research papers
Determination of the magnetic spin direction from the nuclear forwardscattering line intensities
^{a}Instituut voor Kern en Stralingsfysica and INPAC, KU Leuven, Celestijnenlaan 200D, B3001 Leuven, Belgium, and ^{b}Advanced Photon Source, Argonne National Laboratory, Argonne, IL 60439, USA
^{*}Correspondence email: riet.callens@fys.kuleuven.be
An expression is derived for the line intensities in a nuclear forwardscattering energy spectrum that is obtained via a Fourier transformation of the time dependence of the wavefield. The calculation takes into account the coherent properties of the nuclear forwardscattering process and the experimental limitations on the observable time window. It is shown that, for magnetic samples, the spin direction can be determined from the ratios between the different lines in the energy spectrum. The theory is complemented with experimental results on αiron.
Keywords: magnetometry; Mössbauer; nuclear resonant scattering.
1. Introduction
A challenging problem in magnetism is the selective study of different entities in a magnetic heterostructure as a function of, for example, the external field, the temperature or the shape of the magnetic structure. Mössbauer spectroscopy is an isotopeselective technique and, therefore, allows one to study a particular entity selectively, even when this entity is buried in the material. It provides magnetic information by probing the hyperfine fields. Most Mössbauer experiments are carried out using a e.g. samples embedded in a highpressure cell (Mao et al., 2004; Barla et al., 2004, 2005), thin films (Nagy et al., 2002; Röhlsberger et al., 2002, 2004; L'abbé et al., 2004) or nanostructures (Röhlsberger et al., 2001, 2003), the synchrotron source is often a valuable alternative. It has the advantage that it provides a well collimated intense beam. Moreover, the energy of the beam is tunable over a wide range so that many Mössbauer excitations can be accessed (for overviews, see Röhlsberger, 2004; Leupold et al., 1999).
However, for the study of samples containing a small amount of the resonant Mössbauer isotope,Nuclear ; Gerdau et al., 1986). In order to extract the magnetic information from the quantum beat pattern, a substantial effort and experience is required, often inhibiting an online interpretation of the data. Therefore, it has been proposed to record both the norm and the phase of the scattered wavefield (Sturhahn, 2001; Sturhahn et al., 2004; Callens et al., 2005), allowing for a Fourier transformation of the wavefield to the energy domain. The obtained spectra are usually less complex and quite easy to interpret. A practical scheme for the phase determination consists of a moving singleline reference sample that is placed inline with the sample under investigation (Callens et al., 2005). For each nuclear resonant scattered photon, both the time delay and the velocity of the reference sample is recorded. For each time channel, the velocity spectrum is a cosine function, from which the norm and the phase of the wavefield component along the incident polarization can be derived. The energy spectrum is obtained by taking the norm squared of the Fouriertransformed wavefield component.
experiments with synchrotron radiation are generally performed in a timedifferential mode where the is recorded as a function of the time after the excitation by the synchrotron pulse. The quantum beats that are revealed are the fingerprints of the magnetic fields at the position of the Mössbauer nuclei (Trammell & Hannon, 1978In Mössbauer spectra taken with a et al., 1962; Gonser et al., 1966). In this article we will show that, for an energy spectrum obtained with synchrotron radiation, the magnetization direction can also be determined from the line intensities. Explicit expressions for the line intensities of a ^{57}Fe magnetic spectrum recorded with linearly polarized synchrotron radiation are calculated. The calculation is based on the of nuclear (Blume & Kistner, 1968) and takes into account the finite experimental time window. The theory is illustrated by a forwardscattering experiment on an αiron foil.
the line intensities depend on the magnetization direction (Frauenfelder2. Calculation of the line intensities
For the calculation of the line intensities in a reconstructed energy spectrum, we will focus on nuclear ^{57}Fe nuclei submitted to a uniaxial magnetic hyperfine field. The magnetic hyperfine field causes a Zeeman splitting of the nuclear ground and excited states so that several nuclear transitions can be distinguished. For a sufficiently large Zeeman splitting, each allowed transition gives rise to a well resolved resonance line in the energy spectrum. In Appendix A we show that, for a sample with Mössbauer thickness L, the intensity of the line associated with the jth hyperfine transition, P_{j}(L), can be related to the line intensity for a singleline sample, P_{SL}(W_{j}L), having a weighted Mössbauer thickness W_{j}L,
byThe weighting factor W_{j} depends on the direction of the hyperfine field, the polarization of the incident radiation and the multipolarity of the Mössbauer transition.
The expression in equation (1) gives the projection of the polarization state of the incident radiation, e_{in}, on the polarization state e_{j} of the radiation that is resonantly scattered via the jth hyperfine transition. Note that the polarization projection appears to the fourth power in equation (1), owing to the fact that phase reconstruction methods using a singleline reference sample only measure the wavefield component along e_{in} (Sturhahn et al., 2004; Callens et al., 2005). For the M1 transition in ^{57}Fe and incident radiation that is linearly polarized along e_{x}, the factors W_{j} and the polarization projections are given in Table 1.

