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Eur. J. Mineral. 2008, 20, 333–339 Published online April 2008 Special Issue on Diamonds Plastic deformation of lower mantle diamonds by inclusion phase transformations Nicola J. CAYZER 1 ,Shoko ODAKE 2 ,Ben HARTE 1, * and Hiroyuki KAGI 2 1 School of GeoSciences, University of Edinburgh, West Mains Road, Edinburgh EH9 3JW, UK *Corresponding author, e-mail: [email protected] 2 Geochemical Laboratory, Graduate School of Science, University of Tokyo, Tokyo 113-0033, Japan Abstract: The changes in pressure (P) and temperature (T ) conditions experienced by a natural diamond, as it rises to shallower depths and is eventually erupted at the Earth’s surface, give rise to stresses between the diamond and any inclusions it contains as a consequence of their diering changes in volume cell parameters in response to P and T . In the case of diamonds with lower-mantle silicate inclusions such eects will be markedly enhanced by phase transformations by which MgSi-perovskite and CaSi-perovskite transform to pyroxene, pyroxenoid and other phases. Using micro-Raman and SEM-EBSD techniques, we have carried out investigations of the diamond immediately adjacent to inclusions with lower mantle perovskite chemical compositions, and compared them with studies of lower-mantle ferropericlase and upper-mantle inclusions. In the Raman studies evidence of shifts in frequency and width in the diamond Raman spectral band at 1333 cm 1 were sought and found to be greater for perovskite than ferropericlase. However, even for MgSi-perovskite the eects only indicated stored residual pressures of ca 0.35 GPa, which have also been found for typical upper mantle inclusions. The EBSD studies showed a marked contrast between diamond adjacent to former perovskite (now transformed to pyroxene and other phases) and ferropericlase. EBSD mapping shows lattice distortions in diamond adjacent to former MgSi-perovskite and CaSi-perovskite of 4 to 7 ; but adjacent to ferropericlase the distortion was only 1–2 . The results show that the large strain changes accompanying decompression and phase transformation of the perovskite inclusions were largely accommodated by plastic strain within the diamonds at high temperatures, and only small residual pressures were retained in the diamonds. Key-words: diamonds, lower mantle, deformation haloes, residual pressure, inclusion, Raman. 1. Introduction Natural diamonds often contain mineral inclusions whose characteristics provide evidence of the pressure- temperature conditions under which a diamond formed at depth in the Earth’s Mantle (Meyer, 1987; Gurney, 1989; Harris, 1992). With transport of the diamond upward through the mantle and eventual eruption as xenocrysts in kimberlitic or lamproitic magmas, the changes in pressure (P) and temperature (T ) conditions give rise to stresses between the inclusion and the diamond as a consequence of their diering changes in volume cell parameters in re- sponse to diminishing P and T . Such stresses are presumed to be responsible for the cracks often seen around inclu- sions in diamonds. Examination with Raman spectroscopy of some inclusions and their adjacent diamond host has also revealed that residual stresses may be preserved in the in- clusion and diamond. Thus the shift of Raman peaks may be used to provide minimum estimates of the initial pres- sures of diamond and inclusion formation (Izraeli et al., 1999; Sobolev et al., 2000; Nasdala et al., 2005). Barron (2003) has used the concept of a pressure preservation in- dex (PPI) to assess the capacity of dierent mineral inclu- sions to preserve the pressure of diamond formation. In considering these dierential decompression eects, lower mantle diamonds and their inclusions are of particu- lar interest. Cracking of the diamond adjacent to these in- clusions is common but not obviously more extensive than that around diamonds with mineral inclusions formed at lower pressure conditions. Yet the silicate perovskites be- lieved to have been initially present in lower mantle di- amonds have typically undergone phase transformations to pyroxene, pyroxenoid and other lower-pressure silicates with much higher molar volume than the original phase (Harte et al., 1999; Joswig et al., 1999; Kaminsky et al., 2001; Hutchison et al., 2001). In the case of MgSiO 3 the transformation from MgSi-perovskite to enstatite at ca 660 kms would involve an increase of molar volume of 1112% depending on precise temperatures and pressures. The full decompression of MgSi-perovskite in transport to the Earth’s surface, with preservation of equilibrium, would involve five phase transformations and a volume in- crease of 22% (Fei & Bertka, 1999). For CaSi-perovskite the equivalent structural changes (Swamy & Dubrovinsky, 0935-1221/08/0020-1811 $ 3.15 DOI: 10.1127/0935-1221/2008/0020-1811 c 2008 E. Schweizerbart’sche Verlagsbuchhandlung, D-70176 Stuttgart

