Moessbauer Spectroscopy With Quadrature Interferometery

Introduction

Mossbauer spectroscopy aims to probe the absorption spectra of nuclei through gamma radiation. In this experiment we explored the properties of the Fe57 nucleus, with gamma radiation from the decay of Co57. A full absorption spectra of energy can be created with the one resonant energy that Co57 emits by oscillating the source to introduce a Doppler shift about the resonant energy, allowing us to probe the nucleus with many different energies. To get an energy spectrum, it is important to know the velocity, allowing us to use the Doppler effect relation to calculate energy from velocity. We will use a quadrature interferometer to make measurements of the velocity of the source as it oscillates. This allows for very fine energy resolution. We will be using Mossbauer spectroscopy to measure the properties of the Fe57 nucleus, specifically the Zeeman effect, isomer shifts, and quadrupole splitting. We will look at each of these properties with three different iron compounds, an iron sheet, Fe2O3, and Fe3O4. Each of these compounds have different properties expected for Fe57 nuclei.

Mossbauer spectroscopy is famous for its incredible energy resolution, and can be used for many different types of measurement. The theoretical limiting resolution of Mossbauer spectroscopy is 1e-9 eV. To put this in perspective it is the same as stacking paper from the Earth to the moon, and being able to measure the difference if one sheet was removed. Mossbauer spectroscopy was used as an early confirmation of Einstein's General Theory of Relativity by measuring a very small change in the gravitational redshift of Earth. The resolution of this technique is one of few ways to measure the minute changes from relativity here on Earth [1]. Mossbauer spectroscopy is used in several other fields, such as geology, nuclear physics, and bio-inorgainic chemistry [2]. The fine energy resolution and iron compound identification make Mossbauer spectroscopy an important tool used in many disciplines.

Theory

Mossbauer spectroscopy is a high precision technique to examine the properties of the nuclear absorption spectra. It is based in the idea of recoil free absorption and re-emission, in which the lattice structure of the sample negates any energy going to momentum of the nucleus. In our case with Fe57 nuclei, we are able to find precision of one part in one trillion. This accuracy comes from the precise measurements energy shift of source, as will be discussed in this section. With this precision we were able to measure the hyperfine structure of the Fe57 nucleus energy levels.

Mossbauer Effect

Mossbauer spectroscopy is dependent on the Mossbauer effect. This effect is takes advantage of the recoil free emission and absorption in atoms bound in a lattice structure.

In a gas, when a photon is emitted or absorbed, the atom is free to recoil, as it must to conserve momentum. This recoil requires energy, which changes the emission and absorption energy of the transition. In a gas, the emission and absorption energies of transitions are not equal. This makes precision spectroscopy difficult. The Mossbauer effect uses atoms in lattice structures to increase the mass that recoils. This means little energy is transferred into recoil, making emission and absorption energies equivalent [2].

Doppler Shift

The Mossbauer effect allows high precision spectroscopy, but how does Mossbauer spectroscopy achieves this level of precision? This is where Mossbauer spectroscopy utilizes the Doppler shift.

In order to observe an absorption spectra, we hit our iron samples with radiation from Co57, which emits photons at the resonant energy of Fe57 nuclei (14.4keV). We were only interested in the photons at the resonant energy, but a spectra cannot be observed with only one energy. In order to look at a spectrum of energies, we oscillated the source in order to introduce a Doppler shift to the photons incident upon the iron samples. The shift in frequency of the photons from Doppler shift is given by equation 3, where $f_0$ is the frequency of the photons when the source is at rest, v/c is the fraction of the speed of light the source is moving at, and f_shift is by how much to photon's frequency changes.

E_shift = (v/c) E_0

For this energy spread to be appropriate, it must be possible to resolve energy measurements much smaller than 1e-5 eV. To get an understanding of the limiting resolution, we look at the time-energy Hiesenberg uncertainty relation.

When a Fe57 nucleus is the excited state (J=3/2), it will decay back down to the ground state ($J=1/2$) with a mean time of 141 ns. This is the event that happens when Co57 decays to Fe57, and releases a photon from the J=3/2 to J=1/2 transition. The time of this process is short enough that Heisenberg uncertainty relation for energy and time is the limiting factor for the energy resolution [3].

ΔE = h/[2 Δt] = hbar/[2(141e-9)] = 2.33e-9 eV

Following equations 3 and 4, our limiting energy is on the order of 1e-9 eV, and any peaks in our data can (theoretically) be detected if they are wider than 1e-9 eV.

