Determination of the magnetic spin direction from the nuclear forward-scattering line intensities

Callens, R.; L'abbé, C.; Meersschaut, J.; Serdons, I.; Sturhahn, W.; Toellner, T.S.

doi:10.1107/S0909049507023229

research papers

JOURNAL OF
SYNCHROTRON
RADIATION

ISSN: 1600-5775

Volume 14| Part 4| July 2007| Pages 366-371

https://doi.org/10.1107/S0909049507023229

Determination of the magnetic spin direction from the nuclear forward-scattering line intensities

R. Callens,^a ^* C. L'abbé,^a J. Meersschaut,^a I. Serdons,^a W. Sturhahn ^b and T. S. Toellner ^b

^aInstituut voor Kern- en Stralingsfysica and INPAC, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium, and ^bAdvanced Photon Source, Argonne National Laboratory, Argonne, IL 60439, USA
^*Correspondence e-mail: riet.callens@fys.kuleuven.be

(Received 23 October 2006; accepted 11 May 2007)

An expression is derived for the line intensities in a nuclear forward-scattering 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 forward-scattering 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 α-iron.

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 isotope-selective 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 radioactive source. However, for the study of samples containing a small amount of the resonant Mössbauer isotope, e.g. samples embedded in a high-pressure 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 ).

Nuclear resonant scattering experiments with synchrotron radiation are generally performed in a time-differential mode where the nuclear decay 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, 1978 ; 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 on-line 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 single-line reference sample that is placed in-line 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 Fourier-transformed wavefield component.

In Mössbauer spectra taken with a radioactive source, the line intensities depend on the magnetization direction (Frauenfelder 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 ⁵⁷Fe magnetic spectrum recorded with linearly polarized synchrotron radiation are calculated. The calculation is based on the dynamical theory of nuclear forward scattering (Blume & Kistner, 1968 ) and takes into account the finite experimental time window. The theory is illustrated by a forward-scattering experiment on an α-iron foil.

2. Calculation of the line intensities

For the calculation of the line intensities in a reconstructed energy spectrum, we will focus on nuclear forward scattering by ⁵⁷Fe 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 single-line sample, P_SL(W_jL), having a weighted Mössbauer thickness W_jL,

$[P_j(L)=P_{\rm{SL}}(W\!_j\,L)|{\bf{e}}_j^*\!\cdot{\bf{e}}_{\rm{in}}|^4.\eqno(1)]$

The 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 $[{\bf{e}}_j^*\!\cdot{\bf{e}}_{\rm{in}}]$ 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 single-line reference sample only measure the wavefield component along e_in (Sturhahn et al., 2004; Callens et al., 2005). For the M1 transition in ⁵⁷Fe and incident radiation that is linearly polarized along e_x, the factors W_j and the polarization projections $[|{\bf{e}}_j^*\!\cdot{\bf{e}}_x|^2]$ are given in Table 1.

Table 1
The spin projection of the ground state (m_g) and the excited state (m_e) on the hyperfine field direction, the thickness weighting factors W_j and the polarization projections $[|{\bf{e}}_j^*\!\cdot{\bf{e}}_x|^2]$ for the six allowed transitions in ⁵⁷Fe

j	m_g	m_e	W_j	$[\|{\bf{e}}_j^*\!\cdot{\bf{e}}_x\|^2]$
1	−1/2	−3/2	0.375(1 + cos²θ)	(1 − sin²θsin²φ)/(1 + cos²θ)
2	−1/2	−1/2	0.5sin²θ	sin²φ
3	−1/2	1/2	0.125(1 + cos²θ)	(1 − sin²θsin²φ)/(1 + cos²θ)
4	1/2	−1/2	0.125(1 + cos²θ)	(1 − sin²θsin²φ)/(1 + cos²θ)
5	1/2	1/2	0.5sin²θ	sin²φ
6	1/2	3/2	0.375(1 + cos²θ)	(1 − sin²θsin²φ)/(1 + cos²θ)

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 single-line 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 single-line 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 time-window function and is given by

$[S(t)=\left\{\matrix{\exp\left[-(t-75)^2/1682\right]\hfill &{\rm{if}}\,\,t\in[17\semi133]\hfill\cr0\hfill&{\rm{elsewhere,}}\hfill}\right.\eqno(2)]$

where t is in ns. The line intensity P_SL is calculated by integrating the energy spectrum over the interval $[[\hbar\omega_0-3\Gamma;\hbar\omega_0+3\Gamma]]$ where ω₀ 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 forward-scattered 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).

