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Helium gas-bubble superlattice formation in molybdenum
NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 243 (2006) 325–334 www.elsevier.com/locate/nimb
Helium gas-bubble superlattice formation in molybdenum P.B. Johnson
a,b,*
, Fenella Lawson
a
a b
Victoria University of Wellington, School of Chemical and Physical Sciences, P.O. Box 600, Wellington, New Zealand The MacDiarmid Institute for Advanced Materials and Nanotechnology, Victoria University of Wellington, New Zealand Received 24 August 2005 Available online 28 November 2005
Abstract The gas-bubble superlattice is a striking phenomenon where small helium bubbles of uniform size are regularly arranged on a space lattice having the same symmetry as the crystal lattice of the host metal but with a lattice spacing some twenty times greater. Typically, bubble diameters are 2 nm and bubble concentrations are 1 · 1025 m3. Superlattices have been observed in all three metal types: bcc, fcc and hcp. The most common superlattice orientation is parallel with the host lattice. However, for the fcc metals, and the bcc metal vanadium the superlattice also contains structural variants – regions where the ordered bubble array has an orientation that is rational with, but different from, that of the crystal lattice of the host metal. Here we report on the structure of the bubble superlattice in the bcc metal molybdenum. Recently the proposal has been made that this superlattice could act as a photonic crystal for soft X-rays. An assessment is made of the quality of the superlattice in molybdenum in relation to this proposal. 2005 Elsevier B.V. All rights reserved. PACS: 81.07.b; 61.82.d; 61.82.Bq; 68.37.Lp; 68.65.Cd; 61.10.Eq Keywords: Helium gas-bubble superlattice; Molybdenum; Structural variants; Photonic crystal
1. Introduction The spatial ordering of gas bubbles, produced in metals by the ion implantation of inert gases such as helium, has been the subject of many studies based on transmission electron microscopy (TEM) [1–29]. The gas-bubble superlattice stage in most metals occurs at local helium concentrations of 10–15 at.% in the matrix. The superlattice comprises small helium bubbles of uniform size (2 nm diameter) ordered on a space lattice having the same symmetry as the host. The bubble lattice spacing is typically some twenty times that of the crystal lattice. Superlattice formation has been observed in metals representing all three major crystal types fcc, bcc and hcp and so seems to be a universal response for helium implantations at * Corresponding author. Address: Victoria University of Wellington, School of Chemical and Physical Sciences, P.O. Box 600, Wellington, New Zealand. Tel.: +64 4 4635966; fax: +64 4 4635237. E-mail address: [email protected] (P.B. Johnson).
0168-583X/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2005.09.021
temperatures in the region of 0.2Tm where Tm is the melting temperature of the metal. Much of the original impetus for research into bubble ordering came from a recognition that it could provide a unique insight into the micro-structural processes underlying more general cases of materials modification by ion implantation. More recently, interest has been stimulated by the possible technological applications of heliumimplanted surfaces. For example, the highly swollen nanoporous layers that form beyond the superlattice stage show promise for applications in areas such as catalysis and biocompatible materials [30–33]. The unique characteristics of these nanoporous structures are thought to be ‘‘seeded’’ by the highly uniform superlattice stages that precede them. Further, the superlattice stage of bubble development, itself, as a three-dimensionally ordered structure, could find application in optoelectronics [30]. For example it has been proposed that the superlattice in molybdenum could act as a photonic crystal for soft X-rays [34]. In a photonic crystal, the dielectric constant (generally complex-valued) has
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spatial periodicity in one, two or three dimensions. Computations using the electromagnetic transfer matrix method have shown that the superlattice in molybdenum, under the assumption of a homogeneous superlattice oriented parallel with the matrix, should act as a selective reflector of soft X-rays with wavelengths close to 10 nm. Information on the gas-bubble superlattice has come mainly from TEM bright-field micrographs and from TEM selected-area electron diffraction (SAD) patterns taken from the ordered arrays. A review of research on bubble ordering prior to 1990 can be found in [21]; more recent research is covered in [22–27]. Early research centred on the bcc metal molybdenum, see for example [1,2]. The results from these and subsequent studies [23,27] established that the superlattice in molybdenum had bcc symmetry and was at least largely parallel with the host crystal lattice. For the fcc metals the bubble superlattice always contains structural variants – i.e. regions where the ordered bubble array has an orientation that is rational with, but different from, that of the crystal lattice of the host metal. Until recently, the evidence suggested that in the bcc metals structural variants of the type found in the fcc metals did not form and that the bubble lattice aligned solely parallel with the host crystal [23]. However, bubble variant lattices have now been demonstrated to occur in the bcc metal vanadium [29], and this has renewed interest in the detailed structure of the gas-bubble superlattice in the other bcc metals. In the present study, TEM is used to investigate the structure of the gas-bubble superlattice in molybdenum with the particular aim of testing the previous assumption of a homogeneous superlattice oriented solely parallel with the matrix. This is referred to as the m orientation.
surface to different depths by means of ion-beam milling, prior to chemical back-thinning the foil to produce electron transparent specimens [9,27].
2. Experimental procedure
3. Results and discussion
2.1. Implantation conditions
3.1. Planar ordering
Unless otherwise stated the following procedure was used to prepare and examine specimens. Molybdenum foils, 99.8% pure, 25 lm thick, from Goodfellow Metals were diamond polished in stages down to 1 lm diamond paste and annealed at 1100 C for 2 h in vacuum. Most of the samples were implanted with 160 keV helium ions at 60 to the surface normal of the foil. During the implantation the sample temperature was maintained near 300 C (0.2Tm). Typical ion doses were 5 · 1021 He+ ions m2 and dose rates were 2 · 1017 He+ ions m2 s1. The distribution of helium with depth is a broad bellshaped curve with a maximum in the helium concentration occurring at a depth, d0, corresponding to the mean projected range, Rp, for the incident ions. In this case d0 calculated using TRIM 95 [35] is 225 nm. The calculated damage profile (in the form of interstitial–vacancy pairs) has a similar bell-shape. However, it is centred on a depth that is somewhat less than d0. To investigate bubble structures at different depths below the surface a depth-profiling technique is used which is based on eroding the implanted
3.1.1. General The dense-packed planes in a bcc structure are {1 1 0}. Bubble ordering on to these planes was a pronounced feature in all specimens investigated. SAD patterns taken from the ordered bubble arrays were dominated by {1 1 0} reflections. The separation of these reflections was used to calculate the interplanar spacing, d110, of the {1 1 0} bubble planes. Measurements of bubble interplanar spacings made on direct micrographs were then used to check the results from diffraction. For a fully ordered bcc structure the lattice constant, a‘, is related to the interplanar spacing, duvw, by: a‘ = duvw(u2 + v2 + w2)0.5. The subscript ‘ refers to the particular set of planes, here (uvw), whose spacing has been used to determine a‘. In the following the subscripts m and b refer to the crystal matrix and the bubble array respectively.
2.2. TEM examination Thinned specimens prepared parallel with the foil surface were examined using TEM in a Philips EM 420 120 keV electron microscope. Bubbles were generally photographed in over-focus conditions. SAD patterns were taken with the electron beam direction (B) along a principal zone axis in the metal. Results were obtained for the following zone axes covering many different grains in a wide variety of different specimens – [0 0 1], [0 1 1], [1 1 1] and [1 1 3]. A variety of implantation conditions (temperature, helium flux and energy) were included in the study. Also specimens were prepared from several different depths below the implanted surface. The spacings of bubble planes in the ordered bubble arrays were deduced from the SAD patterns. Results were checked by measuring interplanar bubble spacings in bright-field micrographs. 2.3. Method of analysis The patterns of projected bubble images in bright-field micrographs of bubble arrays are generally too complicated to allow the presence of variant structures to be determined directly. The best evidence comes from SAD patterns taken within a single grain. Micrographs can then be used to provide a useful confirmation of structures deduced from diffraction patterns and allow the spatial distribution of any structural variants present in the array to be determined.
3.1.2. B = h1 1 1im Helium bubble images, typical of the many specimens examined with B = h1 1 1im in the crystal matrix, are shown
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It should be noted that the occurrence of 1 1 0 reflections from dense-packed planes, taken in isolation, does not necessarily mean the bubbles are ordered in three-dimensions. Such reflections could simply result from bubbles ordered on to dense-packed planes without necessarily being ordered within the planes themselves. In terms of this model the 1 1 0 reflections in the different directions would then come from different domains in the bubble structure. Within any given domain the bubbles would be ordered on to just one of the set of 1 1 0 planes normal to the electron beam. In contrast, the reflections from low-density planes (since they are defined by the intersection of dense-packed planes in the same domain) show that the local bubble ordering is fully three-dimensional [23,27]. This is taken up in more detail in a later section.
Fig. 1. Bright-field TEM micrograph (over-focus contrast) of the bubble structure in Mo following He ion-implantation (conditions given in the text). The thinned section was taken parallel with the helium-implanted surface. The electron beam is close to [1 1 1] in the Mo matrix. A section of the diffraction pattern (reverse contrast), showing a pair of 1 1 2 matrix spots and three pairs of satellites at 60 around the 0 0 0 transmitted beam, is included in the lower part of the figure. These satellites are shown in the enlarged pattern inset in the top right of the micrograph in approximately the correct orientation relative to the micrograph.
in the bright-field micrograph of Fig. 1. The grain orientation is {1 1 1}. The average bubble diameter measured from the micrograph is (1.81 ± 0.05) nm. Considerable bubble ordering is evident with bubble rows (which are the traces of the dense-packed bubble planes) parallel to each of the three matrix Tr{1 1 0} directions. The average interplanar spacing, d110, measured from the micrograph as the row spacing, is (4.3 ± 0.2) nm. A section of the SAD pattern is shown in the bottom part of the figure. In the original negative satellite reflections are evident around both the 0 0 0 transmitted beam and around the 1 2 1 matrix spot, however the latter satellites have not survived reproduction here. The spots in the superlattice pattern are assigned 1 1 0 indices because they are in line with the 1 1 0 matrix spots (not shown) and in the positions expected for diffraction from the dense-packed {1 1 0} bubble planes seen edge on in the micrograph. The value of d110 deduced from the radial spacing of the satellite reflections around the 0 0 0 transmitted beam is (4.5 ± 0.1) nm. The spacing of the 1 1 0b satellites around the matrix reflections thus gives a value in good agreement with that found from bright-field micrographs. The pattern of projected bubble images in brightfield micrographs suggests that the bubble array is at least partially ordered in three-dimensions in cases such as Fig. 1 but firm diffraction evidence is lacking. If the bubble array were a fully formed bcc lattice then the corresponding lattice constants, a110, based on the above d spacings are (6.1 ± 0.3) nm and (6.4 ± 0.1) nm respectively.
