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Some physical properties of disorder Blume-Emery-Griffiths model. S.L. Yan, H.P Dong Department of Physics Suzhou University CCAST 2007-10-26. Outlines. 1. Introduction 2. Theory 3. Results & discussions 4. Summary. 1. Introduction. Ising model and its development
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S.L. Yan, H.P Dong
Department of Physics
3. Results & discussions
(2) Applications of the BEG model
(3) Methods of research the BEG model
In this report, we give the phase diagrams and magnetic properties of the disorder BEG model
The Hamiltonian for the spin-1 BEG model can be defined as:
The effective Hamiltonian is
and K is the exchange and biquadratic interaction between the nearest-neighbor pairs. Di is a crystal field parameter. h is a uniform magnetic field.
K>0 or K<0.
By intruducing the ratio of biquadratic and exchange interaction, defined to be .
case, it is a single lattice problem.
Within the EFT, by using differential operator technique and the van der Waerden identities, the average magnetization and the quadrupolar moment can be formulated by:
Since order parameter equations deal with multi-spin correlations, we cannot treat exactly all the spin-spin correlations. Therefore, a cutting approximation procedure shall be adopted:
Then equations may be rewritten to be
where z is the coordination number. By solving above-mention equations, the expression can be obtained:
According to Laudau theory, the second-order phase transition equation is determined by:
where and are given by:
The tricritical point at which the second-order phase
transition changes to the first-order one must satisfy
The first-order phase transition equation is determined
For case, it is necessary to adopttwo sublattices. We can discuss staggered quadrupolar (SQ) phase and bicritical point of the BEG model. The Hamiltonian is
The SQ phase satisfies
This is condition of the two-cycle fixed points. In addition, the paramagnetic phase corresponds
Thus, one has:
Obviously, the paramagnetic phase satisfies the single fixed point condition. In order to obtain the SQ and the paramagnetic phase boundary, the single fixed point must bifurcate into two-cycle fixed points. The limit of stability for bifurcation of the fixed point is as follows:
The solution of the SQ-P phase boundary is given by equations (8’) and (9’).
The F-P second-order phase boundary is still given by equations (16) and (17).
The intersection of the SQ-P line and the F-P one is a BCP. The coordinate of the BCP is solution of the set of coupled equations (8’)-(9’) and (16)-(17).
For simplification, a simple cubic lattice is selected as the three-dimensional version
(1) Critical behaviors of the BEG model for
Shrink of Double TCPs
Fig. (a) apparent reentrant phenomenon, Fig.(b) weak reentrant phenomenon
The TCP is depressed monotonically with decreasing crystal field concentration.
(2) Staggered quadrupolar phase and bicritical point
PRB 69 (2004) 064423
Influence of different parameters on the SQ phase and the BCP
(3) Magnetic properties of the BEG model for
Influence of positive and negative ratio onInitial magnetizations
Influence of ratio and bond dilution on Initial magnetizations
Influence of positive and negative ratio on susceptibility
Influence of ratio and bond dilution on susceptibility
The temperature dependence of the magnetization under different parameters
(PRB 71 (2005) 024434)