Nuclear Properties

-Zeeman Splitting

The main feature of the spectra are six absorption lines (as simply demonstrated in figure 1). This is due to Zeeman splitting. A nucleus has angular momentum paired with a magnetic dipole moment. When placed in a magnetic field, the energy states split according to the magnetic dipole moment. This comes through as splitting the transitions between different states. The change in energy between different splittings is represented by Δ0 and Δ1.

Figure 1

-Isomer Shift

Isomer shifts, or chemical shifts in the absorption spectra happen in Fe57 nuclei from interaction with the nucleus and the electrons around the nucleus. The energy shift is represented by ε. This shift changes the resonant energy of the Fe57 nucleus.

Figure 2

-Quadrupole Splitting

Quadrupole splitting is due to a quadrupole moment caused by the deviation of spherical symmetry in the charge distribution around the nucleus. This is described in the above figure as δ.

Quadrature Interferometer

The use of a quadrature interferometer in this experiment is to accurately measure the velocity of the Co57 source. A quadrature interferometer uses two superimposed beams of light with linearly and circularly polarized light to measure path length differences. The two beams add up to make a sum electric field in the x and y direction. What is measured is the intensity in the x and y direction, which is Ix = Ex^2, and Iy = Ey^2. From the way in which the two beams add their electric fields, the x and y intensities can be plotted, and as object of interest moves, the [Ix, Iy] point moves in a circle. The difference in path length and the direction the measurement arm was moving can be determined from how the plotted points [Ix, Iy] move about the circle [3]. Figure 3 shows a diagram of a quadrature interferometer, the paths of the beams, and what optical instruments were used. Figure 4 shows an image of the quadrature interferometer that we used.

Figure 3

Figure 4

Data Collection

Figure 5

Figure 5 shows the general layout of this experiment. A function generator controls the Mossbauer driver, which oscillates the source. The oscillating source then directs the radiation through iron, and into a detector. The signals that come through are then filtered through a single channel analyzer to a range around the energy of interest, the resonant energy (14.4 keV), and each time a photon is found in this range, it triggers the DAQ card to read the velocity (measured in voltage) of the source. The interferometer is used to measure the velocity of the source, to find the conversion to energy shift.

The data is collected in a bin vs counts histogram. These are the spectra, with the bins corresponding to a voltage from the Mossbauer driver. It is necessary to convert form bins to energy shift to get the final results. This is done partly through the use of the quadrature interferometer.

Data Analysis

The data collected from the output of the experimental setup was able to be plotted as counts against a bin value as a histogram. To be able to get the energy shifts, the data needed to be converted from bins to energy. The data binned by the computer was the voltage from the Mossbauer driver. We were able to calibrate the binning of the computer by inputting specific voltages and observing which bin the counts occurred. A linear relationship between bin number and voltage was found, giving an easy equation to convert between the two. The voltage data that was provided by the Mossbauer driver was directly related to the velocity of the source. We were able to directly measure the velocity of the source using the quadrature interferometer. Using both the velocity and voltage data we were able to find a linear relationship for another easy calibration from voltage to velocity. For the final step to converting velocity into energy, we used equation 4 that relates the two through the Doppler effect. This process allowed us to create plots in terms of counts against energy shift.

Once the calibration/conversion from bins to energy is accomplished, we can calculate the nuclear properties of our samples in energies instead of relative bin values.

Origin Software Peak Fitting

We fit the bins vs voltage data in the software Origin. This allows you to fit 6 different Lorentzian peaks simultaneously. This is done in the 'fitting' tab, select 'multiple peaks', then selecting where the peaks are. Once this is done, before you fit the data, go into the advanced settings, and add weights, choosing 'Arbitrary Dataset', so you can input the error in the counts as another column.

Nuclear Effects

Each peak for each spectra is shifted by some value ΔEi, These shifts are parameterized by the zeeman effect, isomer shift, and quadrupole splitting. These equations are shown below.

This system of equations is over constrained, therefore the four unknowns can be solved for in multiple ways. Some of these are shown below.

Results

Conclusions

In this experiment, we set out to measure the nuclear properties of the Fe57 nucleus in several different compounds through Mossbauer spectroscopy. We measured the spectra of an iron sheet, Fe2O3, and Fe3O4. This was done with high resolution, measuring energy steps as small as 10^-9 eV. We have successfully made measurements of the Zeeman effect, isomer shift, and quadrupole splitting, all within one standard deviation from the accepted value. This implies two things: we have made measurements of the Fe57 nucleus very accurately within the uncertainty calculated, and that our uncertainty could have been lower to produce more confident results. Since the error was caused mostly by the conversion from bins to energy, the error in this experiment can be reduced by better calibration and more measurements within the conversion from bins to energy in future experiments.