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Page 1: Plastic deformation of lower mantle diamonds by …pburnley.faculty.unlv.edu/Raman/cayzer et al inclusions...Eur. J. Mineral. 2008, 20, 333–339 Published online April 2008 Special

Eur. J. Mineral.2008, 20, 333–339Published online April 2008 Special Issue on Diamonds

Plastic deformation of lower mantle diamonds by inclusionphase transformations

Nicola J. CAYZER1, Shoko ODAKE2, Ben HARTE1,* and Hiroyuki KAGI2

1 School of GeoSciences, University of Edinburgh, West Mains Road, Edinburgh EH9 3JW, UK*Corresponding author, e-mail: [email protected]

2 Geochemical Laboratory, Graduate School of Science, University of Tokyo, Tokyo 113-0033, Japan

Abstract: The changes in pressure (P) and temperature (T ) conditions experienced by a natural diamond, as it rises to shallowerdepths and is eventually erupted at the Earth’s surface, give rise to stresses between the diamond and any inclusions it containsas a consequence of their differing changes in volume cell parameters in response to P and T . In the case of diamonds withlower-mantle silicate inclusions such effects will be markedly enhanced by phase transformations by which MgSi-perovskite andCaSi-perovskite transform to pyroxene, pyroxenoid and other phases. Using micro-Raman and SEM-EBSD techniques, we havecarried out investigations of the diamond immediately adjacent to inclusions with lower mantle perovskite chemical compositions,and compared them with studies of lower-mantle ferropericlase and upper-mantle inclusions.

In the Raman studies evidence of shifts in frequency and width in the diamond Raman spectral band at 1333 cm−1 were sought andfound to be greater for perovskite than ferropericlase. However, even for MgSi-perovskite the effects only indicated stored residualpressures of ca 0.35 GPa, which have also been found for typical upper mantle inclusions. The EBSD studies showed a markedcontrast between diamond adjacent to former perovskite (now transformed to pyroxene and other phases) and ferropericlase. EBSDmapping shows lattice distortions in diamond adjacent to former MgSi-perovskite and CaSi-perovskite of 4 to 7◦; but adjacent toferropericlase the distortion was only 1–2◦. The results show that the large strain changes accompanying decompression and phasetransformation of the perovskite inclusions were largely accommodated by plastic strain within the diamonds at high temperatures,and only small residual pressures were retained in the diamonds.

Key-words: diamonds, lower mantle, deformation haloes, residual pressure, inclusion, Raman.

1. Introduction

Natural diamonds often contain mineral inclusionswhose characteristics provide evidence of the pressure-temperature conditions under which a diamond formedat depth in the Earth’s Mantle (Meyer, 1987; Gurney,1989; Harris, 1992). With transport of the diamond upwardthrough the mantle and eventual eruption as xenocrysts inkimberlitic or lamproitic magmas, the changes in pressure(P) and temperature (T ) conditions give rise to stressesbetween the inclusion and the diamond as a consequenceof their differing changes in volume cell parameters in re-sponse to diminishing P and T . Such stresses are presumedto be responsible for the cracks often seen around inclu-sions in diamonds. Examination with Raman spectroscopyof some inclusions and their adjacent diamond host has alsorevealed that residual stresses may be preserved in the in-clusion and diamond. Thus the shift of Raman peaks maybe used to provide minimum estimates of the initial pres-sures of diamond and inclusion formation (Izraeli et al.,1999; Sobolev et al., 2000; Nasdala et al., 2005). Barron(2003) has used the concept of a pressure preservation in-

dex (PPI) to assess the capacity of different mineral inclu-sions to preserve the pressure of diamond formation.

