S18_Moessbauer Spectroscopy of Iron Compounds

Abstract

Mossbauer spectroscopy was used to measure the ground and excited Zeeman energy gaps, isomer shift, and quadrupole shifts in Iron compounds by measuring the spectrum created as a ⁵⁷Co source decays to ⁵⁷Fe and the resulting γ passes through an Iron compound; in this experiment pure ⁵⁷Fe, Fe₂O₃, and Fe₃O₄ were studied. The γ energies are Doppler shifted by moving the ⁵⁷Co source to map complete spectra of the relevant energy range.

Introduction

The goal of spectroscopy is to investigate matter’s interaction with electromagnetic radiation. This is done by bombarding the object of interest (called the absorber) with photons and counting how many photons pass through the object and at what energies they pass through. The absorber is placed in between a photon source and a particle detector that allows us to count the photons that pass through the absorber. Mossbauer spectroscopy is particularly useful due to its extremely fine resolution, on the order of 1 part in 10¹². Say we were to stack sheets of paper from the surface of the Earth to the moon, the resolution achievable in Mossbauer spectroscopy is analogous to being able to detect if one of these sheets of paper were to be removed. Mossbauer spectroscopy is also one of the first experiments that allowed direct observation of relativistic effects on Earth. Knowing that change in velocity due to relativistic effects are on the order of (v/c)² from the Lorentz factor, taking into consideration the resolution of Mossbauer spectroscopy. This means that a source needs to be moved at about 300 m/s, achievable in laboratory settings before the relativistic effects in moving a source could be observed with Mossbauer spectroscopy.[1] Based on this, the first definitive measurement of gravitational redshift was performed in 1960, known famously as the Pound-Rebka experiment.[2] To understand the resolution M¨ossbauer spectroscopy can achieve in this particular experiment, an estimation of the energy peak’s natural line-width can be performed based on Heisenberg’s uncertainty principle:

∆E∆t ≥ ħ/2 (1)

The ⁵⁷Fe decay has a mean lifetime of 141 ns. Using the uncertainty relation, t

his corresponds to a ∆E of 4.7 neV. For comparison, the average energy of the electron (≈ k_BT) at room temperature is about 25 meV, meaning M¨ossbauer spectroscopy has a resolution of 1/5,000,000 that of the electron energy. In this experiment, pure ⁵⁷Fe, Fe₂O₃, and Fe₃O₄ were used as absorbers. A radioactive source of 57Co released photons, those with energies in the 14.4 keV range were examined. The ⁵⁷Co source was also moved using a Mossbauer linear motor to shift the emitted photon energies slightly. This resulted in a spectrum in which the Zeeman splitting of the energy level could be examined, along with isomer shift and quadrupole splitting for the iron compounds.

Theory

The counting of photons becomes meaningful when the incident photons correspond to some transitional energy in our object of interest, if this is the case, then the object will absorb the photons and enter some excited state. After some time, the object becomes de-excited and decays back to its original state and emit a photon of the same energy in a random direction. Thus, if the incident photons correspond to such a transitional energy, the photodetector will receive a significantly lower count due to the excitation-decay-emittance process of the absorber. In this experiment, a radioactive 57Co sample was used as the photon source, which decays 100% by electron capture to stable ⁵⁷Fe, a table of common decay energies is shown in Table 1 below. In one of the ⁵⁷Co’s decay processes, it emits a photon of energy in the range of 14.4 keV.

Table of common decay modes for ⁵⁷Co.

The absorber was a sample of a ⁵⁷Fe compound, pure ⁵⁷Fe, Fe₂O₃, and Fe₃O₄ were used. Incident photons in the energy range of 14.4 keV correspond to several interesting excitations in the ⁵⁷Fe nuclei including Hyperfine Zeeman Splitting, Isomer Shift, and Quadrupole Splitting. The energy differences between these phenomena are incredibly small. We were able to probe the ⁵⁷Fe nuclei over an entire range by moving the ⁵⁷Co source back and forth in a manner that slightly redshifts (decreasing velocity) and blue shifts (increasing velocity) the photons incident to the ⁵⁷Fe sample by way of Doppler shift (and by making use of the recoilless nuclear resonance that was described above). The change in energy is proportional to the change in frequency from the Doppler effect, on the order of v/c. Herein lies the truly remarkable power of Mossbauer spectroscopy, due to the fine resolution, this allows for well-resolved peaks in energy which in turn allows us to further examine atomic structures and effects. There are a few effects concerning atomic structure that must be considered in both the measurement and analysis of the energy spectrum.

