Is ferromagnetic to paramagnetic phase transition a reversible process?
If I start with a ferromagnetic material with a spontaneous magnetization below the Curie temperature, and then I start to heat it, it will become paramagnetic above the critical temperature. If I then start to drop the temperature slowly to below the Curie temperature then will I achieve the ferromagnetic behaviour with same spontaneous magnetization as before?
Yes, locally it will have the same magnetization as before due to spontaneous ferromagnetic ordering within magnetic domains. But without an applied magnetic field, the domains will arrange in such a way that the large-scale magnetization will be zero.
I once made this Curie pendulum: https://www.youtube.com/watch?v=CvIGr3wFVgo
This can be compared with melting a nice single crystal of for example silver. Cooling it down will give an ordinary polycrystalline lump.
It will become ferromagnetic again.
The domain pattern and magnetization state will be different to before unless a single state is strongly preferred due to a mixture of other boundary conditions such as:
In general it is extremely unlikely to end up in the same state. However, notatable and even pratical exceptions exist especially for the first two bullets.
Spontaneous symmetry breaking
Unless there is a preferred direction for magnetization, specified, e.g., by the external magnetic field or crystal symmetries, there is no reason for the magnetization to point in the same direction as before. The direction chosen by the magnetization when entering the ferromagnetic phase is an example of spontaneous symmetry breaking.
Domains
Also, as @user137289 has correctly pointed out, unless the crystal is cooled all the way to zero temperature, the magnetization is not homogeneous, but would rather consist of many homogeneously magnetized domains, which are differently oriented in respect to each other.
Let me rephrase your question slightly to make it clearer what (I think) you are asking.
Suppose we start with a ferromagnetic material above the Curie point and we cool it through the Curie point in the absence of any external influence e.g. no externally applied magnetic field. And suppose we repeat this experiment many times. Will the final state of the material always be the same?
The answer is that without any external field the total magnetic field will be zero. This is because the magnetic domains formed as we pass through the Curie point will be randomly oriented and their total magnetic field will sum to zero. In this I agree with Pieter.
But it is only the total field that is the same. If we watched an individual spin in the material as we cycled it through the Curie point it would not have the same orientation each time. And unless there was some controlling factor, like defects in the solid, the magnetic domains would not be the same each time.
In any system we have random thermal fluctuations, and in the ferromagnetic above the Curie point there will be random thermal fluctuations in the alignment of the dipoles. As we cool towards the Curie temperature these fluctuations will get larger and larger, and at some point magnetic domains will nucleate and start to grow. These give rise to the domains we observe in the material at low temperature. But the nucleation process is random and hence the final pattern of the domains will be random. So while the overall field is always zero after the cooling the microstructure will not be.
Going above and below the Curie point, back and forth, in a cyclical manner was done by Giauque in his use of cryogenic cooling of a paramagnetic salt, see magneto caloric effect.
Similar adiabatic cooling effect can also be observed in Gadolinium metal and also in its alloys at room or near room temperature. The effect is used in developing commercial refrigerators. Gschneidner and Pecharsky in MAGNETOCALORIC MATERIALS claim that the transition between ferromagnetic and paramagnetic is a reversible process:
In the case of a ferromagnet near its magnetic ordering temperature, the adiabatic application of a magnetic field reduces the magnetic entropy of a solid and, in turn, it is heated via the increase of its lattice entropy to maintain the entropy of a closed system at a constant value. In a reversible process, a ferromagnet is cooled as the magnetic entropy increases, and the lattice entropy decreases upon adiabatic removal of the magnetic field.
There's no reason for it to show any periodicity. It doesn't depend in a simple way on the outer electron configuration of atoms. The specifics of crystal structure are important in ferrmoganetic (also in antiferromagnetic, and ferrimagnetic) materials
The paramagnetic/ferromagnetic phase transition is an archetypal example of a continuous (or second-order) phase transition. When the temperature T approaches the Curie temperature Tc, the magnetization M(T), which is the order parameter of the transition, continuously goes to zero.