Experimentally, the wavefield is measured only in a certain finite time interval. The properties of this experimental time window influence the line intensities. According to equation (1), we can restrict the discussion to the case of a singleline sample. We will work out the example of the time window that was used in the experiment described below, i.e. a window lasting from 17 to 133 ns. The energy spectra for singleline samples with different Mössbauer thicknesses are calculated using the procedure outlined in Appendix A. In order to reduce truncating wiggles in the energy spectrum, the time dependence of the wavefield is multiplied by a Gaussian before the Fourier transformation is performed. This function is called the timewindow function and is given by
where t is in ns. The line intensity P_{SL} is calculated by integrating the energy spectrum over the interval where ω_{0} is the resonance frequency and Γ is the natural linewidth of the Mössbauer level. The result is shown in Fig. 1(a). In order to intuitively understand this curve, one should have a look at the time dependence of the norm of the forwardscattered wavefield (Fig. 1b). If the Mössbauer thickness varies from 10 to 50, the first dynamical minimum is moving through the experimental time window. This explains the maximum around L = 16 and the minimum around L = 33 in the line intensity of Fig. 1(a).
An expression for the lineintensity ratios will now be derived from equation (1) and the expressions in Table 1,
θ and φ are the polar and the azimuthal angle of the hyperfine field with respect to the propagation direction (zaxis) and the polarization direction (xaxis) of the incident synchrotron radiation. The ratio between the first and the third line [equation (3)] does not depend on the azimuthal angle φ and, therefore, allows one to determine the polar angle θ. Note that, using linearly polarized radiation, one cannot distinguish between a polar angle θ and 180° − θ. The degeneracy could be resolved by using circularly polarized radiation (L'abbé et al., 2004). In Fig. 2(a) the ratio P_{1}/P_{3} is plotted as a function of the polar angle θ for the experimental time window and the nominal Mössbauer thickness of the sample used in the experiment. The highest sensitivity of the ratio P_{1}/P_{3} to the angle θ is around θ = 45°.
Once the polar angle θ is known, the azimuthal angle φ can be determined from the ratio between the first and the second line [equation (4)]. Again, owing to the sin^{2} dependence on the angle φ, different angles give rise to the same line intensities, i.e. ±φ and 180° ± φ. Part of this ambiguity can be resolved by performing a second measurement using a polarizer that tilts the linear polarization direction by 45°. This allows one to distinguish between φ and 180° + φ on the one hand and −φ and 180° − φ on the other.^{1} If the magnetic hyperfine field lies in the plane perpendicular to the propagation direction of the photon, i.e. θ = 90°, equation (4) can be rewritten as
This function is plotted in Fig. 2(b) for the nominal Mössbauer thickness of the sample used in the experiment. Since the intensity of either the first or the second line is negligible near φ = 90° or φ = 0°, respectively, optimal sensitivity is achieved in the intermediate region. Note that performing a second measurement using a polarizer that tilts the linear polarization direction by 45° will improve the sensitivity near φ = 90° and φ = 0°.
3. Experimental results
The formalism will be illustrated using experimental data on an αiron foil. The experiment was performed at the Advanced Photon Source at beamline XOR3ID (Alp et al., 1994), a beamline that is specially designed for nuclear experiments. The ring was operated in the standard topup mode consisting of singlets with 153 ns interval. The energy of the beam was tuned to the 14.4 keV Mössbauer transition in ^{57}Fe. In order to filter a 1 meV bandwidth, a highresolution monochromator consisting of a pair of asymmetrically cut silicon (4 0 0) reflections followed by a pair of asymmetrically cut silicon (10 6 4) reflections was used (Toellner, 2000). The polarization of the beam is known to lie in the plane of the storage ring. The sample was a 50 µmthick natural αiron foil. It was placed perpendicular to the beam and magnetized along four different directions in the plane of the foil (θ = 90°) using a small external field. For the energyresolved measurements, a 95% ^{57}Feenriched stainlesssteel reference foil was placed in line with the αiron foil. The foil was mounted on a velocity drive operating in the sinusoidal mode with a maximum velocity of 16.7 mm s^{−1}. For the photons with a delay between 17 and 133 ns, both the time delay and the velocity of the reference sample were registered. Using the computer code PHASE,^{2} the data were transformed to the energy spectra of Fig. 3(a). This code was developed to automatically perform the procedure outlined by Callens et al. (2005). It does not require any information on the sample parameters and can be used for online visualization of the data.
The energy spectra of Fig. 3(a) provide us directly with information about the direction of the magnetic hyperfine field. Assuming that the magnetic hyperfine field lies in the plane of the foil, i.e. θ = 90°, we can use equation (5) for the determination of the azimuthal angle φ. The experimental lineintensity ratios P_{1}/P_{2} and the values for the azimuthal angle φ_{p} derived from these ratios are tabulated in Table 2. These values are in good agreement with the values φ_{t} obtained from a CONUSS (Sturhahn, 2000) analysis of time spectra taken without reference sample in the beam (Fig. 3b). The polar angle θ can also be derived from the experimental energy spectra. We find that the lineintensity ratio P_{1}/P_{3} ≥ 4.7, from which we can conclude that θ = 84 (6)°. In order to determine the azimuthal angle φ using θ = 84 (6)°, the more general expression of equation (4) was used. Within the error bars, we found the same results as for θ = 90°.

