@article{ijp2018643,
author={{Waddahou, A. and Chahid, M. and Maarouf, A. and Benzouine, F.},
title={Critical Behavior at Surfaces of Strong Coupling Paramagnetic Systems Exhibiting a Paramagnetic-Ferrimagnetic Transition},
journal={International Journal of Physics},
volume={6},
number={4},
pages={116--121},
year={2018},
url={http://pubs.sciepub.com/ijp/6/4/3},
issn={2333-4576},
abstract={The purpose of this work is the investigation of the critical surface effects of two strongly coupled paramagnetic sublattices exhibiting a para-ferrimagnetic transition. The model is of Landau-Ginzburg type, whose bulk free energy is a functional of two kind of order parameters (local magnetizations) &#966; and <img src=image/abs1.png></img> This free energy involves, beside quadratic and quartic terms in both &#966; and <img src=image/abs2.png></img> , a lowest-order coupling, <img src=image/abs3.png></img>  where C<SUB>o</SUB>&lt;0 is the coupling constant measuring the interaction between the two sublattices. Two terms H&#966; and <img src=image/abs4.png></img> are also introduced, to describe the interaction within an external magnetic field <i>H</i>. We introduce a surface free energy expanded in terms of the local order parameters. The magnetization at the surface are &#966;<SUB>s</SUB> and <img src=image/abs5.png></img> We show, in particular, that the model can be reduced to an effective theory written in terms of the overall magnetization <img src=image/abs6.png></img> and the associated fraction of magnetization <img src=image/abs7.png></img>. This formulation leads us to define an effective extrapolation length &#955;<SUB>s</SUB>. We then derive all the critical properties of the system close to the critical temperature <i>T</i><SUB><i>c</i></SUB>. In particular, we determine the critical behavior of the overall surface magnetization <img src=image/abs8.png></img>, in terms of <SUB>b</SUB> above and below <i>T</i><SUB><i>c</i></SUB>. The variations of <SUB>s</SUB> with the magnetic field <i>H</i>, and when a surface field H<SUB>s</SUB> is applied actually at <i>T</i><SUB><i>c</i></SUB>, are derived. We determine, also, the associated susceptibilities at the surface &#967;<SUB>s</SUB> and &#967;<SUB>s</SUB><SUB>.s</SUB>. The determination of the full profile of the magnetization close to the surface will be the subject of a future communication.},
doi={10.12691/ijp-6-4-3}
publisher={Science and Education Publishing}
}