1999 Fullerenes Under High Pressures

A dvances in Physics, 1999, Vol. 48, No. 1, 1 134

Fullerenes under high pressuresB. Sundqvist Department of Experimental Physics, Umea University, S-90187 Umea, Sweden [Revision received 2 December 1997; accepted 11 December 1997] Abstract This paper reviews the properties and phases of fullerenes and their derivatives and compounds under high pressures. For obvious reasons most of the paper deals with C60 but the materials reviewed also include C70, simple derivatives of C60 , carbon nanotubes, and intercalation compounds of C60 with both acceptors and donors, mainly alkali metals. After a brief overview of high-pressure techniques and the structures and properties of C60 at atmospheric pressure, the structural phase diagram of C60 from atmospheric pressure to above 40GPa (400kbar) is reviewed. The evolution with pressure of the orientational and translational structure of `normal molecular C60 in the range up to 1 5 GPa (depending on temperature) is discussed in some detail, as is the appearance of a large number of polymeric phases at higher pressures and temperatures, some of them known to have extreme mechanical properties. At very high static (or shock) pressures or temperatures, C60 transforms into ordered or disordered forms of diamond or graphite. The phase diagramis reasonably well investigated up to near 10GPa, but at higher pressures there are still large gaps in our knowledge. Available experimental data for the physical properties of both monomeric and polymeric C60 under high pressures are reviewed as far as possible. The compression behaviour of C60 has been well investigated and is discussed in detail because of its basic importance, but optical, electrical and lattice properties have also been studied for several of the many structural phases of C60 . Whenever possible, experimental data are compared with the results of theoretical calculations. The phase diagram and properties of C70 are much less known because of the larger complexity caused by the anisotropy of the molecule, and very little is known about most compounds of C60 . However, noble-gas intercalation in C60 has been reasonably well investigated. Finally, the high-pressure properties of superconducting alkali-metalintercalated C60 are brie y reviewed. Contents 1. 2. 3. 4. Introduction Structures and phases of fullerenes at zero pressure Some comments on high-pressure techniques Structures and properties of molecular (monomeric) C60 4.1. Structural phase diagram 4.1.1. High-temperature behaviour and the fcc phase 4.1.2. Fcc-to-sc transition boundary 4.1.3. The orientational structure in the sc phase 4.1.4. The glassy crystal transition and the properties of the glassy phase 4.1.5. Low-pressure phase diagram of C60 4.1.6. Does liquid C60 exist? 4.2. Physical properties of monomeric C60 under pressure 4.2.1. Compressibility and other elastic properties0001 8732/99 $1200

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B. Sundqvist4.2.2. Lattice vibrations 4.2.3. Thermal properties 4.2.4. Electronic band structure 4.2.4.1. Optical studies 4.2.4.2. Positron annihilation 4.2.4.3. Other studies 4.2.5. Electrical resistivity 4.2.6. Nuclear magnetic resonance 5. Structures and properties of polymeric C60 5.1. Photopolymerization 5.2. Early very-high-pressure studies 5.3. Low-pressure polymerized material ( p < 2 GPa) 5.4. Intermediate-pressure range (2 8 GPa) 5.5. High-pressure range ( p > 8 GPa) 5.6. Transformations into other forms of carbon 5.6.1. High static pressures and/or temperatures 5.6.2. Catalytic conversion to diamond 5.6.3. Shock wave experiments 5.6.4. Other studies 5.7. Pressure temperature phase diagram of C60 6. Structures and properties of C70 6.1. Phase diagram 6.2. Physical properties of C70 6.2.1. Thermophysical properties 6.2.2. Lattice vibrations 6.2.3. Electronic properties 7. Other fullerenes and fullerene compounds 7.1. Nanotubes 7.2. C60O, C61 H2, C60Hx and ( C59N) 2 7.3. Other C60 complexes and compounds 7.4. Intercalation of gases into C60 7.5. Endohedral fullerenes 8. Alkali-metal- and alkaline-earth-metal-doped C60 8.1. Normal-state properties 8.1.1. Bulk modulus 8.1.2. Electrical resistivity 8.1.3. Other studies 8.2. Superconducting-state properties 9. Comments, speculations and conclusions Acknowledgments References 1. Introduction 37 44 46 46 48 49 50 53 54 54 54 56 64 73 79 79 81 81 82 82 83 83 89 89 91 92 93 93 94 97 98 100 101 102 102 104 107 108 113 115 115

It is hardly original or controversial to state that the identi cation of the C60 molecule by Kroto et al. [1]and the subsequent discovery by Kratschmer et al. [2]of a simple and inexpensive method to produce large amounts of this and related materials must be counted among the most important scienti c discoveries during the last two decades of this century. Over the last 6 years, research in this eld has developed at a breathtaking rate, exceeded only by that on ceramic high-transitiontemperature superconductors, and the discoverers were recently honoured by being awarded the 1996 Nobel Prize in chemistry. Not surprisingly, a large number of books, reviews and conference proceedings have already been published on this subject. A number of these will be referred to below and it should be stated here that

Fullerenes under high pressures

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the selection to some extent re ects the authors own preferences, but in particular the fact that these particular publications happened to be readily available at the time of writing. For a particularly complete and up-to-date general review of the eld the reader is referred to recent excellent book by Dresselhaus et al. [3] to which , reference will often be made below, but several other very general reviews [4 7] or collections of reviews [8 10] have appeared over the last few years. However, although the whole fullerene eld, various sub elds, and particular properties of various fullerenes have been reviewed in di erent publications, I am not aware of any previous general review of the properties of C60 under high pressures. Only a small number of short, rather specialized reviews [11 15]and some sections on highpressure e ects in general reviews [3, 4, 7, 16] have so far been published. This is somewhat surprising, since high-pressure studies have given very important information on both pure and doped fullerenes and contributed signi cantly to the rapid developments in this eld, and many recent high-pressure studies hint at possibilities for commercial exploitation of the structural phases existing at high pressures. In this paper, I have tried to collect as much as possible of the information available on the high-pressure properties and phases of both pure and doped solid fullerenes, as well as compounds and derivatives. Fullerenes are here de ned in a broad sense to include both quasispherical molecular species such as C60, C70 and C76 and larger complexes such as `buckytubes and `buckyonions. Most high-pressure work so far has of course been carried out on C60 because of the easy availability of this material in pure form, but some information is also available on other fullerenes. `High pressure is also very loosely de ned, but as a general rule I have arbitrarily selected 100MPa (= 1 kbar) as the lower limit for inclusion, although in some cases reference is made to work at lower pressures. The paper will also concentrate on experimental results and discuss theoretical work only brie y, partly because I am myself an experimentalist, partly because only a small number of theoretical papers have explicitly discussed the pressure or volume dependence and partly because I feel that at the present level of understanding the most important task is to collect and evaluate available experimental data to build a reference platform on which to build future work, experimental or theoretical. The rst version of this review attempted to give the reader a view of the state of our knowledge in the eld up to the autumn of 1996. During the refereeing process the eld has developed rapidly in some areas and the Editor has been kind enough to permit me to update the paper. In this nal version I have therefore extended the text, in particular in section 5, to take into account also the most important developments between the initial submission of the paper and the summer of 1997. I make no claim that the paper is complete and I apologize for any oversights or errors that have succeeded in creeping in, but I hope that I have included references to most of the important work that has been carried out under high pressures. The structure of the paper is as follows. A very brief and incomplete overview is rst given of the ambient-pressure structural phases of undoped fullerenes to set the stage for subsequent sections. Since excellent reviews already exist in this area, reference will be made either to original work or to available reviews rather than presenting too many details. Before starting with the actual review, some brief comments on high-pressure studies and techniques are also given in section 3. In sections 4 and 5 the high-pressure phases, structures and properties of C60 will then be discussed in some detail since these have now been extensively investigated by many groups. The discussion will be divided into two main parts, with section 4

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discussing rst the rather well known structures of the phases based on molecular (or monomeric) C60 and then various physical properties of these phases under pressure, and section 5 discussing the structure and properties of the less well studied polymeric forms of C60 . It should be noted that most physical properties of these phases have only been measured on metastable material at `zero pressure, here de ned as any pressure near or below atmospheric. The short sections 6 and 7 then discuss available information on the phase diagrams and physical properties of C70 and other fullerenes, derivatives and compounds respectively, before we turn to doped fullerenes and their properties under high pressures in section 8. Finally, the paper is rounded o by a short section 9 containing some general comments and conclusions.2. Structures and phases of the fullerenes at zero pressure

