Crystal structures and mechanisms

Allotropes of iron

Why do metals adopt the crystal structures that they do? This no longer is a curiosity because the metallic state is so well understood that it is possible to select, from a calculation of the cohesive energies of trial structures, that which should be the most stable. Figure 1.1 shows the cohesive energy as a function of the density and crystal structure. Of all the test structures, hexagonal close- packed (hep) iron is found to have the highest cohesion and therefore should represent the most stable form. This contradicts experience, but the calculations do not account for the ferromagnetism of body-centred cubic iron (ferrite), which would make it more stable than the hep form. There are in fact magnetic transitions in each of the allotropes of iron, details of which are reserved for Chapter 2.

Figure 1.1 Plot of cohesive energy £b for OK and OPa pressure versus the normalised volume per atom for a variety of crystal structures of iron. £u is the binding energy per atom in the crystal relative to that of a free atom. Hexagonal-P and Cubic-P are primitive structures; like the diamond cubic form, they do not exist on earth. Adapted with permission from [1]. Copyrighted by the American Physical Society.

Only three allotropes of iron occur in nature, in bulk form; Figure 1.2 shows the phase diagram for pure iron. Each point on a boundary between the phase fields represents an equilibrium state in which two phases can coexist. The triple point where the three boundaries intersect represents an equilibrium between all three co-existing phases. It is seen that in pure iron, the hexagonal close- packed form is stable only at high pressures, consistent with its high density.

There are two further allotropes which can be created in the form of thin films. Face-centred tetragonal iron can be prepared by coherently depositing iron as a thin film on a {100} plane of a substrate such as copper with which the iron has a mismatch. The atoms in the first deposited layer replicate the positions of those on the substrate. A few monolayers can be forced into coherency in the plane of the substrate with a corresponding distortion normal to the substrate. This gives the deposit a face-centred tetragonal structure which in the absence of any mismatch would be face-centred cubic [2, 3]. Eventually, as the film thickens during the deposition process to beyond about ten mono- layers (on copper), the structure relaxes to the low-energy bcc form, a process accompanied by the formation of dislocation defects which accommodate the misfit with the substrate.

Growing iron on a misfitting {111} surface of a face-centred cubic substrate similarly causes a distortion along the surface normal, giving trigonal iron [4].¹ Graphene is a single layer of carbon atoms that has a hexagonal structure with a lattice parameter of 0.245 nm. The crystal is seldom perfect, often containing holes. Compounds used in its manufacture provide a source of iron atoms that can attach themselves to the edges of these holes, building up into monoatomic layers that are suspended in the graphene films. These “free-standing” single-atom-thick two-dimensional arrays of iron form a square lattice with a parameter 0.265±0.005 nm [5]. This particular structure has been suggested to be a consequence of lattice matching with the “armchair” configuration of carbon atoms at the graphene edges where the bonding between the iron and carbon atoms is stronger than when the iron attaches to the “zig-zag” edges of the graphene [5]. The largest stable monolayer of iron on holes in graphene is found to be about 3 nm² in area.

Figure 1.2 Temperature versus pressure equilibrium phase diagram for pure iron, based on assessed thermodynamic data. The triple point temperature and pressure are 490°C and 11 GPa respectively, a. у and e refer to ferrite, austenite and e-iron respectively. 8 is simply the higher temperature designation of a. Diagram courtesy of Shaumik Lenka, calculated using the database 'TCFE8’, and ThermoCalc version 2015b.

Table 1.1 lists some transformation temperatures and thermodynamic data for the natural allotropes of iron. The free energy changes during the solid-state transformations are really quite small, even when compared with the energy associated with the magnetic disordering of ferrite. In a way, this is a reflection of the fact that their crystal structures are not all that different (Figure 1.3). For example, assuming a hard-sphere model, the crystal structure of austenite and e-iron can be generated by stacking close-packed layers of iron atoms in such a way that each successive layer sits in the depressions of the underlying layer. There are in each layer, two types of depressions, so the close- packed layers can be stacked either in the sequence ... ABCABCABC... or... ABABABABA.... The former sequence, which has a stacking period of three, generates the austenite structure, whereas the latter gives the hep structure of e-iron with a stacking period of two. The best comparison of the relative densities of the phases is made at the triple point where the allotropes are in equilibrium at an identical temperature and pressure and where the sum of all the volume changes is zero:

Ferrite is the least dense of all the allotropes under ambient conditions.

