Deltoidal icositetrahedron faceting on α-Fe crystals found on the surface of the Moon

Crystallographic analysis of scanning electron microscopy images of small μm-sized single crystals of b.c.c. iron found on the surface of the Moon shows that the deltoidal icositetrahedron faceting behaviour clearly seen is best describable as being from planes of the {229} form. While possibly unexpected given the lack of any report of such faceting in terrestrial and meteoritic b.c.c. iron single crystals, this deltoidal icositetrahedron faceting behaviour can be rationalised straightforwardly in terms of the local chemical conditions which will have been experienced by these crystals while growing from the vapour in the part of the lunar environment from which these samples were obtained.


Introduction
Some fifty years ago, Clanton et al. [1][2][3][4] reported scanning electron microscope observations of b.c.c iron crystals found as small μm-sized single crystals in lunar samples from the various Apollo missions to the Moon. Energy dispersive X-ray analysis of these b.c.c. iron crystals showed that levels of nickel, phosphorus, cobalt and sulfur were below the detection limit of the instrumentation, reported as 0.5% [1][2][3][4], indicating that they were therefore of high purity. Interestingly, Clanton et al. reported that these b.c.c. iron crystals exhibited six of the seven possible forms of the holosymmetric class m3m [3][4][5]. In contrast to these observations, observations of iron crystals produced as single crystals on Earth have a more restricted range of morphologies. From surface energy considerations, and also from experimental observations, the morphology of such crystals is expected to be dominated by { 110} and {100} planes (or facets) [6,7]. In addition to these planes bounding b.c.c. iron crystals, large {111} facets and small {112} facets have also been reported recently on plastically deformed and carefully annealed b. c.c. iron nanocrystals [8].
An example of a small group of iron crystals found in a small cavity of recrystallised fragmented rock from the Apollo 15 Hadley-Apennine lunar landing site is shown in Fig. 1 [9]. The largest crystal here is reported to be three micrometres across. These three crystals would all seem to have the same facet morphology. They are seen to be formed of {100} cube planes and {hhl} planes of a deltoidal icositetrahedron form with l > h so that for each crystal the overall shape is describable as a truncated deltoidal icositetrahedron. As a consequence of the presence of the {100} planes, the {hhl} planes each have five edges. Fig. 2 of [1] and also Fig. 2 of [2] where Clanton et al. refer to the {hhl} form as the trapezohedron form are both part of the image shown here in Fig. 1. The trapezohedron nomenclature used by Clanton et al. [1,2] is equivalent to describing this form as the deltoidal icositetrahedron form, as we have chosen to do here, or, alternatively, simply the icositetrahedron form, the nomenclature favoured by Phillips [5].
Interestingly, there is no further crystallographic analysis in the literature of the {hhl} form seen in Fig. 1, or indeed the {hhl} forms in Figs. 2 and 3, reproduced from the two Clanton et al. papers [2,4], respectively. The figure captions accompanying these latter figures in these two papers suggest that for both crystals there are two separate sets of {hhl} forms. While there is clear evidence for this surrounding one of the 〈11 1〉 directions of the iron crystal in Fig. 3, the evidence for two separate sets of {hhl} forms in Fig. 2 here would seem to have been lost in the process of reproduction into the journal image from the original scanning electron microscope image. Even though Fig. 2 is taken from a 45 MB high resolution scan of a printed version of the Clanton et al. paper, this printed version had already lost subtleties in contrast presumed to be apparent in the original image. It is evident from Fig. 3 here that the contrast between the two different sets of {hhl} forms is indeed subtle, implying that the angular difference between these two sets is clearly a few degrees at most.
The problem of determining suitable h and l for the five crystals in Fig. 1 and the further two in Figs. 2 and 3 is a projectional geometry problem where the unknown direction [uvw] along which the crystals are projected, and the precise details of the unknown faces of the form {hhl} for suitable h and l, with l > h need to be determined. This is a classical crystallography problem which makes use of the concept of the law of constancy of angle, and it requires an understanding of the way in which a two-dimensional image in a scanning electron microscope relates to the three-dimensional shape being imaged. It is the purpose of this paper to solve this problem for the deltoidal icositetrahedra {hhl} forms seen in Figs. 1-3 and to comment on the nature of these forms in the wider context of the growth of α-Fe single crystals.

