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Journal of Physics Conference Series

Spin-rotation coupling and Thomas precession of ions in a storage ring

Gaetano Lambiase, G. Papini

Published Jun 10, 2013• 1 citation
Publisher

Abstract

We discuss the role of spin-rotation coupling and Thomas precession in the decay of hydrogen-like heavy ions that are injected in a storage ring. We find that the decay curve has a time modulation that vanishes if the magnetic field also disappears. The results can be applied to the recently reported modulation in the decay of the hydrogen-like ions 140Pr58+, 142Pm60+ and 122I52+, where rotation couples to the spins of electron and nucleus. Spin-rotation coupling and Thomas precession also generate oscillations in the decay of a muon bound to a nucleus rotating in a storage ring.

DOI
10.1088/1742-6596/442/1/012073

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We discuss the role of spin-rotation coupling and Thomas precession in the decay of hydrogen-like heavy ions that are injected in a storage ring. We find that the decay curve has a time modulation that vanishes if the magnetic field also disappears. The results can be applied to the recently reported modulation in the decay of the hydrogen-like ions 140Pr58+, 142Pm60+ and 122I52+, where rotation couples to the spins of electron and nucleus. Spin-rotation coupling and Thomas precession also generate oscillations in the decay of a muon bound to a nucleus rotating in a storage ring.
The theoretical aspects of spin-rotation coupling are presented. The approach is based on the general covariance principle. It is shown that the gyrogravitational ratio of the bare spin- and the spin-1 particles is equal unity. That is why spin couples with rotation as an ordinary angular momentum. This result is the rigorous substantiation of the cranking model. To observe the phenomenon, the experiment with hydrogen-like ions in a storage ring is suggested. It is found that the splitting of the 1 2 S 1/2 , F =1/2 hyperfine state of the 140 Pr 58+ and 142 Pm 60+ ions circulating in the storage ring ESR in Darmstadt along a helical trajectory is about 4.5 MHz. We argue that such splitting can be experimentally determined by means of the ionic interferometry.
We consider a semi-classical model to describe the origin of the spin-orbit interaction in a simple system such as the hydrogen atom. The interaction energy U is calculated in the rest-frame of the nucleus, around which an electron, having linear velocity v and magnetic dipole-moment μ, travels in a circular orbit. The interaction energy U is due to the coupling of the induced electric dipole p=(v/c)×μ with the electric field En of the nucleus. According to quantum mechanics, the radius of the electron’s orbit remains constant during a spin-flip transition. Under such circumstances, our model predicts that the energy of the system changes by ΔE=1/2U, the factor 1/2 emerging naturally as a consequence of equilibrium and the change of the kinetic energy of the electron. The correct 1/2 factor for the spin-orbit coupling energy is thus derived without the need to introduce the well-known Thomas precession in the rest-frame of the electron.
We consider a semi-classical model to describe the origin of the spin–orbit interaction in a simple system such as the hydrogen atom. The interaction energy is calculated in the rest-frame of the nucleus, around which an electron, having linear velocity and magnetic dipole-moment travels in a circular orbit. The interaction energy is due to the coupling of the induced electric dipole with the electric field of the nucleus. Assuming the radius of the electron's orbit remains constant during a spin-flip transition, our model predicts that the energy of the system changes by the factor emerging naturally as a consequence of equilibrium and the change of the kinetic energy of the electron. The correct factor for the spin–orbit coupling energy is thus derived without the need to invoke the well-known Thomas precession in the rest-frame of the electron.
We have applied the density matrix and second-order perturbation theory in order to re-analyze the two-photon decay of hydrogen-like ions for the polarization of emitted light. Special attention is paid to the linear polarization of one of the photons, while the spin state of the second photon is supposed to be unobserved. For such an “angle-polarization” correlation of the two photons we investigate the contributions that arise from non electric-dipole terms in the expansion of the electron-photon interaction. Detailed calculations are performed for the 2s 1/2 →1s 1/2 and 3d 5/2 →1s 1/2 transitions in neutral hydrogen as well as in hydrogen-like uranium.
The behavior of an atom in a nonuniform magnetic field is analyzed, as well as the motion of a classical magnetic dipole (a spinning charged ball) and a rotating charged ring. For the atom it is shown that, while the magnetic field does no work on the electron-orbital contribution to the magnetic moment (the source of translational kinetic energy being the internal energy of the atom), whether or not it does work on the electron-spin contribution to the magnetic moment depends on whether the electron has an intrinsic rotational kinetic energy associated with its spin. A rotational kinetic energy for the electron is shown to be consistent with the Dirac equation. If the electron does have a rotational kinetic energy, the acceleration of a silver atom in a Stern-Gerlach experiment or the emission of a photon from an electron spin flip can be explained without requiring the magnetic field to do work. For a constant magnetic field gradient along the z axis, it is found that the classical objects oscillate in simple harmonic motion along the z axis, the total kinetic energy -- translational plus rotational -- being a constant of the motion. For the charged ball, the change in rotational kinetic energy is associated only with a change in the precession frequency, the rotation rate about the figure axis remaining constant.