This review is focused on solid fullerenes only and, since little information is available for materials other than C60 and C70 under high pressures, most of the text deals with these. In this section a very brief introduction will be given to the structures of the zero-pressure phases as a help to understand the modi cations brought about by the application of a high pressure. Information on the structural and dynamic properties of C60 at zero pressure can be found in a number of reviews [3, 6, 16 21]which will also be referred to where necessary below. The most convenient method to produce fullerenes such as C60 in large amounts is to evaporate graphite electrodes to soot under a low-pressure He gas [2] usually , using an electric arc discharge [22] The soluble fullerenes are then washed out of the . soot using toluene which on evaporation leaves solid fullerene deposits behind. The production, separation and puri cation of fullerenes and related materials have recently been discussed in some detail by Dresselhaus et al. [3] Fullerenes can also be . produced in ames [23], although normally most molecules burn before leaving the ame, and claims have been made for the detection of natural fullerenes in carbonrich minerals (Shungite) [24], in fulgurites (rocks produced by melting minerals during lightning discharges) [25], in sediments containing traces of intense burning from the Cretaceous Tertiary boundary layer [26] and even in trace amounts at , minuscule impact craters on spacecrafts [27]. Whatever the source, solid fullerenes form black or brownish powders or crystals. As discussed recently by Dresselhaus et al. [3] and many others, the most stable crystal structure at high temperatures is the same for all quasispherical fullerenes (C60 , C70, C76 , etc. ). Because the molecules rotate almost completely freely they are e ectively spherical and the molecules form a close-packed face-centred cubic (fcc) structure. The room-temperature ( RT) structure of C60 has been investigated in very close detail by X-ray and neutron di raction. These studies show [3, 21, 28 31] that the molecular rotation is not, in fact, completely free, but that there is a strong intermolecular orientational correlation even in this phase. Close to the orientational transition these correlated clusters may reach 40 A in size [30, 31]and thus consist of several dozen molecules. For our purposes, however, the description of the molecules as freely rotating is good enough. On cooling the rotation slows down and, below an orientational transition temperature [3, 16 19, 29, 32, 33] To = 260K, the rotational motion is replaced by a combination of rapid jumping motion between a number of well de ned orientational states and a librational motion about the equilibrium orientation in each such state. The


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realization by Heiney et al. [32]that in this low-temperature phase the molecules are orientationally ordered was one of the rst of the many surprising and exciting discoveries concerning solid C60 . The basic structure of the low-temperature phase is identical with that of the fcc phase except that the four molecules in the cubic unit cell have di erent orientations, changing the fcc symmetry into a simple cubic (sc) structure. However, there are two possible orientations [3, 29, 33] obtained by rotating the molecules by about 38 and 98 respectively around the [111]axis of the crystal from the `standard orientation. (A recent study [34]showed that the actual angles may in fact be closer to 42 and 102 , values optimizing nearest-neighbour orientations rather than long-range orientational order. ) In the resulting ordered structures, electron-rich double bonds on one molecule face the electron poor centres of hexagons (H orientation) and pentagons (P orientation) respectively on its neighbour, as shown in the simpli ed sketch in gure 1 (A). Since the latter description is easier to visualize for non-specialists (including the author), I shall refer in the following to H- or P-oriented molecules. The di erence in energy is very small and the fraction of molecules in the more stable P orientation is only about 60% near 260K ( gure 1 (B)), increasing to about 84% near Tg = 90 K [33]. Below this the thermal energy becomes too small compared with the energy threshold between the two states for further reorientation to be possible, and below a glass transition (or glassy crystal transition) temperature Tg the remaining orientational disorder becomes frozen in creating an orientational glass. Interestingly, the lattice parameter of H-oriented sc C60 is slightly smaller than for the P-oriented state. This can be observed in gure 2 as an anomalously low thermal expansion coe cient a in the sc phase [29, 35] since for every increase in temperature a certain number of , molecules reorient from the majority P orientation to the minority H orientation, giving a small volume decrease which adds to the normal volume increase due to the increase in temperature. Most physical properties of C60 have already been well studied in these three phases at zero pressure and data can be found in the reviews given above. In addition to the fcc, sc and `glassy states, a fourth state, photopolymerized C60, insoluble in common solvents, can be obtained by submitting thin lms of fcc C60 to high-intensity visible or ultraviolet (UV) light [36]. This treatment is believed to break up the sp2 double bonds joining adjacent hexagons on the molecules and to reform the bonds to form four-membered carbon rings linking neighbouring molecules by two single carbon bonds with a strong sp3 character. Because C60 is a strong absorber of light, only very thin lms can be completely polymerized in this way and this has seriously hampered the study of this interesting material, but it is believed that the polymer chains formed in this way are probably quite disordered and that chain length is highly variable with polymerization conditions. The situation regarding the structure of C70 is much more complicated. Although molecular dynamics calculations [37 40] have shown that the fcc structure should theoretically be the most stable high-temperature structure, in real crystals the other close-packed structure, hexagonal close packed (hcp), very often appears. Because of the small di erence in energy, most crystals contain enough stacking faults to be considered as mixtures of the fcc and hcp phases. Molecular rotation in C70 can also give a much more complicated phase diagram because of the anisotropy of the molecule. `Free molecular rotation (with the same caveats as above) seems to be the rule at high temperatures, where the fcc and hcp structures are both based on the resulting quasispherical molecular shape. Below about 350K, rotation around the

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(A)

(B) Figure 1. (A) Molecules may be oriented with the double bond on the black molecule facing either a pentagon (top) or hexagon (bottom) on the grey neighbouring molecule. (B) Temperature dependence of the fraction of C60 molecules in the more stable Poriented state at atmospheric pressure. Insets (a) and (b) show projections of molecules in the P- and H-oriented states respectively. (Reprinted with permission from David et al. [33]) .

short molecular axes is believed to stop and the molecules rotate around the long axis only. This rotation persists at least down to about 280K. Taking the fcc phase as an example, most workers agree that this phase is stable above 350K. Below this, the axes of the uniaxially rotating molecules can either remain in random directions or line up along random [111]directions, in both cases giving a fcc structure, or line up


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Figure 2. Cubic lattice parameter of C60 plotted against temperature, showing clearly the glass transition temperature Tg near 90K and the orientational transition temperature To near 260K. ( Reprinted with permission from David et al. [33]) .

in parallel along one particular [111] direction giving a rhombohedral (rh) crystal structure. When the uniaxial rotation nally stops below 280K, any of these structures may remain, or molecules in the rh phase may order orientationally relative to their nearest neighbours in much the same way as in C60 to form a monoclinic (mc) phase. However, there are many other possibilities. For example, it has been suggested that in the uniaxially rotating phase (280 350K) the molecules behave as rotating tops, carrying out a nutational movement where the long axis rotates around its equilibrium orientation [40, 41] and two hcp phases have been , reported to occur at RT [42] Recent studies also show that uniaxial rotation . continues down to about 150K [43 46] but it is still unclear whether this , temperature corresponds to the arrest of `free uniaxial rotation or to a glass transition similar to that observed near 90 K in C60. Starting with the less stable hcp structure at high temperatures a di erent structural evolution not discussed here is believed to occur on cooling. It is thus obvious that the structural properties of C70 are much less well known than those of C60 , mainly because most samples are probably mixtures of initially fcc and hcp materials and transitions belonging to both structural sequences are often observed [47]. In fact, it may well be thermodynamically impossible to create a pure single-phase crystal of C70. As will be seen below, this is also re ected in the present status regarding the knowledge of the phase diagram and the physical properties of C70 .3. Some comments on high-pressure techniques

Before discussing the phase diagrams and properties of fullerenes under high pressures a few remarks should be made about high-pressure experiments and techniques. Many experimental techniques have been used in studies of fullerenes and it has sometimes been suggested that the technique used has a large e ect on the

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results obtained. The most obvious problem that may occur is deformation of the sample. If the pressure is transmitted by a uid medium (gas or liquid) with zero shear strength, the deformation will be homogeneous and a cubic single crystal such as C60 will simply su er a decrease in volume without any change in shape. (However, relative changes in atomic positions within the molecule or unit cell may still occur. ) In practice, low-pressure experiments are often carried out under such hydrostatic conditions but, at su ciently high pressures (greater than 12 GPa), only non-hydrostatic solid media are available, and in this case shear stress is unavoidable. The question is then: how important is shear stress? Let us take the compressibility of C60, discussed in detail in section 4.2.1, as an example. Many methods have been devised for such studies and the most common method today is to measure directly the lattice constant using X-ray or neutron di raction methods which give information on the lattice structure in the same experiment. Di raction methods can be used with or without a pressure-transmitting medium, uid or solid. In very high-pressure studies the sample is usually compressed without the use of an additional medium, even if a `soft quasihydrostatic material such as NaCl can be added in the cell to work as a combined pressuretransmitting medium and in-situ pressure calibrant, but at lower pressures a uid pressure medium is often used. The compression properties can also be obtained by measuring the volume of a sample mechanically at known pressures. In the pistonand-cylinder geometry the piston displacement is measured during compression and the maximum pressure is limited to 3 4 GPa. Usually no pressure-transmitting medium is used since the presence of such media makes it necessary to carry out additional corrections for their often not very well de ned compression properties. The compressibility of C60 has been investigated experimentally by a large number of groups using all the above methods and a variety of pressure-transmitting media, ranging from noble gases to relying simply on the very small shear strength of C60 itself [48] to produce quasihydrostatic conditions in the experiment without a pressure-transmitting medium. As will be shown below, measurements on the sc phase below 1 GPa give almost identical results independent of whether noble gases, Freons, NaCl or the C60 sample itself are used as pressure-transmitting medium, showing that in this pressure range this property does not depend on whether the pressure is hydrostatic or not. However, surprisingly, at the lowest pressures, in the fcc phase, the results from the same experiments di er signi cantly, such that the resulting bulk moduli di er by a factor of up to two. Turning to the higher-pressure range, it was observed already in the rst very-high-pressure compression experiments on C60 by Duclos et al. [49] that the compression properties were strongly dependent on the pressure medium used, and it has been veri ed in many experiments that the crystal structures observed can be very di erent under hydrostatic and non-hydrostatic conditions since high non-hydrostatic pressures seem to promote the formation of the polymeric high-pressure phases discussed in section 5. The evaluation of data measured under non-hydrostatic conditions thus needs some care, since shear stress sometimes has a large e ect on the measured properties and sometimes little or no e ect at all. The most important e ect of nonhydrostatic pressure seems to be to promote the formation of the high-pressure polymeric phases of C60 , and the di erences observed between data from experiments under di erent conditions at very high pressures can usually be ascribed to this e ect. The di erences observed between results obtained under di erent


