  Superconducting structures with the size of the order of the superconducting
coherence length xi(T) have a critical temperature T_c, oscillating as a
function of the applied perpendicular magnetic field H (or flux Phi). For a
thin-wire superconducting loop, the oscillations in T_c are perfectly periodic
with H (this is the well-known Little-Parks effect), while for a singly
connected superconducting disk the oscillations are pseudoperiodic, i.e. the
magnetic period decreases as H grows. In the present paper, we study the
intermediate case: a loop made of thick wires. By increasing the size of the
opening in the middle, the disk-like behaviour of T_c(H) with a quasi-linear
background (characteristic for '3-dimensional' (3D) behaviour) is shown to
evolve into a parabolic T_c(H) background ('2D'), superimposed with perfectly
periodic oscillations. The calculations are performed using the linearized
Ginzburg-Landau theory, with the proper normal/vacuum boundary conditions at
both the internal and the external interface. Above a certain crossover
magnetic flux Phi, T_c(Phi) of the loops becomes quasi-linear, and the flux
period matches with the case of the filled disk. This dimensional transition is
similar to the 2D-3D transition for thin films in a parallel magnetic field,
where vortices enter the material as soon as the film thickness t > 1.8 xi(T).
For the loops studied here, the crossover point appears for w = 1.8 xi(T) as
well, with w the width of the wires forming the loop. In the 3D regime, a
"giant vortex state" establishes, where superconductivity is concentrated near
the sample's outer interface. The vortex is then localized inside the loop's
opening.
