During my research I've explored the use of the Bean-Rodbell model in the study of magnetic/magnetocaloric materials. The matlab-based computational tool I developed and used throughout my PhD and post-doc is available here to download. The software serves two main purposes: you can use it to fit/compare simulations to experimental magnetization data, or to explore purely simulated data. I'll give a very brief guide on its use, particularly on how to get started.
First of all, you will need the Matlab Compiler Runtime version 8.0, available for various OS and 32/64 bit architectures. It is freely available through Mathworks here.
Download the Mean-Field Simulation Suite 32-bit and 64-bit Windows binaries, or for 64-bit linux. When running the executable the user interface will appear.
Chose some initial parameters for your magnetic system, such as a spin=2 ; saturation magnetization (Msat) = 100 ; T0 (Curie temperature of the rigid lattice) = 300; and the Bean-Rodbell magnetovolume coupling parameter (eta) = 0. Temperature is in kelvin units, field in Oersted and magnetization in units of Msat value. Press the "Simulation output" to initialize the system, calculating the associated system properties, such as spin density (N), the mean-field exchange parameter (lambda_1), Curie constant (C), the maximum magnetic entropy change (dS Max) and the magnetovolume exchange parameter (lambda_3). A list of applied magnetic field (H) and temperature (T) values need to be loaded. You can use the following H.dat and T.dat data file examples.
You can now simulate the M(H,T) behavior of the system by clicking the "Simulate (H,T)_exp" button. Data plots will now appear, in red lines (simulations). You can load an experimental M.dat data file, which data will appear as black lines.
To simulate a disordered system, a gaussian distribution of T0 is used, and its width is chosen through the "Tc FWHM" value, and smoothness by the number of calculated points of the distribution "points". To simulate a disordered system, chose the distribution parameters, press "show dist" and "Simulate Disordered". Replicating the above figure example should be straightforward. Custom distributions can be loaded using the "Load Tc distribution" button.
It is straightfoward to simulate the behavior of a first-order magnetovolume system, by choosing an eta value > 1. All the nasty calculations are done behind the scenes and the software calculates both equilibrium and non-equilibrium solutions. The equilibrium solutions are the ones shown in the main plots. Change the eta value from zero to 1.6, press the "Simulation output" to update the system parameters, Graphs->Clear plots and then the "Simulate (H,T)_exp". Results should be the following:
By clicking File->Hysteresis Analysis (pure 1st-order), you can quicly see the non-equilibrium solutions (blue and red lines), and a quick hysteresis width/area and critical temperature/field analysis:
You can simulate disorder effects in materials with a first-order phase transition much in the same way as for a second-order system. Calculations considerably longer, typically the number of points of the distribution times the calculation time of the pure system.
I hope this software is useful for the magnetism and magnetocaloric community. Questions and comments are welcome, send me an email! I haven't covered here all the possibities of use of this software. Feel free to explore. All generated data is exported to Matlab workspace data files on the working directory.
If you publish data or results obtained through the use of this software suite, please acknowledge its use by citing the following papers, depending on what features you have used:
General use and fitting experimental magnetization/magnetocaloric data of ferromagnets:
"Spontaneous magnetization above Tc in polycrystalline La0.7Ca0.3MnO3 and La0.7Ba0.3MnO3", J. A. Turcaud, A. M. Pereira, K. G. Sandeman, J. S. Amaral, K. Morrison, A. Berenov, A. Daoud-Aladine and L.F. Cohen, Phys. Rev. B 90 024410 (2014) [DOI].
and
"Strain induced enhanced ferromagnetic behavior in inhomogeneous low doped La0.95Sr0.05MnO3+delta", S. Das, J. S. Amaral, et al.; Appl. Phys. Lett. 102 112408 (2013) [DOI].
Irreversibility in first-order phase transitions and the use of the Maxwell relation:
"The effect of magnetic irreversibility on estimating the magnetocaloric effect from magnetization measurements", J. S. Amaral, V. S. Amaral; Appl. Phys. Lett. 94 042506 (2009) [DOI].
Disorder effects in second-order phase transition materials:
"The effect of chemical distribution on the magnetocaloric effect: A case study in second-order phase transition manganites", J. S. Amaral, P. B. Tavares, M. S. Reis, J. P. Araújo, T. M. Mendonça, V. S. Amaral, J. M. Vieira; Journal of Non-Crystalline Solids 354 (47-51), 5301-5303 (2008) [DOI].
Evolution of magnetic entropy change with Tc:
"On the Curie temperature dependency of the Magnetocaloric effect", J. H. Belo, J. S. Amaral, et al.; Appl. Phys. Lett. 100 242407 (2012) [DOI].
Disorder effects in giant magnetocaloric (first-order) phase transition materials:
"Disorder effects in giant magnetocaloric materials", J. S. Amaral and V. S. Amaral; Phys. Stat. Sol. A 211 971 (2014) [DOI].