Elliptic vector loci of average electron velocity of electron swarm in constant-collision-frequency model gas under ac electric and dc magnetic fields crossed at arbitrary angles

Abstract

The analytic solution of the average electron velocity vector $$\mathbf {W}(t)$$ of an electron swarm in gas under ac electric and dc magnetic fields ( $$\mathbf {E}(t)$$ and $$\mathbf {B}$$ , respectively) in a constant-collision-frequency model is extended to cover not only the known case of a right crossing angle ( $$\mathbf {E}(t) \perp \mathbf {B}$$ ) but also the cases of arbitrary crossing angles ( $$\mathbf {E}(t) \not \perp \mathbf {B}$$ ). The x, y, and z components of $$\mathbf {W}(t)$$ are obtained as explicit formulae with the following parameters: the amplitude E of $$\mathbf {E}(t)$$ , the angle $$\theta $$ between $$\mathbf {E}(t)$$ and $$\mathbf {B}$$ , the angular frequency $$\omega _E$$ of $$\mathbf {E}(t)$$ , the cyclotron angular frequency $$\omega _B$$ (subject to the strength B of $$\mathbf {B}$$ ), and the electron collision frequency $$\nu $$ . A basic feature that $$\mathbf {W}(t)$$ draws elliptic locus in velocity space is unchanged even under $$\mathbf {E}(t) \not \perp \mathbf {B}$$ , but the plane including the locus may tilt when $$\omega _E$$ , $$\omega _B$$ , or $$\nu $$ varies. The derivation of $$\mathbf {W}(t)$$ based on the analytic solution of single electron motion is detailed and fundamental behavior of the $$\mathbf {W}(t)$$ loci is observed to understand the electron transport under $$\mathbf {E}(t) \times \mathbf {B}$$ fields.