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Basic Theoretical Physics: A Concise Overview P33

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Basic Theoretical Physics: A Concise Overview P33. This concise treatment embraces, in four parts, all the main aspects of theoretical physics (I . Mechanics and Basic Relativity, II. Electrodynamics and Aspects of Optics, III. Non-relativistic Quantum Mechanics, IV. Thermodynamics and Statistical Physics). It summarizes the material that every graduate student, physicist working in industry, or physics teacher should master during his or her degree course. It thus serves both as an excellent revision and preparation tool, and as a convenient reference source, covering the whole of theoretical physics. It may also be successfully employed to deepen its readers’ insight and. | 332 42 Phase Changes van der Waals Theory and Related Topics and different coefficients for T Tc and T Tc. The values given above for 1 2 ß y 1 and S 3 are so-called molecular field exponents. Near the critical point itself they must be modified see below . 42.3 Critical Behavior Ising Model Magnetism and Lattice Gas Molecular field theory and thus the van der Waals expressions are no longer applicable in the neighborhood of the critical point because thermal fluctuations become increasingly important such that they may no longer be neglected. The spontaneous magnetization is then given by 1 3 mH T0 T ß with ß not 1 2 for a three-dimensional system and 3 8 in a two-dimensional space. As a consequence of thermal fluctuations instead of the Arrott equation for T T0 the following critical equation of state holds1 2 H a T T0 YmH b m-H 6 . 42.4 where Y 3 and S 5 for d 3 and Y - and 4 S 15 for d 2 . However independent of the dimensionality the following scaling law applies Y ß ß S i.e. the critical exponents are indeed non-trivial and assume values dependent on d but - independently of d - all of them can be traced back to two values e.g. 3 and y 1 Here the interested reader might consider the exercises file 8 problem 2 of winter 1997 2 2 The scaling behavior rests on the fact that in the neighborhood of Tc there is only a single dominating length scale in the system the so-called thermal correlation length ffT see e.g. Chap. 6 in 38 . 42.3 Critical Behavior Ising Model Magnetism and Lattice Gas 333 The not only qualitative similarity between magnetism and phase changes on liquefaction is particularly apparent when it comes to the Ising model. In this model one considers a lattice with sites l and m on which a discrete degree of freedom sl which can only assume one of the values sl 1 is located. The Hamilton function energy function of the system is written H - Ji msism - hisi Ising model . 42.5 The standard interpretation of this system is the magnetic interpretation .