Crystallographic analysis
The μm-sized dimensions of the five iron crystals in Figs. 1-3 is such that it is entirely reasonable to make the assumption that the projective geometry here is that of parallel projection, so that the centre of projection is at infinity This approximation is suggested by Cornille [10] as being suitable for magnifications in the scanning electron microscope above 20,000; it is still likely to be a reasonable approximation for the unknown actual magnifications of the original NASA images from which Figs. 1-3 here were taken. Hence, with this assumption, the projectional geometry seen at this magnification in a scanning electron microscope is that of an orthographic projection of each crystal. It follows that the facets of the crystals seen in Figs. 1-3 are seen as though they are projected onto the plane normal to a direction [uvw] in the coordinates of the b.c.c. unit cell of iron defining the projection direction for each crystal.
Inspection of each of the five iron crystals in Figs. 1-3 shows that furthermore it is reasonable to assume that the cube {100} planes are squares for all five crystals, even though it is evident that for the upper right crystal in Fig. 1 Fig. 3 from Clanton et al. [4]. This is also a scanning electron micrograph. It is evident that there are two sets of {hhl} faceting surrounding one of the 〈111〉 directions of this iron crystal. In addition, there is a much smaller set of {100} faceting. The subtlety in contrast between the two sets of {hhl} faceting is such that the angular difference between these two sets is clearly a few degrees at most.
It is convenient to let the line common to (hhl) and (001) be of unit length. We can then choose the length of the lines common to (i) (hhl) and (hhl) and (ii) (hhl) and (hhl) to be α times in length that of the line common to (hhl) and (001).
Likewise, the lines common to (i) (hhl) and (hlh), and (ii) (hhl) and (lhh) will both be β times in length that of the line common to (hhl) and (001).
Starting from the centre of the (001) face, we therefore have the identity Hence, equating either the x-or the y-coefficients in this equation, it is evident that Suppose now that the origin is shifted to the centre of this crystal with external m3m symmetry, so that the vector to the (001) face from the origin is of the form [0,0,n] for some n. The vector [0,n,0] then specifies the centre of the (010) face. Starting from the centre of the cube, we therefore have the identity Hence, equating either the y-or the z-coefficients in this equation, it is evident that These calculations help to determine how such a crystal with {001} and {hhl} interfaces and external m3m symmetry is seen in an orthographic projection along various possible [uvw] projection directions.
Another useful result is that the projection direction [uvw] for such a crystal with exact external m3m symmetry can be deduced from the external shape of the crystal seen in projection: the projection direction will be at the point where lines linking opposite vertices of the perimeter of this external shape all meet. A close inspection of both the crystal in the upper left of Fig. 1 and the crystal in Fig. 2 shows that this useful result is able to establish narrow bounds both for the projection directions [uvw] of these two crystals and suitable {hhl} forms for the deltoidal icositetrahedron faceting behaviour. Fig. 1 Fig. 3 For these two crystals, the assumption for modelling purposes is that the {hhl} facets would all be identical in the absence of the planes of the {100} form. Therefore, for an (001) plane as a 'reference' facet where the line common to (hhl) and (001) is of unit length, we can let a clearly larger (010) or (100) facet have facet sides of length γ 1 . To compensate for this, it is evident that for this larger facet, α must decrease to a value α 1 , defined by the identity

and the crystal in
since the terms in β in Eq. (1) are unchanged by the assumption that the {hhl} facets would all be identical in shape in the absence of the {100} facets. This therefore is a minor adjustment to the two crystals whose external morphology can be taken to have external m3m symmetry. Furthermore, just like the two crystals which can be taken to be ones with exact external m3m symmetry, as long as the lines joining opposite vertices of the external shape are chosen with care, narrow bounds both for the projection directions [uvw] of these two crystals and suitable {hhl} for the deltoidal icositetrahedron faceting can usefully be determined. Fig. 1 For this crystal, it is evident that the shape of the crystal is far from the ideal external m3m symmetry, and so we have to generalise equation (1) for facets of a more general shape, while keeping the line common to (hhl) and (001) to be of unit length. Suppose the (hhl) face is one of these elongated faces clearly seen on this crystal in Fig. 1 A more general form of equation (1) then becomes