3.1.3. Other examples In the course of this and related work a large number of molybdenum specimens have been investigated. Results representing a variety of zone axes, grain orientations, implantation conditions and depths below the implanted surface have been obtained. Many of these examples exhibit diffraction patterns with satellites only from dense-packed bubble planes as in Fig. 1. The most frequently encountered examples of this type are summarised in the schematic diffraction patterns of Fig. 2. Again, whereas the diffraction results provide conclusive evidence only of ordering on to 1 1 0 planes and not of ordering within these planes, the patterns of bubble images in the associated bright-field micrographs suggest that there is at least some degree of ordering within the planes themselves. 3.2. 3-D bubble ordering 3.2.1. General In a fully three-dimensional lattice the intersection of the {1 1 0} dense-packed planes will define planes in other directions. These planes have lower (areal) density and closer spacing than the {1 1 0} planes so the corresponding allowed reflections in other directions are expected to be of low intensity. In fact allowed reflections in directions other than h1 1 0i have been identified only in the bcc metal molybdenum and in just one foil – a pre-thinned TEM specimen implanted at two different energies (60 and 40 keV) to high fluence (2.0 and 8.0 · 1021 He ions m2) [2,23]. In that case, 1 1 2 reflections, as well as first- and second-order 1 1 0 spots, were observed around the 0 0 0 transmitted beam in an SAD pattern taken with B along h1 1 1im. In the present work we have found many examples of so-called disallowed reflections from low-density planes. See for example, Fig. 3. When these reflections occur they are usually around one (or more) of the matrix reflections rather than around the 0 0 0 transmitted beam. In contrast, the 1 1 0 reflections from bubble planes parallel with the matrix {1 1 0} planes are usually more pronounced around the 0 0 0 spot. Disallowed reflections are those requiring
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B = [111]m
[101]m [0 1 1]m
[110]m
000
121m
B = [001]m
[020]m [1 10]m
[110]m
1 10b
200m
000
[ 1 10] m
B = [001]m [ 200 ]m
11 0m
000
[011 ]m
B = [011]m [211 ]m
200 m
B = [113 ] m
Fig. 3. Bright-field TEM micrograph (over-focus contrast) of heliumbubble structure in Mo. The electron beam is close to [0 0 1] in the Mo matrix. A section of the diffraction pattern (reverse contrast), showing a pair of 2 0 0 matrix spots, and two pairs of satellites at 90 around the 0 0 0 transmitted beam, is included in the lower part of the figure. An enlarged pattern inset in the top right of the micrograph shows these satellites in approximately the correct orientation relative to the micrograph.
[ 2 11 ]m
the corresponding allowed reflections, because they occur at smaller scattering angles.
000 [110]m
110b
2 1 1m
000
Fig. 2. Schematic spot patterns showing diffraction satellites from bubble arrays in Mo. In these examples, which represent the most frequently occurring cases, satellites are observed only from the dense-packed bubble planes ({1 1 0} for a bcc lattice). The electron beam direction and key directions in the matrix are as indicated on each diagram.
indices that are prohibited under the usual selection rules for an ideal bcc lattice (for example, 1 0 0 rather than 2 0 0). There are several factors that can lead to a relaxation of these selection rules. They include double-diffraction [36], inhomogeneities in the bubble lattice (such as variations in the size, faceting and spacing of bubbles) and the limited extent of the bubble domains [15–17,29] in which the ordering occurs. The intensity of electron scattering from bubbles falls off rapidly with increasing scattering angle [37]. Consequently, apart from the residual inhibitions that result from the selection rules, disallowed reflections are favoured over
3.2.2. 3-D ordering – examples A common grain orientation in the molybdenum foils is {0 0 1}. A bright-field micrograph with B = [0 0 1]m in a (0 0 1) grain, is shown in Fig. 3. Again considerable bubble ordering is evident with bubble rows parallel to each of the two matrix Tr{1 1 0} directions. The average bubble diameter measured from the micrograph is (1.32 ± 0.05) nm. The average interplanar spacing, d110, measured as the row spacing, is (3.6 ± 0.1) nm. In the SAD pattern, satellite reflections are evident around both the 0 0 0 transmitted beam and around the matrix spots. The satellite spots in the superlattice pattern around 0 0 0 and the matrix spots that are in the same direction as one or other of the two sets of 1 1 0 matrix spots (not shown) excited at this zone, and in the positions expected for diffraction from the dense-packed {1 1 0} bubble planes seen edge-on in the micrograph, are assigned 1 1 0 indices. The radial spacing of these satellite reflections gives a value for d110 of (3.6 ± 0.1) nm in good agreement with that found from bright-field micrographs. If the bubble lattice is three-dimensional in character then the associated value of the lattice parameter is a‘ = a110 = (5.1 ± 0.2) nm. Also evident in the diffraction pattern is a satellite in the [0 2 0]m direction (i.e. up the page) on the 2 0 0m spot and a satellite in the ½0 2 0m direction (i.e. down the page) on the 2 0 0m spot. Their directions suggest that they should be assigned 2 0 0-type indices. However, their radial spacing
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[020]m
B = [001]m [110]m
[1 10]m
1 00b
2 00m
000
[1 10]m
B = [001]m [ 2 00]m 1 00b
[020]m
010b
11 0m
000
Fig. 4. Schematic spot patterns showing two examples of diffraction satellites from bubble arrays in Mo for an electron beam direction of B = [0 0 1]m. In both cases satellites are observed in h1 1 0im matrix directions around the 0 0 0 transmitted beam, but only in h1 0 0im matrix directions around matrix spots. This is discussed further in the text.
gives a value for the associated interplanar bubble spacing of duvw = (5.1 ± 0.2) nm. If the indices 0 2 0 are assigned to uvw then an associated lattice parameter of a020 = (10.2 ± 0.4) nm is obtained – a value that is not consistent with the value of a110 found above. However, if the normally disallowed 1 0 0-type indices are used instead, the lattice parameter is a010 = (5.1 ± 0.2) nm – a value in excellent agreement with a110. Consequently these satellites are indexed as 0 1 0 and 0 1 0, respectively. The presence of these satellites from low-density bubble planes provides conclusive evidence the bubble ordering is three-dimensional at least in local regions. Many similar demonstrations of three-dimensional ordering have been found for various zone axes and grain orientations. A commonly encountered example is where B = [0 0 1]m in an (0 0 1)m grain. Here the diffraction pattern commonly contains satellite splitting of types represented schematically in the two diagrams of Fig. 4. In both examples, the radial separation of the two pairs of satellites around the matrix reflections again requires the assignment of 1 0 0-type indices (rather than 2 0 0) to obtain consistency with the radial separation of the 1 1 0 satellite reflections around the 0 0 0 transmitted beam. No obvious correlation has been found between the frequency of occurrence of bubble arrays exhibiting three-dimensional structure and the depth below the implanted surface from which the specimen was taken. 4. Structural variants
Fig. 5. Bright-field TEM micrograph (over-focus contrast) of heliumbubble structure in Mo. The electron beam is close to [1 1 3] in the matrix. A section of the diffraction pattern (reverse contrast), showing a pair of 1 1 0 matrix spots and three pairs of satellites around the 0 0 0 transmitted beam is included in the lower part of the figure.
the diffraction of electrons from the gas-bubble superlattice. These anomalies were subsequently explained [12,15– 17,23,29] in terms of the presence of structural variant bubble lattices (also having the same symmetry as the host metal – i.e. fcc bubble lattice in an fcc host etc.) with orientations different from, but rational with, the orientation of the host metal. Guided by this background we have examined many diffraction patterns from ordered bubble arrays in molybdenum, obtained in this and in previous work, for diffraction anomalies that indicate the presence of structural variants. Only two conclusive examples have been found in molybdenum and these are presented below in Figs. 5 and 9. 4.2. Example 1: B ¼ ½1 1 3m 4.2.1. Bright-field TEM imaging Helium bubble images from a particular specimen examined with B ¼ ½1 1 3m in the crystal matrix, are shown in the bright-field micrograph of Fig. 5. The grain orientation is (1 1 3Þm . The average bubble diameter measured from the micrograph is (1.7 ± 0.2) nm. Considerable bubble ordering is evident with bubble rows (which are the traces of the dense-packed bubble planes) parallel to the Tr(1 1 0)m direction. ((1 1 0)m, is the only set of dense-packed matrix planes excited at this zone axis.) The average interplanar spacing, d110, measured from the micrograph as the row spacing, is (3.6 ± 0.1) nm.
4.1. Introduction In the case of the fcc metals and the bcc metal vanadium it was noted in previous work that there were anomalies in
4.2.2. Electron diffraction In the section of the SAD pattern shown in the lower part of Fig. 5, satellite reflections are evident around both
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the 0 0 0 transmitted beam and around both the matrix spots. The satellites around the 0 0 0 spot are shown in approximately the correct orientation with respect to the bright-field micrograph in the inset in the top right of the figure. Again the satellite reflections that are in line with the 1 1 0 matrix spots are assigned 1 1 0 indices. The value of d110 deduced from their radial spacing is (3.6 ± 0.1) nm, a value in good agreement with that found above. If the bubble array were a fully formed bcc lattice then the corresponding lattice constant, a110, is (5.1 ± 0.2) nm. The patterns of satellites around each of the three spots in the SAD pattern of Fig. 5 are shown enlarged in Fig. 6. The strongest satellites in the original negative are shown (each identified by a letter) in Fig. 6 and in the schematic diagram of Fig. 7. The satellite spots around 0 0 0 labelled c and d (the corresponding spots across a diameter through 0 0 0 are also well-defined in the original negative) are in h1 1 2i directions with respect to the host matrix. On the basis of their radial separation they are assigned the normally disallowed indices of 12 ð 21 1Þ and 12 ð 12 1Þ respectively, to give a lattice parameter that is consistent with that found from the radial separation of the two 1 1 0 satellites ((5.1 ± 0.2) nm). The two pairs of satellites labelled c and d are thus from lowdensity bubble planes that are parallel with the corresponding planes in the matrix. This demonstrates that the parts
Fig. 6. Three enlarged sections taken from the diffraction pattern (B ¼ ½ 1 1 3m ) of Fig. 5 showing in more detail the satellites around the matrix spots, 1 1 0 (left) and 1 1 0 (right), and around the 0 0 0 transmitted beam (centre).
