The theory of critical phenomena in systems at equilibrium is reviewed at an introductory level with special emphasis on the values of the critical point exponents α, β, γ,..., and their interrelations. The experimental observations are surveyed and the analogies between different physical systems - fluids, magnets, superfluids, binary alloys, etc. - are developed phenomenologically. An exact theoretical basis for the analogies follows from the equivalence between classical and quantal `lattice gases' and the Ising and Heisenberg-Ising magnetic models. General rigorous inequalities for critical exponents at and below Tc are derived. The nature and validity of the `classical' (phenomenological and mean field) theories are discussed, their predictions being contrasted with the exact results for plane Ising models, which are summarized concisely. Pade approximant and ratio techniques applied to appropriate series expansions lead to precise critical-point estimates for the three-dimensional Heisenberg and Ising models (tables of data are presented). With this background a critique is presented of recent theoretical ideas: namely, the `droplet' picture of the critical point and the `homogeneity' and `scaling' hypotheses. These lead to a `law of corresponding states' near a critical point and to relations between the various exponents which suggest that perhaps only two or three exponents might be algebraically independent for any system.

https://iopscience.iop.org/article/10.1088/0034-4885/30/2/306/pdf

The theory of equilibrium critical phenomena

Large magnetic-field-induced strains1 have been observed in Heusler alloys with a body-centred cubic ordered structure and have been explained by the rearrangement of martensite structural variants due to an external magnetic field1,2,3. These materials have attracted considerable attention as potential magnetic actuator materials. Here we report the magnetic-field-induced shape recovery of a compressively deformed NiCoMnIn alloy. Stresses of over 100 MPa are generated in the material on the application of a magnetic field of 70 kOe; such stress levels are approximately 50 times larger than that generated in a previous ferromagnetic shape-memory alloy4. We observed 3 per cent deformation and almost full recovery of the original shape of the alloy. We attribute this deformation behaviour to a reverse transformation from the antiferromagnetic (or paramagnetic) martensitic to the ferromagnetic parent phase at 298 K in the Ni45Co5Mn36.7In13.3 single crystal.

Magnetic-field-induced shape recovery by reverse phase transformation.

“…. For the truth of the conclusions of physical science, observation is the supreme Court of Appeal….” (Sir Arthur Eddington, The Philosophy of Physical Science.) In this paper we shall review the theoretical and experimental results obtained on simple magnetic model systems. We shall consider the Heisenberg, XY and Ising type of interaction (ferro and antiferromagnetic), on magnetic lattices of dimensionality 1, 2 and 3. Particular attention will be paid to the approximation of these model systems in real crystals, viz. how they can be realized or be expected to exist in nature. A large number of magnetic compounds which, according to the available experimental information, meet the requirements set by one or the other of the various models are considered and their properties discussed. Many examples will be given that demonstrate to what extent experiments on simple magnetic systems support theoretical descriptions of magnetic ordering phenomena and contribute to their understanding. It will a...

Experiments on simple magnetic model systems

This paper compares theory and experiment for behavior very near critical points. The primary experimental results are the "critical indices" which describe singularities in various thermodynamic derivatives and correlation functions. These indices are tabulated and compared with theory. The basic theoretical ideas are introduced via the molecular field approach, which brings in the concept of an order parameter and suggests that there are close relations among different phase transition problems. Although this theory is qualitatively correct it is quantitatively wrong, it predicts the wrong values of the critical indices. Another theoretical approach, the "scaling law" concept, which predicts relations among these indices, is described. The experimental evidence for and against the scaling laws is assessed. It is suggested that the scaling laws provide a promising approach to understanding phenomena near the critical point, but that they are by no means proved or disproved by the existing experimental data.

Static Phenomena Near Critical Points: Theory and Experiment

Magnetocaloric effect: From materials research to refrigeration devices

The temperature dependence of the initial susceptibility ${\ensuremath{\chi}}_{0}$ of nickel above the Curie point ${T}_{c}$ and the field dependence of its magnetization at ${T}_{c}$ are deduced from the data of Weiss and Forrer and found to be at variance with the simple molecular-field model. Instead, the experimental ${{\ensuremath{\chi}}_{0}}^{\ensuremath{-}1}$-versus-$T$ curve just above ${T}_{c}$ is shown to follow the simple relation ${{\ensuremath{\chi}}_{0}}^{\ensuremath{-}1}=A{(T\ensuremath{-}{T}_{c}]}^{\ensuremath{\gamma}}$, with $\ensuremath{\gamma}=1.35\ifmmode\pm\else\textpm\fi{}0.02$, in excellent agreement with the $\frac{4}{3}$-power relation recently predicted from the exact series for the Heisenberg model. From the coefficient $A$, it is deduced that ${\ensuremath{\mu}}_{0}$, the average atomic moment, is 0.642 ${\mathrm{\ensuremath{\mu}}}_{\mathrm{B}}$ and that the individual electron moments are in a state corresponding to $S=\frac{1}{2}$. At higher temperatures, the ${{\ensuremath{\chi}}_{0}}^{\ensuremath{-}1}$-versus-$T$ curve deviates from the Heisenberg-model predictions, possibly because of a gradual rise in ${\ensuremath{\mu}}_{0}$ with increasing temperature. Up to the highest field $H$ of measurement, the magnetization at ${T}_{c}$ is shown to vary as ${H}^{\ensuremath{\epsilon}}$ with $\ensuremath{\epsilon}=0.237$, which is consistent with the exponent values for an analogous empirical relationship between the density and pressure of several different gases at their critical points.