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conditions in the fcc phase have not yet been explained but possible causes are discussed below in section 4.2.1. The upper pressure limits for hydrostatic conditions depend on temperature. At room temperature, most light hydrocarbons, alcohols and oils give hydrostatic conditions only below 0.5 2 GPa. Mixtures of light hydrocarbons or alcohols do not crystallize easily and can be used up to higher pressures. Common examples are the 50 50 mixture of isopentane and n-pentane which can be used up to 5 7 GPa, depending on how we de ne `hydrostatic, and the 4 : 1 mixture of methanol and ethanol which is hydrostatic up to greater than 10 GPa [50]. Although even He solidi es already near 11.5 GPa [51] the use of liquid or solid noble gases allows pressures to remain approximately hydrostatic up to about 60 GPa [52]. However, as will be shown in section 7.4 such media cannot be used in the case of fullerenes. A number of good books on experimental high-pressure techniques have been published recently [53] To give a brief overview, at low pressures (less than 3 GPa), . either piston-and-cylinder or directly pumped cylinder devices are usually used. Depending on the design and material such devices can be used for almost any type of scienti c study, from optical spectroscopy (using quartz, sapphire or other windows) and electrical transport studies to mechanical compressibility measurements and neutron scattering studies, over a temperature range from below 4 K to 2000K. If a higher pressure is desired, large-volume belt-type devices [53], often made from tungsten carbide, can be used over similar ranges in temperature. These can also be used for neutron and X-ray scattering studies and for resistivity measurements. The upper pressure limit is usually 8 10 GPa, but maximum pressures approaching 20 GPa have been achieved [54]. Finally, for the highest pressures, anvil devices are used. Most present-day studies are carried out in diamond anvil cells (DACs) [52, 53] consisting of two natural diamonds, about 1 carat in size, between the surfaces of which the sample is squeezed. These devices have a very convenient size and can usually be compressed by hand using a screw. The sample is usually placed in a small hole (up to 0.5 mm in diameter) in a simple metal gasket between the diamonds, and this hole also gives the possibility of using a uid pressure-transmitting medium. Because DACs in many cases are capable of generating pressures well in excess of 100GPa (1Mbar), techniques have been developed to use liquid noble gases, such as liquid He, as pressure media even at RT and above. If a larger sample but a lower maximum pressure is desired, anvil devices [53]based on sintered diamond, sapphire, tungsten carbide or even steel [55] can be used. Although resistance or neutron scattering measurements are sometimes carried out in anvil devices, most anvil-type devices are constructed from sapphire or diamond and mainly designed for optical and/or X-ray di raction studies.4. Structure and properties of molecular (monomeric) C60

4.1. Structural phase diagram As described in section 2 above, solid C60 forms two crystal phases at zero pressure, a fcc phase with quasifree molecular rotation above To = 260K and a nominally orientationally ordered sc phase with a strongly temperature-dependent orientational disorder below. Near Tg < 90 K, molecular reorientation becomes su ciently slow that the orientational disorder appears `frozen in most experiments, and the resulting low-temperature structure is usually described as an orientational glass. The application of pressure changes the intermolecular distances and thus the

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intermolecular interactions, leading to large changes in the transition temperatures and the orientational order. This section will discuss the e ects of pressure on the translational and orientational structure, starting in the fcc phase and working down towards low temperatures, and map the low-pressure part of the presssure temperature phase diagram. At high pressures, polymeric phases are formed but, in this section, only monomeric C60 , that is materials built up from free molecules without covalent intermolecular bonds, will be discussed. A small number of reviews of the structural evolution with pressure in this range have been published, dealing mainly with the fcc-to-sc transition [13] and the orientational structure in the sc phase [14, 15] . 4.1.1. High-temperature behaviour and the fcc phase A simple phase diagram, drawn on logarithmic scales to bring out clearly the low-pressure behaviour, has been given by Poirier et al. [56]and is shown in gure 3. The diagram shows that solid fcc C60 sublimes on heating without forming a liquid phase at normal pressures. (The possible existence of a liquid phase under pressure will be discussed in section 4.1.6.) The vapour pressure of C60 has been measured by several groups [56 60] and may be extrapolated as in the gure to nd an e ective upper phase boundary for the fcc phase. Vorobev and Eletskii [61, 62]have shown that this curve can be redrawn in reduced coordinates to coincide with that for noble gases. A molecular dynamics simulation of the sublimation of C60 at 2700K and

Figure 3. Phase diagram of C60 drawn on a logarithmic pressure scale to show the lowpressure range: ( ), data from experiment; (---), theoretical curves or extrapolations from experiment. The liquid phase indicated has not yet been observed (see text). (Reprinted with permission from Poirier et al. [56]) .


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50MPa has also been published [63] but it is uncertain to what extent this is realistic , since the C60 molecules have been reported by Sundar et al. [64]to start to break up into amorphous carbon near 700 C (973K). However, Kolodney et al. [65]reported that free molecules are thermally stable to much higher temperatures. It may be speculated that material produced by the standard solvent extraction method has a lower amorphization temperature than sublimed C60 because of the unavoidable presence of remaining solvent molecules in the lattice. Little is known about the detailed e ects of pressure on the dynamic (rotational) properties in the fcc phase, but it is reasonable to assume that the increase in intermolecular interaction with pressure will slow down the molecular rotation and increase the rotational anisotropy observed even at zero pressure [21, 28 31] At high . pressures the rotational motion will probably approach the ratcheting motion observed in the sc phase, where the molecules instead of rotating quasifreely will jump between a number of discrete orientational states. The decrease in the intermolecular distance with increasing pressure and the continuation of molecular rotation increase the probability that double bonds on neighbouring molecules can interact to form intermolecular covalent bonds, and as a result the application of pressures greater than 1 GPa at high temperatures usually leads to the formation of polymeric phases, to be discussed in some detail in section 5. At lower temperatures, the application of pressure and/or cooling at any pressure rst gives a transition into the well known sc phase. 4.1.2. Fcc-to-sc transition boundary The orientational ordering or fcc-to-sc transition was initially suspected [32]to be a continuous transition but thermal and structural studies soon showed it to be rst order with a small hysteresis and connected with rather large volume changes D V < 1% [29, 33, 66] as well as entropy changes D S [67] The transition can thus . be detected fairly easily by a variety of methods, although both the transition temperature and the size and shape of the transition anomalies are very sensitive to deformation and impurities. Since the rst measurements of the compressibility of C60 [49, 68] were carried out almost simultaneously with the discovery of the orientationally ordered sc phase [32] it was immediately realized that the application , of pressure to this soft material should lead to signi cant changes in the intermolecular interactions, which in turn might give signi cant e ects on the transition temperature To. To investigate this idea, Samara et al. [69] and almost simul, taneously Kriza et al. [70] used di erential thermal analysis (DTA) to measure the , fcc-to-sc transition temperature as a function of pressure. The two groups used virtually identical techniques and found similar results for the slope of the phase line, dTo /dp = 104KGPa- 1 [69] and 117KGPa- 1 [70] the very large values re ecting , the large e ects of pressure on the intermolecular potential. However, later investigations have shown that these values are not, in fact, intrinsic to pure C60 for which the true slope of the phase line is even larger. Both groups applied pressure using helium gas which rapidly di uses into the interstitial cavities of solid C60 , changing the physical properties of the material and also the phase diagram. Such e ects are discussed in some detail in section 7.4. The slope of the fcc-to-sc phase line has later been reinvestigated by a large number of techniques, including DTA, nuclear magnetic resonance (NMR), electron paramagnetic resonance (EPR), compression measurements and resistivity measurements, using several di erent pressure-transmitting media. The results from a

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Table 1. Experimental data for the slope dTo /dp of the fcc-to-sc phase boundary. If only one pressure value is given, the experiment was carried out at a single temperature or pressure and a literature value for the zero-pressure value taken as a second reference point. The error limits given are estimates by the original authors Pressure range Pressure (GPa) medium 0 0.8 0 0.5 0 0.8 0.6 0 0.5 0.6 0.6 0 0.3 0 0.5 0 0.8 0 1.4 0 0.5 0 0.5 0 0.5 He He He He He Ne Ar Ar Ar N2 Pentane Isopentane Oil b dTc /dp (K GPa- 1) 104 6 2 117 109 6 4 111 119.5 118 141 174.2 165 164 6 2 159 6 3 160 132 6 3 163

Technique used DTAa DTA DTAa Compressibility Thermal expansion Compressibility Compressibility Thermal expansion Compressibility DTAa DTAa NMR Resistance Compressibility

Reference Samara et al. [69] Kriza et al. [70] Samara et al. [71] Schirber et al. [72] Grube [73] Schirber et al. [72] Schirber et al. [72] Grube [73] Pintschovius et al. [74] Samara et al. [71, 75] Samara et al. [71, 75] Kerkoud and co-workers [76, 77] Matsuura et al. [78] Lundin and Sundqvist [79, 80]