In considering these differential decompression effects,lower mantle diamonds and their inclusions are of particu-lar interest. Cracking of the diamond adjacent to these in-clusions is common but not obviously more extensive thanthat around diamonds with mineral inclusions formed atlower pressure conditions. Yet the silicate perovskites be-lieved to have been initially present in lower mantle di-amonds have typically undergone phase transformationsto pyroxene, pyroxenoid and other lower-pressure silicateswith much higher molar volume than the original phase(Harte et al., 1999; Joswig et al., 1999; Kaminsky et al.,2001; Hutchison et al., 2001). In the case of MgSiO3the transformation from MgSi-perovskite to enstatite at ca660 kms would involve an increase of molar volume of11−12% depending on precise temperatures and pressures.The full decompression of MgSi-perovskite in transportto the Earth’s surface, with preservation of equilibrium,would involve five phase transformations and a volume in-crease of ∼ 22% (Fei & Bertka, 1999). For CaSi-perovskitethe equivalent structural changes (Swamy & Dubrovinsky,

0935-1221/08/0020-1811 $ 3.15DOI: 10.1127/0935-1221/2008/0020-1811 c© 2008 E. Schweizerbart’sche Verlagsbuchhandlung, D-70176 Stuttgart

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334 N.J. Cayzer, S. Odake, B. Harte, H. Kagi

1997) would involve a volume increase of ∼ 33%. How-ever, kinetic factors and the retention of high residual pres-sures within the diamonds may prevent all the decreasingpressure equilibrium reactions from occurring.

For lower mantle diamonds with initial perovskite inclu-sions the question arises of how the potentially exception-ally high differential decompression stresses between in-clusion and host diamond have been accommodated, orwhether they are still largely preserved as exceptionallyhigh residual pressures within the diamonds. A interest-ing comparison may also be made with ferropericlase in-clusions in lower mantle diamonds because this phase isnot expected to undergo lattice transformations on decom-pression; although residual pressures due to volume expan-sion of ferropericlase relative to diamond may be gener-ated (Barron, 2003). Hayman et al. (2005) suggest that thedifference in volume changes between ferropericlase andperovskites may help to preserve ferropericlase-bearing di-amonds and account for the unexpectedly high ratio of fer-ropericlase to perovskites found as inclusions.

The aim of this study has been to seek evidence of the ex-tent to which residual pressures are preserved around initialsilicate perovskite inclusions in diamonds and to investi-gate how the stresses generated by decompression are ac-commodated. In addition to Raman spectroscopy we havemade electron backscatter diffraction (EBSD) studies of thediamond surrounding inclusions and discovered substan-tial evidence of plastic deformation. Four lower-mantle-diamonds with inclusions still in situ have been studied andthese show both MgSi- and CaSi- perovskites as well asferropericlase. We have also carried out preliminary EBSDstudies on some other lower pressure inclusion-bearing di-amonds for comparison.

The lower mantle samples chosen for study come fromthe São Luiz alluvial deposit, which lies in the Mato GrossoState, Juina Province, Brazil. Summaries of the lower man-tle inclusions found within diamonds from São Luiz andadjacent parts of the Juina Province of Brazil are given by:Harte & Harris (1994), Harte et al. (1999), Kaminsky et al.(2001) and Hayman et al. (2005). Although in all cases thesilicate-perovskite phases have undergone phase transfor-mations as a consequence of decompression, the inclusionassociations within the diamonds, and particularly those ofMgSi-perovskite and ferropericlase, provide clear evidenceof a lower mantle origin. For the suite of samples investi-gated here, Harte et al. (1999) proposed diamond and in-clusion crystallisation in the depth range of 660 to 730 km.

Carbon isotope and nitrogen abundances for the São Luizdiamonds may be found in Wilding (1990), Hutchinson(1997) and Hutchinson et al. (1999). The diamonds havegenerally very low but variable nitrogen abundances, asis typical of lower mantle diamonds. The δ13C values aredominantly in the range −2 to −7 and therefore close totypical mantle values.

2. Methods

The São Luiz samples were first imaged with cathodo-luminescence (CL) imaging using the XL30CP Scanning

Electron Microscope (SEM) at the School of GeoSciences,University of Edinburgh, to characterise the diamond andthe location of the inclusion(s) within the samples. Ramanstudies were carried out at the Geochemical Laboratory,University of Tokyo, whilst EBSD studies were done in theSchool of Geosciences, Edinburgh.