B = [ 113 ] m
[ 33 2 ] m [ 211 ] m
a
b
[12 1 ] m d
c
e
f g
110 m
000
h 110 m
Fig. 7. Schematic diagram of the diffraction satellites appearing in Fig. 6. The satellites around the 0 0 0 transmitted beam are consistent with regions of bubble lattice in the matrix (m) orientation whereas the satellites a, b, e, f, g and h are attributed to a structural variant bubble lattice, /, which is rotated with respect to the matrix. (The pair of 1 1 0 satellites around the 0 0 0 beam are consistent with both m and /.)
of the bubble superlattice that are parallel to the host matrix are ordered in three-dimensions. The remaining satellites labelled with letters are clearly not consistent with a bubble lattice in the same orientation as the host metal matrix. The satellites labelled e and g are across a diameter and have the same radial separation from the 1 1 0m matrix spot. They lie at an angle of 45 with respect to [1 1 0]m. Their radial separation is consistent with the lattice parameter of the matrix-oriented bubble lattice if 1 0 0-type indices are assigned. Satellite f is orthogonal to, and at the same radial spacing, as e and g and so too is assigned 1 0 0-type indices. Satellite h is at an angle of 45 with respect to e–g and has a radial separation consistent with 1 1 0-type indices. The satellites e–h are thus consistent with a variant bubble lattice (which will be referred to as variant /) excited with B along [0 0 1]/. For this zone axis, the satellites are assigned indices referred to the variant lattice as follows: a and e – 1 0 0, b and f – 0 1 0, g – 1 0 0 and h – 1 1 0. In summary to explain satellites appearing in the diffraction pattern requires a variant bubble lattice / with the following orientation: [0 0 1]/ coincides with ½ 1 1 3m ; [1 1 0]/ coincides with [1 1 0]m, and ½1 1 0/ coincides with ½3 3 2m . 4.2.3. Projected bubble images Inspection of bright-field images such as Fig. 5 shows that whereas particular bubble rows are often well defined over distances 200 nm or greater along Tr(1 1 0) ð¼ ½3 3 2m Þ, the pattern of bubble images along these rows often changes markedly over distances as short as 20 nm. Schematic diagrams of the pattern of bubble images projected on to the ð1 1 3Þ grain-surface are shown in Fig. 8 for three types of ordered bcc bubble array. We adopt a coordinate system where the x-axis is across the page (positive to the right) and the y-axis is up the page. The z-axis is normal to the plane of the diagram and is directed upward out of the page. Following the usual convention B ¼ ½ 1 1 3m (= [0 0 1]/) is along the (positive) z direction. The upper pattern (shaded circles) is for a bubble lattice in the matrix orientation; the middle pattern (open circles) is for the structural variant bubble lattice in the orientation defined earlier. The x-axis coincides with Trð3 3 2Þm ð¼ ½1 1 0m Þ and with Trð1 1 0Þ/ ð¼ ½1 1 0/ Þ; the y-axis coincides with Trð1 1 0Þm ð¼ ½3 3 2m Þ and with Trð1 1 0Þ/ ð¼ ½1 1 0/ Þ. Comparing the idealised patterns of Fig. 8 with the patterns of projected images in the micrograph of Fig. 5 leads to the following conclusions. There are many regions 20 nm across where the patterns match that for the matrix-oriented bubble lattice alone. These regions can be readily identified in the original negative by the intersection of bubble rows in the primary row direction (i.e. Tr(1 1 0)m), with rows in either or both the Trð2 1 1Þm or Trð1 2 1Þm directions. (In Fig. 8, Trð2 1 1Þm is 25 above the X-axis and Trð1 2 1Þm is 25 below the X-axis; in Fig. 5 Trð2 1 1Þm is 65 (clockwise) from Tr(1 1 0)m and Trð1 2 1Þm is at 115.) The intersection of these bubble rows confirms the previous diffraction evidence that the
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Fig. 9. Bright-field TEM micrograph (under-focus contrast) of heliumbubble structure in Mo. The electron beam is close to [1 1 1] in the matrix. The lower part of the figure is a section of the diffraction pattern (reverse contrast) showing a pair of 1 1 0 matrix spots and three pairs of satellites at 60 around the 0 0 0 transmitted beam.
Fig. 8. Schematic bubble projections on the ð1 1 3Þ grain-surface plane of a bubble lattice in the matrix orientation (upper figure – shaded circles) and of the structural variant / discussed in the text (middle figure – open circles). Tr(1 1 0)m and Tr(1 1 0)b are both parallel with the long side of the diagram. A superposition of the two patterns is shown in the bottom figure.
matrix-oriented bubble lattice is genuinely three-dimensional. Whereas regions where the variant bubble lattice occurs alone have not been found, there are many regions where the pattern of bubble images along bubble rows in Tr(1 1 0) directions exhibits a confused overlapping appearance similar to that in the composite pattern of Fig. 8. In these regions it is concluded that the variant structure does not occur alone but occurs only immediately above or below (in terms of the z-axis) a patch of matrix-oriented bubble lattice. 4.3. Example 2: B ¼ ½ 1 1 1m 4.3.1. Bright-field TEM imaging The bright-field micrograph of Fig. 9 shows a bubble array examined with B (approximately) along ½ 1 1 1m in a ð 1 1 1Þm grain. The average bubble diameter measured from the micrograph is (1.51 ± 0.05) nm. Considerable bubble ordering is evident with bubble rows (which are the traces of the dense-packed bubble planes) parallel to each of the three Tr{1 1 0}m directions lying in the plane of the figure. (For a h1 1 1i zone-axis there are three sets of {1 1 0}m excited.) The average interplanar bubble spacing, d110,
measured from the micrograph, is (4.1 ± 0.3) nm. There are many regions evident where bubble rows in different Tr{1 1 0} directions intersect. This shows that the bubble lattice aligned parallel with the host metal matrix is three-dimensional in character. The associated lattice constant, a110, is (6.0 ± 0.4) nm. 4.3.2. Electron diffraction A section of the SAD pattern is shown in the lower part of Fig. 9. Satellite reflections are evident around both the 0 0 0 transmitted beam and around both the matrix spots. In the inset in the top right of the micrograph the satellites around the 0 0 0 spot are shown in approximately the correct orientation with respect to the bright-field micrograph. Again the satellite reflections that are in line with the 1 1 0 matrix spots are assigned 1 1 0 indices. The value of d110 deduced from their radial spacing is (4.5 ± 0.1) nm and the associated lattice constant, a110, is (6.4 ± 0.2) nm – values which are in satisfactory agreement with those found above. The patterns of satellites around each of the three spots in the SAD pattern of Fig. 9 are shown enlarged in Fig. 10. The directions of the satellites relative to matrix directions are indicated in the schematic diffraction pattern of Fig. 11. The three pairs of satellite spots around 0 0 0 are in h1 1 0i matrix directions and are assigned the appropriate indices (the first three spots, anticlockwise starting from the (positive) x-axis are thus 1 1 0m, 0 1 1m and 10 1m . On the 1 1 0m matrix spot there is a single strong satellite in the ½1 1 2m direction. Similarly there is a single strong satellite in the ½1 1 2m direction, on the 1 1 0m matrix spot. These two satellites have the same radial separation – a spacing which corresponds to an interplanar bubble
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Fig. 10. Three enlarged sections taken from the diffraction pattern (B ¼ ½ 1 1 1) of Fig. 9 showing in more detail the satellites around the matrix spots, 1 1 0m (left) and 1 1 0m (right), and around the 0 0 0 transmitted beam (centre).
B = [ 111] m [ 101 ] m
[1 1 2] m [ 011] m
001 β
001β 1 10 m
000
ordered bcc bubble array, are shown in Fig. 12. Again the coordinate system has the x-axis across the page, the y-axis up the page and the z-axis directed upward out of the page. 1 1 0b ) is Following the usual convention B ¼ ½1 1 1m ð¼ ½ along the (positive) z direction. The upper pattern (shaded circles) is for a bubble lattice in the matrix orientation; the middle pattern (open circles) is for the structural-variant bubble lattice. The x-axis corresponds to Trð 11 2Þm ð¼ ½1 1 0m Þ and to Trð0 0 1Þb ð¼ ½1 1 0b Þ; the y-axis corresponds to Trð1 1 0Þm ð¼ ½1 1 2m Þ and to Trð1 1 0Þb ð¼ ½0 0 1b Þ. Comparing the idealised patterns of Fig. 12 with the patterns of projected images in the micrograph of Fig. 9 leads to the following conclusions. For B ¼ ½1 1 1m there are three sets of dense-packed matrix planes orthogonal to the surface: (1 1 0)m, (0 1 1Þm and (1 0 1Þm . These planes lie at 60 with respect to each other. In Fig. 9 bubble rows are evident in each of the Tr(1 1 0)m, Trð0 1 1Þm and Trð1 0 1Þm directions. Regions 20 nm across can be identified where bubble rows in at least two of these trace directions intersect to indicate the presence of fully threedimensional matrix-oriented bubble lattice. Similarly there
110 m
Fig. 11. Schematic diagram of the diffraction satellites appearing in Fig. 10. The origin and indexing of the satellites are discussed in the text. The three pairs of satellites around the 0 0 0 transmitted beam are consistent with regions of bubble lattice in the matrix orientation, whereas the satellites on the matrix spots 1 1 0m and 1 1 0m (0 0 1b and 0 0 1b respectively) are attributed to a structural variant bubble lattice, b, which is rotated with respect to the matrix.
