# Polarization Analysis in Mössbauer Reflectometry with Synchrotron Mössbauer Source

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{57}FeBO

_{3}monochromator (synchrotron Mössbauer source—SMS) has been installed at the ID18 beamline of the European synchrotron (ESRF) and at the BL11XU beamline of SPring-8 [34,35,36,37]. The key point of the SMS is the pure nuclear (111) or (333) reflection (forbidden for electronic diffraction) of the iron borate

^{57}FeBO

_{3}crystal, which provides a single-line purely π-polarized 14.4 keV radiation within the energy bandwidth of 8 neV. The crystal should be heated at a specific temperature close to the Neél point of 348.35 K. In comparison with the laboratory experiments, the application of the π-polarized beam results in new features of Mössbauer spectra measured with SMS in absorption or reflection geometry [38]. Use of a diamond phase plate in addition to the

^{57}FeBO

_{3}monochromator at the BL11XU beamline gives the new possibilities to perform measurements in forward and grazing-incidence geometries with various (linear, circular, elliptical) polarization states of radiation [39].

## 2. Theory

^{57}Fe $L$ = 1, ${I}_{e}$ = 3/2, ${I}_{g}$ = 1/2, ${m}_{e},{m}_{g}$ are the magnetic quantum numbers, $\langle {I}_{g}{m}_{g}L\Delta m|{I}_{e}{m}_{e}\rangle $ are the Clebsch–Gordan coefficients, ${\mathsf{\sigma}}_{res}$ = 2.56 × 10

^{−4}nm

^{2}is the resonant cross-section, $\mathsf{\lambda}$ = 0.086 nm, $j$ numerate the kinds of the hyperfine splitting (i.e., different multiplets in Mössbauer spectrum), ${f}_{j}^{LM}$ is the Lamb–Mössbauer factor, ${\widehat{h}}_{\Delta m}$ in (1) are the spherical unit vectors in the hyperfine field principal axis ${\overrightarrow{h}}_{x},{\overrightarrow{h}}_{y},{\overrightarrow{h}}_{z}$:

^{57}Fe), the matrices in (3), (4) should be considered for the magnetic field of radiation. Therefore, the vector-column of the magnetic field of radiation for the π-polarized incident radiation from SMS is represented as $\left(\begin{array}{c}1\\ 0\end{array}\right)$, and the first columns in (3), (4) describe the angular dependences and polarization properties of the amplitudes of the nuclear resonant scattering ${f}_{\Delta m}^{nucl}$ in our case. It follows from (3), (4) that for $\Delta m=0$ transitions the rotated π→σ polarization component appears in the scattering intensity only if ${B}_{hf}$ has a non-zero projection on the normal to the surface. For $\Delta m=\pm 1$ transitions the rotated π→σ polarization component is created if ${B}_{hf}$ lies in the surface plane ($\mathsf{\beta}={90}^{\circ}$, but not for $\mathsf{\gamma}={0}^{\circ}$ and maximal for $\mathsf{\beta}={90}^{\circ},\mathsf{\gamma}={90}^{\circ}$). Later we consider such planar magnetic structures, typical for thin films.

^{57}Fe layers the angular and polarization dependencies for the spectrum lines are described by other polarization matrices than (3), (4) (see Ref. [38]).

^{57}Fe(0.8 nm)/Cr(2 nm)]

_{30}multilayer, in

^{57}Fe layers we assume the presence of the hyperfine magnetic field of B

_{hf}= 33 T with ΔB

_{hf}= 1 T distribution. We have considered the two magnetization directions relative to the radiation beam and the two types of the interlayer coupling between adjacent

^{57}Fe layers (ferromagnetic or antiferromagnetic), schematically shown by the blue and green arrows in the left column of Figure 1. Note that 14.4 keV Mössbauer transition is of magnetic dipole M1 type, so the magnetic field of radiation H

^{rad}interacts with

^{57}Fe nuclei. In the case of the π-polarized radiation from SMS the radiation field vector H

^{rad}lies in the sample surface. The hyperfine nuclear transitions (Δm = 0, Δm = ±1) allowed for π-polarized radiation are also indicated by orange lines in these sketches.

