Corrected field enhancement factor for the floating sphere model of carbon nanotube emitter

We have corrected the field enhancement factor for the “floating sphere at emitter-plane potential” model with the finite anode-cathode distance. If $ρ$ is the radius of sphere, $h$ is the distance from cathode to the center of sphere, and $l$ is the distance from the center to the anode, then the field enhancement factor is given as the following expression $βsph=(2+7η−η2)(λ2−2λ+2)/[2η(1−λ)(2−λ)]$⁠, where $η=ρ/h$⁠, $λ=ρ/l$⁠. This expression demonstrates reasonable behavior for three limiting cases: if $h→ρ$⁠, if $l→∞$⁠, and if $l→ρ$⁠. We have compared our factor $βsph$ with the field enhancement factor $βtube$ for the “hemisphere on a post” model and the factor $βell$ for the “hemiellipsoid on plane” model. We have shown realization of the approximate evaluation $βtube≈(βsph+βell)/2$⁠.