spacing of d = (6.6 ± 0.3) nm. It is clear that satellites in a h1 1 2im direction with this radial spacing can only arise from a structural variant (referred to as variant b). To obtain a lattice parameter that is consistent with that based on the 1 1 0 satellite reflections around 0 0 0, requires the assignment of 1 0 0-type indices – here we choose 0 0 1b for the spot in the ½ 11 2m direction and 0 0 1b for the spot in the opposite direction. The lowest index bcc zone axis giving rise to the presence of only one pair of 0 0 1 satellites (as in the structural variant diffraction pattern) is h1 1 0i. For excitation of a lattice at this zone axis a single pair of 1 1 0 satellites would be expected in a direction orthogonal to the 1 0 0 satellites. 1 1 0 are allowed reflections for a bcc lattice and so if present would be expected to be found around the 0 0 0 transmitted beam rather than around the matrix spots. If this were the case then the single pair of 1 1 0 reflections from the structural variant would coincide with the 1 1 0 reflections from the matrix-oriented bubble lattice and so would not be detectable as a separate spot pair. Under this assumption the orientation of the structural variant is as follows: B ¼ ½ 1 1 0b ¼ ½ 1 1 1m ; ½0 0 1b ¼ ½1 1 2m and [1 1 0]b = [1 1 0]m. 4.3.3. Projected bubble images Schematic diagrams of the pattern of bubble images projected on to the ð 1 1 1Þ grain-surface, for three types of
Fig. 12. Schematic bubble projections on the ð1 1 1Þ grain-surface plane of a bubble lattice in the matrix orientation (upper figure – shaded circles) and of the structural variant bubble lattice b discussed in the text (middle figure – open circles). Trð1 1 0Þm and Tr(1 1 0)b are both parallel with the long side of the diagram. The bottom section of the figure is a superposition of the two patterns aligned on the three bubble projections indicated with dots.
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are some regions where the pattern of bubble images is consistent with the presence of the structural variant lattice alone. This is evidenced by the presence of intersecting orthogonal bubble rows – one of the row directions is along Tr(1 1 0)m and the other, which exhibits more widely spaced bubbles, is along Trð 11 2Þm . Finally, there are many regions where the pattern of bubble images is consistent with the composite pattern of Fig. 12. These regions are characterised by well defined bubble rows in the Tr(1 1 0)m (= Tr(1 1 0)b) direction with a high density of confused bubble images along the rows themselves as predicted by Fig 12. In these regions it is concluded that the variant structure occurs immediately above or below (in terms of the z-axis) a patch of matrix-oriented bubble lattice having the same lattice constant. 5. Discussion and conclusions Several types of ordered bubble structures have been identified in helium-implanted molybdenum. Bubbles ordered on 1 1 0 matrix planes are a dominant feature. The common occurrence of diffraction satellites from bubble planes parallel with lower density matrix planes shows that there are many regions where the bubbles are fully ordered on a three-dimensional bcc bubble lattice parallel with the host matrix. Also in two cases the presence in the bubble array of structural variant lattices has been demonstrated. These variant lattices are identical to the bubble lattice parallel with the host matrix except for orientation – the variants have major axes aligned along noncorresponding (but rational) low-index directions in the host matrix. Although many bubble arrays have been examined, the sampling procedure and microscope examination are not sufficiently precise to allow reliable conclusions to be drawn about the frequency of occurrence of the various types of ordered arrays. However, it seems clear that structural variants occur only rarely in molybdenum – only two conclusive examples having been found in the several hundred cases surveyed. The results obtained are consistent with the following model of bubble development. Bubbles initially form in a random array and, with continued implantation, order first on to 1 1 0 planes. At a later stage, bubbles order within these planes (without change in the interplanar spacing) to form a fully three-dimensional bubble lattice parallel with the matrix. It is at this stage that the bcc symmetry and the lattice parameter of the bubble lattice are determined. It is only in a final stage, corresponding to a local helium concentration of 10–15 at.%, that the bubbles in some regions (some five to ten interbubble-spacing across) move to form a lattice in a structural variant orientation. The structural variant lattices retain the same bcc symmetry and lattice parameter as the matrix-oriented lattice. The /(b) and matrix-oriented bubble lattices have a common set of dense-packed {1 1 0} bubble planes. It is probable (but not completely certain) that this common
333
set of 1 1 0 planes is orthogonal to the grain surface in both cases. (The uncertainty arises because of the possibility that an undetected foil curvature might have caused a departure from ‘‘flat foil’’ conditions in the TEM examination.) For both / and b the geometrical transformation that carries the bubble lattice in a local region from an orientation parallel with the matrix to that of the variant is a rotation of order 30 (25.2 and 35.2 respectively) about the normal to the common set of 1 1 0 planes. We propose that the bubble movements needed to effect this transformation take place solely on these planes. (Note that the superficial bubble density on a given 1 1 0 plane in the set remains unchanged in going from the matrix-oriented bubble lattice to either / or b.) It can be shown that individual bubbles in small domains need move only small distances (comparable with the bubble diameter) to effect this transformation. The features of the structural variants observed here in molybdenum bear close similarities with those found previously in the fcc metal copper [12,15–17]. In that case the change in orientation of sections of the bubble lattice to form structural variants was attributed to effects of stress in the implanted layer. Work is in progress to determine whether a similar explanation can be applied here. For use as a photonic crystal, the ideal gas-bubble superlattice in molybdenum would be uniform throughout the depth of the foil and would contain no structural variants. The exact conditions needed to promote the formation of a fully three-dimensional bubble lattice oriented solely parallel with the host matrix are not clear. However some factors can be identified as a guide to future work. In an early study Mazey et al. [2] found the most complete bubble superlattice parallel with the matrix yet found. This case was analysed in more detail in a later paper [23], where it was thought significant that the implantation was into a pre-thinned foil and involved two different helium implantation energies. It was proposed [23] that there were two important factors: (i) the more uniform distribution of helium over the depth of the foil and (ii) the prevention of the build-up of high levels of stress in the implanted layer because of the close proximity of the surfaces. Also in previous work it has been established that there is both a higher temperature limit [2] and a lower temperature threshold [27] for bubble superlattice formation in molybdenum of approximately 0.35Tm and 0.15Tm respectively. These results suggest the following implantation conditions could be the most favourable for producing a fully developed, uniform bubble superlattice parallel to the molybdenum crystal lattice. The implantation should be at a temperature of 0.25Tm (700 K) into an ultra-thin foil (<100 nm in thickness) using a helium implantation protocol (energies and doses) chosen to deposit helium uniformly with depth to a final helium loading of 10–15 at.%. Under these conditions it could be expected that imperfections in the bubble lattice in the form of structural variants will be rare. However, there will still be imperfections in the form of variations in bubble size, in the degree of bubble faceting, and in the degree of bubble ordering.
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These residual imperfections could be expected to reduce the reflection coefficient for soft X-rays obtained in practice below that calculated on the basis of an ideal lattice. Acknowledgements This work was performed under research contracts to the New Zealand Foundation for Research, Science and Technology (e.g. VICX0206). We thank Dr P.W. Gilberd and C. Varoy for useful discussions and research assistance and Y. Morrison and J. Johnson for technical assistance. References [1] S.L. Sass, B.L. Eyre, Phil. Mag. 27 (1973) 1447. [2] D.J. Mazey, B.L. Eyre, J.H. Evans, S.K. Erents, G.M. McCracken, J. Nucl. Mater. 64 (1977) 145. [3] P.B. Johnson, D.J. Mazey, Nature 276 (1978) 595. [4] P.B. Johnson, D.J. Mazey, Nature 281 (1979) 359. [5] P.B. Johnson, D.J. Mazey, J. Nucl. Mater. 93/94 (1980) 712. [6] P.B. Johnson, D.J. Mazey, Radiat. Eff. 53 (1980) 195. [7] W. Ja¨ger, J. Roth, J Nucl. Mater. 93–94 (1980) 756. [8] W. Ja¨ger, J. Roth, Nucl. Instr. and Meth. 182–183 (1981) 975. [9] P.B. Johnson, D.J. Mazey, J. Nucl. Mater. 111–112 (1982) 681. [10] P.B. Johnson, D.J. Mazey, J.H. Evans, Radiat. Eff. 78 (1983) 157. [11] H. Van Swijgenhoven, G. Knuyt, J. Vanoppen, L.M. Stals, J. Nucl. Mater. 114 (1983) 157. [12] P.B. Johnson, D.J. Mazey, J. Nucl. Mater. 127 (1985) 30. [13] D.J. Mazey, J.H. Evans, J. Nucl. Mater. 138 (1986) 16. [14] J.H. Evans, D.J. Mazey, J. Nucl. Mater. 138 (1986) 176. [15] P.B. Johnson, A.L. Malcolm, D.J. Mazey, Nature 329 (1987) 316. [16] P.B. Johnson, A.L. Malcolm, D.J. Mazey, J. Nucl. Mater. 152 (1988) 69. [17] P.B. Johnson, A.L. Diprose, D.J. Mazey, J. Nucl. Mater. 158 (1988) 108.
