That certain iron and nickel alloys exhibit an anomalously low thermal expansion of a wide temperature range has been observed since late 1800s, and this effect is known as the Invar effect. Since then, many theories have been proposed to explain the phenomenon. While it is generally agreed that the effect is related to magnetism, a full explanation of the effect has yet to be found. One recent theory connected the effect to spin-flips in the iron atoms' magnetic moment and that the probability for a spin-flip to occur depends on the atom's local chemical environment. The aim of this thesis is to perform a theoreticalinvestigation into the magneticenergy landscapes for atomic magnetic moments in different local chemical environments in disordered Fe64Ni36 alloys, and the change in pressure upon populating different parts of the magnetic energy landscape. Constrained calculations are performed to obtain the energy landscapes for both iron and nickel atoms in ferromagnetic Fe64Ni64. The calculated nickel atoms all show one global minimum between 0.64 to 0.72μB. The calculated iron atoms all exhibit two local minima: one where the magnetic moment's direction is the same as the ferromagnetic background's direction and has a size between 2 to 3μB, one where the magnetic moment is flipped and has a reversed direction in regards to the ferromagnetic background with a size between -2.5 to -1.9μB. A weak trend is seen for the energy difference between the two local minima: for iron-atoms with iron-rich local environments the energy difference is smaller than for iron-atoms with nickel-rich local environments. The energy landscapes for a moment rotated with respect to the background show that it is energetically favored to rotate the moment from the spin-up local minimum to the spin-flipped local minimum, rather than shrink in size and then increase in size in the opposite direction. This indicates that the negative local minimum might not be a local minimum, but further calculations are needed to determine if the spin-flipped state is a local minimum or just a saddle point in the complete size-and angle magnetic energy landscape. It is observed that the pressure varies little for different magnetic moment sizes for a nickel atom, but shows a larger variation for different magnetic moment sizes for an iron atom. The pressure difference between the magnetic local minima is about 6-9 kbar, and from thermodynamical simulations a small, nonlinear, decline in pressure with increased temperature is observed.

In the development of materials, the understanding of their properties is crucial. For magnetic materials, magnetism is an apparent property that needs to be accounted for. There are multiple factors explaining the phenomenon of magnetism, one being the effect of vibrations of the atoms on longitudinal spin fluctuations. This effect can be investigated by simulations, using density functional theory, and calculating energy landscapes. Through such simulations, the energy landscapes have been found to depend on the magnetic background and the positions of the atoms. However, when simulating a supercell of many atoms, to calculate energy landscapes for all atoms consumes many hours on the supercomputer. In this thesis, the possibility of using machine learning models to accelerate the approximation of energy landscapes is investigated. The material under investigation is body-centered cubic iron in the paramagnetic state at 1043 K. Machine learning enables statistical predictions to be made on new data based on patterns found in a previous set of data. Kernel ridge regression is used as the machine learning method. An important issue when training a machine learning model is the representation of the data in the so called descriptor (feature vector representation) or, more specific to this case, how the environment of an atom in a supercell is accounted for and represented properly. Four different descriptors are developed and compared to investigate which one yields the best result and why. Apart from comparing the descriptors, the results when using machine learning models are compared to when using other methods to approximate the energy landscapes. The machine learning models are also tested in a combined atomistic spin dynamics and ab initio molecular dynamics simulation (ASD-AIMD) where they were used to approximate energy landscapes and, from that, magnetic moment magnitudes at 1043 K. The results of these simulations are compared to the results from two other cases: one where the magnetic moment magnitudes are set to a constant value and one where they are set to their magnitudes at 0 K. From these investigations it is found that using machine learning methods to approximate the energy landscapes does, to a large degree, decrease the errors compared to the other approximation methods investigated. Some weaknesses of the respective descriptors were detected and if, in future work, these are accounted for, the errors have the potential of being lowered further.