Figure 1
(a) Line intensity for a single-line sample as a function of the Mössbauer thickness L for a time window lasting from 17 to 133 ns and a time-window function as defined in equation (2)

. (b) Solid line: time evolution of synchrotron radiation that is nuclear resonant scattered by a single-line sample of Mössbauer thickness L. Dotted line: time-window function as defined in equation (2)

An expression for the line-intensity ratios will now be derived from equation (1) and the expressions in Table 1,

$[{{P_1}\over{P_3}}={{P_{\rm{SL}}[0.375(1+\cos^2\theta)L]}\over{P_{\rm{SL}}[0.125(1+\cos^2\theta)L]}},\eqno(3)]$

$[{{P_1}\over{P_2}}={{P_{\rm{SL}}[0.375(1+\cos^2\theta)L]}\over{P_{\rm{SL}}[0.5(\sin^2\theta)L]}}\left[{{1-\sin^2\varphi\sin^2\theta}\over{\sin^2\varphi\,(1+\cos^2\theta)}}\right]^2.\eqno(4)]$

θ and φ are the polar and the azimuthal angle of the hyperfine field with respect to the propagation direction (z-axis) and the polarization direction (x-axis) 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 P₁/P₃ 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 P₁/P₃ to the angle θ is around θ = 45°.

Figure 2
Line-intensity ratios for a Zeeman-split sample with Mössbauer thickness L = 18.2 for the time-window function S(t) of equation (2)

. (a) Dependence of the line-intensity ratio P₁/P₃ on the polar angle θ. (b) Dependence of the line-intensity ratio P₁/P₂ on the azimuthal angle φ for θ = 90°.

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 sin² 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

$[{{P_1}\over{P_2}}={{P_{\rm{SL}}(0.375L)}\over{P_{\rm{SL}}(0.5L)}}{{1}\over{\tan^4\varphi}}.\eqno(5)]$

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 α-iron foil. The experiment was performed at the Advanced Photon Source at beamline XOR-3-ID (Alp et al., 1994 ), a beamline that is specially designed for nuclear resonant scattering experiments. The ring was operated in the standard top-up mode consisting of singlets with 153 ns interval. The energy of the beam was tuned to the 14.4 keV Mössbauer transition in ⁵⁷Fe. In order to filter a 1 meV bandwidth, a high-resolution 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 µm-thick natural α-iron 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 energy-resolved measurements, a 95% ⁵⁷Fe-enriched stainless-steel reference foil was placed in line with the α-iron 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 on-line visualization of the data.

Figure 3
Nuclear resonant scattering data for an α-iron foil with the hyperfine field B oriented along different directions in the plane perpendicular to the photon direction k (θ = 90°). The angle φ is the angle between the polarization of the radiation e_x and the direction of the magnetic hyperfine field B. The dots are the data and the solid line is a fit obtained using the software package CONUSS (Sturhahn, 2000

). (a) Energy spectra. The energy scale is given in units of the natural linewidth of the Mössbauer level in ⁵⁷Fe, γ = 4.66 neV. (b) Time spectra taken without reference sample in the beam. (c) Schematic picture of the hyperfine field direction with respect to the photon direction.

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 line-intensity ratios P₁/P₂ 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 line-intensity ratio P₁/P₃ ≥ 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°.

Table 2
The line-intensity ratio P₁/P₂ calculated from the experimental energy spectra of Fig. 3(a), the azimuthal angle φ_p deduced from this ratio, and the azimuthal angle φ_t determined from the CONUSS analysis of the time spectra of Fig. 3(b)

φ (°)	P₁/P₂	φ_p (°)	φ_t (°)
90	<0.0035	83 (7)	84 (4)
60	0.0659	62 (1)	62 (2)
30	7.78	29 (1)	28 (2)
0	>400	7 (7)	2 (3)

4. Conclusion

From the line-intensity 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 forward-scattering energy spectrum of a ⁵⁷Fe Zeeman-split 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 nuclear transition frequency ω_j equals (Blume & Kistner, 1968)