[18] B.A. Kalin, S.N. Korshunov, I.I. Chernov, A.V. Markin, I.V. Reutov, V.N. Chernikov, A.P. Zacharov, V.M. Gureev, M.I. Guseva, J. Nucl. Mater. 161 (1989) 231. [19] J.H. Evans, A.J.E. Foreman, R.J. McElroy, J. Nucl. Mater. 168 (1989) 340. [20] P.B. Johnson, D.J. Mazey, J. Nucl. Mater. 170 (1990) 290. [21] P.B. Johnson, Bubble lattices in metals, in: J.H. Evans, S.E. Donnelly (Eds.), Fundamental Aspects of Inert Gases in Solids, Plenum, New York, 1991, p. 167. [22] P.B. Johnson, K.J. Stevens, R.W. Thomson, Nucl. Instr. and Meth. B 62 (1991) 218. [23] P.B. Johnson, D.J. Mazey, J. Nucl. Mater. 218 (1995) 273. [24] P.B. Johnson, K.L. Reader, R.W. Thomson, J. Nucl. Mater. 231 (1996) 92. [25] P.B. Johnson, P.W. Gilberd, Nucl. Instr. and Meth. B 127–128 (1997) 734. [26] K.J. Stevens, P.B. Johnson, J. Nucl. Mater. 246 (1997) 17. [27] F.E. Lawson, P.B. Johnson, J. Nucl. Mater. 252 (1998) 34. [28] P.B. Johnson, R.W. Thomson, K.L. Reader, J. Nucl. Mater. 273 (1999) 117. [29] P.B. Johnson, P.W. Gilberd, Y. Morrison, C.R. Varoy, Mod Phys. Lett. B 15 (28–29) (2001) 1391. [30] P.B. Johnson, A. Markwitz, P.W. Gilberd, Adv. Mater. 13 (2001) 997. [31] A. Markwitz, P.B. Johnson, P.W. Gilberd, G.A. Collins, Nucl. Instr. and Meth. B 190 (2002) 718. [32] P.B. Johnson, V.J. Kennedy, A. Markwitz, C. Varoy, Nucl. Instr. and Meth. B 206 (2003) 1056. [33] V.J. Kennedy, P.B. Johnson, A. Markwitz, C.R. Varoy, N. Dytlewski, K.T. Short, Nucl. Instr. and Meth. B 210 (2003) 543. [34] D.A. Ksenzov, E.P. Petrov, Presented at the Saratov Fall Meeting 2000: Laser Physics and Photonics and Optical Techniques in Biophysics and Medicine, Saratov, Russia, 3–7 October 2000. [35] J.F. Ziegler, P. Biersack, U. Littmark, The Stopping and Range of Ions in Matter, Pergamon, New York, 1985. [36] P.E. Champness, Electron diffraction in the electron microscope, Royal Microscopical Society, Microscopy Handbook, Vol. 47, BIOS Scientific Publishers Ltd., Oxford, 2001. [37] K.J. Stevens, Ph.D. Thesis, Victoria University of Wellington, Wellington, New Zealand, 1993.
Nuclear Instruments and Methods in Physics Research B 243 (2006) 325–334 www.elsevier.com/locate/nimb
Helium gas-bubble superlattice formation in molybdenum P.B. Johnson
a,b,*
, Fenella Lawson
a
a b
Victoria University of Wellington, School of Chemical and Physical Sciences, P.O. Box 600, Wellington, New Zealand The MacDiarmid Institute for Advanced Materials and Nanotechnology, Victoria University of Wellington, New Zealand Received 24 August 2005 Available online 28 November 2005
Abstract The gas-bubble superlattice is a striking phenomenon where small helium bubbles of uniform size are regularly arranged on a space lattice having the same symmetry as the crystal lattice of the host metal but with a lattice spacing some twenty times greater. Typically, bubble diameters are 2 nm and bubble concentrations are 1 · 1025 m3. Superlattices have been observed in all three metal types: bcc, fcc and hcp. The most common superlattice orientation is parallel with the host lattice. However, for the fcc metals, and the bcc metal vanadium the superlattice also contains structural variants – regions where the ordered bubble array has an orientation that is rational with, but different from, that of the crystal lattice of the host metal. Here we report on the structure of the bubble superlattice in the bcc metal molybdenum. Recently the proposal has been made that this superlattice could act as a photonic crystal for soft X-rays. An assessment is made of the quality of the superlattice in molybdenum in relation to this proposal. 2005 Elsevier B.V. All rights reserved. PACS: 81.07.b; 61.82.d; 61.82.Bq; 68.37.Lp; 68.65.Cd; 61.10.Eq Keywords: Helium gas-bubble superlattice; Molybdenum; Structural variants; Photonic crystal
1. Introduction The spatial ordering of gas bubbles, produced in metals by the ion implantation of inert gases such as helium, has been the subject of many studies based on transmission electron microscopy (TEM) [1–29]. The gas-bubble superlattice stage in most metals occurs at local helium concentrations of 10–15 at.% in the matrix. The superlattice comprises small helium bubbles of uniform size (2 nm diameter) ordered on a space lattice having the same symmetry as the host. The bubble lattice spacing is typically some twenty times that of the crystal lattice. Superlattice formation has been observed in metals representing all three major crystal types fcc, bcc and hcp and so seems to be a universal response for helium implantations at * Corresponding author. Address: Victoria University of Wellington, School of Chemical and Physical Sciences, P.O. Box 600, Wellington, New Zealand. Tel.: +64 4 4635966; fax: +64 4 4635237. E-mail address: [email protected] (P.B. Johnson).
0168-583X/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2005.09.021
temperatures in the region of 0.2Tm where Tm is the melting temperature of the metal. Much of the original impetus for research into bubble ordering came from a recognition that it could provide a unique insight into the micro-structural processes underlying more general cases of materials modification by ion implantation. More recently, interest has been stimulated by the possible technological applications of heliumimplanted surfaces. For example, the highly swollen nanoporous layers that form beyond the superlattice stage show promise for applications in areas such as catalysis and biocompatible materials [30–33]. The unique characteristics of these nanoporous structures are thought to be ‘‘seeded’’ by the highly uniform superlattice stages that precede them. Further, the superlattice stage of bubble development, itself, as a three-dimensionally ordered structure, could find application in optoelectronics [30]. For example it has been proposed that the superlattice in molybdenum could act as a photonic crystal for soft X-rays [34]. In a photonic crystal, the dielectric constant (generally complex-valued) has
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spatial periodicity in one, two or three dimensions. Computations using the electromagnetic transfer matrix method have shown that the superlattice in molybdenum, under the assumption of a homogeneous superlattice oriented parallel with the matrix, should act as a selective reflector of soft X-rays with wavelengths close to 10 nm. Information on the gas-bubble superlattice has come mainly from TEM bright-field micrographs and from TEM selected-area electron diffraction (SAD) patterns taken from the ordered arrays. A review of research on bubble ordering prior to 1990 can be found in [21]; more recent research is covered in [22–27]. Early research centred on the bcc metal molybdenum, see for example [1,2]. The results from these and subsequent studies [23,27] established that the superlattice in molybdenum had bcc symmetry and was at least largely parallel with the host crystal lattice. For the fcc metals the bubble superlattice always contains structural variants – i.e. regions where the ordered bubble array has an orientation that is rational with, but different from, that of the crystal lattice of the host metal. Until recently, the evidence suggested that in the bcc metals structural variants of the type found in the fcc metals did not form and that the bubble lattice aligned solely parallel with the host crystal [23]. However, bubble variant lattices have now been demonstrated to occur in the bcc metal vanadium [29], and this has renewed interest in the detailed structure of the gas-bubble superlattice in the other bcc metals. In the present study, TEM is used to investigate the structure of the gas-bubble superlattice in molybdenum with the particular aim of testing the previous assumption of a homogeneous superlattice oriented solely parallel with the matrix. This is referred to as the m orientation.
surface to different depths by means of ion-beam milling, prior to chemical back-thinning the foil to produce electron transparent specimens [9,27].
2. Experimental procedure
3. Results and discussion
2.1. Implantation conditions
3.1. Planar ordering
Unless otherwise stated the following procedure was used to prepare and examine specimens. Molybdenum foils, 99.8% pure, 25 lm thick, from Goodfellow Metals were diamond polished in stages down to 1 lm diamond paste and annealed at 1100 C for 2 h in vacuum. Most of the samples were implanted with 160 keV helium ions at 60 to the surface normal of the foil. During the implantation the sample temperature was maintained near 300 C (0.2Tm). Typical ion doses were 5 · 1021 He+ ions m2 and dose rates were 2 · 1017 He+ ions m2 s1. The distribution of helium with depth is a broad bellshaped curve with a maximum in the helium concentration occurring at a depth, d0, corresponding to the mean projected range, Rp, for the incident ions. In this case d0 calculated using TRIM 95 [35] is 225 nm. The calculated damage profile (in the form of interstitial–vacancy pairs) has a similar bell-shape. However, it is centred on a depth that is somewhat less than d0. To investigate bubble structures at different depths below the surface a depth-profiling technique is used which is based on eroding the implanted
3.1.1. General The dense-packed planes in a bcc structure are {1 1 0}. Bubble ordering on to these planes was a pronounced feature in all specimens investigated. SAD patterns taken from the ordered bubble arrays were dominated by {1 1 0} reflections. The separation of these reflections was used to calculate the interplanar spacing, d110, of the {1 1 0} bubble planes. Measurements of bubble interplanar spacings made on direct micrographs were then used to check the results from diffraction. For a fully ordered bcc structure the lattice constant, a‘, is related to the interplanar spacing, duvw, by: a‘ = duvw(u2 + v2 + w2)0.5. The subscript ‘ refers to the particular set of planes, here (uvw), whose spacing has been used to determine a‘. In the following the subscripts m and b refer to the crystal matrix and the bubble array respectively.
2.2. TEM examination Thinned specimens prepared parallel with the foil surface were examined using TEM in a Philips EM 420 120 keV electron microscope. Bubbles were generally photographed in over-focus conditions. SAD patterns were taken with the electron beam direction (B) along a principal zone axis in the metal. Results were obtained for the following zone axes covering many different grains in a wide variety of different specimens – [0 0 1], [0 1 1], [1 1 1] and [1 1 3]. A variety of implantation conditions (temperature, helium flux and energy) were included in the study. Also specimens were prepared from several different depths below the implanted surface. The spacings of bubble planes in the ordered bubble arrays were deduced from the SAD patterns. Results were checked by measuring interplanar bubble spacings in bright-field micrographs. 2.3. Method of analysis The patterns of projected bubble images in bright-field micrographs of bubble arrays are generally too complicated to allow the presence of variant structures to be determined directly. The best evidence comes from SAD patterns taken within a single grain. Micrographs can then be used to provide a useful confirmation of structures deduced from diffraction patterns and allow the spatial distribution of any structural variants present in the array to be determined.
3.1.2. B = h1 1 1im Helium bubble images, typical of the many specimens examined with B = h1 1 1im in the crystal matrix, are shown
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It should be noted that the occurrence of 1 1 0 reflections from dense-packed planes, taken in isolation, does not necessarily mean the bubbles are ordered in three-dimensions. Such reflections could simply result from bubbles ordered on to dense-packed planes without necessarily being ordered within the planes themselves. In terms of this model the 1 1 0 reflections in the different directions would then come from different domains in the bubble structure. Within any given domain the bubbles would be ordered on to just one of the set of 1 1 0 planes normal to the electron beam. In contrast, the reflections from low-density planes (since they are defined by the intersection of dense-packed planes in the same domain) show that the local bubble ordering is fully three-dimensional [23,27]. This is taken up in more detail in a later section.