The fact that the hep iron is denser than austenite is not expected from the hard sphere approximation, which also predicts an ideal lattice parameter ratio for hep (eja) of 8/3 ~ 1.633. Experimental measurements at high pressures indicate that the ratio is close to ideal, but the results do not

Table 1.1

Transformation temperatures and thermodynamic data for pure iron at ambient pressure (after Hoffman, Tauer, Paskin, Weiss, Chipman and Orr [6-9]). The transformation temperatures are consistent with the International Practical Temperature Scale, which was in 1968 modified by raising the designated melting point of palladium by 2 K. Tq is the Curie temperature for the transition between the ferromagnetic and paramagnetic states of ferrite; the energy and entropy quoted alongside T_c are those required to completely disorder ferromagnetic iron. The approximate Curie and Neel temperatures for the two states of austenite are also listed.

	т/°с	т/К	Enthalpy change ДЯ/Jmol-1	Entropy change AS/JK-'mol-'
	911.5	1184.65	900	0.7548
	1394.0	1667.15	837	0.5025
	1527.0	1800.15
	1538.0	1811.15	13807	7.6325
	769.0	1042.15	8075	9.21
		1800
		55-80

agree with respect to its variation with pressure. Figure l .4. Measurements made at ambient pressure on retained hep phase generated by shock-loading Fe-l4Mn wt% alloys tend to support the data of Mao et al., with a lattice parameter ratio of about l.6l [10]. The fact that hep iron has the highest density of all the common allotropes of iron means that it is likely to be the stable solid phase of iron at the inner core of the Earth. The crystal structure of the Fe-lONi wt% alloy that may represent the core composition is thought to be hep, based on X-ray diffraction data from in-situ measurements at 340 GPa and 4700 К [11]. Simulations using techniques such as ab initio molecular dynamics show that under Earth-like conditions, it is indeed the hep phase that is stable at the inner core [12]. Exoplanets have been discovered outside of our solar system that are much larger than the Earth. This has prompted studies of the iron phase diagram under conditions of extraordinarily large pressures and temperatures. This is because like Earth, such exoplanets and even some so-called gas giants may contain iron-rich cores that determine their magnetic fields. Calculations using ub initio

Figure 1.3 Projections of atom positions on plane normal to the z-axes of the crystal structures of austenite, ferrite and e-iron, drawn to scale with respect to their observed lattice parameters. The black atoms have fractional z-axis coordinates 0 and 1, whereas the white atoms are at jz.

Figure 1.4 The measured lattice parameter ratio c/a for hep iron as a function of pressure (after Clendenen and Drickamer [13] and Mao et al, [14]). The ideal value of the ratio in the hard- sphere approximation is -у/8/3 = 1.633.

methods are shown in Figure 1.5 for temperatures up to 40,000 К and pressures reaching 100 TPa [15]. Counter to intuition, the hep ceases to be stable at very high pressures. This is because the electronic structures of the phases change - the configuration of the nonmagnetic bcc iron atoms under normal conditions is 3d⁶4s² with the core 3s and 3p bands being very narrow and at energy levels where they are tightly bound to the nucleus and hence unable to participate significantly in the metallic bonding process. Extreme pressure changes that, with the core electron bands widening, overlapping and hence participating more effectively in the bonding process leading to changes in the relative stabilities of the allotropes of iron [15, 16]. It is interesting that direct measurements using laser pulses for pressure and X-ray diffraction for crystal structure have shown that while Fe- 6.5Si becomes hexagonal close-packed at pressures in the range 400-1314GPa, Fe-15Si becomes body-centred cubic over the same pressure range [17]. The estimated temperatures involved in these experiments lie in the range 1500-3000 K.

Figure 1.5 Phase diagram of iron under extreme conditions calculated using finite temperature density functional theory. The continuous curves are from experimental data whereas the calculations, that neglect magnetic contributions, are represented by dashed curves. Pressure-temperature conditions at the centre of the Earth, Jupiter and a planet five times the size of the Earth (“superearth”) are also illustrated. The appearance of oc at pressures in excess of 34TPa is a body-centred tetragonal distortion (с/д и 0.975) of the bcc form [16]. Adapted with permission from Strixrude [15]. Copyrighted by the American Physical Society.