$[\eqalignno{{\bf{E}}_{\rm{tr}}(\omega-\omega_j)={}&\exp\left[i(2\pi/k)\rho\,f(\omega-\omega_j)d\right]\cr&\times{\bf{E}}_{\rm{in}},&(6)}]$

where k is the wavenumber, ρ is the concentration of the chemical element, f is the coherent forward-scattering matrix from a single nucleus, d is the sample thickness and E_in = E_ine_in is the incoming wavefield polarized along e_in. For a single-line sample with resonance frequency ω_j, the matrix f is a diagonal matrix explicitly given by (Hannon & Trammell, 1969 )

$[f(\omega-\omega_j)=-{{k}\over{8\pi}}\sigma_0\,f_{LM}\,\chi{{\gamma}\over{\omega-\omega_j+i\gamma/2}}\left[\matrix{1&0\cr0&1}\right],\eqno(7)]$

where γ is the inverse of the lifetime of the excited state, f_LM is the recoilless fraction, χ is the isotopic enrichment and σ₀ is the maximal resonant-scattering cross section. By combining equations (6) and (7) we find that the wavefield transmitted through a single-line sample is given by

$[{\bf{E}}_{\rm{SL}}(\omega-\omega_j,L)=\exp\left(-i\,{{L}\over{4}}{{\gamma}\over{\omega-\omega_j+i\gamma/2}}\right)E_{\rm{in}}{\bf{e}}_{\rm{in}},\eqno(8)]$

where the Mössbauer thickness L is defined as

$[L=\sigma_0\,f_{LM}\,\chi\rho{d}.\eqno(9)]$

In the case of a hyperfine-split sample, there are several resonance frequencies ω_j. For well separated hyperfine levels, the matrix for coherent scattering by a single ⁵⁷Fe nucleus in the vicinity of a transition frequency ω_j is given by (Hannon & Trammell, 1969)

$[f(\omega-\omega_j)=-{{k}\over{8\pi}}\sigma_0\,f_{LM}\,\chi\,{{3}\over{4}}C_j^2\,\zeta_{1\,M}\,{{\gamma}\over{\omega-\omega_j+i\gamma/2}},\eqno(10)]$

where C_j is the Clebsch–Gordan coefficient,

$[C_j=C\left(\,{\textstyle{{1}\over{2}}}\,\,1\,\,{\textstyle{{3}\over{2}}}\,\semi \,m_g\,M\,m_e\right),]$

in the notation of Rose (1957 ) and is given in Table 3. The matrix ζ_1M gives the dependence of the scattering matrix on the direction of the hyperfine field. It is defined in terms of the vector spherical harmonics $[{\bf{Y}}_{1\,M}^{(1)}({\bf{k}})]$ and the polarization vectors e_in and e_sc before and after the scattering process,

$[\zeta_{1\,M}=8\pi{\bf{e}}_{\rm{sc}}^*\cdot{\bf{Y}}_{1 M}^{(1)}({\bf{k}}){\bf{Y}}_{1\,M}^{(1)}({\bf{k}})^*\!\cdot{\bf{e}}_{\rm{in}}.\eqno(11)]$

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,

$[\zeta_{1\,M}={\left[\matrix{\lambda_j&0\cr0&0}\right]}.\eqno(12)]$

Combining equations (6), (10) and (12) yields an expression for the transmitted wavefield in the vicinity of the transition frequency ω_j,

$[{\bf{E}}_{j}(\omega-\omega_j,L)=\exp\left[-i{{W\!_j\,L}\over{4}} \, {{\gamma}\over{\omega-\omega_j+i\gamma/2}}\right]E_{\rm{in}}({\bf{e}}_j^*\!\cdot{\bf{e}}_{\rm{in}}){\bf{e}}_j,\eqno(13)]$

where W_j is defined as

$[W_j={{3}\over{4}}\,C_j^{\,2}\lambda_j\eqno(14)]$

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 | $[{\bf{e}}_j^*\!\cdot{\bf{e}}_{\rm{in}}]$ |² for the case of incident linearly polarized radiation e_in = e_x = 2^−1/2(−e₊ + e₋) are given in Table 1.

