Condensed-Matter Physics (1986) / Chapter Skim
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3 Critical Phenomena and Phase Transitions
3 Critical Phenomena and Phase Transitions
Pages 75-94
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From page 75... ...
A continuous-phase transition, in turn, may be defined as a point at which a substance changes from one state to another without a discontinuity or jump in its density, its internal energy, its magnetization, or similar properties. The critical point or continuous-phase transition may be contrasted with the more familiar case of a first-order phase transition, where the above-mentioned properties do jump discontinuously as the temperature or pressure 75
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From page 76... ...
The challenge has been difficult for experimentalists, because in order to make measurements sufficiently close to a critical point, to test existing theoretical calculations, or to discover directly the laws of critical behavior where no theory exists, it is necessary to have extremely precise control over the sample temperature, and frequently over the pressure and purity as well. The study of critical phenomena has been rewarding in spite of its difficulties, and the understanding gained has proved useful to the understanding of other types of systems-including quantum field theories in elementary-particle physics, analyses of phenomena in long polymer chains, and the description of percolation in macroscopically inhomogeneous systems in which fluctuations play an important and subtle role but where precise direct experiments may be even more difficult than in the case of critical phenomena.
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From page 77... ...
In most magnetic systems (including iron) the magnetization M(D decreases continuously to zero as the temperature approaches TC from below; then we say that there is a continuous phase transition, or critical point, at Tc In some cases, however, the magnetization of a substance approaches a finite, nonzero value, as T approaches TC from below, and the magnetization jumps discontinuously to zero, as the temperature passes through Tc In these cases there is a first-order transition at Tc.
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From page 78... ...
Other evidence for this viewpoint, and hints at the shape that the new theory must take, were provided by various types of numerical calculations (one might call them computer experiments) , which included both computer simulations of thermal fluctuations in simple magnetic models and also numerical extrapolations of the properties of these magnetic systems from temperatures far above the critical temperature, where accurate calculations could be done.
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From page 79... ...
One important factor that affects the critical behavior is the spatial dimensionality of the system- e.g., 3-D systems have different critical behavior than 2-D systems but there are other factors that are relevant, including the symmetry differences between the states at the phase transition, the presence or absence of certain long-range interactions, and other factors that will be discussed below. A proper understanding of the factors that determine the universality class of a system had to await the developments of the 1970s, however, and in fact, a classification in the more difficult cases remains one of the tasks for the 1980s.
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From page 80... ...
Perhaps the most fundamental measurement to make in the vicinity of a critical point is to determine the way in which the magnitude of the order parameter approaches zero, as the critical point is approached from the low-temperature side. According to the classical theories of phase transitions, such as the van der Waals or mean-field theories, the order parameter should approach zero as the square root of the temperature difference from Tc.
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From page 81... ...
The temperature variation of the order parameter on the coexistence curve is certainly not the only quantity that can be studied with experiments in critical phenomena. Another important quantity is the order-parameter susceptibility, defined as the derivative (i.e., the rate of change)
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From page 82... ...
The specific heat is found to become infinite at the critical point in some systems; for some other universality classes one finds that the specific heat is finite but has a sharp cusplike maximum at the critical point. In either case, one may define an exponent ax that characterizes the anomalous behavior of the specific heat at the critical point.
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From page 83... ...
The majority of theoretical studies and of experiments on critical phenomena are concerned with these static measurements, and the usual division of systems into different universality classes is based on these static phenomena. There are other properties of systems, known as dynamic properties, which require a more detailed theoretical
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From page 84... ...
Among the experiments used to study dynamic properties are measurements of sound-wave attenuation and dispersion, widths of nuclear or electron magnetic resonance lines, and inelastic-scattering experiments, in which the energy change of the scattered particle is determined along with the scattering angle. Typically, one finds that the relaxation rate of the order parameter becomes anomalously slow at a critical point.
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From page 85... ...
The Ising model, the XY model, and the Heisenberg model may be said to have order parameters that are, respectively, a 1-D vector, a 2-D vector, and a 3-D vector. We have already seen that spatial dimensionality is crucial in determining the universality class of a system the Ising model on a 2-D lattice has different critical exponents from the 3-D Ising model, for example.
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From page 86... ...
In more complicated systems, with multicomponent order parameters, there is a variety of possible higher-order symmetry breaking terms, which may favor some discrete subset of the possible orientations of the order parameter. In some cases these terms lead to a change in critical behavior; in some others they lead to a small fluctuation-induced first-order transition, even though the classical theory predicts a continuous transition.
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From page 87... ...
EXPERIMENTAL REALIZATIONS OF LOW-DIMENSIONAL SYSTEMS Although the world we live in is three dimensional, theoretical studies of 2-D systems have direct applications to systems in nature. For example, a transition between commensurate phases of a layer of atoms adsorbed on a crystalline substrate, or the melting of a commensurate adsorbate phase, will generally fall into the same universality class as some simple 2-D model with a discrete order parameter, such
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From page 88... ...
. Recent experimental developments, including improved substrates, and the availability of synchrotron x-ray sources have made possible new precise measurements of phase transitions in adsorbed gas systems.
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From page 89... ...
Since this power-law behavior is different from the exponential fallow of the correlation function (shortrange correlations) that one finds at high temperatures in the same systems, there must be a definite temperature separating these two behaviors, which is by definition a phase transition temperature.
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From page 90... ...
In layered phases of certain organic molecules (smectic liquid crystals) , there are phase transitions arising from a change in the order within a layer, which may be considered as generalizations of the 2-D melting transition.
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From page 91... ...
One particularly interesting case occurs when the quenched disorder couples linearly to an Ising-like order parameter, as would be the case if there were a local magnetic field of random sign on each site of the Ising ferromagnet. (This situation has been realized experimentally in an Ising-like antiferromagnet, with a uniform magnetic field and randomly missing magnetic atoms.)
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From page 92... ...
PERCOLATION AND THE METAL-INSULATOR TRANSITION IN DISORDERED SYSTEMS There are a number of problems in condensed-matter physics that bear a qualitative resemblance to systems at a continuous phase transition and that may indeed be understood by methods of analysis similar to those used in the theory of critical phenomena but where the source of disorder is entirely quenched randomness and not thermal fluctuations. Among these are various problems concerned with geometry and transport in disordered systems, including metal-insulator transitions in disordered systems where quantum mechanics plays a critical role, as well as the classical problem of percolation in a mixture of macroscopic conducting and insulating particles.
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From page 93... ...
Indeed, the reader will find that phase transitions are featured in virtually every chapter of the report. OUTLOOK Work on critical phenomena and related problems, in the 1980s, should lead to progress in a number of directions, among which we may expect the following: 1.
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From page 94... ...
For example, it seems likely that in the 1980s considerable progress will be made in our understanding of 2-D melting and related phenomena, through experiments on a variety of systems, including liquid crystals, both in bulk and in suspended films of several layers thickness; adsorbed layers; the electron crystal on the surface of liquid helium; and perhaps synthetic systems, such as a film containing colloidal polystyrene spheres. The role of the substrate in the transition, in the case of adsorbed layers, will be investigated.
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