Fig. 1. Bright-field TEM micrograph (over-focus contrast) of the bubble structure in Mo following He ion-implantation (conditions given in the text). The thinned section was taken parallel with the helium-implanted surface. The electron beam is close to [1 1 1] in the Mo matrix. A section of the diffraction pattern (reverse contrast), showing a pair of 1 1 2 matrix spots and three pairs of satellites at 60 around the 0 0 0 transmitted beam, is included in the lower part of the figure. These satellites are shown in the enlarged pattern inset in the top right of the micrograph in approximately the correct orientation relative to the micrograph.
in the bright-field micrograph of Fig. 1. The grain orientation is {1 1 1}. The average bubble diameter measured from the micrograph is (1.81 ± 0.05) nm. Considerable bubble ordering is evident with bubble rows (which are the traces of the dense-packed bubble planes) parallel to each of the three matrix Tr{1 1 0} directions. The average interplanar spacing, d110, measured from the micrograph as the row spacing, is (4.3 ± 0.2) nm. A section of the SAD pattern is shown in the bottom part of the figure. In the original negative satellite reflections are evident around both the 0 0 0 transmitted beam and around the 1 2 1 matrix spot, however the latter satellites have not survived reproduction here. The spots in the superlattice pattern are assigned 1 1 0 indices because they are in line with the 1 1 0 matrix spots (not shown) and in the positions expected for diffraction from the dense-packed {1 1 0} bubble planes seen edge on in the micrograph. The value of d110 deduced from the radial spacing of the satellite reflections around the 0 0 0 transmitted beam is (4.5 ± 0.1) nm. The spacing of the 1 1 0b satellites around the matrix reflections thus gives a value in good agreement with that found from bright-field micrographs. The pattern of projected bubble images in brightfield micrographs suggests that the bubble array is at least partially ordered in three-dimensions in cases such as Fig. 1 but firm diffraction evidence is lacking. If the bubble array were a fully formed bcc lattice then the corresponding lattice constants, a110, based on the above d spacings are (6.1 ± 0.3) nm and (6.4 ± 0.1) nm respectively.
3.1.3. Other examples In the course of this and related work a large number of molybdenum specimens have been investigated. Results representing a variety of zone axes, grain orientations, implantation conditions and depths below the implanted surface have been obtained. Many of these examples exhibit diffraction patterns with satellites only from dense-packed bubble planes as in Fig. 1. The most frequently encountered examples of this type are summarised in the schematic diffraction patterns of Fig. 2. Again, whereas the diffraction results provide conclusive evidence only of ordering on to 1 1 0 planes and not of ordering within these planes, the patterns of bubble images in the associated bright-field micrographs suggest that there is at least some degree of ordering within the planes themselves. 3.2. 3-D bubble ordering 3.2.1. General In a fully three-dimensional lattice the intersection of the {1 1 0} dense-packed planes will define planes in other directions. These planes have lower (areal) density and closer spacing than the {1 1 0} planes so the corresponding allowed reflections in other directions are expected to be of low intensity. In fact allowed reflections in directions other than h1 1 0i have been identified only in the bcc metal molybdenum and in just one foil – a pre-thinned TEM specimen implanted at two different energies (60 and 40 keV) to high fluence (2.0 and 8.0 · 1021 He ions m2) [2,23]. In that case, 1 1 2 reflections, as well as first- and second-order 1 1 0 spots, were observed around the 0 0 0 transmitted beam in an SAD pattern taken with B along h1 1 1im. In the present work we have found many examples of so-called disallowed reflections from low-density planes. See for example, Fig. 3. When these reflections occur they are usually around one (or more) of the matrix reflections rather than around the 0 0 0 transmitted beam. In contrast, the 1 1 0 reflections from bubble planes parallel with the matrix {1 1 0} planes are usually more pronounced around the 0 0 0 spot. Disallowed reflections are those requiring
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B = [111]m
[101]m [0 1 1]m
[110]m
000
121m
B = [001]m
[020]m [1 10]m
[110]m
1 10b
200m
000
[ 1 10] m
B = [001]m [ 200 ]m
11 0m
000
[011 ]m
B = [011]m [211 ]m
200 m
B = [113 ] m
Fig. 3. Bright-field TEM micrograph (over-focus contrast) of heliumbubble structure in Mo. The electron beam is close to [0 0 1] in the Mo matrix. A section of the diffraction pattern (reverse contrast), showing a pair of 2 0 0 matrix spots, and two pairs of satellites at 90 around the 0 0 0 transmitted beam, is included in the lower part of the figure. An enlarged pattern inset in the top right of the micrograph shows these satellites in approximately the correct orientation relative to the micrograph.
[ 2 11 ]m
the corresponding allowed reflections, because they occur at smaller scattering angles.
000 [110]m
110b
2 1 1m
000
Fig. 2. Schematic spot patterns showing diffraction satellites from bubble arrays in Mo. In these examples, which represent the most frequently occurring cases, satellites are observed only from the dense-packed bubble planes ({1 1 0} for a bcc lattice). The electron beam direction and key directions in the matrix are as indicated on each diagram.
indices that are prohibited under the usual selection rules for an ideal bcc lattice (for example, 1 0 0 rather than 2 0 0). There are several factors that can lead to a relaxation of these selection rules. They include double-diffraction [36], inhomogeneities in the bubble lattice (such as variations in the size, faceting and spacing of bubbles) and the limited extent of the bubble domains [15–17,29] in which the ordering occurs. The intensity of electron scattering from bubbles falls off rapidly with increasing scattering angle [37]. Consequently, apart from the residual inhibitions that result from the selection rules, disallowed reflections are favoured over
3.2.2. 3-D ordering – examples A common grain orientation in the molybdenum foils is {0 0 1}. A bright-field micrograph with B = [0 0 1]m in a (0 0 1) grain, is shown in Fig. 3. Again considerable bubble ordering is evident with bubble rows parallel to each of the two matrix Tr{1 1 0} directions. The average bubble diameter measured from the micrograph is (1.32 ± 0.05) nm. The average interplanar spacing, d110, measured as the row spacing, is (3.6 ± 0.1) nm. In the SAD pattern, satellite reflections are evident around both the 0 0 0 transmitted beam and around the matrix spots. The satellite spots in the superlattice pattern around 0 0 0 and the matrix spots that are in the same direction as one or other of the two sets of 1 1 0 matrix spots (not shown) excited at this zone, and in the positions expected for diffraction from the dense-packed {1 1 0} bubble planes seen edge-on in the micrograph, are assigned 1 1 0 indices. The radial spacing of these satellite reflections gives a value for d110 of (3.6 ± 0.1) nm in good agreement with that found from bright-field micrographs. If the bubble lattice is three-dimensional in character then the associated value of the lattice parameter is a‘ = a110 = (5.1 ± 0.2) nm. Also evident in the diffraction pattern is a satellite in the [0 2 0]m direction (i.e. up the page) on the 2 0 0m spot and a satellite in the ½0 2 0m direction (i.e. down the page) on the 2 0 0m spot. Their directions suggest that they should be assigned 2 0 0-type indices. However, their radial spacing
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[020]m
B = [001]m [110]m
[1 10]m
1 00b
2 00m
000
[1 10]m
B = [001]m [ 2 00]m 1 00b
[020]m
010b
11 0m
000
Fig. 4. Schematic spot patterns showing two examples of diffraction satellites from bubble arrays in Mo for an electron beam direction of B = [0 0 1]m. In both cases satellites are observed in h1 1 0im matrix directions around the 0 0 0 transmitted beam, but only in h1 0 0im matrix directions around matrix spots. This is discussed further in the text.
gives a value for the associated interplanar bubble spacing of duvw = (5.1 ± 0.2) nm. If the indices 0 2 0 are assigned to uvw then an associated lattice parameter of a020 = (10.2 ± 0.4) nm is obtained – a value that is not consistent with the value of a110 found above. However, if the normally disallowed 1 0 0-type indices are used instead, the lattice parameter is a010 = (5.1 ± 0.2) nm – a value in excellent agreement with a110. Consequently these satellites are indexed as 0 1 0 and 0 1 0, respectively. The presence of these satellites from low-density bubble planes provides conclusive evidence the bubble ordering is three-dimensional at least in local regions. Many similar demonstrations of three-dimensional ordering have been found for various zone axes and grain orientations. A commonly encountered example is where B = [0 0 1]m in an (0 0 1)m grain. Here the diffraction pattern commonly contains satellite splitting of types represented schematically in the two diagrams of Fig. 4. In both examples, the radial separation of the two pairs of satellites around the matrix reflections again requires the assignment of 1 0 0-type indices (rather than 2 0 0) to obtain consistency with the radial separation of the 1 1 0 satellite reflections around the 0 0 0 transmitted beam. No obvious correlation has been found between the frequency of occurrence of bubble arrays exhibiting three-dimensional structure and the depth below the implanted surface from which the specimen was taken. 4. Structural variants
Fig. 5. Bright-field TEM micrograph (over-focus contrast) of heliumbubble structure in Mo. The electron beam is close to [1 1 3] in the matrix. A section of the diffraction pattern (reverse contrast), showing a pair of 1 1 0 matrix spots and three pairs of satellites around the 0 0 0 transmitted beam is included in the lower part of the figure.
the diffraction of electrons from the gas-bubble superlattice. These anomalies were subsequently explained [12,15– 17,23,29] in terms of the presence of structural variant bubble lattices (also having the same symmetry as the host metal – i.e. fcc bubble lattice in an fcc host etc.) with orientations different from, but rational with, the orientation of the host metal. Guided by this background we have examined many diffraction patterns from ordered bubble arrays in molybdenum, obtained in this and in previous work, for diffraction anomalies that indicate the presence of structural variants. Only two conclusive examples have been found in molybdenum and these are presented below in Figs. 5 and 9. 4.2. Example 1: B ¼ ½1 1 3m 4.2.1. Bright-field TEM imaging Helium bubble images from a particular specimen examined with B ¼ ½1 1 3m in the crystal matrix, are shown in the bright-field micrograph of Fig. 5. The grain orientation is (1 1 3Þm . The average bubble diameter measured from the micrograph is (1.7 ± 0.2) nm. Considerable bubble ordering is evident with bubble rows (which are the traces of the dense-packed bubble planes) parallel to the Tr(1 1 0)m direction. ((1 1 0)m, is the only set of dense-packed matrix planes excited at this zone axis.) The average interplanar spacing, d110, measured from the micrograph as the row spacing, is (3.6 ± 0.1) nm.
