In 1897 Guillaume1 discovered that face-centred cubic alloys of iron and nickel with a nickel concentration of around 35 atomic per cent exhibit anomalously low (almost zero) thermal expansion over a wide temperature range. This effect, known as the Invar effect, has since been found in various ordered and random alloys and even in amorphous materials2. Other physical properties of Invar systems, such as atomic volume, elastic modulus, heat capacity, magnetization and Curie (or Neel) temperature, also show anomalous behaviour. Invar alloys are used in instrumentation, for example as hair springs in watches. It has long been realized that the effect is related to magnetism2,3; but a full understanding is still lacking. Here we present ab initio calculations of the volume dependences of magnetic and thermodynamic properties for the most typical Invar system, a random face-centred cubic iron–nickel alloy, in which we allow for non-collinear spin alignments—that is, spins that may be canted with respect to the average magnetization direction. We find that the magnetic structure is characterized, even at zero temperature, by a continuous transition from the ferromagnetic state at high volumes to a disordered non-collinear configuration at low volumes. There is an additional, comparable contribution to the net magnetization from the changes in the amplitudes of the local magnetic moments. The non-collinearity gives rise to an anomalous volume dependence of the binding energy, and explains other peculiarities of Invar systems.

Origin of the Invar effect in iron–nickel alloys

A general expression for the directionally anisotropic Poisson's ratio (PR) of cubic materials under external pressure is discussed. It is expressed in terms of the elastic moduli ratios, X = G/K, Y= G/W, where K is the bulk modulus and G, Ware shear moduli, and where the mechanical stability criteria are taken into account. The global maximum and global minimum PR surfaces are shown. In the X, Y-plane, regions of different auxetic behavior are identified, and a straightforward way of classifying any cubic material as auxetic, nonauxetic, and completely auxetic is given. The domains of the different extreme directions, for which the PR shows either a global maximum or a global minimum for given X, Y, are identified and discussed. There are three extreme directions: [100], [110], and a novel, noncrystallographic one denoted as the V3-direction. The main features of the V3 extreme direction and corresponding XY-regions are described. It is found that the most extreme PR values can be achieved exclusively in the very limited domain of the X, Y-plane, corresponding to the V3-direction, and for X, Y → 0, the absolute value of the PR becomes unbounded. Analysis of literature data on real materials demonstrates that the great majority of cubic materials are nonauxetics or auxetics (with the [110] extreme direction). The materials possessing very large (i.e., PR > 2) and very small (i.e., PR < -1) PR values are identified.

Auxeticity of cubic materials

Introduction Many materials which we encounter in everyday life expand on heating. This fact of life is easily highlighted with a simple experiment that we are all familiar with and involves a metallic sphere passing through a metal ring. At room temperature, the sphere is of such a size that it can easily pass through the ring, but after the sphere is heated, it will have expanded with the result that it can no longer pass through the ring.

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Negative thermal expansion

The global maximum and global minimum Poisson's ratio (PR) surfaces, regions of different auxetic behaviour and the domains of the different extreme directions of cubic materials are shown and discussed in terms of the elastic moduli ratios, , where K is the bulk modulus and are shear moduli. A straightforward way is given to classify any cubic material as being auxetic, nonauxetic and partially auxetic, as well as calculating its extreme PR values. A decomposition of the XY plane with the elastic constant ratio is introduced to facilitate the interpretation of cubic materials under hydrostatic pressure, , and isotropic tension, , which are two qualitatively different situations. Using this representation it is demonstrated that cubic materials with can be auxetic only under ‘negative’ pressure (). It is shown that the influence of pressure on the auxetic behaviour is different for the positive and negative cases. In particular, a cubic material with may be auxetic at negative as well as positive pressure. The work demonstrates the crucial role of P in obtaining desired auxetic behaviour. Microscopic mechanisms which tune the cubic system to targeted regions in the XY plane are investigated with a fcc crystal of particles interacting via the pairwise spherically symmetric potential, . It is found that all fcc static models in which the range of interaction is only between nearest neighbour particles are placed on a universal curve in the XY plane. It is shown that auxetic behaviours can be achieved with simple static analytic forms. The influence of the next neighbour interactions on the universal curve is also considered. In order to assess the role of thermal fluctuations, molecular dynamic simulations for two different (the Lennard–Jones (LJ) and the tethered particle potentials) were performed. At low temperatures the studied systems are well represented by the universal curve and on increasing temperature a systematic departure from it is observed.

Auxeticity of cubic materials under pressure

A method of fabricating an acoustic resonator includes forming a ferromagnetic compensator, such as one comprised of a nickel-iron alloy, which at least partially offsets temperature-induced effects introduced by an electrode-piezoelectric stack. The compensator has a positive temperature coefficient of frequency, while the stack has a negative temperature coefficient of frequency. By properly selecting the thickness of the compensator, temperature-induced effects on resonance may be neutralized. Alternatively, the thickness can be selected to provide a target positive or negative composite temperature coefficient of frequency. In order to prevent undue electromagnetic losses in the ferromagnetic compensator, a metallic flashing layer may be added to at least partially enclose the compensator.

Method of making an acoustic wave resonator

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Acoustic-mode vibrational anharmonicity related to the anomalous thermal expansion of Invar iron alloys.

Pressure dependence measurements of ultrasound velocities in the monocrystalline Invar alloy Fe72Pt28 have revealed normal behaviour in the paramagnetic phase but marked longitudinal mode softening in the ferromagnetic phase: below Tc both ( delta C11/ delta P)P=0 and ( delta BS/ delta P)P=0 are negative due to the magnetoelastic interaction. This is the first experimental evidence for negative longitudinal acoustic mode Gruneisen parameters in the ferromagnetic phase-as recently suggested from itinerant electron magnetism theory-and accounts for the negative thermal expansion.

H. Rahdi

Papers

Acoustic-mode vibrational anharmonicity related to the anomalous thermal expansion of Invar iron alloys.

Longitudinal acoustic mode softening and Invar behaviour in Fe72Pt28