Table 3
The spin projection of the ground state (m_g) and the excited state (m_e) on the hyperfine field direction, the change in spin projection (M), the square of the Clebsch–Gordan coefficients (C_j²), the eigenvalues λ_j, and the projections $[{\bf{e}}_+^*\!\cdot{\bf{e}}_j]$ and $[{\bf{e}}_-^*\!\cdot{\bf{e}}_j]$ for the six allowed transitions in ⁵⁷Fe

j	m_g	m_e	M	C_j²	λ_j	$[{\bf{e}}_+^*\!\cdot{\bf{e}}_j]$	$[{\bf{e}}_-^*\!\cdot{\bf{e}}_j]$
1	−1/2	−3/2	−1	1	(1 + cos²θ)/2	$[{{ \sqrt 2 \sin^2(\theta/2)}/{\sqrt{1+\cos^2 \theta}}}]$	$[-{{\sqrt 2 \exp({-2i\varphi}) \cos^2(\theta/2)}/{\sqrt{1+\cos^2 \theta}}}]$
2	−1/2	−1/2	0	2/3	sin²θ	$[{{ 1}/{\sqrt 2}}]$	$[{{ \exp({-2i\varphi}})/{\sqrt 2}}]$
3	−1/2	1/2	+1	1/3	(1 + cos²θ)/2	$[{{ \sqrt 2 \cos^2(\theta/2)}/{\sqrt{1+\cos^2 \theta}}}]$	$[-{{\sqrt 2 \exp({-2i\varphi}) \sin^2(\theta/2)}/{\sqrt{1+\cos^2 \theta}}}]$
4	1/2	−1/2	−1	1/3	(1 + cos²θ)/2	$[{{ \sqrt 2 \sin^2(\theta/2)}/{\sqrt{1+\cos^2 \theta}}}]$	$[-{{\sqrt 2 \exp({-2i\varphi}) \cos^2(\theta/2)}/{\sqrt{1+\cos^2 \theta}}}]$
5	1/2	1/2	0	2/3	sin²θ	$[{{ 1}/{\sqrt 2}}]$	$[{{\exp({-2i\varphi}})/{\sqrt 2}}]$
6	1/2	3/2	+1	1	(1 + cos²θ)/2	$[{{ \sqrt 2 \cos^2(\theta/2)}/{\sqrt{1+\cos^2 \theta}}}]$	$[-{{\sqrt 2 \exp({-2i\varphi}) \sin^2(\theta/2)}/{\sqrt{1+\cos^2 \theta}}}]$

Table 4
The matrix elements of ζ_1M in the basis of circular polarization (e₊, e₋)

	e_sce_in = e₊e₊	e_sce_in = e₊e₋	e_sce_in = e₋e₊	e_sce_in = e₋e₋
M = +1	[(1 + cosθ)/2]²	− $[{{1}\over{4}}\,]$ sin²θexp(2iφ)	− $[{{1}\over{4}}]$ sin²θexp(−2iφ)	[(1 − cosθ)/2]²
M = 0	$[{{1}\over{2}}\,]$ sin²θ	$[{{1}\over{2}}\,]$ sin²θexp(2iφ)	$[{{1}\over{2}}]$ sin²θexp(−2iφ)	$[{{1}\over{2}}\,]$ sin²θ
M = −1	[(1 − cosθ)/2]²	− $[{{1}\over{4}}\,]$ sin²θexp(2iφ)	− $[{{1}\over{4}}]$ sin²θexp(−2iφ)	[(1 + cosθ)/2]²

If we compare the argument of the exponential function in equation (13) with that for the single-line 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 nuclear transition can be written as

$[{\bf{E}}_j(\omega-\omega_j,L)=E_{\rm{SL}}(\omega-\omega_j,W\!_j\,L)({\bf{e}}_j^*\cdot{\bf{e}}_{\rm{in}}){\bf{e}}_j.\eqno(15)]$

Using the phase determination method described by Callens et al. (2005), the component along e_in of this wavefield is measured,

$[\eqalignno{E_j^{\rm{in}}(\omega-\omega_j,L)&={\bf{e}}_{\rm{in}}^*\!\cdot{\bf{E}}_{j}(\omega-\omega_j,L)\cr&=E_{\rm{SL}}(\omega-\omega_j,W\!_j\,L)\,|{\bf{e}}_j^*\!\cdot{\bf{e}}_{\rm{in}}|^2.&(16)}]$

Since the intensity is proportional to the square of the norm of the wavefield, the line intensity for the jth resonance is given by

$[P_j(L)=P_{\rm{SL}}(W\!_j\,L)|{\bf{e}}_j^*\!\cdot{\bf{e}}_{\rm{in}}|^4,\eqno(17)]$

where P_SL(W_jL) is the line intensity for a single-line sample having a Mössbauer thickness W_jL.