Recoil-free Emission and Absorption

In the process of absorption and emission, it is important to consider the motions of the absorber and source. From a very quick mechanical analysis of the absorption and re-emittance process, one can see that when some piece of the object absorbs a photon, there must be some energy loss to recoil, just as in a collision. Therefore, the incident photon must actually have greater energy than the transitional energy to account for the recoil energy loss. Likewise, when the object emits a photon after decay, some energy must be lost due to the recoil of the object and the re-emitted photon will have an energy less than the transitional energy. Thus, it becomes exceedingly difficult to know the exact energies the photons possess. To predict at exactly what energy the photon is emitted at the source is due to recoil-free emission in both the source material and the absorber material. Transitions between nuclear states typically involve recoil emission, governed by the conservation of momentum: the nucleus must carry away momentum equal and opposite direction that which the emitted photon has, taking away some kinetic energy from it. These energy shifts are large compared to the resolution of the emission line and will result in a broadened absorption line. To combat this, the recoil energy is taken up by an entire lattice, this lattice exists as the structure of solid ⁵⁷Co emitter and solid ⁵⁷Fe compound absorber. The lattice is much more massive than the nucleus, so the effect of the recoil energy becomes negligible, and the emitted photon carries away exactly the energy of the transition. [4]

Isomer shift, Quadrupole Splitting, and Zeeman Splitting

To understand the exact location of the observed energy peak, Isomer shift must also be studied. Although it is common in calculations to treat the nucleus as a point charge around which electrons revolve, it is actually not an accurate assumption. In reality, there is a probability that electrons may be found within the nucleus. This changes the Coulomb interaction between the nucleus and electrons and subsequently a shift in energy. This shifts the required energy for a transition, meaning that the observed resonance peak in the spectrum will also be shifted by a small amount, ε, known as the Isomer shift. Quadrupole splitting occurs due to the nuclear charge distribution being not exactly spherically symmetric. A non-spherically symmetric distribution of atoms in the lattice causes an electric field gradient, thus the distribution of charge also becomes asymmetrical. If a nucleus has a quadrupole moment, its energy then depends on how it is oriented with respect to the electric field gradient. The quadrupole splitting parameter is labeled as δ. Finally, the Zeeman effect was measured. The nucleus has an angular momentum J, which is accompanied by a magnetic dipole moment. When the nucleus is placed in a magnetic field caused by the spins of the surrounding electrons, the energy states are split into 2J + 1 components, determined by quantum spin number m_J. Due to selection rules, only transitions for which ∆m_J = −1, 0, +1 are possible, eliminating two of the eight possible transitions, leaving six (see Figure 1). These are also observed as a splitting of energy peaks about the resonant energy. These splitting effects are small, typically on the order of 10⁻⁷ eV in a 57Fe atom [7], making them a difficult phenomenon to observe and measure. However, due to the extremely fine resolution of Mossbauer spectroscopy, they become easily observable. A sample of Zeeman splitting as observed in ⁵⁷Fe is shown in Figure 1, represented by the ∆₀ and ∆₁ parameters. Furthermore, the relative intensities of the transitions are: I₁ = I₆ = 3(1 + cos²θ), I₂ = I₅ = 4sin²θ , and I₃ = I₄= 1 + cos² θ. Where θ is the angle between the magnetic field line and the direction of propagation of the radiation. For an unmagnetized absorber with a single line source, the magnetic domains are randomly orientated and the radiation is unpolarized, therefore the ratio of the intensities for absorption is 3:2:1:1:2:3, from averaging the intensities over all angles.[8]

Setup & Experimental Technique

The source of ⁵⁷Co used in this experiment as of February 2018 was measured to have an activity of 7.9 mCi. The source was contained within lead shielding throughout experimentation. The source was set in oscillatory motion by a M¨ossbauer driver and M¨ossbauer linear DC motor. The 57Co was mounted on a spring-like coil of the linear motor and was driven by the driver, the motor operates by moving the source at velocities on the order of cm/s. This produced the Doppler shift that is required to shift the energy, allowing us to map the entire hyperfine structure spectrum of the 14.4 keV peak. The driver was supplied with a reference frequency output by the HP 33120A function generator. An intricacy of this experiment is ensuring that the linear motor moving the source with a constant acceleration forward, then reversing and moving the source with a constant acceleration backwards, resulting in a velocity profile of a triangle wave. The velocity was determined by the voltage in the pickup coil of the driver motor. 7 To detect the photons that pass through the 57Fe sample, a silicon x-ray detector was used, the Amptek XR-100CR. The detector outputs a signal that is proportional to the energy of the detected photon, which is then acquired by the Maestro program on a computer.