Detailed Magnetic Behavior of Nickel Near its Curie Point

Magnetization and electrical resistivity measurements on an iron-rhodium alloy of approximate composition FeRh, having CsCl-type structure, confirm recent x-ray and neutron diffraction evidence for a first-order antiferromagnetic-ferromagnetic transition at about 350°K. From 350°K down to 77°K, an essentially constant, field-independent susceptiblity of about 1×10−4 emu/g is observed. Above 350°K, the alloy behaves like a normal ferromagnet with a Curie point of 675 °K and a saturation magnetization (extrapolated to 0°K) of 130 emu/g. This σ0 value is equivalent to an average atomic moment of 1.85 μb, or to 3.85 μb per iron atom if the rhodium moment is assumed to be zero. Alternatively, if the iron moment is taken to be 2.2 μb, the rhodium moment has to be about 1.5 μb. Qualitative results are given for the effect of high magnetic fields and pressures on the first-order transition temperature of this alloy.

Anomalous Magnetic Moments and Transformations in the Ordered Alloy FeRh

Magnetic properties are reported of powder compounds isomorphous to hausmannite, but containing partial to nearly complete substitution of diamagnetic ${\mathrm{Zn}}^{2+}$ or ${\mathrm{Mg}}^{2+}$ ions in the tetrahedral sites of ${\mathrm{Mn}}_{3}$${\mathrm{O}}_{4}$. X-ray diffraction of these compounds reveals a single phase, whose structure corresponds to a tetragonally distorted spinel. Unidirectional anisotropy is present and is detected by the observation of hysteresis loops displaced along the field axis when the materials are cooled to low temperatures in magnetic fields of several kilo-oersteds. The unidirectional behavior is stable to reverse field pulses of 140 koe. An exchange anisotropy model is proposed involving interactions between ferrimagnetic and nearly antiferromagnetic regions brought about by the random distribution of the diamagnetic ions among the tetrahedral sites and the consequent magnetic inhomogeneity. This model is similar to that proposed for disordered alloy systems. Magnetic viscosity effects are observed which correspond in part to ordinary magnetic viscosity, but which are influenced by the presence of unidirectional anisotropy. The identity of this anisotropy is not impaired by the ordinary viscosity effects. In addition, observations of certain differences in the hysteresis loops measured dynamically and quasi-statically reveal an "extraordinary" viscosity associated with a gradual breakdown of the unidirectional anisotropy.

Exchange Anisotropy in Mixed Manganites with the Hausmannite Structure

The magnetization and electrical resistivity of the ordered alloy MnRh were measured during a complete temperature cycle between 4.2° and 700°K. At all temperatures the magnetization is proportional to field. From 700°K down to about 170°K, the susceptibility rises according to a Curie‐Weiss relation, χ=C(T−θ)−1, where θ≃−260°K and C gives an average effective atomic moment of about 3.3 μB, and the resistivity decreases slowly, yielding a large spin‐disorder term, about 85 μΩ‐cm, when extrapolated linearly to 0°K. This classical paramagnetic behavior is identified with the ordered cubic (CsCl‐type) phase of this alloy. Below 170°K the susceptibility and resistivity decrease very rapidly and then gradually level off at lower temperatures, reaching values of about 5×10−6 emu/g and 6 μΩ‐cm, respectively, at 4.2°K. The subsequent return of these properties to their original values at higher temperatures reveals a large temperature hysteresis in this transition. It is discovered from x‐ray diffraction measurem...

Magnetic Properties and Crystal‐Structure Transformation of the Ordered Alloy (MnRh)

Single‐crystal and polycrystalline specimens of cobalt were subjected at room temperature to hydrostatic pressures up to about 5 kbars, and the effects on their magnetizations were measured in fields up to 14 kOe. For the single crystal magnetized along its easy axis, complete saturation was achieved at about 3 kOe, and above this field the pressure dependence of the magnetization (per g) remained constant at a saturation value, σs−1 (∂σs/∂p), of −2.18×10−4 kbars−1. This value is shown to be identifiable with the pressure dependence of the atomic moment at 0°K and has the same sign and even roughly the same magnitude as the recent low‐temperature pressure results for iron and nickel. For the cobalt polycrystal, the change in magnetization with pressure varied widely (though smoothly) with field before attaining the above saturation value at about 13 kOe. The lack of saturation at lower fields was attributed entirely to magneto‐crystalline anisotropy, and a graphical integration method applied to the data ...

J. S. Kouvel

Papers

Detailed Magnetic Behavior of Nickel Near its Curie Point

Anomalous Magnetic Moments and Transformations in the Ordered Alloy FeRh

Exchange Anisotropy in Mixed Manganites with the Hausmannite Structure

Magnetic Properties and Crystal‐Structure Transformation of the Ordered Alloy (MnRh)

Pressure Dependence of the Magnetization of Cobalt