For the calculation of the line intensity P_SL(W_jL) for a single-line sample, one has to take into account the experimental time-window. The time dependence of the nuclear resonant wavefield for a single-line sample with Mössbauer thickness L and resonance frequency ω_j is given by (Kagan et al., 1979 )

$[\eqalignno{{\bf{E}}&_{\rm{SL}}(t,\omega_j,L)\simeq\cr&\left\{\matrix{-L\exp[-i\omega_jt-(t/2\tau)]\,J_1[(Lt/\tau)^{1/2}]{\bf{e}}_{\rm{in}}/(Lt/\tau)^{1/2}\hfill&{\rm{if}}\,\,t\geq0\hfill\cr0\hfill&{\rm{if}}\,\,t\,\lt\,\,0,\hfill}\right.\cr&&(18)}]$

where τ is the lifetime of the Mössbauer level and J₁ is the first-order Bessel function. This expression for the wavefield is multiplied by the experimental time-window function S(t) and Fourier transformed to the energy domain,

$[{\bf{E}}_{\rm{SL}}^{S}(\omega-\omega_j,L)\simeq\int_{-\infty}^{+\infty}{\rm{d}}t\,\exp[-i(\omega-\omega_j)t]S(t)E_{\rm{SL}}(t,\omega_j,L){\bf{e}}_{\rm{in}}.\eqno(19)]$

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 single-line sample is given by

$[P_{\rm{SL}}(L)\simeq\int_{\omega_j-3\gamma}^{\omega_j+3\gamma}{\rm{d}}\omega\,|E_{\rm{SL}}^{S}(\omega-\omega_j,L)|^2.\eqno(20)]$

Equations (17)–(20) now allow for the calculation of the line-intensity ratios as a function of the direction of the hyperfine field (see §2).

Footnotes

¹In order to calculate the line-intensity ratios for the set-up with the polarizer tilted by 45°, the angle φ in equation (4) should be replaced by φ − 45°.

²On request, the computer code PHASE is available from the author.

Acknowledgements

This work was supported by the Fund for Scientific Research-Flanders (G.0224.02 and G.0498.04), the Inter-University 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. NMP4-CT-2003-001516 (DYNASYNC). Use of the Advanced Photon Source was supported by the US DOE, Office of Science, under Contract No. W-31-109-Eng-38. RC and CL'a thank the FWO-Flanders for financial support.