0.4 Alcohol 0 0.2 Te on 0 1.26 Talc epoxy 0.5 (Solid) 0.4 ?a b

90 Raman Meletov et al. [81 83] 120 Thermal conductivity Andersson et al. [84] 99 Resistancea Ramasesha and Singh [85] 6 5 Positron annihilation Jean et al. [86] 65 100 6 10 EPR Kempinski et al. [87]

Double transition observed (see text). Pressure applied directly to the C60 sample itself.

number of such studies are collected in table 1, subdivided into three groups. The rst group contains studies carried out in the intercalating noble gases neon and helium, the second the results of studies using non-intercalating hydrostatic media and methods that give well de ned transition pressures, and the third group contains studies carried out in solid media or over small ranges in T (such as RT only). While the results given in the third group show a very large scatter, as might be expected, those given in the rst two groups are relatively well de ned but di er between the groups. This is well illustrated in gure 4, reprinted from Samara et al. [71] and showing the di erence between phase lines obtained in helium and N2 gas respectively. Inspection of table 1 shows that ve of the values in the second group are in almost perfect agreement and thus that the `best value for the phase line slope of pure C60 can be taken as 162 6 2K GPa- 1 , while in helium and neon a signi cantly smaller average slope dTo /dp = 113KGPa- 1 is found. The accepted explanation for the lower slope under these conditions [71, 75]is that the presence of intercalated gas atoms in the lattice interstitials increases the bulk modulus of the lattice. Since the transition is observed to occur at the same intermolecular distance in all noble gases [72] this means that at any given temperature the transition occurs , at a higher pressure in the intercalated materials. The slope dTo /dP can be calculated from simple thermodynamic considerations. Samara et al. [71] used the known volume change D V at the transition [33, 66]


13

Figure 4. Pressure-temperature phase diagram of C60, showing the fcc-to-sc phase boundary as measured in helium and in N2. (Reprinted with permission from Samara et al. [71] ) (1GPa= 10kbar. ) .

together with the Clausius Clapeyron relation dTo /dp = D V /D S to calculate a value for D S at the transition which turned out to be in very good agreement with experimental data [67] The volume change at the transition is close to 1% at zero . pressure [33, 66]but there are reports that indicate that D V decreases with increasing pressure. Lundin and Sundqvist [80] found that, for pure C60 , D V decreased to about 0.7% near 0.5 GPa, while Grube [73] found a similar rate of decrease in an experiment in helium gas to 0.4 GPa. A similar rapid decrease in D V under pressure was also observed at the fcc-to-rh transition in C70 [88], where D V was even found to approach zero near 1.5GPa. Since the slope is independent of pressure, this indicates that D S must also decrease, and Lundin and Sundqvist [80] speculated that in C60 the transition might change character under pressure from rst order to continuous in analogy with the orientational transition in NH4 Cl [89] However, very recent . accurate neutron studies of C60 by Pintschovius et al. [74]showed no such decrease in D V with increasing pressure. Surprisingly, some studies of V against p in C60 fail to observe any transition anomaly at all [90 92] For the C60 C70 mixture studied by Lundin et al. [90]this is . not unexpected, since in such mixtures the transition is often quite smeared in T (or p), but both the `mechanical study of Bao et al. [91] and the neutron work by Blaschko et al. [92] were carried out on pure C60 which should have a clearly detectable volume anomaly. The slope of the phase line has also been calculated theoretically using other methods. Several groups have calculated this slope as a test of di erent intermolecular potentials, basically van der Waals-type potentials with various added electrostatic interactions (to be brie y discussed in section 4.2.1). Although Burgos et al. [93] calculate a very low dTo /dp = 40 KGPa- 1 , better agreement with experiments is found in other calculations with slightly di erent potentials; Lu et al. [94]reported a value of 115KGPa- 1 and Cheng et al. [37]120KGPa- 1 . Lamoen

14

B. Sundqvist

and Michel [95] instead calculated the slope in a model which assumed that the transition originates in a coupling between molecular orientations and acoustic lattice displacements and found dTo /dp = 182KGPa- 1 , which like the two former values is in reasonable agreement with experiments. Finally, it should be pointed out that many researchers have observed anomalies more complicated than would be expected for a simple rst-order transformation. Already in their rst study Samara et al. [69]found that the DTA peaks developed from being slightly anisotropic at zero pressure to having a de nite shoulder, perhaps indicating the presence of more than one peak, at the highest pressures used. Although they could not rule out a direct impurity e ect, Samara et al. suggested that the double-peak structure may result from partial inhibition of the molecular rotation by interstitial impurities, such that the molecules can have two rotational states, hindered and unhindered rotation, about 5 K apart in energy. Rasolt [96]instead discussed the precursor e ects observed by Samara et al. in terms of uctuations in short-range orientational order and deduced that the transition was a uctuation-driven transition. More clearly de ned double transitions were observed in the resistance studies by Ramasesha and Singh [85] who found that the , di erence in transition temperature depended on pressure and approached zero at zero pressure, and very recently a double transition with a splitting of 0.1 0.3 K has also been observed in highly pure C60 at zero pressure in a thermal study using modulated di erential scanning calorimetry [97]. It is thus possible that an intermediate structural state exists in a very narrow range close to To. (The anomaly could in principle be connected with surface `melting, but the surface layers have been observed to be rotationally disordered already at temperatures signi cantly below To [98].) 4.1.3. The orientational structure in the sc phase As discussed in section 2 there are two possible orientational states, the P (pentagon) and H (hexagon) orientations, in the sc phase at zero pressure. Far below 260K the relative fractions of P- and H-oriented molecules are given by the usual thermal distribution 1 (4.1) f (T) = , 1 + exp (- /kT ) where is the energy di erence between the two states. (Close to To uctuations or other e ects tend to bring the actual measured f closer to 50%.) The P-oriented state has a slightly lower energy and is the preferred orientation at zero pressure, but the H state has a slightly smaller molecular volume and should thus be energetically preferable at high pressures. Because is very small, pressure has a large e ect on the details of the intermolecular interaction in C60 . Using the observed di erence in molecular volume and simple thermodynamic arguments, David and Ibberson [99] showed that the relative energies of the two states should cross already at quite low pressures. They also veri ed this experimentally by neutron di raction studies under pressure between 150 and 200K. At 150K, their data showed that the ratio [P]/[H]of P- to H-oriented molecules decreased from about 70/30 at zero pressure to 50/50 at an `equilibrium pressure peq = 191MPa (where obviously = 0). The equilibrium pressure is probably weakly temperature dependent, since the orientational potential should primarily depend on the intermolecular distance and thus the volume rather than the pressure. If we assume that the equilibrium occurs at a certain (temperature-


15

Figure 5. Low-temperature phase diagram of C60 as suggested by Sundqvist et al. [103] .

independent) molecular volume, we can use data for the thermal expansion [29, 35] and the compressibility (section 4.2.1) to nd that peq should decrease to near 165MPa at 100K and to increase to about 217MPa and 242MPa at 200K and 250K respectively. This is, of course, a rst approximation only, but later highpressure studies of the thermal expansion [73] the compressibility [74] and the , thermal conductivity [100] as well as high-pressure Raman studies [101]all indicate , that the equilibrium pressure near 90 100K indeed is closer to 150MPa than to 200MPa. Above peq , the stable orientational state should be the H orientation. At zero pressure the energy di erence between the two states is too small for any completely P-oriented state to be observed at any temperature but, since peq is quite low, it should be possible to apply pressures high enough for to be much greater than kT, and thus an almost completely H-oriented structure. This was rst suggested by Sundqvist et al. [102, 103]to explain anomalies observed in their experimental data for the compression properties [79]and the thermal conductivity [104]of C60 under pressure. By making a simple linear extrapolation of the fraction f of P-oriented molecules against pressure at 150K they deduced an approximate `phase line for the formation of such an H-oriented phase and obtained the low-temperature phase diagram shown in gure 5, as well as an approximate value for f ( T ) at any pressure [14] As discussed in detail elsewhere [14] the phase diagram obtained agreed with . , their observations of anomalies in the measured bulk modulus under high pressures ( gure 19 below) and the thermal conductivity. Soon after, Wolk et al. [101] made Raman scattering studies as a function of pressure and temperature and observed the appearance of new lines due to libron modes. From these they also deduced the existence of a preferably H-oriented phase above peq < 150MPa, as shown in gure 6. Blaschko et al. [92]later carried out high-pressure neutron scattering studies which showed much better agreement with an assumed predominantly H-oriented phase [P]/[H]= 70/30 than with a predominantly P-oriented phase at 1.6 GPa and RT. Although the existence of a predominantly H-oriented phase has thus been veri ed, the form of equation (4.1) in principle forbids a completely P- or H-oriented

16

B. Sundqvist

Figure 6. Low-temperature phase diagram of C60 as suggested by Wolk et al. [101] (reprinted with permission from the authors).

