Abstract
In general, the Ising model is a well-known model of ferromagnetism. Ising solve the model in one-dimension in 1925 eraly.But until in 1944,the partition function of the rectangular-lattice Ising model in two-demension with nearest-neighbor interaction in the absence of external magnetic field was derived by Onsager. The spontaneous magnetization of the model was first obtained by Osanger but he never publish his derivation. C.N. Yang was first to publish a detailed derivation of the spontaneous magnetization on the square lattice and the critical exponentβis 0.125 in 1952 .And then Yang suggests that C.H. Chang generalize the above calculation to the rectangular lattice and the result coincide with before .Therefore Chang noted that the exponentβ=0.125 does not change with varying the ratio of the vertical and the horizontal interactions.Maybe according to the same result, he speculate about all two-dimensional Ising models have the same critical exponents naturally. This is the interesting direction of study people are striving toward .However after a long time research, statistical physicists find regular plan lattices enable to exist exact solution,but two-dimensional Ising model with non-planer lattice or in the external field is unsolved exactly. The low-temperature series expansions for the square lattice Ising model where the nearest-neighbor interaction is the same as the next-nearest-neighbor(cross-interacting) was first studied by Dalton and Wood in1969. They derived the low temperature series expansions of the spontaneous magnetization up to the 17th order .Their result is that 0.122<β<0.134, Lee and Lin extended the series to 24th order and found that 0.124<β<0.131 in1996. Because the generalized square(checkerboard) lattice with next-nearest neighbor interactions in the external magnetic field and this model can not be solved exactly, so we use a powerful method of low-temperature expansions to study the critical behavior of the Ising model .This method applies to any kind of Ising models, we imposed this method to calculate the models without exact solution, because the method of series expansion yields accurate estimate of the critical temperature and exponent. So we consider two dimensional Ising model with checkerboard(non-planer)lattice in the external magnetic field and want to check the valueβof this system is still remain certainly. I think this more general case is conducive to verify the value of universality in existence. The low-temperature series expansions for checkerboard lattice was studied by Shyu and Lin in 1997. He derived the low-temperature series expansions of spontaneous magnetization up to the 13th order and use the Pade□ approximation to estimate the critical exponent, he concluded that 0.126<β<0.151. In this thesis,the main purpose of my research improves the order up to 19th and the universality hypothesis is further convinced .