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Introduction Results • Understanding the magnetic orders in materials is fundamental in understanding their applications. • Two of the most common configurations of magnetic atoms are ferromagnetic, where every electron spins in the same aligned direction, and antiferromagnetic where each spin is in an alternating, antialigned direction. • One way to observe these molecular moments is with the Ising model. This model depicts phase transitions but also exactly calculates the various physical properties which impact the system. This model is relevant to all magnetic systems near the Curie temperature (McCoy & Wu). • Scientifically researched and reported results were replicated for the duration of this project in order to ensure accuracy and understanding of the models. • Indepth observation of atomic spin within a distorted kagome lattice of KCu3As2O7 (OD) 3 material yielded interesting information. Materials and Methods Several different Matlab computer program codes were ran in order to understand the process of producing results for the Ising and Heisenberg Models. These codes included several variables which were carefully observed and changed in order to achieve knowledge of how these differences affect a magnetic material. In addition to these programs, codes from the Paul Scherrer Institute tutorials on magnetism were utilized in dissected in order to research the individual steps of certain magnetic phenomena. The specific phenomenon in this study is that of atom directions within a distorted magnetic kagome lattice of KCu3As2O7 (OD) 3 molecules. Antiferromagnetic Frustration of Superconducting Materials with Ising and Heisenberg Models John Lodico III and Dr. Paul Larson; Physics Department, Viterbo University Figure 1: Examples of Ising Model grids portraying a magnetic lattice with atoms of many opposing spins (left) and matter with atoms of similar spins(right). Each model begins with the atoms in random orientation resulting in regions aligning differently. Discussion • The higher the degree of coupling (direction in either positive or negative) the more intense the energy. Zero coupling produces zero energy. • Antiferromagnetic coupling values of equal to or less than zero produces no net magnetization. Its ordering, alternating updown spins, offers insight to the lowest energy state whereas in ferromagnetic ordering it is constant upup or downdown spins and offers a negative energy. Positive coupling begins to show positive magnetization followed by a drastic change to negative magnetization which remains negative for increased coupling. • Lattice size in model has direct effect on energy and magnetization. Energy remains quite negative while drawing magnetization to zero as size increases. • Twenty tutorials of varying concepts were replicated. Focus put upon number eighteen. • Atomic spin direction and neighbor interactions gradually alter in a distorted kagome lattice of KCu3As2O7 (OD) 3 material. This simulation was essential in determining the lowest energy due to no simple ground state existing. Acknowledgements Research was funded by the Viterbo University 2015 Summer Research Fellowship Program. Specifically, sponsorship stemmed from the Charles D. Gelatt Endowment Fund. Dr. Paul Larson for his invaluable help and guidance throughout the project. Both the work and assistance that he put in facetoface and on his own time was integral in the understanding and completion of this project. References McCoy, Barry M., and Tai Tsun Wu. The TwoDimensional Ising Model. Courier Corporation, 2014. Print. Paul Scherrer Institute (https://www.psi.ch/spinw/tutorials) R. F. Werner, Remarks on a quantum state extension problem, Lett. Math. Phys. 19 , 319326 (1990). K. Binder, Ising model, Encyclopedia of Mathematics. Springer, 2001. Figure 2: • Magnetization of magnetic material in the Ising Model under different circumstances of 1/β, a constant relating to temperature divided by the strength of atomic coupling. • Magnetization remains relatively high until nearing the Curie temperature where it drops to zero. Figure 3: • Steps one, three, and five (top to bottom) of a five step code output which models the atomic direction of the atoms within the distorted kagome lattice made up of KCu3As2O7 (OD) 3 molecules. • Positions of the magnetic moments are defined using the space group operators and the spin Hamiltonian. • Spinning and change in direction of spinning is not immediate but gradual and different. • Variations in neighboring atoms allows speculation on interactions between one another. • Antiferromagnetic state of this molecular section would be impossible to observe without using a simulation. 300x300 Ising model beta = 1.06, M = 0.00, E = 0.99, B = 0.00, i = 178 2D Ising model with 100 by 100 lattice J=100.00, kT = 2.27, M = 0.127, E = 376.320 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 Absolute Value of Magnetization 1/β J J' Ja Jab Jip a MCu2(2) 4 MCu2(2) 5 MCu2(1) 2 MCu2(2) 3 MCu2(2) 6 MCu2(1) 1 b c J J' Ja Jab Jip a MCu2(2) 4 MCu2(2) 5 MCu2(1) 2 MCu2(2) 3 MCu2(2) 6 MCu2(1) 1 b c J J' Ja Jab Jip a MCu2(2) 4 MCu2(2) 5 MCu2(1) 2 MCu2(2) 3 MCu2(2) 6 MCu2(1) 1 b c
