You can see the suggested equation (7) in the paper (http://link.springer.com/article/10.1007/s10967-014-2965-9).
n_h (Z)= (n_(h,max )- n_(h,max ))/(2invtan(Z_max- Z_0)) invtan[α(Z- Z_0 )+ n_h (Z_0 )
in which α = 1.2 is a coefficient characterizes the amplitude of the hydration number-jump in the centre of nf-series and the centre’s average hydration number nh(Z0) = 13.55.
All properties varying in S-shaped versus the atomic number Z of f-series can obey at this proposed equation ! But the problem is what the significance of th e α-parameter ?!!
This is an excellent question with a number of possible answers. In addition to the very helpful answers already given, there is a bit more to add.
In considering the answer to this question, consider the following first principles study of rare earth oxides in
https://arxiv.org/pdf/cond-mat/0503667.pdf
See, also,
https://arxiv.org/ftp/arxiv/papers/1201/1201.5249.pdf
Thank you for all advises.
I have written a modest paper.
I try my best with collaboration to suggest other expressions.
Article On the Modeling of the S-Shaped Thermodynamic and Transport ...
Dear James F. Peters
Thank you for you previous answer, it's useful for me.
I have written a modest paper, I will continue with to those have the same trend or behavior in other discipline, to test the proposed expression.
Article On the Modeling of the S-Shaped Thermodynamic and Transport ...