Micro-Raman experimental procedures

All the spectra were obtained in the confocal mode using aRaman micro-probe spectrometer. The system is composedof a 30 cm single polychromator (250is; Chromex), an op-tical microscope (BX60; Olympus Optical Co. Ltd.,), Ar+

laser (514.5 nm, 5500A, ion laser technology) and a Si-based charge-coupled device (CCD) camera with 1024 ×128 pixels (DU-401-BR-DD SH; Andor Technology). Thelaser power was 3 mW at the sample surface, which isenough below the threshold for the heat-induced spectralchange (Kagi et al., 1994). The entrance slit width of thespectrometer was 130 μm. The CCD camera was electroni-cally cooled at −70 ◦C using a Peltier device. The Rayleighline was removed using a holographic supernotch filter(HIPF-514.5-1.0; Kaiser Optical Systems inc.). The scat-tered light was dispersed using a grating with 1800 groovesper millimetre and the spectral resolution was approxi-mately 1.5 cm−1 per pixel. The practical resolution can beenhanced down to at least 0.15 cm−1 by applying curve fit-ting procedure to the obtained spectral data (Fukura et al.,2006; Yamamoto et al., 2007).

The measurements focussed on the position of the F2gRaman band of diamond, which is nominally observed at1333 cm−1 but shifts to higher frequency with increasingpressure with a gradient of 2.87 cm−1/GPa (Boppart et al.,1985). By using this pressure response, distributions of in-ternal pressure within the sample were mapped in two- andthree-dimensions. Mapping was done using a motorized,software-controlled x-y stage. A Raman spectrum was ac-quired at each sampling point with an increment of 10 μmand was curve-fitted to a Lorentzian curve. The FWHM ofRaman spectra were also plotted to colour-coded imageswith an appropriate smoothing procedure. Contour mapsof the obtained Raman shifts and widths were drawn usingcomputer software Surfer 8 (Golden Software Inc.).

Electron Backscatter Diffraction (EBSD) analysis

EBSD analysis uses the diffraction patterns generated bythe interaction of the SEM electron beam and a crystallinesample to determine the crystallographic orientation of thematerial. The diffraction patterns are imaged on a phos-phor screen by a low-light sensitive camera and analysedby software to determine the crystallographic orientation ofthe material. For this study EBSD analyses were performedusing the HKL Channel 5 EBSD system (Day & Trimby,2004) on the SEM at the School of GeoSciences, Univer-sity of Edinburgh. Maps for the orientation of the diamondcrystal structure around each of the inclusions were gener-ated from point analyses collected every 3–5 μm over an

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Plastic deformation around inclusions in lower mantle diamonds 335

Fig. 1. CL images of samples studied. (a) Sample BZ251 showing the location of the two inclusions (indicated by white arrows), A:ferropericlase and B: MgSi-perovskite. (b) BZ252 showing the location of the CaSi-perovskite inclusion (indicated by the white arrow).(c) BZ254 showing the location of the MgSi-perovskite inclusion (indicated by the white arrow). (d) BZ255 showing the location ofthe ferropericlase inclusion (indicated by the white arrow). The small black points visible on the images are the pits from previous ion-microprobe analyses (see Hutchison et al., 1999).

area approximately 750–1000 μm wide. Conditions of op-eration were an accelerating voltage of 20 kV and a ∼ 1 nAelectron beam, with a tilt angle of 70◦ and an aperture of200 μm. For each point of the map the diffraction patternwas collected and solved by the software for the orientationof the diamond crystal structure to within < 1◦. The result-ing maps were then processed to highlight any variation inorientation within the host diamond.

3. Results

The lower mantle São Luiz samples and their inclusionsare summarised in Table 1. Despite their low and variableN contents (Hutchinson et al., 1999), the diamonds givereasonable CL images (Fig. 1) showing complex octahe-dral growth patterns, which are typical of many natural di-amonds.

Micro Raman data

The samples BZ254 and BZ255 were investigated bymicro-Raman spectroscopy in order to provide comparison

Table 1. Lower mantle diamond samples studied and the nature oftheir inclusions.

Sample name Inclusion Approximate size

of inclusion

BZ251 (A) ferropericlase 200 μm

(B) MgSi-perovskite 200 μm

BZ252 CaSi-perovskite 150 μm

BZ254 MgSi-perovskite 50 μm

BZ255 ferropericlase 100 μm

of former MgSi-perovskite and ferropericlase inclusions.The results are shown in the Raman frequency and band-width maps of Fig. 2, where the white dashed rings showthe positions of the inclusions. Raman spectra of BZ 254were observed on the focal plane 20 μm below the surface,and showed that in addition to the inclusion exposed onthe surface (Fig. 1), another inclusion of probable MgSi-perovskite composition occurred approximately 10 μm be-low the surface of the diamond (Fig. 2a, 2b). Raman spectra(Fig. 2c, 2d) of BZ255 were measured on the surface of thesample.