The radioactive ⁵⁷Co source emits photons of energies other than our desired 14.4 keV photons, seen in Table 1. However, we are interested only in counting the photons which are in the 14.4 keV range. To achieve this, a single channel analyzer (SCA) was used. The SCA was connected between the detector and the DAQ, its purpose to ensure that the DAQ only records the velocity information of photons in the 14.4 keV range. The SCA outputs a pulse only when it receives a signal from the detector within the 14.4 keV range, which had been calibrated. This pulse acts as a trigger to the DAQ, at which point, the DAQ will record the voltage information, and therefore velocity and energy of the photon as described Equation 2, from the Mossbauer driver itself. The velocity and photon count information was recorded through a LabView program. The essential aspects of the setup include the ability to oscillate the ⁵⁷Co sample with a linear velocity profile with the driver and the motor, and ensuring that we are only counting photons that are in the 14.4 keV range. A histogram was produced of the photon counts versus the energy of photons, and analysis was be done using Origin. There is a significant quadratic-like trend in the resulting profile, which was resultant the fact that some percentage of photons arriving at the source has a non-zero incident angle. This trend was accounted for by subtracting the mean values of non-absorbed photons to obtain a corrected value of intensity. Each dip in energy was fitting with a Lorentzian profile.

Data & Analysis

From the raw data, the background was subtracted, as mentioned in the Experimental Technique section, and the bins were converted to energy. This is done by knowing the relation that the energy splitting between the two outermost peaks of pure ⁵⁷Fe spectrum corresponds to a velocity change of 10.657 mm/s. The uncertainty in counts was decided to be the standard deviations of areas in which absorption was not observed, representing uncertainty in the background, and the Poissonian statistical uncertainty, representing the statistical fluctuations in counts of the absorption peaks, added in quadrature. Each peak was fitted with a Lorentzian profile, and from the fitting, the location of the center of each peak could be determined. This process was repeated for all three samples. For Fe₃O₄ doublet peaks were observed in the higher three energy peaks, this is due to the sample of Fe₃O₄ used. Typical Fe₃O₄ samples have two distinct crystalline structures, which is referred to as Fe₃O₄a and Fe₃O₄b in this experiment. At higher energies, the peaks can be distinguished from each other, however, the doublets overlap at the three peaks of lower energies, forming singlets. It was assumed that isomer shift is not observed in the 57Fe foil, and through the comparison of the center of the other spectra to the center of the ⁵⁷Fe spectrum, isomer shift was determined for the Fe₂O₃ and Fe₃O₄ samples. The parameters ∆₀ and ∆₁ can be determined for each sample by subtracting peak 4 center from peak 5 center and peak 5 center from peak 6 center, respectively. The motivation for this can be seen from Figure 1. The quadrupole splitting was found by subtracting peak 5 center from peak 6 center, dividing by two, and peak 1 center from peak 2 center, dividing again by two and taking the weighted average of these values.

A sample of raw data.

Conclusions

Our measurements of hyperfine structures and their parameters largely agreed with accepted values. Larger deviations from accepted were seen in the spectrum of Fe₃O₄, which could be due to the crystalline structure discussed earlier, from which it becomes harder to observe peak locations. Also, the Fe₂O₃ spectra had less defined absorption features, this may be caused by improper sample preparation, and may be improved if a thicker sample was used, which would increase data collection time, but show more significant absorption features. The ratio of intensities measured deviated by the expected value by 1.42 σ, this could be due to a number of reasons. The most likely reason is improper background subtraction, also, it was assumed that the sample was unmagnetized, which may not be entirely the case. An extension of this project may be done on accurately determining background and the reason for the spread of background intensities. Overall, we successfully observed the Zeeman splitting, isomer shift, and the quadrupole splitting in various iron compounds with extremely high resolution. Using Mossbauer spectroscopy, we were able to observe these hyperfine structures that would have been otherwise unobservable and be able to probe the atomic structures of nuclei.