1 Introduction

A glance at the Fe–C binary phase diagram reveals interesting and enigmatic features.[1] The low temperature iron rich phase is not a close packed structure which we are used to seeing in binary phase diagrams at low temperatures (either FCC or HCP) but rather is a phase usually designated as BCC α iron. Even more intriguing is the replacement of this BCC phase at 912 °C (1185 K) by an FCC one (designated as γ iron) and the subsequent reappearance at 1391 °C (1664 K) of the BCC phase, designated as the δ phase. Another interesting feature is that in the modern phase diagrams the phases are labeled α, γ and δ (and ε if high pressures were included in the diagram) but no β phase? Why is this?

In an earlier paper Massalski and Laughlin[2] discussed the thermodynamic reasons for the low temperature phase not being the FCC phase, and why the paramagnetic BCC phase is replaced by the FCC one at higher temperatures. We accounted for this interesting behavior in terms of the magnetic moments associated with the Fe atoms, including the little known fact that at temperatures below ≈ 70 K, metastable FCC γ iron becomes anti-ferromagnetic. The added entropy due to the antiferromagnetic to paramagnetic transition at low temperatures is what enables γ iron to replace the BCC phase at higher temperatures. Massalski and Laughlin also discuss the “missing” β phase and made a plea to reinstate it on the phase diagrams of iron based alloys.

In this paper, I will review some of the history of the β phase of iron and give the stated reasons for its excising from the phase diagram. After criticizing the reasons that have been given for its exclusion from the phase diagram, I will make a case for reinstatement of the β phase to phase diagrams as the paramagnetic BCC phase of iron. Some salient features of including the magnetic state of the phases on the equilibrium Fe–C phase diagram will be discussed.

2 Overview of the History of the β Phase

I have gleaned many of the references in this section from the paper by Cohen and Harris, “The β—Iron Controversy”.[3]

The α and β phases were named by Osmond and Werth in 1885.[4,5] In later papers[6,7,8] Osmond reported on thermal arrest temperatures he called Ar (on cooling) and Ac (on heating). The Ar3 and Ac3 temperatures correspond to the critical temperatures on cooling and heating respectively of the transformation between what we call today the FCC γ phase (paramagnetic) and the BCC paramagnetic phase. The temperatures Ar2 and Ac2 (which were nearly identical) correspond to the transformation of the BCC paramagnetic phase to the ferromagnetic α phase, and vice versa. Today we call this temperature the Curie temperature (770 °C, 1043 K), after Pierre Curie, (1859-1906).

Based on these results the phase diagrams of iron alloys in the early decades of the twentieth century, the solid state phases of iron were delineated as

  • α, for the low temperature ferromagnetic phase,

  • β, for the BCC paramagnetic phase which forms from α at the Curie temperature (Ac2)

  • γ, the FCC paramagnetic phase which forms from β at ~ 912 °C (1185 K), the Ac3.

Later, the high temperature BCC phase which formed from the γ phase on heating was called the δ phase. Thus the solid state phases of iron were delineated as shown in Fig. 1.[9]

Fig. 1
figure 1

Temperature scale (in °C) showing the regions of stability of the various Fe allotropes. PM paramagnetic and FM ferromagnetic

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This delineation of the phases of iron implied that iron existed in three distinct polymorphs or allotropes, namely those designated as the α, β and γ. It was realized that the δ phase was a continuation of the β phase: both were BCC and both were paramagnetic. In fact the plot shown in Fig. 2 of the inverse of the magnetic susceptibility of Fe versus temperature shows that the susceptibility of the δ phase is a continuation of that of the β phase. It also shows that at the Ar2 temperature (Curie temperature) the magnetic properties of β iron changed discontinuously when it is cooled through the Curie temperature and that when β iron transforms to α iron, the iron becomes a ferromagnetic phase.

Fig. 2
figure 2

Inverse susceptibility of Fe as a function of temperature for the various phases. Note: below the Curie temperature the value the inverse susceptibility α is essentially zero. Also note that the β and δ phases can be seen to fall on a continuous curve, showing that they are the same BCC paramagnetic phase

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So why was β extracted from the phase diagram? A little more than five decades ago, Cohen and Harris[3] wrote an article entitled “The β – Iron Controversy”. In a clear and concise fashion the authors reviewed the history of the β – Iron “controversy”. They first discussed some of the properties which had been attributed to β iron, such as its effect on hardness etc. After clearing up these erroneous concepts (which the present paper does not discuss) they moved onto the idea that α and β iron were different allotropes of iron, and made their opinion clear: β should not be included in modern phase diagrams of iron and its alloys because the α to β transition is not a real phase change.

This statement has as its basic underlying assumption that the α and β states of Fe are the same phase. This is said, even though the two states have very different magnetic properties. This implies that properties of the states do not enter into the definition of a phase. Now this is a strange approach as nearly all definitions of thermodynamic phase includes the uniformity and specificity of physical and chemical properties. As far back as 1923 in their highly influential text “Thermodynamics and the Free Energy of Chemical Substances’, Lewis and Randall[10] define a phase as:

a homogeneous region of matter in which its properties are the same or at least vary continuously from point to point.