The crystal in the lower right of
for some β 1 and β 2 to be determined. Equating coefficients of the x-and y-coordinates, we then have respectively to solve for β 1 and β 2 . It is evident from these two simultaneous equations that when α 1 = α 2 , β 1 = β 2 . Expressing equations (9) and (10) in matrix form, we have equations of the form ( For analysing this crystal, the faces (hhl) and (hhl) have α 1 = α 2 , as do the faces on the (010) side of this truncated deltoidal icositetrahedron, while the faces (hhl) and (hhl) have α 1 ∕ = α 2 , with α 2 = 2α 1 ..
The (100) 'face' of this truncated deltoidal icositetrahedron composed of the five facets (100), (lhh), (lhh), (lhh) and (lhh) is not clearly seen in Fig. 1 In comparison with the other four crystals, it is somewhat more difficult to establish narrow bounds for the projection directions [uvw] of this crystal, but the subtleties of the projectional geometry of the possible {hhl} facets for this crystal help in this regard.

Projectional geometry
For calculations to predict how the five crystals seen in Figs. 1-3 are seen in projection, extensive use was made of the formula for the projected direction, r p , of a vector r when viewed along a vector u = [uvw] in an orthographic projection of the crystal: This formula enabled the directions in projection of the {100} and {hhl} facet border vectors to be determined, and therefore the lengths of these facet borders also to be determined. Angles seen in projection between different facet borders could also be calculated and compared with the observed values seen in Figs. 1-3 for different θ and ϕ. For each crystal, two different approaches were examined to determine the likely families of {hhl} facet planes. The first approach was to determine the angles seen in projection between a few prominent facet borders within a crystal and then compare angles determined from the micrographs with those predicted experimentally. This had the advantage of helping to narrow down possible [uvw] projection directions and {hhl}, but it had the obvious disadvantage of not using all the possible information present in the scanning electron micrographs. Furthermore, for a particular crystal under consideration, it highlighted the need to be able to visualise the entire crystal morphology, and to determine what facets would be visible externally in a particular projection [uvw], and what facets would not be seen. Therefore, a second more systematic and more thorough approach was used. Excel spreadsheets were generated to visualise the entire crystal morphology for each of the five crystals seen in Figs Fig. 1 has been used to demonstrate in Fig. 4(b) how the projection direction [uvw] can be deduced from the external shape of this crystal seen in projection, as has been described in Section 2.1.

Initially
The {229} facet geometry is able to capture the geometry in  Fig. 4(e). The difference in appearance between the predicted forms for {112} and {223} in Fig. 4(d) and (c) respectively and the experimental scanning electron micrograph is accentuated in moving from {113} to {112} to {223}.
The difference between the predicted form for {115} and {229} is very subtle. A close examination of Fig. 4(a) and 4(g) might suggest that in Fig. 4 (a) the external perimeter of the projected crystal in the centre top left part of the crystal is slightly more faithfully reproduced than in Fig. 4(g).
Similar comparisons can be made for the crystal in the top right of Fig. 1, the predictions for which are shown in Fig. 5. Here, it is useful to examine evidence in the predictions for the occurrence in the projections of (hlh) and (hlh) planes; both these planes are present to some very slight degree in the crystal in the top right of Fig. 1 A comparison of the experimental image of the crystal in the lower right of Fig. 1 with the predictions in Fig. 6 is best made by examining evidence for the relative absence in the predicted images of (lhh) and (lhh) facets and the slight hint of presence of (hlh) and (hlh) facets. Here, the predicted images for the {114} and {115} forms do better than all the others apart from {229}, but again the differences between {114}, {229} and {115} are nuanced.
For the crystal in Fig. 2, the presence of the (lhh) facet evident in this image and its relative size seen in this image is useful in establishing that the predicted geometry for the {116} form is inconsistent with the experimental evidence, because in Fig. 7(h) this facet is barely visible. By contrast, this (lhh) facet is all much too visible for the {113}, {112} and {223} forms. Once again, while both the predicted projections for  Finally, for a comparison of the experimental predictions in Fig. 8 with the experimental image seen in Fig. 3, it is useful to look at the external perimeter of the crystal in Fig. 3 and see how faithfully this is reproduced in the predictions in Fig. 8 The predicted shapes of the crystals seen in projection were more sensitive to θ than to ϕ for the projections shown in these figures because of the choice made in labelling the {100} facets seen in the scanning electron micrographs: changes in units of 0.5 • for θ were found to be reasonable for the calculations, while changes in units of 1 • were found to be reasonable for ϕ. A major conclusion from these calculations evident from Figs. 4-8 is that the deltoidal icositetrahedral {hhl} facets were clearly not {112}, the minor facet geometry reported by Kovalenko et al. [8] in sub-μm sized α-Fe crystals annealed for 24 hr at 880 • C.

