  We analyse within first-order perturbation theory the instantaneous
transition rate of an accelerated Unruh-DeWitt particle detector whose coupling
to a massless scalar field on four-dimensional Minkowski space is regularised
by a spatial profile. For the Lorentzian profile introduced by Schlicht, the
zero size limit is computed explicitly and expressed as a manifestly finite
integral formula that no longer involves regulators or limits. The same
transition rate is obtained for an arbitrary profile of compact support under a
modified definition of spatial smearing. Consequences for the asymptotic
behaviour of the transition rate are discussed. A number of stationary and
nonstationary trajectories are analysed, recovering in particular the Planckian
spectrum for uniform acceleration.