_{hf}in the adjacent

^{57}Fe layers). Accordingly, the Mössbauer spectrum at the magnetic maximum is determined practically only by π→σ scattering (see Figure 1i). The rotated polarization component appears in the reflected signal only when the hyperfine field B

_{hf}has a finite projection on the beam direction, i.e., when the excitation of the resonant $\Delta m=\pm 1$ transitions leads to the reemission of radiation with some part of the circular polarization. Accordingly, the Mössbauer reflectivity spectra with rotated polarization show not six but only four lines corresponding to the $\Delta m=\pm 1$ transitions as it takes place in Figure 1f,h,i. In the other cases, there is no reflectivity with rotated polarization at all.

^{57}Fe) from the bottom one, we assumed for the top layer the hyperfine magnetic field B

_{hf}= 28 T, and for the deeper

^{57}Fe layer (3 nm thickness) B

_{hf}= 30 T. The calculated Mössbauer reflectivity spectra at several grazing angles near the critical one without and with π→σ polarization selection clearly show that the π→σ reflectivity spectra contain more intense contribution from the 1.5 nm top layer than the Mössbauer reflectivity spectra without polarization selection.

^{57}Fe isotope. Such decreased nuclear resonance susceptibility is typical for real thin films with lower enrichment and broad distribution of the hyperfine magnetic fields. Thereby, the direct comparison between the kinematical approximation (11) and the exact calculations shows the excellent agreement, provided that the resonant contribution to the susceptibility is not too large.

## 3. Experimental Results and Discussion

^{57}Fe isotope. Then X-rays were collimated by the compound refractive lenses down to the angular divergence of a few μrad. The high-resolution monochromator decreases the energy bandwidth of the beam further to ~15 meV. Final monochromatization down to the energy bandwidth of ~8 neV was achieved with the pure nuclear (111) reflection of the

^{57}FeBO

_{3}crystal, and the sweep through the energy range of a Mössbauer resonant spectrum (about ±0.5 μeV) was achieved using the Doppler velocity scan. Radiation from the SMS was focused vertically using the compound refractive lenses down to the beam spot of 50 μm. The intensity of the X-ray beam incident on the sample was about 10

^{4}photons/s. More details on the design of the SMS can be found in Ref [37].

_{B}= 45.1°. According to theoretical calculations, the channel-cut should suppress the π-polarized radiation by about five orders of magnitude, while transmitting about 70% of the ⌠-polarized radiation with the angular acceptance of about 2.6 arc sec (FWHM).

^{57}Fe/Cr]

_{30}samples. The samples were grown with the Katun-C molecular beam epitaxy facility, equipped by 5 thermal evaporators at the Institute of Metal Physics (Ekaterinburg, Russia). The growth was performed under the UHV (Ultra-high vacuum) regime (5 × 10

^{−10}mbar). During the buffer layer deposition, the Al

_{2}O

_{3}substrate temperature was gradually decreased from 300 °C down to 180 °C. The typical deposition rate of Cr and

^{57}Fe layers was about of 0.15 nm/min. The details on the preparations and characterization of the samples can be found elsewhere [38]. The samples were mounted in the cassette holder of the He-exchange gas superconducting cryo-magnetic system. The studies were performed at helium temperature (4.0 K).

^{57}Fe layers and increase the Mössbauer dichroic reflectivity, the external magnetic field of 5 T was applied along the beam direction.

^{57}FeBO

^{3}crystal for this particular measurement was lowered by few degrees. This sacrifices the energy resolution of the SMS to ~230 neV (~5 mm/s), which is not important for the angular dependency measurements, but increases the intensity of the SMS by an order of magnitude [37]. The nuclear resonance reflectivity with the rotated polarization shown in Figure 7 was obtained for each grazing angle θ by integrating over the angular distributions measured with the Si(840) analyzer and drawn in Figure 6.

^{57}FeBO

_{3}nuclear monochromator.

_{hf}

^{(i)}, i = 1, 2, …, 7, and the fitted parameters and distribution probabilities of the fields P(B

_{hf}) are shown in Figure 9a.

