Perturbative theory of grazing-incidence diffuse nuclear resonant scattering of synchrotron radiation

Theoretical description of off-specular grazing-incidence nuclear resonant scattering of synchrotron radiation (synchrotron Mössbauer reflectometry, SMR) is presented. The recently developed SMR, similar to polarized neutron reflectometry (PNR), is an analytical tool for the determination of isotopic and magnetic structure of thin films and multilayers. It combines the sensitivity of Mössbauer spectroscopy to hyperfine interactions and the depth selectivity of x-ray reflectometry. Specular reflection provides information on the depth profile, while off-specular scattering on the lateral structure of scattering layers. Off-specular SMR and PNR intensity formulas of a rather general multilayer with different domains, based on a distorted incident-wave approximation (DIWA), are presented. The distorted-wave Born approximation results are given in an appendix. Physical and numerical implications, of using DIWA are explained. The temporal character of SMR imposes specific differences between SMR and PNR. In order to reveal the limits of DIWA and to compare the two analytical methods, two-dimensional diffuse SMR and PNR maps of an antiferromagnetic multilayer are calculated and critically compared. Experimental “$\omega -2\theta$” SMR map of a periodic ${[\mathrm{Fe}∕\mathrm{Cr}]}_{20}$ multilayer is presented and compared with simulations by the present theory.

Figure 1

Simulated ${}^{57}\mathrm{Fe}$ specular synchrotron Mössbauer reflectograms ($\theta \text{−}2\theta$ scans) of the $\mathrm{Mg}\mathrm{O}∕{[{}^{57}\mathrm{Fe}\left(2.62\mathrm{nm}\right)∕\mathrm{Cr}\left(1.28\mathrm{nm}\right)]}_{20}$ antiferromagnetic multilayer with sublayer magnetizations parallel and antiparallel to the wave vector for different domain bias parameters $\eta$ indicated in the figure.

Figure 2

Simulated off-specular PNR $\omega$ scans $\left(I^{++}\right)$ calculated for the $\mathrm{Mg}\mathrm{O}∕{[{}^{57}\mathrm{Fe}\left(2.62\mathrm{nm}\right)∕\mathrm{Cr}\left(1.28\mathrm{nm}\right)]}_{20}$ antiferromagnetic multilayer with $\lambda =0.4\mathrm{nm}$ and detector position $2\theta$ fixed at the $1∕2$-order antiferromagnetic Bragg peak position for various correlation lengths $\xi$ indicated in the figure. A single domain bias parameter of $\eta =0.1$ was used. Dotted and solid lines show the BA and SDIWA simulations, respectively.

Figure 3

Simulated off-specular SMR $\omega$ scans calculated for the $\mathrm{Mg}\mathrm{O}∕{[{}^{57}\mathrm{Fe}\left(2.62\mathrm{nm}\right)∕\mathrm{Cr}\left(1.28\mathrm{nm}\right)]}_{20}$ antiferromagnetic multilayer with $\lambda =0.086\mathrm{nm}$ of the ${}^{57}\mathrm{Fe}$ Mössbauer radiation and the detector position $2\theta$ fixed at the $1∕2$-order antiferromagnetic Bragg peak position for various correlation lengths $\xi$ indicated in the figure. A single domain bias parameter of $\eta =0.1$ was used. Dotted and solid lines show the BA and SDIWA simulations, respectively.

Figure 4

(Color online) Simulated ${\theta }_{\text{in}}\text{−}{\theta }_{\text{out}}$ PNR diffuse intensity maps for the $\mathrm{Mg}\mathrm{O}∕{[{}^{57}\mathrm{Fe}\left(2.62\mathrm{nm}\right)∕\mathrm{Cr}\left(1.28\mathrm{nm}\right)]}_{20}$ antiferromagnetic multilayer structure. The intensities are normalized and shown on a logarithmic color scale. (a) BA and (b) SDIWA $I^{++}$ intensities are shown. A single domain bias parameter of $\eta =0.1$ was used.

Figure 5

(Color online) Simulated ${\theta }_{\text{in}}\text{−}{\theta }_{\text{out}}$ SMR diffuse intensity maps of the $\mathrm{Mg}\mathrm{O}∕{[{}^{57}\mathrm{Fe}\left(2.62\mathrm{nm}\right)∕\mathrm{Cr}\left(1.28\mathrm{nm}\right)]}_{20}$ antiferromagnetic multilayer structure around the $1∕2$-order antiferromagnetic Bragg peak using (a) the BA and (b) the SDIWA. The intensities are shown on a logarithmic color scale and are normalized. A single domain bias parameter of $\eta =0.1$ was used.

Figure 6

(Color online) (a) Measured and (b) simulated $\omega \text{−}2\theta$ SMR diffuse intensity maps of the $\mathrm{Mg}\mathrm{O}∕{[{}^{57}\mathrm{Fe}\left(2.62\mathrm{nm}\right)∕\mathrm{Cr}\left(1.28\mathrm{nm}\right)]}_{20}$ antiferromagnetic multilayer in the vicinity of the $1∕2$-order antiferromagnetic Bragg peak. The intensities are normalized and shown on a logarithmic color scale. Map (b) was simulated using Eqs. (44, 47, 48), with $\xi =1.0\mu \mathrm{m}$ domain correlation length. The dotted white line shows the 10% level of the maximum intensity.