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A simple theory of the Invar effect in iron-nickel alloys

Certain alloys of iron and nickel (so-called 'Invar' alloys) exhibit almost no thermal expansion over a wide range of temperature. It is clear that this is the result of an anomalous contraction upon heating which counteracts the normal thermal expansion arising from the anharmonicity of lattice vibrations. This anomalous contraction seems to be related to the alloys' magnetic properties, since the effect vanishes at a temperature close to the Curie temperature. However, despite many years of intensive research, a widely accepted microscopic theory of the Invar effect in face-centered-cubic Fe-Ni alloys is still lacking. Here we present a simple theory of the Invar effect in these alloys based on Ising magnetism, ab initio total energy calculations, and the Debye-Gruneisen model. We show that this theory accurately reproduces several well known properties of these materials, including Guillaume's famous plot1 of the thermal expansion coefficient as a function of the concentration of nickel. Within the same framework, we are able to account in a straightforward way for experimentally observed deviations from Vegard's law. Our approach supports the idea that the lattice constant is governed by a few parameters, including the fraction of iron-iron nearest-neighbour pairs.

have seen many more precise measurements on the Fe-Ni series 2, 3 , as well as the observation of a similar effect in other systems such as Fe-Pt 3 and Ni-Mn 3 . Some materials have also been shown to exhibit the opposite: an anomalously large thermal expansion, referred to as the 'anti-Invar' effect 6 .
Naturally there has been a great deal of theoretical work on the origin of the Invar effect in face-centred-cubic (fcc) Fe-Ni alloys. The relevant measured quantity is the linear thermal expansion coefficient, (dL/dT)/L, which we denote by α. This may be decomposed as α = α latt + α anom , where α latt is the contribution from the lattice vibrations.
There appears to be an almost universal consensus that the anomalous term α anom is related to the magnetic properties of the systems in question.
On the issue of which magnetic models are appropriate, however, opinions differ.
One strand in the literature favours a so-called 2γ-state model, where the iron atoms in the alloy switch between two states with different magnetic moments (and thus different volumes) as the temperature is raised 3 . This approach, however, appears to be incompatible with the results of Mössbauer 7 and neutron 8 experiments. A second approach is based on weak itinerant ferromagnetism 3 ; this, in turn, seems incompatible with the experimental observation 9 that substantial local magnetic moments persist in the paramagnetic phase of Fe 65 Ni 35 .
Models based on local moments 3, 10-14 have also been suggested, a point of view that is supported by certain Mössbauer 15 experiments. Within this class of models, there has been some emphasis recently 12 on the importance of non-collinearity of the magnetic moments on the iron sites, though experiments 16 undertaken to detect such noncollinearity do not seem to find it. Other authors, by contrast, conclude 11 that an Ising ('up or down') model with properly chosen exchange constants is adequate to the problem -a view supported by our results, as we shall demonstrate below. 3 An important step towards a quantitative prediction of the thermal expansion of Invar alloys has recently been made using the partially disordered local moment (PDLM) model 17 . This model can be used to calculate ab initio total energies of an A 1-x B x alloy in collinear magnetic states, given the fraction of A atoms whose spins are 'up', x AU , and the fraction of B atoms whose spins are 'up', x BU . A PDLM state for x AU =x BU =1 is fully magnetised, whereas a PDLM state for x AU =x BU =0.5 is completely disordered. By applying this formalism, the Invar effect in disordered Fe-Pt alloys has been found to originate from thermal magnetic disorder.
In this work, we present a simple theory of the relationship between magnetism and thermal expansion in fcc Fe-Ni alloys. X-ray diffraction measurements 18,19 have shown that fcc Fe-Ni alloys are chemically disordered. We treat them as random alloys, calculating the Wigner-Seitz radius by minimising the free energy within the Debye-Grüneisen model at the temperature of interest, T. For this purpose, we use a partially disordered local moment state exhibiting similar magnetic properties to those of the random alloy at T.
