  We theoretically investigate the damping and trapping forces in a
three-dimensional magneto-optical trap (MOT), by numerically solving the
optical Bloch equations. We focus on the case where there are dark states
because the atom is driven on a "type-II" system where the angular momentum of
the excited state, $F'$, is less than or equal to that of the ground state,
$F$. For these systems we find that the force in a three-dimensional light
field has very different behaviour to its one dimensional counterpart. This
differs from the more commonly used "type-I" systems ($F'=F+1$) where the 1D
and 3D behaviours are similar. Unlike type-I systems where, for red-detuned
light, both Doppler and sub-Doppler forces damp the atomic motion towards zero
velocity, in type-II systems in 3D, the Doppler force and polarization gradient
force have opposite signs. As a result, the atom is driven towards a non-zero
equilibrium velocity, $v_{0}$, where the two forces cancel. We find that
$v_{0}^{2}$ scales linearly with the intensity of the light and is fairly
insensitive to the detuning from resonance. We also discover a new
magneto-optical force that alters the normal MOT force at low magnetic fields
and whose influence is greatest in the type-II systems. We discuss the
implications of these findings for the laser cooling and magneto-optical
trapping of molecules where type-II transitions are unavoidable in realising
closed optical cycling transitions.
