Multispace quantum interference in a ${}^{57}$Fe synchrotron Mössbauer source

A physical picture for coherent emission of $\gamma$ rays by a nuclear array excited with synchrotron radiation is given. The particular case of a pure nuclear Bragg reflection from the ${}^{57}$FeBO${}_3$ crystal is analyzed. During free de-excitation of the nuclei, a nuclear exciton polariton is developing inside the crystal and generates at the exit of the crystal a coherent $\gamma$-ray beam. In the crystal the nuclear levels of ${}^{57}$Fe are split because of a combined magnetic and electric interaction. The rich picture of $\gamma$-ray interference is described, which involves geometrical space, energy, and spin domains. In the vicinity of the Néel temperature of the crystal the magnetic splitting of nuclear levels nearly collapses. These conditions lead to drastic changes in the angular, energy, and temporal properties of the emitted radiation. The emission angular function, which in the approximation of a plane incident wave represents the emission intensity for different angular settings of the crystal near Bragg angle, strongly broadens and transforms to a double-hump structure with a central dip between the peaks. The energy and temporal distributions of the emitted radiation crucially depend upon the crystal angular setting. Beyond the central dip, the energy distribution of nuclear scattering acquires a complicated form with several satellites at various energies. In contrast, at the exact angular position of the central dip, the energy spectrum exhibits a single line shape with the line width close to the natural width of the nuclear resonance. The obtained results constitute the theoretical basis for the understanding and for the further elaboration of the ${}^{57}$Fe synchrotron Mössbauer source—the device that provides a collimated beam of intense and polarized $\gamma$ radiation in an energy bandwidth of nanoelectronvolts, the nuclear resonance natural level width.

Figure 1

Amplitudes ${\alpha }_{1-4}$ of contributions of the pure spin states $|\pm \frac{1}{2}\rangle$, $|\pm \frac{3}{2}\rangle$ into the mixed spin states for different energy sublevels ${\epsilon }_{1-4}$ of the excited nuclear state of different nuclei in the unit cell of the IB crystal (see Table ). For parameter $\nu$ see explanation to Eq. (1).

Figure 2

Scattering paths via the pure spin states of the excited nuclear substate having energy ${\varepsilon }_1$ in the first nucleus of the IB crystal unit cell. On the upper panel the transitions, up and down, between the ground and the excited states are displayed. On the lower panel four possible paths are shown, where changes of spin projections in different paths are indicated. $M=m_e-m_g=\pm 1$ are the changes of magnetic quantum number in separate transitions; $m_e,m_g$ are the magnetic quantum numbers for the excited and ground nuclear states respectively.

Figure 3

Scattering geometry of synchrotron radiation from two nuclei in the IB crystal unit cell; ${\mathbf{k}}_{0,1}$ and ${\mathbf{h}}_{0,1}^{\pi ,\sigma }$ are the wave vectors and the magnetic polarization vectors of the incident and scattered waves respectively, $\theta$ is incidence angle (symmetric scattering geometry is considered); ${\mathbf{B}}_{1,2}$ are the magnetic fields at the nuclei in the unit cell; ${\mathbf{u}}_{0,1,-1}^{1,2}$ are the spherical unit vectors related to hyperfine fields at the first and second nuclei in the IB crystal unit cell (see text for definition).

Figure 4

Evolution of the $\gamma$-ray emission landscape. The internal magnetic field at ${}^{57}$Fe nuclei positions is: (a) 330 KOe, (b) 50 kOe, (c) 2 kOe.

Figure 5

The emission angular dependence of $\gamma$ radiation from the IB crystal integrated over the whole resonance range. The angular setting of the IB crystal is changed in the vicinity of Bragg angle for pure nuclear reflection (333). The angular functions are displayed for different values of the internal magnetic field decreasing from the bottom to the top in approaching Néel temperature.

Figure 6

Angular distribution of the emitted $\gamma$ radiation (solid line); angular distribution of the exciting synchrotron radiation (dots) (see text). The first distribution is scaled in intensity for comparison with the second one. The crystal is set at the position of angular dip (top panel in Fig. 5).

Figure 7

Energy distributions of the emitted $\gamma$ radiation at different angles of incidence of the exciting SR in the vicinity of the dip zone of the emission angular function for the (333) reflection and for the internal magnetic field of 2 kOe. The dip zone (taken from Fig. 5) is shown on the upper panel. The energy distributions on the lower panels refer to the left and the right sides of the dip zone, L and R respectively, as shown on the upper panel. The energy distribution at the exact dip position is displayed on both L and R panels (bold line curves). Each next distribution corresponds to an angular shift of the crystal from the dip position by 5 $\mu$rad. For better visualization the energy spectra are equally spaced along the vertical axis.

Figure 8

Time distributions of the emitted $\gamma$ radiation at different angles of incidence of the exciting SR in the vicinity of the dip zone of the emission angular function for the (333) reflection and for the internal magnetic field of 2 kOe. The dip zone (taken from Fig. 5) is shown on the upper panel. The time distributions on the lower panels refer to the left and the right sides of the dip zone, L and R respectively, as shown on the upper panel. The time distribution at the exact dip position is displayed on both L and R panels (bold line curves). Each next distribution corresponds to an angular shift of the crystal from the dip position by 5 $\mu$rad. For better visualization the time distributions are equally spaced along the vertical axis.