Polarisation aspects

One of the unique properties of the synchrotron radiation is its polarisation. The synchrotron radiation is linearly polarized in the plane of the storage ring in the middle of the beam profile. Due to the selection rules the ratio $R$ between the emission lines in the energy domain is defined by equation [33,34,35]

$$R=\frac{1}{3} \frac {\sin \alpha (4-3\cos^{2} \beta)+\cos \alpha} {\sin \alpha \cos^{2} \beta + \cos \alpha}.$$ (52)

The values for $\sin \alpha$ and $\cos \beta$ necessary for the calculation of equation of a vector $V_{zz}$ described by spherical coordinates ($\Theta$ and $\Phi$) can be derived from figure

$$\sin \alpha = \frac{\cos \Theta}{\sqrt{1-\sin ^{2} \Theta \sin^{2} \Phi}}$$

$$\cos \alpha = \frac{\cos \Phi \sin \Theta}{\sqrt{1-\sin ^{2} \Theta \sin^{2} \Phi}}$$

$$\cos \beta = \sin \Phi.$$ (53)

Figure: Definition of angles $\alpha$ and $\beta$ used in equation to calculate the intensity ratio of the Mössbauer emission lines (left) (assuming a linear polarised source). The angles $\Phi$ and $\Theta$ are the common spherical angles defining the orientation of a vector in 3D [36]. The shown definition of angles $\alpha$ and $\beta$ corresponds to [34] yielding the angular dependence of the emission line intensities $R$ shown right. The grey region denotes a region where the quantum beats in a NRS time spectrum are very well pronounced.

In the extreme case only one emission line is allowed and the time spectrum becomes a simple exponential function in the thin sample limit. As an example, the geometry where the main axis of the electric field gradient is perpendicular to the plane of polarisation and perpendicular to the incident radiation can be discussed. Both transitions ( $\Delta m=\pm 1$) are allowed if $V_{zz}$ lies in the plane of the storage ring and the interference results in a quantum beat structure (figure upper part). However if $V_{zz}$ is perpendicular to the storage ring only $\Delta m=\pm 0$ transitions are allowed and no quantum beats can be observed.

Figure: In the case of polarised radiation, e.g. synchrotron radiation, only certain transitions are allowed in the case of quadrupole interaction depending on the relative orientation of the polarisation and the main axis of the electric field gradient $V_{zz}$.