4. Conclusion
From the lineintensity ratios in the reconstructed energy spectrum, the magnetization direction can be deduced. For incident synchrotron light that is linearly polarized, the polar angle θ can be determined from the intensity ratio between the first and the third line. This determination is most sensitive around 45°. Once the polar angle θ is known, the azimuthal angle φ is obtained from the intensity ratio between the first and the second line. Also, for the azimuthal angle φ, the highest sensitivity is obtained around 45°.
APPENDIX A
Detailed calculation of the line intensity
In the following, an expression for the line intensities in a nuclear forwardscattering energy spectrum of a ^{57}Fe Zeemansplit sample is derived. The magnetic hyperfine field is assumed to be uniaxial and sufficiently large so that the lines in the spectrum are completely resolved. The energy dependence of the wavefield transmitted through the sample in the vicinity of the frequency ω_{j} equals (Blume & Kistner, 1968)
where k is the wavenumber, ρ is the concentration of the f is the coherent forwardscattering matrix from a single nucleus, d is the sample thickness and E_{in} = E_{in}e_{in} is the incoming wavefield polarized along e_{in}. For a singleline sample with resonance frequency ω_{j}, the matrix f is a diagonal matrix explicitly given by (Hannon & Trammell, 1969)
where γ is the inverse of the lifetime of the f_{LM} is the recoilless fraction, χ is the and σ_{0} is the maximal resonantscattering By combining equations (6) and (7) we find that the wavefield transmitted through a singleline sample is given by
where the Mössbauer thickness L is defined as
In the case of a hyperfinesplit sample, there are several resonance frequencies ω_{j}. For well separated hyperfine levels, the matrix for coherent scattering by a single ^{57}Fe nucleus in the vicinity of a transition frequency ω_{j} is given by (Hannon & Trammell, 1969)
where C_{j} is the Clebsch–Gordan coefficient,
in the notation of Rose (1957) and is given in Table 3. The matrix ζ_{1M} gives the dependence of the on the direction of the hyperfine field. It is defined in terms of the vector spherical harmonics and the polarization vectors e_{in} and e_{sc} before and after the scattering process,
The matrix elements in the basis of circular polarization are listed in Table 4. The eigenvalues of this matrix are calculated according to equation (59) in Hannon & Trammell (1969) and are given by λ_{j} (Table 3) and 0, resulting in the following expression for the matrix ζ_{1M} in the eigenbasis of the jth transition,
Combining equations (6), (10) and (12) yields an expression for the transmitted wavefield in the vicinity of the transition frequency ω_{j},
where W_{j} is defined as
and e_{j} is the eigenpolarization corresponding to λ_{j}. Explicit expressions for e_{j} in the basis of circular polarization can be calculated using equation (60) of Hannon & Trammell (1969), and are tabulated in Table 3. Expressions for ^{2} for the case of incident linearly polarized radiation e_{in} = e_{x} = 2^{−1/2}(−e_{+} + e_{−}) are given in Table 1.


If we compare the argument of the exponential function in equation (13) with that for the singleline sample in equation (8), we find that they are identical except for the Mössbauer thickness that is scaled with the factor W_{j}. Consequently, the transmitted wavefield for a particular can be written as
Using the phase determination method described by Callens et al. (2005), the component along e_{in} of this wavefield is measured,
Since the intensity is proportional to the square of the norm of the wavefield, the line intensity for the jth resonance is given by
where P_{SL}(W_{j}L) is the line intensity for a singleline sample having a Mössbauer thickness W_{j}L.
For the calculation of the line intensity P_{SL}(W_{j}L) for a singleline sample, one has to take into account the experimental timewindow. The time dependence of the nuclear resonant wavefield for a singleline sample with Mössbauer thickness L and resonance frequency ω_{j} is given by (Kagan et al., 1979)
where τ is the lifetime of the Mössbauer level and J_{1} is the firstorder Bessel function. This expression for the wavefield is multiplied by the experimental timewindow function S(t) and Fourier transformed to the energy domain,
The norm squared of this wavefield corresponds to the intensity in the energy domain. As a measure for the line intensity, we will integrate the intensity over the interval [ω_{j} − 3γ; ω_{j} + 3γ] where γ = 1/τ is the inverse of the lifetime. Thus, the expression for the line intensity for a singleline sample is given by
Equations (17)–(20) now allow for the calculation of the lineintensity ratios as a function of the direction of the hyperfine field (see §2).
Acknowledgements
This work was supported by the Fund for Scientific ResearchFlanders (G.0224.02 and G.0498.04), the InterUniversity Attraction Pole (IUAP P5/1), the Concerted Action of the KULeuven (GOA/2004/02), the Centers of Excellence Programme INPAC EF/05/005 and by the European Community via STREP No. NMP4CT2003001516 (DYNASYNC). Use of the Advanced Photon Source was supported by the US DOE, Office of Science, under Contract No. W31109Eng38. RC and CL'a thank the FWOFlanders for financial support.
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