Figure 7. The function f ( T ) from equation (4.1), showing the quasilinear low-T behaviour (---) and the observed constant f below Tg for C60 (- -). ( Reprinted from Lundin and Sundqvist [80]) .

phase to exist at any temperature above 0K, and no well de ned `phase boundary should exist for an H-oriented phase. It is therefore surprising that anomalies in bulk physical properties have indeed been observed [79, 80]near pressures corresponding to extrapolated `phase transition pressures. Figure 7 shows the theoretical fraction of P-oriented molecules as a universal function of kT / [80] Because of the form of . the equation there is always a quasilinear increase or decrease in f ( T ) over [P]/[H] ratios from about 80/20 to about 98/2, after which there is a nal more gradual exponential decrease in the minority orientation. Any property sensitive to the orientational structure should thus experience a fairly sharp change in its temperature dependence near kT < 0.25. From equation (4.1) together with data for


17

Figure 8. The function f ( T , p) calculated from equation (4.1) and data for D (p) given by David and Ibberson [99] at pressures from 0 to 1 GPa in steps of 100MPa.

against p given by David and Ibberson [99] it is possible to calculate approximate values for the fraction f of P-oriented molecules at any pressure or temperature. The result is shown in gure 8, which veri es that an almost perfectly (H) orientationally ordered phase must exist above some critical pressure. (The corresponding gure in [14]was calculated in a simpli ed model assuming a linear dependence of f on p and is thus less accurate.) Although in principle no perfectly ordered phase exists, in practice any structure with, say, 1% or 0.1% misoriented molecules must be considered very well ordered. Note also that this phase is the only truly orientationally ordered phase existing in the phase diagram of C60 . The pressure dependence of f at constant temperature is shown in gure 9, which has been calculated for pure C60 at a temperature of 150K, that is with 70% Poriented molecules at zero pressure. Near 5 10% there is a fairly sharp bend in the curve, but again it is di cult to see how this rather smooth change in slope can give rise to the sharp anomalies in B(p) reported by Lundin et al. [79, 80] (see above). However, as discussed by Burgos et al. [93] the crystal potential for molecular reorientation is more complicated than usually assumed. These workers calculated the intermolecular energy not only for the rotation of one molecule surrounded by neighbours all in the P orientation, but also for several other cases. As an example, gure 10 shows the intermolecular energy for one molecule as a function of the rotation angle about the [111] crystal axis for these cases. The dotted curve is the standard result assuming all 12 nearest neighbours are in the P-oriented state, while the full curve assumes coherent collective rotation of all molecules and the chain curve assumes all neighbours in the H orientation. In the last curve there is only one minimum; that is, if all (or possibly even a majority?) of the neighbours are H oriented, the molecule studied will also end up in the same state. Burgos et al. concluded that sc C60 will probably contain a large number of well oriented (both P and H) microdomains rather than consist of a mixture of randomly oriented molecules. Since the energy of the H-oriented state also decreases with increasing pressure, it is not unreasonable that a nal `lock-in to a completely H-oriented phase might also occur when the fraction of H-oriented molecules exceeds some critical value which we may guess is not very far from 11 /12 < 0.92 (i.e. on average all

18

B. Sundqvist

Figure 9. The function f calculated from equation (4.1) as a function of pressure p at a nominal temperature of 150K.

Figure 10. Intermolecular energy as a function of rotation angle about the [111]crystal axis: ( ) curve obtained assuming all molecules rotate coherently; ( ) curve assuming neighbouring molecules are all in the 98 (P) orientation; (- -) curve assuming neighbours are in the 38 (H) orientation. (Reprinted with permission from Burgos et al. [93]) .

nearest neighbours except one are in the H orientation). One interesting and testable consequence of this model is that, once formed, the H-oriented state should be very stable, and a `completely H-oriented phase might remain stable down to peq or the fcc transition pressure, whichever is higher, and at low T possibly even at zero pressure. Very recent calculations by L. Pintschovius (1997, private communication), assuming to be a linear function of f , also showed that the combination of such an e ect and the di erent compressibilities of the two states (section 4.2.1) will result in an abrupt nal transition. As hinted at above, the P H reorientation under pressure has been observed to have signi cant e ects on thermophysical properties such as


19

the bulk modulus and the thermal conductivity. This will be discussed further in the relevant parts of section 4.2 below. Before closing this section we note that a thermodynamic model which describes the relative fractions of frozen-in, rotating and ratcheting molecules in the lattice has been developed by Saito et al. [105]but, since this model has not yet been applied to the material under high pressures, it will not be discussed further here. Also, the activation enthalpy D Ha for molecular rotation at constant pressure and other thermodynamic quantities have been measured by NMR in the sc phase under pressure and will be discussed further in section 4.2.6. Finally, it should be noted that anomalies are observed in many physical properties, mechanical [106, 107] , structural [108] acoustic [109] dielectric [110] and others [3], in the sc phase near , , 150K at zero pressure. The cause of these anomalies is still not undisputed, but in some cases it can be shown that they are connected with the glass transition which at high frequencies falls in this temperature range. The only study showing anomalies in this range under pressure is the thermal conductivity study by Andersson et al. [84] . The anomaly shifts to higher temperatures with increasing pressure, indicating either a correlation with the sti ening of the lattice under pressure or with the increase in glass transition temperature Tg because of the decrease in molecular reorientation frequency. 4.1.4. The glassy crystal transition and the properties of the glassy phase The glass transition near Tg = 90K has also been studied in a few experiments under high pressures. In the glassy crystal phase (or orientational glass) below Tg , molecular motion has become slow enough that no reorientational motion can usually be detected during the course of a normal measurement. This means, for example, that, once a sample has been cooled through the transition, the fraction of P-oriented molecules can be considered constant at the limiting value obtained on approaching Tg from above. The pressure dependence of Tg has been found from thermal conductivity studies by Andersson et al. [84, 100, 104] shows de nite anomalies at Tg because of its . sensitivity to orientational disorder. To a rst approximation, the thermal resistivity Wdis due to orientational disorder should be given by Wdis ~ f (1 - f ) and thus have a maximum at f = 0.5 in analogy with the electrical resistivity of concentrated alloys [111] and on cooling towards Tg there must always be a decrease in W dis . When Wdis , suddenly becomes constant at Tg, there will be a sharp anomaly in d /dT. From data such as those shown in gure 11 the slope dTg /dp of the glass transition line has been deduced in the range up to 0.7GPa, with the result dTg /dp = +62 KGPa- 1 [84, 100] (Similar slopes have later been reported by Grube [73]and by Pintschovius et . al. [74] ) Above 0.7 GPa the glass transition anomalies become small because of the . very low fraction of P-oriented molecules (see above). In the range 0.15 0.2 GPa the anomalies are also very small, since the [P]/[H] ratio is close to 50/50 at all temperatures. Again, the molecular reorientation in sc C60 has interesting e ects on the thermodynamic properties. For most glasses, the slope dTg /dp can be determined from the formula [112] D a dTg (4.2) = Tg V g , D cp dp where D a and D cp are the changes in the thermal expansivity and the speci c heat capacity respectively at the transition. For C60, direct use of this formula gives a

20

B. Sundqvist

Figure 11. Thermal conductivity of C60 against temperature at the pressures (in gigapascals) indicated, showing typical glass transition anomalies (arrows). ( Reprinted from Andersson et al. [100] ) .

large negative value dTg /dp < - 130KGPa- 1 [104] because the P-to-H molecular reorientation above Tg leads to a negative D a at the transition (see gure 2 above). Correcting for this e ect a positive slope in better agreement with theory can be found [104] . If the pressure is changed while the sample is kept at a temperature below Tg , no change in the orientational structure can occur. On heating such a sample towards Tg, interesting relaxation anomalies are observed [100] as shown in gure 12. A , comparison with gure 11 shows that the character of the anomalies changes, and both minima and maxima occur in when the molecules relax towards the normal orientational structure at the new pressure. If the sample is cooled at low pressures, a majority of the molecules will be in the P orientation. If the pressure is then increased to a pressure where H orientation should normally dominate, the fraction of Poriented molecules will decrease from its initial high value, through 50% where will have a minimum because of the form of Wdis, towards a low value. This will be observed as an initial decrease in , followed by a nal increase. Close to Tg , such a molecular relaxation can also be observed by measuring as a function of time [100] . Since the H orientation is connected with a smaller molecular volume, the reorientation of each extra P oriented molecule to the H state increases the volume available for other molecules to reorient, and as a result the e ective glass transition temperature becomes lower if the low-temperature phase has a surplus of P-oriented molecules, and vice versa. Wolk et al. [101] tried to determine the glass transition slope by cooling at zero pressure, then applying pressure at low temperature and measuring the glass transition temperature under pressure during heating. As pointed out by Sundqvist et al. [113] the resulting slope, dTg /dp = 35 KGPa- 1 , , shown in gure 6 above, is signi cantly lower than the value of 62 KGPa- 1 obtained by Andersson et al. [84, 100] by measurements at constant pressure ( gure 5) but agrees well with the slope obtained from the position of the minima in observed at


21

Figure 12. Thermal conductivity of C60 against temperature at the pressures (in gigapascals) indicated, showing modi ed glass transition anomalies. In all cases, the samples had been cooled near 0.1GPa and pressurized to the nal pressure below 90K before the measurements were carried out during heating. (Reprinted from Andersson et al. [100]) .