The same may be seen from Professor Christian’s rather complete description of a phase:

“… an assembly of atoms or molecules which has attained equilibrium under specified external constraints consists of one or more homogeneous and physically distinct regions. The regions of each type may be distinguished by a common set of parameters defining such intrinsic properties as density, composition, etc. and they constitute a phase of the assembly. Two phases are distinguishable if they represent different states of aggregation, different structural arrangements in the solid, or have different compositions”.[11]

More recently Soffa and Laughlin[12] define a phase as:

“.. a physically distinct homogeneous portion of a thermodynamic system delineated in space by a bounding surface, called an interphase interface, and distinguished by its state of aggregation (solid, liquid or gas), crystal structure, composition and/or degree of order. Each phase in a material system generally exhibits a characteristic set of physical, mechanical and chemical properties and is, in principle, mechanically separable from the whole.”

So on the basis of the drastic change in magnetic properties, the α to β transformation should be designated as a phase change and thus α and β would be different allotropes of iron.

It is at this point that the real problem seems to have raised its head. Allotropes are said to have different crystal structures and α to β iron were thought to have the same crystal structure, namely BCC. This was believed to have been shown by some very careful high temperature x-ray diffraction of Westgren and Phragmén reported in 1922.[13] The x-ray patterns for the purported α and β phases (at 16 and 800 °C respectively) seemed to be identical except for the shift in spacing due to thermal expansion. Since the crystal structures were the same the phases were said to be the same. Thus, it was concluded there was no phase transformation and no need to invoke a β phase separate from the α phase. Slowly the β phase disappeared from the mainline metallurgical literature.

The following summarizes the demise of the β phase of iron from the metallurgical literature.

  • The 1917 Thum’s Practice Book in Elementary Metallurgy refers to the β modification of iron as the modification that exists between the α and γ modification (Ref 14, p. 126).

  • In 1923, Walter Rosenhain wrote in his Metallurgy; an introduction to the study of physical metallurgy, that “β (iron) is still the subject of discussion, (Ref 15, p. 167).

  • In 1930 Woldman’s Physical Metallurgy, β is included in the Fe–C phase diagram (Ref 16, Fig. 21).

  • In 1935, Doan’s, Principles of Physical Metallurgy does not include the β phase in the phase diagram but does include a horizontal lineFootnote 2 in the α + γ two phase region.[17]

  • In 1936, Van Vlert’s Introduction to Physical Metallurgy does not mention the β phase of iron.[18]

  • In 1940, Sachs’ Practical Metallurgy, states that “the magnetic transformation which is completed (on heating) at the Curie point, ~ 1415 °F (770 °C) is not considered as a phase change, in spite of the marked changes in some of its physical properties”.[19]

The reference books also show this trend. In the 1919 and 1925 editions of “The Making, Shaping and Treating of Steel”,[20,21] the β phase is shown in the phase diagram of Fe–C (Fig. 197, 1925 4th edition), but in the discussion it is mentioned that since the two phases have “exactly the same space lattice” (eluding to the work of Westgren and Phragmén) they can not be different phases “without changing the definition of allotropy” (Ref 21, p. 661). By the 5th edition[22] β is in the phase diagram with an attached “?” (Figure 340) and in the discussion the 1940 text reads “most careful investigators fail to note any difference in the properties of α and β iron except in their magnetism” (emphasis mine). No mention of β iron exists in the 1965 edition. Also the influential Metals Handbook,[23] published by ASM makes no mention of β iron in the 1948 edition.

1n 1939, Williams and Homerberg’s Principles of Metallography, perhaps best sums up the thinking of mid twentieth century metallurgists: “… if allotropy is defined as a change in crystal form, then β-iron, which would occur in the range from 906 to 768 °C, does not exist as an allotropic form”.[24]

But is this really the case? Is the paramagnetic form of BCC iron really the same phase as the ferromagnetic form of iron? Clearly they have different physical properties. What about their structures? The careful work of Westgren and Phragmén did not show a difference, but could their x-ray diffraction experiment have shown a difference between a ferromagnetic structure and a paramagnetic one?

3 The Magnetic Symmetry of “BCC” Ferromagnetic Iron

Let us consider the basic principles of crystal structure. We have all learned in our crystallography classes that crystal structure is lattice plus a basis (or motif). So we start with a BCC lattice, Fig. 3(a). The lattice has the full cubic symmetry and can be represented as \(Im\overline{3} m\). If we add to each lattice point a spherical atom, centered on the lattice points, the overall symmetry (space group) does not change, because the new symmetry is an intersection of the symmetry of the BCC lattice and the spherical symmetry of the atom. See Fig. 3(b). This structure is called the BCC structure with the space group \(Im\overline{3} m\). What happens however when we add an atom that has associated with it a magnetic moment, represented by a circulating electron? Clearly the symmetry has been lowered to that of \(I\frac{4}{m}\), since only one of the orthogonal axes retains its four-fold rotation symmetry, that is the new structure is tetragonal. This latter structure is the structure which best represents a body centered unit cell with magnetic moments on or related to each lattice point. This shows that ferromagnetic α phase is NOT cubic, but rather tetragonal whilst the paramagnetic β phase is cubic since the magnetic vectors are randomly arrayed in a paramagnetic substance.

