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Journal of Physics C Solid State Physics

The critical properties of the magnetised ideal Bose gas

H.O. Frota, M S Silva, S. Goulart Rosa

Published Apr 10, 1984• 4 citations
Publisher

Abstract

The critical behaviour of a d-dimensional ideal gas of uncharged bosons with spin S in an uniform magnetic field is examined. The critical exponents associated with the spontaneous magnetisation, the zero-field magnetic susceptibility, the equation of state, and the density-density correlation function are calculated as well as their relationship to the critical exponents obtained by Gunton and Buckingham (1968) for the spinless Bose gas.

DOI
10.1088/0022-3719/17/10/008

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The critical behaviour of a d-dimensional ideal gas of uncharged bosons with spin S in an uniform magnetic field is examined. The critical exponents associated with the spontaneous magnetisation, the zero-field magnetic susceptibility, the equation of state, and the density-density correlation function are calculated as well as their relationship to the critical exponents obtained by Gunton and Buckingham (1968) for the spinless Bose gas.
In 1968 Gunton and Buckingham pointed out that there is a close relationship between the critical behaviour of the spherical model and that of the ideal Bose gas. In this paper we concentrate on this similarity between the two systems. We show that in the spherical model there is in fact Bose-Einstein condensation into the spin modes with low energy. As in the Bose gas we distinguish between generalised Bose-Einstein condensation and Bose-Einstein condensation. We find that the spherical model in certain cases has two critical temperatures: one temperature T c corresponding to the onset of generalised Bose-Einstein condensation (and of spontaneous magnetisation) and a lower temperature T m at which generalised condensation becomes condensation into the spin mode with the lowest energy
Thermal and magnetic properties of the system of ideal magnetic bosons exposed to a magnetic field are investigated. The Boson particle is assumed to have a spin of magnitude one and a magnetic moment m0. The results are summarized as follows: The grand thermodynamical potential of the system has a certain singularity on the field versus temperature (T-H) plane, in which two sorts of singular lines SL1 and SL3 as shown in Fig. 1 appear. The Bose-Einstein (BE) condensation occurs in a lower temperature and higher field region bounded by the line SL3 or ACB on the T-H plane. A spontaneous magnetization appears below a critical temperature, at which the zero field susceptibility diverges. This nature is called the “BE Ferromagnetism” in this paper. The system exhibits a phase transition of first order on the line SL1 and that of third order on SL3. The transition reduces to the second order one at the critical temperature T0. The critical exponents at T0 are found to be α = α′ = 0, β = γ = 1 and δ = 2, which satisfy the Griffiths inequalities. In conclusion, the system of magnetic bosons which can really exist is also discussed.
Ideal Bose-Einstein condensation (BEC) remains a paradigmatic example of a continuous phase transition and a cornerstone for understanding quantum degenerate bosonic matter. We demonstrate that critical behavior of the ideal Bose gas near the BEC phase transition falls into three distinct classes, determined exclusively by the low-energy scaling of the density of states. Depending on its scaling exponent, which is controlled by dimensionality and confinement, the transition displays either the usual algebraic divergences of thermodynamic susceptibilities, divergent behavior with marginal logarithmic corrections, or a more subtle form of criticality, where only the correlation length diverges. Our work provides a unified framework for criticality in noninteracting bosonic systems. This classification applies broadly to atomic, photonic, polaritonic, and magnonic condensates, where dimensionality, confinement, and spectral engineering can strongly reshape the density of states.
Abstract Some properties of an ideal gas of massive bosons in a mean field potential and, confined between two infinite parallel slabs in a d-dimensional configuration space are investigated systematically. Here, one particle density of states approach is employed to study the critical temperature, shift of density, Casimir effects and critical exponents, starting from the evaluation of the grand canonical free energy in d-dimension. We have found that, the shift of density, Casimir force and the critical temperature depend on the space dimensionality. But the Casimir force decays as an inverse power law of the distance between two slabs in the condensate and, decays exponentially in the non-condensed state situated very close to the point of phase transition. Specifically, this study enabled us to predict the shift of the density of boson in the excited states due to mean field potential, and also the dimensional dependence of the critical exponent. The form of the critical exponent is found to be 1 d − 2 for the imperfect Bose gas. This leads to a value of critical exponent 1 for d = 3. Most interestingly, in the limit when the mean field potential goes to zero the resulting expressions for the above properties coincide with those of the ideal Bose gas.
The thermodynamic properties of a Bose gas of particles with nonzero spin and magnetic moment in states with and without a Bose-Einstein condensate are studied as a function of the external magnetic field. It is shown that there exists a temperature interval where a transition into a state with a Bose-Einstein condensate occurs in a Bose gas with constant temperature and particle number density when the magnetic field is increasing and reaches a critical magnitude. The temperature and magnetic field derivatives of the specific heat at constant volume and the magnetic susceptibility undergo a jump at such a transition.