[P]/[H]= 50/50 in the experiments with a frozen-in orientational distribution [100] . An extrapolation of the `50/50 line to lower pressures shows that it crosses the `normal glass transition line at 0.15 6 0.05GPa, indicating that this is the equilibrium pressure peq near 100K. 4.1.5. L ow-pressure phase diagram of C6 0 The information given in the preceding sections can be collected and condensed in the form of the pressure temperature phase diagram for C60 shown in gure 13. The various phases have already been discussed above. The slope of the fcc-to-sc phase line is taken as 162KGPa- 1 as discussed in section 4.1.2 and that of the glass transition line is taken from Andersson et al. [84, 100] To describe the orientational . states in the sc phase, two `boundaries are given. The almost vertical broken line near 0.2 GPa is a weakly temperature-dependent boundary line corresponding to peq ( = 0) , calculated for a constant transition volume as discussed in section 4.1.3 above. For p < peq the P-oriented state is more stable while above this the H orientation dominates. The shaded area at higher pressures corresponds to the region of the transition into the H-oriented state discussed above. The transition region boundaries shown have been calculated in the following way. The lowerpressure side corresponds to a fractional P state occupancy f = 15%, calculated using a constant peq to obtain a lower limit. The high-pressure side corresponds to f = 5% but calculated using the temperature-dependent peq to ensure that we nd an upper limit. These limits on f were chosen because inspection of the theoretical curve for f (p) in gure 9 indicates that the macroscopic `transition should occur near f = 10%, very close to the value f = 0. 92 for which I speculated (from the theoretical potential given by Burgos et al. [93] and shown in gure 10) that a `lock-in transition to an oriented phase might occur. The observed anomalies in the

22

B. Sundqvist

Figure 13. Phase diagram of monomeric C60 as discussed in the text. The full line [71] denotes the fcc-to-sc transformation and the dotted line the glass transition [100] The . almost vertical broken line delineates the equilibrium line between regions with P (low p) and H (high p) orientations and the shaded area is the region where a transition into an H-oriented phase should occur (see text). Symbols denote anomalies observed in various properties by Lundin and Sundqvist [80] ( j ), Jeon et al. [117](n ), Huang et al. [118](, ), Meletov et al. [115]( r ), Meletov et al. [82](h ), Bao et al. [91] ( d ), Blank et al. [114] ( s ) and Jephcoat et al. [116] ( m ).

macroscopic properties should thus fall between these limits if the assumptions made above are correct. Above the high-pressure side of the shaded area less than 5% of the molecules should be P oriented even in a pessimistic calculation and, if an orientational `lock-in transition occurs as suggested by the potential of Burgos et al., the crystal should to a very good approximation be `completely H oriented. In both cases the dominant type of disorder is probably no longer orientational but probably stacking faults, or vacancies and interstitials. If domains of well ordered material form, as suggested by Burgos et al., spectroscopic anomalies may occur even at pressures below this region. The fcc-to-sc phase line is well known and has been well studied, and the same is true to a smaller extent for the low-pressure part of the glass transition line. However, although the existence of the P H orientational ordering transition is proved beyond doubt [92, 101] the actual transition `line or area is not well , investigated. The symbols in gure 13 denote compression anomalies observed by Lundin and Sundqvist [80] at low temperatures as well as spectroscopic and structural anomalies observed at room temperature. Several groups have reported [81 83, 91, 114, 115]the existence of anomalies in spectroscopic and other properties between 2 and 2.5 GPa at room temperature, and Jephcoat et al. [116]observed the formation of a structurally ordered phase near 2.5 GPa. Surprisingly, this highpressure phase is reported to be completely P oriented, although Jephcoat et al. stated that data at 1.3 GPa are in better agreement with an H-oriented state.


23

Whether the unexpected orientation reported is correct, a mistake or owing to the use of He as pressure medium (section 7.4) is not clear. Anomalies have also been reported in the Raman [117]and infrared (IR) [118]spectra of C60 in the range above 1 GPa. It has sometimes been assumed [81 83] that the anomalies observed near 2 GPa correspond to the glass transition, but the slope of the glass transition line is only 62KGPa- 1 at low pressures and a linear extrapolation thus gives a glass transition pressure of 3.4 GPa at 293K. Since the normal curvature of glass transition lines is in such a direction as to increase this pressure value, it is more likely that the anomalies observed are connected with the formation of an (H) orientationally ordered phase. However, this still remains to be proved by lowtemperature high-pressure experiments. 4.1.6. Does liquid C6 0 exist? Most neutral materials exist in three states: gas, liquid and solid. As already discussed in section 4.1.1 above, C60 sublimes near 700K without forming a liquid phase. As shown in gure 14 [119] the standard graphite diamond phase diagram of , C contains a liquid phase at very high temperatures above 5000K, but no such phase has yet been observed for C60 . It might be argued that it should be possible to observe such a state if sublimation was suppressed by the application of a high pressure, and a number of theoretical studies have been carried out to try to predict the properties [120]and range of stability of such a phase. Interestingly, there is still no consensus even on the question of whether a liquid phase can exist at all. Most

Figure 14. Phase diagram of carbon. Solid lines are equilibrium phase boundaries and the broken curve BFG the threshold for rapid graphite diamond transformation. Commercial diamond synthesis is carried out at A, C denotes rapid diamond-tographite conversion, DE is the area where graphite is converted into hexagonal diamond and, along the dotted line, HIJ graphite reversibly transforms into diamond-like structures. ( Reprinted with permission from Bundy et al. [119]) .

24

B. Sundqvist

investigations have used the same intermolecular potential [121] but even so the , results di er between di erent groups. Hagen et al. [122]found that no liquid phase can exist, since in their calculation the liquid vapour coexistence line always fell at lower temperatures than the solid uid coexistence curve, while almost simultaneously Cheng et al. [123] found a narrow range above 1800K where a liquid phase could possibly exist under a low pressure of a few megapascals (tens of bars). Molecular dynamics simulations by Abramo and Caccamo [124] also show an anomaly near 2250K at 2.2 MPa which might be associated with melting. Although theoretically the C60 molecule should be stable to about 4000K [125] experiments , have shown it to break down at very much lower temperatures (section 4.1.1) and it is not clear whether liquid C60 at 1800 2300K would be stable on a molecular scale long enough to be observed even in a rapid transient heating experiment. Later theoretical studies [126 129]have been even less conclusive and the question is still open. Rascon et al. [130]suggested that the reason for the di erent results obtained by di erent groups might be the form of the intermolecular potential used [121] , which they showed is close to the critical potential for which no liquid phase can occur. (Although Shchelkacheva [128]used a di erent potential due to Yakub [131] and found results very similar to those obtained in previous studies, it should be noted that the Yakub potential is only an approximation to that of Girifalco [121]) . Finally, Vorobev and Eletskii [61, 62] argued from thermodynamic scaling arguments that the similarity between C60 and the noble gases shows that a liquid phase should indeed exist. The question of whether a liquid phase does exist or not has thus not yet been answered but, since the physical properties of such a phase should be quite unusual [120] the search will probably continue until either such a phase is , found or its existence has been conclusively disproved. 4.2. Physical properties of C6 0 under high pressures 4.2.1. Compressibility and other elastic properties Having discussed how changes in temperature and pressure modify the structure of C60, I now turn to the interesting question of how the physical properties of C60 are modi ed by high pressures. As long as the molecules remain stable it is obvious from the enormous strength of the intramolecular bonds that the molecular properties will change little under an applied pressure, but it is equally obvious that the weakness of the intermolecular bonds means that the lattice properties will be quite sensitive to the applied pressure. The pressure in itself is not, of course, important; what is important is the change in volume V (and thus intermolecular distance) brought about by the applied pressure. In this section the important relation between volume and pressure will therefore be discussed in some detail. If the pressure is hydrostatic and the material is isotropic, the compression will also be isotropic, there will be no change in sample shape and the resulting volume decrease can be described completely by the isothermal bulk modulus B B=

-V

dp dV

T

,

(4.3)

or, equivalently, the isothermal compressibility = B- 1. As discussed in section 3, this ideal situation rarely occurs in real experiments. Although C60 itself is isotropic at low pressures because of its cubic structure, the pressure in real experiments is not often perfectly hydrostatic but contains uniaxial or shear stress components. In


25

particular, as mentioned in section 3, all pressure media solidify below 12 GPa [51 53], and soft solid noble gases such as helium cannot be used to extend the quasihydrostatic pressure range [52] because they intercalate into the C60 lattice under high pressures. High-pressure studies of C60 are thus often carried out using liquid media such as pentane, Freons or oils, or by compressing pure C60 without a medium, relying on its small shear strength [48] to produce quasihydrostatic conditions. The discussion below will concentrate on data for the isothermal bulk modulus B obtained by direct measurements of the volume change under pressures, but available data for the adiabatic bulk modulus Bs obtained from acoustic or optical studies will also be discussed brie y. The relation between B and Bs is most clearly written in terms of the associated compressibilities and s through the thermodynamic relation [132, 133] = s +

and Bs = - 1 should thus be expected to be slightly larger than B. s The compressibility or bulk modulus of a material not only is of technical and practical interest but is also important from a basic scienti c point of view. The compression behaviour is determined by the intermolecular potential, and measured data for B are therefore very important in forming a benchmark against which theoretical models for the intermolecular interactions can be tested. A large number of calculations have therefore been carried out for the bulk modulus of C60 and will be brie y discussed below. Many experimental methods have been devised for studies of B, and the most common method today is to measure simultaneously the lattice constant and the structure using X-ray or neutron di raction methods. Unfortunately, this is a relatively slow method which gives only a limited number of experimental points. The most common method is to use X-ray di raction and to pressurize the sample in a DAC but the small sample size (less than 0.001mm3) implies very long exposure times and most studies are therefore carried out using synchrotron sources. With the small aperture available and the low intensity of the scattered radiation, only a small number of di raction lines can usually be seen and it can be di cult to identify small changes in the lattice structure. Neutron scattering studies require larger sample volumes and are limited to the signi cantly lower pressures produced by piston-and-cylinder or large anvil devices. In very-highpressure studies the sample is usually compressed without the use of an additional medium, even if a quasihydrostatic material such as NaCl can be added in the cell to function as a combined pressure-transmitting medium and in-situ pressure calibrant, but at lower pressures a uid pressure medium is often used. An alternative method is to measure the compression properties mechanically in the piston-and-cylinder geometry, where the piston movement can be measured very accurately but the maximum pressure is limited to 3 4 GPa. With large samples the piston-and-cylinder methods give a very high resolution and accuracy provided that the device is carefully calibrated and that accurate corrections for the compression of pistons and gaskets as well as for the radial expansion of the cylinder are applied [134] In . most cases, no pressure transmitting medium is used since it is then necessary to carry out additional corrections for the usually not very well de ned compression of the medium. This method gives a semicontinuous record of V against pressure and can thus be used to detect phase transitions and to map the pressure temperature phase diagram, but the accuracy is sometimes limited because of hysteresis e ects