Discussion
Recognition that the projectional geometry can be taken to be an orthographic projection for μm-sized single crystals observed in the scanning electron microscope, such as these b.c.c. Fe single crystals recovered from lunar samples, has enabled {229} facets to be identified as a 'best fit' compromise on these Fe single crystals. These {229} facets contrast with observations on iron crystals produced as single crystals on Earth, and also with observations of {110} faceting on Fe crystals from the Haverö meteorite [11]. The predicted projection geometries for  The predicted projectional geometry seen in Fig. 8 also confirms that the contrast and projectional geometry seen in Fig. 3, where two sets of {hhl} facets surround one of the 〈111〉 directions of this iron crystal, are together clearly rationalizable in terms of two sets of facets such as {11 4} and {11 5}.
Conditions for crystallization of these fine α-Fe single crystals have been established by Clanton et al. [1][2][3][4]. They quote an abundance of studies which indicate that a wide variety of minerals formed by vapour phase crystallization in the lunar environment [3,4]. Such crystallization conditions differ markedly from those used to produce α-Fe single crystals on Earth. Instead, they are merely indicative of the wide variety of conditions under which minerals are known to grow, such as 'solutions often containing all manner of other substances as impurities', as Phillips observes in Chapter I of his book [5]. It is highly likely that it is the local presence of such 'impurities' which has enabled these remarkable facets to grow on the five α-Fe crystals seen in Figs. 1-3 and to dominate their morphology. As others have noted in connection with growth from the vapour of various materials, a number of kinetic and thermodynamic processes are involved, as well as having favourable sites to enable atoms to be deposited, such as kink sites of atomic steps. The details of such growth is often system-specific and is also very temperature-dependent [14][15][16][17]. While outside the scope of this particular study, it would clearly be worthwhile scientifically to undertake such experiments to try and simulate the lunar environment within which these iron crystals will have been produced through vapour phase crystallization.
In practice, facets labelled {229} on a scale of a few hundred nanometres of more will themselves be structured at the atomic level, providing sites for continued growth. For example, ball models of tungsten (11 3) and tungsten (117) surfaces considered by Biernat and Błaszczyszyn [12] suggest that for these b.c.c. (117) surfaces, (001) terraces dominate the morphology at an atomic level and are linked by steps which have a common [110] vector with the (001) terraces to produce overall (117) surfaces, whereas for the (113) surfaces, there are (112) terrace 'fragments' and (001) steps. Faceting of atomically rough planar surfaces such as for iridium (210) surfaces at the scale of nanometres is known to be driven by chemical considerations caused by absorbates [13]. Given such considerations, it is unsurprising that some of these lunar α-Fe single crystals have these exotic {229}-type morphologies. Clanton et al. found evidence of other α-Fe single crystal lunar morphologies which we have chosen not to analyse here, such as those shown in Figs. 9-11 of [2]. This is because of the dominance in these morphologies of the {100} cube form; this has the effect of making unambiguous analyses of the crystallography of the other forms more challenging because there is less information to be able to extract from these scanning electron microscopy images.

Conclusions
Crystallographic analysis of scanning electron microscopy images of small μm-sized single crystals of b.c.c. iron found on the surface of the Moon has shown that the deltoidal icositetrahedron faceting clearly seen is best described as {229} faceting. As others have noted [4], this faceting behaviour is indicative of the vapour phase crystallization in the lunar environment where the precise nature of the vapour and the temperatures of formation remain elusive. Recent studies of the way in which absorbates enable surfaces to be fashioned at the nanometre level help to rationalise this unusual and rather photogenic faceting, but clearly it would be of interest to determine experimentally how this faceting behaviour arose through vapour phase crystallization in the lunar environment.