4.1. Introduction In the case of the fcc metals and the bcc metal vanadium it was noted in previous work that there were anomalies in
4.2.2. Electron diffraction In the section of the SAD pattern shown in the lower part of Fig. 5, satellite reflections are evident around both
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the 0 0 0 transmitted beam and around both the matrix spots. The satellites around the 0 0 0 spot are shown in approximately the correct orientation with respect to the bright-field micrograph in the inset in the top right of the figure. Again the satellite reflections that are in line with the 1 1 0 matrix spots are assigned 1 1 0 indices. The value of d110 deduced from their radial spacing is (3.6 ± 0.1) nm, a value in good agreement with that found above. If the bubble array were a fully formed bcc lattice then the corresponding lattice constant, a110, is (5.1 ± 0.2) nm. The patterns of satellites around each of the three spots in the SAD pattern of Fig. 5 are shown enlarged in Fig. 6. The strongest satellites in the original negative are shown (each identified by a letter) in Fig. 6 and in the schematic diagram of Fig. 7. The satellite spots around 0 0 0 labelled c and d (the corresponding spots across a diameter through 0 0 0 are also well-defined in the original negative) are in h1 1 2i directions with respect to the host matrix. On the basis of their radial separation they are assigned the normally disallowed indices of 12 ð 21 1Þ and 12 ð 12 1Þ respectively, to give a lattice parameter that is consistent with that found from the radial separation of the two 1 1 0 satellites ((5.1 ± 0.2) nm). The two pairs of satellites labelled c and d are thus from lowdensity bubble planes that are parallel with the corresponding planes in the matrix. This demonstrates that the parts
Fig. 6. Three enlarged sections taken from the diffraction pattern (B ¼ ½ 1 1 3m ) of Fig. 5 showing in more detail the satellites around the matrix spots, 1 1 0 (left) and 1 1 0 (right), and around the 0 0 0 transmitted beam (centre).
B = [ 113 ] m
[ 33 2 ] m [ 211 ] m
a
b
[12 1 ] m d
c
e
f g
110 m
000
h 110 m
Fig. 7. Schematic diagram of the diffraction satellites appearing in Fig. 6. The satellites around the 0 0 0 transmitted beam are consistent with regions of bubble lattice in the matrix (m) orientation whereas the satellites a, b, e, f, g and h are attributed to a structural variant bubble lattice, /, which is rotated with respect to the matrix. (The pair of 1 1 0 satellites around the 0 0 0 beam are consistent with both m and /.)
of the bubble superlattice that are parallel to the host matrix are ordered in three-dimensions. The remaining satellites labelled with letters are clearly not consistent with a bubble lattice in the same orientation as the host metal matrix. The satellites labelled e and g are across a diameter and have the same radial separation from the 1 1 0m matrix spot. They lie at an angle of 45 with respect to [1 1 0]m. Their radial separation is consistent with the lattice parameter of the matrix-oriented bubble lattice if 1 0 0-type indices are assigned. Satellite f is orthogonal to, and at the same radial spacing, as e and g and so too is assigned 1 0 0-type indices. Satellite h is at an angle of 45 with respect to e–g and has a radial separation consistent with 1 1 0-type indices. The satellites e–h are thus consistent with a variant bubble lattice (which will be referred to as variant /) excited with B along [0 0 1]/. For this zone axis, the satellites are assigned indices referred to the variant lattice as follows: a and e – 1 0 0, b and f – 0 1 0, g – 1 0 0 and h – 1 1 0. In summary to explain satellites appearing in the diffraction pattern requires a variant bubble lattice / with the following orientation: [0 0 1]/ coincides with ½ 1 1 3m ; [1 1 0]/ coincides with [1 1 0]m, and ½1 1 0/ coincides with ½3 3 2m . 4.2.3. Projected bubble images Inspection of bright-field images such as Fig. 5 shows that whereas particular bubble rows are often well defined over distances 200 nm or greater along Tr(1 1 0) ð¼ ½3 3 2m Þ, the pattern of bubble images along these rows often changes markedly over distances as short as 20 nm. Schematic diagrams of the pattern of bubble images projected on to the ð1 1 3Þ grain-surface are shown in Fig. 8 for three types of ordered bcc bubble array. We adopt a coordinate system where the x-axis is across the page (positive to the right) and the y-axis is up the page. The z-axis is normal to the plane of the diagram and is directed upward out of the page. Following the usual convention B ¼ ½ 1 1 3m (= [0 0 1]/) is along the (positive) z direction. The upper pattern (shaded circles) is for a bubble lattice in the matrix orientation; the middle pattern (open circles) is for the structural variant bubble lattice in the orientation defined earlier. The x-axis coincides with Trð3 3 2Þm ð¼ ½1 1 0m Þ and with Trð1 1 0Þ/ ð¼ ½1 1 0/ Þ; the y-axis coincides with Trð1 1 0Þm ð¼ ½3 3 2m Þ and with Trð1 1 0Þ/ ð¼ ½1 1 0/ Þ. Comparing the idealised patterns of Fig. 8 with the patterns of projected images in the micrograph of Fig. 5 leads to the following conclusions. There are many regions 20 nm across where the patterns match that for the matrix-oriented bubble lattice alone. These regions can be readily identified in the original negative by the intersection of bubble rows in the primary row direction (i.e. Tr(1 1 0)m), with rows in either or both the Trð2 1 1Þm or Trð1 2 1Þm directions. (In Fig. 8, Trð2 1 1Þm is 25 above the X-axis and Trð1 2 1Þm is 25 below the X-axis; in Fig. 5 Trð2 1 1Þm is 65 (clockwise) from Tr(1 1 0)m and Trð1 2 1Þm is at 115.) The intersection of these bubble rows confirms the previous diffraction evidence that the
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Fig. 9. Bright-field TEM micrograph (under-focus contrast) of heliumbubble structure in Mo. The electron beam is close to [1 1 1] in the matrix. The lower part of the figure is a section of the diffraction pattern (reverse contrast) showing a pair of 1 1 0 matrix spots and three pairs of satellites at 60 around the 0 0 0 transmitted beam.
Fig. 8. Schematic bubble projections on the ð1 1 3Þ grain-surface plane of a bubble lattice in the matrix orientation (upper figure – shaded circles) and of the structural variant / discussed in the text (middle figure – open circles). Tr(1 1 0)m and Tr(1 1 0)b are both parallel with the long side of the diagram. A superposition of the two patterns is shown in the bottom figure.
matrix-oriented bubble lattice is genuinely three-dimensional. Whereas regions where the variant bubble lattice occurs alone have not been found, there are many regions where the pattern of bubble images along bubble rows in Tr(1 1 0) directions exhibits a confused overlapping appearance similar to that in the composite pattern of Fig. 8. In these regions it is concluded that the variant structure does not occur alone but occurs only immediately above or below (in terms of the z-axis) a patch of matrix-oriented bubble lattice. 4.3. Example 2: B ¼ ½ 1 1 1m 4.3.1. Bright-field TEM imaging The bright-field micrograph of Fig. 9 shows a bubble array examined with B (approximately) along ½ 1 1 1m in a ð 1 1 1Þm grain. The average bubble diameter measured from the micrograph is (1.51 ± 0.05) nm. Considerable bubble ordering is evident with bubble rows (which are the traces of the dense-packed bubble planes) parallel to each of the three Tr{1 1 0}m directions lying in the plane of the figure. (For a h1 1 1i zone-axis there are three sets of {1 1 0}m excited.) The average interplanar bubble spacing, d110,
measured from the micrograph, is (4.1 ± 0.3) nm. There are many regions evident where bubble rows in different Tr{1 1 0} directions intersect. This shows that the bubble lattice aligned parallel with the host metal matrix is three-dimensional in character. The associated lattice constant, a110, is (6.0 ± 0.4) nm. 4.3.2. Electron diffraction A section of the SAD pattern is shown in the lower part of Fig. 9. Satellite reflections are evident around both the 0 0 0 transmitted beam and around both the matrix spots. In the inset in the top right of the micrograph the satellites around the 0 0 0 spot are shown in approximately the correct orientation with respect to the bright-field micrograph. Again the satellite reflections that are in line with the 1 1 0 matrix spots are assigned 1 1 0 indices. The value of d110 deduced from their radial spacing is (4.5 ± 0.1) nm and the associated lattice constant, a110, is (6.4 ± 0.2) nm – values which are in satisfactory agreement with those found above. The patterns of satellites around each of the three spots in the SAD pattern of Fig. 9 are shown enlarged in Fig. 10. The directions of the satellites relative to matrix directions are indicated in the schematic diffraction pattern of Fig. 11. The three pairs of satellite spots around 0 0 0 are in h1 1 0i matrix directions and are assigned the appropriate indices (the first three spots, anticlockwise starting from the (positive) x-axis are thus 1 1 0m, 0 1 1m and 10 1m . On the 1 1 0m matrix spot there is a single strong satellite in the ½1 1 2m direction. Similarly there is a single strong satellite in the ½1 1 2m direction, on the 1 1 0m matrix spot. These two satellites have the same radial separation – a spacing which corresponds to an interplanar bubble
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Fig. 10. Three enlarged sections taken from the diffraction pattern (B ¼ ½ 1 1 1) of Fig. 9 showing in more detail the satellites around the matrix spots, 1 1 0m (left) and 1 1 0m (right), and around the 0 0 0 transmitted beam (centre).
