Quantum hard spheres

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The great usefulness of the hard sphere model for representing particles in classical statistical mechanics is very well known and its study has provided guidance in the understanding of classical fluids and solids. This model assumes that pairwise interactions between particles are singular in that they become an infinite repulsion for distances smaller than the diameter of the spheres, being identically zero otherwise. Perhaps, the most remarkable feature of classical hard spheres is that they show a fluid-solid transition, which was first predicted with computer simulation, and confirmed later with experiments on colloidal particles (see Pusey & van Megen). From the thermodynamic point of view the states of this model only need one parameter to be characterized: the (number) density. This classical state of affairs implies that the quantum thermal de Broglie wavelength of the particles is zero. With the use of reduced units (unit length= hard-sphere diameter) the results arising from this singular interaction potential (hard core) can be transferred between systems differing in the size of their spheres. Among the many interesting features displayed by this model one should mention that there is “contact” between particles at distances equal to the diameter (d^+) (they are like hard billiards), which reflects in the fact that the main peak of the pair radial correlation function is located just at that “contact” point.

Nevertheless, the switching on of the quantum conditions upon this system (i.e. non-zero de Broglie wavelengths) changes dramatically the classical properties. To illustrate this three examples will suffice. First, the characterization of the state points requires an additional parameter, the thermal wavelength, which contains the temperature, the mass of the particles, and Planck’s constant (once again, using reduced units allows one to transfer results between situations at the same values of the density and of the de Broglie wavelength). Secondly, the above classical “contact” is forbidden, as quantum hard spheres repel each other before getting into (classical) contact. And thirdly, the fluid-solid phase transition is driven by energy in the quantum limit of zero temperature, this being different from the classical case which is driven by entropy. Furthermore, quantum hard spheres seem appropriate to understand the low temperature properties of ultra-hard materials and colloids. In this endeavour Feynman’s path-integrals combined with computer simulations provide a very powerful tool to undertake the pertinent calculations. Thus, apart from being an appealing mathematical problem, quantum hard spheres can be very useful from a practical standpoint for the design of new materials.

Attention has been focused upon the equilibrium properties, thermodynamic and structural. It is worth realizing that, while there is only one pair radial correlation function in the classical case, the quantum de-localisation brings about three different pair radial correlation functions in the quantum case, each of them possessing a definite physical meaning. Great emphasis has been placed on the study of the different structural functions, in both the r-correlation and the k-Fourier spaces that can be determined in a quantum many-body system, as they open an alternative way to carry out computations leading to the fixing of the equation of state. This effort has helped to clarify the role of the path-integral centroids in (equilibrium) quantum statistical mechanics, and also the possibilities of utilizing Ornstein-Zernike classical frameworks in dealing with quantum fluids.

Isothermal compressibility[edit]

Isothermal compressibility [1].

Crystallization line[edit]

The structural regularities along the crystallization line has been studied by way of path integral Monte Carlo simulations and the Ornstein-Zernike pair equation [2].

References[edit]

Related reading