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Shamim Monjazeb Journal Review CHEN610 The journal article reviewed here is titled “Ising Model on Mixed twodimensional Lattices” by W. Lebrecht, E.E. Vogel, and J.F. Valdes. The paper was published in volume 320 of Physica B in July 2002 pages 343 – 347. The purpose of the paper is to find the ground level of Ising model for Kagome lattices (coordination number of 4) and five-point star lattices (coordination number of 5). The basic principals applied are to randomly assign ferromagnetic (-J) and anti-ferromagnetic (+J) nearest neighbor interactions to each atom in each lattice and vary the concentration of this mixed ferromagnetic and anti-ferromagnetic interactions between zero and one while having a periodic boundary condition for N number of spins. With this setup, two methods are utilized to find frustration segment, energy per interaction, fraction of non-frustrated bonds in the entire ground manifold over total number of bonds, and frustration of plaquettes. The methods used are: 1) numerical method related to multiple replicas and 2) analytical method based on probabilistic analysis of flat or curved plaquettes. The results from both these methods and results from honeycomb (HL), square (SL), and triangular (TL) lattices (previously calculated) are used to form conclusions about the effects of inhomogeneous geometry and coordination numbers. Ising model is defined on a collection of discrete variables called spins. These spins can take on the value 1 or -1. This model is especially helpful in dealing with short range interactions which consequently gives rise to frustration in systems. In this study ferromagnetic (F) interactions (-1 or –J) and anti-ferromagnetic (AF) interactions (+1 or +J) are assigned randomly with x concentration for F and (1 – x) for AF. Two lattices are studied here that have inhomogeneous geometries. The first is the Kagome Lattice (KL). This lattice consists of a hexagonal honeycomb center with six triangular plaquettes around it which has a coordination number of 4. The second lattice is the five-point star lattice (FPSL) which consists of a square plaquette surrounded by four triangular plaquettes with coordination number of 5. For both these lattices an array of L by M cells is created which constitutes a total of N = (C – 1) L * M spins. One method used to study these lattices is a numerical method. It uses the Ising Hamiltonian equation and an algorithm (algorithm not discussed in the paper) to find all lowest energy states. With these lowest energy states known some physical properties can be found. Such physical properties include frustration segment, energy per interaction, and fraction of non-frustrated bonds in the entire ground manifold over total number of bonds. The average energy per interaction is found to be slightly higher for FPSL than KL. This is believed to be a result of higher coordination number of FPSL. These values are -0.66 for FPSL and -0.63 for KL which fall within the range for TL and HL. The fraction of non- frustrated bonds in the manifold is found to be 0.5 for KL and 0.7 for FPSL. The large difference between the two values is due to the difference in coordination numbers. And finally the average frustration segment is found to be 1.34 for KL and 1.22 for FPSL. This difference is a result of the position and orientation of the triangles in each lattice. The second method used in this study is a theoretical approach. Each lattice has a specific number of interactions (KL = 10 and FPSL = 14). The process starts with zero AF bond and weights the probability as a function of x. Then it goes to one AF bond and does the same thing. It goes through all the different configurations of two, three, four, etc. AF bonds and weights those probabilities. The sum of all configurations that results in a frustrated plaquette gives a probabilistic frustration as a function of x. The general equation from this approach is The actual equations for the lattices studies here are When plotted, the theoretical approach for each lattice matches almost exactly the numerical simulation results previously done for each lattice. As further verification of the validity of the equations, it is noted that at x = 0.5 the probability of a frustrated plaquette is exactly 0.5 as is expected and at x = 0 the probability is 2/3 which corresponds to only frustrated triangular plaquettes. Though the two lattices are very different geometrically, they follow about the same tendencies. This is not what is expected since each of the homogenous plaquettes (TL, HL, and SL) behave less similar. Based on the analysis and calculations listed above several conclusions about these types of lattices can be drawn. The two inhomogeneous lattices with their incredibly different overall geometries have very similar distribution of curved plaquettes which greatly differs from homogenous lattices (honeycomb, square, and triangular lattices). They both also have very similar energy per interaction. However, these exceptional similarities are not cause to create a specific class for these lattices. Coordination number plays a big roll in the unfrustrated portion of each lattice. Odd coordination numbers create a larger number of unfrustrated interactions than even coordination numbers. Also, a smaller frustration segment is found in FPSL in comparison to KL which is a direct result of the locations and orientation of the triangles in each lattice. Therefore one can conclude that in spite their uniqueness in some ways, these inhomogeneous lattices still have properties correlated to homogenous lattices.