Fig. 3
figure 3

(a) A BCC Bravais lattice. (b) spherical atoms placed on the BCC Bravias lattice producing the BCC structure of the β and δ phases with the space group \(Im\overline{3} m\) (c) Circulating arrows representing the magnetic spins present on the low temperature α phase. The symmetry of α is seen to be tetragonal, with space group \(I\frac{4}{m}\)

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In the terms discussed above, α, β and γ are three allotropes of iron and should be represented as such on equilibrium phase diagrams. The difference between the ferromagnetic iron and paramagnetic iron is one of the ordering of the magnetic moments: in the cubic β phase, the moments are disordered: in the tetragonal α phase the moments are aligned and thus ordered. A material that changes its symmetry from cubic to tetragonal due to ordering undergoes a phase transformation.

The topic of magnetic symmetry is an interesting one and has been discussed in some of the text books of magnetism and crystal physics for the last 50 years or so.[25,26,27,28,29,30,31] See also Ref 32 and 33.

4 Some Effects of Magnetism on the Fe–C Phase Diagram

Consider the schematic phase diagram of Fe–C shown in Fig. 4.

Fig. 4
figure 4

Schematic of the iron rich end of the Fe–C phase diagram. The dashed lines represent the Curie temperature of the alloy. The horizontal line is the Curie temperature of the alloy with the composition of the dot. The plot is exaggerated as the difference in the value of the Curie temperature of pure Fe and the of saturated C in Fe is less than 1 °C

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First we focus on the Curie temperature of pure iron, TCurie. At this temperature the low temperature ferromagnetic α iron phase has lost its magnetization, that is, the phase has a zero magnetization and it is no longer ferromagnetic, but paramagnetic. At and above the Curie temperature the stable phase is the paramagnetic phase of BCC iron (β). This temperature is unlike the temperature denoted β/γ where two phases may be in equilibrium with each other. Such temperatures are equilibrium transition temperatures of first order structural transitions which follow the Gibbs equilibrium phase rule. (The Gibbs phase rule was derived assuming the transitions were first order transitions.) Since the Curie temperature is not one where two phases coexist in equilibrium, it does not give rise to a sharp rise in the heat capacity on heating, characteristic of first order transitions. Rather, there is a gradual rise in the heat capacity which reaches a maximum at the Curie temperature and subsequently decreases. There is no concomitant release of thermal energy. This higher order phase transition occurs because the loss of magnetization is continuous from below the transition temperature.

The line in Fig. 4 from the Curie temperature of pure iron to X, is displayed as dot-dashed to show that it represents the Curie temperature of Fe–C alloys with C% < X in solution. The curve depicting the Curie temperature decreases slightly until it intersects the BCC solubility curve. If an alloy of Fe–C with less than X % C were to be heated to the temperature where it intersects the dot-dashed line, it would transform from ferromagnetic iron to paramagnetic iron, there would be no thermal arrest in the heating curve. Again this change of phase is not a first order transition.

We now look at the horizontal line designated as XY. This line is horizontal since it represents the Curie temperature of a specific Fe–C alloy, namely one with a composition of X. It is drawn across the two phase field because any alloy of Fe–C with compositions in that area will have the Curie temperature of the low temperature α ferromagnetic phase as TC(X). This horizontal line is not a designation of a three phase equilibrium line: it represents the Curie temperature of the α phase which exists in equilibrium below it. It should be noted that below the line, α (ferromagnetic iron) and γ are in equilibrium, while above the line, β (paramagnetic, BCC) and γ (paramagnetic FCC) are in equilibrium.

Finally we look at the point Y in the diagram. The early representations of the phase diagram of Fe–C showed a marked change of slope of the α/γ solubility curve at this point. This was consistent with the phase diagram construction rules for first order phase transitions. It was thought that since at that point the phase in equilibrium with γ changed from α to β, the slope of the solvus must also change. But the transition of α to β is a continuous one and since just below the temperature T X the α phase has nearly the same free energy as the β phase just above T X , the slope change here should be minimal. It should be noted that there were never any data that were consistent with a change in slope at the point Y.

This in depth look at one region of a phase diagram that contains a ferromagnetic to paramagnetic transition has shown that there are some details which must be taken into account in the construction of such phase diagrams.

5 Closing Thoughts

In his influential monograph “Introduction to phase transformations in condensed systems” Morris Fine says “… comparing the phases before and after transformation, there are three basic types of transformation: (1) change in structure, (2) change in composition, and (3) change in degree of order”.[34] The ferromagnetic to paramagnetic transformation fits both conditions (1) and (3) of a phase change.

6 Summary

The transformation of a ferromagnetic phase to a paramagnetic phase is a phase transformation with changes in crystal symmetry. All such changes should be denoted on phase diagrams and the different phases should be designated with Greek letters and/or primes which show that they are not the same phase! Once again[2] I urge that we bring the β phase back to Fe phase diagrams.