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Full Text Search  Introduction Results • Understanding the magnetic orders in materials is fundamental in understanding their applications. • Two of the most common configurations of magnetic atoms are ferromagnetic, where every electron spins in the same aligned direction, and antiferromagnetic where each spin is in an alternating, antialigned direction. • One way to observe these molecular moments is with the Ising model. This model depicts phase transitions but also exactly calculates the various physical properties which impact the system. This model is relevant to all magnetic systems near the Curie temperature (McCoy & Wu). • Scientifically researched and reported results were replicated for the duration of this project in order to ensure accuracy and understanding of the models. • Indepth observation of atomic spin within a distorted kagome lattice of KCu3As2O7 (OD) 3 material yielded interesting information. Materials and Methods Several different Matlab computer program codes were ran in order to understand the process of producing results for the Ising and Heisenberg Models. These codes included several variables which were carefully observed and changed in order to achieve knowledge of how these differences affect a magnetic material. In addition to these programs, codes from the Paul Scherrer Institute tutorials on magnetism were utilized in dissected in order to research the individual steps of certain magnetic phenomena. The specific phenomenon in this study is that of atom directions within a distorted magnetic kagome lattice of KCu3As2O7 (OD) 3 molecules. Antiferromagnetic Frustration of Superconducting Materials with Ising and Heisenberg Models John Lodico III and Dr. Paul Larson; Physics Department, Viterbo University Figure 1: Examples of Ising Model grids portraying a magnetic lattice with atoms of many opposing spins (left) and matter with atoms of similar spins(right). Each model begins with the atoms in random orientation resulting in regions aligning differently. Discussion • The higher the degree of coupling (direction in either positive or negative) the more intense the energy. Zero coupling produces zero energy. • Antiferromagnetic coupling values of equal to or less than zero produces no net magnetization. Its ordering, alternating updown spins, offers insight to the lowest energy state whereas in ferromagnetic ordering it is constant upup or downdown spins and offers a negative energy. Positive coupling begins to show positive magnetization followed by a drastic change to negative magnetization which remains negative for increased coupling. • Lattice size in model has direct effect on energy and magnetization. Energy remains quite negative while drawing magnetization to zero as size increases. • Twenty tutorials of varying concepts were replicated. Focus put upon number eighteen. • Atomic spin direction and neighbor interactions gradually alter in a distorted kagome lattice of KCu3As2O7 (OD) 3 material. This simulation was essential in determining the lowest energy due to no simple ground state existing. Acknowledgements Research was funded by the Viterbo University 2015 Summer Research Fellowship Program. Specifically, sponsorship stemmed from the Charles D. Gelatt Endowment Fund. Dr. Paul Larson for his invaluable help and guidance throughout the project. Both the work and assistance that he put in facetoface and on his own time was integral in the understanding and completion of this project. References McCoy, Barry M., and Tai Tsun Wu. The TwoDimensional Ising Model. Courier Corporation, 2014. Print. Paul Scherrer Institute (https://www.psi.ch/spinw/tutorials) R. F. Werner, Remarks on a quantum state extension problem, Lett. Math. Phys. 19 , 319326 (1990). K. Binder, Ising model, Encyclopedia of Mathematics. Springer, 2001. Figure 2: • Magnetization of magnetic material in the Ising Model under different circumstances of 1/β, a constant relating to temperature divided by the strength of atomic coupling. • Magnetization remains relatively high until nearing the Curie temperature where it drops to zero. Figure 3: • Steps one, three, and five (top to bottom) of a five step code output which models the atomic direction of the atoms within the distorted kagome lattice made up of KCu3As2O7 (OD) 3 molecules. • Positions of the magnetic moments are defined using the space group operators and the spin Hamiltonian. • Spinning and change in direction of spinning is not immediate but gradual and different. • Variations in neighboring atoms allows speculation on interactions between one another. • Antiferromagnetic state of this molecular section would be impossible to observe without using a simulation. 300x300 Ising model beta = 1.06, M = 0.00, E = 0.99, B = 0.00, i = 178 2D Ising model with 100 by 100 lattice J=100.00, kT = 2.27, M = 0.127, E = 376.320 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 Absolute Value of Magnetization 1/β J J' Ja Jab Jip a MCu2(2) 4 MCu2(2) 5 MCu2(1) 2 MCu2(2) 3 MCu2(2) 6 MCu2(1) 1 b c J J' Ja Jab Jip a MCu2(2) 4 MCu2(2) 5 MCu2(1) 2 MCu2(2) 3 MCu2(2) 6 MCu2(1) 1 b c J J' Ja Jab Jip a MCu2(2) 4 MCu2(2) 5 MCu2(1) 2 MCu2(2) 3 MCu2(2) 6 MCu2(1) 1 b c 