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Investigation of Ising Model in Nano Systems with Long Range Interaction

Taherkhani, Farid | 2011

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 41435 (03)
  4. University: Sharif University of Technology
  5. Department: Chemistry
  6. Advisor(s): Parsafar, Gholam Abbas
  7. Abstract:
  8. Ising model in nano systems are studied, in the presence of a magnetic field. For a one-dimensional(1-D) array of spins interacting via nearest-neighbor and next-nearest-neighbor interactions we calculated the heat capacity , the surface energy, the finite-size free energy, and bulk free energy per site. The heat capacity versus temperature exhibited a common wide peak for systems of any size. A small peak also appears at lower temperatures for small arrays when the ratio of magnetic field- spin interaction energy over the nearest-neighbour spin-spin interaction energy f is within and size. The peak becomes smaller for longer array and eventually vanishes for long arrays, disappearing when the number of spins, N, is greater than 25 when only nearest neighbor interactions are taken into account, and more than 14 when next-nearest-neighbor interactions are included as well. Ising models in which the nearest-neighbor interactions are ferromagnetic, while the next-nearest-neighbor interactions are either ferromagnetic antiferromagnetic are compared, and it is found that the reduced free energy in the former case exhibits larger deviation from the bulk value. Also To solve the controversy on the question raised as the title of this paper, we apply the transfer matrix method to solve the one dimensional Ising model in the presence of a magnetic field, taking both nearest and next-nearest neighbor interactions into account. We show that it is possible to write a transfer matrix only if the number of sites is even. Even in such a case, it is difficult to diagonalize the transfer matrix in an analytic form. Therefore, we employ a numerical method to obtain the eigenvalues of the transfer matrix. Moreover, the heat capacity, magnetization, and magnetic susceptibility versus temperature for different values of the competition factor (the ratio of next-nearest-neighbor to nearest-neighbor interactions) are presented. We have shown that, at high magnetic fields for 1-D Ising model, the surface free energy has an asymptotic limit. Absolute surface energy increases when the value of f (the ratio of magnetic field to nearest-neighbor interactions) increases. For , the surface energy approaches a constant value. Under such a circumstance, the surface free energy is strongly affected by increasing magnetic field. For the values of , finite site free energy increases slowly. It has also been explored the thermodynamic limit in which physical properties approach the bulk value. A new method is developed to calculate the critical line of 2-D anisotropic Ising model considering nearest interactions. This method is based on the real space renormalization group theory by extending block sizes. The reduced temperatures (ka, kb, kc) are calculated for different block sizes. The critical line for different types of lattice namely; triangular, square and honeycomb by extending block size is obtained which shows to be accurate result comparing with results by Onsager. Our result Also shows that for triangular and honeycomb lattices, there are asymptotic limits for critical lines when is plotted versus kb in the case of . Finally the critical exponents are obtained via renormalization group approach. Our survey shows that in isotropic limit, there is good agreement with Onsager results; however. The formation and diffusion of a vacancy in a silver nanocluster are studied via a combination of first principles and statistical mechanics simulations. A 38-atom truncated-octahedral (TO) arrangement and its homologue with 37 Ag atoms and one vacancy are considered, and density-functional calculations are performed to derive the energies of the local minima and the energy barriers connecting them. Data used as an input for a study of the system dynamics via a kinetic Monte Carlo algorithm, evaluating site occupancies, and diffusion coefficient and equilibration time. It is found that vacancy formation and diffusion represents a viable path for atom–atom exchange in these conditions.
  9. Keywords:
  10. Ising model ; Nanosystem ; Renormalization Group ; Magnetic Fields

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