_{hf}

^{(i)}are presented in the left column of Figure 9b. We can see that the hyperfine fields in the top two

^{57}Fe layers are significantly different from the fields in the deeper layers: on average, in the top layers the hyperfine splitting is smaller, and the fraction of the low field contributions is larger. Assuming the same field distribution in the top layers as in the whole periodic part (shown in Figure 9b, middle column), we obtain the theoretical spectra, presented by dashed green lines in Figure 8. They reveal worse correspondence with the experimental spectra. These calculations confirm the enhanced depth selectivity of the spectra of reflectivity with rotated π→σ polarization.

^{57}Fe layer where two hyperfine fields B

_{hf}= 28 T and 30 T take place, the resonant spectra for these two fields are drown in Figure 10b. The angular dependencies of the Mössbauer $\mathsf{\pi}\to \mathsf{\sigma}$ reflectivity, calculated for the photon energies corresponding to two outer lines in these spectra reveal the different angular positions of the critical angle peaks which is clearly seen in Figure 10c. This shift of the peak positions explains the asymmetry of the rotated π→σ polarization component in Mössbauer reflectivity spectra.

_{hf}= 28 T have opposite influences on the refraction index and, accordingly, on the critical angle position for the radiation wavelengths corresponding to the 1st and 6th lines of the broader sextet. Because the maximum of reflectivity for these two lines occurs at slightly different angles, the relative intensities of these two lines are not the same (as they are supposed to be for the absorption case), and they are changed with a small variation of the grazing angle. This provides the observed asymmetry of the spectra depending on the small variation of the grazing angle observed in Figure 8. Note that the studied sample shows even more low-field B

_{hf}

^{(i)}contributions. However, the effect of all smaller fields is, in principle, the same as revealed by the simplest model calculations.

## 4. Summary

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Model calculation of the angular dependencies of the Mössbauer reflectivity (

**a**–

**d**) and Mössbauer reflectivity spectra at the critical angle (

**e**–

**h**) and in the “magnetic” maximum (

**i**) for π-polarized SMS radiation and for different cases of the ferromagnetic and antiferromagnetic coupling between adjacent

^{57}Fe layers, schematically drawn on the left.

**Figure 2.**Comparison of the angular-depth dependence of the standing wave $|{E}_{\mathsf{\pi}}(\mathsf{\theta},z,\mathsf{\omega}){|}^{2}$ (

**left panel**) and squared standing wave $|{{E}_{\mathsf{\pi}}}^{2}(\mathsf{\theta},z,\mathsf{\omega}){|}^{2}$ (

**right panel**), calculated for an iron mirror neglecting the nuclear resonant contribution to the scattering.

**Figure 3.**(

**a**) The used in calculations model: two thin layers with slightly different B

_{hf}(their resonant spectra are on the left), directed along the beam. (

**b**,

**bottom panels**) The angular-depth distribution of the squared standing waves $|{{E}_{\mathsf{\pi}}}^{2}(\mathsf{\theta},z,\mathsf{\omega}){|}^{2}$, calculated for two indicated velocities (v) points in the resonant spectrum, the solid horizontal lines show the layers boundaries, (

**b**,

**upper panels**) the angular curves for π→σ reflectivity, corresponding to this energy points. The solid and dashed lines (nearly indistinguishable) show the results, obtained by Equation (1) and by the exact calculations, performed with the program REFSPC [46]. (

**c**) The Mössbauer reflectivity spectra without the polarization selection (

**left column**) and for π→σ reflectivity (

**right column**), calculated for three indicated grazing angles in vicinity of the critical angle.

**Figure 4.**(

**a**) Mössbauer reflectivity spectrum from Ref. [55], measured without polarization selection for the [

^{57}Fe(0.12 nm)/Cr(1.05 nm)] × 30 sample in remanence. Symbols are the experimental data, three solid (practically undistinguishable) lines are the results of the fit with three different orientations of the hyperfine magnetic field B

_{hf}: for B

_{hf}in the surface plane and perpendicular to the beam (azimuthal angle γ = 0°), B

_{hf}parallel to the beam (γ = 90°), and B

_{hf}in the surface plane with γ = 57° (for this angle the spectrum without polarization analysis is the same as in the case of the random in space orientations of B

