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Canadian Journal of Physics

Critical Examination of a Proposed Charged-Particle Stern–Gerlach Experiment

Thomas F. Knott

Published Feb 1, 1972• 1 citation
Publisher

Abstract

It has been proposed by Enga and Bloom that combined electric and magnetic helical quadrupole fields may be used to perform a Stern–Gerlach experiment on charged particles. A detailed investigation shows that the longitudinal Lorentz force due to coupling of the transverse velocity of the particles to the transverse magnetic field produces an additional focusing effect which masks the Stern–Gerlach force in large regions of initial phase space. Consideration of uncompensated magnetic fields, produced by small random variations in conductor dimensions and location, shows that the tolerances required to preserve spin separation in the useful range of initial conditions are several orders of magnitude higher than can be achieved at this time.

DOI
10.1139/p72-030

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It has been proposed by Enga and Bloom that combined electric and magnetic helical quadrupole fields may be used to perform a Stern–Gerlach experiment on charged particles. A detailed investigation shows that the longitudinal Lorentz force due to coupling of the transverse velocity of the particles to the transverse magnetic field produces an additional focusing effect which masks the Stern–Gerlach force in large regions of initial phase space. Consideration of uncompensated magnetic fields, produced by small random variations in conductor dimensions and location, shows that the tolerances required to preserve spin separation in the useful range of initial conditions are several orders of magnitude higher than can be achieved at this time.
Abstract We present a feasibility study with several magnetic field configurations for creating spin-dependent forces that can split a low-energy ion beam by the Stern–Gerlach (SG) effect. To the best of our knowledge, coherent spin-splittings of charged particles have yet to be realised. Our proposal is based on ion source parameters taken from a recent experiment that demonstrated single-ion implantation from a high-brightness ion source combined with a radio-frequency Paul trap. The inhomogeneous magnetic fields can be created by permanently magnetised microstructures or from current-carrying wires with sizes in the micron range, such as those recently used in a successful implementation of the SG effect with neutral atoms. All relevant forces (Lorentz force and image charges) are taken into account, and measurable splittings are found by analytical and numerical calculations.
Two experiments are described. One is the successful observation of the resonant deflection of a beam of neutral potassium atoms at a frequency of 7.2 Mhz, in agreement with the predictions of the theory of the Transverse Stern-Gerlach (TGS) experiment. The other is a proposal for a charged particle Stern-Gerlach experiment, which is based on an extension of the TSG experiment to time independent, inhomogeneous magnetic fields having the form [formula omitted] If the field B₁(r) is well chosen, the charged particle trajectories are confined in a stable beam by the resulting Lorentz forces for motion generally along the z axis. This is, in fact, the principle of strong focusing which is now widely used in accelerator design. But in such a system it is also possible to satisfy the criterion for a TSG experiment, since in a frame of reference moving with a particle in the z direction, the field B₁(r) is rotating in time.
The classical Stern-Gerlach experiment is analyzed with an emphasis on the spin dynamics. The central question asked is whether there occurs a relaxation of the spin angular momentum during the time the particle passes through the Stern-Gerlach magnet. We examine in particular the transverse relaxation, involving angular momentum exchange between the spin of the particles and the spins of the magnet. A method is presented describing relaxation effects at an individual particle level. This leads to a stochastic equation of motion for the spins. This is coupled to a classical equation of motion for the particle translation. The experimental situation is then modeled through simulations of individual trajectories using two sets of parameter choices and three different sets of initial conditions. The two main conclusions are: (A) if the coupling between the magnet and the spin is solely described by the Zeeman interaction with the average magnetic field the simulations show a clear disagreement with the experimental observation of Stern and Gerlach. (B) If one, on the other hand, also allows for a T(2) relaxation time shorter than the passage time one can obtain a practically quantitative agreement with the experimental observations. These conclusions are at variance with the standard textbook explanation of the Stern-Gerlach experiment.
In 1932 Pauli and Bohr declared that electromagnetic fields cannot be used to separate electrons according to their spin. A Stern–Gerlach magnet can split atomic beams into beams with different polarizations, but the Lorentz force blurs the splitting for charged particles such as electrons. Brillouin suggested that electrons could be separated by applying magnetic gradients along the electron beam, but Pauli disagreed. He wrote in 1958: "It is impossible to observe the spin of the electron, separated fully from its orbital momentum, by means of experiments based on the concept of classical particle trajectories."
A successful transverse Stern–Gerlach experiment has been performed, using a beam of neutral potassium atoms and an inhomogeneous time-dependent magnetic field of the form[Formula: see text]A classical analysis of the Stern–Gerlach experiment is given for a rotating inhomogeneous magnetic field. In general, when space quantization is achieved, the spins are quantized along the effective magnetic field in the reference frame rotating with angular velocity ω about the z axis. For ω = 0, the direction of quantization is the z axis (conventional Stern–Gerlach experiment), while at resonance (ω = −γH 0 ) the direction of quantization is the x axis in the rotating reference frame (transverse Stern–Gerlach experiment). The experiment, which was performed at 7.2 Mc, is described in detail.