1. Chromium Clusters in an External Magnetic Field: Classical Discrete Models Kalin Arsov Third Year Undergraduate Student University of Sofia, Faculty of Physics Adviser: Prof. Dr. Ana Proykova University of Sofia, Department of Atomic Physics

2. Motivation Frustrated systems and their degenerated ground states Frustrated systems and their degenerated ground states A frustrated system has one or more rules for a given quantity suppressed. An example is a non- collinear orientation of half-integer spins coupling: suppression of the Dirac rule A frustrated system has one or more rules for a given quantity suppressed. An example is a non- collinear orientation of half-integer spins coupling: suppression of the Dirac rule Clusters of a specific number of AF (anti- ferromagnetic) interacting atoms – super- paramagnetic states Clusters of a specific number of AF (anti- ferromagnetic) interacting atoms – super- paramagnetic states

3. Previous Studies in the Field Axenovich and Luban, Phys. Rev. B, volume 63, 2001 - ground state properties of - the magnetic moment of this giant icosahedral molecule is linear with B until a critical saturating field B satur. Axenovich and Luban, Phys. Rev. B, volume 63, 2001 - ground state properties of - the magnetic moment of this giant icosahedral molecule is linear with B until a critical saturating field B satur. Proykova and Stauffer, CEJP, v.3, No.2, 2005 - the magnetic moments of the atoms in the ground states of some Cr clusters are planar (no magnetic field; finite temperatures) Proykova and Stauffer, CEJP, v.3, No.2, 2005 - the magnetic moments of the atoms in the ground states of some Cr clusters are planar (no magnetic field; finite temperatures)

4. Studied Configurations of Cr clusters 5-atom Cr configuration 6-atom Cr configuration (a) 6-atom Cr configuration (b)9-atom Cr configuration

5. Studied Configurations of Cr clusters 11-atom Cr configuration 13-atom icosahedral configuration

6. Heisenberg Models: nn & nnn Heisenberg Hamiltonian Heisenberg Hamiltonian B – external field B – external field J1, J2 – coupling constants J1, J2 – coupling constants Important – T=0 K Important – T=0 K

7. Goals Determine the magnetic field value that restores the collinearity of frustrated clusters Determine the magnetic field value that restores the collinearity of frustrated clusters Investigate the dependence of the total magnetization as a function of the field B Investigate the dependence of the total magnetization as a function of the field B Investigate the influence of the next-nearest neighbour interactions J2 in the presence of magnetic field B Investigate the influence of the next-nearest neighbour interactions J2 in the presence of magnetic field B

8. Results: magnetization curves – size dependence Small B – m~ B Small B – m~ B B 1 < B < B 2 B 1 < B < B 2 Magnetization plateaus Magnetization plateaus Vanishing magnetic susceptibility Discontinuity of Large B– saturation Large B– saturation Size effect Size effect

9. Results: Orientation of the spins with the increase of the field

10. Results: Influence of J2/J1 (Cr11) Plateaus Plateaus Appearance of a spin gap if J2<<J1 Appearance of a spin gap if J2<<J1

11. Conclusions Classical spin models exhibit magnetization plateaus, considered a purely quantum- mechanical phenomenon, in finite-size systems Classical spin models exhibit magnetization plateaus, considered a purely quantum- mechanical phenomenon, in finite-size systems The plateaus correspond to different spin orders The plateaus correspond to different spin orders

12. Acknowledgements The members of the “Monte Carlo” group The members of the “Monte Carlo” group Prof. Deitrich Stauffer – for the code of the zero field Ising model Prof. Deitrich Stauffer – for the code of the zero field Ising model