B = [ 111] m [ 101 ] m
[1 1 2] m [ 011] m
001 β
001β 1 10 m
000
ordered bcc bubble array, are shown in Fig. 12. Again the coordinate system has the x-axis across the page, the y-axis up the page and the z-axis directed upward out of the page. 1 1 0b ) is Following the usual convention B ¼ ½1 1 1m ð¼ ½ along the (positive) z direction. The upper pattern (shaded circles) is for a bubble lattice in the matrix orientation; the middle pattern (open circles) is for the structural-variant bubble lattice. The x-axis corresponds to Trð 11 2Þm ð¼ ½1 1 0m Þ and to Trð0 0 1Þb ð¼ ½1 1 0b Þ; the y-axis corresponds to Trð1 1 0Þm ð¼ ½1 1 2m Þ and to Trð1 1 0Þb ð¼ ½0 0 1b Þ. Comparing the idealised patterns of Fig. 12 with the patterns of projected images in the micrograph of Fig. 9 leads to the following conclusions. For B ¼ ½1 1 1m there are three sets of dense-packed matrix planes orthogonal to the surface: (1 1 0)m, (0 1 1Þm and (1 0 1Þm . These planes lie at 60 with respect to each other. In Fig. 9 bubble rows are evident in each of the Tr(1 1 0)m, Trð0 1 1Þm and Trð1 0 1Þm directions. Regions 20 nm across can be identified where bubble rows in at least two of these trace directions intersect to indicate the presence of fully threedimensional matrix-oriented bubble lattice. Similarly there
110 m
Fig. 11. Schematic diagram of the diffraction satellites appearing in Fig. 10. The origin and indexing of the satellites are discussed in the text. The three pairs of satellites around the 0 0 0 transmitted beam are consistent with regions of bubble lattice in the matrix orientation, whereas the satellites on the matrix spots 1 1 0m and 1 1 0m (0 0 1b and 0 0 1b respectively) are attributed to a structural variant bubble lattice, b, which is rotated with respect to the matrix.
spacing of d = (6.6 ± 0.3) nm. It is clear that satellites in a h1 1 2im direction with this radial spacing can only arise from a structural variant (referred to as variant b). To obtain a lattice parameter that is consistent with that based on the 1 1 0 satellite reflections around 0 0 0, requires the assignment of 1 0 0-type indices – here we choose 0 0 1b for the spot in the ½ 11 2m direction and 0 0 1b for the spot in the opposite direction. The lowest index bcc zone axis giving rise to the presence of only one pair of 0 0 1 satellites (as in the structural variant diffraction pattern) is h1 1 0i. For excitation of a lattice at this zone axis a single pair of 1 1 0 satellites would be expected in a direction orthogonal to the 1 0 0 satellites. 1 1 0 are allowed reflections for a bcc lattice and so if present would be expected to be found around the 0 0 0 transmitted beam rather than around the matrix spots. If this were the case then the single pair of 1 1 0 reflections from the structural variant would coincide with the 1 1 0 reflections from the matrix-oriented bubble lattice and so would not be detectable as a separate spot pair. Under this assumption the orientation of the structural variant is as follows: B ¼ ½ 1 1 0b ¼ ½ 1 1 1m ; ½0 0 1b ¼ ½1 1 2m and [1 1 0]b = [1 1 0]m. 4.3.3. Projected bubble images Schematic diagrams of the pattern of bubble images projected on to the ð 1 1 1Þ grain-surface, for three types of
Fig. 12. Schematic bubble projections on the ð1 1 1Þ grain-surface plane of a bubble lattice in the matrix orientation (upper figure – shaded circles) and of the structural variant bubble lattice b discussed in the text (middle figure – open circles). Trð1 1 0Þm and Tr(1 1 0)b are both parallel with the long side of the diagram. The bottom section of the figure is a superposition of the two patterns aligned on the three bubble projections indicated with dots.
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are some regions where the pattern of bubble images is consistent with the presence of the structural variant lattice alone. This is evidenced by the presence of intersecting orthogonal bubble rows – one of the row directions is along Tr(1 1 0)m and the other, which exhibits more widely spaced bubbles, is along Trð 11 2Þm . Finally, there are many regions where the pattern of bubble images is consistent with the composite pattern of Fig. 12. These regions are characterised by well defined bubble rows in the Tr(1 1 0)m (= Tr(1 1 0)b) direction with a high density of confused bubble images along the rows themselves as predicted by Fig 12. In these regions it is concluded that the variant structure occurs immediately above or below (in terms of the z-axis) a patch of matrix-oriented bubble lattice having the same lattice constant. 5. Discussion and conclusions Several types of ordered bubble structures have been identified in helium-implanted molybdenum. Bubbles ordered on 1 1 0 matrix planes are a dominant feature. The common occurrence of diffraction satellites from bubble planes parallel with lower density matrix planes shows that there are many regions where the bubbles are fully ordered on a three-dimensional bcc bubble lattice parallel with the host matrix. Also in two cases the presence in the bubble array of structural variant lattices has been demonstrated. These variant lattices are identical to the bubble lattice parallel with the host matrix except for orientation – the variants have major axes aligned along noncorresponding (but rational) low-index directions in the host matrix. Although many bubble arrays have been examined, the sampling procedure and microscope examination are not sufficiently precise to allow reliable conclusions to be drawn about the frequency of occurrence of the various types of ordered arrays. However, it seems clear that structural variants occur only rarely in molybdenum – only two conclusive examples having been found in the several hundred cases surveyed. The results obtained are consistent with the following model of bubble development. Bubbles initially form in a random array and, with continued implantation, order first on to 1 1 0 planes. At a later stage, bubbles order within these planes (without change in the interplanar spacing) to form a fully three-dimensional bubble lattice parallel with the matrix. It is at this stage that the bcc symmetry and the lattice parameter of the bubble lattice are determined. It is only in a final stage, corresponding to a local helium concentration of 10–15 at.%, that the bubbles in some regions (some five to ten interbubble-spacing across) move to form a lattice in a structural variant orientation. The structural variant lattices retain the same bcc symmetry and lattice parameter as the matrix-oriented lattice. The /(b) and matrix-oriented bubble lattices have a common set of dense-packed {1 1 0} bubble planes. It is probable (but not completely certain) that this common
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set of 1 1 0 planes is orthogonal to the grain surface in both cases. (The uncertainty arises because of the possibility that an undetected foil curvature might have caused a departure from ‘‘flat foil’’ conditions in the TEM examination.) For both / and b the geometrical transformation that carries the bubble lattice in a local region from an orientation parallel with the matrix to that of the variant is a rotation of order 30 (25.2 and 35.2 respectively) about the normal to the common set of 1 1 0 planes. We propose that the bubble movements needed to effect this transformation take place solely on these planes. (Note that the superficial bubble density on a given 1 1 0 plane in the set remains unchanged in going from the matrix-oriented bubble lattice to either / or b.) It can be shown that individual bubbles in small domains need move only small distances (comparable with the bubble diameter) to effect this transformation. The features of the structural variants observed here in molybdenum bear close similarities with those found previously in the fcc metal copper [12,15–17]. In that case the change in orientation of sections of the bubble lattice to form structural variants was attributed to effects of stress in the implanted layer. Work is in progress to determine whether a similar explanation can be applied here. For use as a photonic crystal, the ideal gas-bubble superlattice in molybdenum would be uniform throughout the depth of the foil and would contain no structural variants. The exact conditions needed to promote the formation of a fully three-dimensional bubble lattice oriented solely parallel with the host matrix are not clear. However some factors can be identified as a guide to future work. In an early study Mazey et al. [2] found the most complete bubble superlattice parallel with the matrix yet found. This case was analysed in more detail in a later paper [23], where it was thought significant that the implantation was into a pre-thinned foil and involved two different helium implantation energies. It was proposed [23] that there were two important factors: (i) the more uniform distribution of helium over the depth of the foil and (ii) the prevention of the build-up of high levels of stress in the implanted layer because of the close proximity of the surfaces. Also in previous work it has been established that there is both a higher temperature limit [2] and a lower temperature threshold [27] for bubble superlattice formation in molybdenum of approximately 0.35Tm and 0.15Tm respectively. These results suggest the following implantation conditions could be the most favourable for producing a fully developed, uniform bubble superlattice parallel to the molybdenum crystal lattice. The implantation should be at a temperature of 0.25Tm (700 K) into an ultra-thin foil (<100 nm in thickness) using a helium implantation protocol (energies and doses) chosen to deposit helium uniformly with depth to a final helium loading of 10–15 at.%. Under these conditions it could be expected that imperfections in the bubble lattice in the form of structural variants will be rare. However, there will still be imperfections in the form of variations in bubble size, in the degree of bubble faceting, and in the degree of bubble ordering.
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These residual imperfections could be expected to reduce the reflection coefficient for soft X-rays obtained in practice below that calculated on the basis of an ideal lattice. Acknowledgements This work was performed under research contracts to the New Zealand Foundation for Research, Science and Technology (e.g. VICX0206). We thank Dr P.W. Gilberd and C. Varoy for useful discussions and research assistance and Y. Morrison and J. Johnson for technical assistance. References [1] S.L. Sass, B.L. Eyre, Phil. Mag. 27 (1973) 1447. [2] D.J. Mazey, B.L. Eyre, J.H. Evans, S.K. Erents, G.M. McCracken, J. Nucl. Mater. 64 (1977) 145. [3] P.B. Johnson, D.J. Mazey, Nature 276 (1978) 595. [4] P.B. Johnson, D.J. Mazey, Nature 281 (1979) 359. [5] P.B. Johnson, D.J. Mazey, J. Nucl. Mater. 93/94 (1980) 712. [6] P.B. Johnson, D.J. Mazey, Radiat. Eff. 53 (1980) 195. [7] W. Ja¨ger, J. Roth, J Nucl. Mater. 93–94 (1980) 756. [8] W. Ja¨ger, J. Roth, Nucl. Instr. and Meth. 182–183 (1981) 975. [9] P.B. Johnson, D.J. Mazey, J. Nucl. Mater. 111–112 (1982) 681. [10] P.B. Johnson, D.J. Mazey, J.H. Evans, Radiat. Eff. 78 (1983) 157. [11] H. Van Swijgenhoven, G. Knuyt, J. Vanoppen, L.M. Stals, J. Nucl. Mater. 114 (1983) 157. [12] P.B. Johnson, D.J. Mazey, J. Nucl. Mater. 127 (1985) 30. [13] D.J. Mazey, J.H. Evans, J. Nucl. Mater. 138 (1986) 16. [14] J.H. Evans, D.J. Mazey, J. Nucl. Mater. 138 (1986) 176. [15] P.B. Johnson, A.L. Malcolm, D.J. Mazey, Nature 329 (1987) 316. [16] P.B. Johnson, A.L. Malcolm, D.J. Mazey, J. Nucl. Mater. 152 (1988) 69. [17] P.B. Johnson, A.L. Diprose, D.J. Mazey, J. Nucl. Mater. 158 (1988) 108.
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