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336 N.J. Cayzer, S. Odake, B. Harte, H. Kagi

Fig. 2. Raman frequency (a and c) and band width (b and d) maps of diamond around inclusions in São Luiz diamonds; positions of inclusionsmarked by white dotted rings. (a) and (b) of sample BZ254 taken 20 μm below sample surface and showing MgSi-perovskite inclusions(upper one exposed at surface, lower one below surface. (c) and (d) of BZ 255; taken at sample surface of ferropericlase inclusion. Thehigher wavelengths and band widths correspond to higher stress. Note that the upshift of values near the ferropericlase inclusion in BZ255are considerably smaller than those around the MgSi-perovskite inclusions of BZ254 (see text).

In the absence of stress the prominent F2g Raman band ofdiamond is at 1333 cm−1, and shifts to higher wavenumberwith increasing pressure (stress); the width of the Ramanband increases as the anisotropy of stress increases and im-plies lattice distortion.

Sample BZ254 shows marked changes in Raman fre-quency and band width adjacent to the former MgSi-perovskite inclusion exposed at the surface (centre ofFig. 2a, 2b) and the one buried beneath the surface (bottomof Fig. 2a, 2b). The wide spread of the changing Ramanvalues beyond the position of the inclusions shows that theeffects are present in diamond and not just an artefact ofthe presence of the inclusions. The shift in position of thespectral frequency by approaching 1.0 cm−1 imply residualpressures of up to 0.35 GPa assuming an isotropic stressfield within the host diamond and applying displacementof 2.87 cm−1/GPa (as determined by Boppart et al., 1985).The difference in the residual pressures between the twoinclusions suggests that the residual stress of the exposedinclusion was significantly released by the surface cracks.It may be noted that the half-width mapping (Fig. 2b) for

the two inclusions shows similar contrast, but the residualpressures for the two inclusions differ from each other.

By contrast the ferropericlase inclusion in BZ255(Fig. 2c, 2d) is associated with much smaller changeof frequency and bandwidth in the diamond around theinclusion. Furthermore, in this specimen the changes in pa-rameters are tightly confined to the position of the inclu-sion itself, which suggests that they are essentially artefactscaused by heating of the inclusion during the analyses.

EBSD data

The results of EBSD mapping of the areas around in-clusions within the São Luiz samples are summarised inFig. 3. All maps have been coloured for changes in crys-tallographic orientation relative to that of a fixed referencepoint, which for all maps was chosen as the top left cornerof the image. Any change in orientation relative to this ref-erence point was coloured with a blue (minimum) to red(maximum) rainbow scale. A blue colour indicates little

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Plastic deformation around inclusions in lower mantle diamonds 337

Fig. 3. EBSD maps showing variations in crystallographic orientation of the diamond lattice around silicate-perovskite and ferropericlaseinclusions in São Luiz diamonds. In all cases the maps are coloured blue-red for change in orientation relative to a reference point in thetop left-hand corner of the map; in the case of (a), (b) and (e) this colour range is for 5◦ orientation change, for (c) and (d) it covers 7◦.(a) Sample BZ251(A) with ferropericlase inclusion shows “wings” (seen as green) indicating a 1–2◦ change in orientation of the hostdiamond around the inclusion. (b) BZ251(B) with MgSi-perovskite inclusion; shows a deformation halo with up to 4–5◦ deviation from it’soriginal orientation. (c) BZ252 with CaSi-perovskite inclusion; shows a halo within the host diamond deformed by ca. 7◦. (d) BZ254 withMgSi-perovskite inclusions; shows a deformation halo of up to 7◦ change in orientation around the inclusion exposed at the surface (neartop of image) and also another halo for an inclusion still below the polished surface of the diamond (near base of image). (e) BZ255 withferropericlase inclusion, shows “wings” (seen as green-yellow) of ca. 2◦ change in orientation close to the inclusion.