_{hf}). (

**b**–

**d**) Theoretical Mössbauer reflectivity spectra for these models with the selection of polarization: solid red lines with symbols are the results for the reflectivity with nonrotated (π$\to $π) polarization, solid blue lines are those for the rotated (π$\to $σ) polarization component in the reflectivity. (

**e**–

**g**) The corresponding different field distributions P(B

_{hf}) for these three cases, giving the same result of the fit to the experimental spectrum shown in (

**a**).

**Figure 5.**Experimental set-up for measurements of the Mössbauer reflectivity with the selection of radiation with the rotated (π$\to $σ) polarization. HHLM—high-heat-load monochromator; CRL—compound refractive lenses, HRM—high-resolution monochromator.

**Figure 6.**Angular distributions of the radiation reflected with π$\to $σ rotation of the polarization from the [

^{57}Fe(0.8 nm)/Cr(1.05 nm)]

_{30}sample measured with the Si(840) channel-cut analyzer for various grazing angles θ.

**Figure 7.**Angular dependencies of the Mössbauer reflectivity measured without selection of polarization and with the selection of the rotated $\mathsf{\pi}\to \mathsf{\sigma}$ component in reflectivity, obtained as the integral over scans, presented in Figure 6. The lines with symbols show the experimental data. The red solid lines show the results of theoretical calculations.

**Figure 8.**Mössbauer reflectivity spectra measured from [

^{57}Fe(0.8 nm)/Cr(1.05 nm)]

_{30}multilayer without polarization analysis at the critical angle (

**a**) and with selection of the rotated polarization component at the two angles in the vicinity of the critical angle (

**b**,

**c**). The solid lines with symbols are the experimental data. The red sold and green dashed lines are the fits discussed in the text.

**Figure 9.**(

**a**) Hyperfine field distribution, obtained by the fit of the spectra in Figure 8. (

**b**) Depth distribution of the each kind of B

_{hf}

^{(i)}(i = 1, 2, …, 7) in several top

^{57}Fe layers (the period, repeating 27 times in the deeper part of the multilayer, is marked, the last period contacted with the buffer layer is also different from the main periodic part but is not shown here), obtained by the fit (

**left column**); the initially suggested homogeneous depth distribution of B

_{hf}

^{(i)}in all

^{57}Fe layers (

**middle column**), the calculated reflectivity spectra for this distribution are presented in Figure 8 by the dashed green lines. In the right column, the resonant spectra corresponding to every B

_{hf}

^{(i)}are shown.

**Figure 10.**(

**a**) Model calculations of the rotated π→σ polarization component in Mössbauer reflectivity spectra. Dashed vertical lines mark the energies, for which the angular dependencies of Mössbauer reflectivity in (

**c**) are calculated. (

**b**) The resonant spectra for the B

_{hf}= 30 T and 28 T which we assume in the

^{57}Fe layer. (

**c**) The angular dependencies of the Mössbauer reflectivity with the rotated π→σ polarization, calculated for the 1st (v = −5 mm/s) and 6th lines (v = +5 mm/s) of the resonant spectrum corresponding to B

_{hf}= 30 T. The dashed vertical lines mark the angles, for which the spectra in (

**a**) are calculated.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Andreeva, M.; Baulin, R.; Chumakov, A.; Kiseleva, T.; Rüffer, R.
Polarization Analysis in Mössbauer Reflectometry with Synchrotron Mössbauer Source. *Condens. Matter* **2019**, *4*, 8.
https://doi.org/10.3390/condmat4010008

**AMA Style**

Andreeva M, Baulin R, Chumakov A, Kiseleva T, Rüffer R.
Polarization Analysis in Mössbauer Reflectometry with Synchrotron Mössbauer Source. *Condensed Matter*. 2019; 4(1):8.
https://doi.org/10.3390/condmat4010008

**Chicago/Turabian Style**

Andreeva, Marina, Roman Baulin, Aleksandr Chumakov, Tatiyana Kiseleva, and Rudolf Rüffer.
2019. "Polarization Analysis in Mössbauer Reflectometry with Synchrotron Mössbauer Source" *Condensed Matter* 4, no. 1: 8.
https://doi.org/10.3390/condmat4010008