The following general procedure is used to determine the average lattice constant at temperature T. (See the Methods section for details of the calculations.) 1. We calculate the following characteristics of the magnetic structure at T: the magnetization M(T); the fraction of nearest-neighbour iron-iron pairs whose spins are anti-parallel, X FFAP (T); the analogous quantity for iron-nickel pairs, X FNAP (T); and the same for nickel-nickel pairs, X NNAP (T). As discussed above, an Ising model seems to be sufficient for this step of the calculation; see the Methods section for further details. 4 2. We use in our ab initio code a PDLM state that reproduces the values of X FFAP , X FNAP , X NNAP , and M found in part 1. In our Ising approach, this state is labelled by the fraction of nickel spins that are up (x NU ) and the fraction of iron spins that are up (x FU ). We thus obtain the total energy for that PDLM state as a function of the lattice constant.
3. We fit a Morse function to the results of part 2, using a least-squares procedure.
The parameters of the fit give us the equilibrium lattice constant, the bulk modulus, and the Grüneisen constant 12 for the values of x FU and x NU in question.
4. Using the Debye-Grüneisen model, we calculate the free energy as a function of the lattice constant. This is done by adding the total energy obtained in part 2 to the free energy of the vibrating lattice estimated from the three quantities obtained in part 3 4 . Minimising this free energy allows us to calculate the average lattice constant at temperature T.
A useful way to think of the above procedure is shown in Fig. 1. Imagine that the magnetic configuration were fixed. Then the material would show only "normal" thermal expansion. Let us call the corresponding thermal expansion curve a(x FU , x NU, T); the curve for x=0.35, x FU =0.90, and x NU =1 is the uppermost dashed curve in Fig. 1. In reality, however, raising the temperature causes the material to demagnetise, and the values of X FFAP, X FNAP and X NNAP change accordingly. One may say that the system 'hops' from the curve a(x FU , x NU , T) to the curve a(x' FU , x' NU, T'), resulting in a lattice spacing given by the curve a(x FU (T), x NU (T), T). This is shown as the solid line in Fig. 1. In the case depicted, each 'hop' is to a curve lower than the last, cancelling the upward trend of each individual curve: this is the essence of the Invar effect. 5 Having thus outlined the general procedure to be followed, we now present the details, with some commentary. In Fig. 2a There is one question, however, that our ab initio calculations so far do not answer: namely, why is it that an increase in the number of nearest-neighbour iron-iron ↑↓ pairs, X FFAP, causes a reduction in the lattice spacing? To address this point, we show in Fig. 4a the average separations of nearest-neighbour iron-iron pairs, for both parallel and antiparallel orientations of their local moments: d FFP and d FFAP respectively. It is clear that d FFAP is significantly smaller than d FFP , i.e. that the iron-iron bond contracts when the spin configuration is changed from parallel to anti-parallel. Thus, the bigger X FFAP is, the smaller is the average distance between two nearest-neighbour iron atoms, d FF . 6 Furthermore, as shown in Fig. 4b, there is a robust positive correlation between the average lattice constant, a, and d FF . Thus, a tends to decrease with increasing X FFAP .
Finally, we make two comments. First, the physical origin of the discrepancy between d FFAP and d FFP in random face-centered-cubic Fe 65 Ni 35 alloy has already been discussed 20 . It seems that this difference arises from the dependence of the exchange parameter on the distance between iron-iron nearest-neighbour atoms. Second, we can easily explain the deviation from Vegard's law shown, for example, by the lowtemperature lattice constant of fcc Fe 65 Ni 35 alloy 2, 6 . We obtain ab initio that the lattice constant of fully magnetised fcc Fe 1-x Ni x decreases linearly with increasing the Ni atomic concentration, x, between x=0.3 and x=0.8, in obedience to Vegard's law. However, according to our Ising model, the zero-temperature magnetic structure of fcc Fe 65 Ni 35 alloy is not fully magnetised, but rather collinear ferrimagnetic: there are some antiparallel iron-iron nearest neighbour moments even at zero temperature. Consequently, due to the mechanism discussed above, the lattice constant is smaller for the collinear ferrimagnetic alloy than for the fully magnetised alloy. The predicted relative deviation from Vegard's law is -0.37%, which is in fair agreement with the experimental result of -0.11% 6 .

Methods.
To carry out point 1 of the procedure for Fe 1-x Ni x for x=0. 30