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338 N.J. Cayzer, S. Odake, B. Harte, H. Kagi

(< 1◦) or no difference in crystallographic orientation rela-tive to the reference point. Brighter colours indicate greaterdeviation in orientation to a maximum at red (for 5 or 7◦).

The black areas in the maps are areas where no diamondanalysis was possible. These areas include the inclusionsthemselves as well as cracks, and the pits made during pre-vious ion-microprobe analysis (Hutchinson et al., 1999).Some speckled black areas (e.g. lower left corner of Fig. 3aand lower right corner of Fig. 3c are areas where the surfacepolish was too poor to obtain crystallographic orientationdata on all points.

A contrast may be immediately seen between the inclu-sions of silicate-perovskite composition (Fig. 3b, 3c, 3d)and ferropericlase (Fig. 3a, 3e) inclusions. The ferroperi-clase inclusions show little distortion of the diamond crys-tal structure – only a slight change of 1–2◦ immediatelyadjacent to the inclusion is seen. The maps of the formerMgSi- and CaSi-perovskite inclusions show much greaterdeformation in the host diamond. In specimen BZ 254(Fig. 3d) the former MgSi-perovskite inclusion below thediamond surface is also shown by diamond lattice dis-tortion as well as for the diamond at the surface. TheEBSD data show that the host diamond around both MgSi-perovskite and CaSi-perovskite inclusions has been plas-tically deformed by up to 7◦ from the original crystal-lographic orientation. The deformation haloes extend toa diameter approximately double that of the size of theinclusion.

4. Conclusions

The micro-Raman data show that no exceptionally highresidual stresses are preserved around either silicate-perovskite or ferropericlase inclusions. The estimatedresidual stresses (maximum 0.35 GPa) are similar to thoseof 0.11 to 0.41 GPa recorded in diamond around olivineinclusions by Izraeli et al. (1999). It is noteworthy thatstresses of this magnitude appear to apply to perovskite in-clusions in diamond, which are still fully enclosed in di-amond as well as to those, which have been exposed atthe polished diamond surface. Thus there is no evidencethat the particularly high stresses expected between di-amond and lower mantle diamond inclusions (especiallysilicate perovskites undergoing phase transitions) are pre-served within the diamond. For ferropericlase inclusions,which are expected to develop residual stresses purely todue to volume expansion relative to diamond, the preservedresidual pressures are also far less than the ca. 8 GPa pres-sures predicted thermodynamically (Barron, 2003).

However, the EBSD studies have revealed clear evidencethat the diamond around both former MgSi-perovskite andCaSi-perovskite inclusions has undergone significant plas-tic deformation with lattice misorientations of 5–7◦. Thatthese are particularly due to the phase transformations ofthe high-pressure perovskites to lower pressure and highermolar volume phases is supported by the fact that ferroper-iclase inclusions (sometimes within the same diamond) donot show similar deformation haloes. The plastic deforma-tion around the perovskite inclusions must be presumed to

take place at depth within the upper mantle where high tem-peratures (1600 ◦C or more) would allow such deformationto occur in diamonds without causing brittle fracture.

Finally, for comparison with the above data on lowermantle inclusions, we have performed EBSD maps on di-amonds from the São Luiz locality, which contain inclu-sions (clinopyroxene and spinel and carbonate) of upperrather than lower mantle origin provenance. None of theseshowed deformation haloes exceeding 1◦ lattice distortion.Likewise we have examined some diamonds from Guani-amo Venezuela with clinopyroxene, garnet and coesite in-clusions and found < 1◦ lattice distortion in the surround-ing diamond. Thus the deformation haloes found aroundthe former silicate-perovskite inclusions of lower mantleorigin appear to be truly exceptional.

Acknowledgements: We wish to thank JeffHarris for giv-ing us access to the São Luiz samples following earlierstudies on the same sample suite. The studies were under-taken as part of a JIF development grant from NERC (grantNo. NER/H/S/1998/00008) for which we are very grate-ful. A travel grant to Kagi from the JSPS (Japanese Societyfor the Promotion of Science) aided the collaborative stud-ies. Andrew Taylor of the Diamond Trading Company Re-search Centre, Maidenhead, is thanked for excellent prepa-ration of polished diamond surfaces. Rolf Neuser and ananonymous referee are thanked for their helpful commentson the draft manuscript.

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Received 20 April 2007Modified version received 21 January 2008Accepted 15 February 2008