9TV a 2 B2 , cV

(4.4)

26

B. Sundqvist

arising from friction both in the gaskets and inside the sample itself, which su ers plastic ow in the experiment. After the suggestion by Ruo and Ruo [135, 136] that compressed C60 might have a bulk modulus signi cantly higher than the value of 441GPa for diamond [137] the bulk modulus of C60 has been investigated experimentally by a large , number of groups using all the above methods and a variety of pressure-transmitting media. Surprisingly, although the methods used should all have a similar accuracy, the results from di erent groups often di er by signi cant amounts. The compressibility study by Fischer et al. [68]was the rst high-pressure study of C60 . Using X-ray di raction in a DAC under hydrostatic pressure they obtained one single experimental point at 1.2 GPa, with a pressure calibration based on an empirical pressure against load relation. This method is rarely used with DACs because of the large inherent inaccuracy. The phase diagram in section 4.1.5 shows that 1.2GPa is well into the stability range of the fcc phase and the measured volume change up to 1.2GPa thus included both the continuous volume changes within the two phases involved and the volume drop (about 1%) at the fcc-to-sc transition. The average linear compressibility up to 1.2 GPa was d(ln a) /dp = - 2.3 10- 12 cm2 dyn- 1 = - 2.3 10- 2 GPa- 1 , corresponding to a volume compressibility = 6.74 10- 2 GPa- 1 or a bulk modulus B = 14. 8 GPa. These values con rmed the basic assumption that the intermolecular interaction in C60 must be very similar to the interplanar interaction in graphite, since the c-axis compressibility of graphite is d(ln c) /dp = - 2.8 10- 2 GPa- 1 [138] Because of the very small in-plane compress. ibility of graphite the bulk modulus is close to the inverse of this value, at B = 33.8 GPa [138] . Very soon afterwards Duclos et al. [49]presented the results of a more complete investigation of the compressive properties up to 20GPa, under both hydrostatic and non-hydrostatic conditions as deduced from the deformation of the ruby chip used for pressure calibration. As shown in gure 15, the relative volume V / V o is a smooth function of p at all pressures investigated under hydrostatic conditions (experiment 1) but not under non-hydrostatic conditions (experiments 2 and 3). The data from experiment 1 were tted to both the Vinet et al. [139] and the Murnaghan [140] equations of state with identical results, yielding an initial zero pressure bulk modulus B(0) = 18.1 GPa with a pressure derivative B = dB /dp = 5. 7. Because of the experimental procedure used, no data were obtained below about 0.5 GPa and contrary to statements in the original paper the measured values should thus apply to the sc phase only. Again, no account was taken of the volume change at the fcc-tosc transition which makes the low-pressure value B(0) rather dubious. At very high pressures, Duclos et al. observed repeatable changes in the X-ray di raction diagrams above 20 and 16.5 GPa in experiments 2 and 3 respectively. The compression data also indicate a larger V / V o at very high pressures in these experiments than in the hydrostatic experiment. Since a volume increase at a structural transition is highly unlikely for stability reasons, such a change in molecular volume should be connected with a transition at lower pressures. If the high-pressure compression data are extrapolated to lower pressures, the extrapolations cross the data from experiment 1 between 5 and 7GPa, in good agreement with the observations of the formation of an orthorhombic polymeric phase to be discussed in section 5. From the slopes of the curves it is also possible to deduce very uncertain values B > 200GPa for the average bulk modulus between 5 and 20GPa in this high-pressure phase. These values agree to within an order of


27

Figure 15. RT compression behaviour of C60. Hydrostatic conditions are believed to extend to 20GPa in experiment 1 and to 10GPa in experiment 2. Experiment 3 was carried out without a pressure-transmitting medium. The curve has been tted to the data using the Vinet et al. [139]equation of state. (Reprinted with permission from Duclos et al. [49]) .

magnitude with the predictions of Ruo and Ruo [135, 136]and others (see below) for the compressibility of the C60 molecule itself. Although more accurate numerical data have been obtained in later experiments, these two early studies are still widely cited because they illustrate both the fact that C60 is basically `three-dimensional (3D) graphite in its intermolecular properties and the fact that measurements of the high-pressure properties are complicated by the formation of high-pressure phases. Later experiments have concentrated on either increasing the accuracy of the measurements or on measuring the compression properties over larger ranges in temperature and in other structural phases, or all of these. To give some examples, David and Ibberson [99] carried out a highly accurate neutron di raction study of the compressibility up to an Ar pressure of 0.4GPa at 150 200K, with the primary objective of studying the molecular orientation as a function of pressure (see section 4.1.3). Lundin and Sundqvist [79]were the rst to measure the bulk modulus in the fcc phase at and above RT and also carried out careful studies of the compression properties of sc C60 up to 1 GPa over a wide range in T, from 150 to 430K [80] The . pressure range was extended signi cantly by Ludwig et al. [141] who measured V , against p in both sc and fcc C60 up to above 8 GPa at temperatures down to 70 K. Very careful studies of the compressibility in both the sc and the fcc phases up to 0.6 GPa at RT were carried out using neutron scattering in a gas environment by Schirber et al. [72]. This study also included the e ects of gas intercalation on the structure, bulk modulus and phase diagram on C60 (see section 7.4). Finally, Pintschovius et al. [74] have very recently carried out a neutron scattering study up to a pressure of 0.5GPa over an extended temperature range and answered several questions that remained unanswered in the rst version of this review. To date, over 20 publications have been devoted to measurements of the bulk modulus of C60, some by direct compression as those presented above, some by other methods. Numerical data from most of these are presented in table 2, and digitalized

28

B. Sundqvist

Table 2. Experimental data for the bulk modulus B and for B = dB /dp for C60 under various conditions. The values given are usually the initial bulk modulus B(0) at atmospheric (zero) pressure. RT denotes room temperature; X indicates X-ray diffraction in a DAC, M mechanical measurements and N neutron scattering. T (K) RT RT RT RT RT RT 298 317 336 RT RT RT RT RT RT RT RT 298 RT 290 RT RT RT RT RT RT RT RT RT 262 236 180 170 152 150 Pressure range (GPa) B(0) 0 1.2 0 0 0 0 0.38 0.05 0.2 0 0.13 0 0.25 0 0.3 0 0.35 0 0.3 0 0.25 0 0.25 0 0.2 0 0.2 0 20 14.8a 6.4b 8.4b 11.3b 8.4c 8.5 6.8 6.7 6.8 6.7a a

Phase fcc+ sc fcc fcc fcc fcc fcc fcc fcc fcc fcc fcc fcc fcc fcc fcc sc sc sc sc? sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc

B

Reference Fischer et al. [68] Coufal et al. [142] Kobelev et al. [143, 144] Fioretto et al. [145]

Comments and techniques X Surface acoustic wave technique, thin lm Ultrasound Brillouin scattering, thin lm M M, in Ar M M M X X N, in He N, in Ne N, in Ar N, in Ar X; hydrostatic pressure M M M M X X X X N N, in He N, in Ne N, in Ar N, in Ar N, in Ar M M, in Ar X M N continued

13.4 12.5 12 11.8 9.6 18.1

6.7c Lundin et al. [90] 32 Grube [73] 29 Lundin and Sundqvist [79, 80] 24 Lundin and Sundqvist [79, 80] 21 Lundin and Sundqvist [79, 80] Komori and Miyamoto [146] Ludwig et al. [141] 15 Schirber et al. [72] 12 Schirber et al. [72] 5 Schirber et al. [72] 20 Pintschovius et al. [74] 5.7 Duclos et al. [49] 4.2c Lundin et al. [90] 11.1 Lundin and Sundqvist [79, 80] 6.9 Bao et al. [91] Bashkin et al. [147] 5.7 Wang et al. [148] 10.6 Haines et al. [149] 2.53d Ludwig et al. [141] Komori and Miyamoto [146] Blaschko et al. [92] Schirber et al. [72] Schirber et al. [72] Schirber et al. [72] 14 Pintschovius et al [74] 13 Pintschovius et al. [74] 13.1 Lundin and Sundqvist [79, 80] Grube [73] Ludwig et al. [141] 16.2 Lundin and Sundqvist [79, 80] David and Ibberson [99]

0.38 1.1 11.6c 0.5 1.1 9.5 0 2 0 2.7 0 13.7 0 10 0.3 12 0.35 1.6 0 1.8 0.4 0.6 0.4 0.6 0.3 0.55 0.23 0.5 0 0.5 0 0.5 0 0.1 0 9 0 0.6 15.4 16.4 20.9 14.4 13.4d 28a 18.5a 16.5a 15.3a 13.2a 8.3 9.5 9.6 11.5 14.2d 10.4

