  The finite size and surface roughness effects on the magnetization of NiO
nanoparticles is investigated. A large magnetic moment arises for an
antiferromagnetic nanoparticle due to these effects. The magnetic moment
without the surface roughness has a non-monotonic and oscillatory dependence on
$R$, the size of the particles, with the amplitude of the fluctuations varying
linearly with $R$. The geometry of the particle also matters a lot in the
calculation of the net magnetic moment. An oblate spheroid shape particle shows
an increase in net magnetic moment by increasing oblateness of the particle.
However, the magnetic moment values thus calculated are very small compared to
the experimental values for various sizes, indicating that the bulk
antiferromagnetic structure may not hold near the surface. We incorporate the
surface roughness in two different ways; an ordered surface with surface spins
inside a surface roughness shell aligned due to an internal field, and a
disordered surface with randomly oriented spins inside surface roughness shell.
Taking a variational approach we find that the core interaction strength is
modified for nontrivial values of $\Delta$ which is a signature of
multi-sublattice ordering for nanoparticles. The surface roughness scale
$\Delta $ is also showing size dependent fluctuations, with an envelope decay
$\Delta\sim R^{-1/5}$. The net magnetic moment values calculated using
spheroidal shape and ordered surface are close to the experimental values for
different sizes.
