel number to a corresponding velocity value, and using the Doppler shift relation to an energy value. Then, for each sample, we had three histograms with the peaks of each centered around the zero energy shift value.
We also adjusted the background level of photons, because it had a slight quadratic trend. It was desirable to have the background level be a flat line at 0 counts, so that Lorentzians could be fit more accurately (Origin software likes this). This was done by fitting a quadratic polynomial to the background level of photons, and then subtracting this off from each photon count value to obtain an adjusted count rate. Using these adjusted plots, Lorentzians were fit to all three sample histograms. The results are shown below.
The Lorentzian data provided by Origin gave both the energy shift locations of the peaks and their widths. Using a weighted average of all peak widths, we observed the line width to be (9.67 ± 0.02) * 10−9 eV. Next, we observed for each sample the energy shift values at peaks 1-6. These could be plugged into the appropriate energy shift equation- for example, the energy shift value at Peak 1 was put into Equation 1, and a similar process done for peaks 2-6. By subtracting values of these energy locations, then the values for the four desired parameters could be obtained. The results are shown in the tables below.
The Result columns are all in units of 10-7 eV, and deviation refers to the standard deviation of our result from expected. For a concise table of expected values, see [5].
For example, if we write E1 to be the length of arrow 1 and so on, we calculate these parameters in Fe2O3 to be
where we only have shown some of the possible calculations. By calculating all possible combinations, we were able to reduce our uncertainties (and double check our results).
All figures are original.
References:
[1] Shankar, Ramamurti. Principles of Quantum Mechanics. Springer Science and Business Media, 2012.
[2] Lustig, Harry. The Mossbauer Effect. American Journal of Physics 29, 1 (1961).
[3] Bearden, Alan J., P. L. Mattern, and P. S. Nobel. Mossbauer Effect Apparatus for an Advanced Undergraduate Teaching Laboratory. American Journal of Physics 32, 2 (1964)
[4] Gutlich, Philipp, Eckhard, Bill, and Trautwein, Alfred. Mossbauer Spectroscopy and Transition Metal Chemistry: Fundamentals and Applications. Springer Science & Busi- ness Media, 2010.
[5] Westerdale, Shawn. Mossbauer Spectroscopy of 57Fe. MIT Department of Physics, May 13, 2010. http://web.mit.edu/shawest/Public/jlab/Mossbauer/mossPaper.pdf
[6] Preston, R. S., S. S. Hanna, and J. Heberle. "Mossbauer effect in metallic iron." Physical Review 128, 5 (1962).
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