References

Alp, E. E., Mooney, T. M., Toellner, T. & Sturhahn, W. (1994). Hyperfine Interact. 90, 323–334. CrossRef Web of Science Google Scholar
Barla, A., Derr, J., Sanchez, J. P., Salce, B., Lapertot, G., Doyle, B. P., Rüffer, R., Lengsdorf, R., Abd-Elmeguid, M. M. & Flouquet, J. (2005). Phys. Rev. Lett. 94, 166401. Web of Science CrossRef PubMed Google Scholar
Barla, A., Sanchez, J. P., Haga, Y., Lapertot, G., Doyle, B. P., Leupold, O., Rüffer, R., Abd-Elmeguid, M. M., Lengsdorf, R. & Flouquet, J. (2004). Phys. Rev. Lett. 92, 066401. Web of Science CrossRef PubMed Google Scholar
Blume, M. & Kistner, O. C. (1968). Phys. Rev. 171, 417–425. CrossRef Web of Science Google Scholar
Callens, R., L'abbé, C., Meersschaut, J., Serdons, I., Sturhahn, W. & Toellner, T. S. (2005). Phys. Rev. B, 72, 081402(R). Web of Science CrossRef Google Scholar
Frauenfelder, H., Nagle, D. E., Taylor, R. D., Cochran, D. R. F. & Visscher, W. M. (1962). Phys. Rev. 126, 1065–1075. CrossRef CAS Web of Science Google Scholar
Gerdau, E., Rüffer, R., Hollatz, R. & Hannon, J. P. (1986). Phys. Rev. Lett. 57, 1141–1144. CrossRef PubMed CAS Web of Science Google Scholar
Gonser, U., Grant, R. W., Wiedersich, H. & Geller, S. (1966). Appl. Phys. Lett. 9, 18–21. CrossRef CAS Web of Science Google Scholar
Hannon, J. P. & Trammell, G. T. (1969). Phys. Rev. 186, 306–325. CrossRef CAS Web of Science Google Scholar
Kagan, Y., Afanas'ev, A. M. & Kohn, V. G. (1979). J. Phys. C, 12, 615–631. CrossRef CAS Web of Science Google Scholar
L'abbé, C., Meersschaut, J., Sturhahn, W., Jiang, J. S., Toellner, T. S., Alp, E. E. & Bader, S. D. (2004). Phys. Rev. Lett. 93, 037201. Web of Science PubMed Google Scholar
Leupold, O., Chumakov, A. I., Alp, E. E., Sturhahn, W. & Baron, A. Q. R. (1999). Hyperfine Interact. 123, 611–631. Web of Science CrossRef Google Scholar
Mao, W. L., Sturhahn, W., Heinz, D. L., Mao, H. K., Shu, J. F. & Hemley, R. J. (2004). Geophys. Res. Lett. 31, L15618. CrossRef Google Scholar
Nagy, D. L., Bottyan, L., Croonenborghs, B., Deak, L., Degroote, B., Dekoster, J., Lauter, H. J., Lauter-Pasyuk, V., Leupold, O., Major, M., Meersschaut, J., Nikonov, O., Petrenko, A., Rüffer, R., Spiering, H. & Szilagyi, E. (2002). Phys. Rev. Lett. 88, 157202. Web of Science CrossRef PubMed Google Scholar
Röhlsberger, R. (2004). Nuclear Condensed Matter Physics with Synchrotron Radiation: Basic Principles, Methodology and Applications. Berlin: Springer. Google Scholar
Röhlsberger, R., Bansmann, J., Senz, V., Jonas, K. L., Bettac, A., Leupold, O., Rüffer, R., Burkel, E. & Meiwes-Broer, K. H. (2001). Phys. Rev. Lett. 86, 5597–5600. Web of Science PubMed Google Scholar
Röhlsberger, R., Bansmann, J., Senz, V., Jonas, K. L., Bettac, A., Meiwes-Broer, K. H. & Leupold, O. (2003). Phys. Rev. B, 67, 245412. Google Scholar
Röhlsberger, R., Klein, T., Schlage, K., Leupold, O. & Rüffer, R. (2004). Phys. Rev. B, 69, 235412. Google Scholar
Röhlsberger, R., Thomas, H., Schlage, K., Burkel, E., Leupold, O. & Rüffer, R. (2002). Phys. Rev. Lett. 89, 237201. Web of Science PubMed Google Scholar
Rose, M. E. (1957). Elementary Theory of Angular Momentum. New York: John Wiley and Sons. Google Scholar
Sturhahn, W. (2000). Hyperfine Interact. 125, 149–172. Web of Science CrossRef CAS Google Scholar
Sturhahn, W. (2001). Phys. Rev. B, 63, 094105. Web of Science CrossRef Google Scholar
Sturhahn, W., L'abbé, C. & Toellner, T. S. (2004). Europhys. Lett. 66, 506–512. Web of Science CrossRef CAS Google Scholar
Toellner, T. S. (2000). Hyperfine Interact. 125, 3–28. Web of Science CrossRef CAS Google Scholar
Trammell, G. T. & Hannon, J. P. (1978). Phys. Rev. B, 18, 165–172. [Erratum: Phys. Rev. B, (1979). 19, 3835–3836.] Google Scholar

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JOURNAL OF
SYNCHROTRON
RADIATION

ISSN: 1600-5775

Volume 14| Part 4| July 2007| Pages 366-371

https://doi.org/10.1107/S0909049507023229

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research papers\(\def\hfill{\hskip 5em}\def\hfil{\hskip 3em}\def\eqno#1{\hfil {#1}}\)

Determination of the magnetic spin direction from the nuclear forward-scattering line intensities

1. Introduction

2. Calculation of the line intensities

3. Experimental results

4. Conclusion

APPENDIX A

Detailed calculation of the line intensity

Footnotes

Acknowledgements

References

research papers