0 0.28 12.75a

Fullerenes under high pressuresTable 2 (concluded ) T (K) 150 110 110 70 60 RT 290 RT Pressure range (GPa) B(0) 0 0.5 0 0.1 0 0.5 0 9 0 0.1 0 3 0 2.7 5 20 0 12 12.5 13 14.7d 13.75 9.1 10.5 32.9 > 200a 540b

29

Phase sc sc sc sc sc orhg orhh orh?h

B e e

Reference Pintschovius et al. [74] Grube [73] Pintschovius et al. [74] Ludwig et al. [141] Grube [73] Shimomura et al. [150] Bashkin et al. [147] Duclos et al. [49] Blank et al. [151]

Comments and techniques N, in Hef M, in Hef N, in Hef X M, in He; in glass phase X M X; for polymer phase? (see text) Ultrasound

RT Amorphous 2ha b c d e f g h

Average value over pressure range indicated (linear approximation). Adiabatic bulk modulus BS . Mixture of C60 and C70; no sharp fcc-to-sc transition observed. Non-standard modi ed Birch equation of state. B nonlinear in p; negative initial dB /dp. He reported not to intercalate in C60 below 180 K [73]. Orthorhombic C60 crystal obtained by crystallization from solution. Polymeric material; see section 5.5.

data from most of those carried out at RT have been plotted together to bring out the di erences and similarities. Some of these curves are shown in gures 16 18, plotted on di erent scales to enable comparisons in di erent ranges of pressure. Of the values shown in table 2, some are given in the original papers, while some have been extracted with reduced accuracy from graphs. It is obvious from table 2 that the scatter in the data is surprisingly large, considering the accuracy expected in such experiments. Because of the large number of investigations all these data cannot be discussed in detail, but a number of comments should be made on the results. First, Lundin and Sundqvist [80] noted that an internally consistent set of data now exists for the compression properties of the sc phase up to above 1 GPa. This is clearly seen in gure 16, showing the similarity of the RT data of Lundin and Sundqvist [79, 80] Ludwig et al. [141]and Schirber et al. [72], which all agree very , closely above 0.25 GPa. To these data sets we can add the very recent data of Pintschovius et al. [74]; in the sc phase, the data from [72] [74]and [80]are, in fact, , almost indistinguishable at all points in their common pressure range. The lowtemperature data of Ludwig et al., Lundin and Sundqvist and Pintschovius et al. also agree very well with those of David and Ibberson [99] To give some numerical . examples, the relative volumes V / V o observed agree extremely well between the groups. At 150 152K and 280MPa, Lundin and Sundqvist obtained V / V o = 0.9780, Pintschovius et al. V / V o = 0. 9778, and David and Ibberson V / V o = 0.9776 while at 1 GPa Lundin and Sundqvist obtained V / V o = 0.9460 at 152K and Ludwig et al. V /V o = 0.9464 at 170K. The RT data for the bulk modulus B given in table 2 appear to di er greatly, but this seems to result mainly from di erences in the extrapolations of the data since the data for the volume compression in gure 16 actually agree very well between di erent groups. A

30

B. Sundqvist

Figure 16. Measured volume change against pressure for C60 in the low-pressure range at RT. The `reference set for the sc phase discussed in the text contains data from Schirber et al. [72] (d ), Pintschovius et al. [74] (---), Lundin and Sundqvist [79, 80] ( ) and Ludwig et al. ( j ). Other data are from Duclos et al. [49](n ), Blaschko et al. [92] ( s ) and Bashkin et al. [147] ( h ).

Figure 17. Measured volume change against pressure for C60 up to 3 GPa at RT. Data are from Duclos et al. [49] ( n ), Fischer et al. [68] (, ), Schirber et al. [72] (d ), Lundin and Sundqvist [79, 80] ( ), Bao et al. (---), Blaschko et al. [92] (s ), Ludwig et al. [141] ( j ) and Bashkin et al. [147] (h ).

calculation of the average bulk modulus B over the common pressure range 0.3 0.6 GPa gives values greater than 13.2 GPa from Schirber et al., 14.5 GPa from Ludwig et al., 14.7 GPa from Pintschovius et al. and 14.5GPa from Lundin and Sundqvist, all in very good agreement. Below 260K, on the other hand, the sc phase is stable at zero pressure and no extrapolations are involved in the calculation of B(0) . Data for the zero-pressure bulk modulus B(0) from the latter three data sets, plus the low-temperature data of Grube [73] and David and Ibberson [99] have , therefore been plotted against temperature in gure 19. All data points in the range


31

Figure 18. Measured volume change against pressure for C60 up to 28GPa at RT. Data are from Duclos et al. [49] ( n ), Ludwig et al. [141] ( j ), Wang et al. [148] (---), Haines and Leger [149] (s ) and Nguyen et al. [152] (d ).

Figure 19. Low-temperature zero-pressure bulk modulus B(0) for sc C60. Data are from Grube [73] (n ), Lundin and Sundqvist [79, 80] (, ), Pintschovius et al. [74] ( d ), David and Ibberson [99](h ) and Ludwig et al. [141]( j ). The full line has been tted to the data up to 262K only.

below 260K can be well described by a straight line B(0) = (15.6 - 0.023T ) GPa,

(4.5)

with T in kelvins. An extrapolation of this line to RT gives a value for B(0) of 8.9 GPa which agrees very well with the data of Lundin et al. and Pintschovius et al. (also shown on the plot). This shows again that we have a very good agreement between six recent sets of data which thus seem to form a reliable reference set for sc C60 in the low-pressure range. In particular, it should be noted both in gures 16 and 19 and in table 2 that all recent accurate values for B are signi cantly (25 55%) lower than the value of 18.1 GPa given by Duclos et al. [49], which unfortunately is

32

B. Sundqvist

Figure 20. Measured bulk modulus B as a function of pressure at 150 and 236K. The broken line has been corrected to show B against p for a hypothetical state with constant f . (Reprinted from Lundin and Sundqvist [80]) .

still widely used as a reference standard in calculations (see below). The temperature dependence of the bulk modulus also seems to be unusually strong in C60 . The P H reorientation with pressure and temperature has a large e ect on the measured compression data in the sc phase. First, the two states have recently been shown to have di erent compressibilities [74] with the P-oriented state being about , 10% `harder. Second, every increase in pressure is accompanied by a reorientation of a number of molecules from the P- to the H-oriented state. Because the H orientation has a lower molecular volume, this gives an extra volume decrease which adds to the volume change that would have been observed at a constant [P]/[H]ratio. The sc phase thus has an anomalously low B in the pressure and temperature range where this reorientation occurs, and theoretical calculations assuming a constant [P]/ [H] ratio should be expected to give higher values for B(0) than those observed in experiments. Figure 20 shows the measured B as a function of pressure at 152 and 236K [80] (Because of the high experimental resolution in this experiment, B could . be calculated directly from consecutive measured pairs of V and p.) At both temperatures, anomalies are observed in the pressure range where the reorientational transformation is expected to be approaching completion, as discussed in section 4.1.3. The reorientation in fact mainly a ects the slope B = dB /dp, as shown by the broken line which is the result of a simple approximate calculation of how B would depend on the pressure if the [P]/[H] ratio (see section 4.1.3) had stayed constant. Even more pronounced anomalies, including changes in the sign of B , have recently been observed below 150K by Pintschovius et al. [74] Below the glass transition no . further molecular reorientation occurs, and both Pinschovius et al. and Grube [73] showed that this results in a strong increase in B on cooling through Tg , as would be expected. At pressures well above 1 GPa the di erence between the results of di erent studies become larger but data from all groups tend to fall on or in between the extreme curves found by Duclos et al. [49] for hydrostatic and non-hydrostatic conditions respectively ( gures 17 and 18; see also gure 15). Ludwig et al. [141] found a very strong curvature in V against pressure and noted that no reasonable standard equation of state is able to describe their data over the full range of


33

pressures studied. Considering the C60 molecules as incompressible spheres they suggested a modi ed Birch Murnaghan equation of state in which the constant volume of the spheres is subtracted from the crystal lattice. This modi ed equation is then able to describe the compression of the `compressible fraction of the lattice very well at all temperatures, but the structure of the equation makes it di cult to compare their numerical data with data from other sources. (The actual physical e ect behind the observed rapid increase in B with increasing pressure is probably a continuous polymerization with increasing pressure, as will be discussed further in section 5.) The individual C60 molecules indeed seem to be quite incompressible. This has been shown both theoretically, as will be discussed brie y below, and experimentally. Duclos et al. [49]showed from the disappearance of the (200) di raction peak that the molecular form factor [32] must be almost unchanged near 20 GPa, indicating a decrease in molecular radius of the order of 1% and a `molecular B of at least 670GPa. Although Singh [153] tried to explain the di erences between hydrostatic and non-hydrostatic studies from elasticity theory, the e ects discussed above and in sections 4.1.3 and 5 suggested that the large di erences observed simply re ect the fact that the translational, the orientational and even the molecular structure of C60 may di er between experiments depending on the rate of compression and the pressure medium used. These factors lead to di erent structures, depending on the degree of polymerization, the degree of lattice disorder created by inhomogeneous pressure and possibly pressure quenching of the orientational state, all factors which will a ect the measured compression properties. Returning to the low-pressur

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1999 Fullerenes Under High Pressures