Abstract
Neutral atoms entering an electric field experience a defocusing force in the dipole field direction, which is proportional to the field gradient. If an experiment, such as the search for a permanent electron electric dipole moment (eEDM), requires a very strong electric field ($13.5\text{ }\text{ }\mathrm{MV}/\mathrm{m}$), then this end-field defocusing results in beam blowup and much reduced phase-space acceptance. In this paper we discuss how these defocusing fields arise from the longitudinal changes in the electric dipole field and their dependence on the electrode shape and spacing between lenses. We find that the end-field defocusing comes from strong impulse forces, whose defocusing power was calculated for simple electrodes with rounded ends. To compensate for this end-field defocusing, a triplet of transverse-focusing lenses was added to the pure dipole field plates in the generic eEDM cesium fountain experiment used to study the neutral beam optics. Envelope equations, which calculated the beam sizes of the atom bunch for the linear forces, are used to obtain a set of lens parameters that give a well focused beam in the fountain. Atom trajectory equations allow us to calculate the phase-space acceptance of the lens system with the nonlinear force terms included.
Highlights
A beam of slow neutral atoms in a static electric field gradient experiences a force arising from the interaction of the electric field with the induced electric dipole moment of the atom, which can produce useful focusing [1] as well as undesirable end-field defocusing
The atoms on entering an electric dipole field will be accelerated in the longitudinal direction by the rising field strength, and decelerated when they leave the electric field region
At the entrance to the electrodes the electric field strength increases off axis in the electric field direction (Fig. 1), so the atoms will experience an end-field defocusing force
Summary
A beam of slow neutral atoms in a static electric field gradient experiences a force arising from the interaction of the electric field with the induced electric dipole moment of the atom, which can produce useful focusing [1] as well as undesirable end-field defocusing. At the entrance (and exit) to the electrodes the electric field strength increases off axis in the electric field direction (Fig. 1), so the atoms will experience an end-field defocusing force (Sec. IV) at both ends. Triplet lenses with the DFD configuration produce focusing that gives more control and the smallest beam size in the y direction, where the phasespace acceptance is strongly limited by the physical plate apertures and the end-field defocusing. The eEDM-fountain design optics (Table I: triplet lengths Lp) were obtained using the linear-focusing beam envelope equations (see Sec. VII), and varying the (D1, F2, D3)-lens plate lengths Lp to change the transverse focusing, and obtain the smallest beam sizes in the fountain with maximum phase-space acceptance areas
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