1D toy model for magnetic trapping

Abstract

We study, both classically and quantum mechanically, the problem of a neutral particle with a spin angular momentum S, mass m, and magnetic moment , moving in one dimension in an inhomogeneous magnetic field given by BϭB 0 ẑϩB Ќ Ј xŷ. This problem serves for us as a toy model to study the trapping of neutral particles. We identify Kϵͱ͓S 2 (B Ќ Ј ) 2 /mB 0 3 ͔, which is the ratio between the precessional frequency of the particle and its vibrational frequency, as the relevant parameter of the problem. Classically, we find that when is antiparallel to B, the particle is trapped provided that KϽ0.5. We also find that viscous friction, be it translational or precessional, destabilizes the system. Quantum mechanically, we study the problem of a spin Sϭប/2 particle in the same field. Treating K as a small parameter for the perturbation from the adiabatic Hamiltonian, we find that the lifetime T esc of the particle in its trapped ground state is T esc ϭ(T vib /2) ϫ(1/ͱ8K)exp(2/K), where T vib ϭ2ͱmB 0 /(B Ќ Ј ) 2 is the classical period of the particle when placed in the adiabatic potential Vϭ͉B͉.

Majorana spin-flip transitions in a magnetic trap

“…The cases F = 1/2 or 1 were studied in Sukumar and Brink [7] and the transition rates for atoms to escape from a trap were calculated. Similar calculations have been made by other authors [8]. In the present paper we extend the results of ref.…”

On the spinning motion of the hovering magnetic top

“…To study this we assume the top to be endowed intrincily with spin proportional to its magnetic moment. We find that it may be possible to hover a top having only this spin with no additional angular momentum [18].…”

“…The physical principles underlying the dynamical stability of the hovering magnetic top rely on the so-called 'adiabatic approximation' and have recently been discussed in several papers [12,13,14,15]. In these articles the top was modeled as an axially symmetric shape with its magnetization taken as a dipole pointing along the symmetry axis and situated at the center of mass.…”

“…We therefore set δθ(t) = δθ 0 e λt δφ(t) = δφ 0 e λt δψ(t) = δψ 0 e λt and substitute these into Eqs. (18). The result is the matrix equation for the determination of the eigenvalues λ…”

Non-Contacting Interfaces: A Case Study in Modular Spacecraft Design

“…However, it requires power, interacts detrimentally with spacecraft electronics, induces unwanted, attitudeperturbing torques due to surrounding fi elds (such as the geomagnetic fi eld), and introduces the very real risk that a temporary loss of power or a software failure may cause the assembly to lose structural integrity.  Oscillating and moving magnets, whose quasipassive, periodic motion creates relative equilibria (in the Hamiltonian sense): An entertaining example of this behavior is popular Levitron toy (Gov et al 1999(Gov et al , 2000, in which one spinning magnet levitates several inches above another. Because it depends on bound angular momentum, this principle is not particularly useful for spacecraft, where angular momentum is carefully managed for attitude control.…”